Droplets and Sprays: Simple Models of Complex Processes (Mathematical Engineering) 3030997456, 9783030997458

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Mathematical Engineering

Sergei S. Sazhin

Droplets and Sprays: Simple Models of Complex Processes

Mathematical Engineering Series Editors Jörg Schröder, Institute of Mechanics, University of Duisburg-Essen, Essen, Germany Bernhard Weigand, Institute of Aerospace Thermodynamics, University of Stuttgart, Stuttgart, Germany Advisory Editors Günter Brenn, Institut für Strömungslehre und Wärmeübertragung, TU Graz, Graz, Austria David Katoshevski, Ben-Gurion University of the Negev, Beer-Sheva, Israel Jean Levine, CAS- Mathematiques et Systemes, MINES-ParsTech, Fontainebleau, France Jan-Philip Schmidt, University of Heidelberg, Heidelberg, Germany Gabriel Wittum, Goethe-University Frankfurt am Main, Frankfurt am Main, Germany Bassam Younis, Civil and Environmental Engineering, University of California, Davis, Davis, CA, USA

Today, the development of high-tech systems is unthinkable without mathematical modeling and analysis of system behavior. As such, many fields in the modern engineering sciences (e.g. control engineering, communications engineering, mechanical engineering, and robotics) call for sophisticated mathematical methods in order to solve the tasks at hand. The series Mathematical Engineering presents new or heretofore little-known methods to support engineers in finding suitable answers to their questions, presenting those methods in such manner as to make them ideally comprehensible and applicable in practice. Therefore, the primary focus is—without neglecting mathematical accuracy—on comprehensibility and real-world applicability. To submit a proposal or request further information, please use the PDF Proposal Form or contact directly: Dr. Thomas Ditzinger ([email protected]) Indexed by SCOPUS, zbMATH, SCImago.

More information about this series at https://link.springer.com/bookseries/8445

Sergei S. Sazhin

Droplets and Sprays: Simple Models of Complex Processes

Sergei S. Sazhin School of Architecture Technology and Engineering Advanced Engineering Centre University of Brighton Brighton, East Sussex, UK

ISSN 2192-4732 ISSN 2192-4740 (electronic) Mathematical Engineering ISBN 978-3-030-99745-8 ISBN 978-3-030-99746-5 (eBook) https://doi.org/10.1007/978-3-030-99746-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Elena M. Sazhina (1955–2020), my wife, collaborator and friend

Preface

Mathematical and Engineering Modelling Rainbows are real but not true. There are no actual coloured arcs in the sky. Each person sees their own rainbow. This is not the start of a fairy tale, but an interesting insight from Michael Berry [9]. The difference between ‘true’ and ‘real’ is commonly overlooked not only by the general public but also by the research community. We can draw a parallel between the concepts of ‘true’ and ‘real’ and the concepts of ‘reality’ and ‘modelling’. There are no tools which would enable us to understand ‘true reality’. The only scientific way we can get a limited understanding of the world around us is by modelling. Modelling is real but it is always limited in how much of the truth it can capture. A theory of everything does not exist and it cannot be developed. This is something that needs to be kept in mind when reading this book, which is focused on modelling of specific phenomena related to droplets and sprays. The limitations of models are well known. Physical models are limited in space and time. As written by Francis Bacon, ‘Nature to be commanded, must be obeyed’ [5]. As commented by M. Eigen, ‘A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant’ [17]. The correctness of models and their accuracy are not synonymous. Agreement with experimental data does not imply that the model is factually correct. Correctness does not imply agreement with experimental data. The accuracy of models is more important than their correctness, which cannot be easily established. Models are communication between human beings like art, literature and music. This leads to very special requirements for models: models should be beautiful. At Moscow University, there is a tradition that the distinguished visiting scientists are asked to write on a blackboard a self-chosen inscription. When Paul Dirac visited Moscow in 1956, he wrote ‘A physical law must possess mathematical beauty’ [70]. See [19] for a detailed discussion of the beauty of models and equations. The approaches to modelling droplets and sprays developed so far can be approximately split into two main groups, which are commonly called ‘Mathematical

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modelling’ (or Engineering mathematics) and ‘Engineering modelling’. The first approach is primarily focused on individual processes (e.g. modelling deformation of individual droplets). The second approach is primarily focused on realistic engineering systems (e.g. modelling of the processes in Diesel engines). Historically, these two approaches developed in parallel with not much overlap between them. Engineers have mainly focused on the commercial or research applications of Computational Fluid Dynamics (CFD) codes to solve specific engineering problems, including those linked with droplets and sprays. The users of CFD codes will usually have limited interest in the models of individual processes incorporated in these codes. Mathematicians have primarily engaged in modelling individual processes with new or existing mathematical tools. This focus has limited how much the results can be applied to understanding real-world engineering processes. The approaches described in this book are based on the author’s belief that the analysis of realistic engineering systems requires a combined approach using the methods of engineering mathematics and Computational Fluid Dynamics (CFD) codes. For this to be achieved, mathematical models of individual processes should be suitable for incorporation in CFD codes. We need to find a compromise between accuracy and simplicity when developing any model. This will be the primary focus of the development of the models described in the book. In some cases, the results of the incorporation of the new models in CFD codes will be described. Even when this incorporation has not been performed, the models to be described will essentially complement conventional CFD codes widely used by the engineering community. Some of the modelling methods described in this monograph were originally developed by the author for modelling plasma waves [56, 57]. This shows how similar mathematical tools can be applied to modelling very different processes.

Scope of the Book This book is a substantially revised and extended version of the author’s previous monograph [58]. The main differences are as follows: (a)

(b)

New developments and results obtained since 2014 are included in this book. Part of these results, obtained between 2014 and 2017, are summarised in the author’s review [59]. Most of this review is incorporated into the book. Although [58] was not designed as a textbook for students, it was successfully used by the author as a companion for a master’s level module on ‘Advanced Computational Fluid Dynamics’ at the University of Brighton (UK). Accordingly, many parts of [58] have been revised to make them more easily comprehensible by the students. To make it easier for the students to follow the analysis presented in the book, the author presented detailed derivations of several key new formulae and included some proofs which are well known to mathematicians but not to engineers (e.g. proofs of orthogonality of eigenfunctions). The

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(c)

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author tried to follow his own instructions on technical writing [54]. In the future, the new book, rather than [58], will be recommended to students. The author received numerous comments and suggestions from readers since the publication of [58] in 2014. These were taken into account and/or corrected in the new book.

Both [58] and the new book address the long-standing problem of modelling droplets and sprays. Investigations of this problem have been motivated by many engineering, medical and environmental applications (e.g. [18]). Early results were presented in well-known classical monographs [21, 36, 69]. Among recent monographs and review papers, the author can mention [3, 22, 29, 59, 60, 62, 71]. None of these publications are intended to be comprehensive. Their focus was on specific topics linked with the specific research interests of the authors. The same comment refers to the current book. This book essentially complements the topics covered in monographs [45, 65, 78] and especially the most recent monograph [14]. There is, however, almost no overlap between these publications. The interest in modelling of droplets and sprays by the author has been predominantly motivated by automotive applications. Experimental and numerical studies of the processes in internal combustion engines have been a focus of research at the Advanced Engineering Centre (AEC) for the last two decades. The main motivation for these studies has been the development of efficient and less polluting engines. The author has been involved in the modelling part of these studies which focused on investigating fluid dynamics, heat/mass transfer, autoignition and combustion processes in complex engine enclosures. Additional complexities of these processes come from the fact that the presence of droplets and sprays in the engines needs to be considered. The dynamics of these needs to be modelled as well. The most widely used approach to modelling the above processes in internal combustions engines is based on the application of commercial (e.g. ANSYS FLUENT, VECTIS and PHOENICS) or research (e.g. OpenFOAM and KIVA) Computational Fluid Dynamics (CFD) codes. The challenges of this approach are in selecting appropriate approximations of the engine geometry, choice of available models (e.g. spray break-up and turbulence models) and the choice of numerical schemes for the solution of the underlying equations. Nobody, including the author, is going to question this. The main emphasis of our research has, however, been on different problems. These include investigation of the underlying physics of the processes involved, development of new physical and mathematical models of processes and investigation of the interaction between complex physical processes which are observed in sprays. This ‘physical’ approach to droplet and spray modelling cannot replace the conventional approach but effectively complements it. The structure of the book broadly reflects the sequence of the processes which occur in internal combustion (IC) engines, involving the direct injection of liquid fuel [27, 73]. Liquid fuel injected through a nozzle leads to the formation of fuel sprays and their penetration into the engine enclosure. Modelling of these processes is discussed in Chap. 1. During and after the completion of the process of spray formation and

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penetration, heating and evaporation of droplets take place. The modelling of these processes is described in Chaps. 2–6 in order of increasing complexity of the models. In Chap. 2, the modelling of the process of droplet heating without evaporation is described. In Chap. 3, the interaction between droplet heating and evaporation is described assuming that the droplets are mono-component and gas can be treated as a continuum medium (hydrodynamic approach). In Chap. 4, a more general case of heating and evaporation of multi-component droplets is described within the same hydrodynamic approach as in Chap. 3 assuming that the interaction between the components can be described in terms of the mutual diffusion between them. In Chap. 5, heating and evaporation of more complex composite droplets, containing clusters of another liquid in them, are considered. The most important processes in the latter droplets, which need to be considered in modelling, are their possible puffing and micro-explosion. Chapter 6 focuses on the modelling of droplet heating and evaporation using the Boltzmann equation for fuel vapour and air in the vicinity of the droplets’ surfaces and molecular dynamics simulation of the processes at the liquid– vapour interface. These processes are commonly ignored in conventional approaches to modelling, including those based on the application of commercial CFD codes. During the process of droplet heating and evaporation, chemical reactions between fuel vapour and oxygen in the air are expected to take place. These can eventually lead to the development of autoignition. The approaches to modelling of the interaction between heating and evaporation of droplets and autoignition of fuel vapour/air are described in Chap. 7. The main challenges of future research in this field, from the author’s point of view, are summarised in Chap. 8. Although the structure of the book and the examples used to illustrate the theory are focused mainly on automotive applications, it is anticipated that the models to be described can be applied to spray modelling in other areas, including environment and medicine.

Topics and Assumptions As mentioned earlier, the present monograph is a revised and extended version of the author’s previous monograph [58]. There is almost no overlap between other books on this topic (e.g. [14, 45, 65]) and this one. This refers to both the topics covered and the methods used for their analysis. Although the application of the models is mainly illustrated through examples referring to fuel droplets, most of them can be applied to other liquid droplets if needed. Only subcritical heating and evaporation are considered. Near-critical and super-critical heating and evaporation of droplets are covered in the relatively recent reviews [8, 24] (see also [44, 79]). Analysis of the interaction between droplets, collisions, coalescence, atomization, oscillations (including instabilities of evaporating droplets) and size distribution of droplets is also beyond the scope of this monograph, although all these processes indirectly influence the processes considered [2, 4, 30, 38, 40–42, 46, 48, 66, 68, 72, 77]. The problem of heating and evaporation of

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droplets on heated surfaces is not discussed [15, 34, 63, 76]. Although the phenomena considered in this book can be an integral part of the more general process of spray combustion, the detailed discussion of the latter is also beyond the scope of this monograph [23, 27, 31, 35, 37, 39, 52, 74]. Although the problem of radiative heating of droplets is closely linked with the problem of scattering of thermal radiation, the modelling of the two processes can be separated. The models of the latter process were reviewed in [33] (cf. [53]), and they are not considered in this monograph. The analysis of droplet evaporation in turbulent flows is presented in [10]. This topic is beyond the scope of the monograph. The analysis is focused on liquid droplets and sprays in gaseous medium. The liquid/liquid systems are not considered (e.g. [75]). The Soret and Dufour effects are not considered. The Soret effect describes the flow of matter caused by a temperature gradient (thermal diffusion), while the Dufour effect describes the flow of heat caused by concentration gradients. The two effects occur simultaneously. They are considered to be small in most cases although sometimes their contribution cannot be ignored (see [6, 7, 13, 16, 26, 32, 51, 67]). The interaction of droplets with electric and magnetic fields, including the problem of electromagnetically levitated molten droplets, is not considered in the book (e.g. [20]). This book is intended to be both an introduction to the problem and a comprehensive description of its state of the art. Most of the book is planned to be a self-sufficient text, although in some cases the reader is referred to the original papers, without a detailed description of the models. Experimental results are presented only when they are essential for understanding and/or validation of the models. The analysis of the effects of droplets on ambient gas is not considered (e.g. [49]). The description of new applications of well-established models is limited. The focus is on the models suitable or potentially suitable for implementation in Computational Fluid Dynamics (CFD) codes. These are the public domain (e.g. OpenFOAM) or commercial (e.g. ANSYS FLUENT) codes. The structures of these codes can be very different but basic approaches to droplet and spray modelling used in them are similar. This makes it possible for us to link the models, described in the monograph, with any of these codes, without making specific references. Among commercial CFD codes, our main focus is on ANSYS FLUENT. The results of the implementation of some of the new models in the latter code are described. Several specialised codes focused on modelling droplets and spray have been developed. These include DropletSMOKE++ [55]. These codes are not analysed and/or discussed in the monograph. In [65], the following classification of the models of droplet heating, in order of ascending complexity, was suggested: (1) (2)

models based on the assumption that the droplet temperature is uniform and constant; models based on the assumption that there is no temperature gradient inside droplets (thermal conductivity of the liquid is infinitely large), but droplet temperature can change with time;

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(3) (4)

(5) (6)

Preface

models considering finite liquid thermal conductivity, but not the recirculation inside droplets (conduction limit); models considering both finite liquid thermal conductivity and the recirculation inside droplets via the introduction of a correction factor to the liquid thermal conductivity (effective conductivity models); models describing the recirculation inside droplets in terms of vortex dynamics (vortex models); models based on the full solution to the Navier–Stokes equation.

The first group allows the reduction of the dimension of the system via the complete elimination of the equation for droplet temperature. This appears to be particularly attractive for the analytical studies of droplet evaporation and thermal ignition of fuel vapour/air mixture (e.g. [11, 25, 61]). This group of models, however, appears to be too simplistic for applications in most CFD codes. Groups (5) and (6) have not been used and are not expected to be used in these codes in the foreseeable future due to their complexity. These models are widely used for validation of more basic models of droplet heating and evaporation or for the in-depth understanding of the underlying physical processes (e.g. [1, 12, 28, 50, 65]). The main focus of this monograph will be on groups (2–4), as these are the ones that are actually used in CFD codes, or their incorporation in them is feasible. Most of the results shown and discussed in the book were originally obtained in close collaboration with my colleagues and published in our joint papers. I am deeply grateful to all of them for this collaboration which led to valuable contributions to this book. The author is grateful to Dr. Ekaterina Roberts for her close reading of the final manuscript and helpful comments. Brighton, UK December 2021

Sergei S. Sazhin

References 1. 2. 3. 4. 5. 6.

7. 8.

Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transf., 32, 1605–1618. Alexandrov, D. V. (2018). The steady-state solutions of coagulation equations. Int. J. Heat Mass Transf., 121, 884–886. Ashgriz, V., (Editor) (2011). Handbook of Atomization and Sprays. Springer. Babinsky, E., & Sojka, P. E. (2002). Modelling drop size distribution. Prog. Energy Combust. Sci., 28, 303–329. Bacon, F., https://en.wikiquote.org/wiki/Francis_Bacon (Accessed May 30, 2020). Beg, O. A., Ramachandra Prasad, V., Vasu, B., Bhaskar Reddy, N., Li, Q., & Bhargava, R. (2011). Free convection heat and mass transfer from an isothermal sphere to a micropolar regime with Soret/Dufour effects. Int. J. Heat Mass Transf., 4, 9–18. Bekezhanova, V.B., Goncharova, O. N. (2020). Influence of the Dufour and Soret effects on the characteristics of evaporating liquid flows. Int. J. Heat Mass Transf., 154:119696. Bellan, J. (2000). Supercritical (and subcritical) fluid behavior and modelling: Drops, steams, shear and mixing layers, jets and sprays. Prog. Energy Combust. Sci., 26, 329–366.

Preface 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27. 28. 29.

30. 31. 32.

33. 34.

xiii

Berry, M. (2020). True, but not real. Lateral Thoughts. Physics World, Issue 29 January (2020) Birouk, M., & Goökalp, I. (2006). Current status of droplet evaporation in turbulent flows. Prog. Energy Combust. Sci., 32, 408–423. Bykov, V., Goldfarb, I., Gol’dshtein, V., & Greenberg, J. B. (2002). Thermal explosion in a hot gas mixture with fuel droplets: A two reactants model. Combust. Theory Model., 6, 1– 21. Chiang, C. H., Raju, M. S., & Sirignano, W. A. (1992). Numerical analysis of convecting, vaporizing fuel droplet with variable properties. Int. J. Heat Mass Transf., 35, 1307–1324. Coelho, R. M. L., & Silva Telles, A. (2002). Extended Graetz problem accompanied by Dufour and Soret effects. Int. J. Heat Mass Transf., 45, 3101–3110. Cossali, G. E., & Tonini, S. (2021). Drop Heating and Evaporation: Analytical Solutions in Curvilinear Coordinate Systems. Springer. Crafton, E. F., & Black, W. Z. (2004) Heat transfer and evaporation rate of small liquid droplets on heated horisontal surfaces. Int. J. Heat Mass Transf., 471187–1200. de Groot, S. R., & Mazur, P. (1962). Non-equilibrium Thermodynamics. North-Holland Publishing Company. Eigen, M., https://todayinsci.com/E/Eigen_Manfred/EigenManfred-Quotations.htm (Accessed May 30, 2020). Faghri, A., & Zhang, Y. (2006). Transport Phenomena in Multiphase Systems. Elsevier. Farmelo, G. (Ed.) (2003). It Must be Beautiful. Great Equations of Modern Science, London – New York: Granta Books. Feng, L., & Shi, W.-Y. (2018). Numerical investigation on frequency shift of an electromagnetically levitated molten droplet. Int. J. Heat Mass Transf., 122, 69–77. Fuchs, N. A. (1959). Evaporation and Droplet Growth in Gaseous Media. Pergamon Press. Fujikawa, S., Yano, T., & Watanabe, M. (2011). Vapor-Liquid Interfaces. Springer. Fujita, A., Watanabe, H., Kurose, R., & Komori, S. (2013). Two-dimensional direct numerical simulation of spray flames – Part 1: Effects of equivalence ratio, fuel droplet size and radiation, and validity of flamelet model. Fuel, 104, 515–525. Givler, S. D., & Abraham, J. (1996). Supercritical droplet vaporization and combustion studies. Prog. Energy Combust. Sci., 22, 1–28. Goldfarb, I., Gol’dshtein, V., Kuzmenko, G., & Sazhin, S. S. (1999). Thermal radiation effect on thermal explosion in gas containing fuel droplets. Combust. Theory Model., 3, 769–787. Gopalakrishnan, V., & Abraham, J. (2004). Effects of multicomponent diffusion on predicted ignition characteristics of an n-heptane diffusion flame. Combust. Flame, 136, 557–566. Heywood, J. B. (1988). Internal Combustion Engines Fundamentals. McGraw-Hill Book Company. Haywood, R. J., Nafziger, R., & Renksizbulut, M. (1989). A detailed examination of gas and liquid transient processes in convection and evaporation. J Heat Transfer, 111, 495–502. Holyst, R., Litniewski, M., Jakubczyk, D., Kolwas, K., Kolwas, M., Kowalski, K., et al. (2013). Evaporation of freely suspended single droplets: experimental, theoretical and computational simulations. Rep. Prog. Phys., 76:034601. Imaoka, R. T., & Sirignano, W. A. (2005). A generalized analysis for liquid-fuel vaporization and burning. Int. J. Heat Mass Transf., 48, 4342–4353. Jenny, P., Roekaerts, D., & Beishuizen, N. (2012). Modeling of turbulent dilute spray combustion. Prog. Energy Combust. Sci., 38, 846–887. Jiang, N., Studer, E., & Podvin, B. (2020). Physical modeling of simultaneous heat and mass transfer: species interdiffusion, Soret effect and Dufour effect. Int. J. Heat Mass Transf., 156:119758. Jones, A. R. (1999). Light scattering for particle characterization. Prog. Energy Combust. Sci., 25, 1–53. Kandlikar, S. G., & Steinke, M. E. (2002). Contact angles and interface behavior during rapid evaporation of liquid on a heated surface. Int. J. Heat Mass Transf., 45, 3771–3780.

xiv

Preface

35.

Kitano, T., Nakatani, T., Kurose, R., & Komori, S. (2013). Two-dimensional direct numerical simulation of spray flames — Part 2: Effects of ambient pressure and lift, and validity of flamelet model. Fuel, 104, 526–535. Levich, V. G. (1962). Physiochemical Hydrodynamics. Prentice Hall. Li, S. C. (1997). Spray stagnation flames. Prog. Energy Combust. Sci., 23, 303–347. Loth, E. (2000). Numerical approaches for motion of dispersed particles, droplets and bubbles. Prog. Energy Combust. Sci., 26, 161–223. Luo, K., Fan, J., & Cen, K. (2013). New spray flamelet equations considering evaporation effects in the mixture fraction space. Fuel, 103, 1154–1157. Mashayek, F. (2001). Dynamics of evaporating drops. Part II: Free oscillations. Int. J. Heat Mass Transf., 44, 1527–1541. Mashayek, F., & Pandya, R. V. R. (2003). Analytical description of particle laden flows. Prog. Energy Combust. Sci., 29, 329–378. Mashayek, F., Ashgriz, N., Minkowycz, W. J., & Shotorban, B. (2003). Coalescence collision of liquid drops. Int. J. Heat Mass Transf., 46, 77–89. Meléan, Y., & Sigalotti, L. D. G. (2005). Coalescence of colliding van der Waals liquid drops. Int. J. Heat Mass Transf., 48, 4041–4061. Meng, H., & Yang, V. (2014). Vaporization of two liquid oxygen (LOX) droplets in tandem in convective hydrogen streams at supercritical pressures. Int. J. Heat Mass Transf., 68, 500– 508. Michaelides, E. E. (2006). Particles, Bubbles & Drops. New Jersey: World Scientific. Mihalyko, Cs., Lakatos, B. G., Matejdesz, A., & Blickle, T. (2004). Population balance model for particle-to-particle heat transfer in gas-solid systems. Int. J. Heat Mass Transf., 47, 1325– 1334. Nakoryakov, V. E., & Misyura S. Ya., & Elistratov S. L. (2012). The behavior of water droplets on the heated surface. Int. J. Heat Mass Transf., 55, 6609–6617. Orme, M. (1997). Experiments on droplet collisions, bounce, coalescence and disruption. Prog. Energy Combust. Sci., 23, 65–79. Pakhomov, M. A., & Terekhov, V. I. (2019). Effect of evaporating droplets on flow structure and heat transfer in an axisymmetrical separated turbulent flow. Int. J. Heat Mass Transf., 140, 767–776. Polyanin, A. D., Kutepov, A. M., Vyazmin, A. V., & Kazenin, D. A. (2002). Hydrodynamics, Mass and Heat Transfer in Chemical Engineering. London and New York: Taylor & Francis, pp. 149–214. Postelnicu, A. (2004). Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Int. J. Heat Mass Transf., 47, 1467–1472. Reitz, R. D., & Rutland, C. J. (1995). Development and testing of diesel engine CFD models. Prog. Energy Combust. Sci., 21, 173–196. Rysakov, V. M. (2004). Light scattering by “soft” particles of arbitrary shape and size. J Quant Spectrosc Radiat Transf., 87, 261–287. Sajina, A. S., & Sazhin, S. S. (2019). Online version: Teaching intelligence: how to improve science students writing. Times Higher Education Supplement, 7th February 2019: https://www.timeshighereducation.com/news/teachingintelligence-how-improve-sci ence-students-writing. Print version: Teaching intelligence: Teach them how to tell a good story. Times Higher Education Supplement, 7th February 2019, p. 20. Saufi, A. E., Frassoldati, A., Faravelli, T., & Cuoci, A. (2019). DropletSMOKE++: A comprehensive multiphase CFD framework for the evaporation of multidimensional fuel droplets. Int. J. Heat Mass Transf., 131, 836–853. Sazhin, S. S. (1982). Natural Radio Emissions in the Earth’s Magnetosphere. Nauka. (in Russian). Sazhin, S. S. (1993). Whistler-mode Waves in a Hot Plasma. Cambridge University Press. Sazhin, S. S. (2014). Droplets and Sprays. Springer.

36. 37. 38. 39. 40. 41. 42. 43. 44.

45. 46.

47. 48. 49.

50.

51.

52. 53. 54.

55.

56. 57. 58.

Preface 59. 60.

61.

62. 63. 64. 65. 66. 67.

68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79.

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Sazhin, S. S. (2017). Modelling of fuel droplet heating and evaporation: Recent results and unsolved problems. Fuel, 196, 69–101. Sazhin, S. S. (2020). Classical and novel approaches to modelling droplet heating and evaporation. in ‘Droplet Interactions and Spray Processes’, Fluid Mechanics and Its Applications v. 121, G. Lamanna et al. (eds.), Switzerland: Springer Nature. Sazhin, S. S., Feng, G., Heikal, M. R., Goldfarb, I., Goldshtein, V., & Kuzmenko, G. (2001). Thermal ignition analysis of a monodisperse spray with radiation. Combust. Flame, 124, 684–701. Sazhin, S. S., Shchepakina, E., & Sobolev, V. (2019). Modelling of sprays: simple solutions of complex problems. J. Phys. Conf. Ser., 1368:042059. Shen, S., Bi, F., & Guo, Y. (2012). Simulation of droplets impact on curved surfaces with lattice Boltzmann method. Int. J. Heat Mass Transf., 55, 6938–6943. Shusser, M., & Weihs, D. (2001). Stability of rapidly evaporating droplets and liquid shells. Int. J. Multiph. Flow, 27, 299–345. Sirignano, W. A. (2010). Fluid Dynamics and Transport of Droplets and Sprays – (2nd ed.). Cambridge University Press. Sommerfeld, M., & Pasternak, L. (2019). Advances in modelling of binary droplet collision outcomes in sprays: A review of available knowledge. Int. J. Multiph. Flow, 117, 182–205. Soret, Ch. (1879). Sur l’état d’équilibre que prend au poin de vue de sa concentration une dissolution saline primitivement homogene dont deux parties sont portées a des temperatures différentes. Archives des Science Physiques et Nat. 2:48–61. Sovani, S. D., Sojka, P. E., & Lefebvre, A.H. (2001). Effervescent atomization. Prog. Energy Combust. Sci., 27, 483–521. Spalding, D. B. (1963). Convective Mass Transfer; an Introduction. Edward Arnold Ltd. Stakhov, A. (2009). Dirac’s principle of mathematical beauty, mathematics of harmony and “Golden” scientific revolution. Visual Mathematics Issue: 41 ISSN: 1821–1437. Subramaniam, S. (2013). Lagrangiane-Eulerian methods for multiphase flows. Prog. Energy Combust. Sci., 39, 215–245. Sun, K., Zhang, P., Jia, M., & Wang, T. (2018). Collision-induced jet-like mixing for droplets of unequal-sizes. Int. J. Heat Mass Transf., 120, 218–227. Tomic, M. V., & Petrovic, S. V. (2000). Internal Combustion Engines. Mašinski Fakultet Unuiverziteta u Beogradu, Beograd (in Serbian). Tsai, C.-H., Hou, S.-S., & Lin, T.-H. (2005). Spray flames in a one-dimensional duct of varying cross-sectional area. Int. J. Heat Mass Transf., 48, 2250–2259. Wegener, M., Paul, N., & Kraume, M. (2014). Fluid dynamics and mass transfer at single droplets in liquid/liquid systems. Int. J. Heat Mass Transf., 71, 475–495. Xie, H., & Zhou, Z. (2007). A model for droplet evaporation near Leidenfrost point. Int. J. Heat Mass Transf., 50, 5328–5333. Zaichik, L. I., Alipchenkov, V. M., & Avetissian, A. R. (2006). Modelling turbulent collision rates of inertial particles. Int. J. Heat Fluid Flow, 27, 937–944. Zaichik, L. I., Alipchenkov, V. M., & Sinaiski, E. G. (2008). Particles in Turbulent Flows. Weinheim Wiley-VCH ISBN:9783527407392. Zhu, G.-S., Reitz, R. D., & Aggarwal, S. K. (2001). Gas-phase unsteadiness and its influence on droplet vaporization in sub- and super-critical environments. Int. J. Heat Mass Transf., 44, 3081–3093.

Contents

1 Spray Formation and Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Spray Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Classical WAVE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 TAB and Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Modified WAVE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spray Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Initial Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Effects of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vortex Ring-Like Structures in Sprays . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Conventional Vortex Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Turbulent Vortex Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Translational Velocities of Vortex Rings-Like Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Confined Vortex Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Two-Phase Vortex Ring Flows . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 8 10 17 18 19 23 26 28 31

2 Heating of Non-evaporating Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Convective Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Stagnant Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Moving Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Radiative Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Basic Equations and Approximations . . . . . . . . . . . . . . . . . . 2.2.2 Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Integral Absorption of Radiation in Droplets . . . . . . . . . . . . 2.2.4 Geometric Optics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 52 70 81 81 84 87 89 93

33 35 36 40

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3 Heating and Evaporation of Mono-component Droplets . . . . . . . . . . . . 3.1 Empirical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Maxwell and Stefan–Fuchs Models . . . . . . . . . . . . . . . . . . . . 3.2.2 Abramzon and Sirignano Model . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Yao, Abdel-Khalik and Ghiaasiaan Model . . . . . . . . . . . . . . 3.2.4 Tonini and Cossali Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Fully Transient Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 d2 and d1.5 Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effects of Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effects of the Moving Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Basic Equations and Approximations . . . . . . . . . . . . . . . . . . 3.4.2 Solution for the Case when Rd (t) is a Linear Function . . . . 3.4.3 Solution for the Case of Arbitrary Rd (t) but Td0 (R) = const . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Solution for Arbitrary Rd (t) and Td0 (R) . . . . . . . . . . . . . . . . 3.4.5 A Comparison between Model Predictions . . . . . . . . . . . . . . 3.5 Conventional and Alternative Approaches to Modelling . . . . . . . . . 3.6 Heating and Evaporation of Spheroidal Droplets . . . . . . . . . . . . . . . 3.6.1 Background Research: Non-spherical Droplets . . . . . . . . . . 3.6.2 The Tonini and Cossali Model (Spheroidal Droplets) . . . . . 3.6.3 The Coupled Liquid/Gas Model (Spheroidal Droplets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Miscellaneous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Effect of Droplet Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Modelling Versus Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Monodisperse Droplet Stream . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Suspended Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 108 108 114 117 119 122 122 123 127 127 129

4 Heating and Evaporation of Multi-component Droplets . . . . . . . . . . . . 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Discrete Component Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 An Analytical Solution for Rd = const . . . . . . . . . . . . . . . . . 4.2.2 An Analytical Solution for Rd = const . . . . . . . . . . . . . . . . . 4.2.3 Bi-component Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Biodiesel Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Kerosene Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Drying Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Quasi-discrete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Description of the Quasi-discrete Model . . . . . . . . . . . . . . . . 4.3.2 Application to Diesel and Petrol Fuel Droplets . . . . . . . . . . 4.4 Multi-dimensional Quasi-discrete Model . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Application to Diesel Fuel Droplets . . . . . . . . . . . . . . . . . . . .

175 175 181 181 183 184 197 202 213 217 217 220 229 229 232

131 133 136 140 144 144 145 149 150 151 153 154 161 165

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4.4.3 Application to Petrol Fuel Droplets . . . . . . . . . . . . . . . . . . . . 4.4.4 Heating, Evaporation and Ignition of Fuel Droplets . . . . . . 4.4.5 Biodiesel/Diesel/Ethanol/Petrol Droplets . . . . . . . . . . . . . . . 4.4.6 Auto-selection of Quasi-components/Components . . . . . . . 4.5 Gas Phase Models for Multi-component Droplets . . . . . . . . . . . . . . 4.6 Other Approaches to Modelling Multi-component Droplets . . . . . . 4.7 Heating and Evaporation of Multi-component Liquid Films . . . . . 4.7.1 Mono-component Liquid Film . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Multi-component Liquid Film . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Validation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.5 Verification of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 237 241 242 244 248 249 250 253 258 259 261 265

5 Processes in Composite Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Simple Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Basic Equations and Approximations . . . . . . . . . . . . . . . . . . 5.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Simple Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Key Equations and Approximations . . . . . . . . . . . . . . . . . . . 5.3.2 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Boiling Versus Nucleation Temperature . . . . . . . . . . . . . . . . 5.3.4 Times to Puffing/Micro-Explosion . . . . . . . . . . . . . . . . . . . . . 5.4 Puffing/Micro-Explosion in the Presence of Coal Particles . . . . . . . 5.4.1 Rapeseed Oil/Water Droplets with Coal Micro-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Modelling Versus Experimental Results . . . . . . . . . . . . . . . . 5.5 Puffing/Micro-Explosion in Closely Spaced Droplets . . . . . . . . . . . 5.5.1 Puffing/Micro-Explosion in Two Droplets in Tandem . . . . . 5.5.2 Puffing/Micro-Explosion in a String of Three Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Effects of Thermal Radiation and Support . . . . . . . . . . . . . . . . . . . . . 5.7 Composite Multi-component Droplets . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Diffusion of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Modelling and Experimental Results . . . . . . . . . . . . . . . . . . . 5.8 The Shift Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 277 280 281 283 285 286 289 290 294 300

309 312 315 315 318 320 322

6 Kinetic Modelling of Droplet Heating and Evaporation . . . . . . . . . . . . 6.1 Early Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes . . . . . . . . . 6.2.1 Boltzmann Equations for the Kinetic Region . . . . . . . . . . . . 6.2.2 Vapour Density and Temperature at the Boundaries . . . . . . 6.3 Approximations of the Kinetic Results . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Approximations for Chosen Gas Temperatures . . . . . . . . . .

327 328 336 337 341 344 345

300 302 305 306

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Contents

6.3.2 Approximations for Chosen Initial Droplet Radii . . . . . . . . Effects of Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Kinetic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Molecular Dynamics Simulations (Background) . . . . . . . . . 6.5.2 United Atom Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Evaporation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Quantum-Chemical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Brief Overview of Quantum-Chemical Methods . . . . . . . . . 6.6.2 Evaporation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Interaction between Molecules and Clusters/Nanodroplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Estimation of the Evaporation Coefficient . . . . . . . . . . . . . . 6.7 Results of the Kinetic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Results for βm = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Results for βm < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Kinetic Modelling in the Presence of Three Components . . . . . . . . 6.8.1 Preliminary Testing of the Numerical Code . . . . . . . . . . . . . 6.8.2 Application to Two-Component Droplets: Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Application to Two-Component Droplets: Results . . . . . . . 6.9 A Self-consistent Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4

349 352 353 360 363 364 366 369 372 372 373 375 376 378 378 381 383 383 390 394 397 402

7 Heating, Evaporation and Autoignition of Sprays . . . . . . . . . . . . . . . . . 7.1 Autoignition Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Coupled Solutions: Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Systems with Non-Lipschitzian Non-linearities . . . . . . . . . . 7.2.5 Spray Ignition and Combustion Model with Radiation . . . . 7.3 Coupled Solutions: Dynamic Decomposition . . . . . . . . . . . . . . . . . . 7.3.1 Decomposition Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Application of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413 413 418 418 419 421 423 426 431 431 433 437 445

8 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Fully Lagrangian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Non-spherical Droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Limitations of the ETC/ED Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Effects of the Interaction Between Droplets . . . . . . . . . . . . . . . . . . . 8.5 Heating and Evaporation in Near/Super-Critical Conditions . . . . . 8.6 Effects of the Moving Interface due to Evaporation . . . . . . . . . . . . .

453 453 453 454 454 455 455

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8.7 Complex Multi-component Droplets . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Puffing and Micro-explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Advanced Kinetic and Molecular Dynamics Models . . . . . . . . . . . . 8.10 Effective Approximation of the Kinetic Effects . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456 456 457 457 458

Appendix A: Derivation of Formula (2.86) . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Appendix B: Derivation of Formula (2.89) . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Appendix C: Proof of Orthogonality of vn (R) with the Weight b . . . . . . . 473 Appendix D: Derivation of Formula (3.103) . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Appendix E: The Convergence of the Series in G 1 (t, τ, r) . . . . . . . . . . . . . 485 Appendix F: Numerical Solution of Equation (D.37) . . . . . . . . . . . . . . . . . . 487 Appendix G: Numerical Calculation of the Improper Integrals . . . . . . . . 491 Appendix H: Derivation of Equation (3.150) . . . . . . . . . . . . . . . . . . . . . . . . . 493 Appendix I: Evolution of the Droplet Shape . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Appendix J: Derivation of Expressions (3.162) . . . . . . . . . . . . . . . . . . . . . . . . 501 Appendix K: Derivation of Formula (4.21) . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Appendix L: Derivation of Formula (4.29) . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Appendix M: Derivation of Formula (L.29) . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Appendix N: Approximations for Alkane Fuel Properties . . . . . . . . . . . . . . 525 Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Appendix P: Derivation of Expression (4.91) . . . . . . . . . . . . . . . . . . . . . . . . . 557 Appendix Q: Derivation of Expression (5.8) . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Appendix R: Derivation of Expression (5.15) . . . . . . . . . . . . . . . . . . . . . . . . . 567 Appendix S: Solution of Equation (5.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Appendix T: Calculations of Q n based on (S.56)

. . . . . . . . . . . . . . . . . . . . . 585

Appendix U: Graphical Solutions of (5.45) and (5.46) . . . . . . . . . . . . . . . . . 587 Appendix V: Verification of the Numerical Code . . . . . . . . . . . . . . . . . . . . . . 591 Appendix W: Tikhonov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Chapter 1

Spray Formation and Penetration

1.1 Spray Formation Liquid spray formation is a complex process, many details of which are still not fully understood. One of the most rigorous overviews of this process is presented in [48, 125]. This chapter covers essentially the same topic as [48, 125], but from a different perspective. The focus will be on the physical transparency and engineering relevance of the models, rather than on their in-depth mathematical analysis. Jet formation starts inside the nozzle which, in the simplest case, is a cylinder, through which liquid is supplied to a chamber. The pressure drop across this cylinder is typically very high. For example, in the case of Diesel engines it can reach 1.8 × 108 Pa [61] with nozzle diameters in the range 0.1–0.2 mm [24]. The high velocity of liquid inside the nozzle can lead to a drop in pressure there below the vapour saturation pressure. This causes cavitation (the formation of small vapour-filled cavities in the liquid). Modelling and experimental investigations of this phenomenon affecting the discharge coefficient of the nozzle have been extensively described in the literature (e.g. [37, 72, 79]). The detailed analysis of these topics is beyond the scope of this monograph. A simplified approach to the analysis of cavitating flows based on their hydrodynamic similarity is discussed in [127], while the most comprehensive models are presented in [61, 110]. The authors of [61] claim that ‘cavitation modelling has reached a stage of maturity at which it can consistently identify many of the effects of nozzle design on cavitation, thus making a significant contribution to nozzle performance and optimisation’. The authors of [110] suggested a polydisperse cavitation model based on population balances. This model predicts the transitions from large bubbles to small cavities in regions of massive condensation, or from small bubbles to large cavities in regions of strong evaporation. The phenomenon, closely linked with cavitation, is known as superheated atomisation [70]. The dynamics of internal flow and jet break-up under injection pressures and initial conditions, relevant for Diesel engines, were studied by the authors of [26, 27]. They used a combination of high-speed microscope imaging techniques and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_1

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2

1 Spray Formation and Penetration

CFD numerical simulation with Volume of Fluid (VOF) and RNG k−ε models. Four main types of initial spray fragmentation were identified: (1) compact mushroom-shaped spray head, (2) thin mushroom-shaped spray head, (3) thin mushroom-shaped spray head with drummed plum and (4) thin mushroom-shaped spray tip with the wave-shaped spray. The effects of nozzle structure were investigated. A similar combined experimental and numerical study of internal flow inside the nozzle and spray formation was performed by the authors of [212]. They considered the same injector but with different nozzle taper angles. Fuel pressure and temperature were set at 200 bar and room temperature, respectively. It was found that adjusting the nozzle taper angle did not lead to any noticeable difference in the static flow rate. This rate was proportional to the square root of the pressure difference across the orifice plate. The initial tip penetration was shown to depend on the injection timing and was nearly independent of the nozzle taper angle. For well-developed sprays, larger nozzle taper angle was shown to lead to shorter penetration and faster break-up. The results of various numerical analyses of jet break-up, including the in-nozzle flows, are presented in [131, 169, 220, 221]. The primary break-up of biofuel sprays in pressurised gaseous crossflow was investigated numerically by the authors of [84]. The results of experimental studies of the in- and near-nozzle flow characteristics, referring to the straight-hole and diverging-tapered-hole Gasoline (Petrol) Direct Injection (GDI) nozzles, are presented in [130]. X-ray phase-contrast and absorption imaging techniques were used. It was shown that the diverging-tapered-hole nozzle with proper hole length generated a radially expanding crescent-moon-like flow structure in the hole which facilitated the spray atomisation and its dispersion outside the nozzle. The widely accepted understanding of spray formation after the liquid has left the nozzle is based on identifying the following stages: development of a jet, conversion of a jet into liquid sheets and ligaments, disintegration of ligaments into relatively large droplets (primary break-up) and break-up of large droplets into smaller ones (secondary break-up) [39, 109, 113]. These processes are clearly visible in the image of a typical spray presented in Fig. 1.1. Sometimes, liquid emerges from the nozzle in the form of liquid sheets, which disintegrate into ligaments and droplets, following the above scheme [183]. This scheme (and its various modifications, e.g. [115, 194]), however, is too crude to describe the observed spray formation [24, 48, 125] in most cases, and also too complex to turn it into a quantitative mathematical model. Alternative approaches to modelling these processes were considered in several papers including [83, 105, 126, 186]. The analysis of these processes was often based on the Reynolds-Averaged Navier–Stokes (RANS) equations, using commercial CFD codes including ANSYS FLUENT [58], Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) [20, 45, 55, 129], level-set and Volume-of-Fluid (VOF) methods [75, 141], coupled LES/VOF technique [192], specially developed axisymmetric Boundary Element Method (BEM) [69], fractal concept [96], Combined Level-Set Volume-

1.1 Spray Formation

3

Fig. 1.1 A typical image of a coal water slurry spray, with a coal mass fraction of 50%, injected from an industrial nozzle (see [102] for further details of the experiment). The image is reproduced with the consent of Prof. P. A. Strizhak of the National Research Tomsk Polytechnic University, Russia

of-Fluid (CLSVOF) method [5], and dynamic mesh refinement and step response theory [219]. All quantitative models of spray formation developed so far are based on the assumption that liquid jets emerging from the nozzle disintegrate directly into droplets due to the development of jet instabilities [41]. One of the main problems with the analysis of these instabilities lies in the fact that the disturbances of even two-dimensional flows (axisymmetric or plane) need to be considered as three-dimensional in the general case. For plane jets, this problem can be overcome using Squire’s theorem [191]. According to this theorem, for any unstable three-dimensional disturbance, there is a corresponding two-dimensional disturbance (with zero perturbation in the third dimension) that is more unstable than the threedimensional one [139]. This allows us to seek the stability of the plane jets with a two-dimensional disturbance. Unfortunately, the same approach has been widely applied to round jets, with the assumption that disturbances are also axisymmetric (e.g. [154, 155]). This approach is not necessarily wrong, but it cannot guarantee that the instability captured this way is the strongest one. A rigorous analysis of this problem, considering the three-dimensionality of the round jet disturbances, is described in [85, 111, 119, 165, 214]. The results of experimental studies of jet disintegration are presented in [99].

4

1 Spray Formation and Penetration

When studying jet instabilities, attention has been primarily focused on wellknown modal instabilities [41]. Possible contributions of non-modal instabilities of jets have been largely overlooked (see [13] for details). In what follows, some well-known spray formation models which give insight into the spray break-up processes are described. See [46] for a more comprehensive summary of these models.

1.1.1 Classical WAVE Model One of the most popular models of spray formation is the WAVE model based on the temporal analysis of the Kelvin–Helmholtz instability for a round liquid jet (density ρl ) in an inviscid gas (density ρg ) [153]. The liquid velocity is assumed to be constant inside the jet; it drops to zero at the liquid/gas interface. Assuming that the disturbances are small, axisymmetric (along the flow and in the radial directions) and are proportional to ∝ exp (ikz + ωt) , (1.1) the stability analysis leads to a dispersion equation which was presented as [154]  ω2 + 2νl k 2 ω

I1 (k R j ) 2kL I1 (k R j ) I1 (L R j ) − 2 I0 (k R j ) k + L 2 I0 (k R j ) I0 (L R j )



   L 2 − k 2 I1 (k R j ) σs k  2 2 1 − Rjk = L 2 + k 2 I0 (k R j ) ρl R 2j ρg + ρl

   2 L − k 2 I1 (k R j ) K 0 (k R j ) iω Uj − , k L 2 + k 2 I0 (k R j ) K 1 (k R j )

(1.2)

where U j and R j are the unperturbed jet velocity and radius, k is the real wave number, ω is the complex frequency (positive real part of ω shows the growth of instability), σs is the surface tension, νl is the liquid kinematic viscosity and L 2 = k 2 + νωl . Primes show differentiation. The value of U j was estimated as  Uj = Cj

2Δp , ρl

where C j and Δp are the discharge coefficient and discharge pressure, respectively. The following formulae for the maximum growth rate (Ω = max(Re(ω)) and the corresponding wavelength Λ were inferred from the numerical solution to Eq. (1.2) [153, 188]:

1.1 Spray Formation

5



ρl R 3j

0.5

0.34 + 0.38We1.5 g

,

(1.3)

Λ (1 + 0.45Z 0.5 )(1 + 0.4 T 0.7 ) = 9.02 ,  0.6 Rj 1 + 0.87We1.67

(1.4)

Ω

σs

=

(1 + Z )(1 + 1.4 T 0.6 )

g

where 2 Z = 2Wel0.5 /Rel , T = Z We0.5 g , Wel,g = ρl,g U j R j /σs , Rel = 2U j R j /νl .

Approximations (1.3)–(1.4) are valid for Z ≤ 1 and ρg /ρl ≤ 0.1 [153]. These conditions are acceptable for many engineering applications. Note a typo in the equation corresponding to (1.3) given in [117]. Z is often called the Ohnesorge number: Oh = νl

ρl . R j σs

(1.5)

This number does not depend on velocity but is proportional to viscosity [48]. Sometimes, Oh is defined using the jet diameter (2R j ) in Expression (1.5) [81]. In many engineering applications, one can assume that Rel  1. This leads to the following conditions Z  1, T  1. Assuming that Wel,g  1, Eqs. (1.3)–(1.4) can be rewritten as  0.5 ρl R 3j = 0.38We1.5 (1.6) Ω g , σs Λ 1 = 9.806 . Rj Weg

(1.7)

For a very slow-moving jet, the following conditions are expected Z  1, T  1, Wel,g  1. In this case, Expression (1.4) is simplified to Λ = 9.02R j . This is the well-known Rayleigh result: the most unstable wavelength of the jet satisfies the condition k R j ≈ 0.7 (cf. Fig. 1.5 of [41]). Some of the above-mentioned results referring to jet instabilities were incorporated into the WAVE model. In this model, a jet was approximated by a string of droplets leaving the nozzle with radii Rd greater or equal to R j . The droplet number density was found from the conservation of the liquid flow rate. The droplet velocities had two components: z−component, assumed to be close to U j , and the radial component in the perpendicular direction. The latter was expected to be proportional to the rate of wave growth Ω. Constructing a dimensionless parameter, based on U j and Ω, it was shown that the maximal deviation of the droplets leaving the nozzle from the z−axis, described by the angle Θ/2, follows from the following formula [153]:

6

1 Spray Formation and Penetration

 tan

Θ 2

 = Aj

ΛΩ , Uj

(1.8)

where the fitting constant A j depends on the design of the nozzle. For sharp entrance constant diameter nozzles, with length to diameter ratios in the range 4–8, this constant was assumed equal to 0.188 [153]. The axial angle ϕ was randomly chosen in the range 0–2π . The angle Θ predicted by Eq. (1.8) was identified with the spray cone angle. It was assumed that the angles of droplets leaving the nozzle are initially uniformly distributed in the range 0 to Θ/2. Detailed experimental studies of the values of this angle for plain jet air blast atomisers, widely used in industrial applications, were performed by the authors of [204]. When the wavelength Λ is much greater that R j , then the radii of the droplets leaving the nozzle were estimated from the condition of conservation of mass:   4 3 2 2 2πU j . π Rd = min π R j Λ, π R j 3 Ω

(1.9)

The first term on the right-hand side of (1.9) is the volume of a cylinder with the radius R j and height Λ. The second term in this equation contributes when the jet disintegrates over a distance less than Λ (strongly unstable jet). In many papers, including [153], Ω/(2π ) is identified with the disturbance frequency. This is not correct as this parameter refers to wave growth or damping. Equation (1.9) can be presented as [153]: Rd = min

 0.33  0.33 . 3R 2j Λ/4 , 3π R 2j U j /(2Ω)

(1.10)

If B0 Λ ≤ R j ,

(1.11)

where constant B0 is chosen equal to 0.61 to get agreement with data on droplet sizes in sprays, then the initial radii of droplets leaving the nozzle are assumed to be equal to R j . In contrast to the case when B0 Λ > R j , these droplets are unstable and break up until their radii reach (1.12) Req = B0 Λ. If R j = Req , then droplets leaving the nozzle are marginally stable. Using Eq. (1.7), Condition (1.12) for R j = Req can be rewritten as Weg = Weg(cr) = 9.806 × 0.61 ≈ 6. This is a well-known condition for bag break-up: Weg > Weg(cr) ≡ 6.

(1.13)

1.1 Spray Formation

7

Condition (1.13) refers to the case when Weg is defined based on droplet radius. If Weg is defined based on droplet diameter, then this condition needs to be rewritten as Weg > 12 (e.g. [21]). Some authors estimated Weg(cr) as 5.5 ± 1 (see [222]). To consider the effect of liquid viscosity, Eq. (1.13) was generalised to [222]   Weg(cr) = 6 1 + C1 OhC2 .

(1.14)

The empirical coefficients C1 and C2 , suggested by various authors, are presented and discussed in [222]. The authors of [10] demonstrated experimentally that bag break-up is expected to develop in the range of Weg 10 to 18. This result is consistent with the experimental data presented in [80] based on experiments on the secondary break-up of liquid metal Galinstan droplets exposed to a shock-induced crossflow. Their estimate of the range of Weg in which bag break-up is expected to develop was 11–18. The results of experimental observations of two closely spaced droplets in an air jet in the bag break-up regime are presented in [223]. Criterion (1.13) assumes zero viscosity of the ambient gas. If this assumption is relaxed, then a new criterion for droplet break-up can be derived based on the hypothesis that the gas boundary layer transmits shear stresses to the liquid, and these stresses lead to the break-up process. The criterion of this break-up, known as stripping break-up, can be written as [21, 134] (cf. [152])  Weg / Reg > 0.5.

(1.15)

Although Condition (1.15) does not follow from the classical WAVE model assumptions, the analysis of stripping break-up is widely used alongside the bag break-up analysis within the framework of the classical WAVE model [156, 157]. This tradition will be followed in our presentation of this model. The WAVE model was not developed to describe the details of the break-up process. The only process which it was designed to capture is the decrease with time of the average droplet radius described by the equation: Rd − Rd (eq) d Rd =− , dt tbu

(1.16)

where tbu is the characteristic break-up time, Rd (eq) is the radius of a marginally stable droplet, inferred from Condition (1.13) (bag break-up) or (1.15) (stripping break-up). Keeping in mind the underlying physics of the phenomenon, we can expect that tbu is proportional to Rd /Λ and inversely proportional to Ω. Following [153], these two requirements are combined in the following expression: tbu = 3.726

B1 Rd , ΛΩ

(1.17)

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1 Spray Formation and Penetration

where B1 is the fitting constant. Using (1.3)–(1.4), in the limits We g → 0 and Weg → ∞ Expression (1.17) can be simplified to  tbu = 1.72 B1 tbu =

B1 Rd Ud

ρl Rd3 , 2σs



ρl , ρg

(1.18)

(1.19)

respectively. Expression (1.18) with B1 = π/1.72 gives the characteristic bag break-up time, while Expression (1.19) gives the characteristic stripping break-up time [153]. There is some uncertainty regarding the choice of constant B1 in Expressions (1.18) and [157] considered (1.19). Nichols [134] assumed that B1 = 8, Reitz and Diwakar √ B1 = 20, while O’Rourke and Amsden [137] used B1 = 3. The effects of both coefficients B0 and B1 on performance characteristics of HSDI engine were investigated in [198]. Although the stripping break-up is expected to occur at higher Weg in the general case, since Reg is commonly much greater than 1, the condition Weg → 0 does not strictly speaking refer to bag break-up, which is expected at Weg > 6. The difference between the actual values of tbu and those which follow from the condition Weg → 0 can be accounted for by the choice of fitting constants used in the analysis. To improve the predictive capabilities of bag and stripping break-up models and reduce the requirement of their tuning, an alternative secondary break-up model was suggested by the authors of [9]. Their model is essentially based on the models considered in this section, but it was assumed that different models are activated in different zones. More specifically, it was assumed that only stripping break-up can occur near the nozzle, while bag break-up could only take place sufficiently far from it. Model parameters were treated as functions of the operating conditions.

1.1.2 TAB and Stochastic Models Apart from the WAVE model presented in Sect. 1.1.1, a few other models of spray formation have been described. In this section, two of these models are briefly summarised.

1.1.2.1

TAB Model

The Taylor Analogy Break-up (TAB) model describes the process in terms of the critical deformation of an oscillating–distorting droplet [137]. The external force is caused by the relative droplet motion, the restoring force is the surface tension force,

1.1 Spray Formation

9

and the damping is controlled by the liquid viscosity. It is assumed that break-up occurs when the droplet deformation exceeds Rd /2 (some results of the analysis of droplet deformation at low Weber numbers are shown in [63]). The Sauter Mean Radius (SMR) of the product droplets (volume-to-surface ratio of the ensemble of droplets) at the time instant when break-up occurs is found from the conservation of droplet energy during the break-up process: SMR =

Rd 7 3

+

ρl Rd vdef 4σs

,

where Rd is the parent droplet radius and vdef is the velocity of droplet deformation at the time instant when break-up occurs. In contrast to the classical WAVE model, in the TAB model, after break-up, the product droplets’ radii Rd pr follow the distribution: f (Rd pr ) =

  Rd pr exp − , R R 1

where R = SMR/3. The spray penetration predicted by the TAB model agrees with the results of measurements presented in [78]. At the same time, it over-predicts the rate of droplet break-up, and tends to predict smaller droplets close to the injector (cf. [199]). Despite the above-mentioned problems, the TAB model is widely used for spray computations, and it is a default break-up model in KIVA 2 code. Advanced versions of the TAB model, called improved TAB model and modified Navier–Stokes (M-NS) model, were suggested by the authors of [195].

1.1.2.2

Stochastic Model

As follows from the previous analysis, the WAVE model is essentially a deterministic model, in which the radii of product droplets are predicted by Eq. (1.16). The TAB model has a stochastic element in choosing the radii of product droplets assuming that the distribution function of these droplets is a priori given, but it still focuses on sample droplets rather than on the whole spectrum. The model suggested in [66] is based on a different approach to break-up modelling. The approach used in [66] assumes, following Kolmogorov [97], that the break-up of parent particles into secondary particles does not depend on the instantaneous sizes of the parent particles. This assumption is obviously not valid when Rd is close to Rd(eq) . In highpressure injection sprays, characterised by large Weber numbers, the hydrodynamic mechanism of atomisation due to the velocity difference at the liquid–gas surface, can be complicated by the impact of turbulent fluctuations on jet break-up [66]. Under such conditions, when the specific mechanism of atomisation and the scale of the

10

1 Spray Formation and Penetration

break-up length cannot be clearly defined, stochastic approaches to the modelling of break-up become more appropriate than deterministic ones. It was shown in [66] that in the limit of long times t → ∞, the general equation for the evolution of the droplet number distribution function F(Rd ) can be presented in the form of the Fokker–Planck-type equation:   9 ∂ ∂ F(Rd ) = −3 ln α − ln2 α − Rd ln α ∂t 2 ∂ Rd   2  ∂ 1 ∂ Rd Rd ln α ν F(Rd ), + 2 ∂ Rd ∂ Rd where



 lnn α =



1

(1.20)

lnn α q(α) dα,

0

0 ≤ α ≤ 1 is the parameter linking the radii of product (Rd ) and parent (Rd0 ) droplets (α = Rd /Rd0 ), q(α)dα is the normalised probability that the radius of each product droplet is in the range [α Rd , (α + dα)Rd ], ν = ν0 q0 , ν0 is the break-up frequency of an individual droplet and q0 is the average number of droplets produced after each break-up.   Equation (1.20) includes two unknown constants ln α and ln2 α . To reach an agreement between the predictions of the  model and experimental data [78], it was assumed that ln α = −1/2 and ln2 α = 1. The frequency of break-up ν was obtained from the expression: 1 |U | ν= B1 Rd0



ρg . ρl

(1.21)

√ The value of constant B1 = 3 was chosen in order to match experimental results referring to the stripping break-up of droplets. Further developments of this model were discussed in several papers including [67, 68, 159, 160, 171]. An alternative approach to considering the effects of turbulence on droplet break-up using the Spectrum Analogy Break-up (SAB) model is described by Habchi [71].

1.1.3 Modified WAVE Models Several modifications to the classical WAVE model have been suggested (e.g. [118]). In what follows, some of these modifications are described.

1.1 Spray Formation

1.1.3.1

11

Rayleigh–Taylor Break-up Based Model

The analysis of the Rayleigh–Taylor instability, used in the original WAVE model, did not consider the effects of viscosity and surface tension [41]. This analysis predicted the instabilities for all wavelengths of the initial disturbance; the rate of growth of disturbances grew with decreasing wavelengths. This model was generalised in [7] to consider the effects of viscosity and surface tension. This generalisation of the model led to the prediction of instability in a limited range of wavelengths. In the case when the surface tension was considered but the contribution of viscosity was ignored, the wavelength of the most unstable wave was estimated as  ΛRT = 2π

3σs , aρl

(1.22)

where σs is the surface tension, ρl is the liquid density and a is the acceleration perpendicular to the surface. The rate of wave growth at this wavelength was predicted as ΩRT =

  2a aρl 1/4 . 3 3σs

(1.23)

When deriving Expressions (1.22) and (1.23), it was assumed that ρg  ρl . Expressions (1.22) and (1.23) were used for modelling droplet break-up by several authors. The results are described in several papers, including [108, 144, 162]. The contribution of gravity to a was not taken into account. a was estimated as the ratio of the drag force to the mass of droplets. This leads to the following formula [144]: a=

ρg U 2 3 , CD 8 ρl Rd

(1.24)

where C D is the drag coefficient. Patterson and Reitz [144] suggested that droplet break-up due to the Rayleigh– Taylor instability occurs when (1.25) 2Rd > ΛRT . Using (1.22) and (1.24), this condition can be presented as Rd > aRT

σs , ρg U 2

(1.26)

where aRT = 32π 2 /(3C D ). Since aRT is expected to be well above 6 in most practically important cases, Condition (1.26) is more stringent than the corresponding condition for bag instability (Condition (1.13)). For the Newton flow regime (see [40]) when C D = 0.44, aRT =

12

1 Spray Formation and Penetration

239. In several papers, including [108]) the right-hand side of Eq. (1.22) is multiplied by an adjustable constant CRT in the range 1–9. This makes Condition aRT  6 even more reliable. Alternative approaches to modelling bag break-up and liquid film disintegration during droplet bag break-up mode are described in [62, 64]. A simplified analytical model for droplet break-up was presented in [185].

1.1.3.2

Models Based on the Rigid Core Concept

One of the main weaknesses of the classical WAVE model is that it assumes that the jet disintegrates immediately at the exit of the nozzle. This is not supported by several experimental observations, including those referring to spray penetration, discussed below and in the next section. To overcome this problem, several authors suggested modified versions of the WAVE model, using the assumption that the jet behaves as a solid body at the exit of the nozzle over a certain distance from it. In what follows, two of these models are briefly described. In the model described in [178], it was assumed that parcels constituting the liquid core experience no drag from the gas and move as a rigid jet (core) at a velocity equal to the injection jet velocity U = U j . This assumption was based on the experimental observation that the momentum of the core of a Diesel spray is conserved [172, 182] and is compatible with the results of highly resolved Volume-of-Fluid simulations presented in [3]. Also, Karimi [93] showed experimentally that at early injection times the velocity of the jet, estimated from the mass flow rate, is close to the velocity of the jet tip. This model was incorporated into KIVA II CFD code using a modified version of the collision algorithm of Nordin [135] for droplets in the liquid core and the conventional algorithm by O’Rourke [137] away from the core. The radius of the liquid core was allowed to decrease due to the stripping of droplets from its surface. This process continued until its radius became half the radius of the nozzle. Once this has happened, the conventional WAVE model with modified values of parameters was used. These modifications referred primarily to transient accelerating jets. The decrease in Ω with increasing injection acceleration was considered, but it was assumed that the wavelength of critical instability Λ was not affected by the transient nature of the flow. The decrease in Ω with increasing injection acceleration was related to the observation that flow acceleration is expected to lead to relaminarisation of the flow and thickening of the boundary layer in the gas phase around the jet for a certain range of Reynolds numbers [132]. The increase in the boundary layer thickness was, in turn, expected to stabilise the gas–liquid interface [120]. This implies suppression of instability by the acceleration of the flow. Since tbu ∼ 1/Ω, the effect of flow acceleration was accounted for by modifying the expression for B1 in Expression (1.17). The following formula was used:  c B1 = B1 st + c1 a + 2 ,

(1.27)

1.1 Spray Formation

where

13

√ Rd dUinj a + = 2 Re 2 Uinj dt

is the acceleration parameter considering the effect of flow acceleration; c1 and c2 are adjustable constants. In the steady-state limit, a + is zero and B1 = B1 st . Following Reitz [153], it was assumed that B1 st = 10. The acceleration parameter a + was constructed by analogy with the local pressure gradient parameter p + described in [17], assuming the laminar-type dependence of the local skin friction coefficient on the Reynolds number. In the model described in [202], the length of the rigid core (break-up length) was estimated using the following expression: L b = min (L s , L bu ) , 

where

t

Ls =

U j (t  )dt 

(1.28)

(1.29)

0

is the penetration length of the solid jet,  L bu =

t

cg (t  )dt  ,

(1.30)

t−tbu

cg is the group velocity of the fastest growing disturbance and tbu is the break-up time inferred from Expression (1.19). Assuming that Weg  1, cg /U j was shown to be constant in the range 0.91–0.99 for ρg /ρl in the range 0.1–0.01. This assumption is consistent with the assumption of the validity of (1.19). It is satisfied in many practical engineering applications, including those in Diesel engines. At distances larger than L b the classical WAVE model was used. This approach was shown to be as accurate as the one described in [178]. In contrast to [178], however, it does not require the specification of two fitting constants c1 and c2 . Also, in contrast to [178], it does not use the hypothesis that jet acceleration leads to stabilisation of the jet. As follows from the analysis of the stability of plane jets, described in [200, 201], the effect of acceleration is not expected to be the dominant factor in the development of jet break-up. Also, it was shown in [202] that considering the effects of gas viscosity by modifying the velocity profile in the gas phase allows larger droplets to be predicted at jet break-up. This leads to the prediction of droplet sizes consistent with experimental observations. Models described in [1, 215] are notable models using the rigid core concept.

14

1 Spray Formation and Penetration

Fig. 1.2 The conceptual illustration of the model of primary break-up described in [82]. Reproduced from Fig. 1 of [82] with permission from Begell House

1.1.3.3

A Unified Spray Break-up Model

In the previous sections, basic principles of the WAVE model and its modifications were presented. These principles are applicable to a wide range of sprays, including those used in internal combustion engines and fire extinguishers [187]. Further refinements of this model are essential when quantitative analysis of specific processes is required. These refinements have been focused on the description of the process rather than its individual elements. Thus, the models based on these refinements are generally called ‘unified models’. These unified models generally lose their universality and are applicable for a specific range of parameters including the shapes of the nozzles. In what follows, one of these models, described in [22], focused on internal combustion engine applications, is briefly summarised. The primary break-up model used in [22] was originally described in [82]. The conceptual picture describing the essence of this model is schematically shown in Fig. 1.2. This model considers two main processes: the initial perturbation and wave growth on the jet surface, eventually leading to the detachment of droplets. The model is based on two key assumptions. Firstly, the length scale of atomisation (L A ) is proportional to both turbulence length scale (L t ), describing the initial perturbation, and the wavelength (L w ): L A = C1 L t = C2 L w .

(1.31)

Secondly, the time scale of atomisation (τ A ) can be presented as a linear sum of turbulence (τt ) and wave growth (τw ) time scales: τ A = C3 τt + C4 τw .

(1.32)

1.1 Spray Formation

15

Empirical constants C1 , C2 , C3 and C4 are assumed equal to 2.0, 0.5, 1.2 and 0.5, respectively. Assuming that turbulence can be described by the classical k − ε model, the initial values of L t and τt are found from the following expressions: L t = Cμ

k 3/2 , ε

(1.33)

k τt = Cμ , ε

(1.34)

where Cμ = 0.09, k and ε are the turbulence kinetic energy and dissipation rate, respectively [205]. The initial values of k and ε (k0 and ε0 ) are found from the balance of forces acting on the flow in the nozzle. Only gas inertia and turbulent internal stresses are considered [82]: k0 =

  1 U2 − K c − (1 − s 2 ) , 8 (L/D) C j

U3 ε0 = K ε 2L



(1.35)

 1 2 − K c − (1 − s ) , Cj

(1.36)

where L and D are the nozzle length and diameter, respectively (typically 3 · 10−4 m and 1.5 · 10−3 m), U is the jet velocity at the nozzle (approximately 200 m/s), C j is the discharge coefficient, mentioned in Sect. (1.1.1) (assumed equal to 0.7), K ε is the constant considering the shape of the nozzle exit (for a sharp entrance corner, typically K ε = 0.45) and s is the area ratio at the nozzle contraction (typically s = 0.01 [82]). Assuming that turbulence is homogeneous, the following solution to the equations of the k − ε model can be obtained [82]:  k(t) =

ε0 k0Cε

1/(1−Cε ) (Cε − 1) t +

k01−Cε

ε(t) = ε0 [k(t)/k0 ]Cε ,

,

(1.37)

(1.38)

where Cε = 1.92 [205]. Having substituted (1.37) and (1.38) into (1.35) and (1.36), the following expressions for L t and τt were obtained:   0.0828 t 0.457 , L t (t) = L 0t 1 + τt0

(1.39)

τt (t) = τt0 + 0.0828t,

(1.40)

16

1 Spray Formation and Penetration

where t is time since the start of injection, L 0t and τt0 are the initial values of L t and τt . τw is estimated as (cf. Eq. (1.19)) τw =

Lw U



ρl , ρg

(1.41)

where L w is the wavelength of the fastest growing wave as in the conventional WAVE model. As in the classical WAVE model, the liquid jet is presented as a string of droplet parcels. The break-up rate of individual droplets, however, is estimated not based on Eq. (1.16) but using the following equation: k1 L A d Rd , =− dt 2 τA

(1.42)

where the calibration constant k1 is chosen equal to 0.5 [22]. The drag force, acting on the droplets, emerging from the nozzle, is considered to be the same as the one acting on the cone-shaped liquid core with the drag coefficient equal to 0.3 [22]. The authors of [82] estimated the cone half angle to be tan

Θ L A /τ A = . 2 U

(1.43)

The authors of [22] found that Expression (1.43) tends to under-estimate Θ and suggested that this angle should be doubled, compared with the one predicted by (1.43)   L A /τ A . (1.44) Θ = 4 tan−1 U If the atomiser produces a conical liquid sheet instead of a jet, then θ is controlled by the angle of deflection [22]. The liquid sheet instability atomisation model, described in [183], was recommended for the analysis of the instability in this case [22]. The behaviour of the droplets formed during the primary break-up depends mainly on the Weber number Weg , introduced in Eqs. (1.13)–(1.14). When Weg < 6, the droplets do not break-up, but rather deform to form oblate spheroids [22]. At Weg > 6, the droplets can undergo secondary break-up. The possibility of the break-up and its type are controlled both by the values of Weg and the values of the Ohnesorge number, defined by Expression (1.5). At small Ohnesorge numbers (Oh < 0.1), the transition between break-up regimes depends on We g only [50]. The following breakup regimes were identified in this range of Oh [22, 28, 50]: bag break-up (6 < Weg ≤ 10), multimode break-up (10 < Weg ≤ 40), shear break-up (40 < Weg ≤ 425) and catastrophic break-up (Weg > 425). The threshold values of Weg for these regimes increase as Oh increases. Viscous forces inhibit droplet deformation which is the first

1.1 Spray Formation

17

step in the break-up process [81], except for catastrophic break-up, when the range does not depend on Oh. In all four break-up regimes, atomisation was modelled as a rate process. The detailed analysis of these regimes, incorporation of the relevant models into a numerical code and validation of the results against experimental data for internal combustion engine applications are presented in [22]. Droplet properties after the secondary break-up at Oh < 0.039 were investigated experimentally in [81]. Alternative approaches to the multi-scale analysis of liquid atomisation processes are presented in [38, 42–44, 123]. The authors of [114] combined several previously suggested models into one which they called Wu–Faeth–Kelvin–Helmholtz– Rayleigh–Taylor atomisation model. This model was successfully applied to gasoline (petrol) direct injection (GDI) spray simulations. A review of some primary atomisation models is presented in [76].

1.2 Spray Penetration As follows from the analysis presented in the previous section, spray formation is a complex process the details of which are not yet fully understood. This detailed understanding, however, is not always necessary for practical applications of spray models. In many cases, researchers focus just on one aspect of spray behaviour instead of trying to develop a comprehensive universal model. In most cases, this aspect is spray penetration. Interest in spray penetration has been motivated by three main factors: Firstly, the practical importance of spray penetration (e.g. optimisation of spray penetration in internal combustion engines [77]). Secondly, this parameter is easily measurable, which can be used for validating the models [98]. Thirdly, the correct prediction of the spray penetration can indirectly indicate the correctness of complex models of spray formation. Various phenomenological and semi-phenomenological models of spray penetration and spray cone angle formation have been suggested and described (see [95] for details). Their overview is beyond the scope of this book. Physical and mathematical models of spray penetration which have been developed so far, fall into two categories. These are (1) models based on Computational Fluid Dynamics (CFD) codes with submodels describing jet and droplet break-up processes implemented into them (see Sect. (1.1) and numerous publications, including [107], for more details); (2) simplified models, in which spray penetration has been predicted from first principles. These groups of models are complementary and are sometimes used in parallel. This section is focused on the second group of models which usually allow one to get better insight into the physical background of the processes. Only axisymmetric jets are considered. The analysis of the influence of cross winds on the dynamics of sprays is described by Ghosh and Hunt [60].

18

1 Spray Formation and Penetration

The problem of spray penetration is closely linked with the problem of induced air velocity within droplet-driven sprays. The latter problem was extensively studied by Ghosh and Hunt [59] who considered 3 spray zones: zone 1, where the initial velocities of droplets are much greater than that of the air stream and are not much affected by it; zone 2, where the droplets slow down and their velocities become comparable with the air velocity; zone 3, where the droplet velocities become comparable or even less than droplet terminal velocities. These zones have the same physical meanings as mixing, transition and fully developed regions considered by Borman and Ragland [12]. All simplified models suggested so far have been restricted to zones 1 (initial stage) and 3 (fully developed region, which is referred to as the two-phase flow). In what follows, the models developed for these zones are described.

1.2.1 The Initial Stage The model for the initial stage of spray penetration suggested in [174, 176] is based on the analysis of trajectories of individual droplets formed at the exit of the nozzle. It was assumed that the only force acting on the droplets is the drag force. The contributions of other forces, including gravity and added mass forces (see [147]), were not taken into account. This was justified by the small size of the droplets and the fact that droplet density is much greater than ambient air density. The effects of droplet break-up, evaporation and air entrainment were considered. As a result, explicit or implicit expressions for spray penetration as functions of time were obtained for the cases of Stokes (Red ≤ 2), Allen (2 < Red ≤ 500) and Newton (500 < Red ≤ 105 ) flows, where Red is the droplet Reynolds number based on droplet diameter. These expressions were based on the formulae for the drag coefficient for spheres in these flows [40]: ⎫ when Red ≤ 2 C D = 24/Red ⎬ C D = 18/Re0.6 when 2 < Red ≤ 500 (1.45) d ⎭ when 500 < Red ≤ 105 . C D = 0.44 Note a typo in [40] where the transition from the Stokes to Allen flow is identified with Red = 0.2. An alternative formula for this coefficient, valid for 1 < Red < 800 for isothermal flows, was suggested in [181]: CD =

 1  1 + 0.15Re0.687 . d Red

(1.46)

1.2 Spray Penetration

19

This formula was generalised to the case of non-isothermal flows by the authors of [49]. Formula (1.46) or its generalisation could be used to develop an alternative model for the initial stage of spray penetration. One of the main weaknesses of the model developed in [174, 176] lies in the modelling of air entrainment at the initial stage of spray formation. As in [59], it was assumed that air velocity at this stage is much less than droplet velocity. This assumption leads to the prediction of strong drag when the jet leaves the nozzle. As follows from experimental observations, however, mass fraction of air in the vicinity of the nozzle is much less than the mass fraction of liquid (see [208] for detailed measurements of the near-nozzle air entrainment in high-pressure Diesel sprays). As a result, air is expected to be almost instantly entrained by liquid fuel in this region. This enables liquid leaving the nozzle to maintain the injection velocity while it is still close to the nozzle. This was considered in the models of spray formation described in Sect. (1.1) and allows us to predict the initial spray penetration as  s=

t

vinj (t)dt,

(1.47)

t0

where vinj (t) is the time-dependent injection velocity. Despite its simplicity, Eq. (1.47) is expected to predict the initial spray penetration more accurately than the models described in [174, 176] in most cases. Roisman et al. [161] drew attention to the fact that jet velocity at the exit from the nozzle can exceed the speed of sound in air ca . This is expected to lead to the formation of a shock wave in front of the jet. This shock wave was assumed to be one-dimensional and normal to the spray axis. This can be valid only for times less than D0 /ca , where D0 is the nozzle diameter. It is not clear how the model described in [161] can be generalised for longer times.

1.2.2 Two-Phase Flow The analysis of spray penetration is simplified by the fact that zone 3, where droplet velocities are almost equal to ambient air velocities, occupies most of spray volume. The analysis of droplet and air dynamics in this case can be based on the assumption that droplet and air velocities are equal. This allows us to treat the system dropletsambient air as a two-phase flow. Most models predicting spray penetration, which approximate spray as a twophase flow, are based on the analysis of the conservation of mass and momentum at various spray cross-sections (e.g. [35, 36, 161, 174]). These models differ in some underlying assumptions, but all√ of them predict that the dependence of spray penetration s on time t is close to s ∼ t. In what follows, one of the earlier models, developed for fuel sprays by the authors of [174], is presented. Although this model was developed more than two decades ago, its predictions still fit experimental data marginally better than the predictions of other similar models (e.g. [206]).

20

1 Spray Formation and Penetration

From the equation of conservation of mass of droplets, we obtain ρd A0 vin = ρm Am vm − (1 − αd )Am ρg vm ,

(1.48)

where A0 is the cross-sectional area of the nozzle, vin the initial velocity of droplets, ρm the density of the mixture of droplets and gas, Am the cross-sectional area of a spray and vm the velocity of the mixture. When deriving Eq. (1.48) and the following equations, the effects of the gradient of droplet number densities and velocities inside the spray in the direction perpendicular to the spray axis were ignored. These effects were considered in a number of papers, including [23]. Note that the state of droplets is not important in (1.48) and αd considers the contribution of the gaseous fuel as well. The left-hand side of Eq. (1.48) is the mass flow rate of fuel at the nozzle. The second term on the right-hand side of (1.48) considers the contribution of entrained air. The first term is the mass flow rate of the mixture of fuel and air. The following relation between Am and A0 was used: Am = A0 + π D0 s tan θ + π s 2 tan2 θ,

(1.49)

where s is the distance from the nozzle along the axis of the spray, θ = Θ/2 the spray half angle and D0 the diameter of the nozzle. When deriving (1.49), it was assumed that θ is constant. This assumption is similar to the one made by other authors (e.g. [142]). It was relaxed in [173]. The condition of conservation of momentum can be presented as 2 = ρm Am vm2 . ρd A0 vin

(1.50)

Equations (1.48)–(1.50) lead to the system of equations: ⎫ + (1 − αd )ρ˜a ⎬

ρ˜r =

v˜ A˜

ρ˜r =

v˜ 2 , A˜

(1.51)

⎭

where Am 4 s tan θ 4 s 2 tan2 θ A˜ = =1+ + ; A0 D0 D02

ρ˜r = ρm /ρd ;

ρ˜a = ρa /ρd ;

v˜ = vin /vm .

Having eliminated ρ˜r from (1.51), the following physically meaningful solution was obtained:    1 ˜ 1 + 1 + 4(1 − αd )ρ˜a A . (1.52) v˜ = 2

1.2 Spray Penetration

21

As follows from (1.52), in the case of no entrained air (αd = 1) we have v˜ = 1, which means that vm = vin . This solution, however, does not have a physical meaning since the formation of a spray always includes entrained air. In a realistic spray environment αd  1. ˜ Solution (1.52) can be presented as Remembering the definitions of v˜ and A,  2vin ds  = , √ dt m 1 + a + bs + cs 2

(1.53)

where a = 1 + 4(1 − αd )ρ˜a ;

b=

16(1 − αd )ρ˜a tan θ ; D0

c=

16(1 − αd )ρ˜a tan2 θ , D02

  is the velocity of the mixture. subscript m indicates that ds dt m Integration of (1.53) gives √ 2cs + b  b a a + bs + cs 2 − 4c 4c    2 c(a + bs + cs 2 ) + 2cs + b 4ac − b2 ln = 2vin t. + √ 8c3/2 2 ac + b s+

(1.54)

Two limiting cases of Eq. (1.54) were investigated: small s (a  bs  cs 2 ) and large s (a  bs  cs 2 ). For a  bs  cs 2 (immediate vicinity of the nozzle), Eq. (1.54) was rewritten as s=

2vin t  ≈ vin t. 1 + a + 8cb√a (a − 1)

(1.55)

ρ˜a  1 and a − 1  1 was assumed in deriving (1.55). Expression (1.55) shows that close to the nozzle vm ≈ vin . The condition a  bs  cs 2 is satisfied when s is large and/or D0 is small. In these cases, Eq. (1.54) can be rewritten as s+

√ √     b a a + 2bs 4ac − b2 4cs s2 c = 2vin t. (1.56) − 1+ + ln √ 2 2cs 2 4c 8c3/2 2 ac + b

Equation (1.56) can be further simplified if we remember that:  b √ ≈ 2 ρ˜a  1. 2 ac Condition (1.57) allows us to reduce (1.56) to

(1.57)

22

1 Spray Formation and Penetration

√   √  1 a 2 cs s2 c +s+ √ + ln = 2vin t. 2 a 2 c 2

(1.58)

Since x  ln x for large x, we can keep the two highest order terms on the lefthand side of Eq. (1.58). This allows us to reduce (1.58) to √   2 s2 c 1+ √ . 4vin s c

t≈

(1.59)

Equation (1.59) can be rearranged to √ s=



vin D0 t

√ (1 − αd )1/4 ρ˜a 1/4 tan θ

√

D0

1− √ √ √ 4 vin (1 − αd )1/4 ρ˜a 1/4 tan θ t

 .

(1.60)

Expression (1.60) can be further simplified if the second term on the right-hand side is ignored. This allows us to present (1.60) as √ s=

(1 −

vin D0 t

αd )1/4 ρ˜a 1/4

. √ tan θ

(1.61)

The structure of Expression (1.61) for spray penetration s is similar to the one suggested by several authors (e.g. [34]). From this point of view, Eqs. (1.60) and (1.54) can be considered as generalisations of previously suggested formulae. The combination of Eqs. (1.55) and (1.61) gives the expression for spray penetration similar to the ones presented by Lefebvre [109] and Borman and Ragland [12]. Formulae (1.60) and (1.61) can be compared with the empirical correlation for biodiesel sprays, suggested in [11] and used in [133], which predicts that s ∼ t 0.5432 . For practical applications, it seems more appropriate to use the general Eqs. (1.54) and (1.60) rather than their simplified versions (1.55) and (1.61). The main advantage of Eq. (1.54) is that it can accurately predict a smooth transition from the immediate vicinity of the nozzle to the two-phase flow in the region where the spray is fully developed. Separate solutions for the near zone and the far zone described in [12, 109] inevitably lead to a physically unrealistic jump in the velocity between these zones. The value of vin is controlled by the pressure drop in the nozzle (Δp):  vin = C j 2Δp/ρd ,

(1.62)

where C j is the discharge coefficient. There is some uncertainty regarding the value of C j . Chehroudi and Bracco [18] recommend C j ≈ 0.7, while Lefebvre [109] and Borman and Ragland [12] believe that this value is close to 0.39 (see Eq. (7.7) in [109] and Eq. (9.22a) in [12]). Note that there is a typo in Eq. (7.7) of [109]: ρ A in this equation should be replaced by ρl , which is the same as ρd in our notation.

1.2 Spray Penetration

23

√ The proportionality of spray penetration to t has been predicted by a number of other simplified models of spray penetration including the ones described in [19, 224]. The spray penetrations predicted by Eqs. (1.54), (1.60) and (1.61) were compared with the experimental results presented in [4] (Case 1) and [197] (Case 2). The parameter θ (half cone angle Θ/2) can be estimated based on available theoretical formulae [109] or obtained experimentally from the data in the original papers. The second approach was chosen, as it was considered to be more reliable. This gave the following values: θ ≈ 13◦ (Case 1) and θ ≈ 19◦ (Case 2). It was shown (see Fig. 1 of [174]) that all three Eqs. (1.54), (1.60) and (1.61) give reasonably accurate predictions of the observed spray penetration. The spray penetration predicted by Eqs. (1.54) and (1.60), however, was noticeably closer to the experimental data than the spray penetration predicted by the simplified Expression (1.61). Since the results predicted by Eqs. (1.54) and (1.60) were very close, it was recommended that Eq. (1.60) should be used for modelling the spray penetration. The predictions of the model described above were shown to be close to the penetration of Diesel fuel spray observed at Brighton University (UK) and a high-pressure dimethyl ether spray penetration observed at Chungbuk National University (Korea) [177]. This gives additional support to the viability of this model. Also, experimental results reported √ in [145] show that spray penetration length is approximately proportional to t. Results of a detailed experimental study of the dependence of liquid-phase penetration on the type of fuel are presented in [143]. A simplified model for a Diesel fuel spray described in [124] is based on the same continuity and momentum equations as the model described above based on [174]. Good agreement between predicted and observed liquid spray penetration and cone angle were demonstrated by these authors. Some other semi-empirical models of spray penetration and comparisons of their predictions with the results of experimental measurements are reviewed by the authors of [213, 218]. Despite the encouraging results above, the agreement between model predictions and experimental data turned out to be far from universal. For example,√Kostas et al. [98] demonstrated that in their experiments, s ∼ t 3/2 rather than s ∼ t. This and other similar results led researchers to look for alternative approaches to the analysis of spray penetration. In what follows, one of these approaches is described.

1.2.3 Effects of Turbulence In Sect. 1.2.2, analytical expressions for spray penetration were derived using the equations for conservation of mass and momentum for a two-phase flow. Several simplifying assumptions were made when deriving these expressions. These include the assumption that the density of the mixture of gas and droplets in the planes perpendicular to the spray axis is constant inside the spray and zero outside it. The shape of the spray boundary was controlled exclusively by the spray cone angle. These assumptions are reasonable if the effects of turbulence are not considered.

24

1 Spray Formation and Penetration

In more realistic cases, when the effects of turbulence are considered, their validity becomes questionable. In what follows we describe, following [151], an approach in which the effects of turbulence are considered and these assumptions are relaxed. The previous assumption that the density of the mixture of droplets and gas (ρm ) is constant in the plane perpendicular to the spray axis inside the spray is replaced with the assumption that it depends on the distance from the spray axis r as [151]   V0 r 2 , ρm = ρm0 (z) exp − 4Dt z

(1.63)

where ρm0 (z) is the mixture density at the axis of the spray, Dt is the turbulent diffusion coefficient and V0 is the initial velocity which is equal to vin (cf. Eq. (1.48)). Assuming that ρm0 (z) is a weak function of z, Expression (1.63) predicts that √ the curves of constant ρm correspond to r ∝ z. This parabolic form of the spray shape was observed experimentally [173]. Assuming axial symmetry of the spray and supposing that the velocity of the mixture vm is constant for given z, we calculate the mass flow rate of the mixture of droplets and gas at level z as [151] 

∞

m˙ = 2π 0

vm ρm r dr =

4π Dt ρm0 vm z = ρm0 Am vm , V0

(1.64)

where Am = 4π Dt z/V0 is the effective cross-section of the spray. Expression (1.64) predicts that the mass flow rate is zero when z → 0. This means that this expression is not applicable for small z. One would be interested in constructing a model which could predict accurate results for large z, but still reasonable ones for z → 0. This was achieved by replacing the effective cross-section introduced above with Am defined as Am = A0 + 4π Dt z/V0 ,

(1.65)

where A0 is the cross-sectional area of the nozzle (the same as in Expression (1.49)). In the limit z → ∞, the contribution of A0 is small, while in the limit z → 0 Expression (1.65) reduces to the physically correct statement that Am = A0 . Ignoring the contribution of air outside of the area Am and assuming that the volume fraction of droplets αd is small (this assumption is valid everywhere except the immediate vicinity of the nozzle), we can present the equation of conservation of mass in the form almost identical to Eq. (1.48): ρd A0 V0 = ρm0 Am vm − (1 − αd )Am ρg vm ,

(1.66)

where Am is defined by Expression (1.65), αd is the volume fraction of droplets, as in Eq. (1.48). Similarly, we can present the equation for conservation of momentum in a form almost identical to Eq. (1.50):

1.2 Spray Penetration

25

ρd A0 V02 = ρm0 Am vm2 .

(1.67)

The combination of Eqs. (1.65), (1.66) and (1.67) gives the equation for the velocity of the mixture (cf. Eq. (1.53)):  dz  2V0 , = vm ≡ √ dt |m 1 + a + bz

(1.68)

where a = 1 + 4[1 − αd ]ρ˜a ;

b=

64[1 − αd ]Dt ρ˜a . V0 D02

ρ˜a = ρg /ρd .

Integration of Eq. (1.68) gives 3bz + 2[a + bz]3/2 = 6V0 bt.

(1.69)

For large z when bz  a, Eq. (1.69) is simplified to  z=

9V02 b

1/3 t 2/3 =

V0 4



9d02 (1 − αd )ρ˜a Dt

1/3 t 2/3 .

(1.70)

The dependence of z on t is different from the one described by Expression (1.61). When αd → 0, Expressions (1.70) and (1.61) predict the same penetration if √ 3/2 9 V0 (tan θ )3/2 d0 t Dt = . 1/4 64 ρ˜a

(1.71)

For the values of parameters inferred from experiments [174] (case 1), V0 = 318.3 m/s, d0 = 0.2 mm, θ = 13◦ , t = 1 ms, ρ˜a = 19.7/760 = 0.025, we obtain Dt = 0.1 m2 /s. The turbulent diffusivity coefficient can be estimated from the Tchen formula Dt = σu2 TL∗ (strictly valid only in homogeneous turbulence for long times), where σu2 = 2k/3 and TL = O(k/). There is much uncertainty regarding the experimental measurements of spray penetration (cf. [173]). The authors of [173] concluded that the shapes of low-pressure injection√ sprays are close to conical and spray penetration is approximately proportional to t. The shapes of high-pressure injection sprays were shown to be close to parabolic, described by Eq. (1.63). In this case, the spray penetration is expected to be reasonably well described by Expression (1.70) and is proportional to t 2/3 . These results were compared with experimental data inferred from a high-speed video recording of a Diesel spray injected at 100 MPa into air with a density of 49 kg/m3 . It was shown that the experimental data fit the curve ∝ t 2/3 noticeably better than the ∝ t 1/2 plot. The maximum and mean deviations for the plot ∝ t 2/3 (37.8% and 9.6%) were clearly less than the maximum and mean deviations for the plot ∝ t 1/2 (65.0%

26

1 Spray Formation and Penetration

and 21.1%) [151]. This supports our approach to explaining spray penetration by turbulent dispersion of droplets, although more research in this direction is required. The effects of turbulence on the initial stage of spray penetration were also studied in [151], using the analysis of turbulent diffusion of the liquid phase (Eulerian approach; see [25]) or turbulent perturbation of individual droplet trajectories (Lagrangian approach; see [149, 150]). In both approaches, the analysis assumed that spray at the initial stage can be approximated as an array of non-interacting droplets. As mentioned in Sect. (1.2.1), this assumption is questionable. Note that spray penetration is closely linked with its momentum flux. Results of detailed numerical and experimental investigation of momentum fluxes in highpressure Diesel fuel sprays are described in [148]. Effects of cavitation and nozzle geometry on spray penetration were investigated in [189, 190]. Results of experimental studies of supersonic flow effect on Diesel fuel spray penetration are presented in [170]. A simplified model of the effects of turbulent structures on droplet dynamics, leading to their grouping in the oscillating velocity field, was developed by D Katoshevski and his colleagues (e.g. [94, 179]). A different type of grouping attributed to different drag forces acting on two droplets in tandem was considered in [56]. The analysis of spray dynamics presented so far has been based mainly on the Eulerian–Lagrangian approach (Eulerian approach for the carrier phase and Lagrangian approach for the dispersed phase). An alternative approach to the analysis of this process could be based on vortex-based algorithms (e.g. [106, 203]). In the latter paper, this algorithm was combined with the fully Lagrangian approach to modelling the dispersed phase. This approach is sometimes known as the Osiptsov method [73]. In the next section, some of these approaches are applied to the analysis of vortex ring-like structures in sprays and droplet dynamics in them.

1.3 Vortex Ring-Like Structures in Sprays Our analysis so far has been focused on the basic processes leading to spray formation and penetration, ignoring several important details. These details include the oscillations of sprays and the formation/dynamics of vortex ring-like structures near the spray leading edges. The former process was considered in several papers, including [158]. The focus of this section is on the latter phenomenon. The vortex ring-like structures are not observed for all sprays. For example, they are not observed for sprays in Diesel engine conditions, where liquid fuel is injected into a high-pressure gas, except at the very initial stage of the process [24, 100]. At the same time, these structures are typical for petrol engine sprays, where liquid fuel is injected into gas at pressures close to atmospheric pressure. Sprays with vortex ring-like structures are sometimes referred to as vortex sprays [211]. A typical spray image in petrol engine-like conditions [6] is presented in Fig. 1.3. As follows from Fig. 1.3, the shape of the spray looks rather chaotic, but vortex ring-like structures can be easily identified. There are some recognisable similarities

1.3 Vortex Ring-Like Structures in Sprays

27

Fig. 1.3 A typical image acquired in petrol engine conditions using G-DI injector. Positions of the points where radial and axial components of the velocity are equal to zero are indicated as crosses. Reprinted from [92], Copyright Elsevier (2010)

between these structures and the conventional well-organised vortex rings formed, for example, during the injection of water into water with the help of a round piston (e.g. [116, 184]). However, early attempts to apply the theory of conventional vortex rings to the analysis of the above-mentioned vortex ring-like structures were not successful [175]. In [91, 92], conventional vortex ring theory was generalised to consider the effect of turbulence. The new model described in these papers was successful in predicting some features of the vortex ring-like structures presented in Fig. 1.3, including their translational velocities. Some interesting features of air-assisted n-heptane spray formation are described in [210]. Two distinct spray shapes were observed: ‘spindle-shaped’ sprays (without clearly seen vortex ring-like structures) and ‘anchor-shaped’ sprays (with clearly seen vortex ring-like structures). The penetration lengths of ‘anchor-shaped’ sprays were smaller than those of ‘spindle-shaped’ sprays, but they had larger dispersion areas. ‘Spindle-shaped’ and ‘anchor-shaped’ sprays were formed during high and relatively low speeds of injection, respectively. Vortex ring-like structures in the ‘anchor-shaped’ sprays were shown to substantially suppress fuel wall impingement and promote spray spreading. The latter is expected to facilitate mixture formation and combustion in real-life internal combustion engines. A detailed overview of recent developments in modelling vortex rings is presented in monograph [33]. In what follows in this section, a short summary of the analysis and the results presented in [33] with emphasis on their relevance to sprays is presented. In many cases relevant to engineering applications the mass fraction of droplets is very small compared with that of the ambient gas. This allows us to model the vortex ring-like structures in sprays focusing only on the dynamics of the ambient gas, ignoring the contribution of the droplets altogether. This section will give a brief

28

1 Spray Formation and Penetration

Fig. 1.4 A schematic view of a vortex ring. Reproduced from [180] with permission from JUMV—Society of Automotive Engineers in Serbia

overview of the history of conventional vortex ring theory. Then recent developments presented in [91, 92] are briefly described. Some predictions of the models developed [91, 92] are compared with experimental results referring to vortex ring-like structures similar to those shown in Fig. 1.3. This is followed by an overview of models for confined vortex rings. Finally, the focus will be on the case when the interaction between droplets and vortex ring-like structures in the ambient gas needs to be considered. The results of the analysis of two-phase vortex ring flows, using the fully Lagrangian (Osiptsov) approach, will be described.

1.3.1 Conventional Vortex Rings The results described in this section are mainly reproduced from [92]. A schematic sketch of a vortex ring is shown in Fig. 1.4. R0 in this figure is the radius of the vortex ring (distance from the vortex ring axis to the area of zero vorticity);  is the characteristic vortex ring thickness. Two approaches have been developed in theoretical studies of vortex ring translational velocities and energies. In the first approach, the relation between velocity and vorticity was used to obtain the formulae for thin-cored rings:  = /R0  1 [47, 51]. A more general approach valid for arbitrary , described in [74] (see also [103]), is based on the Helmholtz–Lamb formula for the ring’s translational velocity U:  ∞ ∞  ∂Ψ π Ψ − 6x ζ d xdr, (1.72) U= 2M ∂r 0 −∞

where ζ and Ψ are the vorticity and stream function, respectively, and M = I /ρ the momentum of vorticity per unit density. Using Eq. (1.72), Saffman [167] (see also [168]) derived the following explicit expression for the translational velocity of a thin-cored viscous vortex ring:

1.3 Vortex Ring-Like Structures in Sprays

Us =

29

      νt 4R0 Γ0 νt − 0.558 + O ln √ , ln 4π R0 R0 2 R0 2 νt

(1.73)

where Γ0 is the initial circulation of the ring, t time and ν the fluid kinematic viscosity. The vorticity distribution inside this ring corresponds to the Lamb–Oseen vortex filament [103]. This asymptotic formula is applicable for the description of the initial stage of viscous vortex ring development when νt > R0 2 ) is described based on the Phillips self-similar solution for the vorticity (ζ f ) and stream function (Ψ f ) distributions [146]: ζf = M Ψf = 4π





erf

s∗ √ 2

Mr 16π 3/2 (νt)5/2

 − s∗

  −s∗ 2 r2 2 exp  3/2 , π 2 r2 + x2

where s∗ =

 2 s exp − ∗ , 2

(1.74)

(1.75)

r2 + x2 , 2νt

x, r are cylindrical coordinates for the axisymmetric vortex ring. The derivation of the translational velocity in this case is not straightforward. Since Formula (1.72) was derived based on the full Navier–Stokes equation, the substitution of Expressions (1.74) and (1.75) into (1.72) leads to inconsistency. Attempts to account for the second-order effects of the non-linear convective terms in the vorticity equation were made by Kambe and Oshima [87]. Their results, however, are not uniformly valid. Rott and Cantwell [163, 164] investigated this case considering the flow dynamics in the potential flow region surrounding the vortical region. They showed that the asymptotic translational velocity of the ring can be described by the following formula: Uf =

7M I /ρ = 0.0037038 . 15 (8π νt)3/2 (νt)3/2

(1.76)

Another approach to this problem is described in [8, 86, 89, 90]. These authors obtained a first-order solution to the Navier–Stokes equation with the origin in the centre of the vortex centroid, valid in the limit of small Reynolds numbers Re defined as Re = ζ0 L 2 /ν, where ζ0 = At λ is the vorticity scale; constant A is to be specified from the conservation of M. The translational velocity of the viscous vortex ring was presented as [89]

30

1 Spray Formation and Penetration

√   2  2      θ θ 3 3 θ2 5 Mθ π 2 3 exp − I1 + , , , 3 , −θ U= F2 2 2 12 2 2 2 2 4π 2 R03 −

3θ 2 F2 5 2



    7 3 5 , , 2, , −θ 2 , 2 2 2

(1.77)

where θ = R0 / =  −1 , I1 is the first-order Bessel function and 2 F2 the generalised hypergeometric function [128]. Similarly, the kinetic energy and circulation were presented as [90] E=

√       3 3 5 M 2θ π 1 2 , , , , 3 , −θ F 2 2 2 2 2π 2 R03 12 2   2  θ M 1 − exp − Γ = . 2 2 π R0

(1.78)

(1.79)

Apart from the definition of Re given above, at least two other definitions of Re have been suggested: Reu = U p D/ν, (1.80) based on the ejection velocity U p and nozzle diameter D, and ReΓ0 = Γ0 /ν,

(1.81)

based on the initial circulation carried by the ring Γ0 . The closed-form representations (1.77)–(1.79) enable us to investigate the asymptotic behaviour of these parameters. In the limit of small θ , these equations reduce to     11θ 2 Mθ 3 √ 7 − + O θ4 , (1.82) π Uf = 3 2 30 140 4π R0 M 2θ 3 √ Ef = π 2π 2 R03



Γf =

θ2 1 − 12 40



  + O θ4 ,

Mθ 2 . 2π R02

(1.83)

(1.84)

In the limit of large θ , they reduce to Us =

√     1 M π 2 log (θ ) + 3 − γ − 2ϕ (3/2) , + O 2 θ4 4π 2 R03

(1.85)

√   M2 π 1 , Es = − γ /2 − ϕ + O (log (θ ) (3/2)) 3 2 θ4 2π R0

(1.86)

1.3 Vortex Ring-Like Structures in Sprays

31

Γs =

M , π R02

(1.87)

where γ ≈ 0.57721566 is the Euler constant. ϕ the di-gamma function is defined as ϕ=

d log Γ (x) , dx

where Γ (x) is the Gamma function. Stanaway et al. [193] performed a direct numerical simulation of the Navier– Stokes equation for an axisymmetric vortex ring at small and moderate Reynolds numbers. They showed that Formula (1.77) compares well with their result for a small Reynolds number [52]. The large Reynolds number asymptotics were discussed in [54]. An alternative approach to estimate the temporal evolution of the vortex ring translational velocity was described by Saffman [167], using simple dimensional analysis. The following formula was derived: U=

−3/2 M 2 R0 + k  νt , k

(1.88)

where k and k  are fitting constants. To obtain these constants, Weingand and Gharib [209] compared their experimental data for 830 < ReΓ0 < 1650 with those predicted by Expression (1.88). This comparison allowed them to obtain the following values: k = 14.4 and k  = 7.8. Later k = 10.15 and k  = 8.909 were obtained theoretically by Fukumoto and Kaplanski [52].

1.3.2 Turbulent Vortex Rings In contrast to the aforementioned laminar vortex ring models, the theory of turbulent vortex rings is far less developed. To the best of the author’s knowledge, the first attempt to study turbulent vortex ring flow structures was made by Lugovtsov [121, 122] who based his analysis on the introduction of the time-dependent, turbulent (eddy) viscosity (cf. [101, 104]): (1.89) ν∗ ∝  , where  = d/dt and  ∝ t 1/4 . Formula (1.89) follows from a simple dimensional analysis ( has the dimension of length,  has the dimension of velocity) [16]. Using Formula (1.89), Lugovtsov [121, 122] developed a turbulent vortex ring model with turbulent viscosity ν∗ . √ Expressions (1.77)–(1.79) were originally derived for  = 2νt (laminar vortex ring). In [91], it was shown that they remain valid in a more general case when

32

1 Spray Formation and Penetration

 = at b , where a and b are constants (1/4 ≤ b ≤ 1/2). The model based on this presentation of  was called the generalised vortex ring model. This model incorporates both √ the laminar model for b = 1/2 and the fully turbulent model for b = 1/4. For a = 2ν, b = 1/2 and for large times (small θ ), the leading order term of (1.77) is identical to the one predicted by Expression (1.76). For small times, νt 0. Re was taken equal to 100. For the first type of flow, the position of the vortex centroid was calculated based on integrating (1.77) d xvc , xvc (0) = 0, Vvc = (1.109) dt and from the solution to the equation ∂ζ /∂r = 0. The latter was presented as 2 + 2t I0 (0.5Re rvc /t) Re rvc . = Re rvc I1 (0.5Re rvc /t)

(1.110)

As follows from the analysis of [166], at the initial stage, the vortex ring propagates with higher velocity and almost constant radius. The decay of the vortex ring starts at t ≈ 75; it is characterised by low velocity and slow increase of√its radius. This is √ consistent with the prediction of Solution (1.98) in the limits of t˜ = t/Re → 0 √ √ and t˜ = t/Re → ∞. It was shown that for Stk = 8, the droplet number density is almost uniform with variations not exceeding 10%, while for Stk = 0.32, this number density becomes visibly non-uniform. In the latter case, the highest droplet concentration was observed in the cap region, where the dimensionless droplet number density reached its maximum. In both cases, the droplet clouds overtook the vortex centroid and moved ahead of it. The effect of gravity led to a shift of the cloud of droplets keeping its shape. It was shown that the locations of the vortex ring-like structures identified with the help of the droplet number density fields are rather different from the locations

1.3 Vortex Ring-Like Structures in Sprays

39

of the vortex rings formed in the gas. Thus, the location of the vortex rings in the gas cannot be identified based on the analyses of the distribution of droplet number densities in the general case (cf. our analyses presented in Sect. 1.3.3). The effect of gravity led to a shift of the cloud without affecting its shape. To investigate the effect of inhomogeneous initial distribution n d (r, x, 0) of droplet number density, it was assumed that n d (r, x, 0) =

1 . 1 + exp[10(r − 0.9)]

(1.111)

It was shown that the initial inhomogeneity of droplet number density distribution leads to its non-uniform distribution with radius along the centre line of the jet. The highest droplet number density was observed in the periphery of the cap, similar to the case of uniform distribution. The values of the number density were close in both cases. √ The effect of droplet inertia was studied for 0.02 ≤ ρ/ρd ≤ 0.04 and 0.01 ≤ Stk ≤ 100, which corresponds to droplet diameters from a few µm to a few hundred µm. It was shown that the shape of the jet only slightly depends on ρd . The effect of ρd was shown to be the most pronounced for Stk in the range 1–10. The cloud of large droplets (Stk ∼ 10 − 100) travelled almost without deformation. When considering the vortex ring propagation through a cloud of droplets, the interactions of the vortex ring with droplets of high and low inertia were considered first. The following initial conditions were used: t0 = 0.2, rd = rd0 ,

xd = xd0 , u d = 0,

vd = 0, n d = 1,

qi j = 0,

Ji j = δi j .

(1.112) It was shown that for high-inertia droplets, the initially rectangular-shaped cloud deformed into a mushroom-like structure. The droplets initially located at the periphery moved towards the centre line, forming a stalk. The droplets initially located closer to the axis of symmetry and closer to the vortex ring were pushed to the periphery. The cloud turned inside out and a fold was developed. The droplet number density formed a singularity at the edge of the fold. This singularity was integrable and the assumption of the collisionless flow of droplets remained valid when their volume fraction was not too large. Low-inertia droplets, initially located close to the axis of symmetry, were accumulated near a two-dimensional surface. The originally rectangular-shaped cloud deformed into a mushroom-like structure, similar to the previous case. The regions with the highest droplet number density developed at the cap sides. The main limitation of the conventional fully Lagrangian approach (FLA) used in the analysis of [166] is that it assumes that the ambient gas flow is laminar, while typical flows in practical engineering applications are turbulent. An attempt to generalise this approach to turbulent gas flows was made in [140]. The authors of this paper focused on the investigation of the effects of droplet inertia and turbulent mixing on the distribution of droplet number density in turbulent flows. A new

40

1 Spray Formation and Penetration

formulation of the turbulent convective diffusion equation for the droplet number density, based on the modified FLA, was proposed. Droplets with moderate inertia were transported and dispersed by large-scale structures of a filtered Large Eddy Simulation field. It was considered that turbulent fluctuations, not seen in the filtered solution for the droplet velocity field, induce an additional diffusion mass flux and dispersion of droplets. The spatial derivatives for the droplet number density were calculated by projecting the solution obtained from the FLA on the Eulerian mesh. This led to a hybrid Lagrangian–Eulerian approach. The results of the calculations for droplet mixing in decaying homogeneous and isotropic turbulence were validated by the results of Direct Numerical Simulations for several Stokes numbers. The results of calculating the dynamic and distribution of droplets in realistic gas velocity fields with vortex ring-like structures (like the one shown in Fig. 1.3) using the FLA are presented and discussed in [216]. Also, in this paper, the results of the implementation of the FLA into ANSYS FLUENT CFD code are presented and discussed. The results of the implementation of a more general version of the FLA, taking into account droplet heating and evaporation, into ANSYS FLUENT are presented in [217]. The authors of [112] developed a generalised version of the FLA to analyse polydisperse flows with evaporating droplets. This was achieved via the extension of the space of Lagrangian variables to include droplet radii. This version of the FLA was incorporated into OpenFOAM. The authors of [207] presented an overview of the previously suggested approaches to modelling particle-laden flows, including the fully Lagrangian approach, and suggested their new technique to analyse these flows, which they called a Globally Eulerian Locally Lagrangian (GELL) method. The following stages of spray formation were considered in this chapter: instability of a jet emerging from the nozzle, break-up of droplets and spray penetration considering and not considering the effect of turbulence. In the case of petrol direct injection engines, the development of sprays was typically accompanied by the formation of vortex ring-like structures. Some new approaches to modelling these structures, not including and including the effects of confinement, were described. The predicted velocities of displacement of the regions of maximal vorticity in typical petrol engines were compared with available experimental data where possible. The results of the investigation of two-phase vortex ring flows using the fully Lagrangian approach were presented.

References 1. Abdelghaffar, W. A., Elwardany, A. E., & Sazhin, S. S. (2010). Modeling of the processes in Diesel engine-like conditions: Effects of fuel heating and evaporation. Atomization Sprays, 20, 737–747.

References

41

2. Afanasyev, Y. D., & Korabel, V. N. (2004). Starting vortex dipoles in a viscous fluid: Asymptotic theory, numerical simulation, and laboratory experiments. Physics of Fluids, 16(11), 3850–3858. 3. Agarwal, A., & Trujillo, M. F. (2018). A closer look at linear stability theory in modeling spray atomization. International Journal of Multiphase Flow, 109, 1–13. 4. Allocca, L., Belardini, P., Bertoli, C., Corcione, F. E., & De Angelis, F. (1992). Experimental and numerical analysis of a diesel spray. SAE Report 920576. 5. Arienti, M., & Sussman, M. (2014). An embedded level set method for sharp-interface multiphase simulations of Diesel injectors. International Journal of Multiphase Flow, 59, 1–14. 6. Begg, S., Kaplanski, F., Sazhin, S. S., Hindle, M., & Heikal, M. (2009). Vortex ring structures in gasoline engines under cold-start conditions. International Journal of Engine Research, 10, 195–214. 7. Bellman, R., & Pennington, R. H. (1954). Effects of surface tension and viscosity on Taylor instability. Quarterly of Applied Mathematics, 12, 151–162. 8. Berezovski, A., & Kaplanski, F. (1995). Vorticity distributions for thick and thin vortex pairs and rings. Archives of Mechanics, 47(6), 1015–1026 (in Russian). 9. Berni, F., Sparacino, S., Riccardi, M., et al. (2022). A zonal secondary break-up model for 3D-CFD simulations of GDI sprays. Fuel, 309, 122064. 10. Boggavarapu, P., Ramesh, S. P., Avulapati, M. M., & Ravikrishna, R. V. (2021). Secondary breakup of water and surrogate fuels: Breakup modes and resultant droplet sizes. International Journal of Multiphase Flow, 145, 103816. 11. Bohl, T., Tian, G., Smallbone, A., & Roskilly, A. P. (2017). Macroscopic spray characteristics of next-generation bio-derived diesel fuels in comparison to mineral diesel. Applied Energy, 186(3), 562–573. 12. Borman, G. L., & Ragland, K. W. (1998). Combustion Engineering. New York: McGraw-Hill. 13. Boronin, S. A., Healey, J. J., & Sazhin, S. S. (2013). Non-modal stability of round viscous jets. Journal of Fluid Mechanics, 716, 96–119. 14. Brasseur, J. G. (1979). Kinematics and dynamics of vortex rings in a tube. NASA Tech. Rep. JIAA TR-26. 15. Brasseur, J. G. (1986). Evolution characteristics of vortex rings over a wide range of Reynolds numbers. In Proceedings of 4th AIAA/ASME Fluid Mechanics, Plasma Dynamics and Lasers Conference, May 12–14 1986, Atlanta, GA. AIAA Paper 86-1097. 16. Cantwell, B. (2002). Introduction to Symmetry Analysis. Cambridge: Cambridge University Press. 17. Cebeci, T., & Smith, A. M. O. (1974). Analysis of Turbulent Boundary Layers. Applied mathematics and mechanics (Vol. 15). NY: Academic. 18. Chehroudi, B., & Bracco, F. V. (1988). Structure of a transient hollow cone spray. SAE Report 880522. 19. Cheng, Q., Tuomo, H., Kaario, O. T., & Martti, L. (2019). Spray dynamics of HVO and EN590 diesel fuels. Fuel, 245, 198–211. 20. Chesnel, J., Menard, T., Reveillon, J., & Demoulin, F.-X. (2011). Subgrid analysis of liquid jet atomization. Atomization Sprays, 21, 41–67. 21. Chigier, N., & Reitz, R. D. (1998). Regimes of jet breakup and breakup mechanisms (physical aspects). In K. K. Kuo (Ed.), Recent Advances in Spray Combustion: Spray Atomization and Drop Burning Phenomena (pp. 109–135). Published by American Institute of Aeronautics and Astronautics, Inc. 22. Chryssakis, C., & Assamis, D. N. (2008). A unified fuel spray breakup model for internal combustion engine applications. Atomization Sprays, 18, 375–426. 23. Cossali, G. E. (2001). An integral model for gas entrainment into full cone sprays. Journal of Fluid Mechanics, 439, 353–366. 24. Crua, C., Shoba, T., Heikal, M., Gold, M., & Higham, C. (2010). High-speed microscopic imaging of the initial stage of Diesel spray formation and primary breakup. SAE Technical Report 2010-01-2247.

42

1 Spray Formation and Penetration

25. Csanady, G. T. (1973). Turbulent Diffusion in the Environment. Dordrecht-Holland: D. Reidel Publishing Comp. 26. Dai, X., Wang, Z., Liu, F., Lee, C.-F., Sun, Q., & Li, Y. (2020). The effect of bubbles on primary breakup of diesel spray. Fuel, 263, 116664. 27. Dai, X., Wang, Z., Liu, F., et al. (2020). The effect of nozzle structure and initial state on the primary breakup of diesel spray. Fuel, 280, 118640. 28. Dai, Z., & Faith, G. M. (2001). Temporal properties of secondary drop breakup in the multimode breakup regime. International Journal of Multiphase Flow, 27, 217–236. 29. Danaila, I., & Helie, J. (2008). Numerical simulation of the postformation evolution of a laminar vortex ring. Physics of Fluids, 20, 073602. 30. Danaila, I., Kaplanski, F., & Sazhin, S. S. (2015). Modelling of confined vortex rings. Journal of Fluid Mechanics, 774, 267–297. 31. Danaila, I., Kaplanski, F., & Sazhin, S. S. (2017). A model for confined vortex rings with elliptical core vorticity distribution. Journal of Fluid Mechanics, 811, 67–94. 32. Danaila, I., Luddens, F., Kaplanski, F., Papoutsakis, A., & Sazhin, S. S. (2018). The formation number of confined vortex rings. Physical Review Fluids, 3, 094701. 33. Danaila, I., Kaplanski, F., & Sazhin, S. S. (2021). Vortex Ring Models. Springer Nature. 34. Dent, J. C. (1971). A basic for the comparison of various experimental methods for studying spray penetration. SAE Report 710571. 35. Desantes, J. M., Payri, R., Salvador, F. J., & Gil, A. (2006). Development and validation of a theoretical model for diesel spray penetration. Fuel, 85, 910–917. 36. Desantes, J. M., Payri, R., Garcia, J. M., & Salvador, F. J. (2007). A contribution to the understanding of isothermal diesel spray dynamics. Fuel, 86, 1093–1101. 37. Desantes, J. M., Payri, R., Salvador, F. J., & De la Morena, J. (2010). Influence of cavitation phenomenon on primary break-up and spray behavior at stationary conditions. Fuel, 89, 3033–3041. 38. Devassy, B. M., Habchi, C., & Daniel, E. (2013). A new atomization model for high speed liquid jets using a turbulent, compressible, two-phase flow model and a surface density approach. In Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 32. 39. Dombrowski, N., & Johns, W. R. (1963). The aerodynamic instability and disintegration of viscous liquid sheets. Chemical Engineering Science, 18, 203–214. 40. Douglas, J. F., Gasiorek, J. M., Swaffield, J. A., & Jack, L. B. (2005). Fluid Mechanics (5th ed.). Singapore: Pearson. 41. Drazin, P. G., & Reid, W. H. (2004). Hydrodynamic Stability (2nd ed.). Cambridge: Cambridge University Press. 42. Dumouchel, C., & Grout, S. (2009). Application of the scale entropy diffusion to describe a liquid atomization process. International Journal of Multiphase Flow, 35, 952–962. 43. Dumouchel, C., & Grout, S. (2011). On the scale diffusivity of a 2-D liquid atomization process analysis. Physica A, 390, 1811–1825. 44. Dumouchel, C., & Blaisot, J.-B. (2013). Multi-scale analysis of liquid atomization processes and sprays. In Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 25. 45. Duret, B., Menard, T., Reveillon, J., & Demoulin, F. X. (2013). Improving primary atomization modelling through DNS of two-phase flows. In Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 110. 46. Duronio, F., De Vita, A., Allocca, L., & Anatone, M. (2020). Gasoline direct injection engines – A review of latest technologies and trends. Part 1: Spray breakup process. Fuel,265, 116948. 47. Dyson, F. W. (1893). The potential of an anchor ring-Part II. Philosophical Transactions of the Royal Society of London A, 184, 1041–1106. 48. Eggers, J., & Villermaux, E. (2008). Physics of liquid jets. Reports on Progress in Physics, 616, 79.

References

43

49. Ellendt, N., Lumanglas, A. M., Moqadam, S. I., & Mädler, L. (2018). A model for the drag and heat transfer of spheres in the laminar regime at high temperature differences. International Journal of Thermal Sciences, 133, 98–105. 50. Faeth, G. M., Hsiang, L.-P., & Wu, P.-K. (1995). Structure and breakup properties of sprays. International Journal of Multiphase Flow, 21(Suppl.), 99–127. 51. Fraenkel, L. E. (1972). Examples of steady vortex rings of small cross-section in an ideal fluid. Journal of Fluid Mechanics, 51, 119–135. 52. Fukumoto, Y., & Kaplanski, F. (2008). Global time evolution of an axisymmetric vortex ring at low Reynolds numbers. Physics of Fluids, 20, 053103. 53. Fukumoto, Y., & Moffatt, H. K. (2000). Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. Journal of Fluid Mechanics, 417, 1–45. 54. Fukumoto, Y., & Moffatt, H. K. (2008). Kinematic variational principle for motion of vortex rings. Physica D: Nonlinear Phenomena, 237, 2210–2217. 55. Fuster, D., Bagué, A., Boeck, T., Le Moyne, L., Leboissetier, A., Popinet, S., et al. (2009). Simulation of primary atomization with an octree adaptive mesh refinement and VOF method. International Journal of Multiphase Flow, 35, 550–565. 56. Geyari, Y., Greenberg, J. B., Arad, A., Katoshevski, D., Vaikuntanathan, V., & Roth, N. (2021). Some new insights into droplet grouping dynamics. In Proceedings of 15th Triennial International Conference on Liquid Atomization and Spray Systems, Edinburgh, UK, 29 Aug. 2 Sept. 2021. 57. Gharib, M., Rambod, E., Kheradvar, A., Sahn, D. J., & Dabiri, J. O. (2006). Optimal vortex formation as an index of cardiac health. Proceedings of the National Academy of Sciences of the United States of America, 103, 6305–6308. 58. Ghasemi, A., Barron, R. M., & Balachandar, R. (2014). Spray-induced air motion in single and twin ultra-high injection diesel sprays. Fuel, 121, 284–297. 59. Ghosh, S., & Hunt, J. C. R. (1994). Induced air velocity within droplet driven sprays. Proceedings of the Royal Society of London A, 444, 105–127. 60. Ghosh, S., & Hunt, J. C. R. (1998). Spray jets in a cross-flow. Journal of Fluid Mechanics, 365, 109–136. 61. Giannadakis, E., Gavaises, M., & Arcoumanis, C. (2006). Modelling of cavitation in diesel injector nozzles. Journal of Fluid Mechanics, 616, 153–193. 62. Girin, A. G. (2012). On the mechanism of inviscid drop breakup at relatively small Weber numbers. Atomization Sprays, 22, 921–934. 63. Girin, A. G. (2013) Deformation and acceleration of drop. In Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 43. 64. Girin, A. G., & Ivanchenko, Y. A. (2012). Model of liquid film disintegration at ‘bag’ mode of drop breakup. Atomization Sprays, 22, 935–949. 65. Glezer, A., & Coles, D. (1990). An experimental study of a turbulent vortex ring. Journal of Fluid Mechanics, 211, 243–283. 66. Gorokhovski, M. A., & Saveliev, V. L. (2003). Analysis of Kolmogorov’s model of breakup and its application into Lagrangian computation of liquid sprays under air-blast atomisation. Physics of Fluids, 15, 184–192. 67. Gorokhovski, M. A., & Herrmann, M. (2008). Modeling primary atomization. Review of Fluid Mechanics, v. 40. 68. Gorokhovski, M. A., & Saveliev, V. L. (2008). Statistical universalities in fragmentation under scaling symmetry with a constant frequency of fragmentation. Journal of Physics. D. Applied Physics, 41, 085405. 69. Grout, S., Dumouchel, C., Cousin, J., & Nuglisch, H. (2007). Fractal analysis of atomizing liquid flows. International Journal of Multiphase Flow, 33, 1023–1044. 70. Günther, A., & Wirth, K.-E. (2013). Evaporation phenomena in superheated atomization and its impact on the generated spray. International Journal of Heat and Mass Transfer, 64, 952–965.

44

1 Spray Formation and Penetration

71. Habchi, C. (2011). The energy spectrum analogy breakup (SAB) model for the numerical simulation of sprays. Atomization Sprays, 21, 1033–1057. 72. Habchi, C. (2013). A Gibbs free energy relaxation model for cavitation simulation in Diesel injectors. In: Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 93. 73. Healy, D. P., & Young, J. B. (2005). Full Lagrangian methods for calculating the particle concentration and velocity fields in dilute gas-particle flows. Proceedings of the Royal Society of London Series A, 461, 2197–2225. 74. Helmholtz, H. (1858). On integrals of the hydrodynamical equations which express vortexmotion. Transl. P. G. Tait with a letter by Lord Kelvin (W. Thompson) in London Edinburgh and Dublin Philosophy Magazine and Journal of Science. Fourth Series, 33, 485–512. 75. Herrmann, M. (2011). On simulating primary atomization using the refined level set grid method. Atomization and Sprays, 21, 283–301. 76. Herrmann, M. (2013). On simulating primary atomization. Atomization and Sprays, 23(11– 12), v–ix. 77. Heywood, J. B. (1988). Internal Combustion Engines Fundamentals. New York: McGraw-Hill Book Company. 78. Hiroyasu, H., & Kadota, T. (1974). Fuel droplet size distribution in a diesel combustion chamber. SAE Paper 740715. 79. Hong, J. G., Ku, K. W., & Lee, C.-W. (2011). Numerical simulation of the cavitating flow in an elliptical nozzle. Atomization and Sprays, 21, 237–248. 80. Hopfes, T., Petersen, J., Wang, Z., Giglmaier, M., & Adams, N. A. (2021). Secondary atomization of liquid metal droplets at moderate Weber numbers. International Journal of Multiphase Flow, 143, 103723. 81. Hsiang, L.-P., & Faeth, G. M. (1993). Drop properties after secondary breakup. International Journal of Multiphase Flow, 19, 721–735. 82. Huh, K. Y., Lee, E., & Koo, J.-Y. (1998). Diesel spray atomization model considering nozzle exit turbulence conditions. Atomization and Sprays, 8, 453–459. 83. Jiang, X., Siamas, G. A., Jagus, K., & Karayiannis, T. G. (2010). Physical modelling and advanced simulations of gas-liquid two-phase jet flows in atomization and sprays. Progress in Energy and Combustion Science, 36, 131–167. 84. Jiao, D., Jiao, K., Zhang, F., Bai, F., & Du, Q. (2019). Primary breakup of power-law biofuel sprays in pressurized gaseous crossflow. Fuel, 258, 116061. 85. Juniper, M. P. (2008). The effect of confinement on the stability of non-swirling round jet/wake flows. Journal of Fluid Mechanics, 605, 227–252. 86. Kaltaev, A. (1982). Investigation of dynamic characteristics of a vortex ring of viscous fluid. In Continuum Dynamics (pp. 63–70). Kazah State University, Alma-Ata (in Russian). 87. Kambe, T., & Oshima, Y. (1975). Generation and decay of viscous vortex rings. Journal of the Physical Society of Japan, 38, 271–280. 88. Kaplanski, F., Fukumoto, Y., & Rudi, U. (2012). Reynolds-number effects on vortex ring evolution in a viscous fluid. Physics Fluids, 24, 033101. 89. Kaplanski, F., & Rudi, U. (1999). Dynamics of a viscous vortex ring. International Journal of Fluid Mechanics Research, 26, 618–630. 90. Kaplanski, F., & Rudi, Y. (2005). A model for the formation of ‘optimal’ vortex rings taking into account viscosity. Physics of Fluids, 17, 087101. 91. Kaplanski, F., Sazhin, S. S., Fukumoto, Y., Begg, S., & Heikal, M. (2009). A generalised vortex ring model. Journal of Fluid Mechanics, 622, 233–258. 92. Kaplanski, F., Sazhin, S. S., Begg, S., Fukumoto, Y., & Heikal, M. (2010). Dynamics of vortex rings and spray induced vortex ring-like structures. European Journal of Mechanics B/Fluids, 29(3), 208–216. 93. Karimi, K. (2007). Characterisation of Multiple-injection Diesel Sprays at Elevated Pressures and Temperatures. Ph.D. Thesis, University of Brighton, Brighton, United Kingdom. 94. Katoshevski, D., Shakked, T., Sazhin, S. S., Crua, C., & Heikal, M. R. (2008). Grouping and trapping of evaporating droplets in an oscillating gas flow. International Journal of Heat and Fluid Flow, 29, 415–426.

References

45

95. Kegl, B., & Lešnik, L. (2018). Modeling of macroscopic mineral diesel and biodiesel spray characteristics. Fuel, 222, 810–820. 96. Kolakaluri, R., Li, Y., & Kong, S.-C. (2010). A unified spray model for engine spray simulation using dynamic mesh refinement. International Journal of Multiphase Flow, 36, 858–869. 97. Kolmogorov, A. N. (1941). On the log-normal distribution of particle sizes during the breakup process. Doklady Akademii Nauk SSSR, 31, 99–101. 98. Kostas, J., Honnery, D., & Soria, J. (2009). Time resolved measurements of the initial stages of fuel spray penetration. Fuel, 88, 2225–2237. 99. Kasyap, T. V., Sivakumar, D., & Raghunandan, B. N. (2009). Flow and breakup characteristics of elliptical liquid jets. International Journal of Multiphase Flow, 35, 8–19. 100. Koukouvinis, P., Vidal-Ronceroa, A., Rodriguez, C., Gavaises, M., & Pickett, L. (2020). High pressure/high temperature multiphase simulations of dodecane injection to nitrogen: Application on ECN Spray-A. Fuel, 275, 116622. 101. Kovasznay, L. S. G., Fujita, H., & Lee, R. L. (1974). Unsteady turbulent puffs. Advances in Geophysics, 18B, 253–263. 102. Kuznetsov, G. V., Strizhak, P. A., Valiullin, T. R., & Volkov, R. S. (2022). Atomization behavior of composite liquid fuels based on typical coal processing wastes. Fuel Processing Technology, 225, 107037. 103. Lamb, H. (1932). Hydrodynamics. New York: Dover Publishers. 104. Lavrentiev, M. A., & Shabat, B. V. (1973). Problems of Hydrodynamics and Mathematical Models. Moscow (in Russian): Nauka Publishing House. 105. Lebas, R., Menard, T., Beau, P. A., Berlemont, A., & Demoulin, F. X. (2009). Numerical simulation of primary break-up and atomization: DNS and modelling study. International Journal of Multiphase Flow, 35, 247–260. 106. Lebedeva, N. A., Osiptsov, A. N., & Sazhin, S. S. (2013). A combined fully Lagrangian approach to mesh-free modelling of transient two phase flows. Atomization and Sprays, 23, 47–69. 107. Lee, C. H., & Reitz, R. D. (2013). CFD simulations of diesel spray tip penetration with multiple injections and with engine compression ratios up to 100:1. Fuel, 111, 289–297. 108. Lee, C. S., & Park, S. W. (2002). An experimental and numerical study on fuel atomization characteristics of high-pressure diesel injection sprays. Fuel, 81, 2417–2423. 109. Lefebvre, A. H. (1989). Atomization and Sprays. Bristol, PA: Taylor & Francis. 110. Li, J., & Carrica, P. M. (2021). A population balance cavitation model. International Journal of Multiphase Flow, 138, 103617. 111. Li, X. (1995). Mechanism of atomisation of a liquid jet. Atomization and Sprays, 5, 89–105. 112. Li, Y., & Rybdylova, O. (2021). Application of the generalised Fully Lagrangian Approach to simulating polydisperse gas-droplet flows. International Journal of Multiphase Flow, 142, 103716. 113. Li, Y., & Umemura, A. (2014). Two-dimensional numerical investigation on the dynamics of ligament formation by Faraday instability. International Journal of Multiphase Flow, 60, 64–75. 114. Li, Y., Huang, Y., Luo, K., Liang, M., & Lei, B. (2021). Development and validation of an improved atomization model for GDI spray simulations: Coupling effects of nozzle-generated turbulence and aerodynamic force. Fuel, 299, 120871. 115. Lightfoot, M. (2009). Fundamental classification of a atomization processes. Atomization and Sprays, 19, 1065–1104. 116. Lim, T., & Nickels, T. (1995). Vortex rings. In S. I. Green (Ed.), Fluid Vortices (pp. 95–153). Dordrecht: Kluwer. 117. Lin, S. P., & Rietz, R. D. (1998). Droplet and spray formation from a liquid jet. Annual Review of Fluid Mechanics, 30, 85–105. 118. Liu, F.-S., Zhou, L., Sun, B.-G., Li, Z.-J., & Schock, H. J. (2008). Validation and modification of WAVE spray model for diesel combustion simulation. Fuel, 87, 3420–3427. 119. Liu, Z., & Liu, Z. (2006). Linear analysis of three-dimensional instability of non-Newtonian liquid jets. Journal of Fluid Mechanics, 559, 451–459.

46

1 Spray Formation and Penetration

120. Lozano, A., Barreras, F., Hauke, G., & Dopazo, C. (2001). Longitudinal instabilities in an air-blasted liquid sheet. Journal of Fluid Mechanics, 437, 143–173. 121. Lugovtsov, B. A. (1970). On the motion of a turbulent vortex ring and its role in the transport of passive contaminant. In Some Problems of Mathematics and Mechanics (Dedicated to the 70th Anniversary of M.A. Lavrentiev) (pp. 182–187). Leningrad: Nauka Publishing House (in Russian). 122. Lugovtsov, B. A. (1976). On the motion of a turbulent vortex ring. Archives of Mechanics, 28, 759–766. 123. Malaguti, S., Fontanesi, S., Cantore, G., Montanaro, A., & Allocca, L. (2013). Modelling of primary breakup process of a gasoline direct engine multi-hole spray. Atomization and Sprays, 23, 861–888. 124. Marcic, S., Marcic, M., Wensing, M., et al. (2018). A simplified model for a diesel spray. Fuel, 222, 485–495. 125. Marmottant, P., & Villermauz, E. (2004). On spray formation. Journal of Fluid Mechanics, 498, 73–111. 126. Martinez, L., Benkenida, A., & Cuenot, B. (2010). A model for the injection boundary conditions in the context of 3D simulation of diesel spray: Methodology and validation. Fuel, 89, 219–228. 127. Martynov, S. B., Mason, D. J., & Heikal, M. R. (2006). Numerical simulation of cavitation flows based on their hydrodynamic similarity. International Journal of Engine Research, 7, 1–14. 128. MATHEMATICA Book version 6.0.0, Wolfram Research Inc. (2007). Available at http:// functions.wolfram.com. Retrieved from 25 July 25 2008. 129. Ménard, T., Tanguy, S., & Berlemont, A. (2007). Coupling level set/VOF/ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet. International Journal of Multiphase Flow, 33, 510–524. 130. Moon, S., Huang, W., & Wang, J. (2020). Spray formation mechanism of diverging-taperedhole GDI injector and its potentials for GDI engine applications. Fuel, 270, 117519. 131. Movaghar, A., Linne, M., Herrmann, M., et al. (2018). Modeling and numerical study of primary breakup under diesel conditions. International Journal of Multiphase Flow, 98, 110– 119. 132. Narasimha, R., & Sreenivasan, K. R. (1979). Relaminarization of fluid flows. Advances in Applied Mechanics, 19, 221–301. 133. Navaneeth, P. V., Suraj, C. K., Mehta, P. S., & Anand, K. (2021). Predicting the effect of biodiesel composition on the performance and emission of a compression ignition engine using a phenomenological model. Fuel, 293, 120453. 134. Nichols, J. (1972). Stream and droplet breakup by shock waves. In D. T. Harrje & F. H. Reardon (Eds.), NASA SP-194, Liquid Propellant Rocket Combustion Instability (pp. 126–128). 135. Nordin, N. (2001). Complex Chemistry Modeling of Diesel Spray Combustion, Ph.D. Thesis, Chalmers University of Technology, Gothenburg, Sweden. 136. O’Rourke, P. J. (1981). Collective Drop Effects on Vaporizing Liquid Sprays. Ph.D. thesis. Princeton University. 137. O’Rourke, P. J., & Amsden, A. A. (1987). The TAB method for numerical calculation of spray droplet breakup. SAE report 872089. 138. Osiptsov, A. N. (2000). Lagrangian modelling of dust admixture in gas flows. Astrophysics and Space Science, 274(1–2), 377–386. 139. Panton, R. L. (1996). Incompressible Flow. New York, Chichester: Wiley. 140. Papoutsakis, A., Rybdylova, O. D., Zaripov, T. S., Danaila, L., Osiptsov, A. N., & Sazhin, S. S. (2018). Modelling of the evolution of a droplet cloud in a turbulent flow. International Journal of Multiphase Flow, 104, 233–257. 141. Park, K. S., & Heister, S. D. (2010). Nonlinear modeling of drop size distributions produced by pressure-swirl atomizers. International Journal of Multiphase Flow, 36, 1–12. 142. Pastor, J. V., López, J. J., García, J. M., & Pastor, J. M. (2008). A 1D model for the description of mixing-controlled inert diesel sprays. Fuel, 87, 2871–2885.

References

47

143. Pastor, J. V., García-Oliver, J. M., Nerva, J.-G., & Giménez, B. (2011). Fuel effect on the liquid-phase penetration of an evaporating spray under transient diesel-like conditions. Fuel, 90, 3369–3381. 144. Patterson, M. A., & Reitz, R. D. (1998). Modelling of the effects of fuel spray characteristics on Diesel engine combustion and emission. SAE Report 980131. 145. Payri, R., Salvador, F. J., Gimeno, J., & Zapata, L. D. (2008). Diesel nozzle geometry influence on spray liquid-phase fuel penetration in evaporative conditions. Fuel, 87, 1165–1176. 146. Phillips, O. M. (1956). The final period of decay of non-homogeneous turbulence. Proceedings of the Cambridge Philosophical Society, 252, 135–151. 147. Poole, D. R., Barenghi, C. F., Sergeev, Y. A., & Vinen, W. F. (2005). The motion of tracer particles in helium II. Physical Review B, 71, 064514-1–16. 148. Postrioti, L., Mariani, F., & Battistoni, M. (2012). Experimental and numerical momentum flux evaluation of high pressure Diesel spray. Fuel, 98, 149–163. 149. Pozorski, J., & Minier, J.-P. (1998). On the Lagrangian turbulent dispersion models based on the Langevin equation. International Journal of Multiphase Flow, 24, 913–945. 150. Pozorski, J., & Minier, J.-P. (1999). PDF modeling of dispersed two-phase turbulent flows. Physical Review E, 59, 855–863. 151. Pozorski, J., Sazhin, S. S., Wacławczyk, M., Crua, C., Kennaird, D., & Heikal, M. R. (2002). Spray penetration in a turbulent flow. Flow, Turbulence and Combustion, 68(2), 153–165. 152. Ranger, A. A., & Nicholls, J. A. (1969). The aerodynamic shattering of liquid drops. AIAA Journal, 3, 285–290. 153. Reitz, R. D. (1987). Modelling atomization processes in high-pressure vaporizing sprays. Atomisation and Spray Technology, 3, 09–337. 154. Reitz, R. D., & Bracco, F. V. (1982). Mechanism of atomization of a liquid jet. Physics of Fluids, 25, 1730–1742. 155. Reitz, R. D., & Bracco, F. V. (2009). Mechanisms of breakup of round jets. Encyclopedia of Fluid Mechanics, 3, Chapter 10, 012101. 156. Reitz, R. D., & Diwakar, R. (1986). Effect of drop breakup on fuel sprays. SAE Report 860469. 157. Reitz, R. D., & Diwakar, R. (1987). Structure of high-pressure fuel sprays. SAE Report 870598. 158. Rewse-Davis, Z., Nouri, J., Gavaises, M., & Arcoumanis, C. (2013). Near-nozzle instabilities in gasoline direct injection sprays. In Proceedings of ILASS – Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 1–4 September 2013, paper 141. 159. Rimbert, N. (2010). Simple model for turbulence intermittencies based on self-avoiding random vortex stretching. Physical Review E, 81, 056315. 160. Rimbert, N., & Séro-Guillaume, O. (2004). Log-stable laws as asymptotic solutions to a fragmentation equation: Application to the distribution of droplets in a high Weber-number spray. Physical Review E, 69, 056316. 161. Roisman, I. V., Araneo, L., & Tropea, C. (2007). Effect of ambient pressure on penetration of a diesel spray. International Journal of Multiphase Flow, 33, 904–920. 162. Rotondi, R., Bella, G., Grimaldi, C., & Postrioti, L. (2001). Atomization of high-pressure Diesel spray: Experimental validation of a new breakup model. SAE Report 2001-01-1070. 163. Rott, N., & Cantwell, B. (1993). Vortex drift. I: Dynamic interpretation. Physics of Fluids A, 5, 1443–1450. 164. Rott, N., & Cantwell, B. (1993). Vortex drift. II: The flow potential surrounding a drifting vortical region. Physics of Fluids A, 5, 1451–1455. 165. Ruo, A. C., Chen, F., & Chang, M. H. (1992). Linear instability of compound jets with nonaxisymmetric disturbances. Physics of Fluids, 21, 681–689. 166. Rybdylova, O., Sazhin, S. S., Osiptsov, A. N., Kaplanski, F. B., Begg, S., & Heikal, M. (2018). Modelling of a two-phase vortex-ring flow using an analytical solution for the carrier phase. Applied Mathematics and Computation, 326, 159–169. 167. Saffman, P. G. (1970). The velocity of viscous vortex rings. Studies in Applied Mathematics, 49, 371–380.

48

1 Spray Formation and Penetration

168. Saffman, P. G. (1992). Vortex Dynamics. Cambridge: Cambridge University Press. 169. Salvador, F. J., Ruiz, S., Crialesi-Esposito, M., & Blanquer, I. (2018). Analysis on the effects of turbulent inflow conditions on spray primary atomization in the near-field by direct numerical simulation. International Journal of Multiphase Flow, 102, 49–63. 170. Salvador, F. J., De la Morena, J., Taghavifar, H., & Nemati, A. (2020). Scaling spray penetration at supersonic conditions through shockwave analysis. Fuel, 260, 116308. 171. Sarimeseli, A., & Kelbaliev, G. (2004). Modelling of the break-up of deformable particles in developed turbulent flow. Chemical Engineering Science, 59, 1233–1240. 172. Sakaguchi, D., Yamamoto, S., Ueki, H., & Ishdia, M. (2010). Study of heterogeneous structure in Diesel fuel spray by using micro-probe L2F. Journal of Fluid Science and Technology, 5, 75–85. 173. Savich, S. (2001). Spray Dynamics and In-cylinder Air Motion. Ph.D. Thesis, The University of Brighton. 174. Sazhin, S. S., Feng, G., & Heikal, M. R. (2001). A model for fuel spray penetration. Fuel, 80(15), 2171–2180. 175. Sazhin, S. S., Kaplanski, F., Feng, G., Heikal, M. R., & Bowen, P. J. (2001). A fuel spray induced vortex ring. Fuel, 80(13), 1871–1883. 176. Sazhin, S. S., Crua, C., Kennaird, D., & Heikal, M. R. (2003). The initial stage of fuel spray penetration. Fuel, 82(8), 875–885. 177. Sazhin, S. S., Crua, C., Hwang, J.-S., No, S.-Y., & Heikal, M. (2005). Models of fuel spray penetration. Proceedings of the Estonian Academy of Sciences. Engineering, 11(2), 154–160. 178. Sazhin, S. S., Martynov, S. B., Kristyadi, T., Crua, C., & Heikal, M. R. (2008). Diesel fuel spray penetration, heating, evaporation and ignition: Modelling versus experimentation. International Journal of Engineering Systems Modelling and Simulation, 1, 1–19. 179. Sazhin, S. S., Shakked, T., Sobolev, V., & Katoshevski, D. (2008). Particle grouping in oscillating flows. European Journal of Mechanics B/Fluids, 27, 131–149. 180. Sazhin, S. S., Kaplanski, F., Begg, S., & Heikal, M. (2009). Vortex ring-like structures in gasoline fuel sprays. In Proceedings of the JUMV International Automotive Conference and Exhibition (XXII Science and Motor Vehicles 2009), Belgrade 14–16 April 2009, paper 31 (CD). Published by the Society of Automotive Engineers of Serbia. 181. Schiller, L., & Naumann, A. (1935). A drag coefficient correlation. Zeitschrift des Vereins Deutscher Ingenieure, 77, 318–320. 182. Schugger, C., Meingast, U., & Renz, U. (2000). Time-resolved velocity measurements in the primary breakup zone of a high pressure diesel injection nozzle. In Proceedings of ILASSEurope, Darmstadt, Germany. 183. Senecal, P. K., Schmidt, D. P., Nouar, I., Rutland, C. J., Reitz, R. D., & Corradini, M. L. (1999). Modeling high-speed viscous liquid sheet atomization. International Journal of Multiphase Flow, 25, 1073–1097. 184. Shariff, K., & Leonard, A. (1992). Vortex rings. Annual Review of Fluid Mechanics, 24, 235–279. 185. Sher, I., & Sher, E. (2011). Analytical criterion for droplet breakup. Atomization and Sprays, 21, 1059–1063. 186. Shinjo, J., & Umemura, A. (2004). Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation. International Journal of Multiphase Flow, 36, 513–532. 187. Snegirev, A., Talalov, V., Sheinman, I., & Sazhin, S. S. (2011). An enhanced spray model for flame suppression simulations. In Proceedings of the Fifth European Combustion Meeting ECM, Cardiff, 29th June to 1st July 2011 paper 072-1 (CD). 188. Som, S., & Aggarwal, S. K. (2009). Assessment of atomization models for Diesel engine simulations. Atomization and Sprays, 19, 885–903. 189. Som, S., & Aggarwal, S. K. (2010). Effects of primary breakup modeling on spray and combustion characteristics of compression ignition engines. Combustion and Flame, 157, 1179–1193. 190. Som, S., Ramirez, A. I., Longman, D. E., & Aggarwal, S. K. (2011). Effect of nozzle orifice geometry on spray, combustion, and emission characteristics under diesel engine conditions. Fuel, 90, 1267–1276.

References

49

191. Squire, H. B. (1933). On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proceedings of the Royal Society of London, 142, 621–628. 192. Srinivasan, V., Salazar, A. J., & Saito, K. (2008). Numerical investigation on the disintegration of round turbulent liquid jets using LES/VOF techniques. Atomization and Sprays, 18, 571– 617. 193. Stanaway, S., Cantwell, B. J., & Spalart, P. R. (1988). A numerical study of viscous vortex rings using a spectral method, NASA Technical Memorandum 101041. 194. Stapper, B. E., Sowa, W. A., & Samuelson, G. S. (1992). An experimental study of the effects of liquid properties on the breakup of a two-dimensional liquid sheet. ASME Journal of Engineering for Gas turbines and Power, 114, 39–45. 195. Stefanitsis, D., Strotos, G., Nikolopoulos, N., Kakaras, E., & Gavaises, M. (2019). Improved droplet breakup models for spray applications. International Journal of Heat and Fluid Flow, 76, 274–286. 196. Stewart, K., Niebel, C., Jung, S., & Vlachos, P. (2012). The decay of confined vortex rings. Experiments in Fluids, 53, 163–171. 197. Su, T. F., Patterson, M. A., Reitz, R. D., & Farrell, P. V. (1996). Experimental and numerical studies of high pressure multiple injection sprays. SAE Report 960861. 198. Taghavifar, H., Anvari, S., & Mousavi, S. H. (2020). Assessment of varying primary/secondary breakup mechanism of diesel spray on performance characteristics of HSDI engine. Fuel, 262, 116622. 199. Tanner, F. X. (2004). Development and validation of a cascade atomization and drop breakup model for high-velocity dense sprays. Atomization and Sprays, 14, 211–242. 200. Turner, M. R., Healey, J. J., Sazhin, S. S., & Piazzesi, R. (2011). Stability analysis and break-up length calculations for steady planar liquid jets. Journal of Fluid Mechanics, 668, 384–411. 201. Turner, M. R., Healey, J. J., Sazhin, S. S., & Piazzesi, R. (2012). Wave packet analysis and break-up length calculations for accelerating planar liquid jets. Fluid Dynamics Research, 44(1), 015503. 202. Turner, M. R., Sazhin, S. S., Healey, J. J., Crua, C., & Martynov, S. B. (2012). A breakup model for transient Diesel fuel sprays. Fuel, 97, 288–305. 203. Uchiyama, M. (2009). Numerical simulation of non-evaporating spray jet by the vortex method. Atomization and Sprays, 19, 917–928. 204. Urbán, A., Katona, B., Malý, M., Jedelský, J., & Józsa, V. (2020). Empirical correlation for spray half cone angle in plain-jet airblast atomizers. Fuel, 277, 118197 205. Versteeg, H. K., & Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics. The finite volume method (2nd edn.) Harlow: Pearson Prentice Hall. 206. Vogel, T., Rieβ, S., Lutz, M., Webzing, M., & Leipertz, A. (2011). Comparison of current spray models under high pressure and high temperature engine relevant conditions. In Proceedings of the 24th European Conference on Liquid Atomization and Spray Systems (ILASS) – Europe 2011, Estoril, Portugal, 5–7 September. 207. Wang, S., & Loth, E. (2014). A globally Eulerian locally Lagrangian particle concentration scheme. Powder Technology, 253, 614–625. 208. Wei, Y., Li, T., Zhou, X., & Zhang, Z. (2020). Time-resolved measurement of the near-nozzle air entrainment of high-pressure diesel spray by high-speed micro-PTV technique. Fuel, 268, 117343. 209. Weigand, A., & Gharib, M. (1997). On the evolution of laminar vortex rings. Experiments in Fluids, 22, 447–457. 210. Wu, H., Zhang, F., Zhang, Z., Guo, Z., Zhang, W., & Gao, H. (2020). On the role of vortex-ring formation in influencing air-assisted spray characteristics of n-heptane. Fuel, 266, 117044. 211. Wu, H., Zhang, F., Zhang, Z., & Gao, H. (2020). Experimental investigation on the spray characteristics of a self-pressurized hollow cone injector. Fuel, 266, 117710. 212. Wu, S., Gandhi, A., Li, H., & Meinhart, M. (2020). Experimental and numerical study of the effects of nozzle taper angle on spray characteristics of GDI multi-hole injectors at cold condition. Fuel, 275, 117888.

50

1 Spray Formation and Penetration

213. Wu, S., Meinhart, M., Petersen, B., Yi, J., & Wooldridge, M. (2021). Breakup characteristics of high speed liquid jets from a single-hole injector. Fuel, 289, 119784. 214. Yang, H. Q. (1992). Asymmetric instability of a liquid jet. Physics of Fluids A, 4, 681–689. 215. Yi, Y., & Reitz, R. D. (2004). Modelling the primary breakup of high-speed jets. Atomization and Sprays, 14, 53–80. 216. Zaripov, T. S., Gilfanov, A. K., Begg, S. M., Rybdylova, O., Sazhin, S. S., & Heikal, M. R. (2017). The fully Lagrangian approach to the analysis of particle/droplet dynamics: Implementation into ANSYS FLUENT and application to gasoline sprays. Atomization and Sprays, 27(6), 493–510. 217. Zaripov, T. S., Rybdylova, O., & Sazhin, S. S. (2018). A model for heating and evaporation of a droplet cloud and its implementation into ANSYS Fluent. International Communications in Heat and Mass Transfer, 97, 85–91. 218. Zhai, C., Jin, Y., Nishida, K., & Ogata, Y. (2021). Diesel spray and combustion of multihole injectors with micro-hole under ultra-high injection pressure - Non-evaporating spray characteristics. Fuel, 283, 119322. 219. Zhang, K., Wang, Z., Wang, J., & Wang, Z. (2012). Spray model based on step response theory. Fuel, 95, 499–503. 220. Zhang, W., Liu, H., Liu, C., Jia, M., & Xi, X. (2019). Numerical investigation into primary breakup of diesel spray with residual bubbles in the nozzle. Fuel, 250, 265–276. 221. Zhang, W., Liu, H., Liu, C., Jia, M., & Xi, X. (2019). Numerical analysis of jet breakup based on a modified compressible two-fluid-LES model. Fuel, 254, 115608. 222. Zhao, H., Li, H.-F., Cao, X.-K., Li, W.-F., & Xu, J.-L. (2011). Breakup characteristics of liquid drops in bag regime by a continuous and uniform air jet flow. International Journal of Multiphase Flow, 37, 530–534. 223. Zhao, H., Wu, Z., Li, W., et al. (2019). Interaction of two drops in the bag breakup regime by a continuous air jet. Fuel, 236, 843–850. 224. Zhou, X., Li, T., Lai, Z., & Wei, Y. (2019). Modeling diesel spray tip and tail penetrations after end-of-injection. Fuel, 237, 442–456.

Chapter 2

Heating of Non-evaporating Droplets

Someone wishing to model heating of non-evaporating droplets would be required to consider several important processes. These include the deformation of droplets in the air stream and the inhomogeneity of the temperature distribution at the droplet surface. Rigorous solution of this general problem, however, would have been not only difficult, but also would have rather limited practical applications. Indeed, in realistic engineering applications simultaneous modelling of heating of many droplets would be required. Moreover, this modelling would have to be performed alongside gas dynamics, turbulence and chemical modelling. This leads to the situation where the parameters of gas around droplets may be estimated with substantial errors, which are often difficult to control. This is why the focus has to be on finding a reasonable compromise between accuracy and computer efficiency of the models, rather than on the accuracy of the models alone. The most used assumption is that the droplets retain their spherical forms even as they move. This assumption is used in this chapter as well. The generalisations of the models to droplets of arbitrary shapes were discussed in a number of books and articles (e.g. [71, 130, 178]). These generalisations could be applied to simple models of droplet heating, but it is not obvious how they can be applied to the more sophisticated models described below. Another simplification widely used in droplet heating models is the assumption that the temperature over the whole droplet surface is the same (although it can vary with time). This assumption effectively allows the separation of the analysis of heat transfer in gaseous and liquid phases. It is expected to be a good approximation in the case of a stationary or very fast-moving droplet, when the isotherms almost coincide with streamlines [6]. The errors introduced by this assumption in intermediate conditions are considered to be acceptable without rigorous justification of this. In the following analysis, the models for convective and radiative heating of droplets are considered separately (Sects. 2.1 and 2.2).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_2

51

52

2 Heating of Non-evaporating Droplets

2.1 Convective Heating 2.1.1 Stagnant Droplets 2.1.1.1

Coupled Analytical Solutions

In stagnant droplets, there is no bulk motion of gas relative to the droplets, and the problem of their heating by the ambient gas reduces to a conduction problem. The heat conduction equation can be solved separately in the droplet and the gas, and the solutions are matched at the droplet surface. Assuming the spherical symmetry of the problem, its mathematical formulation is based on the solution of the equation [157, 158]:  2  ∂ T 2 ∂T ∂T =κ , (2.1) + ∂t ∂ R2 R ∂R where  κ=

κl = kl /(cl ρl ) when R ≤ Rd κg = k g /(c pg ρg ) when Rd < R ≤ ∞,

(2.2)

κl(g) , kl(g) , cl( pg) and ρl(g) are the liquid (gas) thermal diffusivity, thermal conductivity, specific heat capacity and density, respectively; R is the distance from the centre of the droplet, and t time. Equation (2.1) is based on Fourier’s law. Although the limitations of this law, including the lack of causality between the heat flux and the temperature gradient (see [72] for details), are well known, it is commonly assumed to be valid in most engineering applications. The model based on this equation is known as diffusion model [101]. Other heat conduction models and their generalisations, including thermal waves described by the hyperbolic heat equation, are discussed in [33, 75, 98, 99, 101, 106, 121, 138, 185, 195]. The problem of reflection and refraction of a thermal wave at an ideal interface was considered in [125]. The results of experimental investigation of non-Fourier heat conduction characteristics in the oil-in-water emulsions were presented in [105]. The author of [111] presented critical analysis of various approaches to experimental validations of predictions of non-Fourier models. Other approaches to the analysis of non-Fourier models are discussed in [22, 93, 103, 136, 194, 198, 199]. Most of the results referring to Fourier’s law are applicable to Fick’s law describing the mass transfer process [164]. Possible anisotropy of thermal conductivity will not be considered in our analysis (see [70] for details). Equation (2.1) was solved subject to the following initial and boundary conditions:  T |t=0 =

Td0 (R) Tg0 (R)

when R ≤ Rd when Rd < R ≤ ∞,

(2.3)

2.1 Convective Heating

53

T | R=Rd −0 = T | R=Rd +0 ; kl

  ∂ T  ∂ T  = k ; g ∂ R  R=Rd −0 ∂ R  R=Rd +0

T | R=∞ = Tg∞ .

(2.4) (2.5)

Numerical solutions of Eq. (2.1) are widely discussed in the literature (e.g. [137]). The following analysis focuses on analytical solutions to (2.1) and their implementations into numerical codes. Assuming that Td0 (R) = Td0 = const1 , and Tg0 (R) = Tg∞ = const2 , Cooper [37] solved this problem analytically, using the Laplace transform method, and obtained the following expression: T (R ≤ Rd ) = Tg∞ +

∞

× 0

T (R ≥ Rd ) = Tg∞ +

2kl π kg



 Rd κg  Tg∞ − Td0 κl R

    R κl 2 sin u , duΦ(u) exp −u Fo κg Rd

2kl π kg



 Rd κg  Tg∞ − Td0 κl R



∞

du 0

  Φ(u) κl exp −u 2 Fo u κg

   κg kl × cos(γc u) sin u + sin γc (u cos u − sin u) + sin u , κl kg where Φ(u) =

u 2 sin2 u +

κg κl

(2.6)

(2.7)

(u cos u − sin u)

2 , kl (u cos u − sin u) + sin u kg

Fo = tκg /Rd2 (Fourier number),  γc =

κl κg



 R − Rd u. R

As expected, this solution predicts diffusion of heat from gas to droplets, if Tg∞ > Td0 . As a result, droplet and gas temperatures approach each other in any finite domain. Initially, the heat flux from gas to droplets predicted by the expression q˙ = −k g

 ∂ T  ∂ R  R=Rd +0

(2.8)

is infinitely large. It approaches zero at t → ∞. This solution was originally applied to the problem of heating a stationary liquid sodium sphere in UO2 atmosphere [37]. It is questionable if this solution can be

54

2 Heating of Non-evaporating Droplets

applied to more general problems, involving the time variations of gas temperature due to external factors and evaporation effects. The only practical approach to solve this general problem is currently based on Computational Fluid Dynamics (CFD) codes. In this case, one would need to consider the distribution of temperature inside droplets at the beginning of each timestep and the finite size of computational cells. These effects were considered in the solution suggested in [157, 158]. In the models described in these papers, the condition R ≤ ∞ in (2.2) and (2.3) was replaced with the condition R ≤ Rg < ∞, where Rg is the external radius of the gaseous sphere. The temperature T was assumed to be constant for all R ≥ Rg . The value of Rg − Rd was interpreted as the thickness of the thermal boundary layer (see [158] for details). The analysis of [157] assumed that the initial distribution of temperature inside droplets is a function of the distance from the droplet centre, while the initial gas temperature was assumed to be constant. In a more general model, described in [158], both initial droplet and gas temperatures were assumed to be functions of the distance from the droplet centre when R ≤ Rg . In what follows, the main findings of [158] are described. Having replaced the condition R ≤ ∞ in (2.2) and (2.3) with the condition R ≤ Rg < ∞, the solution to Eq. (2.1) can be presented as [158] (see the solution presented in Appendix B without considering the contribution of thermal radiation) T (R, t) = Tg0 (Rg ) +

 Rd ∞

  1 1  exp −λ2n t (−(Tg0 (Rg ) − Td0 (R))Rvn (R)cl ρl d R R n=1 ||vn ||2 0 +

Rg

 (−(Tg0 (Rg ) − Tg0 (R))Rvn (R)c pg ρg d R

vn (R),

(2.9)

Rd

where  vn (R) =

||vn ||2 =

sin(λn al R) sin(λn al Rd ) sin(λn ag (R−Rg )) sin(λn ag (Rd −Rg ))

when R < Rd when Rd ≤ R ≤ Rg ,

(2.10)

c pg ρg (Rg − Rd ) kl − k g cl ρl Rd + − . 2Rd λ2n 2 sin2 (λn al Rd ) 2 sin2 (λn ag (Rd − Rg ))

A countable set of positive eigenvalues λn was obtained from the solution to the equation:   kl − k g , kl cl ρl cot(λal Rd ) − k g c pg ρg cot(λag (Rd − Rg )) = Rd λ where

(2.11)

2.1 Convective Heating

55

 al =

cl ρl , ag = kl



c pg ρg . kg

The values of λn were arranged as 0 < λ1 < λ2 < · · · . Using the dimensionless variables Td0 (R) T˜d = , Tg0 (Rg )

T (R, t) T˜ = , Tg0 (Rg )

Tg0 (R) Rg R , rg = , T˜g = , r= Tg0 (Rg ) Rd Rd

Expression (2.9) was presented as  1 ∞    1 Rd  exp −λ2n t T˜ = 1 + (−(1 − T˜d )r vn (Rd r )cl ρl dr r n=1 ||vn ||2 0

rg

+

(−(1 − T˜g )r vn (Rd r )c pg ρg dr

 vn (Rd r ).

(2.12)

1

If Tg0 (R) = Tg0 (Rg ) = const and Td0 does not depend on R then Expression (2.9) is simplified to √ ∞

 2  (Tg0 − Td0 ) kl cl ρl 1  exp −λn t T (R, t) = Tg0 + R n=1 λn ||vn ||2   1 vn (R). × Rd cot(λn al Rd ) − λn al

(2.13)

This solution was studied in detail in [157]. Although the above solutions are strictly valid for stationary droplets only, they are applicable to moving droplets. The droplet movement affects the values of Rg − Rd , as discussed in the next section.

2.1.1.2

Thermal Film Thickness

The physical meaning of Rg − Rd can be interpreted in terms of the so-called ‘film’ theory [6]. The key concept of this theory is thermal film thickness δT , the expression for which was derived from the requirement that the rate of a purely molecular transport by thermal conduction through the film is equal to the actual intensity of the convective heat transfer between the external flow and the droplet surface. This requirement was presented as [84] 

qs =

k g ΔT Rd −

Rd2 Rd +δT 0

= hΔT,

(2.14)

56

2 Heating of Non-evaporating Droplets 

where qs = |q˙s |/(4π Rd2 ) is the value of the heat flux at the surface of the droplet, ΔT = Tg − Ts . Index 0 indicates that the effects of the Stefan flow are not considered; see Sect. 3.2.2 for the discussion of the case when the contribution of this flow is considered. The convective heat transfer coefficient is h. Using the definition of the Nusselt number for a non-evaporating droplet in a gas flow (Nu), we can rearrange Eq. (2.14) to 2 1−

=

Rd Rd +δT 0

2(Rd + δT 0 ) 2h Rd = ≡ Nu. δT 0 kg

(2.15)

From Eq. (2.15), it follows that δT 0 =

2Rd . Nu − 2

(2.16)

The value of Rg − Rd in the model is identified with δT 0 . Following [6], Nu is estimated as   (2.17) Nu = 1 + [1 + RePr]1/3 max 1, Re0.077 , where Re and Pr are Reynolds and Prandtl numbers, respectively. Other approximations for Nu are described in [44]. If an additional requirement that the initial heat rate inside the ‘film’ does not depend on R is imposed, the following equation for Tg0 (R) can be obtained:   4π k g Tg0 (Rg ) − Td0 1 Rd

−

1 Rg

=

  4π k g Tg0 (R) − Td0 1 Rd

−

1 R

.

(2.18)

It follows from (2.17) that for stationary droplets Nu = 2. Remembering the definition of Nu (Eq. (2.15)), this implies that for stationary droplets: h = k g /Rd .

(2.19)

The introduction of non-zero Re affects our earlier assumption about the spherical symmetry of the problem and Expression (2.19). This can be overcome if we replace k g by k g, eff = k g Nu/2 to satisfy Eq. (2.14). Also, kl would need to be replaced by the effective liquid thermal conductivity, following the Effective Thermal Conductivity model. This is discussed later in this chapter. Note that the ‘film theory’ based on Eqs. (2.14)–(2.18) was developed under the assumption that droplet heating is quasi-steady. This obviously contradicts the unsteady formulation of the problem (2.1)–(2.4). This contradiction, however, seems to be unavoidable, as the value of Rg − Rd in our model needs to be imposed ‘exter-

2.1 Convective Heating

57

nally’ as an input parameter. In [157], a range of values of Rg − Rd was investigated without any attempt to link them with the underlying physics of the phenomenon.

2.1.1.3

Analysis of the Coupled Solutions

Let us consider a Diesel fuel droplet with an initial homogeneous temperature of 300 K and radius Rd = 10 µm. It is assumed to be injected into a gas at temperature 900 K and pressure 30 atm (conditions typical for Diesel engines [150]). The following thermodynamic and transport properties are used [157, 158]: ρl = 600 kg/m3 kl = 0.145 W/(mK) cl = 2830 J/(kgK) ρg = 23.8 kg/m3 k g = 0.061 W/(mK) c pg = 1120 J/(kgK). This gives us the following values of κl and κg (cf. Expression (2.2)): κl = 8.54 × 10−8 m2 /s;

κg = 2.29 × 10−6 m2 /s.

Note that the gas temperature 900 K is slightly higher than the one considered in [157], where it was assumed that Tg0 (Rg ) = 800 K. The difference in the values of these coefficients for these two temperatures was not taken into account as in [157]. Pr was assumed to be equal to 0.7 and two values of Re were considered: 1 and 5. Having estimated Nu from Expression (2.17), Expression (2.16) allows us to obtain the following values of Rg : Rg1 = 3.301 Rd

and

Rg2 = 11.337 Rd

for Re = equal to 5 and 1, respectively. Two initial distributions of gas temperature in the range Rd < R ≤ Rg are considered following [158]. Firstly, it is assumed that Tg0 (R) satisfies Eq. (2.18), which leads to the following formula:   Tg0 (R) = Td0 + Tg0 (Rg ) − Td0

1 Rd 1 Rd

− −

1 R 1 Rg

.

(2.20)

Secondly, it is assumed that Tg0 (R) = Tg0 (Rg ).

(2.21)

The second case is identical to the one investigated in [157]. The effect of thermal radiation is ignored for the time being. This is described in Sect. 2.2. Following [158], the analysis is focused on dimensionless time (Fourier number), distance and temperature:

58

2 Heating of Non-evaporating Droplets

Fig. 2.1 Plots of Tˆ ≡ (Tg0 (Rg ) − T (R, t))/(Tg0 (Rg ) − Td0 ) versus r = R/Rd for r g ≡ Rg /Rd = 3.301 and four Fo (indicated near the curves). Solid curves refer to the initial distribution (2.21), while dashed curves refer to the initial distribution (2.20). The thickness of the curves is inversely proportionate to Fo. Reprinted from [158], Copyright Elsevier (2011)

Fo = tκg /Rd2 ,

r = R/Rd ,

Tˆ(s) = (Tg0 (Rg ) − T(s) (R, t))/(Tg0 (Rg ) − Td0 ).

The calculations were performed using Wolfram Mathematica v. 6.0. 100 terms of the series were taken. Plots of Tˆ versus r for Rg = 3.301 Rd and four values of Fo are presented in Fig. 2.1. The plots are presented for both initial distributions of gas temperature in the range Rd < R ≤ Rg , defined by Expressions (2.20) and (2.21). As can be seen from this figure, for Fo = 0.1 most of the interior of the droplet is not affected by high gas temperature for both initial distributions of Tg0 (R), but the droplet temperatures near the surface are affected stronger by gas temperature for distribution (2.21), compared with distribution (2.20). The difference in gas temperatures (r > 1) for these initial distributions of Tg0 (R) is clearly visible as expected. For Fo = 1 and Fo = 10, distribution (2.21) shows more rapid heating of the droplet compared with distribution (2.20). This is seen more clearly than for the case Fo = 0.1. Gas temperatures predicted by both distributions are much closer for Fo = 1 and Fo = 10 compared with Fo = 0.1. For Fo = 100, for both initial temperature distributions, droplet and gas temperatures become very close to Tg0 (Rg ). The plots, similar to those shown in Fig. 2.1 but for Rg = 11.337 Rd , are presented in Fig. 2.2. Comparing Figs. 2.1 and 2.2, one can see that the trends of the curves in both figures are essentially the same, although the difference in the droplet heat-up for Fo = 1 and Fo = 10, predicted for distributions (2.20) and (2.21), is more clearly visible in Fig. 2.2 than in Fig. 2.1. Plots of Tˆs versus Fo for Rg = 3.301 Rd , Rg = 11.337 Rd and both initial distributions of Tg0 (R) are shown in Fig. 2.3. As follows from this figure, the droplet surface is always heated quicker for distribution (2.21) compared with distribution (2.20), as expected. Also, the droplet is heated quicker for Rg = 3.301 Rd than for Rg = 11.337 Rd . These results are consistent with those shown in Figs. 2.1 and 2.2.

2.1 Convective Heating

59

Fig. 2.2 The same as Fig. 2.1 but for r g = 11.337. Reprinted from [158], Copyright Elsevier (2011)

Fig. 2.3 Plots of Tˆs ≡ (Tg0 (Rg ) − Ts (R, t))/(Tg0 (Rg ) − Td0 ) versus Fo for r g = 3.301 and distribution (2.21) (solid), r g = 11.337 and distribution (2.21) (dashed-dotted), r g = 3.301 and distribution (2.20) (thick dashed), r g = 11.337 and distribution (2.20) (thin dashed). Reprinted from [158], Copyright Elsevier (2011)

Ignoring the effects of the droplet movement, the heat flux arriving at its surface can be estimated as   ∂ T  ∂ T   = kl . (2.22) q = kg ∂ R  R=Rd +0 ∂ R  R=Rd −0 On the other hand, from Newton’s law of cooling follows that    q N = h Tg0 (Rg ) − Ts , k

(2.23)

where h = Rgd (see Expression (2.19)). Note that Newton’s law is also called the Newton–Richmann law of cooling [41]. A formulation of this law based on generalised statistics was suggested in [40]. Strictly

60

2 Heating of Non-evaporating Droplets

speaking, the formulation of Newton’s law of cooling in the form of Expression (2.23) was suggested not by Newton but much later by Fourier (see [15]).   In the steady-state limit, q = q N . In the general transient case, however, they are linked by the equation [157]:   (2.24) q = χq N ,   Rd ∂∂ TR  R=R +0 k g ∂∂ TR  R=R +0 d d . χ = kg  = Tg0 (Rg ) − Ts T (R ) − T g0 g s Rd

where

(2.25)

An explicit expression for χ follows from Cooper’s solution [37] (see Expression (2.7) ) in the limit when Rg → ∞: χ ≡ χt =

− kkgl

∞ 0

(u cos u − sin u)Φ(u) exp(−u 2 Fo κκgl )du ∞ , κl 2 0 sin uΦ(u) exp(−u Fo κg )du

(2.26)

where Φ(u) is the same as in Expressions (2.6) and (2.7). If Newton’s law is valid then χ = 1. The plots of χ versus Fo for various r g ≡ Rg /Rd , and both initial distributions of Tg0 (R) are shown in Fig. 2.4. The solid plots referring to distribution (2.21) are identical to those presented in [157]. The solid plot referring to r g = 50 is practically indistinguishable from the one which follows from the analysis by Cooper [37], obtained in the limit Rg = ∞ using an approach totally different from the one considered in [157, 158] (see Eq. (2.26)). This coincidence demonstrates the validity of both approaches. The dashed curves, obtained for the initial distribution (2.21), coincide with the solid curves referring to distribution (2.20) in the limit of large Fo. For small Fo,

Fig. 2.4 Plots of χ versus Fo for four r g (indicated near the curves). Solid curves refer to the initial distribution (2.21), while dashed curves refer to the initial distribution (2.20). Reprinted from [158], Copyright Elsevier (2011)

2.1 Convective Heating

61

the deviations between the curves corresponding to distributions (2.20) and (2.21) is clearly seen. For distribution (2.21), χ rapidly increases with decreasing Fo (χ → ∞ when Fo → 0). For distribution (2.20), χ approaches finite values when Fo → 0, being always less than predicted for distribution (2.21). The values of χ in the limit Fo → 0 can be estimated analytically from (2.21) and (2.24) as χ=

R˜ g R˜ g − 1

.

(2.27)

Note that the values of χ for Fo = 0.1 can differ by up to about 8% from those predicted by (2.27) (although the values of temperature were calculated with errors less than about 0.5%). This is related to the very slow convergence of the corresponding series in (2.9) for the derivative of the temperature in the vicinity of the droplet surface (up to 3000 terms in this series were taken). Although the solutions described in this section so far clearly show the importance of coupling between the liquid and gas phases, it remains unclear how they can be used in most practical engineering applications. All the available CFD codes are based on separate solutions for gas and liquid phases, followed by their coupling [78, 109, 139]. Hence, some kind of separation of the solutions for gas and liquid phases would be essential to make them compatible with these codes. The required separation between the solutions could be achieved based on comparing thermal diffusivities of gas and liquid. Let us consider previously used typical values of parameters for Diesel fuel droplets and assume that these droplets have initial temperature 300 K and are injected into a gas at temperature 800 K and pressure 30 atm [150]. For these values of parameters, we obtain κl = 8.53 × 10−8 m2 /s and κg = 2.28 × 10−6 m2 /s. This leads to the following condition: κl κg .

(2.28)

Condition (2.28) tells us that gas responds much quicker to changes in the thermal environment than liquid. As a zeroth approximation, we can ignore the changes in liquid temperature altogether, and assume that the droplet surface temperature remains constant in time. This allows us to decouple the solution to Eq. (2.1) for the gas phase from the trivial solution to this equation for the liquid phase (T (R ≤ Rd ) = const). The former solution is described in the next section.

2.1.1.4

Analytical Solution for the Gas Phase

Assuming that T (R ≤ Rd ) = Td = const, the following solution for the gas phase can be obtained [151]:

  R − Rd Rd T (R > Rd ) = Tg∞ + (Td − Tg∞ ) 1 − erf , √ R 2 κg t

(2.29)

62

where

2 Heating of Non-evaporating Droplets

2 erf(x) = √ π



x

exp(−t 2 )dt.

0

For R = Rd , Expression (2.29) gives T = Td . For t → 0, but R = Rd , this expression gives T = Tg∞ . Having substituted (2.29) into (2.8), the following expression for the heat flux from gas to droplets is obtained [172]:   k g (Td − Tg∞ ) Rd 1+ √ . q˙ = Rd π κg t

(2.30)

The same expression follows from the analysis presented later in [66, 151], who were apparently not aware of the original paper [172]. Moreover, this expression might have been derived even earlier, as in 1971 it was referred to in [34] as the ‘well-known conduction solution’ without giving references. For t  td ≡ Rd2 /(π κg ), Expression (2.30) can be simplified to q˙ = −h(Tg∞ − Td ),

(2.31)

where h is the convection heat transfer coefficient given by Expression (2.19). q˙ is assumed positive when directed away from the droplet. Expression (2.31) is identical to (2.23) describing Newton’s law. Solution (2.31) could be obtained directly from Eq. (2.1) if the time derivative of temperature is ignored (steady-state solution). Its derivation was implicitly based on the assumption that the boundary layer around droplets has had enough time to develop. For the values of parameters mentioned above, td = 3.5 µs. That shows that except at the very start of droplet heating, the analysis of this process can be based on Expression (2.31). This expression is widely used in CFD codes. Comparing Expressions (2.30), (2.31) and (2.19), one can see that Newton’s law (Expressions (2.31) and (2.19)) can be used to describe the transient process discussed above, if the gas thermal conductivity k g is replaced by the ‘time-dependent’ gas thermal conductivity k g(t) defined as [151, 172] k g(t) = k g (1 + ζt ) , 

where ζt = R d

c pg ρg . π kg t

(2.32)

(2.33)

This is applicable at any time including the very start of droplet heating (at the start of calculations when a droplet is injected into the gas). Unless abrupt changes in gas temperature occur, one may assume that the boundary layer around the droplet has had enough time to adjust to varying gas temperatures. This justifies the application of Newton’s law in its original formulation (Expressions (2.31) and (2.19)).

2.1 Convective Heating

63

In the limit t → ∞, Expression (2.29) can be presented as ΔT ≡ T − Tg∞ =

Rd (Td − Tg∞ ), R

(2.34)

where ΔT shows the local changes in gas temperature after the boundary layer around the droplet has been formed. The change of gas enthalpy, due to the presence of the droplet, in this case can be estimated as ΔHe =

∞

ρg c pg ΔT 4π R 2 d R = 4π Rd (Td0 − Tg∞ )ρg c pg

Rd

∞

Rd R = ∞.

Rd

(2.35) Thus, the formation of the required boundary layer around a single droplet leads to an infinitely large change in the enthalpy of the gas. Todes [172] seems to have been the first to draw attention to this fact. One can see, however, that if the value of ΔT was calculated for any t = ∞ from Eq. (2.29) then ΔHe would have remained finite. This is related to the fact that at any t < ∞, the second term in the square √ brackets in Expression (2.29) cannot be ignored at R − Rd > 2 κg t. This means that Expression (2.34) is not valid at these radii, and ΔHe cannot be calculated using Expression (2.35). In practice ΔHe has never been calculated from Expression (2.35) to the best of the author’s knowledge. Instead, the amount of heat gained or lost by the droplet during a certain period Δt is calculated based on the values of Nu. This gives reasonable results provided that the assumption that the droplet surface temperature remains constant in time is valid. To summarise this section, Expressions (2.31) and (2.19) are the most useful for engineering application in CFD codes. At the initial stage of heating, the corrections described by Expression (2.32) can be introduced if required. The range of applicability of these expressions for transient heating, however, has not been rigorously justified (cf. Expression (2.35)). In what follows, the processes in the liquid phase are considered.

2.1.1.5

Analytical Solutions for the Liquid Phase

If kl → ∞, which infers that Td (R ≤ Rd ) = const, the droplet temperature can be obtained from the energy balance equation: dTd 4 π Rd3 ρl cl = 4π Rd2 h(Tg∞ − Td ). 3 dt

(2.36)

This equation shows that all the heat supplied from gas is spent on raising the droplet temperature. Its solution is straightforward:   3ht , Td = Tg∞ + (Td0 − Tg∞ ) exp − cl ρl Rd

(2.37)

64

2 Heating of Non-evaporating Droplets

where Td (t = 0) = Td0 . Equation (2.36) and its solution (2.37) are widely used in various applications. They were used to determine experimentally the heat transferred by convection from gas to droplets [29]. Other applications are described in [1, 18, 23, 43, 82, 95, 134, 187, 193]. They are almost universally used in Direct Numerical Simulations (DNS) [179, 180] and Large Eddy Simulations (LES) [42, 62, 86, 88, 89, 145, 171] of the processes in sprays. Solution (2.37) is widely used in most Computational Fluid Dynamics (CFD) codes including ANSYS Fluent [166]. In some cases, this is justified by the fact that liquid thermal conductivity is much higher than that of gas. However, the main parameter which controls droplet transient heating is not its conductivity, but its diffusivity. As shown earlier, in the case of Diesel engine sprays the diffusivity for liquid is more than an order of magnitude less than that for gas. This raises the question of applicability of the assumption kl → ∞ to modelling fuel droplets in these engines. To answer this question, we must consider the model which takes into account the effect of finite liquid thermal conductivity. The model considering the effect of finite kl is based on the solution of Eq. (2.1) inside the droplet with the Robin boundary condition at its surface: h(Tg − Ts ) = kl

 ∂ T  , ∂ R  R=Rd −0

(2.38)

the condition at the droplet centre  ∂ T  = 0, ∂ R  R=0

(2.39)

T (t = 0) = Td0 (R),

(2.40)

and the initial condition where Ts = Ts (t) is the droplet’s surface temperature, Tg = Tg (t) is the ambient gas temperature (which is time-dependent in the general case), the subscript ∞ has been omitted. Note that Condition (2.39) can be replaced by a requirement that T (R, t) is a twice continuously differentiable function at R ≤ Rd . Both approaches would lead to the same final solution. If h = const, the solution to (2.1), subject to the above-mentioned boundary and initial conditions, was presented as [79, 154] (see the solution given in Appendix A in which the effects of thermal radiation are not considered) ∞      sin λn Rd  qn exp −κ R λ2n t − μ0 (0) exp −κ R λ2n t − T (R, t) = R n=1 || vn ||2 λ2n

2.1 Convective Heating

65

     R dμ0 (τ ) + Tg (t), exp −κ R λ2n (t − τ ) dτ sin λn dτ R d 0 (2.41) where λn are solutions to the equation: −

sin λn || vn ||2 λ2n



t

λ cos λ + h 0 sin λ = 0, || vn ||2 =

(2.42)

    1 1 sin 2λn h0 = , 1− 1+ 2 2 2λn 2 h 0 + λ2n

1 qn = Rd || vn ||2



Rd 0

  R ˜ d R, T0 (R) sin λn Rd hTg (t)Rd , kl

κR =

kl , cl ρl Rd2

μ0 (t) =

h0 =

h Rd − 1, kl )

RTd0 (R) . T˜0 (R) = Rd

The solution to Eq. (2.42) gives a set of positive eigenvalues λn numbered in ascending order (n = 1, 2, . . .). Proof of the convergence of the series in (2.41) is presented in [154]. For kl → ∞, Solution (2.41) reduces to Solution (2.37) [148] (see Appendix A for details). Solution (2.41) was generalised to the case of almost constant convection heat transfer coefficient h [154]. For the general time-dependent h, the solution of the differential Eq. (2.1) in the liquid phase can be reduced to the solution of the Volterra integral equation of the second kind [154]. The usefulness of these solutions to applications in CFD codes, however, was limited [155]. The benefits of considering the finite thermal conductivity of fuel droplets in the CFD modelling of combustion processes in Diesel engines were first demonstrated in [17]. These authors performed the calculations using KIVA II CFD code with the conventional model for droplet heating based on Expression (2.37) (called the Spalding model), and with the model considering the finite thermal conductivity of droplets. The effect of finite thermal conductivity of droplets was considered based on the direct numerical solution of Eq. (2.1) in the liquid phase, rather than on the analytical solution to this equation (see Solution (2.41)) applied at each timestep. Their results clearly showed that considering the effects of finite thermal conductivity of droplets leads to better agreement with experimental data, compared with the conventional model. Further details are described in Sect. 3.8. The first direct experimental evidence of the need to consider the effect of finite kl seems to have been provided in [112]. Some key results of this paper are discussed in Sect. 3.8. Although nobody seems to contest the benefits of considering the finite thermal conductivity of droplets in CFD modelling, the developers of CFD codes do not embrace this modelling opportunity too eagerly. The main reason for this is the addi-

66

2 Heating of Non-evaporating Droplets

tional CPU time involved. This has stimulated efforts to develop models considering the effect of finite thermal conductivity of droplets, but with minimal extra demands on CPU time. The idea of one of these new models may have been prompted by looking at the character of the plots T (R) predicted by Expression (2.41). Assuming that initially T (R) is constant, one can see that, except at the very start of heating, the shape of the curve T (R) looks very close to a parabola. This allows us to approximate T (R) as [51]  2 R , (2.43) T (R, t) = Tc (t) + [Ts (t) − Tc (t)] Rd where Tc and Ts are the temperatures at the centre (R = 0) and on the surface (R = Rd ) of the droplet, respectively. Approximation (2.43) is obviously not valid at the very beginning of the heating process when T = Tc except in the immediate vicinity of the surface of the droplet. Substitution of Expression (2.43) into the boundary condition at R = Rd (Eq. (2.38)) gives [51] ζ (2.44) Ts − Tc = (Tg − Ts ), 2 k

g where ζ = Nu , Nu is the Nusselt number introduced earlier. Expression (2.43) 2 kl does not satisfy Eq. (2.1), but it is expected to satisfy the equation of thermal balance, obtained from integrating Eq. (2.1) along R:

ρl cl

  Rd dT = h Tg − Ts , 3 dt

where T =

3 Rd3



Rd

R 2 T (R)d R

(2.45)

(2.46)

0

is the average droplet temperature. Note that Eq. (2.45) is identical to Eq. (2.36) describing droplet heating in isothermal (infinite liquid conductivity) approximation, if h is replaced by h ∗ inferred from the modified Nusselt number Nu∗∗ = Nu

Tg − Ts Tg − T

.

(2.47)

Having substituted Expression (2.43) into Expression (2.46) and remembering Eq. (2.44), we obtained [51] T =

2Tc + 3Ts = Ts − 0.2ζ (Tg − Ts ). 5

(2.48)

2.1 Convective Heating

67

Substitution of Expression (2.48) into (2.47) gives [51] Nu∗∗ =

Nu . 1 + 0.2ζ

(2.49)

The combination of (2.44) and (2.48) gives [51] T + 0.2ζ Tg , 1 + 0.2ζ

(2.50)

(1 + 0.5ζ )T − 0.3ζ Tg . 1 + 0.2ζ

(2.51)

Ts =

Tc =

The combination of Expressions (2.49)–(2.51) gives the full solution to the problem of convective droplet heating as predicted by Expression (2.43). At first, Eq. (2.36) is solved with h replaced by h ∗ from Expression (2.49). Then the values of Ts and Tc are obtained from Expressions (2.50) and (2.51). These two parameters are then substituted into Expression (2.43) to give the radial distribution of temperature inside the droplet. For most practical applications, we are primarily interested in the values of Ts , which determine the rate of evaporation and break-up of droplets. The model based on Expression (2.43) was called the parabolic temperature profile model [51]. This model shows good accuracy at long times, but can differ considerably from the results based on the numerical solution to Eq. (2.1) for short times [51]. The period of time when this deviation happens is usually short and can be ignored in most practical calculations. The actual implementation of this model into a customised version of the CFD code VECTIS is described in [152]. The predictions of the model were shown to be reasonably close to those predicted by Expression (2.41), and its additional CPU requirements were shown to be small and could be acceptable in most engineering applications. The parabolic temperature profile model was generalised to consider the initial heating of droplets in [51], but the implementation of this generalised model into CFD codes has not yet been investigated. Note that the parabolic temperature profile model can be developed rigorously using Expression (2.41) if only the first term in this series is considered and the initial temperature inside the droplet is assumed to be constant. This approach was suggested in [191]. In the limit t → ∞, Eq. (18) of [191] is identical with Eq. (2.48). For t → 0 the accuracy of Equation (18) of [191] becomes questionable since in this case all terms in Expression (2.41) need to be taken into account. Well-known limitations of the parabolic model and the complexity of the model based on the rigorous analytical/numerical solutions to the heat transfer equation inside droplets stimulated efforts to develop new models. These were more accurate than the parabolic model and simpler than models based on rigorous solutions to the above-mentioned equation. One such model, known as the power law approximation, was suggested by Brereton [25] and further investigated by Snegirev [163]. This

68

2 Heating of Non-evaporating Droplets

model is based on the following approximation of the temperature profile inside the droplet:  p R , (2.52) T (R) = c p0 + c pp Rd where p is the fitting parameter adjusted to replicate temperature profile at small times, constants c p0 and c pp are determined from the values of the heat flux at the surface of the droplets and their average temperature. For p = 2, Expression (2.52) describes the parabolic temperature profile. To describe the droplet transient heating, it was assumed that p depends on time. At the initial stage of heating, the values of p were shown to be very high (typically in the range 10–100), and then these values rapidly decreased. Since the parabolic temperature profile model is known to describe adequately the temperature distribution inside a droplet in a long time limit, it was assumed that p ≥ 2 at all times. The power law approximation was shown to describe the heating of non-evaporating droplets in gas with fixed temperature, both in short and long time limits, reasonably accurately. The model developed in [183] is also based on (2.52) but with c p0 = Tc and c pp = Ts − Tc , where Tc and Ts are temperatures at the centre of the droplet and at its surface, respectively. An empirical formula for n was obtained based on the temperature distribution predicted by the rigorous one-dimensional model. The main limitation of this approximation is that it can predict only monotonic temperature profiles. This limitation was overcome by the so-called polynomial approximation, originally suggested in [168] (although for a different problem), and investigated in detail in [163]. In this approximation, the temperature profile inside the droplet is presented as  T (R) = c p0 + c p2

R Rd

2

 + c pp

R Rd

p ,

(2.53)

where p > 2. Approximation (2.53) allows one to overcome the above-mentioned limitation of the model based on (2.52). It was shown that the polynomial approximation can adequately describe non-evaporating droplet heating for arbitrary dependence of the external heat flux on time [163] (this was demonstrated for the cases of stepwise and smooth periodic surface heat flux). Intrinsic limitations of power law (including the parabolic temperature profile model) and polynomial approximations stem from the assumption that the temperature profile (either parabolic or higher order polynomial) is instantly established in the whole droplet volume. At the same time, one would expect (and this is confirmed by a rigorous analytical/numerical solution to the heat transfer problem inside droplets) that initially only a thin layer close to the droplet surface is affected by the external heat supplied to the droplet; then this heat gradually penetrates inside the

2.1 Convective Heating

69

droplet up to its centre. These processes are considered in the model based on the heat balance integral method, originally suggested in [162, 163]. This method is based on the introduction of the thermal layer of thickness δ(t), which is assumed to be time-dependent in the general case. Inside this layer, temperature is approximated by the parabolic profile, and the initial temperature is assumed constant outside the layer:  T (t)

ch0 + ch2 T0



R−(Rd −δ) Rd

2

,

Rd − δ < R ≤ Rd 0 ≤ R ≤ Rd − δ.

(2.54)

Expression (2.54) describes the parabolic temperature profile model when δ = Rd . Having found the droplet average temperature from the heat balance equation for the whole droplet, the thickness δ can be estimated by iterations of the following equation:

δ (i)

     −1   2keff T0 − T 1 δ (i−1) 2 1 δ (i−1)  = Rd + , 1− qs R d 2 Rd 10 Rd

(2.55)

where i = 1, 2, 3, . . . is the iteration number, qs is the heat flux at the droplet surface, keff is the droplet effective thermal conductivity, considering recirculation inside droplets (see Expression (2.81)). For δ (i) = δ (i−1) = Rd , Eq. (2.55) reduces to T0 = T +

3qs Rd . 10keff

(2.56)

This expression coincides with the one predicted by the parabolic temperature profile model for R = Rd . It was shown that the thermal layer expands to δ = Rd when the Fourier number Fo = keff t/(cl ρl Rd2 ), where cl and ρl are specific liquid heat capacity and density, respectively, becomes equal to 0.1. As shown in [163], the heat balance integral method can predict the time evolution of the surface temperature of a non-evaporating droplet more accurately than the parabolic temperature profile model. This method, as well as the power law and polynomial approximations, however, were verified in [163] based on the analytical solution to the heat transfer equation inside a droplet using the Neumann boundary condition (fixed external heat flux). The limits of applicability of this solution have not been investigated. This could be done based on comparing this solution with the solution using the Robin boundary condition (see Eq. (2.38)). A new liquid phase model based on the assumption that the third-order polynomial of the distance from the droplet centre can approximate temperature distribution in the droplet in the presence of the Hill vortex was described in [196] and applied to the analysis of R134a droplets in [197]. The main advantage of this model is that it is able to capture the local minima of the temperature distribution expected to

70

2 Heating of Non-evaporating Droplets

develop at the centre of the Hill vortex in the moving droplet. The limitation of this approach is that it attempts to approximate the expected two-dimensional structure of the distribution of temperature inside the moving droplet by a one-dimensional distribution, similar to the one assumed in the Effective Thermal Conductivity (ETC) model to be discussed later in Sect. 2.1.2.2. Although the authors of [196] claim that the predictions of their model lead to better agreement with experimental data compared with other models and the CPU requirement for the new model is far lower than that for the ETC model, the range of applicability of this model needs to be investigated before it can be recommended for applications. The analysis of Eq. (2.1) presented in this section was essentially based on the assumption that thermal conductivities, specific heat capacities and densities of gas and liquid are constant. This assumption is reasonable for CFD applications, where the calculations are performed over small timesteps when the variations of these parameters can be ignored. They can be updated from one step to another. Attempts to consider the temperature dependence of these parameters (i.e. to solve non-linear heat conduction equations) started more than 50 years ago and are still continuing (e.g. [14, 26, 27, 58, 76, 80, 92, 190]). Other approaches to solving non-linear heat conduction problems are described in [64, 83, 114, 124]. The heat transfer problem in an orthotropic sphere is described in [146]. The orthotropic sphere allows us to consider difference in heat conductivities in different directions. This generalisation seems not to be relevant to the problems considered in this book.

2.1.2 Moving Droplets The analysis of heat exchange between gas and stationary droplets was simplified by the fact that this problem is essentially one-dimensional in space. The complexity of the problem of heat transfer between gas and moving droplets lies in the fact that this problem is at least two-dimensional in space. This leads to the need to replace Eq. (2.1) by a more general equation for the temperatures of both liquid and gas and present it as [116] ∂T + ∇ · (u(t, x)T ) = κ∇ 2 T, (2.57) ∂t where u(t, x) is the fluid velocity, which depends on time t and position in space x in the general case. The Laplacian ∇ 2 is three-dimensional in the general case; its two-dimensional approximation has been almost universally used. In most practical applications, the time dependence of u is ignored, even if the problem of transient heat transfer is studied. In the general case, this equation is solved both in droplets and in the surrounding gas. Equation (2.57) does not contain the so-called history terms. These will be briefly described later. Application of Eq. (2.57) instead of Eq. (2.1) leads to qualitative differences between the mechanisms of heat transfer in stationary and moving droplets. In

2.1 Convective Heating

71

the case of stationary droplets, the heat transfer takes place via conduction, and is described by the heat conduction equation in both liquid and gas. In the case of moving droplets, however, convective heat transfer takes place, which incorporates effects of bulk fluid motion (advection) and diffusion effects (conduction). The modelling of heat transfer in this case needs to be based on the Navier–Stokes equations for enthalpy and momentum [81]. For stationary droplets, the thickness of the boundary layer around droplets is infinitely large. In the case of moving droplets, however, this thickness is always finite (see Eq. (2.16)). All these differences between the heat transfer processes in stationary and moving droplets require the development of different methods of analysis. While in the case of stationary droplets we started with the general transient solution and ended up with the analysis of the limiting steady-state case, in the case of moving droplets the starting point will be the simplest steady-state case.

2.1.2.1

Steady-State Heating

For steady-state droplet heating, the first term in Eq. (2.57) can be assumed equal to zero, and this leads to considerable simplification of the solution of this equation. In some practically important cases, this solution can be avoided altogether, and the analysis can be performed at a semi-qualitative level, backed by experimental observations. The first simplification of the problem assumes that there is no spatial gradient of temperature along the surface of a droplet. This allows us to separate the analysis of the gas and liquid phases. If the surface temperature of the droplet is fixed, then using dimensional analysis it can be shown that the Nusselt number (Nu) depends only on Reynolds and Prandtl numbers (Re and Pr). If Re = 0 (stagnant droplet), then Nu = 2. The qualitative analysis of the problem for large Re, but in the laminar boundary layer region, shows that Nu is proportional to Re1/2 Pr1/3 [71]. Thus, the general expression for Nu can be written as Nu = 2 + βc Re1/2 Pr 1/3 ,

(2.58)

where the coefficient βc cannot be derived from this simplified analysis. It should be obtained either from experimental observations or from more rigorous numerical analysis of the basic equations. The most widely used value of this coefficient, supported by experimental observations, is βc = 0.6 (see, for example, [19]). In several papers (see, for example, [6, 160]), βc = 0.552 was used. Sometimes, the power 1/3 of the Prandtl number was replaced by 0.4 [116]. For stationary droplets heated or cooled due to natural convection, the value of Nu was estimated as [39, 189] Nu = 2 + 0.43 (Gr Pr)1/4 , where Grashof’s number is defined as

(2.59)

72

2 Heating of Non-evaporating Droplets

Gr =

gβG (Ts )(2Rd )3 |Tg − Ts | ,  2 νg (Ts )

βG =

(2.60)

ρg (Tg ) − ρg (Ts ) ρg (Ts )[Ts − Tg ]

is the coefficient of thermal expansion, approximated for an ideal gas as βG = 1/Ts , and νg is gas kinematic viscosity. Formula (2.59) is valid for 1 < Gr Pr < 100,000. Whitaker [181] drew attention to the fact that in the wake region, for large Re, Nu is expected to be proportional to Re2/3 Pr1/3 . Also, the difference in gas dynamic viscosities in the vicinity of the droplet surface (μgs ) and in the bulk flow (μg∞ ) had to be taken into account. The following correlation for Nu was suggested   Nu = 2 + aW Re1/2 + bW Re2/3 Pr 1/3



μg∞ μgs

cW ,

(2.61)

where constants aW , bW and cW had to be calculated based on experimental data. The analysis of these data led to aW = 0.4, bW = 0.06 and cW = 0.25. Expression (2.61) is expected to be more accurate than Expression (2.58) (at least due to the larger number of fitting constants), but it has rarely been used in applications. Note that the ratio μg∞ /μgs is expected to be close to unity. As demonstrated in [6, 35], Expression (2.58) over-estimates the heat transfer rate at low Reynolds numbers (Re≤ 10). Also, both Expressions (2.58) and (2.61) predict the physically incorrect super-sensitivity of the heat transfer rate to small velocity fluctuations near Re = 0, since ∂Nu/∂Re → ∞ when Re → 0 [160]. As an alternative to Expression (2.58), the following correlation was recommended [35, 160] (2.62) Nu = 1 + [1 + RePr]1/3 f c (Re), where f c (Re) = 1 at Re ≤ 1 and f c (Re) = Re0.077 at 1 < Re ≤ 400. Expression (2.62) approximated the numerical results obtained by several authors for 0.25 < Pr < 100 with errors less than 3% [160]. Analyses of the process of heating of a spherical droplet in a flow of fluid were often restricted to the case of small Re when the flow can be considered as Stokesian (cf. Formula (1.45)). In this case, considering droplets as solid bodies without internal recirculation, the radial and azimuthal components of external flow velocity are described as [34, 135]     3  1 3Re Rd 3Rd 1+ + cos θ 2− ur = u 0 2 16 R R 

2 3 4    Rd Rd 3Re 3Rd Rd − + 3 cos2 θ − 1 − 2− + 64 R R R R

2.1 Convective Heating

73

  +O Re2 log Re ,  uθ = u0

(2.63)

   3  3Re Rd 1 3Rd 1+ − − sin θ 4− 4 16 R R

   4   3   2 Rd 3Re 3Rd Rd −2 cos θ sin θ + O Re log Re , + + 4− 64 R R R (2.64) where u 0 is the flow velocity unperturbed by the droplet, u r and u θ are the radial (away from the centre of the droplet) and circumferential flow velocity components (angle θ is estimated relative to the direction of the velocity in the unperturbed flow). As follows from Expressions (2.63) and (2.64), both components of flow velocity are zero at R = Rd as expected (no slip condition). In the limit Re→ 0, Expressions (2.63) and (2.64) are simplified to  ur = u 0

1 3Rd + 1− 2R 2



 u θ = −u 0

1 3Rd − 1− 4R 4

Rd R



3  cos θ,

Rd R

(2.65)

3  sin θ.

(2.66)

The main advantages of Expressions (2.63) and (2.64) compared with Expressions (2.65) and (2.66) can be seen at R close to Rd . In the limit R → ∞, Expressions (2.65) and (2.66) predict the components of the unperturbed flow velocity u 0 . For droplets in which recirculation is allowed, Expressions (2.65) and (2.66) can be generalised to [127] (p. 697), [66] 

A v Rd + 2Bv 1− R

ur = u 0

 u θ = −u 0 where Av =



A v Rd − Bv 1− 2R

3λv + 1 , 2(λv + 1)

Rd R



Bv =

3 

Rd R

cos θ,

(2.67)

3  sin θ,

(2.68)

λv , 4(λv + 1)

λv = μl /μg is the ratio of dynamic viscosities of liquid and gas (note that this definition of λv is different from the one used in [66, 127] but consistent with the definition of this parameter used later). For λv = ∞ (solid body), Expressions (2.67) and (2.68) refer to the case when R ≥ Rd , and they reduce to Expressions (2.65) and (2.66). Similar expressions for the volume inside the droplets could be derived from

74

2 Heating of Non-evaporating Droplets

Expression (21.14.5b) of [127]. These describe Hill’s spherical vortex [127]. A more in-depth analysis of recirculation within a droplet is presented in [12]. If the effects of gas viscosity are ignored then for both solid and liquid moving droplets, the components of velocity have particularly simple forms valid for R ≥ Rd (see [127], p. 562):   3  Rd ur = u 0 1 − cos θ, (2.69) R  u θ = −u 0

1 1+ 2



Rd R

3  sin θ.

(2.70)

The components of fluid velocity given by Expressions (2.69) and (2.70) can be used in Eq. (2.57). Expressions (2.65) and (2.66) were used in [8] for the asymptotic analysis of the process of heating of a spherical body in a flow of fluid. The authors of this paper assumed that both Reynolds number (Re) and Peclet number (Pe ≡ Re · Pr) are small, and used the technique originally developed in [135], which led to the derivation of Eqs. (2.63) and (2.64). They derived the following expression for Nu:   1 1 1 Nu = 2 1 + Pe + Pe2 ln Pe + 0.01702Pe2 + Pe3 ln Pe + O(Pe3 ) . 4 8 32 (2.71) As follows from (2.71), Nu → 2 when Pe → 0. Although this expression was derived using the assumptions that Re 1 and Pe < 1, it can still be applied for Re in the range 0–0.7 [116]. Sometimes, Re is defined based on Rd and not 2Rd (as in this book). This leads to a different form of Expression (2.71) [140]. Using Expressions (2.63) and (2.64) but considering Pe→ ∞, an alternative expression for Nu was obtained [8]:  Nu = 0.991Pe

1/3

 3 1 2 2 Re ln Re + O(Re ) . 1 + Re + 16 160

(2.72)

The generalisation of the analysis by [8] for the case when Re < 1 and Pe < 1 by taking extra terms in the velocity field (that is replacing Expressions (2.65) and (2.66) with Expressions (2.63) and (2.64)) is described in [140]. The generalisations of this expression to the case of bodies of arbitrary shape are described in [7, 13, 24]. Numerically analysing the transient heat transfer from a sphere at high Reynolds and Peclet numbers, the following steady-state correlation was suggested [68]: Nu = 0.922 + Pe1/3 + 0.1Re1/3 Pe1/3 .

(2.73)

For Re = 0, Expression (2.73) reduces to the solution described in [7]. The prediction of (2.73) is close to that of Expression (2.72) at large Pe, but small Re.

2.1 Convective Heating

75

In [115], a mathematical model was developed to describe the heat transfer process when a melting sphere is immersed in a moving fluid. The following correlation for Nu was obtained in this case Nu = 2 + 0.47Re0.5 Pr 0.36 ,

(2.74)

where 102 ≤ Re ≤ 5 × 104 .

0.003 ≤ Pr ≤ 10,

This model was validated against experimental results involving metals and water. Nu described by Expression (2.74) is reasonably close to Nu predicted by Expression (2.58). Expressions (2.71)–(2.74) do not consider recirculation inside the droplets. The latter could be accounted for based on asymptotic or rigorous numerical analysis of coupled fluid dynamics and heat transfer equations. Levich [102] (see p. 408) was perhaps the first to provide an asymptotic solution for small Re, but large Pe in the form (see also [116]):  1 4 Pe. (2.75) Nu = 3π 1 + λv As this formula is valid for very large Pr, its applicability to the analysis of droplet heating in gas is limited. Earlier studies in this direction were reviewed by Feng and Michaelides [69]. Also, in this paper a comprehensive numerical analysis of heat and mass transfer coefficients for viscous moving spheres (droplets) is described. Their analysis was based on a number of conventional assumptions. Namely, it was assumed that droplets retain their spherical forms and there are no temperature gradients along the droplet surfaces. In what follows their main results are described. One of the aims of [69] was to investigate the dependence of Nu on Re, Pe ≡ Re · Pr, λv = μl /μg and ρl /ρg , where μl(g) and ρl(g) are dynamic viscosities and densities of liquid (gas), respectively. As follows from their analysis, the dependence of Nu on ρl /ρg is negligibly small for ρl /ρg in the range 0.1–10 (bubbles and droplets), the range of Pe 1 to 500, Re = 10 and λv = 1. It was not expected to be important for other ranges and values of parameters. Hence, the analysis focused on the investigation of the dependence of Nu on Re, Pe and λv . The results were presented in a table, covering the range of Re from Re→ 0 to 500, the range of Pe from 1 to 1000, and of λv from 0 to ∞. Using the information shown in this table, the following working correlations were suggested [69]. At small but finite Re (0 < Re < 1) and 10 ≤ Pe ≤ 1000, the following expression for Nu was suggested [69]:

Nu(λv , Pe, Re) =

0.651 Pe1/2 + 1 + 0.95λv

76

2 Heating of Non-evaporating Droplets

 

0.991λv 1/3 1.65(1 − αv (Re)) λv , Pe + [1 + αv (Re)] + 1 + λv 1 + 0.95λv 1 + λv where αv (Re) =

(2.76)

0.61Re + 0.032. Re + 21

In the limit Re → 0 and λv → ∞, Nu predicted by Expression (2.76) does not reduce to 2, as expected for stationary droplets. In this limit, Pr should be infinitely large to satisfy the condition that Pe > 10. For 1 ≤ Re ≤ 500 and 10 ≤ Pe ≤ 1000, the following formulae for Nu were obtained [69]: Nu(λv , Pe, Re) =

4λv 2 − λv Nu(0, Pe, Re) + Nu(2, Pe, Re) 2 6 + λv

(2.77)

4 λv − 2 Nu(∞, Pe, Re) Nu(2, Pe, Re) + 2 + λv λv + 2

(2.78)

for 0 ≤ λv ≤ 2, and Nu(λv , Pe, Re) = for 2 ≤ λv ≤ ∞, where



 0.61Re 0.61Re Nu(0, Pe, Re) = 0.651Pe1/2 1.032 + + 1.60 + , Re + 21 Re + 21   Nu(2, Pe, Re) = 0.64Pe0.43 1 + 0.233Re0.287 + 1.41 − 0.15Re0.287 ,   Nu(∞, Pe, Re) = 0.852Pe1/3 1 + 0.233Re0.287 + 1.3 − 0.182Re0.355 . These expressions could be potentially incorporated into any CFD code and used in practical applications. The prediction of various correlations for Nu was compared in [60]. A detailed analysis of heat and mass exchange between a moving sphere and ambient fluid is given in [16]. While the analysis in this paper was formulated in terms of mass transfer, it is also applicable to heat transfer, with concentration replaced by temperature and the Sherwood number by the Nusselt number. The authors of [16] investigated the impact of viscosity on the steady-state mass transfer from a sphere at low Peclet numbers, using singular perturbation technique, to find approximate temperature profiles. They suggested simple analytical formulae that can be used to calculate the local and average Nusselt numbers.

2.1 Convective Heating

2.1.2.2

77

Transient Heating

The complexity of the problem of transient heating of moving droplets lies in the fact that both variations in temperature and flow velocity need to be accounted for in the general case. This is something that is performed in most commercial CFD codes. This analysis, however, is usually case-dependent and the results might have limited general applications. At the same time, in many applications the characteristic times of variation of flow parameters are much longer than the characteristic times of droplet heating. This allows us to consider the problem of droplet heating assuming that the flow is fixed. This assumption is almost universally used in the analysis of this problem. For small Re, we can further assume that flow is Stokesian, with the velocities described by Expressions (2.63) and (2.64), or their simplified versions (Expressions (2.65) and (2.66)). Any effects of droplet oscillations on the heating process are generally ignored. This could be justified when the droplet Weber number We = 2u 0 Rd ρl /σs (where σs is the interfacial surface tension) is less than 4 [182]. This result was obtained in [182] based on experimental studies of falling droplets at Re in the range 138–971. Most approaches to investigating transient heating of moving droplets performed so far can be presented in two groups. The first approach assumes that the temperature of the droplet surface remains constant throughout the whole heating process. The second approach allows for changes in droplet temperature with time. In what follows, these two approaches are considered separately. Fixed Surface Temperature Choudhoury and Drake [34] and Konopliv and Sparrow [96] were perhaps the first to solve the problem of transient heating of a solid sphere at small Reynolds and Peclet numbers, assuming that the surface temperature of the sphere is fixed. In [34], the steady-state velocity distribution in fluid described by Expressions (2.63) and (2.64) was used and it was assumed that initially the temperatures of the sphere and surrounding fluid were equal. Then at t = 0, the temperature of the sphere abruptly changed to another constant value. The solution obtained for small t in the limit of very small Re reduced to the one predicted by Expression (2.30). In the limit of large t, the solution predicts values of Nu close to 2. This is as expected since this solution assumed small Re and Pe. Konopliv and Sparrow [97] generalised the analysis of their earlier paper [96] to the case of large Pe. As in [96], the analysis was based on Expressions (2.65) and (2.66). They drew attention to the fact that for high Pe, the temperature field is confined to a boundary layer which is very thin compared with the velocity field boundary layer. This allowed these authors to simplify both the temperature Eq. (2.57) (if heat conduction takes place in the radial direction only) and Expressions (2.65) and (2.66) (if R is close to Rd ). Two methods were used to obtain the solution to the simplified temperature equation. One method involved a numerical inversion of an integral transform, while the second was based on an asymptotic expansion in the Laplace transform s plane. For small t both methods predicted similar results, while

78

2 Heating of Non-evaporating Droplets

for larger t, the first method was recommended. In the limit of large times, the results of [97] reduced to those predicted by Expression (2.72) with Re = 0. A similar problem, but for the heating of a liquid droplet, was considered in [66]. These authors assumed that both Re and Pe are small, used Expressions (2.67) and (2.68) and obtained analytical expressions for Nu, valid in short and long time limits. In the short time limit, their expression reduced to Expression (2.30). In the long time limit (Fo = O(Pe−2 )), they derived the following expression: 

Pe Nu = 2 1 + 2 +



 √ Fo 1 erf Pe 2 4

  Pe2 Fo 2 exp − + √ 16 Pe π Fo

  3 + 2λv Pe + O(Pe1+ ), Pe2 ln 24(1 + λv ) 2

(2.79)

where Fo is the Fourier number introduced in Expressions (2.6) and (2.7). The predictions of Expression (2.79) for a solid sphere, Pe = 0.5 and long times, differ by less than 3% from the results, predicted by the numerical analyses for the Stokesian flow [68]. Also, it was shown that at short times the predicted values of Nu do not practically depend on Re [68]. This dependence was shown to be always rather weak for small Pe (Pe = 2), but significant for Pe = 200 at large times [68]. The models described so far have been focused predominantly on spherical droplets, while the observed droplets in sprays are typically far from being spherical [38, 104]. Several papers have considered the problem of transient heat transfer between non-spherical bodies and ambient fluid. In [67], a model for the heat transfer processes from an oblate and a prolate spheroid at the limit of very small Pe was developed. Other approaches to this and similar problems are discussed in [65, 85, 91, 94, 123, 165]. In [132], a more general problem of heat transfer from a particle with an arbitrary shape suspended in a fluid was investigated. Variable Surface Temperature Chao [30] was perhaps the first to solve the problem of conjugate heat transfer between a moving droplet and ambient gas. He took into account the recirculation inside a droplet but ignored the effects of viscosity. This allowed him to approximate the velocity profile outside the droplet by Expressions (2.69) and (2.70). A similar set of expressions for the internal flow (Hill vortex) was used. This approximation is expected to lead to substantial errors at small Re (Stokesian flow), but could be justified for Re > 200. Also, the assumption that a thermal boundary layer is thin was made, which was justified for large Pe. This allowed further simplification of Expressions (2.69) and (2.70), if |R − Rd | Rd and using Eq. (2.57) (assuming that heat conduction takes place in the radial direction only (cf. [97])). The simplified temperature equation was solved outside and inside the droplet, assuming that the temperature and heat fluxes on both sides of the interface are equal. The solution for the average Nusselt number was obtained in an explicit analytical form. As for

2.1 Convective Heating

79

stationary droplets, this solution predicted large Nu for short t. For long times, it asymptotically approached a steady-state solution in the form: 2Pe1/2 , Nu = √ π (1 + βv ) 

where βv =

(2.80)

k g ρg c pg . kl ρl cl

For liquid droplets moving in gas, βv is much less than 1. Chao and Chen [32] solved the same problem as [30] using an analytical technique originally developed in [31]. The results previously obtained in [97] for a sphere with fixed temperature were generalised to the case when sphere heating is considered provided that the thermal conductivity of this sphere is infinitely large. Abramzon and Borde [2] presented the results of numerical analysis of transient heat transfer to slowly moving droplets (Stokesian approximation) in a wide range of Peclet numbers 0 ≤ Pe ≤ 1000. They investigated the conjugate problem so that the heat transfer processes in gas and droplets were considered. They used the flow velocity distribution predicted by Expressions (2.67) and (2.68) (note a printing mistake in their Eq. (2)). It was demonstrated that the droplet heating process can be divided into two stages. In the first stage, the heat is transferred primarily by conduction, and the isotherms are almost spherical. In the second stage, convective heat transfer becomes dominant. During this stage, the internal isotherms were close to streamlines. For large Pe, Nu was shown to oscillate in time about its asymptotic value. The fact that initially the droplet is heated predominantly by conduction agrees with the result presented in [68], where it was shown that Nu practically does not depend on Re at the initial stage of heating (see the discussion in the previous subsection). It was demonstrated that for large Pe (100 ≤ Pe ≤ 1000), the heating can be considered as quasi-stationary after a short initial time lapse. During this initial period, the analytical solution of [30] was shown to predict acceptable results. A detailed comparison of the asymptotic values of Nu, predicted by the numerical analysis, and previously reported results of asymptotic analyses were presented for λv = 1 (fluids with similar hydrodynamic properties) and λv = ∞ (solids). Results of further analysis of transient heat transfer between an ambient fluid and a single sphere were presented in [3]. This analysis was restricted to solid spheres, but considered a wide range of Pe (1 ≤ Pe ≤ 10000). It investigated the ratios of volumetric heat capacities of the fluid to that of a sphere in the range 0 ≤ (ρc)12 ≡ ρg c pg /(ρl cl ) ≤ 2. The thermal conductivity of the sphere was assumed to be large, so that the temperature gradients inside it could be ignored. The flow field was assumed to be Stokesian and the velocity distribution outside the sphere was given by Expressions (2.65) and (2.66). It was demonstrated that for (ρc)12 = 0 and Pe close to zero, the values of Nu asymptotically approach 2, as predicted by Expression (2.30). However, for (ρc)12 ≥ 0.2 and the same Pe, the asymptotic values of Nu were

80

2 Heating of Non-evaporating Droplets

shown to be close to zero. This is consistent with the prediction of Fig. 2.4. Note that this figure is presented for (ρc)12 = 0.0156, for which the calculations were not performed in [3]. One of the main implicit conclusions made in this paper ‘at (ρc)12 ≥ 0.2 the asymptotic values of the Nusselt number appears to be considerably less than the corresponding steady-state value for the case of constant sphere temperatures’ could be generalised to any (ρc)12 > 0. In [90], the analysis presented in [3] was generalised to consider flow recirculation in droplets, although the range of Pe was reduced to 10 ≤ Pe ≤ 1000. Whenever the results of [3] and [90] were compared, the difference between them was less than 1%. The energy equation for a rigid sphere of infinitely large thermal conductivity in a viscous fluid subject to an unsteady flow and temperature field under the assumption of small Pe was investigated in [117]. The values of the temperature inside the sphere were shown to depend on instantaneous fluid temperatures and the ‘histories’ of both fluid and sphere temperatures. These histories were accounted for via the so-called history integral, which, in the energy equation, is analogous to the history force (Basset force) in the equation for the motion of a sphere. The presence of the history integral effectively turns the differential equation for the temperature in the sphere into an integro-differential equation, the analysis of which can be computer-intensive. The authors of [73] presented a more detailed analysis of the effect of the history term on the transient energy equation in a sphere. More specifically, they attempted to clarify when this term can be ignored if a predetermined calculation error can be accepted. They considered the transient heat transfers related to three cases: step temperature change, ramp temperature change and sinusoidal temperature change. It was demonstrated that the importance of the history term is mainly controlled by the parameter (ρc)12 . For 0.002 ≤ (ρc)12 ≤ 0.2 (the range of typical values for fuel droplets in air), this term is important and needs to be retained. More detailed analysis of the history term is presented in [11, 116]. Talley and Yao [170] drew attention to the fact that heat transfer to a moving droplet can be correctly predicted by Eq. (2.1), with appropriate boundary conditions, if the liquid thermal conductivity kl is replaced by the so-called effective thermal conductivity keff . This approach was further developed by Abramzon and Sirignano [6] who defined keff as (2.81) keff = χT kl , where the coefficient χT increases from about 1 (at droplet Peclet number Ped(l) = Red(l) Prl < 10) to 2.72 (at Ped(l) > 500). χT was approximated as    χT = 1.86 + 0.86 tanh 2.245 log10 Ped(l) /30 .

(2.82)

Liquid fuel transport properties and the maximum surface velocity inside droplets Us were used in Ped(l) . Us was calculated as [6] Us =

1 ΔU 32



μg μl

 Red C F ,

(2.83)

2.1 Convective Heating

81

where ΔU ≡ Ug − Ud is the relative velocity between gas and droplets, μg(l) the dynamic viscosity of gas (liquid), Red the droplet Reynolds number based on the droplet diameter (introduced as Re∞ in [6]) and C F the friction drag coefficient [6]. This model predicts the droplet average surface temperature reasonably accurately (especially at small and large Red(l) ). It is weaker in predicting the distribution of temperature inside droplets. The droplet average surface temperature is particularly important for applications as it controls droplet evaporation and break-up. This justified the relevance and usefulness of this model. This model is commonly known as ‘the Effective Thermal Conductivity (ETC) model’. The simplicity of this model makes it particularly attractive for application in CFD codes. For example, it allows the application of Solution (2.41) to moving droplets by replacing kl with keff , if h = const. The details of the derivation of Expression (2.83), starting with the velocity distribution corresponding to the Hill vortex inside droplets, are presented in [28]. In the analysis presented so far, the effect of turbulence on droplet heating was not taken into account. Among recent papers focused on the later problem, we can mention [141] where this effect was investigated using direct numerical and large eddy simulations for Reynolds numbers in the range 103 –104 . More recent analyses of the problem of transient heating of moving droplets were more case-specific. In most cases, they were linked with evaporation processes. They are described in Chap. 4.

2.2 Radiative Heating The importance of modelling radiative heat transfer in sprays, including those in Diesel engines, is widely recognised [128]. This modelling includes the interaction of thermal radiation with gas and liquid. Our analysis of this process is focused on the interaction of thermal radiation with individual droplets.

2.2.1 Basic Equations and Approximations Since thermal radiation propagates with a velocity close to the velocity of light, droplet motion does not affect the process of radiative heating. Hence, the analysis in this section is equally applicable to stationary and moving droplets. The simplest model for radiative heating of droplets uses the assumption that droplets are opaque grey spheres with emissivity ε. In this case, the effect of radiative heating of droplets could be described via replacing the heat flux given by Expression (2.31), with the expression   |q| ˙ = h(Tg∞ − Ts ) + σ ε θ R4 − Ts4 ,

(2.84)

82

2 Heating of Non-evaporating Droplets

where θ R is the so-called radiative temperature, σ the Stefan–Boltzmann constant and ε emissivity (0 ≤ ε ≤ 1). For optically thick gas, θ R can be assumed equal to the ambient gas temperature Tg∞ , while for optically thin gas it can be identified with the external temperature Text (e.g. temperature of remote flame) [122, 159]. The simplicity of this model makes it particularly attractive for applications, including application in CFD codes (e.g. [113, 143, 144, 149, 161, 167, 177, 186, 192]) and even in detailed unsteady numerical simulations focused on accurate description of the liquid phase during the evaporation process [120]. The value of ε in this equation can be considered as a free parameter, which can be specified based on more rigorous calculations or inferred from experimental data. If there are no temperature gradients inside droplets and θ R = Text , Expression (2.84) allows us to generalise Eq. (2.36) and its Solution (2.37) to account for the effect of thermal radiation by replacing Tg∞ with   4 − Ts4 σ ε Text Tg∞ + . h Similarly, radiative heating can be accounted for in Solution (2.41). In both cases, it is implicitly assumed that the external radiation is that of a grey body. In most practically important cases of fuel droplet heating, Ts Text and the contribution of the terms proportional to Ts4 can be safely ignored. If this is not the case (Text is of the order of or less than Ts ), then Ts is still below the critical temperature of fuel (about 660 K for n-dodecane), and the contribution of radiative heating can almost always be ignored altogether. The approach based on the assumption that droplets are opaque grey spheres, however, overlooks the fact that droplet radiative heating takes place not at their surface (as in the case of convective heating) but via the absorption of thermal radiation penetrating inside the droplets. Thus, the droplets should be considered as semitransparent rather than grey opaque [4, 5, 46, 49, 52–56, 74, 100, 108, 118, 119, 153, 174] unless their optical thickness (τ0 ; see Expression (2.102)) is much greater than 1 [142]. The focus of this section is on the models of thermal radiation absorption in semitransparent droplets, without considering the contribution of radiation from droplets (see [108, 118, 119] for the investigation of the latter effect). Also, the analysis of thermal radiation propagation in ambient non-grey gas or gas-droplets mixture is beyond the scope of this monograph (e.g. [169, 173, 188]). A general introduction to the radiative heat transfer in disperse systems can be found in books [45, 50]. The analysis of this section assumes that droplets are spherical and their irradiation is spherically symmetric. The problem of the interaction of radiation with non-spherical droplets was considered in several papers including [87]. The model developed in the latter paper was used for the analysis of the radiation absorption inside prolate and oblate water droplets under infrared laser irradiation. Equation (2.1) for semi-transparent spherical droplets in the presence of spherically symmetric thermal radiation can be generalised to

2.2 Radiative Heating

83

∂T =κ ∂t



∂2T 2 ∂T + ∂ R2 R ∂R

 + P(R),

(2.85)

where P(R) in K/s accounts for the radiative heating of droplets. The solution to Eq. (2.85) inside the droplet subject to boundary conditions (2.38) and (2.39) and the initial condition (2.40) is a straightforward generalisation of Solution (2.41) and was presented as [154] (see Appendix A for details):   ∞    pn Rd  pn 2 T = + exp −κ R λn t qn − R n=1 κ R λ2n κ R λ2n − −

sin λn || vn ||2 λ2n

where

  sin λn μ0 (0) exp −κ R λ2n t || vn ||2 λ2n



    R dμ0 (τ ) + Tg (t), exp −κ R λ2n (t − τ ) dτ sin λn dτ Rd 0 (2.86) Rd 1 pn = 2 R P(R) sin (λn R/Rd ) d R, Rd || vn ||2 0

t

and other notations are the same as in (2.41). Ignoring the effect of the temperature gradient inside the droplet but considering the effect of thermal radiation absorption, Eq. (2.36) can be generalised to dTd 4π 3 Rd ρl cl = 4π Rd2 h(Tg∞ − Td ) + ρl cl Ptotal , 3 dt where

Ptotal = 4π

Rd

P(R)R 2 d R

(2.87)

(2.88)

0

is the total amount of thermal radiation absorbed in the droplet. A general approach to the solution of the heat conduction problem with internal heat sources was considered in [61]. Solution (2.9) to Eq. (2.85) subject to initial and boundary conditions (2.3)–(2.5) was generalised considering the contribution of thermal radiation. The final formula for T (R, t) was presented as (see Appendix B for details) T (R, t) = Tg0 (Rg ) +

 Rd ∞

  1 1  exp −λ2n t (−(Tg0 (Rg ) − Td0 (R))Rvn (R)cl ρl d R R n=1 ||vn ||2 0

84

2 Heating of Non-evaporating Droplets



Rg

+

 (−(Tg0 (Rg ) − Tg0 (R))Rvn (R)c pg ρg d R

Rd



t

+ 0

   exp −λ2n (t − τ ) pn (τ )dτ vn (R),

where pn =

cl ρl ||vn ||2



Rd

R P(t, R)vn (R)d R.

(2.89)

(2.90)

0

All other notations are the same as in Solution (2.9). The application of Solutions (2.86) and (2.89) to practical problems is complicated by the fact that the value of P(R) is not known. Estimation of this function, using various approximations, is one of the main focuses of the rest of this section.

2.2.2 Mie Theory The most rigorous approach to the calculation of P(R) is based on the solution to Maxwell’s equations, with boundary conditions at the droplet’s surface. These boundary conditions are (1) continuity of the normal component of the wave electric field; (2) the jump in its tangential components controlled by the complex index of refraction of the liquid [20, 176]. This is the well-known Mie theory. In the case of interaction between a plane linearly polarised electromagnetic wave and a semi-transparent sphere (droplet), this theory gives the following relations for the complex amplitudes of the components of the wave electric field inside the droplet [20, 45, 176, 184]: Er =

∞ E 0 cos ϕ  k i (2k + 1)dk ψk (m λ ρλ )Pk1 (μθ ), m 2λ ρλ2 k=1

(2.91)

∞  E 0 cos ϕ  ik (2k + 1)  ck ψk (m λ ρλ )πk (μθ ) − idk ψk (m λ ρλ )τk (μθ ) , m λ ρλ k=1 k(k + 1) (2.92) ∞  E 0 sin ϕ  ik+1 (2k + 1)  ick ψk (m λ ρλ )τk (μθ ) + dk ψk (m λ ρλ )πk (μθ ) , Eϕ = m λ ρλ k=1 k(k + 1) (2.93) where mλi , ck =  m λ ζk (xλ )ψk (m λ xλ ) − ζk (xλ )ψk (m λ xλ )

Eθ =

2.2 Radiative Heating

85

dk =

ζk (xλ )ψk (m λ xλ )

mλi , − m λ ζk (xλ )ψk (m λ xλ )

! d P 1 (μθ ) τk (μθ ) = − 1 − μ2θ k , dμθ

P 1 (μθ ) πk (μθ ) = ! k , 1 − μ2θ

m λ = n λ − iκλ is the complex index of refraction, ρλ = 2π R/λ = Rxλ /Rd (xλ is size parameter defined as xλ = 2π Rd /λ), μθ = cos θ , the prime denotes differentiation with respect to the argument, θ = 0 is the direction of wave propagation, ϕ is the azimuthal angle measured from the plane of electric field oscillations of the incident wave (as viewed along the direction of wave propagation), E 0 is the amplitude of the electric field in the incident wave, ψk and ζk are the Riccati–Bessel functions [45]: ψk (z) =

 π z 1/2 2

Jk+ 21 (z),

ζk (z) =

 π z 1/2 2

(2) Hk+ 1 (z),

(2.94)

2

(2) Jk+ 21 are the Bessel functions, Hk+ 1 the Hankel functions of the second kind and 2

Pk1 (μθ ) the associated Legendre polynomials. Note that there is a printing mistake in the expression for E ϕ given in [46] (cf. his Eq. (5)). Expressions for Er , E θ and E ϕ given in [100] differ from those given above by the signs and the powers of i and signs of κλ . This difference is related to the fact that we follow the definitions of complex amplitudes and the azimuthal angle given in [133, 176], while Lage and Rangel [100] followed the definitions given in [20, 59]. This obviously does not influence the values of |Er |, |E θ | and |E ϕ | used in the monograph. Expressions similar to (2.91)–(2.93) can be obtained for the components of the wave magnetic field [59], but these are not be used in our analysis. The distribution of thermal radiation power density absorbed inside the droplet for a given wavelength of incident radiation is given by the expression [59]:  2 4π n λ κλ ext |Er |2 + |E θ |2 +  E ϕ  Iλ , pλ (R, μθ , ϕ) = λ |E 0 |2

(2.95)

where Iλext is the intensity of external thermal radiation in a given direction. This expression was obtained using the assumption that magnetic permeability of liquid is equal to 1 (this is a natural assumption for fuel droplets). For unpolarised external thermal radiation, coming from one direction θ = 0, we should set ϕ = π/4 in the right-hand side of Expression (2.95) [59]. If the illumination of the droplet by unpolarised thermal radiation is spherically symmetric, then the radial profile of the absorbed power is found via integration of (2.95) over all angles ϕ and θ , i.e. from the expression [53, 110, 133, 175]: pλ (R) = 2π

1

−1

pλ (R, μθ , ϕ)dμθ =

16π 2 n λ κλ ext Iλ S(R), λ

(2.96)

86

2 Heating of Non-evaporating Droplets

where S(R) =

∞   1 (2k + 1) k(k + 1) |dk ψk (m λ ρλ )|2 4 4 2|m λ | ρλ k=1

  2  . +|m λ |2 ρλ2 |ck ψk (m λ ρλ )|2 + dk ψk (m λ ρλ )

(2.97)

P(R) is inferred from Expression (2.96) via its integration over all wavelengths λ: P(R) =

1 ρl cl



∞

pλ (R)dλ.

(2.98)

0

The division of the integral in (2.98) by ρl cl converts the units of the radiation power density absorbed in droplets from W/m3 to K/s. The total power absorbed inside droplets (ρl cl Ptotal ) is obtained from Expression (2.88). The calculations of P(R) from Expression (2.98) and Ptotal from Expression (2.88) require the knowledge of Iλ0 , the spectral index of absorption and the index of refraction. Assuming that the external radiation is that of a black body (cf. the derivation of Eq. (2.84)), we can write [122, 159] Iλext = Bλ (Text ) ≡ where C1 = 3.742 × 108

π λ5

Wµm4 , m2

C1  , exp (C2 /(λText )) − 1

C2 = 1.439 × 104

(2.99)

µm · K,

λ is the wavelength in µm. Function Bλ (Text ) is the Planck function. For optically thick gas, the external temperature Text needs to be replaced with ambient gas temperature. The index of absorption κλ can be inferred from experimental measurements (e.g. [10]), or calculated using measured values of spectral reflectance at near-normal incidence [129]. The details of such measurements are described in many papers including [55, 56, 153, 154, 156] focused on Diesel fuels. In all cases, the dependence of κλ on the type of Diesel fuel was noticeable, especially in the ranges of semitransparency λ < 3 µm and 4 µm < λ < 6 µm. Peaks of absorption for all types of Diesel fuel almost coincided. The results of measurements of κλ could allow us to calculate the spectral distribution of the index of refraction n λ , using the Kramers–Krönig equations [122]. This, however, is expected to lead to big errors due to the limited range of wavelengths for which the measurements of κλ are usually performed. This problem can be overcome if we use the subtractive Kramers–Krönig analysis, using the measurements of n λ at one particular wavelength, and its calculation for other wavelengths [9, 56]. For Diesel fuels, it was demonstrated that the results of these measurements and calculations are well approximated by the following formula [54]:

2.2 Radiative Heating

87

n λ = n 0 + 0.02

λ − λm , (λ − λm )2 + 0.001

(2.100)

where λ is in µm, n 0 = 1.46, λm = 3.4 µm. In many applications, the dependence of n λ on λ can be ignored and n λ can be assumed equal to 1.46 [56]. The results of experimental measurements of the index of refraction of biodiesel– Diesel blends are described in [36]. Formally, Expression (2.98), combined with the measured values of κλ and measured or calculated values of n λ , provides the required input parameters for Expressions (2.86) and (2.88) describing droplet heating in the presence of thermal radiation. Direct application of Expressions (2.98) and (2.88) is limited by the complexity of relevant calculations. Their incorporation into CFD codes is not feasible. In what follows, several simple but practically important models are described.

2.2.3 Integral Absorption of Radiation in Droplets A convenient and widely used approach to characterise the integral absorption of thermal radiation in droplets is based on the application of the efficiency factor of absorption Q a . This is defined as the ratio of radiation power absorbed in a droplet to the power of thermal radiation illuminating it. Using Expression (2.96), the explicit formula for Q a was obtained as [53] Pλ(abs) = Qa = Pλ(ins)

 Rd 0

4π R 2 pλ (R)d R 8n λ κλ = 2 0 2 4π Rd Iλ xλ2



xλ 0

S(ρλ )ρλ2 dρλ ,

(2.101)

where S is given by Expression (2.97). As follows from Expressions (2.101) and (2.97), for a given λ, Q a depends on n λ , κλ and Rd . Since n λ is almost constant over the whole range of λ under consideration n ≡ n λ ≈ 1.46, it was demonstrated that Q a is most sensitive towards the optical thickness of droplets τ0 = 2κλ xλ = aλ Rd , where aλ is the absorption coefficient [55]. The results of calculations using Expression (2.101) were approximated by the following expression [47]: Qa =

  4n 1 − exp(−4τ0 ) , (n + 1)2

(2.102)

where τ0 = aλ Rd (hereafter, the subscript λ at n is omitted). This is an improved version of the formula given in [55] (Q a = 1 − exp(−4τ0 )). Expression (2.102) is much simpler than Expression (2.101), and this is particularly useful for application in CFD codes (e.g. [57]). Additional errors introduced by this equation are expected to be less than the errors introduced by other approximations used in the model (for example, the assumption that droplets are spherical).

88

2 Heating of Non-evaporating Droplets

If the thermal radiation illuminating the droplet is that of a black body and n is constant, the average efficiency factor of absorption of thermal radiation in the range of λ from λ1 to λ2 was presented as [55, 153] Qa =

×

⎧ ⎨ ⎩

⎡ 1−⎣

λ2 λ1

  exp − 8π κλλ Rd

4n (n + 1)2

  dλ λ5 exp (C2 /(λText )) − 1

'

λ2

λ1

⎤⎫ ⎬ dλ  ⎦ . λ5 exp (C2 /(λText )) − 1 ⎭

(2.103) Considering the values of κλ inferred from experiments, it was shown that the best approximation for Q a in the ranges 5 ≤ Rd ≤ 50 µm and 1000 ≤ Text ≤ 3000 K is given by the formula [55, 153]: (2.104) Λ0 = a Rdb , where Rd is in µm, a and b are functions of Text approximated as a = a0 + a1 (Text /1000) + a2 (Text /1000)2 , b = b0 + b1 (Text /1000) + b2 (Text /1000)2 ,

(2.105)

Text is in K. The values of the coefficients in the expressions for a and b depend on the type of fuel under consideration. For unboiled yellow Diesel fuel, Expression (2.105) was presented in a more explicit form [153]: a = 0.10400 − 0.054320Text /1000 + 0.008 (Text /1000)2 , b = 0.49162 + 0.098369Text /1000 − 0.007857 (Text /1000)2 .

(2.106)

Similar results, but with different numerical values of the coefficients, were obtained for the yellow boiled Diesel fuel, imitating the ageing process due to long storage, and pink Diesel fuel (unboiled and boiled), used in off road equipment, in which dye was added for legislative purposes. When presenting Expressions (2.105) and (2.106), it was assumed that gas is optically thin. In the general case, Text in these formulae need to be replaced with θ R . The typical plots of Q a and Λ0 versus Rd for a range of Text from 1000 to 3000 K are presented in Fig. 2.5. As can be seen from this figure, the values of Q a and Λ0 are very close to each other in the whole range of temperatures under consideration. This justifies the application of approximation (2.104) in CFD codes. When the range of temperatures is extended to 500 ≤ Text ≤ 3000 K, the values of a and b need to be approximated by more complex polynomials of the 4th power [153, 156]. These, however, seem to be of limited practical importance, as the contribution of radiation at external temperatures less than 1000 K is expected to be small.

2.2 Radiative Heating

89

Fig. 2.5 Plots of Q a , predicted by Expression (2.103), versus Rd (solid) and Λ0 , predicted by Expression (2.104), versus Rd (dashed) for Text = 1000 K, Text = 1500 K, Text = 2000 K, Text = 2500 K and Text = 3000 K (shown near the curves) for yellow unboiled Diesel fuel. Plots for Λ0 were obtained using the values of the coefficient given by Formula (2.106) for external gas temperatures in the range 1000 ≤ Text ≤ 3000 K and droplet radii in the range 5 ≤ Rd ≤ 50 µm. Reproduced from Figure 2a of [153] with permission from ASME

In practical applications, the range of applicability of Expression (2.104) should always be kept in mind. The values of Λ0 can never exceed 1. Remembering (2.104), the explicit expression for P(R) in Equation (2.85) can be presented as (2.107) P(R) = 3Λ0 σ θ R4 (Rd cl ρl ) , where Λ0 is defined by Expression (2.104). For very large droplets (when all incident thermal radiation is absorbed in them), Λ0 = 1. Note a mistake in Expression (3.94) of [147]. The values of pn used in Expression (2.89) can be presented as cl ρl ||vn ||2



3Λ0 σ θ R4 [1 − λn al Rd cot(λn al Rd )] . Rd ||vn ||2 λ2n al2 0 (2.108) The effects of the thermal radiation, predicted by Solution (2.89), on nonevaporating droplet heating in the limit when Tg0 (R) = const at Rd < R ≤ Rg were investigated in [157]. The authors of the latter paper showed that the radiative effects are negligible at small Fourier numbers (Fo) but can be dominant at Fo > 50. pn =

Rd

R P(t, R)vn (R)d R =

2.2.4 Geometric Optics Analysis So far, two extreme cases were considered: the distribution of thermal radiation absorption in droplets predicted by Mie theory (Sect. 2.2.2) and a simplified model

90

2 Heating of Non-evaporating Droplets

predicting the overall absorption of thermal radiation in droplets (Sect. 2.2.3). In this section, we return to the problem of finding the distribution of thermal radiation absorption in droplets, but solve it using a much simpler model than in Sect. 2.2.2. The analysis of this section reproduces the original results presented in [52]. Firstly, we assume that the size parameter xλ = 2π Rd /λ is much greater than 1. This allows us to replace the Mie theory analysis with a geometric optics approximation. As in the previous sections, it is assumed that the illumination of the droplet by external radiation is spherically symmetrical, the angular distribution of this radiation is known, and the droplet is spherical. This allows us to write the radiation transfer equation as [126] μθ

1 − μ2θ ∂ Iλ ∂ Iλ + + aλ Iaλ = 0, ∂R R ∂μθ

(2.109)

where Iλ (r, μθ ) is the spectral radiation intensity at a given point integrated along the azimuthal angle, θ is measured from the R-direction. This equation does not consider the effects of scattering and the contribution of the internal source of thermal radiation (see [77, 131]). The boundary conditions for Iλ were written as [126] Iλ (0, −μθ ) = Iλ (0, μθ ),   Iλ (Rd , −μθ ) = Rref (n, μθ )Iλ (Rd , μθ ) + 1 − Rref (1/n, −μθ ) n 2 Iλext (−μθ ), (2.110) ! where μθ = 1 − n 2 (1 − μ2θ ), Rref is the reflection coefficient [21] and Iλext is the spectral intensity of external radiation. Note that Rref (1/n, −μθ ) = Rref (n, μθ ). The assumption that the illumination of the droplet by external radiation is spherically symmetrical does not imply the isotropy of the distribution of thermal radiation inside the droplet, except in its centre. The first boundary condition in (2.110) is the symmetry condition at the droplet centre. The second boundary condition in (2.110) shows that the value of Iλ (Rd , −μθ ) at the surface of the droplet is the sum of the intensity of the reflected radiation (the first term in the right-hand side of this equation) and the intensity of the refracted radiation (the second term in the right-hand side of this equation). The radiation power absorbed per unit volume of the droplet is found as

∞

p(R) =

pλ (R)dλ,

(2.111)

0

where

pλ (R) = aλ Iλ0 (R),

Iλ0 (R) =

1 −1

Iλ (R, μθ )dμθ ,

and Iλ0 (R) is the spectral radiation power density. Note that p(R) (in contrast to P(R)) has units of W/m3 .

2.2 Radiative Heating

91

Calculation of p(R) based on Eqs. (2.109)–(2.111) is rather difficult. Following [46, 48], this problem is simplified using the so-called MDP0 approximation. In this approximation, it is assumed that in the droplet core (R ≤ R∗ ≡ Rd /n), radiation intensity is constant in the ranges −1 ≤ μθ < 0 and 0 < μθ ≤ 1. At the droplet periphery (R∗ < R ≤ Rd ), however, constant values of the radiation  intensity are assumed when −1 ≤ μθ < −μ∗ and μ∗ < μθ ≤ 1, where μ∗ = 1 − (R∗ /R)2 . External radiation cannot penetrate into the droplet peripheral zone at −μ∗ < μθ < μ∗ . Numerical solutions to Eq. (2.109) with boundary conditions (2.110) demonstrated that the MDP0 adequately predicts the angular dependence of the radiation intensity [46]. This approximation is based on the analysis of the following function:  0 R ≤ R∗ I (R), (2.112) g0 (R) = λ0 Iλ (R)/(1 − μ∗ ), R∗ < R ≤ Rd . Integration of Eq. (2.109) over μθ in the ranges −1 ≤ μθ < 0 and 0 < μθ ≤ 1 (droplet core) and in the ranges −1 ≤ μθ < −μ∗ and μ∗ < μθ ≤ 1 (droplet periphery) leads to the following equation for g0 (R) [46, 48, 52]: 1  2   C g1 − 1  R g0 + g0 = C g2 aλ2 g0 , 4R 2 2R 

where C g1 =

1 (1 − μ∗ )/2μ∗

 C g2 =

1 (1 + μ∗ )−2

(2.113)

when R ≤ R∗ when R∗ < R ≤ Rd , when R ≤ R∗ when R∗ < R ≤ Rd ,

the differentiation is over R. The boundary conditions for Eq. (2.113) are presented as g0 = 0

when R = 0

  1+μc  2naλ 2 0(ext) n g = I − g 0 2 λ 0 2 n +1

where μc =



when R = Rd ,

1−

(1/n 2 ),

Iλ0(ext)



=

1 −1

(2.114)

Iλext (μ)dμ.

When deriving Eq. (2.113) and boundary conditions (2.114), it was assumed that aλ does not depend on R and the average values of Rref (n, μθ ) in the ranges −1 ≤ μθ < −μc and μc < μθ ≤ 1 are equal to Rref (n, 1). The spectral power of the radiation absorbed per unit volume inside the droplet was found as (2.115) pλ (R) = aλ [1 − μ∗ Θ(R − R∗ )] g0 (R),

92

2 Heating of Non-evaporating Droplets

where Θ is the Heaviside unit step function:  Θ(x) =

0 1

when when

x 440.00 K.



Another approximation for pvs was used in [3]:

pvs



300 = exp 8.1948 − 7.8099 Ts





300 − 9.0098 Ts

2 (bar).

(3.11)

3.1 Empirical Correlations

107

Expression (3.11) leads to the following estimate of the mass fraction of vapour:  

p Ma −1 Yvs = 1 + −1 , pvs Mv

(3.12)

where p is the total pressure of the mixture of vapour and air. The prediction of Expression (3.6) agrees with the results of numerical analysis [57] (cf. Expression (3.1) for the Nusselt number). Although Formulae (3.1) and (3.6) were obtained more than 25 years ago, they are still widely used in engineering applications (e.g. [173]). An alternative formula for Sh was offered by the authors of [74]:   Sh = 2.009 + 0.514Re1/2 Sc1/3 .

(3.13)

Correlation (3.13) was obtained for the reference temperature Tr estimated as follows: Tr = Ts +

 1 Tg + Ts , 3

(3.14)

where Ts and Tg are the temperatures at the surface of the droplet and ambient gas, respectively (Correlation (3.13) is valid only when the contribution of the Stefan flow (introduced later in this chapter) is small). Expression (3.14) is commonly used in other approximations for Nu and Sh, although its limitations are well known [45]. Several other experimental correlations for Nu and Sh were described [1, 16, 168– 170]. Reference [71] offers a comparative analysis of earlier correlations. These were mainly focused on the investigation of the effects of turbulence on droplet evaporation. The effects of droplet size and turbulence on their evaporation at high temperatures and pressures were investigated experimentally by the authors of [161]. The results of the theoretical investigation of the effect of pressure oscillations on evaporating sprays are presented in [8]. Results of the investigation of the dependence of Sh on Re, Sc and viscosity ratio λv = μl /μg , based on the numerical investigation of the transient mass transfer from a single moving droplet, were presented by the authors of [108]. The effect of Re on Sh was found to be small for Re < 2 in agreement with Expression (3.6). Sensitivity of Sh to λv was noticeable only for 0.1 < λv < 100. It was shown that for smaller λv the interface moves with the same velocity as the ambient gas, while for larger λv the droplet behaves similar to a rigid sphere. This agrees with the prediction of the dependence of Nu on λv described by Expressions (2.76)–(2.78).

108

3 Heating and Evaporation of Mono-component Droplets

3.2 Classical Models 3.2.1 Maxwell and Stefan–Fuchs Models The simplest model for droplet evaporation was suggested by Maxwell back in 1877 [50]. According to this model, the rate of droplet evaporation is controlled exclusively by the diffusion process and is given by the following expression: m˙ d = 4π R 2 Dv

dρv , dR

(3.15)

where Dv is the binary diffusion coefficient of the vapour in ambient gas (air), ρv vapour density, R ≥ Rd the distance from the droplet centre and Rd the droplet radius. Since m˙ d < 0 does not depend on R in a steady-state case, this expression can be integrated from R = Rd to R = ∞ to give the following: m˙ d = −4π Rd Dv (ρvs − ρv∞ ) ,

(3.16)

where ρvs and ρv∞ are the same as in Expression (3.7). Expression (3.16) is known as the Maxwell equation [50]. Its limitation lies in the fact that it considers the diffusion process but ignores the effect of convective flow of the mixture of gas (air) and vapour away from the surface of the droplet (Stefan flow). Since there is no net gas (air) mass flux towards the droplet, we obtain the following equation: dρg , (3.17) Uρg = Dg dR where Dg is the binary diffusion coefficient of the gas (air) in vapour (Dg = Dv for a single-component droplet), ρg density of the gas (air) and U the net velocity of the mixture of gas (air) and vapour away from the droplet. Using Eq. (3.17), the expression for the droplet evaporation rate in the presence of vapour diffusion and Stefan flow is presented as follows:  m˙ d = 4π R

2

dρv Dv − ρv U dR



 = 4π R

2

dρv ρv dρg Dv − Dv dR ρg d R

 .

(3.18)

The condition Dg = Dv was used when deriving (3.18). Expression (3.18) can be rearranged to  m˙ d = 4π R 2 Dv

ρtotal ρtotal − ρv



dρv , dR

(3.19)

where ρtotal = ρg + ρv . Expression (3.19) can be considered as an equation for ρv as a function R. To solve this equation, we could assume, following [50], that the total pressure and

3.2 Classical Models

109

overall molar concentration of the mixture of gas (air) and vapour are constant. A more widely used approach assumes that the total density of the mixture of gas and vapour is constant (ρtotal = ρg + ρv = const) (e.g. [42]). None of these assumptions can be rigorously justified. Following [4], our further analysis is based on the second assumption. This allows us to separate variable ρv and R in (3.19) and present it as follows: dR dρv . (3.20) m˙ d 2 = 4π Dv ρtotal R ρtotal − ρv Integration of the left-hand side of (3.20) over R from Rd to R∞ = ∞ and its righthand side over ρv from ρvs to ρv∞ gives  m˙ d

R∞ Rd

dR = 4π Dv ρtotal R2



ρv∞

ρvs

dρv . ρtotal − ρv

(3.21)

Both integrals in (3.21) are elementary and the latter equation can be rewritten as follows:     1 1 ρtotal − ρv∞ = −4π Dv ρtotal ln . (3.22) − m˙ d Rd R∞ ρtotal − ρvs 1 Remembering that R1∞ = ∞ = 0 and the definition of the Spalding mass transfer number B M (Expression (3.7)), the following well-known expression for m˙ d is obtained: m˙ d = −4π Rd Dv ρtotal ln (1 + B M ) . (3.23)

The evaporation model based on Expression (3.23) is sometimes called the Stefan– Fuchs model (e.g. [155]) to recognise the facts that this model was first introduced by Fuchs [50] and it is based on the concept of the Stefan flow. The generalisation of Expression (3.23) to the case of a droplet located in a finite domain is discussed in [109]. Formula (3.23) can be presented as follows [140]: m˙ d = − (m d /τd ) (Sh/Sc) ln (1 + B M ) ,

(3.24)

where τd = 4ρd Rd2 /(18μg ) is the Stokes time scale and Sh and Sc are the Sherwood and Schmidt numbers, respectively (cf. Formula (3.13)); Sh = 2 for stationary droplets. If Yvs  1 and Yv∞  1, then B M  1 and Expression (3.23) reduces to Maxwell’s equation (3.16). These conditions are typically satisfied except in the case when the droplet surface temperature approaches boiling temperature. This can justify the application of Expression (3.16) in asymptotical studies of droplet evaporation and the ignition of fuel vapour/air mixture (see [18, 52–54, 94, 129] and Chap. 7). If the droplet surface temperature is constant, then B M is constant is well. In this case, Expression (3.23) reduced to the statement that d Rd2 /dt is constant. This is a

110

3 Heating and Evaporation of Mono-component Droplets

well-known d 2 -law (d = 2Rd ) (see [9, 25, 61, 68, 86] and Sect. 3.2.6 for further details). Note that Dv in Expression (3.23) is the mass diffusion coefficient which is different from the molar diffusion coefficient (see [38] for the details). An alternative expression for m˙ d can be obtained based on the analysis of the energy balance equation. Assuming that the evaporating droplet is stationary, this equation can be presented as follows [141]: 4π R 2 k g

dT = −m˙ d c pv (T − Ts ) − m˙ d L(Ts ) + |q˙d |, dR

(3.25)

where R > Rd . As in the previous analyses, m˙ d ≤ 0. The left-hand side of Eq. (3.25) describes the heat supplied from the surrounding gas to the droplet. The first term on the right-hand side describes the heat required to increase vapour temperature from Ts to T = T (R) (the gas temperature at the distance R from the centre of the droplet). The second and third terms on the right-hand side describe the heat spent on droplet evaporation and raising its temperature, respectively. Equation (3.25) can be rearranged to 4π k g

m˙ d d R dT =− . c pv (T − Ts ) + L(Ts ) − (|q˙d |/m˙ d ) R2

(3.26)

Integration of the left- and the right-hand sides of this equation from T = Ts to T = Tg and from R = Rd to R = R∞ = ∞, respectively, leads to the following expression: 4π k g Rd m˙ d = − ln(1 + BT ), (3.27) c pv where BT =

c pv (Tg − Ts ) L(Ts ) − (|q˙d |/m˙ d )

(3.28)

is the Spalding heat transfer number [4]. Expression (3.28) can be rearranged to BT =

c pv (Tg − Ts ) c pv (Tg − Ts ) =  |q˙d | L L 1 + |q˙s |−|q˙d | =



|q˙s | − |q˙d | |q˙s |

  c pv (Tg − Ts ) |q˙d | 1− , L |q˙s |



(3.29)

where |q˙s | is heat reaching the droplet surface. Expression (3.29) is identical to Expression (3.2) for B f in the absence of thermal radiation. From Expressions (3.23) and (3.27) follows the expression for BT as a function of B M [4]:

3.2 Classical Models

111

BT = (1 + B M )ϕ − 1, 

where ϕ=

c pv c pg



1 , Le

(3.30)

(3.31)

Le = k g /(c pg ρtotal Dv ) is the Lewis number. If Le = 1 and c pv = c pg , then BT = B M . This condition is sometimes used in the analysis of droplet heating and evaporation (e.g. [78]). In some papers, Expression 1 (e.g. [5, 87, 107]), which can be justified only when (3.30) is used with ϕ = Le c pv = c pg . This difference in the presentation of these equations can be traced to the form of the energy conservation equation (Eqs. (3.25) or (3.26)) used in some reviews (e.g. Eq. (3.4) in [43]) where c pv was replaced by c p , and the latter was implicitly identified with the heat capacity of the ambient gas. Expressions (3.23) and (3.27) are not valid at boiling point. In the latter case, the overall droplet evaporation rate can be presented as a sum of the superheat evaporation rate and the evaporation rate due to external heat transfer. See [182] for further details. The generalisation of Expression (3.31) to moving droplets is discussed in Sect. 3.2.2. From Expressions (3.27) and (3.28) follows the equation: |q˙s | = |q˙ins |

ez

z , −1

(3.32)

where |q˙s | = −m˙ d L + |q˙d | is the heat which reaches the droplet’s surface (the same as in Expression (3.29)), |q˙ins | = 4π Rd2 (k g /Rd )(Tg − Ts ) the heat transferred from   gas to droplets and z = −m˙ d c pv /(4π k g Rd ) = m˙ d Rd c pv /k g (m˙ d = |m˙ d |/(4π Rd2 )). Expression (3.32) was suggested in [41] and has been widely used in Computational Fluid Dynamics (CFD) codes (e.g. [70]). Having introduced the convective heat transfer coefficients describing heat reaching the surface of the droplet and heat transferred from gas to droplet as h and h 0 and the corresponding Nusselt numbers as Nu and Nu0 , Expression (3.32) can be rearranged to |q˙s | = 4π Rd2

Nu k g Nu0 k g z 4π Rd2 (Tg − Ts ) = z (Tg − Ts ). 2 Rd e −1 2 Rd

(3.33)

This allows us to obtain the following expression: Nu =

ln(1 + BT ) z Nu0 = Nu0 . ez − 1 BT

(3.34)

Recalling that for stationary droplets Nu0 = 2, the combination of Eqs. (3.34) and (3.27) gives kg  m˙ d = Nu BT . (3.35) 2c pv Rd

112

3 Heating and Evaporation of Mono-component Droplets

In the absence of evaporation, Nu0 = Nu. In the presence of evaporation, Nu0 is always greater than Nu as part of the heat transferred from gas to droplets is spent on heating the vapour produced during the evaporation process (cf. Eq. (3.25)). The latter heat does not reach the droplet surface. The value of Nu0 is not affected by the evaporation process. Comparing Eqs. (3.3) and (3.16), it can be seen that when the contribution of the Stefan flow is ignored and the droplet is stationary, Sh = Sh0 = 2. In the presence of the Stefan flow, the analysis is expected to be based on Expression (3.23). This expression can be presented as follows: m˙ d =

Dv ρtotal Sh B M , 2Rd

(3.36)

Sh =

ln(1 + B M ) Sh0 . BM

(3.37)



where

Both Nu0 and Sh0 can consider the movement of droplets (see Expressions (2.58)– (2.62) for Nu0 and similar expressions for Sh0 with Pr replaced by Sc). Expression (3.35) can be applied to modelling moving evaporating droplets if the dependence of Nu on Re and Pr is considered. Various models for this were discussed in Sect. 2.1.2. The models which are most widely used in applications in CFD codes are usually the simplest ones, i.e. those using Expressions (2.58) or (2.62). All expressions presented in Sect. 2.1.2 are equally applicable to the analysis of the Sherwood number Sh for moving droplets if Nu is replaced with Sh and the Prandtl number Pr is replaced with Sc. In this case, the Peclet number is defined as Pe = ReSc. This allows one to use Expression (3.36) for the analysis of moving evaporating droplets alongside Expression (3.35). Using these comments about the values of Sh for moving droplets, Expression (3.36) is almost universally used for the investigation of droplet evaporation processes in CFD codes. In most cases, this expression is linked with the equation describing droplet heating in the absence of internal temperature gradients (see Eq. (2.36) in Chap. 2). Equation (2.36) can be generalised to consider the effect of droplet evaporation and presented as follows: 3 dTs = (qs − jv L) , dt Rd ρl cl

(3.38)

where ρl , cl and L are liquid density, specific heat capacity and latent heat of evap oration, respectively, and jv ≡ m˙ d = |m˙ d /(4π Rd2 )| is the vapour mass flux from the surface of the droplet. As mentioned earlier in Sect. 2.1.2, ignoring temperature gradients inside droplets cannot always be justified, and a more general model considering these gradients need to be used. This model could be based on direct numerical solutions of Eq. (2.1) (in the absence of thermal radiation), Eq. (2.85) (in the presence of thermal radiation) or

3.2 Classical Models

113

analytical solutions to these equations (Expressions (2.41) and (2.86)). To consider the effect of droplet evaporation in Solutions (2.41) and (2.86), the gas temperature needs to be replaced by the so-called effective temperature: Teff = Tg +

ρl L R˙ d , h

(3.39)

where R˙ d (the derivative of the droplet radius with respect to time) can be estimated from the previous time step. This approach uses the observation that gas temperature changes more quickly than droplet radius during the evaporation process. The latter was assumed constant in the analytical solutions, but was updated at the end of each time step. Also, the modelling of droplet heating in the presence of evaporation can be based on the parabolic model and its generalisations described in Sect. 2.1.1. An alternative version of the parabolic model specifically designed for the analysis of droplet evaporation was described in [37]. The latter model assumed that the droplet surface temperature is constant. This can be justified when the heat-up period has been completed, which makes this model rather restrictive for applications. The authors of [131] describe results of detailed comparison between the performances of the algorithms using the analytical solution (2.86), the algorithm based on the numerical solution of the discretised Eq. (2.85) and the parabolic model (Eqs. (2.43)–(2.51)). As follows from the analysis presented in this paper, the algorithm using the analytical solution is more effective (from the points of view of accuracy and CPU time requirement) than that based on the numerical solution of the discretised heat conduction equation inside the droplet and more accurate than the algorithm based on the parabolic temperature profile model. Thermal radiation makes a relatively small contribution to droplet heating and evaporation in many engineering applications. This allows us to describe the effects of this radiation using a simplified model, which considers the semi-transparency of droplets, but not the spatial variations of radiation absorption inside them (see Sect. 2.2). This result was confirmed in [2, 3], where the predictions of the models considering and not considering the distribution of radiative heating inside droplets were compared. Note that the model for droplet heating and evaporation similar to the one developed in [130, 131] (that is based on the analytical solution to the heat transfer equation inside a droplet over the time step) but without considering thermal radiation is described in [98]. In realistic CFD codes designed for modelling spray combustion (e.g. in Diesel engines), the above-mentioned models for droplet heating and evaporation are complemented by models of droplet dynamics, break-up, gas dynamics and heating and ignition of fuel vapour/air mixture (see Chaps. 1 and 7). Although the models described so far in this section have been successfully used in many engineering problems, there is still much scope for improvement. In what follows, possible refinements are described.

114

3 Heating and Evaporation of Mono-component Droplets

3.2.2 Abramzon and Sirignano Model The classical model discussed in the previous section was further developed by Abramzon and Sirignano [4]. To consider the effect of convective transport caused by the droplet motion relative to the gas, the authors of [4] used the so-called ‘film’ theory, described in several well-known books (e.g. [15, 48]). Some ideas of this theory referring to thermal films were described in Sect. 2.1.1. Considering both heat and mass transfer, film thicknesses δT and δ M were introduced. The expressions for δT and δ M were derived from the requirements that the rates of a purely molecular transport by thermal conduction or diffusion through the film should be equal to the actual intensity of the convective heat or mass transfer between the droplet surface and the external flow. For heat conduction above the surface of a non-evaporating droplet, this requirement was presented as Eq. (2.14). This equation led to Formula (2.16) for δT 0 . A similar analysis for mass transfer at the droplet surface led the authors of [4] to the following formula for δ M0 : δ M0 =

2Rd , Sh0 − 2

(3.40)

where index 0 indicates that the effect of the Stefan flow is ignored. The latter effect for δT 0 and δ M0 was considered by introducing the following correction factors: FT = δT /δT 0 ;

FM = δ M /δ M0 ,

(3.41)

which describe the relative change of the film thickness due to the Stefan flow. To find these factors, a model problem of the laminar boundary layer of the flow past an evaporating wedge was investigated. The range of parameters 0 ≤ (B M , BT ) ≤ 20; 1 ≤ (Sc, Pr) ≤ 3 was investigated with the wedge angle in the range (0, 2π ). It was shown that for an isothermal surface and constant thermophysical properties of the fluid, the problem has a self-similar solution and FT and FM do not depend on the local Re and are very weak functions of Sc, Pr and the wedge angle. They were approximated as follows: 0.7 ln(1 + BT (M) )  . FT (M) = 1 + BT (M) BT (M)

(3.42)

FT (M) , predicted by (3.42), increases from 1 to 1.285 when BT (M) increases from 0 to 8. For BT (M) greater than 8, FT (M) remains almost constant. From (3.41) and (3.42) follow the expressions for δT and δ M considering the contribution of the Stefan flow. If we replace δT 0 and δ M0 with δT and δ M in the left-hand sides of Eqs. (2.16) and (3.40), we need to replace Nu0 and Sh0 with the other Nusselt and Sherwood numbers. These new numbers were called ‘modified’ Nusselt and Sherwood numbers by the authors of [4]. Rearranging Eqs. (2.16) and (3.40) with δT 0 and δ M0 replaced with δT and δ M , and Nu0 and Sh0 with the modified Nusselt and Sherwood numbers (Nu∗ and Sh∗ ), the following formulae are obtained:

3.2 Classical Models

115

Nu∗ = 2 +

2Rd 2Rd Nu0 − 2 =2+ =2+ , δT FT δT 0 FT

(3.43)

Sh∗ = 2 +

2Rd 2Rd Sh0 − 2 =2+ =2+ . δM FM δ M0 FM

(3.44)

Nu∗ and Sh∗ should replace Nu0 and Sh0 in Eqs. (3.35) and (3.36), respectively. The term ‘modified’ was introduced for Nu∗ and Sh∗ since Nu∗ → Nu0 when FT → 1 and Sh∗ → Sh0 when FM → 1. This term, however, might be misleading, as the actual Nusselt and Sherwood numbers, describing heat and vapour mass fluxes at the droplet surface, are inferred from Eqs. (3.34) and (3.37) with Nu0 and Sh0 replaced by Nu∗ and Sh∗ , respectively (see [4]). The authors of [102] demonstrated good match between the values of the Nusselt and Sherwood numbers predicted by a direct numerical method with the high-fidelity interface capturing methodology and those predicted by Expressions (3.43) and (3.44). The values of Nu0 and Sh0 in the latter expressions were taken from Expression (2.58) with βc = 0.6 and the corresponding expression with Pr replaced with Sc. This match was demonstrated in the ranges 1.5 ≤ Re ≤ 12.8 and 500 K ≤ T ≤ 800 K. The introduction of Nu∗ and Sh∗ leads to the generalisation of Formula (3.31), to consider the effect of the moving droplets, to [4]  ϕ=

c pv c pg



Sh∗ Nu∗



1 . Le

(3.45)

Expression (3.45) is much more complex than Expression (3.31) and its application requires the iteration process to solve the problem of droplet heating and evaporation [4]. In many applications, including Diesel engines, the results of modelling using Expressions (3.45) and (3.31) are practically indistinguishable [39]. This justifies the applications of Expression (3.31) for moving droplets as was done in several papers including [130–132]. The predictions of the experimental correlation for Nu (Eq. (3.1)), Stefan–Fuchs model (Eq. (3.34)) and Abramzon and Sirignano [4] model (Eqs. (3.34), (3.42) and (3.43)) were compared. To achieve consistency between these models, it was assumed that the heat-up period is completed (|q˙d | = 0), the effect of thermal radiation is ignored and Nu0 is given by Expression (2.58) with βc = 0.57. It was assumed that Pr = 0.7 and Re = 100, and three values of BT were used. The results are shown in Table 3.1. As follows from Table 3.1, the values of Nu predicted by the Abramzon and Sirignano model are noticeably closer to those predicted by the experimental correlation than those predicted by the Stefan–Fuchs model. A more detailed comparison between the models would be required to further support this conclusion. Note that Correlation (3.1) was obtained in a rather limited range of parameters (droplet radii between 0.5 and 3 mm), and its applicability to modelling of droplets with radii less than 20 µm, observed in Diesel engines, is not at first evident.

116

3 Heating and Evaporation of Mono-component Droplets

Table 3.1 The values of Nu predicted by three models and three values of BT Model Formulae BT = 0.1 BT = 1 BT = 3 Experimental correlation Stefan–Fuchs model Abramzon and Sirignano

(3.1)

Nu = 6.605

4.347

2.676

(3.34)

Nu = 6.732

4.894

3.263

(3.34), (3.42), (3.43)

Nu = 6.641

4.501

2.975

The performance of several droplet evaporation models, including the classical and Abramzon and Sirignano models, in the analysis of a turbulent droplet-laden flame was assessed in [104]. The effect of internal circulation on heat transfer in a droplet was investigated in [4] using two models: the ‘extended model’ which directly solves the energy equation inside the droplet and the ‘Effective Thermal Conductivity’ (ETC) model, described earlier. In [2, 3], these models were generalised to consider the contribution of thermal radiation and the temperature dependence of liquid properties. In both models, the contribution of thermal radiation was considered using the models for thermal radiation absorption presented in Sect. 2.2.4, considering and not considering the spatial distribution of radiation absorption in droplets. In the latter case, the analysis was based on the total radiation heat absorption in a droplet divided by its volume. These models were applied to the analysis of Diesel fuel and n-decane droplet heating and evaporation in Diesel engine-like conditions. Transport and thermodynamic properties of Diesel fuel were approximated by those for n-dodecane. It was shown that the radiation absorption in Diesel fuel is generally stronger than in n-decane, and it needs to be considered in modelling combustion processes in Diesel engines. Comparison between calculations, performed using the ‘extended model’ with distributed radiation absorption heat source and those using the ‘Effective Thermal Conductivity’ model with uniformly distributed radiation absorption, showed very good agreement between the results. This allowed the authors of [2, 3] to recommend using the ETC model with uniform radiation absorption for spray combustion calculations, including applications in Diesel engines. Note that in the absence of thermal radiation, the mono-component droplet surface temperature approaches an equilibrium or ‘wet-bulb’ temperature. At that temperature, all heat coming to the droplet surface from the gas is spent on evaporation (latent heat). The net heat penetrating to the liquid phase becomes zero q˙d = 0. In the presence of thermal radiation, however, the droplet surface temperature continues to rise above the wet-bulb temperature [2, 3]. As the surface droplet temperature increases, the heat coming to the droplet surface through convection decreases, but the heat used for vapourisation increases. As a result, the value of q˙d becomes negative, as confirmed by calculations presented in [2, 3]. At the end of the evaporation period, the total radiation absorption decreases very quickly with the droplet radius (see Sect. 2.2.3), while the heat transferred through the droplet surface, |q˙d |, decreases

3.2 Classical Models

117

relatively slowly. Thus, at a certain stage at the end of the evaporation process, the contribution of thermal radiation can be ignored altogether. This leads to the situation when at the very end of the evaporation process the droplet surface temperature approaches an equilibrium or ‘wet-bulb’ temperature both in the presence and absence of thermal radiations. This is confirmed by the calculations presented in [2, 3]. From the point of view of underlying physics, this resembles the situation in which a droplet suspended at room temperature is heated by internal heat sources. Since the evaporation is relatively slow, the droplet temperature approaches a steadystate value, which is higher than the regular wet-bulb temperature. If the internal heat sources are suddenly ‘turned-off’, the droplet temperature starts to decrease to the wet-bulb temperature [2, 3]. These results were confirmed by calculations reported in [132]. The analysis in the latter paper was based on the Stefan–Fuchs evaporation model. The importance of considering the radiation effects in modelling droplet evaporation was demonstrated in [172].

3.2.3 Yao, Abdel-Khalik and Ghiaasiaan Model As in the case of the Abramzon and Sirignano model, thermal and mass thicknesses δT and δ M were introduced by Yao, Abdel-Khalik and Ghiaasiaan [174] in such a way that T = Tg at R = Rd + δT and Y F = Y F∞ at R = Rd + δ M . The effect of the Stephan flow was considered, but in a different way to [4]. The boundary conditions at the droplet surface were the same as in the two models considered earlier. Although the key equations presented in this section are the same as in [174], some details of their analysis are slightly different from those presented in the original paper [174]. As in the previous models, the analysis of [174] is based on Eq. (3.26), but the right-hand side of this equation was integrated not from R = Rd to R = ∞, but from R = Rd to R = Rd + δT to consider the finite thickness of the thermal boundary layer. As a result, the following expression for BT was derived: BT = exp where

ΩY δ T Rd + δ T

− 1,

(3.46)



m˙ c pv Rd ΩY = d . kg

The condition that heat reaching the droplet’s surface is spent on droplet heating and evaporation can be presented as follows:   h Tg − Ts 4π Rd2 = L|m˙ d | + |q˙d |,

(3.47)

where h is the convection heat transfer coefficient in the presence of the Stefan flow, introduced earlier. From Eqs. (3.46) and (3.47) and remembering the definition of BT , we obtain

118

3 Heating and Evaporation of Mono-component Droplets 

m˙ c pv  d 

h= exp

ΩY δ T Rd +δT

−1

.

(3.48)

This expression is the same as derived in [174] using a different approach. In the  limit m˙ d → 0, Expression (3.48) simplifies to h=

k g (Rd + δT ) . Rd δ T

(3.49)

This expression is equivalent to Expression (2.16) obtained in the limit when the contribution of the Stephan flow is ignored. In the limit when δT → ∞, Expression (3.49) gives the trivial result h = k g /Rd . Equation (3.46) can be presented as follows: kg m˙ d = c pv Rd 

  kg Rd 1+ ln(1 + BT ) = NuBT , δT 2c pv Rd

  Rd ln(1 + BT ) . Nu = 2 1 + δT BT

where

(3.50)

(3.51)

Expression (3.51) is a generalisation of Expression (3.34) to consider the contribution of the finite thickness of the thermal boundary layer. It reduces to Expression (3.34) if δT is replaced by δT 0 defined by Expression (2.16). The effect of the mass boundary layer was considered when analysing Eq. (3.20) (like the effect of the thermal boundary layer). This equation was integrated not from R = Rd to R = ∞, as in the classical model, but from R = Rd to R = Rd + δ M to consider the finite thickness of the mass boundary layer. The following expression  for m˙ d was derived: Dg ρtotal m˙ d = Rd 



Rd 1+ δM

 ln (1 + B M ) .

(3.52)

This expression can be presented in the form (3.36) with the following Sherwood number:   Rd ln(1 + B M ) . (3.53) Sh = 2 1 + δM BM Expression (3.53) is a generalisation of Expression (3.37) to consider the contribution of the finite thickness of the mass boundary layer. It reduces to Expression (3.37) if δ M is replaced by δ M0 as defined by Expression (3.40). Expressions (3.51) and (3.53) reduce to those derived by the Abramzon and Sirignano [4] model if δT and δ M are inferred from Eq. (3.41). Note that in Expressions (3.51) and (3.53) the dependence of Nu and Sh on Re, Pr and Sc takes place via the dependence of δT and δ M on Re, Pr and Sc. The value of δ M in [174] was determined via Expression (3.53), assuming that Sh is known.

3.2 Classical Models

119

The latter was taken from the empirical correlation (3.6). Once the mass evaporation flux has been found, then the value of ΩY can be determined. In this case, Eq. (3.50) was rearranged to 2ΩY . (3.54) BT = Nu Using the empirical correlation for Nu (Expression (3.1)), BT can be found. Finally, using the definition of BT (Expression (3.28)), the value of |q˙d | can be found. The calculation of the values of δT is not required in this approach. The model relies on the separately measured or calculated values of Nu and Sh. This constitutes one of the main limitations of [174].

3.2.4 Tonini and Cossali Model The simplicity of the models of droplet heating and evaporation described so far was achieved by introducing several simplifying assumptions for the analysis of the underlying mass, momentum and energy conservation equations. These models imposed constant total density in the flow field and ignored the effects of gas temperature on gas density. This is untenable when the difference between droplet and gas temperatures is large and it is inconsistent with the concept of diffusion of the evaporating components [33, 155]. The limitations of these assumptions were discussed in Sect. 3.2.1. Tonini and Cossali [155] developed a new model for the heat and mass transfer from suspended mono-component spherical droplets evaporating in stagnant air, which relaxes the above-mentioned assumptions. The focus of their model was on the vapour phase; they assumed that the thermal conductivity of the liquid phase is infinitely large and there is no temperature gradient inside the droplets. Their analysis was based on steady-state mass, momentum and energy balance equations for the vapour and gas (air) mixture surrounding a droplet: d dR d dR

 R 2 ρv U − R 2 Dv ρtotal

dYv dR

R 2 ρa U − R 2 Dv ρtotal

dYa dR



d ptotal dU ρtotal U = + μmix dR dR ρtotal U c p,mix

dT = kmix dR





 = 0,

(3.55)

= 0,

(3.56)



d2 U 2 dU + 2 dR R dR

d2 T 2 dT + 2 dR R dR

 ,

(3.57)

 ,

(3.58)

120

3 Heating and Evaporation of Mono-component Droplets

where U is the Stefan flow velocity described in Sect. 3.2.1. All transport and thermodynamic properties refer to the mixture of vapour (v) and air (a), and they are assumed to be constant. R ≥ Rd is the distance from the droplet centre. In contrast to the previously described models, both ρtotal and T are assumed to be functions of R. Partial ( pv and pa ) and total ( ptotal ) pressures are calculated from the ideal gas and Dalton laws. The following boundary conditions for Eqs. (3.55)–(3.58) are used: T (R = Rd ) = Ts , pv (R = Rd ) = pvs (Ts ).

T (R = ∞) = Ta,∞ ,

Yv (R = ∞) = Yv,∞ ,

Combining Eqs. (3.55)–(3.56) yields the following equation: ρtotal U =

|m˙ d | , 4π R 2

(3.59)

where m˙ d is the droplet evaporation rate. Equation (3.59) is the statement of the conservation of mass flux. It allows us to decouple Eq. (3.58) for gas temperature from the other equations and present its solution as follows [155]: T = Ta,∞ ATC (R) + (1 − ATC (R)) Ts , where

    ˆd d Rd − exp − mLe exp − mˆLeR   ATC (R) = , ˆd 1 − exp − mLe

(3.60)

(3.61)

Le ≡ Sc/Pr = kmix /(Dv ρa,∞ c p,mix ) and mˆ d = |m˙ d |/(4π Rd Dv ρa,∞ ) is the normalised evaporation rate. Note that Expression (3.60) refers to a steady-state distribution of gas temperature, which is different from its transient distributions described in Sect. 2.1.1. New variables were introduced for the further analysis of Eqs. (3.55)–(3.58): ξTC = Rd /R,

G TC = ln(Ya ).

These variables allowed Tonini and Cossali [155] to obtain the following formula for the normalised evaporation rate mˆ d : mˆ d = − where ρ˜ = ρtotal (R)/ρtotal (R = ∞).

dG TC ρ, ˜ dξTC

(3.62)

3.2 Classical Models

121

The problem was further simplified if ΛTB =

Ru Ta, ∞ Rd2

1. Mv Dv2

(3.63)

Condition (3.63), applicable to many practical applications, allowed Tonini and Cossali [155] to obtain an analytical solution for G TC which led to the following implicit equation for mˆ d : ⎛

⎞ 

 m ˆ Ts pˆ v, cr − pˆ vs d ⎝ ⎠  − Le  , (3.64) mˆ d + − 1 = − pˆ v, cr ln ˆd Ta pˆ v, cr − Yv ∞ 1 − exp − mLe where pˆ v, cr = 1 +

θTC M v − Ma , θTC = , 1 − Yv ∞ Ma

pˆ vs =

pvs Mv , Ru Ta ∞ ρa ∞

pvs (Ts ) is the saturation surface vapour pressure, Ru is the universal gas constant and Mv and Ma are molar masses of vapour and air. The approaches to experimental measurements of vapour concentration around evaporating droplets, which could be potentially used for comparison with Yv predicted by the model, are discussed by the authors of [177]. The results of these measurements, however, refer to droplets on a solid substrate and cannot be directly used for validation of the droplet evaporation models. The lifetimes of evaporating droplets described by the model developed in [155] were shown to be always larger than those predicted by the Maxwell and Stefan– Fuchs models and the difference increases with air temperature. The original Tonini and Cossali model [155] was generalised to consider the dependence of gas density and diffusion coefficient on temperature [29]. Further development of this model, described in [29], accounted for the temperature dependence of density, diffusivity, thermal conductivity and specific heat capacity of the gas species. This was taken into account by assuming power law dependences of these parameters on temperature and was justified using available data. An analysis based on this model for various vapour species (water, ethanol, acetone, n-hexane, n-octane and n-dodecane) demonstrated that a proper choice of the reference temperature as a function of the operating conditions can significantly improve the predictions of the classical models. The gas model developed by the authors of [175] is complementary to the abovedescribed Tonini and Cossali model. In contrast to [155], the authors of [175] focused their attention only on the equations for vapour mass fraction and temperature, but considered the fully transient versions of these equations, assuming their spherical symmetry. The implicit analytical solutions to these equations were obtained and they were used to determine mass fraction of fuel vapour and temperature at the droplet surface with the help of iterations.

122

3 Heating and Evaporation of Mono-component Droplets

Sometimes, the analysis of the effect of droplet evaporation was separated from the analysis of the effect of droplet heating by assuming that all heat supplied to the droplet is spent on its evaporation. This ensures that the droplet surface temperature remains constant (Wet-Bulb Temperature) (e.g. [76]). This approach is considered in more detail in Sect. 3.2.6.

3.2.5 Fully Transient Models A number of authors developed fully transient models of droplet heating and evaporation (see [12, 46] and the references therein). These models are essentially based on the numerical solutions to the generalised system of Eqs. (3.55)–(3.58), in which the terms proportional to the partial derivatives with respect to time are taken into account, and the transient heat conduction equations in the liquid phase are considered. In the absence of evaporation, these models reduce to the transient droplet heating model described in Sect. 2.1.1.1. An approach to fully coupled numerical simulation of droplet heating and evaporation, using the Arbitrary Lagrangian–Eulerian formulation, is presented in [171].

3.2.6 d2 and d1.5 Laws Expression (3.35) can be simplified if we assume that the droplet surface temperature does not change with time (we ignore the initial heat-up period). In this case, all heat supplied from the gas is spent on droplet evaporation. This leads to the following expression for the rate of change of the droplet radius [144]: k g (Tg − Ts ) Nu, R˙ d = − 2Lρl Rd

(3.65)

where, using Expressions (3.34) and (2.62), the expression for the Nusselt number (Nu) is presented as follows:  1 + BT  , Nu = 1 + (1 + Re Pr)1/3 f c (Re) BT

(3.66)

f c (Re) = 1 at Re ≤ 1 and f c (Re) = Re0.077 at 1 < Re ≤ 400, and BT is the Spalding heat transfer number inferred from (3.28) with q˙d = 0. For small Re, (3.66) simplifies to Nu = 2

ln(1 + BT ) , BT

(3.67)

3.2 Classical Models

123

and the solution to (3.65) can be presented as follows [144]:

2k g (Tg − Ts ) ln(1 + BT ) t. − Lρl BT

Rd2

=

2 Rd0

(3.68)

This is a well-known d 2 law (d = 2Rd ). In the limit of large Re (but Re < 400), (3.66) is simplified to Nu = Re0.4 Pr 1/3

ln(1 + BT ) . BT

(3.69)

In this case, the solution to (3.65) can be presented as follows [144]:

Rd1.6

0.8k g (Tg − Ts ) 1.6 = Rd0 − Lρl



2vd νg

0.4

ln(1 + B ) T Pr 1/3 t. BT

(3.70)

This is the analogue of the d 2 law for large Re. Note that if we used not Expression (3.66) (the Clift et al. correlation) for Nu but Expression (2.59) (the Ranz and Marchall correlation), predicting that Nu is proportional to Re1/2 , then Solution (3.70) would need to be replaced with the following solution [144]:

Rd1.5

=

1.5 Rd0

0.75k g (Tg − Ts ) − βc Lρl



2vd νg

0.5 Pr

1/3 ln(1

+ BT ) t. BT

(3.71)

We believe that Terekhov et al. [152] were the first to draw attention to this d 1.5 law. Note that for very small droplets when their evaporation is controlled by the kinetic processes, to be discussed in Sect. 6, Rd decreases linearly with time (see [162] for the details).

3.3 Effects of Real Gases In most models of droplet heating and evaporation, including those discussed in Sect. 3.2, it is assumed that ambient gas is ideal. This assumption becomes questionable when the pressures and temperatures are high enough, as observed in internal combustion engines. This section discusses the main approaches to considering ‘real gas’ effects. Real gases show deviations from the ideal gas law due to molecular interactions, via repulsive and attractive forces, and finite volumes of molecules. These effects are commonly described using a compression factor [10, 118]: Z = Vm /Vm0 ,

(3.72)

124

3 Heating and Evaporation of Mono-component Droplets

where Vm is the actual molar volume of gas, and Vm0 is the molar volume predicted by the ideal gas law. Since the molar volume of an ideal gas is equal to Ru T / p, (3.72) can be presented as follows [10, 118]: Z=

pVm . Ru T

(3.73)

In the limit p → 0, Z = 1. For most substances, for small p, Z < 1 (attractive forces dominate when the molecules are well separated), and for large p, Z > 1 (repulsive forces dominate when the molecules are close enough). Various methods of estimation of Z have been discussed in a few classical textbooks [10, 118]. The most widely used approach is based on the solution to the equations of state for real gases. Several of these equations have been discussed, including the Van der Waals and Redlich–Kwong equations [10, 118]. The latter equation was modified by Soave [142], and its modified version is sometimes referred to as the Soave–Redlich–Kwong equation [113]. The real gas equations reduce to the ideal gas law in the limit p → 0 and ideally should satisfy the thermodynamic stability criteria at the critical temperature Tc [118]: 



∂p ∂ Vm ∂2 p ∂ Vm2

 = 0,

(3.74)

= 0.

(3.75)

Tc

 Tc

In what follows, one of these equations, originally suggested by Peng and Robinson [110], is described. This equation is widely believed to be particularly convenient for application to modelling droplet evaporation [36, 58, 59, 67]. It was presented as follows [110]: p=

a Ru T − , Vm − b Vm (Vm + b) + b(Vm − b)

(3.76)

where a was assumed to be a function of T , while b was assumed to be constant. The explicit expressions for these coefficients were found from the requirements (3.74) and (3.75). Having substituted Expression (3.76) into Equations (3.74) and (3.75), the following equations were obtained: a(Tc ) =

Ru Tc (V˜m2 + 2 V˜m − 1)2 b, 2(V˜m + 1)(V˜m − 1)2

  (V˜m + 1)3 = 2 V˜m3 − 1 ,

(3.77)

(3.78)

3.3 Effects of Real Gases

125

where V˜m = Vm /b. Analytical or numerical solution to Eq. (3.78) gives V˜m = 3.946. In this case, Eqs. (3.76) and (3.77) give us the following explicit expressions [110]: a(Tc ) = 0.45724 b = 0.0778

Ru2 Tc2 , pc

(3.79)

Ru Tc , pc

(3.80)

Z c ≡ Z (Tc ) = 0.307.

(3.81)

In the case when T = Tc , the value of a was found from the relation [110]:





a(T ) = a(Tc ) 1 + κω 1 −

T Tc

 2 ,

(3.82)

where κω = 0.37464 + 1.54226ω − 0.26992ω2 , ω is the acentric factor (a measure of the non-sphericity (centricity) of molecules) [118]:   p(0.7Tc ) − 1. (3.83) ω = − log pc For mono-atomic gases, ω = 0 and a(T ) = a(Tc ). In the general case, this parameter is obtained from tables provided by various authors (see [118] and the references therein). For example, ω = 0.562 for n-dodecane, ω = 0.04 for nitrogen and ω = 0.021 for oxygen. Expression (3.76) can be presented in a form convenient for finding the parameter Z [110]: Z 3 − (1 − B)Z 2 + (A − −3B 2 − 2B)Z − (AB − B 2 − B 3 ) = 0, where

(3.84)

A=

ap , Ru2 T 2

(3.85)

B=

bp . Ru T

(3.86)

The analysis of real gases is simplified if a new variable, known as fugacity f , is introduced via the following relation [10, 118]: dG = N Ru T d(ln f ),

(3.87)

126

3 Heating and Evaporation of Mono-component Droplets

where G is the Gibbs function. Expression (3.87) can be obtained from the ideal gas law when f is replaced by pressure p. The main attractive feature of Expression (3.87) is that it is valid for real gases. The main problem, however, lies in the fact that the relation between fugacity and pressure, via the so-called fugacity coefficient ϕ = f / p, needs to be found. It can be shown that (see page 129 in [10])  ln ϕ =

p

0

Z −1 d p. p

(3.88)

The calculation of the integral in (3.88) requires the knowledge of the function Z ( p). If the latter function is obtained from Eq. (3.84), then ln ϕ is presented as follows [59, 110]: A ln ϕ = Z − 1 − ln(Z − B) − √ ln 2 2B



 √ 2)B . √ Z − (1 − 2)B Z + (1 +

(3.89)

In engineering applications, fugacities and fugacity coefficients for individual species in the mixture are commonly introduced: ϕi = f i / (X i p) , where X i are molar fractions of species. Expression (3.89) in this case is generalised to [110] bi (Z − 1) − ln(Z − B) b      √ 2 j x j a ji bi Z + (1 + 2)B A − − √ , ln √ a b 2 2B Z − (1 − 2)B ln ϕi =

where a=

 i

b=

(3.90)

X i X j ai j ,

(3.91)

X i bi ,

(3.92)

j

 i

√ ai j = (1 − ζi j ) ai a j ,

(3.93)

3.3 Effects of Real Gases

127

ζi j are empirically determined binary interaction coefficients (they were assumed equal to zero in [59]). The specific enthalpy of evaporation for each species Δh v,i is inferred from the expression [59]:   f i,g Ru T 2 ∂ , (3.94) ln Δh v,i = − Mi ∂ T f i,l where Mi is the molar mass of the ith species, and subscripts l and g refer to liquid and gas phases, respectively. Note a printing mistake in Eq. (22) of [59] where the fugacity coefficients were used instead of fugacities. In the case of ideal gases, when fugacities reduce to pressures, Eq. (3.94) reduces to the classical Clausius–Clapeyron equation (see Eq. (216) of [75]). In [59], the model described above was used to simulate droplet evaporation at high pressures. The model showed noticeably better agreement with experimental data than the model based on the ideal gas law. Applications of the real gas model to the analysis of droplet evaporation under Rapid Compression Machine (RCM) and Diesel engine conditions were described in [63] and [84], respectively. The results of generalising some real gas equations of state to nano-scale confined fluids are presented in [176]. Perhaps one of the most comprehensive analyses of real gas models is presented in [72]. None of the heating and evaporation models described so far consider the motion of the droplet surface (interface) during the evaporation and swelling processes. It was assumed constant during each time step. This effect is taken into account in the models described in the next section.

3.4 Effects of the Moving Interface Considering the effect of receding droplet radius on droplet heating and evaporation leads to the well-known Stefan problem, which has been widely discussed in the literature (e.g. [17, 19, 20, 56, 66, 99, 100, 126, 163, 181]), but has been rarely applied to engineering sprays, due to their complex structure. Hence, a substantial gap has developed between mathematical and engineering research in this field. This gap was partly filled in [134, 136]. In what follows, the main findings presented in these papers are described. To simplify the analysis, the ideal gas approximation is assumed to be valid.

3.4.1 Basic Equations and Approximations As in Sect. 3.2, it is assumed, following [134, 136], that a stationary evaporating droplet is immersed into a homogeneous hot gas at temperature Tg . The droplet is

128

3 Heating and Evaporation of Mono-component Droplets

heated by convection with convection heat transfer coefficient h depending on time t and cooled due to evaporation. In contrast to Sect. 3.2, the droplet radius is allowed to change with time. Rd (t) is assumed to be a continuously twice differentiable function of time. Both Rd (t) and h(t) are assumed to be known. Effects of thermal radiation are ignored, which is justified for typical Diesel fuel droplets and gas temperatures about or less than 1000 K, when the effect of radiation on droplet evaporation time is typically less than 1% [3]. Droplet temperatures are inferred from the heat conduction Eq. (2.1) for the liquid phase (κ = kl /(cl ρl )), which is solved subject to boundary condition (2.38) with Tg replaced with Teff given by Expression (3.39). The initial condition is taken as T (t = 0) = Td0 (R), where 0 ≤ R ≤ Rd0 , Rd0 is the initial droplet radius. Boundary condition (2.38) with Teff defined by Expression (3.39) can be presented as follows [134, 136]: 

  ∂T h h h ρl + T  = Teff = Tg + L R˙ d (t) ≡ M(t). ∂R kl k k kl l l R=Rd (t)

(3.95)

Introducing variable u = T R, Eq. (2.1) can be rewritten as ∂ 2u ∂u =κ 2 ∂t ∂R

(3.96)

with the boundary conditions 

  ∂u + H (t)u  = μ(t), ∂R R=Rd (t) u| R=0 = 0,

where H (t) =

1 h(t) , − kl Rd (t)

(3.97) (3.98)

μ(t) = M(t)Rd (t).

The initial condition is presented as follows: u(R)|t=0 = RTd0 (R)

(3.99)

for R ≤ Rd0 . As in the case of models discussed in Sect. 3.2, Eq. (3.96) is solved at each time step together with the relevant equation for droplet radius to describe the process of droplet heating and evaporation. In what follows, several solutions to Eq. (3.96) for various Rd (t) and Td0 (R), leading to the distributions of temperature inside the droplet, are described.

3.4 Effects of the Moving Interface

129

3.4.2 Solution for the Case when Rd (t) is a Linear Function Assuming that Rd (t) is a linear function of t during the time step, we can write the following [134]: (3.100) Rd (t) = Rd0 (1 + αt), where Rd0 is the droplet radius at the beginning of the time step. Following [134, 136], we make a simplifying assumption that Le = 1 and c pv = c pg leading to identity B M = BT . Ignoring the effect of swelling, this allows us to obtain the expression for α as follows (cf. Expression (3.23)): α=−

k g ln (1 + B M ) , 2 ρl c pg Rd0

(3.101)

where B M = Yvs /(1 − Yvs ), Yvs is the mass fraction of fuel vapour at the droplet surface and k g and c pg are gas thermal conductivity and specific heat capacity, respectively. A more rigorous expression for α, not using the above-mentioned assumptions, can be presented as follows: ⎡ ⎡ ⎤⎤ 1/3  R˙ d0 ρ(T ρ ln + B ) D 1 ˙ R 1 ) (1 v total M d0 0 ⎣ ⎣ Rde + R˙ ds = − α= ≡ − − 1⎦⎦ , Rd Rd Rd0 ρl Rd0 Δt ρ(T 1 )

(3.102) where R˙ de and R˙ ds are the rates of change of droplet radius due to evaporation and swelling (or contraction), respectively. Subscripts 0 and 1 refer to the values of average temperatures at the beginning and at the end of the time step. The following analysis is based on Expression (3.101), but can be generalised to the case when Expression (3.102) is used. For Rd (t) defined by (3.100), the solution to Eq. (3.96), subject to the abovementioned boundary and initial conditions, allows us to obtain the following expression for T (R, t) [134] (see Appendix D):

α Rd0 R 2 1 exp − T (R, t) = √ 4κ Rd (t) R Rd (t) ×

∞ 

 Θn (t) sin λn

n=1

R Rd (t)

where Θn (t) = Θn (0) exp



κλ2n 2 α Rd0

μ0 (t) R + , 1 + h 0 Rd (t) 



1 −1 1 + αt

(3.103)

130

3 Heating and Evaporation of Mono-component Droplets



t

+ fn 0

 

1 κλ2n 1 dμ0 (τ ) exp − dτ, 2 dτ 1 + αt 1 + ατ α Rd0

 Rd (t)Rd (t) μ0 (t) ≡ μ(t) ˜ Rd (t) exp , 4κ !

μ(t) ˜ = M(t)Rd2 (t), M(t) =

h kl

Tg +

(3.104)

ρl kl

(3.105)

L R˙ d (t), f n = − ||vsinn ||λ2nλ2 , n

vn (r ) = sin λn r

(n = 1, 2, . . .),

    1 sin 2λn h0 1 2 = , || vn || = 1− 1+ 2 2 2λn 2 h 0 + λ2n

(3.106) (3.107)

r = R/Rd , λn are positive solutions to the equation λ cos λ + h 0 sin λ = 0

(3.108)

(λ1 < λ2 < λ3 < · · · ). 

h0 =

R (t)Rd (t) h(t) Rd (t) − 1 − d kl 2κ

is assumed to be constant during the time step (Rd (t) = Rd0 ), Θn (0) = qn + μ0 (0) f n , 1 qn = || vn ||2



1

W0 (r )vn (r )dr,

0

W0 (r ) =

(3.109)

3/2 Rd0 r T0 (r Rd0 ) exp

 Rd (0)Rd0 2 r . 4κ

(3.110)

Note that condition h 0 = const can be violated at the final stage of droplet evaporation. This imposes a rather stringent restriction on the choice of the time steps just before the droplet fully evaporates. Condition (3.100) allows us to use the above solution for individual time steps but not for the whole period of droplet heating and evaporation. In what follows, this assumption is relaxed and we consider arbitrary Rd (t) provided that this function is twice continuously differentiable (natural condition keeping in mind the physical background to the problem).

3.4 Effects of the Moving Interface

131

3.4.3 Solution for the Case of Arbitrary Rd (t) but Td0 (R) = const The following analysis uses the assumption that Td0 (R) = Td0 = const [136]. Introducing new variable v = u − RTd0 , Eq. (3.96) can be presented as follows [136]: ∂ 2v ∂v = κ 2, ∂t ∂R

(3.111)

with boundary conditions 

  ∂v = μ0 (t), + H (t)v  ∂R R=Rd (t)

(3.112)

v| R=0 = 0,

(3.113)

v|t=0 = 0

(3.114)

and the initial condition for R ≤ Rd (t), where μ0 (t) = −Td0 − H (t)Rd (t)Td0 + μ(t) = −

h(t) Rd (t)Td0 + μ(t), kl

μ(t) is the same as in Eq. (3.97). We look for the following solution to Problem (3.111)–(3.114):  v(R, t) =

t

ν(τ )G(t, τ, R)dτ,

(3.115)

0

where G(t, τ, R) = G 0 (t − τ, R − Rd (τ )) − G 0 (t − τ, R + Rd (τ )), √

κ x2 . G 0 (t, x) = √ exp − 4κt 2 πt The expression for G(t, τ, R) can be presented as follows: √



 " (R + Rd (τ ))2 (R − Rd (τ ))2 κ G(t, τ, R) = √ − exp − , exp − 4κ(t − τ ) 4κ(t − τ ) 2 π(t − τ ) (3.116) where G(t, τ, R = 0) = 0. ν(t) is an unknown continuous function to be found from one of the boundary conditions. Function v(R, t) is known as a single layer heat potential and it has the following properties for every continuous function v(t) [66, 154]:

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3 Heating and Evaporation of Mono-component Droplets

(1) It satisfies Eq. (3.111) for 0 < R < Rd (t); (2) It satisfies Conditions (3.113) and (3.114); (3) It is continuous at R → Rd − 0; (4) The following formula is valid: 

  t ∂v(R, t)  ν(t) ∂G(t, τ, R)  + = ν(τ ) dτ.  ∂ R  R→Rd (t)−0 2 ∂R 0 R=Rd (t)

(3.117)

Thus, for any continuous function ν(t), the potential v(R, t) satisfies Eq. (3.111) and boundary and initial conditions (3.113) and (3.114). The choice of function ν(t), satisfying integral Eq. (3.117), should be made in such a way that the remaining boundary condition (3.112) is satisfied as well. It follows from Eq. (3.116) that  1 ∂G(t, τ, R)  =− √  ∂R 4 π κ(t − τ )3/2 R=Rd (t)

" (Rd (t) − Rd (τ ))2 × (Rd (t) − Rd (τ )) exp − 4κ(t − τ )

 (Rd (t) + Rd (τ ))2 . −(Rd (t) + Rd (τ )) exp − 4κ(t − τ )

(3.118)

Since Rd (t) is a continuously differentiable function, the following condition is valid in the limit τ → t − 0:    1 ∂G(t, τ, R)  . (3.119) ∝ O √  ∂R t −τ R=Rd (t) Condition (3.119) shows that there is an improper integral on the right-hand side of Eq. (3.117). Remembering Eqs. (3.117) and (3.115), we can present the boundary condition (3.112) as follows:

  t  t ν(t) ∂G(t, τ, R)  + ν(τ ) dτ + H (t) ν(τ )G(t, τ, Rd (t))dτ = μ0 (t),  2 ∂R 0 0 R=Rd (t)

or $

 ∂G(t, τ, R)  ν(τ ) + H (t)G(t, τ, Rd (t)) dτ = μ0 (t),  ∂R 0 R=Rd (t) (3.120) where G(t, τ, R) and its derivative with respect to R are defined by Expressions (3.116) and (3.118). ν(t) + 2



t

#

3.4 Effects of the Moving Interface

133

As follows from Expression (3.116), in the limit τ → t − 0, G(t, τ, Rd (t)) has the singularity:   1 G(t, τ, Rd (t)) ∝ O √ t −τ (cf. Eq. (3.119)). Thus, the integral in Eq. (3.120) is defined as an improper integral. Equation (3.120) is an integral equation of Volterra type. It has a unique continuous solution. A scheme for its numerical solution is like the one described in Appendix F (see also Appendix A in [136]). This solution is then substituted into Eq. (3.115). The final distribution of temperature inside the droplet is obtained from the following formula: √ "

 t (R − Rd (τ ))2 κ ν(τ ) exp − T (t, R) = Td0 + √ √ 4κ(t − τ ) 2R π 0 t −τ

 (R + Rd (τ ))2 dτ. − exp − 4κ(t − τ )

(3.121)

Details of the numerical calculation of the integral on the right-hand side of Eq. (3.121) are presented in Appendix G.

3.4.4 Solution for Arbitrary Rd (t) and Td0 (R) It is assumed that an arbitrary continuously twice differentiable function Td0 (R) is defined for 0 ≤ R ≤ Rd0 . This definition is extended for Rd0 < R < ∞ [136]: ⎧ ⎨ Td0 (R) when 0 ≤ R ≤ Rd0 Td0 (R) = Tout (R) when Rd0 < R ≤ Reff (3.122) ⎩ 0 when R > Reff , where 1 Tout (R) = R

"



  Rd0 Td0 (Rd0 ) + (R − Rd0 ) (RTd0 (R)) R 





R=Rd0

,

Reff is the effective outer radius such that Reff > Rd0 . The function Td0 (R) defined by Expression (3.122) is continuously differentiable in the range 0 ≤ R ≤ Reff . Let us now introduce a new function U (t, R) defined as follows [136]:  U (t, R) = 0

Reff

(ζ Tdo (ζ ))G 1 (t, R, ζ )dζ,

(3.123)

134

3 Heating and Evaporation of Mono-component Droplets

where G 1 (t, R, ζ ) =

1 [G 0 (t, R − ζ ) − G 0 (t, R + ζ )] , κ

G 0 (t, x) is the same as in Eq. (3.115). Based on the latter equation, G 1 (t, R) is presented as follows: "



 (R − ζ )2 (R + ζ )2 1 exp − − exp − . G 1 (t, R) = √ 4κt 4κt 2 π tκ

(3.124)

It can be noted that G 1 (t, R = 0) = 0. The function U (t, R) has the following properties [66, 154]: (1) It satisfies Eq. (3.111) for 0 < R < ∞; (2) It satisfies the boundary condition (3.113); (3) It satisfies the initial condition " U (t, R)|t=0 =

RTd0 (R) when 0 ≤ R ≤ Reff 0 when R > Reff .

(3.125)

The latter relation follows from the presentation of the delta-function as lim

αdelta →∞

αdelta 2 x 2 ) = δ(x). √ exp(−αdelta π

(3.126)

We look for the solution to Eq. (3.96) in the following form: u(t, R) = U (t, R) + v(t, R).

(3.127)

Having substituted Expression (3.127) into Eq. (3.96) and boundary and initial conditions (3.97)–(3.99), we obtain Problems (3.111)–(3.114) for v(t, R) in which     μ0 (t) = − U R (t, R) + H (t)U (t, R) 

R=Rd (t)

+ μ(t).

(3.128)

The solution to the latter problem was discussed in the previous section. The expression for μ0 (t) contains   U R (t, R)





R=Rd (t)

= 0

Reff

 ∂G 1 (t, R, ζ )  (ζ Tdo (ζ )) dζ,  ∂R R=Rd (t)

(3.129)

where 

  1 ∂ ∂G 1 (t, R, ζ )   = (t, R − ζ ) − G (t, R + ζ )) (G 0 0   ∂R κ ∂R R=Rd (t) R=Rd (t)

3.4 Effects of the Moving Interface

1 κ

=



135

  ∂ , (−G 0 (t, R − ζ ) − G 0 (t, R + ζ ))  ∂ζ R=Rd (t)

(3.130)

G 0 is the same as in  Eq. (3.115). Using the latter equation, we can rewrite the expres∂G 1 (t,R,ζ )  as follows: sion for  ∂R R=Rd (t)

 ∂G 1 (t, R, ζ )   ∂R

"

(R − ζ )2 1 (R − ζ ) exp − =− √ 4κt 4 π (κt)3/2

R=Rd (t)



(R + ζ )2 −(R + ζ ) exp − 4κt

   

R=Rd (t)

.

This allows us to obtain an explicit expression for μ0 (t): 1 μ0 (t) = √ 4 π (κt)3/2



Reff 0

" (Rd (t) − ζ )2 (ζ Td0 (ζ )) (Rd (t) − ζ ) exp − 4κt

 (Rd (t) + ζ )2 dζ −(Rd (t) + ζ ) exp − 4κt H (t) − √ 2 π κt



Reff

0



 (Rd (t) + ζ )2 (Rd (t) − ζ )2 − exp − dζ (ζ Td0 (ζ )) exp − 4κt 4κt "

+ M(t)Rd (t).

(3.131)

In the limit t → 0, the expression for μ0 (t) simplifies to [136]

  μ0 (0) = − (ζ Tdo (ζ ))ζ 





ζ =Rd0

+ H (0)Rd0 Td0 (Rd0 ) + μ(0).

(3.132)

Combining Eqs. (3.115) and (3.127), the final solution to the problem can be presented as follows [136]: T (R, t) =

√  t "

(R − Rd (τ ))2 κ ν(τ ) 1 exp − U (R, t) + √ √ R 4κ(t − τ ) 2 π 0 t −τ



(R + Rd (τ ))2 dτ , − exp − 4κ(t − τ )

(3.133)

where ν(τ ) is the solution to Eq. (3.120) with μ0 (t) defined by Expression (3.131) and U (R, t) defined by Expression (3.123). Note that considering the initial distribution of temperature along R is essential when the solution is applied to individual time steps. In the solutions described in the

136

3 Heating and Evaporation of Mono-component Droplets

last two sections, however, the same formulae describe the time evolution of droplet temperatures during the whole period of their evaporation. It is anticipated that in this case the effect of the initial distribution of droplet temperatures is not important in most engineering applications. Although the analysis presented in this section refers to stationary droplets, it can be generalised in a straightforward way to the case of moving droplets, based on the Effective Thermal Conductivity (ETC) model discussed in Sect. 2.1.2. In the solutions presented in the last two sections, it was assumed that Rd (t) is known. From the point of view of the physical background to the problem, however, Rd (t) depends on the time evolution of the droplet temperature T (R, t), which is the solution to the problem. Hence, an iterative process is required. Firstly, the time evolution of droplet radius Rd (t) is obtained using the conventional approach, when it remains constant during the time step, but changes from one time step to another due to the evaporation process. Then these values of Rd (t) are used in the new solutions to obtain updated values of the time evolution of the distribution of temperatures inside the droplet and on its surface T (R, t). These new values of droplet temperature are used to update the function Rd (t). This process is repeated until convergence is achieved, which typically occurs after about 15 iterations [136].

3.4.5 A Comparison between Model Predictions In Figs. 3.1 and 3.2, the results of calculations of droplet surface temperatures and radii are compared, considering the effects of evaporation. The results based on the integral solution for arbitrary Rd (t) but constant Td0 (Eq. (3.121)), those based on the linear approximation of Rd (t) (Expression (3.103)) and the conventional approach using the assumption that droplet radius does not change during the time steps are shown. Droplets are assumed to be those of n-dodecane (M f = 170 kg/kmole) with initial radii Rd0 = 5 µm; ambient air temperature and pressure are assumed equal to 1000 K and 3 MPa, respectively (typical Diesel engine-like conditions). One-thousand time steps were used in the conventional approach and the approach which was based on Solution (3.103). In the integral solution using Eq. (3.121), Integral (F.1) was approximated as the sum of 100 terms, and up to 15 iterations were used. At the first iteration, the time evolution of the droplet radius was assumed to be the same as predicted by the conventional model. As follows from Figs. 3.1 and 3.2, the results predicted by the integral solution (3.121) and linear solution (3.103) almost coincide, which suggests that both approaches are correct. Both these solutions predict lower droplet temperatures and longer evaporation times than those predicted by the conventional solution. Deviations between the predictions of the integral and linear solutions were observed in the immediate vicinity of the instant of time when the droplet completely evaporates. There were numerical problems when this instant of time was approached since the time derivative of Rd became very large. The extrapolation, using the assumption that the second derivative of Rd (t) is constant, was used in this case. This led to small

3.4 Effects of the Moving Interface

137

Fig. 3.1 Plots of Ts versus time using the conventional model (thick solid), integral model based on Eq. (3.121) (dashed) and linear model (thin solid) for a stationary n-dodecane (M f = 170 kg/kmole) droplet with an initial radius 5 µm, evaporating in ambient air at a pressure of p = 3 MPa and temperature 1000 K. Reprinted with minor modifications from [136], Copyright Elsevier (2011) Fig. 3.2 The same as Fig. 3.1 but for the droplet radius versus time. Reprinted with minor modifications from [136], Copyright Elsevier (2011)

Fig. 3.3 The same as Fig. 3.1 but for different numbers of iterations in the integral solution. Reprinted from [136], Copyright Elsevier (2011)

deviations between the predicted evaporation times. In the case shown in Figs. 3.1 and 3.2, the evaporation times predicted by the conventional model, linear solution and integral solution were 0.595 ms, 0.622 ms and 0.628 ms, respectively. That means that the difference between the evaporation times predicted by the linear and integral solutions was less than 1% and can be safely ignored in most engineering applications (this error can be reduced further if required). The same comment applies to other cases described in [136]. The effect of the number of iterations on the prediction of the integral solution is shown in Figs. 3.3 and 3.4 for the same case as presented in Figs. 3.1 and 3.2.

138

3 Heating and Evaporation of Mono-component Droplets

Fig. 3.4 The same as Fig. 3.3 but for the droplet radius versus time. Reprinted from [136], Copyright Elsevier (2011)

This effect is illustrated only for the times when the deviation between the results predicted by the linear and integral solutions is maximal. For the first iteration, the time evolution of droplet radius is assumed to be the same as predicted by the conventional model. The deviation of the corresponding droplet temperatures predicted by the integral and linear solutions can be clearly seen. For the fifth iteration, the droplet surface temperatures predicted by the integral and linear solutions are almost the same up to t ≈ 0.45 ms. The corresponding plots of Rd (t), predicted by the integral solution, are closer to those predicted by the linear solution than those predicted by the conventional model. The plots predicted by the linear and integral solutions became closer as the number of iterations increased. However, even for the 15th iteration, the deviation between the results remains noticeable, although not important for practical applications (cf. Figs. 3.1 and 3.2). For higher iterations, the results were almost indistinguishable from those predicted by the 15th iteration. Interestingly, odd iterations predicted smaller Rd (t) while even iterations predicted larger Rd (t) compared with those predicted by the linear solution. At the qualitative level, this could be related to the fact that a faster evaporation rate, assumed for the first iteration (conventional model), leads to a lower droplet surface temperature. At the second iteration, this lower droplet surface temperature leads to a slower evaporation rate, etc. As to the computational efficiency of the integral model, for a PC Xeon 3000 Hz (the calculations were processed on one kernel only) with 2 GB RAM, the conventional approach requires 3586 s to calculate 1191 steps. Once these calculations have been completed, the integral model requires 453 s to calculate 15 iterations. This makes this model potentially suitable for incorporation into Computational Fluid Dynamics (CFD) codes. Calculations similar to those presented in Figs. 3.1 and 3.2 were performed for droplets with initial radii 50 and 100 µm [136]. Time evolution of droplet surface temperatures and radii were largely unaffected by the initial droplet radii. This agrees with similar results presented in [134] (see Figs. 4–6 in that paper). The effect of non-constant initial distribution of droplet temperature on the time and space evolution of this distribution is shown in Fig. 3.5. Two cases are presented in this figure. In both cases, the initial droplet radii are assumed to be equal to 5 µm, and gas temperature is assumed to be constant and equal to Tg = 1000 K. In the first

3.4 Effects of the Moving Interface

139

Fig. 3.5 Plots of T versus r = R/Rd for a stationary n-dodecane (M f = 170 kg/kmole) droplet with initial radius 5 µm, evaporating in ambient air at a pressure of p = 3 MPa and temperature 1000 K. The instants of time are shown near the curves. The calculations for the ‘Constant’ curve were performed using Expression (3.121). The calculations for the ‘General’ curve were performed using Expression (3.134). Reprinted with minor modification from [136], Copyright Elsevier (2011)

case, the initial temperatures were assumed to be the same for all R (or r = R/Rd ) and equal to 300 K. In this case, the analysis based on Expression (3.121) was used. In the second case, the following initial distribution of droplet temperature Td0 (R) = 300 + 10(R/Rd0 )2 = 300 + 10r 2

(3.134)

and Eq. (3.133) were used. Comparing the plots presented in Fig. 3.5, it can be seen that these plots converge with time. This is related to the fact that increased droplet surface temperatures lead to decreased convective heating of droplets. Hence, the droplet surface temperatures increase more slowly in the general case than in the case of constant initial temperatures inside the droplet. In the case when the initial temperatures inside the droplet are constant, the predictions of Expressions (3.121) and (3.133) are the same [136]. This demonstrates the consistency of both solutions to the problem. The results predicted by the linear model were verified using the results of calculations based on the direct numerical solution to Eq. (3.96) [97]. The authors of [97] used the boundary immobilisation method coupled to a Keller Box discretisation scheme of the one-phase one-dimensional time-dependent governing equations. This algorithm is implicit, therefore, not having any limitation on the time step size and was in addition shown to be second-order accurate in time and space variables. Note that the difference between the predictions of the models considering and not considering the change in droplet radii during individual time steps does not depend on the duration of the time steps when these time steps are short enough. The results shown in Figs. 3.1, 3.2, 3.3 and 3.4 would remain the same if the time steps are further reduced. The underlying physics of the exchange of heat between

140

3 Heating and Evaporation of Mono-component Droplets

ambient gas and droplets in the presence of moving and stationary interface can be related to the exchange of energy between a moving and stationary lorry, and a ball hitting its back (see Sect. 8.6 for further discussion of this matter). It is appreciated that the errors associated with the conventional assumption that the droplet radii remain constant during the time step are comparable with or even smaller than those associated with other effects, including uncertainties in gas temperature measurements, approximations of the convection heat transfer coefficient and effect of interactions between droplets in realistic sprays. The importance of the latter effect was discussed in [73, 133], but its analysis lies beyond the scope of this book.

3.5 Conventional and Alternative Approaches to Modelling In this section, the focus will be on two approaches to modelling mono-component droplet heating and evaporation discussed in [137]. Using (3.28) and (3.30), the heat rate supplied to the droplet to change its temperature can be obtained as follows [137]:





c pv (Tg − Ts ) c pv (Tg − Ts ) − L(T − L(Ts ) = −m˙ d ) . s BT (1 + B M )ϕ − 1 (3.135) Our analysis is restricted to the case of stationary heated droplets (|q˙d | > 0; ϕ is defined by (3.31). Note that in this case q˙d < 0. Since the pioneering paper by Abramzon and Sirignano [4], Expression (3.135) has been commonly used for modelling the heating of evaporating droplets, in combination with Expression (3.23) for modelling droplet evaporation. An obvious limitation of Expression (3.135) is that the value of q˙d is not affected by the thermal conductivity of liquid, which is not compatible with the physical nature of q˙d . An alternative approach to the estimation of |q˙d | is based on the analysis of temperature distribution inside droplets, inferred from the direct analysis of convective heating of evaporating droplets (see Expression (2.41) with Tg replaced with Teff using Expression (3.39)). This approach is limited to the case when liquid thermal conductivity is finite, which is expected for realistic liquids. It allows us to estimate |q˙d | as follows:  ∂ T  2 |q˙d | = 4π Rd kl , (3.136) ∂R |q˙d | = −m˙ d

R=Rd −0

where T (R) is inferred from the analytical solution to the heat transfer equation inside the droplet for a fixed convection heat transfer coefficient given by Expression (3.34). Having substituted (2.41) into (3.136), the following expression is obtained:

3.5 Conventional and Alternative Approaches to Modelling ∞ "    qn exp −κ R λ2n t − |q˙d | = 4π Rd kl n=1

−

sin λn || vn ||2 λ2n

 0

t

141

  sin λn μ0 (0) exp −κ R λ2n t − 2 2 || vn || λn

   dμ0 (τ ) exp −κ R λ2n (t − τ ) dτ [−1 − h 0 ] sin λn , dτ

(3.137)

where λn are solutions to Eq. (2.42) which give a set of positive eigenvalues, λn , numbered in ascending order; other notations are the same as in (2.41). Once |q˙d | has been found, the mass evaporation rate (m˙ d ) follows from Expression (3.27). Using the definition of BT (Expression (3.28)), this expression can be rewritten as follows:   4π k g Rd c pv (Tg − Ts )m˙ d . (3.138) ln 1 + m˙ d = − c pv L(Ts )m˙ d − |q˙d | Thus, we have two approaches to modelling heating and evaporation of stationary droplets. The first one is based on Expressions (3.30) and (3.135) (conventional approach originally suggested in [4], Model 1) and the approach based on Expressions (3.137) and (3.138) (Model 2). In what follows, the predictions of these approaches are compared following [137]. Expression (3.137) is applicable to any time step. t = 0 refers to its beginning; t refers to the end of the time step. The values of |q˙d | at the beginning of each time step are equal to the values of |q˙d | at the end of the previous time step or the start of the heating process. Hence, without loss of generality, we can assume that t = 0 in Expression (3.137). The values of |q˙d | predicted by Expression (3.137) coincided, within  the accuracy of plotting, with those predicted by Expression (2.86), in which ∂T  was calculated using direct numerical differentiation of T (R) given by ∂ R R=Rd −0 Expression (2.41). In contrast to Expression (3.23), Eq. (3.138) is a non-linear equation for m˙ d . It has two solutions, m˙ d = 0 (non-evaporating droplet) and m˙ d < 0 (evaporating droplet), when 4π k g Rd (Tg − Ts ) > 1, (3.139) |q˙d | and only one trivial solution m˙ d = 0 (non-evaporating droplet) when Condition (3.139) is not valid. In the limiting case when BT  1, Eq. (3.138) has the analytical solution: m˙ d =

 1  |q˙d | − 4π k g Rd (Tg − Ts ) . L(Ts )

(3.140)

This solution has no physical meaning unless Condition (3.139) is valid. Expression (3.23) can still be used in this approach together with (3.30). The latter equation can be rewritten as B M = (1 + BT )1/ϕ − 1. All thermodynamic and transport properties for liquid and gas were assumed constant during each time step but their changes from one time step to another due to

142

3 Heating and Evaporation of Mono-component Droplets

Fig. 3.6 Plots of absolute values of q˙d versus time predicted by Model 1 and Model 2 for an evaporating n-dodecane droplet heated in air at a pressure of 30 bar and temperature 700 K. The initial droplet temperature and radius are assumed to be equal to 300 K and 10 µm, respectively. Reprinted from [137], Copyright Elsevier (2014)

the corresponding changes in temperature were considered. The effects of thermal swelling were also considered. The model was applied to the analysis of heating of an evaporating n-dodecane droplet in air at a pressure of 30 bar and temperature 700 K. Thermodynamic and transport properties of n-dodecane were taken from [40], except the diffusion coefficient for n-dodecane vapour in air which was taken from [3]. The initial droplet temperature and radius were taken equal to 300 K and 10 µm, respectively. The predictions of Model 2 were compared with predictions of Model 1. In both cases, the finite thermal conductivity of liquid was considered. The values of |q˙d |, predicted by these models, are presented in Fig. 3.6. At the very final stage of droplet evaporation, the values of |q˙d | predicted by Model 2 became negative (although close to zero) which eventually led to the situation where Eq. (3.138) had no real solutions. To avoid this situation, the distribution of temperature inside droplets was frozen at the time instant when |q˙d | = 0. Also, at the very final stage of droplet evaporation, the predicted droplet temperature could approach the critical temperature and even exceed it. This was partly remedied by assuming that once Teff has reached its minimal value it remains at this level until the droplet fully evaporates. These assumptions cannot be rigorously justified but they are expected to produce only minor effects on the predicted surface temperatures and radii of droplets which are not important for practical applications. The volumes of droplets when these assumptions were used were less than 0.1% of their initial volumes in most cases. Modelling droplet heating and evaporation at the final stages of their lifetime when d Rd /dt → ∞ were discussed previously (e.g. [136]). As can be seen from Fig. 3.6, the shapes of the curves |q˙d | versus time predicted by both approaches are rather close but the actual values of |q˙d | are noticeably different. This difference leads to large differences in the corresponding values of droplet radii and surface temperatures, as can be seen in Fig. 3.7. As follows from Fig. 3.7, Model 2 predicts lower droplet surface temperatures and shorter evaporation times than Model 1. Lower droplet surface temperatures predicted by Model 2 compared with Model 1 are expected to lead to lower values of the heat fluxes at the droplet’s surface. This is consistent with the predicted values of |q˙d | presented in Fig. 3.6.

3.5 Conventional and Alternative Approaches to Modelling

143

Fig. 3.7 The same as in Fig. 3.6 but for droplet surface temperatures Ts and radii Rd . Reprinted from [137], Copyright Elsevier (2014)

Similar trends in time evolution of the parameters predicted by both models allow us to use them for qualitative analysis of droplet heating and evaporation, but their reliability for quantitative analysis of these processes remains unclear. One of the reasons for the differences between the predicted results might be related to the fact that both approaches to the calculation of the evaporation rate are based on the quasi-steady-state approximation. The limitations of this approximation for the case of non-evaporating droplet heating are described in Sect. 2.1.1.1. The model described in [143, 150] is based on the coupled solutions to heat and mass transfer equations in the gas and liquid phase for stationary droplets (this model was called by the authors ‘a complete model’). This approach is rather like the one considered in Sect. 2.1.1.1 for non-evaporating droplets. In contrast to Sect. 2.1.1.1, the focus of the investigation by the authors of [143, 150] was not on the solutions to the coupled equations, but on finding simplified solutions to these equations in several limiting cases. A simplified version of the complete model, called the quasihomogeneous model, was developed using an asymptotic analysis in the limiting case 2 /αl ; αl is liquid diffusivity) when the so-called homogenisation time (τhom, l = Rd0 is much shorter than the droplet evaporation time [150]. The quasi-homogeneous model was further reduced to the fully quasi-steady-state model (time dependence of the liquid surface temperature is ignored). A simple formula was obtained to estimate the relative difference between the droplet evaporation times predicted by the complete model and those predicted by the fully quasi-steady-state model. The application of this formula together with the fully quasi-steady-state model allows one to estimate the droplet evaporation time predicted by the complete model. The models described so far have focused on spherical droplets. In what follows, developments in modelling the heating and evaporation of non-spherical droplets, approximated by spheroids in most cases, are described.

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3 Heating and Evaporation of Mono-component Droplets

3.6 Heating and Evaporation of Spheroidal Droplets 3.6.1 Background Research: Non-spherical Droplets The models considered so far assumed that droplets are perfect spheres. At the same time, the shapes of most droplets observed in various applications are not spherical (see [34, 95]). It is not possible to develop a general theory for heating and evaporation of droplets of arbitrary shapes except by performing direct numerical simulations. In most applications, the effects of non-sphericity of droplets have assumed that droplet shapes can be approximated by oblate or prolate spheroids. The heat conduction equation inside a spheroid was first (to the best of the author’s knowledge) solved analytically more than 140 years ago [103]. This solution, however, is too complex for most applications. In most cases, this problem (and the related problem of mass transfer) has been investigated using numerical solutions to the heat transfer (and mass diffusion) equations [62, 81]. Complementary analytical solutions to the problem of steady-state temperature distribution around a prolate spheroid were obtained in [65]. This distribution was inferred from the solution to the Laplace equation with the Neumann (uniform flux) and Dirichlet (isothermal) boundary conditions. The generalisation of these solutions to the corresponding mass transfer problem would be straightforward. The problem of heat and mass transfer inside spheroids, considered in the abovementioned papers, is complementary to the problem of heat and mass transfer from/to ambient fluid (gas) to/from a spheroid, considering the relative velocity between the gas and the spheroid, in the general case. The latter problem has been investigated in many papers using numerical solutions to momentum and heat transfer equations in the ambient fluid (gas) in the ellipsoidal coordinate system. The analysis of [6, 69, 121, 145, 146] assumed that the spheroid surface was fixed. Juncu [64] considered changes in spheroid temperature with time, while assuming that there is no temperature gradient inside the spheroid (the thermal conductivity of the spheroid was assumed infinitely high). Most of these approaches are applicable to solid bodies and droplets. In the case of droplets, however, apart from heating, the evaporation processes should also be considered in the general case. Grow [55] was perhaps the first to solve the problem of heat and mass transfer in the vicinity of spheroidal particles assuming that their relative velocities are equal to zero, although she considered coal chars rather than droplets. One of the main limitations of this paper is that both mass and heat transfer equations were presented in the form of Laplace equations, which implies that the effects of the Stefan flow from the surface of the particles were ignored. The latter effects were described using the exact solutions to the mass and heat transfer equations in the gas phase around a spheroidal droplet in [156]. In that paper, it was assumed that the temperatures at all points on the surface of a droplet are the same and constant, and the droplet’s shape remains spheroidal. A combined problem of spheroidal droplet heating and evaporation, like the one studied in [156], was discussed in [80]. As in [156], the authors of [80] based their

3.6 Heating and Evaporation of Spheroidal Droplets

145

study on the solution to the species conservation equation in the gas phase and assumed that the thermal conductivity of droplets is infinitely large. In contrast to [156], the authors of [80] considered the relative velocities of droplets, assuming that the dependencies of the Nusselt and Sherwood numbers on the Reynolds and Prandtl numbers are the same as those for the spherical droplets. Also, they considered the time dependence of droplet temperatures and sizes, although their analysis focused on oblate droplets only. The unsteady conjugate mass and heat transfer from/to a prolate droplet suspended in a creeping flow was investigated numerically by the authors of [82]. An empirical correlation predicting the Sherwood number with errors less than about 12% was proposed. Strotos et al. [149] described the results of CFD analysis of the evaporation of nearly spherical suspended droplets. They solved the Navier–Stokes equations, the energy conservation equation and the species transport equations; the Volume-ofFluid (VOF) method was used to capture the droplet surface. As follows from a brief overview of the models described above, the general problem of heating and evaporation of spheroidal droplets is far from resolved. We believe, however, that the results described in [156] could be a starting point for constructing a model at least for spheroidal droplets (see Sect. 3.6.3). The main ideas of the model suggested in [156] are described in the next section (cf. [33] for an overview of this model and other similar models developed by these authors).

3.6.2 The Tonini and Cossali Model (Spheroidal Droplets) The authors of [156] focused on exact solutions to the mass and heat transfer equations in the gas phase around a spheroidal droplet. The droplet was considered to be monocomponent, and the following steady-state equation for the vapour mass fraction (Yv ) was investigated: ∇ (ρtot UYv − ρtot Dv ∇Yv ) = 0, (3.141) where ρtot is the density of the mixture of vapour and ambient gas, U is the Stefan velocity of this mixture and Dv is the diffusion coefficient of vapour in ambient gas. Equation (3.141) was solved in ellipsoidal coordinates ξ, u, ϕ keeping in mind that all processes are axially symmetric; z was chosen as the axis of symmetry. These ellipsoidal coordinates are linked with Cartesian coordinates x, y, z as follows: ⎫ x = aΦ− (ξ ) sin(u) cos(ϕ) ⎬ y = aΦ− (ξ ) sin(u) sin(ϕ) ⎭ z = aΦ+ (ξ ) cos(u) where Φ± (ξ ) =

eξ ± s(ε)eξ , s(ε) = sign(ε − 1), ε = az /ar , 2

(3.142)

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3 Heating and Evaporation of Mono-component Droplets

2az and 2ar are the sizes of the spheroid along and perpendicular to z-axis, respectively (ε > 1 and s = 1 for prolate spheroids; ε < 1 and s = −1 for oblate spheroids).1 Assuming that the volume of a spheroid is equal to that of a perfect sphere of radius R0 , the following formulae were derived, valid for both prolate and oblate spheroids:   + 1 − ε2 1/2 εs + 1 ξ0 = ln , (3.143) ; a = R0 εs − 1 ε1/3 where ξ = ξ0 at the surface of the spheroid. Tonini and Cossali [156] solved Eq. (3.141) assuming that the values of Yv and all other scalar properties are the same along the whole surface of the droplet and equal to Yv = Yvs , and the Stefan velocity and diffusive fluxes are perpendicular to the droplet surface (U = (Uξ , 0, 0)). These assumptions allowed them to simplify Eq. (3.141) to

Dv d dYv dYv = ρtot Φ− (ξ ) , (3.144) ρtot Uξ dξ aS 2 dξ dξ where  1/2 S 2 ≡ S 2 (ξ, u) = Φ− (ξ ) Φ−2 (ξ ) cos2 u + Φ+2 (ξ ) sin2 u .

(3.145)

The equation of conservation of mass  d  2 S ρtot Uξ = 0, dξ

(3.146)

where S 2 is defined by Expression (3.145), was used for the derivation of Eq. (3.144). Equation (3.144) is different from the corresponding equation on which the analysis of [80] was based (see their Equation (10)). The latter equation is the Laplace-type equation which is valid only when the Stefan flow is not considered. Although in the original paper [156] Eq. (3.144) was derived under the assumption that Yv is the same along the whole surface of the droplet, it can be shown that this equation remains valid under a less stringent assumption that the directions of the gradients of Yv are close to the ξ -directions. This assumption is expected to be valid in the case when the sphericity of the droplet ε is reasonably close to 1. The original assumption that Yv is the same along the whole surface of the droplet implied that there is no temperature gradient at the surface of the droplet. The latter follows from another assumption made in [156] that droplet thermal conductivity is infinitely large. Assuming that ρtot = ρv + ρg = const and that at large distances from the droplet Yv = Yv∞ = const, the solutions to Eqs. (3.144) and (3.146) were obtained as follows [156]:

Note that in [145, 146] prolate and oblate spheroids were defined as those with ε < 1 and ε > 1, respectively; the same definition is used in Figure 1 of [69]

1

3.6 Heating and Evaporation of Spheroidal Droplets

Yv = 1 − (1 − Yvs )

147

⎧   arctan eξ −arctanξeξ0 ⎪ ⎪ ⎨ 1−Yv∞ (π/2)−arctan e 0

oblate

 1− (ξ ) ( ξ ) ⎪ ⎪ ln(e 0 +1)−ln(e 0 −1) ⎩ 1−Yv∞

prolate

1−Yvs

(3.147)

ln eξ +1 −ln eξ −1

1−Yvs

dm˙ ev ρtot Dv 1 − Yv∞ = ρtot Uξ (ξ0 ) = (ε) ln , dA R0 1 − Yvs

(3.148)

where (ε) =

1  ⎧  1 − ε2 1/2 ⎨ π−2 arctan√ 1+ε 

ε1/3

⎩

ln

√

oblate

1−ε

 1 √  1+ε −ln ε−1 −1

1+ε ε−1 +1

prolate

(3.149)

and ddm˙Aev is the evaporation flux (assumed to me positive). When deriving Expression (3.148), it was assumed that ξ -directions near the droplet surface are close to the radial directions. This can be acceptable for spheroids with ε close to 1. In [156], it was assumed that Yvs = const. In the general case, Formulae (3.147) and (3.148) can be applied to the case when Yvs = Yvs (u) provided that the spheroid can be considered as a slightly deformed sphere (ε is close to 1). This implies that the evaporation flux can be a function of u in the general case, while in the original paper [156], this flux did not depend on u. Also, in [156], the expression for the evaporation rate, rather than the evaporation flux, was presented. This evaporation rate could be found only in the case when the temperatures at all points on the droplet surface are the same. The validity of (3.148) does not depend upon this assumption. The assumption ρtot = ρv + ρg = const for spherical droplets was relaxed in [155]. The generalisation of the approach described in [155] to the case of spheroidal droplets has not been investigated to the best of the author’s knowledge. The analysis of Eq. (3.141) in [156] was complemented by the analysis of the heat transfer equation (see Appendix H for the details of its derivation): ρtot Uc pv ∇T = k g ∇ 2 T,

(3.150)

where c pv is the specific heat capacity of vapour at constant pressure and k g is the thermal conductivity of the mixture of fuel vapour and ambient gas (air). In [156], Eq. (3.150) was solved using assumptions and boundary conditions like those which were used for the solution to Eq. (3.141). Namely, it was assumed that the temperatures at all points along the droplet surface are the same and the temperature gradients are perpendicular to the surfaces ξ = const. These assumptions, alongside the assumption that the temperature at a large distance from the droplet is equal to T∞ = const, allowed Tonini and Cossali [156] to simplify Eq. (3.150) and present its solution as follows: T =

 T∞ − Ts  ζ (ξ,ε) η − η + Ts , 1−η

(3.151)

148

where

3 Heating and Evaporation of Mono-component Droplets

1 − Yv∞ 1 , ln η = exp − Lev 1 − Yvs

ζ (ξ, ε) =

⎧ π−2 arctan eξ ( ) ⎪ ⎨ π−2 arctan√ 1+ε 

(3.152)

oblate

1−ε ξ

(3.153)

ln(e +1)−ln(e −1) ⎪  √  prolate ⎩ 2 ξ

ln ε+ ε −1

  and Lev = k g / ρtot c pv Dv is the fuel vapour Lewis number. If the assumption that Ts = const made in [156] is relaxed, then η becomes a function of u in the general case (note that Yvs = Yvs (Ts )). Expression (3.151) allows us to find the local convective heat transfer coefficient h:       −k g ∇T ξ =ξ0  . (3.154) h= |T∞ − Ts | Although the value of h was not explicitly calculated in [156], this calculation follows in a straightforward way from the analysis by these authors. Hence, this calculation is considered as part of the Tonini and Cossali model. Having substituted (3.151) into (3.154), the following expression for h is obtained [183]: ⎧ ⎪  ⎪ ⎨

√

ε1/3 +  

1+ε π−2 arctan kg η 1−ε h=− 1/3 ε + R0 (1 − η) ⎪ ⎪ ⎩ lnε+√ε2 −1 

1 1−ε 2

1 −sin2 ε 2 −1

−sin2 u 



oblate prolate

,

(3.155)

u

where η is defined by Expression (3.152). Formula (3.155) can be considered complementary to the empirical correlations for the Nusselt number (Nu) for cuboidal and ellipsoidal particles in crossflow discussed in [121]. !  Note that u is linked with θ = arctan x 2 + y 2 /z via the following equation: tan u = ε tan θ,

(3.156)

valid for both oblate and prolate spheroids. A model of heating and evaporation of mono-component spherical and spheroidal (oblate and prolate) droplets in a gaseous quiescent environment, considering the effect of non-uniform distribution of temperature at the droplet surface, is described in [159]. The species conservation equations were solved analytically in spherical and spheroidal coordinate systems. The energy equation was solved numerically in these systems. It was shown that for moderate differences between the temperatures at the droplet pole and equator (less than 20◦ ), the effect of curvature of the droplet surface

3.6 Heating and Evaporation of Spheroidal Droplets

149

is more important than the temperature difference. In this case, the expression for the heat transfer coefficient obtained for the uniform temperature case (Expression (3.155)) can still be used with acceptable errors, at least for certain substances (e.g. water and n-dodecane). Note that the earlier mentioned model developed in [31] for spherical droplets could be extended, in analytic form, to any droplet shape for which an analytical solution to the Laplace equation with uniform Dirichlet boundary conditions is available. An example for the case of a spheroidal droplet was shown in [31].

3.6.3 The Coupled Liquid/Gas Model (Spheroidal Droplets) A coupled liquid/gas model for heating and evaporation of spheroidal droplets was developed by the authors of [183]. The analytical solutions to the heat and mass transfer equations for the gas phase surrounding a spheroidal droplet, described in the previous section, were used as boundary conditions for the solutions to these equations in the liquid phase. The temperature gradients inside and at the surface of the droplets and the changes in their shape during the heating and evaporation process were considered, although the droplet shape was assumed to be close to that of a sphere. The effects of surface tension and droplet motion on droplet heating and evaporation were ignored. The following heat transfer equation in the liquid phase was used [183]: ρl cl

∂T − ∇(kl ∇T ) = 0, ∂t

(3.157)

where kl , ρl and cl are liquid thermal conductivity, density and specific heat capacity, respectively. This equation was solved subject to the following boundary condition: − n(−kl ∇T ) = qev + h(T∞ − Ts )

at

ξ = ξ0 ,

(3.158)

where n is the unit vector normal to the droplet surface, the convective heat transfer coefficient h is defined by (3.155) and qev is the heat flux due to evaporation to be specified later. The decrease in the droplet size due to evaporation was considered but not the effect of thermal swelling. The shape of the droplet was recalculated at each time step assuming that the droplet remains spheroidal (for isothermal droplets, this can be proven rigorously; see Appendix I for the details). The changes in the sizes of the droplet along and perpendicular to the z-axis in time were estimated as ar (t)

 1 dm˙ ev  =− , ρl d A u=π/2

az (t)

 1 dm˙ ev  =− , ρl d A u=0

(3.159)

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3 Heating and Evaporation of Mono-component Droplets

where evaporation mass flux ddmA˙ is allowed to change along the droplet surface. The following initial conditions for the solution to (3.159) were used: ar (0) = ar 0 , az (0) = az0 .

(3.160)

The temperature and vapour density, ρv , at the droplet surface were linked by the ideal gas law. It was assumed that ρv∞ = 0. The normal velocity of the recession of the surface of the evaporating droplet vn was estimated as follows (see Appendix J for the details): vn = vr n r + vz n z ,

(3.161)

vr = r (r 2 ar /ar3 + z 2 az /az3 ),

vz = z(r 2 ar /ar3 + z 2 az /az3 ),

(3.162)

n r = r/ar2 r 2 /ar4 + z 2 /az4 ,

n z = z/az2 r 2 /ar4 + z 2 /az4 .

(3.163)

The model was applied to the analysis of heating and evaporation of an n-dodecane droplet in Diesel engine-like conditions. The effect of droplet non-sphericity on its heating and evaporation was shown to be relatively weak for droplets with initial eccentricities in the range from 2/3 to 1.5. The results of further studies of the problem discussed in [156] are presented in [157]. In that paper, the evaporation process of a liquid spheroidal droplet was investigated, accounting for the oscillation between oblate and prolate states. The exact solutions to the heat and mass transfer equations derived in [156] were extended to investigate the effect of oscillation on droplet evaporation under the assumption of quasi-steady-state conditions. The results were compared with the predictions of the approximate model [91, 92]. The validity of the quasi-steady-state assumption, on which the study of [157] was based, was investigated. The model described in [157] was able to capture various evaporating mechanisms from oblate and prolate droplets. It predicted an increase in the average evaporation and heat transfer rates due to droplet oscillations, of up to 20% for a maximum excess surface area equal to 100%. The model was shown to be valid for small, highly volatile liquid droplets, evaporating in gas at high temperatures. In [160], it was shown that evaporation leads to an increase in the frequency of droplet oscillations.

3.6.4 Miscellaneous Models The results of the generalisation of the model presented in [157] to the case of triaxial ellipsoidal droplets are described in [158]. In that paper, an analytical model for heat and mass transfer from deformed droplets was developed, using the solutions to the species and energy conservation equations under steady-state conditions.

3.6 Heating and Evaporation of Spheroidal Droplets

151

Explicit expressions for the vapour mass fraction and temperature distribution, the local vapour and heat fluxes and evaporation rates were derived. It was demonstrated that the droplet deformation enhances both the total and local mass and heat transfer. The evaporation rate from deformed droplets, having the same volume and surface, was shown to be maximal for the prolate droplet and minimal for the oblate droplet, while intermediate values of evaporation rate were found for triaxial ellipsoidal droplets. For these droplet shapes, the local vapour flux was found to be proportional to the fourth root of the surface curvature. The results of numerical analysis of fluid flow and heat transfer from heated spheroids are presented in [124]. The governing thermal boundary layer equations for the rotating prolate and oblate spheroids were derived in [138]. Flow and heat transfer of merging and bouncing non-evaporating droplets were studied using Direct Numerical Simulation in [151]. A coupled level-set and volume-of-fluid (CLSVOF) method was used to capture the topology of the droplets. The problem of heating and evaporation of non-spherical droplets in homogeneous gas is complementary to the problem of heating and evaporation of spherical droplets in non-homogeneous gas. When the heating and evaporation take place in the presence of a temperature gradient in ambient gas, the well-known Marangoni effect (the mass transfer along an interface between two fluids due to a gradient of the surface tension) can be observed. The results of the investigation of this effect on droplet heating and evaporation have been described in many papers, including [7, 13, 51, 123]. Asymmetric liquid–liquid droplet heating in a laminar boundary layer was considered in [167]. The analysis of these authors focused on the influences of Weber, Prandtl and Reynolds numbers on the system evolution. They performed simulations using a coupled Eulerian–Lagrangian interface capturing methodology, alongside a Eulerian solver for the Navier–Stokes equations. They predicted the spatial and temporal evolution of the temperature and velocity fields in the droplet and the surrounding fluid.

3.7 Effect of Droplet Support In most experiments with heated and evaporating droplets, these droplets are placed at the end of a holder (support). General and side views and scheme of a droplet supported by a hollow metal rod in the experiments described in [147] are shown in Fig. 3.8. As follows from the experimental observations described in [147], a droplet’s support plays an important role in its heating and evaporation and needs to be considered in modelling. A rigorous approach to the modelling of droplet heating in the presence of this support would need to be based on three-dimensional modelling of the fluid dynamics/heat transfer problem, considering both the droplet and its support. Although possible in principle, this approach is not viable in most practical

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3 Heating and Evaporation of Mono-component Droplets

Fig. 3.8 General and side views and scheme of a droplet supported by a hollow metal rod. Reprinted from [147], Copyright Elsevier (2018)

applications. Following [147], we focus on a much simpler model which allows us to obtain a less accurate but simple and reliable estimate of this effect. We ignore the effect of the support on the shape and surface area of the droplet (e.g. [164]). This assumption has been widely used in previous studies and its validity is inferred from the experimental observations described in [147]. The results of detailed experimental and theoretical investigations of the effect of the support on the shape of the droplet are presented in [180]. The rate at which heat is supplied by the support to the droplet is estimated using the following simple expression [147]: q=

kl (Tsup − Tc ) Sc , Rd

(3.164)

where kl is the thermal conductivity of liquid at the average droplet temperature, Tc the temperature at the centre of the droplet, Tsup the surface temperature of the supporting cylindrical pipe (rod), measured experimentally [147], Rd the droplet radius and Sc the contact area between the droplet and pipe estimated as follows [147]: (3.165) Sc = π dsr Rd , where dsr is the outer diameter of the supporting rod (see [180] for more accurate estimate of this parameter). All parameters in (3.164) are time dependent and this equation can be applied during short time steps. The value Tc follows from Expression (2.89). It is assumed that heat supplied to droplets through the supporting rod is homogeneously and instantaneously distributed throughout the whole droplet volume. In this case, heat supplied per unit droplet volume can be estimated as follows [147]: qv =

3kl (Tsup − Tc ) Sc . 4π Rd4

(3.166)

3.7 Effect of Droplet Support

153

This allows us to formally consider the additional effect of heating droplets via fibres (rods) in terms of an additional source term in Solution (2.89) by assuming that P(R) =

3kl (Tsup − Tc ) Sc . 4π cl ρl Rd4

(3.167)

The results of the application of this approach to modelling heating and evaporation of suspended water droplets are described in [147]. The predictions based on this approach can be compared with the experimental correlation   for the dependence of the droplet evaporation rate, defined as K e = −4d Rd2 /dt, on the cross-section area of the supporting fibre (proportional to Sc ) presented in [26]. It can be shown that in the absence of fibre K e is constant when droplet surface and gas temperatures are constant (the heating-up period has been completed). It follows from (3.164) that in the presence of fibre in the same conditions, K e should be proportional to Sc /Rd2 . Thus, the effect of the fibre should reduce for larger droplets. In the correlations presented in [26, 166], K e is proportional to Sc . This is consistent with the predictions of the model described above when K e is considered for given Rd . Expression (3.167) does not include the properties of the material the support is made from. Thus, it cannot be used for analysing experimentally observed effects of these properties on droplet evaporation (e.g. [115]). These properties affect the values of Tsup but this effect is not considered in the book. A comprehensive overview of supports used in various experiments with droplets with initial diameters in the range 0.40–2.78 mm is presented in [178] (see Table 1 in that paper).

3.8 Modelling Versus Experimental Data The validation of the Abramzon and Sirignano model, presented in Sect. 3.2.2, has been discussed in many papers, including [111, 112]. In most cases, the validation of this model was focused on comparing predicted and observed time evolution of droplet diameters d. This model predicts the linear dependence of the time evolution of droplet diameters squared d 2 after the completion of the heat-up period if the ambient vapour pressure is fixed and the droplet is stationary or slow moving (cf. Sect. 3.2.6). This makes this time evolution particularly convenient for comparison with experimental data. In the comparison presented in [112], the most visible deviation between the experimental data and predicted time evolutions of d 2 was observed at the final stage of droplet evaporation. This was attributed to the contribution of ambient vapour pressure which was not considered in the version of the model which the authors of [112] used. The validation of the models considering the gradients of temperature inside droplets is not easy. As mentioned earlier, the first (to the best of the author’s knowledge) experimental evidence of the need to consider temperature gradients in Diesel

154

3 Heating and Evaporation of Mono-component Droplets

fuel droplets was demonstrated by Bertoli and Migliaccio [14]. These authors compared pressure traces in a Diesel engine predicted by the models considering and not considering the effects of the above-mentioned temperature gradients and demonstrated that the prediction of the former model agrees better with experimental data than that of the latter one. This result, however, can provide only indirect support for the model, and not its proper experimental validation.

3.8.1 Monodisperse Droplet Stream The first direct experimental validation of the models described in this chapter, considering the effects of temperature gradient inside droplets, was provided by a series of experiments performed at the Université de Lorraine [39, 77, 88–90], to the best of the author’s knowledge. In these experiments, linear mono-disperse droplet streams were produced by the Rayleigh disintegration of a liquid jet undergoing vibrations generated in a piezoelectric ceramic (see Fig. 3.9). The voltage applied to the piezoceramic was a square wave, whose amplitude determined the position of the break-up zone for a given fuel at a given temperature. The fuel was pre-heated in the injector by externally heated circulating water. The temperature of the fuel was measured exactly at the injection point with a K-type thermocouple. For specific frequencies of forced mechanical vibration, the liquid jet broke up into almost equally spaced and mono-sized droplets [49]. By adjusting the liquid flow rate and the piezoceramic frequency, it was possible to increase the droplet spacing up to about 6 times the droplet diameter. This, however, was accompanied by a modification of droplet sizes.

Fig. 3.9 Generation of a mono-dispersed stream of droplets. Reprinted from [90], Copyright Elsevier (2008)

3.8 Modelling Versus Experimental Data

155

Downstream distance from the injector x was converted into time t using the space evolution of the droplet velocity:  t= 0

x

dx . Udrop (x)

(3.168)

Droplets were injected into a quiescent atmosphere at room temperature. Their temperatures were measured using the technique described in [89, 90]. The input parameters for the models were the initial droplet temperature (assumed to be homogeneous), ambient gas temperature (assumed to remain constant at any specific position x), the distance parameter (ratio of the distances between droplets and their diameters) and the droplet velocities (assumed to remain constant at any specific position x). The analysis of this section is focused on pure acetone and pure ethanol droplets, although several other substances and their mixtures were studied (see [73, 90]). The measured time evolution of the droplet velocities in the axial direction was close to a linear function. The relevant approximations of the experimental results were obtained in [39] and are reproduced in Table 3.2. Figure 3.10 demonstrates a comparison between experimentally observed velocities for acetone droplets and their approximation presented in Table 3.2. As can be seen from this figure, the linear approximation of the experimental data, shown in Table 3.2, is reasonably accurate for practical applications. The same agreement was obtained for ethanol droplets.

Table 3.2 Approximations of acetone and ethanol droplet velocities (t is measured from the moment of injection) Substance Approximation of Udrop in m/s (t is in ms) Acetone Ethanol

Fig. 3.10 Experimentally observed velocities for pure acetone droplets (triangles). These were approximated by the straight line Ud = 12.81 − 0.316 t taken from Table 3.2 (solid line). Reprinted from [133], Copyright Elsevier (2010)

12.81 – 0.316 t 12.30 – 0.344 t

156

3 Heating and Evaporation of Mono-component Droplets

Table 3.3 The measured initial values of droplet temperature and diameter, ambient gas temperature and initial distance parameter for the same cases as in Table 3.2 Substance Droplet Diameter Gas temperature Distance temperature parameter Acetone Ethanol

35.1 ◦ C 38.0 ◦ C

21.5 ◦ C 22.0 ◦ C

143.4 µm 140.8 µm

7.7 7.1

The measured initial values of droplet temperature and diameter, ambient gas temperature and initial distance parameter C (ratio of the distance between droplets to their diameters) for the same cases as in Table 3.2 are presented in Table 3.3. The changes in C from the previous to the current time step were considered using the following expression: Udrop, new Rd, old Cnew = Cold , (3.169) Udrop, old Rd, new where subscripts new and old refer to the values of variables at the previous time step and one time step behind, respectively. In this case, the values of Rd, old and Rd, new are known at the current time step. Note that the changes in C during the experiments were observed in all cases and cannot be ignored, as was done in [133]. Parameter C was used for quantifying the effect of interaction between droplets on the values of Nusselt and Sherwood numbers (Nu and Sh) [133]: η(C) =

  Sh Nu 1 − exp [−0.13(C − 6)] . = = 1 − 0.57 1 − Shiso Nuiso 1 + exp [−0.13(C − 6)]

(3.170)

Note a mistake in the corresponding formula presented in [133] which was corrected in [39]. Other approximations for η(C) are described in [28, 44, 47, 79]. As demonstrated in [35], the above-mentioned ratios of Sherwood and Nusselt numbers are controlled not only by the distance parameter but also by the nature of the evaporation process of a given substance. To consider the latter effect, the following dimensionless time t ∗ was introduced: t∗ =

δf , vr

(3.171)

where f is the frequency of droplet production (in Hz) (set up for each particular experiment and directly linked with the distance parameter), δ the film thickness, which is different for mass and thermal boundary layers (δ M and δT ), and vr the radial velocity of the vapour with density ρv released at the droplet surface estimated as    m˙ d  .  (3.172) vr =  4πρv Rd2 

3.8 Modelling Versus Experimental Data

157

Thus, parameter t ∗ considers the inputs of both the distance parameter (via f ) and the volatility of the substance (via vr ). The values for δ M and δT for mono-component droplets were estimated using Eq. (3.41). The definitions of these thicknesses and vr were generalised to the case of multi-component droplets, via introduction of the average density and mass averaged values of transport coefficients. Using the concept of t ∗ , Deprédurand et al. [35] Nu for acetone, ethanol, 3-pentanone, nsuggested the following correlations for Nu iso Sh decane, n-dodecane and n-heptane, and Shiso for ethanol, 3-pentanone and n-heptane Nu ( ShShiso for acetone was approximated by the same correlation as for Nu ): iso Nu 1 − AT = + AT , Nuiso (G T t ∗ + 1)C T

(3.173)

1 − AM Sh = + AM , Shiso (G M t ∗ + 1)C M

(3.174)

where the coefficients in these equations are presented in Table 1 of [135]. Although Formulae (3.173) and (3.174) were derived for a limited number of substances, they are applicable for a wider range of substances and their mixtures (e.g. [135]). The authors of [24] solved the Navier–Stokes and vapour transport equations for periodical arrangements of droplets. The results of their calculations were approximated by the following correlations for Sherwood and Nusselt numbers [24]:   058+γ δ

z Re Sc ln (1 + B M ) κ +α , Sh = BM 2Rd FMC

(3.175)

  058+γ δ

z Re ln (1 + BT ) Pr κ +α , BT 2Rd FT C

(3.176)

Nu = where

FMC(T C)

  0.655 ln 1 + B M(T )  = 1 + B M(T ) , B M(T )

(3.177)

B M(T ) is Spalding mass (heat) transfer numbers; Re, Sc and Pr are Reynolds, Schmidt and Prandtl numbers, respectively, and z is the distance from the current droplet to the leading droplet in the row. Coefficients α, β, γ , κ and δ in (3.175) and (3.176) were presented in tabular forms as the functions the distance parameter C. The expressions for FMC(T C) are close to the corresponding expressions for FM(T ) (see Expressions (3.42)), but the exponent 0.7 in the formulae for FM(T ) was replaced with 0.655. The latter value of the exponent allowed the authors of [24] to improve the quality of the estimations of the unknown parameters in Eqs. (3.175) and (3.176). Correlations (3.175) and (3.176) with appropriate values of the coefficients were

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shown to be able to approximate the numerical results with errors not more than 1.415% for 1.1 ≤ C ≤ 18. An alternative approach to considering the effect of interaction between droplets on their heating and evaporation is discussed in [30]. These authors suggested an analytical model to describe heat and mass transfer processes from neighbouring spherical droplets considering the dependence of the gas density on temperature and composition. The conservation equations for mono-component non-identical spherical droplets evaporating in quiescent gas were solved in a bi-spherical coordinate system. Analytical expressions for the local heat and mass fluxes on the surfaces of the droplets were obtained. The results were compared with those obtained using the assumption of constant gas density for various evaporating species and operating conditions. Comparison of the results with those obtained for isolated droplets allowed the authors to quantify the screening effect on droplet heating and evaporation. The results of the generalisation of the analysis by the authors of [30] to an arbitrary number of interacting droplets floating in a gaseous environment are presented in [32]. As in the case of [30], the analysis of [32] was restricted to steady-state conditions. The temperature dependence of the thermophysical properties of the gaseous mixture was considered, but not the processes in the liquid phase. The most rigourous approach to considering the effects of the interaction between droplets would require Direct Numerical Simulation (DNS) (e.g. [85, 165]). In the experiments performed at Université de Lorraine (see Fig. 3.9), droplet temperatures were measured using a technique based on two-colour laser-induced fluorescence (LIF). The possibility of measuring the droplet temperature using LIF was discussed in [21, 77]. The liquid in the injector was initially seeded with a very low concentration of a fluorescent dye, and a laser beam, having a wavelength tuned to the absorption band of the dye, illuminated the droplets downstream of the injector. The fluorescent emissions from two different spectral bands with different temperature sensitivities were used. The ratio of the intensities of these two bands allowed the authors to eliminate the effects of parameters that are unknown or difficult to control, such as variations in laser intensity, tracer concentration and measurement volume during the acquisitions [77]. At the same time, this ratio depends on the temperature, making possible the temperature measurement. Although fluorescent tracers like Rhodamine B and Kitton red are well known for the temperature sensitivity of their fluorescent signal and their high quantum yield, the fluorescence spectrum of these molecules can vary significantly with the composition of the liquid mixture in which they are dissolved. To account for this effect, a special method was developed, using the inclusion of a third spectral band [88]. The probe volume (intersection between the laser beams and the detection field of view) was about 150 × 150 × 1200 µm3 . It was larger than the droplet in order to provide a global excitation of the whole droplet volume. The signal was averaged over the total time of droplet transit in the probe volume. Possible errors in these measurements are discussed in [77, 88]. The measurement method requires calibration and a reference temperature which can lead to systematic errors. These errors, however, are not expected to be significant. Calibration was performed over an extended range of temperatures. The reference value used to

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convert the fluorescence ratio into temperature was obtained from a thermocouple placed at the exit of the injector, which limits the risk of bias. The uncertainty related to the statistical convergence of the data was small (typically the RMS did not exceed 0.5 ◦ C). Given the periodicity of the droplet streams, the fluorescence signal could be averaged over thousands of droplets; this was achieved easily through acquisition times of several seconds. The overall errors of the technique were estimated to be about ±1.2 ◦ C [77, 88]. In the presence of thermal gradients inside droplets, the temperatures obtained using the fluorescence signal could differ from the volume-averaged droplet temperatures. This is related to the fact that the laser intensity (and thus the fluorescence field) is not distributed uniformly inside the droplet, mainly due to light refraction at the droplet surface. Also, the receiving optical equipment collected the fluorescent signal from the droplet interior with a spatial distortion. The optical system was designed to obtain a measurement volume much larger than the droplet, which reduces this adverse effect (this refers to the size of the laser beam and focal lengths of the lenses). Nevertheless, even if the measurements allowed the authors to estimate the volume-averaged droplet temperature, there was systematically more weighting in the zones near the centre of the droplet than at its edge. A complete modelling of the interactions between the laser beams and the droplet would be required to account for this effect. The evaluation of the differences between the volume-averaged droplet temperature and the measured temperature would also be related to the preconceived temperature distribution within the droplet (i.e. the local temperature gradients). Examples of such calculations are presented in [21, 23]. Geometrical optics and ray tracing were used in combination with a two-dimensional description of heat transfer inside the droplet. One-dimensional models were used. These models may be good at predicting the droplet heating in terms of volume-averaged temperature but they are inappropriate for estimating the local gradients. It will be assumed that the values of the measured temperatures are between the temperatures at the centre of the droplets and the average droplet temperatures, being closer to the droplet average temperatures than to the temperatures at the centre of the droplets. In what follows, the focus is on the comparison of the results of calculations using the analytical solution to the heat transfer Eq. (2.1) (Eq. (2.41)) inside droplets (Solution A), those based on the numerical solutions to Eq. (2.1) (Solution B) and experimental data. The code for the numerical solution of Eq. (2.1) was developed by Castanet et al. [22]. The effects of the movement of the droplet surface due to evaporation and thermal swelling/contraction during individual time steps are not considered in both solutions. To capture these effects in the validation process, more refined experiments would be required. Only one-way solutions were used, so the effects of droplets on gas were ignored. The latter effects would modify the results slightly, but would not change the trends of the predicted results [135]. The effects of interaction between droplets were considered using Expression (3.170). The plots of the time dependence of the surface, average and centre temperatures (Ts , Tav and Tc ) for acetone droplets, obtained using Solutions A and B, and experimentally observed average droplet temperatures are presented in Fig. 3.11. The calculations started with the first observed droplets approximately 1 ms after

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3 Heating and Evaporation of Mono-component Droplets

Fig. 3.11 The time evolution of droplet surface, average and centre temperatures (Ts , Tav and Tc ), predicted by Solutions A and B, and experimentally observed temperatures for acetone droplets with the values of the initial parameters, droplet velocity and gas temperature presented in Tables 3.2 and 3.3. Reproduced from Fig. 3 of [39] with permission of Begell House

the start of injection. At earlier times, the liquid fuel formed an unstable jet whose temperature evolution cannot be predicted based on the models described earlier. It was assumed that there is no initial temperature gradient inside the droplets. As can be seen in Fig. 3.11, all three temperatures are well separated. Hence, the difference between them needs to be considered in the analysis of experimental data. The results predicted by Solutions A and B coincide within the accuracy of plotting, which gives us confidence in the correctness of both solutions. The observed temperatures are close to the predicted average temperatures. In the case of ethanol (figure is not presented), the observed temperature values are close to or below the surface temperature of the droplets. The reason for this rather poor quantitative agreement between the predicted and observed temperatures for ethanol is not clear to us. Detailed experimental measurements of temperatures inside droplets, using the setup shown in Fig. 3.9, were described by Castanet et al. [23]. The temperature field within the droplets was measured using the two-colour laser-induced fluorescence technique. Experiments were undertaken on droplets made of various substances including acetone, ethanol, 3-pentanone, n-heptane, n-decane and n-dodecane which have different thermophysical properties. In some cases, the isotherms appeared circular and concentric suggesting that thermal conduction is the dominant mechanism of heat transfer in the droplets. In other cases, measurements showed rather significant temperature differences between the leading and the trailing edges of the moving droplets. In the latter case, the experimental results cannot be interpreted using simplified models described in this section, and the effects of recirculation inside droplets (Hill vortex) need to be considered. This was done in the numerical solution described in [23].

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3.8.2 Suspended Droplets Results of the analysis of the temperature fields in suspended droplets of water, kerosene, Diesel fuel and petroleum oil (petrol) are described and discussed by the authors of [148]. In the experiments described in this paper, single droplets were heated in a flow of hot air. The initial droplet radii were between 1 and 2 mm, air temperatures were between 20 ◦ C and 500 ◦ C, and velocities of air flow were in the range of 3–3.5 m/s. The experimental results presented in [148] were interpreted using the Effective Thermal Conductivity (ETC) model described in Chap. 2. In what follows, this interpretation is presented following [148]. In the approach described in this paper, it was assumed that droplets are spherical, and the effects of the droplet support were ignored. The version of ANSYS Fluent with the model based on Solution (2.41) to the transient heat transfer equation at each time step implemented into it was used. This implementation is described in [122]. The multi-component compositions of kerosene, Diesel fuel and petroleum oil were ignored, and they were approximated by n-decane (C10 H22 ), n-dodecane (C12 H26 ) and iso-octane (C8 H18 ), respectively. Fifty terms in the series in Expression (2.41) and time steps 1 ms were used; 100 cells along the droplet radius were used to calculate integrals for the parameters used in the above-mentioned series. Eigenvalues λn were found using the bisection method with absolute accuracy of 10−6 . The droplet initial temperatures were assumed equal to 300 K. The time period between the introduction of the droplets into the chamber and the first measurements was assumed equal to 1.25 ± 0.1 s. Air temperature and velocity were assumed equal to 300 ◦ C and 3 m/s, respectively. The plots of surface temperatures Ts versus time for water, n-decane, n-dodecane and iso-octane droplets are shown in Fig. 3.12. The experimentally observed values of Ts are shown in the same figure. The initial droplet radii for water, kerosene, Diesel fuel and petroleum oil droplets, observed experimentally and used in modelling, were 1.54 ± 0.024 mm, 1.16 ± 0.024 mm, 1.175 ± 0.024 mm and 1.12 ± 0.024 mm, respectively. At every time instant, the temperatures were measured at seven points at the droplet surface. Temperatures at these points could differ up to 15–20 ◦ C. The comparison between modelling and experimental results was focused both on individual measurements at these points (empty circles) and on the average temperatures (filled stars). As can be seen in Fig. 3.12, at the initial stage of droplet heating, the ETC model predicts higher droplet temperatures than those predicted by the ITC model. This result is expected as in this case the heat supplied to the droplet surface is spent only on heating the near-surface area of the droplet in the ETC model; in the ITC model, it is spent on heating the whole droplet. The agreement between modelling and experimental results is rather poor in all cases although the model predicts correct trends in the evolution of surface temperatures. This could be attributed to both the limitations of the model and the uncertainty

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3 Heating and Evaporation of Mono-component Droplets

Fig. 3.12 Plots of droplet surface temperatures versus time predicted by the Effective Thermal Conductivity (ETC) model (solid curves) and Infinite Thermal Conductivity (ITC) model (dashed curve). Empty circles refer to the actual experimental results, while filled stars show the averaged values. Parts a, b, c and d refer to water, kerosene, Diesel fuel and petrol fuel droplets, respectively. The initial radii for them were 1.54 mm, 1.16 mm, 1.175 mm and 1.12 mm, respectively. Reproduced from [148], Copyright Elsevier

of the experimental data. The latter is particularly visible in the case of water droplets. The application of the ETC model instead of the ITC model does not lead to better agreement between the model predictions and experimental data. A more detailed discussion of the reasons for the difference between the modelling and experimental results is given in [148]. The authors of [144] applied the Effective Thermal Conductivity (ETC) model based on Solution (2.41) to the transient heat transfer equation inside the droplet and the Abramzon and Sirignano model for droplet evaporation to the analysis of suspended water droplets. The model presented in Sect. 3.7 was used to describe the effect of the support. In their experiments, water droplets with initial diameters close to 2 mm were suspended on an asbestos thread with a diameter of 0.3 mm. The experiments were performed in a dry air flow with velocities 1–3 m/s and temperatures 26 ◦ C–29 ◦ C.

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Fig. 3.13 a Tg − Ts for pure water versus time. Symbols refer to experimental data (Case 1 (squares): gas velocity 1 m/s, initial droplet diameter 2.47 mm; Case 2 (circles): gas velocity 1.5 m/s, initial droplet diameter 2.31 mm; Case 3 (triangles): gas velocity 2 m/s, initial droplet diameter 2.42 mm; Case 4 (diamonds): gas velocity 3 m/s, initial droplet diameter 2.67 mm; gas and initial droplet temperatures were 299 K in all cases). Solid (dashed) curves refer to predictions of the models for the same values of input parameters considering (ignoring) the effect of the thread. The thickest (thinnest) curves refer to Case 1(4). b Zoomed part referring to the initial stage of the process. Reproduced from [144], Copyright Elsevier

The experimentally measured and predicted values of the differences between ambient gas temperature Tg (constant during the experiments) and droplet surface temperature Ts are shown in Fig. 3.13 for four cases. As can be seen from this figure, for all gas velocities, the observed Tg − Ts initially rises very quickly and then stays at about the same level of around 15 K. Similar behaviours of Tg − Ts are predicted by the models which consider and ignore the contribution of the thread. The values predicted by the model in which the effects of the thread are considered are visibly closer to the experimental data than those predicted by the model ignoring the thread. The predictions of the latter model are about the same for all gas velocities. The model in which the contribution of the thread is considered predicts slight decreases in Ts

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3 Heating and Evaporation of Mono-component Droplets

Fig. 3.14 The same as Fig. 3.13, but for (d/d0 )1.5 . Reproduced from [144], Copyright Elsevier

after t greater than about 50 s; this is attributed to a reduction in droplet sizes due to evaporation, which is not clearly visible in the experimental data. Keeping in mind that in all cases in the experiments described in [144] the droplet Reynolds numbers (Re) were greater than 100 and less than 400, the authors of [144] took into account that Expressions (3.70) and (3.71) are valid during most of the droplet heating and evaporation process. This allowed the authors of [144] to describe the time evolution of droplet diameters as the normalised droplet diameters

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(d/d0 )1.5 versus time. The time evolutions of experimentally observed and predicted values of this ratio, for the same input parameters as in Fig. 3.13, are presented in Fig. 3.14. As can be seen in Fig. 3.14, both observed and predicted values of (d/d0 )1.5 decrease almost linearly with time, as expected. The rate of this decrease visibly increases when flow velocity increases, from 1 to 1.5 m/s, which was also expected. No clear decrease in this rate was seen in the experiments described in [144] when air velocity was further increased. In all cases, the contribution of the initial heat-up stage can be ignored. The predictions of the model which considers the effect of the thread better match the experimental values of (d/d0 )1.5 . Note that the predicted values of (d/d0 )1.5 tend to be larger than those observed experimentally, especially for larger values of air velocity. The models used for the analysis of mono-component droplet heating and evaporation described in this chapter assume that vapour in the vicinity of the droplet surface is saturated. Hence, the rate of droplet evaporation was equal to the rate of vapour diffusion from its surface to ambient gas. These are known as the hydrodynamic models of droplet evaporation. The analysis started with empirical correlations which are not directly linked with any evaporation model. Then classical hydrodynamic models of droplet evaporation were described. All of these models assumed that the droplet’s radius remains constant during the time step but changes from one time step to another due to droplet thermal swelling and evaporation. Then the effects of droplet radii change during individual time steps on the heating process were investigated. Approaches to modelling heating and evaporation of spheroidal droplets were presented. Previously developed tools for modelling radiative heating of droplets were adapted to modelling the effects of support on droplet heating and evaporation. Comparisons with experimental results were described.

References 1. Abou Al-Sood, M., & Birouk, M. (2008). Droplet heat and mass transfer in a turbulent hot airstream. International Journal of Heat and Mass Transfer, 51, 1313–1324. 2. Abramzon, B., & Sazhin, S. S. (2005). Droplet vaporization model in the presence of thermal radiation. International Journal of Heat and Mass Transfer, 48, 1868–1873. 3. Abramzon, B., & Sazhin, S. S. (2006). Convective vaporization of fuel droplets with thermal radiation absorption. Fuel, 85, 32–46. 4. Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. International Journal of Heat and Mass Transfer, 32, 1605–1618. 5. Aggarwal, S. K., & Chitre, S. (1991). Computations of turbulent evaporating sprays. Journal of Propulsion and Power, 7, 213–220. 6. Alassar, R. S. (2005). Forced convection past an oblate spheroid at low to moderate Reynolds numbers. ASME Journal of Heat Transfer, 127, 1062–1070. 7. Albernaz, D. L., Amberg, G., & Do-Quang, M. (2016). Simulation of a suspended droplet under evaporation with Marangoni effects. International Journal of Heat and Mass Transfer, 97, 853–860.

166

3 Heating and Evaporation of Mono-component Droplets

8. Anani, K., Prud’homme, R., & Hounkonnou, M. N. (2018). Dynamic response of a vaporizing spray to pressure oscillations: Approximate analytical solutions. Combustion and Flame, 193, 295–305. 9. Archambault, M. R. (2010). A maximum entropy approach to modeling the dynamics of a vaporizing spray. Atomization and Sprays, 20, 1017–1031. 10. Atkins, P., & de Paula, J. (2002). Atkins’ Physical Chemistry (7th ed.). Oxford: Oxford University Press. 11. Awasthi, I., Gogos, G., & Sundararajan, T. (2013). Effects of size on combustion of isolated methanol droplets. Combustion and Flame, 160, 1789–1802. 12. Azimi, A., Arabkhalaj, A., Ghassemi, H., & Markadeh, R. S. (2017). Effect of unsteadiness on droplet evaporation. International Journal of Thermal Sciences, 120, 354–365. 13. Barash, L. (2016). Marangoni convection in an evaporating droplet: Analytical and numerical descriptions. International Journal of Heat and Mass Transfer, 102, 445–454. 14. Bertoli, C., & Migliaccio, M. (1999). A finite conductivity model for diesel spray evaporation computations. International Journal of Heat and Fluid Flow, 20, 552–561. 15. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena. Chichester: Wiley. 16. Birouk, M., & Gökalp, I. (2002). A new correlation for turbulent mass transfer from liquid droplets. International Journal of Heat and Mass Transfer, 45, 37–45. 17. Brenn, G. (2005). Concentration fields in evaporating droplets. International Journal of Heat and Mass Transfer, 48, 395–402. 18. Bykov, V., Goldfarb, I., Gol’dshtein, V., & Greenberg, J. B. (2002). Thermal explosion in a hot gas mixture with fuel droplets: A two reactants model. Combustion Theory and Modelling, 6, 1–21. 19. Caldwell, J., & Kwan, Y. Y. (2003). On the perturbation method for the Stephan problem with time-dependent boundary conditions. International Journal of Heat and Mass Transfer, 46, 1497–1501. 20. Caldwell, J., Savovi´c, S., & Kwan, Y. Y. (2003). Nodal integral and finite difference solution of one-dimensional Stefan problem. ASME Journal of Heat Transfer, 125, 523–527. 21. Castanet, G., Lavieille, P., Lebouché, M., & Lemoine, F. (2003). Measurement of the temperature distribution within monodisperse combusting droplets in linear stream using two color laser-induced fluorescence. Experiments in Fluids, 35, 563–571. 22. Castanet, G., Maqua, C., Orain, M., Grisch, F., & Lemoine, F. (2007). Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: Comparison with modelling. International Journal of Heat and Mass Transfer, 50, 3670–3683. 23. Castanet, G., Labergue, A., & Lemoine, F. (2011). Internal temperature distributions of interacting and vaporizing droplets. International Journal of Thermal Sciences, 50, 1181–1190. 24. Castanet, G., Perrin, L., Caballina, O., & Lemoine, F. (2016). Evaporation of closely-spaced interacting droplets arranged in a single row. International Journal of Heat and Mass Transfer, 93, 788–802. 25. Chauveau, C., Birouk, M., & Gökalp, I. (2011). An analysis of the d 2 −law departure during droplet evaporation in microgravity. International Journal of Multiphase Flow, 37, 252–259. 26. Chauveau, C., Birouk, M., Halter, F., & Gökalp, I. (2019). An analysis of the droplet support fiber effect on the evaporation process. International Journal of Heat and Mass Transfer, 128, 885–891. 27. Chiang, C. H., Raju, M. S., & Sirignano, W. A. (1992). Numerical analysis of convecting, vaporizing fuel droplet with variable properties. International Journal of Heat and Mass Transfer, 35, 1307–1324. 28. Chiang, C. H., & Sirignano, W. A. (1993). Interacting convecting vaporising fuel droplets with variable properties. International Journal of Heat and Mass Transfer, 36, 875–886. 29. Cossali, G. E., & Tonini, S. (2018). Modelling the effect of variable density and diffusion coefficient on heat and mass transfer from a single component spherical drop evaporating in high temperature air streams. International Journal of Heat and Mass Transfer, 118, 628–636.

References

167

30. Cossali, G. E., & Tonini, S. (2018). Variable gas density effects on transport from interacting evaporating spherical drops. International Journal of Heat and Mass Transfer, 127, 485–496. 31. Cossali, G. E., & Tonini, S. (2019). An analytical model of heat and mass transfer from liquid drops with temperature dependence of gas thermo-physical properties. International Journal of Heat and Mass Transfer, 138, 1166–1177. 32. Cossali, G. E., & Tonini, S. (2020). Analytical modelling of drop heating and evaporation in drop clouds: Effect of temperature dependent gas properties and cloud shape. International Journal of Heat and Mass Transfer, 162, 120315. 33. Cossali, G. E., & Tonini, S. (2021). Drop Heating and Evaporation: Analytical Solutions in Curvilinear Coordinate Systems. Berlin: Springer. 34. Crua, C., Shoba, T., Heikal, M., Gold, M., & Higham, C. (2010). High-speed microscopic imaging of the initial stage of Diesel spray formation and primary breakup. SAE Report, 2010-01-2247 35. Deprédurand, V., Castanet, G., & Lemoine, F. (2010). Heat and mass transfer in evaporating droplets in interaction: Influence of the fuel. International Journal of Heat and Mass Transfer, 53, 3495–3502. 36. Desantes, J. M., López, J. J., García, J. M., & Pastor, J. M. (2007). Evaporative Diesel spray modelling. Atomization and Sprays, 17, 193–231. 37. Dombrovsky, L. A., & Sazhin, S. S. (2003). A simplified non-isothermal model for droplet heating and evaporation. International Communications in Heat and Mass Transfer, 30, 787– 796. 38. Ebrahimian, V., & Habchi, C. (2011). Towards a predictive evaporation model for multicomponent hydrocarbon droplets at all pressure conditions. International Journal of Heat and Mass Transfer, 54, 3552–3565. 39. Elwardany, A. E., Gusev, I. G., Castanet, G., Lemoine, F., & Sazhin, S. S. (2011). Mono- and multi-component droplet cooling/heating and evaporation: Comparative analysis of numerical models. Atomization and Sprays, 21, 907–931. 40. Elwardany, A. E., & Sazhin, S. S. (2012). A quasi-discrete model for droplet heating and evaporation: Application to Diesel and gasoline fuels. Fuel, 97, 685–694. 41. El Wakil, M. M., Uyehara, O. A., & Myers, P. S. (1954). A theoretical investigation of the heating-up period of injected fuel droplets vaporizing in air. NACA Technical Note, 3179. 42. Faeth, G. M. (1977). Current status of droplet and liquid combustion. Progress in Energy and Combustion Science, 3, 191–224. 43. Faeth, G. M. (1983). Evaporation and combustion of sprays. Progress in Energy and Combustion Science, 9, 1–76. 44. Fieberg, C., Reichelt, L., Martin, D., Renz, U., & Kneer, R. (2009). Experimental and numerical investigation of droplet evaporation under diesel engine conditions. International Journal of Heat and Mass Transfer, 52, 3738–3746. 45. Finneran, J. (2021). On the evaluation of transport properties for droplet evaporation problems. International Journal of Heat and Mass Transfer, 181, 121858. 46. Finneran, J., Garner, C. P., & Nadal, F. (2021). Deviations from classical droplet evaporation theory. Proceedings of the Royal Society A, 477, 20210078. 47. Frackowiak, B., Lavergne, G., Tropea, C., & Strzelecki, A. (2010). Numerical analysis of the interactions between evaporating droplets in a monodisperse stream. International Journal of Heat and Mass Transfer, 53, 1392–1401. 48. Frank-Kamenetskii, D. A. (1969). Diffusion and Heat Transfer in Chemical Kinetics. New York: Plenum Press. 49. Frohn, A., & Roth, N. (2000). Dynamics of Droplets. Berlin: Springer. 50. Fuchs, N. A. (1959). Evaporation and Droplet Growth in Gaseous Media. London: Pergamon Press. 51. Ghata, N., & Shaw, B. D. (2014). Computational modeling of the effects of support fibers on evaporation of fiber-supported droplets in reduced gravity. International Journal of Heat and Mass Transfer, 77, 22–36.

168

3 Heating and Evaporation of Mono-component Droplets

52. Goldfarb, I., Gol’dshtein, V., Kuzmenko, G., & Greenberg, J. B. (1998). On thermal explosion of a cool spray in a hot gas. In Proceedings of the 27th International Symposium on Combustion, Colorado, USA (Vol. 2, pp. 2367–2374). 53. Goldfarb, I., Gol’dshtein, V., Kuzmenko, G., & Sazhin, S. S. (1999). Thermal radiation effect on thermal explosion in gas containing fuel droplets. Combustion Theory and Modelling, 3, 769–787. 54. Goldfarb, I., Sazhin, S. S., & Zinoviev, A. (2004). Delayed thermal explosion in flammable gas containing fuel droplets: Asymptotic analysis. International Journal of Engineering Mathematics, 50, 399–414. 55. Grow, D. A. (1990). Heat and mass transfer to an elliptical particle. Combustion and Flame, 80, 209–213. 56. Gupta, S. C. (2003). The Classical Stefan Problem: Basic Concepts, Modelling and Analysis. Amsterdam: Elsevier. 57. Haywood, R. J., Nafziger, R., & Renksizbulut, M. (1989). A detailed examination of gas and liquid transient processes in convection and evaporation. ASME Journal of Heat Transfer, 111, 495–502. 58. He, X., Feng, H., Liu, Z., Wang, H., Li, X., Zeng, F., Fe, L., & Liu, F. (2020). Numerical simulation of the evaporation characteristics of a dimethyl ether droplet in supercritical environment. Fuel, 267, 117120. 59. Hohmann, S., & Renz, U. (2003). Numerical simulation of fuel sprays at high ambient pressure: The influence of real gas effects and gas solubility on droplet vaporisation. International Journal of Heat and Mass Transfer, 46, 3017–3028. 60. Incropera, F. P., & DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer. New York: Wiley. 61. Javed, I., Baek, S. W., Waheed, K., Ali, G., & Cho, S. O. (2013). Evaporation characteristics of kerosene droplets with dilute concentrations of ligand-protected aluminum nanoparticles at elevated temperatures. Combustion and Flame, 160, 2955–2963. 62. Jog, M. A. (1997). Transient heat transfer to a spheroidal liquid drop suspended in an electric field. International Journal of Heat and Fluid Flow, 18, 411–418. 63. Johnson, M. V., Zhu, G. S., Aggarwal, S. K., & Goldsborough, S. S. (2010). Droplet evaporation characteristics due to wet compression under RCM conditions. International Journal of Heat and Mass Transfer, 53, 1100–1111. 64. Juncu, G. (2010). Unsteady heat transfer from an oblate/prolate spheroid. International Journal of Heat and Mass Transfer, 53, 3483–3494. 65. Kang, S. J., Dehdashti, E., & Masoud, H. (2019). Conduction heat transfer from oblate spheroids and bispheres. International Journal of Heat and Mass Transfer, 139, 115–120. 66. Kartashov, E. M. (2001). Analytical Methods in the Heat Transfer Theory in Solids. Moscow: Vysshaya Shkola. (in Russian). 67. Kim, H., & Sung, N. (2003). The effect of ambient pressure on the evaporation of a single droplet and a spray. Combustion and Flame, 135, 261–270. 68. Kim, H., Baek, S. W., & Chang, D. (2014). A single n-heptane droplet behavior in rapid compression machine. International Journal of Heat and Mass Transfer, 69, 247–255. 69. Kishore, N., & Gu, S. (2011). Momentum and heat transfer phenomena of spheroid particles at moderate Reynolds and Prandt l numbers. International Journal of Heat and Mass Transfer, 55, 2595–2601. 70. Klingsporn, M., & Renz, U. (1994). Vaporization of a binary unsteady spray at high temperature and high pressure. International Journal of Heat and Mass Transfer, 37(Suppl. 1), 265–272. 71. Kolaitis, D. I., & Founti, M. A. (2006). A comparative study of numerical models for EulerianLagrangian simulations of turbulent evaporating sprays. International Journal of Heat and Fluid Flow, 27, 424–435. 72. Kontogeorgis, G. M., & Folas, G. K. (2010). Thermodynamic Models for Industrial Applications. Chippenham (UK): Wiley.

References

169

73. Kristyadi, T., Deprédurand, V., Castanet, G., Lemoine, F., Sazhin, S. S., Elwardany, A., Sazhina, E. M., & Heikal, M. R. (2010). Monodisperse monocomponent fuel droplet heating and evaporation. Fuel, 89, 3995–4001. 74. Kulmala, M., Vesala, T., Schwarz, J., & Smolik, J. (1995). Mass transfer from a drop-II. Theoretical analysis of temperature dependent mass flux correlation. International Journal of Heat and Mass Transfer, 38, 1705–1708. 75. Kuo, K.-K. (1986). Principles of Combustion. Chichester: Wiley. 76. Labowsky, M., Rosner, D. E., & Arias-Zugasti, M. (2011). Turbulence effects on evaporation rate-controlled spray combustor performance. International Journal of Heat and Mass Transfer, 54, 2683–2695. 77. Lavieille, P., Lemoine, F., Lebouché, M., & Lavergne, G. (2001). Evaporating and combusting droplet temperature measurement using two color laser induced fluorescence. Experiments in Fluids, 31, 45–55. 78. Lefebvre, A. H. (1989). Atomization and Sprays. Bristol: Taylor & Francis. 79. Li, C., Lv, Q., Wu, Y., Wu, X., & Tropea, C. (2020). Measurement of transient evaporation of an ethanol droplet stream with phase rainbow refractometry and high-speed microscopic shadowgraphy. International Journal of Heat and Mass Transfer, 146, 118843. 80. Li, J., & Zhang, J. (2014). A theoretical study of the spheroidal droplet evaporation in forced convection. Physics Letters A, 378(47), 3537–3543. 81. Lima, D. R., Farias, S. N., & Lima, G. B. (2004). Mass transport in spheroids using the Galerkin method. Brazilian Journal of Chemical Engineering, 21, 667–680. 82. Liu, A., Chen, J., Wang, Z., Mao, Z.-S., & Yang, C. (2019). Unsteady conjugate mass and heat transfer from/to a prolate spheroidal droplet in uniaxial extensional creeping flow. International Journal of Heat and Mass Transfer, 134, 1180–1190. 83. Liu, Y., Yu, J., Chen, L., & Jin, S. (2018). Theoretical investigation and experimental verification of a mathematical model for counter-flow spray separation tower. International Journal of Heat and Mass Transfer, 120, 316–327. 84. Luijten, C. C. M., & Kurvers, C. (2010). Real gas effects in mixing-limited Diesel spray vaporization models. Atomization and Sprays, 20, 595–609. 85. Lupo, G., Gruber, A., Brandt, L., & Duwig, C. (2020). Direct numerical simulation of spray droplet evaporation in hot turbulent channel flow. International Journal of Heat and Mass Transfer, 160, 120184. 86. Mandal, D. K., & Bakshi, S. (2012). Internal circulation in a single droplet evaporating in a closed chamber. International Journal of Multiphase Flow, 42, 42–51. 87. Mao, C.-P., Szekely, G. A., & Faeth, G. M. (1980). Evaluation of a locally homogeneous flow model of spray combustion. Journal of Energy, 4, 78–87. 88. Maqua, C., Castanet, G., Lemoine, F., Doué, N., & Lavergne, G. (2006). Temperature measurements of binary droplets using three-color laser-induced fluorescence. Experiments in Fluids, 40, 786–797. 89. Maqua, C., Castanet, G., & Lemoine, F. (2008). Bi-component droplets evaporation: Temperature measurements and modelling. Fuel, 87, 2932–2942. 90. Maqua, C., Castanet, G., Grisch, F., Lemoine, F., Kristyadi, T., & Sazhin, S. S. (2008). Monodisperse droplet heating and evaporation: Experimental study and modelling. International Journal of Heat and Mass Transfer, 51, 3932–3945. 91. Mashayek, F. (2001). Dynamics of evaporating drops. Part 1: Formulation and evaporation model. International Journal of Heat and Mass Transfer, 44, 1517–1526. 92. Mashayek, F. (2001). Dynamics of evaporating drops. Part II: Free oscillations. International Journal of Heat and Mass Transfer, 44(8), 1527–1541. 93. Maxwell, J. B. (1950). Data Book on Hydrocarbons: Application to Process Engineering. New York: D. van Nostrand Company INC. 94. McIntosh, A. C., Gol’dshtein, V., Goldfarb, I., & Zinoviev, A. (1998). Thermal explosion in a combustible gas containing fuel droplets. Combustion Theory and Modelling, 2, 153–165. 95. Michaelides, E. (2006). Particles, Bubbles and Drops. New Jersey: World Scietific.

170

3 Heating and Evaporation of Mono-component Droplets

96. Miliauskas, G., & Garmus, V. (2009). The peculiarities of hot liquid droplets heating and evaporation. International Journal of Heat and Mass Transfer, 52, 3726–3737. 97. Mitchell, S. L., Vynnycky, M., Gusev, I. G., & Sazhin, S. S. (2011). An accurate numerical solution for the transient heating of an evaporating spherical droplet. Applied Mathematics and Computation, 217, 9219–9233. 98. Mukhopadhyay, A., & Sanyal, D. (2005). A semi-analytical model for evaporating fuel droplets. ASME Journal of Heat Transfer, 127, 199–203. 99. Myers, T. G. (2010). Optimal exponent heat balance and refined integral methods applied to Stefan problems. International Journal of Heat and Mass Transfer, 53, 1119–1127. 100. Naaktgeboren, C. (2007). The zero-phase Stefan problem. International Journal of Heat and Mass Transfer, 50, 4614–4622. 101. Nafziger, R. (1988). Convective Droplet Transport Phenomena in High Temperature Air Streams. M.A.Sc: Thesis, University of Waterloo, Ontario, Canada. 102. Ni, Z., Hespel, C., Han, K., & Foucher, F. (2021). Numerical simulation of heat and mass transient behavior of single hexadecane droplet under forced convective conditions. International Journal of Heat and Mass Transfer, 167, 120736. 103. Niven, C. (1880). On the conduction of heat in ellipsoids of revolution. Philosophical Transactions of the Royal Society of London, 171, 117–151. 104. Noh, D., Gallot-Lavallée, S., Jones, W. P., & Navarro-Martinez, S. (2018). Comparison of droplet evaporation models for a turbulent, non-swirling jet flame with a polydisperse droplet distribution. Combustion and Flame, 194, 135–151. 105. Pal, M. K. (2019). Effect of ambient fuel vapour concentration on the liquid length and air-fuel mixing of an evaporating spray. Fuel, 256, 115945. 106. Park, S., & Son, G. (2020). Numerical investigation of acoustically-triggered droplet vaporization in a tube. International Journal of Heat and Mass Transfer, 148, 119029. 107. Park, T. W., Aggarwal, S. K., & Katta, V. R. (1995). Gravity effects on the dynamics of evaporating droplets in a heated jet. Journal of Propulsion and Power, 11, 519–528. 108. Paschedag, A. R., Piarah, W. H., & Kraume, M. (2005). Sensitivity study for the mass transfer at a single droplet. International Journal of Heat and Mass Transfer, 48, 3402–3410. 109. Pathak, A., & Raessi, M. (2018). Steady-state and transient solutions to drop evaporation in a finite domain: Alternative benchmarks to the d2 − law. International Journal of Heat and Mass Transfer, 127, 1147–1158. 110. Peng, D.-Y., & Robinson, D. B. (1976). A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15, 59–64. 111. Pinheiro, A., & Vedovoto, J. M. (2019). Evaluation of droplet evaporation models and the incorporation of natural convection effects. Flow, Turbulence and Combustion, 102, 537–558. 112. Pinheiro, A., Vedovoto, J. M., da Silveira, Neto A., & van Wachem, B. G. M. (2019). Ethanol droplet evaporation: Effects of ambient temperature, pressure and fuel vapor concentration. International Journal of Heat and Mass Transfer, 143, 118472. 113. Poblador-Ibanez, J., & Sirignano, W. A. (2018). Transient behavior near liquid-gas interface at supercritical pressure. International Journal of Heat and Mass Transfer, 126, 457–473. 114. Rachih, A., Legendre, D., Climent, E., & Charton, S. (2020). Numerical study of conjugate mass transfer from a spherical droplet at moderate Reynolds number. International Journal of Heat and Mass Transfer, 157, 119958. 115. Radhakrishnan, S., Srivathsan, N., Anand, T. N. C., & Bakshi, S. (2019). Influence of the suspender in evaporating pendant droplets. International Journal of Thermal Sciences, 140, 368–376. 116. Rayapati, N. P., Panchagnula, M. V., Peddieson, J., Short, J., & Smith, S. (2011). Eulerian multiphase population balance model of atomizing, swirling flows. International Journal of Spray and Combustion Dynamics, 3111–136. 117. Rayapati, N. P., Panchagnula, M. V., & Peddieson, J. (2013). Application of population balance model to combined atomization and evaporation processes in dense sprays. Atomization and Sprays, 23, 505–523.

References

171

118. Reid, R. C., Prausnitz, J. M., & Sherwood, T. K. (1977). The Properties of Gases and Liquids (3rd ed.). New York: McGraw-Hill. 119. Renksizbulut, M., & Yuen, M. C. (1983). Experimental study of droplet evaporation in a high-temperature air stream. ASME Journal of Heat Transfer, 105, 384–388. 120. Renksizbulut, M., & Yuen, M. C. (1983). Numerical modeling of droplet evaporation in a high-temperature air stream. ASME Journal of Heat Transfer, 105, 389–397. 121. Richter, A., & Nikrityuk, P. A. (2012). Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal particles in cross flow at sub-critical Reynolds numbers. International Journal of Heat and Mass Transfer, 55, 1343–1354. 122. Rybdylova, O., Al Qubeissi, M., Braun, M., et al. (2016). A model for droplet heating and its implementation into ANSYS Fluent. International Communications in Heat and Mass Transfer, 76, 265–270. 123. Samareh, B., Mostaghimi, J., & Moreau, C. (2014). Thermocapillary migration of a deformable droplet. International Journal of Heat and Mass Transfer, 73, 616–626. 124. Sasmal, C., & Nirmalkar, N. (2016). Momentum and heat transfer characteristics from heated spheroids in water based nanofluids. International Journal of Heat and Mass Transfer, 96, 582–601. 125. Saufi, A. E., Frassoldati, A., Faravelli, T., & Cuoci, A. (2021). Interface-resolved simulation of the evaporation and combustion of a fuel droplet suspended in normal gravity. Fuel, 287, 119413. 126. Savovi´c, S., & Caldwell, J. (2003). Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. International Journal of Heat and Mass Transfer, 46, 2911–2916. 127. Sazhin, S. S. (2006). Advanced models of fuel droplet heating and evaporation. Progress in Energy and Combustion Science, 32(2), 162–214. 128. Sazhin, S. S., & Heikal, M. R. (2012). Droplet heating and evaporation - Recent results and unsolved problems. Computational Thermal Sciences, 4(6), 485–496. 129. Sazhin, S. S., Feng, G., Heikal, M. R., Goldfarb, I., Goldshtein, V., & Kuzmenko, G. (2001). Thermal ignition analysis of a monodisperse spray with radiation. Combustion and Flame, 124, 684–701. 130. Sazhin, S. S., Krutitskii, P. A., Abdelghaffar, W. A., Mikhalovsky, S. V., Meikle, S. T., & Heikal, M. R. (2004). Transient heating of diesel fuel droplets. International Journal of Heat and Mass Transfer, 47, 3327–3340. 131. Sazhin, S. S., Abdelghaffar, W. A., Krutitskii, P. A., Sazhina, E. M., & Heikal, M. R. (2005). New approaches to numerical modelling of droplet transient heating and evaporation. International Journal of Heat and Mass Transfer, 48, 4215–4228. 132. Sazhin, S. S., Abdelghaffar, W. A., Sazhina, E. M., & Heikal, M. R. (2005). Models for droplet transient heating: Effects on droplet evaporation, ignition, and break-up. International Journal of Thermal Sciences, 44, 610–622. 133. Sazhin, S. S., Elwardany, A., Krutitskii, P. A., Castanet, G., Lemoine, F., Sazhina, E. M., & Heikal, M. R. (2010). A simplified model for bi-component droplet heating and evaporation. International Journal of Heat and Mass Transfer, 53, 4495–4505. 134. Sazhin, S. S., Krutitskii, P. A., Gusev, I. G., & Heikal, M. R. (2010). Transient heating of an evaporating droplet. International Journal of Heat and Mass Transfer, 53, 2826–2836. 135. Sazhin, S. S., Elwardany, A., Krutitskii, P. A., Deprédurand, V., Castanet, G., Lemoine, F., Sazhina, E. M., & Heikal, M. R. (2011). Multi-component droplet heating and evaporation: Numerical simulation versus experimental data. International Journal of Thermal Sciences, 50, 1164–1180. 136. Sazhin, S. S., Krutitskii, P. A., Gusev, I. G., & Heikal, M. (2011). Transient heating of an evaporating droplet with presumed time evolution of its radius. International Journal of Heat and Mass Transfer, 54, 1278–1288. 137. Sazhin, S. S., Al Qubeissi, M., & Xie, J.-F. (2014). Two approaches to modelling the heating of evaporating droplets. International Communications in Heat and Mass Transfer, 57, 353–356.

172

3 Heating and Evaporation of Mono-component Droplets

138. Shah, R., & Li, T. (2018). The thermal and laminar boundary layer flow over prolate and oblate spheroids. International Journal of Heat and Mass Transfer, 121, 607–619. 139. Shanthanu, S., Raghuram, S., & Raghavan, V. (2013). Transient evaporation of moving water droplets in steam-hydrogen-air environment. International Journal of Heat and Mass Transfer, 64, 536–546. 140. Shinjo, J., & Umemura, A. (2019). Fluid dynamic and autoignition characteristics of early fuel sprays using hybrid atomization LES. Combustion and Flame, 203, 313–333. 141. Sirignano, W. A., & Law, C. K. (1978). Transient heating and liquid phase mass diffusion in droplet vaporization. In J. T. Zung (Ed.), Evaporation-Combustion of Fuels. Advances in Chemistry Series (Vol. 166, pp. 1–26). Washington, DC: American Chemical Society. 142. Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science, 27(6), 1197–1203. 143. Sobac, B., Talbot, P., Haut, B., Rednikov, A., & Colinet, P. (2015). A comprehensive analysis of the evaporation of a liquid spherical drop. Journal of Colloid and Interface Science, 438, 306–317. 144. Starinskaya, E. M., Miskiv, N. B., Nazarov, A. D., Terekhov, V. V., Terekhov, V. I., Rybdylova, O. D., & Sazhin, S. S. (2021). Evaporation of water/ethanol droplets in an air flow: Experimental study and modelling. International Journal of Heat and Mass Transfer, 177, 121502. 145. Sreenivasulu, B., & Srinivas, B. (2015). Mixed convection heat transfer from a spheroid to a Newtonian fluid. International Journal of Thermal Sciences, 87, 1–18. 146. Sreenivasulu, B., Srinivas, B., & Ramesh, K. (2014). Forced convection heat transfer from a spheroid to a power law fluid. International Journal of Heat and Mass Transfer, 70, 71–80. 147. Strizhak, P. A., Volkov, R. S., Castanet, G., Lemoine, F., Rybdylova, O., & Sazhin, S. S. (2018). Heating and evaporation of suspended water droplets: Experimental studies and modelling. International Journal of Heat and Mass Transfer, 127, 92–106. 148. Strizhak, P. A., Volkov, R. S., Antonov, D. V., Castanet, G., & Sazhin, S. S. (2020). Application of the laser induced phosphorescence method to the analysis of temperature distribution in heated and evaporating droplets. International Journal of Heat and Mass Transfer, 127, 92– 106. 149. Strotos, G., Malgarinos, I., Nikolopoulos, N., & Gavaises, M. (2016). Predicting the evaporation rate of stationary droplets with the VOF methodology for a wide range of ambient temperature conditions. International Journal of Thermal Sciences, 109, 253–262. 150. Talbot, P., Sobac, B., Rednikov, A., Colinet, P., & Haut, B. (2016). Thermal transients during the evaporation of a spherical liquid drop. International Journal of Heat and Mass Transfer, 97, 803–817. 151. Talebanfard, N., Nemati, H., & Boersma, B. J. (2019). Heat transfer in deforming droplets with a direct solver for a coupled level-set and volume of fluid method. International Communications in Heat and Mass Transfer, 108, 104272. 152. Terekhov, V. I., Terekhov, V. V., Shishkin, N. E., & Bi, K. C. (2010). Heat and mass transfer in disperse and porous media: Investigation of non-stationary evaporation of liquid droplets. Journal of Engineering Physics and Thermophysics, 83(5), 883–890. 153. Tian, J., Zhao, T., Zhou, Z., Chen, B., Wang, J., & Xiong, J. (2021). Isolated moving charged droplet evaporation characteristics in electrostatic field with highly volatile R134a. International Journal of Heat and Mass Transfer, 178, 121583. 154. Tikhonov, A. N., & Samarsky, A. A. (1972). Equations of Mathematical Physics. Moscow: Nauka Publishing House. (in Russian). 155. Tonini, S., & Cossali, G. E. (2012). An analytical model of liquid drop evaporation in gaseous environment. International Journal of Thermal Sciences, 57, 45–53. 156. Tonini, S., & Cossali, G. E. (2013). An exact solution of the mass transport equations for spheroidal evaporating drops. International Journal of Heat and Mass Transfer, 60, 236–240. 157. Tonini, S., & Cossali, G. E. (2014). An evaporation model for oscillating spheroidal drops. International Communications in Heat and Mass Transfer, 51, 18–24.

References

173

158. Tonini, S., & Cossali, G. E. (2016). One-dimensional analytical approach to modelling evaporation and heating of deformed drops. International Journal of Heat and Mass Transfer, 97, 301–307. 159. Tonini, S., & Cossali, G. E. (2018). Modelling of heat and mass transfer from spheroidal drops with non-uniform surface temperature. International Journal of Heat and Mass Transfer, 121, 747–758. 160. Tonini, S., Varma RajaKochanattu, G., & Cossali, G. E. (2020). The effect of evaporation on the oscillation frequency of an inviscid liquid drop. International Communications in Heat and Mass Transfer, 116, 104609. 161. Verwey, C., & Birouk, M. (2018). Fuel vaporization: Effect of droplet size and turbulence at elevated temperature and pressure. Combustion and Flame, 189, 33–45. 162. Vlasov, V. A. (2021). On a theory of mass transfer during the evaporation of a spherical droplet. International Journal of Heat and Mass Transfer, 178, 121597. 163. Voller, V. R., & Falcini, F. (2013). Two exact solutions of a Stefan problem with varying diffusivity. International Journal of Heat and Mass Transfer, 58, 80–85. 164. Vu, T. V. (2018). Deformation and breakup of a pendant drop with solidification. International Journal of Heat and Mass Transfer, 122, 341–353. 165. Wang, B., Kronenburg, A., Tufano, G. L., & Stein, O. T. (2018). Fully resolved DNS of droplet array combustion in turbulent convective flows and modelling for mixing fields in inter-droplet space. Combustion and Flame, 189, 347–366. 166. Wang, J., Huang, X., Qiao, X., Ju, D., & Sun, C. (2020). Experimental study on effect of support fiber on fuel droplet vaporization at high temperatures. Fuel, 268, 117407. 167. Wenzel, E., Kulacki, F., & Garrick, S. (2016). Modeling and simulation of liquid-liquid droplet heating in a laminar boundary layer. International Journal of Heat and Mass Transfer, 97, 653–661. 168. Wu, J.-S., Hsu, K.-H., Kuo, P.-M., & Sheen, H.-J. (2003). Evaporation model of a single hydrocarbon fuel droplet due to ambient turbulence at intermediate Reynolds numbers. International Journal of Heat and Mass Transfer, 46, 4741–4745. 169. Wu, J.-S., Liu, Y.-J., & Sheen, H.-J. (2001). Effects of ambient turbulence and fuel properties on the evaporation rate of single droplets. International Journal of Heat and Mass Transfer, 44, 4593–4603. 170. Xu, G., Ikegami, M., Honma, S., Ikeda, K., Ma, X., Nagaishi, H., Dietrich, D. L., & Struk, P. M. (2003). Inverse influence of the initial diameter on droplet burning rate in cold and hot ambiences: A thermal action of flame in balance with heat loss. International Journal of Heat and Mass Transfer, 46, 1155–1169. 171. Yang, K., Hong, F., & Cheng, P. (2014). A fully coupled numerical simulation of sessile droplet evaporation using Arbitrary Lagrangian-Eulerian formulation. International Journal of Heat and Mass Transfer, 70, 409–420. 172. Yang, J.-R., & Wong, S.-C. (2001). On the discrepancies between theoretical and experimental results for microgravity droplet evaporation. International Journal of Heat and Mass Transfer, 44, 4433–4443. 173. Yang, J.-R., & Wong, S.-C. (2002). An experimental and theoretical study of the effects of heat conduction through the support fiber on the evaporation of a droplet in a weakly convective flow. International Journal of Heat and Mass Transfer, 45, 4589–4598. 174. Yao, G. F., Abdel-Khalik, S. I., & Ghiaasiaan, S. M. (2013). An investigation of simple evaporation models used in spray simulations. ASME Journal of Heat Transfer, 240, 179– 182. 175. Yu, D., & Chen, Z. (2021). Theoretical analysis on droplet vaporization at elevated temperatures and pressures. International Journal of Heat and Mass Transfer, 164, 120542. 176. Zhang, K., Tontiwachwuthikul, P., Jia, N., & Li, S. (2019). Four nanoscale-extended equations of state: Phase behaviour of confined fluids in shale reservoirs. Fuel, 250, 88–97. 177. Zhang, P., Xu, Z., Wang, T., & Che, Z. (2021). A method to measure vapor concentration of droplet evaporation based on background oriented Schlieren. International Journal of Heat and Mass Transfer, 168, 120880.

174

3 Heating and Evaporation of Mono-component Droplets

178. Zhang, Y., Huang, R., Zhou, P., Huang, S., Zhang, G., Hua, Y., & Qian, Y. (2021). Numerical study on the effects of experimental parameters on evaporation characteristic of a droplet. Fuel, 293, 120323. 179. Zhifum, Z., Guoxiang, W., Bin, C., Liejin, G., & Yueshe, W. (2003). Evaluation of evaporation models for single moving droplet with a high evaporation rate. Powder Technology, 125, 95– 102. 180. Zhou, F., Zhuang, D., Lu, T., & Ding, G. (2020). Observation and modeling of droplet shape on metal fiber with gravity effect. International Journal of Heat and Mass Transfer, 161, 120294. 181. Zhou, Y., Wang, Y., & Bu, W. (2014). Exact solution for a Stefan problem with latent heat a power function of position. International Journal of Heat and Mass Transfer, 69, 451–454. 182. Zhou, Z.-F., Lu, G.-Y., & Chen, B. (2018). Numerical study on the spray and thermal characteristics of R404A flashing spray using OpenFOAM. International Journal of Heat and Mass Transfer, 117, 1312–1321. 183. Zubkov, V. S., Cossali, G. E., Tonini, S., Rybdylova, O., Crua, C., Heikal, M., & Sazhin, S. S. (2017). Mathematical modelling of heating and evaporation of a spheroidal droplet. International Journal of Heat and Mass Transfer, 108(B), 2181–2190.

Chapter 4

Heating and Evaporation of Multi-component Droplets

The droplet heating and evaporation models described so far have assumed that liquid consists of one-component only. This assumption is not valid for most practically important fuels, including Diesel, petrol and kerosene fuels. The application of the single-component assumption in this case was justified not by the physical nature of the problem but by the fact that it led to considerable simplification of the modelling of the processes involved. Early developments of the models of multi-component droplet heating and evaporation are reviewed in Sect. 4.1. Section 4.2 is focused on the new approach to the development of the Discrete Component (DC) model using the analytical solution to the component diffusion equation in the liquid phase. This approach is applied to the analysis of bi-component, biodiesel and kerosene droplets and to the analysis of the process of droplet drying. The quasi-discrete model, suitable for modelling heating and evaporation of droplets consisting of a large number of components belonging to the same group (e.g. alkanes), is described and analysed in Sect. 4.3. A generalisation of the quasi-discrete model to the case when various groups of components are present in the system is presented and discussed in Sect. 4.4. Gas phase evaporation models for multi-component droplets and other approaches to modelling these droplets are summarised in Sects. 4.5 and 4.6, respectively. Section 4.7 focuses on the modelling of heating and evaporation of multi-component liquid films (limiting case of prolate droplets).

4.1 Background In multi-component droplets, different components evaporate at different rates, leading to concentration gradients in the liquid phase. The latter leads to component diffusion described by the diffusion equation for the mass fractions of components. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_4

175

176

4 Heating and Evaporation of Multi-component Droplets

simplest form of this equation for stationary droplets, when only the radial diffusion is accounted for, is written as [107, 178]: ∂Yli = Dl ∂t



∂ 2 Yli 2 ∂Yli + ∂ R2 R ∂R

 ,

(4.1)

where subscripts l and i refer to the liquid phase and components, respectively. It is assumed that the diffusion coefficient Dl is the same for all liquid components. For moving droplets, the situation becomes much more complicated, and recirculation inside the droplets would need to be considered [178]. This, however, would make the whole model far too complicated for most engineering applications, as in the case of single-component droplets. The analysis of moving droplets can be greatly simplified when the actual recirculation inside them is accounted for by an increase of diffusivity in them by a factor like χT (see Eq. (2.82)) used for the analysis of heat transfer (with Pr replaced by Sc) [48]. This leads to replacing Dl in (4.1) with the so-called effective diffusivity defined as [178]: Deff = χY Dl ,

(4.2)

where the coefficient χY varies from about 1 (at PeY d(l) ≡ Red(l) Scl < 10) to 2.72 (at PeY d(l) > 500) and can be approximated as    χY = 1.86 + 0.86 tanh 2.245 log10 PeY d(l) /30 ,

(4.3)

where Scl = νl /Dl is the Schmidt number and νl is the liquid kinematic viscosity. The model based on Expressions (4.2) and (4.3) is known as the Effective Diffusivity (ED) model. Analysis of Eq. (4.1) together with the heat transfer equation, in which the contribution of thermal radiation is considered, was discussed in [106]. A simplified model, based on the assumption that the absorption of thermal radiation is homogeneous, was used. In most cases, including this monograph, the value of Dl is estimated using the laminar model. The effects of turbulence on Dl are discussed in [3]. An analysis of the diffusion equation for Yli without the spherical symmetry assumption is described in [25, 155, 179]. If we further assume that the diffusion coefficients referring to all components in the gas phase are the same (this assumption is relaxed in Sect. 4.5; cf. [66]), then the evaporation rate of the ith component can be presented as m˙ i = εi m˙ d = −4π εi Rd Dv ρtotal ln (1 + B M ) ,

(4.4)

where εi describes the component’s evaporation rate. If component concentrations in the ambient gas are equal to zero, the values of εi can be obtained from the following formula [37, 58, 168, 196]: Yvis , (4.5) εi =  i Yvis

4.1 Background

177

where subscript v refers to the vapour phase. When deriving Expression (4.4) Eq. (3.23) was used. The latent heat of evaporation of multi-component droplets follows from the expression: L=



εi L i ,

(4.6)

i

where L i is the latent heat of evaporation of the ith component. If there were no supply of the ith component to the surface of the droplet from its interior then: (4.7) Ylis = εi , where the additional subscript s refers to the ‘surface’. In the general case, this supply of the ith component can be described by the following component diffusion rate: m˙ i(suppl) =





. ∂ R R=Rd −0

∂Yli −4π Rd2 Dl ρtotal

(4.8)

In [67], Eq. (4.8) was generalised to consider the effect of change of droplet radius during the timestep. This effect is discussed later in this chapter, using an approach different from that used in [67]. Note that the analysis of evaporation of multi-component droplets by some authors (e.g. [70]) led to the following expression for m˙ d : m˙ d = −2π Rd Div ρtotal B Mi Shiso (i) ,

(4.9)

where B Mi is the component Spalding mass transfer number defined as B Mi =

Yvis − Yvi∞ , εi − Yvis

(4.10)

Div is the diffusion coefficient of the ith component in air, and Shiso (i) is defined by (3.37) replacing B M with B Mi . Remembering (4.5) one can see that B M = B Mi . Hence, for stationary droplets this leads to a paradox that the same value of m˙ d is predicted by Expression (4.9) for different Div . This paradox is resolved by the fact that, although Expression (4.9) is correct, the value of Shiso (i) cannot be approximated by the analogue of the corresponding equation presented in Chap. 3 (Eq. (3.37)) which implicitly assumes that the evaporating components do not affect each other. In our analysis it is implicitly assumed that Div are the same for all components (see [52, 182] for an in-depth discussion of this issue). We cannot agree with the assumptions made by some authors that the Spalding mass transfer number for a mixture is equal to a sum of Spalding mass transfer numbers for individual components (cf. [80]). In the steady-state case, m˙ i(suppl) is equal to the difference between the actual evaporation flux and the evaporation flux which would take place in the absence of supply. This allows us to write the components’ balance equation at the surface of

178

4 Heating and Evaporation of Multi-component Droplets

the droplet as:

Dv ρtotal ln (1 + B M ) ∂Yli

= (Yli − εi ) . ∂ R R=Rd −0 Dl ρl Rd

(4.11)

This equation was derived in [178] following a different procedure. It can be considered as a boundary condition for Eq. (4.1) at the droplet surface. This equation needs to be supplemented by the condition at the droplet centre:

∂Yli

=0 ∂ R R=0

(4.12)

Yli (t = 0) = Yli0 (R).

(4.13)

and the initial condition:

As for the equation for temperature inside the droplet, Condition (4.12) can be replaced by a more general requirement that Yli (R, t) is a continuously twice differentiable function at R ≤ Rd . Some authors (e.g. [163]) considered the effect of component diffusion and heat transfer inside droplets using a two-layer model: droplet core was assumed to be homogeneous and described by temperature Td and component mass fraction Yil , which are different from temperature and component mass fraction at the surface of the droplet. This model had to be based on the introduction of semi-empirical constants describing heat and component transfer processes in the droplet. It was expected to predict correct trends in time evolution of temperature and component mass fractions in the droplet but cannot be used for accurate quantitative description of the processes. In the equilibrium state, the partial pressure of the ith vapour component at the droplet’s surface can be found from the expression: ∗ , pvi = γi X lis pvi

(4.14)

∗ where X lis is the molar fraction of the ith liquid component at the droplet surface, pvi is the partial vapour pressure of the ith component when X lis = 1, γi is the activity coefficient. In some applications the latter coefficient is taken equal to 1. In this case Expression (4.14) leads to Raoult’s law: ∗ . pvi = X lis pvi

(4.15)

A molecular interpretation of Expression (4.15) is described in [17] (pp. 169–70). Deviations from the Raoult law are discussed in [19, 178]. Using the Clausius-Clapeyron equation for the ith component we can write [104]  ln

∗ pvi p



Li = R u Mi



1 1 − Tbi Ts

 ,

(4.16)

4.1 Background

179

where Mi is the molar mass of the ith component, Tbi is the boiling temperature of the ith component, and p is the ambient pressure. ∗ is equal to the ambient pressure When deriving Eq. (4.16) it was considered that pvi when Ts = Tbi . Using Eq. (4.15), Eq. (4.16) can be rewritten as [178]:    1 Li 1 X vi −1 , = p˜ v exp − X li Ru Mi Tbi Ts

(4.17)

where p˜ v = pv / p. The numerical solution of Eq. (4.1) can be too CPU intensive when more than two components are considered. This approach has been applied mainly to the case of bi-component droplets [25, 107, 134, 135]. One of relatively few exceptions is [138], the authors of which applied this approach to modelling heating and evaporation of nine-component blended Diesel droplets. Two limiting cases were described, in which the solution of this equation becomes unnecessary. These are (a) the rapid regression or zero-diffusivity limit and (b) the uniform concentration or infinite-diffusivity limit [62, 178]. In the first case, the composition of droplets remains constant during the whole evaporation process. The analysis of this case would be like the analysis of onecomponent droplets. In the second case, the droplet composition changes with time in such a way that the mass fraction of its less volatile components increases. The second limit is strictly applicable to the description of slow evaporation processes only [178] but it has been used in a wide range of applications (e.g. [62, 96, 129]). The model based on Eq. (4.1) is known as the Discrete Component (DC) model. This model is typically applicable only when the number of components in the droplets is small (e.g. [97, 200]). In realistic cases, such as Diesel or petrol fuels, when the number of components in a droplet is measured in hundreds, the usefulness of this approach becomes questionable. Alternative approaches are based on the probabilistic analysis of a large number of components (e.g. Continuous Thermodynamics approach [1, 15, 75, 116, 143, 161, 195, 230] and the Distillation Curve Model [27]). In these approaches, several additional simplifying assumptions have been used, including the assumption that components inside droplets mix infinitely quickly. The Continuous Thermodynamics approach is based on the introduction of the distribution function f m (I ) such that

I2

f m (I )dI = 1,

(4.18)

I1

where I is the property of the component (e.g. the molar mass), f m characterises relative contribution of the components having this property in the vicinity of I , I1 and I2 are limiting values of this property. f m (I ) has been approximated by relatively simple functions. For example, the following function was considered in [15, 116, 143, 230]:

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4 Heating and Evaporation of Multi-component Droplets

   I −γ (I − γ )α−1 , f m (I ) = Cm (I1 , I2 ) α exp − β Γ (α) β

(4.19)

where Γ (α) is the Gamma function, α and β are parameters that determine the shape of f m (I ) , γ determines the original shift, and Cm (I1 , I2 ) is the normalisation constant. Cm (I1 , I2 ) = 1 for I1 = γ and I2 = ∞. The authors of [78] used a more general function which was the weighted sum of the functions (4.19) (double-Gamma-PDF). The main attractive feature of this approach to modelling multi-component droplet heating and evaporation is that the composition of fuel can be characterised by just a few key parameters (e.g. parameters α and β for given γ in Function (4.19)), instead of hundreds of mole fractions of individual components, required in the approach based on the solution to Eq. (4.1). As in the conventional approach, it is assumed that the gas diffusion coefficients for all the components are the same (see [15, 76–78, 116, 143, 227, 230] for further details). In the Distillation Curve Model, the distillation curves of the actual multicomponent fuels are considered. The most important feature of this model is the description of fractional boiling during the droplet evaporation process as a function of a single variable: the mean molar mass of fuel inside the droplet. An additional advantage is that the model is based on algebraic equations, which brings advantages from the point of view of CPU efficiency. The influence of diffusion resistance in droplets on evaporation is described by the liquid Peclet number [27]: Pel =

|m˙ d | . 4π Rd Dl ρl

(4.20)

See [27] for further details. Models containing features of discrete and continuous models were described in [110–112, 176, 226]. In [226], a mixtures of multi-component (Diesel, petrol or biodiesel fuels) and mono-component substances were investigated using a combination of Discrete Components and Continuous Thermodynamics models. As in the Continuous Thermodynamics approach, it was assumed that the mixing processes inside droplets are infinitely fast for both components and temperature. The analysis of [110–112] was based on the application of the Quadrature Method of Moments (QMoM), originally developed in [105]. This method allows one to use 2 or 3 pseudo-components for each group of components instead of dozens of real components for the whole mixture. The normal boiling point of each pseudocomponent was allowed to change during the evaporation process. This approach was shown to be particularly useful if the condensation process needs to be considered alongside the evaporation. As in the Continuous Thermodynamics approach, it was assumed that the droplets are well mixed. A model, called the discrete/continuous multi-component (DCMC) model and focused primarily on petrol fuel, was described in [214]. In this model petrol was assumed to consist of five families of hydrocarbons: n-paraffins, i-paraffins, naphthenes, aromatics and olefins. Each family was described using a probability density function (PDF), and the mass fraction of each family of hydrocarbons; the mean

4.1 Background

181

and variance of each PDF were tracked. Compared with the DC model the DCMC model saves computer time. Compared with the continuous multi-component (CMC) model, the DCMC model has much higher accuracy. As in the Continuous Thermodynamics and combined approaches, described earlier, the droplets are assumed to be well mixed. The authors of [153] applied the Discrete Component (DC) model for investigation of mixtures of substances containing many components (Diesel and petrol fuels) by approximating these fuels with relatively small numbers of components (sixcomponents for Diesel fuel and seven components for petrol). They considered the effect of temperature gradient inside droplets but assumed that the component transfer processes inside droplets are infinitely fast. As mentioned earlier, the applicability of this assumption is questionable. One of the main limitations of the Discrete Component (DC) models, suggested so far, is that in most cases they are based on the numerical solution of Eq. (4.1) which can be a serious limitation when the models are implemented in a CFD code. The main limitation of the Continuous Thermodynamics and Distillation Curve approaches is that both models assume infinitely fast diffusivity of components inside droplets. These limitations were overcome in the version of the DC model, based on the analytical solution to Eq. (4.1). This version is described in Sect. 4.2, following [54, 168, 169].

4.2 Discrete Component Model The model for multi-component droplet heating and evaporation, developed in [73, 168, 169], is based on the same assumptions as the Discrete Component (DC) model, described in the previous section, with the only difference that the numerical solution to the component diffusion equation (4.1) is replaced with its analytical solution. This solution is used in the numerical code at each timestep of calculations. The temperature gradient and recirculation inside droplets are considered using the Effective Thermal Conductivity (ETC) model (see Expressions (2.81) and (2.82)) and the Effective Diffusivity (ED) model (see Expressions (4.2) and (4.3)), considering or not considering the effect of the moving interface during individual timesteps (see Sects. 3.4 and 4.2.2). To consider the effect of the moving interface in the DC model, a model using the assumption that the droplet radius is a linear function of time during each timestep was used (cf. Sect. 3.4). In what follows, the main ideas of the model developed in [73, 168, 169] and its applications are described.

4.2.1 An Analytical Solution for Rd = const Assuming that Rd = const, the analytical solution to Eq. (4.1) at each individual timestep, subject to boundary conditions (4.11)–(4.12) and initial condition (4.13)

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4 Heating and Evaporation of Multi-component Droplets

was obtained in the form [168, 169] (see Appendix K): 1 Yli = εi + R +

∞









exp Dl





exp −Dl

n=1

λn Rd

λ0 Rd

2    R t [qY i0 − Q Y 0 εi ] sinh λ0 Rd

2    R t [qY in − Q Y n εi ] sin λn , Rd

(4.21)

where λ0 and λn (n ≥ 1) are solutions to the equations: tanh λ0 = − tan λn = −  hY 0 = − 1 +

αm Rd Dl

QY n =



1 ||vY 0 ||2

1 ||vY n ||2

qY in =



 Rd λn

||2

Rd

||vY 0 || = 2

0

||vY n || = 0

n ≥ 1,

(4.23)

(1 + h Y 0 ) sinh λ0 when n = 0

(1 + h Y 0 ) sin λn

1 ||vY n

2

Rd λ0 2

Rd

Rd

when n ≥ 1,

RYli0 (R)vY n (R)d R,

n ≥ 0,

(4.24)

(4.25)

0

  R vY 0 (R) = sinh λ0 , Rd

2

(4.22)

,

⎧ ⎨− ⎩

λn hY 0

λ0 , hY 0

  R vY n (R) = sin λn , n ≥ 1, Rd  hY 0 Rd , 1+ 2 =− 2 h Y 0 − λ20

(4.26)

 hY 0 Rd , n ≥ 1, 1+ 2 = 2 h Y 0 + λ2n

(4.27)

vY2 0 (R)d R

vY2 n (R)d R

Ylis = Ylis (t) are liquid components’ mass fractions at the droplet’s surface, αm =

|m˙ d | , 4πρl Rd2

(4.28)

m˙ d is the droplet evaporation rate. Note that Expressions (4.26) and (4.27) include the term Rd (cf. the corresponding expression for ||vn ||2 used in Expression (2.41)).

4.2 Discrete Component Model

183

Since we are interested only in a solution which is continuously differentiable twice in the whole domain, Yli should be bounded for 0 ≤ R ≤ Rd . Moreover, the physical meaning of Yli , as the mass fraction, implies that 0 ≤ Yli ≤ 1. Note that Solutions (4.21) and (2.41) look rather different, although Eq. (4.1) has essentially the same structure as Eq. (2.1). The former equation can be obtained from the latter if we replace T with Yli , κ with Dl , Tg with εi , Ts with Ylis and kl / h with −Dl /αm . The reason for this is that Solution (2.41) is valid only for h 0 > −1. At the same time the boundary condition for Eq. (4.1) at R = Rd (Eq. (4.11)) implies that h Y 0 < −1. The condition αm = const is applicable when the timesteps are sufficiently small. Solution (4.21) has been used at each timestep in the numerical codes developed for the analysis of droplet heating and evaporation. For example, this solution and the corresponding solution to the heat transfer equation inside droplets (Solution (2.41)) were used by the authors of [224] for the analysis of heating and evaporation of ethanol/iso-octane droplets in conditions relevant to direct injection internal combustion engines. The code based on this solution has been implemented into CFD code ANSYS Fluent [162]. Solution (4.21) is essentially based on the assumption that Rd = const. This assumption is relaxed in the next section.

4.2.2 An Analytical Solution for Rd  = const To consider the effect of the moving interface due to evaporation, it was assumed that Rd (t) is a linear function of time defined by Expression (3.100). In this case, the solution to Eq. (4.1) for an individual timestep, subject to boundary conditions (4.11)–(4.12) and initial condition (4.13) can be presented as [73] (see Appendix L):

Yli =

αm εi exp



α Rd0 4Dl

αm + ∞



Rd0 Rd (t)−R 2 Rd (t)

 5/2

Rd0

α Rd0 2

5/2

Rd (t)

+

 1 α Rd0 R 2 × exp − 4Dl Rd (t) R Rd (t) √

  R Dl λ2n t sin λn + + f Y n μY 0 (0)] exp − Rd0 Rd (t) Rd (t) 

[qY n

n=1

 [qY 0 + f Y 0 μY 0 (0)] exp

  R Dl λ20 t sinh λ0 , Rd0 Rd (t) Rd (t)

(4.29)

where fY n

Rd = || vY n ||2

1 0

 f Y (r )vY n (r )dr =

Rd sinh λ0 ||vY 0 ||2 λ20 Rd − ||vY n ||2 λ2 sin λn n

when n = 0 when n ≥ 1

,

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4 Heating and Evaporation of Multi-component Droplets

qY n

Rd = || vY n ||2

1

WY 0 (r )vY n (r )dr, n ≥ 0,

0

f Y (r ) ≡ −r/(1 + h Y 0 ) =

∞

f Y n vY n (r ),

(4.30)

n=0

WY 0 (r ) =

∞

qY n vY n (r ),

(4.31)

n=0

   Rd (t)Rd (t) αm εi (Rd (t))5/2 μY 0 (t) ≡ − exp , Dl 4Dl

(4.32)



HY 0 (t) ≡ h Y 0 = −

R (t)Rd (t) αm Rd (t) − 1 − d = const < −1, Dl 2Dl

vY 0 (r ) = sinh (λ0 r ) ,

vY n (r ) = sin (λn r )

(4.33)

n ≥ 1,

λ0 and λn (n ≥ 1) are solutions to Eqs. (4.22) and (4.23), respectively, r = R/Rd (t). The norms ||vY 0 ||2 and ||vY n ||2 are defined by Expressions (4.26) and (4.27). Note that Solution (4.29) is presented as a function of R to make it easier to compare it with Solution (4.21), while the input parameters and functions in (4.29) are presented as functions of r following [73]. Solution (4.29) was obtained assuming that parameter h Y 0 is a constant and less than −1 during each timestep (it can vary from one timestep to another). In the general case of a time dependent h Y 0 , the solution was reduced to the solution of the Volterra integral equation of the second kind [73] (see Appendix L for the details). It is essential to retain both α and αm in Solution (4.29) to be able to compare the prediction of this solution with the prediction of the conventional approach to modelling component diffusion when α = 0 but αm = 0 (see Solution (4.21)). In the latter case, for Rd (t) = 0 Solution (4.29) reduces to Solution (4.21).

4.2.3 Bi-component Droplets This section focuses on the analysis of the bi-component droplet heating and evaporation using the models based on the analytical solutions to the component diffusion equation, described in Sects. 4.2.1 and 4.2.2. This analysis is performed separately for the models based on the assumptions that the droplet radius is constant during the timestep, droplet radius is a linear function of time during the timestep, but the droplet temperature is frozen (the process of component diffusion is separated from the process of droplet heating), droplet radius is a linear function of time during the timestep

4.2 Discrete Component Model

185

and both processes of component diffusion and droplet heating are considered. The modelling results are validated against experimental data where appropriate.

4.2.3.1

Heating and Evaporation when Rd = const

The Discrete Component (DC) model described in Sect. 4.2.1 was applied to the analysis of heating and evaporation of droplets of two mixtures. These are the mixture of acetone and ethanol and the mixture of water and ethanol. In what follows, the results referring to these two cases are presented and discussed. Droplets of Acetone and Ethanol The results predicted by the DC model based on the solution to the component diffusion equation presented in Sect. 4.2.1 were validated against the experimental data obtained using the same setup as shown in Fig. 3.9 but applied for the analysis of droplets of a mixture of acetone and ethanol. The results of this validation were described in [54, 168, 169]. In what follows they are reproduced following [54]. The analysis was based on Expressions (4.14) and (4.15) (Raoult’s law) with the activity coefficients predicted by the following expression [29, 124]:   1 ∂ n l, total G E , ln γi = Ru T ∂n li

(4.34)

 where Ru is the universal gas constant, T is the temperature in K, n l, total = i n li , n li is the molar concentration of the ith component in the liquid phase, and G E is the total excess Gibbs energy. The following estimation of G E for the ethanol (i = 1) – acetone (i = 2) mixture, based on fitting experimental data, was used [29]: n l, total G E = Ru T

n l1 n l2 n l1 + n l2



αn l2 βn l1 δn l1 n l2 , + − n l1 + n l2 n l1 + n l2 (n l1 + n l2 )2

where α=

546.3 − 0.9897, T

β=

543.3 − 0.9483, T

δ=

15.63 + 0.0759. T

(4.35)

Having substituted (4.35) into (4.34) the following expressions for γ1 = γeth and γ2 = γac were obtained:

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4 Heating and Evaporation of Multi-component Droplets

Table 4.1 Approximations of the acetone and ethanol mixture droplet velocities Udrop Mixture

Udrop in m/s (t is in ms)

25% ethanol + 75% acetone 50% ethanol + 50% acetone 75% ethanol + 25% acetone

12.75 − 0.370 t 12.71 − 0.448 t 12.28 − 0.306 t

Table 4.2 The measured initial droplet temperatures and diameters, ambient gas temperatures and distance parameters for the same cases as in Table 4.1 Mixture Droplet Diameter Gas temperature Distance temperature parameter 25% 75% 50% 50% 75% 25%

ethanol + acetone ethanol + acetone ethanol + acetone

32.5 ◦ C

133.8 µm

21.1 ◦ C

8.7

37.5 ◦ C

142.7 µm

20.8 ◦ C

7.53

38.6 ◦ C

137.1 µm

21.6 ◦ C

7.53

  2  2 α + 2 (β − α − δ) X l1 + 3δ X l1 γ1 = exp X l2  2   . 2 γ2 = exp X l1 β + 2 (α − β − δ) X l2 + 3δ X l2

(4.36)

Expressions (4.36) were used in (4.14). As in the cases of pure acetone and ethanol, described in Sect. 3.8, the approximation for the measured time evolution of the droplet velocities for various mixtures of acetone and ethanol were obtained in [54] and summarised in Table 4.1. Percentages of the components refer to their volume fractions. The measured initial droplet temperatures and diameters, ambient gas temperatures and distance parameters C for the same cases as in Table 4.1 are presented in Table 4.2. The distance parameters were used for estimating the corrections for Nu and Sh to consider the interaction between droplets. Expression (3.170) was used in our analysis. Note that the effects of neighbouring droplets on the evaporation of individual droplets in realistic spray-like conditions were investigated in [125]. In the latter paper, a transient approach was used to simulate the evaporation of a multi-component droplet in a hypothetical spherical bubble. It was shown that the effects of neighbour droplets become significant for spacing parameters less than 55. In what follows we focus on the comparison of the results of calculations using two models and experimental data. The first model was based on the analytical solutions to the heat transfer equation inside droplets (2.1) (Expression (2.41)) and the equations for component diffusion inside them (4.1) (Expression (4.29)) (Solution A). The second model was based on the numerical solutions to Eqs. (2.1) and (4.1) (Solution B). The numerical codes for the solutions of Eqs. (2.1) and (4.1) were developed by Castanet et al. [30]. The effects of the movement of the droplet interface due

4.2 Discrete Component Model

187

Fig. 4.1 The time evolution of droplet surface, average and centre temperatures (Ts , Tav and Tc ), predicted by one-way Solution A for ideal and non-ideal models and experimentally observed temperatures for the 25% ethanol – 75% acetone mixture droplets with the values of the initial parameters, droplet velocity and gas temperature given in Tables 4.1 and 4.2. Reproduced (with minor modification) from Fig. 5a of [54] with permission by Begell House

to evaporation and thermal swelling/contraction during individual timesteps were ignored in both solutions. Only the one-way solutions were considered so that the effects of droplets on gas were ignored. The latter effects would modify the results slightly but would not change the trends of the predicted results [169]. The plots like those presented in Fig. 3.11, but for mixtures of ethanol and acetone, are shown in Figs. 4.1, 4.2, 4.3 and 4.4. The calculations were performed for the cases of the ideal mixture (γi = 1 in Eq. (4.14)) (see Eq. (4.15)) and the non-ideal mixture (γi in Eq. (4.14) was estimated based on Expressions (4.36)). As can be seen from these figures, in all cases the predictions of the temperatures by the ideal and non-ideal models are noticeably different (by up to several degrees), especially at later times. Both these models, however, predict the same trends in the evolution of temperature with time. The ideal model can be used if the prediction errors of several degrees can be tolerated. This seems to be our case where the random errors of the estimates of droplet temperatures appear to be about 2–3 degrees. As in the cases shown in Fig. 3.11, the results predicted by Solutions A and B coincide within the accuracy of plotting, which gives us confidence in the results predicted by both solutions. This closeness of the results predicted by Solutions A and B is illustrated for the 25% ethanol – 75% acetone mixture droplets in Fig. 4.2. For other mixtures the closeness of the curves is like what is shown in Fig. 4.2 (the plots are shown in [54]). For an acetone dominated mixture (25% ethanol – 75% acetone: see Fig. 4.1), agreement between the observed and predicted average droplet temperatures, for both ideal and non-ideal models, was good. Most of the observed temperatures lie

188

4 Heating and Evaporation of Multi-component Droplets

Fig. 4.2 The time evolution of droplet surface, average and centre temperatures (Ts , Tav and Tc ), predicted by Solutions A and B for the non-ideal model for the 25% ethanol – 75% acetone mixture droplets with the values of the initial parameters, droplet velocity and gas temperature given in Tables 4.1 and 4.2. Reproduced (with minor modification) from Fig. 5b of [54] with permission by Begell House

Fig. 4.3 The same as Fig. 4.1 but for the 50% ethanol – 50% acetone mixture droplets. Reproduced (with minor modification) from Fig. 6a of [54] with permission by Begell House

4.2 Discrete Component Model

189

Fig. 4.4 The same as Figs. 4.1 and 4.3 but for the 75% ethanol – 25% acetone mixture droplets. Reproduced (with minor modification) from Fig. 7a of [54] with permission by Begell House

between average and central temperatures, although the scatter of experimental data in this case is more noticeable than for pure acetone (see Fig. 3.11). For the 50% ethanol – 50% acetone mixture (see Fig. 4.3), the experimentally observed temperatures lie close to the average temperatures predicted by the non-ideal model. For the 75% ethanol – 25% acetone mixture (see Fig. 4.4), the experimentally observed temperatures are close to the surface temperatures predicted by the non-ideal model. As with pure acetone and ethanol, the reason for this deviation between the modelling and experimental results is not clear to us. The non-ideal behaviour of ethanol/heptane droplet evaporation for ambient temperatures and pressures in the ranges 473 K to 673 K and 1 to 10 bar, respectively, was investigated by the authors of [139]. It was shown that the effect of non-ideal behaviour is particularly important for ethanol dominated droplets. To consider the effect of coupling between droplets and ambient gas, in [54, 169] it was assumed that a droplet exchanges heat and mass with a certain volume Vg , surrounding the droplet, which is called the region of influence. The shape of this region can be either spherical, in the case of isolated droplets, or cylindrical, for droplet streams (see Fig. 3.9). The initial value of the ratio r V = Vg /Vd , where Vd = 43 π Rd3 , was chosen to be 3,500 in calculations performed in [54, 169]. At this value of r V , the difference between the one-way and coupled solutions was visible, but the vapour pressure in the region of influence did not reach saturation level during the droplet life time. The values of Vg were recalculated at each timestep depending on the volume of the droplet. Since the changes in Vd were small for the cases under consideration, the values of Vg remained close to their initial values. Although this approach is not suitable for rigorous quantitative analysis of heat/mass exchange between droplets and surrounding gas, it can be used for the qualitative analysis

190

4 Heating and Evaporation of Multi-component Droplets

of this effect. Only Solution A was used for the comparison between the results of coupled and one-way solutions, since the predictions of Solutions A and B were shown to be almost identical. A similar problem of the region of influence, but using more rigorous analysis, was considered in [229]. The radius of this region was found to be about 30 droplet radii, which would lead to the ratio of volumes 27,000. This is larger than the ratio of volumes considered in [54, 169] (see previous paragraph). The region radius was shown to increase with increasing droplet radius, temperature difference between droplet and ambient gas, and with decreasing operating pressure [229]. For the 25% ethanol – 75% acetone mixture droplets, the coupled solutions showed a slower reduction in droplet surface temperature at the end of the evaporation process compared with the prediction of the one-way solution. For the 50% ethanol – 50% acetone mixture droplets, the effects of the coupled solution on the results were like those for the 25% ethanol – 75% acetone mixture droplets. The experimentally observed droplet temperatures in both cases were close to the average droplet temperatures predicted by both one-way and coupled solutions. For the 75% ethanol – 25% acetone mixture droplets, the effects of the coupled solution were similar to those for the previous two cases. In the latter case, however, the deviation between the predicted and observed temperatures was larger for the coupled than for the one-way solution. The grid dependence of evaporation rates in the context of Euler-Lagrange simulations of dilute sprays was investigated by the authors of [181]. It was found that the error of the steady-state evaporation rate depends on the ratio of cell size to droplet diameter and the cell Peclet number. The error of the evaporation time depended on the initial ratio of liquid to gaseous mass in the computational cell. An alternative approach to modelling evaporation of non-ideal liquid mixtures, using the COSMO-RS based CFD model, was discussed in [90, 91]. In contrast to the approach described above, the liquid mixture was considered to be well mixed. Droplets of Water and Ethanol The authors of [184] applied the DC model with the analytical solution to the component diffusion equation described in Sect. 4.2.1 to the analysis of suspended water/ethanol droplets. The setup used in the experiments was the same as used for pure water droplets (see Figs. 3.13 and 3.14). The model described in Sect. 3.7 was used to consider the effect of support. Raoult’s law (see Expression (4.15)) was assumed to be valid. The plots of the experimentally observed and predicted values of Tg − Ts for water/ethanol droplets are presented in Fig. 4.5. Comparing this figure and Fig. 3.13 one can see that the addition of ethanol leads to a qualitative change in the function Tg − Ts . As in the case of water droplets, Tg − Ts rapidly increases initially. In contrast to pure water droplets, however, this increase is followed by a decrease in Tg − Ts for water/ethanol droplets. The minimum surface temperatures of water/ethanol droplets were always less than those of water droplets. The values of these minimum temperatures decrease as the mass fraction of ethanol increases.

4.2 Discrete Component Model

191

Fig. 4.5 a Tg − Ts for water/ethanol droplets versus time. Symbols refer to experimental data (Case 1 (filled squares): pure water, initial droplet diameter 2.15 mm; Case 2 (filled circles): mixture with mass fraction of ethanol 9%, initial droplet diameter 2.06 mm; Case 3 (filled triangles): mixture with mass fraction of ethanol 23%, initial droplet diameter 1.95 mm; Case 4 (empty squares): mixture with mass fraction of ethanol 47%, initial droplet diameter 1.87 mm; Case 5 (empty circles): mixture with mass fraction of ethanol 70%, initial droplet diameter 1.77 mm; Case 6 (empty triangles): mixture with mass fraction of ethanol 94%, initial droplet diameter 1.82 mm; in all cases gas velocity was 1.5 m/s and gas and initial droplet temperatures were 302 K). Solid (dashed) curves refer to predictions of the models for the same values of input parameters considering (ignoring) the effect of the thread. The thickest (thinnest) curves refer to Case 1(6). b Zoomed part referring to the initial stage of the process. Note that we halved the number of experimental points shown in part (a) to improve the presentation of the results. Reproduced from [184], Copyright Elsevier (2021)

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4 Heating and Evaporation of Multi-component Droplets

Fig. 4.6 The same as Fig. 4.5, but for (d/d0 )1.5 . Reproduced from [184], Copyright Elsevier (2021)

As in the case of pure water droplets, the predictions of the model in which the contribution of the thread is taken into account are always closer to experimental data than those of the model in which it is ignored. In fact, the modelling and experimental results are closer for the water/ethanol droplets than for pure water droplets. Maximal deviations between observed and modelling results are seen for the mixture with 94% ethanol. Even in this case this deviation was only about 2 K. The plots like those presented in Fig. 3.14, but for water/ethanol droplets, are shown in Fig. 4.6. As one can see from the latter figure, in all cases the values of (d/d0 )1.5 decrease almost linearly with time. This is the same as in the case of pure water droplets. The rate of this decrease increases when the mass fraction of ethanol increases. This is in full agreement with the model predictions. In contrast to the cases shown in the previous figures, however, the improvement in the predictions of the

4.2 Discrete Component Model

193

model when the effects of the thread are considered is less clear. This improvement can be seen when ethanol is present in the mixture but is not seen for pure water. Note that Case 1, shown in Fig. 4.6, differs from Case 2, shown in Fig. 3.14, in terms of the values of initial droplet diameter and ambient/droplet temperature. Useful uncertainty quantification of some of the models of multi-component droplet evaporation described so far is presented in [122]. The authors of [122] drew the readers’ attention to the fact that these models rely on several model parameters, which can be subject to uncertainty. The effect of the parameter uncertainties was studied by performing uncertainty analysis and Monte Carlo variance-based sensitivity analysis. The individual contributions of the model parameters to the variance were ranked in order of importance. The results are expected to provide useful guidelines for the use of these models in various practical calculations.

4.2.3.2

Heating and Evaporation when Rd = const

As demonstrated in Sect. 3.4 the assumption that the droplet radius is fixed during the timestep can lead to noticeable deviations from the results predicted by the models that consider the changes in this radius during the individual timesteps. The cases tested in that section refer to droplet heating and evaporation in a hot gas and the moving interface was linked only with droplet evaporation. The effects of thermal swelling/contraction were ignored. Also, only mono-component droplets were considered, in which case the moving interface had an effect only on the heat conduction equation inside droplets. In this section the general case of multi-component droplets is investigated, and the effects of the moving interface, described by Expression (3.100), on both heat transfer and component diffusion are considered using Expressions (3.103) and (4.29), mainly following [54]. The model is first applied to the case of the acetone–ethanol mixture droplets, considered in the previous section. Then other related cases are considered. The analysis of this section is limited to the one-way Solution A. The plots of time evolutions of the temperatures at the centre and the surface of the droplets and the average droplet temperatures, predicted by the models not considering the effect of the moving interface and considering this effect for both temperature and component diffusion for the 25% ethanol – 75% acetone mixture droplets, are shown in Fig. 4.7. As can be seen from this figure, the effect of the moving interface on the predicted temperatures can be safely ignored in the analysis of experimental data presented in the previous section. The same conclusion can be drawn for the cases of the 50% ethanol – 50% acetone and 75% ethanol – 25% acetone mixture droplets (figures are not shown). In Fig. 4.8 a hypothetical case is presented in which the 50% ethanol – 50% acetone mixture droplets are cooled or heated and evaporated until complete evaporation takes place. This example allows us to achieve good visualisation of the effect of the moving interface on droplet heating and evaporation. Plots for both the droplet surface temperature and droplet radius are presented. The same values as shown in Table 4.2 for the initial droplet temperature, diameter, distance parameter and gas temperature

194

4 Heating and Evaporation of Multi-component Droplets

Fig. 4.7 The time evolution of droplet surface, average and centre temperatures (Ts , Tav and Tc ), predicted by the one-way Solution A using the non-ideal model, considering and not considering the effects of the moving interface during individual timesteps (moving and stationary interfaces) on the solutions to both heat transfer and component diffusion equations. The 25% ethanol – 75% acetone mixture droplets are considered. The values of the initial parameters, droplet velocity and gas temperature are presented in Tables 4.1 and 4.2. Reproduced (with minor modifications) from Fig. 11a of [54] with permission by Begell House

Fig. 4.8 The time evolution of droplet surface temperature (Ts ) and radius (Rd ), predicted by the one-way Solution A using the non-ideal model, considering and not considering the effects of the moving interface during individual timesteps on (i) the solutions to the heat transfer equation only, (ii) component diffusion equation only and (iii) both heat transfer and component diffusion equations. The 50% ethanol – 50% acetone mixture droplets are considered. The values of the initial parameters, and gas temperature are presented in Table 4.2; it was assumed that the droplet velocity is constant and equal to 12.71 m/s. Reproduced (with minor modifications) from Fig. 12 of [54] with permission by Begell House

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Fig. 4.9 The plots of Ts and Rd versus time for the same parameters as in Fig. 4.8 but for the gas temperature equal to 1000 K. Reproduced (with minor modifications) from Fig. 13 of [54] with permission by Begell House

were used. In contrast to the case shown in Table 4.1, it was assumed that the droplet velocity remains constant and equal to 12.71 m/s. This is the initial velocity of the droplet in the experiment described earlier. The following cases are shown: (i) the case of the stationary interface during individual timesteps, (ii) the cases when the effects of the moving interface are considered for the heat transfer and component diffusion equations separately during individual timesteps, and (iii) the case when these effects are considered simultaneously for heat transfer and component diffusion equations. As can be seen from this figure, the plots considering the effects of the moving interface on the heat transfer equation only and ignoring this effect altogether almost coincide. That means that this effect can be safely ignored for this case. Also, the plots considering the effects of the moving interface on the solution to the component diffusion equation and considering it for both solutions to the heat transfer and component diffusion equations practically coincide. The difference between both these curves and the ones ignoring this effect altogether, however, can be clearly seen after about 0.1 s. The effect of the moving interface leads to a reduction of the predicted droplet surface temperature at times between about 0.1 s and 0.6 s. During this period the droplet surface temperature is below the ambient gas temperature. Hence, the reduction of the droplet surface temperature is expected to increase the heat flux from the ambient gas to the droplets, leading to the acceleration of droplet evaporation. This agrees with the predicted time evolution of the droplet radius, considering and not considering the effect of the moving interface, presented in Fig. 4.8. In Fig. 4.9 the plots for gas temperature equal to 1000 K are presented. In this case, the droplet surface temperature increases during the whole period of droplet

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4 Heating and Evaporation of Multi-component Droplets

Fig. 4.10 The same as Fig. 4.9 but for the mass fraction of ethanol at the surface of the droplet. Reproduced (with minor modifications) from Fig. 14 of [54] with permission by Begell House

heating and evaporation, in contrast to the case presented in Fig. 4.8. As can be seen from Fig. 4.9, the plots considering the effects of the moving interface on the solution to the heat transfer equation, and ignoring this effect altogether almost coincide, as in the case presented in Fig. 4.8. Also, similarly to the case presented in Fig. 4.8, the plots considering the effects of the moving interface on the solution to the component diffusion equations and considering it for both heat transfer and component diffusion equations almost coincide. The difference between both these curves and the ones ignoring this effect altogether can be clearly seen after about 5 ms. This difference between the curves is much more visible than in the case presented in Fig. 4.8. As in the case shown in Fig. 4.8, the effect of the moving interface leads to a reduction in the predicted droplet surface temperature. This, in its turn, leads to an increase in the heat flux from the ambient gas to the droplets and acceleration of droplet evaporation. This agrees with the predicted time evolution of droplet radius, considering and not considering the effect of the moving interface, presented in Fig. 4.8. The plots of time evolution of the surface mass fraction of ethanol Yls,eth for the same case as presented in Fig. 4.9 are shown in Fig. 4.10. Similarly to the case presented in Fig. 4.9, the main effect of the moving interface in the solution to the component diffusion equation is on the values of Yls,eth . This effect leads to clearly seen reductions of the values of Yls,eth until complete evaporation of the droplet. The model described in this section was generalised in [68] to consider the effects radiation and chemical reactions in the gas phase. This generalised model was applied to the investigation of combustion of an isolated nonane/hexanol droplet. A more complex model of bi-component droplet heating and evaporation, considering the deformation of droplet surfaces and formation of vortices inside droplets, is described in [21, 189].

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4.2.4 Biodiesel Droplets In this section, the results of application of the Discrete Component (DC) model to the analysis of biodiesel fuel droplet heating and evaporation are presented and discussed following mainly [10, 171]. Interest in biodiesel fuels has been stimulated by various factors, including depletion of fossil fuels and the need to reduce carbon dioxide emissions that contribute toward climate change [108]. The term ‘biodiesel’ typically refers to ‘a fuel comprised of mono-alkyl esters of long-chain fatty acids derived from vegetable oils or animal fats’ [83]. Using biodiesel fuel as an alternative to conventional hydrocarbon fuels has several advantages. This fuel mixes well with Diesel fuels, is less polluting, has higher lubricity, higher flash point, is cost effective, and can be used in Diesel engines with minimal modifications (e.g. [144, 156]). The analysis described in this section focuses on modelling of biodiesel fuel droplet heating and evaporation using the Discrete Component (DC) model. The temperature gradients and component diffusion inside droplets are considered using the analytical solutions to heat transfer and component diffusion equations, described earlier. These solutions were incorporated into a numerical algorithm describing these processes. The application of the DC model can be justified by the fact that biodiesel is composed of a relatively small (less than 14) number of fatty acid ethyl and methyl esters [101, 131] (only biodiesels composed of methyl esters is investigated). The preliminary results of modelling biodiesel fuel droplet heating and evaporation, using the above-mentioned approach, were described in [171]. In [10], the results of the analysis, like the one presented in [171] but for a much wider range of biodiesel fuels (19 types) and more realistic engine conditions, were described. The analysis in this section will follow [10].

4.2.4.1

Biodiesel Composition

These types of biodiesel fuels were used in [10]: Tallow Methyl Ester (TME), Lard Methyl Ester (LME), Butter Methyl Ester (BME), Coconut Methyl Ester (CME), Palm Kernel Methyl Ester (PMK), Palm Methyl Ester (PME), Safflower Methyl Ester (SFE), Peanut Methyl Ester (PTE), Cottonseed Methyl Ester (CSE), Corn Methyl Ester (CNE), Sunflower Methyl Ester (SNE), Tung Methyl Ester (TGE), Hempoil Methyl Ester, produced from Hemp seed oil in the Ukraine (HME1), Soybean Methyl Ester (SME), Linseed Methyl Ester (LNE), Hemp-oil Methyl Ester, produced in the European Union (HME2), Canola seed Methyl Ester (CAN), Waste cookingoil Methyl Ester (WME) and Rapeseed Methyl Ester (RME). The molar fractions of the components of these fuels (in percentages), inferred from data provided in [49, 65, 103, 126, 145, 172, 203], are presented in Table 4.3. The meaning of the symbols of the components, shown in Table 4.3, and their acid codes, molecular formulae, molar masses and boiling temperatures are presented in Table 4.4 (the values of boiling temperatures in this table are taken from [84, 171]).

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Table 4.3 Types of biodiesel fuels, their abbreviations, acid codes and molar fractions of the components (pure methyl esters) used in [10]. Symbols M for the acid codes are omitted. Reprinted from [10]. Copyright Elsevier (2015)

The symbols of the components in Tables 4.3 and 4.4 indicate the numbers of carbon atoms in fatty acids (nacid) and numbers of double bonds (DB). For example, C18:1M has nacid = 18 and DB = 1. The addition of one more carbon atom gives the total number of carbon atoms in methyl esters (nacid + 1). There are other names used for some methyl esters shown in Table 4.4. For example, ‘Methyl dodecanoate’ is also known as ‘Methyl laurate’, ‘Methyl tetradecanoate’ is also known as ‘Methyl myristate’, and ‘Methyl decosanoate’ is also known as ‘Methyl behenate’ [102, 140, 171]. The molar fractions of unidentified additives in biodiesel fuels, which could vary from 0 to around 8.7%, are presented in Table 4.3 as ‘Others’. Since the exact nature of these additives has not been identified, there is a certain freedom in selecting their transport and thermodynamic properties. In [171] these properties were calculated as the arithmetic weighted averages of the corresponding values for all remaining components (C12:0M to C18:3M in the case considered in [171]). In what follows these properties are identified with those of C18:1M; these turned out to be close to those obtained in [171]. The transport and thermodynamic properties of all components presented in Tables 4.3 and 4.4 are shown in Appendix B of [171]. These properties were extrapolated to the cases of other fatty acids shown in Table 4.4, not considered in [171]. 4.2.4.2

Results of Calculations (Biodiesel Droplets)

Following [10], we compare the predictions of the following models: (1) Discrete Component (DC) model in which the contributions of all components and the recirculation within the droplet are taken into account using the Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) model (‘ME’ model);

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Table 4.4 Names, acid codes, molecular formulae, molar masses and boiling points of the components (pure methyl esters) used in [10]. Reprinted from [10]. Copyright Elsevier (2015)

(2) Infinite Thermal Conductivity/Infinite Diffusivity (ITC/ID) model, considering the contributions of all components (‘MI’ model); (3) Infinite Thermal Conductivity (ITC) model, in which all components are treated as a single-component with properties depending on temperature (updated at each timestep) (‘SI’ model). Following [10], the initial droplet radius and temperature are taken equal to Rd0 = 12.66 µm and Td0 = 360 K, respectively. The chosen droplet radius is compatible with the results of Diesel spray observations presented in [60]. The droplet is assumed

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4 Heating and Evaporation of Multi-component Droplets

Fig. 4.11 Plots of time evolution of a droplet’s surface temperature (Ts ) and radius (Rd ) for Butter Methyl Ester predicted by ME, MI and SI models. The values of the input parameters are described in the text. Reprinted from [10], Copyright Elsevier (2015) Fig. 4.12 Plots of surface mass fractions of C8:0M, C12:0M, C14:0M, C16:0M, C18:0M and C22:1M for a Butter Methyl Ester droplet versus time for the same input parameters as in Fig. 4.11. Reprinted from [10], Copyright Elsevier (2015)

to be moving at constant velocity of Ud = 28 m/s. Ambient air temperature and pressure are assumed equal to 700 K and 3.2 MPa, respectively. The plots of surface temperature (Ts ) and radius (Rd ) versus time for a Butter Methyl Ester droplet are shown in Fig. 4.11. As can be seen from this figure, the ME model predicts longer evaporation times compared with the MI and SI models. The deviation of the droplet lifetimes predicted by the SI model from those predicted by the ME model was found to be 25.2%, and this needs to be considered in engineering applications. To provide better understanding of the processes taking place during droplet heating and evaporation, in Figs. 4.12, 4.13 and 4.14 the following plots are presented: surface mass fractions of some components versus time, mass fractions of some components versus normalised distance from the droplet centre at three time instants, and

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Fig. 4.13 Plots of mass fractions of C12:0M and C22:1M versus normalised distance from the droplet centre at time instants 0.03 ms, 0.5 ms and 1 ms for a Butter Methyl Ester droplet for the same input parameters as in Figs. 4.11 and 4.12. Reprinted from [10], Copyright Elsevier (2015)

Fig. 4.14 Plots of temperature versus normalised distance from the droplet centre at time instants 0.03 ms, 0.3 ms, 0.5 ms and 1 ms for a Butter Methyl Ester droplet for the same input parameters as in Figs. 4.11, 4.12 and 4.13. Reprinted from [10], Copyright Elsevier (2015)

temperatures versus normalised distance from the droplet centre at four time instants. As one can see from Fig. 4.12, the surface mass fractions of the lightest components (C8:0M, C12:0M and C14:0M) decrease with time. The surface mass fraction of the heaviest component (C22:1M) increases with time. The surface mass fractions of the intermediate components (C16:0M and C18:0M) first increase and then decrease with time. At the end of the evaporation process, only the heaviest and least volatile component remains at the droplet surface. This component contributes for prolonged droplet lifetime predicted by the ME model compared with the SI model, and higher surface temperatures at the final stage of droplet evaporation. As can be seen in Fig. 4.13, the decrease in the surface mass fraction of the light component (C12:0M) with time is linked with a corresponding decrease in its mass fraction inside the droplet. The rate of this decrease reduces close to the droplet centre. Thus a negative gradient for the mass fraction of this component is formed inside the droplet, which leads to its diffusion from the centre of the droplet to its surface. As can be seen in the same figure, the increase in the surface mass fraction of the heaviest component (C22:1M) with time is accompanied by a corresponding increase in the mass fraction of this component in the body of the droplet; the rate of

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4 Heating and Evaporation of Multi-component Droplets

this increase reduces in the regions close to the droplet centre. Thus, positive gradients for this mass fraction are formed inside the droplet. These lead to the diffusion of this component from the droplet surface to its centre leading to the formation of a droplet which includes mainly the heaviest component (C22:1M) at the end of the droplet lifetime. It can be clearly seen from Fig. 4.13 that gradients of mass fractions of the components inside the droplet increase with time. This observation shows the limitations of the well mixed models, including the MI model, which are widely used for the analysis of the processes in multi-component droplets. As can be seen in Fig. 4.14, at the initial stage of droplet heating (0.03 ms after the start of the process) large temperature gradients inside the droplet close to the droplet surface develop. In contrast to the case of component molar fractions, however, these temperature gradients decrease with time. They are reasonably small at 1 ms after the start of the process. This means that the Infinite Thermal Conductivity model can be applied to the analysis of droplet heating and evaporation, except at the initial stage of the process. To summarise the results shown in Figs. 4.11, 4.12, 4.13 and 4.14 and similar results for other biodiesel fuels shown in Tables 4.3 and 4.4, we were able to conclude that the evaporation times predicted by the MI model are reasonably close to those predicted by the ME model (considered as the most reliable one). The MI model under-predicts this time by not more than 4.3% except for Rapeseed Methyl Ester for which this under-prediction was 15.1%. In most cases, the ME model predicts higher droplet surface temperatures at the final stages of evaporation and longer evaporation times (up to about 26%) than the SI model. This is attributed to the fact that at the final stages of droplet evaporation the mass fractions of heavier components (which evaporate slower than the lighter components) increases at the expense of lighter components. The ME model is recommended to be used for the analysis of biodiesel fuel droplet heating and evaporation in engineering applications. Among other approaches to modelling heating/evaporation of biodiesel droplets we can mention the one described in [85]. In this paper, it was shown that Palm Methyl Ester (PME) droplets tend to evaporate much more slowly than Diesel fuel droplets in the same conditions. The authors of [159] presented the results of rigorous numerical simulations of evaporation of biodiesel droplets of Indian origin in air at atmospheric pressure under normal gravity conditions. They solved transport equations in the liquid and vapour phases considering the interface coupling conditions. The influence of suspension bead and internal recirculation in droplets were considered.

4.2.5 Kerosene Droplets The interest in the problem of heating and evaporation of kerosene and kerosene surrogate droplets has been mainly linked with the importance of these processes in kerosene combustion in propulsion systems [177]. Amongst recent studies of these phenomena, we can mention [23, 99, 207, 212] (experimental investigations of the evaporation of kerosene gel droplets and kerosene ignition and combustion

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Table 4.5 Molar fractions of the kerosene components (in percent), adapted from Table 6 of [117]. CN stands for carbon number of a component, n-Alk – n-alkanes, iso-Alk – isomers of alkanes, Cyclo – cycloalkanes, iso-Cyclo – isomers of cycloalkanes, AlkyB – alkylbenzenes, NaphtoB – naphtobenzenes, Naphtha – naphthalenes. Reprinted from [152], Copyright Elsevier (2020) CN n-Alk iso-Alk Cyclo iso-Cyclo AlkyB NaphtoB Naphtha C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17

0.3092 0.2712 0.623 0.8023 0.7824 1.1009 0.7695 0.7315 0.4376 0.036 0

0.3743 0.5568 2.1869 4.6876 5.5606 6.0985 4.4756 4.2249 4.322 1.5195 0

0.2823 0.9155 2.5316 5.4172 3.9842 3.9428 2.7631 2.9976 1.2012 0.5376 0.3282

0 0 0.5296 1.3601 3.3395 3.2259 1.2519 0.1495 0.093 0 0

0.1592 0.9366 2.1157 3.3035 2.4082 3.0139 2.6909 1.4904 0.2792 0 0

0 0 0.2984 1.2874 1.9904 3.4961 0.8322 0.2056 0 0 0

0 0 0 0.1144 0.2865 0.3129 0.0574 0 0 0 0

characteristics), [213] (numerical investigation of kerosene sprays), [160] (numerical investigation of an oblique detonation wave in a kerosene-air mixture), and [74] (numerical investigation of a single kerosene droplet ignition). Although these and related studies are important for our understanding of the processes, their attention has been mainly on the gas phase; rather basic models have been used for the liquid phase. The temperature and component concentration gradients in droplets have been ignored, although the limitations of these assumptions have been widely discussed in the literature [164]. This section will focus on the modelling of the effects of these gradients on kerosene and kerosene surrogate fuel droplet heating and evaporation using the Discrete Component (DC) model, mainly following [152]. We start with the discussion of the composition of kerosene and its surrogates. Then a comparison between kerosene surrogate and kerosene droplet heating and evaporation characteristics are presented. In most cases, the same conditions as in the experiment of [92] are used in the analysis.

4.2.5.1

Composition of Kerosene and Its Surrogates

The composition of 57 components of kerosene analysed by Lissitsina et al. [117], based on ‘comprehensive two-dimensional gas chromatography coupled with time of flight mass spectrometry’, was used in the analysis by [152]. This composition, inferred from Table 6 of [117], is reproduced in Table 4.5. The thermophysical properties of the components in Table 4.5 are presented in Section 1 of the Supplementary Material of [152].

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4 Heating and Evaporation of Multi-component Droplets

Table 4.6 Mass fractions of the kerosene components, inferred from a simplified version of Table 4.5. CN are carbon numbers, Par – paraffins, Na/Ol – naphthenes/olefins, Alk – alkylbenzene, Napht – naphtobenzene, Dia – diaromatics. Reprinted from [152], Copyright Elsevier (2020) CN Par Na/Ol Alk Napht Dia C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17

0.4100 0.5800 2.2100 4.7900 6.0900 7.5200 5.9300 6.0300 6.2100 2.1600 0

0.1700 0.6300 2.3700 5.8300 6.9300 7.4000 4.4900 3.7800 1.6700 0.7400 0.4800

0.0900 0.6100 1.5600 2.7200 2.1900 3.0000 2.9100 1.7400 0.3500 0 0

0 0 0.2200 1.0600 1.8100 3.4800 0.9000 0.2400 0 0 0

0 0 0 0.0900 0.2500 0.3000 0.0600 0 0 0 0

A simplified version of Table 4.5, using 40 components, is shown in Table 4.6. The following simplifying assumptions were made when presenting the results of [117] in Table 4.6. The contributions of n-alkanes and iso-alkanes were not distinguished. The contributions of mono-, di- and tri-naphthenes/olefins were considered to be the same. Naphthalenes, biphenyls and fluorenes were treated as the same components as diaromatics. These assumptions were based on the closeness of the thermodynamic and transport properties of these components. The composition of kerosene surrogates used in [152] is shown in Table 4.7, and the names of these surrogates and the relevant references are presented in Table 4.8. 11 surrogates taken from the literature and 2 original surrogates were used in [152]. The number of components in these surrogates varied between 1 and 6. The selection of surrogates was based mainly on the similarity between the most important properties of the surrogates and kerosene, including molar mass, the ratio of the numbers of hydrogen and carbon atoms (H/C), basic composition, ignition delay, laminar flame velocity, density, heat of evaporation, cetane number, viscosity, surface tension, distillation curve, and production of soot precursors [38, 100, 225]. See [36, 46, 87, 152, 180, 187, 206] for further details. Approximations of the thermophysical properties of the components used in the surrogates presented in Tables 4.7 and 4.8 are given in Section 2 of the Supplementary Material of [152]. A surrogate fuel for emulating the physical and chemical properties of RP-3 kerosene was suggested in [119]. The authors of this paper targeted six properties for their surrogate: hydrogen/carbon ratio, molar mass, cetane number, calorific value, density and viscosity. The surrogate was developed using only components for which kinetic mechanisms are readily available: n-decane, n-dodecane, isocetane, methylcyclohexane and toluene. The molar fractions of these components were obtained as 14%, 10%, 30%, 36% and 10%, respectively.

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Table 4.7 Mass fractions of the components of kerosene surrogates (Surr.). ‘Ind, Tet and Naph’ stands for indane and tetraline (C10 H12 ) or naphtalenes (C11 H10 ). n-alkanes are shown in italic, and iso-alkanes are shown in bold. Reprinted from [152], Copyright Elsevier (2020) Surr. n− or iso-alkanes Cycloalkanes Alkylbenzene Ind, Tet and Naph 1 2 3 4 5 6 7

8

9 10 11

12

13

100% of C10 H22 57.6923% of C12 H26 46.4229% of C10 H22 30.8611% of C10 H22 76.9231% of C10 H22 77.6398% of C12 H26 28.7613% of C12 H26 19.5321% of C14 H30 10.1368% of C16 H34 40.1989% of C12 H26 26.6884% of C16 H34 9.1% of C6 H14 72.7% of C10 H22 87.0841% of C10 H22 10.9974% of C8 H18 18.3902% of C12 H26 30.2090% of C16 H34 28.1328% of C10 H22 19.2691% of C12 H26 15.2804% of C16 H34 18.6725% of C10 H22 38.3681% of C12 H26

0 19.7436% of C7 H14 26.0095% of C10 H20 33.5642% of C10 H20 0

0 22.5641% of C8 H10 27.5676% of C10 H14 35.5748% of C10 H14 23.0769% of C6 H6 22.3602% of C9 H12 0

0 0

14.3968% of C7 H14

0

18.7159% of C11 H10

0

18.2% of C6 H6

0

0

12.9159% of C6 H6 0

0

19.7829% of C7 H14

11.3045% of C8 H10

6.2303% of C10 H12

26.1542% of C10 H20

16.8052% of C6 H6

0

0 19.6855% of C7 H14 15.6845% of C10 H20

9.7223% of C9 H18

0 0 0 0 6.1997% of C10 H12

30.6811% of C11 H10

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Table 4.8 Names by which surrogates presented in Table 4.7 are known and the relevant references. ‘O’ indicates that these surrogates were developed at Samara National Research University. Reprinted from [152], Copyright Elsevier (2020) Surr. Name Reference 1 2 3 4 5 6 7 8 9 10 11 12 13

Normal decane Surrogate C Surrogate D Surrogate E Aachen surrogate Modified Aachen surrogate Modified Utah surrogate Drexel surrogate 2 Strelkova surrogate Lindstedt surrogate Slavinskaya surrogate SU1 SU2

[42, 43] [87] [36] [36] [46] [36] [206] [36] [187] [115] [180] O O

The results of development and testing of five naval jet fuel JP-5 surrogates are presented in [40]. These were prepared from components representing the classes of components found in JP-5: linear and branched alkanes, aromatics and cycloalkanes. A three-component surrogate of jet fuel (n-dodecane, n-propylbenzene and 1,3,5trimethylcyclohexane) was investigated experimentally and numerically for various temperatures, fuel equivalence ratios and pressures by the authors of [120]. A unified surrogate (JI) to mimic both the physical and chemical characteristics of kerosene (Jet-A) fuel was suggested in [33]. The fuel composition was determined by an inverse batch distillation methodology to match an experimental distillation curve. The ignition delay times were predicted using the CANTERA package. A multicomponent skeleton reaction mechanism was validated using chemical surrogate UM2 and experimental data. JI was applied to model droplet heating, evaporation and ignition processes. The results were shown to be compatible with available kerosene droplet ignition data. Pyrolysis of a three-component jet fuel surrogate was studied experimentally at temperatures 850 K to 1150 K under atmospheric pressure. by the authors of [95]. The focus of the analysis of the authors of [147] was on 12 Jet-A (aviation kerosene) surrogates suggested in the literature. The results of the analysis by these authors of heating and evaporation of droplets of these surrogates are summarised later in this section.

4.2 Discrete Component Model

4.2.5.2

207

Results of Calculations (Kerosene and Kerosene Surrogate Droplets)

In this section, the results of the application of the Discrete Component (DC) model to the analysis of kerosene droplet heating and evaporation, in the conditions of the experiments described in [92], are presented following [152]. In these experiments, heating and evaporation of droplets of kerosene with diameters 0.9 mm to 1.1 mm, initial temperature 298 K, supported by SiC fibre of 100 µm diameter were considered. Gas temperatures 400 ◦ C to 800 ◦ C and ambient atmospheric pressure were used in the experiments. The effect of support was modelled using the approach described in Sect. 3.7. The Nusselt number, considering the effects of natural convection and droplet evaporation, was estimated using a slightly modified Churchill correlation [35]:   0.589Ra 1/4 ln(1 + BT ) , (4.37) Nune = 2 +   9/16 4/9 BT 1 + (0.469/Pr) where Pr and Ra are the Prandtl and Rayleigh numbers, respectively; BT is the Spalding heat transfer number defined by Expression (3.28). The results of a comparison between the observed values of relative squared diameters of droplets d 2 /d02 , where d0 is the initial droplet diameter, versus normalised time t/d02 and those predicted by the model for ambient gas temperature 500 ◦ C and for the kerosene composition presented in Table 4.5 is shown in Fig. 4.15. This normalisation was used to make it easier to compare the model predictions with the results of the experiments. The following models were used: the Discrete Component (DC) model considering the contributions of all kerosene components and both natural convection and supporting fibre (plots M); the same model as used for plot M, but not considering the contribution of the supporting fibre (plots M1); the same model as used for plot M, but not considering the contribution of the supporting fibre and natural convection (plots M2). As in [92], d0 is assumed equal to 1 mm. As in [188], we assumed that the fibre temperature was equal to half the gas temperature measured in ◦ C (the fibre temperature was taken equal to 250 ◦ C for the case shown in Fig. 4.15). As can be seen in Fig. 4.15, the predicted values of d 2 /d02 are reasonably close to those measured experimentally, when the effects of natural convection and supporting fibre are considered. The effect of supporting fibre is small and can be not taken into account when analysing these experiments. Ignoring the contribution of natural convection, however, leads to noticeable overestimation of the droplet evaporation time. There could be several reasons for the deviation between the modelling and experimental results. The kerosene composition which we used in the modelling is not the same as used in the experiments. Our model is rather simplistic. The errors in measurement of gas temperature, taken with a thermocouple, could exceed 8 ◦ C for gas temperature 500 ◦ C [64]. Finally, droplet diameters could be different from the 1 mm used in our analysis. The calculations like those shown in Fig. 4.15 were performed

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4 Heating and Evaporation of Multi-component Droplets

Fig. 4.15 Plots of relative squared diameters of droplets d 2 /d02 versus t/d02 predicted by the Discrete Component (DC) model for ambient gas temperature equal to 500 ◦ C. The kerosene composition is presented in Table 4.5. Curve M is the prediction of the DC model considering the effect of supporting fibre and natural convection (Eq. (4.37). Curve M1 (M2) shows the prediction of the same model when the effects of supporting fibre (supporting fibre and natural convection) are not considered. The droplet initial diameter d0 was assumed equal to 1 mm. The experimental data are presented by the circles. Reprinted from [152], Copyright Elsevier (2020)

for droplet initial diameters 0.9 mm and 1.1 mm. It was shown that the experimental curves lie between these new curves [152]. The same level of agreement between the modelling results and experimental data was observed for gas temperature 600 ◦ C. In the case of gas temperature 700 ◦ C, however, the agreement between the modelling results and experimental data was poorer than for the case shown in Fig. 4.15. This is attributed to several factors, including the limitations of the gas phase model used in modelling. As shown earlier (see Chap. 3), the main assumptions of the gas phase model that the total density of the mixture of vapour and air is constant is likely to lead to large errors in model predictions for high gas temperature. As in the case of gas temperature 500 ◦ C, however, the experimental curves for this temperature lay between the curves for droplet initial diameters 0.9 mm and 1.1 mm. This gives us reasonable confidence in good predictability of the DC model in a rather wide range of gas temperatures [152]. A comparison between evaporation times predicted by the models and those observed experimentally for gas temperatures 400 ◦ C to 800 ◦ C is shown in Table 4.9. Note that we compared not the total evaporation times, but the time instants when

4.2 Discrete Component Model

209

Table 4.9 Time instants when droplet diameters reached the minimal observable values (second column) for five ambient gas temperatures, inferred from the experiments of [61] (first column) and predicted by models M, M1 and M2 (third to fifth columns). Droplet diameters were assumed equal to 1 mm in all calculations. Reprinted from [152], Copyright Elsevier (2020) Temperature Experimental M M1 M2 results [61] 400 ◦ C 500 ◦ C 600 ◦ C 700 ◦ C 800 ◦ C

0.127 mm, 6.540 s 0.134 mm, 3.989 s 0.169 mm, 2.587 s 0.172 mm, 1.920 s 0.199 mm, 1.308 s

5.022 s

5.093 s

6.258 s

3.574 s

3.658 s

4.507 s

2.713 s

2.791 s

3.443 s

2.214 s

2.282 s

2.810 s

1.842 s

1.901 s

2.336 s

droplet diameters reached the minimal observable values. These minimal droplet diameters are presented in the second column of this table. As follows from Table 4.9, the evaporation times predicted by model M at ambient temperatures 500 ◦ C to 700 ◦ C are close to those observed experimentally in agreement with Fig. 4.15. This table, however, shows considerable deviations between the modelling and experimental results for gas temperatures 400 ◦ C and 800 ◦ C. A possible reason for discrepancies between these results for 800 ◦ C was discussed earlier. The reasons for the discrepancies for gas temperature 400 ◦ C are less clear [152]. Analysis similar to the one described above was performed for simplified kerosene composition shown in Table 4.6. The predictions based on Tables 4.5 and 4.6 turned out to be close. This allows us to use either of these tables for the analysis of heating and evaporation of kerosene droplets [152]. Plots similar to those presented in Fig. 4.15, but for droplet surface temperatures, showed that these temperatures strongly depend on the choice of the model, as in the case of droplet radii. Considering the effects of supporting fibre and natural convection led to prediction of higher droplet surface temperatures compared with the case when these effects were ignored. It was shown that the predicted droplet surface temperatures increased when gas temperature increased [152]. It was shown that for initial droplet diameter and gas temperature equal to 1 mm and 700 ◦ C, respectively, the surface mass fraction of the least volatile component (cycloalkane) monotonically increased with time. Mass fractions of most other components initially increased and then decreased with time. In the case of naphtobenzene, this mass fraction initially decreases with time, then increases and then decreases again until droplet fully evaporated. The composition of kerosene at the surface of the droplets at any given time instant appeared to be rather different for different gas temperatures. The effects of this change in droplet composition on diffu-

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4 Heating and Evaporation of Multi-component Droplets

sion of kerosene vapour components could be considered using the model developed in [198]. This is beyond the scope of our analysis. In what follows we refocus on the comparison between the heating and evaporation characteristics of droplets of the surrogate fuels presented in Tables 4.7 and 4.8 and those of kerosene, with a composition presented in Table 4.5, for the same conditions as used in Fig. 4.15. Two groups of surrogates shown in Tables 4.7 and 4.8 were considered: one with decane as the main component or one of the main components (Group 1) and another with dodecane as the main component or one of the main components (Group 2). The plots of normalised droplet diameters versus normalised time for kerosene and kerosene surrogates are presented in Fig. 4.16. The results for surrogates of Group 1 are presented in the top part of this figure, while those for surrogates of Group 2 are shown in its bottom part. As one can see from Fig. 4.16 (top), the evaporation characteristics of droplets of surrogate SU1 (surrogate 12) are the closest to those of kerosene among the surrogates of Group 1. The evaporation of all other surrogates in this group is visibly slower than that of kerosene. As one can see from Fig. 4.16 (bottom), the evaporation characteristics of droplets of the modified Utah surrogate (surrogate 7) are the closest to those of kerosene among the surrogates of Group 2. The droplets of Drexel surrogate 2 (surrogate 8) and Slavinskaya surrogate (surrogate 11) evaporate slower than kerosene droplets, while droplets of all other surrogates of this group evaporate quicker than kerosene droplets. The plots of relative evaporation times τ = [(te((K) − te((S) )/te((K) ] × 100% versus surrogate numbers for gas temperatures 500 ◦ C and 700 ◦ C are presented in Fig. 4.17. These times were estimated for the time instants when droplet radii reached their minimal values (cf. Table 4.9). As follows from this figure, the minimal absolute values of τ are predicted for Surrogates 7 (modified Utah surrogate) and 12 (SU1). In most cases, τ is negative, and this shows that kerosene droplets tend to evaporate faster than surrogate droplets. The values of τ for both temperatures are rather close in most cases. Plots similar to those presented in Fig. 4.16 but for droplet surface temperatures are shown in Fig. 4.18. As follows from the latter figure, in Group 1 the heating characteristics of droplets of SU1 (surrogate 12) are closest to those of kerosene droplets, as in the case shown in Fig. 4.16 (Top). In Group 2, the heating characteristics of droplets of surrogate 7 (modified Utah surrogate) are closest to those of kerosene droplets, as in the case shown in Fig. 4.16 (Bottom). Considering the balance between the heating and evaporation characteristics of droplets of 13 surrogates investigated in [152] it can be concluded that those of SU1 and the modified Utah surrogates are the closest to those of kerosene droplets. Note that the authors of [152] did not make a distinction between the different types of kerosene and this limits the applicability of their analysis. Additionally, some simplifications used in the analysis presented in [152] turned out to be too crude for accurate investigation of droplet heating and evaporation. The authors of [147] addressed essentially the same problem as considered in [152], but using

4.2 Discrete Component Model

211

Fig. 4.16 (Top) Plots of relative droplet diameters (d 2 /d02 ) versus t/d02 (in s/mm2 ) for kerosene (curve K) and Group 1 kerosene surrogate (decane dominated) droplets heated and evaporating in air at temperature 500 ◦ C and atmospheric pressure. Numbers near the curves show the numbers of surrogates presented in Tables 4.7 and 4.8. (Bottom) the same as (Top) but for Group 2 kerosene surrogate (dodecane dominated) droplets. Reprinted from [152], Copyright Elsevier (2020)

a different approach to its solution. Firstly, their analysis was restricted to Jet-A (aviation kerosene) and surrogates developed specifically for this fuel. Secondly, a number of improvements to the model used in [152] were made. Thirdly, the model was implemented in the in-house code MFSim, in which the Discrete Component (DC) model was implemented, opening the way for modelling the droplet heating and evaporation process alongside other spray processes.

212

4 Heating and Evaporation of Multi-component Droplets

Fig. 4.17 Plots of τ = (te((K) − te((S) )/te((K) (in percent) versus surrogate number. e refers to the evaporation time, K and S refer to kerosene and surrogates, respectively. Circles and triangles refer to the cases when the ambient gas temperatures are equal to 500 ◦ C and 700 ◦ C, respectively. Reprinted from [152], Copyright Elsevier (2020)

In [147], heating and evaporation characteristics of droplets of Jet-A surrogates were compared with those of Jet-A droplets using the same input parameters as in the experiments described in [207]. It was shown that the evaporation time of a droplet of the Surrogate from [2] (Jameel’s surrogate, 2 components) was the closest to that predicted for a Jet-A droplet, while droplets of the Surrogate from [50] (Dooley’s 2nd generation surrogate, 4 components) and the Surrogate from [221] (Improved Dooley’s 2nd generation surrogate, 4 components) presented the best results when droplet evaporation rate was the target parameter. Furthermore, the maximum temperature predicted for droplets of the Surrogate from [207] (Won’s 4th surrogate, 9 components) was shown to be the closest to that predicted for a Jet-A droplet. Sensitivity analysis of various numerical parameters was also conducted in [147]. These included: the number of concentric layers used to discretise the droplet volume, the absolute accuracy of the bisection method used for finding the eigenvalues in the analytical solutions for temperature and species mass fractions, the number of terms in the series in the analytical solutions for temperature and species mass fractions, and the duration of timesteps used in calculations. It was shown that the number of layers should not be less than 250. Using a less restrictive accuracy for finding the eigenvalues did not significantly reduce the computational cost and resulted in greater errors of predicted temperatures and component mass fractions. Also, it was shown that the number of terms of the series for temperature and component mass fractions can be reduced from 200 to 20 without affecting the accuracy of calculations. It was observed that droplet temperature is more sensitive than evaporation time to variations in these numerical parameters. To summarise the results presented in this section, it can be concluded that the Discrete Component (DC) model is an effective modelling tool for the analysis of heating and evaporation of both kerosene and kerosene surrogate droplets.

4.2 Discrete Component Model

213

Fig. 4.18 The surface temperature versus normalised time t/d02 predicted by the same model as in Fig. 4.16. Top and bottom plots refer to the results for surrogates of Groups 1 and 2, respectively. Reprinted from [152], Copyright Elsevier (2020)

4.2.6 Drying Droplets This section focuses on the application of the Discrete Component (DC) model to the analysis of the process of droplet drying. The importance of this modelling in pharmaceutical and engineering applications has been widely discussed in the literature, and the results have been summarised in various review papers including [146]. In some applications, drying of droplets can be considered as an integral

214

4 Heating and Evaporation of Multi-component Droplets

part of more complex processes, including combustion of droplets with aluminium nanoparticles [61]. Mathematical models of the process were developed by many authors, including [34, 44, 113, 114, 128, 130, 137, 150, 151, 205]. The latter model was implemented in ANSYS Fluent CFD code; the results of calculations were validated using in-house experimental data [150]. The model described in [136] was focused on the analysis of evaporation and thermal decomposition of a single iron(III) nitrate nonahydrate/ethanol droplet. The models described in these papers, however, were based on several simplifying assumptions. These included the assumption of the absence of temperature gradient inside droplets. The applicability of these assumptions has not been well investigated to the best of the author’s knowledge. In contrast to the above-mentioned papers, the authors of [93] developed a threedimensional model for the analysis of drying a single evaporating droplet. Their model was able to describe the agglomeration process, including droplet shrinkage, particle accumulation on droplet surface and buckling. A Lagrangian approach was used for tracking suspended solid particles, and the Eulerian approach was used for the liquid phase. The model allowed the authors to further investigate the effect of the ratio of solid particle size to suspension droplet size, liquid surface tension and rate of evaporation on the buckling and deformation of droplets. Although the model described in [93] provides useful insight of the underlying physics of the drying process, the complexity of this model limits its engineering applications. The authors of [175] developed a new model of the process taking into consideration the gradients of dissolved non-evaporating substance mass fractions (including solid particles) and temperature gradients inside droplets in a self-consistent way. Their model was essentially based on further development and application of the DC model in which the mass fraction of one of the components in the gas phase at the droplet surface was assumed equal to zero. In Sects. 4.2.6.1 and 4.2.6.2 the key ideas of the model developed by the authors of [175] and its application are described. Note that in our analysis we do not consider the problem of the formation of cenospheres (hollow spheres) during the drying process [158, 209].

4.2.6.1

Description of the Model of Droplet Drying

The model developed in [175], and summarised in this section, is essentially the DC model described earlier. This model was used for the investigation of heating and evaporation of a two-component droplet consisting of evaporating liquid (water) and a non-evaporating component (solid particles or polymer). The relevant values of εi , defined by Expression (4.5), were εl = 1 and εs = 0 for evaporating and nonevaporating substance, respectively. The vapour pressure at the droplet’s surface was estimated by Raoult’s law (Expression (4.15)). In this expression, the liquid molar fraction at the droplet’s surface is expected to be close to 1 at the beginning of the evaporation process and close to zero at the end of this process. The thermal swelling was considered.

4.2 Discrete Component Model

215

The dissolved non-evaporating substance was analysed similarly to non-dissolved substances with masses of particles equal to molecular masses. This allowed the authors of [175] to find the liquid diffusion coefficient using the Wilke-Chang formula:  7.4 · 10−15 M v T , (4.38) Dl = μl Vv0.6 where M v is the average liquid molar mass Mv =

 i=2

−1 (Yi /Mi )

,

(4.39)

i=1

i = 1 = l and i = 2 = s refer to liquid and non-evaporating matter, respectively, Vv = (σv /1.18)3 ,

(4.40)

σv (the Lennard-Jones length (in Å)) was found from the formula 0.297

σv = 1.468 M v

.

(4.41)

If dissolved non-evaporating substance is approximated as an array of spherical particles with identical radii R p then Dl can be estimated using the Stokes-Einstein formula [109]: kB T , (4.42) Dl = 6π μR p where k B is the Boltzmann constant and T and μ are the liquid temperature (in K) and dynamic viscosity, respectively. The average values of liquid density, specific heat capacity and thermal conductivity, considering the contributions of both evaporating and non-evaporating components were estimated following the averaging procedures described in [148, 173]. The viscosity of the mixture of evaporating and non-evaporating components was assumed to be the same as that of evaporating component. The dependence of all properties on temperature was considered.

4.2.6.2

Application of the Model of Droplet Drying

The analysis of [175] focused on the modelling of drying water droplets with dissolved polymer (chitosan). Since the shapes of these polymer molecules are expected to be very far from spherical, Expression (4.38) will be used to estimate Dl . The initial mass fraction of chitosan (Y p0 ) was 0.004 and that of water was 0.996. The average ambient gas temperature was taken equal to 393 K. Ambient pressure was atmospheric. The effects of flow velocity on droplet heating and evaporation pro-

216

4 Heating and Evaporation of Multi-component Droplets

Fig. 4.19 Droplet radius (in µm) versus time for input parameters described in Sect. 4.2.6.2. Reprinted from [175], Copyright Elsevier (2018)

Fig. 4.20 Water and chitosan surface mass fractions versus time for input parameters described in Sect. 4.2.6.2. Reprinted from [175], Copyright Elsevier (2018)

cess were not considered. The average initial droplet diameter and temperature were assumed equal to 20 µm and 293 K, respectively [127]. The plot of droplet radius versus time is presented in Fig. 4.19. As can be seen from this figure, a well-known droplet evaporation process is clearly seen until approximately t = 0.127 s. Then, the evaporation process stops and the droplet turns into a solid ball of radius 1.51 µm. The plots of surface mass fractions of water and chitosan versus time are presented in Fig. 4.20. As can be seen from this figure, the contribution of chitosan in the mixture is negligible at t 0.12 s, however, the mass fractions of water and chitosan become close. At t ≈ 0.127 s, the mass fraction of chitosan is larger

4.2 Discrete Component Model

217

than that of water. These properties are consistent with the result shown in Fig. 4.19. Possible links between the results shown in Figs. 4.19 and 4.20 and experimental data are discussed in [175].

4.3 Quasi-discrete Model Although the Discrete Component (DC) model, introduced in the Sect. 4.3, can potentially be used for any number of components, its practical applicability is limited only to the case when the number of these components is relatively small. As follows from the review presented in Sect. 4.1, for large number of components different approaches, using the Continuous Thermodynamics and Distillation Curve models, were suggested. The main limitation of those models is that they are not capable in principle of considering the diffusion of liquid components in droplets. As follows from the results presented in Sect. 4.2, this process needs to be considered in many engineering applications. An approach to modelling heating and evaporation of multi-component droplets, focused on the case when a large number of components is present in the droplets, was described in [53, 170]. In contrast to the previously suggested models, designed for the analysis of droplets with large numbers of components, the approach described in [53, 170] considers the diffusion of liquid components and thermal diffusion as in the classical Discrete Component (DC) model. This model was called the ‘quasi-discrete model’. In this section, it will be presented and discussed following [53]. It is possible to draw a parallel between the approach to modelling the heating and evaporation of multi-component droplets described in [53, 170] and the modelling of absorption of thermal radiation in molecular gases, using the weighted-sum-ofgrey-gases method [132, 165]. In this method the medium is assumed to consist of several fractions of grey gases with different (but grey) absorption coefficients. The accuracy of this method turned out to be sufficient for many engineering applications [41, 72]. At the same time its application is much more CPU efficient compared with the rigorous approach in which all or most of the molecular absorption bands are considered.

4.3.1 Description of the Quasi-discrete Model As in the Continuous Thermodynamics approach, the quasi-discrete model is based on the distribution function f m (I ), described in Sect. 4.1 (cf. Expression (4.19)). An obvious limitation of this approach is that it is applicable only in the case when all properties of the components depend on only one parameter I . Although molar mass M is almost universally used to describe the property I , this choice is certainly far from being a unique one. Remembering that most practically important hydrocarbon fuels consist mainly of molecules of the type Cn H2n+2 (alkanes), where n ≥ 1 in the

218

4 Heating and Evaporation of Multi-component Droplets

general case or n ≥ 5 for liquid fuels, we can present f m as a function of the carbon number n rather than M. These two parameters are linked by the relation: M = 14n + 2,

(4.43)

where M is in kg/kmole. Assuming that f m (n) can be approximated by (4.19), the latter expression can be rearranged to:    M(n) − γ (M(n) − γ )α−1 f m (n) = Cm (n 0 , n f ) exp − , β α Γ (α) β where n 0 ≤ n ≤ n f , subscripts 0 and the largest) values of n,  Cm (n 0 , n f ) =

nf n0

f

(4.44)

stand for initial and final (the smallest and

   −1 M(n) − γ (M(n) − γ )α−1 exp − dn . β α Γ (α) β

(4.45)

Expression (4.45) for Cm is the normalisation condition leading to

nf

f m (n)dn = 1.

n0

The other parameters in (4.44) are the same as in (4.19). Note that most real life hydrocarbon fuels, including Diesel and petrol fuels, apart from alkanes, contain significant amounts of alkenes, alkynes, naphthenes and aromatics. The contribution of these components is not considered in the quasidiscrete model developed by the authors of [53, 170], and this is one of the main limitations of this model. The Antoine equation for the dependence of the saturation vapour pressure of alkanes p sat (in MPa) on n was used [15]:  B(n) , p (n) = exp A(n) − T − C(n) 

sat

(4.46)

where A(n) = 6.318 n 0.05091 ,

B(n) = 1178 n 0.4652 , C(n) = 9.467 n 0.9143 ,

T is in K. These approximations for A(n), B(n), C(n) were derived for 4 < n < 17. They were used for n ≥ 17 assuming that the contribution of the components with these n is small. Following [15], we can formally find L from the Clausius-Clapeyron equation as:

4.3 Quasi-discrete Model

219

Fig. 4.21 Plots of L(n) versus n for several T inferred from Expression (4.48). Reprinted from [170], Copyright Elsevier (2011)

L=−

Ru d ln p sat (n) , M(n) d(1/T )

(4.47)

where Ru is the universal gas constant. Note that this approach to finding L is strictly speaking not self-consistent in the general case as the Clausius-Clapeyron equation is based on the assumption that L does not depend on T . Remembering (4.46), Formula (4.47) can be written as: L=

Ru B(n)T 2 . M(n)(T − C(n))2

(4.48)

Plots of L versus n for temperatures 300 K to 500 K, as predicted by Expression (4.48), are shown in Fig. 4.21. As can be seen from this figure, L increases with decreasing temperature as expected. For n < 10, L slowly decreases with increasing n, while at larger n it increases with increasing n; this increase is particularly strong at low temperatures. This is consistent with the expectation that heavier components are generally less volatile than lighter ones. The values of L for n = 10 and n = 12 (n-decane and n-dodecane), inferred from Expression (4.48), are close to those used in [5]. The approximations of the dependence of liquid density, viscosity, specific heat capacity and thermal conductivity on n are described in Appendix N. As follows from (4.46), (4.48) and the results shown in Appendix N, the transport and thermodynamic properties of the fuels under consideration are relatively weak functions of n. In this case, it is sensible to assume that the properties of hydrocarbons in a certain narrow range of n are close and replace the continuous distribution (4.44) with a discrete one, consisting of N f quasi-components with carbon numbers  nj n

j−1 n j =  nj

n j−1

n f m (n)dn f m (n)dn

.

(4.49)

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4 Heating and Evaporation of Multi-component Droplets

The corresponding molar X j and mass Y j fractions are obtained from the following formulae

n j

Xj =

f m (n)dn,

(4.50)

n j−1

M(n j )X j Y j =  j=N f  , M(n j )X j j=1

(4.51)

where j is an integer in the range 1 ≤ j ≤ N f . Note that

j=N f

j=1



j=N f

Xj =

Y j = 1.

(4.52)

j=1

The choice of n j can be arbitrary. As the first approximation it was assumed that the step sizes n j − n j−1 are equal, i.e., all quasi-components have the same range of n. For N f = 1 this approach reduces the analysis of multi-component droplets to mono-component ones. These quasi-components are not the actual physical hydrocarbon components (n j are not integers in the general case). Hence, this model is called a quasi-discrete model. These quasi-components are considered as actual components in the DC model, including considering diffusion of liquid components in droplets. This model is expected to be particularly useful when N f is much less than the number of actual components in the mixture. One can draw an analogy between quasi-components and pseudo-components introduced in [18]. The mixtures are considered as ideal (Raoult’s law is valid). This allows us to estimate the partial pressures of individual quasi-components as: pv (n j ) = X ls j (n j ) p sat (n j ),

(4.53)

where X ls j are the molar fractions of liquid quasi-components at the droplet’s surface, p sat (n j ) are inferred from Expression (4.46). Having replaced n in Expression (4.48) with n j the values of L for all quasicomponents are obtained.

4.3.2 Application to Diesel and Petrol Fuel Droplets Following [15] we consider the values of parameters α, β, γ , n 0 and n f in Function (4.44) for Diesel and petrol fuels presented in Table 4.10.

4.3 Quasi-discrete Model

221

Table 4.10 The values of α, β, γ , n 0 and n f for Diesel and petrol fuels Fuel Diesel Petrol

α 18.5 5.7

β (kg/kmole) γ (kg/kmole) 10 15

0 0

n0 5 5

nf 25 18

Fig. 4.22 Plots of f m (n) versus n inferred from Expression (4.44) for Diesel (thin solid) and petrol (thick solid) fuels with the values of parameters presented in Table 4.10. Reprinted from [53], Copyright Elsevier (2012)

The plots of f m versus n for Diesel and petrol fuels for the values of parameters shown in Table 4.10 are presented in Fig. 4.22. As can be seen in this figure, these plots are rather different. The values n for which f m is maximal are equal to 12.4 and 5 for Diesel and petrol fuels, respectively. The average values of n (n) for these fuels are 12.56 and 7.05, respectively. The results of the analysis of droplet heating and evaporation for both fuels are presented in the following sections following [53, 170].

4.3.2.1

Diesel Fuel

It is assumed that the initial droplet temperature is the same in the whole droplet volume and equal to 300 K. Gas temperature and pressure are taken equal to 880 K and 3 MPa, respectively. The initial composition is described by distribution function (4.44) for Diesel fuel, shown in Fig. 4.22, with the values of parameters presented in Table 4.10. The plots of droplet surface temperatures Ts and radii Rd versus time for the initial droplet radius equal to 10 µm are shown in Fig. 4.23. The droplet velocity is assumed to be constant and equal to 1 m/s. The calculations were performed for the cases of N f = 1 (one quasi-component, n = 12.56) and N f = 20 (twenty quasicomponents), using the ETC/ED and ITC/ID models. In the same figure, the plots of Ts and Rd versus time for N f = 20, using the ETC/ED model, but assuming that

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4 Heating and Evaporation of Multi-component Droplets

Fig. 4.23 Plots of Ts and Rd , predicted by four models, versus time. The initial droplet radius and temperature are taken equal to 10 µm and 300 K, respectively; the droplet velocity is assumed to be constant and equal to 1 m/s; gas temperature is taken equal to 880 K. These are the models used for calculations: Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) model using one quasi-component; ETC/ED model using twenty quasi-components and Infinite Thermal Conductivity/Infinite Diffusivity (ITC/ID) model using twenty quasi-components. The approximations for density, viscosity, heat capacity and thermal conductivity of liquid components are presented in Appendix N. The fourth model is the ETC/ED model using twenty quasi-components with the density, viscosity, heat capacity and thermal conductivity of these components taken equal to those of n-dodecane, following [170]. Reprinted from [53], Copyright Elsevier (2012)

the density, viscosity, heat capacity and thermal conductivity of all components are the same and equal to those of n-dodecane, are presented. As can be seen from Fig. 4.23, the droplet radii Rd and surface temperatures Ts , predicted by the ETC/ED model, using one and twenty quasi-components are noticeably different, especially at the final stages of droplet heating and evaporation. The model using twenty quasi-components predicts higher Ts and longer evaporation times compared with the model using one quasi-component. This is related to the fact that at the final stages of droplet evaporation the components with large n become the dominant, as shown later. These components evaporate slower, and their surface temperatures can reach higher values than in the case of the components with lower n. As follows from Fig. 4.23, Ts and Rd predicted by the model used in [170] (density, viscosity, heat capacity and thermal conductivity of the liquid components assumed to be equal to those on n-dodecane) are noticeably different from those predicted by the more rigorous model used in [53] (the approximations for thermodynamic and transport coefficients of liquid components [217] are presented in Appendix N). This demonstrates the limitations of the model described in [170]. Also, Fig. 4.23 shows rather different values of Ts and Rd predicted by the ETC/ED and ITC/ID models, using twenty quasi-components. These differences are particularly noticeable for Ts at the initial stage of heating and evaporation. Accurate

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223

Fig. 4.24 Plots of Ts (top) and Rd (bottom) versus the number of quasi-components N f for the same conditions as in Fig. 4.23 at time instant 0.5 ms as predicted by the ETC/ED model (squares) and ITC/ID model (triangles). Reprinted from [53], Copyright Elsevier (2012)

prediction of this temperature is important for the prediction of auto-ignition timing in Diesel engines [167]. This questions the reliability of the models for heating and evaporation of multi-component droplets, based on the ITC/ID approximations, almost universally used in various engineering applications and incorporated into commercial CFD codes, including ANSYS Fluent (see Sect. 4.1 for further discussions). The plots of droplet surface temperatures Ts and radii Rd at time equal to 0.5 ms versus the number of quasi-components N f , predicted by the ETC/ED and ITC/ID models, are presented in Fig. 4.24 for the same input parameters as in Fig. 4.23. Symbols refer to those N f for which calculations were performed. The approximations for thermodynamic and transport properties of liquid components, given in Appendix N, were used. As follows from this figure, for N f ≥ 10 the predicted Ts and Rd no longer depend on N f . In fact the differences between the surface temperatures and radii, predicted for N f = 5 and N f = 20, are negligible compared with the differences between the temperatures, predicted by the ETC/ED and ITC/ID models. Hence, heating and evaporation of Diesel fuel droplets can be safely modelled using only 5 quasi-components, in agreement with our earlier results [170], obtained for

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time instant 0.25 ms using a simplified version of the quasi-discrete model. The errors due to the ITC/ID approximation for N f ≥ 3 are significantly larger than those due to the choice of a small number of quasi-components, especially for the surface temperature. These errors cannot be ignored in most engineering applications, and this questions the applicability of the models using the ITC/ID approximation, including the widely used Continuous Thermodynamics models. The results shown in Fig. 4.24 are consistent with those obtained for other instants of time, including time equal to 1 ms [53]. The closeness of the temperatures predicted by ETC/ED and ITC/ID models at the later stages of droplet heating and evaporation is related to the fact that at this stage the droplet temperature becomes almost homogeneous (see Fig. 9 of [170]), and the effects of the temperature gradient inside droplets can be ignored. In agreement with [170], smaller droplet radii are predicted by the ITC/ID model, compared with the ETC/ED model, at the final stages of droplet heating and evaporation. Note that at early stages of droplet heating and evaporation (t = 0.5 ms), the predicted Rd reduces slightly with the increase in the number of quasi-components used, while at a later stage (t = 1 ms) the opposite effect can be seen, in agreement with the results presented in [170]. This is attributed to the fact that at the early stages, and droplet evaporation is controlled by the most volatile quasi-components, while at the later stages it is controlled by less volatile quasi-components. When the number of quasi-components increases then the volatility of the most volatile component increases and that of the least volatile decreases.

4.3.2.2

Petrol Fuel

Plots like those presented in Fig. 4.23, but for petrol fuel, are shown in Fig. 4.25. The maximal number of quasi-components for petrol fuel is 13. The initial conditions are considered to be the same as in the case of Diesel fuel droplets to enable us to compare heating and evaporation characteristics of Diesel and petrol fuel droplets in identical conditions. As in the case presented in Fig. 4.23, the droplet velocity is considered to be constant and equal to 1 m/s during the whole process. The calculations were performed for N f = 1 (one quasi-component, n = 7.05) and N f = 13 (thirteen quasi-components), using the ETC/ED and ITC/ID models. The thermodynamic and transport properties of all liquid components are presented in Appendix N. As in the case of Diesel fuel droplets, the droplet radii Rd and surface temperatures Ts , predicted by the ETC/ED model, using one and thirteen quasi-components are noticeably different, especially at the final stages of droplet heating and evaporation. The model using thirteen quasi-components predicts higher surface temperatures and longer evaporation time compared with the model using one quasi-component. As for Diesel fuel droplets, this is related to the fact that at the final stages of droplet evaporation the components with large n become the dominant, as shown later. These components evaporate slower than the components with lower n and their surface temperatures reach higher values.

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Fig. 4.25 The same as Fig. 4.23 for the first three curves, but for the petrol fuel with the maximal number of quasi-components N f = 13. All plots are based on the approximations for density, viscosity, heat capacity and thermal conductivity of liquid components presented in Appendix N. Reprinted from [53], Copyright Elsevier (2012)

The differences in predictions of the ETC/ED and ITC/ID models, using thirteen quasi-components, are more clearly seen than in the case of Diesel fuel droplets. These differences are observed not only at the initial stage of droplet heating and evaporation but also at the later stages of these processes. This provides additional support for our questioning of the reliability of the models for heating and evaporation of multi-component droplets, based on the ITC/ID approximations. The plots of Ts and Rd at time equal to 0.75 ms versus the number of quasicomponents N f , predicted by the ETC/ED and ITC/ID models for petrol fuel droplets, are presented in Fig. 4.26 for the same conditions as in Fig. 4.25. As can be seen in Fig. 4.26, for N f ≥ 6 the predicted Ts and Rd do not depend on N f . In fact the difference between the values for temperatures and radii, predicted for N f = 3 and N f = 13, is negligible compared with the difference between the values for temperatures and radii, predicted by the ETC/ED and ITC/ID models. Hence, heating and evaporation of petrol fuel droplets can be safely modelled using just 3 quasicomponents. As for Diesel fuel droplets, the errors due to the ITC/ID approximation for N f ≥ 3 are significantly larger than those due to the choice of a small number of quasi-components, especially for the surface temperature. Note that at an early stage of droplet heating and evaporation (t = 0.2 ms), the predicted droplet radii reduce slightly with the increase in the number of quasicomponents, while at a later stage (t = 0.75 ms) the opposite effect is observed, in agreement with the results presented in [170], and those discussed earlier for Diesel fuel droplets.

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Fig. 4.26 Plots of Ts (top) and Rd (bottom) versus the number of quasi-components N f for the same conditions as in Fig. 4.25 at time 075 ms as predicted by the ETC/ED model (squares) and ITC/ID model (triangles). Reprinted from [53], Copyright Elsevier (2012)

Plots similar to those presented in Fig. 4.25, but for more realistic conditions in petrol engines, are shown in Fig. 4.27. Following [22], we assume that ambient gas temperature is equal to 450 K, gas pressure is equal to 0.3 MPa, and droplet constant velocity is equal to 10 m/s. As in the case presented in Fig. 4.25, we assume that the initial droplet temperature is the same in the whole volume of the droplet and is equal to 300 K. The droplet initial radius is taken equal to 10 µm. Comparing Figs. 4.25 and 4.27, in the latter case the difference between the predicted temperatures and droplet radii for one and thirteen quasi-components is much more visible than in the former. The same conclusion applies to the predictions of the ETC/ED and ITC/ID models. This can be attributed to slower evaporation for the case presented in Fig. 4.27, compared with the case presented in Fig. 4.25. The general trends of the curves presented in Fig. 4.27 are similar to the ones presented in Fig. 4.25. In the case when thirteen quasi-components are used, at the end of the evaporation process, mainly the heavier components in the droplets remain. These can reach higher temperatures and evaporate slower compared with the light and middle-range components. The plots of Ts and Rd at time equal to 0.5 ms versus the number of quasicomponents N f , predicted by the ETC/ED and ITC/ID models for petrol fuel

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Fig. 4.27 The same as Fig. 4.25 but for droplet constant velocity equal to 10 m/s, ambient gas temperature and pressure equal to 450 K and 0.3 MPa, respectively. Reprinted from [53], Copyright Elsevier (2012) Fig. 4.28 The plots of Ts (top) and Rd (bottom) versus the number of quasi-components N f for the same conditions as in Fig. 4.27 at time 0.5 ms as predicted by the ETC/ED model (squares) and ITC/ID model (triangles). Reprinted from [53], Copyright Elsevier (2012)

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Fig. 4.29 The plots of Ysi versus time for four quasi-components (i = 1, 2, 3, 4) for the same case as shown in Fig. 4.27. Reprinted from [53], Copyright Elsevier (2012)

droplets, are presented in Fig. 4.28 for the same conditions as in Fig. 4.27. As follows from Fig. 4.28, for N f ≥ 6 the predicted Ts and Rd do not depend on N f (cf. the cases shown in Fig. 4.26). This range can be extended to N f ≥ 3 at least for the ETC/ED model. In contrast to the cases presented in Fig. 4.26, the temperatures and radii, predicted by the ETC/ED and ITC/ID models, appear to be very close for small numbers of quasi-components. This is related to the fact that in this case the temperature reaches saturation level by 0.5 ms, when one or two quasi-components are considered. The results presented in Fig. 4.28 are consistent with those obtained for time instant equal to 2 ms. In both cases, droplet surface temperatures and radii are well predicted by the ETC/ED model with only three quasi-components. In contrast to the case presented in Fig. 4.26, at time equal to 2 ms the temperatures and radii, predicted by the ETC/ED and ITC/ID models, are very close only for the case when one quasi-component is used. The plots of Ysi versus time for the four quasi-components for the same case as presented in Fig. 4.27 are shown in Fig. 4.29. The results shown in Fig. 4.29 are consistent with those presented in Fig. 8 of [170] for Diesel fuel droplets using a simplistic approach to approximate thermodynamic and transport properties of the liquid components. The values of Ys1 monotonically decrease with time, while those of Ys4 monotonically increase with time. The values of Ys2 and Ys3 initially increase with time, but at later times they rapidly decrease with time. At times close to the time instant when the droplet completely evaporates, only the quasicomponent Ys4 remains. Since this quasi-component is the most slowly evaporating one and its surface temperature reaches the highest values, the model based on four quasi-components is expected to predict longer evaporation times and larger droplet surface temperatures at the final stages of droplet evaporation, compared with the model using one quasi-component. This result can be generalised to the case when the number of quasi-components is greater than four. These results are consistent with those presented in Fig. 4.27.

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Effects of petrol fuel approximations on heating and evaporation of petrol fuel droplets were investigated in [55]. The following approximations were considered: 3 quasi-components introduced in the quasi-discrete model (molar fractions: 83% nC6.26 H14.58 + 15.6% n-C10.26 H22.48 + 1.4% n-C14.42 H30.84 ) and their approximations: Surrogate I (molar fractions: 83% n-C6 H14 + 15.6% n-C10 H22 + 1.4% n-C14 H30 ) and Surrogate II (molar fractions: 83% n-C7 H16 + 15.6% n-C11 H24 + 1.4% n-C15 H32 )), surrogate approximations of petrol fuel based on its ignition characteristics: Surrogate A (molar fractions: 56% n-C7 H16 + 28% iso-C8 H18 + 17% C7 H8 ) and Surrogate B (molar fractions: 63% n-C7 H16 + 20% iso-C8 H18 + 17% C7 H8 ). Surrogates I and II allowed the authors of [55] to investigate the sensitivity of the results to the choice of the values of the number of carbon atoms for each of three quasi-components. Also, this rounding up or down of the values of the carbon numbers allows one to use these approximations of quasi-components in Computational Fluid Dynamics (CFD) codes, which do not recognise substances with non-integer n. Surrogates A and B were considered in [63]. The results were compared with the predictions of the model using the approximation of petrol fuel by 13 and 1 quasi-components [55]. It was shown that the predictions of the quasi-discrete model using the approximation of petrol fuel with three quasi-components, especially Surrogate II, are more accurate than those based on other surrogate approximations. These predictions were compared with the predictions of the quasi-discrete model based on 13 quasi-components. This shows the limitation of using fuel surrogates, suggested based on fuel ignition characteristics [63], for the analysis of petrol fuel droplet heating and evaporation.

4.4 Multi-dimensional Quasi-discrete Model 4.4.1 Description of the Model As follows from the analysis presented in Sect. 4.3 the quasi-discrete model is useful and efficient for many practical applications. At the same time, it has several serious limitations. The most important of these limitations is the assumption that multicomponent liquids (Diesel and petrol fuels) consist only of alkanes. At the same time, in the case of Diesel fuel the total molar fraction of alkanes (n-alkanes and isoalkanes) in it is only about 40%. Thus, the contribution of other components apart from alkanes cannot be ignored when analysing droplets of this fuel. Also, even if we restrict our analysis to alkanes alone, it does not appear to be easy to approximate their distribution using a simple function f m (n), similar to the one described by Formula (4.44). In [173], the quasi-discrete model was generalised to address both these issues. This generalisation was focused on Diesel fuels, but the principles on which it was based are expected to have much larger range of applications.

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Fig. 4.30 Molar fractions of various hydrocarbons in a representative sample of Diesel fuel. Reprinted from [71], Copyright Elsevier (2013)

A realistic composition of Diesel fuel, schematically presented in Fig. 4.30, was used in the analysis of [173]. In what follows the key features of the model described in [173], called the Multi-dimensional Quasi-discrete Model, are summarised. The results shown in Fig. 4.30 were simplified, remembering that the properties of n-alkanes and iso-alkanes are close. Since the contributions of tricycloalkanes, diaromatics and phenanthrenes to Diesel fuel are small (less than about 1.6% for each of these components) we can ignore the dependence of the properties of these components on the number of carbon atoms n and replace these three groups with three components, tricycloalkane, diaromatic and phenanthrene, with arbitrary n in the range shown in Fig. 4.30. The molar fractions of tricycloalkanes, diaromatics and phenanthrenes were estimated to be 1.5647%, 1.2240% and 0.6577%, respectively. Transport and thermodynamic properties of the components are presented in Appendices 1-7 of [173] (cf. estimates of pressure (up to 350 MPa) and temperature (up to 160 ◦ C) dependence of volume and viscosity of Diesel fuels in [20]). In the Multi-dimensional Quasi-discrete Model, the focus is shifted from the analysis of the distribution function to the direct analysis of molar fractions of the components. These are described by the matrix X nm , where n refers to the number of carbon atoms, and m refers to the groups (e.g. alkanes) or individual components (tricycloalkane, diaromatic and phenanthrene). The link between the values of m and the groups or components is presented in Table 4.11. For each m the values of n jm were estimated as [173]:

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Table 4.11 The relation between parameter m and groups (m = 1-6) and components (m = 7-9). Reprinted from [173], Copyright Elsevier (2014) m Component 1 2 3 4 5 6 7 8 9

Alkanes Cycloalkanes Bicycloalkanes Alkylbenzenes Indanes and tetralines Naphthalenes Tricycloalkane Diaromatic Phenanthrene

n 1m = n 2m =

n=n (ϕm +1)m

(n X nm ) n=n 1m , n=n (ϕm +1)m X nm n=n 1m

n=n (2ϕm +2)m

n=n (ϕm +2)m (n X nm ) , n=n (2ϕm +2)m n=n (ϕm +2)m X nm

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ n 3m = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ...................................... ⎪ ⎪ ⎪ n=n km ⎪ n=n ((−1)ϕm +)m (n X nm ) ⎪ ⎭ n m = n=n ,⎪ n=n (3ϕm +3)m

(4.54)

n=n (2ϕm +3)m (n X nm ) n=n (3ϕm +3)m n=n (2ϕm +3)m X nm

km n=n ((−1)ϕm +)m

X nm

where n 1m =n m(min) is the minimal n for which X nm = 0, n km =n m(max) is the maximal n for which X nm = 0,  = integer ((km + ϕm )/(ϕm + 1)). Parameter ϕm is assumed to be integer; ϕm + 1 is equal to the number of components to be included into quasicomponents, except possibly the last one in the group. ϕm is considered to be the same for all quasi-components in group m. If ϕm = 0 then  = km and the number of quasi-components is equal to the number of actual components. ϕm and km depend on m in the general case. If ((km + ϕm )/(ϕm + 1)) is an integer then n m = n km m . As in the case of the quasi-discrete model, n im are not integers in the general case. Due to the additional dimensions introduced by the subscript m in Expression (4.54), this model was called the Multi-dimensional Quasi-discrete Model (MDQDM). MDQDM reduces to the single-component model in the limiting case when all components are replaced with a single quasi-component. The maximal number of quasi-components/components in this model, providing the most accurate approximation of Diesel fuel, is equal to the actual number of components (98 in the case considered in [173]). In this case, this model reduces to the conventional Discrete Component (DC) model. The quasi-components in the MDQDM are treated in the same way as the quasi-components in the quasi-discrete model. Also,

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the temperature and quasi-component gradients inside droplets are considered as in the quasi-discrete model. In [173] the selection of quasi-components for Diesel fuel which leads to errors in calculations acceptable for engineering applications was investigated. Some key findings of that paper are described in Sect. 4.4.2.

4.4.2 Application to Diesel Fuel Droplets In [173] the model described in Sect. 4.4.1 was applied to the analysis of heating and evaporation of a Diesel fuel droplet with initial radius Rd0 = 10 µm in air with density, temperature, and pressure equal to ρa = 11.9 kg/m3 , Ta = 880 K, pa = 30 bar, respectively. Properties of the components presented in Appendix O were used in the analysis. The plots of the droplet surface temperatures Ts and radii Rd versus time for a stationary droplet and four approximations of Diesel fuel composition are presented in Fig. 4.31. These are the approximations used: the contributions of all 98 components are considered (indicated as (98)); the contributions of only 20 alkane components are considered (standard approximation used in the quasi-discrete model (indicated as (20A)); the contribution of all 98 components is approximated by a single quasi-component (indicated as (S)); and

Fig. 4.31 The plots of the droplet surface temperatures Ts and radii Rd versus time for four approximations of Diesel fuel composition (see the text of the section). Reprinted from [173], Copyright Elsevier (2014)

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the contributions of only 20 alkane components are considered and these are approximated by a single quasi-component with the average value of the carbon number (C14.763 H31.526 ; indicated as (SA)). When only the contribution of alkanes was considered, the mass fractions of the components were recalculated to ensure that the total mass fraction of all alkanes is equal to 1. The same applies to the cases when the components are removed from the analysis. As can be seen in Fig. 4.31, the approximation of 98 components by a single quasicomponent leads to a noticeable under-estimation of the droplet surface temperature and an under-estimation of the evaporation time by about 17%. In the case when Diesel fuel was approximated by 20 alkane components, the predicted droplet surface temperatures appeared to be higher and the evaporation time shorter by about 23% than in the case of approximation of Diesel fuel by 98 components. This means that the approximation of Diesel fuel by alkanes, a widely used assumption in the modelling of Diesel fuels, leads to results which are less accurate, compared with the approximation of Diesel fuel by a single quasi-component. The approximation of Diesel fuel by a single alkane quasi-component (C14.763 H31.526 ) leads to underprediction of the evaporation time by about 37% which is not acceptable even for qualitative analysis of the process. This leads us to questioning the validity of the results of numerous papers where Diesel fuel was approximated by a single alkane component (n-dodecane in most cases). Note that in all cases presented in Fig. 4.31 the droplet surface temperatures keep increasing with time until the droplets evaporate. This is consistent with our earlier investigations of this process (e.g. [164]). This result questions the applicability of the assumption that the droplet surface temperature remains constant during the evaporation process which is widely used in simplified models. The well-known d 2 -law is implicitly based on this assumption (see Sect. 3.2.6). The plots of the droplet surface temperatures Ts and radii Rd versus time for the same conditions as in Fig. 4.31 but for a wider range of approximations of Diesel fuel are presented in Figs. 4.32 and 4.33. Only the final stage of droplet heating and evaporation is shown in these figures. As can be seen from Figs. 4.32 and 4.33, the curves S and S7 (ignoring the contribution of diaromatic and phenanthrene) for surface temperatures and radii are almost indistinguishable. Also, plots 9 and 7 (ignoring the contribution of diaromatic and phenanthrene) are close. The same applies to plots 23 and 21 (ignoring the contribution of diaromatic and phenanthrene). This means that the contribution of diaromatic and phenanthrene can be safely ignored in the approximation of Diesel fuel when modelling the heating and evaporation of droplets in realistic Diesel engine-like conditions. Both for droplet surface temperatures and radii, the accuracy of approximations improves as the number of QC/Cs increases. For 15 QC/Cs the droplet evaporation time was estimated with an error of about 2.5%. For 21 QC/Cs, this error reduces to about 1.5%. This error is comparable with that for the approximation of Diesel fuel with 40 QC/Cs. Thus, when balancing simplicity with accuracy of the model the approximation of Diesel fuel with 21 QC/Cs can be recommended if errors less than about 2% can be tolerated. This number of QC/Cs can be reduced

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Fig. 4.32 Plots of the droplet surface temperatures Ts versus time for ten approximations of Diesel fuel composition: 98 components (indicated as (98)); 23, 21, 17, 15, 12, 9 and 7 quasicomponents/components (numbers near the curves); the contributions of all groups are approximated by single quasi-components, to which the contribution of tricycloalkane is added, leading to 7 quasi-components/components (indicated as (S7)); the contribution of all 98 components is considered as that of a single-component as in the case shown in Fig. 4.31 (indicated as (S)) (see [173] for the details); the contributions of only 20 alkane components are considered and these are treated as a single-component, with the average value of the carbon number (C14.763 H31.526 ; indicated as (SA)). The same ambient and initial conditions as in Fig. 4.31 were considered; only the final stage of droplet heating and evaporation is presented. Reprinted from [173], Copyright Elsevier (2014) Fig. 4.33 The same as Fig. 4.32 but for the droplet radii Rd . Reprinted from [173], Copyright Elsevier (2014)

to 15 if errors less than about 3% can be tolerated. The latter model requires about 6 times less CPU time compared with the model considering the contributions of all 98 components. The authors of [82] used a discrete continuous multi-component (DCMC) evaporation model for the analysis of Diesel fuel droplet heating and evaporation. 180 individual components from 11 hydrocarbon families were identified. Four continuous distribution functions for normal paraffins, mono-naphthenics, mono-aromatics,

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235

and naphthenic-mono-aromatics were derived. These were expected to cover 80% of the total Diesel fuel composition. As in the conventional Continuous Thermodynamics model, discussed in Sect. 4.1, the effects of temperature and component mass fraction gradients inside the droplets could not be taken into account in this model.

4.4.3 Application to Petrol Fuel Droplets The application of the Multi-dimensional Quasi-discrete Model (MDQDM) to petrol fuel droplets is described in [11]. In what follows the main results presented in that paper are summarised. The analysis of [11] focused on FACE-C petrol fuel (Fuel for Advanced Combustion Engines – C) droplets. Composition of this fuel was simplified by replacing groups of similar components with single-components (with averaged properties, based on averaged molar masses; or the ones with the highest molar contributions in the groups with molar fractions up to 1.5%). This approach allowed the authors of [11] to reduce the number of components in petrol fuel to 20. These components were allocated to 3 groups, n-alkanes (5 components), iso-alkanes (8 components), and aromatics (4 components); and 3 components approximating groups with small molar fractions (indanes/naphthalenes, cycloalkanes and olefins). Typical petrol engine conditions were used. The initial droplet radius and temperature were assumed equal to 12 µm and 296 K, respectively. The droplet relative velocity was assumed to be constant and equal to 24 m/s. Ambient gas (air) pressure and temperature were taken equal to p = 9 bar and Tg = 545 K, respectively. The plots of the droplet surface temperatures Ts and radii Rd versus time for FACE-C petrol fuel droplet heating and evaporation are shown in Fig. 4.34. Four cases are presented in this figure: (1) the contributions of all 20 components are considered using the ETC/ED model (indicated as (ME)); (2) the contributions of 20 components are considered using the ITC/ID model (indicated as (MI)); (3) the thermodynamic and transport properties of 20 components are averaged to form a single-component and temperature gradient inside the droplet is ignored (ITC model) (indicated as (SI)); and (4) the ITC model in which petrol fuel is approximated by iso-octane (2,2,4trimethylpentane; indicated as (IO)) is used. As follows from Fig. 4.34, the errors in droplet surface temperatures and evaporation times, predicted by the SI model are 13.6% and 67.5%, respectively. For the IO model these errors reduce to 6.3% and 47.1%, respectively, and reduce further to 4.8% and 8%, respectively, when the MI model is used. Although the accuracy of the latter model might be acceptable in some applications, this model cannot describe adequately the underlying physics of the processes inside droplets, including heat conduction and component diffusion.

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Fig. 4.34 The droplet surface temperatures Ts and radii Rd versus time for the cases when: (1) the contributions of all 20 components are considered using the ETC/ED model (ME); (2) the contribution of 20 components are considered using the ITC/ID model (MI); (3) the 20 components are approximated by a single-component with average thermodynamic and transport properties in combination with the ITC model (SI); (4) petrol fuel is approximated by iso-octane in combination with the ITC model (IO). Reprinted from [11], Copyright Elsevier (2015)

The same plots as in Fig. 4.34 but for the cases when 20 components of petrol fuel are approximated by 15 (3 QC/Cs of n-alkanes, 6 QC/Cs of iso-alkanes, 3 QC/Cs of aromatics, 1 indane/naphthalene, 1 cycloalkane and 1 olefin), 11 (2 QC/Cs of n-alkanes, 4 QC/Cs of iso-alkanes, 2 QC/Cs of aromatics, 1 indane/naphthalene, 1 cycloalkane and 1 olefin) and 7 (2 QCs of alkanes, 3 QC/Cs of iso-alkanes, and 2 QC/Cs of aromatics) QC/Cs, using the ETC/ED model, are presented in Fig. 4.35. As follows from Fig. 4.35, the errors in surface temperatures and evaporation times predicted by the model using 15 QC/Cs are 0.3% and 1.3%, respectively. These errors increase to 0.5% and 4%, respectively, when petrol fuel is approximated by 11 QC/Cs, and further increase to 0.8% and 6.4%, respectively, when it is approximated by 7 QC/Cs. Even in the latter case, however, these errors can be tolerated in many applications. This model is more accurate than the MI model, and it describes adequately the underlying physics of the processes in droplets. It was demonstrated that the approximation of the composition of petrol fuel by 6 quasi-components/components (2 QCs of n-alkanes, 2 QCs of iso-alkanes, and 2 QCs of aromatics), using the MDQDM, leads to errors in estimated droplet surface temperatures and evaporation times of about 0.9% and 6.6%, respectively, for typical engine conditions. This can be tolerated in some applications. It was demonstrated that the application of the latter model leads to about 70% reduction in CPU time compared to the model considering the contributions of all 20 components of petrol fuel.

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Fig. 4.35 The same as Fig. 4.34 but for the cases when the ETC/ED model was used considering the contributions of all 20 components of petrol fuel (indicated as ME) and assuming that these components are approximated by 15, 11 and 7 quasi-components/components (QC/Cs) (numbers are indicated near the curves). Reprinted from [11], Copyright Elsevier (2015)

4.4.4 Heating, Evaporation and Ignition of Fuel Droplets The original version of the Multi-dimensional Quasi-discrete Model (MDQDM) was specifically designed to model Diesel and petrol fuel droplet heating and evaporation but not their ignition characteristics. Several surrogates of these fuels have been developed, specifically to model these characteristics. For example, the authors of [186] used three Diesel fuel surrogates (2 components, 6 components and 8 components). The authors of [223] formulated their Diesel fuel surrogate of five-components: n-dodecane, iso-octane, isocetane, decalin and toluene. The authors of [209] also used five surrogates to approximate Diesel fuel: n-hexadecane, n-octadecane, isocetane, 1-methylnaphthalene and decalin. The same authors used five surrogates to approximate petrol: n-heptane, iso-octane, toluene, diisobutylene and cyclohexane. A six-component surrogate of Diesel fuel from direct coal liquefaction (DDCL) is described in [86]. This surrogate was shown to describe well the chemical and physical processes taking place in fuel sprays, including the ignition delay. A threecomponent Diesel fuel surrogate (n-cetane, 2,2,4,4,6,8,8-heptamethylnonane and 1-methylnaphthalene) was investigated by the authors of [222]. See [28, 31, 47, 79, 149, 154, 208, 221] for the discussion of other similar surrogates. The results of the development of a Diesel Surrogate Fuel Library containing 18 fuels to approximate Diesel chemical and physical properties are presented in [194]. An approach for surrogate fuel formulation by matching target fuel functional groups, while minimising the number of surrogate components, was suggested in

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[89]. Surrogates were developed for six FACE (Fuels for Advanced Combustion Engines) petrol fuels: FACE A, C, F, G, I and J. In these cases, the analysis of fuel droplet heating and evaporation could be performed using a much simpler Discrete Component (DC) model. The main problem with this approach is that the applicability of most of these surrogates to the analysis of fuel droplet heating and evaporation has not been carefully investigated to the best of the author’s knowledge [55]. A new formulation of physical surrogates of FACE-A petrol fuel, based on droplet heating and evaporation characteristics, was offered by the authors of [56]. These surrogates were developed using an approach similar to that used in the development of the MDQDM. They were shown to be suitable for modelling both heating/evaporation of fuel droplets and the ignition characteristics of a fuel vapour/air mixture. The main ideas of the model suggested in [56] are described below. The analysis of [56] focused on FACE-A petrol fuel. Firstly, four surrogates of FACE-A developed earlier were described. These surrogates include the fivecomponent surrogate chosen for its ability to match the ignition delay time of the FACE-A petrol fuel (called Surr1), the primary reference fuel surrogate (PRF84) that matches the research octane number (RON) of FACE-A, the one that matches hydrogen-to-carbon ratio (H/C), RON, density and distillation curve with FACE-A (Surr2), and the one that matches the RON based on molar fraction linear blending (Surr3). It was demonstrated that these surrogates cannot predict adequately the time evolution of surface temperatures and radii of FACE-A droplets. New ‘physical’ surrogates with 8, 7 and 6 components (Surr4, Surr5 and Surr6) were suggested to match the heating and evaporation characteristics of these droplets. FACE-A petrol fuel has the following mass fractions 10.57% n-paraffins, 86.12% iso-paraffins, 0.37% aromatics, 2.49% naphthalenes and 0.45% olefins which represent 66 components. To design surrogates Surr4, Surr5 and Surr6, these 66 components were replaced by 19 components to represent FACE-A. This reduction in the number of components was based on merging components from the same chemical groups and having the same chemical formula, which have close thermodynamic and transport properties; the components with the highest initial mass fractions were chosen to be the representative components. The heating and evaporation characteristics of all previously suggested and new surrogates were verified against the results predicted by this 19-component model. New ‘physical’ surrogates were developed via further simplifications of this 19component model by retaining the most important components and ignoring the contributions of other components. Mass fractions of components in these surrogates are presented in Table 4.12. Firstly, an 8-component surrogate, Surr4, retaining the same mass fractions of nbutane, n-heptane, iso-pentane and iso-octane as in FACE-A, was described. These components contribute more than 70% of the total mass of FACE-A petrol fuel. It was demonstrated that 2-methylpentane, 3-methylhexane and 2,3-dimethylpentane have similar evaporation behaviour; they were replaced by 3-methylhexane which contributes 25.875% of Surr4. The remaining minor components were replaced by 2,6-dimethyloctane, 1t,2 dimethylcyclopentane and 1-methyl-2-propylcyclohexane.

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239

Table 4.12 Mass fractions (in %) of three ‘physical’ surrogates of FACE-A petrol fuel. Reprinted from [56], Copyright Elsevier (2016) # Component Surr4 (8 Comp) Surr5 (7 Comp) Surr6 (6 Comp) 1 2 3 5 7 9 14 16

n-butane 3.919 n-heptane 6.652 iso-pentane 12.784 3 methyl hexane 25.875 iso-octane 46.869 2,61.194 dimethyloctane 1t,2 dimethylcy- 1.585 clopentane 1-methyl-21.121 propylcyclohexane

3.919 8.238 12.784 25.875 46.869 1.194

3.919 8.238 12.784 25.875 48.063 0.000

0.000

0.000

1.121

1.121

Table 4.13 H/C ratio, molar mass and RON of FACE-A fuel and seven surrogates. Reprinted from [56], Copyright Elsevier (2016) Target FACE A PRF 84 Surr1 Surr2 Surr3 Surr4 Surr5 Surr6 H/C ratio 2.29 M 97.8 (kg/kmol) RON 83.5

2.26 112

2.28 101.5

2.28 102

2.26 106.5

2.29 98.6

2.3 98.64

2.3 98.44

84

85.3

86.6

85.6

80.3

79

79.5

The composition of Surr4 was further simplified in a 7-component surrogate, Surr5, in which n-heptane and 1t,2 dimethylcyclopentane were replaced by n-heptane. Finally in a 6-component surrogate, Surr6, the composition of Surr5 was simplified by replacing iso-octane and 2,6-dimethyloctane with iso-octane. It was demonstrated that the heating and evaporation characteristics of droplets of the new surrogates are much closer to those inferred from the 19 component model, compared to those of the previously suggested surrogates PRF84, Surr1, Surr2 and Surr3. The evaporation time for the Surr6 droplet was demonstrated to be almost identical to that of the FACE-A fuel droplet, while the maximal error in the prediction of the droplet surface temperature did not exceed 2%, which is acceptable in most applications. The evaporation times for Surr4 and Surr5 droplets were predicted to be longer than those of the FACE-A droplets by 5%. The difference between the predicted droplet surface temperatures for Surr4 and Surr5 and that for FACE-A did not exceed 13%. It was suggested that Surr5 could be considered an optimal physical surrogate. To illustrate the suitability of the new surrogates to represent FACE-A fuel in engine applications, three additional properties were considered: the H/C ratio, molar mass and RON. These are shown in Table 4.13 alongside similar properties predicted for surrogates PRF84, Surr1, Surr2 and Surr3. Matching molar masses and H/C ratios

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4 Heating and Evaporation of Multi-component Droplets

of the target fuels indicate both matching diffusivity and flame speed, while matching RONs indicate matching ignition delay time. The values of these properties for FACE-A, PRF84, Surr1, Surr2 and Surr3 were taken from the literature (see [56]). The RONs for Surr4, Surr5 and Surr6 were calculated following the previously suggested procedure using the detailed composition of fuels (see [56]). As follows from Table 4.13, Surr4, 5 and 6 have RON, molar masses and H/C ratios marginally closer to those of FACE-A, than these characteristics of the previously suggested surrogates. Therefore, surrogates Surr4, 5 and 6 not only improved the predictions of heating and evaporation of droplets but also have better representations of RON, molar masses and H/C ratios. The authors of [4, 190] drew attention to the fact that there was no comprehensive petrol surrogate that could mimic both the evaporation and combustion of the target petrol simultaneously with a limited number of components. They suggested new approaches to the design of such surrogates, in a different way to that suggested in [56]. In [4], surrogate components were selected to emulate the H/C ratio, hydrocarbon class distribution, heating value, research and motor octane numbers (RON and MON), density and distillation curve. Also, their choice was constrained by the availability of kinetic mechanisms. Thus, their approach to the selection of surrogates was more stringent than that of [56], where chemical characteristics were concerned, and less stringent than that of [56], where physical characteristics were concerned (the analysis of the physical characteristics of surrogates performed in [4] focused only on the distillation curve). The five-component petrol surrogate suggested by the authors of [215] (n-heptane, iso-octane, 1-hexene, iso-hexane and toluene) was focused on chemical properties of the fuel. The analysis of [4] led to the development of a seven-component petrol surrogate for emulation of the physical and chemical properties of USA non-oxygenated petrol fuel RD387. It was shown that the surrogate successfully reproduces the distillation curve, H/C, density and heating value. It was also shown to be able to adequately reproduce the first stage and total ignition delay times. Finally, it led to reproduction of the RCM pressure traces with acceptable errors. The laminar flame speeds of the surrogate were also simulated and compared with experimental data for a wide range of pressures and equivalence ratios. Good agreement between the surrogate and petrol laminar flame speeds was demonstrated, especially for lean to stoichiometric conditions. It was suggested that the surrogate can be used in internal combustion engine modelling. Petrol fuel six-component surrogates developed by the authors of [69] were focused on reproduction of real fuel physical and chemical properties. The following target properties were chosen for surrogate optimisation: the true boiling point curve, research octane number, liquid density, carbon-to-hydrogen (C/H) ratio and the oxygenate (ethanol) content. As in the case of [4], the choice of ‘physical’ surrogate in [190] was based on matching the distillation curves. A new 6-component surrogate mixture composed of n-pentane/n-heptane/toluene/iso-octane/n-propyl cyclohexane/iso-undecane was developed in [190] to match the targeted petrol in terms of thermodynamic and trans-

4.4 Multi-dimensional Quasi-discrete Model

241

port properties and distillation curve. This new surrogate also covered the toluene reference fuel components, which are the three basic components for ignition modelling. This allowed the authors to match both physical and chemical characteristics of petrol fuel using only one surrogate. In [190] the temperature distribution inside droplets was described by two parameters: surface and core temperatures. This approach is much simpler and less accurate than the one used in MDQDM, where the detailed distribution of temperature inside droplets was considered. In contrast to [56], the authors of [190] considered the effects of turbulence on liquid thermal diffusivity. In [191] a multi-component droplet evaporation model was integrated with detailed fuel chemistry and soot models for simulating biodiesel/Diesel spray combustion. The model was validated against available experimental results. Gas phase chemical reactions were simulated using a detailed reaction mechanism including PAH reactions leading to the production of soot precursors. The model was applied to predict combustion, soot and NOx emissions from a Diesel engine. The hybrid multi-component (HMC) model, described in [218], can be considered a simplified version of the MDQDM. In the HMC model, the multi-component fuels (petrol fuels were considered in [218]) were modelled as several discrete classes, each of which was described by a separate distribution function. An alternative approach to formulation of complex fuel surrogates was suggested by the authors of [7]. Their approach was focused on petrol, Diesel and biodiesel fuels and blends of biodiesel/Diesel and ethanol/petrol fuels. Their approach, called a ‘Complex Fuel Surrogates Model’, was based on a modified version of the MDQDM. In contrast to MDQDM all components in their surrogates had integer carbon numbers. They managed to reduce the full composition of fuel to a much smaller number of components based on their mass fractions. The formulated surrogates matched the data of the full compositions of the fuels referring to droplet evaporation time, time evolutions of surface temperature, density, vapour pressure, H/C ratio, molar mass and research octane number (RON).

4.4.5 Biodiesel/Diesel/Ethanol/Petrol Droplets The application of the Multi-dimensional Quasi-discrete Model (MDQDM) to the analysis of heating and evaporation of blended biodiesel (soybean methyl ester, SME)/Diesel fuel droplets in typical Diesel engine-like conditions is described in [12]. Fuel considered by these authors contained up to 105 components, which were replaced with a smaller number of components and quasi-components (C/QCs), similarly to pure Diesel and petrol fuels considered in the previous sections. Transient diffusion of these C/QCs in the liquid phase, temperature gradient and recirculation inside droplets due to relative velocities between droplets and ambient gas were considered using the Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) model. It was demonstrated that the approximation of the full composition of the fuel by 17 C/QCs of B50 (50% SME and 50% Diesel) using this model led to deviations

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4 Heating and Evaporation of Multi-component Droplets

in estimated droplet surface temperatures and evaporation times of up to 1% and 4%, respectively. This can be tolerated in many applications. The selection of 17 C/QCs of B5 (5% SME and 95% Diesel), however, led to 9% error in estimated evaporation time. It was demonstrated that the application of this model to B50 and B5 with 17 C/QCs led to more than 83% reduction in CPU time compared to the model which considers the contributions of all 105 components. Thermophysical properties of several other biodiesel/Diesel blended fuels are presented in [88]. The results of modelling of blended ethanol/petrol fuel droplet heating and evaporation in conditions typical for internal combustion engines are described in [13]. The effects of ambient pressure, gas and radiation temperatures and ethanol/petrol fuel blend ratios on droplet heating and evaporation were investigated based on the Discrete Component (DC) model described earlier. The ambient pressures, gas and radiation temperatures, and ethanol/petrol fuel ratios were considered in the ranges of 3–30 bar, 400–650 K, 1000–2000 K and 0–100% (0% refers to pure petrol), respectively. It was shown that the combination of ethanol and petrol has a noticeable effect on droplet heating and evaporation. For pure ethanol, the droplet surface temperature was predicted to be 24.3% lower, and lifetime 33.9% higher, than those for petrol fuel under the same conditions. In [9], the results of application of the MDQDM to the analysis of heating and evaporation of droplets of mixtures of E85 (with volume fractions of 85 % of ethanol and 15 % of petrol) with Diesel fuel, known as E85-diesel blends, are described. The UNIFAC (UNIversal quasi-chemical Functional group Activity Coefficients) model for the calculation of vapour pressure was used. The contribution of 119 components of E85-diesel fuel blends was considered. These components were replaced with smaller number of components/quasi-components (C/QCs) using the MDQDM. Conditions typical for Diesel engines were used. It was shown that high fractions of E85 in these blends have a significant impact on the evolutions of droplet radii and surface temperatures. The application of the MDQDM was shown to improve considerably the computational efficiency with minimal sacrifice to accuracy. The impact of the activity coefficient on heating and evaporation of ethanol/petrol fuel droplets was discussed in [8]. The authors of [216] demonstrated the importance of considering the effect of the activity coefficient, calculated using the UNIFAC model, in the analysis of heating/evaporation of ethanol/iso-octane and iso-octane/n-pentane/n-decane mixture droplets. The results of experimental investigations of evaporation of binary alkane-ethanol blends are presented in [81]. These authors combined a high-resolution Raman setup with a fast trigger system to quantify vapour composition behind free falling droplets.

4.4.6 Auto-selection of Quasi-components/Components The selection of components/quasi-components (C/QCs) in the Multi-dimensional Quasi-discrete Model (MDQDM), used so far, was based on a trial-and-error-

4.4 Multi-dimensional Quasi-discrete Model

243

approach, which made it difficult to implement it into Computational Fluid Dynamics (CFD) codes. A new algorithm for automatic selection of C/QCs during the droplet evaporation process was developed by the authors of [14]. This new algorithm opens the way to the implementation of the MDQDM into any CFD code. This algorithm is briefly described below following [14]. In the algorithm suggested in [14], the contribution of all components in the multicomponent liquid are considered at the first timestep. At this stage, this algorithm is essentially based on the Discrete Component (DC) model considered in Sect. 4.2. Then the mass fractions of all groups of components G i at the droplet surface are calculated: Ni Gi = Yni , (4.55) n=1

where Yni are the mass fractions of individual components n in group i and Ni is the number of components in this group. The mass fractions were considered  as they are more sensitive to transient effects than the molar fractions. Note that i G i = 1. After the first-timestep, G i for all groups are expected to change due to component evaporation (increase or decrease). This change is quantified as: ΔG i =

|G i − G iold | . Gi

(4.56)

If ΔG i is greater than an a priori chosen small number K (in [14] K = 0.1 was used) in a particular group, the number of C/QCs within this group Ni is reduced from the previous number Niold by a certain factor F: Ni = [F Niold ] .

(4.57)

where [] indicates rounding up or down to the nearest integer (e.g. [7.5] = 8 and [7.4] = 7). The authors of [14] recommended F = 0.75, using a trial-and-error approach. The QCs in the algorithm described in [14] are formed of the components with the smallest molar fractions in any group i which typically correspond to the components with the largest carbon numbers. The selection is based on merging the least contributing components in that group first, starting with the components with the largest carbon numbers and ending with the components with the smallest carbon numbers. The number of C/QCs selected to form the new QCs is taken equal to [Ni /2]. In the case where [Ni /2] is an even number, the QCs are formed of each 2 components in the half of components with the largest carbon numbers. If [Ni /2] is odd, however, then the nearest component that is not selected is added to this group to form an even number of components and each two QC/Cs in this group are merged to form a new QC.

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4 Heating and Evaporation of Multi-component Droplets

For example, in the case of the alkanes, which include 20 of 98 Diesel fuel components, at the initial stage Niold = 20. If ΔG i > K , then Ni = [0.75 × 20] = 15 QC after the reduction process has been completed. The first 10 components remain unchanged, and the remaining 10 components form 5 QCs (each 2 components form 1 QC). If a certain group contains 11 components, these reduce to Ni = [0.75 × 11] = 8 QC. The first 5 components remain unchanged, while the last 6 components form 3 QCs containing 2 components each. In the new algorithm, users are allowed to define the minimum number of C/QCs. This option is built into the final stage of the algorithm when further auto-reduction in the number of C/QCs is blocked after this number reaches a certain minimum value. For example, if the minimum number of C/QCs is defined by the user as 10 and the remaining number of C/QCs is 15, the auto-reduction leads to 11 C/QCs ([0.75 × 15]). The further reduction of 11 C/QCs, however, would lead to less than 10 C/QCs. Hence, 11 C/QCs are auto-reduced to 10 C/QCs only. This algorithm was applied to the analysis of a wide range of E85 fuel (85% ethanol and 15% petrol) and Diesel fuel blends (E85, E85-5, E85-20, E85-50, and pure Diesel fuel). It was shown that using the new algorithm allows one to reduce the full compositions of E85-Diesel mixtures from their initial 119 components to 5 quasi-components/components at the end of the heating and evaporation process with less than 1.9% errors in predicted droplet lifetimes and temperatures. These predictions were shown to be more accurate than those obtained using the original version of the Multi-dimensional Quasi-discrete Model (MDQDM). The CPU time needed to run the new algorithm was shown to be 80% less than that needed by the Discrete Component (DC) model using the full composition of fuel [14]. The analysis of heating and evaporation of multi-component droplets thus far described has focused primarily on the liquid phase. It has been assumed that all vapour components in the gas phase behave as a single-component. This assumption is relaxed in Sect. 4.5 where some gas phase evaporation models for multi-component droplets are described.

4.5 Gas Phase Models for Multi-component Droplets In the classical Stefan-Fuchs model, Eq. (3.23) for evaporation of mono-component droplets was derived considering the conservation of vapour mass flux at any point around a stationary droplet. In the case of multi-component droplets, a similar condition can be imposed for all components in the gas phase. Following [197], this condition can be written as:   dYk d 2 2 (k,m) R ρtotal U Yk − R D = 0, (4.58) ρtotal dR dR where subscript k refers to ambient gas (k = 0) or fuel vapour components (k = 1, ....., n, n is the total number of vapour components), R ≥ Rd is the distance from

4.5 Gas Phase Models for Multi-component Droplets

245

the centre of the droplet in the gaseous phase, D (k,m) the mass diffusion coefficient for the component k in the mixture, Yk the mass fraction of component k, U the Stefan velocity defined as n |m˙ (k) | (4.59) U = k=12 d , 4π R ρtotal m˙ (k) d evaporation rate of component k, ρtotal the total density of the mixture, including ambient gas. The analysis of Eq. (4.58) is very difficult due to the fact that both ρtotal and D (k,m) are unknown functions of R. The analysis presented in this section follows mainly [197] and is based on the assumption that ρtotal and D (k,m) remain constant for all R (the assumption that ρtotal is constant was made when deriving Eq. (3.23)). The values of D (k,m) were estimated in the reference conditions as (Blanc’s law): ⎛

D (k,m)

⎞−1 n Y j (ref) ⎠ =⎝ , D (k, j)

(4.60)

j=0; j =k

where Y j (ref) =

2Y j (s) + Y j (∞) , 3

(4.61)

Y j (s) and Y j (∞) are the mass fractions of component j at the surface of the droplet and in ambient gas, respectively. Formula (4.61) allows us to consider ρtotal under the reference conditions as well (ρtotal = ρref ). Using new variable ζ = Rd /R, the general analytical solution to Eq. (4.58) can be presented as [197]: 

 | |m˙ (total) d ζ + εk , Yk = αk exp − 4πρtotal Rd D (k,m) where |m˙ (total) |= d

n

(4.62)

|m˙ (k) d |

k=1

,

|m˙ (k) | εk = n d (k) ˙d | k=1 |m

(4.63)

is the evaporation rate of component k, αk are unknown constants. Recalling that Yk (ζ = 0) = Yk∞ , it can be shown that αk = Yk∞ − εk . This allows us to present Expression (4.62) for the surface of the droplet (ζ = 1) as:

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4 Heating and Evaporation of Multi-component Droplets



| |m˙ (total) d Yks = (Yk∞ − εk ) exp − 4πρtotal Rd D (k,m)

 + εk .

(4.64)

Expression (4.63) was rearranged to [197]:   |m˙ (total) | Yks − Yk∞ exp − 4πρtotaldRd D(k,m) εk =  .  |m˙ (total) | 1 − exp − 4πρtotaldRd D(k,m)

(4.65)

This leads to the following equation: n k=1

Yks − Yk∞ Yk∞ .   = 1 = (total) |m˙ | 1 − exp − 4πρtotaldRd D(k,m) k=1 n



(4.66)

Equation (4.66) is non-linear. It was used in [197] to estimate the total evaporation rate |m˙ (total) | assuming that all other parameters are known. Once |m˙ (total) | was d d calculated, the values of εk were obtained from Expression (4.65). The Stefan-Fuchs equation (4.58) could be formulated in terms of molar rather than mass fluxes [198]. The latter equation could be solved under the assumption that the molar density of the mixture does not depend on the distance from the droplet surface. The solution to this equation would be rather like (4.65) and (4.66) and its explicit form was presented in [198]. These two equations and their solutions predict slightly different evaporation rates since the conditions of constant total mass density and constant molar density of the mixture are not equivalent. To consider the effects of multi-component droplet movement on droplet heating and evaporation, in [197] (as well as in a number of other papers, including [123]) it was assumed that there is no interaction between evaporating components. For each of these components the Abramzon and Sirignano model [6], described in Sect. 3.2.2, was applied. The validity of this assumption is not at first evident. This is the reason why, in many papers and books, including [164], the effect of relative motion between components in the gas phase has been ignored altogether. | inferred from Expression (4.66) and A comparison between the values of |m˙ (total) d obtained using a simplified model based on Eq. (3.23) for stationary droplets was also performed in [197]. In the latter formula it was assumed that ρtotal = ρref ; B M was calculated based on the summations of mass fractions of vapour components at the droplet’s surface and ambient conditions. Two approaches to calculating the diffusion coefficient in (3.23) were used. Firstly, this coefficient was calculated based on the Wilke and Lee formula (see Formula (45) in [54]) with all input parameters averaged over all components present in the system (this averaging did not consider different mass fractions of components). Secondly, the coefficient was calculated based on direct averaging of the values of D (k,m) considering mass fractions of components:

4.5 Gas Phase Models for Multi-component Droplets

n Yk (ref) D (k,m) n Dv = k=1 . k=1 Yk (ref)

247

(4.67)

The models using the first and second approaches were referred to in [197] as | inferred from Models 1 and 3. In [197], the predictions of the values of |m˙ (total) d (4.66) were compared with the predictions of Models 1 and 3 and the predictions of a simplified model described in [26], referred to as Model 2. It was shown that Models | which are almost identical to those inferred from 1 and 3 predict values of |m˙ (total) d (4.66), while Model 2 clearly underestimates the evaporation rate. It is anticipated that the predictions of Model 1 would be improved if the averaging of input parameters in the Wilke and Lee formula considered mass fractions of individual components as in Eq. (4.67). This invites a simplified model, where multi-component gas is treated as mono-component, when describing the heating and evaporation processes in multi-component droplets (cf. [164]). The authors of [45, 91, 142, 198] drew attention to the fact that more accurate description of multi-component diffusion, compared with Eq. (4.58), should be based on the Maxwell-Stefan equations. Ignoring the Soret effects, diffusion due to pressure gradients and external forces, these equations were written as [24, 198]: ∇ X ( p) =

n k=0

 ( p) (k)  1 X N − X (k) N( p) , Cm D pk

(4.68)

where X (k) is the molar fraction of the kth component, Cm is the molar density of the mixture, D pk = Dkp is the binary diffusion coefficient of the pth component into the kth component, N( p) is the molar flux of the pth component, k = 0 refers to ambient gas. For a multi-component spherical droplet only the radial components of the component molar fluxes is retained. In this case, Eq. (4.68) was presented in a similar format to that inferred from Eq. (4.58). This allowed the authors of [198] to present the solution to (4.68) in a similar format to (4.65) and (4.66), but for molar fractions, assuming that the total molar density does not depend on the distance from the droplet surface. The generalisation of the above analysis based on the Stefan-Maxwell equations to the case of the evaporation of multi-component spheroidal droplets is described in [199]. The local vapour fluxes of evaporating components on the droplet surfaces were correlated with the local Gaussian curvatures. The model predictions were compared with experimental measurements of tetradecane-hexadecane droplets evaporating in stagnant air. It was shown that the predictions using Eq. (4.58) (Stefan-Fuchs equation) underestimate the total evaporation rate, especially at high ambient gas temperatures, for various droplet compositions. The largest deviation of the absolute values of the evaporation rate, predicted by the Stefan-Fuchs and Maxwell-Stefan equations, was found when none of the component mass fractions was dominant.

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4 Heating and Evaporation of Multi-component Droplets

4.6 Other Approaches to Modelling Multi-component Droplets A quasi-dimensional multi-component heating and evaporation model for multicomponent fuel droplets is described in [219]. In contrast to the Discrete Component (DC) model (in [219] this model is referred to as the one-dimensional model) described in Sects. 4.1 and 4.2, the model suggested in [219] is based not on the rigorous solution to heat transfer and component diffusion equations inside droplets, but on the polynomial (quadratic) approximations of the temperature and component distributions inside droplets (in the case of temperature this approach is similar to the one used in the parabolic model described in Sect. 2.1.1). As in the case of the DC model, the analysis of the quasi-dimensional model is based on the ETC/ED model. Both the ideal gas approach (Raoult’s law is valid) and the real gas approach were used in the analysis of [219]. The Peng-Robinson equation of state and the van der Waals mixing rule were used. The radiative heating of droplets was considered. The authors of [219] believed that their model could be a reasonable compromise between a rigorous DC model and a simplistic model in which the gradients of temperature and component mass fractions inside droplets were ignored (this model was called the zero-dimensional model in [219]). The model was extensively validated against experimental measurements, and good agreement with these measurements was demonstrated (compared with the predictions of the zero-dimensional model). When modelling ethanol-blended petrol fuel droplet evaporation, the authors of [91] relaxed the assumption that Rault’s law is valid and estimated the vapour-liquid equilibrium using quantum-chemical ab initio approach (see Chap. 6). The MaxwellStefan diffusion and convection theory (see Eq. (4.68)) was used for the calculation of gas phase transport characteristics of the components. A simple relation providing the multi-component diffusion matrix as a power series in terms of the N − 1 independent mole fractions in the mixture, where N is the total number of molar fractions, was derived from the kinetic gas theory in [16]. This power series converged quickly for gas mixtures with one dominant component. A reduced multi-component diffusion model for the gas phase is described in [210] for application to premixed flames. The main ideas of this model could be applied to the problem of diffusion of components in a multi-component fuel although this has not yet been investigated to the best of the author’s knowledge. A simplified model for the analysis of multi-component droplet heating and evaporation, called Coupled Algebraic-Direct Quadrature Method of Moments (CADQMoM), was suggested and described in [39]. This model considers the temperature and component mole fraction gradients inside the droplets without solving the heat transfer and component diffusion equations. Both temperature and component mole fraction distributions inside droplets were assumed to be parabolic (cf. Expression (2.43)). The authors of this paper believe that the combination of accuracy and computational efficiency of this model makes it attractive for incorporation into Computational Fluid Dynamics (CFD) codes.

4.6 Other Approaches to Modelling Multi-component Droplets

249

A model for multi-component droplet evaporation, called analogical multicomponent (AMC) vaporization model, was suggested and described by the authors of [220]. This model was specifically focused on Diesel fuel droplets. The proposed AMC model was extensively validated using available experimental results. It was claimed that this model can provide a reasonable compromise between computational efficiency and accuracy.

4.7 Heating and Evaporation of Multi-component Liquid Films The models discussed so far in this chapter were focused on perfectly spherical droplets. Unfortunately, a model for spheroidal multi-component droplets, similar to the one described in Sect. 3.6 for mono-component droplets, has not been developed. However, some progress has been made in developing a model for the limiting case of a multi-component oblate spheroid when this spheroid reduces to liquid film [174]. The main ideas presented in the latter paper are described in this section. Observations and practical importance of modelling heating and evaporation of multi-component liquid films have been widely discussed in the literature (e.g. [59, 192, 193, 202, 228] and the references therein). The modelling of these processes has been described in many papers some of which are reviewed in [228]. The simplest models use the assumption that the liquid is well mixed and the thermal diffusion inside it is infinitely fast (zero-dimensional model) [133]. This assumption allowed the authors of [32] to develop a detailed model for the thermodynamically non-ideal vapour phase above the liquid film. In their model, evaporation was initiated and maintained by a spatial chemical potential gradient, while its rate was limited by the component diffusion fluxes across the vapour–liquid interface. In [141] the temperature gradients inside the liquid film were considered but using a rather simplistic assumption that the temperature distribution is a piecewise linear function (linear temperature model). The model using the assumption of a more complex polynomial distribution of temperature inside the liquid film is known as the quasi-dimensional model [228]. Finally, the models based on the rigorous solution to the heat conduction equation inside the liquid film and the assumption that temperature gradients in the direction perpendicular to the wall are much larger than those along the wall are described in [118, 228]. In contrast to most previously developed models of the phenomenon, in the model suggested in [174] the presence of multiple components in the liquid film was taken into account. This is particularly important for the analysis of automotive fuel films [51]. In the model described in [174], both thermal and component diffusion inside the liquid film were considered. As in the case of the analysis of multi-component droplet heating and evaporation (described earlier in this chapter), the model is based on the analytical solutions to the heat transfer and component diffusion equations. The film was assumed to be thin which allowed the authors of [174] to use the one-

250

4 Heating and Evaporation of Multi-component Droplets

dimensional model in which both temperature and liquid component mass fractions depend only on the distance from the wall. Following [174], the analysis starts with the case of mono-component liquid film (Sect. 4.7.1). In Sect. 4.7.2 this analysis is generalised to the case of multi-component liquid films. The solution algorithm is described in Sect. 4.7.3. The validation of the model against experimental data and its application to the analysis of bi-component fuel film heating and evaporation is described in Sect. 4.7.4. The verification of the results predicted by the solution algorithm described in Sect. 4.7.3, based on their comparison with the results predicted by an algorithm using the numerical solutions to the heat transfer and component diffusion equations, is presented and discussed in Sect. 4.7.5.

4.7.1 Mono-component Liquid Film 4.7.1.1

Liquid Phase

Assuming that the gradients of temperature in the film in the direction perpendicular to the wall are much larger than those in the direction parallel to it, the heat conduction equation inside the film can be presented as: ∂2T ∂T = κl 2 , ∂t ∂x

(4.69)

where t is time, x the distance from the wall, κl = kl /(cl ρl ) the liquid thermal diffusivity, kl , cl and ρl are the liquid thermal conductivity, specific heat capacity and density, respectively. Equation (4.69) describes the heat conduction process in this film except in the vicinity of its edges. Note that in thin films, the contribution of the convective term, ignored in Eq. (4.69), is negligible. Following [121], it is assumed that the liquid temperature at the wall is constant and equal to wall temperature Tw : T (x = 0, t) = Tw (Dirichlet boundary condition). The authors of [211] specified heat flux rather than temperature at the wall (Neumann boundary condition). This heat flux was estimated as qw = kw (Tout − Tin )/δw , where kw is the thermal conductivity of the wall, δw is the wall thickness, Tout and Tin are wall temperatures at the outer and inner boundaries. This approach to the estimation of the heat flux is applicable only in the case of steady-state problems, which is not compatible with the modelling of the transient process in the liquid film. The rigorous approach to this problem would require a coupled solution for the liquid film and the wall like the one considered earlier for spherical layers. To the best of the author’s knowledge, this approach to the problem of liquid film heating and evaporation has not been investigated. Following [121], the boundary condition at the surface of the liquid film (x = δ0 ) is presented as:

4.7 Heating and Evaporation of Multi-component Liquid Films

h(Teff − Ts ) = kl where Teff = Tg +

251

∂ T

, ∂ x x=δ0 −0

(4.70)

ρl L δ˙0e , h

(4.71)

the value of δ˙0e (the time derivative of the film thickness), controlled by film evaporation (indicated by the additional subscript e ), is taken from the previous timestep, L is the specific heat of evaporation, Tg and Ts are ambient gas and film surface temperatures, respectively. The value of δ˙0e is estimated in Sect. 4.7.1.2 (see Formula (4.76)). Equation (4.70) is the energy balance equation at the film’s surface: heat transferred by convection from ambient gas to the surface is spent on the evaporation of the film and its heating. This equation is the Robin boundary condition for Eq. (4.69) at the surface of the film. This boundary condition and one at the wall are supplemented by the initial condition T (t = 0) = T0 (x). Equation (4.69), with relevant boundary and initial conditions, in most cases has been solved numerically (e.g. [183]). The authors of [121] obtained an analytical solution to this equation subject to the above-mentioned boundary and initial conditions. A simplified and slightly corrected version of this solution is described as [174] (using notations different from those used in [121]):   X h0 (Teff − Tw ) + exp −κδ0 λ2n t [qn + f n h 0 (Teff − Tw )] sin(λn X ), 1 + h0 ∞

T (X, t) = Tw +

n=1

(4.72)

  where X = x/δ0 , h 0 = hδ0 /kl , κδ0 = kl / cl ρl δ02 , qn =

1 || vn ||2

1 fn = || vn ||2

1

1

(T0 (X ) − Tw ) sin(λn X )dX,

0

f (X ) sin(λn X )dX = −

0

  f (X ) = −X/(1 + h 0 ), || vn ||2 = 21 1 − sin2λ2λn n = λn are non-trivial solutions to the equation

1 2

λ cos λ + h 0 sin λ = 0.



sin λn , || vn ||2 λ2n

1+

h0 h 20 +λ2n



,

(4.73)

Note that there is a typo in Expression (7) of [121]: the sign before f n in their formula should be a minus. When deriving Solution (4.72) we took into account that

252

4 Heating and Evaporation of Multi-component Droplets

h 0 and Teff are constant during the timestep. With these assumptions, the last term in Expression (7) of [121] should be zero. Expression (4.72) is used at each timestep in the numerical code. The values of temperature predicted by this solution at the end of the previous timestep are used as the initial condition for the following timestep, with updated values of other input parameters, including the thickness of the film and gas temperature. Note that Solution (4.72) could be obtained from the corresponding solution for droplet heating and evaporation described in 2004 (see [166] and Appendix A).

4.7.1.2

Gas Phase

To complete the analysis presented in the previous section, two parameters, h and δ˙0e , need to be estimated. These parameters can be inferred from the gas phase model or from experimental data. As in [121, 174], it is assumed that the fuel vapour at the film surface is always saturated and the analysis of the evaporation process reduces to the analysis of vapour diffusion from the film surface to the ambient gas. In the case of evaporating droplets, it is typically assumed that vapour mass flow rate does not depend on the distance from the surface of the droplet and vapour mass fraction at an infinitely large distance from the droplet surface is zero (see Sect. 3.2). In the case of an evaporating liquid film, however, this assumption would lead to unphysical infinitely large evaporation rates and cannot be used in the model. As in [174], the value of the convection heat transfer coefficient h is imposed using experimental data. The results of experimental investigations of h in Diesel engine-like conditions showed that h can vary from about 500 W/(m2 K) to 5500 W/(m2 K) [157]. It is assumed, following [211], that h = 2000 W/(m2 K). Once the value of h has been estimated, the value of the mass transfer coefficient is estimated based on the Chilton-Colburn analogy as [185]: hm =

h Le−2/3 , ρg c pg

(4.74)

where Le = Sc/Pr is the gas Lewis number, Sc = μg /(ρg Dg ) is the Schmidt number, Dg is gas diffusivity, and Pr is the gas Prandtl number. The results of experimental validation of (4.74) are presented in [57]. For multicomponent vapour, the diffusivities of all components are considered to be the same, leading to a common Sc. The convection mass transfer coefficient predicted by Eq. (4.74) leads to the following expression for the evaporation mass flux from the film surface [201]: m˙ f = h m (ρvs − ρva ),

(4.75)

where ρvs and ρva are the vapour density at the surface of the film and in ambient gas, respectively (m˙ f ≥ 0). In our analysis, it is assumed that ρva = 0.

4.7 Heating and Evaporation of Multi-component Liquid Films

253

Note that Expression (3.23), derived for spherical droplets with finite radii, cannot be used for the analysis of evaporation from the surface of a film (cf. the model described in [204]). The value of δ˙0e is estimated as: δ˙0e = −

m˙ f ρ(T 0 )

,

(4.76)

where T 0 is the average temperature in the film. This formula is used in Expression (4.71) to obtain the value of Teff .

4.7.2 Multi-component Liquid Film All equations referring to heating mono-component films, described in the previous section, remain valid for multi-component films. However, the effect of mutual diffusion of components needs to be considered, as in the case of heating and evaporation of multi-component droplets. The modelling of the diffusion of components in the liquid film is discussed in Sect. 4.7.2.1. The implications of this diffusion for the processes in the gas phase are described in Sect. 4.7.2.2.

4.7.2.1

Liquid Phase

Assuming that the gradients of component mass fractions in the film in the direction perpendicular to the wall are much larger than those in the direction parallel to the wall (cf. similar assumption about temperature gradients in Sect. 4.7.1.1), the component diffusion equation inside the film can be presented as: ∂ 2 Yli ∂Yli = Dl , ∂t ∂x2

(4.77)

where t is the time, x is the distance from the wall (as in the case of Eq. (4.69)), and Dl is the liquid diffusion coefficient (assumed to be the same for all components (cf. similar assumption made for the analysis of multi-component droplets in Sect. 4.2). Equation (4.77) is to be solved subject to the following boundary conditions at the outer surface of the film and at the wall:

  ∂Yli

(4.78) = |δ˙0e | Yli |x=δ0 − εi , Dl

∂ x x=δ0 −0

∂Yli

= 0, ∂ x x=0

(4.79)

254

4 Heating and Evaporation of Multi-component Droplets



 where δ˙0e = |m˙ f /ρl | = h m i=N i=1 ρvsi /ρl is the same as in (4.76), ρl is the density of the mixture of liquid components, N is the total number of components in the film, Yvsi ρvsi εi = i=N = i=N , (4.80) i=1 Yvsi i=1 ρvsi Yvsi and ρvsi are the mass fraction and vapour density of the ith vapour component at the outer surface of the film, respectively. The initial condition for Eq. (4.77) is Yli (t = 0) = Yli0 .

(4.81)

Note the sign convention: m˙ f ≥ 0, while δ˙0e ≤ 0. The physical meaning of Eq. (4.78) is the same as that of Eq. (4.11) used for the analysis of heating and evaporation of multi-component droplets in Sect. 4.7.1.1. Equation (4.79) shows that components cannot penetrate through the wall. To simplify the analysis, we assume that εi = const. This assumption is supported by the fact that Eq. (4.77) is solved over a short timestep. The main parameter which could influence the values of εi is the rate of change of temperature during the timestep. This change in temperature is expected to be greatest during the heating up period of the film when its evaporation is the weakest. During the strongest evaporation at high temperatures, this change is expected to be small. Introducing a variable: (4.82) u = Yli − εi , Equation (4.77), boundary conditions (4.78) and (4.79) and initial condition (4.81) are presented as: ∂ 2u ∂u = Dl 2 , (4.83) ∂t ∂x



δ˙0e

∂u − u

= 0, (4.84) ∂x Dl

x=δ0 −0

∂u

= 0. ∂ x x=0

(4.85)

u(t = 0) = Yli0 (x) − εi = u 0 (x),

(4.86)

where the subscript 0 refers to initial values. One can see a similarity between Eqs. (4.83)–(4.86) and Eqs. (K.7)–(K.9) in Appendix K describing the component diffusion in multi-component droplets, except that for the droplets it was assumed that u at the centre of the droplet is zero (Dirichlet boundary condition) which is different from the corresponding boundary condition

4.7 Heating and Evaporation of Multi-component Liquid Films

255

for the film (Eq. (4.85), Neumann boundary condition). Hence, the need to find a new solution to Eq. (4.83) rather than to adapt the one described in Appendix K. As in Appendix K we look for the solution to (4.83) in the form: u ≡ u(t, x) =

∞

Θn (t)vn (x),

(4.87)

n=0

where vn (x) is the full set of non-trivial solutions to the equation: ∂ 2v + pv = 0, ∂x2

(4.88)

subject to the boundary conditions:

∂v

=0 ∂ x x=0 



δ˙0e

∂v − v

∂x Dl

(4.89)

= 0.

(4.90)

x=δ0

Equation (4.88) with boundary conditions (4.89) and (4.90) is the well-known SturmLiouville problem. Our first task is to find eigenvalues p for this problem. The case p = 0 leads to the trivial solution v = 0 (cf. Appendix K). The cases p < 0 and p > 0 are considered in Appendix P. The following set of eigenfunctions was obtained (see Eqs. (P.4) and (P.9)): ⎧   ⎨ cosh λ0 x δ 0  vn (x) = ⎩ cos λn x δ0

n=0

(4.91)

n ≥ 1,

where λn (n ≥ 0) are the solutions to the following equations (see (P.3) and (P.8)): λ0 D l coth λ0 = ,

δ˙0e δ0

λn D l cot λn = −

δ˙0e δ0

(n ≥ 1).

The substitution of (4.87) into (4.83) leads to the following equation: ∞



Θn (t)vn (x) = Dl

n=0

where 

Θn (t) =

∞



Θn (t)vn (x),

n=0

dΘn (t) , dt



vn (x) =

d2 vn (x) . dx 2

(4.92)

256

4 Heating and Evaporation of Multi-component Droplets

Since the expansion in a series with respect to vn (Fourier series) is unique, Eq. (4.92) is satisfied only when it is satisfied for each term in this expansion. Remembering that  2  2 λ0 λn   v0 = v0 and vn = − vn (n ≥ 1), δ0 δ0 the following equations are obtained: 



Θ0 (t) = Dl 



Θn (t) = −Dl

λn δ0

2

λ0 δ0

Θ0 (t),

(4.93)

2 Θn (t), n ≥ 1.

(4.94)

To find initial conditions for (4.93) and (4.94) we recall that u(t = 0) = u 0 (x) (see (4.86)). Expansion of u 0 (x) in the Fourier series with respect to vn gives: u 0 (x) =

∞

qY n vn (x),

(4.95)

n=0

where qY n =

1 ||vn ||2

δ0

u 0 (x)vn (x)dx.

0

Comparing (4.92) and (4.95) it can be seen that the initial conditions for (4.93) and (4.94) can be presented as: Θn (t = 0) = qn , n ≥ 0.

(4.96)

Thus, the following solutions to Eqs. (4.93) and (4.94) are obtained: 



Θ0 (t) = qY 0 exp Dl 



Θn (t) = qY n exp −Dl

λn δ0

λ0 δ0

2  t ,

2  t

n ≥ 1.

(4.97)

(4.98)

Having substituted (4.97) and (4.98) into (4.87), and remembering (4.91), the solution to (4.83) is obtained as:  u = qY 0 exp Dl



λ0 δ0

   2    2   ∞ λn x x t cosh λ0 qY n exp −Dl t cos λn + . δ0 δ0 δ0 n=1

(4.99)

4.7 Heating and Evaporation of Multi-component Liquid Films

257

From the definition of u (see (4.82)), the final expression for Yli (t, x), satisfying boundary conditions (4.78), (4.79) and initial condition (4.81), is presented as: 



Yli (t, x) = qY 0 exp Dl

     2     ∞ x x λ0 2 λn + + εi . t cosh λ0 qY n exp −Dl t cos λn δ0 δ0 δ0 δ0

(4.100)

n=1

Growth of Yli (t, x) over time is restricted by the physical condition 0 ≤ Yli (t, x) ≤ 1 (cf. Eq. (4.21), describing component diffusion in multi-component droplets). Equation (4.100) is valid only for short timesteps when δ˙0e , and all other input parameters can be considered constant.

4.7.2.2

Gas Phase

Once the mass fractions of the liquid components at the film surface have been found, the partial pressures of vapour components at this surface are inferred from Raoult’s law: ∗ , (4.101) pvsi = X lsi pvi where X lsi is the molar fraction of the ith component in the liquid at the film surface, ∗ the partial vapour pressure of the ith component when X li = 1. pvi The values of X lsi and Ylsi are linked as: Ylsi Mi

X lsi =    , Ylsi i

(4.102)

Mi

where Mi is the molar mass of component i. Equation (4.102) follows from the definition of the mass fraction: X lsi Mi . Ylsi =  i X lsi Mi

(4.103)

∗ The values of pvi depend on gas temperatures at the surface of the film and can be obtained from the Antoine equation. Once the values of pvsi have been obtained, the values of vapour density at the film surface ρvsi are inferred from the ideal gas law:

ρvsi =

pvsi Mi , Ru Ts

(4.104)

where Ru is the universal gas constant, and Ts is the temperature at the surface of the film. Formula (4.104) allows us to find εi from Expression (4.80).

258

4 Heating and Evaporation of Multi-component Droplets

Assuming that the convection mass transfer coefficient, given by Expression (4.74), is the same for all components and the contribution of fuel vapour in the ambient gas is negligible, the evaporation flux of component i from the film surface is obtained as (cf. Formula (4.75)): m˙ fi = h m ρvsi .

(4.105)

Using Expression (4.105) and considering the effect of thermal swelling, the change of film thickness during the timestep Δt is obtained as: Δδ0 = −Δt

|m˙ fi | ρ(T 0 )

 +

ρ(T 0 ) ρ(T 1 )

 − 1 δ0 ,

(4.106)

where ρ(T 0 ) is the liquid density estimated at the average temperature of the film at the beginning of the timestep, ρ(T 1 ) is the same density but calculated at the end of the timestep. Note that Expression (4.106) predicts that the change in film thickness due to swelling is proportional to ρ(T 0 )/ρ(T 1 ) (cf. the change in droplet radius due to 1/3  swelling which is proportional to ρ(T 0 )/ρ(T 1 ) ). The film thickness at the end of the timestep is obtained as: (4.107) δ1 = δ0 + Δδ0 . All liquid thermodynamic and transport properties were calculated at film average temperatures and compositions. The contribution of vapour to ambient air thermodynamic and transport properties was not taken into account. These properties were calculated at the reference temperature (Tref = (2/3)Ts + (1/3)Ta , where Ts is the temperature at the film surface, Ta is the temperature in the ambient air). Partial pressures of vapour components and specific heats of their evaporation were obtained at film surface temperatures.

4.7.3 Solution Algorithm These are the main steps of the numerical algorithm: 1. Assume the initial distribution of temperature and mass fractions of components across the liquid film, or use the distributions obtained at the previous timestep (in our analysis all initial distributions are assumed homogeneous). 2. Calculate component partial pressures and molar fractions in the gas phase based on Eq. (4.101). 3. Calculate the component evaporation rates (εi ) based on Eq. (4.80). 4. Calculate the liquid thermal conductivities and other properties for the mixture if the liquid is multi-component.

4.7 Heating and Evaporation of Multi-component Liquid Films

259

5. Calculate the distribution of temperature inside the film using Eq. (4.72), and 40 terms in the series. 6. Calculate the distribution of components inside the film using Eq. (4.100), and 200 terms in the series. 7. Calculate the change in film thickness based on Eq. (4.106); recalculate the film thickness at the end of the timestep based on Eq. (4.107). 8. For dimensional x, recalculate the distributions of temperature and compo˜ where δ0,1 are nents for the new film thickness (e.g. T (x) = T (x δ1 /δ0 ) = T (x), film thicknesses at the beginning and the end of the timestep, x˜ is the new x used at the second timestep). For dimensionless x/δ0 , used in our analysis, no recalculation is required. 9. Return to Step 1 and repeat the calculations for the next timestep.

4.7.4 Validation of the Model The results obtained using the algorithm described in Sect. 4.7.3, based on the model described in Sects. 4.7.1 and 4.7.2, were validated against the experimental results presented in [98]. These results refer to the evaporation of a film composed of mixtures of iso-octane/3-methylpentane (3MP). The comparison between the prediction of the algorithm/model (hereafter in this section referred to as model) and experimental results for three cases, pure iso-octane, pure 3MP, and a 50%/50% mixture of iso-octane and 3MP, are presented in Fig. 4.36. The same input parameters as in [98] (Tg = Tw = 302.25 K, T0 (x) = 293.15 K, δ0 = 602.72 µm) and h = 14 W/(m2 K) were used. The value of h was used as a fitting parameter and was close to the one used in [228] where h = 10 W/(m2 K) was considered. As follows from Fig. 4.36, the results predicted by the numerical algorithm are close to the experimental results for all three cases. The results of the comparison between the predictions of the model and experimental results for time evolution of the average mass fraction of 3MP in the 50%/50% mixture of iso-octane and 3MP are presented in Fig. 4.37. As follows from this figure, the agreement between the model predictions and experimental results is good at the initial stage of film heating and evaporation, but at a later stage (after approximately 40 s) the model and experimental data show different trends. The loss of linearity of the experimental plot presented in Fig. 4.37 was attributed by the authors of [98] to the violation of Raoult’s law (see Eq. (4.101)). Since our model was based on this law, it cannot be used for these times. We anticipate that considering the contribution of the activity coefficients to the value of the partial pressures of the vapour components would improve the agreement between the predictions of the model and experimental results at these times. The results presented in Fig. 4.36 and those referring to short times in Fig. 4.37 give us confidence in application the model to the analysis of other liquid films.

260 Fig. 4.36 Time evolution of the normalised film thickness. Circles, triangles and squares show the values taken from [98]; solid and dashed curves show the corresponding predictions of the model. Three cases were considered: pure iso-octane, pure 3MP, and a 50%/50% mixture of iso-octane and 3MP. Reprinted from [174], Copyright Elsevier (2018)

Fig. 4.37 Time evolution of the average mass fraction of 3MP during evaporation of a 3MP and iso-octane mixture film. Solid curve refers to the prediction of the model, dots refer to experimental results. Reprinted from [174], Copyright Elsevier (2018)

4 Heating and Evaporation of Multi-component Droplets

4.7 Heating and Evaporation of Multi-component Liquid Films

261

4.7.5 Verification of the Model The aim of this final section of the chapter is to perform a verification of the algorithm for the analysis of multi-component film heating and evaporation, using the model described in Sect. 4.7.2. This is done by comparing the predictions of this algorithm with the prediction of the algorithm using the numerical solution to the equations for heat transfer and component diffusion in the film (Eqs. (4.69) and (4.77)) at each timestep. The analysis presented in this section is reproduced from [94]. In what follows the details of the latter algorithm are presented. This is followed by the comparison of the results predicted by both algorithms.

4.7.5.1

Algorithm Based on Numerical Solutions of Eqs. (4.69) and (4.77)

Details of the algorithm based on numerical solutions to Eqs. (4.69) and (4.77) are presented in [228]. These are the key steps of this algorithm. Firstly, the distributions of temperature and component mass fractions in the liquid film are initialised using the initial and boundary conditions. The partial pressures of components and their vapour densities at the film surface are calculated based on Expressions (4.15) and (4.104). The component evaporation rate is determined by Expression (4.105). Then the transport and thermodynamic properties of the mixtures of the liquid components are calculated. Furthermore, the temperature and component mass fractions in the film for the next timestep are obtained by numerically solving Eqs. (4.69) and (4.77). Finally, the film thickness at the next timestep is inferred from Expression (4.106). For the numerical solution of Eqs. (4.69) and (4.77) an implicit finite difference approach was used. This approach used the central-differencing scheme for space and the forward-Euler scheme for time. A uniform grid of 100 cells across the film and a timestep of 10−5 s was used. The results remained almost the same when these parameters were further refined. As the number of cells did not change during calculations, the grid size continually shrank when the film evaporated. In the following sections the predictions of the algorithms using analytical and numerical solutions to Eqs. (4.69) and (4.77) are compared for one specific example.

4.7.5.2

Results of Calculations

Our comparison focuses on the same problem of heating and evaporation of a 50%/50% heptane-hexadecane film as presented in Section 5.2 of [174]. Calculations are based on the input parameters shown in Table 4.14. The same thermophysical properties as in [174] are used. Firstly the predictions of the two algorithms for the simplest case, when the effects of evaporation are not considered, are compared. The liquid film thickness and temperature versus time predicted by both algorithms are presented in Figs. 4.38 and 4.39. As follows from these figures, the agreement between the predictions of

262

4 Heating and Evaporation of Multi-component Droplets

Table 4.14 Values of parameters used in the calculations. Reprinted with minor modifications from [94] with permission by Avestia Ambient temperature, Tg Ambient pressure, p Wall temperature, Tw Initial film temperature, T0 Initial film thickness, δ0 Convection heat transfer coefficient, h

900 K 60 bar 500 K 363 K 20 µm 2000 W/(m2 K)

Fig. 4.38 The film thickness predicted by the algorithms using the numerical (dashed curve) and analytical (circles) solutions to Eqs. (4.69) and (4.77) at each timestep for the parameters presented in Table 4.14 (non-evaporating film). Reprinted from [94] with permission by Avestia

Fig. 4.39 The surface (solid) and average (dashed) temperature predicted by the algorithms based on the numerical (curves) and analytical (triangles and circles) solutions to Eqs. (4.69) and (4.77) at each timestep for the parameters presented in Table 4.14 (non-evaporating case). Reprinted from [94] with permission by Avestia

the algorithms is almost perfect which gives us confidence in the reliability of these algorithms for this particular case. Note that unrealistically high film temperatures, predicted by both algorithms and presented in Fig. 4.39, are attributed to the absence of evaporation. The increase in film thickness presented in Fig. 4.38 is the result of thermal swelling.

4.7 Heating and Evaporation of Multi-component Liquid Films

263

Fig. 4.40 The film thickness predicted by the algorithms using the numerical (dashed curve) and analytical (circles) solutions to Eqs. (4.69) and (4.77) at each timestep for the values of the parameters presented in Table 4.14 (evaporating film). Reprinted from [94] with permission by Avestia

Now we show the results of comparison for a realistic case when evaporation of the components and their mutual diffusion are considered. The film thicknesses and temperatures versus time predicted by both algorithms are presented in Figs. 4.40 and 4.41. As follows from the figures, the predictions of these algorithms are very close for most times. In contrast to the results presented in Figs. 4.38 and 4.39, however, there is a noticeable difference between the predictions of the algorithms at times close to 0.02 s. Note that there are some slight differences between the assumptions used in [174] and those used in [94] from where the results presented in Figs. 4.40 and 4.41 were reproduced. Firstly, the liquid diffusivity model used in [94] and in this section was based on Expressions (51) and (52) of [173] for viscosity, while a rather crude model for this parameter was used in [174]. Secondly, the analysis of this section assumes that Le=1, to be consistent with the corresponding assumption used in [228]. These differences led to slight differences between the plots shown in Figures 3 and 5 of [174] and the above-mentioned Figs. 4.40 and 4.41. Also, the lifetime of the film predicted by the algorithm based on the numerical solutions to Eqs. (4.69) and (4.77) is slightly longer than the one predicted by the algorithm using the analytical solutions to these equations. The plots of mass fraction of heptane for the same input parameters as in Figs. 4.40 and 4.41 are presented in Fig. 4.42. As in the cases shown in Figs. 4.40 and 4.41, the predictions of both algorithms presented in Fig. 4.42 are rather close, although there is visible deviation between them at times close to 0.015 s. Figure 4.42 shows that both algorithms predict a complete evaporation of heptane by about 0.02 s. This leads to a rapid decrease in the film thickness presented in Fig. 4.40, after the initial thermal swelling. Slight deviations between the predictions of two algorithms are not considered to be important for engineering applications. These deviations are much smaller than the typical measurement errors of these parameters. Thus, both algorithms using analytical and the numerical solutions to Eqs. (4.69) and (4.77) can be recommended for practical applications.

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Fig. 4.41 The surface/average temperature predicted by the algorithms based on the numerical (solid/dashed curve) and analytical (triangles/circles) solutions to Eqs. (4.69) and (4.77) at each timestep for the parameters presented in Table 4.14 (evaporating film). Reprinted from [94] with permission by Avestia

Fig. 4.42 Average heptane mass fraction predicted by the algorithms using the numerical (dashed curve) and analytical (circles) solutions to Eqs. (4.69) and (4.77) at each timestep for the values of the parameters presented in Table 4.14 (evaporating film). Reprinted from [94] with permission by Avestia

Early developments of the models for multi-component droplet heating and evaporation were reviewed. A model for multi-component droplet heating and evaporation, based on the analytical solution to the equation for component diffusion inside liquid droplets, was described. An original analytical solution to the component diffusion equation derived under the assumption that droplet radius is constant was described. The generalisation of this solution to the case that the droplet radius is a linear function of time during each individual timestep was discussed. The results of the application of the model to the analysis of biodiesel and kerosene droplets, and to the analysis of droplet drying were described. The predictions of the model were validated against experimental data where possible. Basic principles of quasidiscrete and multi-dimensional quasi-discrete models, based on the introduction of quasi-components, and their applications to modelling Diesel and petrol fuel droplets and various mixtures were described. A model of heating and evaporation of multicomponents liquid films was presented and discussed.

References

265

References 1. Abdel-Qader, Z., & Hallett, W. L. H. (2005). The role of liquid mixing in evaporation of complex multicomponent mixtures: Modelling using continuous thermodynamics. Chemical Engineering Science, 60, 1629–1640. 2. Abdul Jameel, A. G., Naser, N., Emwas, A.-H., & Sarathy, S. M. (2019). Surrogate formulation for diesel and jet fuels using the minimalist functional group (MFG) approach. Proceedings of the Combustion Institute, 37(4), 4663–4671. 3. Abianeh, O. S., & Chen, C. P. (2012). A discrete multicomponent fuel evaporation model with liquid turbulence effects. International Journal of Heat and Mass Transfer, 55, 6897–6907. 4. Abianeh, O. S., Oehlschlaeger, M. A., & Sung, C.-J. (2015). A surrogate mixture and kinetic mechanism for emulating the evaporation and autoignition characteristics of gasoline fuel. Combustion and Flame, 162(10), 3773–3784. 5. Abramzon, B., & Sazhin, S. S. (2006). Convective vaporization of fuel droplets with thermal radiation absorption. Fuel, 85, 32–46. 6. Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. International Journal of Heat and Mass Transfer, 32, 1605–1618. 7. Al-Esawi, N. H. I., & Al Qubeissi, M. A. (2021). A new approach to formulation of complex fuel surrogates. Fuel, 283, 118923. 8. Al-Esawi, N. H. I., Al Qubeissi, M. A., Sazhin, S. S., & Whitaker, R. (2018). The impacts of the activity coefficient on heating and evaporation of ethanol/gasoline fuel blends. International Communications in Heat and Mass Transfer, 98, 177–182. 9. Al-Esawi, N. H. I., Al Qubeissi, M. A., Whitaker, R., & Sazhin, S. S. (2019). Blended E85diesel fuel droplet heating and evaporation. Energy & Fuels, 33(3), 2477–2788. 10. Al Qubeissi, M. A., Sazhin, S. S., Crua, C., Turner, J., & Heikal, M. R. (2015). Modelling of biodiesel fuel droplet heating and evaporation: Effects of fuel composition. Fuel, 154, 308–318. 11. Al Qubeissi, M. A., Sazhin, S. S., Turner, J., Begg, S., Crua, C., & Heikal, M. (2015). Modelling of gasoline fuel droplets heating and evaporation. Fuel, 159, 373–384. 12. Al Qubeissi, M. A., Sazhin, S. S., & Elwardany, A. (2017). Modelling of blended Diesel and biodiesel fuel droplet heating and evaporation. Fuel, 187, 349–355. 13. Al Qubeissi, M. A., Al-Esawi, N. H. I., Sazhin, S. S., & Ghaleeh, M. (2018). Ethanol/gasoline droplet heating and evaporation: Effects of fuel blends and ambient conditions. Energy & Fuels, 32(6), 6498–6506. 14. Al Qubeissi, M. A., Al-Esawi, N. H. I., & Sazhin, S. S. (2021). Auto-selection of quasicomponents/components in the multi-dimensional quasi-discrete model. Fuel, 294, 120245. 15. Arias-Zugasti, M., & Rosner, D. E. (2003). Multicomponent fuel droplet vaporization and combustion using spectral theory for a continuous mixture. Combustion and Flame, 135, 271–284. 16. Arias-Zugasti, M., Garcia-Ybarra, P. L., & Castillo, J. L. (2016). Efficient calculation of multicomponent diffusion fluxes based on kinetic theory. Combustion and Flame, 163, 540– 556. 17. Atkins, P., & de Paula, J. (2002). Atkins’ Physical Chemistry (7th ed.). Oxford: Oxford University Press. 18. Azimi, A., Arabkhalaj, A., Markadeh, R. S., & Ghassemi, H. (2018). Fully transient modeling of the heavy fuel oil droplets evaporation. Fuel, 230, 52–63. 19. Bader, A., Keller, P., & Hasse, C. (2013). The influence of non-ideal vapor-liquid equilibrium on the evaporation of ethanol/iso-octane droplets. International Journal of Heat and Mass Transfer, 64, 547–558. 20. Bair, S. (2014). The pressure and temperature dependence of volume and viscosity of four diesel fuels. Fuel, 135, 112–119. 21. Banerjee, R. (2013). Numerical investigation of evaporation of a single ethanol/iso-octane droplet. Fuel, 107, 724–739.

266

4 Heating and Evaporation of Multi-component Droplets

22. Basshuysen, R. V. (2009). Gasoline Engine with Direct Injection: Processes, Systems, Development, Potential (1st ed.). Wiesbaden: GWV Fachverlage GmbH. 23. Bello, M. N., Hill, K. J., Pantoya, M. L., Jouet, R. J., & Horn, J. M. (2018). Surface engineered nanoparticles dispersed in kerosene: The effect of oleophobicity on droplet combustion. Combustion and Flame, 188, 243–249. 24. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena. New York: Wiley. 25. Brenn, G. (2005). Concentration fields in evaporating droplets. International Journal of Heat and Mass Transfer, 48, 395–402. 26. Brenn, G., Deviprasath, L. J., Durst, F., & Fink, C. (2007). Evaporation of acoustically levitated multi-component liquid droplets. International Journal of Heat and Mass Transfer, 50(25– 26), 5073–5086. 27. Burger, M., Schmehl, R., Prommersberger, K., Schäfer, O., Koch, R., & Wittig, S. (2003). Droplet evaporation modelling by the distillation curve model: Accounting for kerosene fuel and elevated pressures. International Journal of Heat and Mass Transfer, 46, 4403–4412. 28. Cai, L., & Pitsch, H. (2015). Optimized chemical mechanism for combustion of gasoline surrogate fuels. Combustion and Flame, 162(5), 1623–1637. 29. Campbell, S. W., Willsak, R. A., & Thodos, G. (1987). Vapor-liquid equilibrium measurements for the ethanol-acetone system at 372.7, 397.7 and 422.6 K. Journal of Chemical & Engineering Data, 32, 357–362. 30. Castanet, G., Maqua, C., Orain, M., Grisch, F., & Lemoine, F. (2007). Investigation of heat and mass transfer between the two phases of an evaporating droplet stream using laser-induced fluorescence techniques: comparison with modelling. International Journal of Heat and Mass Transfer, 50, 3670–3683. 31. Chang, Y., Jia, M., Li, Y., Liu, Y., Xie, M., Wang, H., et al. (2015). Development of a skeletal mechanism for diesel surrogate fuel by using a decoupling methodology. Combustion and Flame, 162(10), 3785–3802. 32. Chatwell, R., Heinen, M., & Vrabec, J. (2019). Diffusion limited evaporation of a binary liquid film. International Journal of Heat and Mass Transfer, 132, 1296–1305. 33. Chen, X., Khani, E., & Chen, C. (2016). A unified jet fuel surrogate for droplet evaporation and ignition. Fuel, 182, 284–291. 34. Cheong, H. W., Jeffreys, G. V., & Mumford, C. J. (1986). A receding interface model for the drying of slurry droplets. AIChE Journal, 32(8), 1334–1346. 35. Churchill, S. W. (1983). Free convection around immersed bodies. In E. U. Schlünder (Ed.), Heat Exchanger Design Handbook. New York: Hemisphere Publishing; Section 2.5.7. 36. Colket, M., Edwards, T., Williams, S., Cernansky, P., Miller, D. L., Egolfopoulos, F., & Lindstedt, P. (2007). Development of an experimental database and kinetic models for surrogate jet fuels. In 45th AIAA Aerospace Sciences Meeting and Exhibit (no. 770, pp. 1–21). American Institute of Aeronautics and Astronautics. 37. Continillo, G., & Sirignano, W. A. (1991). Unsteady, spherically-symmetric flame propagation through multicomponent fuel spray clouds. In G. Angelino, L. De Luca, & W. A. Sirignano (Eds.), Modern Research Topics in Aerospace Propulsion (pp. 173–198). Springer. 38. Cooke, J. A., Bellucci, M., Smooke, M. D., Gomez, A., Violi, A., Faravelli, T., & Ranzi, E. (2005). Computational and experimental study of JP-8, a surrogate, and its components in counterflow diffusion flames. Proceedings of the Combustion Institute, 30, 439–446. 39. Cooney, A. Y., & Singer, S. L. (2018). Modeling multicomponent fuel droplet vaporization with finite liquid diffusivity using Coupled Algebraic-DQMoM with delumping. Fuel, 212, 554–565. 40. Cowart, J., Foley, M. P., & Prak, D. L. (2019). The development and testing of Navy jet fuel (JP-5) surrogates. Fuel, 249, 80–88. 41. da Silva, R. M., Fraga, G. C., & França, F. H. R. (2021). Improvement of the efficiency of the superposition method applied to the WSGG model to compute radiative transfer in gaseous mixtures. International Journal of Heat and Mass Transfer, 179, 121662.

References

267

42. Dagaut, P., Reuillon, M., Boettner, J., & Cathonnet, M. (1994). Kerosene combustion at pressures up to 40 atm: Experimental study and detailed chemical kinetic modeling. Physics Chemistry Biology, 25, 919–926. 43. Dagaut, P., Reuillon, M., Cathonnet, M., & Voisin, D. (1995). High pressure oxidation of normal decane and kerosene in dilute conditions from low to high temperature. Twenty Fifth Symposium (International) on Combustion/The Combustion Institute, 92, 47–76. 44. Dalmaz, N., Ozbelge, N., Eraslan, H. O., & Uludag, Y. (2007). Heat and mass transfer mechanisms in drying of a suspension droplet: A new computational model. Drying Technology, 25(2), 391–400. 45. Dal’Toé, A. T. O., Padoin, N., Ropelato, K., & Soares, C. (2015). Cross diffusion effects in the interfacial mass and heat transfer of multicomponent droplets. International Journal of Heat and Mass Transfer, 85, 830–840. 46. Dean, A. J., Penyazkov, O. G., Sevruk, K. L., & Varatharajan, B. (2007). Autoignition of surrogate fuels at elevated temperatures and pressures. Proceedings of the Combustion Institute, 31, 2481–2488. 47. Del Pecchia, M., & Fontanesi, S. (2020). A methodology to formulate multicomponent fuel surrogates to model flame propagation and ignition delay. Fuel, 279, 118337. 48. Delplanque, J.-P., Rangel, R. H., & Sirignano, W. A. (1991). Liquid-wast incineration in a parallel-stream configuration: effect of auxiliary fuel. Progress in Aeronautics Astronautics, 132, 164–184. 49. Dirbude, S., Eswaran, V., & Kushari, A. (2012). Droplet vaporization modeling of rapeseed and sunflower methyl esters. Fuel, 92, 171–179. 50. Dooley, S., Won, S. H., Chaos, M., et al. (2010). A jet fuel surrogate formulated by real fuel properties. Combustion and Flame, 157(12), 2333–2339. 51. Drake, M. C., Fansler, T. D., Solomon, A. S., & Szekely Jr, G. A. (2003). Piston fuel films as a source of smoke and hydrocarbon emissions from a wall-controlled spark-ignited directinjection engine. SAE Technical Report 2003-01-0547. 52. Ebrahimian, V., & Habchi, C. (2011). Towards a predictive evaporation model for multicomponent hydrocarbon droplets at all pressure conditions. International Journal of Heat and Mass Transfer, 54, 3552–3565. 53. Elwardany, A. E., & Sazhin, S. S. (2012). A quasi-discrete model for droplet heating and evaporation: Application to Diesel and gasoline fuels. Fuel, 97, 685–694. 54. Elwardany, A. E., Gusev, I. G., Castanet, G., Lemoine, F., & Sazhin, S. S. (2011). Mono- and multi-component droplet cooling/heating and evaporation: Comparative analysis of numerical models. Atomization and Sprays, 21, 907–931. 55. Elwardany, A. E., Sazhin, S. S., & Farooq, A. (2013). Modelling of heating and evaporation of gasoline fuel droplets: A comparative analysis of approximations. Fuel, 111, 643–647. 56. Elwardany, A. E., Sazhin, S. S., & Im, H. G. (2016). A new formulation of physical surrogates of FACE A gasoline fuel based on heating and evaporation characteristics. Fuel, 176, 56–62. 57. Enayatollahi, R., Nates, R. J., & Anderson, T. (2017). The analogy between heat and mass transfer in low temperature crossflow evaporation. International Communications in Heat and Mass Transfer, 86, 126–130. 58. Faeth, G. M. (1983). Evaporation and combustion of sprays. Progress in Energy and Combustion Science, 9, 1–76. 59. Faghri, A., & Zhang, Y. (2006). Transport Phenomena in Multiphase Systems. Amsterdam: Elsevier. 60. Feng, Z., Tang, C., Yin, Y., Zhang, P., & Huang, Z. (2019). Time-resolved droplet size and velocity distributions in a dilute region of a high-pressure pulsed diesel spray. International Journal of Heat and Mass Transfer, 133, 745–755. 61. Ferrão, I. A. S., Silva, A. R. R., Moita, A. S. O. H., Mendes, M. A. A., & Costa, M. M. G. (2021). Combustion characteristics of a single droplet of hydroprocessed vegetable oil blended with aluminum nanoparticles in a drop tube furnace. Fuel, 302, 121160. 62. Gauthier, J. E. D., Bardon, M. F., & Rao, V. K. (1991). Combustion characteristics of multicomponent fuels under cold starting conditions in a gas turbine. In Proceedings of the American Society of Mechanical Engineers, Orlando, Florida, Paper 91-GT-109.

268

4 Heating and Evaporation of Multi-component Droplets

63. Gauthier, B. M., Davidson, D. F., & Hanson, R. K. (2004). Shock tube determination of ignition delay times in full-blend and surrogate fuel mixtures. Combustion and Flame, 139, 300–311. 64. Ghassemi, H., Baek, S. W., & Khan, Q. S. (2006). Experimental study on evaporation of kerosene droplets at elevated pressures and temperatures. Combustion Science and Technology, 178, 1669–1684. 65. Giakoumis, E. G. (2013). A statistical investigation of biodiesel physical and chemical properties, and their correlation with the degree of unsaturation. Renewable Energy, 50, 858–878. 66. Gopalakrishnan, V., & Abraham, J. (2004). Effects of multicomponent diffusion on predicted ignition characteristics of an n-heptane diffusion flame. Combustion and Flame, 136, 557– 566. 67. Gopireddy, S. R., & Gutheil, E. (2013). Numerical simulation of evaporation and drying of a bi-component droplet. International Journal of Heat and Mass Transfer, 66, 404–411. 68. Grover, N. K. (2020). Transient combustion of a multi-component fuel droplet with gas radiation. International Communications in Heat and Mass Transfer, 117, 104729. 69. Grubinger, T., Lenk, G., Schubert, N., & Wallek, T. (2021). Surrogate generation and evaluation of gasolines. Fuel, 283, 118642. 70. Gu, X., Basu, S., & Kumar, R. (2012). Dispersion and vaporization of biofuels and conventional fuels in a crossflow pre-mixer. International Journal of Heat and Mass Transfer, 55, 336–346. 71. Gun’ko, V. M., Nasiri, R., Sazhin, S. S., Lemoine, F., & Grisch, F. (2013). A quantum chemical study of the processes during the evaporation of real-life diesel fuel droplets. Fluid Phase Equilibria, 356, 146–156. 72. Guo, J., Shen, L., He, X., Liu, Z., & Im, H. G. (2021). Assessment of weighted-sum-of-graygases models for gas-soot mixture in jet diffusion flames. International Journal of Heat and Mass Transfer, 181, 121907. 73. Gusev, I. G., Krutitskii, P. A., Sazhin, S. S., & Elwardany, A. (2012). A study of the species diffusion equation in the presence of the moving boundary. International Journal of Heat and Mass Transfer, 55, 2014–2021. 74. Giusti, A., Sitte, M., Borghesi, G., & Mastorakos, E. (2018). Numerical investigation of kerosene single droplet ignition at high-altitude relight conditions. Fuel, 225, 663–670. 75. Hallett, W. L. H. (2000). A simple model for the vaporization of droplets with large numbers of components. Combustion and Flame, 121, 334–344. 76. Hallett, W. L. H., & Beauchamp-Kiss, S. (2010). Evaporation of single droplets of ethanol-fuel oil mixtures. Fuel, 89, 2496–2504. 77. Hallett, W. L. H., & Legault, N. V. (2011). Modelling biodiesel droplet evaporation using continuous thermodynamics. Fuel, 90, 1221–1228. 78. Harstad, K., & Bellan, J. (2004). Modeling evaporation of Jet A, JP-7, and RP-1 drops at 1 to 15 bars. Combustion and Flame, 137, 163–177. 79. He, Z., Li, J., Mao, Y., et al. (2019). A comprehensive study of fuel reactivity on reactivity controlled compression ignition engine: Based on gasoline and diesel surrogates. Fuel, 255, 115822. 80. He, R., Yi, P., & Li, T. (2020). Evaporation and condensation characteristics of n-heptane and multi-component diesel droplets under typical spray relevant conditions. International Journal of Heat and Mass Transfer, 163, 120162. 81. Hillenbrand, T., & Brüggemann, D. (2020). Evaporation of free falling droplets of binary alkane-ethanol blends. Fuel, 274, 117869. 82. Hinrichs, J., Shastry, V., Junk, M., Hemberger, Y., & Pitsch, H. (2020). An experimental and computational study on multicomponent evaporation of diesel fuel droplets. Fuel, 275, 117727. 83. Hoekman, S. K., Broch, A., Robbins, C., Ceniceros, E., & Natarajan, M. (2012). Review of biodiesel composition, properties, and specifications. Renewable and Sustainable Energy Reviews, 16, 143–169. 84. Hopfe, D. (1990). Data Compilation of FIZ CHEMIE, Germany, (p. 20).

References

269

85. Hashimoto, N., Nomura, H., Suzuki, M., Matsumoto, T., Nishida, H., & Ozawa, Y. (2015). Evaporation characteristics of a palm methyl ester droplet at high ambient temperatures. Fuel, 143, 202–210. 86. Huang, Z., Xia, J., Ju, D., et al. (2018). A six-component surrogate of diesel from direct coal liquefaction for spray analysis. Fuel, 234, 1259–1268. 87. Humer, S., Frassoldati, A., Granata, S., Faravelli, T., Ranzi, E., & Seiser, R. (2007). Experimental and kinetic modeling study of combustion of JP-8, its surrogates and reference components in laminar nonpremixed flows. Proceedings of the Combustion Institute, 31, 393–400. ´ R., Veljkovi, C. ´ B., & Kijev´canin, M. Lj. (2016). Thermodynamic 88. Ivani, Š. R., Radovi, C. properties of biodiesel and petro-diesel blends at high pressures and temperatures: Experimental and modeling. Fuel, 184, 277–288. 89. Jameel, A. G. A., Naser, N., Issayev, G., et al. (2018). A minimalist functional group (MFG) approach for surrogate fuel formulation. Combustion and Flame, 192, 250–271. 90. Járvás, G., Quellet, C., & Dallos, A. (2011). COSMO-RS based CFD model for flat surface evaporation of non-ideal liquid mixtures. International Journal of Heat and Mass Transfer, 54, 4630–4635. 91. Járvás, G., Kontos, J., Hancsok, J., & Dallos, A. (2015). Modeling ethanol-blended gasoline droplet evaporation using COSMO-RS theory and computation fluid dynamics. International Journal of Heat and Mass Transfer, 84, 1019–1029. 92. Javed, I., Baek, S. W., Waheed, K., Ali, G., & Cho, S. O. (2013). Evaporation characteristics of kerosene droplets with dilute concentrations of ligand-protected aluminum nanoparticles at elevated temperatures. Combustion and Flame, 160, 2955–2963. 93. Javid, S. M., Moreau, C., & Mostaghimi, J. (2020). A three-dimensional analysis of drying of a single suspension droplet in high rate evaporation processes. International Journal of Heat and Mass Transfer, 157, 119791. 94. Jia, M., Zhang, Y., Rybdylova, O., & Sazhin, S. S. (2020). Two approaches to mathematical modelling of heating/evaporation of a multi-component liquid film. In Proceedings of the International Conference on Fluid Flow and Thermal Science (ICFFTS’20), Virtual Conference - September 9–10, 2020. Paper No. 130. Published by AVESTIA International ASET Inc., Canada https://avestia.com/about/https://avestia.com/ICFFTS2020− Proceedings/files/papers.html. 95. Jin, Z.-H., Chen, J.-T., Song, S.-B., Tian, D.-H., Yang, J.-Z., & Tian, Z.-U. (2021). Pyrolysis study of a three-component surrogate jet fuel. Combustion and Flame, 226, 190–199. 96. Ju, D., Xiao, J., Geng, Z., & Huang, Z. (2014). Effect of mass fractions on evaporation of a multi-component droplet at dimethyl ether (DME)/n-heptane-fueled engine conditions. Fuel, 118, 227–237. 97. Kazakov, A., Conley, J., & Dryer, F. L. (2003). Detailed modeling of an isolated, ethanol droplet combustion under microgravity conditions. Combustion and Flame, 134, 301–314. 98. Kelly-Zion, P., Jelf, C., Pursell, C., & Oxley, S. (2006). Measuring the changing composition and mass of evaporating fuel films. In ASME Proceedings ICEF2006, (pp. 77–86). 99. Kim, D. M., Baek, S. W., & Yoon, J. (2016). Ignition characteristics of kerosene droplets with the addition of aluminum nanoparticles at elevated temperature and pressure. Combustion and Flame, 173, 106–113. 100. Kim, D., Martz, J., & Violi, A. (2014). A surrogate for emulating the physical and chemical properties of conventional jet fuel. Combustion and Flame, 161, 1489–1498. 101. Knothe, G. (2010). Biodiesel and renewable diesel: A comparison. Progress in Energy and Combustion Science, 36, 364–373. 102. Kolodnytska, R. V. (2010). Analytical study for atomization of hemp oil biodiesel. Visnik East-Ukrainian National University, 6, 41–46. 103. Kolodnytska, R. V., Al Qubeissi, M., & Sazhin, S. S. (2013) Biodiesel fuel droplets: Transport and thermodynamic properties. In 25th European Conference on Liquid Atomization and Spray Systems, Crete, Greece: 2013. Paper No. 7 (CD). 104. Kuo, K.-K. (1986). Principles of Combustion. Chichester: Wiley.

270

4 Heating and Evaporation of Multi-component Droplets

105. Lage, P. L. C. (2007). The quadrature method of moments for continuous thermodynamics. Computers & Chemical Engineering, 31, 782–799. 106. Lage, P. L. C., Hackenberg, C. M., & Rangel, R. H. (1995). Nonideal vaporization of dilating binary droplets with radiation absorption. Combustion and Flame, 101, 36–44. 107. Landis, R. B., & Mills, A. F. (1974). Effect of internal diffusional resistance on the evaporation of binary droplets. In Proceedings of the Fifth International Heat Transfer Conference, Tokyo, Japan, paper, B7, 9. 108. Lapuerta, M., Armas, O., & Rodrigues-Fernandez, J. (2008). Effect of biodiesel fuels on diesel engine emissions. Progress in Energy and Combustion Science, 34, 198–223. 109. Larbi, Z., Sadoun, N., Si-Ahmed, E., & Legrand, J. (2021). An efficient numerical prediction of the crust onset of a drying colloidal drop. International Journal of Heat and Mass Transfer, 165, 120613. 110. Laurent, C. (2008). Dévelopment et Validation de Modèles d’Évaporation Multi-composant. Thèse de l’Institut Supériour de l’Aéronautique et de l’Espace de Toulouse. 111. Laurent, C., Lavergne, G., & Villedieu, P. (2009). Continuous thermodynamics for droplet vaporization: Comparison between Gamma-PDF model and QMoM. C.R. Mechanique, 337, 449–457. 112. Laurent, C., Lavergne, G., & Villedieu, P. (2010). Quadrature method of moments for modeling multi-component spray vaporization. International Journal of Multiphase Flow, 36, 51–59. 113. Lee, A., & Law, C. K. (1991). Gasification and shell characteristics in slurry droplet burning. Combustion and Flame, 85(1), 77–93. 114. Levi-Hevroni, D., Levy, A., & Borde, I. (1995). Mathematical modeling of drying of liquid or solid slurries in steady state one-dimensional flow. Drying Technology, 13(5–7), 1187–1201. 115. Lindstedt, R., & Maurice, L. (2000). Detailed chemical-kinetic model for aviation fuels. Journal of Propulsion and Power, 16, 187–195. 116. Lippert, A. M., & Reitz, R. D. (1997). Modelling of multicomponent fuels using continuous distributions with application to droplet evaporation and sprays. SAE Technical Paper 972882. 117. Lissitsyna, K., Huertas, S., Quintero, L. C., & Polo, L. M. (2014). PIONA analysis of kerosene by comprehensive two-dimensional gas chromatography coupled to time of flight mass spectrometry. Fuel, 116, 716–722. 118. Liu, H., Yan, Y., & Jia, M. (2017). An analytical solution for wall film heating and evaporation. International Communications in Heat and Mass Transfer, 87, 125–131. 119. Liu, J., Hu, E., Zeng, W., & Zheng, W. (2020). A new surrogate fuel for emulating the physical and chemical properties of RP-3 kerosene. Fuel, 259, 116210. 120. Liu, Y.-X., Richter, S., Naumann, C., et al. (2019). Combustion study of a surrogate jet fuel. Combustion and Flame, 202, 252–261. 121. Liye, S., Weizheng, Z., Tien, Z., & Zhaoju, Q. (2015). A new approach to transient evaporating film heating modeling based on analytical temperature profiles for internal combustion engines. International Journal of Heat and Mass Transfer, 81, 465–469. 122. Lupo, G., & Duwig, C. (2020). Uncertainty quantification of multispecies droplet evaporation models. International Journal of Heat and Mass Transfer, 154, 119697. 123. Ma, X., Zhang, F., Han, K., & Song, G. (2016). Numerical modeling of acetone-butanolethanol and diesel blends droplet evaporation process. Fuel, 174, 206–215. 124. Maqua, C., Castanet, G., & Lemoine, F. (2008). Bi-component droplets evaporation: Temperature measurements and modelling. Fuel, 87, 2932–2942. 125. Markadeh, R. S., Arabkhalaj, A., Ghassemi, H., & Azimi, A. (2020). Droplet evaporation under spray-like conditions. International Journal of Heat and Mass Transfer, 148, 119049. 126. Mata, T. M., Cardoso, N., Ornelas, M., Neves, S., & Caetano, N. S. (2010). Sustainable production of biodiesel from tallow, lard and poultry fat and its quality evaluation. Chemical Engineering Transactions, 19, 13–18. 127. Merchant, Z., Taylor, K. M. G., Stapleton, P., Razak, S. A., Kunda, N., Alfagih, I., Sheikh, K., Saleem, I. Y., & Somavarapu, S. (2014). Engineering hydrophobically modified chitosan for enhancing the dispersion of respirable microparticles of levofloxacin. European Journal of Pharmaceutics and Biopharmaceutics, 88, 816–829.

References

271

128. Mezhericher, M., Levy, A., & Borde, I. (2007). Theoretical drying model of single droplets containing insoluble or dissolved solids. Drying Technology, 25(6), 1025–1032. 129. Mikhil, S., Bakshi, S., & Anand, T. N. C. (2021). A computational model for the evaporation of urea-water-solution droplets exposed to a hot air stream. International Journal of Heat and Mass Transfer, 168, 120878. 130. Minoshima, H., Matsushima, K., Liang, H., & Shinohara, K. (2001). Basic model of spray drying granulation. Journal of Chemical Engineering of Japan, 34(4), 472–478. 131. Mishra, S., Bukkarapu, K. R., & Krishnasamy, A. (2021). A composition based approach to predict density, viscosity and surface tension of biodiesel fuels. Fuel, 285, 119056. 132. Modest, M. F. (2003). Radiative Heat Transfer (2nd ed.). Amsterdam, Boston, London: Academic Press. 133. Mouvanal, S., Lamiel, Q., Lamarque, N., Helie, J., Burkhardt, A., Bakshi, S., & Chatterjee, D. (2019). Evaporation of thin liquid film of single and multi-component hydrocarbon fuel from a hot plate. International Journal of Heat and Mass Transfer, 141, 379–389. 134. Mukhopadhyay, A., & Sanyal, D. (2001). A spherical cell model for multi-component droplet combustion in a dilute spray. International Journal of Energy Research, 25, 1275–1294. 135. Mukhopadhyay, A., & Sanyal, D. (2001). A parametric study of burning of multicomponent droplets in a dilute spray. International Journal of Energy Research, 25, 1295–1314. 136. Narasu, P., Keller, A., Kohns, M., Hasse, H., & Gutheil, E. (2021). Numerical study of the evaporation and thermal decomposition of a single iron(III) nitrate nonahydrate/ethanol droplet. International Journal of Thermal Sciences, 170, 107133. 137. Nesic, S., & Vodnik, J. (1991). Kinetics of droplet evaporation. Chemical Engineering Science, 46(2), 527–537. 138. Ni, Z., Han, K., Zhao, C., Chen, H., & Pang, B. (2018). Numerical simulation of droplet evaporation characteristics of multicomponent acetone-butanol-ethanol and diesel blends under different environments. Fuel, 230, 27–36. 139. Ni, Z., Hespel, C., Han, K., & Foucher, F. (2021). The non-ideal evaporation behaviors of ethanol/heptane droplets: Impact on diameter, temperature evolution and the light scattering by droplet at the rainbow angle. International Journal of Heat and Mass Transfer, 164, 120401. 140. NIST. Natl. Inst. Stand. Technol. (2014). http://webbook.nist.gov/chemistry/fluid/. Retrieved 28 February 2014. 141. O’Rourke, P., & Amsden, A. (1996). A particle numerical model for wall film dynamics in port-injected engines, SAE Paper 961961. 142. Padoin, N., Dal’Toé, A. T. O., Rangel, L. P., Ropelato, K., & Soares, C. (2014). Heat and mass transfer modeling for multicomponent multiphase flow with CFD. International Journal of Heat and Mass Transfer, 73, 239–249. 143. Pagel, S., Stiesch, G., & Merker, G. P. (2002). Modelling of evaporation of a multicomponent fuel. In J. Taine (Ed.), Proceedings of the Twelfth International Heat Transfer Conference, V. 1. Grenoble, (August 18–23, 2002). Paris: Editions Scientifique et Medicale Elsevier SAS; CD ROM. 144. Pan, K.-L., Li, J.-W., Chen, C.-P., & Wang, C.-H. (2009). On droplet combustion of biodiesel fuel mixed with diesel/alkanes in microgravity condition. Combustion and Flame, 156, 1926– 1936. 145. Park, J.-Y., Kim, D.-K., Wang, Z.-M., Lu, P., Park, S.-C., & Lee, J.-S. (2008). Production and characterization of biodiesel from tung oil. Applied Biochemistry and Biotechnology, 148, 109–117. 146. Paudel, A., Worku, Z. A., Meeus, J., Guns, S., & Mooter, G. V. (2013). Manufacturing of solid dispersions of poorly water soluble drugs by spray drying: Formulation and process considerations. International Journal of Pharmaceutics, 453(30), 253–284. 147. Pinheiro, A. P., Rybdylova, O., Zubrilin, I. A., Sazhin, S. S., Filho, F. L. S., & Vedovotto, J. M. (2021). Modelling of aviation kerosene droplet heating and evaporation using complete fuel composition and surrogates. Fuel, 305, 121564. 148. Poling, B. E., Prausnitz, J. M., & O’Connell, J. (2001). The Properties of Gases and Liquids (5th ed.). New York: McGraw-Hill.

272

4 Heating and Evaporation of Multi-component Droplets

149. Poon, H. M., Pang, K. M., Ng, H. K., Gan, S., & Schramm, J. (2016). Development of multicomponent diesel surrogate fuel models - Part II: Validation of the integrated mechanisms in 0-D kinetic and 2-D CFD spray combustion simulations. Fuel, 181, 120–130. 150. Poozesh, S., Lu, K., & Marsac, P. J. (2018). On the particle formation in spray drying process for bio-pharmaceutical applications: Interrogating a new model via computational fluid dynamics. International Journal of Heat and Mass Transfer, 122, 863–876. 151. Porowska, A., Dosta, M., Fries, L., Gianfrancesco, A., Heinrich, S., & Palzer, S. (2016). Predicting the surface composition of a spray-dried particle by modelling component reorganization in a drying droplet. Chemical Engineering Research and Design, 110, 131–140. 152. Poulton, L., Rybdylova, O., Zubrilin, I. A., Matveev, S. G., Gurakov, N. I., Al Qubeissi, M., AlEsawi, N., Khan, T., Gun’ko, V. M., & Sazhin, S. S. (2020). Modelling of multi-component kerosene and surrogate fuel droplet heating and evaporation characteristics: a comparative analysis. Fuel, 269, 117115. 153. Ra, Y., & Reitz, R. D. (2009). A vaporization model for discrete multi-component fuel sprays. International Journal of Multiphase Flow, 35, 101–117. 154. Ra, Y., & Reitz, R. D. (2015). A combustion model for multi-component fuels using a physical surrogate group chemistry representation (PSGCR). Combustion and Flame, 162(10), 3456– 3481. 155. Raghuram, S., Raghavan, V., Pope, D. N., & Gogos, G. (2013). Two-phase modeling of evaporation characteristics of blended methanol-ethanol droplets. International Journal of Multiphase Flow, 52, 46–59. 156. Rajak, U., Nashine, P., Chaurasiya, P. K., Verma, T. N., Patel, D. K., & Dwivedif, G. (2021). Experimental and predicative analysis of engine characteristics of various biodiesels. Fuel, 285, 119097. 157. Rakopoulos, C., & Mavropoulos, G. (2008). Experimental evaluation of local instantaneous heat transfer characteristics in the combustion chamber of air-cooled direct injection diesel engine. Energy, 33, 1084–1099. 158. Ranjbar, N., & Kuenzel, C. (2017). Cenospheres: A review. Fuel, 207, 1–12. 159. Ray, S., & Raghavan, V. (2020). Numerical study of evaporation characteristics of biodiesel droplets of Indian origin. Fuel, 271, 117637. 160. Ren, Z., Wang, B., Xiang, G., & Zheng, L. (2019). Numerical analysis of wedge-induced oblique detonations in two-phase kerosene/air mixtures. Proceedings of the Combustion Institute, 37, 36273635. 161. Rivard, E., & Brüggemann, D. (2010). Numerical investigation of semi-continuous mixture droplet vaporization. Chemical Engineering Science, 65, 5137–5145. 162. Rybdylova, O., Poulton, L., Al Qubeissi, M., Elwardany, A. E., Crua, C., Khan, T., & Sazhin, S. S. (2018). A model for multi-component droplet heating and evaporation and its implementation into ANSYS Fluent. International Communications in Heat and Mass Transfer, 90, 29–33. 163. Samimi Abianeh, O., Chen, C. P., & Mahalingam, S. (2014). Numerical modeling of multicomponent fuel spray evaporation process. International Journal of Heat and Mass Transfer, 69, 44–53. 164. Sazhin, S. S. (2017). Modelling of fuel droplet heating and evaporation: Recent results and unsolved problems. Fuel, 196, 69–101. 165. Sazhin, S. S., & Sazhina, E. M. (1996). The effective emissivity approximation for the thermal radiation transfer problem. Fuel, 75, 1646–1654. 166. Sazhin, S. S., Krutitskii, P. A., Abdelghaffar, W. A., Sazhina, E. M., Mikhalovsky, S. V., Meikle, S. T., & Heikal, M. R. (2004). Transient heating of diesel fuel droplets. International Journal of Heat and Mass Transfer, 47, 3327–3340. 167. Sazhin, S. S., Abdelghaffar, W. A., Sazhina, E. M., & Heikal, M. R. (2005). Models for droplet transient heating: Effects on droplet evaporation, ignition, and break-up. International Journal of Thermal Sciences, 44, 610–622. 168. Sazhin, S. S., Elwardany, A., Krutitskii, P. A., Castanet, G., Lemoine, F., Sazhina, E. M., & Heikal, M. R. (2010). A simplified model for bi-component droplet heating and evaporation. International Journal of Heat and Mass Transfer, 53, 4495–4505.

References

273

169. Sazhin, S. S., Elwardany, A., Krutitskii, P. A., Deprédurand, V., Castanet, G., Lemoine, F., Sazhina, E. M., & Heikal, M. R. (2011). Multi-component droplet heating and evaporation: Numerical simulation versus experimental data. International Journal of Thermal Sciences, 50, 1164–1180. 170. Sazhin, S. S., Elwardany, A., Sazhina, E. M., & Heikal, M. R. (2011). A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets. International Journal of Heat and Mass Transfer, 54, 4325–4332. 171. Sazhin, S. S., Al Qubeissi, M., Kolodnytska, R., Elwardany, A., Nasiri, R., & Heikal, M. R. (2014). Modelling of biodiesel fuel droplet heating and evaporation. Fuel, 115, 559–572. 172. Sazhin, S. S., Al Qubeissi, M., & Heikal, M. R. (2014). Modelling of biodiesel and diesel fuel droplet heating and evaporation. In 15th International Heat Transfer Conference IHTC-15, paper IHTC15-8936, Kyoto, Japan. 173. Sazhin, S. S., Al Qubeissi, M., Nasiri, R., Gun’ko, V. M., Elwardany, A. E., Lemoine, F., Grisch, F., & Heikal, M. R. (2014). A multi-dimensional quasi-discrete model for the analysis of Diesel fuel droplet heating and evaporation. Fuel, 129, 238–266. 174. Sazhin, S. S., Rybdylova, O., & Crua, C. (2018). A mathematical model for heating and evaporation of a multi-component liquid film. International Journal of Heat and Mass Transfer, 117, 252–260. 175. Sazhin, S. S., Rybdylova, O., Pannala, A. S., Somavarapu, S., & Zaripov, S. K. (2018). A new model for a drying droplet. International Journal of Heat and Mass Transfer, 122, 451–458. 176. Singer, S. L., Hayes, M. P., & Cooney, A. Y. (2021). A hybrid droplet vaporization-adaptive surrogate model using an optimized continuous thermodynamics distribution. Fuel, 288, 119821. 177. Siouris, S., Blakey, S., & Wilson, C. W. (2013). Investigation of deposition in aviation gas turbine fuel nozzles by coupling of experimental data and heat transfer calculations. Fuel, 106, 79–87. 178. Sirignano, W. A. (2010). Fluid Dynamics and Transport of Droplets and Sprays (2nd ed.). Cambridge: Cambridge University Press. 179. Sirignano, W. A., & Wu, G. (2008). Multicomponent-liquid-fuel vaporization with complex configuration. International Journal of Heat and Mass Transfer, 51, 4759–4774. 180. Slavinskaya, N. A., Zizin, A., & Aigner, M. (2010). On model design of a surrogate fuel formulation. Journal of Engineering for Gas Turbines and Power, 132(111501), 1–11. 181. Sontheimer, M., Kronenburg, A., & Stein, O. T. (2021). Grid dependence of evaporation rates in Euler-Lagrange simulations of dilute sprays. Combustion and Flame, 232, 111515. 182. Spille-Kohoff, A., Preuß, E., & Böttcher, K. (2012). Numerical solution of multi-component species transport in gases at any total number of components. International Journal of Heat and Mass Transfer, 55, 5373–5377. 183. Stanton, D., & Rutland, C. (1996). Multi-dimensional modeling of thin liquid films and spraywall interactions resulting from impinging sprays. International Journal of Heat and Mass Transfer, 41, 3037–3054. 184. Starinskaya, E. M., Miskiv, N. B., Nazarov, A. D., Terekhov, V. V., Terekhov, V. I., Rybdylova, O. D., & Sazhin, S. S. (2021). Evaporation of water/ethanol droplets in an air flow: Experimental study and modelling. International Journal of Heat and Mass Transfer, 177, 121502. 185. Steeman, H.-J., Janssens, A., & Paepe, M. D. (2009). On the applicability of the heat and mass transfer analogy in indoor air flows. International Journal of Heat and Mass Transfer, 52, 1431–1442. 186. Srivastava, S., & Jaberi, F. (2017). Large eddy simulations of complex multicomponent diesel fuels in high temperature and pressure turbulent flows. International Journal of Heat and Mass Transfer, 104, 819–834. 187. Strelkova, M., Kirillov, I., Potapkin, B., Safonov, A., Sukhanov, L., Umanskiy, S., Deminsky, M., Dean, A., Varatharaja, B., & Tentner, A. (2008). Detailed and reduced mechanisms of jet a combustion at high temperatures. A Combustion Science and Technology, 180, 1788–1802. 188. Strizhak, P. A., Volkov, R. S., Castanet, G., Lemoine, F., Rybdylova, O., & Sazhin, S. S. (2018). Heating and evaporation of suspended water droplets: Experimental studies and modelling. International Journal of Heat and Mass Transfer, 127, 92–106.

274

4 Heating and Evaporation of Multi-component Droplets

189. Strotos, G., Gavaises, M., Theodorakakos, A., & Bergeles, G. (2011). Numerical investigation of the evaporation of two-component droplets. Fuel, 90, 1492–1507. 190. Su, M., & Chen, C. (2015). Heating and evaporation of a new gasoline surrogate fuel: a discrete multicomponent modeling study. Fuel, 161, 215–221. 191. Sukumaran, S., & Kong, S.-C. (2016). Modelling biodiesel-diesel spray combustion using multicomponent vaporization coupled with detailed fuel chemistry and soot models. Combustion Theory and Modelling, 20(5), 913–940. 192. Sykes, D., Turner, J., Stetsyuk, V., de Sercey, G., Gold, M., Pearson, R., & Crua, C. (2021). Quantitative characterisations of spray deposited liquid films and post-injection discharge on diesel injectors. Fuel, 289, 119833. 193. Sykes, D., Stetsyuk, V., Turner, J., de Sercey, G., Gold, M., Pearson, R., & Crua, C. (2022). A phenomenological model for near-nozzle fluid processes: Identification and qualitative characterisations. Fuel, 10(A), 122208. 194. Szymkowicz, P. G., & Benajes, J. (2018). Development of a Diesel surrogate fuel library. Fuel, 222, 21–34. 195. Tamim, J., & Hallett, W. L. H. (1995). Continuous thermodynamics model for multicomponent vaporization. Chemical Engineering Science, 50, 2933–2942. 196. Tong, A. Y., & Sirignano, W. A. (1986). Multicomponent transient droplet vaporization with internal circulation: Integral equation formulation. Numerical Heat Transfer, 10, 253–278. 197. Tonini, S., & Cossali, G. E. (2015). A novel formulation of multi-component drop evaporation models for spray applications. International Journal of Thermal Sciences, 89, 245–253. 198. Tonini, S., & Cossali, G. E. (2016). A multi-component drop evaporation model based on analytical solution of Stefan-Maxwell equations. International Journal of Heat and Mass Transfer, 92, 184–189. 199. Tonini, S., & Cossali, G. E. (2022). An analytical model for the evaporation of multicomponent spheroidal drops based on Stefan-Maxwell equations. International Journal of Thermal Sciences, 171, 107223. 200. Torres, D. J., O’Rourke, P. J., & Amsden, A. A. (2003). Efficient multi-component fuel algorithm. Combustion Theory and Modelling, 7, 67–86. 201. Tsilingiris, P. T. (2010). Modeling heat and mass transport phenomena at higher temperatures in solar distillation systems - the Chilton-Colburn analogy. Solar Energy, 84, 308–317. 202. Turner, J. E., Stetsyuk, V., Crua, C., Pearson, R. J., & Gold, M. R. (2015). The effect of operating conditions on post-injection fuel discharge in an optical engine. In ICLASS 13th Triennial International Conference, Tainan, Taiwan, August 23–27. 203. Tyson, K. S., Bozell, J., Wallace, R., Petersen, E., & Moens, L. (2004). Biomass Oil Analysis: Research Needs and Recommendations. National Renewable Energy Lab., Golden, CO (US). https://doi.org/10.2172/15009676. 204. Ullal, A., & Ra, Y. (2021). Analytical model for multicomponent wall film evaporation with non-unity Lewis number. International Journal of Heat and Mass Transfer, 176, 121485. 205. Vehring, R., Foss, W. R., & Lechuga-Ballesteros, D. (2007). Particle formation in spray drying. Journal of Aerosol Science, 38, 728–746. 206. Violi, A., Yan, S., Eddings, E., Sarofim, A., Granata, S., Faravelli, T., & Ranzi, E. (2002). Experimental formulation and kinetic model for JP-8 surrogate mixtures. Combustion Science and Technology, 174, 399–417. 207. Wang, F., Liu, R., Li, M., et al. (2018). Kerosene evaporation rate in high temperature air stationary and convective environment. Fuel, 211, 582–590. 208. Wang, S., Feng, Y., Qian, Y., et al. (2020). Experimental and kinetic study of diesel/gasoline surrogate blends over wide temperature and pressure. Combustion and Flame, 213, 369–381. 209. Wu, B.-H., & Chung, C. A. (2020). Modeling and simulation of solid-containing droplet drying and different-structure particle formation. International Journal of Heat and Mass Transfer, 152, 119469. 210. Xin, Y., Liang, W., Liu, W., Lu, T., & Law, C. K. (2015). A reduced multicomponent diffusion model. Combustion and Flame, 162(1), 68–74.

References

275

211. Yan, Y., Liu, H., Jia, M., Xie, M., & Yin, H. (2015). A one-dimensional unsteady wall film evaporation model. International Journal of Heat and Mass Transfer, 88, 138–148. 212. Yang, D., Xia, Z., Huang, L., Ma, L., Feng, Y., & Xiao, Y. (2018). Exprimental study on the evaporation characteristics of the kerosene gel droplet. Experimental Thermal and Fluid Science, 93, 171–177. 213. Yang, S., Hsu, T., & Wu, M. (2016). Spray combustion characteristics of kerosene/bio-oil. Part II: Numerical study. Energy, 115, 458–467. 214. Yang, S., Ra, Y., Reitz, R. D., VanDerWege, B., & Yi, J. (2010). Development of a realistic multi-component evaporation model. Atomization and Sprays, 20, 965–981. 215. Yang, S., Wang, Q., Curran, H. J., & Jia, M. (2021). Development of a 5-component gasoline surrogate model using recent advancements in the detailed H2 /O2 /CO/C1 -C3 mechanism for decoupling methodology. Fuel, 283, 118793. 216. Yang, W., Xia, J., Wang, X. Y., Wan, K. D., Megaritis, A., & Zhao, H. (2021). Predicting evaporation dynamics of a multicomponent gasoline/ethanol droplet and spray using nonideal vapour-liquid equilibrium models. International Journal of Heat and Mass Transfer, 168, 120876. 217. Yaws, C. L. (2008). Thermophysical Properties of Chemicals and Hydrocarbons. William Andrew Inc. 218. Yi, P., Long, W., Jia, M., Feng, L., & Tian, J. (2014). Development of an improved hybrid multicomponent vaporization model for realistic multi-component fuels. International Journal of Heat and Mass Transfer, 77, 173–184. 219. Yi, P., Long, W., Jia, M., Tian, J., & Li, B. (2016). Development of a quasi-dimensional vaporization model for multi-component fuels focusing on forced convection and high temperature conditions. International Journal of Heat and Mass Transfer, 97, 130–145. 220. Yi, P., Long, W., & Jia, M. (2018). An analogical multi-component vaporization model for single diesel droplets. International Journal of Thermal Sciences, 127, 158–172. 221. Yu, J., Ju, Y., & Gou, X. (2016). Surrogate fuel formulation for oxygenated and hydrocarbon fuels by using the molecular structures and functional groups. Fuel, 166, 211–218. 222. Yu, L., Mao, Y., Li, A., et al. (2019). Experimental and modeling validation of a large diesel surrogate: Autoignition in heated rapid compression machine and oxidation in flow reactor. Combustion and Flame, 202, 195–207. 223. Yu, W., & Zhao, F. (2020). Formulating of model-based surrogates of jet fuel and diesel fuel by an intelligent methodology with uncertainties analysis. Fuel, 268, 117393. 224. Zanutto, C. P., Paladino, E. E., & Nazareth, T. C. (2019). Heating and evaporation of ethanol/iso-octane droplets considering nonideal mixtures and finite rate transfer. Fuel, 256, 115811. 225. Zhang, H. R., Eddings, E. G., & Sarofim, A. F. (2007). Criteria for selection of components for surrogates of natural gas and transportation fuels. Proceedings of the Combustion Institute, 31, 401–409. 226. Zhang, L., & Kong, S.-C. (2010). Vaporization modeling of petroleum-biofuel drops using a hybrid multi-component approach. Combustion and Flame, 157, 2165–2174. 227. Zhang, L., & Kong, S.-C. (2012). Multicomponent vaporization modeling of bio-oil and its mixtures with other fuels. Fuel, 95, 471–480. 228. Zhang, Y., Jia, M., Yi, P., Liu, H., & Xie, M. (2017). An efficient liquid film vaporization model for multi-component fuels considering thermal and mass diffusions. Applied Thermal Engineering, 112, 534–548. 229. Zhao, F., Liu, Q., Zhao, C., & Bo, H. (2019). Influence region theory of the evaporating droplet. International Journal of Heat and Mass Transfer, 129, 827–841. 230. Zhu, G.-S., & Reitz, R. D. (2002). A model for high-pressure vaporization of droplets of complex liquid mixture using continuous thermodynamics. International Journal of Heat and Mass Transfer, 45, 495–507.

Chapter 5

Processes in Composite Droplets

5.1 Background The models of heating and evaporation of multi-component droplets, described in Chap. 4, assumed that all components are well mixed locally. This allowed us to reduce the analysis of the interaction between the components to that of their mutual diffusion. In many practical applications, however, this assumption cannot be used, and we need to take into account the presence of clusters of certain liquid components within the droplets. In the latter case, the droplets are commonly called composite. Perhaps the most well-known examples of composite droplets are automotive fuel droplets with water sub-droplets inside them [34, 60]. When these droplets are heated in automotive (Diesel) engine conditions the temperatures inside them can reach and exceed water boiling or nucleation temperatures. When this happens, the fuel droplets start disintegrating via puffing and/or micro-explosion process [29, 30, 54]. Puffing is the partial ejection of some of the water dispersed within a fuel droplet outside that droplet, while the term micro-explosion refers to the complete break-up of the droplet [58]. The idea of micro-explosions in water-fuel emulsions dates to 1965 [25]. The authors of that paper observed that the evaporation of a fuel droplet accelerates on the addition of water. Since the publication of that pioneering paper, the results of numerous experimental studies of this phenomenon have been widely discussed by many authors, including [3, 4, 24, 27, 29, 30, 54, 55, 58]. In the experiments described in [26], the external energy of a pulse laser was introduced into the emulsion droplet, after which its dynamic response was examined. The authors of [65] investigated the effect of small addition of water (2–6% by weight) to biodiesel fuel on combustion and emission characteristics of a medium speed Diesel engine. The focus was on in-cylinder pressure and temperature, rate of heat release, brake power, thermal efficiency, and specific fuel consumption, NOx , soot, CO and CO2 emissions. It was found that water addition can reduce NOx , soot, CO and CO2 emissions. The optimal water addition level for optimal engine emission and performance parameters was found to be at 4% by weight. In contrast to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_5

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[65], the authors of [32] found that addition of water to Diesel fuel does not lead to improvement of CO emissions. At the same time, this addition led to considerable decrease in NOx and HC emissions, particulate matter (PM) and smoke. The improvement of performance of Diesel engines after addition of 10% of water to Diesel fuel is described in [53]. The authors of [59] drew attention to the difference between petrol and water-in-petrol mixture spray characteristics, including the shape and penetration of sprays. In some applications and experiments puffing/micro-explosions were observed not in fuel/water droplets but in droplets composed of different fuels. For example, the authors of [10] studied experimentally single Diesel-biodiesel-ethanol droplets located at the tip of a 75 µm gauge thermocouple in a furnace where air temperature was 500 ◦ C. The results of investigation of micro-explosion characteristics of biodiesel, aviation fuel RP-3 and ethanol mixed droplets are presented as discussed in [31]. The authors of [13] focused their attention on Diesel-ethanol micro-emulsions, while the authors of [16] investigated micro-explosions in free-falling and suspended droplets of biodiesel/alcohol blended fuels. Also, micro-explosions were observed in some composite fuel droplets such as jatropha vegetable oil droplets [57]. The authors of [18, 56] presented experimental evidence of micro-explosions triggered by ceria nanoparticles. Note that the presence of water-in-fuel droplets does not necessarily lead to the development of micro-explosions. For example, the authors of [63] investigated stability of laminar premixed flames of water-in-fuel emulsion sprays. It was shown that the presence of water-in-droplets reduces the flame velocity and temperature. Some parts of the analysis of [63] were generalised in [64]. In the latter paper, the influence of micro-explosions of composite droplets on the premixed flame characteristics and stability was investigated. A typical development of the puffing/micro-explosion phenomenon is illustrated in Fig. 5.1. The first frame in this figure shows the state of the droplet just before puffing/micro-explosion starts (only heating, evaporation and swelling processes take place but these are not visible in the figure). The second frame shows the start of the puffing/micro-explosion when individual child droplets can be clearly seen. In the third frame, the fully developed micro-explosion is shown (a cloud of small child droplets is visible as the black spot in the middle of the figure). Perhaps the most advanced models of puffing/micro-explosion are described in [19, 45, 47, 48]. In these studies, high-fidelity Direct Numerical Simulations (DNS) were used to investigate micro-explosion from first principles. The authors attempted to identify dominant mechanisms of micro-explosion and puffing and obtain modelling insights for Eulerian–Lagrangian simulation of fuel sprays under micro-explosion conditions. Using their multiphase flow code [44], the gas–liquid equations were solved directly. To obtain detailed information, a single emulsion droplet at a characteristic scale in a fuel spray was considered and the evolution of micro-explosion and/or puffing was investigated. The authors of [51] focused their DNS analysis on the comparison between puffing characteristics of sessile (‘sitting’ on the substrate) and free droplets.

5.1 Background

279

Fig. 5.1 Typical dynamics of puffing/micro-explosion in the experiments performed at National Research Tomsk Polytechnic University. A droplet of Diesel fuel with volume fraction of water Vw0 = 10% and initial radius Rd0 = 0.75 mm was placed in an air flow at atmospheric pressure, temperature Tg = 723 K and constant velocity Ug = 3 m/s. t = 0 s refers to the time instant when the droplet was placed in the air flow. Reprinted from [41], Copyright Elsevier (2020)

Although the models described in [45, 47, 48] are crucial for understanding the underlying physics of the phenomena leading to puffing and micro-explosion in droplets, their applicability for solving practical engineering problems is questionable. Among simplified models of these processes, those suggested in [20, 21, 33, 40, 41, 62] can be mentioned. The model suggested in [21] focuses on the final stage of the puffing process when water inside fuel droplets has evaporated, and the problem of the development of puffing refocuses on the interaction between water vapour and the liquid fuel shell around this vapour. The stability analysis of the vapour/liquid shell was performed assuming that the fuel shell is initially spherically symmetric. This analysis allowed the author of [21] to predict the number of droplets generated after the shell rupture and their average diameters. These were functions of the initial fuel shell thickness. It was shown that the most likely diameter of the child droplets (generated during rupture) is equal to about a quarter (0.27) of the thickness of the fuel shell before the start of the process. We could not obtain these values from the results of experiments described in [3]. We could anticipate, however, that this thickness is proportional to the thickness of the fuel shell at the start of the process. In this case, assuming that the volume fraction of water is fixed, based on this model we could anticipate that diameters of the droplets generated during the shell rupture are proportional to the initial fuel droplet radii. Unfortunately, this was not supported by the experimental data presented in [3]. The models developed by the authors of [20, 62] focused on heating of composite droplets, bubble generation, and their subsequent growth leading to the final explosion. In [62], the finite thermal conduction and species diffusion equations were

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solved numerically inside the multicomponent droplets, if the processes described by these equations are spherically symmetric. Bubble generation and growth was described based on the model previously developed by the authors of [66]. In the model described in [20], a fitting procedure was used to approximate the results of the heating of composite droplets inferred from CFD calculations. Bubble growth time in [20] was estimated using the model presented in [50]. The total time to puffing/micro-explosion was estimated in [20, 62] as the sum of time of heating of the fuel/water interface to superheated temperature (spinodal) and time of bubble growth. In contrast to [21], the focus of the model described in [33, 40, 41] is on the prediction of the time to puffing/micro-explosion. This parameter is believed to be particularly important for applications to engineering problems. In contrast to [20, 62], bubble generation and growth were not explicitly considered by the authors of [33, 40, 41]. The approaches developed in the latter papers are described in the following section and Sect. 5.3.

5.2 A Simple Analytical Model In this section, a simple analytical model predicting the time instant when puffing/ micro-explosion is initiated is described following [40]. The model assumes that a spherical water sub-droplet is located in the centre of a fuel (n-dodecane) droplet as illustrated in Fig. 5.2. The constant temperature Ts is used, and evaporation is not considered. The droplet is considered to be stationary and all processes are spherically symmetric. The heat transfer is described by the transient heat conduction equation. This equation is solved analytically with the Dirichlet boundary condition at the droplet surface. This solution describes a timedependent distribution of temperature inside the droplet. If Ts is larger than the water boiling temperature, at a certain time instant Tw will reach this temperature. This time instant is considered as the start of the puffing process leading to micro-explosion. It is called the ‘time to puffing/micro-explosion’, or ‘micro-explosion delay time’. The maximal value of Ts is taken equal to the n-dodecane boiling temperature. For this Ts , the time to puffing/micro-explosion is expected to be the shortest.

Fig. 5.2 Scheme showing a water sub-droplet of radius Rw inside the fuel droplet of radius Rd . Tw is the temperature at the interface between water and fuel, Ts is the surface temperature of the droplet. Reprinted from [40], Copyright Elsevier (2019)

5.2 A Simple Analytical Model

281

5.2.1 Basic Equations and Approximations The transient heat conduction equation in a composite droplet, schematically presented in Fig. 5.2, is written as [14, 28]: ∂T =κ ∂t



2 ∂T ∂2T + 2 ∂R R ∂R

 + P(t, R),

(5.1)

where  κ=

κw = kw /(cw ρw ) when R ≤ Rw κ f = k f /(c f ρ f ) when Rw < R ≤ Rd ,

(5.2)

κw( f ) is the water (fuel) thermal diffusivity, kw( f ) the water (fuel) thermal conductivity, cw( f ) the water (fuel) specific heat capacity, ρw( f ) the water (fuel) density, R the distance from the droplet’s centre, t time. P(t, R) describes the volumetric droplet heating (e.g. heating by thermal radiation). The following initial and boundary conditions were used to solve Eq. (5.1):  T |t=0 = T|

R=Rw−

= T|

Tw0 (R) when R ≤ Rw T f 0 (R) when Rw < R ≤ Rd ,

R=Rw+

,

  ∂ T  ∂ T  kw = kf , ∂ R  R=Rw− ∂ R  R=Rw+

   ∂ T  h Tg − T (Rd ) = k f , ∂ R  R=Rd −0

(5.3)

(5.4)

(5.5)

where h is the convection heat transfer coefficient, describing the droplet’s heating. Equation (5.5) is the widely used Robin boundary condition. This equation can be re-written as  k f ∂ T  . (5.6) T (Rd ) ≡ Ts = Tg − h ∂ R  R=Rd −0 Assuming that the right-hand side of Eq. (5.6) is constant Condition (5.5) is reduced to the Dirichlet boundary condition: T (Rd ) ≡ Ts .

(5.7)

To be consistent with the previous assumptions of the model, several further simplifying assumptions were made. 1. The droplet surface temperature is assumed to be constant during the whole period of droplet heating. This would have been a natural assumption for the times after the droplet surface temperature reaches boiling temperature. The assumption

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that the droplet surface temperature is equal to the boiling temperature of the liquid fuel allows us to predict the minimal time to puffing/micro-explosion. Realistic values of this temperature are expected to be below this temperature. 2. Droplet swelling and evaporation are not considered. This means that we assume that puffing/micro-explosion occurs before droplet evaporation becomes noticeable, which is consistent with observations. The effects of swelling are expected to be small and can be safely ignored keeping in mind the crudeness of the model. 3. Tw0 and T f 0 are assumed equal to Td0 . They do not to depend on R. 4. The source term (thermal radiation) is considered when deriving the formulae, but eventually it is ignored. This allows us to apply the model to the cases when the contribution of thermal radiation is important, if we wish. 5. The thermophysical properties (thermal conductivity, heat capacity and density) are taken at Td0 and assumed to be constant. These assumptions lead to the following analytical solution to Eq. (5.1), subject to initial and boundary conditions (5.3), (5.4) and (5.5) (see Appendix Q for the details): T (R, t) = Ts +

t ∞ 



 1  exp −λ2n (t − τ ) pn (τ )dτ vn (R), exp −λ2n t (n1 + n2 ) + R 0

(5.8)

n=1

where n1 = T0 c f ρ f n2 =  2 ||vn ||2 λn a f



T0 cw ρw [λn aw Rw cot (λn aw Rw ) − 1] , ||vn ||2 (λn aw )2   λn a f Rw cot λn a f (Rd − Rw ) −

 λn a f Rd  +1 ,  sin λn a f (Rd − Rw )

T0 = Ts − Tw0 = Ts − T f 0

vn (R) =

||vn ||2 =

⎧ sin(λn aw R) ⎨ ± sin(λ a R )

when R < Rw

⎩ ± sin(λn a f (R−Rd )) sin(λn a f (Rw −Rd ))

when Rw ≤ R ≤ Rd ,

n w

w

(5.9)

(5.10)

(5.11)

(5.12)

c f ρ f (Rd − Rw ) kw − k f cw ρw Rw + − , 2 2 2Rw λ2n 2 sin (λn aw Rw ) 2 sin (λn a f (Rw − Rd )) pn (t) =

cw ρw ||vn ||2



Rw

R P(t, R)vn (R)d R.

0

A countable set of positive eigenvalues λn (0 < λ1 < λ2 < · · · .) is obtained from the solution to the equation:

5.2 A Simple Analytical Model

  kw − k f kw cw ρw cot(λaw Rw ) − k f c f ρ f cot(λa f (Rw − Rd )) = . Rw λ

283

(5.13)

  cf ρf . aw = cwkρw w , a f = kf Sign ‘−’ in Expression (5.12) is taken to obtain a physically meaningful solution for T ≤ Ts .

5.2.2 Analysis The model presented in Sect. 5.2.1 was used to analyse the experimental results described in [24]. The experiments described by these authors were performed in a high temperature chamber (700 K) at atmospheric pressure, with optical access. The times to puffing/micro-explosion were obtained with the help of a high-speed camera fitted with a long-distance microscope. The images obtained using this camera were processed to infer the initial droplet diameters and the times to puffing/microexplosion. The experiments were performed with a resolution of 256 pixels×800 pixels, 12,000 frames per second (fps) and exposure time of 2 µs. The image scale factor was 0.0185 mm/pixel. The uncertainty for the time to puffing/micro-explosion (±167 µs) was linked to the identification of the frames for droplet creation and start of puffing/micro-explosion. The measurement error of the droplet diameter (±37 µm) was related to the detection of the droplet perimeter. Droplets with diameters less than 50 µm were not used in the analysis due to the large experimental uncertainty. Typical initial values of parameters for composite n-dodecane/water droplets (droplet surface temperatures and ambient gas parameters), used in the analysis, are presented in Table 5.1. Substituting these values into Expression (5.8) allows us to obtain the distribution of temperature inside droplets at various time instants. Examples of these distributions for a droplet of initial radius 25 µm, initial homogeneous temperature 434 K and surface temperature 489.47 K (boiling temperature of n-dodecane) are presented in Fig. 5.3. In Fig. 5.3, the temperatures inside the composite droplet are presented as functions of the normalised distance from the droplet centre (R/Rw ). Note that Rd /Rw = 1.88. The curves 1, 2, 3, 4 and 5 refer to time instants 1.1 µs, 11 µs, 0.11 ms, 0.25 ms and 0.5 ms, respectively. As can be seen from this figure, for curve 5, corresponding to time instant 0.5 ms, the predicted temperature at R = Rw is very close to 373.15 K (the boiling temperature of water). At this time instant, evaporation of water is expected to lead to puffing/micro-explosion above the surface of the water sub-droplet. This time instant is referred to as ‘time to puffing/micro-explosion’. It is assumed that the micro-explosion occurs shortly after the start of puffing. Plots like those shown in Fig. 5.3 were produced for all droplet radii, and initial and surface droplet temperatures presented in Table 5.1. For all these plots, the times to puffing were obtained. These times are presented in Fig. 5.4 as functions of initial

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Table 5.1 Gas and droplet properties Parameter

Value

Parent droplet radii (Rd ) [µm] Droplet composition [vol] n-dodecane density (ρ f ) [kg/m3 ] Gas composition Droplet surface temperatures (Ts ) [K] Initial droplet temperature (Td0 ) [K] Gas (air) pressure [MPa]

Fig. 5.3 The temperatures inside the composite droplet (T ) versus normalised distance from the droplet centre R/Rw at five instants of time: 1.1 µs (curve 1), 11 µs (curve 2), 0.11 ms (curve 3), 0.25 ms (curve 4) and 0.5 ms (curve 5). Reprinted from [40], Copyright Elsevier (2019)

25; 50; 100 0.15 water + 0.85 n-dodecane 825 Air 489.47 (boiling temperature), 470, 450 343 and 363 0.1

T,500 K

460

5

420

4 3

380 Tb = 373.15 K

2 1 340 0

1 R /Rw

1.5

Rd /Rw

10 8 Time to puffing, ms

Fig. 5.4 Times to puffing/micro-explosion, predicted by the simple analytical model, versus droplet diameters for six combinations of initial and surface droplet temperatures. Times to puffing inferred from experimental data are presented as empty circles. Reprinted from [40], Copyright Elsevier (2019)

0.5

6 4 Td0 Ts 363 489.47 343 489.47 363 470 343 470 363 450 343 450 Experiment

2 0 0

50

100 150 200 Initial droplet diameter, µm

250

droplet diameters for six combinations of initial and surface droplet temperatures. Times to puffing for various droplet diameters inferred from experimental data are presented in the same Fig. 5.4.

5.2 A Simple Analytical Model

285

As follows from Fig. 5.4, the predicted times to puffing/micro-explosion when droplet surface temperature is equal to the n-dodecane boiling temperature are always less than those inferred from experimental data. The orders of magnitude of the predicted times to puffing/micro-explosion for other values of droplet surface temperature are close to those observed experimentally in most cases. Note that for a small droplet with diameter 50 µm only one value of time to puffing/micro-explosion was obtained. The predicted increase in times to puffing/micro-explosion as the droplet diameters increase is consistent with experimental data, despite considerable scatter of the latter. The model, however, cannot explain visibly shorter predicted times to puffing/micro-explosion for droplets with diameters less than 100 µm compared with experimental observations. The decrease in droplet surface temperature leads to an increase in the time to puffing/micro-explosion, as expected. To summarise our analysis, we can conclude that the analytical model, despite its simplicity, can in some cases describe the processes leading to puffing/micro-explosions, both qualitatively and quantitatively.

5.3 A Simple Numerical Model Although the model presented in Sect. 5.2 leads to correct crude prediction of the time to puffing/micro-explosion and correct dependence of this time on droplet sizes, it is not suitable for precise quantitative prediction in the general case. This is primarily attributed to numerous assumptions on which this model is based. In [41], some of the assumptions of the model described in Sect. 5.2 were relaxed and a new numerical model of the phenomenon was suggested. In what follows, the model suggested in [41] is described. The model presented in [41] uses the same assumption about a spherical water sub-droplet in the centre of a spherical fuel droplet as in the previously described model (see Fig. 5.2). In contrast to the model presented in Sect. 5.2, however, the analytical solution to the transient heat transfer equation inside the composite droplet was obtained based on the Robin (rather than Dirichlet) boundary condition. This analytical solution was applied not for the whole period preceding puffing/microexplosion but during short time steps. This solution was used in a numerical code in which the effects of evaporation, thermal swelling and the temperature dependence of all thermophysical properties were taken into account. The effects of evaporation were considered based on the Abramzon and Sirignano model (see Sect. 3.2.2). The droplet surface was assumed to be stationary during each time step. Finally. in contrast to the model described in Sect. 5.2, puffing/micro-explosion is assumed to start when the temperature at the water–fuel interface reaches the water nucleation (rather than boiling) temperature. The description of the model in Sect. 5.3.1 starts with a summary of equations and approximations on which it is based.

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5 Processes in Composite Droplets

5.3.1 Key Equations and Approximations The model uses the same geometry of a fuel/water droplet as shown in Fig. 5.2. Similar to the model presented on Sect. 5.2, the heat conduction process in the model of [41] is based on Eq. (5.1) with the initial condition (5.3) and the continuity condition at the water–fuel interface (5.4). In contrast to the model of Sect. 5.2, however, the Robin boundary condition, considering the effect of evaporation, is used at the fuel droplet surface: h (Teff − T (Rd )) = k f where Teff = Tg +

 ∂ T  , ∂ R  R=Rd −0

(5.14)

ρl L R˙ d(e) , h

h is the convection heat transfer coefficient, the term R˙ d(e) = d Rd /dt considers the effect of evaporation (see Expression (3.23)). The coefficient h is presented as h = k g Nu/(2Rd ), where the Nusselt number for stationary evaporation droplets is described by Eq. (3.34). The analytical solution to Eq. (5.1) subject to initial and boundary conditions (5.3), (5.4) and (5.14) was presented as (see Appendix R for the details) [41]:  ∞ μ0 r 1  n (t)vn (r ) + T = , (5.15) Rd r n=1 1 + h0 where  2   2  pn λ t λ t , n (t) = (qn + f n μ0 ) exp − n2 + 2 1 − exp − n2 λn Rd Rd fn = =

1 ||vn ||2

 rw 0

1 ||vn ||2

 rw 0



1

f (r )vn (r )bdr

0

 1 −r sin(λn aw r ) −r sin(λn a f r + βn ) 2 dr + k w aw k f a 2f dr 1 + h 0 sin(λn aw rw ) rw 1 + h 0 sin(λn a f rw + βn )

qn = =

1 ||vn ||2

Rd r T0 (Rd r )

1 ||vn ||2



1

F0 (r )vn (r )bdr

0

 1 sin(λn a f r + βn ) sin(λn aw r ) 2 dr + k w aw k f a 2f dr Rd r T0 (Rd r ) sin(λn aw rw ) sin(λn a f rw + βn ) rw

5.3 A Simple Numerical Model

287

1 pn = ||vn ||2 1 = ||vn ||2

 rw 0



1 0

˜ )vn (r )bdr Rd3 r P(r

1 sin(λn a f r + βn ) 2 dr + ˜ ) sin(λn aw r ) kw aw ˜ ) k f a 2f dr Rd3 r P(r Rd3 r P(r sin(λn aw rw ) sin(λn a f rw + βn ) rw



∞

f (r ) ≡

F0 (r ) ≡ Rd r T0 (Rd r ) =

 −r = f n (t)vn (r ), 1 + h0 n=1

∞ 

˜ )= qn (t)vn (r ), Rd3 r P(r

n=1

 vn (r ) =

sin(λn aw r ) sin(λn aw rw ) sin(λn a f r +βn ) sin(λn a f rw +βn )

∞ 

pn (t)vn (r ),

n=1

when 0 ≤ r ≤ rw when rw < r ≤ 1,

βn is β(λ = λn ),

1

||vn ||2 = 0

+

vn2 bdr =

 cfρfkf



λn sin2 (λn a f rw + βn )

β = cot

−1



√

sin(2aw λn rw ) a w λn r w cw ρw kw − 2 4 λn sin2 (λn aw rw )

sin(2λn a f + 2βn ) − sin(2λn a f rw + 2βn ) a f λn (1 − rw ) − 2 4



k f − kw k w aw + cot (aw λrw ) + iπ − a f λrw , k f a f rw λ k f a f

i = 0, 1, 2, 3, . . ., the analysis is restricted to the case when i = 0 (the values of v are the same for other i), a countable set of positive eigenvalues λn was obtained from Condition (5.14): λn a f cos(λn a f + β) + h 0 sin(λn a f + β) = 0.

(5.16)

The following parameters were used: H=

 1 Rd  h ˜ ) = P(r Rd ), hTg + ρl L f R˙ d , r = R/Rd , P(r − , μ= kf Rd kf F(t, r ) = u(t, R) ≡ T (R, t)R, h 0 = H Rd =

h Rd − 1, kf

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5 Processes in Composite Droplets

μ0 = μRd =

 Rd2  hTg + ρl L f R˙ d . kf

In the limit when thermophysical parameters of water and fuel are the same, Solution (5.15) reduces to Expression (16) of [38] derived for homogeneous droplets (see Appendix R for the details). The effect of thermal swelling was considered using the requirement of conservation of masses of water and fuel during the heating process. For each time step, this requirement leads to the following relations:  Rw1 = Rw0

ρw0 ρw1

1/3 (5.17)

  4  3 4  3 3 3 π Rd0 − Rw0 ρ f 0 = π Rd1 ρ f 1, − Rw1 3 3

(5.18)

where subscripts 0 and 1 show the beginning and end of the time step, respectively. Equations (5.17) and (5.18) lead to the following formula for the change in droplet radius due to swelling at the end of the time step:

Rd(s) ≡ Rd1 − Rd0

  

1/3 3  3 ρf0 Rw0 ρw0 Rw0 1− 3 = Rd0 + 3 −1 . Rd0 ρ f 1 Rd0 ρw1

(5.19)

where subscript (s) shows the change due to thermal swelling. Expression (5.19) is more accurate than that used in [43]. The total change of droplet radius due to evaporation and swelling follows from the expression: (5.20)

Rd(t) = Rd(e) + Rd(s) , where Rd(s) is given by (5.19), while the change of droplet radius due to evaporation ( Rd(e) ) follows from Eq. (3.23): 

Rd(e) =

m˙ d 2 4π Rd0 ρf0



t,

(5.21)

where t − t1 − t0 . Note that the first attempt to obtain the solution to (5.1) with the initial and boundary conditions (5.3), (5.4) and (5.14) was made in [33]. Unfortunately, there was a mistake in the analysis presented in this paper which was corrected in [41] where Solution (5.15) was first obtained.

5.3 A Simple Numerical Model

289

5.3.2 Preliminary Analysis Calculations using the model presented in Sect. 5.3.1 were performed in parallel based on Mathematica 12.1 and Matlab R2020a. The complete agreement between the results gave us confidence in both approaches. No difference in the results was noticed when 50 and 100 terms were used in series (5.15). Time steps 1 µs and 0.1 s were used in calculations for the smallest and largest droplets, respectively; 300 cells along the droplet radius were used to calculate integrals in (5.15). The roots of Eq. (5.16) were obtained based on the bisection method with absolute accuracy of 10−16 . Firstly, the predictions of the model presented in Sect. 5.3.1 were compared with the results of the experiment described in [33]. Pressure was assumed to be atmospheric and ambient gas temperature Tg was constant. The effects of fuel vapour on ambient gas thermophysical properties were not considered. In contrast to [33], the temperature dependence of thermodynamic and transport properties of water and fuel (n-dodecane) were taken into account following [1, 61]. The droplet surface temperatures (Ts ) and average temperatures of water and liquid fuel (Tavg(w) and Tavg( f ) ) were calculated at each time step. The specific heat of evaporation and saturated fuel vapour mass fractions were calculated at Ts . The thermophysical properties of water and fuel were obtained at Tavg(w) and Tavg( f ) , respectively. The ambient air thermodynamic and transport properties were estimated using the 2/3 rule (see Formula (3.14)). These properties were estimated for the initial temperatures or the temperatures obtained at the end of the previous time step. The same refers to d Rd /dt used for finding Teff . The droplet initial temperature and ambient gas temperature were assumed equal to 300 K and 700 K, respectively. The droplet initial radius was assumed equal to 5 µm. The ambient pressure was set to 101.325 kPa and the initial volume fraction of water Vw0 was assumed equal to 15% (the same values of parameters as used for Fig. 5.3 of [33]). Figure 5.5 shows the distribution of temperature in the droplet at eight time instants predicted by the model. As follows from Fig. 5.5, rw shifts and tends to merge with the droplet surface during the droplet heating and evaporation process. The changes in rw are linked with shrinking of droplets during evaporation. At t = 0.25 µs, the droplet radius reduced to about 4.7 µm. The effects of swelling are ignored to be consistent with the approximation used to obtain the results presented in Fig. 5.3 of [33]. Firstly, as in the simple analytical models presented in Sect. 5.2, the start of the puffing/micro-explosion is identified as the time instant when the temperature at the fuel/water interface reaches the water boiling temperature. This allows us to estimate the time to puffing/micro-explosion for the case shown in Fig. 5.5 as 0.135 ms. Plots like those shown in Fig. 5.5, but for bigger droplets used in the experiments at Tomsk Polytechnic University are shown in Figs. 5.6 (ignoring swelling) and 5.7 (with swelling considered). In these experiments, kerosene (Jet A-1) was used as the fuel. In the modelling, kerosene was approximated by n-dodecane (as in Fig. 5.5). A composite fuel/water droplet at initial temperature Td0 = 300 ± 3 K was introduced into a hot air. Air temperatures 573, 673 and 773 K were used. The calculations, the

290

5 Processes in Composite Droplets

Fig. 5.5 Plots of T (r ) at eight time instants for Rd0 = 5 µm, Td0 = 300 K, Tg = 700 K, p = 101.325 kPa, Vw0 = 15%. Reprinted from [41], Copyright Elsevier (2020)

results of which are shown in Figs. 5.6 and 5.7, were performed for 773 K and at atmospheric pressure. The initial droplet radius Rd0 = 1.33 ± 0.03 mm and water volume fraction Vw0 = 20% were used. Comparing Figs. 5.5 and 5.6 it can be seen that the evolutions of temperatures inside small and large droplets in space and time are rather similar, but the predicted time to puffing/micro-explosion is much longer for large droplets (8 s) than for small ones. Comparing Figs. 5.6 and 5.7 one can see that considering the effect of swelling leads to longer times to puffing (8.2 s) compared to the case when this effect is ignored (8.0 s). This is related to the fact that swelling leads to an increase in the thickness of the fuel shell through which heat is transferred from the droplet surface to the fuel/water interface. As follows from the analysis presented in [41], the predicted droplet radius reduced from 1.33 mm to about 1 mm over 25 s when swelling was not considered. When swelling was considered, this radius was reduced to only 1.1 mm in the same time. This demonstrates that the effect of swelling is worthy of note and should be considered in the investigation of the phenomenon.

5.3.3 Boiling Versus Nucleation Temperature As described in Sect. 5.2, the simple analytical model suggested in [40] uses the assumption that puffing/micro-explosion starts when the temperature at the fuel/water

5.3 A Simple Numerical Model

291

Fig. 5.6 Plots of T (r ) at seven time instants for Rd0 = 1.33 mm, Td0 = 300 K, Tg = 773 K, p = 101.325 kPa, Vw0 = 20%. Swelling was ignored. Reprinted from [41], Copyright Elsevier (2020)

Fig. 5.7 Plots of T (r ) at seven time instants for Rd0 = 1.33 mm, Td0 = 300 K, Tg = 773 K, p = 101.325 kPa, Vw0 = 20%. Swelling was considered. Reprinted from [41], Copyright Elsevier (2020)

292

5 Processes in Composite Droplets

interface becomes equal to the water boiling temperature TB . The same assumption was used in Sect. 5.3.2. It is well known, however, that in the transient case when temperature increases with time quickly enough, boiling is expected to take place not at the boiling temperature TB , but at the nucleation temperature TN which is higher than TB . The link between these temperatures has been studied by many authors (e.g. [12, 50]). The following correlation was suggested by the authors of [12] for large heating rates: TN = TB + 0.37TB · T˙ 10/J a H N

105 ≤ T˙ ≤ 109 K/s,

(5.22)

where T˙ is the rate of temperature increase, T is in K, J a H N = 626 for water. Using the experimental results described in [12, 50], an alternative correlation was suggested in [41]: TN = 385 + 160 × tanh(T˙ /105 )

102 ≤ T˙ ≤ 106 K/s.

(5.23)

Both correlations ((5.22) and (5.23)) infer from the experiments performed at atmospheric pressure. The results of comparison of the nucleation temperatures predicted by Expressions (5.22) and (5.23) and those observed experimentally is presented in Fig. 5.8. The experimental results published in [17, 22, 23, 36, 37, 41, 50] are used in this figure. In the experiments described in [41], the two-colour Laser-Induced Fluorescence (LIP) was used to estimate the temperatures at the fuel/water interface immediately before and at the time instant when puffing started. The temperature at the time instant when puffing started was identified as the nucleation temperature, while the changes in temperature before this time instant allow one to obtain the values of T˙ used in Correlations (5.22) and (5.23). This approach is suitable for heating rates up to about 250–300 K/s. As can be seen from Fig. 5.8, the agreement between Correlations (5.22) and (5.23) and experimental data is reasonably good. This allows us to recommend these correlations for obtaining the values of nucleation temperatures in practical applications. In the experiments described in [41] T˙ were less than 250-300 K/s, and on many occasions, they were less than 100 K/s. This limits the applicability of Expression (5.23) to the estimation of TN . This was the motivation behind the development of a new correlation applicable to low heating rates [41]: TN = TB + 12 × tanh(T˙ /50)

0 ≤ T˙ ≤ 300 K /s.

(5.24)

The degrees of superheating ( TS H = TN − TB ) inferred from the experiments described in [41], those described by Su et al. [50], and those inferred from Expression (5.24) are shown in Fig. 5.9. Good agreement between the results of experiments and prediction of (5.24) is clearly demonstrated this figure.

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Fig. 5.8 Nucleation temperatures versus heating rates—a comparison between experimentally observed values and those obtained from Correlations (5.23) (solid curve) and (5.22) (dashed curve). The experimental points referred to as Iida (1994), Glod (2002), Rosenthal (1957), Ching (2014), Sakurai (1977), and Su (2016) were taken from [17, 22, 23, 36, 37, 50], respectively. The points ‘Current study’ refer to experimental data presented in [41]. Reprinted from [41], Copyright Elsevier (2020) Fig. 5.9 Superheating values of TS H = TN − TB observed in the experiment described in [41] (filled stars), those by Su et al. [50] (filled and empty circles) and those predicted by Expression (5.24) (solid curve) for the input parameters described in the text. Reprinted from [41], Copyright Elsevier (2020)

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Error bars are included in Figs. 5.8 and 5.9. Average errors of measurement described in [22, 23, 36] were ±5 K. Those described in [41, 50] were ±1.5% and ±4 K, respectively. In [37] the errors of measurements were not specified. The development of a single correlation for TN instead of the above-mentioned three correlations is not straightforward as the physical nature of the onset of nucleate boiling is different for different heating rates [41]. Note that in the model developed in [20, 62] the start of the puffing/microexplosion was associated not with the time instant when the fuel/water interface temperature reaches TN but with that when this temperature reaches the superheat limit. The latter temperature, also called spinodal, is the absolute thermodynamic limit of stability of superheated (meta-stable) liquids [11]. For non-ideal systems, this limit can be reached only for very fast heating rates (above 1011 K/s for water at atmospheric pressure) [11], which is well above heating rates observed during most micro-explosion experiments.

5.3.4 Times to Puffing/Micro-Explosion The results described in Sect. 5.3.3 allow us to estimate the times to puffing/microexplosion more accurately than in Sect. 5.3.2. This is achieved based on the observation that the model presented in Sect. 5.3.2 predicts not only the values of the temperature at the fuel/water interface, but also the rate of increase of this temperature at this location (T˙ ). The latter allows us to obtain the water nucleation temperature at this interface at each time instant using one of the correlations described in Sect. 5.3.3. This leads to more accurate estimates of the start time of puffing/micro-explosion: the time instant when the interface temperature is equal to the water nucleation temperature. This approach to the estimation of this time is illustrated for the cases shown in Figs. 5.5, 5.6 and 5.7. Plots of T˙ at the fuel/water interface versus time are presented in Fig. 5.10. The same parameters as in Fig. 5.5 are used. As can be seen in this figure, for almost the whole time period under consideration the values of T˙ are in the range 102 ≤ T˙ ≤ 106 K/s which allowed us to use Correlation (5.23) to find the value of TN . The plot of time evolution of the nucleation temperature (TN ), predicted by Correlation (5.23) and T˙ presented in Fig. 5.10, is shown in Fig. 5.11. The time evolutions of the droplet surface temperature (Ts ), the temperature at the fuel/water interface (Tw ) and the boiling temperature of water (TB ) are also presented in Fig. 5.11. As follows from Fig. 5.11, the curves TN (t) and Tw (t) intersect at 0.41 ms. This time is identified with the time to puffing/micro-explosion if the time instant when Tw = TN is considered as the start of this process. This time is compared with the time to puffing/micro-explosion equal to 0.135 ms when the start of this process is identified with the time instant when Tw = TB . The difference between these times needs to be considered when modelling puffing/micro-explosion phenomena.

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Fig. 5.10 The plot of time evolution of T˙ ≡ dT /dt at the fuel/water interface, for the same parameters as in Fig. 5.5; the cross and circle show the time instants when Tw = TB and Tw = TN , respectively (see Fig. 5.8). Reprinted from [41], Copyright Elsevier (2020)

Fig. 5.11 Plots of Tw (thick solid), Ts (thin solid), TN (dashed-dotted), and TB (horizontal dashed line) versus time; the vertical dashed lines indicate the time instants when Tw = TB and Tw = TN ; the same parameters as in Fig. 5.5 were used. Reprinted from [41], Copyright Elsevier (2020)

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Fig. 5.12 Plots of time evolution of T˙ at the fuel/water interface for the same parameters as in Figs. 5.6 and 5.7, considering swelling (solid) and ignoring it (dashed); the crosses and circles show the time instants when Tw = TB and Tw = TN , respectively (see Figs. 5.13 and 5.14 for the details). Reprinted from [41], Copyright Elsevier (2020)

Note that at t = 0.41 ms the predicted droplet radius becomes equal to 3.86 µm. This radius is well above the radius of the water sub-droplet (2.66 µm) for Vw0 = 15%. Plots of time evolution of T˙ at the fuel/water interface for the same parameters as in Figs. 5.6 and 5.7 (large droplet heating and evaporating with and without considering swelling) are shown in Fig. 5.12. As can be seen in this figure, the values of T˙ are less than 11 K/s in this case. This allows us to use Formula (5.24) for the nucleation temperature TN . Note that the maximal values of T˙ are slightly smaller for the case with swelling than for the case without swelling. Plots of time evolution of the nucleation temperature (TN ), predicted by Formula (5.24) and the values of T˙ presented in Fig. 5.12 for the case without swelling, are presented in Fig. 5.13. The plots of time evolution of the droplet surface temperature (Ts ), the temperature at the fuel/water interface (Tw ) and the boiling temperature of water (TB ) are presented in the same figure. As follows from Fig. 5.13, in contrast to the case presented in Fig. 5.11, the curves TN (t) and Tw (t) intersect at 8.2 s. This time is identified with the time to puffing/micro-explosion: the time instant when Tw = TN . This is considered the start of puffing/micro-explosion. This time can be compared with the time to puffing/micro-explosion equal to 8.0 s predicted by the model described in Sect.

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Fig. 5.13 The same as Fig. 5.11 but for the same parameters and approximations as in Fig. 5.6; vertical dashed lines indicate time instants when Tw = TB and Tw = TN . Reprinted from [41], Copyright Elsevier (2020)

5.3.2, where the start of puffing/micro-explosion is identified with the time instant when Tw = TB (also shown in Fig. 5.13). Plots similar to those presented in Fig. 5.13 but considering the effect of swelling are shown in Fig. 5.14. The shapes of the curves in Figs. 5.14 and 5.13 are rather similar, although in Fig. 5.14 the time to puffing/micro-explosion predicted by the model with swelling (8.5 s) is slightly longer than the one predicted by the model without swelling. This value of time to puffing/micro-explosion can be considered the most accurate one within the restrictions of the model. It is slightly longer than the one which follows from the model presented in Sect. 5.3.2, where the start of puffing/micro-explosion is identified with the time instant when Tw = TB (8.2 s). Note that at t = 8.2 s the droplet radius reduces, due to the combined effect of evaporation and swelling, to 1.31 mm. This radius is well above the radius of the water sub-droplet (0.78 mm) for Vw0 = 20%. As follows from our analysis, the model predicts a strong effect of TN on the onset of puffing/micro-explosion for small droplets (Fig. 5.11) and relatively weak effect for large droplets (Figs. 5.13 and 5.14). In what follows, the model is applied to investigate the effects of the difference between TN and TB on the time to puffing/microexplosion for intermediate droplets. As in [33], it is assumed that the initial n-dodecane droplet radius and temperature are equal to 100 µm and 300 K, respectively. The ambient gas temperature and pressure are taken equal to 700 K and 101.325 kPa, respectively. The initial volume fraction of water Vw0 is taken equal to 15%. The temperature dependence of

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Fig. 5.14 The same as Fig. 5.13 but considering swelling. Reprinted from [41], Copyright Elsevier (2020)

thermophysical properties of the components is assumed to be the same as used in the previous figures. The plot of time evolution of T˙ at the fuel/water interface considering swelling is presented in Fig. 5.15. The cross and circle indicate the time instants when the temperature at the fuel/water interface reaches TB and TN . In both cases, T˙ was in the range 102 ≤ T˙ ≤ 106 K/s, which allowed us to use Correlation (5.23) to find TN . Plots of the time evolution of TN , predicted by Formula (5.23) and T˙ presented in Fig. 5.15, are shown in Fig. 5.16. The time evolutions of the droplet surface temperature (Ts ), the temperature at the fuel/water interface (Tw ) and the boiling temperature of water (TB ) are shown in the same figure, as in the case of Figs. 5.11, 5.13 and 5.14. As follows from Fig. 5.16, the curves TN (t) and Tw (t) intersect at 67 ms. As in the previous cases, this time is identified with the time to puffing/micro-explosion. This time can be compared with the time to puffing/micro-explosion equal to 56 ms predicted by the model in which the start of puffing/micro-explosion is identified with the time instant when Tw = TB . The difference between these times needs to be considered when modelling puffing/micro-explosion phenomena. Note that at t = 67 ms the droplet radius increases, due to the combined effect of swelling and evaporation, to 101.65 µm which is well above the radius of the water sub-droplet (53.13 µm) for Vw0 = 15%. The experimental measurements of the time to puffing/micro-explosion, for droplets with the same parameters as shown in Fig. 5.14, led to the value of

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299

Fig. 5.15 The plot of time evolution of T˙ ≡ dT /dt at the fuel/water interface, for a droplet with initial radius 100 µm; the cross and circle show the time instant when Tw = TB and Tw = TN , respectively (see Fig. 5.16 for the details). Reprinted from [41], Copyright Elsevier (2020)

Fig. 5.16 Plots of Tw (thick solid), Ts (thin solid), TN (dashed-dotted), and TB (horizontal dashed line) versus time; the vertical dashed lines indicate the time instants when Tw = TB and Tw = TN ; a droplet with initial radius 100 µm was considered. Reprinted from [41], Copyright Elsevier (2020)

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3.075 ± 0.2 s. The experimentally observed values for droplets with initial radius 100 µm and the same parameters as in Figs. 5.15 and 5.16 were found to be shorter than 12 ms [33, 40]. In both cases, the predicted values of this time are longer than those inferred from experimental data. As in the case described in [3], this difference is attributed to the main assumption of our model: the water sub-droplet is located in the centre of the fuel droplet. This assumption is questionable for the intermediate and relatively large droplets used in the experiments. To summarise the analysis in this section, it can be concluded that the difference in the times to puffing/micro-explosion, inferred from the assumptions that puffing/micro-explosion starts at water boiling and nucleation temperatures, needs to be considered if accurate estimation of this time is required. In what follows, the model described above is applied to the investigation of puffing/micro-explosion in several specific cases of engineering importance.

5.4 Puffing/Micro-Explosion in the Presence of Coal Particles This section is focused on the application of the model presented in Sect. 5.3 to the analysis of experimental data referring to puffing/micro-explosions in composite rapeseed oil/water droplets in the presence of lignite and bituminous coal microparticles in water. The most important original results presented in [5] are reproduced.

5.4.1 Rapeseed Oil/Water Droplets with Coal Micro-Particles In the experiments described in [5], rapeseed oil/water droplets, with solid coal micro-particles in water, with radii in the range 1–2 mm were introduced into a chamber with air velocities 3–7 m/s and temperatures up to 600 ◦ C. Two types of coal micro-particles were used in the experiments: lignite and bituminous coal. The selection of these coals was based on the differences between their thermodynamic, transport and hydrophilic/hydrophobic properties (lignite is more hydrophilic than bituminous coal). Snapshots and schematics of droplet evaporation, puffing and micro-explosion are shown in Fig. 5.17a–c. The numbers of child droplets with various radii, generated during puffing and micro-explosion, are presented in Fig. 5.17d. The droplet radii gradually reduced during the evaporation process. During puffing, isolated fragments of liquid were detached from the droplet; up to 20 large liquid fragments were formed. An increase in droplet diameters usually preceded the micro-explosions. The latter led to the formation of a cloud of more than 200 child droplets with diameters of up to 0.01 mm. Micro-explosions tended to be observed at temperatures higher than those at which puffing and evaporation were observed. As follows from Fig. 5.17d,

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Fig. 5.17 Snapshots and schematics of droplet evaporation, puffing, and micro-explosion for droplet initial radii equal to 1.21 mm (a–c) and plots of the numbers of child droplets produced during puffing and micro-explosions versus child droplet radii (d). Mass fractions of rapeseed oil, water and coal micro-particles were 0.7, 0.27 and 0.03, respectively. a evaporation of a droplet, with lignite micro-particles, in air at temperature Ta = 200 ◦ C; b puffing of a droplet, with lignite micro-particles, in air at temperature Ta = 300 ◦ C; c micro-explosion of a droplet, with bituminous coal micro-particles, in air at temperature Ta = 400 ◦ C. Reprinted from [5], Copyright Elsevier (2021)

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no child droplets with radii over 0.16 mm were generated during micro-explosion; no child droplets with radii under 0.06 mm were generated during puffing. The relative air-droplets velocity in the experiments performed in [5] was linked with the air temperature Ta in ◦ C via the following correlation: Ua = 0.0097Ta + 1.3357.

(5.25)

5.4.2 Modelling Versus Experimental Results We focus on the results referring to times to puffing/micro-explosion and attempt to interpret them in terms of a generalised version of the model presented in Sect. 5.3. One of the most important limitations of the latter model is that it assumes spherical symmetry of all processes. At the same time, the processes investigated in [5] took place in the presence of air flow. This makes it difficult to apply this assumption to their analysis. To address this problem, a non-self-consistent generalisation of this model was suggested in [5]. In the generalised version of this model, the convection heat and mass transfer coefficients were taken from the Abramzon and Sirignano model described in Sect. 3.2.2 with the correlation for Nu given by Expression (2.62) and the corresponding correlation for Sh given by the same expression with Pr replaced with Sc (Schmidt number). At the same time, the recirculation of the liquid inside the droplet, caused by its motion, was not considered (this makes the model nonself-consistent). The effect of the droplet support was not considered. Following the procedure described in Sect. 5.3, the times to puffing/microexplosion were initially obtained assuming that this process is initiated when the temperature at the fuel/water interface (Tw ) has reached the boiling temperature of water (TB ). Nucleation temperatures (TN ) were obtained, and new times were calculated assuming that this process starts when Tw = TN . Plots of time to puffing/micro-explosion versus air temperature, using the same input parameters as in the experiments described in [5], without taking into account the contribution of coal particles, and assuming that puffing is initiated when Tw = TB , are shown in Fig. 5.18. The results of measurements are shown in the same figure. The differences in times to puffing/micro-explosions for the same conditions as used for the plots in Fig. 5.18 in the presence and absence of coal micro-particles (lignite (a) and bituminous coal (b)) without the contribution of Ua ( τ ) are presented in Fig. 5.19. As follows from Fig. 5.18, the dependence of predicted times to puffing/microexplosion on Ua is particularly strong at low ambient air temperatures. At air temperature 200 ◦ C the model predicts times to puffing/micro-explosion that are five times shorter when the contribution of the relative droplet-air velocity is considered compared to the case when its contribution is ignored. This shows the importance of this effect. It follows from the same figure that the model correctly predicts decreasing time to puffing/micro-explosion with increasing ambient air temperature.

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Fig. 5.18 Times to puffing/micro-explosion of composite droplets (τ ) versus air temperatures, predicted by the model and inferred from experimental data. It was assumed that puffing starts when Tw = TB . The contribution of coal particles was not considered in simulations. Curves 0S and AS show simulation results ignoring and considering the contribution of relative air-droplet velocities Ua , respectively. Symbol 0E refers to the experimental results without coal micro-particles; symbols 1E, 3E and 5E refer to the cases when the mass fractions of coal micro-particles (lignite (a) and bituminous coal (b)) were 0.01, 0.03 and 0.05, respectively. Reprinted from [5], Copyright Elsevier (2021)

Fig. 5.19 The difference in times to puffing ( τ ) predicted by the same conditions as in Fig. 5.18 in the presence and absence of coal micro-particles (lignite (a) and bituminous coal (b)) without considering Ua . Curves 1, 3 and 5 refer to the cases when the mass fractions of particles were 0.01, 0.03 and 0.05, respectively. Reprinted from [5], Copyright Elsevier (2021)

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Fig. 5.20 The plot of T˙ at the rapeseed oil/water interface versus time; Ta = 500 ◦ C, initial water volume fraction 0.3, and initial droplet radius Rd0 = 1.21 mm. The contribution of coal micro-particles and Ua was not considered. The cross and circle indicate the time instants when Tw = TB and Tw = TN , respectively. Reprinted from [5], Copyright Elsevier (2021)

As follows from Fig. 5.19 showing the case without the contribution of Ua , the effect of the type of coal and coal micro-particle mass fraction on the time to puffing/micro-explosion, predicted by the model, is small. In all cases, the difference between these times for the cases with and without coal particles for both types of coal is less than 0.3 s. It decreases when ambient air temperature increases. This difference is marginally larger for droplets with bituminous coal micro-particles than for those with lignite micro-particles. When the contribution of Ua is considered these times become less than 0.1 s in all cases [2]. The plot of T˙ ≡ dT /dt at the fuel/water interface versus time is shown in Fig. 5.20. The plot is presented for Ta = 500 ◦ C, initial water volume fraction 0.3, and initial droplet radius Rd0 = 1.21 mm. It was shown in [5] that for these values of input parameters the puffing/micro-explosion was most stable. The meanings of the cross and circle are explained later in this section. As follows from Fig. 5.20, the values of T˙ are always less than 13. This allows us to use Correlation (5.24). The plot of TN versus time for the same parameters as in Fig. 5.20, using Expression (5.24) and the values of T˙ taken from Fig. 5.20, are presented in Fig. 5.21. In the same figure, the predicted values of Tw and the line T = TB are presented. The intersection of the curves Tw and TB takes place at the time to puffing/microexplosion, if puffing/micro-explosion starts when Tw = TB . The time instant when the curves Tw and TN intersect is the time to puffing/micro-explosion, if puffing/microexplosion starts when Tw = TN . In the case shown in Fig. 5.21, these times are 6.9 s and 7.1 s, respectively. T˙ for these times are indicated in Fig. 5.20 as the cross and circle, respectively. The results of experimental study and modelling of puffing/micro-explosion in composite water/rapeseed oil mixed with coal micro-particles droplets were presented in [9]. It was shown that observed and predicted times to puffing/microexplosion decreased with increasing ambient gas temperature and mass fractions of

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Fig. 5.21 The plot of Tw (solid), TB (dashed), and TN (dashed-dotted) versus time for the same conditions as in Fig. 5.20. The values of TN were taken from Formula (5.24) with T˙ shown in Fig. 5.20. Vertical dashed lines refer to time instants when Tw = TB and Tw = TN . Reprinted from [5], Copyright Elsevier (2021)

coal particles in rapeseed oil Y p in the range 0 to 60% for ambient gas temperatures Tg = 850 K and Tg = 1100 K. The maximal deviations between the experimental and modelling results were observed for Tg = 850 K and Y p = 60%. This was related to the fact that, for large mass fractions of coal and low ambient gas temperatures, bubbles can be initiated not only at the water/oil interface but also where the coal micro-particles are located. This effect was not taken into account in the model used in the analysis of [9]. To summarise the results presented in this section, the prediction of the model using the assumptions that puffing/micro-explosion starts when Tw = TB is close to the prediction of the model using the assumption that puffing/micro-explosion starts when Tw = TN . The contribution of Ua needs to be considered [2].

5.5 Puffing/Micro-Explosion in Closely Spaced Droplets The main limitation of the experimental and modelling results presented in the previous sections of this chapter is that they focus on individual droplets without considering possible interactions between droplets. These interactions, however, cannot be ignored when puffing/micro-explosion happens in realistic spray-like conditions. ‘Grouping effects’ on the development of puffing/micro-explosion in composite droplets were investigated using Direct Numerical Simulation in [46, 48] and relatively simple models of the phenomenon are described in [6, 8]. In what follows, the key features of the models developed in [6, 8] are summarised.

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5.5.1 Puffing/Micro-Explosion in Two Droplets in Tandem In Fig. 5.22, typical video clips illustrating the processes of puffing/micro-explosion in two droplets in tandem for two values of the distance parameter C = L/(2Rd0 ), where L is the distance between droplets, are presented. As can be seen from this figure, puffing/micro-explosion of the lead droplets can be followed by evaporation, puffing and micro-explosion of the downstream droplet depending on the values of C and ambient gas temperatures. For sufficiently high ambient gas temperatures, micro-explosion of the lead droplet was followed by puffing or micro-explosion of the downstream droplet [6]. At lower gas temperatures, puffing or evaporation of the lead droplet was followed by puffing or evaporation of the downstream droplet [6]. If the lead droplet evaporated without puffing/micro-explosion, so did the downstream droplet. No visible effects of downstream droplets on the puffing/micro-explosion in the lead droplets were noticed. As follows from Fig. 5.22a, for C = 7.5 the difference between times to puffing/ micro-explosion of the downstream and lead droplets ( τp ) was rather short (about 0.44 s). On the other hand, it can be seen in Fig. 5.22b that for C = 3.25 this difference increases to τp = 3.23 s.

Fig. 5.22 Typical video clips demonstrating the development of puffing/micro-explosion in the lead (Droplet 1) and downstream (Droplet 2) composite droplets (with initial volume fractions of rapeseed oil and water 90% and 10%, respectively), introduced into air with velocity 0.1 ± 0.05 m/s. a C = 3.25 and Ta = 485 K; b C = 7.5 and Ta = 497 K. Reprinted from [6]. Copyright Elsevier (2021)

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The model used in the analysis of puffing/micro-explosion of two closely spaced composite droplets in tandem is essentially the same as used in Sect. 5.4, except that the effects of the interaction between droplets on Nusselt (Nu) and Sherwood (Sh) numbers were taken into account. To find the new correlations for Nu and Sh, considering this interaction, a numerical investigation of the heating and evaporation of two droplets in tandem was performed using several simplifying assumptions. These are the final formulae obtained from this study [6]. Case of the lead droplet   1 B1 −1 (5.26) Nu1 = Nu0 (Re1 , Pr) − A1 Re1 + G 1 · rT  Sh1 = Sh0 (Re1 , Sc) − A1 Re1B1 + G 1 · G1 =

1 −1 rc



F11 F12 + + F13 , 2 Re1 Re1

(5.27)

(5.28)

Nu0 and Sh0 are Nu and Sh of the isolated droplets: Nu0 (Re, Pr) = 1 + (1 + αRe Pr)γ · (1 + δRe)β

(5.29)

Sh0 (Re, Sc) = 1 + (1 + αRe Sc)γ · (1 + δRe)β ,

(5.30)

where rc =

c S,1 − c∞ , c S,2 − c∞

(5.31)

rT =

TS,1 − T∞ , TS,2 − T∞

(5.32)

α = 1.0024, γ = 0.3419, β = 0.0989 and δ = 0.2046, subscript S refers to the surface of the droplet, c is fuel vapour concentration (in mole/m3 ). The case of the isolated droplet was addressed separately as a deviation of a few percent could be observed for both Nu and Sh between the results of the numerical simulations and those predicted by classical models (see Chaps. 2 and 3). Expressions (5.29) and (5.30) are valid for 2 ≤ Re ≤ 25 and 0.775 ≤ Sc, Pr ≤ 4. In Formulae (5.26)–(5.28), parameters A1 , B1 , F11 , F12 and F13 are functions of C and r D = Rd01 /Rd02 ; subscripts 1 and 2 refer to the lead and downstream droplets, respectively. The results of calculation of the coefficients for Pr = 0.775 and Sc = 3.676 are presented in [6]. Case of the downstream droplet Nu2 = Nu0 (Re2 , Pr) − A2 Re2B2 + G 2 · (r T − 1)

(5.33)

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Fig. 5.23 The predicted and experimentally observed times to puffing/micro-explosion (τ p ) for the lead and downstream droplets versus the initial droplet radii. Circles and diamonds refer to experimental results for the lead and downstream droplets, respectively. Dotted and dashed-dotted curves refer to the corresponding modelling results if rc and r T do not change during the heating/evaporation process and remain equal to those at t = 0. Solid and dashed curves refer to the corresponding modelling results considering the changes in rc and r T during these processes (the values of the coefficients are given in [6]). The experiments and calculations were performed at atmospheric pressure, Ta = 430 K, ambient air velocity 0.1 m/s, L = 4.2 mm, and volume fraction of water-in-droplets equal to 10%. Reprinted from [6]. Copyright Elsevier (2021)

Sh2 = Sh0 (Re2 , Sc) − A2 Re2B2 + G 2 · (rc − 1)

(5.34)

G 2 = F21 Re2 + F22 ,

(5.35)

where Nu0 and Sh0 are given by Formulae (5.29) and (5.30). Parameters A2 , B2 , F21 , F22 are functions of C and r D . The results of their calculation for Pr = 0.775 and Sc = 3.676 are presented in [6]. The predicted and experimentally observed times to puffing/micro-explosion for the lead and downstream droplets versus the initial droplet radii (Rd0 ) are presented in Fig. 5.23. As follows from this figure, as in the case of isolated droplets, both predicted and experimentally observed times to puffing/micro-explosion (τ p ) increase with increasing Rd0 . Both predicted and experimentally observed τ p are longer for the downstream droplet than those for the lead droplet. Changes in rc and r T during droplet heating/evaporation have relatively small effects on the values of τ p , especially for the lead droplet. The predicted differences in τ p for the lead and downstream droplets are close to those observed experimentally. As in the case of isolated

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droplets, predicted τ p tend to be longer than those observed experimentally. This is related to the key assumption of the model that the water sub-droplet is located in the centre of the fuel droplet. Realistic shifts of this sub-droplet from the centre of the fuel droplet are expected to lead to prediction of shorter τ p .

5.5.2 Puffing/Micro-Explosion in a String of Three Droplets In Sect. 5.5.1 the results of modelling and experimental studies of the mutual effect of droplets on their puffing/micro-explosion in a flow, using an example of two composite rapeseed oil/water droplets in tandem, were presented. This section is focused on the generalisation of the results presented in Sect. 5.5.1, following [8]. This generalisation is performed in the following three directions. Firstly, three droplets, one behind the other are considered. Secondly, Diesel fuel/water droplets in a wide range of fuel and water volume fractions are considered. Thirdly, the focus is not only on the time instant when puffing and micro-explosion are initiated, but also on the time evolution of temperature at the fuel/water interface before the start of these processes. The case of three droplets instead of two made it possible to perform more realistic analysis of the effects of surrounding droplets on any particular droplet in a spray. An overview of the experimental setup with three droplets is shown in Fig. 5.24. The droplets shown in this figure were illuminated by the laser beam which led to yellow light emission around them generated by Rhodamine B dye. Ambient air temperature and velocity used in the experiments were in the range 589–629 K, and 0.1 m/s, respectively. The temperature profiles in the droplets were obtained using the Planar Laser-Induced Fluorescence (PLIF) method. The time instant when puffing/micro-explosion started was identified from the frame in which the first fragment separated from the main droplet.

Fig. 5.24 An image of the group of three droplets used in the experiments of [8]. Reprinted from [8]. Copyright Elsevier (2021)

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5 Processes in Composite Droplets

The same model as described in Sect. 5.5.1, but with different correlations for Nu and Sh was used in the analysis. These correlations were obtained from a numerical investigation of the heating/evaporation of three droplets. In Fig. 5.25 the predicted and experimentally observed temperatures at the fuel/water interface Tw are presented for the lead, middle and downstream droplets for several initial droplet radii Rd0 , distances between droplets L and distance parameters C = L/(2Rd0 ). As follows from Fig. 5.25, the general trends of the time evolution of Tw observed experimentally and predicted by the model are similar for all three values of C. These include almost linear increase in Tw with time for most of the heating period and smaller rates of increase in Tw for the middle and downstream droplets compared with the lead droplets. This difference between the rates of increase in Tw increased when C decreased. Although the difference between predicted and observed Tw lies within the experimental error bars in most cases, the observed Tw tend to be larger than those predicted by simulations. This is attributed to the main assumption of the model that a water sub-droplet is located exactly in the centre of the fuel droplet. If we consider a possible shift of the water sub-droplet from the centre then we would expect quicker heating of the fuel/water interface in agreement with experimental observations. Predicted and experimentally observed times to puffing/micro-explosion (τ p ) for the lead, middle and downstream droplets of 90% Diesel fuel and 10% water versus ambient gas (air) temperature Ta are presented in Fig. 5.26. As can be seen in Fig. 5.26, the general trends of the predicted and experimentally observed dependence of τ p on Ta are similar for all three droplets. The shortest τ p is observed and predicted for the lead droplet. Both experimentally observed and predicted values of τ p decrease with increasing Ta . Observed and predicted differences in τ p for the lead and middle droplets and for the middle and downstream droplets decrease with increasing Ta . Note that the observed τ p tend to be shorter than the predicted ones. This is compatible with the trends in Tw presented in Fig. 5.25 and is attributed to the main assumption of the model that the water sub-droplet is placed exactly in the centre of the fuel droplet. In Fig. 5.27, predicted and experimentally observed times to puffing/microexplosion (τ p ) versus distance parameter are shown for the same droplets as in Fig. 5.26. As follows from Fig. 5.27, the general trends of the dependence of τ p on the distance parameter C observed experimentally and predicted by the model are similar for all three droplets. The shortest τ p is predicted and observed for the lead droplet. Both predicted and experimentally observed τ p for the middle and downstream droplets decrease with increasing C. Observed and predicted differences in τ p for the lead and middle droplets and for the middle and downstream droplet decrease with increasing C. As in the cases presented in Figs. 5.25 and 5.26, the tendency for the observed τ p to be shorter than predicted is attributed to the main limitation of the model used in the analysis: the assumption that the water sub-droplet is located in the centre of the fuel droplet.

5.5 Puffing/Micro-Explosion in Closely Spaced Droplets Fig. 5.25 The temperature at the fuel-water interface (Tw ) observed experimentally (predicted by the simulation) versus time for the lead (LE(LS)), middle (ME(MS)) and downstream (DE(DS)) droplets of 90% Diesel fuel and 10% water. Top, L = 9.86 ± 0.06 mm and Rd0 = 0.88 ± 0.04 mm (C = 5.54); middle, L = 6.48 ± 0.04 mm and Rd0 = 0.89 ± 0.02 mm (C = 3.64); bottom, L = 3.55 ± 0.05 mm and Rd0 = 0.88 ± 0.03 mm (C = 2.01). In all cases, ambient gas (air) temperature was 604 ± 10 K. Reprinted from [8]. Copyright Elsevier (2021)

311

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Fig. 5.26 Time to puffing/micro-explosion (τ p ) observed experimentally (predicted by the simulation) versus ambient gas (air) temperature Ta for the lead (LE(LS)), middle (ME(MS)) and downstream (DE(DS)) composite droplets (90% volume fraction of Diesel fuel; 10% volume fraction of water). The distance between droplets (L) was 9.86 ± 0.08 mm and the initial droplet radii Rd0 were 0.89 ± 0.04 mm. Reprinted from [8]. Copyright Elsevier (2021)

Fig. 5.27 Time to puffing/micro-explosion observed experimentally (predicted by the simulation) versus the distance parameter (C) for the lead (LE(LS)), middle (ME(MS)) and downstream (DE(DS)) composite droplets (90% volume fraction of Diesel fuel; 10% volume fraction of water). The ambient gas temperature was 604 ± 10 K and the initial droplet radii Rd0 were 0.89 ± 0.04 mm. Reprinted from [8]. Copyright Elsevier (2021)

5.6 Effects of Thermal Radiation and Support Although the effect of thermal radiation is incorporated into Solution (5.15), it has not been considered in the analysis of composite droplet puffing/micro-explosion. The main reason for this is that, so far, the models have been focused on the investigation of droplet puffing/micro-explosion in gases at temperatures below 1000 K when the effects of thermal radiation are expected to be small. In what follows, the focus is shifted to the analysis of droplet puffing/micro-explosion in flames where gas temperatures exceed 1000 K and the effects of thermal radiation cannot be ignored. Apart from thermal radiation the effect of the support on droplet puffing/micro-

5.6 Effects of Thermal Radiation and Support

313

explosion is investigated using the model described in Sect. 3.7. The results described in this section are reproduced from [7]. In the experiments described in [7] composite rapeseed oil/water droplets were introduced into flames generated by either an ethanol (with temperatures up to 1,120 K) or propane/butane (with temperatures up to 1,450 K) mixture burner. The initial volume fractions of rapeseed oil and water in droplets were 90% and 10%, respectively. The analysis of puffing/micro-explosion in these droplets was based on Solution (5.15). As in Sect. 5.3, it is assumed that puffing/micro-explosion is initiated when the temperature at the fuel/water interface reaches the nucleation temperature of water. The source term P(R) in Solution (5.15) is proportional to the external power density absorbed at a certain point inside the droplet. If this power density is provided by external thermal radiation and that radiation is instantaneously and homogeneously absorbed inside the droplet (see [1] for the discussion of validity of this assumption), the dependence of P on R can be ignored, and P(R) can be presented as [7]: P(R) ≡ Pr =

3σ Q¯ a θ R4 , Rd cl ρl

(5.36)

where θ R is the radiative temperature (gas temperature when gas is optically thick and external temperature when it is optically thin); σ is the Stefan–Boltzmann constant; subscript l refers to liquid; Q¯ a is the efficiency factor of absorption. Expression (5.36) was derived for homogeneous droplets. Keeping in mind that the initial volume fraction of rapeseed oil in the experiments described in [7] was 90% a crude estimate of the effect of thermal radiation is to assume that radiation absorption in composite droplets is the same as in rapeseed oil droplets. Q¯ a is estimated as [7, 52]: exp (−2τi ) 1 − exp (−2τi ) − , Q¯ a = 1 + τi 2τi2

(5.37)

where τi = 2κλ xdλ is the optical thickness of the droplet, κλ = kλ λ/(4π ) and xdλ = 2π Rd0 /λ are the index of the absorption and diffraction parameter of the droplets, respectively. kλ and λ are the absorption coefficient and wavelength, respectively. Using the definitions of κλ and xdλ , the expression for the optical thickness is presented as τi = kλ Rd0 . The following estimate of the values of Q¯ a was obtained for the conditions of the experiments described in [7]: 0.97 ≤ Q¯ a ≤ 1.

(5.38)

This estimate allowed the authors of [7] to assume that Q¯ a ≈ 1. Note that this assumption is different from the approximation of droplets by black spheres, for which Q¯ a = 1 but all radiation is absorbed at the surface of the droplets. The effect of the support (holder) on composite droplet heating was considered by the authors of [7] using the model presented in Sect. 3.7. In this model, the effect

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5 Processes in Composite Droplets

Fig. 5.28 The times to puffing/micro-explosion of a composite rapeseed oil/water droplet introduced into the ethanol (top) and propane/butane mixture (bottom) flames versus initial droplet radii. Curves 1 show the experimental data; curves 2 show the predictions of the model when the effects of radiation and support were ignored; curves 3 show the predictions of the model when the effects of the support were considered, but the effects of radiation were ignored; curves 4 show the predictions of the model when the effects of radiation and support were considered. Reprinted from [7]. Copyright Elsevier (2021)

of the support is considered via the introduction of an additional source term in Expression (5.15) given by Formula (3.167). The contact area Sc in this formula is given by Expression (3.165). As in the case of radiative heating, Formula (3.167) was used assuming that the contribution of water in the composite droplet is small. For practical application of (3.167) one needs to assess the values of Tsup . Following [7], it is assumed that Tsup is the same as the droplet surface temperature. This assumption is compatible with the basic assumption of the model that there is no temperature gradient along the droplet surface, which enabled us to use the one-dimensional model. The effects of thermal radiation and support are additive. The combined contribution of thermal radiation Pr and the holder Ph to droplet heating is described by the following expression: (5.39) P(R) = Pr + Ph .

5.6 Effects of Thermal Radiation and Support

315

A comparison between predicted and experimentally observed times to puffing/ micro-explosion (τ p ) for droplets placed in the ethanol and propane/butane mixture flames is presented in Fig. 5.28. Ambient gas temperatures were 1,120 and 1,400 K for ethanol and propane/butane mixture flames, respectively. It was assumed that θ R = Ta (ambient air temperature) [7]. As can be seen in Fig. 5.28, the effect of the support is relatively small, while the effect of thermal radiation leads to a considerable reduction in the values of τ p for both flames. When the effects of both radiation and support are considered, the agreement between the predicted and experimentally observed τ p turns out to be surprisingly good. In agreement with the results discussed earlier, both modelling and experimental data show an increase in τ p with increasing Rd0 . Note that the effects of thermal radiation for large droplets are much stronger than for smaller droplets when Q¯ a is expected to be much smaller than 1. For example, for Diesel fuel droplets with radii close to 10 µm, typical for Diesel engine-like conditions, Q¯ a is close to 0.1 (cf. Fig. 2.5). In all cases presented in Fig. 5.28 the predicted values of τ p are longer than those observed in the experiments. This is related to the main assumption of the model: the water sub-droplet is located in the centre of the rapeseed oil droplet. Considering any shifting of the water sub-droplet from the centre of the rapeseed oil droplet would lead to a reduction in τ p .

5.7 Composite Multi-component Droplets The models of puffing/micro-explosion considered so far in this chapter have been based on the assumption that fuel is mono-component. The limitations of this assumption in the case of realistic multi-component droplets have been discussed in detail in Chap. 4. This section describes the main ideas of a new simple model of puffing/micro-explosion in which the effects of multiple components in liquid fuel are considered, following [42]. Key equations and approximations used in the model are summarised in Sect. 5.7.1. The application of the model to the analysis of puffing/micro-explosion in kerosene droplets is described in Sect. 5.7.2.

5.7.1 Diffusion of Components The model of puffing/micro-explosion in composite multi-component droplets, suggested in [42], is based on the same equations and approximations as the model for mono-component droplets described in Sect. 5.3. In contrast to mono-component droplets, however, the effect of diffusion of components in droplets is taken into account in the new model. The effects of thermal radiation and support are considered using the model described in Sect. 5.6.

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5 Processes in Composite Droplets

Considering a spherically symmetric problem with a water sub-droplet located exactly in the centre of fuel droplet the following equations for mass fractions of the components Yli ≡ Yli (t, R) inside the liquid multi-component liquid fuel shell surrounding water sub-droplets were used [42]: ∂Yli = Dl ∂t



∂ 2 Yli 2 ∂Yli + 2 ∂R R ∂R

 ,

(5.40)

where i ≥ 1, Dl is the liquid fuel component diffusivity; this is assumed to be constant for all components (the additional subscripts i of Dl is omitted to simplify the notation). Equation (5.40) is identical to Eq. (4.1) derived for homogeneous multi-component droplets. It was solved, however, for boundary and initial conditions different from those for homogeneous droplets. The following conditions at the outer and inner boundaries of the fuel shell were used:  ∂Yli  , (5.41) αm (εi − Ylis ) = −Dl ∂ R  R=Rd −0  ∂Yli  = 0, ∂ R  R=Rw +0

(5.42)

where Ylis = Ylis (t) are mass fractions of components i at the droplet’s surface, αm and εi are defined by Expressions (4.28) and (4.5), respectively. The following initial condition was used: (5.43) Yli (R, t)|t=0 ≡ Yli0 (R). The analytical solution to Eq. (5.40) subject to Conditions (5.41), (5.42) and (5.43) was obtained as (see Appendix S for the details): 1 Yli = εi + R



 exp Dl



λ0 Rd

2  t [qi0 − εi Q 0 ] v0

   2  ∞ λn 1  + t exp −Dl [qin − εi Q n ] vn , R n=1 Rd

(5.44)

where eigenvalues λ0 and λn (n ≥ 1) are positive solutions to the following equations: ⎛

2 ⎞     Rw Rλ0d  λ0 λ0 ⎟ ⎜ F ≡ ⎝K −  =  (1 − Rw K ), (5.45) ⎠ tanh  Rd Rd

5.7 Composite Multi-component Droplets

⎛



⎜ G ≡ ⎝K +

Rw

2 ⎞

λn  Rd



⎟ ⎠ tan

 = Rd − Rw , 

vn = sin

qin =

   λn λn  =  (1 − Rw K ), (n ≥ 1) Rd Rd

αm Rd 1 1+ . K = Rd Dl

(5.46) (5.47)

   λ0 λ0 R w λ0 (R − Rw ) + cosh (R − Rw ) , Rd Rd Rd

(5.48)

   λn λn R w λn (R − Rw ) + cos (R − Rw ) , (n ≥ 1) Rd Rd Rd

(5.49)

v0 = sinh 



317

1 ||vn ||2





(R + Rw )Yli0 (R + Rw )vn (R)dR,

(n ≥ 0)

(5.50)

0

Qn =

1 ||vn ||2





(R + Rw )vn (R)dR,

(n ≥ 0),

(5.51)

0

R = R − Rw , εi are the initial values of this parameter at each time step, expressions for norms ||vn ||2 (n ≥ 0) are given by (S.47) and (S.48). Solution (5.44) was applied to both stationary and moving droplets considering the effects of droplet motion on droplet heating and evaporation but not on recirculation inside the droplets (cf. non-self-consistent model of moving mono-component composite droplet heating and evaporation considered in Sect. 5.4). The partial pressure of vapour component i at the droplet surface was found from Raoult’s law (4.15). Both the analytical solution to the equation for component diffusion (5.44) and the analytical solution to the equation for heat transfer (5.15) were implemented into the numerical code and used at each time step of calculations. Calculations were performed using Matlab R2020a. One hundred terms were used in series (5.44) and 200 terms were used in series (5.15). Time steps 50 µs and 0.1 s were used for the smallest and largest droplets, respectively; 10,000 cells along the droplet radius were used to calculate integrals for the parameters in (5.44) and (5.15). The roots of Eqs. (5.45) and (5.46) and the corresponding equations for temperature (5.16) were obtained using the bisection method with absolute accuracy of 10−15 . The graphical solutions of (5.45) and (5.46) are illustrated in Appendix U. The results of calculations using the new code were verified by comparing their predictions with those inferred from ANSYS Fluent, in which the analytical solution (5.44) in the limits Rw = 0 was implemented via User Defined Functions (UDF). The details are shown in Appendix V.

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5.7.2 Modelling and Experimental Results The kerosene composition presented in Table 4.5 was used for the analysis [42]. The diffusion coefficient for liquid species was inferred from the Wilke–Chang formula with the molar mass equal to the average molar mass of all species [39]. It was assumed that kerosene vapour can be approximated by the dominant component (cycloundecane), and the following formula for the vapour diffusion coefficient (in m2 /s) was used for atmospheric pressure [35]:   Dv = −0.04025 + 2.4907 × 10−4 × T + 3.1411 × 10−7 × T 2 × 10−4 , (5.52) where T is the reference temperature of gas in K. Liquid thermodynamic and transport properties were estimated at the droplet average temperatures at specific time steps. The latent heat of evaporation and saturated vapour pressure were estimated at the surface temperatures of the droplets. The observed times to puffing/micro-explosion τ p versus ambient gas temperatures are shown in Fig. 5.29. The analysis was focused on kerosene/water composite droplets with initial temperatures 300 K, radii 0.85 mm and volumetric water content 10% (Rw = 0.395 mm). The experiments were performed at atmospheric pressure. In the same figure, the values of τ p predicted by the numerical code are shown. As in Sect. 5.3, it was assumed that puffing and micro-explosion start when the tem-

Fig. 5.29 Times to puffing/micro-explosion of water/kerosene droplets (τ p ) versus gas temperatures Tg 1) observed experimentally (stars) 2) predicted by the model assuming that puffing/microexplosion starts when the temperature at the water/kerosene interface becomes equal to the water nucleation temperature (Tw = TN ) (curves). The solid curve shows the case when the contributions of all kerosene components were considered. The dashed curve shows the case when kerosene was approximated by cycloundecane. Reprinted from [42]. Copyright Elsevier (2021)

5.7 Composite Multi-component Droplets

319

Fig. 5.30 The same as Fig. 5.29 but for times to puffing or micro-explosion versus the initial droplet radii. Reprinted from [42]. Copyright Elsevier (2021)

perature at the water/kerosene interface (Tw ) becomes equal to the water nucleation temperature (TN ). Two cases were investigated. Firstly, the contributions of all kerosene components were considered using the model described in Sect. 5.7.1. Secondly, kerosene was approximated by cycloundecane and a much simpler model (described in Sect. 5.3) was used. As can be seen from Fig. 5.29, in both cases the predictions of the numerical code show the same trend of the reduction of τ p with an increase in ambient gas temperatures. The predictions of the code using both approximations of kerosene are reasonably close to experimental data. The code considering the contributions of all kerosene components predicts longer τ p than that in which kerosene was approximated by cycloundecane. The experimentally observed times to puffing/micro-explosion τ p versus initial droplet radii are shown in Fig. 5.30. The analysis was focused on kerosene/water composite droplets with volumetric water content 10% placed in gas at temperature 548 K. As in Fig. 5.29, in all cases the initial droplet temperatures were 300 K. Also as in Fig. 5.29, the cases when the contributions of all kerosene components were taken into account and when kerosene was approximated by cycloundecane were considered. As can be seen from Fig. 5.30, in both cases the prediction of the numerical code shows the same increase in τ p with an increase in droplet initial radii as observed experimentally. As in the case of Fig. 5.29, the predictions of the code using both approximations of kerosene are reasonably close. The code considering the contributions of all kerosene components predicts longer τ p than in the case when kerosene was approximated by cycloundecane. Note that the model tends to predict longer times to puffing/micro-explosion compared with those observed experimentally as it uses the assumption that the water sub-droplet is located in the centre of the kerosene droplet. See Sects. 5.3 and 5.8 for further discussion of this matter.

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5.8 The Shift Model Despite considerable progress achieved in modelling puffing/micro-explosion in composite droplets, the approaches described in the previous sections have several important weaknesses outlined by the authors of [15]. Firstly, the verification of the model was limited by the development of two separate numerical codes, using Wolfram Mathematica v 12.1 and Matlab R2020a, in which the analytical solution to the heat transfer equation in a fuel-water droplet was implemented. Although these codes predicted the same results, both results could be not correct if there is something wrong with the analytical solution. Secondly, the sensitivity of the results to the shifting of the water sub-droplet away from the centre of the fuel droplet has not been investigated, and if, for example, the prediction of the model can be applied only to cases when this shift is less than 5%, this model would not be applicable to most practical engineering problems. Thirdly, no quantitative estimates of the effect of this shift on predicted and observed times to puffing/micro-explosion were made. It was only observed that in most cases these times predicted by the model described in Sect. 5.3 and the following sections are longer than those observed experimentally. This was related to the shift of the water sub-droplet from the centre of the fuel droplet. However, no quantitative estimates were made. All these issues were addressed in the model developed by the authors of [15], which was called the shift model. The basis of a new approach to the verification of the model described in Sect. 5.3 was a comparison of the predictions of this model with those of a numerical code for solving the same heat transfer problem in the fuel-water droplet (using the heat transfer module available from COMSOL software) [15]. Almost perfect agreement between the predictions of both codes supported the validity of both approaches to the problem. The numerical code was generalised to consider a shift of the water sub-droplet away from the centre of the fuel droplet. This generalised model (called the shift model) was based on the numerical solution to the heat transfer equation in a fuelwater droplet. The geometry used in the shift model is shown in Fig. 5.31. The heat transfer equation in the cylindrical coordinate system (r , ϑ, z), with the z-axis being the line joining the centre of the fuel droplet to that of the water sub-droplet, was solved using the simplifying assumption that the surface temperature of the fuel droplet is uniform although it can change with time [15]. The size of the shift L (see Fig. 5.31) was inferred from experimental observations. It was demonstrated that in most cases the time to puffing/micro-explosion predicted by the shift model is closer to experimental results than that predicted by the model based on the assumption that water sub-droplet is located exactly in the centre of the fuel droplet. The shift was quantified by the normalised shift defined as S = L/L max , where L max = Rd0 − (Rw0 /2) (Rd0(w0) is the initial radius of the fuel droplet (water sub-droplet)) is the maximal value of L. It was demonstrated that for typical values of

5.8 The Shift Model

321

Fig. 5.31 Schematic presentation of the geometry used in the shift model. Reprinted from [15]. Copyright Elsevier (2021)

input parameters for S ≤ 0.2 the predictions of the original (based on the assumption that water sub-droplet is located exactly in the centre of the fuel droplet) and shift models differ by less than 1% [15]. This chapter focused on analysing the processes in fuel droplets with water subdroplets inside them. When these droplets were heated (e.g. in automotive engine conditions) the temperature at the water-fuel interface could reach water nucleation temperature. When this happened, the fuel droplet started disintegrating via puffing and/or micro-explosion processes leading to rapid enhancement of their evaporation rates. Two simple models, designed to predict time to puffing/micro-explosion, were described. Both models assumed that a spherical water sub-droplet was in the centre of a fuel (e.g. n-dodecane) droplet. The first model used an analytical solution to the heat conduction equation in the composite droplet with the Dirichlet boundary condition at its surface, assuming that puffing/micro-explosion is initiated when the temperature of this interface reaches the water boiling temperature. The second model used an analytical solution to this equation with the Robin boundary condition at each time step; this solution was incorporated into a numerical code considering droplet evaporation and swelling. Puffing and/or micro-explosion in the latter model was initiated at the time instant when the temperature at the water–fuel interface became equal to the water nucleation temperature. Further developments of these models and their validation and verification were discussed.

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References 1. Abramzon, B., & Sazhin, S. S. (2006). Convective vaporization of fuel droplets with thermal radiation absorption. Fuel, 85(1), 32–46. 2. Antonov, D.V. (2021). Private communication. 3. Antonov, D. V., Kuznetsov, G. V., Strizhak, P. A., Rybdylova, O., & Sazhin, S. S. (2019). Micro-explosion and autoignition of composite fuel/water droplets. Combustion and Flame, 210, 479–489. 4. Antonov, D. V., Piskunov, M., Strizhak, P. A., Tarlet, D., & Bellettre, J. (2020). Dispersed phase structure and micro-explosion behavior under different schemes of water-fuel droplets heating. Fuel, 259, 116241. 5. Antonov, D. V., Strizhak, P. A., Fedorenko, R. M., Nissar, Z., & Sazhin, S. S. (2021). Puffing and microexplosions in rapeseed oil/water droplets: The effects of coal micro-particles in water. Fuel, 289, 119814. 6. Antonov, D. V., Fedorenko, R. M., Strizhak, P. A., Castanet, G., & Sazhin, S. S. (2021). Puffing/microexplosion of two closely spaced composite droplets in tandem: Experimental results and modelling. International Journal of Heat and Mass Transfer, 176, 121449. 7. Antonov, D. V., Fedorenko, R. M., Strizhak, P. A., Nissar, Z., & Sazhin, S. S. (2021). Puffing/micro-explosion in composite fuel/water droplets heated in flames. Combustion and Flame, 233, 111599. 8. Antonov, D. V., Volkov, R. S., Fedorenko, R. M., Strizhak, P. A., Castanet, G., & Sazhin, S. S. (2021). Temperature measurements in a string of three closely spaced droplets before the start of puffing/micro-explosion: Experimental results and modelling. International Journal of Heat and Mass Transfer, 181, 121837. 9. Antonov, D. V., Kuznetsov, G. V., Sazhin, S. S., & Strizhak, P. A. (2022). Puffing/microexplosion in droplets of rapeseed oil with coal micro-particles and water. Fuel, 316, 123009. 10. Avulapati, M. M., Megaritis, T., Xia, J., & Ganippa, L. (1999). Experimental understanding on the dynamics of micro-explosion and puffing in ternary emulsion droplets. Fuel, 239, 1284– 1292. 11. Bar-Kohany, T. (2021). Minimal heating rate for isobaric nucleation at the spinodal in liquids. International Journal of Heat and Mass Transfer, 165, 120636. 12. Bar-Kohany, T., & Amsalem, Y. (2018). Nucleation temperature under various heating rates. International Journal of Heat and Mass Transfer, 126, 411–415. 13. Boggavarapu, P., & Ravikrishna, R. V. (2019). Evaporating spray characteristics of a dieselethanol micro-emulsion. Fuel, 246, 104–107. 14. Carslaw, H. S., & Jaeger, J. C. (1986). Conduction of Heat in Solids. Oxford: Clarendon Press. 15. Castanet, G., Antonov, D. V., Strizhak, P. A., Sazhin, S. S. (2022). Effects of water subdroplet location on the start of puffing/micro-explosion in composite fuel-water droplets. International Journal of Heat and Mass Transfer 186, 122466. 16. Chao, C.-Y., Tsai, H.-W., Pan, K.-L., & Hsieh, C.-W. (2019). On the microexplosion mechanisms of burning droplets blended with biodiesel and alcohol. Combustion and Flame, 205, 397–406. 17. Ching, E. J., Avedisian, C. T., Carrier, M. J., Cavicchi, R. C., Young, J. R., & Land, B. R. (2014). Measurement of the bubble nucleation temperature of water on a pulse-heated thin platinum film supported by a membrane using a low-noise bridge circuit. International Journal of Heat and Mass Transfer, 79, 82–93. 18. Dai, M., Wang, J., Wei, N., et al. (2019). Experimental study on evaporation characteristics of diesel/cerium oxide nanofluid fuel droplets. Fuel, 254, 115633. 19. Fostiropoulos, S., Strotos, G., Nikolopoulos, N., & Gavaises, M. (2020). Numerical investigation of heavy fuel oil droplet breakup enhancement with water emulsions. Fuel, 278, 118381. 20. Fostiropoulos, S., Strotos, G., Nikolopoulos, N., & Gavaises, M. (2021). A simple model for breakup time prediction of water-heavy fuel oil emulsion droplets. International Journal of Heat and Mass Transfer, 164, 120581.

References

323

21. Girin, O. G. (2017). Dynamics of the emulsified fuel droplet micro-explosion. Atomization and Sprays, 27, 407–422. 22. Glod, S., Poulikakos, D., Zhao, Z., & Yadigaroglu, G. (2002). An investigation of microscale explosive vaporization of water on an ultrathin Pt wire. International Journal of Heat and Mass Transfer, 45, 367–379. 23. Iida, Y., Okuyama, K., & Sakurai, K. (1984). Boiling nucleation on a very small film heater subjected to extremely rapid heating. International Journal of Heat and Mass Transfer, 37, 2771–2780. 24. Ismael, M. A., Heikal, M. R., Aziz, A. R. A., Crua, C., El-Adawy, M., Nissar, Z., Baharom, M. B., Zainal, E. Z. A., & Firmansyah. (2018). Investigation of puffing and micro-explosion of water-in-diesel emulsion spray using shadow imaging. Energies, 11(9), 2281. 25. Ivanov, V. M., & Nefedov, P. I. (1965). Experimental investigation of the combustion process of natural and emulsified liquid fuels. NASA TT F-258. 26. Jang, G. M., & Kim, N. I. (2020). Characteristics of a free-falling single-droplet of water-in-oil emulsion broken up by a pulse laser. Fuel, 264, 116863. 27. Kadota, T., & Yamasaki, H. (2002). Recent advances in the combustion of water fuel emulsion. Progress in Energy and Combustion Science, 28(5), 385–404. 28. Kartashov, E. M. (2001). Analytical Methods in the Heat Transfer Theory in Solids. Moscow: Vysshaya Shkola (in Russian). 29. Law, C. K. (2006). Combustion Physics. Cambridge: Cambridge University Press. 30. Law, C. K., Lee, C. H., & Srinivasan, N. (1980). Combustion characteristics of water-in-oil emulsion droplets. Combustion and Flame, 37, 125–143. 31. Meng, K., Fu, W., Lei, Y., et al. (2019). Study on micro-explosion intensity characteristics of biodiesel, RP-3 and ethanol mixed droplets. Fuel, 256, 115942. 32. Mondal, P. K., & Mandal, B. K. (2019). A comprehensive review on the feasibility of using water emulsified diesel as a CI engine fuel. Fuel, 237, 937–960. 33. Nissar, Z., Rybdylova, O., Sazhin, S. S., Heikal, M., Aziz, A. R. B. A., & Ismael, M. A. (2020). A model for puffing/microexplosions in water/fuel emulsion droplets. International Journal of Heat and Mass Transfer, 149, 119208. 34. Patidar, S. K., & Raheman, H. (2020). Performance and durability analysis of a single-cylinder direct injection diesel engine operated with water emulsified biodiesel-diesel fuel blend. Fuel, 273, 117779. 35. Poulton, L., Rybdylova, O., Zubrilin, I. A., Matveev, S. G., Gurakov, N. I., Al Qubeissi, M., Al-Esawi, N., Khan, T., Gun’ko, V. M., & Sazhin, S. S. (2020). Modelling of multi-component kerosene and surrogate fuel droplet heating and evaporation characteristics: A comparative analysis. Fuel, 269, 117115. 36. Rosenthal, M. W. (1957). An experimental study of transient boiling. Nuclear Science and Engineering, 2(5), 640–656. 37. Sakurai, A., Shiotsu, M. (1977). Transient pool boiling heat transfer. I—Incipient boiling superheat. ASME Transactions Journal of Heat Transfer, November 1977. 38. Sazhin, S. S., Krutitskii, P. A., Abdelghaffar, W. A., Sazhina, E. M., Mikhalovsky, S. V., Meikle, S. T., & Heikal, M. R. (2004). Transient heating of diesel fuel droplets. International Journal of Heat and Mass Transfer, 47, 3327–3340. 39. Sazhin, S. S., Al Qubeissi, M., Kolodnytska, R., Elwardany, A., Nasiri, R., & Heikal, M. R. (2014). Modelling of biodiesel fuel droplet heating and evaporation. Fuel, 115, 559–572. 40. Sazhin, S. S., Rybdylova, O., Crua, C., Heikal, M., Ismael, M. A., Nissar, Z., & Aziz, A. R. B. A. (2019). A simple model for puffing/micro-explosions in water-fuel emulsion droplets. International Journal of Heat and Mass Transfer, 131, 815–821. 41. Sazhin, S. S., Bar-Kohany, T., Nissar, Z., Antonov, D., Strizhak, P. A., & Rybdylova, O. (2020). A new approach to modelling micro-explosions in composite droplets. International Journal of Heat and Mass Transfer, 161, 120238. 42. Sazhin, S. S., Shchepakina, E., Sobolev, V. A., Antonov, D., & Strizhak, P. A. (2022). Puffing/micro-explosion in composite multi-component droplets. International Journal of Heat and Mass Transfer, 184, 122210.

324

5 Processes in Composite Droplets

43. Shen, S., Che, Z., Wang, T., Yue, Z., Sun, K., & Som, S. (2020). A model for droplet heating and evaporation of water-in-oil emulsified fuel. Fuel, 266, 116710. 44. Shinjo, J., & Umemura, A. (2010). Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation. International Journal of Multiphase Flow, 36, 513–532. 45. Shinjo, J., Xia, J., Ganippa, L. C., & Megaritis, A. (2014). Physics of puffing and microexplosion of emulsion fuel droplets. Physics of Fluids, 25, 103302. 46. Shinjo, J., Xia, J., Ganippa, L. C., & Megaritis, A. (2016). Puffing-enhanced fuel/air mixing of an evaporating n-decane/ethanol emulsion droplet and a droplet group under convective heating. Journal of Fluid Mechanics, 793, 444–476. 47. Shinjo, J., Xia, J., Megaritis, A., Ganippa, L. C., & Cracknell, R. F. (2016). Modeling temperature distribution inside a emulsion fuel droplet under convective heating: A key to predicting microexplosion and puffing. Atomization and Sprays, 26, 551–583. 48. Shinjo, J., & Xia, J. (2017). Combustion characteristics of a single decane/ethanol emulsion droplet and a droplet group under puffing conditions. Proceedings of the Combustion Institute, 36, 2513–2521. 49. Scriven, L. (1959). On the dynamics of phase growth. Chemical Engineering Science, 10, 1–13. 50. Su, G.-Y., Bucci, M., McKrell, T., & Buongiorno, J. (2016). Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling. International Journal of Heat and Mass Transfer, 97, 667–684. 51. Tanimoto, D., & Shinjo, J. (2019). Numerical simulation of secondary atomization of an emulsion fuel droplet due to puffing: Dynamics of wall interaction of a sessile droplet and comparison with a free droplet. Fuel, 252, 475–487. 52. van de Hulst, H. C. (1957). Light Scattering by Small Particles. New York: Wiley. 53. Vigneswaran, R., Balasubramanian, D., & Sastha, B. D. S. (2021). Performance, emission and combustion characteristics of unmodified diesel engine with titanium dioxide (TiO2 ) nano particle along with water-in-diesel emulsion fuel. Fuel, 285, 119115. 54. Wang, C. H., Liu, X. Q., & Law, C. K. (1984). Combustion and microexplosion of freely falling multicomponent droplets. Combustion and Flame, 56(2), 175–197. 55. Wang, J., Wang, X., Chen, H., Jin, Z., & Xiang, K. (2018). Experimental study on puffing and evaporation characteristics of jatropha straight vegetable oil (SVO) droplets. International Journal of Heat and Mass Transfer, 119, 392–399. 56. Wang, X., Dai, M., Wang, J., et al. (2019). Effect of ceria concentration on the evaporation characteristics of diesel fuel droplets. Fuel, 236, 1577–1585. 57. Wang, X., Dai, M., Yan, J., et al. (2019). Experimental investigation on the evaporation and micro-explosion mechanism of jatropha vegetable oil (JVO) droplets. Fuel, 258, 115941. 58. Watanabe, H., & Okazaki, K. (2013). Visualization of secondary atomization in emulsified-fuel spray flow by shadow imaging. Proceedings of the Combustion Institute, 34(1), 1651–1658. 59. Wu, Z., Xie, W., Yu, Y., Li, L., & Deng, J. (2021). Comparison of spray characteristics of gasoline and water-in-gasoline mixture at elevated fuel temperature conditions. Fuel, 304, 121409. 60. Wuethrich, D., von Rotz, B., Herrmann, K., & Boulouchos, K. (2019). Spray, combustion and soot of water-in-fuel (n-dodecane) emulsions in a constant volume combustion chamber; Part II: Effects of low temperature conditions and oxygen concentrations. Fuel, 248, 104–116. 61. Yaws, C. L. (2015). Chapter 3—Properties water. The Yaws Handbook of Physical Properties for Hydrocarbons and Chemicals (2nd ed., pp. 811–814). Boston: Gulf Professional Publishing. 62. Yi, P., Li, T., Fu, Y., & Xie, S. (2021). Transcritical evaporation and micro-explosion of ethanoldiesel droplets under diesel engine-like conditions. Fuel, 284, 118892. 63. Yokev, N., & Greenberg, J. B. (2018). Linear stability analysis of laminar premixed water-infuel emulsion spray flames. Fuel, 222, 733–742. 64. Yokev, N., & Greenberg, J. B. (2018). Influence of micro-explosions on the stability of laminar premixed water-in-fuel emulsion spray flames. Combustion Theory and Modelling, 23(2), 310– 336.

References

325

65. Zhang, Z., Jiaqiang, E., Chen, J., et al. (2019). Effects of low-level water addition on spray, combustion and emission characteristics of a medium speed diesel engine fueled with biodiesel fuel. Fuel, 239, 245–262. 66. Zeng, Y., & Lee, C. E. (2007). Modeling droplet breakup processes under micro-explosion conditions. Proceedings of the Combustion Institute, 31, 2185–2193.

Chapter 6

Kinetic Modelling of Droplet Heating and Evaporation

The analysis presented so far has implicitly assumed that both liquid and gas can be considered as a continuum and the vapour/liquid interface remains in the equilibrium state. This means that the details of the velocity distribution of molecules do not affect the physical properties of these phases. The latter are controlled by the number density of molecules, and their average masses and velocities. In most cases, this assumption is valid as long as the droplets are much larger than the mean molecular free paths. The validity of this assumption, however, is not obvious when the interface between liquid droplets and the ambient gas is modelled, even when the gas pressure is well above atmospheric. The experimental evidence of this was demonstrated in [101]. This means that although the liquid phase can always be treated as a continuum in most applications, the properties of gas in the vicinity of the gas/liquid interface can depend not only on the average velocities of molecules but also on the distribution of molecules by velocities. The latter is described using distribution functions of molecules. This chapter focuses mainly on the kinetic modelling of droplet heating and evaporation in dense gases. This modelling will be used in combination with the conventional hydrodynamic approach. The approaches specifically focused on modelling processes in rarefied gases are considered in many well-known monographs (e.g. [52]) and recent papers (e.g. [100]). Early results are briefly described in Sect. 6.1. Section 6.2 focuses on the description of the basic kinetic algorithm used in the analysis. Approximations of the kinetic results are described in Sect. 6.3. A mathematical model and kinetic algorithm for considering the effects of inelastic collisions are discussed in Sect. 6.4. The molecular dynamics algorithm used for the estimation of the evaporation coefficient for n-dodecane droplets is described in Sect. 6.5. The quantum-chemical models of the processes at the surfaces of the droplets are described in Sect. 6.6. The results of calculations, considering the effects of inelastic collisions between molecules and previously obtained values of the evaporation coefficient, are described in Sect. 6.7.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_6

327

328

6 Kinetic Modelling of Droplet Heating and Evaporation

In Sect. 6.8 some results of kinetic modelling in the presence of three components (air and two components of vapour) are presented and discussed. Key ideas of a self-consistent kinetic model of droplet heating and evaporation are described in Sect. 6.9.

6.1 Early Results The choice of the model required for the analysis of droplet evaporation is commonly made based on the Knudsen number Kn = lcoll /Rd , where lcoll is the mean free path of molecules between collisions. It is commonly considered that in the case when Kn < 0.01 the gas may be approximated as a continuum. In another limiting case when Kn > 3 the mean free path of molecules is large compared with the dimensions of the system, and a free molecular regime is expected. In the transitional regime 0.01 ≤ Kn ≤ 3 the mean free path of molecules is comparable with the dimensions of the system. This is known as the slip regime. In this case gas can be regarded as a continuum at several mean paths away from the droplet surface, but kinetic effects need to be considered in the vicinity of the droplet surface [62]. For fuel droplets in many engineering applications, including internal combustion engines, Kn is expected to be well below 0.01, and this justifies the application of the continuum approximation for their analysis. The validity of this approximation in the immediate vicinity of the droplet surfaces, however, is not at first obvious, as was demonstrated in [88]. Hence, the application of kinetic models of droplet heating and evaporation can be justified even in the case of high-pressure gases. This section focuses on the general introduction to the problem and the earlier results of kinetic modelling published mainly before 2006. This is essentially an extended reproduction of Sect. 3.4 of [132]. The rest of this chapter focuses mainly on the results obtained by the author and his colleagues after 2006. The most general approach to the analysis of the velocity distributions of gas molecules is based on the so-called multi-particle distribution functions which consider not only the positions and velocities of individual molecules but also the correlations between them. This eventually leads to the infinite chain of Bogolubov– Born–Green–Kirkwood–Yvon (BBGKY) equations for multi-particle distribution functions [2]. This chain describes the exact dynamics of the fluid but needs to be truncated. Restricting particle interactions to include only isolated binary collisions, it simplifies to the Boltzmann kinetic equation which is presented as [2, 41] ∂f ∂f ∂f +v· +F· = ∂t ∂x ∂v



∂f ∂t

 ,

(6.1)

coll

where f = f (t, v, x) is the molecular distribution function; v, x and F are velocity, position and force acting on individual molecules, respectively.

6.1 Early Results



∂f ∂t

329



is the collision integral which considers collisions between molecules. coll The physical meaning of f infers from the product: Δf = f (t, v, x)ΔxΔv, where Δx is a small element of volume around x, Δv a small range of velocities around v and Δf the number of molecules in the element of volume Δx around x, having velocities between v and v + Δv. Integration of both parts of Eq. (6.1) over velocities leads to the continuity equation (conservation of mass). Integration of both parts of this equation with weights (components of momenta and energy of molecules) gives the equations of conservation of momentum (the Navier–Stokes equation) and energy. In most cases, the contribution of F in Eq. (6.1) is not considered, and this allows us to simplify this equation to ∂f ∂f +v· = ∂t ∂x



∂f ∂t

 .

(6.2)

coll

This could not be done in the case of ionised gases (plasma) even in the absence of external fields [2, 41, 131], but in non-ionised gases F could only come from the gravitational force, which is negligibly small for molecules. For elastic collisions, the collision integral is written as [18, 19] 

∂f ∂t

 = coll

dm2 2





+∞ −∞

dv1 0

π



2π

sin θ dθ 0

    dφ f f 1 − f f 1 |v − v1 | ,

(6.3)

where dm is the diameter of colliding molecules, θ and φ are relative angular coordi nates of molecules, superscript refers to the velocities and the distribution functions     after collisions, f = f (v), f 1 = f (v1 ), f = f (v ) and f 1 = f (v1 ). The first integral in the right-hand side of Eq. (6.3) is calculated in the three-dimensional velocity space. When deriving Eq. (6.3) molecules were assumed to be rigid elastic spheres. For evaporating droplets, the collision integral includes the sum of two terms: the one referring to collisions between molecules of vapour, and the other referring to collisions between molecules of vapour and molecules of ambient gas (air). A general analysis considering the contribution of both processes is described in [9, 89, 90, 175, 176]. A more rigorous derivation of the expression for the collision integral, considering the probability of elastic scattering of molecules after collisions, is discussed in Sect. 13 of [3]. A generalisation of Eq. (6.2) to the case of a multi-species gas mixture leads to the Wang–Chang–Uhlenbeck equations [34]. In these equations, the translational and internal degrees of freedom are considered using classical and quantum mechanics approaches, respectively. The analysis of these equations is beyond the scope of this book.

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6 Kinetic Modelling of Droplet Heating and Evaporation

The general solutions to Eqs. (6.2) and (6.3) are possible using numerical methods (see [18, 19, 156, 179, 180]). The applications of these methods to the investigation of the processes during droplet evaporation are described in [9, 89, 90, 175, 176]. The investigation of Eq. (6.2) is simplified when the collision integral  is not  (6.3) ∂f were calculated but modelled. Several explicit analytical approximations for ∂t coll described (e.g. [2, 16, 130]). An alternative simplification of the Boltzmann equation is based on focusing on the perturbation of the velocity distribution function from the equilibrium state (linearisation of the Boltzmann equation) [35]. The analysis of Eq. (6.2) largely depends on the ratio of the mean free path of molecules between collisions and the size of the system. For droplets, this ratio is described by the Knudsen number Kn introduced earlier. In what follows, some approaches to the analysis of this equation are reviewed mainly following [88]. For Kn  1 the contribution of collisions can be ignored altogether. In this case, the right-hand side of Eq. (6.2) is zero and this equation describes a free molecular flow. Equation (6.2) with zero right-hand side has analytical solutions. Assuming that molecular fluxes entering and leaving the area above evaporating droplets are Maxwellian with temperatures Ts and Tg , respectively, the solution to Eq. (6.2) leads to the Hertz–Knudsen–Langmuir formula for mass flux of vapour leaving the surface of the droplet [63, 79, 81, 116, 153]: |m˙ d | βm =√ jlg ≡ m˙ d = 4π Rd2 2π Rv





pvs pv∞ √ − Ts Tg

,

(6.4)

where |m˙ d | is the droplet evaporation rate, βm the evaporation coefficient (assumed to be the same as the condensation coefficient), Rv gas constant for vapour, pvs the saturated vapour partial pressure at Ts and pv∞ the ambient vapour partial pressure. The evaporation coefficient βm ≤ 1 is the ratio of the actual mass flux leaving the droplet surface (before the first collision) jes and the maximal mass flux [139]: βm =

ρvs

jes

Rv Ts 2π

,

(6.5)

where ρvs is the saturated fuel vapour density at Ts = Tls . The condensation coefficient, defined by the same Expression (6.5), describes the fraction of vapour molecules striking the liquid surface which is absorbed at this surface. (1 − βm ) describes the fraction of reflected molecules. The values of βm depend on the properties of surfaces [148] and can be estimated based on experimental data [65, 87, 115]. Molecular dynamics (MD) and quantum-chemical calculations have also been used for estimating this coefficient (e.g. Sect. 6.5.3). Early approaches to this problem are described in many papers, including [32, 40, 185, 194]. A detailed review of theoretical and experimental investigations of this coefficient for water described up to 2001 is presented in [109] (see also [87]).

6.1 Early Results

331

Expression (6.4) has been widely used for the investigation of evaporation/ condensation processes even when the condition for its validity (Kn >> 1) has not been strictly satisfied (e.g. [54, 60, 106, 117]). A more general approach to the evaporation/condensation problem considers the effects of collisions. This leads to the introduction of the concept of the Knudsen layer, separating the liquid surface from the bulk of the vapour which is described using the continuum equations. The thickness of the Knudsen layer l K is typically estimated as several molecular mean free pass lengths lcoll for small drift vapour velocities vdr , 10lcoll for vdr = 0.5vsound , and 100–200lcoll for vdr close to vsound [170]. The value of lcoll is determined at the surface temperature Ts . In the case of Diesel fuel approximated by n-dodecane C12 H26 (cf. reservations discussed in Sect. 4.3), MC12 H26 = 170.3 kg/kmol and the vapour constant is estimated as Rv = 48.88 J/(kg K). For vapour temperature equal to Ts = 600 K, vsound = 202.6 m/s. For this Ts , pvs = 6.4 × 105 Pa. Using Expression (6.4), for pv∞ = 0 and βm = 0.5, it can be obtained that jlg = 745 kg/(m2 s). From the gas law it follows that ρvs = 21.8 kg/m3 . Hence, vdr = jlg /ρsv = 34 m/s. Note that vsound and vdr are weak functions of Ts . For example, for Ts = 400 K, vsound = 165 m/s and vdr = 28 m/s. More careful investigation of the problem (see Eq. (6.8)) would predict a slightly larger vdr , but the assumption that vdr  vsound would still be valid. For liquid evaporating into its own vapour, the following estimation was obtained [81]: lcoll =

√

2π dm2 (ρvs N A /M

−1

,

(6.6)

where N A is the Avogadro number, M the molar mass and dm the diameter of the molecule. For dm ≡ dnd = 10−9 m, Ts = 600 K ρvs = 21.8 kg/m3 and M = 170.3 kg/kmol (typical parameters for n-dodecane), it follows from Expression (6.6) that lcoll = 2.9 × 10−9 m. For l K < 5lcoll , l K for Diesel engines-like conditions is estimated as l K < 1.5 × 10−8 m = 0.015 µm. This is about 2 orders of magnitude smaller than the droplet radii. As demonstrated later, following [88], the contribution of the Knudsen layer cannot be a priori ignored, even in this case. Note that Eq. (6.2) is applicable only when lcoll >> dnd [104]. When lcoll is close to dnd the model is expected to predict the trends of the processes, rather than give reliable quantitative estimates. Also, the solutions to the Boltzmann equation in most cases do not consider the contribution of the processes inside the vapour molecules (a possible approach to the analysis of the effects of inelastic collisions between molecules is described in Sect. 6.4). The velocity distribution of the molecules in the Knudsen layer is affected by collisions and can be obtained from the solution to Eq. (6.2). Schrage [148] drew attention to the fact that the effects of collisions lead to the formation of the shifted Maxwellian distribution of molecules near the outer boundary of the Knudsen layer. The authors of [93] assumed that this distribution is formed in the whole Knudsen

332

6 Kinetic Modelling of Droplet Heating and Evaporation

layer up to the liquid surface. Considering the case when vdr  vsound (this is consistent with the earlier estimates) and using matching boundary conditions in this layer, the modified expression for jlg , valid for Kn 1 and (Tg − Ts )/Ts  1, was derived [93, 148]: 2βm jlg = √ (2 − βm ) 2π Rv



pvs pv∞ √ − Ts Tg

.

(6.7)

Expression (6.7) considered the convection of vapour and collision processes. For βm = 1 this expression predicts jlg which is twice as large as the one which follows from Expression (6.4). The details of a simple derivation of Expression (6.7) are presented [62]. This expression is also known as the Hertz–Knudsen–Schrage formula [148, 202], Schrage model [149] and Schrage relationship [26]. As noticed by the authors of [123], Formula (6.7) predicts the evaporation flux much larger than what is found in experiments. It was shown that this formula works well only when pressure and temperature are taken in the upper boundary of the Knudsen layer. Since this layer is very thin it is very difficult to make the measurements in this location. Further developments in the kinetic theory of evaporation/condensation are described by many authors, including [94, 95, 97, 112]. The problem was investigated using two forms of the collision term in the Boltzmann equation: the conventional one and the one suggested by Bhatnagar, Gross and Krook [16]. The latter form guarantees the conservation of particles during the collision process. It has been widely used to model collisions in gases and plasma (see e.g. [130]). The authors of [112] derived several equations useful for the analysis of evaporation/condensation processes, including the one describing the mass flux of vapour leaving the droplet and moving into a dense medium (Kn  1): jlg =

βm ( pvs − p Rd ) , √ (1 − 0.4βm ) 2π Rv Ts

(6.8)

where p Rd is the vapour pressure at the outer boundary of the Knudsen layer. Expression (6.8) has been used in many applications (e.g. [83]). It is applicable for the analysis of weak evaporation/condensation expected when (TRd − Ts )/Ts  1. This expression is also known as the Hertz–Knudsen–Schrage formula [148, 202]. The results of further developments in the theory of weak evaporation are described in [29, 31, 80, 150, 168, 169, 181, 196, 198]. These developments, however, do not undermine the usefulness of Expression (6.8). This expression is more accurate than Expressions (6.4) and (6.7) and has been widely used for various applications (e.g. [88]). The increase in intensity of evaporation is expected to lead to deformation of the molecular distribution function in the Knudsen layer. At a certain stage, the linearisation of this function (which was used when deriving Expression (6.8)) is no longer possible. This leads to the situation when vdr becomes comparable with the velocity

6.1 Early Results

333

of sound and the theory of weak evaporation and condensation cannot be used. A theory of intense evaporation has been developed by many authors including [1, 6, 7, 28, 47, 78, 82, 98, 99, 111, 127, 160, 165–167, 171, 172, 174, 177, 183, 197, 199]. Computational and experimental results referring to intensive condensation, with the emphasis on estimating βm , are reviewed in [126]. In [99] the mass flux of vapour from the surface of the droplet was approximated as   (6.9) jlg = 0.6 2Rv Ts (ρvs − ρ Rd ) ρ Rd /ρvs , where ρ Rd is the vapour density at the outer boundary of the Knudsen layer. Expression (6.9) is valid for both weak and strong evaporation and has been derived for βm = 1. For βm = 1, the value of jlg can be obtained via replacing ρvs in Expression (6.9) with (see [82, 99])  √ ρβ = 1 − 2 π

jlg 1 − βm √ ρvs 2Rv Ts βm

 ρvs .

(6.10)

The solution to Eqs. (6.9) and (6.10) with two unknowns jlg and ρβ allows us to find jlg for any βm ≤ 1. For weak evaporation (ρvs − ρ Rd )/ρvs  1, Expression (6.9) reduces to Expression (6.8) for βm = 1 [88]. One of the fundamental differences in predictions of kinetic and hydrodynamic models lies in the behaviour of temperatures and velocities in the immediate vicinity of the liquid surface. It is well known that the hydrodynamic models predict continuous changes of all these parameters in this area, while the kinetic models predict jumps in the values of these parameters over distances of the order of several mean free paths between intermolecular collisions (temperature jumps and velocity slips). These jumps and slips are usually estimated based on the analysis of the Boltzmann equation or its momenta [52, 62, 75, 151, 152, 195]. Once they have been found, the analysis of the processes in the gas phase, including evaporation, can be performed using the hydrodynamic theory. The boundary conditions in this case consider the jumps in the values of these parameters at the boundaries [38, 42, 122, 133]. The effects of temperature jump and velocity slip on the values of the Nusselt number (Nu) for nanoparticles were studied in [43] based on the asymptotical analysis. It was shown that the velocity slip at the interface does not affect significantly the Nusselt numbers. The temperature slip, however, significantly affects this number. Thus, the contribution of the temperature jump at the interface to Nu must be considered in the general case. If this jump is large enough, the droplets can become almost adiabatic. The kinetic model using Expression (6.8) and the hydrodynamic model using Expression (3.23) were applied to the investigation of fuel droplet heating and evaporation in typical Diesel engines-like conditions [88]. The applicability of Expression (6.8), describing weak evaporation, was justified by the predicted results. The

334

6 Kinetic Modelling of Droplet Heating and Evaporation

values of drift vapour velocities were much smaller than the velocity of sound. It was assumed that fuel vapour leaving the Knudsen layer was removed from the outer boundary of this layer via diffusion and convection to the surrounding hydrodynamic region. The mass flux of fuel vapour leaving the droplets was equal to the diffusion and convection mass flux from the outer boundary of the Knudsen layer jdiff . This approach, known as the flux matching method, is widely used in kinetic modelling of droplet evaporation and condensation (e.g. [119]). The location of the interface between the Knudsen layer and the hydrodynamic region was chosen heuristically. Hence, it would be more appropriate to refer in this and similar models discussed later not to the Knudsen layer (the thickness of which is usually well determined, see the earlier discussion) but to the kinetic region (the region in which the analysis is based on the Boltzmann equations rather than hydrodynamic equations). The condition jlg = jdiff allowed the authors of [88] to determine the value of the mass fraction of fuel vapour at the outer boundary of the Knudsen region (Yv Rd ). This equation was solved using the assumption that TRd = Ts . The value of Yv Rd was less than or equal to the value of Yv at the droplet surface where fuel vapour was assumed to be saturated (Yvs ). The value of jdiff was estimated using the hydrodynamic approach (see Chap. 3). The gas pressure and the initial droplet temperature were assumed equal to 30 bar and 300 K, respectively. It was assumed that there is no temperature gradient inside droplets. βm were taken equal to 0.5 and 0.04. These are the average and minimal values of βm for water [65, 115]. βm = 0.04 is close to βm = 0.06 predicted by Shusser et al. [159] for butane. The analysis presented in [88] was performed for a droplet with an initial radius of 5 µm and initial ambient air temperature of 750 K. Droplets with this initial radius are expected to be present in Diesel engines [45]. The initial air temperature used in the analysis is typical for the end of the compression stroke in these engines [45, 137]. As follows from the results described in [88], kinetic effects lead to a small increase in the evaporation time and droplet temperature for βm = 0.5. This increase turned out to be larger for smaller βm . The shapes of the droplet diameter versus time plots, predicted by hydrodynamic and kinetic models, were rather similar. A similar analysis was performed for the initial ambient air temperature of 2000 K and for both these temperatures and the droplet initial radii equal to 20 µm. In all the cases the kinetic effects were demonstrated to be more visible for smaller droplets than for larger ones, and for higher initial ambient air temperatures. As expected, the kinetic model predicted longer evaporation times and higher droplet temperatures compared with the hydrodynamic model. The droplet evaporation times and temperatures increased when the evaporation coefficient decreased. The main recommendation made by the authors of [88] is that kinetic effects should be considered when analysing droplet evaporation in Diesel engine-like conditions.

6.1 Early Results

335

The kinetic effects were even more visible for droplets with an initial radii of 1 µm. In this case, however, the kinetic model used for the analysis was less reliable, as it did not consider the effects of surface tension. For droplets with initial radii close to 1 µm, these effects are expected to be important [108]. Although the results described in [88] demonstrate the importance of considering the kinetic effects when modelling droplet heating and evaporation even in a highpressure environment, the simplifying assumptions used in [88] make it difficult to recommend this model for quantitative analysis of the processes. In the model described in [88], it was assumed that the evaporation takes place into the fuel’s own vapour. This assumption is questionable for Diesel engine-like conditions when fuel evaporates into the air. Among early approaches, considering the presence of two components in the kinetic region [10, 177] can be mentioned. In [177] the two-surface problems of a multi-component mixture of vapour and non-condensable gases in the continuum limit were investigated using the asymptotic analysis of the Boltzmann equation. Results of an asymptotic analysis of the linearised Boltzmann equation for the binary mixture are described in [10]. A more rigorous approach to modelling fuel droplet heating and evaporation in Diesel engine-like conditions was developed in [139, 157]. This approach was based on the rigorous calculation of the collision term in the Boltzmann equation using numerical methods, developed in [156, 157]. The algorithm described in [139, 157] was designed to model the evaporation and condensation processes in binary mixtures of molecules with different radii and masses. This algorithm was used to solve the same problem of evaporation of Diesel fuel into high-pressure air as considered earlier in [88], but considering the contribution of air molecules. As in [88], two regions above the surface of the evaporating droplet were introduced. These are the kinetic region, where the investigation was based on the Boltzmann equations for vapour and air, and the hydrodynamic region. The mass fluxes leaving the kinetic region and the corresponding diffusion fluxes in the hydrodynamic region were matched. The vapour mass flux leaving the droplet’s surface was the maximal one. This value of βm = 1, predicted by the approximate model described in [113], was used in calculations (see Sect. 2.1 of [139] for a more detailed discussion). The kinetic effects predicted by the model described in [139, 157] were noticeable, and were larger than those predicted by the approximate analysis, if the contribution of air in the kinetic region was considered. It is recommended that the kinetic effects are considered when accurate knowledge of fuel droplet heating and evaporation in Diesel engine-like conditions is essential [88]. One of the important limitations of the models suggested in [139, 157] is that their authors did not consider the contribution of the heat flux in the kinetic region, assuming that there is no temperature gradient in it. This assumption cannot be justified without solving a more general problem, considering this effect. This was done in [134] and the results of that paper are summarised in Sect. 6.2.

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6 Kinetic Modelling of Droplet Heating and Evaporation

6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes The kinetic algorithm of the solution of the Boltzmann equation, developed in [134], in which both mass and heat transfer in the kinetic region are considered, is the same as described in [139, 157]. Note that the consideration of the heat flux led to an additional non-trivial problem of formulating the boundary conditions for temperature. Also, a revised version of the hydrodynamic model, and more accurate expressions for the binary diffusion and the convection heat transfer coefficients than in [139] were used. The temperature dependence of the fuel vapour specific heat capacity and thermal conductivity was considered. In what follows the key ideas of the model described in [134] are summarised. As in [139, 157], two regions above the surface of the evaporating droplet, the kinetic and hydrodynamic, are considered. These are presented in Fig. 6.1. As in [139], gas was assumed to consist of vapour and background air in both regions. Chemical reactions between vapour and oxygen were not considered. As in the previous papers, vapour and air dynamics in the kinetic region were considered using the Boltzmann equations, while the hydrodynamic equations were used in the second region. As in [139], the contributions of droplet movement, thermal radiation and heat transfer inside droplets were not considered. None of these assumptions can be rigorously justified in Diesel engine-like conditions, where the model was applied, but they allowed the authors of [134] to separate these effects from the kinetic effects. In a comprehensive model these effects need to be considered alongside the kinetic effects. The model, considering the kinetic effects and heat transfer inside droplets, is discussed later in this chapter.

Fig. 6.1 Liquid phase, kinetic and hydrodynamic regions close to the surface of the droplet. Ts is the droplet surface temperature, ρs is the vapour density at the droplet surface, TRd and ρ Rd are the vapour temperature and density at the outer boundary of the kinetic region. δ Rd shows the thickness of the kinetic region, jv and q indicate the directions of the vapour mass and heat fluxes, respectively. Reprinted from [143], Copyright Elsevier (2013)

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337

6.2.1 Boltzmann Equations for the Kinetic Region 6.2.1.1

Formulation of the Problem

The Boltzmann equations for the molecular velocity distribution functions for air f a ≡ f a (r, t, v) and vapour f v ≡ f v (r, t, v) in the kinetic region (see Fig. 6.1) can be presented as ∂ fa ∂t

⎫ + va ∂∂rfa = Jaa + Jav ⎬

∂ fv ∂t

+ vv ∂∂rfv = Jva + Jvv

⎭

,

(6.11)

where collision integrals Jαβ (α = a, v; β = a, v) are defined as (cf. Eq. (6.3)) Jαβ =

2 dαβ

2





+∞

−∞

π

dv1

 sin θ dθ

0

0

2π

     dφ f α f β1 − f α f β1 vα − vβ1  ,

(6.12)

dαβ = (dα + dβ )/2, dα and dβ are the diameters of molecules α and β, respectively, θ and φ are angular coordinates of molecules β relative to molecules α after the colli sion and superscript indicates the velocities and the molecular velocity distribution functions after collisions (these functions are referred to as ‘molecular distribution functions’). Subscript 1 indicates that molecules of type β collide with molecules of type α and because of this interaction the function f α is modified. The first integral on the right-hand side of (6.12) is calculated in the three-dimensional velocity space. The derivation of (6.11) and (6.12) was based on the assumption that molecules are rigid spheres and that body forces acting on them are negligible. The evaporation coefficient βm was assumed to be equal to 1. The temperature gradient in the kinetic region was considered.

6.2.1.2

Numerical Algorithm

The main structural elements of the numerical algorithm used in the calculations are discussed in [91, 156, 157]. In what follows these elements are briefly summarised and the new elements of the algorithm are described, following [139]. Physical space and time are discretised as in standard computational fluid dynamics (CFD) codes [186]. The discretisation of the velocity space is performed similar to that of the physical space. Continuous values of v are replaced by a discrete set  k M , where k refers to the position of a velocity cell and M is the total numv √ ber of cells. A homogeneous grid in the range vimax − vimin = 7 2Rv Ts , where i = x, y, z, Rv is the vapour gas constant and Ts the droplet surface temperature, was used. Twelve cells for each velocity component were used, leading to the total number of cells in the velocity space equal to 123 . This grid was used for the calculation of the dimensionless Maxwellian distribution with normalised density n = 1

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6 Kinetic Modelling of Droplet Heating and Evaporation

and normalised temperature T = Tg /Ts = 1. The following values were obtained: T = 0.9998 and n = 0.9999 [157]. It was concluded that the velocity grid used in the analysis leads to the calculation of macroscopic variables with errors much less than 0.1% when the molecular distribution function is close to the Maxwellian one. In the general case, it is anticipated that these errors are, at worst, just less than 1%. This is considered to be acceptable for most applications. The choice of the boundaries of the velocity domain in vx , v y , vz directions ensured that the contribution of molecules with velocities outside this range is negligible. For each vk the corresponding value of f k was specified, and System (6.11) was presented in a discretised form as Δf k Δf αk k k + vαk α = Jαα + Jαβ , Δt Δr

(6.13)

k where k is in the range [1, M]. Calculation of Jαβ for each velocity cell vαk allows us to reduce the non-linear system of integral–differential equations (6.11) to the linear system of algebraic equations (6.13). Following [156], the numerical solution of System (6.13) for gaseous components is performed in two steps. Firstly, molecular displacements are calculated without k k = Jαβ = 0). Secondly, the collisional relaxconsidering the effect of collisions (Jαα ation is calculated assuming spatial homogeneity. The boundary conditions for the distribution function of molecules are considered in the first step. The numerical solution of Eq. (6.13) is performed following the explicit approach, assuming that the Courant condition Δt max(|vx |, |v y |, |vz |) < min(Δx, Δy, Δz) is valid. Between 100 and 200 cells in the physical space were used in calculations, leading to errors of less than 1%. In the second step, the displacement of molecules stops and they start colliding. Using the explicit approach, the solution to each simplified equation in System (6.11) in each cell in the physical space is presented as

f αk,n

 k,n−1 k,n−1 + Nαβ f˜αk,n−1 + Δt Nαα ,  = k,n−1 k,n−1 1 + Δt ναα + ναβ

(6.14)

where 



2  π  2π dαβ +∞ k  ˜k,n−1 vαk − vβ1  −∞ dv1 0 sin θ dθ 0 dφ f β1 2  2  π  2π  dαβ   +∞ k,n−1 k,n−1 Nαβ = 2 −∞ dv1 0 sin θ dθ 0 dφ f˜αk,n−1 f˜β1 vαk

k,n−1 = ναβ

⎫ ⎬

 , (6.15) k ⎭ − vβ1 

˜ refers to the value of the distribution function of molecules calculated in the first step; additional superscripts n−1 and n show consecutive timesteps. k,n−1 k,n−1 and Nαβ in Expressions (6.15) is a major The calculation of integrals ναβ challenge from the point of view of CPU requirements. In the algorithm developed in [139] the standard approach to calculating these integrals is replaced by integration

6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes

339

using random cubature formulae. This allowed the authors of [139] to obtain the k,n−1 k,n−1 and Nαβ : following expressions for ναβ 2 ˜k,n−1 V dαβ  K 0 f β1l |vα −vβ1l | sin θl l=1 K0 2 p(wl )   k,n−1 d 2  K 0 f˜αlk,n−1 f˜β1l |vα −vβ1l | sin θl k,n−1 Nαβ = KV0 2αβ l=1 p(wl )

k,n−1 ναβ =

⎫ ⎬ ⎭

,

(6.16)

where V is the volume of the five-dimensional space, wl ≡ wl (vβ1l , θl , φl ) a point in this space, p(wl ) the value of the probability density function at these points, K 0 the total number of these points (number of collisions in a five-dimensional cell in physical and velocity spaces), and the summation is performed over all these points. √ k,n−1 k,n−1 and Nαβ is proportional to 1/ K 0 The relative error of calculation of ναβ and does not depend on the dimension of space [12, 156]. Homogeneous distribution of wl was considered in calculations, which implies that p(wl ) = 1. The efficiency of the application of Eq. (6.16) depends on the choice of random nodes wl . One of the most popular approaches to selecting these nodes is based on Korobov sequences [84, 91, 124, 178]. The condition p(wl ) = 1 for these sequences is satisfied. For piecewise constant functions, the errors of calculations based on Korobov sequences are proportional to 1/K 0 . This approach was used in the algorithm described in [139] with K 0 = 200, leading to errors of about 1%. The expression for V can be presented as [139]     V = 2π 2 vx(max) − vx(min)  v y(max) − v y(min)  vz(max) − vz(min)  .

(6.17)

Initially, the collisions are assumed to be elastic (momentum and energy are conserved). The directions of momenta of molecules in the coordinate system, linked with their centres of inertia, are assumed to be random. There were several difficulties with the numerical implementation of this model. Randomly chosen directions of molecular velocities after collisions are likely to lead to values of these velocities lying between the nodes of the discretised velocity space. This can result in nonconservation of momenta and energies during the collision process. Some authors solved this problem by introducing corrections to the molecular distribution function after collisions [156]. Although these corrections ensured the conservativeness of the system, they led to additional sources of errors. In the projection method, described in [180], the actual molecular velocities after the collisions were replaced by pairs of velocities referring to the nearest nodes. These velocities were appropriately weighted to ensure the conservation of momenta and energy during collision processes. This, however, inevitably increased complexity of the algorithm. The approach developed by the authors of [139] is different from those discussed above. This approach uses the discretisation of the velocities, not only for the analysis of molecular motion but also in the description of the collision processes. Two colliding molecules, with velocities v and v1 , enter a certain interaction zone. We are not interested in the details of the collision process, but we assume that after the collision these molecules acquire new velocities v and v1 for which: (1) the total

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6 Kinetic Modelling of Droplet Heating and Evaporation

Fig. 6.2 Scheme of the collision process between two molecules in the frame of reference linked   with their centre of inertia. p and p1 are momenta of molecules before the collision; p and p1 are their momenta after the collision. The sizes of the grid in this plane Δp are assumed to be the same   in px and p y directions. Both components of all four vectors p, p1 , p and p1 are integers of 0.5Δp. The absolute values of these vectors are equal to the radius of the circumference. The subscripts c , referring to centre of inertia of colliding molecules, are omitted. Reprinted from [139], Copyright Elsevier (2007)

momentum and energy of both molecules are conserved (collisions are elastic); (2) vectors v and v1 belong to an a priori chosen set of velocities. A conservative scheme, using a special choice of collision parameters, is described in [22, 67, 125]. In this scheme, the molecular velocity components before and after collisions are assumed to only be able to take the values corresponding to the nodes in the discretised velocity space [139]. The application of this approach can best be demonstrated if we focus on the collision process in the frame of reference linked with the centre of inertia of both molecules. The dynamics of the system is described in terms of momenta rather than velocities. In this frame of reference, the momenta of oncoming molecules have equal values but opposite directions. For the two-dimensional case these are schematically presented in Fig. 6.2, where px = −1.5Δp and p y = 1.5Δp, Δp is the grid size in the momentum space (considered to be the same in all directions). It can be demonstrated that the components of momenta in the frame of reference linked with the centre of inertia of colliding molecules are always multiples of 0.5Δp. The collision process leads to the rotation of momenta of both molecules in such a way that their absolute values remain the same, but the directions are opposite. All possible momenta satisfying these conditions lie on the circumference as shown in Fig. 6.2. In contrast to most previous investigations of this process, we do not consider all possible values of momenta after the collision, but limit ourselves to the cases when the components of these momenta are multiples of 0.5Δp. In the two-dimensional case presented in Fig. 6.2, these correspond to the points of intersection of the circumference with the nodes in the momentum space. In the case shown in Fig. 6.2 there are four such points which correspond to four combinations of momenta of molecules after collision. In the three-dimensional case the circumference presented in Fig. 6.2 becomes the surface of a sphere and the number of possible

6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes

341

combinations of momenta after collision increases to 12. It has been observed that the maximal number of these combinations in the three-dimensional space is 24 (although this has not been proven). In the practical implementation of this model, the calculations were performed for all possible values of θ and φ for each collision and then the results were averaged over these variables. This is expected to improve the accuracy of the results compared with the random selection of θ and φ from the set of possible values of these variables. This approach provides consistency in the analyses of discretisation processes applied to describe the molecular dynamics and collision processes. It has been tested on many problems, one of which is described in Sect. 6.7.

6.2.2 Vapour Density and Temperature at the Boundaries In [139], the problem of the thickness of the kinetic region was not considered. This approach was based on the observation that the mass flux at the outer boundary of the kinetic region is proportional to the difference between the vapour densities at the surface of the droplet and the outer boundary of the kinetic region. The vapour density at the outer boundary of the kinetic region was obtained from the requirement that mass fluxes in the kinetic and hydrodynamic regions coincide, assuming that the temperature at this boundary is the same as that at the droplet’s surface. Thus the ‘true’ values of vapour density at this boundary and the corresponding values of the vapour mass flux were obtained. This approach cannot be used when neither the vapour density nor the temperature at the outer boundary of the kinetic region (ρ Rd and TRd ) are known. This section addresses the problem of calculating the thickness of the kinetic region. It also presents a new approach to calculating the values of ρ Rd and TRd following [134]. After these values have been found, the Boltzmann equations (6.2) are solved in the kinetic region, subject to the new conditions at its outer boundary, following the procedure discussed in Sect. 6.2.1. Many coupled solutions of the Boltzmann and Navier–Stokes hydrodynamic equations for various gas mixtures have been described (e.g. [92]). It has been demonstrated that in the case where the thickness of the kinetic region is chosen to be about 10 mean free molecular path lengths (λc ), the kinetic and hydrodynamic solutions match well. In this case the combined kinetic and hydrodynamic solution agreed well with the solution based on the kinetic equations in the whole domain. The solutions of both Navier–Stokes and Boltzmann equations were transient in the general case. These solutions were matched at each timestep. The solutions to the Boltzmann equations were used as the boundary conditions for the hydrodynamic equations. In [134] it was shown that this estimate of the thickness of the kinetic region can be applied to the mixture of n-dodecane and air at temperatures and pressures typical for Diesel engine-like conditions. After the minimal thickness of the kinetic region was estimated to be δ Rd = 10λc , the values of the fuel vapour density and temperature at the boundaries of this region

342

6 Kinetic Modelling of Droplet Heating and Evaporation

were found. The droplet surface temperature (Ts ) was found from the Robin boundary condition at the droplet surface presented as (3.38). The vapour density at the droplet surface (ρvs ) was found from the corresponding partial vapour pressure described by the Antoine equation (e.g. Eq. (3.10)), assuming that the ideal gas law is valid. Estimating these parameters at the outer boundary of the kinetic region is not trivial. In [139] it was assumed that TRd = Ts . This assumption, however, led to physical inconsistency of the formulation of the problem as it implied that there is no heat flux in the kinetic region. Following [134], this assumption is relaxed which leads to the problem of finding both ρ Rd and TRd . The values of these parameters depend on the processes on the droplet surface and in the hydrodynamic region. They need to be found from the conservation of heat and mass fluxes at the interface: qk = qh = h(Tg − Ts ),

(6.18)

jk = jh = m˙ d /(4π Rd2 ),

(6.19)

where subscripts k and h indicate kinetic and hydrodynamic regions, respectively. The first step in finding the values of ρ Rd and TRd is focused on an investigation of mass and heat transfer in the kinetic region for a set of realistic values of these parameters. Remembering the physical background of the problem under consideration (heating and evaporation of droplets in a hot gas), these parameters are assumed to be in the ranges: ρ Rd < ρs and TRd > Ts . After the values of ρ Rd and TRd have been found, the solution to the Boltzmann equations (6.11) in the kinetic region allows us to find the normalised heat and mass fluxes at the outer boundary of this region:  q˜k = qk /( p0 Rv T0 ),  j˜k = jk /(ρ0 Rv T0 ), where Rv is the vapour gas constant, T0 the reference temperature arbitrarily chosen equal to 600 K, p0 = pv (T0 ) is the saturated vapour pressure at T = T0 and vapour density ρ0 is obtained from the ideal gas law. As a starting point, the following values of parameters were selected: ρ Rd = 0.95ρs and Ts = T0 . The values of q˜k were calculated for T˜Rd = TRd /T0 in the range 1 to 1.5. The results for T˜Rd from 1 to 1.1 are presented in Fig. 6.3. As follows from this figure, the dependence of q˜k on T˜Rd closely follows a linear function. The same functional dependence of q˜k on T˜Rd was observed in the whole range of T˜Rd up to 1.5. The same result was obtained for other values of Ts from 300 K (room temperature) to 659 K (critical temperature of n-dodecane). The effect of ρ Rd on q˜k was studied. The plots of q˜k versus ρ˜ Rd = ρ Rd /ρs for four T˜Rd are presented in Fig. 6.4. As can be seen in this figure, the dependence of q˜kin on ρ˜ Rd is weak for all T˜Rd . For 1 ≤ T˜Rd ≤ 1.1 (the most important range for applications) the dependence of q˜k on ρ˜ Rd can be safely ignored. This result allowed the authors of [134] to decouple the effects of T˜Rd and ρ˜ Rd on q˜k .

6.2 Kinetic Algorithm: Effects of the Heat and Mass Fluxes

343

Fig. 6.3 Plots of normalised heat flux in the √ kinetic region q˜k = qk /( p0 Rv Ts ) versus normalised temperature T˜Rd = TRd /Ts , assuming that Ts = 600 K. This figure shows how the value of T˜Rd is found using the previously calculated value of q˜h . Reproduced from Fig. 4b of [134] with permission granted by Begell House

Fig. 6.4 Plots of normalised heat flux in the √ kinetic region q˜k = qk /( p0 Rv Ts ) versus normalised fuel vapour density ρ˜ Rd = ρ Rd /ρs for four values of T˜Rd , shown near the plots. Reproduced from Fig. 5 of [134] with permission granted by Begell House

The next stage was focused on the investigation of the dependence of j˜k on ρ˜ Rd under the assumption that T˜Rd = 1.05. The results are presented in Fig. 6.5 for 0.5 ≤ ρ˜ Rd ≤ 0.8. As follows from this figure, j˜k is almost a linear function of ρ˜ Rd . Essentially the same linear dependence of j˜k on ρ˜ Rd was obtained for other values of T˜Rd from 1 to 1.5 and Ts from 300 K to 659 K. Having established the above-mentioned properties of q˜kin and j˜kin , the following algorithm for finding ρ Rd and TRd was developed [134]: 1. Calculate qh based on Expression (6.18); 2. Using a realistic value of ρ˜ Rd (0.95 for Diesel engine-like conditions) prepare the plot of q˜k versus T˜Rd as presented in Fig. 6.3 (the results are not expected to change if a different but reasonable value of ρ˜ Rd has√been chosen; cf. Fig. 6.4); 3. Find the intersection of the line q˜h = qh /( p0 Rv Ts ) with the line q˜k versus T˜Rd as presented in Fig. 6.3. This intersection identifies the required value of T˜Rd ;

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6 Kinetic Modelling of Droplet Heating and Evaporation

Fig. 6.5 The plot of normalised vapour mass flux in the kinetic√region j˜k = jk /(ρs Rv Ts ) versus normalised density ρ˜ Rd = ρ Rd /ρs for T˜Rd = 1.05. This plot illustrates how ρ˜ Rd was found, using the previously calculated value of j˜hyd . Reproduced from Fig. 6 of [134] with permission granted by Begell House

4. For the value of T˜Rd obtained in the previous step, calculate TRd and the value of jh from Eq. (6.19); 5. Draw the plot of j˜k versus ρ˜ Rd as presented√in Fig. 6.5; 6. Find the intersection of the line j˜h = jh /(ρs Rv Ts ) with the line j˜k versus ρ˜ Rd as presented in Fig. 6.5. This intersection identifies the required value of ρ˜ Rd . The model described in this section was applied to calculating heating and evaporation of a fuel droplet in Diesel engine-like conditions [134]. It was shown that in the case of droplet heating in a relatively cool gas (air) (Tg = 750 K), the effect of non-zero heat flux in the kinetic region is very small. This effect, however, becomes important in the cases where air temperature rose to 1000 K and 1500 K. The application of the kinetic model, considering the heat flux in the kinetic region, as described in this section, is recommended when accurate predictions of droplet surface temperatures and evaporation times are important.

6.3 Approximations of the Kinetic Results The model presented in Sect. 6.2 is based on the numerical solution of the Boltzmann equations for vapour and background gas (air) inside the kinetic region. The CPU requirements of this solution make it impractical for engineering computational fluid dynamics (CFD) software [45, 147]. Analysing the kinetic effects in droplet heating and evaporation within CFD software can be based on approximate kinetic calculations using simple approximate analytical formulae. This approach to the problem could be like the one described in [37], where the results of complex Mie calculations, describing the radiative heating of semi-transparent droplets, were approximated by simple analytical expres-

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345

sions (see Sect. 2.2.3), or the one described in [36] where a parabolic approximation was offered for the temperature profile inside a symmetrically heated droplet (see Sect. 2.1.1.5). This section shows ways of approximating the kinetic calculations of droplet heating and evaporation from Sect. 6.2. There is a trade-off between accuracy and simplicity. The most important ranges of the values of fuel droplet radii and gas temperatures are considered; the initial values of droplet temperatures are assumed to be equal to 300 K. Although the applicability of the approximations described in this section is restricted to very specific problem of modelling the processes in Diesel engine-like conditions, it is expected that the suggested approaches can be generalised to a much wider range of applications. The rest of this section is mainly focused on the results presented in [142] (cf. [140, 141]).

6.3.1 Approximations for Chosen Gas Temperatures 6.3.1.1

Droplet Radii

Directly comparing the results of kinetic and hydrodynamic calculations for gas temperatures Tg = 750 K, Tg = 1000 K and Tg = 1500 K and initial droplet radii R0 = 5 µm and R0 = 20µm, the droplet radii can be approximated by  ΔR = a1R exp − where ΔR =

R t1R





+ a2R exp −

Rk − R h , R0

R=

R t2R

,

(6.20)

Rh , R0

Rk and Rh are droplet radii predicted using the kinetic and hydrodynamic approaches, respectively, R0 is the initial droplet radius, a1R , a2R , t1R and t2R are fitting constants. This will be referred to as Approximation I. The effects of droplets on gas were not considered, and the model presented in Sect. 6.2 was used. A simpler although less accurate approximation is  ΔR = a3R exp −

R t3R

,

(6.21)

where a3R and t3R are new fitting constants. This will be referred to as Approximation II. The constants, used in Expressions (6.20) and (6.21), are shown in Table 6.1 for several initial droplet radii and gas (air) temperatures.

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6 Kinetic Modelling of Droplet Heating and Evaporation

Table 6.1 The fitting constants a1R , t1R , a2R , t2R , a3R and t3R , used in Expressions (6.20) and (6.21), for several R0 and Tg R0 , μm

Tg , K

a1R

t1R

a2R

t2R

a3R

t3R

20 20 20 5 5 5

750 1000 1500 750 1000 1500

0.03449 0.03525 0.04359 0.02096 0.13070 0.54256

0.08439 0.08542 0.09221 0.08142 0.29023 0.50583

0.06401 0.06424 0.06605 0.14800 0.02404 −0.13500

0.25315 0.25857 0.26951 0.02802 0.08400 1.72373

0.09453 0.10158 0.30955 0.14532 0.16415 0.44323

0.17772 0.19470 0.25741 0.25205 0.26185 0.30402

Fig. 6.6 Plots of ΔR = (Rk − Rh )/R0 versus R = Rh /R0 for R0 = 5 µm and Tg = 1000 K, prepared using the kinetic model (circles), and Approximation I, Formula (6.20) (curve). Reproduced from Fig. 1 of [142] with permission by Inderscience

The plots of ΔR = (Rk − Rh )/R0 versus R = Rh /R0 , predicted by the kinetic model (calculation) and Formula (6.20) (Approximation I) for R0 = 5 µm and Tg = 1000 K are presented in Fig. 6.6. As follows from this figure, Approximation I looks almost ideal for this choice of input parameters. Some clearly visible deviations between the predictions of Formula (6.21) (Approximation II) and the results of kinetic calculations are shown in [142]. These deviations are immaterial in many applications. Similar closeness between the results of calculations and approximations was shown for other combinations of R0 and Tg presented in Table 6.1 [142]. Approximations (6.20) and (6.21) with coefficients presented in Table 6.1 were interpolated for gas temperatures between 750 K and 1500 K. For Approximation I, the following interpolation formula for the coefficients a1R , a2R , t1R and t2R was suggested [142]: (6.22) s R = b0R + b1R T˜ + b2R T˜ 2 , where T˜ = Tg /Tc , Tc = 659 K is the critical temperature of n-dodecane C12 H26 (approximation of Diesel fuel), s R stands for a1R , a2R , t1R or t2R . The coefficients bi R (i = 0, 1, 2) for Rd0 = 20 µm and 5 µm are presented in Table 6.2.

6.3 Approximations of the Kinetic Results

347

Table 6.2 The coefficients bi R in Formula (6.22) for R0 = 20 µm and R0 = 5 µm R0 = 20 μm 20 μm 20 μm 5 μm 5 μm 5 μm sR a1R t1R a2R t2R

b0R 0.04585 0.09076 0.06602 0.25469

b1R −0.01897 −0.01183 −0.00355 −0.00535

b2R 0.00790 0.00548 0.00156 0.00521

b0R 0.07649 -0.94906 0.69765 2.91559

b1R −0.30234 1.17172 −0.60010 −4.55079

Table 6.3 The constants a1R , t1R , a2R and t2R predicted by Formula (6.22) R0 , μm Tg , K a1R t1R a2R 20 20 20 5 5 5

750 1000 1500 750 1000 1500

0.03492 0.03555 0.04360 0.020971 0.13071 0.54258

0.08439 0.08543 0.09222 0.081425 0.29024 0.50584

0.06400 0.06422 0.06602 0.14800 0.02404 −0.13500

b2R 0.22279 −0.23396 0.10293 1.76927

t2R 0.25535 0.25856 0.26950 0.02803 0.08401 1.72374

The coefficients a1R , a2R , t1R and t2R , predicted by Formula (6.22) for R0 = 20 µm and R0 = 5 µm and Tg = 750 K, 1000 K and 1500 K are presented in Table 6.3. Comparing Tables 6.1 and 6.3 it can be seen that the maximal difference between the values of fitting constants does not exceed about 1%. Thus Formula (6.22) is adequate for practical calculations.

6.3.1.2

Droplet Temperatures

Using the direct comparison of the results of kinetic and hydrodynamic calculations for the same gas temperatures and initial droplet radii as in Sect. 6.3.1.1, it was demonstrated that a good approximation for the droplet temperatures predicted by the kinetic model is given by the following expression [142]: 

ΔT = a1T where ΔT =

T exp − t1T

Tk − Th , Tc − T0



T =

+ a2T ,

(6.23)

Th − T0 , Tc − T0

Tk and Th are droplet temperatures predicted by the kinetic and hydrodynamic models, respectively, T0 is the initial droplet temperature and the coefficients a1T , t1T and a2T

348

6 Kinetic Modelling of Droplet Heating and Evaporation

Table 6.4 The constants a1T , t1T and a2T , used in Formula (6.23), for several R0 and Tg R0 , μm 20 20 20 5 5 5

Tg , K 750 1000 1500 750 1000 1500

a1T · 10−8

1.25792 1.28986 · 10−8 2.48344 · 10−5 1.66779 · 10−6 4.54553 · 10−6 2.88456 · 10−4

t1T

a2T

−0.06989 −0.11769 −0.17572 −0.02012 −0.07213 −0.2292

1.87241 · 10−4 1.71118 · 10−4 4.09783 · 10−5 3.62998 · 10−4 5.78527 · 10−4 −9.16881 · 10−4

Fig. 6.7 Plots of ΔT = (Tk − Th )/(Tc − T0 ) versus T = (Th − T0 )/(Tc − T0 ) for R0 = 5 µm and Tg = 1000 K, obtained using the kinetic model (solid), and Formula (6.23) (dashed-dotted). Reproduced from Fig. 2 of [142] with permission by Inderscience

are fitting constants shown in Table 6.4 for the same initial droplet radii and gas temperatures as in Sect. 6.3.1.1. The plots of ΔT = (Tk − Th )/(Tc − T0 ) versus T = (Th − T0 )/(Tc − T0 ) predicted by the kinetic model (calculation) and Formula (6.23) for R0 = 5 µm and Tg = 1000 K are presented in Fig. 6.7. As can be seen in this figure, Formula (6.23) is reasonably accurate and can be used in applications. Similar closeness between the results of calculations and approximations was shown for other combinations of R0 and Tg presented in Table 6.1 [142]. As in the case of droplet radii, Formula (6.23) with the coefficients presented in Table 6.4 can be interpolated for the whole range of gas temperatures from 750 K to 1500 K, using Formula (6.22) for the coefficients a1T , a2T and t1T , replacing subscript R with T . As a result, Formula (6.22) is replaced by [142]: sT = b0T + b1T T˜ + b2T T˜ 2 ,

(6.24)

where sT stands for a1T , a2T or t1T . The values of these coefficients are presented in Table 6.5. The coefficients a1T , t1T and a2T , predicted by Formula (6.24) for

6.3 Approximations of the Kinetic Results

349

Table 6.5 The coefficients bi T in Formula (6.24) for R0 = 20 µm and R0 = 5 µm sT R0 b0T b1T b2T a1T t1T a2T

20 µm 5 µm 20 µm 5 µm 20 µm 5 µm

4.9653 · 10−5 5.49341 · 10−4 0.14865 0.02981 3.98252 · 10−5 −0.00414

−7.6331 · 10−5 −8.47827 · 10−4 −0.24154 0.02604 2.58552 · 10−4 0.00649

2.87444 · 10−5 3.22124 · 10−4 0.04351 −0.06143 −1.13368 · 10−4 −0.00223

Table 6.6 The fitting constants a1T , t1T and a2T predicted by Formula (6.24) R0 , μm Tg , K a1T t1T a2T 20 20 20 5 5 5

750 1000 1500 750 1000 1500

1.2620 · 10−8 1.2951 · 10−8 2.4834 · 10−5 1.6685 · 10−6 4.5463 · 10−6 2.8845 · 10−4

−0.06989 −0.11769 −0.17571 −0.02012 −0.07212 −0.22918

1.8724 · 10−4 1.7112 · 10−4 4.0978 · 10−5 3.5779 · 10−4 5.7333 · 10−4 −9.2119 · 10−4

R0 = 20 µm and R0 = 5 µm, and for Tg = 750 K, 1000 K and 1500 K are presented in Table 6.6. Comparing Tables 6.4 and 6.6 it can be seen that the maximal difference between the values of fitting constants for the selected R0 and Tg does not exceed about 1%. This justifies the applicability of Formula (6.24).

6.3.2 Approximations for Chosen Initial Droplet Radii This section approximates droplet radii and temperatures for a range of initial droplet radii. Approximations are found for droplets with an initial radius of 10 µm in addition to those previously found for droplets with the initial radii 5 and 20 µm. The predictions are compared with the results of the rigorous kinetic calculations for R0 = 15 µm. The results were previously published in [142].

6.3.2.1

Droplet Radii

Using trial-and-error Formula (6.20) can be interpolated in the whole range of R0 from 5 µm to 20 µm. The following approximation for the coefficients a1R , a2R , t1R and t2R was suggested [142]:

350

6 Kinetic Modelling of Droplet Heating and Evaporation

Table 6.7 The values of ci R (i = 1, 2, 3) in Formula (6.25) for the coefficients a1R , t1R , a2R and t2R in Formula (6.20) (Approximation I) Tg a1R t1R a2R t2R c0R c1R c2R c0R c1R c2R c0R c1R c2R

750 K 750 K 750 K 1000 K 1000 K 1000 K 1500 K 1500 K 1500 K

−0.08057 0.27109 29.4067 µm 0.04522 10.6517 1.03659 µm 0.21792 0.79250 4.26859 µm

0.03860 0.64352 5.12421 µm 0.04676 12.7347 1.26437 µm 0.24514 2.70877 1.87024 µm

0.08288 −0.05476 10.93604 µm 0.10022 −3.56529 1.21772 µm 0.11245 −13.27703 1.12864 µm

  R0 , s R = c0R + c1R exp − c2R

0.25985 −1.68683 2.44184 µm 0.26271 −9.64237 1.31078 µm 0.26081 −18.4742 1.10535 µm

(6.25)

where s R is the same as in Formula (6.22), and R0 is in µm. The coefficients ci R (i = 0, 1, 2) for three gas temperatures are presented in Table 6.7. The main limitation of Formula (6.20) is that it can only predict the droplet radii up to the instant when the hydrodynamic model predicts complete droplet evaporation. Since the evaporation time described by the kinetic model is always longer than that described by the hydrodynamic model, an additional approximation for the final stage of droplet evaporation is needed. The following algorithm was suggested to obtain this additional approximation. 1. Use Formula (6.20) to find droplet radii until the hydrodynamic model predicts complete droplet evaporation. 2. Select a time interval 5–10% longer than the complete evaporation time predicted by the hydrodynamic model. 3. Approximate the time dependence of R˜ = Rk /R0 in this interval as R˜ = a + b t + c t 2 ,

(6.26)

where t is the time (measured from the start of the evaporation process) in µs, and a, b and c are fitting constants. 4. Calculate the evaporation time based on the solution to Eq. (6.26): t1,2 =

−b ±

√ b2 − 4ac , 2a

(6.27)

where you must choose the root closest to the time of complete droplet evaporation predicted by the hydrodynamic model.

6.3 Approximations of the Kinetic Results

351

Note that Formula (6.25), with the coefficients presented in Table 6.7, was obtained by analysing the kinetic results for initial droplet radii 5 µm, 10 µm and 20 µm. To assess the effectiveness of Formula (6.25), it was tested for the initial droplet radius equal to 15 µm and gas temperature equal to 1000 K. The following values of fitting constants in Eq. (6.26) were found: a = −0.51412, b = 5.77135 µs−1 and c = −15.78277 µs−2 . The agreement between the predictions of the kinetic model and the approximation was very good [142]. The difference between the evaporation times predicted by the kinetic model (6.433 µs) and the approximation (6.505 µs) was only about 1%. This analysis was performed for other gas (air) temperatures, and the conclusions were similar to those for Tg = 1000 K. For example, for Tg = 750 K the difference between the evaporation time predicted by the kinetic model (11.535 μs) and the approximation (11.627 μs) was 0.8% [142].

6.3.2.2

Droplet Temperatures

As follows from the analysis presented in Sect. 6.3.1.2, Formula (6.23) for ΔT is reasonably good for practical applications. Another approximation of ΔT can be presented as:  T ΔT = d1T exp , (6.28) τ1T where d1T and τ1T are fitting constants. Using trial-and-error it was demonstrated that Formula (6.28) is more useful than Formula (6.23) for predicting droplet temperatures for the initial droplet radii in the range of 5–20 µm. Formula (6.28) includes fewer fitting constants than Formula (6.23) without affecting the accuracy of the prediction. The dependence of coefficients d1T and τ1T on the initial droplet radii R0 was approximated as   R0 , sT = f 0T + f 1T exp − f 2T

(6.29)

where sT stands for d1T or τ1T , R0 is in μm; f 0T , f 1T and f 2T are fitting constants presented in Table 6.8. Formula (6.28) is less accurate than the corresponding approximation of the droplet radii, although its accuracy improves at the final stage of droplet evaporation. In many applications, however, the difference in temperatures predicted by the hydrodynamic and kinetic models is not large, and the relative errors of temperatures predicted by Formula (6.28) and kinetic calculations do not exceed about 2%.

352

6 Kinetic Modelling of Droplet Heating and Evaporation

Table 6.8 The constants f i R (i = 0, 1, 2) in Formula (6.29) for the coefficients d1T and τ1T in Formula (6.28) Tg d1T τ1T f 0T f 1T f 2T f 0T f 1T f 2T f 0T f 1T f 2T

750 K 750 K 750 K 1000 K 1000 K 1000 K 1500 K 1500 K 1500 K

4.5322 · 10−10 5.62537 · 10−4 0.91832 µm 5.6932 · 10−8 7.6745 · 10−4 1.07509 µm 0.0 7.6745 · 10−4 1.07509 µm

0.05015 2.66871 1.29156 µm 0.07299 2.70172 1.27130 µm 0.18478 0.03506 8.46655 µm

We were not able to find a universal approximation of the kinetic results which is suitable for a wide range of initial droplet radii and gas temperatures (cf. the approximation of the results of Mie calculations of radiative heating of semi-transparent droplets suggested in [37] and described in Sect. 2.2.3).

6.4 Effects of Inelastic Collisions Two important assumptions were used in the model presented in Sect. 6.2: (1) The contribution of inelastic collisions was disregarded; (2) The evaporation coefficient was equal to 1. The first assumption could have been safely used in the analysis of mono-atomic molecules but is highly questionable in the case of complex molecules (e.g. C12 H26 , n-dodecane). Even if the analysis of the dynamics of these molecules is simplified by using the united atom model (see [23, 190] and Sect. 6.5), the number of internal degrees of freedom of them would still exceed 100. There is no reason to ignore the contribution of these degrees of freedom. In this section a simple model is described in which this assumption is relaxed. Section 6.5 is focused on the investigation of the evaporation coefficient which will allow us to relax the second assumption. To the best of the author’s knowledge, an early phenomenological model for binary collisions in a gaseous mixture including the contribution of the internal degrees of freedom of molecules was suggested in [21]. This model was used in Monte Carlo simulations of rarefied gas flows. Since the publication of that paper, many models of inelastic collisions have been described (e.g. [13, 14, 17, 44, 46, 85, 86, 102, 129, 163, 182] and [50]). These do not include papers where the models for collisions of specific atoms and molecules were developed, such as H2 –H2 collisions [201], N–N2 collisions [76] and H–N2 collisions [173]. Also, this list does not include publications in which the effects of inelastic collisions on transport properties are discussed (e.g. [128]).

6.4 Effects of Inelastic Collisions

353

The model described in [154] is different from the models described in the abovementioned publications, although it uses some well-known assumptions, such as the approximation of molecules by inelastic hard spheres (IHS). Although this model was tested only for some specific problems, it is expected to be applicable to any molecules with arbitrarily large numbers of internal degrees of freedom. It is expected to be particularly effective for the analysis of complex molecules, including n-dodecane. The next section presents this model, following [154].

6.4.1 Mathematical Model 6.4.1.1

A Model for Inelastic Collisions

As in Sect. 6.2, it is assumed that molecules can be approximated as hard spheres. In contrast to Sect. 6.2, however, the inelastic effects during the collisions between these spheres are considered, using the inelastic hard spheres (IHS) model [129]. Only the effects of binary collisions are considered. This approach can be supported by the fact that the Boltzmann equation is solved in a very thin (about 10 mean free paths between collisions) layer. Regardless of the nature of the collision between two molecules, their centre of mass is not affected. The state of the molecules after the collision is considered relative to the centre of mass. In this frame of reference, each molecule has three translational and a certain number of internal degrees of freedom. The total number of degrees of freedom of both molecules is assumed to be equal to N . During the collisions, the energy of each molecule is redistributed between the degrees of freedom, but the total number of degrees of freedom remains unchanged. Additionally, it is assumed that none of these degrees of freedom has any preference over the others. This implies that our focus is on the systems close to thermodynamic equilibrium. The model cannot be applied to systems far from thermodynamic equilibrium (e.g. gas lasers, [135, 136]). The possible generalisation of the model to the case when this assumption is relaxed is described later in this section. The assumption that none of the degrees of freedom has any preference over the others allows us to consider the redistribution of energy between these degrees of freedom during the collision process as random. N degrees of freedom are represented by an N -dimensional sphere with its centre at the point where energies of all degrees of freedom are equal to zero. The radius of this sphere can be found as  i=N  r =  Ei ,

(6.30)

i=1

where E i is the energy of the ith degree of freedom (translational or internal). Remembering that r 2 is the total energy of the system (E f ), Eq. (6.30) is the statement that E f at the surface of the sphere is conserved.

354

6 Kinetic Modelling of Droplet Heating and Evaporation

Fig. 6.8 A scheme of the rotation of vector X in the three-dimensional space (e1 , e2 , e3 ). Reprinted from [154]. Copyright Elsevier (2013)

√ Using the new coordinates xi = ± E i , an N -dimensional vector X = (x1 , x2 , ... x N ) is introduced. The basis and the norm of this vector are (e1 , e2 , ..... eN ) and ||X|| =  r = E f , respectively. The norms of the mutually perpendicular vectors ei are equal to 1. The redistribution of energy between the degrees of freedom during the collision process is described as the rotation of vector X in the N -dimensional space. For N = 3 this is schematically presented in Fig. 6.8. If none of the degrees of freedom has a preference over the others, then this rotation of the vector X can be considered as random, and described as [154] X = AX,

(6.31)

where X is the new location of vector X after rotation, and A is the rotation matrix: ⎤ ⎡ a11 a12 .... a1N ⎢ a21 a22 .... a2N ⎥ ⎥ (6.32) A=⎢ ⎣ ... ... .... ... ⎦ . a N 1 a N 2 .... a N N The conservation of the total energy during the collision implies that vector X remains at the surface of the sphere of radius r . This is possible if, and only if, AT A = E, where AT is the transpose of the matrix A,

(6.33)

6.4 Effects of Inelastic Collisions

355

⎡

1 ⎢0 E=⎢ ⎣ ... 0

0 1 ... 0

.... .... .... ....

⎤ 0 0⎥ ⎥ ... ⎦ 1

(6.34)

is the unit matrix. Equation (6.33) is presented as the combination of the following systems of equations [154]: ⎫ 2 2 + a21 + .... + a 2N 1 = 1 ⎪ a11 ⎪ ⎬ 2 2 + a22 + .... + a 2N 2 = 1 a12 ⎪ ... ... .... ... ⎪ ⎭ 2 2 + a2N + .... + a 2N N = 1 a1N

(6.35)

⎫ a11 a12 + a21 a22 + .... + a N 1 a N 2 = 0 ⎪ ⎪ ⎬ a11 a13 + a21 a23 + .... + a N 1 a N 3 = 0 ... ... .... ... ⎪ ⎪ ⎭ a11 a1N + a21 a2N + .... + a N 1 a N N = 0

(6.36)

⎫ a12 a13 + a22 a23 + .... + a N 2 a N 3 = 0 ⎪ ⎪ ⎬ a12 a14 + a22 a24 + .... + a N 2 a N 4 = 0 ... ... .... ... ⎪ ⎪ ⎭ a12 a1N + a22 a2N + .... + a N 2 a N N = 0

(6.37)

..................................................................................................... a1(N −1) a1N + a2(N −1) a2N + .... + a N (N −1) a N N = 0.

(6.38)

In the systems of Eqs. (6.36)–(6.38) identical equations are not included. The total number of equations in System (6.36)–(6.38) is (N − 1) + (N − 2) + (N − 3) + .... + 1 =

N (N − 1) 2

for N 2 unknown coefficients ai j . This allows us to select randomly N 2 − N2 (N − 1) = N2 (N + 1) of these coefficients keeping in mind the restriction imposed by Eq. (6.35) (normalisation condition). The following algorithm for the construction of the matrix A was proposed [154]: 1. The coefficients a11 , a21 , ..., a N 1 are arbitrarily selected but normalised using the first equation in System (6.35). 2. The coefficients a12 , a22 , ..., a(N −1)2 are arbitrarily selected, but the coefficient a N 2 is obtained from the first equation of System (6.36):

356

6 Kinetic Modelling of Droplet Heating and Evaporation

aN 2 = −

1 aN 1

! a11 a12 + a21 a22 + .... + a(N −1)1 a(N −1)2 .

(6.39)

Then all coefficients are normalised using the second equation in System (6.35). 3. The coefficients a13 , a23 , ..., a(N −2)3 are arbitrarily selected, but the coefficients a(N −1)3 and a N 3 are obtained from the solution to the second equation in System (6.36) and the first equation in System (6.37). These equations are rearranged as " a(N −1)1 a(N −1)3 + a N 1 a N 3 = b13 , a(N −1)2 a(N −1)3 + a N 2 a N 3 = b23 where

(6.40)

! b13 = − a11 a13 + a21 a23 + .... + a(N −2)1 a(N −2)3 , ! b23 = − a12 a13 + a22 a23 + .... + a(N −2)2 a(N −2)3 .

Then all coefficients are normalised using the third equation in System (6.35). Following the same procedure, all other components of matrix A are obtained, and this allows us to calculate X using Eq. (6.31). An example of the evolution of the system with 100 degrees of freedom with time is presented in Fig. 6.9. The energies E i of individual degrees of freedom Ni (i ∈ [1, 100]) are shown in the ordinate axis in this figure. Initially, all degrees of freedom are assumed to have energies equal to 1 (E i = 1). This is presented in the top part of this figure (initial state). After the first step, energies E i acquire random values in the range  from zero to five, with the sum of all E i remaining the same as at the initial state ( i=100 i=1 E i = 100). The distribution of E i after steps 2 and 3 remains random although with different values of individual energies. One E i can potentially reach the maximal value of 100, provided that all other E i are equal to zero. This, however, has never been observed in the results of the calculations. The model described in this section could be generalised to the case when the probabilities of excitation of various degrees of freedom are not the same. This could be done by the introduction of the weighting function, and/or by limiting the range of degrees of freedom to be activated. It is not easy, however, to specify this weighting function for such complex molecules as n-dodecane, for which the model has been primarily developed, or even justify the need to introduce this function in this case.

6.4.1.2

The Solution of the Boltzmann Equation

As in Sect. 6.2, the Boltzmann equation (for one or several components) is numerically solved in two steps. Firstly, molecular displacements are calculated ignoring the effect of collisions. Secondly, the collisional relaxation is calculated under the assumption of spatial homogeneity. In this section the details of this solution are presented following [154].

6.4 Effects of Inelastic Collisions Fig. 6.9 Energies of individual degrees of freedom (E i ) versus Ni at the initial state and after Steps 1, 2 and 3. Reprinted from [154]. Copyright Elsevier (2013)

357

358

6 Kinetic Modelling of Droplet Heating and Evaporation

Ignoring the contribution of collisions, the discretised form of the Boltzmann equation, describing molecular displacements, for each component is presented as Δf Δf +v = 0, Δt Δr

(6.41)

where f ≡ f (v, r, t) is the distribution function for the velocities (v) and locations in the physical space (r). In the absence of collisions, the total internal energy in each velocity range is conserved. Thus Δ(E int f ) Δ(E int f ) +v = 0, (6.42) Δt Δr where E int ≡ E int (v, r, t) are internal energies of molecules at certain v and r at time t. A model for inelastic collisions, described in Sect. 6.4.1.1, allows us to obtain the energies of all degrees of freedom after individual collisions as shown in Fig. 6.9. We are interested, however, only in the net changes of the kinetic energies of molecules after collisions in the centre of mass frame of reference. These changes in the kinetic energies can be described in terms of the changes in the radii of the three-dimensional spheres, which are the projections of the N -dimensional spheres (see Fig. 6.8), on the three-dimensional space. These spheres show the kinetic energies of colliding molecules in three directions. This is schematically shown in Fig. 6.10 for the case of the projection of the N -dimensional sphere on the two-dimensional plane. As in Sect. 6.2, this space is presented in terms of the components of momenta px and p y of both colliding molecules. Points p and p∗ indicate the positions of molecules before the collision. If the collisions were elastic, then the values of momenta after collisions would lie on the dashed circle shown in Fig. 6.10, being separated by 180◦ . The possible values of these momenta after the collisions are indicated as empty circles. As described in Sect. 6.2, randomly chosen directions of molecular momenta after collisions are likely to lead to the values of these momenta lying between the values in the nodes of the discretised momenta space. This can lead to non-conservation of momenta and energies during the collision process. To overcome this problem, as in Sect. 6.2 the momenta are discretised not only in the description of molecular motion but also in the analysis of the collision process. Namely, the momenta after the collisions are assumed to belong to an a priori chosen set of momenta, which are nodes in the momenta space presented in Fig. 6.10. This is achieved by moving the actual point on the surface of the sphere to the nearest node. Since the nodes are not uniformly distributed on the surface of the sphere in the general case, this leads to the partial loss of randomness of the distribution of momenta after collisions. This is one of the weaknesses of the model under consideration, but this does not significantly limit its applicability as illustrated in Sect. 6.7.

6.4 Effects of Inelastic Collisions

359

Fig. 6.10 A projection of the surface of an N -dimensional sphere, describing the translational and internal energies of two colliding molecules, into a two-dimensional space describing two translational degrees of freedom. p and p∗ are the locations of the molecular momenta before the collision. The dashed circle indicates possible locations of molecular momenta after the collision if the contribution of internal degrees of freedom is not considered. The thin solid circle indicates possible locations of molecular momenta after the collision if molecular internal energy increases during the collision. The thick solid circle indicates possible locations of molecular momenta after the collision if molecular internal energy decreases during the collision. p and p∗ indicate the allowed locations of the molecular momenta after the collision if molecular internal energy decreases during the collision. Reprinted from [154]. Copyright Elsevier (2013)

In the case presented in Fig. 6.10, in the absence of inelastic collisions, there are four such nodes corresponding to four combinations of momenta of molecules after collision. The maximal number of these combinations for the plane is eight. In the three-dimensional case, the circumference shown in Fig. 6.10 becomes the surface of a sphere and the maximal number of possible intersection points becomes 24. This corresponds to the maximal total number of combinations of momenta after collision. This approach ensures consistency in the discretisation processes used for the description of molecular motion and collisions. If during the collision the net internal energy of molecules increases, this has to be compensated by a decrease in the kinetic energies of molecules, and the radius of the corresponding circle in Fig. 6.10 is decreased. If the net internal energy of molecules decreases, this needs to be compensated by an increase in the kinetic energies of molecules, and the radius of the corresponding circle in Fig. 6.10 is increased. Both scenarios are illustrated in Fig. 6.10. When the net kinetic energy of molecules increases, the possible values of the momenta of both molecules after the collision are presented as grey circles. These include points p  and p∗ . The changes in radii of the circles after the collision were calculated using Korobov’s sequences [84, 124], enhanced by the randomisation of individual points. Similar to elastic collisions, the points p  and p∗ were chosen to coincide with the nodes of the discretised momenta space.

360

6 Kinetic Modelling of Droplet Heating and Evaporation

The application of the model described in this section was illustrated for three test problems: shock wave structure in nitrogen, one-dimensional heat transfer through a mixture of n-dodecane and nitrogen and one-dimensional evaporation of n-dodecane into nitrogen [154]. In the first problem, the predictions of the model, considering the contribution of the rotational degrees of freedom, were demonstrated to be close to experimental data and the predictions of the earlier developed model, using a different approach to account for the effects of inelastic collisions. This problem was generalised to a hypothetical case when the number of internal degrees of freedom of nitrogen (Nint ) is in the range of 0–10. It was demonstrated that the results noticeably changed when Nint increased from 0 to 2, but stayed almost the same for Nint ≥ 6. The predicted heat flux for the second problem did not depend on the number of internal degrees of freedom of the mixture Nint when this number exceeds about 15. In the third problem, the predicted mass flux of n-dodecane also remained almost the same for Nint ≥ 15. These results allow us to consider systems with large numbers of internal degrees of freedom by replacing the analysis of these systems by the analysis of systems with relatively small numbers of internal degrees of freedom [154].

6.4.2 Solution Algorithm This section describes the algorithm to solve the Boltzmann equations (Eqs. 6.11) in the kinetic region, considering the effects of inelastic collisions, following [143]. As in Sect. 6.2, the first step in the solution of Eq. (6.11) is focused on the investigation of mass and heat transfer processes in the kinetic region for a certain range of ρ Rd and TRd . As in Sect. 6.2, these parameters are considered in the ranges: ρ Rd < ρs and TRd > Ts . Once the values of ρ Rd and TRd have been obtained, the solution to the Boltzmann equations (6.11) in the kinetic region allows us to obtain the normalised mass and heat fluxes at the outer boundary of this region:  j˜k = jk /(ρ0 Rv T0 ),

 q˜k = qk /( p0 Rv T0 ),

where Rv is the vapour gas constant, T0 is the reference temperature assumed equal to 600 K, p0 and ρ0 are the saturated vapour pressure and density for T = T0 ; ρ0 is inferred from the ideal gas law and subscript k refers to kinetic. Following [143], it was assumed that ρ Rd = 0.9ρs and Ts = T0 . The values of q˜k were obtained for up to 50 internal degrees of freedom (Nint ) and T˜Rd = TRd /Ts in the range 1 to 1.4. The results are presented in Fig. 6.11. As follows from this figure, for all Nint the dependence of q˜k on T˜Rd is well described by a linear function, as in the case discussed in Sect. 6.2. For T˜Rd under consideration the values of q˜k decrease with increasing Nint . The rate of this decrease, however, is small for Nint > 10 and negligible for Nint > 20. This allows us to limit our analysis to the case of Nint = 20, in agreement with the results described in [154].

6.4 Effects of Inelastic Collisions

361

Fig. 6.11 Plots of normalised heat flux in the kinetic region√ q˜k ≡ qk /( p0 Rv Ts ) versus normalised temperature T˜Rd = TRd /Ts for several numbers of internal degrees of freedom Nint ; it is assumed that Ts = 600 K and ρ Rd = 0.9ρs . Reprinted from [143]. Copyright Elsevier (2013)

Fig. 6.12 Plots of normalised heat flux in the kinetic region q˜k versus normalised density ρ˜ Rd = ρ Rd /ρs for Nint = 20 and T˜Rd = 1.1 and 1.2. Reprinted from [143]. Copyright Elsevier (2013)

The plots of q˜k versus ρ˜ Rd ≡ ρ Rd /ρs for T˜Rd = 1.1 and 1.2 and Nint = 20 are presented in Fig. 6.12. As follows from this figure, the plots for these values of T˜Rd are the lines almost parallel to the ρ˜ Rd axis. This allows us not to consider the dependence of q˜k on ρ˜ Rd in agreement with the similar result obtained in Sect. 6.2 for N√ int = 0. The plots of q˜k versus T˜Rd for ρ˜ Rd = 1, Nint = 0 and 20, and q˜h = qh /( p0 Rv Ts ) versus T˜Rd (horizontal line) are presented in Fig. 6.13. As in Sect. 6.2, it was assumed that TRd in Eq. (6.18) can be replaced with Ts . Also, it was assumed that Tg = 1000 K, Ts = 600 K, Rd = 5 µm. The intersections between the horizontal and inclined lines allow us to obtain the required values of T˜Rd . When only elastic collisions were considered (Nint = 0) it was obtained that T˜Rd = 1.014. When the contribution of inelastic collisions with Nint = 20 is considered T˜Rd = 1.026. This shows that the contribution of internal degrees of freedom leads to a noticeable increase in T˜Rd .

362 Fig. 6.13 Plots of q˜k versus T˜Rd for Nint = 0 and 20, and the plot of √ q˜h ≡ qh /( p0 Rv Ts ) versus T˜Rd . It is assumed that Ts = 600 K, Tg = 1000 K, Rd0 = 5 µm and ρ˜ Rd = 1. The value of T˜Rd is found from the intersections between the plots of q˜k and q˜h . Reprinted from [143]. Copyright Elsevier (2013)

Fig. 6.14 Plots √ of j˜k ≡ jk /(ρ0 Rv Ts ) versus ρ˜ Rd for Nint = 0 and 20, and the plot of √ j˜h ≡ jh /(ρ0 Rv Ts ) versus ρ˜ Rd . It is assumed that Ts = 600 K, Tg = 1000 K, Rd0 = 5 µm and T˜Rd = 1.026. The value of ρ˜ Rd is found from the intersections between the plots of j˜k and j˜h . Reprinted from [143]. Copyright Elsevier (2013) Fig. 6.15 Plots of j˜k versus ρ˜ Rd for Nint = 20 and βm = 1 and 0.36 versus ρ˜ Rd . It is assumed that Ts = 600 K, Tg = 1000 K, Rd0 = 5 µm and T˜Rd = 1.026. Reprinted from [143]. Copyright Elsevier (2013)

6 Kinetic Modelling of Droplet Heating and Evaporation

6.4 Effects of Inelastic Collisions

363

The√plots of j˜k versus ρ˜ Rd for T˜Rd = 1.026, Nint = 0 and 20, and j˜h = jh /( p0 Rv Ts ) versus ρ˜ Rd (horizontal line) are presented in Fig. 6.14. This figure was prepared for the same parameters as Fig. 6.13. As in Sect. 6.2, ρ Rd in Eq. (6.19) was replaced with ρs . The intersections between the horizontal and inclined lines allow us to obtain the required values of ρ˜ Rd . When only elastic collisions were considered (Nint = 0), ρ˜ Rd = 0.968; when the contribution of inelastic collisions with Nint = 20 was considered, ρ˜ Rd = 0.926. This shows that the contribution of internal degrees of freedom leads to a noticeable decrease in ρ˜ Rd . Similar values of T˜Rd and ρ˜ Rd were obtained for other values of Tg and Rd relevant for Diesel engine applications (Tg = 750 K and Rd = 20 µm) and the values of Ts from 300 K to the critical temperature of n-dodecane. The corresponding values of T˜Rd and ρ˜ Rd were used for the analysis of heating and evaporation of n-dodecane droplets in realistic Diesel engine-like conditions [143]. The results are shown in Sect. 6.7. For realistic βm < 1 the values of q˜k were almost the same as those predicted by the model for βm = 1. The plots of j˜k versus ρ˜ Rd for T˜Rd = 1.026, Nint = 20 and βm = 1 and 0.36 are presented in Fig. 6.15. The plot for βm = 1 is identical to the one presented in Fig. 6.14 but shown in a wider range of ρ˜ Rd . The value of βm = 0.36 is close to the one predicted for Ts = 600 K (see Eq. (6.52). As can be seen in Fig. 6.15, the linear dependence of j˜k on ρ˜ Rd is also predicted for βm = 0.36, but the values of j˜k inferred from the calculations based on the model with βm = 0.36 are lower than those predicted by the model with βm = 1. The same procedure of calculating ρ˜ Rd as described earlier for βm = 1 and presented in Fig. 6.14 was followed. As can be seen in Fig. 6.15, this leads to the prediction of smaller ρ˜ Rd for βm = 0.36 than for βm = 1. Thus the reduction of βm leads to the enhancement of the kinetic effects.

6.5 Kinetic Boundary Condition As noticed at the beginning of Sect. 6.4, the model discussed in Sect. 6.2 uses two important simplifications. Firstly, the contribution of inelastic collisions is not considered and secondly, the evaporation coefficient is taken to be equal to 1. The model presented in Sect. 6.4 is focused on relaxing the first assumption. To relax the second assumption the values of the evaporation coefficient need to be obtained. These values would allow us to formulate the kinetic boundary conditions at the droplet surface as (vx > 0), (6.43) f out = βm f e + (1 − βm ) f r where f out is the distribution function of evaporating and reflected molecules leaving the interface from the liquid phase, βm the evaporation coefficient, f e the distribution function of evaporated molecules, f r the distribution function of reflected molecules and vx the velocity component normal to the interface. In the state of equilibrium the

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6 Kinetic Modelling of Droplet Heating and Evaporation

evaporation and condensation coefficients are equal. In what follows, this coefficient is referred to as the evaporation coefficient. One of the most common approaches to estimating the evaporation coefficient is based on the application of the molecular dynamics technique. The background of simulations using this technique is described in Sect. 6.5.1. A new model developed to perform the molecular dynamics simulation of n-dodecane is described in Sect. 6.5.2. In Sect. 6.5.3 the results of applying the latter model to the estimation of the evaporation coefficient are presented. The contributions of quantum mechanical effects to this coefficient are discussed later in Sect. 6.6.

6.5.1 Molecular Dynamics Simulations (Background) A molecular dynamics (MD) model must consider attraction forces between molecules when the distance between them is large, and the repelling forces when molecules are close to one another. The model almost universally used in MD simulations is based on the Lennard-Jones 12-6 potential [20]: # V (Ri j ) = 4εi j

σi j Ri j



12 −

σi j Ri j

6 $ ,

(6.44)

where Ri j is the distance between molecules i and j, εi j and σi j are the minimal energy and the zero-energy separation distance for the pair of molecules, respectively. An alternative form of this potential was considered in [192, 194]. For cross interactions between different types of molecules mixing rules, such as the Lorenz–Berthelot rules, are widely used [4, 32]: εi j =

√

εi ε j ,

σi j = (σi + σ j )/2,

(6.45) (6.46)

where εi and σi refer to specific molecules; the values of these parameters are available in most cases [20, 64, 103]. For example, for nitrogen molecules [32]: εN2 = 95.9k B J and σN2 = 0.371 nm, where k B = 1.38066 × 10−23 J/K is the Boltzmann constant. If these parameters are not available they can be estimated from the properties of the fluid at the critical point, liquid at the normal boiling point or solid at the melting point [20]. At Ri j > Rm , where Rm is the value of Ri j when V = −εi j , V decreases rapidly when Ri j increases. At Ri j = 3σi j the value of |V | is less than 0.01εi j [20]. This justifies ignoring the interaction between the molecules separated by more than about Ri j = 3σi j . The following threshold values were used: Ri j = 3σi j [32], Ri j = 3.5σi j [8], Ri j = 2.5σi j , Ri j = 5σi j and Ri j = 10σi j [187]. In [8] this threshold value of Ri j was considered by presenting the expression for V (Ri j ) as

6.5 Kinetic Boundary Condition

V (Ri j ) =

⎧ ⎨

4εi j

⎩

0

(

365

σi j Ri j

12

−



σi j Ri j

6 )

+ Vshift if if

#

where Vshift = −4εi j

σi j Rcut

12

 −

σi j Rcut

0 < Ri j ≤ Rcut

(6.47)

Ri j > Rcut , 6 $ .

A generalisation of Expression (6.47) leading to the Stockmeyer potential was considered in [77]. Once the values of V (Ri j ) have been obtained, the trajectory of the ith molecule is inferred from Newton’s second law [187]: mi

j=N  d 2 Ri = ∇V (Ri j ), dt 2 j=1

(6.48)

where Ri and m i are the position and mass of the ith molecule, respectively; the summation is performed for all molecules in the sphere with the radius Rcut . Potentially, this approach could allow us to solve the problem of droplet evaporation in a self-consistent way without any additional assumptions. Kinetic theory is not required when this approach is used. This approach, however, has several important limitations. Firstly, the number of molecules which can be calculated is limited by available computing power. To the best of the author’s knowledge, the largest number of molecules simulated before 2000 was 24,000 [39]. Secondly, the molecules calculated using this approach are expected to be spherical. Ideal candidates for this would be monoatomic molecules, such as Ar or Xe [32, 185, 187], although the model has been applied to N2 [32]. The applicability of this approach to simulating evaporation of complex liquids such as Diesel fuel is highly questionable. Even if the latter is approximated by n-dodecane (C12 H26 ), the shape of this molecule cannot be assumed spherical. More advanced approaches to molecular dynamics simulation were described by Tsuruta and Nagayama [184]. Two models for intermolecular potential were used by these authors: the Carravetta-Clementi model [24] and the extended simple point charge model [15]. In both models, the intermolecular interactions were considered as a combination of the short-range pairwise potential of atoms and the long-range Coulombic interaction. The predictions of the extended simple point charge model were demonstrated to be in better agreement with experimental data. It was shown that the translational motion is of primary importance for the evaporation/condensation process, whereas the effects of the rotational motion are small. The evaporation/condensation coefficients, inferred from this approach, were demonstrated to be in good agreement with the values of this coefficient inferred from the transition state theory earlier described in [113] (see Sect. 6.5.3). Among previous molecular dynamics simulations on n-alkane liquid/vapour interfaces, focused on interfacial properties such as phase equilibria, interface tension and

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thermodynamic parameters, those described in [5, 61, 66, 120, 121, 162, 200] can be mentioned. None of these publications, however, focused on the molecular dynamics simulations of the vapour/liquid equilibria of n-dodecane (the closest approximation to Diesel fuel). An alternative approach to molecular dynamics simulation of the droplet evaporation processes could be based on quantum-chemical methods (e.g. [58, 59]). The calculation of βm for evaporating droplets, based on this approach, is not a trivial problem and it has not yet been performed to the best of the author’s knowledge. The development of the evaporation models is closely linked with the development of salvation models. The latter are reviewed in [33]. Also, a method of estimating the difference in the free energies of molecules in the liquid and gaseous states is described in [33]. A more detailed analysis of quantum-chemical approaches to the problem of droplet heating and evaporation is presented in Sect. 6.6.

6.5.2 United Atom Model As mentioned in Sect. 6.5.1, the models discussed in that section are not directly applicable to the analysis of evaporation and condensation of chain-like hydrocarbon molecules, such as those of n-dodecane. Several models have been suggested to describe the dynamics of these molecules, including the OPLS (optimised potential for liquid simulation), originally suggested by Jorgensen et al. [73], and the de Pablo and Toxvaerd models. These models were reviewed by Smit et al. [164], who also suggested an original approach, based on the OPLS. This was claimed to be more accurate. All these approaches are based on the observation that the C–H bond in complex hydrocarbon molecules is much shorter and stronger than the C–C bond, and also stronger than the van der Waals forces between molecules. Thus, the methyl (CH3 ) or methylene (CH2 ) groups can be considered as separate atom-like structures in a relatively simple united atom model (UAM). In this model, these groups can be considered as separate atoms [161]. The underlying physics of these approaches is essentially the same, but they differ by the values of energy parameters ε for CH3 and CH2 , diameters of these groups, and bond bending and torsion potentials. Smit et al. [164] used all the above-mentioned approaches to the analysis of complex hydrocarbons to determine their vapour/liquid coexistence curves. They used the simulation based on the Gibbs-ensemble technique and the configuration-bias Monte Carlo method. All these approaches, based on the UAM, led to almost identical results at standard conditions. They predicted, however, critical temperatures which differed by up to 100 K (the critical temperature is mainly controlled by the ratio εCH3 /εCH2 ). It was concluded that the original approach suggested in [164] gives a good description of the phase behaviour of this curve over a large temperature range. The Toxvaerd model was applied in [161] to the molecular dynamic simulation of n-octane. Other applications of the OPLS approach are described in [74].

6.5 Kinetic Boundary Condition

367

Fig. 6.16 Schematic structure of an n-dodecane molecule (a) and its approximation based on the united atom model (UAM) (b). The bending angles between neighbouring bonds (∼114◦ ) (zigzag structure of the molecule) are considered. Reprinted from [23], Copyright AIP (2011)

Fig. 6.17 Schematic presentation of the bonds in the n-dodecane chain, focusing on four united atoms. Reprinted from [23], Copyright AIP (2011)

This section is focused on an overview of the united atom model (UAM), its application to the investigation of the n-dodecane (C12 H26 ) liquid/vapour interfaces and the description of the simulation method, following [23, 190]. The united atom model applied to the n-dodecane molecule, in which methyl (CH3 ) and methylene (CH2 ) groups are considered as separate united atoms, is schematically presented in Fig. 6.16. The interactions between n-dodecane molecules are modelled based on the optimised potential for liquid simulation (OPLS) considering the Lennard-Jones, bond bending and torsion potentials with the bond length constrained [23, 190]. The bonds in the n-dodecane chain are schematically presented in Fig. 6.17. The non-bonded interactions between atoms were described by the truncated Lennard-Jones (L-J) 12-6 potential defined by Expression (6.44). The following energy parameters of CH2 and CH3 groups (atoms) were used: εCH2 /k B =47 K and εCH3 /k B =114 K, respectively. The energy parameter between CH2 and CH3

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6 Kinetic Modelling of Droplet Heating and Evaporation

√ groups was estimated as εCH3 −CH3 /k B = εCH2 εCH3 /k B =73.2 K (k B is the Boltzmann constant). The diameters of the methylene and methyl groups were assumed to be equal and estimated as σ =3.93×10−10 m. The L-J 12-6 interaction was truncated at 13.8×10−10 m. Note that the above-mentioned Lennard-Jones parameters for alkanes were inferred from accurate full-dimensional intermolecular potentials (see [72] for the details). The interactions within the chains include bond bending and torsion with the bond length constrained at 1.53×10−10 m. The following bond bending potential between the three atoms was used: u bend (θ ) =

1 kθ (θ − θ0 )2 , 2

(6.49)

where kθ /k B =62,500 K/rad2 , and the equilibrium angle is θ0 =114◦ (see Fig. 6.17). The bending coefficients kθ for some other complex molecules are presented in [193]. The following torsion potential between the two atoms with two atoms between them was used: u tors (ϕ) = c0 + 0.5 c1 (1 + cos ϕ) + 0.5 c2 (1 − cos 2ϕ) + 0.5 c3 (1 + cos 3ϕ) , (6.50) where c0 /k B = 0 K, c1 /k B = 355 K, c2 /k B = −68.19 K, c3 /k B = 791.3 K, and ϕ is the dihedral angle with ϕ = 180◦ for the equilibrium state. 720 n-dodecane molecules (8640 CH2 and CH3 groups) were considered in a threedimensional rectangular simulation box of L x × L y × L z = 64.24σ × 16.48σ × 16.48σ , where σ is the rescaled length [190]. This size of the box corresponded to 25.25 nm ×6.48 nm ×6.48 nm. These molecules were initially oriented along the x-axis and placed in the middle of the simulation box. They had zigzag configurations, and the number of molecules chosen was 5 in the x-direction and 12 in both y and z directions. The equations of motion of the atoms were integrated using the Verlet leapfrog method [4]. The bond lengths were constrained by the SHAKE scheme [4]. The timestep 5 fs was used in all simulations. Periodic boundary conditions were used in all directions [190]. The system was relaxed with a constraint of fixed homogeneous and isotropic temperature. The molecules started to relocate within the liquid phase and evaporate. Typically, the system needed 15,000 ps to reach an equilibrium state. Finally, the liquid film was sandwiched between the layers of vapour. Then data were sampled for another 5,000 ps. The positions of the two liquid/vapour interfaces were found by density profiles. The interface parameters, such as density and evaporation/condensation coefficient, were estimated by averaging them over these 5,000 ps. A molecular dynamics investigation on transport properties and structure at the liquid/vapour interfaces of alkanes, using essentially the same model as presented above, was described in [30]. The focus of the analysis by these authors was on decane, tetracosane and hexatriacontane. The authors of [53] applied the UAM in

6.5 Kinetic Boundary Condition

369

their molecular dynamics (MD) simulations of a mixture of isooctane, n-dodecane and n-hexadecane. A generalised version of the UAM described above, considering methyl (CH3 ), methylene (CH2 ) and hydroxyl (OH) groups, was used in [188] for MD simulation of sub/super-critical evaporation of n-butanol/ n-heptane blends. The results of the application of the MD simulation to the analysis of difluoromethane (CH2 F2 ) nanodroplet evaporation are presented in [189]. In [107] MD simulations were used for the analysis of droplet break-up.

6.5.3 Evaporation Coefficient The simplest approach to the estimation of the evaporation/condensation coefficient is based on the transition state theory [113]. In this approach, condensation/evaporation process at the liquid/vapour interface is described as a kind of chemical reaction and the general theory of rate processes [51] is used. As a result, the following expression for this coefficient was obtained [113]: # βm = 1 −



l

V Vg

1/3 $

⎤

⎡ 1 ⎢ 1 exp ⎣−  1/3 2 Vl Vg

−1

⎥ ⎦,

(6.51)

where V l and V g are liquid and gas specific volumes, respectively. Since V l  V g for n-dodecane, except when its temperature is close to the critical temperature, Expression (6.51) predicts that βm is close to 1. This agrees with the assumption that βm = 1 used in Sect. 6.2. A more rigorous justification of this assumption was made in [23, 190], using the molecular dynamics (MD) simulations. In what follows, some key findings described in the latter paper are summarised. Using the UAM and the numerical algorithm presented in Sect. 6.5.2, the evaporation/condensation coefficient was found using Expression (6.5). Since evaporation and condensation coefficients were assumed to be equal we focused on the evaporation coefficients. The evaporation coefficient of n-dodecane was estimated for liquid temperatures Tl = 400 K, 450 K, 500 K and 550 K [190]. The results of calculations of βm versus the reduced temperature, presented in [190], are reproduced in Fig. 6.18 by open circles. As can be seen in this figure, βm decreases from about 0.93 at Tl = 400 K to about 0.45 at Tl = 550 K. The evaporation coefficients of n-dodecane, obtained by the authors of [23] using MD simulation in the system with 400 molecules, are also presented in Fig. 6.18 (filled circles). The values of this coefficient inferred from the transition state theory (Expression 6.51) are shown by the dotted curve. These values were found for the densities of the liquid and vapour phases inferred from MD simulations performed by the authors of [190]. As can be seen in this figure, there is a general agreement between the results presented by the authors of [190], those inferred from the transition state theory and the results of MD simulations

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6 Kinetic Modelling of Droplet Heating and Evaporation

Fig. 6.18 The values of the evaporation coefficient versus reduced temperature as predicted by the molecular dynamics (MD) simulation for n-dodecane performed by the authors of [190] (Present MD data), inferred from published data for n-dodecane (Cao et al. [23], Mizuguchi et al. [110]), argon (Tsuruta et al. [185], Ishiyama et al. [68]), water (Ishiyama et al. [69], Nagayama et al. [184]) and methanol (Ishiyama et al. [69]), and inferred the transition state theory (Expression (6.51)). Reproduced with modifications from Fig. 10 of [190]. Copyright AIP (2011)

for n-dodecane presented in [110], although the reasons for the noticeable deviation between these results are still to be studied. One of the reasons could be attributed to the limited number of molecules used in calculations, the results of which are described in [190]. Also, the united atom model (UAM), used in the calculations performed by the authors of [190], is expected to be not reliable at reduced temperatures close to 1. The evaporation coefficients obtained by other authors, using MD simulations, are also shown in Fig. 6.18 (filled [185] and open [68] squares for argon, filled [69] and open [184] triangles for water and filled diamonds for methanol [69]). In all the cases the evaporation coefficients decreased when the liquid temperature increased. This is consistent with the interface thickness predicted in [190]. The thicker interfaces at higher temperatures are expected to lead to smaller evaporation/condensation coefficients. The evaporation coefficient for such complex polyatomic molecules as n-dodecane would be expected to be lower than for simpler molecules due to the constraint imposed by the rotational motion of molecules in the liquid phase [49]. In MD simulations, the results of which are presented in Fig. 6.18, however it was found that the rotational energy has no visible effect on this coefficient. Its value is mainly controlled by the translational energy in agreement with the results presented in [23, 184]. On the other hand, the transition state theory predicts that this coefficient is

6.5 Kinetic Boundary Condition

371

close to unity at low temperatures when (V g /V l )(1/3) is large, and decreases with increasing temperature due to the decrease in the ratio (V g /V l )(1/3) . These trends are consistent with those inferred from MD simulations the results of which are presented in Fig. 6.18. The values of βm for n-dodecane predicted by MD calculations by the authors of [190], shown in Fig. 6.18, were approximated as [190]: βm (Ts ) = 7 × 10−6 Ts2 − 9.8 × 10−3 Ts + 3.7215,

(6.52)

where the droplet surface temperature Ts is in K. The values βm predicted by Expression (6.52) can be substituted into Expression (6.43) to obtain the value of f out [143]. This is performed later in this chapter. The authors of [71] used the united atom model and all-atom model (considering the contributions of all atoms), and estimated the values of the evaporation/condensation coefficient for octane. The values of this coefficient at room temperature were shown to be close to 1. This agrees with the results presented in Fig. 6.18. Note that another boundary condition implicitly used in the kinetic modelling refers to the distribution of molecules leaving the droplet surface. This distribution is commonly assumed to be close to Maxwellian. The validity of this assumption was investigated in [191]. It was demonstrated by these authors that the distribution functions of evaporated and reflected molecules for the velocity component normal to the surface can deviate considerably from the Maxwellian. At the same time, the distribution function for all molecules leaving this surface (evaporated and reflected) was shown to be close to Maxwellian. This problem of the molecular distribution functions infers from the fact that the processes at the surface of the droplet and in the kinetic region were considered separately, using different approaches (molecular dynamics and kinetic) and these processes were linked only via the evaporation/condensation coefficient. New preliminary ideas of how to overcome this problem are described in [18]. In this model, the traditional link between the processes in the liquid and the kinetic region via the evaporation/condensation coefficient was abandoned. Instead, a new method of solving transport problems in a two-phase (condensate-vapour) system accounting for specific features of the atom-interface interactions was developed. As a result, the distribution function for the evaporated and reflected molecules was obtained. The traditional approach to modelling the evaporation process based on the molecular dynamics simulations was replaced with a reasonable approximation of the dependence of the potential energy of surface molecules on temperature. Finding this approximation is far from trivial. Once this approximation was found, however, then the results appeared to be very close to those inferred from the conventional molecular dynamics (MD) simulations, and experimental data. In [18] this approximation was found for Ar, N2 and CH4 molecules. The model described above was further developed in [156]. In that paper a new approach to coupled solution of liquid and vapour kinetic equations was suggested

372

6 Kinetic Modelling of Droplet Heating and Evaporation

and tested. It was shown that this approach can be used as a single algorithm for the analysis of liquid and vapour phases without focusing on the liquid/vapour interface (in contrast to the conventional molecular-kinetic approaches).

6.6 Quantum-Chemical Models One of the key limitations of the united atom model, on which Formula (6.52) is based, is that in this model the interaction between individual molecules is described using the force field (FF) methods. These methods simplify both inter- and intramolecular interactions by ignoring electrons per se. The applicability of this approach is not obvious, as the dynamics of molecules in the vicinity of droplet surfaces are essentially quantum mechanical processes. The molecular quantum mechanical (quantum-chemical (QC)) models focused on the analysis of the processes at and in the vicinity of Diesel fuel droplet surfaces are described in [55–57, 114, 145]. Note that papers [55–57, 114] primarily address the quantum chemistry community. The importance of the results presented in these papers might have been overlooked by a wider engineering audience. The main objective of [145] was to summarise the key results presented in [55–57, 114] in a format that can be easily understood by the engineering community interested in modelling the droplet heating and evaporation. The key results presented in [145] are reproduced in this section. Further references can be found in [145].

6.6.1 Brief Overview of Quantum-Chemical Methods Although the solution to the Schrödinger equation for the wave function ψ in some simple cases (e.g. isolated hydrogen atoms) is presented in well-known quantum mechanics textbooks, its general solution when many atoms need to be studied simultaneously is still a challenge for quantum mechanics modelling. One of the most widely used simplified approaches to solving this equation is known as the Hartree–Fock (HF) method. There are two strategies for the application of the HF method in practical calculations. In the semi-empirical methods, the integrals used in the HF method are estimated based on experimental data or using a series of rules which allow one to set certain integrals to zero. In the ab initio approach, an attempt is made to calculate all these integrals. Although the HF method is widely used in practical calculations, this method is still an approximate one and requires considerable computational effort. This led to the development of alternative approaches to the calculation of electronic systems, including the density functional theory (DFT). The latter approach focuses on the electron density (ρe ) rather than on the wave function ψe . It is assumed that the energy of a molecule is a function of electron density. Since the electron density is a function of position ρe (r), this energy appears to be a function of a function, that

6.6 Quantum-Chemical Models

373

is functional of density. This approach appears to be not only much less demanding computationally compared with the HF method, but in some cases it can lead to more accurate results compared with the HF method. Several approximations of the energy functional in the DFT, incorporating parts of the HF theory, have been suggested. One such approach is known as B3LYP (Becke, 3-parameter, Lee-Yang-Parr). Various semi-empirical quantum chemistry methods are important for dealing with large molecules where the full HF method without the approximations (ab initio approach) and DFT are too expensive. In these methods a range of fitting parameters is typically used to produce the results that best agree with experimental data or with ab initio results (e.g. PM7 method). The parameters in the PM7 method were calibrated to obtain results consistent with experimental and ab initio data for more than 9000 compounds. The accuracy of the PM7 method is close to that of the ab initio and DFT methods used with the 6-31G(d) basis set. The main differences between the classical MM/MD, semi-empirical PM7 ab initio and DFT methods are due to the way in which the contributions of electrons are considered. The contribution of all electrons is considered in ab initio and DFT with self-consistent field (SCF); only valence electrons are considered in semi-empirical quantum-chemical methods (QCMs) with SCF, and no electrons per se are considered in classical MM/MD methods without SCF. A new continuum solvation model using the quantum mechanical charge density of a solute molecule interacting with a continuum description of a solvent was named the SMD (D stands for density which refers to the full solute electron density) model. The term continuum shows that the solvent is represented as a dielectric medium with surface tension at the solute/solvent boundary. The SMD model was parameterised with a training set of almost 3000 solvation data. For modelling of the transient processes, the dynamic reaction coordinates (DRC) method has been commonly used. The main concept of this method is the dynamic reaction coordinate which is the path followed by all the atoms in a system assuming that the energy is conserved. In contrast to conventional molecular dynamic (MD) approaches, the contributions of the processes at the electronic level are considered. The models summarised above have been implemented in several programmes, including Gaussian 09, WinGAMESS 2013 R1 and MOPAC2012. Some results of the applications of these models, using these three programmes, are described in the following sections.

6.6.2 Evaporation Rate To the best of the author’s knowledge, the first attempt to perform a quantum chemical study of the processes during the evaporation of Diesel fuel droplets was described in [55]. The composition of Diesel fuel used in their analysis is presented in Fig. 4.30. The analysis was focused on the evaporation from the surface of a Diesel fuel droplet into a vacuum, described by the evaporation rate:

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6 Kinetic Modelling of Droplet Heating and Evaporation

γ =

    n ev (t) 1 , ln t n0

(6.53)

where n ev (t) is the time-dependent number of molecules leaving the droplet, n 0 is the initial number of molecules and t is the duration of the process. Considering the evaporation rate of the ith molecule from a cluster (or nanodroplet) i + j (γi(i+ j) ), the value of γi(i+ j) was found as [96, 118] γi(i+ j) = bi j

  ΔG i+ j − ΔG i − ΔG j p , exp k B T n0 kB T

(6.54)

where ΔG i+ j , ΔG i and ΔG j are the Gibbs free energies of formation of the molecules (clusters/nanodroplets) from monomers (molecules) at the reference pressure p. An additional assumption that clusters or nanodroplets are so small that their interaction with molecules can be described by the kinetic gas theory was made for the estimation of bi j . Although these assumptions are restrictive for most engineering applications, they allowed the authors of [55] to gain insight into the physics of some of the processes at the surface of the droplets. The SMD/HF and SMD/DFT models with the same 6-31G(d,p) basis set were used to estimate changes in the Gibbs free energy during the transfer of a molecule from a liquid medium into a gas phase. Such solvents as n-dodecane, tetraline, benzene and isopropyltoluene were used to analyse the effects of surroundings on the evaporation rate of the components of Diesel fuel: normal, iso and cyclic alkanes, 1–3 ring aromatics, tetralines and indanes (in the C12 –C20 range). Note that compounds C14 –C16 are the main contributors in Diesel fuel under consideration. An increase in the molecular size of alkanes from n-octane to n-heptacosane or in the aromaticity of compounds resulted in a decrease in the evaporation rate. In contrast to [55], the analysis of [56] focused only on alkanes as the main components of Diesel fuels, and particularly on n-dodecane, the component widely used as a representative of this fuel. It was shown that the evaporation rate decreases with increasing cluster/nanodroplet diameter and decreasing temperature. The relative number of evaporated molecules, however, did not depend on cluster/nanodroplet diameters, and increased with increasing temperature. At certain temperatures, the clusters/nanodroplets were expected to fully evaporate. The relative number of residual molecules in clusters/nanodroplets for n-alkanes in the range C8 –C27 was shown to increase with temperature and with the carbon numbers in the molecules. Thus the evaporation process of a mixture of n-alkanes was expected to lead to an increased concentration of heavy n-alkanes in droplets, which is consistent with the results described in Chap. 4. Steady-state evaporation from a liquid surface into a vacuum was also modelled in [25, 105] by non-equilibrium molecular dynamics simulations of a Lennard-Jones fluid, without considering quantum-chemical effects.

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6.6.3 Interaction between Molecules and Clusters/Nanodroplets The analysis presented in Sect. 6.6.2 referred to the integral characteristics of the processes at the surface of the droplets. In this section, the details of the analysis of the collision processes between n-dodecane molecules and clusters/nanodroplets, using the dynamic reaction coordinate (DRC) method, are presented, following [56] and [145]. These processes are expected to lead to scattering or sticking of the molecules. In the DRC calculations, the total kinetic energy includes the kinetic energy of random thermal bond vibrations and rotations and the kinetic energy of the translational motion of the whole molecules. In [56], the DRC method was applied to study the dependence of sticking/scattering of n-dodecane molecules on their angles of attack, temperature and cluster/nanodroplet size. The DRC calculations were performed for molecules interacting with a cluster (7 molecules) or a nanodroplet (64 or 128 molecules) of n-dodecane molecules. The results are presented in Fig. 6.19. As follows from Fig. 6.19, at large angles of attack, absorption of a molecule by a cluster or nanodrop of relatively small size (d = 2–7 nm) can be clearly seen if the kinetic energy is low and the attacking molecule is not oriented exactly towards one of the surface molecules. At Θ ≈ 1◦ an almost perfectly elastic collision can be seen if the molecule has relatively high kinetic energy (∼10 kJ/mol or larger) and is oriented directly towards one of the surface molecules. In the DRC calculations the

Fig. 6.19 Interaction of an n-dodecane molecule (temperature ∼ 1100 K) with a cluster of seven n-dodecane molecules (initial temperature 473 K; it increases due to the interaction with a hot molecule) at the angles of attack Θ ≈ a 1◦ , b 60◦ and c 90◦ . The DFT B3LYP was used in calculations. Reprinted from [145], Copyright Elsevier (2016)

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kinetic energy of the molecules in the clusters or nanodroplets was low and thermal vibrations and bond rotations corresponded to 300–400 K. The kinetic energy of the attacking molecule was high (its effective temperature was in the range of 500– 1200 K). Further analyses, like those the results of which are shown in Fig. 6.19, allowed the authors of [56] to conclude that the probability of the attacking molecule sticking to a droplet is maximal if the molecular plane is parallel or almost parallel to the droplet surface. This corresponds to multi-point interactions of relatively long n-dodecane molecules with the droplet surface. If the kinetic energy of the attacking molecules is greater than that of boiling temperature, then they are expected to scatter and be removed from the cluster/nanodroplet surface. Molecule/nanodroplet interaction results (sticking or scattering) depend on the kinetic energy and orientations of the attacking and surface molecules. It was demonstrated that the mechanisms of evaporation of microdroplets and nanodroplets are likely to involve rather different processes. In the case of microdroplets, individual carbon molecules are evaporated from their surfaces, while nanodroplets are expected to disintegrate into clusters and individual molecules. The decrease in the likelihood of evaporation/condensation with temperature, predicted by the above-mentioned analysis, agrees with the prediction of the classical model based on the MD simulations of n-dodecane molecules [190]. At the same time, the analysis of this section does not allow us to predict the evaporation coefficient, as was done in [190] using the classical FF analysis. The analysis of each collision process, presented in Fig. 6.19, required a powerful PC. To study these processes using DFT/DRC methods for larger systems with dozens or hundreds of molecules, a supercomputer would be needed. The latter was used for some calculations to study the conformerisation effects for n-dodecane (95 conformers). To quantify the values of the evaporation/condensation coefficient, using the above-mentioned analysis, one would need to repeat these calculations for a wide range of angles of attack, orientation of molecules and energies for various conformers and various conditions of clusters and nanodroplets (the effects of the size of the clusters/nanodroplets would need to be investigated as well). Since this does not look feasible, an alternative approach to calculating the above-mentioned evaporation coefficient, considering quantum chemical effects, needs to be developed. Such an approach is described in Sect. 6.6.4, following [57] and [145].

6.6.4 Estimation of the Evaporation Coefficient The analysis of [57] was based on the transition state theory (TST) and quantumchemical DFT methods. These were applied to several ensembles of n-dodecane conformers. There was a similarity between the approach used in [57] and the one used in the classical approach (see Sect. 6.5.3). In contrast to the classical approach, however, in the analysis of [57] the TST was based on a QC DFT approach considering the conformerisation of n-dodecane molecules (considered as a representative of

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Fig. 6.20 Comparison of the values of the evaporation coefficient β = βe , predicted by MD, FF (symbols 1–4, curves 5–8) and Formula (6.55) (curve 9), versus normalised temperature (T /Tc , where Tc is the critical temperature). Symbols (1–4) refer to the models for structureless LJ fluids with various input parameters [105, 110], curves 5 and 7 show the results obtained using the united atom model (UAM) described in [23, 190], respectively, curve 6 shows the results of calculations using the TST model reproduced from [23], and curve 8 is based on the results of calculations using the model described by Mizuguchi et al. [110]. QC calculations were performed using DFT ωB97XD/cc-pVTZ and SMD/ωB97X-D/cc-pVTZ. Reprinted from [145], Copyright Elsevier (2016)

Diesel fuel). It was demonstrated that the most accurate expression for the evaporation coefficient is the one averaged over the states of various conformers transferred between two phases [57]: ⎧ ⎧ ⎤−1 ⎫ ⎡# # + $1/3 ⎫ + $1/3 * * ⎪ ⎪ ⎨ ⎬ ⎬ ⎨ ΔG g→l ΔG g→l ρg ρg ⎦ ⎣ βV  = 1 − exp exp −1 , exp −0.5 ⎩ ⎪ ⎭ ⎪ ρl Ru T ρl Ru T ⎭ ⎩

(6.55) where Ru is the universal gas constant, ρg(l) the gas (liquid) density, ΔG g→l the change in the Gibbs free energy during the condensation process. Subscript V shows that the expression for βV explicitly depends on the specific volumes,  shows averaging over the states of various conformers transferred between two phases. The process under consideration was assumed to be quasi-steady-state and the condensation coefficient was assumed to be equal to the evaporation coefficient. The effects of both the conformerisation and cross-conformerisation (changes in conformer state during transfer into another phase) of n-dodecane molecules (CDM effects), which contributed to the Gibbs free energies of evaporation and solvation, were considered. Based on the changes in the Gibbs free energy, 95 stable conformers were chosen.

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A comparison between the results of calculations of βe = βV using Expression (6.55) and those obtained previously is presented in Fig. 6.20. As follows from this figure, considering the QC effects leads to marginal modifications of the evaporation/condensation coefficient, except at temperatures close to the critical temperature (where these modifications are significant). Thus, although the analysis of the QC effects considers many new effects ignored in the conventional FF approach, the contribution of these effects to the values of the evaporation/condensation coefficient turned out to be marginal, unless the processes at temperatures close to the critical temperature were analysed.

6.7 Results of the Kinetic Calculations The models presented in the previous sections were applied to the analysis of droplet heating and evaporation by the authors of [143]. This section is focused on summarising the most important results presented in that paper.

6.7.1 Results for βm = 1 The algorithm presented in Sect. 6.4 was used for the analysis of the heating and evaporation of Diesel fuel droplets in gas at temperatures 750 K and 1000 K. The initial droplet temperature and gas pressure in all cases were taken equal to 300 K and 30 bar, respectively. The initial droplet radii 5 µm were used. Droplets were stationary, but the effect of swelling was considered. The infinite thermal conductivity (ITC) and effective thermal conductivity (ETC) models were used for the liquid phase. The Abramzon and Sirignano model was used for the gas phase (see Sect. 3.2.2). Two versions of the kinetic model, considering and not considering the effects of inelastic collisions, were used. The evaporation coefficient was considered equal to one in both kinetic models. The radii and surface temperatures of heated and evaporating droplets introduced into gas at temperature 750 K are presented in Figs. 6.21 and 6.22, respectively. The results of calculations using six models are compared. As can be seen in Fig. 6.22, at the initial stage of droplet heating and evaporation, the kinetic effects on surface temperature are very small, but the difference in surface temperatures, predicted by the hydrodynamic models considering and not considering the contribution of liquid finite thermal conductivity, can be clearly seen. This is consistent with the results presented in [138]. Initially the droplet surface temperature, predicted by the ETC model is higher than the one predicted by the ITC model, as expected. The larger droplet surface temperature predicted by the ETC model leads to the reduction of the heat flux supplied to the droplet. This eventually leads to the reduction of the droplet surface temperature predicted by the ETC model and it becomes smaller than the one predicted by the ITC model. This happens at the time instant close to 0.4 ms. Smaller

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Fig. 6.21 Plots of Rd versus time t for an n-dodecane droplet, predicted by the hydrodynamic ITC model (curve 1), the kinetic ITC model, ignoring the effects of inelastic collisions (curve 2), the kinetic ITC model, considering the effects of inelastic collisions (curve 3), the hydrodynamic ETC model (curve 4), the kinetic ETC model, ignoring the effects of inelastic collisions (curve 5), and the kinetic ETC model, considering the effects of inelastic collisions (curve 6) (a). Zoomed part of a showing the plots at the final stage of the evaporation process (b). There is no exact match between Fig. b and the area A shown in Fig. a. Tg = 750 K, Td0 = 300 K, and Rd0 = 5 µm. Reprinted from [143], Copyright Elsevier (2013)

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Fig. 6.22 The same as Fig. 6.21 but for Ts . Reprinted from [143], Copyright Elsevier (2013)

temperatures, predicted by the ETC model at this stage, lead to a slower evaporation rate and longer evaporation time of droplets compared with those predicted by the ITC model, as shown in Fig. 6.21. As can be seen in Fig. 6.21, the kinetic models, considering inelastic collisions, predict longer evaporation times compared with those predicted by the hydrodynamic ETC and ITC models. This agrees with the predictions of the model not considering the effects of inelastic collisions, as discussed in Sects. 6.2 and 6.3 (cf. [134]). The kinetic model, considering inelastic collisions, predicts longer evaporation times compared with the model ignoring them, in agreement with the prediction

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of Fig. 6.14. Similar enhancement of the kinetic effects due to the contribution of inelastic collisions can be seen from the temperature curves presented in Fig. 6.22. The curves similar to those shown in Figs. 6.21 and 6.22 but for Tg = 1000 K were presented in [143]. The properties of those curves had many similarities with those shown in Figs. 6.21 and 6.22. As in the case of gas temperature equal to 750 K, for gas temperature equal to 1000 K at the initial stage of droplet heating and evaporation, the kinetic effects on surface temperature were negligible, but the difference in surface temperatures, predicted by the hydrodynamic models considering and not considering the effects of liquid finite thermal conductivity, was clearly seen (see [143] for the details). Initially, the droplet surface temperature predicted by the ETC model was higher than the one predicted by the ITC model as expected. At times longer than about 0.25 ms the surface temperature predicted by the ETC model became lower than the one predicted by the ITC model. In contrast to the case shown in Fig. 6.21, in the case of gas temperature equal to 1000 K the ETC model predicts a shorter evaporation time than the ITC model. This is due to the net effect of the increase in the surface temperature predicted by the ETC model before 0.25 ms and its decrease at times after 0.25 ms. Similar to the case presented in Figs. 6.21 and 6.22, for Tg = 1000 K the kinetic models predicted longer evaporation times and higher temperatures at the final stage of droplet evaporation, compared with the hydrodynamic ETC and ITC models. Also, similar to the case presented in Figs. 6.21 and 6.22, the kinetic model, considering inelastic collisions, predicted longer evaporation times and higher temperatures at the final stage of droplet evaporation, compared with the model ignoring them. These kinetic effects were more visible for gas temperature equal to 1000 K than in the case presented in Figs. 6.21 and 6.22.

6.7.2 Results for βm < 1 This section is focused on investigating predictions of the kinetic model when β = 1. The analysis is based on βm approximated by Expression (6.52). For Ts = 600 K, this expression gives βm = 0.36. This value of βm was used for the plot shown in Fig. 6.15. The results of calculations of the radii and surface temperatures of droplets with initial radii equal to 5 µm, introduced into gas at temperature 1500 K, using the ITC model and kinetic models with βm = 1 and βm = 0.36 are presented in Fig. 6.23. As can be seen in this figure, the effect of βm < 1 is clearly visible. This effect leads to an increase in droplet evaporation time, thus enhancing the kinetic corrections to the prediction of the hydrodynamic model. Note that only the results for the ITC model are shown in Fig. 6.23. The kinetic effects for the ETC model were shown to be similar to those for the ITC model. For gas temperature equal to 1000 K, the effect of βm < 1 on droplet radius and temperature was shown to be small and can be safely ignored in most applications.

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Fig. 6.23 Plots of Rd and Ts versus time t for an n-dodecane droplet, predicted by the hydrodynamic ITC model (curve 1), the kinetic ITC model, ignoring the effects of inelastic collisions and non-unity of β (curve 2), the kinetic ITC model, considering the effects of inelastic collisions but ignoring the non-unity of β (curve 3) and the kinetic ITC model, considering the effects of inelastic collisions and the non-unity of β, which is obtained using Expression (6.52) (curve 4) (a). Zoomed part of a showing the values of Rd at the final stage of the evaporation process (b). There is no exact match between Fig. b and the area A shown in Fig. a. Tg = 1500 K, Td0 = 300 K and Rd0 = 5 µm. Reprinted from [143], Copyright Elsevier (2013)

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For gas temperature equal to 750 K this effect was shown to be even smaller than in the case of gas temperature equal to 1000 K. Thus, the results of the analysis for βm = 1 presented in Sect. 6.4 are applicable.

6.8 Kinetic Modelling in the Presence of Three Components One of the limitations of the solution algorithm discussed in Sect. 6.4.2 and applied in Sect. 6.7 to the analysis of Diesel fuel droplets is that it is based on the assumption that Diesel fuel can be approximated by n-dodecane. A detailed analysis of Diesel fuel composition, presented in Chap. 4, however, demonstrated that it includes about 100 hydrocarbon components. Two main groups of these components can be considered: alkanes and aromatics. The assumption that n-dodecane is a good approximation for alkanes has been widely used (see [55]). Aromatics, on the other hand, can be approximated by p-dipropylbenzene [55]. Thus, a more accurate approximation of Diesel fuel, compared with that by n-dodecane, could be its approximation by a mixture of n-dodecane and p-dipropylbenzene. Mass fractions of n-dodecane in this mixture can be in the range of 0.8–0.7 [55]. Chemical formulae, molar masses and molecular diameters of n-dodecane, pdipropylbenzene and nitrogen (approximation of air) are presented in Table 6.9. The accuracy of prediction of kinetic modelling of Diesel fuel droplet heating and evaporation is expected to improve if the contributions of both n-dodecane and p-dipropylbenzene are considered. The first step in this modelling is the development of a kinetic algorithm for the solution of the system of three Boltzmann equations for n-dodecane, p-dipropylbenzene and nitrogen (air), considering the effects of inelastic collisions between molecules. The preliminary results of the testing of such an algorithm are described in Sect. 6.8.1, following [158].

6.8.1 Preliminary Testing of the Numerical Code The molecular distribution functions of air f a ≡ f a (r, t, v) (approximated by nitrogen), n-dodecane f n ≡ f n (r, t, v) and p-dipropylbenzene f p ≡ f p (r, t, v) were inferred from the solution to the following Boltzmann equations:

Table 6.9 Chemical formulae, molar masses and molecular diameters of n-dodecane, pdipropylbenzene and nitrogen Component Chemical formula Molar mass Molecular diameter (kg/kmol) (Å) n-dodecane p-dipropylbenzene nitrogen

C12 H26 C12 H18 N2

170.3 162.27 28.97

7.12 6.73 3.617

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Fig. 6.24 A scheme of the setup for the preliminary testing of the numerical code for three mixtures. The distance between the walls L is taken equal to ten mean free molecular paths of n-dodecane molecules  at the temperature of the second wall (Tw2 ). Reprinted from [158], Copyright Elsevier (2014)

∂ fa ∂t ∂ fn ∂t ∂ fp ∂t

⎫ + va ∂∂rfa = Jaa + Jan + Jap ⎪ ⎪ ⎬ ∂ fn + vn ∂r = Jna + Jnn + Jnp , ⎪ ⎪ ∂f ⎭ + v p ∂rp = J pa + J pn + J pp

(6.56)

where Jαβ (α = a, n, p; β = a, n, p) are collision integrals, considering the contribution of the collisions between molecules. Triple collisions were not considered. Jαβ were calculated considering the contribution of internal degrees of freedom (inelastic collisions) as discussed in Sect. 6.4. The numerical code solving System (6.56) was tested for mixtures confined by two parallel walls kept at constant normalised temperatures Tw1 = Tw2 = 1 (no heat transfer between the walls) as shown in Fig. 6.24. Both temperatures are normalised by the temperature at the second wall. The distance between the walls L was taken equal to ten mean free paths for n-dodecane n at the wall temperature Tw2 . Space between the walls was filled with: (1) a three-component mixture of n-dodecane, p-dipropylbenzene and nitrogen; (2) a two-component mixture of n-dodecane and nitrogen; (3) pure n-dodecane or p-dipropylbenzene. Initially, normalised molar fractions of all components were considered equal to 1. It was assumed that the distribution functions of n-dodecane and p-dipropylbenzene at both walls are half-Maxwellian with normalised densities n s1 = 2 and n s2 = 1, approximating the evaporation of the components. n-dodecane and p-dipropylbenzene molecules reaching the walls were taken out of the domain (condensation process), while the flux of incoming nitrogen molecules was assumed to be equal to that of molecules reflected from the walls. All densities were normalised by saturated density of n-dodecane vapour at temperature Tw2 . The space grid 0.5 n and 12 × 12 × 12 velocity grid points were used in the numerical analysis. The number densities of n-dodecane for three- and two-component mixtures and pure n-dodecane versus the normalised distance from the first plate, predicted by the code, are presented in Fig. 6.25. Two cases were studied: the contribution of inter-

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Fig. 6.25 Plots of n-dodecane number density n n versus the normalised distance from the first plate x predicted by the model, considering the contributions of: three components, n-dodecane, pdipropylbenzene and nitrogen (thick solid curves), two components: n-dodecane and nitrogen (thin solid curves), and a single component (n-dodecane) (dashed curves). The distance x is normalised by n . The contribution of internal degrees of freedom is either ignored (a) or considered with Nint = 20 for n-dodecane and p-dipropylbenzene and Nint = 2 for nitrogen (b). Reprinted from [158], Copyright Elsevier (2014)

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nal degrees of freedom was ignored (a); the numbers of these degrees of freedom for n-dodecane, p-dipropylbenzene and nitrogen were taken equal to 20, 20 and 2, respectively (b). As follows from the results of investigations presented in Sect. 6.4, only the first 20 internal degrees of freedom of complex molecules like n-dodecane and p-dipropylbenzene affect the predictions of the kinetic calculations (the actual number of internal degrees of freedom of n-dodecane and p-dipropylbenzene, considering translational, rotational and vibrational movements of atoms, is well above 20). As can be seen in Fig. 6.25, the density distributions for three- and two-component mixtures are very close. They coincide within the accuracy of plotting when the contribution of internal degrees of freedom is considered (see Fig. 6.25b). Both these distributions differ considerably from the distribution for the one-component substance. In all these cases the expected concentration jumps near the walls are clearly seen. These jumps are noticeably larger for the one-component substance compared with three- or two-component mixtures. Their physical nature has been described by various authors, starting with a pioneering book [48]. In [133] it was demonstrated that the introduction of these jumps into conventional CFD codes (e.g. FLUENT) enables them to be used for the analysis of the processes in rarefied gases. As follows from Fig. 6.25, the internal degrees of freedom have a limited effect on number densities of n-dodecane. They contribute to an increase in the number density jumps for three- and two-component mixtures and a slight decrease in these jumps for a single component. The number densities of p-dipropylbenzene for a three-component mixture and pure p-dipropylbenzene versus the normalised distance from the first plate, predicted by the code, are presented in Fig. 6.26. Comparing Figs. 6.25 and 6.26 it can be seen that the distributions of n-dodecane and p-dipropylbenzene are rather similar which is attributed to the closeness of masses and sizes of the molecules of these substances as shown in Table 6.9. The number densities of nitrogen for three- and two-component mixtures versus the normalised distance from the first plate, predicted by the code, are presented in Fig. 6.27. Comparing Fig. 6.27 with Figs. 6.25 and 6.26 it can be seen that in contrast to n-dodecane and p-dipropylbenzene, the number density of nitrogen increases with x. Thus, nitrogen molecules move from regions of high concentration of n-dodecane and p-dipropylbenzene to regions of low concentration of these molecules. This is consistent with the corresponding thermodynamics equations for the mixtures. In contrast to the case presented in Fig. 6.25, the distribution of nitrogen is visibly different for three- and two-component mixtures. The number density jump at the walls is not relevant for nitrogen, since this number density at the walls is not specified. As in the cases presented in Figs. 6.25 and 6.26, the internal degrees of freedom produce only minor effects on the number density of nitrogen. The number fluxes of n-dodecane and p-dipropylbenzene for the three-component mixture and pure n-dodecane and p-dipropylbenzene versus the number of internal degrees of freedom Nint are presented in Fig. 6.28. Nint for n-dodecane and pdipropylbenzene are assumed to be equal. Nint for nitrogen was taken to be equal to 2, except when the internal degrees of freedom for n-dodecane and p-dipropylbenzene

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Fig. 6.26 The same as Fig. 6.25 but for p-dipropylbenzene, considering the contributions of three components: n-dodecane, p-dipropylbenzene and nitrogen (thick solid curves) and a single component (p-dipropylbenzene) (dashed curves). Reprinted from [158], Copyright Elsevier (2014)

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Fig. 6.27 The same as Figs. 6.25 and 6.26 but for nitrogen, considering the contributions of three components, n-dodecane, p-dipropylbenzene and nitrogen (thick solid curves); and two components, n-dodecane and nitrogen (thin solid curves). Reprinted from [158], Copyright Elsevier (2014)

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Fig. 6.28 Plots of number fluxes of n-dodecane and p-dipropylbenzene versus the number of internal degrees of freedom Nint , predicted by the model, considering the contributions of three components, n-dodecane, p-dipropylbenzene and nitrogen and those for pure n-dodecane and pdipropylbenzene. Reprinted from [158], Copyright Elsevier (2014)

were not considered. In the latter case Nint for nitrogen was assumed to be equal to 0. As can be seen from this figure, although the number densities of n-dodecane and p-dipropylbenzene are weak functions of Nint , the dependence of number fluxes of these components on this number is clearly visible, at least at Nint ≤ 20. At larger Nint these fluxes do not depend on Nint , in agreement with our earlier results. Mass fluxes of n-dodecane and p-dipropylbenzene are close for the three-component mixture, but the difference between them can be clearly seen in the case of single components. Mass fluxes of n-dodecane and p-dipropylbenzene are much larger for the pure n-dodecane and p-dipropylbenzene than for the three-component mixture as expected. Functionality testing, as shown above, was performed for pure heat transfer and combined heat and mass transfer between the walls [158]. The results obtained in this case were similar to the ones presented above for the case of pure mass transfer. For pure heat transfer between the walls the distributions of temperatures of various components between the walls turned out to be qualitatively similar. It was shown that the distribution functions of n-dodecane remain almost Maxwellian at all x when both heat and mass transfer processes are considered. Noticeable deviations from the Maxwellian distributions were predicted for the corresponding distribution functions of nitrogen.

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The kinetic algorithm described in this section is expected to be applicable for the analysis of a wider range of heat and mass transfer phenomena in which the contribution of three components in the mixture is essential. This includes the problem of Diesel fuel droplet heating and evaporation. The latter is considered in the following two sections.

6.8.2 Application to Two-Component Droplets: Solution Algorithm In this section the details of the solution algorithm used for the investigation of heating and evaporation of droplets of a mixture of n-dodecane and p-dipropylbenzene are presented following [144]. The contributions of both mass and heat transfer in the kinetic region are considered following the same approach as presented in Sect. 6.4.2. Since the dependence of the evaporation coefficient for p-dipropylbenzene on the surface temperature and the effect of p-dipropylbenzene on the evaporation coefficient of n-dodecane have not been investigated, it is assumed that the evaporation coefficients for both components are equal to 1. The details of the following calculations are presented: pure n-dodecane droplets, and the droplets of the n-dodecane and p-dipropylbenzene mixtures with the following molar fractions: 80% n-dodecane and 20% p-dipropylbenzene; 70% n-dodecane and 30% p-dipropylbenzene. Chemical formulae, molar masses and molecular diameters of these species and nitrogen (approximation of air) are presented in Table 6.9. As in Sect. 6.4.2, the solution of Eq. (6.56) starts with an investigation of heat and mass transfer processes in the kinetic region for a set of values of TRd and ρ Rd (for both vapour species). Restricting the analysis to the problem of heating and evaporation of droplets in a hot gas (Diesel engine-like conditions) the following ranges of parameters are considered: ρ Rd < ρs and TRd > Ts . For the selected values of ρ Rd and TRd , the solution to the Boltzmann equations (6.56) in the kinetic region allows us to obtain the normalised mass and heat fluxes at the outer boundary of this region: jk (n,p) , j˜k (n,p) = √ ρ0 Rv T0 q˜k =

qk , √ p0 Rv T0

where T0 is the reference temperature assumed as equal to 600 K, p0 and ρ0 are the saturated n-dodecane vapour pressure and density corresponding to T0 , respectively, Rv is the n-dodecane gas constant, ρ0 is inferred from the ideal gas law, and subscript k refers to ‘kinetic’. As demonstrated in Sect. 6.4.2 for n-dodecane droplets q˜k is a very weak function of ρ˜ Rd ≡ ρ Rd /ρs in a certain range of ρ˜ Rd , and j˜k is a very weak function of T˜Rd ≡

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Fig. 6.29 Plots of normalised heat flux q˜k versus ρ˜ Rd for T˜Rd = 1.05 and T˜Rd = 1.10 for an 80% n-dodecane and 20% p-dipropylbenzene mixture at droplet surface temperature 600 K. Reprinted from [144], Copyright Elsevier (2014)

TRd /Ts in a certain range of T˜Rd relevant to Diesel engines-like conditions. In what follows, it is demonstrated how these results can be generalised to the analysis of droplets of a mixture of n-dodecane and p-dipropylbenzene. Molar fractions of n-dodecane and p-dipropylbenzene in the droplet are assumed to be equal to 80% and 20%, respectively The droplet surface temperature is assumed equal to 600 K, and ρ˜ Rd (n) = ρ˜ Rd ( p) ≡ ρ˜ Rd . The plots of q˜k versus ρ˜ Rd for T˜Rd = 1.05 and 1.1 are presented in Fig. 6.29. As follows from this figure, q˜k are very weak functions of ρ˜ Rd for these values of T˜Rd (the functions q˜k (ρ˜ Rd ) are the lines which are almost parallel to the ρ˜ Rd axis). Similar results were obtained when ρ˜ Rd (n) = ρ˜ Rd ( p) , and for the 70% n-dodecane and 30% p-dipropylbenzene mixture. This allows us to assume that q˜k does not depend on ρ˜ Rd in agreement with the similar result inferred from the analysis presented in Sect. 6.4.2 for n-dodecane droplets. Plots of mass fluxes of n-dodecane and p-dipropylbenzene ( j˜k (n) and j˜k (p) ) versus ˜ TRd for ρ˜ Rd = 0.7 for the same parameters as in Fig. 6.29 are presented in Fig. 6.30. As follows from this figure, both j˜k (n) and j˜k (p) are weak functions of T˜Rd in the range of T˜Rd under consideration (the functions j˜k (n) (T˜Rd ) and j˜k (p) (T˜Rd ) are the lines which are almost parallel to the T˜Rd axis). In contrast to the case presented in Fig. 6.29, a weak dependence of jk on T˜Rd can be noticed, but this can be ignored in most applications. Similar results were obtained when ρ˜ Rd (n) = ρ˜ Rd ( p) , and for the 70% n-dodecane and 30% p-dipropylbenzene mixture. This allows us to assume that j˜k does not depend on T˜Rd . Thus, the results presented in Figs. 6.29 and 6.30 allow

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Fig. 6.30 Plots of normalised mass fluxes of n-dodecane j˜k (n) and p-dipropylbenzene j˜k (p) versus T˜Rd for ρ˜ Rd = 0.7 for the same conditions as in Fig. 6.29. Reprinted from [144], Copyright Elsevier (2014)

us to decouple the analysis of heat and mass fluxes in the kinetic region for ρ˜ Rd in the range 0.7–1 and T˜Rd in the range 1–1.1. The values of T˜Rd for two-component droplets were obtained following the same approach as presented in Sect. 6.4.2 for mono-component droplets. The procedure is shown in Fig. 6.31 for an 80% n-dodecane and 20% p-dipropylbenzene mixture droplet of radius 5 µm, and surface temperature 600 K for gas (air) temperature 1000 K. ρ˜ Rd was assumed to be equal to 0.7. As follows from Fig. 6.29, the results are not expected to depend on ρ˜ Rd . The plots of the heat flux in the kinetic√region, q˜k , versus T˜Rd , and the heat flux in the hydrodynamic region, q˜h = qh /( p0 Rv T0 ), versus T˜Rd (horizontal line), for these values of parameters are presented in Fig. 6.31. T˜Rd = 1.022 infers from the intersection between q˜k and q˜h (the inclined and horizontal lines). ˜ Plots of the mass flux in the kinetic region, j˜k , versus √ ρ˜ Rd for TRd = 1.05 and the mass flux in the hydrodynamic region, j˜h = jh /(ρ0 Rv T0 ), for n-dodecane ( j˜k (n) and j˜h (n) ) and p-dipropylbenzene ( j˜k (p) and j˜h (p) ) versus ρ˜ Rd (ρ˜ Rd (n) or ρ˜ Rd ( p) ) are presented in Fig. 6.32. As follows from Fig. 6.30, the results are not expected to depend on T˜Rd . The mass fluxes in the hydrodynamic region are shown as horizontal lines. The same parameters as in Fig. 6.31 were used in Fig. 6.32. As in Sect. 6.4.2, it is assumed that ρ Rd = ρs . The required values of ρ˜ Rd : ρ˜ Rd(n) = 0.983 for n-dodecane and ρ˜ Rd( p) = 0.987 for p-dipropylbenzene infer from the conditions j˜k (n) = j˜h (n) and j˜k (p) = j˜h (p) (the intersections between the horizontal and inclined lines).

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˜ Fig. 6.31 Plots of the normalised heat flux predicted by the kinetic √ model, q˜k , versus TRd and the heat flux predicted by the hydrodynamic model, q˜h = qh /( p0 Rv T0 ), versus T˜Rd (horizontal line) for ρ˜ Rd = 0.7 for an 80% n-dodecane and 20% p-dipropylbenzene mixture droplet of radius 5 µm, surface temperature 600 K and for gas (air) temperature 1000 K. T˜Rd = 1.022 follows from the intersection between the q˜k and q˜h . Reprinted from [144], Copyright Elsevier (2014)

˜ Fig. 6.32 Plots of the mass fluxes of the components predicted √ by the kinetic model, jk , versus ρ˜ Rd for T˜Rd = 1.05 and the hydrodynamic model, j˜h = jh /( p0 Rv T0 ), for n-dodecane ( j˜h (n) ) and p-dipropylbenzene ( j˜h (p) ) versus ρ˜ Rd (ρ˜ Rd (n) and ρ˜ Rd ( p) for the same parameters as in Fig. 6.31. j˜h (n) and j˜h (p) are horizontal lines. The required values of ρ˜ Rd(n) = 0.983 for n-dodecane and ρ˜ Rd( p) = 0.987 for p-dipropylbenzene infer from the intersections between the horizontal and inclined lines. Reprinted from [144], Copyright Elsevier (2014)

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Similar values of T˜Rd and ρ˜ Rd were obtained for other values of Tg , Ts and Rd relevant for Diesel engine-like conditions (Tg = 750 K, Tg = 700 K; for values of Ts in the range of 300 K to temperatures close to the critical temperature), and several molar fractions of n-dodecane and p-dipropylbenzene. The corresponding values of T˜Rd and ρ˜ Rd were used for the investigation of heating and evaporation of mono- and di-component droplets in realistic Diesel engine-like conditions (see Sect. 6.8.3). Following the procedure illustrated in Figs. 6.31 and 6.32, a set of ρ˜ Rd and T˜Rd was obtained for several pairs of values of droplet surface temperatures and radii, predicted by the hydrodynamic model. Once the values of ρ˜ Rd have been obtained, ρs was replaced by ρ Rd in the kinetic modelling. For the intermediate values of these parameters the values of ρ˜ Rd and T˜Rd were interpolated. Since ρ Rd < ρs , B M predicted by the kinetic model (B M, k ) is always less than B M predicted by the hydrodynamic model (B M, h ). Hence, the evaporation rate predicted by the kinetic model is expected to be always less than that predicted by the hydrodynamic model for the same droplet surface temperatures. The decrease in B M predicted by the kinetic model is expected to lead to a corresponding decrease in BT and in the convection heat transfer coefficient h. This leads to a decrease in Teff . On the other hand, the slowing down of the evaporation process predicted by the kinetic model leads to a decrease in | R˙ d |, and ultimately an increase in Teff . The balance between these two processes leads to either a decrease or an increase in the predicted droplet surface temperatures. Note that the effect of changes in the droplet evaporation rates due to the changes in droplet surface temperatures, predicted by the kinetic model, were not considered. If these changes were considered we would need to use the corrected values of jvi for the mass fluxes predicted by the hydrodynamic model. This would lead to new ρ˜ Rd and the new droplet surface temperature etc. This effect is investigated in Sect. 6.9. In Sect. 6.8.3, some results of modelling of specific droplets are presented and discussed.

6.8.3 Application to Two-Component Droplets: Results This section focuses on the results of application of the kinetic algorithm for three components, described in Sect. 6.8.2, to the analysis of Diesel fuel droplet heating and evaporation. The results to be presented are reproduced from [144]. The radii and surface temperatures of a droplet with an initial radius and temperature equal to 5 µm and 300 K, respectively, immersed in gas with a temperature 1000 K are presented in Fig. 6.33. Three types of droplets were considered: pure n-dodecane, a mixture of 80% n-dodecane and 20% p-dipropylbenzene, and a mixture of 70% n-dodecane and 30% p-dipropylbenzene (the contributions of components refer to their molar fractions). The effects of temperature and species mass fraction gradients in droplets and recirculation in them were considered. In kinetic calculations both heat and mass transfer in the kinetic region and the effects of inelastic collisions were considered. The evaporation coefficient was taken equal to one for both components.

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Fig. 6.33 Plots of Rd and Ts versus time, predicted by the kinetic and hydrodynamic models for droplets with initial radii and temperature equal to 5 µm and 300 K, respectively, introduced into a gas (air) with a temperature of 1000 K; pure n-dodecane and mixtures of n-dodecane and p-dipropylbenzene are considered; plots (1) refer to the prediction of the kinetic model for ndodecane; plots (2) refer to the prediction of the kinetic model for 80% n-dodecane and 20% pdipropylbenzene mixture; plots (3) refer to the prediction of the kinetic model for 70% n-dodecane and 30% p-dipropylbenzene mixture; plots (4) refer to the prediction of the hydrodynamic model for n-dodecane; plots (5) refer to the prediction of the hydrodynamic model for 80% n-dodecane and 20% p-dipropylbenzene mixture; plots (6) refer to the prediction of the hydrodynamic model for 70% n-dodecane and 30% p-dipropylbenzene mixture. A shows the zoomed part of the figure for droplet surface temperatures; B shows the zoomed part of the figure for droplet radii. Reprinted from [144], Copyright Elsevier (2014)

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Fig. 6.34 The same as Fig. 6.33, but for the gas (air) temperature of 700 K. Reprinted from [144], Copyright Elsevier (2014)

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Since the values of ρ Rd were found as perturbations of ρs predicted by the hydrodynamic model, the calculations of the droplet radii using the kinetic model had to be stopped before the evaporation time predicted by the hydrodynamic model (the latter is always less than that predicted by the kinetic model). Then the droplet radii predicted by the kinetic model were extrapolated until the evaporation process completed. As follows from Fig. 6.33, both the addition of p-dipropylbenzene and kinetic effects lead to an increase in the droplet evaporation times. The same plots as in Fig. 6.33 but for gas (air) temperature equal to 700 K are presented in Fig. 6.34. Comparing Figs. 6.33 and 6.34 it can be seen that the decrease in gas temperature from 1000 K to 700 K leads to more than doubling of the evaporation time and reduction of the kinetic effects for all three mixtures. The kinetic effects are expected to be even more noticeable at temperatures greater than 1000 K. In this case, however, the droplet surface temperatures could approach the critical temperature well before its complete evaporation. The model used in the analysis in this case becomes less reliable as it assumes that droplet surface temperature is not close to the critical temperature. One of the problems with the approximation of Diesel fuel by a mixture of ndodecane and p-dipropylbenzene is that the accuracy of this approximation for modelling droplet heating and evaporation has not been investigated for a wide range of available Diesel fuels, to the best of the author’s knowledge.

6.9 A Self-consistent Kinetic Model In the kinetic models described so far in this chapter, the boundary condition at the interface between the kinetic and hydrodynamic regions was inferred using the requirement of the conservation of heat and mass fluxes at this interface. The hydrodynamic heat and mass fluxes were calculated using the simplifying assumptions that the temperature at the outer boundary of the kinetic region is equal to the droplet surface temperature and vapour pressure at this boundary is equal to the saturated vapour pressure at this temperature. The requirement for the conservation of heat and mass fluxes at this interface led to finding the corrected values of temperature and vapour density at this interface. The key problem with this approach is that the heat and mass fluxes in the hydrodynamic region, calculated using these corrected values of temperature and vapour density, are not equal to the heat and mass fluxes in the hydrodynamic region used to find these corrected values, in the general case. The only feasible way to overcome this problem is to perform iterations; that is, to use the corrected values of temperature and vapour density (or densities in the case of bi-component droplets) at the outer boundary of the kinetic region to calculate the corrected values of hydrodynamic heat and mass fluxes etc. A self-consistent model using this approach was developed by the authors of [146]. In what follows, key ideas of this model are described.

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√ Fig. 6.35 Plots of normalised mass fluxes j˜ ≡ j/(ρ0 Rv T0 ) predicted by the kinetic (line ‘k’) and hydrodynamic (lines ‘h’) models for an n-dodecane droplet moving with a constant velocity of 10 m/s versus αρ ≡ ρ Rd /ρs (normalised vapour density at the outer boundary of the kinetic region). Droplet surface and gas (air) temperatures were taken equal to 600 K and 1000 K, respectively. Reprinted from [146], Copyright Elsevier (2016)

As in the approaches discussed so far, the investigation in [146] was focused on finding the temperature and vapour density at the outer boundary of the kinetic region. These were obtained from the requirement that both heat flux and mass flux of vapour (or vapour components) in the kinetic and hydrodynamic regions at the interface between these regions are equal. In the first step, these fluxes in the hydrodynamic region were calculated using the values of temperature and vapour density at the surface of the droplet. Then the values of temperature and vapour density at the outer boundary of the kinetic region, obtained following the procedure described in the previous sections of this chapter, were used to calculate the corrected heat and mass fluxes based on the hydrodynamic model. These corrected fluxes led to new corrected values of temperature and vapour density at the outer boundary of the kinetic region. One would expect that if this process converges then one would obtain self-consistent values for both heat and mass fluxes. The hydrodynamic and kinetic models used in [146] are similar to the ones described earlier with the evaporation coefficient equal to one for both n-dodecane and p-dipropylbenzene (the effect of the evaporation coefficient on the results was shown to be small). As in the model described in Sect. 6.8.3, the effects of temperature and component mass fraction gradients in droplets and recirculation in them were considered alongside the effect of inelastic collisions in the kinetic modelling. The above-mentioned iterative procedure is shown in Figs. 6.35 and 6.36 for the calculation of density and temperature at the interface between the kinetic and hydrodynamic regions. Diesel fuel was approximated by n-dodecane, a droplet was moving

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√ Fig. 6.36 Plots of normalised heat fluxes q˜ ≡ q/( p0 Rv T0 ) predicted by the kinetic (line ‘k’) and hydrodynamic (lines ‘h’) models for an n-dodecane droplet moving with a constant velocity of 10 m/s versus αT ≡ TRd /Ts (normalised temperature at the outer boundary of the kinetic region). Droplet surface and gas (air) temperatures were taken equal to 600 K and 1000 K, respectively. Reprinted from [146], Copyright Elsevier (2016)

with a constant relative velocity of 10 m/s and its surface temperature was equal to 600 K. Gas temperature and pressure 30 bar, respec√ were taken equal to 1000 K and√ tively. The plots of j˜k ≡ jk n /(ρ0 Rv T0 ) versus αρ and q˜k ≡ qk /( p0 Rv T0 ) versus αT are shown in Figs. 6.35 √ and 6.36, respectively (lines ‘k’).√In the same figures, the plots of j˜h ≡ jh n /(ρ0 Rv T0 ) versus αρ and q˜h ≡ qh /( p0 Rv T0 ) versus αT for αρ = 1 and αT = 1, are also shown (lines ‘h’, iteration 1). The intersection between these two pairs of lines allows us to obtain: αρ = 0.979 and αT = 1.031. In the analysis presented in the previous sections of this chapter, these corrections were used for the calculation of mass and heat fluxes, considering the kinetic effects. In the model developed by the authors of [146], these corrections were used to update the values of j˜h and q˜h . The updated values of these parameters are presented in Figs. 6.35 and 6.36 as the lines marked ‘h’, iteration 2. The intersections of these new lines with lines j˜k and q˜k allow us to obtain updated αρ = 0.969 and αT = 1.029. Further iterations up to iteration 50 lead to visible changes in these corrections, as presented in the same Figs. 6.35 and 6.36. The difference in the values of αρ and αT inferred from consecutive iterations decreases with increasing iteration number so that the differences between the values inferred from the 49th iteration were almost indistinguishable from those inferred from the 50th iteration. Note that, in contrast to the non-self-consistent model described in Sect. 6.2, the approach described above does not use the observation that q˜k is a very weak function of αρ and j˜k is a very weak function of αT .

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Fig. 6.37 Plots of αρ versus Ts for an n-dodecane droplet moving with a constant velocity of 10 m/s in gas (air) at a temperature of 1000 K. Reprinted from [146], Copyright Elsevier (2016)

The same analysis as shown in Figs. 6.35 and 6.36 was used for other droplet surface temperatures in the range 300–650 K and gas temperatures 800 K, 1000 K and 1200 K. Also, the same analysis was used for bi-component droplets (mixture of 80% n-dodecane and 20% p-dipropylbenzene), for droplet surface temperatures in the range of 300–650 K and gas (air) temperature equal to 1000 K. The predictions of the kinetic model at temperatures close to the critical temperature of n-dodecane (Tcr = 659 K) were not reliable. It was assumed that αρ and αT at Ts > 650 K are the same as at Ts = 650 K. This assumption affects the very final stage of droplet evaporation but has limited effect on the overall picture of droplet heating and evaporation. The results of this analyses lead to the plots of αρ and αT versus droplet surface temperatures Ts presented in Figs. 6.37 and 6.38. Gas temperature and pressure were taken equal to 1000 K and 30 bar, respectively, and n-dodecane droplet velocity was assumed 10 m/s. As follows from Figure 6.37, αρ decreases with increasing Ts . The values of αρ obtained from iteration 2 are lower than those obtained from iteration 1; these values from the third iteration are almost the same as those obtained from all the following iterations. The behaviour of the curve αT versus Ts , presented in Fig. 6.38, is more complex than that of αρ versus Ts . For low temperatures αT increases with increasing Ts , at intermediate temperatures αT decreases with increasing Ts , and at temperatures close to 650 K, αT again increases with increasing Ts . As in the case of αρ , the values of αT obtained from iteration 3 and higher iterations are almost the same. These values are slightly higher than those obtained from iteration 1 for low Ts and slightly lower than those obtained from iteration 1 for high Ts . In the analysis described in [146], the values obtained from iteration 50 are assumed to give the self-consistent heat and mass fluxes for heated and evaporating droplets.

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Fig. 6.38 Plots of αT versus Ts for an n-dodecane droplet moving with a constant velocity of 10 m/s in gas (air) at a temperature of 1000 K. Reprinted from [146], Copyright Elsevier (2016)

The shapes of the curves for other values of gas temperatures and approximations of Diesel fuel were similar to those presented in Figs. 6.37 and 6.38. The selfconsistent model described in this section was used for the investigation of heating and evaporation of a Diesel fuel droplet with initial radii and temperature equal to 5 µm and 300 K, respectively, placed into gas with temperatures 800 K, 1000 K and 1200 K, and pressure 30 bar. Droplets were stationary or moving with a constant velocity of 10 m/s. It was demonstrated that in all cases the kinetic effects led to a decrease in droplet surface temperature and an increase in the evaporation time. This increase was particularly visible for high gas temperatures and moving droplets. The addition of p-dipropylbenzene led to a decrease in the kinetic effects on the droplet evaporation time. The validity of the assumption that both liquid and gas phases can be treated as a continuum, used in the previous chapters, was shown to be questionable when the interface between liquid and gas is modelled, even when the gas pressure is well above atmospheric. The chapter started with a review of early kinetic models of droplet evaporation. Then more rigorous models, using numerical solutions to the Boltzmann equations for vapour and air, were described. Two regions above the surface of an evaporating droplet were considered: the kinetic and hydrodynamic regions. Vapour and air dynamics in the first region were described by the Boltzmann equations, while the conventional hydrodynamic approach was used in the second region. Collisions between molecules were assumed to be inelastic in the general case. The evaporation coefficient was estimated using molecular dynamics analysis of n-dodecane molecules, based on the united atoms model. The applicability of quantum-chemical models to finding this coefficient was investigated.

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References 1. Abramov, A. A., Kogan, M. N., & Makashev, N. K. (1981). Numerical analysis of the processes in strongly non-equilibrium Knudsen layers. Reports of the Academy of Sciences of USSR. Mechanics of Liquids and Gases, 3, 72–81 (in Russian). 2. Akhiezer, A. I., Akhiezer, I. A., Polovin, R. V., Sitenko, A. G., & Stepanov, K. N. (1975). Plasma Electrodynamics. Oxford: Pergamon. 3. Alexandrov, A. F., Bogdankevich, L. S., & Rukhadze, A. A. (1978). Fundamentals of Plasma Electrodynamics. Moscow (in Russian): Vysshaya Shkola. 4. Allen, M. P., & Tidesley, D. J. (1984). Computer Simulation of Liquids. Oxford: Oxford University Press. 5. Amat, M. A., & Rutledge, G. C. (2010). Liquid-vapor equilibria and interfacial properties of n-alkanes and perfluoroalkanes by molecular simulation. The Journal of Chemical Physics, 132(11), 114704. 6. Anisimov, S. I. (1968). Evaporation of metals under the influence of laser radiation. Journal of Experimental and Theoretical Physics, 54, 339–342 (in Russian). 7. Anisimov, S. I., Imas Ya, A., Romanov, G. S., & Khodyko, Yu. V. (1970). Effect of High Intensity Radiation on Metals. Moscow (in Russian): Nauka Publishing House. 8. Anisimov, S. I., Dunikov, D. O., Zhakhovskii, V. V., & Malyshenko, S. P. (1999). Properties of a liquid-gas interface at high-rate evaporation. The Journal of Chemical Physics, 110, 8722–8729. 9. Aoki, K., Takata, S., & Kosuge, S. (1998). Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas. Physics of Fluids, 10, 1519–1533. 10. Aoki, K., Bardos, C., & Takata, S. (2003). Knudsen layer for gas mixtures. Journal of Statistical Physics, 112, 629–655. 11. Aristov, V. V., & Tcheremissine, F. G. (1992). Direct Numerical Solution of the Boltzmann Equation. Moscow: Computer Centre of Russian Academy of Sciences. 12. Bakhvalov, N. S., Zhidkov, N. P., & Kobelkov, G. M. (1989). Numerical Methods. Moscow (in Russian): Nauka Publishing House. 13. Banasiak, J., & Groppi, M. (2003). Solvability of linear kinetic equations with multi-energetic inelastic scattering. Reports on Mathematical Physics, 52, 235–253. 14. Benedetto, D., & Caglioti, E. (1999). The collapse phenomenon in one-dimensional inelastic point particle systems. Physica D: Nonlinear Phenomena, 132, 457–475. 15. Berendsen, H. J. C., Grigera, J. R., & Straatsma, T. P. (1987). The missing term in effective pair potential. Journal of Physical Chemistry, 91, 6269–6271. 16. Bhatnagar, P. L., Gross, E. P., & Krook, M. (1954). A model for collision processes in gases. 1. Small amplitude processes in charged and neutral one component systems. Physical Review, 94, 511–525. 17. Biben, T., Martin, Ph. A., & Piasecki, J. (2002). Stationary state of thermostated inelastic hard spheres. Physica A, 310, 308–324. 18. Bird, G. A. (1976). Molecular Gas Dynamics. Oxford: Oxford University Press. 19. Bird, G. A. (1994). Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Oxford University Press. 20. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena. Chichester: Wiley. 21. Borgnakke, C., & Larsen, P. S. (1975). Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. Journal of Computational Physics, 18, 405–420. 22. Buet, C., Cordier, S., & Degond, P. (1998). Regularized Boltzmann operators. Computers & Mathematics with Applications, 35, 55–74. 23. Cao, B.-Y., Xie, J.-F., & Sazhin, S. S. (2011). Molecular dynamics study on evaporation and condensation of n-dodecane at liquid-vapour phase equilibria. The Journal of Chemical Physics, 134, 164309.

References

403

24. Carravetta, V., & Clementi, E. (1984). Water-water interaction potential: An approximation of the electron correlation contribution by a functional of the SCF density matrix. The Journal of Chemical Physics, 81, 2646–2651. 25. Chakraborty, P. R., Hiremath, K. R., & Sharma, M. (2017). Evaluation of evaporation coefficient for micro-droplets exposed to low pressure: A semi-analytical approach. Physics Letters A, 381, 413–416. 26. Chandra, A., Keblinski, P., Sahni, O., & Oberai, A. A. (2019). A continuum framework for modeling liquid-vapor interfaces out of local thermal equilibrium. International Journal of Heat and Mass Transfer, 144, 118597. 27. Chapman, S., & Cowling, T. G. (1990). The Mathematical Theory of Non-uniform Gases. Cambridge: Cambridge University Press. 28. Cercignani, C. (1981). Strong evaporation of a poliatomic gas. In S. S. Fisher (Ed.), Rarefied Gas Dynamics Part 1 (pp. 305–310). New York: AIAA. 29. Chernyak, V. (1995). The kinetic theory of droplet evaporation. Journal of Aerosol Science, 26, 873–885. 30. Chilukoti, H. K., Kikugawa, G., & Ohara, T. (2013). A molecular dynamics study on transport properties and structure at the liquid-vapor interfaces of alkanes. International Journal of Heat and Mass Transfer, 59, 144–154. 31. Cipola, J. W., Lang, J. H., & Loyalka, S. K. (1974). Kinetic theory of evaporation and condensation. In Proceedings of 8th International Symposium on Rarefied Gas Dynamics (pp. 1773–1776). New York–London: Academic. 32. Consoline, L., Aggarwal, S. K., & Murad, S. (2003). A molecular dynamics simulation of droplet evaporation. International Journal of Heat and Mass Transfer, 43, 3179–3188. 33. Cramer, C. J., & Truhlar, D. G. (1999). Implicit salvation models: Equilibria, structure, spectra, and dynamics. Chemical Physics, 99, 2161–2200. 34. Dellacherie, S. (2003). Coupling of the Wang-Chang-Uhlenbeck equations with the multispecies Euler system. Journal of Computational Physics, 189, 239–276. 35. Doi, T. (2011). Numerical analysis of the time-dependent energy and momentum transfers in a rarefied gas between two parallel planes based on the linearized Boltzmann equation. ASME Journal of Heat Transfer, 133, 022404-1. 36. Dombrovsky, L. A., & Sazhin, S. S. (2003). A parabolic temperature profile model for heating of droplets. ASME Journal of Heat Transfer, 125, 535–537. 37. Dombrovsky, L. A., Sazhin, S. S., Sazhina, E. M., Feng, G., Heikal, M. R., Bardsley, M. E. A., & Mikhalovsky, S. V. (2001). Heating and evaporation of semi-transparent diesel fuel droplets in the presence of thermal radiation. Fuel, 80, 1535–1544. 38. Dongari, N., & Agrawal, A. (2012). Modeling of Navier-Stokes equations for high Knudsen number gas flows. International Journal of Heat and Mass Transfer, 55, 4352–4358. 39. Dunikov, D. O., Malyshenko, S. P., & Zhakhovskii, V. V. (2000). Liquid-gas interface during transient condensation: Molecular dynamics investigation. In A. Leontiev, & A. Klimenko (Eds.), Proceedings of 3rd Russian National Heat and Mass Transfer Conference (Vol. 4, pp. 257–260). Moscow (in Russian): Moscow Power Engineering Institute Publishing House. 40. Dunikov, D. O., Malyshenko, S. P., & Zhakhovskii, V. V. (2001). Corresponding states law and molecular-dynamics simulations of the Lennard-Jones fluid. The Journal of Chemical Physics, 115, 6623–6631. 41. Elliott, R. J. (1995). Plasma kinetic theory. In Plasma Physics: An Introductory Course. Edited by R Dendy. (pp. 29–53). Cambridge: Cambridge University Press. 42. Elperin, T., & Krasovitov, B. (1995). Radiation, thermal diffusion and kinetic effects in evaporation and combustion of large and moderate size droplets. International Journal of Heat and Mass Transfer, 38, 409–418. 43. Feng, Z.-G., & Michaelides, E. E. (2012). Heat transfer from a nano-sphere with temperature and velocity discontinuities at the interface. International Journal of Heat and Mass Transfer, 55, 6491–6498. 44. Ferrari, L., & Carbognani, A. (1998). Differential kinetic equations for a Rayleigh gas with inelastic collisions. Physica A, 251, 452–468.

404

6 Kinetic Modelling of Droplet Heating and Evaporation

45. Flynn, P. F., Durrett, R. P., Hunter, G. L., zur Loye, A. O., Akinyemi, O. C., Dec, J. E., et al. (1999). Diesel combustion: An integrated view combining laser diagnostics, chemical kinetics, and empirical validation. SAE Report 1999-01-0509. 46. Fournier, N., & Mischler, S. (2005). A spatially homogeneous Boltzmann equation for elastic, inelastic and coalescing collisions. Journal de Mathématiques Pures et Appliquées, 84, 1173– 1234. 47. Frezzotti, A. (1986). Kinetic theory study of the strong evaporation of a binary mixture. In V. Boffi, & C. Cercignani (Eds.), Rarefied Gas Dynamics (Vol. 2, pp. 313–22). Stuttgart: Teubner. 48. Fuchs, N. A. (1959). Evaporation and Droplet Growth in Gaseous Media. London: Pergamon Press. 49. Fujikawa, S., & Maerefat, M. (1990). A study of the molecular mechanism of vapour condensation. JSME International Journal Series 2, 33(4), 634–641. 50. Furioli, G., Pulvirenti, A., Terraneo, E., & Toscani, G. (2010). Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules. The Annales Henri Poincaré AN, 27, 719–737. 51. Gladstone, G., Laidler, K. J., & Eyring, H. (1941). The Theory of Rate Processes. New York: McGraw-Hill. 52. Gombosi, T. I. (1994). Gaskinetic Theory. Cambridge: Cambridge University Press. 53. Gong, Y., Xiao, G., Ma, X., Luo, K. H., Shuai, S., & Xu, H. (2021). Phase transitions of multi-component fuel droplets under sub- and supercritical conditions. Fuel, 287, 119516. 54. Gumerov, N. A. (2000). Dynamics of vapour bubbles with non-equilibrium phase transitions in isotropic acoustic fields. Physics of Fluids, 12, 71–88. 55. Gun’ko, V. M., Nasiri, R., Sazhin, S. S., Lemoine, F., & Grisch, F. (2013). A quantum chemical study of the processes during the evaporation of real-life Diesel fuel droplets. Fluid Phase Equilibria, 356, 146–156. 56. Gun’ko, V. M., Nasiri, R., & Sazhin, S. S. (2014). A study of the evaporation and condensation of n-alkane clusters and nanodroplets using quantum chemical methods. Fluid Phase Equilibria, 366, 99–107. 57. Gun’ko, V. M., Nasiri, R., & Sazhin, S. S. (2015). Effects of the surroundings and conformerisation of n-dodecane molecules on evaporation/condensation processes. The Journal of Chemical Physics, 142(3), 034502. 58. Gun’ko, V. M., & Turov, V. V. (1999). Structure of hydrogen bonds and 1 H NMR spectra of water at the interface of Oxides. Langmuir, 15, 6405–6415. 59. Gun’ko, V. M., Zarko, V. I., Leboda, R., Marciniak, M., Janusz, W., & Chibowski, S. (2000). Highly dispersed X/SiO2 and C/X Si O2 (X = Aluminia, Titania, Aluminia/Titania) in the gas liquid media. Journal of Colloid and Interface Science, 230, 396–409. 60. Hanchak, M. S., Briones, A. M., Ervin, J. S., & Byrd, L. W. (2013). One-dimensional models of nanoliter droplet evaporation from a hot surface in the transition regime. International Journal of Heat and Mass Transfer, 57, 473–483. 61. Harris, J. G. (1992). Liquid-vapor interfaces of alkane oligomers: Structure and thermodynamics from molecular dynamics simulations of chemically realistic models. Journal of Physical Chemistry, 96, 5077–5086. 62. Harvie, D. J. E., & Fletcher, D. F. (2001). A simple kinetic theory treatment of volatile liquidgas interfaces. ASME Journal of Heat Transfer, 123, 486–491. 63. Hertz, H. (1882). Über die Verdunstung der Flussigkeiten, inbesondere des Quecksilbers, im lufteeren Raume. Annalen der Physik, 17, 177–200. 64. Hirschfelder, J. O., Curtiss, C. F., & Bird, R. B. (1967). Molecular Theory of Gases and Liquids (4th ed.). New York: Wiley. 65. Hirth, J. P., & Pound, G. M. (1963). Condensation and Evaporation, Nucleation and Growth Kinetics. Oxford - London - Paris - Frankfurt: Pergamon Press. 66. Ibergay, C., Ghoufi, A., Goujon, F., Ungerer, P., Boutin, A., Rousseau, B., & Malfreyt, P. (2007). Molecular simulations of the n−alkane liquid-vapor interface: Interfacial properties and their long range corrections. Physical Review E, 75, 051602.

References

405

67. Imamuro, T., & Sturtevant, B. (1990). Numerical study of discrete-velocity gases. Physics of Fluids, 2, 2196–2203. 68. Ishiyama, T., Yano, T., & Fujikawa, S. (2004). Molecular dynamics study of kinetic boundary conditions at an interface between argon vapor and its condensed phase. Physics of Fluids, 16(8), 2899–2907. 69. Ishiyama, T., Yano, T., & Fujikawa, S. (2004). Molecular dynamics study of kinetic boundary conditions at an interface between a polyatomic vapor and its condensed phase. Physics of Fluids, 16(12), 4713–4726. 70. Ishiyama, T., Fujikawa, S., Kurz, W., & Lauterborn, W. (2013). Nonequilibrium kinetic boundary condition at the vapor-liquid interface of argon. Physical Review E, 88, 042406. 71. Iskrenova, E. K., & Patnaik, S. S. (2017). Molecular dynamics study of octane condensation coefficient at room temperature. International Journal of Heat and Mass Transfer, 115, 474– 481. 72. Jasper, A. W., & Miller, J. A. (2014). Lennard-Jones parameters for combustion and chemical kinetics modeling from full-dimensional intermolecular potentials. Combustion and Flame, 161, 101–110. 73. Jorgensen, W. L., Madura, J. D., & Swenson, C. J. (1984). Optimized intermolecular potential functions for liquid hydrocarbons. Journal of the American Chemical Society, 106, 6638– 6646. 74. Jorgensen, W. L., & Tirado-Rives, J. (1988). The OPLS potential functions for proteins. Energy minimizations for crystals of cyclic peptides and crambin. Journal of the American Chemical Society, 110, 1657–1666. 75. Kennaird, E. H. (1938). Kinetic Theory of Gases. New York: McGraw-Hill. 76. Kharchenko, V., Balakrishnan, N., & Dalgarno, A. (1998). Thermalization of fast nitrogen atoms in elastic and inelastic collisions with molecules of atmospheric gases. Journal of Atmospheric and Solar-Terrestrial Physics, 60, 95–106. 77. Kieu, H. T., Tsang, A. Y. C., Zhou, K., & Law, A.W.-K. (2020). Evaporation kinetics of nano water droplets using coarse-grained molecular dynamic simulations. International Journal of Heat and Mass Transfer, 156, 119884. 78. Knight, S. J. (1979). Theoretical modelling of rapid surface vaporization with back pressure. AIAA Journal, 17, 519–523. 79. Knudsen, M. (1915). Die maximale Verdampfungsgeschwindigkeit des Quecksilbers. Annalen der Physik, 47, 697–708. 80. Koffman, L. D., Plesset, M. S., & Lees, L. (1984). Theory of evaporation and condensation. Physics of Fluids, 27, 876–880. 81. Kogan, M. N. (1969). Rarefied Gas Dynamics. New York: Plenum. 82. Kogan, M. N., & Makashev, N. K. (1971). On the role of the Knudsen layer in the theory of heterogeneous reactions and flows with the surface reactions. Reports of the Academy of Sciences of USSR. Mechanics of Liquids and Gases, 6, 3–11 (in Russian). 83. Korabelnikov, A. V., Nakoryakov, V. E., & Shreiber, I. R. (1981). Taking nonequilibrium vaporization into account in problems of vapour-bubble dynamics. High Temperatures, 19, 586–590. 84. Korobov, N. M. (1989). Trigonometric Sums and their Applications. Moscow (in Russian): Nauka Publishing House. 85. Koura, K. (1998). Improved null-collision technique in the direct simulation Monte Carlo method: Application to vibrational relaxation of nitrogen. Computers & Mathematics with Applications, 35, 139–154. 86. Kremer, G. M., Silva, A. W., & Alves, G. M. (2010). On inelastic reactive collisions in kinetic theory of chemically reacting gas mixtures. Physica A, 389, 2708–2718. 87. Kryukov, A. P., & Levashov, V. Yu. (2011). About evaporation-condensation coefficients on the vapor-liquid interface of high thermal conductivity matters. International Journal of Heat and Mass Transfer, 54, 3042–3048. 88. Kryukov, A. P., Levashov, V. Yu., & Sazhin, S. S. (2004). Evaporation of diesel fuel droplets: Kinetic versus hydrodynamic models. International Journal of Heat and Mass Transfer, 47, 2541–2549.

406

6 Kinetic Modelling of Droplet Heating and Evaporation

89. Kryukov, A. P., Levashov, V. Yu., & Shishkova, I. N. (2004). Non-equilibrium recondensation in a dusty medium. In S. A. Zhdanok et al. (Eds.), Proceedings of the 5th Minsk International Heat and Mass Transfer Forum, May 24–28 (Vol. 2, pp. 138–139). Minsk: Research Institute of Heat and Mass Transfer Publishing House (in Russian). 90. Kryukov, A. P., Levashov, V. Yu., & Shishkova, I. N. (2005). Condensation in the presence of a non-condensable component. Journal of Engineering Physics and Thermophysics, 78, 15–21 (in Russian). 91. Kryukov, A. P., Levashov, V. Yu., Shishkova, I. N., & Yastrebov, A. K. (2005). The Numerical Solution of the Boltzmann Equation for Engineering Applications. Moscow (in Russian): Moscow Power Engineering Institute Publishing House. 92. Kryukov, A. P., Levashov, V. Yu., & Shishkova, I. N. (2006). Flows of vapour-gas mixtures in micro- and nano-systems in the presence of evaporation and condensation. Proceedings of the 5th National Russian Heat Transfer Conference, Moscow, 78(3), 164–167 (in Russian). 93. Kucherov, R. Ya., & Rickenglaz, L. E. (1959). On hydrodynamic boundary conditions for modelling vaporization and condensation processes. Journal of Experimental and Theoretical Physics, 37, 125–126 (in Russian). 94. Kucherov, R. Ya., & Rickenglaz, R. E. (1960). On the measurement of the coefficient of condensation. Reports of the Academy of Science of USSR (Doklady), 133, 1130–1131 (in Russian). 95. Kucherov, R. Ya., Rickenglaz, R. E., & Tzulaya, T. S. (1963). Kinetic theory of additional condensation in the presence of the small temperature jump. Soviet Physics - Technical Physics, 7, 1027–1030. 96. Kupiainen, O., Ortega, I. K., Kurtèn, T., & Vehkamäki, H. (2012). Amine substitution into sulfuric acid-ammonia clusters. Atmospheric Chemistry and Physics, 12, 3591–3599. 97. Labuntsov, D. A. (1967). Analysis of the evaporation and condensation processes. High Temperatures, 5, 579–585. 98. Labuntsov, D. A., & Kryukov, A. P. (1977). Processes of intensive evaporation. Thermal Power Energy (Teploenergetika), 4, 8–11 (in Russian). 99. Labuntsov, D. A., & Kryukov, A. P. (1979). Analysis of intensive evaporation and condensation. International Journal of Heat and Mass Transfer, 22, 989–1002. 100. Leite, V. B., Kalempa, D., & Graur, I. (2021). Kinetic modelling of evaporation and condensation phenomena around a spherical droplet. International Journal of Heat and Mass Transfer, 166, 120719. 101. Lamanna, G., Steinhausen, C., Weigand, B., et al. (2018). On the importance of nonequilibrium models for describing the coupling of heat and mass transfer at high pressure. International Communications in Heat and Mass Transfer, 98, 49–58. 102. Lambiotte, R., Ausloos, M., Brenig, L., & Salazar, J. M. (2007). Energy and number of collision fluctuations in inelastic gases. Physica A, 375, 227–232. 103. Lee, L. L. (1988). Molecular Nonideal Fluids. Boston: Butterworths. 104. Lifshitz, E. M., & Pitaevski, L. P. (1979). Physical Kinetics. Moscow (in Russian): Nauka Publishing House. 105. Lotfi, A., Vrabec, J., & Fischer, J. (2014). Evaporation from a free liquid surface. International Journal of Heat and Mass Transfer, 73, 303–317. 106. Lu, G., Duan, Y.-Y., Wang, X.-D., & Lee, D.-J. (2012). Internal flow in evaporating droplet on heated solid surface. International Journal of Heat and Mass Transfer, 54, 4437–4447. 107. Ludwig, K. F., & Micci, M. (2011). Molecular dynamics simulations of Rayleigh and first wind-induced breakup. Atomization and Sprays, 21, 275–281. 108. Malyshenko, S. P., & Dunikov, D. O. (2002). On the surface tension corrections in non-uniform and nonequilibrium liquid-gas systems. International Journal of Heat and Mass Transfer, 45, 5201–5208. 109. Marek, R., & Straub, J. (2001). Analysis of the evaporation coefficient and the condensation coefficient of water. International Journal of Heat and Mass Transfer, 44, 39–53. 110. Mizuguchi, H., Nagayama, G., & Tsuruta, T. (2010). Molecular dynamics study on evaporation coefficient of biodiesel fuel. In Seventh International Conference on Flow Dynamics (pp. 386– 387). Sendai, Japan.

References

407

111. Murakami, M., & Oshima, K. (1974). Kinetic approach to the transient evaporation and condensation problem. In M. Becker, & M. Fiebig (Eds.), Rarefied Gas Dynamics. PorsWahn: DFVLR Press, paper F6. 112. Muratova, T. M., & Labuntsov, D. A. (1969). Kinetic analysis of the evaporation and condensation processes. Thermal Physics of High Temperatures, 7, 959–967 (in Russian). 113. Nagayama, G., & Tsuruta, T. (2003). A general expression for the condensation coefficient based on transition state theory and molecular dynamic simulation. The Journal of Chemical Physics, 118, 1392–1399. 114. Nasiri, R., Gun’ko, V. M., & Sazhin, S. S. (2015). The effects of internal molecular dynamics on the evaporation/condensation of n-dodecane. Theoretical Chemistry Accounts, 134(7), 83. 115. Neizvestny, A. I., & Onishenko, L. I. (1979). Experimental determination of the condensation coefficient for distilled water. Physics of Atmosphere and Ocean, 15, 1052–1078 (in Russian). 116. Nigmatilin, R. I. (1991). Dynamics of Multiphase Media (Vol. 1). New York: Hemisphere Publishing Corporation. 117. Odukoya, A., & Naterer, C. F. (2013). Droplet evaporation and de-pinning in rectangular microchannels. International Journal of Heat and Mass Transfer, 56, 127–137. 118. Ortega, I. K., Kupiainen, O., Kurtèn, T., Olenius, T., Wilkman, O., McGrath, M. J., et al. (2012). From quantum chemical formation free energies to evaporation rates. Atmospheric Chemistry and Physics, 12, 225–235. 119. Peeters, P., Luijten, C. C. M., & van Dongen, M. E. H. (2001). Transitional droplet growth and diffusion coefficients. International Journal of Heat and Mass Transfer, 44, 181–193. 120. Petrilla, B. A. (2008). Droplet Vaporization of n-heptane using Molecular Dynamics. A Master Thesis in Aerospace Engineering, The Pennsylvania State University, USA. 121. Petrilla, B. A., Trujillo, M. F., & Micci, M. (2010). N-heptane droplet vaporization using molecular dynamics. Atomization and Sprays, 20, 581–593. 122. Polikarpov, A. Ph., & Graur, I. A. (2018). Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases. International Journal of Heat and Mass Transfer, 124, 967–979. 123. Polikarpov, A. Ph., Graur, I. A., Gatapova E. Ya., & Kabov, O. A. (2019). Kinetic simulation of the non-equilibrium effects at the liquid-vapor interface. International Journal of Heat and Mass Transfer, 136, 449–456. 124. Popov, S. P., & Tcheremissine, F. G. (1999). Conservative method for the solution of the Boltzmann equation for centrally symmetrical interaction potentials. Computational Mathematics and Mathematical Physics, 39, 163–176. 125. Rogier, F., & Schneider, J. A. (1994). A direct method for solving the Boltzmann equation. Transport Theory and Statistical Physics, 23, 1–3. 126. Rose, J. W. (1998). Interphase matter transfer, the condensation coefficient and dropwise condensation. In Lee J. S. (Ed.), Heat Transfer 1998. Proceedings of 11th International Heat Transfer Conference August 23–28, 1998, Kyongji, Korea (Vol. 1, pp. 89–104). Seoul: The Korean Society of Mechanical Engineers. 127. Rose, J. W. (2000). Accurate approximate equations for intensive sub-sonic evaporation. International Journal of Heat and Mass Transfer, 43, 3869–3875. 128. Santos, A. (2003). Transport coefficients of d-dimensional inelastic Maxwell models. Physica A, 321, 442–466. 129. Santos, A. (2006). A simple model kinetic equation for inelastic Maxwell particles. In M. S. Ivanov, & A. K. Rebrov (Eds.), Proceedings of the 25th International Symposium on Rarefied Gas Dynamics, July 21–28th 2006, St. Petersburg, Russia (pp. 191–196). Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences. 130. Sazhin, S. S. (1978). Cyclotron whistler-mode instability in a collisional plasma. In Geomagnetic Research (Vol. 23, pp. 105–107). Moscow (in Russian): Soviet Radio. 131. Sazhin, S. S. (1993). Whistler-Mode Waves in a Hot Plasma. Cambridge: Cambridge University Press. 132. Sazhin, S. S. (2006). Advanced models of fuel droplet heating and evaporation. Progress in Energy and Combustion Science, 32(2), 162–214.

408

6 Kinetic Modelling of Droplet Heating and Evaporation

133. Sazhin, S. S., & Serikov, V. V. (1997). Rarefied gas flows: Hydrodynamic versus Monte Carlo modelling. Planetary and Space Science, 45, 361–368. 134. Sazhin, S. S., & Shishkova, I. N. (2009). A kinetic algorithm for modelling the droplet evaporation process in the presence of heat flux and background gas. Atomization and Sprays, 19, 473–489. 135. Sazhin, S. S., Wild, P., Leys, C., Toebaert, D., & Sazhina, E. M. (1993). The three temperature model for the fast-axial-flow CO2 laser. Journal of Physics D: Applied Physics, 26, 1872– 1883. 136. Sazhin, S. S., Wild, P., Sazhina, E. M., Makhlouf, M., Leys, C., & Toebaert, D. (1994). The three dimensional modelling of the processes in the fast-axial-flow CO2 laser. Journal of Physics D: Applied Physics, 27, 464–469. 137. Sazhin, S. S., Feng, G., Heikal, M. R., Goldfarb, I., Goldshtein, V., & Kuzmenko, G. (2001). Thermal ignition analysis of a monodisperse spray with radiation. Combustion and Flame, 124, 684–701. 138. Sazhin, S. S., Kristyadi, T., Abdelghaffar, W. A., & Heikal, M. R. (2006). Models for fuel droplet heating and evaporation: Comparative analysis. Fuel, 85, 1613–1630. 139. Sazhin, S. S., Shishkova, I. N., Kryukov, A. P., Levashov, V. Yu., & Heikal, M. R. (2007). Evaporation of droplets into a background gas: Kinetic modelling. International Journal of Heat and Mass Transfer, 50, 2675–2691. 140. Sazhin, S. S., Shishkova, I. N., Kryukov, A. P., Levashov, V. Yu., & Heikal, M. R. (2008). Evaporation of droplets into a background gas in the presence of heat flux: Kinetic and hydrodynamic modelling. In Proceedings of the 19th International Symposium on Transport Phenomena, 17–20 August, 2008, Reykjavik, Iceland, paper 48. 141. Sazhin, S. S., Shishkova, I. N., Kryukov, A.P., Levashov, V. Yu., & Heikal, M. R. (2009). A simple algorithm for kinetic modelling of Diesel fuel droplet evaporation. In Progress in Computational Heat and Mass Transfers; Proceedings of 6th ICCHMT, May 18–21, 2009, Guangzhou, China, paper 151 (pp. 386–391). 142. Sazhin, S. S., Shishkova, I. N., & Heikal, M. (2010). Kinetic modelling of fuel droplet heating and evaporation: Calculations and approximations. International Journal of Engineering Systems Modelling and Simulation, 2, 169–176. 143. Sazhin, S. S., Xie, J.-F., Shishkova, I. N., Elwardany, A. E., & Heikal, M. R. (2013). A kinetic model of droplet heating and evaporation: Effects of inelastic collisions and a non-unity evaporation coefficient. International Journal of Heat and Mass Transfer, 56, 525–537. 144. Sazhin, S. S., Shishkova, I. N., & Al Qubeissi, M. A. (2014). Heating and evaporation of a two-component droplet: Hydrodynamic and kinetic models. International Journal of Heat and Mass Transfer, 79, 704–712. 145. Sazhin, S. S., Gun’ko, V. M., & Nasiri, R. (2016). Quantum-chemical analysis of the processes at the surfaces of Diesel fuel droplets. Fuel, 165, 405–412. 146. Sazhin, S. S., Shishkova, I. N., & Al Qubeissi, M. A. (2016). A self-consistent kinetic model for droplet heating and evaporation. International Journal of Heat and Mass Transfer, 93, 1206–1217. 147. Sazhina, E. M., Sazhin, S. S., Heikal, M. R., Babushok, V. I., & Johns, R. (2000). A detailed modelling of the spray ignition process in Diesel engines. Combustion Science and Technology, 160, 317–344. 148. Schrage, R. W. (1953). A Theoretical Study of Interphase Mass Transfer. New York: Columbia University Press. 149. Shan, L., Ma, B., Li, J., Dogruoz, B., & Agonafer, D. (2019). Investigation of the evaporation heat transfer mechanism of a non-axisymmetric droplet confined on a heated micropillar structure. International Journal of Heat and Mass Transfer, 141, 191–203. 150. Shankar, P. N., & Marble, F. M. (1971). Kinetic theory of transient condensation and evaporation at a plane surface. Physics of Fluids, 14, 510–516. 151. Sharipov, F., & Kalempa, D. (2005). Velocity slip and temperature jump coefficients for gaseous mixtures. IV. Temperature jump coefficient. International Journal of Heat and Mass Transfer, 48, 1076–1083.

References

409

152. Shen, C. (1983). The concentration-jump coefficient in a rarefied binary gas mixture. Journal of Fluid Mechanics, 137, 221–231. 153. Shidlovskiy, V. P. (1967). Introduction to the Dynamics of Rarefied Gases. New York: American Elsevier Publishing Company. 154. Shishkova, I. N., Sazhin, S. S., & Xie, J.-F. (2013). A solution of the Boltzmann equation in the presence of inelastic collisions. Journal of Computational Physics, 232, 87–99. 155. Shishkova, I. N., Kryukov, A. P., & Levashov, V. Yu. (2017). Study of evaporationcondensation problems: From liquid through interface surface to vapor. International Journal of Heat and Mass Transfer, 112, 926–932. 156. Shishkova, I. N., Kryukov, A. P., & Levashov, V. Yu. (2019). Vapour-liquid joint solution for the evaporation-condensation problem. International Journal of Heat and Mass Transfer, 141, 9–19. 157. Shishkova, I. N., & Sazhin, S. S. (2006). A numerical algorithm for kinetic modelling of evaporation processes. Journal of Computational Physics, 218, 635–653. 158. Shishkova, I. N., & Sazhin, S. S. (2014). A solution of the Boltzmann equations in the presence of three components and inelastic collisions. International Journal of Heat and Mass Transfer, 71, 26–34. 159. Shusser, M., Ytrehus, T., & Weihs, D. (2000). Kinetic theory analysis of explosive boiling of a liquid droplet. Fluid Dynamics Research, 27, 353–367. 160. Sibold, D., & Urbassek, H. M. (1991). Kinetic study of evaporating flows from cylindrical jets. Physics of Fluids, 3, 870–878. 161. Simon, J.-M., Kjelstrup, S., Bedeaux, D., & Hafskjold, B. (2004). Thermal flux through a surface of n-octane. A non-equilibrium molecular dynamics study. Journal of Physical Chemistry, 108, 7186–7195. 162. Simon, J.-M., Bedeaux, D., Kjelstrup, S., Xu, J., & Johannessen, E. (2004). Integral relations verified by non-equilibrium molecular dynamics. Journal of Physical Chemistry, 110, 18528– 18536. 163. Sizhuk, A., & Yezhov, S. (2006). The dynamic theory for the inelastically colliding particles. Journal of Molecular Liquids, 127, 84–86. 164. Smit, B., Karaborni, S., & Siepmann, J. I. (1995). Computer simulations of vapour-liquid phase equilibria of n-alkanes. The Journal of Chemical Physics, 102, 2126–2140. 165. Sone, Y. (2000). Kinetic theoretical studies of the half-space problem of evaporation and condensation. Transport Theory and Statistical Physics, 29, 227–260. 166. Sone, Y., Aoki, K., Sugimoto, H., & Yamada, T. (1988). Steady evaporation and condensation on a plane condensed phase. Theoretical and Applied Mechanics (Bulgaria), 19, 89–93. 167. Sone, Y., Sugimoto, H., & Aoki, K. (1999). Cylindrical Couette flows of a rarefied gas with evaporation and condensation: Reversals and bifurcation of flows. Physics of Fluids, 11, 476–490. 168. Sone, Y., & Onishi, Y. (1973). Kinetic theory of evaporation and condensation. Journal of the Physical Society of Japan, 35, 1773–1776. 169. Sone, Y., & Onishi, Y. (1978). Kinetic theory of evaporation and condensation - hydrodynamic equation and slip boundary condition. Journal of the Physical Society of Japan, 44, 1981– 1994. 170. Sone, Y., & Sugimoto, H. (1990). Strong evaporation from a plane condensed phase, In G. E. A. Meier, & P. A. Thompson (Eds.), Adiabatic Waves in Liquid-Vapour Systems, Proceedings of IUTAM Symposium, Gottingen, Germany 1989, (pp. 293–304). Berlin: Springer. 171. Sone, Y., & Sugimoto, H. (1993). Kinetic theory analysis of steady evaporating flows from a spherical condensed phase into a vacuum. Physics of Fluids, 5, 1491–1511. 172. Sone, Y., & Sugimoto, H. (1995). Evaporation of rarefied gas from a cylindrical condensed phase into a vacuum. Physics of Fluids, 7, 2072–2085. 173. Stoecklin, T., & Voronin, A. (2007). H-N2 inelastic collision dynamics on new potential energy surface. Chemical Physics, 331, 385–395. 174. Sugimoto, H., & Sone, Y. (1992). Numerical analysis of steady flows of a gas evaporating from its cylindrical condensed phase on the basis of kinetic theory. Physics of Fluids, 4, 419–440.

410

6 Kinetic Modelling of Droplet Heating and Evaporation

175. Taguchi, S., Aoki, K., & Takata, S. (2003). Vapor flows condensing at incidence onto a plane condensed phase in the presence of noncondensable gas. I. Subsonic condensation. Physics of Fluids, 15, 689–705. 176. Taguchi, S., Aoki, K., & Takata, S. (2004). Vapor flows condensing at incidence onto a plane condensed phase in the presence of noncondensable gas. I. Supersonic condensation. Physics of Fluids, 16, 79–92. 177. Takata, S., & Aoki, K. (1999). Two-surface problems of a multi-component mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory. Physics of Fluids, 11, 2743–2756. 178. Tcheremissine, F. G. (1997). Conservative method for the solution of the Boltzmann equation for centrally symmetrical interaction potentials. Reports of Russian Academy of Science, 357, 53–56 (in Russian). 179. Tcheremissine, F. G. (1998). Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation. Computers & Mathematics with Applications, 35, 215–21. 180. Tcheremissine, F. G. (2000). Discrete approximation and examples of the solution of the Boltzmann equation. Computational Dynamics of Rarefied Gases (pp. 37–74). Moscow: Computer Centre of Russian Academy of Sciences. 181. Thomas, J. P., Chang, T. S., & Siewert, S. E. (1974). Reverse temperature gradient in the kinetic theory of evaporation. Physical Review Letters, 33, 680–682. 182. Tilinin, I. S. (1996). Impact-parameter dependence of inelastic energy losses in slow atomatom collisions. Nuclear Instruments and Methods in Physics Research B, 115, 102–105. 183. Titarev, V. A., & Shahov, E. M. (2002). Heat loss and evaporation from a flat surface after a rapid increase of its temperature. Reports of the Russian Academy of Sciences: Mechanics of Liquids and Gases No., 1, 141–153 (in Russian). 184. Tsuruta, T., & Nagayama, G. (2004). Molecular dynamics studies on the condensation coefficient of water. Journal of Physical Chemistry B, 108, 1736–1743. 185. Tsuruda, T., Tanaka, H., & Masuika, T. (1999). Condensation/evaporation coefficient and velocity distribution at liquid-vapour interface. International Journal of Heat and Mass Transfer, 42, 4107–4116. 186. Versteeg, H. K., & Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics (2nd ed.). Harlow (UK): Longman. 187. Walther, J. H., & Koumoutsakos, P. (2001). Molecular dynamics simulation of nanodroplet evaporation. ASME Journal of Heat Transfer, 123, 741–748. 188. Wei, M., Yang, S., Sun, H., Wang, Y., & Guo, G. (2021). Molecular dynamics simulation of sub/supercritical evaporation with n-butanol/n-heptane blended fuel. Fuel, 294, 120556. 189. Wu, X., Yang, Z., & Duan, Y. (2020). Molecular dynamics simulation on evaporation of a suspending difluoromethane nanodroplet. International Journal of Heat Mass Transfer, 158, 120024. 190. Xie, J.-F., Sazhin, S. S., & Cao, B.-Y. (2011). Molecular dynamics study of the processes in the vicinity of the n-dodecane vapour/liquid interface. Physics of Fluids, 23, 112104. 191. Xie, J.-F., Sazhin, S. S., & Cao, B.-Y. (2012). Molecular dynamics study of condensation/evaporation and velocity distribution of n-dodecane at liquid-vapour phase equilibria. Journal of Thermal Science and Technology, 7, 288–300. 192. Xu, X., Cheng, C., & Chowdhury, I. H. (2004). Molecular dynamics study of phase change mechanisms during fentosecond laser ablation. ASME Journal of Heat Transfer, 126, 727–734. 193. Yanagihara, H., Stankovi´c, I., Blomgren, F., Rosén, A., & Sakata, I. (2014). A molecular dynamics simulation investigation of fuel droplet in evolving ambient conditions. Combustion and Flame, 161, 541–550. 194. Yang, T. H., & Pan, C. (2005). Molecular dynamics simulation of a thin water layer evaporation and evaporation coefficient. International Journal of Heat and Mass Transfer, 48, 3516–3526. 195. Young, J. B. (2011). Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods. International Journal of Heat and Mass Transfer, 54, 2902– 2912.

References

411

196. Young-ping, P. (1971). Application of kinetic theory to the problem of evaporation and condensation. Physics of Fluids, 14, 306–311. 197. Ytrehus, T. (1977). Theory and experiments on gas kinetics in evaporation. In J. L. Potter (Ed.), Rarefied Gas Dynamics Part 2 (pp. 1197–1212). New York: AIAA. 198. Ytrehus, T., & Aukrust, T. (1986), Mott-Smith solution for weak condensation. In V. Boffi, & C. Cercignani (Eds.), Rarefied Gas Dynamics (Vol. 2, pp. 271–80). Stuttgart: Teubner. 199. Ytrehus, T., & Ostmo, S. (1996). Kinetic approach to interphase processes. International Journal of Multiphase Flow, 22, 133–135. 200. Zahn, D. (2008). Length-dependent nucleation mechanisms rule the vaporization of n-alkanes. Chemical Physics Letters, 467, 80–83. 201. Zenevich, V. A., Billing, G. D., & Jolicard, G. (1999). Vibrational-rotational energy transfer in H2 -H2 collisions II. The relative roles of the initial rotational excitation of both diatoms. Chemical Physics Letters, 312, 530–535. 202. Zhang, Y., Zhang, L., Dong-Ming, M., Wu, C.-M., & Li, Y.-R. (2020). Numerical investigation on flow instability of sessile ethanol droplets evaporating in its pure vapor at low pressure. International Journal of Heat and Mass Transfer, 156, 119893.

Chapter 7

Heating, Evaporation and Autoignition of Sprays

In many important engineering applications, heating and evaporation of fuel droplets leads to autoignition of fuel vapour/air mixture. Temperature rise of this mixture during the autoignition process leads to the enhancement of droplet heating and evaporation. Hence, the analysis of fuel droplet heating and evaporation in realistic engine conditions should be coupled with the analysis of autoignition. Possible approaches to coupled solutions of the equations describing these processes are described in Sects. 7.2 and 7.3. Section 7.1 focuses on a brief review of several approaches to the modelling of autoignition (in the absence of droplets or in their presence).

7.1 Autoignition Modelling Autoignition is defined as the onset of combustion in a reactive medium raised above a certain temperature and pressure [64, 111]. Sometimes autoignition is called spontaneous ignition, self-ignition or homogeneous ignition [64]. It can be triggered by rapid compression of the fuel-oxidizer mixture, as in conventional Diesel engines, or by pressure waves due to very fast heat release, as in spark ignition engines. In early publications, autoignition was considered as a one-step chemical reaction [30, 120]. In the simplest case, the chemical power released in the gas phase per unit volume was inferred from the Arrhenius formula [64]:   E , Pch = M f Q f C f A exp − Ru Tg

(7.1)

where M f is the molar mass of fuel, Q f the specific combustion energy, C f the fuel vapour molar concentration, A the pre-exponential factor (in 1/s), E the activation energy (in J/kmol). Equation (7.1) assumes that there is no deficiency of oxygen in the gas phase. The simplicity of Expression (7.1) and its straightforward generalisations © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_7

413

414

7 Heating, Evaporation and Autoignition of Sprays

(e.g. [64]) make it attractive for qualitative study of the autoignition process (e.g. [18, 38–40, 73, 97]). In most practical applications, however, the autoignition chemistry is much more complex and its modelling cannot be based on Eq. (7.1). An in-depth review can be found in [2]. In what follows, only a brief review of autoignition models is given, focusing their applicability to the study of autoignition in Diesel engine-like conditions. The autoignition process is characterised by two key parameters: the lowest initial ambient gas temperature above which autoignition can develop, Tg0 , and the time delay before the start of autoignition τd [14]. These parameters depend on pressure, fuel composition, condition of internal surface of the engine and other factors [8, 9, 31, 74, 76, 114]. The detailed kinetic mechanism (DKM) of the autoignition process typically involves up to about 1000 chemical reactions and hundreds of species [16, 27, 32, 62, 75, 87, 90, 134]. Additional problems in the construction of DKMs are related to the lack or scarcity of kinetic data for many of the reactions considered in this approach. Even quantum chemical models (see Sect. 6.6) do not always lead to accurate prediction of rate constants due to the fact that the decisive factors in this calculation are the small value differences between the high energy levels of reacting molecules [6]. The incorporation of DKMs into CFD codes used for the modelling of the autoignition process would be an almost impossible task. This stimulated the development of several reduced chemical mechanisms (e.g. [7, 8, 11, 48–51, 54, 69, 88, 93, 116]). The one described in [48] was used in [115] for modelling autoignition of isolated n-heptane droplets. The general approach to simplification of DKMs was discussed in [70]. Poppe et al. [89] discussed the approach using thirty reactions and twenty one species. The authors of [7] reduced the number of reactions to twenty one and the number of species to thirteen. Even this simplification, however, was not sufficient to describe the chemical reactions together with the flow and heat/mass transfer processes in realistic 3D geometries. Müller et al. [77] simplified the model further to a four-step model. The applicability of this approach was questioned by Griffiths in the discussion of this paper. He drew attention to the fact that in this model, ignition was brought about entirely by thermal feedback. The chain branching process was not considered. This cannot be justified in a model which intends to capture the main features of the autoignition process. The authors of [128] developed a reduced kinetic mechanism for kerosene combustion in air. Their model uses 30 species and 77 irreversible reactions. It was developed to accurately reproduce key flame parameters while being sufficiently small to be used in CFD codes, including those based on Direct Numerical Simulations. A reduced mechanism of 144 species and 653 reactions for the analysis of combustion and soot emission of gasoline-ethanol blends is described in [129]. This mechanism was derived from a detailed chemical mechanism using the gasoline surrogate mixture of n-heptane, iso-octane, toluene and 1-hexene. Several authors described turbulent autoignition models to consider the contribution of flow characteristics [12, 13, 15, 35, 57, 66, 121, 130, 131]. Although these approaches are very useful for the in-depth understanding of the process, the com-

7.1 Autoignition Modelling

415

plexity of turbulence modelling leads to considerable simplification of the chemical kinetics. For example, the model for turbulent combustion described in [121, 131] uses the above-mentioned four-step model [77]. The results of turbulent ignition modelling alongside complex chemistry are described in [84]. The authors of [133] used Direct Numerical Simulation to investigate n-dodecane spray autoignition with an emphasis on low temperature reactions. A review of key and representative developments in the area of turbulent spray combustion with a focus on spray-chemistry-turbulence interactions was presented by the authors of [132]. The effects of turbulence-chemistry, spray-turbulence and spray-chemistry interactions on the spray dynamics, ignition, flame stabilisation and emission at high pressures and temperatures are described and discussed in that review. The approach developed by researchers from Shell Research Ltd focused on capturing the essential features of the process rather than on the construction of a chemically and physically rigorous model [55]. This was achieved by introducing generic species with chemical reaction constants inferred from experimental data. As a result, they developed a model using the eight step chain branching reaction scheme incorporated into four processes: Initiation: RH + O2 → 2R∗ Propagation :

Branching:

Termination:

R∗ → R∗ + P R∗ → R∗ + B R∗ → R∗ + Q R∗ + Q → R∗ + B B → 2R∗ R∗ → out 2R∗ → out

where RH is hydrocarbon fuel (Cn H2m ), R∗ the radical, B the branching agent, Q the intermediate agent and P the product, including CO, CO2 and H2 O. Following [9] the branching agent is related to hydroperoxide (RO2 H) at low temperatures and to hydrogen peroxide (H2 O2 ) at high temperatures. The intermediate species were related to aldehydes (RCHO). This model, called the Shell model, was validated and further developed by many authors [29, 56, 63, 78, 96, 102, 103, 105, 117] and is widely used in automotive applications. In [96], the equations of the Shell model were revisited with a view to their more effective implementation into Computational Fluid Dynamics (CFD) codes. The simplification of the solution procedure without compromising accuracy was achieved

416

7 Heating, Evaporation and Autoignition of Sprays

by replacing time as an independent variable with the fuel vapour depletion, which is the difference between the initial fuel vapour molar density and the current one. All the other variables of the model, including temperature, molar density of oxygen, radicals, intermediate and branching agents were presented as functions of fuel vapour depletion. Equations for the temperature and molar density of the intermediate agent were of the first order and allowed analytical solutions. The molar densities of oxygen and fuel vapour were related via a straightforward algebraic equation. The numerical solution of five coupled first-order ordinary differential equations was reduced to the solution of only two coupled first-order differential equations for the molar density of radicals and the branching agent. It is possible to rearrange these equations even further so that the equation for the molar density of the radicals is uncoupled from the equation for the branching agent. In this case, the equation for the molar density of radicals becomes a second-order ordinary differential equation. This equation is solved analytically in two limiting cases and numerically in the general case. The solution to the first-order ordinary differential equation for the molar density of the branching agent and the solution to the first-order differential equation for time are presented in the form of integrals containing the molar density of the radicals obtained earlier. This approach allows the Central Processing Unit (CPU) time to be more than halved and makes the calculation of the autoignition process using the Shell model much more effective compared with the one based on the original Shell model equations. The Shell model was applied to modelling autoignition in gasoline and Diesel engines by the authors of [102]. The complexities of modelling autoignition in Diesel sprays were highlighted. In contrast to autoignition in gasoline engines, autoignition of Diesel fuel sprays takes place at a wide range of equivalence ratios and temperatures. This makes it necessary to impose flammability limits to restrict the range of equivalence ratios in which the autoignition model is used. The autoignition chemical delay for n-dodecane was demonstrated to be much shorter than the physical delay due to the droplet transit time, atomization, heating, evaporation and mixing (see the analysis later in this chapter). This justified the application by the authors of [102] of the less accurate (than DCM) but more computer efficient Shell model. The mathematical formulation developed in [96] was used for the analysis of Diesel fuel autoignition chemical delay. Since experimental data for the autoignition chemical delay for n-dodecane are not available, the modelling of the Diesel fuel autoignition process, using the Shell model, was based on data for n-heptane. The autoignition time delays for premixed n-heptane predicted by calculations using the kinetic rate parameters for the primary reference fuel, RON70, showed good agreement with experimental results when A f 4 (pre-exponential factor in the rate of production of the intermediate agent) was chosen in the range 3 × 106 to 6 × 106 . It was demonstrated that the difference between the end-of-compression temperature, predicted by the adiabatic law and the actual end-of-compression temperature, considering the exothermic reactions at the end of compression, needs to be taken into account. The relation between the two temperatures was approximated by a linear function. This approach was used for n-dodecane [102].

7.1 Autoignition Modelling

417

Using the analysis of the autoignition of a monodisperse n-dodecane spray in Diesel engines-like conditions, it was demonstrated that for droplets with initial radii (Rd0 ) about or larger than 6 µm the physical autoignition delay (due to droplet heating and evaporation) dominates over the chemical autoignition delay (time required for chemical reactions), while for the droplets with Rd0 ≤ 2.5 µm the chemical autoignition process takes most of the time [103]. A detailed investigation of physical and chemical ignition delay times for sprays of biodiesel-Diesel blends in CompressionIgnition (CI) engines is presented by the authors of [86]. The Shell model, using the mathematical formulation developed in [96], was applied to modelling spray autoignition processed in Diesel engines [103]. The start of the autoignition process was predicted near the periphery of both monodisperse and polydisperse sprays in agreement with common understanding of this phenomenon. A more rigorous approach to the analysis of evaporating polydisperse sprays using the direct quadrature-based sectional method of moments is described in [52]. The autoignition stage of the polydisperse Diesel spray combustion predicted by the Shell autoignition model in combination with conventional models for droplet heating and evaporation (ignoring the temperature gradient inside droplets) was demonstrated to agree with available experimental data for a medium duty truck Diesel engine [103]. Since most of the droplets in Diesel engines have radii larger than 6 µm, accurate modelling of droplet heating and evaporation is more important than accurate modelling of the chemical autoignition process. This gives additional support for the application of the Shell model for modelling the latter process when compared with more accurate, but more complex models, such as DCM. This stimulated the development of more accurate models for droplet heating and evaporation. Note that the chemical autoignition delay is strongly influenced by the droplet surface temperature, the prediction of which depends on the chosen model of droplet heating and evaporation (see Chaps. 3 and 4 for the details). Analysis of the combustion of fuel sprays, following the autoignition process, is beyond the scope of the book (see [26, 33, 34, 36, 65, 92, 122, 125, 126]). Analysis of experimental studies of spray autoignition and combustion in internal combustion engines can be found in [3, 4]. As mentioned earlier, considering finite thermal conductivity and recirculation in droplets increases the modelling accuracy of the autoignition process in Diesel engine-like conditions [10]. This effect was investigated in [99]. In that paper, the effect of the temperature gradient and recirculation inside fuel droplets on droplet evaporation, break-up and the autoignition of fuel vapour/air mixture was considered using a zero-dimensional code. This code considered the coupling between the liquid and gas phases and described the autoignition process using the Shell model with the mathematical formulation developed in [96]. The effect of temperature gradient and recirculation inside droplets was studied by comparing the predictions of Effective Thermal Conductivity (ETC) model and the Infinite Thermal Conductivity (ITC) model, both of which were implemented in this code. It was demonstrated that in the absence of break-up, the effect of the temperature gradient and recirculation in droplets on droplet evaporation in Diesel engine-like conditions is small in most cases (a few percent).

418

7 Heating, Evaporation and Autoignition of Sprays

In the presence of the droplet break-up processes, however, the temperature gradient and recirculation inside the droplets was shown to lead to a significant decrease in evaporation time. This was attributed to the fact that the effect of temperature gradient inside droplets leads to a substantial increase in droplet surface temperature at the initial stage of its heating. This is translated into a decrease in surface tension and the threshold radii of the unstable droplets. Even in the absence of break-up, the effect of the temperature gradient inside the droplets was demonstrated to lead to a noticeable decrease in the total autoignition delay. In the presence of break-up, this effect was shown to enhance substantially, leading to more than halving of the total autoignition delay. It was suggested that the effect of the temperature gradient and recirculation inside droplets is considered in CFD codes focused on the analysis of droplet break-up and evaporation processes, and the autoignition of the fuel vapour/air mixture. Modelling autoignition in sprays is different from modelling the interaction between sprays and developed flames. The latter problem was investigated in several papers, including [58, 123]. The results presented in [10, 99] are based on the coupled solutions of ordinary differential equations for droplets and chemical reactions. A review of the mathematical aspects of these coupled solutions is given in Sect. 7.2.

7.2 Coupled Solutions: Simple Models In this section the link between droplet heating, evaporation and the autoignition of the mixture of fuel vapour with air is described using relatively simple models, following [97, 100, 101]. A summary of physical (Sect. 7.2.1) and mathematical (Sect. 7.2.2) formulations of one of these models is presented. The results of preliminary analysis of the equations presented in Sect. 7.2.2 is given in Sect. 7.2.3. Section 7.2.4 focuses on two approaches to the analysis of the systems with Non-Lipschitzian non-linearities. The results of application of mathematical tools described in Sect. 7.2.4 to the analysis of spray ignition and combustion with radiation are presented in Sect. 7.2.5.

7.2.1 Physical Model Following [42], spray autoignition and combustion are described as an explosion problem, where droplets are considered as the source of endothermicity. The endothermic versus exothermic competition controls explosion regimes and their dependence on the physical and chemical parameters of the system. Spray is considered to be monodisperse and droplets are assumed to be stationary. The first assumption is justified by the fact that a well-defined maximum in droplet distribution by radii is observed in most experiments, although in this case some

7.2 Coupled Solutions: Simple Models

419

crucial effects linked with the polidispersity of realistic sprays are overlooked [46, 47]. The second assumption is not valid near the nozzles, but it is commonly accepted in the regions away from the nozzles where most droplets are entrained by the ambient air (see Sect. 1.2.2 and [98]). The evaporation process is described by the hydrodynamic model using the assumption that fuel vapour at the droplet surfaces is saturated [94]. The autoignition process is described based on the Arrhenius approximation (Eq. (7.1)). The radiative heat exchange between droplets and ambient air is considered, assuming that droplets are grey opaque spheres, and the radiative temperature is equal to ambient air temperature (air is optically thick). This approximation is applicable for relatively large droplets (with radii more than about 0.5 mm). In the general case, the effects of semi-transparency of droplets would need to be considered [94]. Spalding mass (B M ) and heat (BT ) transfer numbers are assumed to be small (B M,T 0. Since ν  γ /β < 1 it can be expected that the variations of p and s are small compared with variations of q, at least until a steady state is reached. Hence, it can be assumed that p˙ = 0 and s˙ = 0, which means that p and s conserve their initial values: (7.18) p ≡ ln(1 + βθ0 ), s ≡ νψ + η0 . The validity of this assumption was demonstrated in [97], using numerical integrating Eqs. (7.12)–(7.14) (see Fig. 2 of [97]). Expressions (7.18) lead to the presentation of Eq. (7.17) as dq = −A q 1/3 − Bq 2/3 , d τ¯

(7.19)

424

7 Heating, Evaporation and Autoignition of Sprays

where   A = ε1 θ0 (1 + βθ0 )1/2 , B = ε1 ε3 (1 + βθ0 )4 − 1 , q(0) = 1. The closeness of the solutions to Eq. (7.17) (in combination with Eqs. (7.12)–(7.13)) and (7.19) was demonstrated using direct integration of these equations. Assumption 2 follows from the fact that q = 0 is the only root of the equation − A q 1/3 − Bq 2/3 = 0

(7.20)

for q ≥ 0. To check Assumption 3, function V (q) = q 2 /2 was considered. It was shown that this is a Lyapunov function of Eq. (7.19) for the solution q = 0. Indeed, V (q) > 0 when q = 0, V (q) = 0 when q = 0 and dV = −A q 4/3 − Bq 5/3 < 0, q > 0. d τ¯ Thus function V (q) = q 2 /2 is the Lyapunov function and Solution q = 0 is asymptotically stable. The domain of attractivity is uniform since the above-mentioned Lyapunov function exists for all q > 0. Hence, Assumption 3 is satisfied for q ≥ 0. Having substituted Solution q = φ( p, s) = 0 into Systems (7.12)–(7.14), the latter is presented as   p e −1 γ dp , = s exp β dτ βe p   p ds e −1 . = −s exp dτ βe p

(7.21) (7.22)

System (7.21)–(7.22) corresponds to Eq. (W.5). This system is Lipschitzian and has a unique solution [28]. Hence, Assumption 4 is satisfied. Assumption 5 is satisfied since any point q0 > 0 belongs to the domain of attractivity of the steady state q = 0. This means that (W.8) and (W.9) are satisfied for q > 0 and t > 0, and the solution to Eq. (7.19) converges uniformly to q = 0 for 0 < t ≤ T . For t > T , q = 0 is the solution to this equation. Hence, the line q = 0 is a positively invariant manifold. This allows us to perform the order reduction of the original non-Lipschitzian Systems (7.12)–(7.14) and reduce it to Systems (7.21)–(7.22) with positively invariant manifold q = 0. The analysis of Systems (7.21) and (7.22), describing the ignition of the mixture of fuel vapour and air, and equation ν

dq = −A q 1/3 − Bq 2/3 , dτ

(7.23)

7.2 Coupled Solutions: Simple Models

425

for the initial reduction of droplet radius, was performed earlier in [97] without correct mathematical justification of the validity of these equations.

7.2.4.2

Approach Based on the Lyapunov Function Theory

One of key limitations of the analysis in Sect. 7.2.4.1 is that it was based on the assumption that ε  1. In what follows a more general analysis of the problem focused on the system x˙ = f (x, y), y˙ = g(x, y) (7.24) is presented following [101]. In contrast to System (7.15), System (7.24) is not singularly perturbed. Recall that a surface y = ℵ(x) (x ∈ R m , y ∈ R n ) is a positively invariant manifold of System (7.24) if any trajectory x = x(t), y = y(t) predicted by this system that has at least one point (x0 , y0 ) in common with the surface y = ℵ(x), i.e. y0 = ℵ(x0 ) at t = 0, lies entirely on this surface for all t > 0. Attractive positively invariant manifolds were used in [101] for non-Lipschitzian system reduction. In their analysis of non-Lipschitzian systems Lyapunov function theory was used. The reduction of non-Lipschitzian systems was justified by the attractivity of positively invariant manifolds. The authors of [101] focused their analysis on the following system: x˙ = f (x, y) y˙ = ψ(y)g(x, y),

 (7.25)

where x and y are a vector and scalar, respectively; scalar function ψ(y) is nonLipschitzian. It was assumed that g(x, y) is bounded (0 < ℘1 ≤ g(x, y) ≤ ℘2 ) for sufficiently small non-negative values of y, f (x, y) and g(x, y) are continuous functions. Firstly, a simple case: ψ = −y α (0 < α < 1) was considered. Since the righthand side of the equation for y in (7.25) is zero at y = 0, any trajectory, described by (7.25), with initial point (x0 , 0) on the surface y = 0 lies on this surface for all t ≥ 0. This surface, however, is not invariant since not all trajectories of System (7.25) which have at least one point in common with this surface lie entirely on it, as trajectories can leave this manifold when t decreases. This surface, however, is positively invariant and any solution to System (7.25) with initial point (x0 , y0 ) with sufficiently small positive y0 reaches this surface during a finite time interval. Moreover, it is attractive. To prove this the approach suggested in [85] was used. This approach is based on considering the Lyapunov function V (y) = y 2 /2 with the derivative V˙ (x, y) = −y 1+α g(x, y). This derivative is negative for y > 0 for all values of x under consideration. This implies the asymptotic stability of y = 0 with respect to variable y, i.e. y → 0 as

426

7 Heating, Evaporation and Autoignition of Sprays

t increases. The same analysis could be performed for more complex functions as discussed in [101]. In the same paper, the closeness of the results predicted by the theory of positively invariant manifolds and direct numerical simulations was demonstrated. In Sect. 7.2.5, this approach is used for the analysis of spray autoignition and combustion in the presence of thermal radiation using the formulation of the equations originally suggested in [42]. Our analysis will follow the one described in [101].

7.2.5 Spray Ignition and Combustion Model with Radiation Following [42], a spatially homogeneous mixture of an optically thin, combustible gas (mixture of fuel vapour and air) with a monodispersed spray of evaporating spherical fuel droplets is considered. Both convective and radiative heating of droplets are considered as in the model described in Sects. 7.2.1–7.2.3. Also, as in the case described in Sects. 7.2.1–7.2.3 the distortion of the incident radiation by surrounding droplets and the effects of droplet movement are ignored (the Nusselt (Nu) and Sherwood (Sh) numbers are taken equal to 2). It is assumed that the incident radiation has a black-body spectrum. The system is assumed to be adiabatic. With a view to the application of the results to Diesel engine-like conditions, gas pressure is assumed to be constant. The thermal conductivity of droplets is assumed to be infinitely large. The volume fraction of the liquid phase is assumed to be much less than that of the gaseous phase. Thus, the heat transfer coefficient of the mixture of droplets and gas is controlled by the gas thermal properties. In contrast to the model described in Sects. 7.2.1–7.2.3 the radiation is assumed to be absorbed inside the droplets and not at their surfaces. The combustion process is described by the first-order exothermic reaction in the gaseous phase. The effects of the Stefan flow on droplet heating and evaporation are not considered (Spalding heat and mass transfer numbers are assumed to be much less than 1). The range of applicability of these assumptions is discussed in Chap. 3. In contrast to the model described in Sects. 7.2.1–7.2.3 the presence of oxygen in the system is assumed to be limited (the equation for oxygen molar density is considered in the model), and the changes in droplet temperatures are considered. Using these assumptions the process is described by the following equations [42]: c pg ρg ϕg

dTg = ωM ˙ f Q f ϕg − 4π Rd2 n d qc , dt

(7.26)

dC f (qc + qr ) = −ν f ω˙ + 4π Rd2 n d (1 − ζ (Td )) , dt L M f ϕg

(7.27)

dCox = −νox ω, ˙ dt

(7.28)

7.2 Coupled Solutions: Simple Models

427

dTd = 4π Rd2 (qc + qr )ζ (Td ), dt   (qc + qr ) d 4 3 π Rd ρ f = −4π Rd2 (1 − ζ (Td )) , dt 3 L c f md

where ω˙ =

a f bx C f Cox

  E , A exp − Ru Tg

ζ (Td ) = qc = h(Tg − Td ), h =

(7.29)

(7.30)

(7.31)

Tb − Td , Tb − Td0

kg 4 , qr = k1 σ Text , k1 = a Rdb , (Rd is in µm) Rd 

a = a0 + a1  b = b0 + b1

Text 103 Text 103



 + a2



 + b2

Text 103 Text 103

2 ,

(7.32)

,

(7.33)

2

ν f and νox are stoichiometric coefficients for fuel vapour and oxidiser, coefficients a f and b f were taken from [124]. All other notations are the same as used previously. Expression (7.31) is similar to Expression (7.73) discussed later in Sect. 7.3.3, although presented in a slightly different format. The following initial conditions are used: Td (0) = Td0 , Tg (0) = Tg0 , Rd (0) = Rd0 , C f (0) = C f 0 , Cox (0) = Cox0 . Gas is considered to be optically thin and the radiation absorption in droplets is controlled by the external temperature Text . It is assumed that ρg ϕg = const (the process takes place at constant pressure). Note that Expression (7.31) used in the model is more accurate than the Arrhenius approximation (7.1). Using the following dimensionless variables: θg =

Cf E Tg − Td0 E Td − Td0 Rd Cox , θd = , rd = , η= , ξ= , Ru Td0 Td0 Ru Td0 Td0 Rd0 Cff Cox0 τ=

t treact

, treact =

1 a f −0.5

AC f f

bx−0.5 Cox0

  Ru Td0 1 , β= , exp β E

428

7 Heating, Evaporation and Autoignition of Sprays

γ =

c pg Td0 ρg β 1 4π 3 , Cff = (1 + ω f ), ω f  1, Rd0 ρ f n d 0.5 (Cox0 C f f ) Q f M f 3 Mf

0.5    Q f ϕg M f Cox0 C f f 1 , ε2 = ε1 = a f exp , bx β ρ f Lϕ f C f f Cox0 AQ f ϕg M f 4π Rd0 n d k g0 Td0 β

ε3 =

3 σ Rd0 k10 4Td0 c f Td0 β , , ε4 = λg0 L

1 ν˜ f = νf

Cff 1 , ν˜ ox = Cox0 νox



Cox0 , Cff

additional subscripts ‘0’ show the initial values, we can rewrite Expressions (7.26)– (7.30) as  dθg 1 = P1 (θg , η, ξ ) − P2 (θg , θd , rd ) , (7.34) dτ γ 1 dη = dτ ν˜ f

   ψ −P1 (θg , η, ξ ) + P23 (θg , θd , rd ) 1 − ζ (θd ) , νf dξ 1 P1 (θg , η, ξ ), =− dt ν˜ ox

(7.35)

(7.36)

ε2 dθd = P23 (θg , θd , rd )ζ (θd ), dτ ε4 r 3     d rd3 = −ε2 P23 (θg , θd , rd ) 1 − ζ (θd ) , dτ

(7.37)

(7.38)

where  P1 (θg , η, ξ ) = ηa f ξ bx exp

P3 (rd ) =

θg 1 + βθg



 , P2 (θg , θd , rd ) = ε1 rd

Td0 (1 + βθg ) (θg − θd ), Tg0

4 ε1 ε3 2+β  rd 1 + βθgext , P23 (θg , θd , rd ) = P2 (θg , θd , rd ) + P3 (rd ), 4β

θgext =

1 Text − Td0 Tb − Td0 (1 + βθd ) , ζ (θd ) = , β Td0 Tb − Td0

with the initial conditions: θg (0) = θg0 = 0, θd (0) = θd0 = 0,

7.2 Coupled Solutions: Simple Models

429

r (0) = r0 = 1, η(0) = η0 , ξ(0) = ξ0 = 1. Introducing a new variable q = rd3 , Eqs. (7.34)–(7.38) are rewritten as  dθg 1 = P1 (θg , η, ξ ) − P2 (θg , θd , q 1/3 ) , dτ γ dη 1 = dτ ν˜ f

   ψ 1/3 −P1 (θg , η, ξ ) + P23 (θg , θd , q ) 1 − ζ (θd ) , νf

(7.39)

(7.40)

dξ 1 P1 (θg , η, ξ ), =− dt ν˜ ox

(7.41)

dθd ε2 = P23 (θg , θd , q 1/3 )ζ (θd ), dτ ε4 q

(7.42)

  dq = −ε2 P23 (θg , θd , q 1/3 ) 1 − ζ (θd ) . dτ

(7.43)

System (7.39)–(7.43) is identical to the one derived in [42]. The analysis of this system, performed by its authors, was correct except they referred to invariant manifolds instead of positively invariant manifolds. In what follows this part of their analysis is corrected and some assumptions made in [42] are relaxed, following [101]. System (7.39)–(7.43) has the positively invariant manifold q ≡ 0. To prove this, the system is represented in the form of (7.25) in which q plays the role of y and the vector with coordinates θg , η, ξ, θd plays the role of vector x. Moreover, the right-hand side of (7.43) can be presented as −q 1/3 g(x, q) for  g(x, q) = ε2 ε1

 4   Td0 (1 + βθg ) ε1 ε3 (1+β)/3  q 1 + βθgext (θg − θd ) + 1 − ζ (θd ) . Tg0 4β

To prove the attractivity of this manifold the same approach as in Sect. 7.2.4.2 was used, and Lyapunov function V (q) = q 2 /2 was considered:   V˙ (q) = −qε2 P23 (θg , θd , q 1/3 ) 1 − ζ (θd ) .

(7.44)

V˙ (q) is negative for all θg and θd under consideration. This implies the asymptotic stability of q = 0 with respect to q, i.e. q → 0 as t → +∞. Equations (7.39)–(7.43) have the partial integral [42]: ε4  q = eθd (ζ (θd ))θdb .

(7.45)

430

7 Heating, Evaporation and Autoignition of Sprays

As follows from (7.45), ζ (θd ) → 0 as q → 0 , i.e. the droplet surface temperature approaches the boiling temperature (θd → θdb ) when q → 0. Having substituted q = 0 and θd = θdb 1 into System (7.39)–(7.43), the latter is simplified to dθg 1 = P1 (θg , η, ξ ), dτ γ

(7.46)

1 dη = − P1 (θg , η, ξ ), dτ ν˜ f

(7.47)

1 dξ =− P1 (θg , η, ξ ). dt ν˜ ox

(7.48)

γ θg + ν˜ f η = γ θg0 + ν˜ f η0

(7.49)

γ θg + ν˜ ox ξ = γ θg0 + ν˜ ox

(7.50)

Two integrals

and of System (7.46)–(7.48) allow us to exclude equations for η and ξ from the following investigation and to obtain the final equation for θg :   dθg 1 γ γ = P1 θg , η0 − (θg − θg0 ), 1 − (θg − θg0 ) . dτ γ ν˜ f ν˜ ox

(7.51)

Thus, using positively invariant manifold q = 0 for System (7.39)–(7.43) the order reduction of the original non-Lipschitzian system (7.34)–(7.38) is performed and scalar Eq. (7.51) for θg is obtained. Remembering the definition of P1 and Eqs. (7.49) and (7.50) Expression (7.51) is rewritten in the form:   a f  bx  θg γ γ 1− . η0 − (θg − θg0 ) (θg − θg0 ) exp ν˜ f ν˜ ox 1 + βθg (7.52) Equation (7.52) has two steady states dθg 1 = dτ γ

θ¯g = θg0 + η0 ν˜ f /γ and θ¯¯g = θg0 + ν˜ ox /γ . Remembering that for the nearest approximation of Diesel fuel available in [124] (C10 H22 ), we have a f = 0.25 and bx = 1.5. Hence, the right-hand side of Eq. (7.52) is non-Lipschitzian in the neighbourhoods of these steady states. 1

From the point of view of the physical background of the problem this implies that the droplets are assumed to have evaporated; note that, in contrast to [100], q is not assumed to be the fastest variable.

7.2 Coupled Solutions: Simple Models

 0.25 γ η0 − (θg − θg0 ) and ν˜ f

431

 1.5 γ 1− (θg − θg0 ) ν˜ ox

have physical meaning only when: η0 −

γ (θg − θg0 ) ≥ 0 and ν˜ f

1−

γ (θg − θg0 ) ≥ 0. ν˜ ox

Thus the dimensionless gas temperature θg increases and attains the steady state value min{θ¯g , θ¯¯g }. The examples, considered in this section allow us to understand the underlying physics of the process, but their applicability to realistic engineering fuel sprays is expected to be limited. The main problem with modelling these sprays lies in a large number of ordinary differential equations which need to be solved. These equations, describe droplet heating and evaporation and chemical reactions in individual computational cells, the number of which can exceed thousands or even millions. Direct numerical solutions of these equations is not feasible. Two possible approaches to their investigation are described in Sect. 7.3.

7.3 Coupled Solutions: Dynamic Decomposition 7.3.1 Decomposition Techniques Decomposition of complex systems, describing heating and evaporation of individual droplets (or droplet parcels) in sprays and the autoignition of fuel vapour/air mixture, into simpler subsystems is de facto almost universally used in engineering and physics applications. It allows the numerical simulation to focus on the subsystems, thus avoiding substantial difficulties and instabilities related to numerical simulation of the original systems. Special rules are usually introduced to incorporate the results of numerical simulation of the subsystems into the general scheme of the simulation of the whole system. The hierarchy of the decomposition process has been in most cases fixed for the duration of the process. As an example of such decomposition, the solutions of ordinary and partial differential equations (ODEs and PDEs) describing sprays in Computational Fluid Dynamics (CFD) codes can be mentioned. Numerical simulation of sprays is traditionally based on the Lagrangian approach coupled with the Eulerian representation of the gas phase. This allows one to decompose complex and highly nonlinear systems of Partial Differential Equations (PDEs) (describing interactions between computational cells) and the systems of Ordinary Differential Equations (ODEs) (describing the processes in individual computational cells). The latter processes include liquid/gas

432

7 Heating, Evaporation and Autoignition of Sprays

phase mass, momentum and energy exchange and chemical kinetics. The systems of ODEs are typically integrated using much shorter timesteps δt (typically 10−6 s) than the global time steps used for calculating the gas phase (solving PDEs) Δt (typically 10−4 s). Thus the decomposition of ODEs and PDEs is de facto used although its basis has not been rigorously studied to the best of the author’s knowledge [103, 118]. Further decomposition of the system of ODEs, describing droplet dynamics and heat and mass exchange inside individual computational cells, is widely used. The simplest decomposition of this system is based on the sequential solution of individual subsystems comprising this system (split operator approach). In this approach, the solution of each individual subsystem for a given subset of variables uses the assumption that all the other variables are known. The sequence of the solutions of individual subsystems is often arbitrarily chosen and the results sometimes vary substantially depending on the order in which these subsystems are solved. In the case when multiple scales are present in the system, the reliability of this approach becomes questionable, as demonstrated later in this section. To overcome these problems, the multi-scale nature of ODEs needs to be investigated before an attempt to solve them is made. This idea could be prompted by the approaches used in [95] for the analysis of the processes in CO2 lasers and the one used in [96] for the analysis of equations describing the autoignition of Diesel fuel (the Shell model; see Sect. 7.1). Before solving a system of five stiff ODEs describing five temperatures in CO2 lasers, the characteristic time scales of these equations were analysed [95]. It was shown that two of these equations describe rapid relaxation of two temperatures to the third one. This allowed the authors of [95] to replace the solution of five stiff differential equations with that of the system of just three non-stiff equations without any noticeable loss of accuracy. The approach used in [96] was different from the one used in [95], but the ultimate result of reduction of the number of ODEs to be solved, and elimination of the stiffness of the system of ODEs, was the same. In mathematical terms in both cases the dimension of the ODE system was reduced. A similar system decomposition into lower dimensional subsystems was used in constructing reduced chemical mechanisms based on Intrinsic Low-Dimensional Manifolds (ILDM) [60, 71, 91] and Computational Singular Perturbation (CSP) [53, 67, 72, 83, 119]. There are many similarities between these approaches. They are based on a rigorous scale separation such that ‘fast’ and ‘slow’ subspaces of the chemical source term are defined, and mechanisms of much reduced stiffness are constructed. These approaches, however, were developed with a view of specific application to modelling chemical kinetics. Their generalisation to general CFD codes has not been investigated to the best of the author’s knowledge. A useful analytical tool for the analysis of stiff systems of ODEs, used for modelling of spray heating, evaporation and autoignition, could be based on the geometrical asymptotic approach to singularly perturbed systems (integral manifold method) described in Sect. 7.2 [37, 43, 44]. This approach overcomes some of the abovementioned problems and is, essentially, focused on systems of ordinary differential equations of the form (7.15). In practical implementations of the integral manifold

7.3 Coupled Solutions: Dynamic Decomposition

433

method a number of simplifying assumptions were made. These include the assumption that the slow variable is constant during the fast processes. This assumption opened the way to analytical/asymptotic study of the processes [18, 39, 41, 97] (cf. the discussion in Sect. 7.2.3). These approaches to decomposing systems of ODEs were suggested and studied with a view to application to rather special problems and were based on many assumptions. These include fixing the decomposition over the duration of the process, and not allowing its hierarchy to change with time. The underlying philosophy of these approaches, however, is attractive for applications to the analysis of a wide range of physical and engineering problems including spray modelling in Computational Fluid Dynamics (CFD) codes. In what follows an alternative approach to decomposition of the system of ODEs is described, following [19, 20, 22, 104]. In contrast to the above-mentioned approaches, the approach described in Sect. 7.3.2 allows the change of the nature of decomposition with time (dynamic decomposition).

7.3.2 Description of the Method As in the original integral manifold method, the formal approach to the decomposition of the system of ODEs in the method described in this section is based on the division of system variables into ‘slow’ and ‘fast’. This leads to the division of this system into ‘slow’ and ‘fast’ subsystems. In contrast to the original version of the integral manifold method, however, linearised variations of slow variables during the time evolution of the fast variables are considered as a first-order approximation to the fast manifold. The usefulness of this division depends on whether the ‘fast’ subsystem has lower dimension compared with the ‘slow’ subsystem. The procedure can be iterative and result in a hierarchical division of the original system. For example, the ‘slow’ subsystem can, in its turn, be subdivided into ‘slow’ and ‘very slow’ subsystems. The approach described in [22] was initially focused on the simplest possible subdivision of the original system into two subsystems and applied to spray combustion modelling. Note that ‘fast’ - ‘slow’ decomposition in this case was different for different phase space regions [39, 97] and for different time intervals. Wider range of applications of this approach is anticipated. In what follows, the details are described, following [22]. Let us consider a system, the state of which is characterised by n dimensionless parameters Z i (i = 1, 2, . . . , n). The value of each parameter for a given point in space depends on time t, i.e. Z i = Z i (t). This dependence can be found from the solution to the system of n ODEs presented in a vector form: dZ = Φ (Z), dt

(7.53)

434

7 Heating, Evaporation and Autoignition of Sprays

where Z = (Z 1 , Z 2 , ......., Z n ),

Φ = (Φ1 , Φ2 , ......., Φn ).

In the general case, a rigorous coupled numerical solution can be found. This may be not practical, when too many equations need to be solved. A more efficient approach to this problem could be based on reducing the dimensions of this system as discussed in Sect. 7.3.1. This could be based on organising Eqs. (7.53) in terms of decreasing parameter Yi defined as    Φi (tk )  ,  Yi =  Z i (tk ) 

(7.54)

where Φi (tk ) ≡ Φik ≡Φi (Z 1 (tk ), Z 2 (tk ), ... Z i (tk ), ... Z n (tk )) and Z i (tk ) ≡ Z ik are the right-hand sides of Eqs. (7.53) and the values of Z i are taken at the time tk for the timestep Δt : tk → tk+1 , i = 1, 2, ....n. The case when Z ik is close to zero requires additional analysis. In most cases, it can be assumed that Z ik is large. The Taylor expansion of the right-hand side of (7.53) gives Φ |Z=Zk (Zk+1 − Zk ) + o (Zk+1 − Zk ) , Φ (Z(tk+1 )) ≡ Φ (Zk+1 ) = Φ (Zk ) + DΦ (7.55) Φ is the Jacobian matrix of the vector field Φ . where DΦ In the general case, the value of Φ (Zk+1 ) is controlled mainly by Φ (Zk ). However, in the special case where Φ (Zk ) = 0, the second term on the right-hand side of (7.55) becomes dominant. Only in this or similar cases, under certain conditions, Eq. (7.53) can be simplified to dZ i (tk ) = λi (tk )Z i (tk ), (7.56) dt and the values of Yi coincide with λi . The analysis of [19, 20, 22, 104], however, was focused not on this special case, but on the general case when Yi are not directly related to λi . If Yi is greater than a certain a priori chosen positive number α < 1, then the corresponding equation can be considered ‘fast’, and solved rigorously. If the number of ‘fast’ equations is f = 0, then the system is called multi-scale and this procedure reduces the dimension of the system of Eqs. (7.53) from n to f . This dimension reduction is particularly attractive when f is small (1 or 2). In an alternative approach, which is more useful than the previous one for practical engineering applications, Yi are reorganised in descending order as Yi1 ≥ Yi2 ≥ ..... ≥ Yi j ≥ ...... ≥ Yin . If j = f can be found such that

Yi f +1 < ε, Yi f

(7.57)

(7.58)

7.3 Coupled Solutions: Dynamic Decomposition

435

where ε is another a priori chosen small parameter, then it can be concluded that the system can be decomposed locally (Δt : tk → tk+1 ) into two subsystems: ‘fast’ and ‘slow’. Note that the subscript of i j shows the order in which the parameters are organised (there is no summation over f in the right-hand side of (7.58)). Equations for these subsystems are presented in vector form as dU =Φf dt dV = Φs dt

 

U V U V

 ,

(7.59)

,

(7.60)



where Φ f = (Φi 1 , ..., Φi f ), Φ s = (Φi f +1 , ..., Φi n ),

U = (Z i 1 , ..., Z i f ),

V = (Z i f +1 , ..., Z i n ).

The transformation from the original order of variables Z = (Z 1 , Z 2 , ......., Z n ) to the new order of the same variables 

Z = (Z i1 , ....., Z i f , Z i f +1 , ....., Z in ) is performed using the transformation matrix Q = Q i,i j , where Q i,i j = 1 when i corresponds to the original position of the variable. i j is the final position of the variable, and Q i,i j = 0 for all other i and i j . Thus, Z is presented as 

U Z = Q(Z0 )Z = Q(Z0 ) V 



   U  = Q f (Z0 ) Qs (Z0 ) , V

(7.61)

where Q is calculated for the values of Z at the initial time instant t0 or the beginning of the timestep (Z0 ). The first f columns of the matrix Q refer to the fast subsystem, while the remaining n − f columns refer to the slow subsystem. This is shown by introducing additional matrices Q f and Qs . Note that so far the simplest form of matrices Q and Q−1 performing the change of the order of variables is considered. More complex forms of these matrices could potentially perform the decomposition of the originally non-multiscale system into the multiscale one. Analysis of the latter decomposition, however, was beyond the scope of the analysis of [19, 20, 22, 104] and is not discussed in this section. Having introduced a new small positive parameter ε  1 and remembering the definitions of Q f and Qs , the system of Eqs. (7.59) and (7.60) can be presented in a form similar to the one used in the Integral Manifold Method:

436

7 Heating, Evaporation and Autoignition of Sprays

ε

     dU U U Φ , = εQ−1 (Z )Φ Q(Z ) ≡ Φ 0 0 fε f V V dt

(7.62)

     dV U U −1 Φ Φ ≡ s , = Qs (Z0 )Φ Q(Z0 ) V V dt

(7.63)

Φ f . In this presentation, the right-hand sides of Eqs. (7.62) and (7.63) where Φ f ε = εΦ are expected to be of the same order of magnitude over a specified period (timestep). Equations (7.62) and (7.63) can be integrated over the time period Δt : tk → tk+1 . The zeroth order solution to Eq. (7.63) is a constant value of the slow variable: 0 = Vk = ((Z i f +1 k , ....., Z in k), where superscript 0 shows the zeroth approxiVk+1 mation, while subscripts k and k+1 show the points in time. The zeroth order for the fast variable is found from Eq. (7.62) with V = Vk . This is interpreted as the equation for the slow variable on the fast manifold. Thus, Eq. (7.62) (or (7.59)) is approximated by the following system: dU =Φf dt



U Vk

 .

(7.64)

0 The solution to Eq. (7.64) at t = tk+1 (Uk+1 ) is the zeroth-order approximation of the fast motion on the fast manifold at t = tk . Note that System (7.64) can be stiff in the general case, but with a reduced level of stiffness, compared with the original system (7.53). Hence, the suggested approach is expected to reduce the level of stiffness of the system and not to eliminate the stiffness altogether. Under the same zeroth-order approximation. the slow variables would remain constant over the same timestep. This assumption was used in the original formulation of the method of integral manifolds [44]. This, however, might lead to an unphysical result when slow variables remain constant for any time t > t0 . Hence, the need to calculate slow variables using at least the first-order approximation. In the case when ε is not asymptotically small, further, or higher order, approximations need to be considered. In this case, the new time scale τ = 1/ε can be introduced, and the slow and fast variables are presented as

 V (τ ) = V (0) + εV (1) (τ ) + ε2 V (2) (τ ) + ............... . U (τ ) = U (0) (τ ) + εU (1) (τ ) + ε2 U (2) (τ ) + ..........

(7.65)

Having substituted Formulae (7.65) into (7.63) it is obtained: d(V (0) + εV (1) (τ ) + ε2 V (2) (τ ) + ...... ) dτ   (0) U (τ ) + εU (1) (τ ) + ε2 U (2) (τ ) + ...... Φs . = εΦ V (0) + εV (1) (τ ) + ε2 V (2) (τ ) + ......

(7.66)

7.3 Coupled Solutions: Dynamic Decomposition

437

Equation (7.66) allowed the authors of [19, 20, 22, 104] to obtain the first-order solution for the slow variable as  (0)  Uk+1 (0) (1) Φs Δτ. (7.67) + εVk+1 = Vk(0) + εΦ Vk+1 = Vk+1 Vk(0) Returning to the original variables, the expression for V(tk+1 ) ≡ Vk+1 is rewritten as  (0)  Uk+1 Δt Vk+1 = Vk(0) + Φ s Vk(0)   (0) (0) = (Z i f +1 k + Φi f +1 Uk+1 , Vk(0) Δt, ..... Z in k + Φi f +1 Uk+1 , Vk(0) Δt.

(7.68)

To increase accuracy of calculations one could continue the process to consider the first-order solution for the fast motion. Then the second-order solution for the slow motion could be obtained, etc. Since the above-mentioned approach to decomposition of the original system of equations is allowed to vary with time, it was suggested to call it dynamic decomposition approach [22].

7.3.3 Application of the Method In this section the approach described in Sect. 7.3.2 is applied to the analysis of heating, evaporation and autoignition of an array of droplets, following [22]. Several processes, including droplet dynamics, break-up and coalescence, and the effects of temperature gradient inside droplets are not considered (see [94] and Chaps. 2 and 3). This is justified by the fact that the main emphasis of this section is on the investigation of the applicability and efficiency the new method of the solution of the systems of ODEs relevant to spray combustion modelling rather than providing a detailed analysis of the processes involved (see [22] for further details).

7.3.3.1

Equations and Approximations

Expression (3.27) is used to describe the evaporation of individual droplets, assuming, following [68], that ϕ = 1 (which implies that BT = B M ). Also, following [68], the expression for the partial fuel vapour pressure at the droplet surface p f s is presented as 

βf , (7.69) p f s = exp α f − Ts − 43

438

7 Heating, Evaporation and Autoignition of Sprays

where α f and β f are constants specified for specific fuels, Ts is the droplet surface temperature in K; p f s is in kPa [22]. The values α f = 15.5274 and β f = 5383.59 recommended for Diesel fuel [68] are used. The following equation for stationary droplet temperature is used: m d cl

dTd = 4π Rd2 h(Tg − Td ) − m˙ d L , dt

(7.70)

where cl is liquid specific heat capacity, convective heat transfer coefficient h is obtained from the expression for Nu: Nu =

2h Rd kg

=2

ln(1 + BT ) . BT

(7.71)

Equation (7.70) is the generalisation of Eq. (2.36) to consider the effects of droplet evaporation (see Sect. 2.2.3). The temperature dependence of L is approximated as [68]:   Tc − Ts L = L Tbn Tc − Tbn where L Tbn is the value of L at the droplet boiling temperature Tbn . Tc is the critical temperature. The following values of parameters are used: Tbn = 536.4 K, Tc = 725.9 K and L Tbn = 254000 J/kg [68]. The liquid density is assumed to be constant. The equations for individual droplets are used to describe droplet parcels, following the approach commonly used in Computational Fluid Dynamics (CFD) codes. The equation for fuel vapour density (ρ f v ) follows from the conservation of fuel vapour:     dρ f v = −αg CT + αg m˙ di V , (7.72) dt i where αg is the volume fraction of gas assumed equal to 1, the summation is assumed over all droplets in volume V, CT is the chemical term describing fuel depletion (in kg/(m3 s); see Expression (7.74)). As in [124], the following expression of the rate of reaction (ω˙ in mole/(cm3 ·s) is used: (7.73) ω˙ = A[fuel]a f [O2 ]b f exp [−E/(Ru T )] , where the molar densities of fuel vapour [fuel] and oxygen [O2 ] are in mole/cm3 . The values of the coefficients for C10 H22 were used (these are the closest to n-dodecane (C12 H26 ) which is widely used for approximating of Diesel fuel) [124]: A = 3.8 × 1011

1 s



mole cm3

1−a f −b f

= 2.137 × 109

1 s



kmole m3

1−a f −b f

;

7.3 Coupled Solutions: Dynamic Decomposition

E = 30 For A in expression:

1 s

439

J kcal = 1.255 × 108 ; mole kmole  kmole 1−a f −b f

and E in

m3

a f = 0.25;

J kmole

b f = 1.5.

CT was found from the following

0.25 1.5 M 0.75 CT = A M O−1.5 f ρ f v ρ O2 exp [−E/(Ru T )] , 2

(7.74)

where M O2 = 32 kg/kmole, and M f = 170 kg/kmole are molar masses of oxygen and fuel (n-dodecane), respectively, ρ O2 is the density of oxygen. Expression (7.73) is similar to Expression (7.31), but presented in a slightly different format. The following one-step global reaction for n-dodecane combustion is used C12 H26 + 18.5O2 =⇒ 12 CO2 + 13H2 0. The rate of change of the density of oxygen is found from the expression: M O2 dρ O2 = −18.5 CT = −3.48235 CT. dt Mf

(7.75)

The equivalence ratio is defined as ϕR =

3.48ρ f v Fuel/Air Fuel/Oxygen 18.5 × 32 ρ f v = = = , 170 ρ O2 ρ O2 (Fuel/Air)stoich (Fuel/Oxygen)stoich

where (Fuel/Air)stoich is the stoichiometric ratio of the densities of fuel and air, is used. The requirement of energy balance leads to the following equation for gas temperature: ⎡ ⎤    dTg dTdi ⎣ = αg Q f CT − + cmix ρmix m di cl m˙ di L + m˙ di c p F (Tg − Tdi )⎦ V, dt dt i

i

i

(7.76) where Q f is the heat released per unit mass of burnt fuel vapour (in J/kg). The rate of change of the density of the mixture of fuel vapour and air (ρmix ) is found from the following expression:  dρi dρmix = , dt dt i where ρi are the densities of individual species.

(7.77)

440

7.3.3.2

7 Heating, Evaporation and Autoignition of Sprays

Application

In this section, equations and approximations described in Sects. 7.3.2 and 7.3.3.1 are applied to the analysis of polydisperse fuel spray heating, evaporation and autoignition following [22]. Three droplets with initial radii 5, 9 and 13 µm were considered. The initial temperatures of these droplets were assumed equal to 400 K. The gas temperature was assumed equal to 880 K [103]. The gas volume was chosen such that if the droplets are fully evaporated without combustion then the equivalence ratio of fuel vapour/air mixture was equal to 4. The initial density of oxygen and air pressure were assumed equal to 2.73 kg/m3 and 3 MPa, respectively. It was assumed that there is no fuel vapour in the gas phase initially. These values of the parameters are considered as approximations of Diesel engine-like conditions [103]. Despite its simplicity, the approach presented in Sect. 7.3.2 can capture the key features of the processes. See [5] for an alternative approach to a similar problem. The calculations were performed until gas temperature reached 1100 K. It was assumed that at this temperature the autoignition process is completed [103]. Since equations for droplet mass and temperature were solved for 3 droplets, the maximal number of equations to be solved was equal to 10. Note that the density of the fuel vapour/air mixture follows from the mass conservation algebraic equation. In the approach presented in this section, however, this parameter was found from the solution of the corresponding ODE. This made it possible to monitor the mass conservation in the system as an additional check for the numerical code (see [22]). After the smaller droplets evaporated the number of equations was reduced. These coupled equations were solved using three methods. Firstly, following widely used practice in CFD codes, the system of equations was divided into subsystems which were solved sequentially. This approach is commonly referred to as the operator splitting technique [112]. These subsystems include equations for mass and temperature of each droplet (three subsystems) and equations for gas temperature, fuel vapour, oxygen and mixture density (additional fourth subsystem). When each of these subsystems was solved, the remaining variables were assumed to be constant over the timestep. In the analysis presented in [22], this approach is referred to as the ‘fixed decomposition approach’ to distinguish it from the ‘dynamic decomposition approach’ described in Sect. 7.3.2. More specifically, at the first step, equations for the mass and the temperature of each droplet were solved simultaneously (3 systems of equations). Then the results were used to calculate the density of fuel vapour and mixture, and temperature of gas without considering the chemical term (see Eqs. (7.72) (7.77) and (7.76)). Then the chemical term was found using Eq. (7.74). The result was used to calculate the density of oxygen (Eq. (7.75)). Finally, the values of the density of fuel vapour and mixture, and gas temperature were updated using the chemical term. This approach is equivalent to the simplest form of (A–B) splitting described in [112]. Secondly, these equations were solved rigorously using DLSODAR stiff solver from ODEPACK developed in LLNL laboratory. In this case, all coupled equations were solved simultaneously.

7.3 Coupled Solutions: Dynamic Decomposition

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Fig. 7.1 Plots of the total number of equations solved (dashed) and the number of equations for fast variables (solid) for the values of parameters described in Sect. 7.3.3.2. The calculations stopped when Tg = 1100 K when the autoignition process is expected to be completed. Reprinted from [22], Copyright Elsevier (2007)

The third approach was based on decomposing the original system following the procedure presented in Sect. 7.3.2 with ε = 0.25. Note that this parameter is not related to the parameter ε used in Eqs. (7.15). The total number of equations solved, and the number of equations for fast variables changed with time as expected. The corresponding plots of the numbers of these equations versus time are presented in Fig. 7.1. As can be seen in this figure, initially all 10 equations were solved, when the first or second approaches were used. Then this number was reduced to 8 when the smallest droplet evaporated, and to 6 when two smallest droplets evaporated. Initially, the number of equations for fast variables to be solved was equal to 4, then it dropped to just one equation describing fuel vapour density. Between about 0.25 and 0.5 ms the number of equations for fast variables was equal to two (equations for fuel vapour density and the radius of the smallest droplet). Then again just the equation for fuel vapour density was solved. Between about 0.6 and 0.8 ms the number of fast equations is comparable with the total number of equations. During this period decomposition of the system is not expected to be useful for practical applications. After about 0.8 ms and until about 1.8 ms only one equation (fuel vapour density) or two equations (fuel vapour density and the radius of the second droplet) were used. In this case, the decomposition technique presented in Sect. 7.3.2 is expected to be particularly important for applications. The time evolution of gas temperature (Tg ), predicted by the above-mentioned three approaches, is presented in Fig. 7.2 [22]. This figure shows the total ignition delay time, i.e. the time needed for the gas temperature to reach 1100 K. As can be seen in Fig. 7.2, the first approach is very sensitive towards the timestep chosen. If the timestep 10−4 s is used then the predicted total ignition delay is almost four times longer than the one predicted using the second approach (coupled solution of the

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Fig. 7.2 Plots of gas temperature versus time, predicted based on the first approach (fixed decomposition) (dashed), second approach (coupled solution) (solid), and the third approach (dynamic decomposition) (dotted). Plots ‘1’, ‘2’ and ‘3’ refer to calculations using the timesteps 10−4 , 5 × 10−5 and 10−5 s, respectively. Reprinted from [22], Copyright Elsevier (2007)

whole system). If the timestep is decreased to 5 × 10−5 and 10−5 s then calculations using the first method appear to be more accurate than in the case when this timestep is equal to 10−4 , but still the accuracy of computations is hardly acceptable for engineering applications. Even for a rather small timestep, 10−5 s, the predicted total ignition delay is more than 20% longer than predicted by the rigorous coupled solution of this system of equations (second approach). The application of the third approach to the solution of this system gives a rather different picture. Even in the case of the largest timestep (10−4 s) the error of calculations of the total ignition time delay was just 13%. In the case of smaller timesteps, the time delay predicted by solving the decomposed system almost coincided with the one obtained based on the rigorous solution of the whole system with errors not exceeding 2%. Essentially the same conclusion regarding the benefits of the third approach, using the dynamic decomposition of the original system of equations, follows from similar figures showing the time evolution of the instantaneous equivalence ratio and the radius of the largest droplets [22]. Similar conclusions were obtained from the solution of equations for a different set of parameters, typical for the peripheral region of fuel sprays in Diesel engine conditions [22]. To compare the CPU efficiencies of the dynamic decomposition and fixed decomposition (conventional CFD) approaches, a series of runs for various timesteps were performed. A polydisperse spray including three droplet parcels, 10,000 droplets each, was injected at the start of the calculation. For fixed timesteps, the CPU requirements of both approaches were about the same. As shown above, however, the accuracy of the new approach was always higher than that of the conventional approach. Thus, the comparison of CPU requirements of both methods for fixed timesteps is misleading. An alternative approach needs to consider the accuracy of calculations.

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443

As followed from calculations the results of which are presented in [22], the autoignition delays predicted by the dynamic decomposition approach coincided with those predicted by the full coupled solution of the system of equations for the timestep of 10−6 s. This value of the autoignition delay was considered as the true value for further comparisons. For example, for a certain set of parameters errors less than 1.5% were achieved for timestep of 1.3 × 10−5 s for the fixed decomposition approach, and for timestep of 2.4 × 10−5 s for the dynamic decomposition approach. When this effect was considered then in all cases under consideration, the CPU time for the dynamic decomposition approach was always smaller than that for the conventional approach. In some cases, the CPU reduction for the dynamic decomposition approach was as high as factor of 3. The CPU time was estimated based on the customised function DATE− AND− TIME. The standard function GETTIME did not give consistent results for small CPU times [22]. Several additional tests were conducted to compare the performances of the fixed decomposition approach (used in CFD codes) and the dynamic decomposition method [22]. Both programs were run sequentially on two workstations (Silicon Graphics, Intel 64bit processor) and using two Fortran Compilers (Intel–Fortran 95, GNU–Fortran 77) with the additional option of code optimisation. The abovementioned customised function DATE− AND− TIME was used to estimate the CPU time required for system integration. It was found that, for a given timestep and for a low level of code optimisation, the CPU time is less for the fixed decomposition approach than for the dynamic decomposition approach. Further optimisation led to comparable times required for program execution. For example, for the timestep Δt = 5 × 10−5 , the CPU time for the test case with the code optimisation level set to O0 (no code optimisation) was approximately equal to 0.12 s for the fixed decomposition approach and 0.21 s for the dynamic decomposition approach. For the optimisation level set to O2 (this option is the default one, it enables optimisations for speed, including global code scheduling, software pipelining, prediction, speculation, etc.) these times become 0.15 and 0.13 s respectively. Therefore further improvements in realisation of the dynamic decomposition method are possible and can be implemented, leading to optimising the realisation structure of the code and numerics [22]. The coupled solution of the system of equations using the stiff solver is always more accurate than the solutions based on dynamic and fixed decompositions, and is usually more CPU efficient. Hence, in the case when the number of equations is relatively small (as in the case considered in this section) there is no need to develop any decomposition technique at the first place. However, in realistic engineering calculations, when the number of droplet parcels could be tens of thousands [103], no stiff solver can cope with the full coupled system of ODEs describing them. The real competition in this case is between fixed and dynamic decomposition approaches [22]. For some problems there is no marked difference in time scales between the original variables. This is a typical situation in modelling reacting flows due to strong coupling of thermo-chemical processes. In this case, the decomposition into fast and slow variables may fail. Also, the decomposition in the original coordinates can pro-

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duce a relatively large subsystem (the number of governing equations of the reacting system can reach or exceed hundreds), and the modelling needs to be performed over very small timesteps to capture the fine features of, for example, an autoignition process. For these problems, the application of the dynamic decomposition method may not be efficient. A decomposition technique specifically designed for such systems was developed in [24]. This method represents the realisation of the general framework of Singularly Perturbed Vector Fields developed in [21, 23]. It allows finding the decomposition and approximating it efficiently for system reduction purposes. Then, depending on a particular asymptotic limit, either slow or fast subsystems are integrated according to the standard singular perturbation theory [37, 44]. In this technique, the same decomposition for the whole time interval under consideration, and for a given set of initial conditions, is applied. Hence it was called the Global Quasi-Linearisation (GQL) technique. Although this approach is less accurate than the dynamic decomposition method, it can potentially be much more efficient for implementation into CFD codes due to the reduced number of dimensions involved. The efficiency of this approach was demonstrated for a relatively simple, but typical combustion chemistry model [17]. The application of the Global Quasi-Linearisation technique to the analysis of the cyclohexane/air mixture autoignition in a rapid compression machine environment is discussed in [25]. A simplified autoignition mechanism including 50 species was used in the analysis based on the customised version of the Computational Fluid Dynamics (CFD) code ANSYS FLUENT with the GQL method implemented into it. The results predicted by ANSYS FLUENT were shown to be very close to those predicted by the zero-dimensional code SPRINT, developed at the University of Leeds. The application of the GQL method showed that further substantial reduction of the autoignition mechanism is possible without significant loss of accuracy. The method was shown to be an efficient tool for identifying and approximating low-dimensional invariant system manifolds. The applicability of this approach to modelling spray heating, evaporation and autoignition has not yet been studied. A probabilistic model of thermal explosion in polydisperse fuel sprays is described in a series of papers by Nave et al. [79–82]. In the approach described in these papers, polydispersity was modelled using a probability density function (PDF) corresponding to the initial distribution of fuel droplet sizes. This approximation of a polydisperse spray was considered to be more accurate than the traditional ‘parcel’ approximation and allowed an analytical treatment using a simplified model. Since the system of the governing equations represented a multi-scale problem, the method of invariant (integral) manifolds was used. An explicit expression of the critical condition for thermal explosion limit was derived analytically. Numerical simulations demonstrated the dependence of this thermal explosion condition on the PDF type. Thus, this approach can be considered as a generalisation of the thermal explosion condition of the classical Semenov theory discussed earlier in this chapter. Coupled solutions of the equations describing heating and evaporation of fuel droplets leading to autoignition of a fuel vapour/air mixture were described. A review

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445

of various approaches to the modelling of autoignition (in the presence or in the absence of droplets) was presented. Simplified models for coupling between droplet heating, evaporation and the autoignition of the fuel vapour/air mixture, based on integral manifolds and positively invariant manifolds were described. This was followed by a description of the numerical solution of the coupled system of ordinary differential equations (ODEs), referring to droplet heating and evaporation, and the autoignition process, using the dynamic decomposition technique. As in the original integral manifold method, the formal approach to the decomposition of the system of ODEs was based on the division of system variables into ‘fast’ and ‘slow’. In contrast to the original version of the integral manifold method linearised variations of slow variables during the time evolution of the fast variables were considered as the first-order approximation to the fast manifold.

References 1. Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. International Journal of Heat Mass Transfer, 32, 1605–1618. 2. Aggarwal, S. K. (1998). A review of spray ignition phenomena: Present status and future research. Progress in Energy Combustion Science, 24, 565–600. 3. Aleiferis, P. G., Serras-Pereira, J., van Romunde, Z., Caine, J., & Wirth, M. (2010). Mechanisms of spray formation and combustion from a multi-hole injector with E85 and gasoline. Combustion and Flame, 157, 735–756. 4. Alhikami, A. F., Yao, C.-E., & Wang, W.-C. (2021). A study of the spray ignition characteristics of hydro-processed renewable diesel, petroleum diesel, and biodiesel using a constant volume combustion chamber. Combustion and Flame, 223, 55–64. 5. Amani, E., & Nobari, M. R. H. (2013). A calibrated evaporation model for the numerical study of evaporation delay in liquid fuel sprays. International Journal of Heat Mass Transfer, 56, 45–58. 6. Basevich, V. Y. (1990). Chemical kinetics in the combustion processes. In N. P. Cheremisinoff (Ed.), Handbook of Heat and Mass Transfer, V. 4. Advances in Reactor Design and Combustion Science (pp. 769–819). Houston: Gulf Publishing Company. 7. Basevich, V. Y., & Frolov, S. M. (1994). A reduced kinetic scheme for autoignition modelling of iso-octane and n−heptane/air mixtures during the induction period for internal combustion engines. Chemical Physics, 13, 146–156. (in Russian). 8. Basevich, V. Y., Beliaev, A. A., Branshtater, V., Neigauz, M. G., Tashl, R., & Frolov, S. M. (1994). Modelling of ISO-octane and n−heptane autoignition with reference to IC engines. Physics of Combustion and Explosion, 30, 15–24. (in Russian). 9. Benson, S. W. (1981). The kinetics and thermochemistry of chemical oxidation with application to combustion and flames. Progress in Energy Combustion Science, 7, 125–134. 10. Bertoli, C., & Migliaccio, M. (1999). A finite conductivity model for diesel spray evaporation computations. International Journal of Heat Fluid Flow, 20, 552–561. 11. Blin-Simiand, N., Rigny, R., Viossat, V., Circan, S., & Sahetchian, K. (1993). Autoignition of hydrocarbon/air mixtures in a CFR engine: Experimental and modeling study. Combustion Science and Technology, 88, 329–348. 12. Borghesi, G., Mastorakos, E., & Cant, R. S. (2013). Complex chemistry DNS of n-heptane spray autoignition at high pressure and intermediate temperature conditions. Combustion and Flame, 160, 1254–1275.

446

7 Heating, Evaporation and Autoignition of Sprays

13. Boudier, P., Henriot, S., Pinsot, T., & Baritaud, T. (1992). A model for turbulent flame ignition and propagation in spark ignition engines. In Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute (pp. 503–510). 14. Brady, R. N. (1996). Modern Diesel Technology. London: Prentice-Hall. 15. Bruel, P., Rogg, B., & Bray, K. N. C. (1990). On auto-ignition in laminar and turbulent nonpremixed systems. In Twenty-Third Symposium (International) on Combustion/The Combustion Institute (pp. 759–766). 16. Buda, F., Bounaceur, R., Warth, V., Glaude, P. A., Fournet, R., & Battin-Leclerc, F. (2005). Progress toward a unified detailed kinetic model for the autoignition of alkanes from C4 to C10 between 600 K and 1200 K. Combustion and Flame, 142, 170–186. 17. Bykov, V., & Maas, U. (2009). Investigation of the hierarchical structure of kinetic models in ignition problems. Zeitschrift für Physikalische Chemie, 223, 461–479. 18. Bykov, V., Goldfarb, I., Gol’dshtein, V., & Greenberg, J. B. (2002). Thermal explosion in a hot gas mixture with fuel droplets: A two reactants model. Combustion Theory and Modelling, 6, 1–21. 19. Bykov, V., Goldfarb, I., Goldshtein, V., Sazhin, S. S., & Sazhina, E. M. (2004). System decomposition technique: Application to spray modelling in CFD codes. In 20th Annual Symposium of the Israeli Section of the Combustion Institute. Book of Abstracts (p. 16). Beer-Sheva, Israel: Ben-Gurion University. 20. Bykov, V., Goldfarb, I., Gol’dshtein, V., Sazhin, S. S., & Sazhina, E. M. (2005). An asymptotic approach to numerical modelling of spray autoignition. In Proceedings of the Fourth Mediterranean Combustion Symposium, MCS4, Paper V.6, Lisbon (Portugal), October 6–10 (CD-ROM). 21. Bykov, V., Goldfarb, I., & Gol’dshtein, V. (2006). Singularly perturbed vector fields. Journal of Physics: Conference Series, 55, 28–44. 22. Bykov, V., Goldfarb, I., Gol’dshtein, V., Sazhin, S. S., & Sazhina, E. M. (2007). System decomposition technique for spray modelling in CFD codes. Computers and Fluids, 36, 601– 610. 23. Bykov, V., & Gol’dshtein, V. (2008). On a decomposition of motions and model reduction. Journal of Physics: Conference Series, 138, 012003. 24. Bykov, V., Gol’dshtein, V., & Maas, U. (2008). Simple global reduction technique based on decomposition approach. Combustion Theory and Modelling, 12, 389–405. 25. Bykov, V., Griffiths, J. F., Piazzesi, R., Sazhin, S. S., & Sazhina, E. M. (2013). The application of the global quasi-linearisation technique to the analysis of the cyclohexane/air mixture autoignition. Applied Mathematics and Computation, 219, 7338–7347. 26. Chen, C.-K., & Lin, T.-H. (2012). Streamwise interaction of burning drops. Combustion and Flame, 159, 1971–1979. 27. Chevalier, C., Pitz, W. J., Warnatz, J., Westbrook, C. K., & Mellenk, H. (1992). Hydrocarbon ignition: Automatic generation of reaction mechanisms and applications to modeling of engine knock. In Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute (pp. 1405–1414). 28. Coddington, E. A., & Levinson, N. (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill. 29. Cox, R. A., & Cole, J. A. (1985). Chemical aspects of the autoignition of hydrocarbon—air mixtures. Combustion and Flame, 60, 109–123. 30. Crespo, A., & Liñan, A. (1975). Unsteady effects in droplet evaporation and combustion. Combustion Science and Technology, 11, 9–18. 31. Crua, C., Kennaird, D. A., Sazhin, S. S., & Heikal, M. R. (2004). Diesel autoignition at elevated in-cylinder pressures. International Journal of Engine Research, 5, 365–374. 32. Curran, H. J., Gaffuri, P., Pitz, W. J., & Westbrook, C. K. (1998). A comprehensive modeling study of n-heptane oxidation. Combustion and Flame, 114, 149–177. 33. De, S., Lakshmisha, K. N., & Bilger, R. W. (2011). Modeling of nonreacting and reacting turbulent spray jets using a fully stochastic separated flow approach. Combustion and Flame, 158, 1992–2008.

References

447

34. Demoulin, F. X., & Borghi, R. (2003). Modeling of turbulent spray combustion with application to diesel like experiment. Combustion and Flame, 129, 281–293. 35. Dopazo, C., & O’Brien, E. E. (1974). An approach to the autoignition of a turbulent mixture. Acta Astronautica, 1, 1239–1266. 36. Durand, P., Gorokhovski, M., & Borghi, R. (1999). An application of the probability density function model to diesel engine combustion. Combustion Science Technology, 144, 47–78. 37. Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations, 31, 53–98. 38. Goldfarb, I., Gol’dshtein, V., Kuzmenko, G., & Greenberg, J. B. (1998). On thermal explosion of a cool spray in a hot gas. In Proceedings of the 27th International Symposium on Combustion (Colorado, USA) (Vol. 2, pp. 2367–2374). 39. Goldfarb, I., Gol’dshtein, V., Kuzmenko, G., & Sazhin, S. S. (1999). Thermal radiation effect on thermal explosion in gas containing fuel droplets. Combustion Theory and Modelling, 3, 769–787. 40. Goldfarb, I., Sazhin, S. S., & Zinoviev, A. (2004). Delayed thermal explosion in flammable gas containing fuel droplets: Asymptotic analysis. International Journal of Engineering Mathematics, 50, 399–414. 41. Goldfarb, I., Gol’dshtein, V., & Maas, U. (2004). Comparative analysis of two asymptotic approaches based on integral manifolds. IMA Journal of Applied Mathematics, 69, 353–374. 42. Goldfarb, I., Gol’dshtein, V., Katz, D., & Sazhin, S. S. (2007). Radiation effect on thermal explosion in a gas containing evaporating fuel droplets. International Journal of Thermal Science, 46(4), 358–370. 43. Gol’dshtein, V., & Sobolev, V. (1988). Qualitative Analysis of Singularly Perturbed Systems. Novosibirsk (in Russian): Siberian Branch of USSR Academy of Science, Institute of Mathematics. 44. Gol’dshtein, V., & Sobolev, V. (1992). Integral manifolds in chemical kinetics and combustion. In Singularity Theory and Some Problems of Functional Analysis (pp. 73–92). American Mathematical Society. 45. Gorban, A. N., Kazantzis, N., Kevrekidis, I. G., Öttinger, H. C., & Theodoropoulos, C. (Eds.). (2006). Model Reduction and Coarse-Graining Approaches for Multi-scale Phenomena. Springer. 46. Gorokhovski, M., & Saveliev, V. L. (2003). Analyses of Kolmogorov’s model of breakup and its application into Lagrangean computation of liquid sprays under air-blast atomization. Physics of Fluids, 15, 184–192. 47. Greenberg, J. B. (2002). Stability boundaries of laminar premixed polydisperse spray flames. Atomization and Sprays, 12, 123–144. 48. Griffith, J. F. (1993). Kinetic fundamentals of alkane autoignition at low temperatures. Combustion and Flame, 93, 202–206. 49. Griffith, J. F. (1995). Reduced kinetic models and their application to practical combustion systems. Progress in Energy Combustion Science, 21, 25–107. 50. Griffiths, J. F., Jiao, Q., Schreiber, M., Meyer, J., & Knoche, K. F. (1992). Development of thermokinetic models for autoignition in a CFD code: Experimental validation and application of the results to rapid compression studies. In Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute (pp. 1809–1815). 51. Griffiths, J., Piazzesi, R., Sazhina, E. M., Sazhin, S. S., Glaude, P. A., & Heikal, M. R. (2012). CFD modelling of cyclohexane auto-ignition in an RCM. Fuel, 96, 192–203. 52. Gumprich, W., & Sadiki, A. (2013). Direct quadrature based sectional method of moments for the simulation of evaporating polydisperse sprays. In Proceedings of ILASS—Europe 2013, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, September 1–4, 2013, paper 43. 53. Hadjinicolaou, M., & Goussis, D. M. (1999). Asymptotic solutions of stiff PDEs with the CSP method: The reaction diffusion equation. SIAM Journal of Scientific Computing, 20, 781–910. 54. Halstead, M., Prothero, A., & Quinn, C. P. (1973). Modeling the ignition and cool-flame limits of acetaldehyde oxidation. Combustion and Flame, 20, 211–221.

448

7 Heating, Evaporation and Autoignition of Sprays

55. Halstead, M. P., Kirsch, L. J., & Quinn, C. P. (1977). The autoignition of hydrocarbon fuels at high temperatures and pressures—fitting of a mathematical model. Combustion and Flame, 30, 45–60. 56. Hamosfakidis, V., & Reitz, R. D. (2003). Optimization of a hydrocarbon fuel ignition model for two single component surrogates of diesel fuel. Combustion and Flame, 132, 433–450. 57. Hilbert, R., & Thevenin, D. (2002). Autoignition of turbulent non-premixed flames investigated using direct numerical simulations. Combustion and Flame, 128, 22–37. 58. Hou, S.-S., Lin, J.-C., & Hsuan, C.-Y. (2014). The interaction between internal heat gain and heat loss on compound-drop spray flames. International Journal of Heat Mass Transfer, 71, 503–514. 59. Incroperra, F. P., & DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer. New York, Chichester: Wiley. 60. Kaper, H. G., & Kaper, T. J. (2001). Asymptotic analysis of two reduction methods for systems of chemical reactions. Argonne National Lab, preprint ANL/MCS-P912-1001. 61. Knobloch, H. W., & Aulback, B. (1984). Singular perturbations and integral manifolds. Journal of Mathematical Physics Science, 18, 415–424. 62. Kojima, S. (1994). Detailed modeling of n−butane autoignition chemistry. Combustion and Flame, 99, 87–136. 63. Kong, S.-C., Han, Z., & Reitz, R. D. (1996). The development and application of a Diesel ignition and combustion model for multidimensional engine simulation. SAE Technical Paper 950278. 64. Kuo, K.-K. (1986). Principles of Combustion. Chichester: Wiley. 65. Kuznetzov, V. R., & Sabel’nikov, V. A. (1990). Turbulence and Combustion. New York: Hemisphere. 66. Lakshmisha, K. N., Zhang, Y., Rogg, B., & Bray, K. N. C. (1992). Modelling auto-ignition in a turbulent medium. In Twenty-Fourth Symposium (International) on Combution/The Combustion Institute (pp. 421–428). 67. Lam, S. H., & Goussis, D. M. (1994). The GSP method for simplifying kinetics. International Journal of Chemical Kinetics, 26, 461–486. 68. Lefebvre, A. H. (1989). Atomization and Sprays. New York: Taylor & Francis. 69. Liu, Z., Yang, L., Song, E., Wang, J., Zare, A., Odisco, T. A., & Brown, R. J. (2021). Development of a reduced multi-component combustion mechanism for a diesel/natural gas dual fuel engine by cross-reaction analysis. Fuel, 293, 120388. 70. Maas, U., & Pope, S. B. (1992). Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds. In Twenty-Fourth Symposium (International) on Combution/The Combustion Institute (pp. 103–112). 71. Maas, U., & Pope, S. B. (1992). Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Combustion and Flame, 117, 99–116. 72. Masias, A., Diamantis, D., Mastorakos, E., & Goussis, D. E. (1999). An algorithm for the construction of global reduced mechanisms with CSP data. Combustion and Flame, 117, 685–708. 73. McIntosh, A. C., Gol’dshtein, V., Goldfarb, I., & Zinoviev, A. (1998). Thermal explosion in a combustible gas containing fuel droplets. Combustion Theory and Modelling, 2, 153–165. 74. Minkoff, G. J., & Tipper, C. F. (1962). Chemistry of Combustion Reactions. London: Butterworth. 75. Minetti, R., Ribaucour, M., Carlier, M., Fittschen, C., & Sochet, L. R. (1994). Experimental and modelling study of oxidation and autoignition of butane at high pressure. Combustion and Flame, 96, 201–211. 76. Minetti, R., Ribaucour, M., Carlier, M., & Sochet, L. R. (1996). Autoignition delays of a series of linear and branched chain alkanes in the intermediate range of temperatures. Combustion Science and Technology, 113–114, 179–192. 77. Müller, U. C., Peters, N., & Li˜nan, A. (1992). Global kinetics for n−heptane ignition at high pressures. In Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute (pp. 777–784).

References

449

78. Natarajan, B., & Bracco, F. V. (1984). On multidimensional modelling of auto-ignition in spark-ignition engines. Combustion and Flame, 57, 179–197. 79. Nave, O., Bykov, V., & Gol’dshtein, V. (2010). A probabilistic model of thermal explosion in polydisperse fuel spray. Applied Mathematics and Computation, 217, 2698–2709. 80. Nave, O., Bykov, V., Gol’dshtein, V., & Lehavi, Y. (2011). Numerical simulations applying to the analysis of thermal explosion of organic gel fuel in a hot gas. Fuel, 90, 3410–3416. 81. Nave, O., & Gol’dshtein, V. (2011). The flammable spray effect on thermal explosion of a combustible gas-fuel mixture. Combustion Science and Technology, 183, 519–539. 82. Nave, O., Gol’dshtein, V., & Dan, E. (2011). The delay phenomena in thermal explosion of polydisperse fuel spray—using the method of integral manifolds. Atomization and Sprays, 21, 69–85. 83. Neophytou, M. K., Goussis, D. A., van Loon, M., & Mastorakos, E. (2004). Reduced chemical mechanism for atmospheric pollution using Computational Singular Perturbation analysis. Atmospheric Environment, 38, 3661–3673. 84. Neophytou, A., Mastorakos, E., & Cant, R. S. (2012). The internal structure of igniting turbulent sprays as revealed by complex chemistry DNS. Combustion and Flame, 159, 641– 664. 85. Oziraner, A. S., & Rumyantsev, V. V. (1972). The method of Liapunov functions in the stability problem for motion with respect to a part of the variables. Journal of Applied Mathematics and Mechanics, 36, 341–362. 86. Pham, P. X., Pham, N. V. T., Pham, T. V., Nguyen, Vu. H., & Nguyen, K. T. (2021). Ignition delays of biodiesel-diesel blends: Investigations into the role of physical and chemical processes. Fuel, 303, 121251. 87. Pilling, M. J., Robertson, S. H., & Seakins, P. W. (1995). Elementary radical reactions and autoignition. Journal of the Chemical Society, Faraday Transactions, 91, 4179–4188. 88. Pitsch, H., & Peters, N. (1998). Investigation of the ignition process of sprays under Diesel engine conditions using reduced n-heptane chemistry. SAE paper 2464. 89. Poppe, C., Schreiber, M., & Griffiths, J. F. (1993). Modelling of n-heptane autoignition and validation of the results. In Proceedings of Joint Meeting of British and German Sections of the Combustion Institute, Cambridge (pp. 360–363). 90. Ranzi, E., Faravelli, T., Gaffuri, P., Pennati, G. C., & Sogaro, A. (1994). A wide range modeling of propane and n−butane oxidation. Combustion and Flame, 100, 299–330. 91. Rhodes, C., Morari, M., & Wiggins, S. (1999). Identification of the low order manifolds: Validating the algorithm of Maas and Pope. Chaos, 9, 108–123. 92. Sabel’nikov, V., Gorokhovski, M., & Baricault, N. (2006). The extended IEM mixing model in the framework of the composition PDF approach; application to diesel spray combustion. Combustion Theory and Modelling, 10, 155–169. 93. Sahetchian, K., Champoussin, J. C., Brun, M., Levy, N., Blin-Simiand, N., Aligrot, C., Jorand, F., Socoliuc, M., Heiss, A., & Guerassi, N. (1995). Experimental study and modeling of dodecane ignition in a Diesel engine. Combustion and Flame, 103, 207–220. 94. Sazhin, S. S. (2006). Advanced models of fuel droplet heating and evaporation. Progress Energy and Combustion Science, 32(2), 162–214. 95. Sazhin, S. S., Wild, P., Leys, C., Toebaert, D., & Sazhina, E. M. (1993). The three temperature model for the fast-axial-flow CO2 laser. Journal of Physics D: Applied Physics, 26, 1872– 1883. 96. Sazhin, S. S., Sazhina, E. M., Heikal, M. R., Marooney, C., & Mikhalovsky, S. V. (1999). The Shell autoignition model: A new mathematical formulation. Combustion and Flame, 117, 529–540. 97. Sazhin, S. S., Feng, G., Heikal, M. R., Goldfarb, I., Goldshtein, V., & Kuzmenko, G. (2001). Thermal ignition analysis of a monodisperse spray with radiation. Combustion and Flame, 124, 684–701. 98. Sazhin, S. S., Feng, G., & Heikal, M. R. (2001). A model for fuel spray penetration. Fuel, 80, 2171–2180.

450

7 Heating, Evaporation and Autoignition of Sprays

99. Sazhin, S. S., Abdelghaffar, W. A., Sazhina, E. M., & Heikal, M. R. (2005). Models for droplet transient heating: Effects on droplet evaporation, ignition, and break-up. International Journal of Thermal Science, 44, 610–622. 100. Sazhin, S. S., Shchepakina, E. A., & Sobolev, V. A. (2010). Order reduction of a nonLipschitzian model of monodisperse spray ignition. Mathematical and Computer Modelling, 52, 529–537. 101. Sazhin, S. S., Shchepakina, E., & Sobolev, V. (2018). Order reduction in models of spray ignition and combustion. Combustion and Flame, 187, 122–128. 102. Sazhina, E. M., Sazhin, S. S., Heikal, M. R., & Marooney, C. (1999). The shell autoignition model: Application to gasoline and Diesel fuels. Fuel, 78, 389–401. 103. Sazhina, E. M., Sazhin, S. S., Heikal, M. R., Babushok, V. I., & Johns, R. (2000). A detailed modelling of the spray ignition process in Diesel engines. Combustion Science and Technology, 160, 317–344. 104. Sazhina, E. M., Bykov, V., Goldfarb, I., Goldshtein, V., Sazhin, S. S., & Heikal, M. R. (2005). Modelling of spray autoignition by the ODE system decomposition technique. In Proceedings of HEFAT2005 (4th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics), Cairo, Egypt; Paper number SE2. 105. Schäpertöns, H., & Lee, W. (1985). Multidimensional modelling of knocking combustion in SI engines. SAE Technical Paper 850502. 106. Semenov, N. N. (1935). Chemical Kinetics and Chain Reactions. Oxford: Oxford University Press. 107. Shchepakina, E., Sobolev, V. A., & Mortell, M. P. (2014). Singular Perturbations: Introduction to System Order Reduction Methods with Applications. Springer Lecture Notes in Mathematics (Vol. 2114). Springer. 108. Sirignano, W. A. (2010). Fluid Dynamics and Transport of Droplets and Sprays (2nd ed.). Cambridge: Cambridge University Press. 109. Sobolev, V. A. (1984). Integral manifolds and decomposition of singularly perturbed systems. System and Control Letters, 5, 169–179. 110. Sobolev, V. A., & Shchepakina, E. A. (2010). Reduction of the Models and Critical Phenomena in Micro-kinetics. Moscow (in Russian): Fizmarlit. 111. Spalding, D. B. (1979). Combustion and Mass Transfer. New York: Pergamon Press. 112. Sportisse, B. (2000). An analysis of operator splitting techniques in the stiff case. Journal of Computational Physics, 161, 140–168. 113. Strygin, V. V., & Sobolev, V. A. (1988). Separation of Motions by the Integral Manifold Method. Moscow (in Russian): Nauka. 114. Suppes, G. J., Srinivasan, B., & Natarajan, V. P. (1995). Autoignition of biodiesel, methanol, and 50:50 blend in a simulated Diesel engine environment. SAE Paper 952758. 115. Tanabe, M., Kono, M., Sato, J., Koenig, J., Eigenbrod, C., Dinkelacker, F., & Rath Zarm, H. J. (1995). Two stage ignition of n-heptane isolated droplets. Combustion Science and Technology, 108, 103–119. 116. Tanaka, S., Ayala, F., & Keck, J. C. (2003). A reduced chemical kinetic model for HCCI combustion of primary reference fuels in a rapid compression machine. Combustion and Flame, 133, 467–481. 117. Theobald, M. A. (1986). Numerical Simulation of Diesel Autoignition, Ph.D. Thesis, MIT. 118. Utyuzhnikov, S. V. (2002). Numerical modeling of combustion of fuel-droplet-vapour releases in the atmosphere. Flow, Turbulence and Combustion, 68, 137–152. 119. Valorani, M., & Goussis, D. M. (2001). Explicit time-scale splitting algorithm for stiff problems: Auto-ignition of gaseous mixtures behind a steady shock. Journal of Computational Physics, 169, 44–79. 120. Varshavski, G. A., Fedoseev, D. V., & Frank-Kamenetskii, A. D. (1968). Autoignition of a fuel droplet. In V. A. Fedoseev (Ed.), Problems of Evaporation, Combustion and Gas Dynamics in Disperse Systems. Proceedings of the Sixths Conference on Evaporation, Combustion and Gas Dynamics in Disperse Systems (October 1966) (pp. 91–95). Odessa: Odessa University Publishing House (in Russian).

References

451

121. Viggiano, A., & Magi, V. (2004). A 2-D investigation of n-heptane autoignition by means of direct numerical simulation. Combustion and Flame, 137, 432–443. 122. Wang, C., Dean, A. M., Zhu, H., & Kee, R. J. (2013). The effects of multicomponent fuel droplet evaporation on the kinetics of strained opposed-flow diffusion flames. Combustion and Flame, 160, 265–275. 123. Wang, H., Luo, K., & Fan, J. (2014). Effects of turbulent intensity and droplet diameter on spray combustion using direct numerical simulation. Fuel, 121, 311–318. 124. Westbrook, C. K., & Dryer, F. L. (1981). Simplified reaction mechanism for the oxidation of hydrocarbon fuels in flames. Combustion Science and Technology, 27, 31–43. 125. Wu, G., & Sirignano, W. A. (2010). Transient burning of a convective fuel droplet. Combustion and Flame, 157, 970–981. 126. Wu, G., & Sirignano, W. A. (2011). Transient convective burning of interactive fuel droplets in double-layer arrays. Combustion and Flame, 158, 2395–2407. 127. Zadiraka, K. V. (1957). On the integral manifold of a system of differential equations containing a small parameter. Doklady Akademii Nauk SSSR, 115, 646–649. (in Russian). 128. Zettervall, N., Fureby, C., & Nilsson, E. J. K. (2020). A reduced chemical kinetic reaction mechanism for kerosene-air combustion. Fuel, 269, 117446. 129. Zhang, L., & Qi, Q. (2021). A reduced mechanism for the combustion of gasoline-ethanol blend on advanced engine combustion modes. Fuel, 300, 120951. 130. Zhang, Y., Rogg, B., & Bray, K. N. C. (1993). Modeling of autoignition in nonpremixed turbulent systems: Closure of the chemical-source terms. Progress in Astronautics and Aeronautics, 152, 87–102. 131. Zhang, Y., Rogg, B., & Bray, K. N. C. (1995). 2-D simulation of turbulent autoignition with transient laminar flamelet source term closure. Combustion Science and Technology, 105, 211–227. 132. Zhou, L., Zhao, W., Luo, K. H., Jia, M., Wei, H., & Xie, M. (2021). Spray-turbulence-chemistry interactions under engine-like conditions. Progress in Energy and Combustion Science, 86, 100939. 133. Zhou, T., Zhao, P., Ye, T., Zhu, M., & Tao, C. (2020). Direct numerical simulation of low temperature reactions affecting n-dodecane spray autoignition. Fuel, 280, 118453. 134. Zhukov, V. P., Sechenov, V. A., & Starikovskii, A. Yu. (2005). Self-ignition of a lean mixture of n-pentane and air over a wide range of pressures. Combustion and Flame, 140, 196–203.

Chapter 8

Concluding Comments

The results presented in the previous chapters show notable achievements in the development of the models of droplet and spray dynamics, heating and evaporation. Nevertheless, many important issues are still not resolved. The focus of this concluding part of the book is not on a summary of the most important results but on some of these unsolved problems. The list is subjective and motivated by the author’s personal research interests. There is an overlap between the discussion in this chapter and the one in [8].

8.1 The Fully Lagrangian Approach Although the advantages of the Fully Lagrangian Approach (FLA) over the conventional Lagrangian approach for calculation of the time and space evolution of the droplet number density have been clearly demonstrated in the publications summarised in Sect. 1.3.5, the applications of this approach to solving practical engineering problems are still very limited. There are at least two main reasons for this. Firstly, most practically important engineering flows are turbulent, while the approaches developed so far for the applications of FLA in turbulent flows use many important assumptions. Their applicability to realistic flows is not obvious (e.g. [7]). Secondly, a model for two-way coupling of droplet clouds with the ambient gas within the FLA is still to be developed. Recent developments in this field are described in [5].

8.2 Non-spherical Droplets Most of the models described in the book are applicable only to spherical droplets. Most droplets observed in applications, including those in internal combustion engines, are far from spherical [3]. Some preliminary results referring to the modelling of heating and evaporation of spheroidal droplets are described in Sect. 3.6. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5_8

453

454

8 Concluding Comments

The main limitation of the approach described in that section is that it is applicable only to weakly deformed spherical droplets. Even in this case, however, it was not possible to develop models as elegant as those developed for spherical droplets. The perturbation methods might be used if the deviation of the shape of the droplets from the spherical is small, or these shapes can be approximated by long cylinders. In the general case, however, these problems would most likely need to be analysed using complex and CPU intensive numerical methods, which cannot be incorporated into conventional Computational Fluid Dynamics (CFD) codes.

8.3 Limitations of the ETC/ED Model The analyses of heating and evaporation of moving spherical droplets in most CFD codes assume that the Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) models are applicable. The validity of these models was investigated based on direct comparison of their predictions and the predictions of a more general vortex model for a limited range of parameters (see this comparison for the ETC model described in [1]). The applicability of the models outside this range is not at all obvious. In any case, the errors linked with the application of these models could only be estimated using direct comparison between their predictions and the predictions of the vortex model for a wide range of parameters. This has not yet been done. Another problem with the application of the ETC/ED models lies in the assumption that a realistic inhomogeneous distribution of surface temperatures in moving droplets can be approximated by the homogeneous distribution in the analysis of droplet heating and evaporation. The limits of applicability of this assumption require special investigation.

8.4 Effects of the Interaction Between Droplets As shown in Chaps. 3 and 4, even in the simple case of droplets moving in line one after another, the effect of interaction between droplets on their heating and evaporation cannot be ignored when the distance parameter (ratio of the distance between droplets and their diameter) is less than about 10. See [2] for further discussions of this issue. Various semi-empirical formulae considering these interactions have been described. In realistic applications, including internal combustion engines, the mutual positions of moving droplets can be very complex [3]. The number of droplets affecting any particular droplet in a dense spray can be large. It is not clear how this complex interaction between droplets can be incorporated when analysing individual droplet heating and evaporation in a flow of gas. Some preliminary investigations of this problem were presented in [6]. The results of aerodynamic breakup of a cluster of Diesel fuel droplets (without considering the effects of their heating and evaporation) moving parallel to the gas flow are presented and discussed in [9].

8.5 Heating and Evaporation in Near/Super-critical Conditions

455

8.5 Heating and Evaporation in Near/Super-Critical Conditions Most of the models presented in this book have been developed for those cases when both temperature and pressure at droplet surfaces do not approach the critical point. This restriction of the models is crucial for many engineering applications including those in realistic Diesel engine conditions, where both temperature and pressure are likely to reach and exceed critical values. Taking pressure as an example, Diesel and biodiesel fuels are injected into a cylinder pressurised to about 25 atm, which may then increase to more than 60 atm after ignition. At the same time, critical pressure for most hydrocarbon fuels is expected to be in the range of 15–30 atm. Although temperatures at droplet surfaces are generally lower than those in the ambient gas, they too can reach critical values. Therefore, the processes in fuel sprays, including droplet heating and evaporation, can occur in near-, trans- and super-critical regimes. Under these regimes, the properties of liquid fuels change significantly. The latent heat of evaporation becomes zero; consequently, the solubility of one fluid into the other becomes important [4] instead of evaporation. The sharp distinction between the liquid and gas phase disappears, and the surface tension vanishes. Transport and thermodynamic properties may vary significantly even with small changes in temperature and pressure due to strong thermodynamic nonideality and nonlinearity [4].

8.6 Effects of the Moving Interface due to Evaporation The effects of evaporation and thermal swelling lead to moving droplet interface. This affects droplet heating and diffusion of components inside droplets, as described in Chaps. 3 and 4. From the point of view of classical mechanics, one would expect that the exchange of energy between a lorry, and a ball hitting the back of that lorry, would decrease in the case when the lorry moves away from the approaching ball, compared with the case of a stationary lorry. The same decrease in energy is predicted by the model presented in Chap. 3. The problem, however, lies in the quantification of this effect. Since the velocity of the moving interface due to evaporation is many orders of magnitude less than the velocity of molecules, this effect would be expected to be negligible in contrast to the prediction of the model. Hence, the investigation of the physical background of this effect still needs to be performed. The analysis of droplet heating and evaporation in the presence of the moving interface, described in Chaps. 3 and 4, also did not consider the effect of thermal radiation. This should be straightforward if we assume that droplets are semi-transparent, and the radiative heating is spherically symmetric (cf. the effect of thermal radiation on droplet heating with a stationary interface discussed in Chap. 2).

456

8 Concluding Comments

8.7 Complex Multi-component Droplets The development of the Multi-dimensional Quasi-discrete Model (MDQDM), presented in Sect. 4.4, was an important step forward in the development of a model for heating and evaporation of complex multi-component droplets (e.g. droplets of Diesel and petrol fuels and their blends). The version of this model described in Sect. 4.4, however, is preliminary. The choice of quasi-components and components in this model was based on trial and error and no ‘universal’ algorithm for their selection was developed. The approach to auto-selection of quasi-components/components described in Sect. 4.4.6 partly solves this problem although further research in this direction, including the optimisation of this approach, is required before it can be recommended for the implementation in Computational Fluid Dynamics (CFD) codes. Another limitation of the MDQDM lies in the fact that this model deals with quasicomponents with non-integer carbon numbers which cannot be used for modelling chemical processes. The approaches to overcoming this problem are discussed in Sect. 4.4.4 but further research in this direction is essential. As in the case of the Discrete Component Model, the MDQDM assumes that the diffusion coefficients of all species are the same and are controlled only by the composition of the droplet. This assumption introduces errors which cannot be quantified at present. The effects of droplet motion on the liquid diffusion coefficient were estimated using the average composition of droplets. The validity of this assumption has never been investigated to the best of the author’s knowledge. Finally, this model was developed for modelling heating and evaporation of spherical droplets. Its generalisation to the case of non-spherical droplets (even slightly deformed spheres) has not been investigated to the best of the author’s knowledge.

8.8 Puffing and Micro-explosion The model of puffing and micro-explosion described in Sect. 5.3, with its extensions and modifications discussed in Sects. 5.4–5.8, proved to be efficient for the qualitative analyses of the phenomena. This model, however, has several important limitations which need to be overcome before it can be recommended for quantitative analyses of these processes. Firstly, this model needs to consider the effect of bubble formation before puffing and/or micro-explosion starts. The time required for this bubble formation would increase the time to puffing/micro-explosion predicted by this model. Secondly, the effects of the shift of the water subdroplet from the centre of the fuel droplet need to be investigated. This effect would reduce the time to puffing/microexplosion predicted by the model without the shift. Some preliminary results of the investigation of this effect are described in Sect. 5.8. Finally, the deformations of the shapes fuel droplet and water subdroplet need to be considered using an approximate model without Direct Numerical Simulation (DNS).

8.8 Puffing and Micro-explosion

457

Another problem with modelling puffing and micro-explosion is that no model suggested so far, to the best of the author’s knowledge, can predict size distribution of child droplets. A possible approach to the construction of such model could be based on the methodology used in the development of the stochastic model of droplet break-up described in Sect. 1.1.2.

8.9 Advanced Kinetic and Molecular Dynamics Models Although there has been considerable progress in the development of kinetic and molecular dynamics models, described in Chap. 6, many important effects during droplet heating and evaporation have not been considered. The effect of the complexity of Diesel fuel composition was considered by approximating this fuel with a mixture of n-dodecane and p-dipropylbenzene. The approximation of Diesel fuel by just two components is certainly too crude even for engineering applications. We cannot use the 98-component approximation, as in the Multi-dimensional Quasidiscrete Model, in kinetic modelling, but it would be essential to consider the heaviest components of this fuel which are expected to be dominant at the final stage of Diesel fuel droplet heating and evaporation. If this is not done then errors due to ignoring the kinetic effects altogether could be less than those due to the simplified approximation of Diesel fuel. Crude estimates of quantum-chemical effects on the value of the evaporation coefficient of n-dodecane droplets were made, and these effects were shown to be small except at temperatures close to the critical temperature. Despite this conclusion that quantum-chemical effects on the values of the evaporation coefficient are weak, this issue cannot be considered closed. Firstly, this conclusion was drawn based on the application of a rather simplistic model. The quantum-chemical effects on the values of the evaporation coefficient, analysed by more advanced models are still to be investigated. Also, the quantum-chemical effects on the evaporation coefficient for realistic multi-component Diesel fuel droplets have not been studied at all to the best of the author’s knowledge.

8.10 Effective Approximation of the Kinetic Effects As noticed in Sect. 6.3, the only feasible way to apply the results of kinetic modelling to the analysis of realistic droplet heating and evaporation within Computational Fluid Dynamics (CFD) codes would be to approximate these results using simple analytical formulae. The first attempts to do this are discussed in Sect. 6.3, where simple approximate formulae describing the temporal evolution of Diesel fuel droplet radii and temperatures predicted by the kinetic model are described. These formulae, however, are valid for only a limited range of gas temperatures and fixed values of initial droplet radii, or for a limited range of initial droplet radii and fixed values of

458

8 Concluding Comments

gas temperature. A more general approximation of these results has yet to be found. An alternative approach to approximating the kinetic results is described in Sect. 6.9. The approach described in that section is based on the approximation of the vapour density and temperature at the outer boundary of the kinetic region, rather than on the direct approximation of the values of the droplet radii and surface temperatures. The usefulness of this approach to engineering CFD modelling is still to be investigated.

References 1. Abramzon, B., & Sirignano, W. A. (1989). Droplet vaporization model for spray combustion calculations. International Journal of Heat and Mass Transfer, 32, 1605–1618. 2. Castanet, G., Perrin, L., Caballina, O., & Lemoine, F. (2016). Evaporation of closely-spaced interacting droplets arranged in a single row. International Journal of Heat and Mass Transfer, 93, 788–802. 3. Crua, C., Heikal, M. R., & Gold, M. (2015). Microscopic imaging of the initial stage of diesel spray formation. Fuel, 157, 140–150. 4. Kontogeorgis, G. M., & Folas, G. K. (2010). Thermodynamic Models for Industrial Applications. New York: Wiley. 5. Li, Y., & Rybdylova, O. (2021). Application of the generalised fully Lagrangian approach to simulating polydisperse gas-droplet flows. International Journal of Multiphase Flow, 142, 103716. 6. Markadeh, R. S., Arabkhalaj, A., Ghassemi, H., & Azimi, A. (2020). Droplet evaporation under spray-like conditions. International Journal of Heat Mass Transfer, 148, 119049. 7. Papoutsakis, A., Rybdylova, O. D., Zaripov, T. S., Danaila, L., Osiptsov, A. N., & Sazhin, S. S. (2018). Modelling of the evolution of a droplet cloud in a turbulent flow. International Journal of Multiphase Flow, 104, 233–257, 119049. 8. Sazhin, S. S. (2017). Modelling of fuel droplet heating and evaporation: Recent results and unsolved problems. Fuel, 196, 69–101, 119049. 9. Stefanitsis, D., Strotos, G., Nikolopoulos, N., & Gavaises, M. (2019). Numerical investigation of the aerodynamic breakup of a parallel moving droplet cluster. International Journal of Multiphase Flow, 121, 103123.

Appendix A

Derivation of Formula (2.86)

In this appendix, the details of the derivation of Formula (2.86) are described, following [2]. Introduction of the new variable u = T r = T R/Rd allows us to present Eq. (2.85) as ∂ 2u ∂u ˜ ) (A.1) = κ R 2 + P(r ∂t ∂r with the boundary and initial conditions: ⎫ + H (t)u = M(t) when r = 1 ⎬ u=0 when r = 0 , ⎭ u(t = 0) = r T0 (r Rd ) ≡ T˜0 (r ) when 0 ≤ r ≤ 1 ∂u ∂r

(A.2)

where H (t) =

h(t)Rd − 1, kl

M(t) =

h(t)Teff (t)Rd , kl

˜ ) = r P(r Rd ). P(r

Assuming that h(t) ≡ h = const, we can write H (t) ≡ h 0 = (h Rd /kl ) − 1 = const. Introducing a new parameter μ0 (t) =

hTeff (t)Rd , kl

boundary and initial conditions (A.2) can be rewritten as ⎫ + h 0 u = M(t) = μ0 (t) when r = 1 ⎬ u=0 when r = 0 . ⎭ u(t = 0) = r T0 (r Rd ) ≡ T˜0 (r ) when 0 ≤ r ≤ 1 ∂u ∂r

(A.3)

Remembering that h 0 > −1, we look for the solution to Eq. (A.1) in the form © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

459

460

Appendix A: Derivation of Formula (2.86)

1 r μ0 (t) + W (r, t). 1 + h0

u(r, t) =

(A.4)

Substitution of Expression (A.4) into Eq. (A.1) leads to the following equation for W : ∂W ∂2W ˜ ) − r dμ0 (t) = κ R 2 + P(r (A.5) ∂t ∂r 1 + h 0 dt with the boundary and initial conditions:    W |r =0 = ∂∂rW + h 0 W r =1 = 0 . r μ0 (0) W |t=0 = T˜0 (r ) − 1+h 0

(A.6)

We look for the solution to Eq. (A.5) with the boundary and initial conditions (A.6) in the form ∞ W (r, t) = cn (t)vn (r ), (A.7) n=1

where functions vn (r ) form the full set of non-trivial solutions to the equation d2 v + λ2 v = 0, dr 2

(A.8)

subject to boundary conditions

v|r =0 =

  dv + h 0 v  = 0. dr r =1

The general solution to Eq. (A.8) v(r ) = A cos λr + B sin λr satisfies the boundary condition when A = 0 and λ cos λ + h 0 sin λ = 0.

(A.9)

The solution to Eq. (A.9) gives a set of positive eigenvalues λn numbered in ascending order (n = 1, 2, ...). If h 0 = 0, then λn = π(n − 21 ). For B = 1, expressions for eigenfunctions vn are presented as vn (r ) = sin λn r

(n = 1, 2, ...).

The value of B is implicitly accounted for by the coefficients cn (t) in series (A.7). Functions vn (r ) form a full set of eigenfunctions which are orthogonal for r ∈ [0, 1]. The orthogonality of functions vn follows from the formula:

Appendix A: Derivation of Formula (2.86)



1

461

vn (r )vm (r )dr = δnm || vn ||2 ,

(A.10)

0

where

δnm =

0 n = m , 1n=m

 



1 sin 2λn h0 1 = . 1− 1+ 2 || vn || = 2 2λn 2 h 0 + λ2n 2

Eigenvalue λ0 = 0 describes the trivial eigenfunction v0 (r ) = 0. The orthogonality of vn allows us to expand known functions in Eqs. (A.5) and (A.6) in the series ˜ )= P(r

∞

pn vn (r ),

(A.11)

qn vn (r ),

(A.12)

n=1

T˜0 (r ) =

∞ n=1

f (r ) ≡ −r/(1 + h 0 ) =

∞

f n vn (r ),

(A.13)

n=1

where 1 ˜ )vn (r )dr, pn = ||v1n ||2 0 P(r 1 1 qn = ||vn ||2 0 T˜0 (r )vn (r )dr, 1 λn f n = ||v1n ||2 0 f (r )vn (r )dr = − ||vsin 2 2 n || λ

⎫ ⎪ ⎬ ⎪ ⎭

.

(A.14)

n

Having substituted (A.7), (A.11) and (A.13) into Eq. (A.5), we obtain ∞

dcn (t) n=1

dt

  ∞

dμ0 (t) pn + f n + cn (t)κ R λ2n vn (r ) = vn (r ). dt n=1

(A.15)

Both sides of Eq. (A.15) are Fourier series of functions vn (r ). Two Fourier series are equal if, and only if, their coefficients are equal. Thus dcn (t) dμ0 (t) + cn (t)κ R λ2n = pn + f n . dt dt

(A.16)

The initial condition for cn (t) follows from the initial condition for W cn (0) = qn + f n μ0 (0).

(A.17)

The solution to Eq. (A.16), with the initial condition (A.17), is presented as

462

Appendix A: Derivation of Formula (2.86)



  pn pn 2 + exp −κ λ t q + f μ (0) − R n n n 0 κ R λ2n κ R λ2n

cn (t) =

+ fn 0

t

  dμ0 (τ ) exp −κ R λ2n (t − τ ) dτ. dτ

(A.18)

Having substituted functions (A.18) and vn (r ) into series (A.7), the solution to Eq. (A.5) subject to (A.6) is obtained as 

∞   pn pn 2 + W (r, t) = + exp −κ R λn t qn + f n μ0 (0) − κ R λ2n κ R λ2n n=1

t

+ fn 0

   dμ0 (τ ) exp −κ R λ2n (t − τ ) dτ sin λn r, dτ

(A.19)

where pn , qn , f n and λn are given by Eqs. (A.14) and (A.9), and u(r, t) =

 ∞  

pn 1 pn 2t + r μ0 (t) + + exp −κ λ q + f μ (0) − n n 0 R n 1 + h0 κ R λ2n κ R λ2n n=1

+ fn 0

t

   dμ0 (τ ) exp −κ R λ2n (t − τ ) dτ sin λn r. dτ

(A.20)

Remembering (A.14) and the definition of u, the final solution to Eq. (2.85) is written as T (r, t) =



∞     pn sin λn 1 pn 2 − + exp −κ λ t q − μ0 (0) exp −κ R λ2n t − R n n r κ R λ2n κ R λ2n || vn ||2 λ2n n=1

−

sin λn || vn ||2 λ2n

0

t

   dμ0 (τ ) exp −κ R λ2n (t − τ ) dτ sin λn r + Teff (t). dτ

(A.21)

This formula is the same as (2.86). In the limit of P = 0, it reduces to (2.41). When deriving (A.21), we considered that Teff (t) = kl μ0 (t)/(h Rd ). If T0 (r ) is twice differentiable, then the series in (A.19), (A.20), (A.21) converge absolutely and uniformly for all t ≥ 0 and r ∈ [0, 1] since | pn | < const,

| qn |
1. It can be demonstrated that λn > π(n − 1). Thus, for n 1 

1 > nπ/2 > n. (A.22) λn > nπ 1 − n

Appendix A: Derivation of Formula (2.86)

463

Equation (A.21) (or (2.86)) is expected to reduce to Eq. (2.37) in the limit when kl → ∞. In what follows, it is proven that this indeed happens, following [1]. Let us first restrict our analysis to the droplet’s surface and present Eq. (A.21) for R = Rd as (A.23) Ts = T∞ + (Ts0 − T∞ )(S1 + S2 ), where S1 = 

S2 = 

  sin2 λ1 exp −κ R λ21 t , 2 || v1 ||

λ21

∞

  sin2 λn exp −κ R λ2n t , 2 || vn ||

λ2 n=2 n

(A.24)

(A.25)

 ≡ k g /kl 1. In this case h 0 =  − 1 in (A.3). In what follows, the value of S2 is estimated and the expression for S1 is rearranged in the limit  → 0. Since λn ≥ λ2 > π/2 for n = 2, 3, ..., we have    sin 2λn  1 1 1    2λ  ≤ 2λ ≤ 2λ < π , n n 2

 1 π −1 1 || vn || > 1− = , 2 π 2π 2

1 2π . < 2 ||vn || π −1

(A.26)

On the other hand, we can see from the solution to the equation for λ that λn > π n/2,

n = 2, 3, ...

This implies that  21 1  . < λn n≥2 πn

(A.27)

Remembering (A.26) and (A.27), S2 is estimated as 2π | S2 |<  π −1

    ∞ ∞ 1 2 2 1 8 = →0 π n2 π(π − 1) n=2 n 2 n=2

(A.28)

when  → 0 for all t. It can be can seen that for  → 0: h 0 =  − 1 → −1 + 0 , and λ1 → +0. Thus, tan λ1 can be expanded in a Taylor series and Eq. (A.9) can be rewritten as ∞

1 2 1 1 + λ21 + λ41 + · · · = k . = 3 15 1− k=0

(A.29)

464

Appendix A: Derivation of Formula (2.86)

We look for the solution to Eq. (A.29) in the form of the series λ21 = c1  + c2  2 + · · ·

(A.30)

Having substituted Formula (A.30) into Eq. (A.29), we can write  2 2  1 c1  + c2  2 + · · · + c1  + c2  2 + · · · + · · · = 1 +  +  2 + · · · 3 15 (A.31) Equation (A.31) is satisfied when the coefficients before equal powers of  are equal. This leads us to the following values of the first two coefficients: c1 = 3 and c2 = −3/5. In the limit  → 0, we can ignore all terms in Eq. (A.30) apart from the first one and write (A.32) λ21 ≈ 3. 1+

Having substituted (A.32) into (A.24) and remembering that in the limit λ1 → +0: (sin(2λ1 ))/(2λ1 ) ≈ 1 − (2λ1 )2 /6 and || v1 ||2 ≈ λ21 /3, we obtain 



3k g t 3ht = exp − . S1 = exp − cl ρl Rd cl ρl Rd2

(A.33)

Remembering that S2 → 0 and having substituted (A.33) into (A.23), we obtain 

3ht . Ts = T∞ + (Ts0 − T∞ ) exp − cl ρl Rd

(A.34)

The same equation follows from the energy balance equation at the droplet surface, if there are no temperature gradients inside the droplet (see Eq. (2.37)) 4 dTs π Rd3 ρl cl = 4π Rd2 h(T∞ − Ts ). 3 dt

(A.35)

This analysis can be generalised to the case when R = Rd and it can be demonstrated that in the case when kl → ∞ and R = 0, the temperature inside droplets is equal to the surface temperature satisfying Eq. (A.34). The temperature at the point R = 0 is the same as at the points R = 0 since ∂ T /∂ R | R=0 = 0. For practical applications, the series in Expression (A.21) needs to be truncated. It was found that the best way to choose the number of terms in this series is to compare the surface temperature which it predicts at t = 0 and R = Rd with Ts0 . If the allowed error of calculations is less than ε, then the required number of terms N need to satisfy the following condition: 

  N   T∞ ka sin λ2n   (N ) =  1 − 1−  < ε. 2 2  Ts0 kl n=1 λn || Vn || 

(A.36)

Appendix A: Derivation of Formula (2.86)

465

References 1. Sazhin SS, Krutitskii PA (2003) A conduction model for transient heating of fuel droplets. In Progress in Analysis. Proceedings of the 3d International ISAAC (International Society for Analysis, Applications and Computations) Congress (August 20–25, 2001, Berlin). Eds. H.G.W. Begehre, R.P. Gilbert, M.W. Wong. World Scientific, Singapore, Vol. II, 1231–1239. 2. Sazhin SS, Krutitskii PA, Abdelghaffar WA, Mikhalovsky SV, Meikle ST, Heikal MR (2004) Transient heating of diesel fuel droplets. Int. J Heat Mass Transfer 47:3327–3340.

Appendix B

Derivation of Formula (2.89)

In this appendix, the details of the derivation of Formula (2.89) are described, following [2]. Introduction of the new variable   u = T − Tg0 (Rg ) R allows us to simplify Eq. (2.85), but with initial and boundary conditions used for Eq. (2.1) (see Eqs. (2.3) and (2.4)), to ∂ 2u ∂u = κ 2 + R P(t, R), ∂t ∂R

(B.1)

with the following initial and boundary conditions: u|t=0 = −T0 R     u| R=Rd− = u| R=Rd+ , kl Rd u R − u 

where T0 ≡ T0 (R) =

R=Rd−

    = k g Rd u R − u 

(B.2) R=Rd+

, u| R=Rg = 0, (B.3)

Tg0 (Rg ) − Td0 (R) when R ≤ Rd . Tg0 (Rg ) − Tg0 (R) when Rd < R ≤ Rg

Conditions (B.3) need to be amended by the boundary condition at R = 0. Since T − Tg0 is finite at R = 0, then u| R=0 = 0. We look for the solution to Eq. (B.1) in the form u=

∞

n (t)vn (R),

(B.4)

n=1

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

467

468

Appendix B: Derivation of Formula (2.89)

where functions vn (R) form the full set of non-trivial solutions to the eigenvalue problem d2 v + a 2 λ2 v = 0 (B.5) d R2 subject to boundary conditions: ⎫ ⎪ v| R=0 = v| R=Rg = 0 ⎬ − + v| R=Rd = v| R=Rd ,       ⎪ kl Rd v R − v  R=R − = k g Rd v R − v  R=R + ⎭ d

(B.6)

d

where ⎧ ⎨ cl ρl ≡ al when R ≤ Rd 1 k a = √ =  c pgl ρg ⎩ κ ≡ ag when Rd < R ≤ Rg . kg

(B.7)

√ Note that λ has dimension 1/ time. We look for the solution to Eq. (B.5) in the form

when R ≤ Rd A sin(λal R) (B.8) v(R) = B sin(λag (R − Rg )) when Rd < R ≤ Rg . Function (B.8) satisfies boundary conditions (B.6) at R = 0. Having substituted (B.8) into boundary conditions (B.6) at R = Rd , we obtain A sin(λal Rd ) = B sin(λag (Rd − Rg )),

(B.9)

Akl [Rd λal cos(λal Rd ) − sin(λal Rd )]   = Bk g Rd λag cos(λag (Rd − Rg )) − sin(λag (Rd − Rg )) .

(B.10)

Condition (B.9) is satisfied when  A = [sin(λal Rd )]−1  −1 . B = sin(λag (Rd − Rg ))

(B.11)

Having substituted Expressions (B.11) into (B.10), we obtain   kl [Rd λal cot(λal Rd ) − 1] = k g Rd λag cot(λag (Rd − Rg )) − 1 .

(B.12)

Remembering the definitions of al and ag , Eq. (B.12) is rearranged to 

kl cl ρl cot(λal Rd ) −

 kl − k g k g c pg ρg cot(λag (Rd − Rg )) = . Rd λ

(B.13)

Appendix B: Derivation of Formula (2.89)

469

The solution to Eq. (B.13) is a countable set of positive eigenvalues λn which can be arranged in ascending order 0 < λ1 < λ2 < ..... The negative solutions −λn also satisfy Eq. (B.13) as both sides of this equation are odd functions of λ. λ = 0, however, does not satisfy it. Having substituted these λn into Expression (B.8) and remembering Expressions (B.11), the following expressions for eigenfunctions vn are obtained:  sin(λ a R) n l when R ≤ Rd n al Rd ) vn (R) = sin(λ (B.14) sin(λn ag (R−Rg )) when Rd < R ≤ Rg . sin(λn ag (Rd −Rg )) It is proven in Appendix C that functions vn (R) are orthogonal with weight

kl al2 = cl ρl when R ≤ Rd k g ag2 = c pg ρg when Rd < R ≤ Rg .

b= This means that



Rg

vn (R)vm (R)bd R = δnm ||vn ||2 ,

0

where δnm =

1 when n = m 0 when n = m.

The proof of completeness of this set of functions is much more complex (it is based on the methods of functional analysis and properties of Banach spaces [3]). Implicitly, the fact that this set is complete, could be supported by the agreement between the predictions of Formula (2.89) without thermal radiation, in the limit Rg = ∞ and Tg0 =const and the corresponding formula obtained by Cooper [1]. The norm of vn with weight b is calculated as

Rg

||vn || = 2

0

+

Rg Rb

vn2 bd R



= 0

Rb



sin(λn al R) sin(λn al Rd )

sin(λn ag (R − Rg )) sin(λn ag (Rd − Rg ))

2 cl ρl d R

2 c pg ρg d R

  cl ρl sin(2λn al Rd ) = Rd − 2λn al 2 sin2 (λn al Rd ) +

  sin(2λn ag (Rd − Rg )) c pg ρg R − R + g d 2λn ag 2 sin2 (λn ag (Rd − Rg )) =

c pg ρg (Rg − Rd ) kl − k g cl ρl Rd + − . 2Rd λ2n 2 sin2 (λn al Rd ) 2 sin2 (λn ag (Rd − Rg ))

(B.15)

470

Appendix B: Derivation of Formula (2.89)

When deriving (B.15), we used Eq. (B.13). Since all functions vn satisfy boundary conditions (B.6), function u defined by Expression (B.4) satisfies boundary conditions (B.3). Expanding R P(t, R) in a series over vn gives ∞

R P(t, R) =

pn (t)vn (R),

(B.16)

n=1

where 1 ||vn ||2

pn (t) =



Rg

R P(t, R)vn (R)bd R.

(B.17)

0

Since P(t, R) = 0 at R > Rd , Formula (B.17) is simplified to pn (t) =

cl ρl ||vn ||2



Rd

R P(t, R)vn (R)d R.

(B.18)

0

Having substituted Expressions (B.4) and (B.16) into Eq. (B.1), we obtain ∞



n (t)vn (R) = −

n=1

∞

n (t)λ2n vn (R) +

n=1

∞

pn (t)vn (R).

(B.19)

n=1

When deriving Eq. (B.19), we took into account that functions vn (R) satisfy Eq. (B.5) for λ = λn . Equation (B.19) is satisfied if, and only if, 

n (t) = −λ2n n (t) + pn (t).

(B.20)

The initial condition for n (t) can be obtained after substituting Expression (B.4) into the initial condition (B.2) for u ∞

n (0)vn (R) = −T0 R.

(B.21)

n=1

Remembering that vn are orthogonal with the weight b, we obtain from Eq. (B.21) n (0) =

1 ||vn ||2



Rg

cl ρl =− ||vn ||2 sin(λn al Rd ) −

c pg ρg 2 ||vn || sin(λn ag (Rd − Rg ))

(−T0 R)vn (R)bd R.

0





Rd

T0 (R)R sin(λn al R)d R

0 Rg Rd

T0 (R)R sin(λn ag (R − Rg ))d R.

Appendix B: Derivation of Formula (2.89)

471

If Tg0 (R) = Tg0 (Rg ) =const, and Td0 (R) =const, then

T0 ≡ T0 (R) =

Tg0 (Rg ) − Td0 (Rd ) when R ≤ Rd 0 when Rd < R ≤ Rg .

In this case, the expression for n (0) is further simplified to cl ρl T0 n (0) = − 2 ||vn || sin(λn al Rd )



Rd

R sin(λn al R)d R 0

√   1 T0 kl cl ρl R . cot(λ a R ) − = d n l d λn ||vn ||2 λn al

(B.22)

The solution to Eq. (B.20) subject to the initial condition (B.22) is presented as   n (t) = exp −λ2n t n (0) +



t 0

  exp −λ2n (t − τ ) pn (τ )dτ.

(B.23)

Equation (2.89) follows from the definition of u and Expressions (B.4) and (B.23). Equation (2.9) follows from Eq. (2.89) when P = 0.

References 1. Cooper F (1977) Heat transfer from a sphere to an infinite medium. Int. J Heat Mass Transfer 20:991–993. 2. Sazhin SS, Krutitskii PA, Martynov SB, Mason D, Heikal MR, Sazhina EM (2007) Transient heating of a semitransparent spherical body. Int. J Thermal Science 46(5):444–457. 3. Vladimirov VS (1971) Equations of Mathematical Physics. Marcel Dekker, N.Y.

Appendix C

Proof of Orthogonality of vn (R) with the Weight b

Using Expressions (B.14) for vn (R), we obtain for n = m [1]

Rg

Inm ≡

vn (R)vm (R)bd R =

0

+

kl al2 sin(λn al Rd ) sin(λm al Rd )

k g ag2 sin(λn ag (Rd − Rg )) sin(λm ag (Rd − Rg ))



Rg



Rd

sin(λn al R) sin(λm al R)d R 0

sin(λn ag (R − Rg )) sin(λm ag (R − Rg ))d R

Rd

  kl al2 sin((λn − λm )al Rd ) sin((λn + λm )al Rd ) = − 2 sin(λn al Rd ) sin(λm al Rd ) (λn − λm )al (λn + λm )al − 

k g ag2 2 sin(λn ag (Rd − Rg )) sin(λm ag (Rd − Rg ))

sin((λn − λm )ag (Rd − Rg )) sin((λn + λm )ag (Rd − Rg )) − × (λn − λm )ag (λn + λm )ag =

+



  k g ag sin((λn − λm )ag (Rd − Rg )) kl al sin((λn − λm )al Rd ) 1 − 2(λn − λm ) sin(λn al Rd ) sin(λm al Rd ) sin(λn ag (Rd − Rg )) sin(λm ag (Rd − Rg ))

  k g ag sin((λn + λm )ag (Rd − Rg )) kl al sin((λn + λm )al Rd ) 1 − + 2(λn + λm ) sin(λn al Rd ) sin(λm al Rd ) sin(λn ag (Rd − Rg )) sin(λm ag (Rd − Rg ))

   kl al (cot(λm al Rd ) − cot(λn al Rd )) − k g ag cot(λm al (Rd − Rg )) − cot(λn al (Rd − Rg )) = 2(λn − λm )    −kl al (cot(λm al Rd ) + cot(λn al Rd )) + k g ag cot(λm al (Rd − Rg )) + cot(λn al (Rd − Rg )) + . 2(λn + λm )

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

473

Appendix C: Proof of Orthogonality of vn (R) with the Weight b

474

Using Eq. (B.13), we obtain [1] Inm

1 = 2(λn − λm )



kl − k g kl − k g − R d λm R d λn =



1 − 2(λn + λm )



kl − k g kl − k g + R d λm R d λn



kl − k g kl − k g − = 0. 2Rd λn λm 2Rd λn λm

Thus, orthogonality of vn (R) with the weight b is proven.

Reference 1. Sazhin SS, Krutitskii PA, Martynov SB, Mason D, Heikal MR, Sazhina EM (2007) Transient heating of a semitransparent spherical body. Int. J Thermal Science 46(5):444–457.

Appendix D

Derivation of Formula (3.103)

In this appendix, the details of the derivation of Formula (3.103) are shown, following [4].

Preliminary analysis Following [1], the new variable r and function F(t, r ) are introduced: r = R/Rd (t), F(t, r ) = u(t, R). This new variable allows us to reduce the problem with a moving interface to the one with a stationary interface. Note that 0≤r ≤1

when

0 ≤ R ≤ Rd (t).

Since 











u t = Ft + Fr rt = Ft − r





Rd (t)  Fr F     Fr ; u R = Fr r R = ; u R R = 2rr , Rd (t) Rd (t) Rd (t)

Equation (3.96) can be rewritten as 







Rd2 (t)Ft = κ Frr + r Rd (t)Rd (t)Fr .

(D.1)

Equation (D.1) is identical to the one studied in [2], where the distribution of temperature in the melting region was considered (plane problem).

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

475

476

Appendix D: Derivation of Formula (3.103)

Equation (D.1) is solved for 0 ≤ r ≤ 1 with the following initial and boundary conditions: F|t=0 = Rd0 r T0 (r Rd0 ), 

F|r =0 = 0,

   Fr + H˜ (t)F 

r =1

= μ(t), ˜

where H˜ (t) = H (t)Rd (t), μ(t) ˜ = M(t)Rd2 (t); the solution is obtained for each timestep (t ∈ [t0 , t0 + t]). Following [1], the new function W (t, r ) is introduced via the formula    Rd (t)Rd (t) 2 1 r W (t, r ). exp − F(t, r ) = √ 4κ Rd (t)

(D.2)

From Expression (D.2), the following expressions for the derivatives are obtained:  





(Rd (t))2 + Rd (t)Rd (t) 2 1 −3/2  −1/2 Rd (t)Rd (t) + Rd (t) r Ft = − 2 4κ 

 −1/2 +Rd (t)Wt (t, r )

"

! W (t, r )



  Rd (t)Rd (t) 2 r , exp − 4κ



#     2r Rd (t)Rd (t) Rd (t)Rd (t) 2 1  r , W (t, r ) + √ W (t, r ) exp − Fr = − √ 4κ 4 Rd (t)κ Rd (t) r 

⎧ ⎨

⎡  R (t)Rd (t) 2  ⎣− d + 2r 2 Frr = √ ⎩ Rd (t) 4κ



Rd (t)Rd (t) 4κ

!2 ⎤ ⎦ W (t, r )

#     Rd (t)Rd (t) 2 1 4r Rd (t)Rd (t)   Wr (t, r ) + √ r . W (t, r ) exp − −√ 4κ 4κ Rd (t) Rd (t) rr Remembering Expression (3.100), it can be shown that d2 Rd /dt 2 = 0. Hence, the substitution of the above-mentioned expressions for the derivatives into Eq. (D.1) gives   (D.3) Rd2 (t)Wt (t, r ) = κ Wrr (t, r ). For non-zero d2 Rd /dt 2 and considering the contribution of P(R), Eq. (D.3) would need to be replaced by the following equation (cf. Eq. (8.149) in [1]):  Rd2 (t)Wt (t, r )

 Rd3 r d 2 Rd r2 Rd3 2 W (t, r ) + P(Rd r ), (D.4) = κ Wrr (t, r ) + 4κ dt q K (r, t) 



Appendix D: Derivation of Formula (3.103)

477

   Rd (t)Rd (t) 2 1 r . exp − q K (r, t) = √ 4κ Rd (t)

where

Equation (D.4) reduces to Eq. (D.3) when d2 Rd /dt 2 = 0 and P(R) = 0. Equation (D.3) needs to be solved subject to the initial and boundary conditions  W (t, r )|t=0 = W0 (r ) ≡

3/2 Rd0 r T0 (r Rd0 ) exp

  Rd (0)Rd0 2 r , 4κ

W (t, r )|r =0 = 0, 

  Wr (t, r ) + H0 (t)W (t, r )  

r =1

where

(D.5) (D.6)



  Rd (t)Rd (t) = μ0 (t) ≡ μ(t) ˜ Rd (t) exp , 4κ (D.7) 





R (t)Rd (t) R (t)Rd (t) h(t) = . H0 (t) = H˜ (t) − d Rd (t) − 1 − d 2κ kl 2κ

Analytical Solution Let us now assume that H0 (t) ≡ h 0 = const > −1. Remembering that h = k g /Rd (t) for stationary droplets, the term h(t) Rd (t) kl is simplified to k g /kl which does not depend on t. Except at the final stage of droplet evaporation, in Diesel engine-like conditions this term is much larger than 

Rd (t)Rd (t) . 2κ This allows us to ignore the time dependence of H0 (t) during each timestep. This assumption is relaxed later. Introducing function V (t, r ) via the relation W (t, r ) = V (t, r ) + Equation (D.3) is rearranged to

μ0 (t) r, 1 + h0

(D.8)

478

Appendix D: Derivation of Formula (3.103) 

 Rd2 (t)Vt (t, r )

μ (t) 2 = κ Vrr (t, r ) − 0 R (t)r. 1 + h0 d 

(D.9)

The following initial and boundary conditions for Eq. (D.9) are used: V (t, r )|t=0 = W0 (r ) −

μ0 (0) r, 1 + h0

0 ≤ r ≤ 1,

V (t, r )|r =0 = 0, 

   Vr (t, r ) + h 0 V (t, r ) 

r =1

= 0.

Without the contribution of thermal radiation, Eq. (D.9) is the same as Eq. (12) in [3]. Using the approach described in that paper, we look for the solution of Eq. (D.9) in the form ∞ V (t, r ) = n (t)vn (r ), (D.10) n=1

where functions vn (r ) form the full set of non-trivial solutions to the equation d2 v + λ2 v = 0, dr 2

(D.11)

(0 ≤ r ≤ 1) with the boundary conditions

v|r =0 =

  dv + h 0 v  = 0. dr r =1

(D.12)

The general solution to Eq. (D.11) v(r ) = A cos λr + B sin λr

(D.13)

satisfies Conditions (D.12) when A = 0 and λ cos λ + h 0 sin λ = 0.

(D.14)

The solution to Eq. (D.14) gives a set of positive eigenvalues λn numbered in ascending order (n = 1, 2, ...). The values of v(r ) for these eigenvalues are known as eigenfunctions. If h 0 = 0, then λn = π(n − 21 ). If B = 1 in (D.13), expressions for eigenfunctions vn (r ) are simplified to: vn (r ) = sin λn r

(n = 1, 2, ...).

(D.15)

The solution λ = 0 is not considered as it leads to a trivial solution vn (r ) = 0.

Appendix D: Derivation of Formula (3.103)

479

The value of B is implicitly accounted for by the coefficients n (t) in Series (D.10). The functions vn (r ) form a complete (see the previous discussion about this matter) set of eigenfunctions which are orthogonal in the range 0 ≤ r ≤ 1. The orthogonality of vn (r ) follows from the formula:

1

vn (r )vm (r )dr = δnm || vn ||2 ,

(D.16)

0

where δnm = || vn ||2 =

0 n = m , 1n=m

 



1 sin 2λn h0 1 = . 1− 1+ 2 2 2λn 2 h 0 + λ2n

(D.17)

The expression for V (t, r ) given by (D.10), with vn (r ) defined by (D.15), satisfies the boundary conditions for Eq. (D.9). The orthogonality and completeness of functions vn (r ) allows us to expand known functions in Eq. (D.9), and the initial conditions in the series f (r ) ≡ −r/(1 + h 0 ) =

∞

f n vn (r ),

(D.18)

n=1

W0 (r ) =

∞

qn vn (r ),

(D.19)

n=1

where fn =

1 || vn ||2 qn =



1

f (r )vn (r )dr = −

0

1 || vn ||2



1

sin λn , || vn ||2 λ2n

W0 (r )vn (r )dr.

0

Although both Series (D.18) and (D.19) converge, the speed of convergence of series (D.18) can be slow. This can create obstacles with applications of this approach to engineering problems. Note that W0 (r ) is a twice continually differentiable function since this property was earlier assumed for T0 (r Rd0 ), Hence, it can be shown that [3] |qn |
1 (see [3]). and uniformly, since λ−2 n −1 and assuming that: H0 (t) = h 0 + h 1 (t),

(D.32)

where h 0 = const > −1, the boundary condition at r = 1 for Eq. (D.3) is presented as:     = μ0 (t) − h 1 (t)W (t, 1) ≡ μˆ 0 (t). (D.33) Wr (t, r ) + h 0 W (t, r )  r =1

If μˆ 0 (t) is known, we can formally use the previously obtained solutions (D.8) and (D.10) to present the solution to Problem (D.3)–(D.7) as:   ∞ μˆ0 (t) κλ2n t W (t, r ) = r + V (t, r ) = sin(λn r )qn exp − 1 + h0 Rd0 Rd (t) n=1 −

∞

sin(λn r )

f n κλ2n

n=1

t

0

  1 1 κλ2n μˆ 0 (τ ) − dτ. exp α Rd0 Rd (t) Rd (τ ) Rd2 (τ )

(D.34)

In contrast to the case of H0 (t) =const, Eq. (D.34) does not give us an explicit solution for W (t, r ) since μˆ 0 (t) depends on W (t, 1). An alternative form of Eq. (D.34) can be presented as:

t

W (t, r ) = V (t, r ) −

μˆ 0 (τ )G(t, τ, r )dτ,

(D.35)

0

where V (t, r ) =

∞

 sin(λn r )qn exp −

n=1

κλ2n t Rd0 Rd (t)



  1 1 κ sin(λn ) κλ2n − . sin(λn r ) 2 exp G(t, τ, r ) = − α Rd0 Rd (t) Rd (τ ) Rd (τ ) || vn ||2 n=1 ∞

Remembering (D.33), Eq. (D.35) is rewritten as:

Appendix D: Derivation of Formula (3.103)



t

W (t, r ) = V (t, r ) −

483

[μ0 (τ ) − h 1 (τ )W (τ, 1)] G(t, τ, r )dτ.

(D.36)

0

This is an integral presentation of a solution to Problem (D.3)–(D.7) for time dependent H0 (t) presented as (D.32). For r = 1, (D.36) reduces to the Volterra integral equation of the second kind for function W (t, 1): W (t, 1) = V (t, 1) −

t

[μ0 (τ ) − h 1 (τ )W (τ, 1)] G(t, τ, 1)dτ.

(D.37)

0

It can be shown that (cf. Eq. (28) of [3]): sin2 λn =

λ2n . λ2n + h 20

(D.38)

Remembering Eqs. (D.17) and (D.38) the following expression for G(t, τ, 1) is obtained:

  ∞ 1 2κ 1 λ2n κλ2n G(t, τ, 1) = − 2 − . (D.39) exp α Rd0 Rd (t) Rd (τ ) Rd (τ ) n=1 h 20 + h 0 + λ2n A new function G 1 (t, τ, r ) defined as: G 1 (t, τ, r ) = Rd2 (τ )G(t, τ, r ),

(D.40)

where 0 ≤ τ ≤ t, is introduced. The convergence of the series in G 1 (t, τ, r ) for (t − τ ) ∈ [δ, −1/α), where δ is an arbitrary small positive number, is proven in Appendix E. Moreover, in the same appendix the validity of the estimate √ |G 1 (t, τ, r )| ≤ c/ ˜ t − τ,

t − τ ∈ (0, t0 ]

(D.41)

is demonstrated, where c˜ is a constant, and t0 is a fixed time instant within the timestep (0, t). The results presented in Appendix E are applicable to the first series in Eq. (D.34) if τ = 0 is assumed. Equation (D.37) has a unique solution, although this solution cannot be found in an explicit form. The details of a possible numerical solution of (D.37) are presented in Appendix F. Once the solution to this equation has been found it can be substituted into integral representation (D.36). This leads to the required solution to the initial and boundary value problem (D.3)–(D.7). The distribution of T is presented as:    Rd (t)R 2 1 exp − W (t, R/Rd (t)). T (R, t) = √ 4κ Rd (t) R Rd (t)

(D.42)

484

Appendix D: Derivation of Formula (3.103)

For h 1 (t) = 0 and αt 1 this solution reduces to that given by Expression (16) of [3]. For h 0 = 0 we have λn = π(n − 21 ) and ||vn ||2 = 21 in all equations.

References 1. Kartashov EM (2001) Analytical Methods in the Heat Transfer Theory in Solids. Moscow: Vysshaya Shkola (in Russian). 2. Savovi´c S, Caldwell J (2003) Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. Int. J Heat Mass Transfer 46:2911–2916. 3. Sazhin SS, Krutitskii PA, Abdelghaffar WA, Mikhalovsky SV, Meikle ST, Heikal MR (2004) Transient heating of diesel fuel droplets. Int. J Heat Mass Transfer 47:3327–3340. 4. Sazhin SS, Krutitskii PA, Gusev IG, Heikal MR (2010) Transient heating of an evaporating droplet. Int. J Heat Mass Transfer 53:2826–2836.

Appendix E

The Convergence of the Series in G 1 (t, τ, r)

The convergence of the Series in G 1 (t, τ, r ) is investigated in this appendix, following [1]. A function:

 1 1 t −τ 1 − = (E.1) f (t, τ ) ≡ − α Rd0 Rd (t) Rd (τ ) Rd (t)Rd (τ ) is introduced, where 0 ≤ τ ≤ t < t,

t is a timestep. As in the previous analysis, it is assumed that t0 = 0. Note that f (t, τ ) ≥

t −τ 2 Rd0

(E.2)

since Rd (t) ≤ Rd0 . From (D.17) and the estimate λn > n for n > 1 it follows that || vn ||2 > 14 for n > 1. Thus: (E.3) || vn ||2 ≥ c0 , n ≥ 1, ) ( where c0 = min || v1 ||2 , 41 is a positive constant. The following estimate follows from Condition (E.2):     2 2t −τ , exp −κλn f (t, τ ) ≤ exp −κn 2 Rd0

n > 1,

(E.4)

where we took into account that λn > n for n > 1. Using (E.4) it can be shown that the series in G 1 (t, τ, r ) converges absolutely and uniformly to a continuous function for (t − τ, r ) ∈ [δ, t) × [0, 1] for any small δ > 0 since: © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

485

Appendix E: The Convergence of the Series in G 1 (t, τ, r )

486

    t −τ δ ≤ exp −κn 2 2 , |sin λn r | ≤ 1. exp −κn 2 2 Rd0 Rd0

(E.5)

Indeed, each term with n > 1 in the series in G 1 (t, τ, r ) for (t − τ, r ) ∈ [δ, t) × [0, 1] can be majorised by the corresponding term of the convergent number series 

δ κc0−1 exp −κn 2 2 . Rd0 Conditions (E.3) and (E.4) lead to the following estimate for t − τ > 0:  |G 1 (t, τ, r )| ≤

c0−1 κ

∞

1+

#   2 exp −κn f (t, τ )

n=2

 ≤ c0−1 κ 1 +

∞

#   2 ˜ − τ ). ≡ G(t exp −κn 2 (t − τ )/Rd0

(E.6)

n=2

  2 exp −κn 2 (t − τ )/Rd0 can be considered as a sum  of areas of polygons 2 . This sum is less than of unit width placed under the curve exp −κ y 2 (t − τ )/Rd0 the area under this curve. Hence, *∞

n=2

∞

  2 < exp −κn 2 (t − τ )/Rd0

n=2

< 0

∞

1

∞

  2 dy exp −κ y 2 (t − τ )/Rd0

  Rd0 2 dy = √ exp −κ y 2 (t − τ )/Rd0 κ(t − τ )



∞

  exp −z 2 dz

0

√ Rd0 π = √ 2 κ(t − τ )

(E.7)

Substitution of (E.7) into (E.6) gives: √   √ ˜ − τ ) < c0 κ 1 + √Rd0 π |G 1 (t, τ, r )| ≤ G(t < c/ ˜ t − τ, 2 κ(t − τ )

(E.8)

t − τ ∈ (0, t00 ], for any fixed t00 ∈ (0, t). The new constant c˜ depends on t00 . Condition (E.8) is uniform for any 0 ≤ r ≤ 1.

Reference 1. Sazhin SS, Krutitskii PA, Gusev IG, Heikal MR (2010) Transient heating of an evaporating droplet. Int. J Heat Mass Transfer 53:2826–2836.

Appendix F

Numerical Solution of Equation (D.37)

An approach to the numerical solution of Eq. (D.37) is described in this Appendix, following [1]. Let ψ(t) ≡ W (t, 1) and rewrite Eq. (D.37) as:

t

ψ(t) = V (t, 1) −

[μ0 (τ ) − h 1 (τ )ψ(τ )] G(t, τ, 1)dτ.

(F.1)

0

We look for the solution of Eq. (F.1) for t ∈ [0, tˆ], where tˆ is a constant, tˆ < t. Let δt = tˆ/N and tn = nδt, where N is the total number of timesteps, n = 0, 1, .....N is the number of the current timestep, t N = tˆ. As in the previous analysis, t0 = 0. A discretised form of Eq. (F.1) is presented as: ψ(tn ) = V (tn , 1) −

n j=1

tj

[μ0 (τ ) − h 1 (τ )ψ(τ )] G(tn , τ, 1)dτ.

(F.2)

t j−1

Note that ψ(t0 ) = ψ(0) = V (0, 1) = W0 (1) is a known constant. The first (n − 1) integrals in this sum are approximated as:

tj

[μ0 (τ ) − h 1 (τ )ψ(τ )] G(tn , τ, 1)dτ

t j−1

(   ) ≈ μ0 (τ j ) − h 1 (τ j ) ψ(t j ) + ψ(t j−1 ) /2 G(tn , τ j , 1)δt,

(F.3)

where j = 1, 2, ...., n − 1, τ j = t j − 21 δt. Approximation (F.3) is valid since all functions in the integrand are continuous and we look for a solution in the class of continuous functions. In (F.3) the known functions are taken at τ = τ j (middle of the range [t j−1 , t j ]), while the unknown functions are taken as the averaged values at times t j−1 and t j .

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

487

488

Appendix F: Numerical Solution of Equation (D.37)

The last term in the sum in (F.2) needs to be investigated separately since the kernel G(tn , τ, 1) in the integrand becomes singular when τ → tn − 0 (see Estimate (E.8)). All other functions in this integrand, including the unknown function ψ(t), are assumed continuous. Hence, we can present this term as: tn [μ0 (τ ) − h 1 (τ )ψ(τ )] G(tn , τ, 1)dτ tn−1

 tn

ψ(tn ) + ψ(tn−1 ) G(tn , τ, 1)dτ. ≈ μ0 (τn ) − h 1 (τn ) 2 tn−1

(F.4)

Remembering Series (D.39) the last integral can be presented as:

tn

G(tn , τ, 1)dτ

tn−1 ∞

λ2m = −2κ h 2 + h 0 + λ2m m=1 0



tn

tn−1

  2

1 1 κλm 1 − dτ exp α Rd0 Rd (tn ) Rd (τ ) Rd2 (τ )

τ =tn  2

1 1 λ2m 1 κλm  = −2κ − exp  2 2 2 α Rd0 Rd (tn ) Rd (τ )  h + h 0 + λm κλm m=1 0 ∞

τ =tn−1

 2

 κλm 1 1 λ2m 1 1 − exp − = −2κ α Rd0 Rd (tn ) Rd (tn−1 ) h 2 + h 0 + λ2m κλ2m m=1 0 ∞

  −κλ2m δt 1 1 − exp = −2 Rd (tn )Rd (tn−1 ) h 2 + h 0 + λ2m m=1 0 ∞

=−

  ∞ 1 −κλ2m δt 1 ≡ gn . +2 exp 1 + h0 Rd (tn )Rd (tn−1 ) h 2 + h 0 + λ2m m=1 0

(F.5)

If h 0 = 0 then λm = π(m − (1/2)) in (F.5). Formulae (F.3)–(F.5) allow us to rewrite the discretised form of Eq. (F.1) (Eq. (F.2)) as: (   ) ψ(tn ) = V (tn , 1) − μ0 (τn ) − h 1 (τn ) ψ(tn ) + ψ(tn−1 ) /2 gn −

n−1 (

  ) μ0 (τ j ) − h 1 (τ j ) ψ(t j ) + ψ(t j−1 ) /2 G(tn , τ j , 1)δt,

j=1

where n = 1, 2, ...., N , and gn is defined by (F.5).

(F.6)

Appendix F: Numerical Solution of Equation (D.37)

489

Expression (F.6) can be rewritten to the form suitable for the numerical analysis:

  1 h 1 (τn )ψ(tn−1 ) V (tn , 1) − μ0 (τn ) − gn ψ(tn ) = 1 − 0.5h 1 (τn )gn 2 ⎫ n−1 ⎬ (   ) μ0 (τ j ) − h 1 (τ j ) ψ(t j ) + ψ(t j−1 ) /2 G(tn , τ j , 1)δt . − ⎭

(F.7)

j=1

For n = 1 the sum in Expression (F.7) is equal to zero and ψ(t0 ) is a known constant (see above). This allows us to calculate ψ(t1 ) explicitly using Expression (F.7). Once ψ(t1 ) has been calculated Expression (F.7) can be used for calculation of ψ(t2 ) etc. At the nth step, Expression (F.7) is applied for finding ψ(tn ) using the values of ψ(t0 ), ψ(t * 1 ), ... ψ(tn−1 ) inferred from the previous steps. At this step all terms in the sum n−1 j=1 are known. Once the numerical solution to the integral equation (D.37) has been obtained, function W (t, r ) can be calculated numerically using its integral representation (D.35), where: μˆ 0 (t) = μ0 (t) − h 1 (t)ψ(t). Using the same discretisation by t and τ as above, the discretised form of this presentation can be written as: W (tˆ, r ) = V (tˆ, r ) −

N j=1

= V (tˆ, r ) −

N −1 μˆ 0 (t j−1 ) + μˆ 0 (t j ) j=1

2 ×

tN

tj

μˆ 0 (τ )G(tˆ, τ, r )dτ

t j−1

G(tˆ, τ j , r )δt −

μˆ 0 (t N −1 ) + μˆ 0 (t N ) 2

G(t N , τ, r )dτ.

(F.8)

t N −1

Note that t N = tˆ. If N = 1 then the sum in Eq. (F.8) is equal to zero. The last integral in Eq. (F.8) is improper and needs to be calculated separately. Remembering the definition of G(t, τ, r ), and almost repeating the derivation of Equation (F.5), we obtain: tN ∞ h 20 + λ2m sin λm sin λm r G(t N , τ, r )dτ = −2 λ2m h 2 + h 0 + λ2m t N −1 m=1 0

 × 1 − exp

−κλ2m δt Rd (t N )Rd (t N −1 )



490

Appendix F: Numerical Solution of Equation (D.37)

  ∞ h 20 + λ2m sin λm sin λm r r −κλ2m δt . =− +2 exp 1 + h0 λ2m Rd (t N )Rd (t N −1 ) h 2 + h 0 + λ2m m=1 0 Having substituted the latter expression into (F.8), and remembering the definition of μˆ 0 (t j ), the required W (tˆ, r ) is obtained.

Reference 1. Sazhin SS, Krutitskii PA, Gusev IG, Heikal MR (2010) Transient heating of an evaporating droplet. Int. J Heat Mass Transfer 53:2826–2836.

Appendix G

Numerical Calculation of the Improper Integrals

The integrals in Eqs. (3.121) and (3.133) have the same type of integrable singularity as the integral in Eq. (3.115). The following analysis focuses on the latter equation, which enables us to simplify the notation, following [1]. The discretised form of Eq. (3.115) can be rewritten as: v(R, tˆ) =

N j=1

tj

ν(τ )G(tˆ, τ, R)dτ,

(G.1)

t j−1

where tˆ = t N , tn = nδt, n = 0, 1, 2, ....N , δt = tˆ/N . In all integrals we can replace ν(τ ) with the average values over the corresponding time interval (ν(t j−1 ) + ν(t j ))/2. Moreover, in all integrals, except the last one, we can replace G(tˆ, τ, R) with G(tˆ, τ j , R), where τ j = (t j−1 + t j )/2. This allows us to present Eq. (G.1) as: v(R, tˆ) =

N −1 ν(t j−1 ) + ν(t j ) j=1

2

ν(t N −1 ) + ν(t N ) G(tˆ, τ j , R)δt + 2



tN

G(tˆ, τ, R)dτ.

t N −1

(G.2) Let us assume that an a priori chosen R is not equal to Rd (tˆ). In this case G(tˆ, τ, R), as defined by Expression (3.116), approaches 0, when τ → tˆ − 0. Hence the singularity in the integrand is not present and the last timestep can be treated as all the previous timesteps. This leads to a simplification of Formula (G.2) to: v(R, tˆ) =

N ν(t j−1 ) + ν(t j ) j=1

2

G(tˆ, τ j , R)δt.

(G.3)

When R = Rd (tˆ) the first  exponent in Formula (3.116) tends to 1 when τ → tˆ − 0. This leads to a singularity 1/ tˆ − τ in the integrand in (G.2). As a result, the integral in this equation can be rewritten as:

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

491

492

Appendix G: Numerical Calculation of the Improper Integrals

√   tN  κ dτ (R − Rd (τ N ))2  G(tˆ, τ N , R)dτ = √ exp − 2 π 4κ(tˆ − τ N ) t N −1 t N −1 tˆ − τ



tN

#   δt (R + Rd (τ N ))2  − exp − 4κ(tˆ − τ N ) tˆ − τ N √     √ (R + Rd (τ N ))2 κ √ (R − Rd (τ N ))2 − exp − =√ 2 exp − δt. (G.4) 2κδt 2κδt 2π This formula allows us to simplify the expression for v(R, tˆ) at R = Rd (tˆ) to: v(R, tˆ) =

N −1 ν(t j−1 ) + ν(t j ) j=1

2

G(tˆ, τ j , R)δt

√   (R − Rd (τ N ))2 (ν(t N −1 ) + ν(t N )) κ √ 2 exp − + √ 2κδt 2 2π   √ (R + Rd (τ N ))2 − exp − δt. 2κδt

(G.5)

Reference 1. Sazhin SS, Krutitskii PA, Gusev IG, Heikal M (2011) Transient heating of an evaporating droplet with presumed time evolution of its radius. Int. J Heat Mass Transfer 54:1278–1288.

Appendix H

Derivation of Equation (3.150)

The details of the derivation of (3.150) are presented in this appendix following [3]. A steady-state energy transport process in a mixture of N components is considered. The equation describing this process, considering inter-diffusional terms, but not work and dissipation terms, is written as [1]: N   ∇ j ρtot U j Htot = −∇ j q˜ j − ∇ j Hi J j(i) ,

(H.1)

i=0

where Hi is the specific enthalpy of component i , J j(i) is the j th component of the enthalpy, thermal the diffusive mass flux of component i; Htot , ktot and ρtot are * N Hi Y (i) ); U j and conductivity and density of the mixture, respectively (Htot = i=0 th q˜ j = −ktot ∇ j T are the j components of the velocity vector of the mixture and the heat flux, respectively. Equation (H.1) can be presented as:  ∇ j ρtot U j

N

Hi Y (i) +

i=0

N

 Hi J j(i)

  = ∇ j ktot ∇ j T .

(H.2)

i=0

The left-hand side of Eq. (H.2) can be presented as  ∇ j ρtot U j

N i=0

Hi Y

(i)

+

N i=0

 Hi J j(i)

= ∇j

N

  Hi ρtot U j Y (i) + J j(i) .

(H.3)

k=0

Introducing the total (diffusive and convective) mass flux: F j(i) = ρtot U j Y (i) + J j(i) ,

(H.4)

the right-hand side of (H.3) can be rewritten as: © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

493

494

Appendix H: Derivation of Equation (3.150)

∇j

N

N   Hi ρtot U j Y (i) + J j(i) = ∇ j F j(i) Hi .

i=0

(H.5)

i=0

Thus, (H.1) can be presented as ∇j

N

  F j(i) Hi = ∇ j ktot ∇ j T .

(H.6)

i=0

*N *N ¯ This is Equation ‘H ’ in Table 19.2–4 of [1] ( i=0 F j(i) Hi is the same as i=0 N (i) j hi in [1], h¯ i is the partial molar enthalpy of component i). For a binary mixture of a vapour and an ambient gas (F(v) = F(1) and F(g) = F(0) ) Eq. (H.6) reduces to     (g) ∇ j F j(v) Hv + F j Hg = ∇ j ktot ∇ j T .

(H.7)

Remembering that the net (convective and diffusion) ambient gas flux is zero   (g) F = 0 , we obtain: F j(v) = ρtot U j Y (v) + J j(v) . (H.8) Remembering that

J j(v) = ρtot U j Y (g) ,

(H.9)

  ∇ j ρtot U j Hv = ∇ j ktot ∇ j T .

(H.10)

Hv = c pv T + Hv0

(H.11)

(H.7) can be rewritten as:

Remembering that and using the equation of conservation of mass (∇U = 0) we have   ρtot U j c pv ∇ j T = ∇ j ktot ∇ j T .

(H.12)

Equation (H.12) is identical to (3.150). It is the same as given in [2] if c in that paper is replaced with c pv .

References 1. Bird R, Stewart W, Lightfoot E (2002) Transport Phenomena, Second ed., John Wiley and Sons. 2. Tonini S, Cossali G (2013) An exact solution of the mass transport equations for spheroidal evaporating drops, Int. J Heat Mass Transfer 60:236–240.

Appendix H: Derivation of Equation (3.150)

495

3. Zubkov VS, Cossali GE, Tonini S, Rybdylova O, Crua C, Heikal M, Sazhin SS (2017) Mathematical modelling of heating and evaporation of a spheroidal droplet, Int. J Heat Mass Transfer 108:2181–2190.

Appendix I

Evolution of the Droplet Shape

In this Appendix, our assumption that the shape of an evaporating isothermal spheroidal droplet remains spheroidal is proven following [1].  Consider a point on a spheroidal surface (r1 , z 1 ), where r1 = mate the time derivative of the following function B=

x12 + y12 , and esti-

z 12 r12 + . ar2 az2

(I.1)

Assuming that initially B = 1, and differentiating (I.1) with respect to time we obtain z 1 z˙ 1 az2 − z 12 az a˙ z 1 dB r1r˙1 ar2 − r12 ar a˙ r = + . (I.2) 2 dt ar4 az4 For a spheroid we have the following expressions: r1 = R0 ε−1/3 sin θ1 ; z 1 = R0 ε2/3 cos θ1 , ar = R0 ε−1/3 ; az = R0 ε2/3 . The time derivative of (r1 , z 1 ) is (˙r1 , z˙ 1 ) = −

m˙ ev K (θ1 ) (n r , n z ) , 4π R02 ρ f

(I.3)

where (n r , n z ) are the r - and z-component of the unit vector normal to the surface:   2/3 ε sin θ, ε−1/3 cos θ ε2/3 , K (θ1 ) =  , (n r , n z ) =     1/2 ε−2/3 cos2 θ + ε4/3 sin2 θ 1 + ε2 − 1 sin2 (θ1 ) m˙ ev 4π R02

is the evaporation flux at the droplet surface.

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

497

498

Appendix I: Evolution of the Droplet Shape

Equation (I.3) can be rewritten as:  5/3  ε sin θ1 , ε2/3 cos θ1 m˙ ev (˙r1 , z˙ 1 ) = −    .  4π R02 ρ f 1 + ε2 − 1 sin2 θ1 3/2 The time derivatives of the half-axes are found from the following expressions:  v θ= a˙ r = − ρf a˙ z = −

π 2

 =−

π  m˙ ev m˙ ev =− K ε−1/3 , 2 2 4π R0 ρ f 4π R02 ρ f

v (θ = 0) m˙ ev m˙ ev =− K (0) = − ε2/3 . 2 ρf 4π R0 ρ f 4π R02 ρ f

Thus the values and time derivatives of ar , az , r1 and z 1 can be presented as: ar = R0 ε−1/3 ; az = R0 ε2/3 ; a˙ r = −Z 1 ε−1/3 ; a˙ z = −Z 1 ε2/3 r1 = R0 ε−1/3 sin θ1 ; z 1 = R0 ε2/3 cos θ1 ; r˙1 = −Z 2 ε5/3 sin θ1 ; z˙ 1 = −Z 2 ε2/3 cos θ1 , where Z1 =

1 m˙ ev m˙ ev   .  ; Z2 = 2 2 2 4π R0 ρ f 4π R0 ρ f 1 + ε − 1 sin2 θ1

(I.4)

Substituting these expressions into Eq. (I.2) gives: z 1 z˙ 1 az2 − z 12 az a˙ z 1 dB r1r˙1 ar2 − r12 ar a˙ r = + 2 dt ar4 az4 −Z 2 ε4/3 + ε−2/3 Z 1 −Z 2 + Z 1 + ε2 cos2 θ1 −1 R0 ε R0 ε 2 !   sin2 θ1 1 − ε2 − 1 m˙ ev 1    + 1 = 0.  = R0 4π R02 ρ f 1 + ε2 − 1 sin2 θ1

= ε−1/3 sin2 θ1

Since B = 1 at the beginning of the process, it stays equal to 1 at any time: B=

r12 z 12 + = 1. ar2 az2

Thus the droplet remains spheroidal. Note that this conclusion refers to isothermal spheroids. These are different to those on which the analysis of Chap. 3 is focussed in the general case.

Appendix I: Evolution of the Droplet Shape

499

Reference 1. Zubkov VS, Cossali GE, Tonini S, Rybdylova O, Crua C, Heikal M, Sazhin SS (2017) Mathematical modelling of heating and evaporation of a spheroidal droplet. Int. J Heat Mass Transfer 108:2181–2190.

Appendix J

Derivation of Expressions (3.162)

The details of the derivation of Expressions (3.162) are presented in this appendix following [1]. The normal components of the velocity of the recession of the surface of an evaporating droplet, described by (3.161)–(3.162), can be presented as: vn = vr n r + vz n z , where

vr = r (r 2 ar /ar3 + z 2 az /az3 ),

Note that: vn (u) = −

(J.1)

vz = z(r 2 ar /ar3 + z 2 az /az3 ).

(J.2)

1 dm˙ . ρf dA

(J.3)

(cf. Expressions (3.159)). It is assumed that the shape of an evaporating droplet remains spheroidal. At each timestep, the sizes (az and ar ) and their derivatives (az and ar ) along and perpendicular to the z-axis, are defined using (3.159). The evaporation fluxes at other points on the surface of the spheroidal droplet are calculated, using interpolation. The derivation of Expression (3.161) is based on following a point at the droplet surface (r1 , z 1 ) at time t1 that moves to a point (r2 , z 2 ) at t2 when the droplet deforms due to evaporation. t1 is assumed to be close to t2 . Remembering that points (r1 , z 1 ) and (r2 , z 2 ) are at the surface of a spheroid with semi-axes ar 1 , az1 and ar 2 , az2 , respectively, we can write:

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

501

502

Appendix J: Derivation of Expressions (3.162)

 

r1 ar 1 r2 ar 2

2 2



+ +

ar 2 ar 1



z1 az1

2 2

⎫ ⎪ = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

= 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ az2 ⎭ = az1 . z2 az2

(J.4)

If ar = ar 2 − ar 1 and az = az2 − az1 are small, and ignoring higher order terms in (J.4), we have

   ⎫  2 2 ⎪

az

ar r1 z1 ,⎪ r2 − r1 = r1 + ⎪ ⎪ ar 1 ar 1 az1 az1 ⎬ (J.5)

   ⎪  2 2 ⎪ ⎪

az

ar r1 z1 ⎪ .⎭ z2 − z1 = z1 + az1 ar 1 ar 1 az1 Dividing both parts of (J.5) by t2 − t1 and remembering that t2 − t1 → 0, the following expressions are obtained:  3 /az1 ), vr |(r1 ,z1 ) = r1 (r12 ar 1 /ar31 + z 12 az1

 3 vz |(r1 ,z1 ) = z 1 (r12 ar 1 /ar31 + z 12 az1 /az1 ). (J.6) These are identical to those introduced in (3.162).

Reference 1. Zubkov VS, Cossali GE, Tonini S, Rybdylova O, Crua C, Heikal M, Sazhin SS (2017) Mathematical modelling of heating and evaporation of a spheroidal droplet. Int. J Heat Mass Transfer 108:2181–2190.

Appendix K

Derivation of Formula (4.21)

In this appendix, the details of the solution of Eq. (4.1) for Yli (R, t), t ≥ 0 and 0 ≤ R < Rd are shown, following [1]. Rd is assumed to be constant during each timestep. We look for a solution which is continuously differentiable twice in the whole domain. The boundary condition (4.11) can be presented as:

  ∂Yli αm αm i (t) − Yli  =− , ∂R Dl Dl R=Rd −0

where αm =

Dv ρtotal ln (1 + B M ) . ρl Rd

(K.1)

(K.2)

The following initial condition Yli (t = 0) = Yli0 (R) is used We look for a solution to Eq. (4.1) in the form: Yli (t, R) = y(t, R) + (t),

(K.3)

where the subscripts i at y and  are omitted. Having substituted (K.3) into Eq. (4.1) and the boundary condition (K.1), Eq. (4.1) and the corresponding boundary and initial conditions can be presented as: ∂y = Dl ∂t



∂2 y 2 ∂y + ∂ R2 R ∂R

 −

d(t) , dt

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

(K.4)

503

504

Appendix K: Derivation of Formula (4.21)



 ∂y αm  − y  = 0, ∂R Dl R=Rd

y|t=0 = Yli0 (R) − (0) ≡ Yli0 (R) − 0 .

(K.5) (K.6)

Introduction of the new variable u(t, R) = y(t, R)R allows us to present (K.4)–(K.6) as: ∂ 2u d(t) ∂u = Dl 2 − R , ∂t ∂R dt

 ∂u h 0  u| R=0 = u  = 0, + ∂R Rd R=Rd −0 u|t=0 = R (Yli0 (R) − 0 ) ,

where:

αm Rd h0 = − 1 + Dl

(K.7)

(K.8) (K.9)

 .

The change of the variable from y to u leads to the requirement for the second boundary condition at R = 0. Our assumption that the solution is continuously differentiable twice implies that y is finite everywhere in the domain 0 ≤ R < Rd . Hence, the boundary condition (K.8) at R = 0. The solution to Problem (K.7)–(K.9) for h 0 > −1 was discussed in Appendix A. Here the focus is on the case h 0 < −1. This condition is satisfied for the problem of diffusion of components in droplets. We look for the solution to Eq. (K.7) in the form: u=

∞

n (t)vn (R),

(K.10)

n=0

where vn (R) is the full set of non-trivial solutions to the equation: ∂ 2v + pv = 0 ∂ R2

(K.11)

 h 0  ∂v + v = 0. ∂R Rd  R=Rd −0

(K.12)

with the boundary conditions:

v| R=0 =

Appendix K: Derivation of Formula (4.21)

505

Equation (K.11) with the boundary conditions (K.12) is the well known SturmLiouville problem. Our first task is to find eigenvalues p for this problem. Note that p has units of 1/m2 . In what follows, the cases p = 0, p < 0 and p > 0 are considered.

The Sturm-Liouville Problem for p = 0 For p = 0 the general solution to Eq. (K.11) is presented as v = A + B R. The condition v| R=0 = 0 implies that A = 0. The boundary condition at R = Rd leads to the following equation B(1 + h 0 ) = 0. Since h 0 ∈ (−∞, −1), the latter equation is satisfied only when B = 0. Thus we have the trivial solution v = 0 which is not considered in the analysis. Hence, Eq. (K.11) has no non-trivial solutions for p = 0.

The Sturm-Liouville Problem for p < 0 If p = −(λ/Rd )2 < 0 we write the general solution to Eq. (K.11) as: 



R R + B sinh λ , v(R) = A cosh λ Rd Rd

(K.13)

where A and B are arbitrary constants, λ is dimensionless. The boundary condition at R = 0 (see (K.12)) implies that A = 0. The boundary condition at R = Rd leads to the following equation: B (λ cosh λ + h 0 sinh λ) = 0. Rd

(K.14)

B in this equation is not equal to zero as we do not consider the trivial solution v = 0. Hence, Eq. (K.14) can be rewritten as: tanh λ = −

λ . h0

(K.15)

It can be shown that for h 0 ∈ (−∞, −1) Eq. (K.15) has three solutions λ = 0; ±λ0 , where λ0 ∈ (0, +∞), and it has no solutions for h 0 > −1. The solution λ = 0 leads to the trivial solution v = 0, which is not considered in the analysis. The solutions λ = ±λ0 lead to solutions (K.13) (eigenfunctions) which differ only by the sign of B. Since the value of the coefficient B is determined by the normalisation condition only (see below), the solution λ = −λ0 can be ignored. Hence, the solution to Eq. (K.15) gives only one eigenvalue λ = λ0 > 0 and the corresponding eigenfunction

506

Appendix K: Derivation of Formula (4.21)



R , v0 (R) = sinh λ0 Rd

(K.16)

where the normalisation B = 1 was used. From (K.16) and Condition (K.15) the following formula is obtained:

Rd

||v0 ||2 = 0

v02 (R)d R = −

  h0 Rd . 1+ 2 2 h 0 − λ20

(K.17)

The Sturm-Liouville Problem for p > 0 If p = (λ/Rd )2 > 0 the general solution to Eq. (K.11) can be presented as:

R v(R) = A cos λ Rd



 R + B sin λ , Rd

(K.18)

where A and B are arbitrary constants. The boundary condition at R = 0 (see (K.12)) implies that A = 0. The boundary condition at R = Rd leads to the following equation: B (λ cos λ + h 0 sin λ) = 0. Rd

(K.19)

B in this equation is not equal to zero as we do not consider the trivial solution v = 0. Hence, Eq. (K.19) can be presented as: tan λ = −

λ . h0

(K.20)

As in the case p < 0 the solutions to this equation corresponding to zero and negative λ are not considered. A countable set of positive solutions to this equation (positive eigenvalues) λn are arranged in ascending order: 0 < λ1 < λ2 < λ3 < .... The corresponding eigen functions are given by the expressions: 

R , vn (R) = sin λn Rd

(K.21)

where the normalisation B = 1 was used as for p < 0. From (K.21) and Condition (K.20) follows the expression for the norm of vn for n ≥ 1:

Appendix K: Derivation of Formula (4.21)



Rd

||vn || = 2

0

507

vn2 (R)d R

  h0 Rd . 1+ 2 = 2 h 0 + λ20

(K.22)

The norm (K.22) differs from the norm derived in Appendix A by the factor Rd . This does not affect the final solution.

Orthogonality of the Eigen Functions The orthogonality of functions vn (n ≥ 1) was proven in Appendix A. To prove that functions v0 and vn (n ≥ 1) are orthogonal, the following integral needs to be calculated: 



Rd R R sin λn d R, (K.23) I = sinh λ0 Rd Rd 0 where n ≥ 1. Using integration by parts twice when calculating the integral on the right-hand side of Eq. (K.23) we obtain:   Rd λ0 λ20 sinh λ0 cos λn − I =− cosh λ0 sin λn + I , λn λn λn R d

(K.24)

where I in the right-hand side of this equation is the same as in (K.23). Equation (K.24) can be rewritten as:

I =−

Rd λn



sinh λ0 cosh λ0

−

λ0 sin λn λn cos λn

1+



cosh λ0 cos λn .  2 λ0 λn

(K.25)

Remembering Eqs. (K.15) and (K.20), it can be seen that I defined by Eq. (K.25) is equal to zero. This implies that functions vn are orthogonal for n ≥ 0. Hence:

Rd

vn (R)vm (R)d R = δnm ||vn ||2 ,

(K.26)

0

where n ≥ 0 and m ≥ 0, ||vn ||2 is defined by (K.17) when n = 0 and (K.22) when n ≥ 1.

508

Appendix K: Derivation of Formula (4.21)

Expansion of R in a Fourier Series with Respect to Functions vn The expansion of R in a Fourier series with respect to functions vn is presented as: R=

∞

Q n vn (R),

(K.27)

n=0

where Qn =

1 ||vn ||2



Rd

Rvn (R)d R.

(K.28)

0

Calculation of the integrals in the right-hand side of (K.28) leads to the following expression for Q n :

Qn =

⎧ ⎨− ⎩

1 ||v0 ||2

1 ||vn ||2





Rd λn

2

Rd λ0 2

(1 + h 0 ) sinh λ0 when n = 0

(1 + h 0 ) sin λn

when n ≥ 1.

(K.29)

Calculation of Coefficients n (t) in Expansion (K.10) Having substituted (K.10) and (K.27) into Eq. (K.7), the latter is presented as: ∞

n (t)vn (R)

= Dl

n=0

∞

n (t)vn (R)



−  (t)

n=0

where n =

∞

Q n vn (R),

(K.30)

n=0

d n d2 vn ; vn (R) = ; dt d R2

  (t) =

d ≡ . dt

Since the expansion in series with respect to vn (Fourier series) is unique, Eq. (K.30) is satisfied if, and only if, it is satisfied for each term in this expansion. Recalling that v0 =



λ0 Rd

2 v0

and

vn = −



λn Rd

2 vn (n ≥ 1),

this leads to the following equations for 0 (t) and n (t) (n ≥ 1): 0 (t) = Dl



λ0 Rd

2

0 (t) −   Q 0 ,

(K.31)

Appendix K: Derivation of Formula (4.21)

n (t)

= −Dl

509

2

λn Rd

n (t) −   Q n (n ≥ 1),

(K.32)

where Q n (n ≥ 0) are defined by Expressions (K.28) or (K.29). Equations (K.31) and (K.32) are to be solved subject to the initial conditions for n (t) (n ≥ 0). These initial conditions are found via substitution of Expression (K.10) into the initial condition (K.9) and expansion of RYli0 (R) into a Fourier series with respect to vn . Remembering that the expansion with respect to vn is unique, the following expression for n (0) is obtained: n (0) = qin − (0)Q n , where qin =

1 ||vn ||2



Rd

RYli0 (R)vn (R)d R,

(K.33)

(K.34)

0

n ≥ 0. The solutions to Eqs. (K.31) and (K.32) with the initial condition (K.33) are presented as:    λ0 2 t [qi0 − (0)Q 0 ] 0 (t) = exp Dl Rd

t

− Q0 0

   λ0 2 d(τ ) exp Dl (t − τ ) dτ, dτ Rd 

n (t) = exp −Dl

t

− Qn 0



λn Rd

(K.35)

2  t [qin − (0)Q n ]

 

2 λn d(τ ) exp −Dl (t − τ ) dτ, dτ Rd

(K.36)

where n ≥ 1.

The Final Solution for Yl i Having substituted Expressions (K.16), (K.21), (K.35) and (K.36), into Expression (K.10), the final solution to Eq. (4.21), satisfying the boundary condition (4.11) and the corresponding initial condition is presented as:

510

Appendix K: Derivation of Formula (4.21)



1 Yli = i + R

t

−Q 0 0

 exp Dl

∞







exp −Dl

n=1 t

−Q n 0

λ0 Rd

2  t [qi0 − i (0)Q 0 ]

    

λ0 2 di (τ ) R exp Dl (t − τ ) dτ sinh λ0 dτ Rd Rd

+



λn Rd

2  t [qin − i (0)Q n ]

  

2 #

λn di (τ ) R exp −Dl (t − τ ) dτ sin λn , dτ Rd Rd

(K.37)

where Q n , qin , λ0 and λn (n ≥ 1) are defined by Eqs. (K.29), (K.34), (K.15) and (K.20), respectively; the subscript i at  is restored. Note that Formula (K.37) has the term which exponentially increases with time. This, however, does not lead to an unphysical solution to Eq. (4.21), since this equation is valid only for 0 ≤ Yli ≤ 1. Once Yli reaches one of its limiting values it remains equal to this value. If Expression (K.37) is applied to individual short timesteps, the time dependence i (τ ) during these timesteps can be ignored. It is assumed that of ddτ  di (t)  di (t) = ≡ i . dt dt t=0 In this case, Solution (K.37) is simplified to: 1 Yli = i + R







exp Dl

λ0 Rd

2  

 Rd2  t qi0 − Q 0 i (0) +  Dl λ20 i



 Rd2  R +Q 0  sinh λ0 Rd Dl λ20 i +

∞ n=1







exp −Dl

λn Rd

−Q n

2  

 Rd2  t qin − Q n i (0) −  Dl λ2n i



 Rd2  R sin λ .  n Dl λ2n i Rd

(K.38)

If the time dependence of i is ignored then Solution (K.38) is further simplified to

Appendix K: Derivation of Formula (4.21)

1 Yli = i + R +

∞ n=1







exp Dl

 exp −Dl



λn Rd

λ0 Rd

511

2  

R t [qi0 − i (0)Q 0 ] sinh λ0 Rd

2  #

R t [qin − (0)Q n ] sin λn . Rd

(K.39)

The time dependence of i during individual timesteps can be ignored in most applications. Thus, the applicability of Solution (K.39) is justified. Solution (K.39) is identical to Expression (4.21)

Reference 1. Sazhin SS, Elwardany A, Krutitskii PA, Deprédurand V, Castanet G, Lemoine F, Sazhina EM, Heikal MR (2011) Multi-component droplet heating and evaporation: numerical simulation versus experimental data. Int. J Thermal Science 50:1164–1180.

Appendix L

Derivation of Formula (4.29)

In this appendix, the details of the derivation of Expression (4.29) are presented, following [1]. As in Appendix D, the following parameters and functions are used: r = R/Rd (t), (0 ≤ r ≤ 1),

F(t, r ) = RYli (t, r Rd0 ),

   Rd (t)Rd (t) 2 1 exp − r W (t, r ). F(t, r ) = √ 4Dl Rd (t)

(L.1)

In this case, Eq. (4.1), and the corresponding initial condition and boundary condition (4.11) are presented as: 



Rd2 (t)Wt (t, r ) = Dl Wrr (t, r ),

(L.2)

where t ≥ 0,  W (t, r )|t=0 = W0 (r ) ≡

3/2 Rd0 r Yli0 (r Rd0 ) exp

  Rd (0)Rd0 2 r , 4Dl

W (t, r )|r =0 = 0, 

(L.3) (L.4)

   Wr (t, r ) + H0 (t)W (t, r ) 

r =1

   Rd (t)Rd (t) αm i (Rd (t))5/2 = μ0 (t) ≡ − exp , Dl 4Dl where:

(L.5)



H0 (t) = −

R (t)Rd (t) αm Rd (t) − 1 − d . Dl 2Dl

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

513

514

Appendix L: Derivation of Formula (4.29)

Condition (L.4) is an additional boundary condition, which follows from the requirement that Yli (t, R) is a twice continuously differentiable function. When deriving (L.2) the condition d2 Rd /dt 2 = 0 was used. Further simplification of Eq. (L.2) and the corresponding initial and boundary conditions is possible when this equation is applied to a short timestep. In this case the time dependence of H0 (t) can be ignored: H0 (t) ≡ h 0 =const. For Presentation (3.100) it is obtained: h0 = −

αm α Rd0 αm α Rd0 Rd (t) − Rd (t) − 1 ≈ − Rd0 − Rd0 − 1 < −1. Dl 2Dl Dl 2Dl

(L.6)

It is important to keep both αm and α in (L.6) to enable us to compare the results of the present analysis with the results of the conventional analysis when α = 0, but αm = 0. The introduction of the new function V (t, r ) W (t, r ) = V (t, r ) +

μ0 (t) r. 1 + h0

(L.7)

allows us to replace the inhomogeneous boundary condition (L.5) with the homogeneous one. Using Formula (L.7), Eq. (L.2) is rearranged to: 





Rd2 (t)Vt (t, r ) = Dl Vrr (t, r ) −

μ0 (t) 2 R (t)r. 1 + h0 d

(L.8)

The initial and boundary conditions for Eq. (L.8) are presented as: μ0 (0) r, 1 + h0     Vr (t, r ) + h 0 V (t, r ) 

V (t, r )|t=0 = W0 (r ) − V (t, r )|r =0 = 0,

r =1

= 0.

As in Appendix D, we look for the solution to Eq. (L.8) in the form: V (t, r ) =

∞

n (t)vn (r ),

(L.9)

n=0

where functions vn (r ) form a full set of non-trivial solutions to the equation: d2 v + pv = 0, 0 ≤ r ≤ 1, dr 2 subject to boundary conditions:

(L.10)

Appendix L: Derivation of Formula (4.29)

v|r =0 =

515

  dv + h 0 v  = 0. dr r =1

(L.11)

For p = 0, Eq. (L.10) has no non-trivial solutions, satisfying (L.11). For p ≡ −λ2 < 0, this equation has the solution: v0 (r ) = sinh (λ0 r ) ,

(L.12)

where λ0 is the solution to the equation tanh λ = −

λ . h0

(L.13)

The latter equation has three solutions (positive, negative and zero) remembering that h 0 < −1. We are interested only in the positive solution to this equation (cf. Appendix K). Note that the heat conduction equation does not have this solution since for this equation h 0 > −1. For p ≡ λ2 > 0, Eq. (L.10) has the solutions: vn (r ) = sin (λn r )

(L.14)

for n ≥ 1, where λn are the solutions to the equation tan λ = −

λ . h0

(L.15)

As in the case of p < 0 we ignore the solutions to this equation corresponding to zero and negative λ. A countable set of positive solutions to (L.15) (positive eigenvalues) λn are arranged in ascending order: 0 < λ1 < λ2 < λ3 < .... It can be shown that functions vn (r ), n ≥ 0 are orthogonal for 0 ≤ r ≤ 1 (cf. Appendix K). The completeness of the set of functions vn (r ) for n ≥ 0 has been tested. Namely, we considered various functions not belonging to this set, and found that Fourier expansions of these functions on the set of {vn (r )}∞ n=0 coincide with the functions themselves. If the set of functions is not complete, then a Fourier expansion of an arbitrary function, constructed based on this set, does not coincide with this function. The norms of functions vn (r ) for n ≥ 0 are found as ||vn ||2 = 0

1

vn2 (r )dr =

  h0 (−1)δn0 . 1+ 2 2 h 0 + (−1)δn0 λ2n

where δn0 is Kronecker’s delta symbol.

(L.16)

516

Appendix L: Derivation of Formula (4.29)

Note that the norm in (L.16) differs from those in (K.17) and (K.22) by the factor Rd , but coincides with the one chosen in Appendix A. This does not affect the final results. Note that Formula (4.29) is presented for the norms used in Expressions (K.17) and (K.22). Remembering that functions vn (r ) for n ≥ 0 are orthogonal and the set of these functions is complete, it is obtained: f (r ) ≡ −r/(1 + h 0 ) =

∞

f n vn (r ),

(L.17)

n=0

W0 (r ) =

∞

qn vn (r ),

(L.18)

n=0

where: 1 fn = || vn ||2



1

 f (r )vn (r )dr =

0

1 qn = || vn ||2



1

1 sinh λ0 ||v0 ||2 λ20 − ||vn ||1 2 λ2 sin λn n

when n = 0 , when n ≥ 1

W0 (r )vn (r )dr.

0

The expressions for f n and qn differ from expressions for f Y n and qY n in Formula (4.29) by the factor Rd , as the latter formula was derived using Formulae (K.17) and (K.22). Remembering Eqs. (L.9) and (L.17), Eq. (L.8) can be presented as: ∞

n=0

d n (t) Rd2 (t) dt

+ (−1)

δn0

 n (t)Dl λ2n

vn (r ) =

∞



dμ0 (t) f n Rd2 (t)

n=0

dt

 vn (r ).

(L.19) Both sides of Eq. (L.19) are Fourier series with respect to vn (r ). Two Fourier series are equal if, and only if, their coefficients are equal. Thus: Rd2 (t)

d n (t) dμ0 (t) + (−1)δn0 n (t)Dl λ2n = f n Rd2 (t) . dt dt

(L.20)

Equation (L.20) needs to be solved with the initial condition: n (0) = qn + μ0 (0) f n .

(L.21)

The general solution to the homogeneous equation: Rd2 (t)

d n (t) + (−1)δn0 n (t)Dl λ2n = 0 dt

(L.22)

Appendix L: Derivation of Formula (4.29)

is written as: ln ( n (t)/ n (0)) = −(−1)δn0 Dl λ2n

517

0

t

dt . Rd2 (t)

(L.23)

If Rd (t) is a linear function of t given by Expression (3.100), Solution (L.23) can be written as:

  1 (−1)δn0 Dl λ2n n (t) = n (0) exp −1 . (L.24) 2 1 + αt α Rd0 It can be shown that the function:

  t 1 1 dμ0 (τ ) Dl λ2n exp (−1)δn0 − dτ (L.25) n (part) (t) = f n 2 dτ 1 + αt 1 + ατ α Rd0 0 satisfies Eq. (L.20). Hence, this function can be considered as a particular solution to Eq. (L.20). Integration by parts in Expression (L.25) allows us to present n (part) (t) as:

  Dl λ2n t n (part) (t) = f n μ0 (t) − μ0 (0) exp −(−1)δn0 Rd0 Rd (t)  − exp (−1)δn0

Dl λ2n α Rd0 Rd (t)





μ0 (τ )Dl λ2n Dl λ2n (−1)δn0 exp −(−1)δn0 2 α Rd0 Rd (τ ) Rd (τ ) 0 t



# dτ .

(L.26)

Remembering Eqs. (L.24) and (L.26), the solution to Eq. (L.20) can be written as:

  1 Dl λ2n − 1 n (t) = n (0) exp (−1)δn0 2 1 + αt α Rd0

t

+ fn 0

  2

1 1 dμ0 (τ ) δn0 Dl λn exp (−1) − dτ. 2 dτ 1 + αt 1 + ατ α Rd0

(L.27)

Remembering Eqs. (L.26) and (L.21), Formula (L.27) is rewritten as:   2 δn0 Dl λn t n (t) = qn exp −(−1) + f n μ0 (t) Rd0 Rd (t) − f n (−1)δn0 Dl λ2n

0

t

  2

1 1 μ0 (τ ) δn0 Dl λn − dτ. (L.28) exp (−1) α Rd0 Rd (t) Rd (τ ) Rd2 (τ )

518

Appendix L: Derivation of Formula (4.29)

Changes of μ0 (τ ) in the integrand before the exponential term can be ignored since Solution (L.28) is applied to a short timestep. Thus, Solution (L.28) is simplified to (see Appendix M):   2 δn0 Dl λn t + f n μ0 (t) − f n μ0 (0). (L.29) n (t) = [qn + f n μ0 (0)] exp −(−1) Rd0 Rd (t) Note that n (t) in the form (L.27) satisfies Eq. (L.20), while n (t) in the form (L.29) does not satisfy it (see further discussion in Appendix D). Recalling Expressions (L.17) and (L.29), Eq. (L.9) is presented as: ∞

μ0 (0) R μ0 (t) R + , 1 + h 0 Rd (t) 1 + h 0 Rd (t)

(L.30)

  Dl λ2n t ˇ n (t) = [qn + f n μ0 (0)] exp −(−1)δn0 . Rd0 Rd (t)

(L.31)

V (t, r ) =

ˇ n (t)vn (r ) −

n=0

where

The final expression for mass fraction of the ith components inside the droplet is presented as:    

∞ α Rd0 R 2 R 1 ˇ Yli (R) = √ n (t) sin λn + exp − 4Dl Rd (t) Rd (t) R Rd (t) n=1

ˇ 0 (t) sinh λ0

R Rd (t)



 μ0 (0) R + , 1 + h 0 Rd (t)

(L.32)

ˇ n are defined by Formula (L.31). where Having substituted Formula (L.31) into (L.32) the latter is rearranged for a short timestep to:

Yli (R) =

αm i exp



α Rd0 4Dl

αm +



Rd0 Rd (t)−R 2 Rd (t)

α Rd0 2

 5/2

Rd0 5/2

Rd (t)

+

  1 α Rd0 R 2 × exp − 4Dl Rd (t) R Rd (t) √

∞



  R Dl λ2n t sin λn + [qn + f n μ0 (0)] exp − Rd0 Rd (t) Rd (t) n=1 

 R Dl λ20 t sinh λ0 . [q0 + f 0 μ0 (0)] exp Rd0 Rd (t) Rd (t) 

Solution (L.33) is the same as given by (4.29).

(L.33)

Appendix L: Derivation of Formula (4.29)

519

Note that Expression (L.33) was obtained using the assumption that H0 (t) ≡ h 0 = const. Both α and αm should be kept in Expression (L.33) if we intend to compare the prediction of this expression with the prediction of the conventional approach to modelling component diffusion when α = 0 but αm = 0 during the timestep. In the latter case Expression (L.33) can be rewritten as Yli (R) = i +

1

∞

 R Rd (t) n=1



 

R Dl λ2n t + sin λn [qn + f n μ0 (0)] exp − Rd0 Rd (t) Rd (t)



 R Dl λ20 t sinh λ0 . [q0 + f 0 μ0 (0)] exp Rd0 Rd (t) Rd (t) 

(L.34)

This expression is identical to Expression (4.21) (see Appendix K). Note that the norm of vn (||vn ||2 ) in Appendix K is dimensional. Relaxing the assumption that H0 (t) ≡ h 0 =const it is assumed that: H0 (t) = h 0 + h 1 (t),

(L.35)

where h 0 =const< −1. Presentation (L.35) allows us to write the boundary condition at r = 1 for Eq. (L.2) as:     = μ0 (t) − h 1 (t)W (t, 1) ≡ μˆ 0 (t). (L.36) Wr (t, r ) + h 0 W (t, r )  r =1

If μˆ 0 (t) is known, the previously obtained Solutions (L.7) and (L.9) are used to present the solution to Problem (L.2)–(L.5) as:   ∞ 2 μˆ0 (t) δn0 Dl λn t W (t, r ) = r + V (t, r ) = vn (r )qn exp −(−1) 1 + h0 Rd0 Rd (t) n=0 −

∞

vn (r )(−1)δn0 f n Dl λ2n

n=0



t

× 0

  2

1 1 μˆ 0 (τ ) δn0 Dl λn − dτ, exp (−1) α Rd0 Rd (t) Rd (τ ) Rd2 (τ )

(L.37)

where n (t) is defined by Expression (L.28). In contrast to the case of H0 (t) =const, Formula (L.37) is not an explicit solution for W (t, r ) since μˆ 0 (t) depends on W (t, 1). Formula (L.37) can be presented in an alternative form:

t

W (t, r ) = V (t, r ) − 0

μˆ 0 (τ )G(t, τ, r )dτ,

(L.38)

520

Appendix L: Derivation of Formula (4.29)

where V (t, r ) =

∞

 vn (r )qn exp −(−1)

δn0

n=0

G(t, τ, r ) = −

∞

 Dl λ2n t , Rd0 Rd (t)

vn (r )

n=0

  2

1 1 Dl vn (1) δn0 Dl λn − . exp (−1) × 2 α Rd0 Rd (t) Rd (τ ) Rd (τ ) || vn ||2 Recalling Eq. (L.36), Expression (L.38) can be rewritten as:

t

W (t, r ) = V (t, r ) −

[μ0 (τ ) − h 1 (τ )W (τ, 1)] G(t, τ, r )dτ.

(L.39)

0

This is an integral representation for a solution to Problem (L.2)–(L.5) for time dependent H0 (t) presented as (L.35). For r = 1, (L.39) reduces to the Volterra integral equation of the second kind for function W (t, 1): W (t, 1) = V (t, 1) −

t

[μ0 (τ ) − h 1 (τ )W (τ, 1)] G(t, τ, 1)dτ.

(L.40)

0

It can be shown that: vn2 (r = 1) 2(−1)δn0 λ2n = . 2 || vn || (−1)δn0 λ2n + h 20 + h 0

(L.41)

Equation (L.41) leads to the following formula: ∞

G(t, τ, 1) = −

2Dl (−1)δn0 λ2n Rd2 (τ ) n=0 h 20 + h 0 + (−1)δn0 λ2n

  1 1 Dl λ2n − . × exp (−1)δn0 α Rd0 Rd (t) Rd (τ )

(L.42)

Equation (L.40) has a unique solution, although this solution cannot be obtained in an explicit form. The scheme of its numerical solution is similar to the one presented in Appendix F. Once the solution to this equation has been found it can be substituted into integral representation (L.39). As a result, the required solution to the initial and boundary value Problem (L.2)–(L.5) is found. The final distribution of Yli is obtained as:    Rd (t)R 2 1 exp − Yli (R, t) = √ (L.43) W (t, R/Rd (t)). 4Dl Rd (t) R Rd (t)

Appendix L: Derivation of Formula (4.29)

521

Reference 1. Gusev IG, Krutitskii PA, Sazhin SS, Elwardany A (2012) A study of the species diffusion equation in the presence of the moving boundary. Int. J Heat Mass Transfer 55:2014–2021.

Appendix M

Derivation of Formula (L.29)

Remembering that Solution (L.28) is applied to a short timestep, changes of μ0 (τ ) in the integrand before the exponential term are not considered. This allows us to present Solution (L.28) as [1]:   (−1)δn0 Dl tλ2n + f n μ0 (t)− n (t) = qn exp − Rd0 Rd (t)    2

1 1 1 δn0 Dl λn − dτ , exp (−1) 2 α Rd0 Rd (t) Rd (τ ) 0 Rd (τ ) (M.1) where μ0 (τ ) in the integrand of Solution (L.28) was taken at the beginning of the timestep. Remembering that R d(Rd (τ )−1 ) = − 2 d dτ Rd (t)  f n μ0 (0) (−1)δn0 Dl λ2n

t

the last term in Expression (M.1) is rearranged to [1]: (−1)

δn0

Dl λ2n

0

t

  2

1 1 1 δn0 Dl λn − dτ = exp (−1) Rd (τ )2 α Rd0 Rd (t) Rd (τ )

  2

(−1)δn0 Dl λ2n 1 δn0 Dl λn exp (−1) Rd α Rd0 Rd (t)

t

× 0

  2

Rd 1 δn0 Dl λn dτ = − exp (−1) Rd (τ )2 α Rd0 Rd (τ )

  2

(−1)δn0 Dl λ2n 1 δn0 Dl λn exp (−1) Rd α Rd0 Rd (t) © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

523

524

Appendix M: Derivation of Formula (L.29)

α Rd0 × (−1)δn0 Dl λ2n

  2

1 δn0 Dl λn exp (−1) − − α Rd0 Rd (t)



 Dl λ2n 1 exp (−1)δn0 − α Rd0 Rd (0)



 1 1 Dl λ2n − = 1 − exp (−1)δn0 α Rd0 Rd (t) Rd (0)   2 δn0 Dl λn t . = 1 − exp −(−1) Rd0 Rd (t)

(M.2)

Presentation (3.100) was considered in the derivation of (M.2). Having substituted Eq. (M.2) into Formula (M.1), Expression (L.29) is obtained.

Reference 1. Gusev IG, Krutitskii PA, Sazhin SS, Elwardany A (2012) A study of the species diffusion equation in the presence of the moving boundary. Int. J Heat Mass Transfer 55:2014–2021.

Appendix N

Approximations for Alkane Fuel Properties

In this appendix, the approximations for the temperature dependencies of density, viscosity, heat capacity and thermal conductivity for liquid alkanes (Cn H2n+2 ) with 5 ≤ n ≤ 25, as inferred from published approximations and data, are described, following [1, 4].

Critical and Boiling Temperatures Using data given in [3] the following approximations for the critical Tc and boiling Tb temperatures of alkanes were suggested [4]: Tc (n) = ac + bc n + cc n 2 + dc n 3 ,

(N.1)

Tb (n) = ab + bb n + cb n 2 + dc n 3 ,

(N.2)

where the coefficients are shown in Table N.1. The values of Tc and Tb , given in [3], and their Approximations (N.1) and (N.2) for 5 ≤ n ≤ 25 are presented in Fig. N.1 [4]. As follows from this figure, both Tc and Tb monotonically increase with n.

Liquid density Following [6], the temperature dependence of the density of liquid alkanes with 5 ≤ n ≤ 25 is presented as: ρl (T ) =

Cρ  − 1− TTc 1000 Aρ Bρ ,

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

(N.3) 525

526

Appendix N: Approximations for Alkane Fuel Properties

Table N.1 Values of coefficients in Expressions (N.1) and N.2) Coefficient ac bc cc Value 242.3059898052 55.9186659144 −2.1883720897 Coefficient ab bb cb Value 118.3723701848 44.9138126355 −1.4047483216

dc 0.0353374481 db 0.0201382787

Fig. N.1 Plots of Tc and Tb , and their Approximations (N.1) and (N.2), versus n. Reprinted from [4]. Copyright Elsevier (2011)

where Aρ , Bρ and Cρ were approximated as [1, 6]: ⎧ ⎨ Aρ = 0.00006196104n + 0.234362 Bρ = 0.00004715697n 2 − 0.00237693n + 0.2768741 ⎩ Cρ = 0.000597039n + 0.2816916

(N.4)

The range of applicability of Approximation (N.3) depends on n. For n = 5 this range was found to be 143.42–469.65 K; for n = 10 it was found to be 243.49–618.45 K; for n = 25 it was found to be 315.15–850.13 K [6] (the upper limits are critical temperatures of the components). Remembering that the mass fraction of alkanes with n close to 25 is small, it is assumed that Formulae (N.3) and (N.4) can be used in the whole range from room temperature until close to the critical temperature. Plots of ρl versus n for T = 300 K and T = 450 K, predicted by Approximation (N.3) with coefficients Aρ , Bρ and Cρ given by [6] (filled squares and filled triangles), and approximated by Formulae (N.4) (solid and dashed curves) are presented in Fig. N.2. As can be seen in this figure, the agreement between the values of ρl predicted by Formula (N.3), with the values of the coefficients given in [6] and approximated by Formulae (N.4), is very good.

Appendix N: Approximations for Alkane Fuel Properties

527

Fig. N.2 Plots of ρl versus n for T = 300 K and T = 450 K, predicted by Approximation (N.3) with coefficients Aρ , Bρ and Cρ given by [6] (filled squares for T = 300 K and filled triangles for T = 450 K), and approximated by Formulae (N.4) (solid curve for T = 300 K and dashed curve for T = 450 K). Reprinted from [1]. Copyright Elsevier (2012)

Liquid Viscosity Following [2], the temperature dependence of the dynamic viscosity of liquid alkanes for 4 ≤ n ≤ 44 is approximated as:

where

  b(n) μl (n, T ) = 10−3 10[100(0.01 T ) ] − 0.8 ,

(N.5)

b(n) = −5.745 + 0.616 ln(n) − 40.468 n −1.5 .

(N.6)

The temperature range of the applicability of Approximations (N.5) and (N.6) was not explicitly specified in [2], but the author of that paper demonstrated good agreement between the predictions of these approximations and experimental data in the range of temperatures from 10 to 100 ◦ C. Plots of μl versus n for T = 300 K and T = 450 K predicted by Approximation (N.5) (solid and dashed curves), and the corresponding values of μl in the range 5 ≤ n ≤ 12, taken from [8] (filled squares (T = 300 K) and filled triangles (T = 450 K)), are presented in Fig. N.3. As can be seen in this figure, the agreement between the values of μl predicted by Formula (N.5) and the results shown on the NIST website [8] is very good. Note that the values of μl affect droplet heating and evaporation only via the corrections to the values of thermal conductivity and diffusivity in the Effective Thermal Conductivity and Effective Diffusivity (ETC/ED) models. In most engineering applications, the effect of viscosity on droplet heating and evaporation is expected to be weak.

528

Appendix N: Approximations for Alkane Fuel Properties

Fig. N.3 Plots of μl versus n predicted by Formula (N.5) (solid (T = 300 K) and dashed (T = 450 K) curves), and the corresponding values of μl in the range 5 ≤ n ≤ 12, given in [8] (filled squares (T = 300 K) and filled triangles (T = 450 K)). Reprinted from [1]. Copyright Elsevier (2012)

Specific Heat Capacity Following [5], the temperature dependence of the heat capacity of liquid alkanes for 2 ≤ n ≤ 26 is approximated as:  cl (n, T ) = 1000

 43.9 + 13.99(n − 1) + 0.0543(n − 1)T , M(n)

(N.7)

where M(n) = 14n + 2 is the molar mass of alkanes. The authors of [5] did not specify range of temperatures in which Formula (N.7) can be used, although they mentioned that this formula is not valid at temperatures close to the temperature of fusion. For n = 16 and n = 17 these ranges were identified as 340–400 K and 335–400, respectively. For n = 16 and n = 25 the temperatures of fusion are equal to 295.1 K and 329.25 K, respectively. Since the contribution of

Fig. N.4 Plots of cl versus n predicted by Approximation (N.7) (solid (T = 300 K) and dashed (T = 450 K) curves), and the corresponding experimental values of cl for T = 300 K in the range 5 ≤ n ≤ 18, given in [8] (filled squares) and [5] (filled circles •). Reprinted from [1]. Copyright Elsevier (2012)

Appendix N: Approximations for Alkane Fuel Properties

529

the alkanes with n > 16 is expected to be small, it is assumed that Formula (N.7) can be used in the whole range of temperatures from room temperature onwards. Plots of cl versus n for T = 300 K and T = 450 K predicted by Formula (N.7) (solid and dashed curves), and the corresponding experimental values of cl for T = 300 K in the range 5 ≤ n ≤ 18, given in [8] (filled squares) and [5] (filled circles •), are presented in Fig. N.4. As can be seen in this figure, the agreement between the values of the liquid heat capacity predicted by Formula (N.7) and the experimental data for T = 300 K is reasonably good.

Thermal Conductivity Following [7], the temperature dependence of thermal conductivity of liquid alkanes for 5 ≤ n ≤ 20 is approximated as: 

kl (n, T ) = 10

 2/7  Ak +Bk 1− TTc

,

(N.8)

where Tc are critical temperatures as in Formula (N.3). The values of Ak and Bk for specific n are presented in [7] and approximated as:

Ak = 0.002911n 2 − 0.071339n − 1.319595 Bk = −0.002498n 2 + 0.058720n + 0.710698.

(N.9)

Although Formulae (N.8) and (N.9) were obtained for 5 ≤ n ≤ 20, they were used in the whole range 5 ≤ n ≤ 25. Possible errors imposed by these approximations for 21 ≤ n ≤ 25 on predicted fuel droplet heating and evaporation rates are expected to be small as the mass fractions of alkanes in this range of n are very small in Diesel fuel, and negligible in gasoline fuel. The range of applicability of Formula (N.8) depends on the values of n. For n = 5 this range was found to be 143–446 K; for n = 10 it was found to be 243–588 K; for n = 20 it was found to be 310–729 K [7]. Remembering that the contribution of alkanes with n ≥ 20 is small, it was assumed that Formulae (N.8) and (N.9) can be used in the whole range from room temperature until close to the critical temperature, as in the case of Approximations (N.3) and (N.4). Plots of kl versus n predicted by Formula (N.8) with coefficients Ak and Bk taken from [7] (filled squares (T = 300 K) and filled triangles (T = 450 K)) and approximated by (N.9) (solid and dashed curves) are presented in Fig. N.5. In the same figure the values of kl given in [8] (squares (T = 300 K) and triangles  (T = 450 K)) are presented. As can be seen in this figure, the agreement between the values of thermal conductivity predicted by Formula (N.8) with the values of the coefficients taken from [7] and approximated by Formulae (N.9), is reasonably good. Both these values agree well with the data presented in [8].

530

Appendix N: Approximations for Alkane Fuel Properties

Fig. N.5 Plots of kl versus n predicted by Formula (N.8) with coefficients Ak and Bk taken from [7] (filled squares (T = 300 K) and filled triangles (T = 450 K)) and approximated by Formulae (N.9) (solid (T = 300 K) and dashed (T = 450 K) curves); the values of kl given in [8] (squares (T = 300 K) and triangles  (T = 450 K)). Reprinted from [1]. Copyright Elsevier (2012)

Note that a small number of lighter components inside droplets reach temperatures exceeding their critical temperatures during calculations [1]. In this case, the values of saturation pressure, latent heat of evaporation, density, viscosity, heat capacity and thermal conductivity were assumed equal to those at T = Tc [1]. This assumption allowed the authors of [1] to avoid the analysis of heat and mass transfer in in nearand supercritical conditions, without imposing significant errors due to the fact that the number of components affected by this assumption is very small.

References 1. Elwardany AE, Sazhin SS (2012) A quasi-discrete model for droplet heating and evaporation: application to Diesel and gasoline fuels. Fuel 97:685–694. 2. Mehrotra AK (1994) Correlation and prediction of the viscosity of pure hydrocarbons. The Canadian Journal of Chemical Engineering 72:554–557. 3. Poling BE, Prausnitz JM, O’Connell J (2000) The Properties of Gases and Liquids. McGrawHill, New York. 4. Sazhin SS, Elwardany A, Sazhina EM, Heikal MR (2011) A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets. Int. J Heat Mass Transfer 54:4325–4332. 5. van Miltenburg JC (2000) Fitting the heat capacity of liquid n-alkanes: new measurements of n-heptadecane and n-octadecane. Thermochimica Acta 343:57–62. 6. Yaws CL (Editor) (2008) Thermophysical Properties of Chemicals and Hydrocarbons. William Andrew Inc, Beaumont, TX, USA. 7. Yaws CL (1995) Handbook of Thermal Conductivity. Vol. 2 (Organic Compounds, C5 to C7 ) and Vol. 3 (Organic Compounds, C8 to C28 ). Gulf Publishing Company, Houston, London, Paris, Zurich, Tokyo. 8. http://webbook.nist.gov/chemistry/

Appendix O

Thermophysical Properties of Hydrocarbons in Diesel Fuels

In this appendix, approximations of thermophysical properties of hydrocarbons in Deisel fuels are presented following [19]. SI units are used for all parameters. All approximations presented in this appendix can be used for the limited ranges of temperatures and carbon numbers as specified later. Two methods of extrapolating the values of the parameters beyond these ranges have been commonly used. Firstly, it has been assumed that these values at T < Tmin and n < n min are the same as those at T = Tmin and n = n min , and the values at T > Tmax and n > n max are the same as those at T = Tmax and n = n max . Secondly, the approximations have been used beyond the range of temperatures and carbon numbers for which they were originally obtained. In both cases, these extrapolations can lead to uncontrollable errors, and their choice depends on the physical nature of the properties and the availability of experimental data beyond the range of temperatures for which they were obtained (see [15, 24]). The methods of extrapolation are specified for specific properties of particular components. The values of properties at T > Tcr are assumed to be the same as those at T = Tcr (this approximation allows us to avoid rigorous analysis of the case when the droplet temperatures exceed the critical temperatures of the lightest components; the errors imposed by this approximation are expected to be small). Unless stated otherwise, the approximations of the properties of the liquid components are assumed to be valid in the range between melting and boiling temperatures. On some occasions these properties are used at temperatures above the boiling temperatures, although in these cases the approximations might be less reliable.

Thermodynamic and Transport Properties of Alkanes This section is complementary to Appendix N. We focus on the range of applicability of the approximations. Also, we add approximations for saturated vapour pressure and enthalpy of evaporation which were not included in Appendix N. © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

531

532

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Molecular Structure, Boiling and Critical Temperatures The chemical formula of alkanes is Cn H2n+2 . The chemical structures of iso-alkanes with n = 20 and n = 16 are presented in Fig. O.1 (n-icosane and 3, 6, 9, 10-methyldodecane). The difference in thermodynamic and transport properties of n-alkanes and iso-alkanes is ignored in the analysis of heating and evaporation of Diesel fuel droplets performed in [19]. For example, the difference between the boiling temperatures of n-alkanes and iso-alkanes for n = 8 − 20 does not exceed 4.5 K [24], which is less than about 1% of the value of this temperature. Using data given in [24], the dependence of critical and boiling temperatures on n was approximated by Formulae (N.1) and (N.2) valid in the range 5 ≤ n ≤ 25 [18]. The range of applicability of the coefficients in (N.1) and (N.2) was extended to 26 ≤ n ≤ 27, keeping in mind that the molar fractions of alkanes with these n is less than 0.05%. The validity of this was checked by finding new correlations valid in the range 8 ≤ n ≤ 27. The predictions of these new correlations turned out to be very close to those which follow from Formulae (N.1) and (N.2).

Liquid Density, Viscosity, Heat Capacity, and Thermal Conductivity The temperature dependence of the density of liquid alkanes for 8 ≤ n ≤ 27 was approximated, using data given in [24], by Expression (N.3). Following [10], the temperature dependence of the dynamic viscosity of liquid alkanes for 4 ≤ n ≤ 44 was approximated by Expression (N.5). The authors of [6] showed good agreement between the predictions of Expression (N.5) and experimental data in the range of temperatures from 10 ◦ C to 100 ◦ C. Also, they showed that the agreement between the values of liquid viscosity predicted by Formula (N.5) and data shown in the NIST website [13] is almost ideal. Following [21], the temperature dependence of the heat capacity of liquid alkanes for 2 ≤ n ≤ 26 was approximated by Expression (N.7). Bearing in mind that molar fraction of alkanes with n = 27 is about 0.3% it is assumed that Formula (N.7) can be used in the whole range 8 ≤ n ≤ 27. Following [23], the temperature dependence of thermal conductivity of liquid alkanes was approximated by Formula (N.8). Although Approximation (N.8) was derived for 5 ≤ n ≤ 20, it was used in the whole range 8 ≤ n ≤ 27. Possible errors imposed by these approximations in the range 21 ≤ n ≤ 27 are expected to have a very small effect on the final results as the molar fractions of alkanes in this range of n are less that 1.5%. The ranges of applicability of Formula (N.8) depend on the values of n: For n = 8 this range was found to be 216–540 K; for n = 9 – 243–588 K; for n = 10 – 248–607 K; for n = 11 – 230–625 K;

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels Fig. O.1 Molecular structures of selected hydrocarbons. Reprinted from [19]. Copyright Elsevier (2014)

533

534

for n for n for n for n for n for n for n for n for n

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

= 12 – 264–625 K; = 13 – 268–642 K; = 14 – 279–658 K; = 15 – 283–671 K; = 16 – 291–685 K; = 17 – 327–732 K; = 18 – 301–708 K; = 19 – 305–718 K; = 20 – 310–729 K [23].

It is assumed that the temperature range for n > 20 is the same as for n = 20. As demonstrated in [6], the values of kl predicted for T = 300 K and T = 450 K agree well with the data presented in [13].

Saturated Vapour Pressure and Enthalpy of Evaporation The following approximation (Antoine equation) for the dependence of the saturation vapour pressure (in Pa) on n was used [5]:   log10 0.001 · p sat (n) = A(n) −

B(n) , T + C(n)

(O.1)

where A(n) = 0.022 n + 5.8474,

B(n) = 52.807 n + 981.92,

C(n) = −5.0431n − 31.205, T is in K. The above approximations for A(n), B(n), C(n) were derived for 8 < n < 27. They are valid in the temperature ranges: 298–423 K for n = 8; 315–449 K for n = 9; 338–468 K for n = 10; 356–499 K for n = 11; 367–520 K for n = 12; 384–540 K for n = 13; 399–559 K for n = 14; 413–577 K for n = 15; 426–594 K for n = 16; 438–610 K for n = 17; 449–625 K for n = 18; 462–639 K for n = 19; 475–652 K for n = 20; 393–630 K for n = 21;

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

402–642 K for n 411–653 K for n 419–664 K for n 427–675 K for n 434–685 K for n 442–695 K for n

535

= 22; = 23; = 24; = 25; = 26; = 27.

Note that the structure of Expression (O.1) is similar to that of Expression (4.46) but with different approximations of the coefficients in different ranges of n. At low temperatures (close to room temperature), the values of pressure are expected to be small and realistic errors in its estimate are not expected to produce noticeable effects on droplet heating and evaporation. By the time the droplet surface temperatures reach values higher than the above-mentioned temperature ranges, their radii have become very small in most cases. In this case the errors in determination of the vapour pressure are also expected to produce a small effect on droplet heating and evaporation. This approximation is consistent with one presented in [6, 18]; the approximation given in these papers is valid for n < 17. Following [24] the specific enthalpies of evaporation for alkanes were approximated as A(1 − Tr ) B · 106 , (O.2) L= Mn where Mn are molar masses, the values of A for specific values of n [24] were approximated as A ≡ A L = 0.0066 n 2 + 4.697 n + 20.258 for n ≤ 20 and

A ≡ A H = −0.1143 n 2 + 7.853 n − 8.8344

for n > 20. The original values of B provided by [24] were used: B = 0.439 for n = 8, B = 0.377 for n = 9, B = 0.451 for n = 10, B = 0.413 for n = 11, B = 0.407 for n = 12, B = 0.416 for n = 13, B = 0.418 for n = 14, B = 0.419 for n = 15, B = 0.422 for n = 16, B = 0.433 for n = 17, B = 0.451 for n = 18, B = 0.448 for n = 19, B = 0.409 for n = 20,

536

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Table O.1 The values of the coefficients in Expressions (O.3) and (O.4) ((c) stands for cycloalkanes) Coefficient acc bcc ccc dcc Value n ≤ 10 667 0 0 0 Value n > 10 425.28 31.442 −0.9002 0.0125 Coefficient abc bbc cbc Value 176.51 32.312 −0.4776

B = 0.380 for n ≥ 21. The closeness of the above-mentioned approximations of A by A L and A H is illustrated in Fig. A2 of [19].

Thermodynamic and Transport Properties of Cycloalkanes Molecular Structure, Boiling and Critical Temperatures The chemical formula of cycloalcanes is Cn H2n . Their typical chemical structure is illustrated in Fig. O.1 (1-propyl-3-hexyl-cycloheptan). Using data shown in [24], the dependence of critical and boiling temperatures on n was approximated as: Tc(c) (n) = acc + bcc n + ccc n 2 + dcc n 3 ,

(O.3)

Tb(c) (n) = abc + bbc n + cbc n 2 ,

(O.4)

where the coefficients are presented in Table O.1. Expressions similar to those given in (O.3) and (O.4) could be obtained using the analysis described in [4, 7, 9].

Liquid Density The temperature dependence of the liquid density of cycloalkanes for 5 ≤ n ≤ 25 was approximated by Expression (N.3) with the coefficients given by the following expressions:

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

⎫ Aρ = 0.00003 n 2 − 0.0016 n + 0.278 ⎬ Bρ = 0.00003 n 2 − 0.00237693 n + 0.2823 ⎭ Cρ = 0.28571

537

(O.5)

and critical temperatures given by Expression (O.3)). For 11 ≤ n ≤ 25 the densities approximated by Expression (N.3) with the coefficients given in Expression (O.5) almost exactly coincided with those presented in [24]. For n ≥ 26 the deviation between these results could reach 3%. This deviation can be ignored in most applications since the molar fractions of cycloalkanes for these n in Diesel fuel are less than 0.03%.

Liquid Viscosity Following [10], the temperature dependence of the dynamic viscosity of liquid cycloalkanes for 10 ≤ n ≤ 94 was approximated by (N.5) with b(n) given by the following expression: b(n) = −9.001 + 2.350 log10 (14n).

Liquid Heat Capacity Following [16, 17], the temperature dependence of the heat capacity of liquid cycloalkanes for 10 ≤ n ≤ 27 was approximated as: 



   T T 2 Ru + cc cl (T ) = ac + bc , Mn 100 100 where Ru is the universal gas constant, Mn is the molar mass in kg/kmole, ac = 33.75209 + 2.7345 (n − 10), bc = −5.21095283 + 0.122732 (n − 10) K−1 , cc = 2.78089 − 0.123482 (n − 10) K−2 .

(O.6)

538

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Liquid Thermal Conductivity The following expression (derived from the combination of the boiling-point method and Riedel formula) was used to estimate temperature dependence of thermal conductivities of liquid cycloalkanes for 10 ≤ n ≤ 27 [24]: 2.64 · 10−3 3 + 20(1 − Tr ) 3 kl (T ) = × √ 2 , Mn 3 + 20(1 − Tbr ) 3 2

(O.7)

where Mn = 14n is the molar mass, Tr = T /Tc , Tbr = Tb /Tc . Tc and Tb are approximated by (O.3) and (O.4), respectively.

Saturated Vapour Pressure and Enthalpy of Evaporation Following [5], the saturated vapour pressure was approximated by the Antoine equation (O.1) with A(n) = 0.0201n + 5.8268, B(n) = 47.34n + 1115.2, C(n) = −5.4145n − 23.03. Expression (O.1) with these values of coefficients can be used in the ranges of temperature: 340–484 K for n = 10; 359–508 K for n = 11; 376–530 K for n = 12; 393–551 K for n = 13; 367–399 K for n = 14; 423–589 K for n = 15; 429–606 K for n = 16; 450–622 K for n = 17; 458–637 K for n = 18; 474–651 K for n = 19; 486–665 K for n = 20; 496–677 K for n = 21; 507–689 K for n = 22; 414–664 K for n = 23; 422–675 K for n = 24; 430–686 K for n = 25.

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

539

As for alkanes, these approximations for A(n), B(n), C(n) were used outside the range of temperatures for which they were obtained, up to the critical temperatures and below the above-mentioned minimal temperatures. As for alkanes, following [24] the values of L for cycloalkanes were given by Formula (O.2). The values of A for individual n given in [24] were approximated as A = −0.0085 n 3 + 0.4134 n 2 − 2.556 n + 56.345,

B = 0.38

for n = 16. A = 101.3122 and B = 0.49 for n = 16.

Thermodynamic and Transport Properties of Bicycloalkanes Molecular Structure, Boiling and Critical Temperatures The chemical formula of bicycloalkanes is Cn H2n−2 Their typical chemical structures are illustrated in Fig. O.1 (diethylbicycloheptan and ethylcycloheptan-cyclononane). Using data given in [24] for 10 ≤ n ≤ 25, the dependence of critical and boiling temperatures on n was approximated as:

for 13 ≤ n ≤ 24,

Tc(b) (n) = 134.85 ln(n) + 395.85,

(O.8)

Tb(b) (n) = 217.41 ln(n) − 32.662,

(O.9)

for 10 ≤ n ≤ 25. Tc(b) (10) = 703.60 K, Tc(b) (11) = 752.51 K, Tc(b) (12) = 762.49 K. (b) stands for bicycloalkanes. Following [24], in the correlations shown later in this section Tc(b) (10) was replaced with Tc(b) (10) = 702.25 K. Since Tc(b) (25) for bicycloalkanes is not available, Tc(b) (25) = 833.34 K for 1,1 dicyclohexyltridecane was used instead of Tc(b) (25) for bicycloalkanes.

Liquid Density Using data given in [22], the temperature dependence of the density of liquid bicycloalkanes for 10 ≤ n ≤ 25 was approximated by Formula (N.3) with Cρ = 0.28571 and Aρ and Bρ approximated as:

540

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Aρ = −0.0034 n + 0.3231, Bρ = −0.0031 n + 0.3022



for 11 ≤ n ≤ 12, Aρ = 0.0002 n 2 − 0.0072 n + 0.3529, Bρ = −0.0008 n 2 − 0.0278 n + 0.4966



for 13 ≤ n ≤ 18, Aρ = 5 × 10−5 n 2 − 0.0032 n + 0.3168, Bρ = 0.0004 n 2 − 0.0179 n + 0.4965



for 19 ≤ n ≤ 25. Since the molar fraction of bicycloalkanes for n = 10 is less than 0.7% it was assumed that ρl(b) (T )(n = 10) = ρl(b) (T )(n = 11).

Liquid Viscosity Following [10], the temperature dependence of the dynamic viscosity of liquid bicycloalkanes for 10 ≤ n ≤ 94 was approximated by Formula (N.5) with b(n) given by the following expression: b(n) = −9.513 + 2.248 log10 (14n − 2).

Liquid Heat Capacity Following [16, 17], the temperature dependence of the heat capacity of liquid bicycloalkanes for 10 ≤ n ≤ 25 was approximated by Formula (O.6) with the following coefficients: ac = 19.2782 + 2.7345 (n − 11), bc = 4.722955 + 0.122732 (n − 11) K−1 , cc = 0.08912 + 0.123482 (n − 11) K−2 .

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

541

Liquid Thermal Conductivity Following [15], the temperature dependence of thermal conductivity of liquid bicycloalkanes for 10 ≤ n ≤ 25 was estimated by Formula (O.7) with Mn = 14n − 2.

Saturated Vapour Pressure and Enthalpy of Evaporation Following [15, 24], the saturated vapour pressure for bicycloalkanes for 10 ≤ n ≤ 25 was approximated as: (O.10) ln( pvb / pcb ) = f 0 + ωb f 1 , where f 0 = 5.92714 −

6.09648 − 1.28862 ln Tr + 0.169347Tr6 , Tr

f 1 = 15.2518 −

15.6875 − 13.4721 ln Tr + 0.43577Tr6 , Tr

Tr = T /Tc(b) , Tc(b) is the critical temperature estimated from (O.8), pcb is the critical pressure of bicycloalkanes estimated as   pcb = 105 0.0711 n 2 − 3.8116 n + 60.998 Pa, ω ≡ ωb = −0.001 n 2 + 0.0679 n − 0.3039. As for alkanes and cycloalkanes, the values of L for bicycloalkanes were estimated by Formula (O.2) with A given in [24] approximated as: A = −0.1405 n 2 + 8.1341 n − 3.2083, B = 0.434 for n = 10 and B = 0.38 otherwise. In contrast to previously considered hydrocarbons, however, it was found that a more accurate approximation for L for bicycloalkanes is given by the following expression (cf. [15, 14]):  L=

−

Ru Tc (b) 1 (Tr , ωb ) Mn

Ru Mn

when Tr < 0.6 Tc (b) 2 (Tr , ωb ) when Tr ≥ 0.6,

where the units of Ru and Mn are the same as in (O.6), 1 = [(−6.09648 − 15.6875ωb ) + (1.28862 + 13.4721ωb ) Tr

(O.11)

542

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

 −6 (0.169347 + 0.43577ωb ) Tr7 , 2 = 7.08 (1 − Tr )0.354 + 10.95 ωb (1 − Tr )0.456 . Predictions of Approximation (O.11) were compared with the data given in [24] for C10 H18 , C11 H20 , C12 H22 and C25 H48 . The deviation between the results predicted by (O.11) and these data did not exceed 5%. Approximation (O.11) was used in [19].

Thermodynamic and Transport Properties of Alkylbenzenes Molecular Structure, Boiling and Critical Temperatures The chemical formula of alkylbenzenes is Cn H2n−6 . Their chemical structures are presented in Fig. O.1 (1, 4-dipropylbenzene and 1, 4-dipentylbenzene). Using data given in [1, 2, 4, 7, 9, 11, 12, 20, 23, 24], the dependence of critical and boiling temperatures on n was approximated as: Tc(ab) (n) = acab + bcab n + ccab n 2 ,

(O.12)

Tb(ab) (n) = abab + bbab n + cbab n 2 ,

(O.13)

where the coefficients are shown in Table O.2, (ab) refers to alkylbenzenes.

Liquid Density Using data given in [24], the temperature dependence of the density of liquid alkylbenzenes for 5 ≤ n ≤ 25 was approximated as: ρl(ab) (T ) = Aρab + Bρab T + Cρab T 2 + Dρab T 3 ,

Table O.2 Values of the coefficients in Formulae (O.12) and (O.13) Coefficient acab bcab Value 427.89 27.408 Coefficient abab bbab Value 171.6 33.426

ccab −0.4388 cbab −0.5252

(O.14)

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

543

where Aρab , Bρab , Cρab and Dρab were approximated as: ⎫ Aρab = −32.04 n + 1422.6 ⎪ ⎪ ⎬ Bρab = 0.1831 n − 2.824 Cρab = −0.0005 n + 0.0056 ⎪ ⎪ ⎭ Dρab = 6 × 10−7 n − 7 10−6

(O.15)

⎫ Aρab = −0.0477 n 2 − 0.4141 n + 1082.6 ⎬ Bρab = 0.0004 n 2 − 0.0062 n − 0.7017 ⎭ Cρab = Dρab = 0

(O.16)

for n = 8 and n = 9, and

for n = 11 − 20. It was assumed that ρ(n = 10) = 21 (ρ(n = 9) + ρ(n = 11)) and ρ(n > 20) = ρ(n = 20). The latter assumption is justified by the fact that molar fractions of components with n > 20 are about or less than 0.2%. Formulae (O.14)–(O.16) are supported by experimental results given in [22].

Liquid Viscosity Following [10], the temperature dependence of the dynamic viscosity of liquid alkylbenzenes was approximated by Formula (N.5) with b(n) approximated as: b(n) = −9.692 + 2.261 log10 (14n − 6).

Liquid Heat Capacity Following [16, 17], the temperature dependence of the heat capacity of liquid alkylbenzenes for 8 ≤ n ≤ 24 was approximated by Formula (O.6) with the following coefficients: ac = 15.1109 + 2.7345 (n − 7), bc = 0.68109 + 0.122732 (n − 7) K−1 , cc = 1.96346 − 0.123482 (n − 7) K−2 .

544

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Liquid Thermal Conductivity Following [15, 24], Formula (O.7) with Mn = 14n − 6 was used for the approximation of the temperature dependence of thermal conductivity of liquid alkylbenzenes for 8 ≤ n ≤ 24.

Saturated Vapour Pressure and Enthalpy of Evaporation Following [5], Eq. (O.1) with A(n) = 0.0007 n 2 − 0.0064 n + 6.0715, B(n) = 51.811 n + 1049.1, C(n) = 0.1215 n 2 − 9.6892 n + 11.161 was used for approximating the saturated vapour pressure for alkylbenzenes. Expression (O.1) with the coefficients presented above for alkylbenzenes was used in the following ranges of temperature: 306–420 K for n = 8; 323–455 K for n = 9; 343–486 K for n = 10; 361–510 K for n = 11; 378–531 K for n = 12; 394–553 K for n = 13; 406–571 K for n = 14; 421–590 K for n = 15; 438–606 K for n = 16; 450–622 K for n = 17; 458–636 K for n = 18; 473–651 K for n = 19; 485–665 K for n = 20; 495–677 K for n = 21; 505–688 K for n = 22; 414–664 K for n = 23; 423–675 K for n = 24. As for alkanes and cycloalkanes, the values of L for alkylbenzenes were estimated by Formula (O.2) [24]. The values of A and B given in [24] were approximated as A = 0.0007124 n 5 − 0.05315 n 4 + 1.4963 n 3 − 19.83 n 2 + 128.65 n − 276.8,

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

B B B B

545

= −0.007 n 2 + 0.1172 n − 0.0989 when 8 ≤ n ≤ 10, = −0.0062 n 2 + 0.1829 n − 0.9093 when 11 ≤ n ≤ 14, = −0.0013315 n 3 + 0.0634 n 2 − 0.9842 n + 5.3794 when 15 ≤ n ≤ 19, = 0.38 when n ≥ 20.

Thermodynamic and Transport Properties of Indanes and Tetralines Molecular Structure, Boiling and Critical Temperatures The chemical formula of indanes and tetralines is Cn H2n−8 . Their chemical structures are presented in Fig. O.1 (pentylindane, as an example of indanes, and 1-propyl-(1, 2, 3, 4-tetrahydronaphthalene), as an example of tetralines). Indanes and tetralines differ by the numbers of carbon atoms in the second ring. For indanes, this number is equal to 5; for tetralines, it is equal to 6. Their properties are very close and are not distinguished in this section. Using data given in [24], the dependence of critical and boiling temperatures on n was approximated as: Tc(i) (n) = aci + bci n + cci n 2 ,

(O.17)

Tb(i) (n) = abi + bbi n + cbi n 2 ,

(O.18)

where the coefficients are shown in Table O.3, (i) refers to indanes & tetralines.

Liquid Density Following [24], the temperature dependence of density of liquid indanes and tetralines for 10 ≤ n ≤ 22 was approximated by Formula (N.3) with the following approximations for Aρ , Bρ and Cρ :

Table O.3 Values of the coefficients in Formulae (O.17) and (O.18) Coefficient aci bci Value (n ≤ 10) 720.15 0 Value (n > 10) 555.59 17.898 Coefficient abi bbi Value 249.21 25.894

cci 0 −0.2486 cbi −0.3319

546

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

⎫ Aρi = 0.0002 n 2 − 0.0079 n + 0.3622, ⎬ Bρi = −7 · 10−5 n 3 + 0.0031 n 2 − 0.0438 n + 0.4608, (O.19) ⎭ Cρi = 0.2677 for n = 10; Cρi = 0.28571 for 11 ≤ n ≤ 16; for 10 ≤ n ≤ 16, and ⎫ Aρi = 0.0002 n 2 − 0.0079 n + 0.3622, ⎬ Bρi = 6 · 10−5 n 2 − 0.0025 n + 0.2908, ⎭ Cρi = 0.28571.

(O.20)

for 17 ≤ n ≤ 20. It was assumed that ρ(n > 20) = ρ(n = 20) which was justified by the fact that molar fractions of these components are less than 0.2%. Approximations (O.19) and (O.20) were obtained in the following ranges of temperatures: 237.4 − 720.15 K for n = 10; 232.3 − 722 K for n = 11; 243.6 − 735 K for n = 12; 254.8 − 745 K for n = 13; 266.1 − 756 K for n = 14; 277.4 − 767 K for n = 15; 288.6 − 777 K for n = 16; 299.9 − 788 K for n = 17; 311.2 − 797 K for n = 18; 322.5 − 805 K for n = 19; 333.7 − 814 K for n ≥ 20.

Liquid Viscosity Following [10], the temperature dependence of the dynamic viscosity of liquid indanes & tetralines for 10 ≤ n ≤ 94 was approximated by Formula (N.5) with b(n) = −9.411 + 2.217 log10 (14n − 8).

Liquid Heat Capacity Following [16, 17], the temperature dependence of the heat capacity of liquid indanes & tetralines for 10 ≤ n ≤ 22 was approximated by Formula (O.6) with the following coefficients: ac = 14.136 + 2.7345 (n − 11),

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

547

bc = 6.43698 + 0.122732 (n − 11) K−1 , cc = 14.136 − 0.123482 (n − 11) K−2 .

Liquid Thermal Conductivity Following [15, 24], Formula (O.7) with Mn = 14n − 8 was used for the approximation of the temperature dependence of thermal conductivity of liquid indanes & tetralines for 10 ≤ n ≤ 22.

Saturated Vapour Pressure and Enthalpy of Evaporation Following [15, 24], the saturated vapour pressure for indanes & tetralines was approximated by Formula (O.10) in which f 0 and f 1 were defined earlier, Tr = T /Tc(i) , Tc(i) is the critical temperature estimated from (O.17) (cf. Formula (O.10) for bicycloalkanes), pci is the critical pressure approximated as   pci = 105 0.0693 n 2 − 3.8821 n + 63.771 Pa, ωi (the acentric factor) was approximated as ωi = 0.617 ln(n) − 1.11. As for alkanes, cycloalkanes and alkylbenzenes, L for indanes & tetralines was approximated by Formula (O.2). The values of A given in [24] were approximated as A = −0.0793 n 2 + 6.3293 n + 5.7796. The following values of B given in [24] were used: B = 0.303 for n ≤ 10 and B = 0.38 for n > 10.

Thermodynamic and Transport Properties of Naphthalenes Molecular Structure, Boiling and Critical Temperatures The chemical formula of naphthalenes is Cn H2n−12 . Their typical chemical structure is presented in Fig. O.1 (1-propylnaphthalene).

548

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Table O.4 Values of the coefficients in Formulae (O.21) and (O.22) Coefficient acn bcn Value 655.14 9.7878 Coefficient abn bbn Value 350.37 15.218

Using data given in [24], the dependence of critical and boiling temperatures on n was approximated as: (O.21) Tc(n) = acn + bcn n, Tb(n) = abn + bbn n,

(O.22)

where the coefficients are shown in Table O.4, (n) refers to naphthalenes.

Liquid Density Using data given in [22], the temperature dependence of the density of naphthalenes for 10 ≤ n ≤ 20 was presented as: ρl(n) (T ) = Aρn + Bρn T,

(O.23)

where Aρn and Bρn are approximated as:

Aρn = 1.45 n 2 − 55.715 n + 1671.9 Bρn = 0.0087 n − 0.8084.

(O.24)

Liquid Viscosity Following [10], the temperature dependence of the dynamic viscosity of liquid naphthalenes for 10 ≤ n ≤ 94 was approximated by Expression (N.5) with b(n) = −9.309 + 2.185 log10 (14n − 12).

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

549

Liquid Heat Capacity Following [16], the temperature dependence of the heat capacity of naphthalenes for 10 ≤ n ≤ 20 was approximated by Formula (O.6) with the coefficients: ac = 9.67805 + 2.7345 (n − 11), bc = 5.982952 + 0.122732 (n − 11) K−1 , cc = 0.2688 + 0.123482 (n − 11) K−2 .

Liquid Thermal Conductivity Following [15, 24], the temperature dependence of thermal conductivity of naphthalenes for 10 ≤ n ≤ 20 was approximated by Formula (O.7) with Mn = 14n − 12.

Saturated Vapour Pressure and Enthalpy of Evaporation Following [15, 24], the saturated vapour pressure for naphthalenes for 10 ≤ n ≤ 20 was approximated by Expression (O.10) (as for bicycloalkanes and indanes & tetralines) but for   pc ≡ pcn = 105 0.2009 n 2 − 8.443 n + 104.09 Pa, ω ≡ ωn = −0.0018 n 2 + 0.0997 n − 0.5082. As for alkanes, cycloalkanes, alkylbenzenes and indanes & tetralines, L for naphthalenes was estimated by Formula (O.2). The values of A given in [24] were approximated as A = 0.2607 n 2 − 2.1791 n + 66.218 for 10 ≤ n ≤ 16, A = −0.1929 n 2 + 10.926 n − 37.384 for n ≥ 17. The values of B given [24] were approximated as B = −0.0003165 n 3 + 0.01545 n 2 − 0.2495 n + 1.722 for all n.

550

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

Thermodynamic and Transport Properties of Tricycloalkanes, Diaromatics and Phenanthrenes The molar fractions of tricycloalkanes, diaromatics and phenanthrenes in Diesel fuel under consideration are 1.5647, 1.2240 and 0.6577%, respectively. Their ranges of n are rather narrow: 14 ≤ n ≤ 20 for tricycloalkanes, 13 ≤ n ≤ 16 for diaromatics and 14 ≤ n ≤ 18 for phenanthrenes. This allows us to ignore the dependence of the properties of these substances on n and consider the properties for just one n for which the properties are available. Thus, the analysis of these three groups is reduced to the analysis of three representative components referred to as tricycloalkane, diaromatic and phenanthrene.

Molecular Structure, Boiling and Critical Temperatures The chemical formula of tricycloalkanes is Cn H2n−4 . The properties described in this section refer to n = 19 (C19 H34 ). The chemical formula of diaromatics is Cn H2n−14 . The properties described in this section refer to n = 13 (C13 H12 ). The chemical formula of phenanthrenes is Cn H2n−18 . The properties described in this section refer to n = 14 (C14 H10 ). Typical chemical structures of diaromatics and phenanthrenes are presented in Fig O.1 (di(3-ethyl-phenyl)methane and ethylphenanthrene, respectively). Using data given in [8], the boiling and critical temperatures for tricycloalkane were approximated as: Tb(t) = 692.33 K, Tc(t) = Tb(t) × 0.738686−1 = 937.25 K. Using data given in [24], these temperatures for the remaining two components were approximated as: Tb(d) = 537.42 K, Tc(d) = 760.00 K for diaromatics, and Tb(p) = 610.00 K, Tc(p) = 869.00 K for phenanthrenes. (t) refers to tricycloalkanes, (d) refers to diaromatics, and (p) refers to phenanthrenes.

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

551

Liquid Density Using data given in [22], the temperature dependence of the density of liquid tricycloalkane and phenanthrene was estimated as: ρl(t) (T ) = Aρt + Bρt T,

(O.25)

where Aρt = 1151.17, Bρt = −0.694690 for tricycloalkanes, and Aρt = 1374.16, Bρt = −0.819355 for phenanthrenes. The approximation for tricycloalkane can be used in the range of temperatures (273.15–372.05) K, while the approximation for phenanthrenes can be used in the range (490.70–557.80) K. Using data given in [22], the temperature dependence of the density of liquid diaromatic was approximated as: ρl(d) (T ) = Aρd + Bρd T + Cρd T 2 + Dρd T 3 , where

(O.26)

Aρd = 1.22498 · 103 , Bρd = −7.21739 · 10−1 , Cρd = −8.65342 · 10−5 , Dρd = 1.63332 · 10−9 .

This approximation can be used in the range of temperatures (284.15–523.15) K. In contrast to the previously considered components, it was assumed that the values at T < Tmin are the same as the values at T = Tmin , and the values at T > Tmax are the same as the values at T = Tmax . This ‘cautious’ approach can be justified by the very small contribution of these components to heating and evaporation of Diesel fuel droplets.

Liquid Viscosity Following [3, 8], the temperature dependence of the liquid dynamic viscosity was approximated as:

μl = M exp

 ηa − 597.82 + ηb − 11.202 , T

(O.27)

552

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

where M is molar mass, M = 262.4733

kg , ηa = 3107.93, ηb = −9.936 kmole

for tricycloalkanes, M = 168.23

kg , ηa = 2199.18, ηb = −5.395 kmole

for diaromatics, and M = 178.23

kg , ηa = 1613.54, ηb = −3.372 kmole

for phenanthrenes. Approximation (O.27) can be used up to 0.7 Tc .

Liquid Heat Capacity Following [16, 17], the temperature dependence of the heat capacity of liquid tricycloalkanes and polycyclic aromatics was approximated by Formula (O.6) with the following coefficients: ac = 32.9773, bc = 8.243707 K−1 , cc = 0.93225 K−2 for tricycloalkanes, ac = 17.9997, bc = 3.230018 K−1 , cc = 0.5203 K−2 for diaromatics, and ac = 2.43092, bc = 12.11225 K−1 , cc = 0.80569 K−2 for phenanthrenes.

Liquid Thermal Conductivity Following [15], the temperature dependence of thermal conductivity of liquid tricycloalkanes and polycyclic aromatics was approximated using Formula (O.7) with Mn = 14n − 4 for tricycloalkanes, Mn = 14n − 14 for diaromatics and Mn = 14n − 18 for phenanthrenes.

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

553

Saturated Vapour Pressure and Enthalpy of Evaporation Following [5], the saturated vapour pressures for tricycloalkane, diaromatic and phenanthrene were approximated by Eq. (O.1) with A(n) = 15.14702, B(n) = 6103.355, C(n) = 0 for tricycloalkane in the range of temperatures 301–321 K, A(n) = 6.38684, B(n) = 2334.129, C(n) = −92.028 for tricycloalkane in the range of temperatures 333–464 K, A(n) = 9.79557, B(n) = 3740.286, C(n) = 0 for diaromatic in the range of temperatures 273–298 K, A(n) = 6.19796, B(n) = 1885.888, C(n) = −88.292 for diaromatic in the range of temperatures 333–647 K, A(n) = 11.631, B(n) = 4873.4, C(n) = 0.05 for phenanthrene in the range of temperatures 306–321 K, A(n) = 6.37081, B(n) = 2329.54, C(n) = −77.87 for phenanthrene in the range of temperatures 356–650 K. The upper bounds of these temperatures are very close to the critical temperatures of the components. These bounds were identified with critical temperatures. At temperatures below Tmin and above Tmax it was assumed that pv (T < Tmin ) = pv (T = Tmin ) and pv (T > Tmax ) = pv (T = Tmax ). At intermediate temperatures when pv (T ≤ T1 ) and pv (T ≥ T2 ) are known but pv (T1 < T < T2 ) are not known, these pressures were approximated by interpolation as: pv (T ) = pv (T1 ) +

pv (T2 ) − pv (T1 ) (T − T1 ). T2 − T1

Following [14, 15], L for tricycloalkane, diaromatic and phenanthrene was estimated by the equation, inferred from the Clausius-Clapeyron equation: L=−

Ru d ln p sat (n) , M(n) d(1/T )

(O.28)

554

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

where Ru is the universal gas constant. Remembering (O.1), Expression (O.28) was presented as: Ru B(n)T 2 . (O.29) L= M(n)(T + C(n))2

References 1. Ambrose D, Broderick BE, Townsend R (1967) The vapour pressures above the normal boiling point and the critical pressures of some aromatic hydrocarbons. Journal of the Chemical Society A 633–641. 2. Bagga OP, Rattan VK, Singh S, Sethi BPS, Raju KSN (1987) Isobaric vapour-liquid equilibria for binary mixtures of ethylbenzene and p-xylene with dimethylformamide. Journal of Physical and Chemical Reference Data 32:198–201. 3. Constantinou L, Gani R (1994) New group contribution method for estimating properties of pure compounds. American Institute of Chemical Engineers (AIChE) Journal 40:1697–710. 4. Daridon JL, Lagrabette A, Lagourette B (1998) Speed of sound, density, and compressibilities of heavy synthetic cuts from ultrasonic measurements under pressure. J Chem Thermodynam 30:607–623. 5. Dykyj J, Svoboda J, Wilhoit RC, Frenkel M, Hall KR (1999) In: LANDOLT-BORNSTEIN: numerical data and functional relationships in science and technology. Vapor pressure of chemicals. subvolume A: Vapor pressure and Antoine constants for hydrocarbons, vol 20, Springer. 6. Elwardany AE, Sazhin SS (2012) A quasi-discrete model for droplet heating and evaporation: application to Diesel and gasoline fuels. Fuel 97:685–694. 7. Hopfe D (1990) Thermophysical Data of Pure Pubstances. Data compilation of FIZ CHEMIE Germany, 1. 8. Joback KG, Reid RC (1987) Estimation of pure-component properties from group-contributions. Chem Eng Commun 57:233–243. 9. Liessmann G, Schmidt W, Reiffarth S (1995) Recommended Thermophysical Data. Data compilation of the Saechsische Olefinwerke Boehlen Germany, 1. 10. Mehrotra AK (1994) Correlation and prediction of the viscosity of pure hydrocarbon. Can J Chem Eng 72:554–557. 11. Nesterova TN, Nesterov IA, Pimerzin AA (2000) The thermodynamics of the sorption and evaporation of alkylbenzene. II. Critical temperatures. Izvestia Vysshikh Uchebnykh Zavedeniy. Khimia i Khicheskaya Tekhnologia (Reports of Higher Education Establishments, Chemistry and Chemical Technology) 434:14–22. 12. Nikitin ED, Popov AP, Bogatishcheva NS (2002) Vapour-liquid critical properties of nalkylbenzenes from toluene to 1-phenyltridecane. Journal of Chemical and Engineering Data 47:1012–1016. 13. NIST Chemistry WebBook, Thermophysical Properties of Fluid Systems. http://webbook.nist.gov/chemistry /Last online-access on 19th May 2020. 14. Poling BE, Prausnitz JM, O’Connell J (2000) The Properties of Gases and Liquids. McGrawHill, New York. 15. Reid RC, Prausnitz JM, Poling BE. (1987) The Properties of Gases and Liquids. 4th ed. New York: McGraw-Hill. 16. Ruzicka JRV, Domalski ES (1993) Estimation of the heat capacities of organic liquids as a function of temperature using group additivity. I. Hydrocarbon compounds. J Phys Chem Ref Data 22:597–618.

Appendix O: Thermophysical Properties of Hydrocarbons in Diesel Fuels

555

17. Ruzicka Jr V, Domalski ES (2005) Estimation of the heat capacities of organic liquids as a function of temperature using group additivity: an amendment. J Phys Chem Ref Data 33:1071–1081. 18. Sazhin SS, Elwardany A, Sazhina EM, Heikal MR (2011) A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets, Int. J Heat Mass Transfer 54:4325–4332. 19. Sazhin SS, Al Qubeissi M, Nasiri R, Gun’ko VM, Elwardany AE, Lemoine F, Grisch F, Heikal MR (2014) A multi-dimensional quasi-discrete model for the analysis of Diesel fuel droplet heating and evaporation, Fuel 129:238–266. 20. Simmrock KH, Janowsky R, Ohnsorge A (1986) Critical Data Of Pure Substances, In ‘Dechema Chemistry Data Series’, Volume II. Frankfurt. 21. van Miltenburg JC (2000) Fitting the heat capacity of liquid n-alkanes: new measurements of n-heptadecane and n-octadecane, Thermochimica Acta 343:57–62. 22. Wilhoit RC, Hong X, Frenkel M, Hall KR In: Hall KR, Marsh KN, Editors (1998) LANDOLTBORNSTEIN: numerical data and functional relationships in science and technology. Thermodynamic properties of organic compounds and their mixtures. Subvolume F: Densities of polycyclic hydrocarbons, vol 8. Springer. 23. Yaws CL (1995) Handbook of Thermal Conductivity. Vol. 2 (Organic Compounds, C5 to C7 ) and Vol. 3 (Organic Compounds, C8 to C28 ). Gulf Publishing Company, Houston, London, Paris, Zurich, Tokyo. 24. Yaws CL (Editor) (2008) Thermophysical Properties of Chemicals and Hydrocarbons. William Andrew Inc, Beaumont, TX, USA.

Appendix P

Derivation of Expression (4.91)

The Sturm-Liouville Problem for p < 0 If p = −λ2 < 0 the general solution to Eq. (4.88) can be presented as: 



x x + B sinh λ , v(x) = A cosh λ δ0 δ0

(P.1)

where A and B are arbitrary constants. The boundary condition at x = 0 (see (4.89)) implies that B = 0. The boundary condition at x = δ0 (see (4.90)) leads to the following equation: A δ0

λ sinh λ −

  δ˙0  δ0 Dl

! cosh λ = 0,

(P.2)

where A is not equal to zero as the trivial solution v = 0 is not considered. Hence, the solution to (P.2) can be presented as: λDl coth λ =   . δ˙0  δ0

(P.3)

It can be shown that Eq. (P.3) has two solutions ±λ0 for all |δ˙D|lδ . These solutions 0 0 lead to Solutions (P.1) (eigenfunctions) which differ only by the sign of A. Since the value of A is determined by the normalisation condition only (see below), the solution λ = −λ0 can be not considered. Hence, the solution to Eq. (P.3) gives only one eigenvalue λ = λ0 > 0 and the corresponding eigenfunction 

x , v0 (x) = cosh λ0 δ0

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

(P.4)

557

558

Appendix P: Derivation of Expression (4.91)

where the normalisation leading to A = 1 is used. The calculation of the integral leads to the following expression for ||v0 ||2 :

δd

||v0 || = 2

0

v02 (x)dx

  sinh(2λ0 ) δ0 . 1+ = 2 2λ0

(P.5)

The Sturm-Liouville Problem for p > 0 If p = λ2 > 0 the general solution to Eq. (4.88) can be written as: 



x x + B sin λ , v(x) = A cos λ δ0 δo

(P.6)

where A and B are arbitrary constants. The boundary condition at x = 0 (see (4.89)) implies that B = 0. The boundary condition at x = δ0 leads to the equation: A δ0

!   δ0 δ˙0  cos λ = 0. −λ sin λ − Dl

(P.7)

A in this equation is not equal to zero as the trivial solution v = 0 is not considered. Hence, Eq. (P.7) is simplified to: cot λ = −

λDl  . δ0 δ˙0 

(P.8)

As in the case p < 0 the solutions to this equation corresponding to negative λ are not considered. A countable set of positive solutions to this equation (positive eigenvalues) λn are arranged in ascending order: 0 < λ 1 < λ2 < · · · < λn . The corresponding eigenfunctions are written as: 

x , vn (x) = cos λn δ0

(P.9)

where the normalisation leading to A = 1 is used as for p < 0. The calculation of the integral, considering Expression (P.8), leads to the following expression for ||vn ||2 for n ≥ 1:

Appendix P: Derivation of Expression (4.91)



δ0

||vn || = 2

0

vn2 (x)dx

559

  sin(2λn ) δ0 . 1+ = 2 2λn

(P.10)

The orthogonality of functions vn is proven in the next section. The completeness of this set of functions implicitly follows from the agreement of the results inferred from Formula (4.87) with those predicted by direct numerical solution of the component diffusion equation. Note that we refer to λn , rather than p, as eigenvalues of the Sturm-Liouville problem.

Orthogonality of vn (x) for n ≥ 0 For n ≥ 1 the calculation of the integrals leads to the following expression: Inm ≡

δ 0 0

vn (x)vm (x)dx =

δ 0 0





1 x x cos λm dx = δ0 cos λn cos (λn y) cos (λm y) dy. δ0 δ0 0

The integral on the right-hand side is calculated using a simplified version of Formula 2.533(5) of [1]:   δ0 sin(λn + λm ) sin(λn − λm ) + Inm = . (P.11) 2 λn + λm λn − λm For λn = λm , Formula (P.11) reduces to (P.10). For λn = λm it can be rewritten as: Inm =

δ0 [λn sin λn cos λm − λm sin λm cos λn ] . λ2n − λ2m

Remembering (P.8), this formula can be presented as: Inm =

δ˙0 δ 2  0  [cos λn cos λm − cos λm cos λn ] = 0. Dl λ2n − λ2m

Thus functions vn (x) for n ≥ 1 are orthogonal. To prove the orthogonality of v0 (x) and vn (x) for n ≥ 1 the following integral I0n = 0

δ0





x x cos λn dx, cosh λ0 δ0 δ0

(P.12)

where n ≥ 1, is calculated. Using integration by parts twice, Expression (P.12) is presented as: I0n

  δ0 λ0 λ20 cosh λ0 sin λn − = sinh λ0 cos λn − I0n , λn λn λ n δ0

(P.13)

560

Appendix P: Derivation of Expression (4.91)

where I0n on the right-hand side of this equation is the same as in (P.12). Equation (P.13) is rewritten as:

I0n =

δd λn

  cosh λ0 sin λn + λλn0 sinh λ0 cos λn .  2 λ0 1 + λn

(P.14)

Remembering Eqs. (P.3) and (P.8), it can be seen that I0n inferred from Expression (P.14) is zero. Thus, functions vn are orthogonal for n ≥ 0:

δ0

vn (x)vm (x)dx = δnm ||vn ||2 ,

(P.15)

0

where n ≥ 0 and m ≥ 0, ||vn ||2 is defined by Expression (P.5) when n = 0 and Expression (P.10) when n ≥ 1.

Reference 1. Gradshtein I, Ryzhik I (1962) Tables of Integrals, Sums, Series and Products, Fizmatgiz, Moscow (in Russian)

Appendix Q

Derivation of Expression (5.8)

In this appendix the details of the derivation of Expression (5.8) are presented following [2]. Using a new variable u = (T − Ts ) R Equation (5.1) and initial and boundary conditions (Eqs. (5.3), (5.4) and (5.7)) are simplified to: ∂ 2u ∂u = κ 2 + R P(t, R), (Q.1) ∂t ∂R u|t=0 = −T0 R when 0 ≤ R ≤ Rd ,

(Q.2)

        u| R=R − = u| R=R + , kw Rw u R − u  = k f Rw u R − u  , u| R=Rd = 0, − w w R=Rw R=Rw+

(Q.3) where Ts (droplet surface temperature) and T0 ≡ Ts − Td0 (R) are assumed to be constant (T0 does not depend on R). Since T − Ts is finite at R = 0 the boundary condition at R = 0 is presented as: u| R=0 = 0. We look for the solution to Eq. (Q.1) in the form: u=

∞

n (t)vn (R),

(Q.4)

n=1

where functions vn (R) form the full set of non-trivial solutions to the eigenvalue problem: d2 v + a 2 λ2 v = 0 (Q.5) d R2 © The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

561

562

Appendix Q: Derivation of Expression (5.8)

with the boundary conditions: ⎫ v| R=0 = v| R=Rd = 0 ⎬ − = v| + v| R=R , R=R  w    w   ⎭ kw Rw v R − v  R=Rw− = k f Rw v R − v  R=Rw+

(Q.6)

where ⎧ ⎨ cw ρw ≡ aw when R ≤ Rw 1 k a = √ =  c f wρ f ⎩ κ ≡ a f when Rw < R ≤ Rd . kf

(Q.7)

√ Note that λ has dimension 1/ time. We look for the solution to Eq. (Q.5) in the form:

when R ≤ Rw A sin(λaw R) (Q.8) v(R) = B sin(λa f (R − Rd )) when Rw < R ≤ Rd . Function (Q.8) satisfies the boundary conditions (Q.6) at R = 0. Substitution of function (Q.8) into the boundary conditions (Q.6) at R = Rw gives: A sin(λaw Rw ) = B sin(λa f (Rw − Rd )),

(Q.9)

Akw [Rw λaw cos(λaw Rw ) − sin(λaw Rw )]   = Bk f Rw λa f cos(λa f (Rw − Rd )) − sin(λa f (Rw − Rd )) .

(Q.10)

Condition (Q.9) is satisfied when:  A = ± [sin(λaw Rw )]−1  −1 . B = ± sin(λa f (Rw − Rd ))

(Q.11)

The signs in Formulae (Q.11) are selected to ensure that u in Expression (Q.4) is negative. Note that either ‘+’ or ‘−’ need to be chosen in both expressions. Having substituted Expressions (Q.11) into (Q.10) the latter is presented as:   kw [Rw λaw cot(λaw Rw ) − 1] = k f Rw λa f cot(λa f (Rw − R f )) − 1 .

(Q.12)

Recalling the definitions of aw and a f , Eq. (Q.12) is simplified to: 

kw cw ρw cot(λaw Rw ) −



k f c f ρ f cot(λa f (Rw − Rd )) =

kw − k f . Rw λ

(Q.13)

The solutions to Eq. (Q.13) lead to a countable set of positive eigenvalues λn arranged in ascending order 0 < λ1 < λ2 < ..... Note that the negative solutions

Appendix Q: Derivation of Expression (5.8)

563

−λn also satisfy Eq. (Q.13) as both sides of this equation are odd functions of λ. λ = 0, however, does not satisfy this equation. Having substituted these λn into Expression (Q.8) and remembering Expressions (Q.11), eigenfunctions vn are obtained as:  vn (R) =

sin(λn aw R) ± sin(λ when R ≤ Rw n aw R w ) sin(λn a f (R−Rd )) ± sin(λn a f (Rw −Rd )) when Rw < R ≤ Rd .

(Q.14)

It can be shown [1] that functions vn (R) are orthogonal with weight

b= Thus:



kw aw2 = cw ρw when R ≤ Rw k f a 2f = c f ρ f when Rw < R ≤ Rd . Rd

vn (R)vm (R)bd R = δnm ||vn ||2 ,

0

where δnm =

1 when n = m 0 when n = m.

The completeness of this set of functions is discussed in [1]. The norm of vn with weight b is presented as:

Rd

||vn ||2 = 0

vn2 bd R =

Rw 0



sin(λn aw R) sin(λn aw Rw )

2

cw ρw d R +

Rd Rw



sin(λn a f (R − Rd )) sin(λn a f (Rw − Rd ))

2 c f ρ f dR

  cw ρw sin(2λn aw Rw ) Rw − = 2λn aw 2 sin2 (λn aw Rw ) +

  cfρf sin(2λn aw (Rw − Rd )) R − R + d w 2λn ag 2 sin2 (λn a f (Rw − R f ))

=

c f ρ f (Rd − Rw ) kw − k f cw ρw Rw + − . 2Rw λ2n 2 sin2 (λn aw Rw ) 2 sin2 (λn a f (Rw − Rd ))

(Q.15)

When deriving Expression (Q.15), Eq. (Q.13) was used. Since all functions vn satisfy boundary conditions (Q.6), function u defined by Expression (Q.4) satisfies boundary conditions (Q.3). R P(t, R) can be expanded in a series over vn : R P(t, R) =

∞ n=1

pn (t)vn (R),

(Q.16)

564

Appendix Q: Derivation of Expression (5.8)

where: 1 ||vn ||2

pn (t) =



Rd

R P(t, R)vn (R)bd R.

0

Having substituted Expansions (Q.4) and (Q.16) into Eq. (Q.1) the latter is written as:

∞



n (t)vn (R) = −

n=1

∞

n (t)λ2n vn (R) +

n=1

∞

pn (t)vn (R).

(Q.17)

n=1

When deriving (Q.17) we considered that functions vn (R) satisfy Eq. (Q.5) for λ = λn . Equation (Q.17) is satisfied if, and only if, 

n (t) = −λ2n n (t) + pn (t).

(Q.18)

The initial condition for n (t) is obtained after substituting Expression (Q.4) into the initial condition (Q.2) for u: ∞

n (0)vn (R) = −T0 R when

0 ≤ R ≤ Rd .

(Q.19)

n=1

Remembering the orthogonality of vn with weight b, Eq. (Q.19) leads to the following formula for n (0): 1 n (0) = ||vn ||2



Rd

(−T0 R)vn (R)bd R.

0

Remembering that T0 is constant this expression can be rewritten as: n (0) = n1 + n2 , where n1 =

(Q.20)

T0 cw ρw [λn aw Rw cot (λn aw Rw ) − 1] , ||vn ||2 (λn aw )2

    T0 c f ρ f λn a f R d   +1 , n2 =  2 λn a f Rw cot λn a f (Rd − Rw ) − sin λn a f (Rd − Rw ) ||vn ||2 λn a f

T0 = Ts − Tw0 = Ts − T f 0 . The solution to Eq. (Q.18) subject to the initial condition (Q.20) is presented as: t     exp −λ2n (t − τ ) pn (τ )dτ. (Q.21) n (t) = exp −λ2n t n (0) + 0

Equation (5.8) follows from the definition of u and Eqs. (Q.4) and (Q.21).

Appendix Q: Derivation of Expression (5.8)

565

References 1. Sazhin SS, Krutitskii PA, Martynov SB, Mason D, Heikal MD, Sazhina EM (2007) Transient heating of a semitransparent spherical body. Int. J Therm. Sci. 46:444–457. 2. Sazhin SS, Rybdylova O, Crua C, Heikal M, Ismael MA, Nissar Z, Aziz ARBA (2019) A simple model for puffing/micro-explosions in water-fuel emulsion droplets. Int. J Heat and Mass Transfer 131:815–821.

Appendix R

Derivation of Expression (5.15)

In this appendix the details of the derivation of Expression (5.15) are presented following [3]. Using a new variable u =TR Equation (5.1) and its initial condition are simplified to: ∂ 2u ∂u = κ 2 + R P(t, R), ∂t ∂R

u|t=0 =

(R.1)

RTw0 (R) when 0 ≤ R ≤ Rw RT f 0 (R) when Rw < R ≤ Rd .

(R.2)

The boundary conditions for Eq. (R.1) are presented as:     u| R=Rw− = u| R=Rw+ , kw Rw u R − u 

R=Rw−

    uR + Hu  where H=

h 1 − ; kf Rd

μ=

R=Rd

= kf



   Rw u R − u 

R=Rw+

,

=μ

(R.3) (R.4)

 Rd  hTg + ρl L f R˙ d . kf

Remembering the definition of Teff , the expression for μ can be presented as μ = Rkdfh Teff . The same applies to μ0 which is introduced in Eq. (R.8). Using the new variable u and functions ˜ ) = P(R) and F(t, r ) = u(t, R) P(r

(r = R/Rd ),

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

567

568

Appendix R: Derivation of Expression (5.15)

and assuming that Rd = const during the individual timesteps except in the expression for μ, (R.1) can be rewritten as Rd2

∂F ∂2 F ˜ ) = κ 2 + Rd3 r P(r ∂t ∂r

(0 ≤ r ≤ 1).

(R.5)

The initial and boundary conditions for (R.5) are written as:

F|t=0 =

Rd r Tw0 (Rd r ) when 0 ≤ r ≤ rw Rd r T f 0 (Rd r ) when rw < r ≤ 1,

    F|r =rw− = F|r =rw+ , kw rw Fr − F 

    = k f rw Fr − F 

r =rw−



   Fr + h 0 F 

r =1

(R.6)

r =rw+

,

= μ0 ,

(R.7) (R.8)

where rw =

Rw ; Rd

h 0 = H Rd =

h Rd − 1; kf

μ0 = μRd =

 Rd2  hTg + ρl L f R˙ d . kf

The convection heat transfer coefficient is presented as h = k g Nu/(2Rd ), where k g is thermal conductivity of ambient gas, the Nusselt number Nu for stationary evaporating droplets is inferred from (3.34). To reduce the boundary condition (Eq. R.8) into a homogeneous one, function V (t, r ) is introduced via the equation: μ0 r. 1 + h0

F(t, r ) = V (t, r ) +

(R.9)

Expression (R.9) allows us to rewrite Eq. (R.5) and initial and boundary conditions (R.6)–(R.8) as ∂2V ∂V ˜ ) Rd2 = κ 2 + Rd3 r P(r (R.10) ∂t ∂r V (t = 0, r ) = Rd r T0 (Rd r ) −

T0 (Rd r )|t=0 =

(R.11)

Tw0 (Rd r ) when 0 ≤ r ≤ rw T f 0 (Rd r ) when rw < r ≤ 1,

    V |r =rw− = V |r =rw+ , kw rw Vr − V 

r =rw−



μ0 r, 1 + h0

   Vr + h 0 V 

(R.12)

    = k f rw Vr − V 

r =1

r =rw+

=0

,

(R.13) (R.14)

Appendix R: Derivation of Expression (5.15)

569

We look for the following solution to Eq. (R.10): V =

∞

n (t)vn (r ),

(R.15)

n=1

where vn (r ) is the full set of non-trivial solutions to the eigenvalue problem1 : d2 v + a 2 λ2 v = 0 dr 2

(R.16)

  ⎫ v|r =0 = vr + h 0 v r =1 = 0 ⎬ − = v| + v|r =r , r =r w     w  ⎭ kw rw vr − v r =rw− = k f rw vr − v r =rw+

(R.17)

with the boundary conditions

⎧ ⎨ cw ρw ≡ aw when 0 ≤ r ≤ rw 1 k a = √ =  c f wρ f ⎩ κ ≡ a f when rw < r ≤ 1. kf

(R.18)

√ Note that λ has dimensions of length/ time in this appendix. The general solution to (R.16) can be written as:

v(r ) =

A1 cos(λaw r ) + B1 sin(λaw r ) when 0 ≤ r ≤ rw A2 cos(λa f r ) + B2 sin(λa f r ) when rw < r ≤ 1.

(R.19)

Note that: A2 cos(λa f r ) + B2 sin(λa f r ) =

 A22

+

B22



A2 −1 .(R.20) sin λa f r + tan B2

To satisfy the first equation in System (R.17) A1 should be equal to zero. The introduction of new coefficients:

  A2 B = B1 , A = A22 + B22 and β = tan−1 B2 allows us to present (R.19) as:

The term a 2 was not considered in Eq. (A1.20) of [1]; this led to the situation in which n (t) depended on r via κ which is not compatible with the original assumption used in presentation (R.15).

1

570

Appendix R: Derivation of Expression (5.15)

v(r ) =

when 0 ≤ r ≤ rw B sin(λaw r ) A sin(λa f r + β) when rw < r ≤ 1.

(R.21)

The following formula follows from the middle equation in (R.17): B sin(λaw rw ) = A sin(λa f rw + β).

(R.22)

This equation is satisfied when: B=

1 1 , and A = . sin(λaw rw ) sin(λa f rw + β)

β follows from the flux conservation at the interface described in the last equation in (R.17): β = cot −1



 k f − kw k w aw + cot (aw λrw ) + iπ − a f λrw , k f a f rw λ k f a f

(R.23)

where i = 0, 1, 2, 3, .....2 . The focus is on i = 0. The values of v are the same for other i. A countable set of positive eigenvalues λn is obtained from the boundary condition at r = 1 (the first equation in (R.17)): λn a f cos(λn a f + β) + h 0 sin(λn a f + β) = 0.

(R.24)

The values of v for λ = λn are known as eigenfunctions and referred to as vn . For homogeneous droplets, when β = 0, Eq. (R.24) is equivalent to Eq. (2.42) if λn a f in (R.24) is replaced with λ. The formulae for vn for λn inferred from (R.24) follow from (R.21), remembering the expressions for A and B:  vn (r ) =

sin(λn aw r ) sin(λn aw rw ) sin(λn a f r +βn ) sin(λn a f rw +βn )

when 0 ≤ r ≤ rw when rw < r ≤ 1,

where βn is β(λ = λn ). Functions vn (r ) are orthogonal with weight (see [3])

b= Thus



kw aw2 = cw ρw when 0 ≤ r ≤ rw k f a 2f = c f ρ f when rw < r ≤ 1. 1

vn (r )vm (r )bdr = δnm ||vn ||2 ,

0 2

For homogeneous droplets, with the same properties of water and fuel, β = 0.

(R.25)

Appendix R: Derivation of Expression (5.15)

where δnm =

571

when n = m when n = m.

1 0

The norm of vn with weight b can be found as:

1

||vn ||2 =

vn2 bdr

0



rw

=



0

=

sin(λn aw r ) sin(λn aw rw )

cw ρw 2 sin (λn aw rw )



rw

2



1

cw ρw dr +



rw

sin(λn a f r + βn ) sin(λn a f rw + βn )

cfρf 2 sin (λn a f rw + βn )

sin2 (λn aw r )dr +

0



1

2 c f ρ f dr

sin2 (λn a f r + βn )dr

rw

√   sin(2aw λn rw ) a w λn r w cw ρw kw − = 2 4 λn sin2 (λn aw rw )  +

cf ρf kf

λn sin2 (λn a f rw + βn )

 a f λn (1 − rw ) sin(2λn a f + 2βn ) − sin(2λn a f rw + 2βn ) − . 2 4

(R.26) The orthogonality of functions vn with weight b allows us to use the expansions: ∞

f (r ) ≡

−r = f n (t)vn (r ), 1 + h0 n=1

F0 (r ) ≡ Rd r T0 (Rd r ) =

∞

qn (t)vn (r ),

(R.27)

(R.28)

n=1 ∞

˜ )= Rd3 r P(r

pn (t)vn (r ),

(R.29)

n=1

fn = 1 = ||vn ||2

rw 0

1 ||vn ||2



1

f (r )vn (r )bdr

0

# 1 −r sin(λn aw r ) −r sin(λn a f r + βn ) 2 2 kw aw dr + k f a f dr 1 + h 0 sin(λn aw rw ) rw 1 + h 0 sin(λn a f rw + βn )

1 qn = ||vn ||2

0

(R.30) 1

F0 (r )vn (r )bdr

572 =

Appendix R: Derivation of Expression (5.15) 1 ||vn ||2

rw 0

Rd r T0 (Rd r )

# 1 sin(λn a f r + βn ) sin(λn aw r ) 2 dr + k w aw k f a 2f dr Rd r T0 (Rd r ) sin(λn aw rw ) sin(λn a f rw + βn ) rw

1 pn = ||vn ||2 =

1 ||vn ||2

0

rw

˜ ) Rd3 r P(r

(R.31)



1 0

˜ )vn (r )bdr Rd3 r P(r

sin(λn aw r ) kw aw2 dr + sin(λn aw rw )



1

rw

˜ ) Rd3 r P(r

 sin(λn a f r + βn ) k f a 2f dr . sin(λn a f rw + βn )

(R.32)

Having substituted (R.15) into (R.10) and remembering (R.16) we obtain: ∞  n=1

Rd2

 ∞ d + λ2n vn (r ) = pn vn (r ). dt n=1

(R.33)

This equation is satisfied if, and only if, the coefficients on the two sides are equal. Thus: d + λ2n = pn . Rd2 (R.34) dt Equation (R.34) is solved for the set of countable values of λ, leading to discrete values of ( n ) subject to the initial condition: n (t = 0) = qn + f n μ0 .

(R.35)

Condition (R.35) follows from (R.11), (R.15), (R.28) and (R.29). If pn is constant during the timestep, the solution to (R.34) can be presented as: 

2 

2  pn λ t λ t . n (t) = (qn + f n μ0 ) exp − n2 + 2 1 − exp − n2 λn Rd Rd

(R.36)

Keeping in mind the definitions of u, F and V , the following final formula for T is obtained:  ∞ 1 μ0 r T = n (t)vn (r ) + . (R.37) Rd r n=1 1 + h0 This expression is the same as (5.15).

Appendix R: Derivation of Expression (5.15)

573

Comparison of Expression (R.37) with Expression (16) of [2] for Homogeneous Droplets To compare the predictions of the model described above and the model developed in [2], it is assumed that the properties of water and fuel are the same: kw = k f ;

κw = κ f ; a w = a f .

In what follows it is assumed that all properties are those of fuel. Note that ρl L f R˙ d μ0 r = Tg + ≡ Teff , Rd (1 + h 0 ) h

(R.38)

(R.39)

where Teff is used in Eq. (16) of [2]. Having substituted (R.38) into (R.23) it is obtained that β = 0. In this case, the expressions for the eigenfunctions vn (see Expression (R.25)) are presented as: vn (r ) =

sin(λ˜ n r ) sin(λ˜ n rw )

when

0 ≤ r ≤ 1,

(R.40)

where λ˜ n = a f λn . Remembering that β = 0, Eq. (R.24) for eigenvalues can be written as: λ˜ n cos λ˜ n + h 0 sin λ˜ n = 0. (R.41) This equation is identical to Eq. (A.3) of [2]. The substitution of (R.40) into (R.26) leads to the following formula:

1

||vn ||2 = 0

cfρf = sin2 (λ˜ n rw )



1

sin (λ˜ n r )dr = 2

0

vn2 c f ρ f dr  c f ρ f ||vn ||2 2004 , sin2 (λ˜ n rw )

(R.42)

 where ||vn ||2 2004 is the value of ||vn ||2 obtained in [2]. Having substituted (R.40) into (R.30)–(R.32) we obtain, remembering (R.42): 1 −r 1 vn (r )bdr fn = 2 ||vn || 0 1 + h 0 =

1 cfρf −r sin2 (λ˜ n rw )  sin(λ˜ n r )dr c f ρ f ||vn ||2 2004 sin(λ˜ n rw ) 0 1 + h 0 = sin(λ˜ n rw ) f n |2004 ,

(R.43)

574

Appendix R: Derivation of Expression (5.15)

1 qn = ||vn ||2 pn =



1 ||vn ||2

1

Rd r T0 (Rd r )vn (r )bdr = Rd sin(λ˜ n rw ) qn |2004 ,

(R.44)

0



1 0

˜ )vn (r )bdr = Rd3 sin(λ˜ n rw ) pn |2004 , Rd3 r P(r

(R.45)

Having substituted (R.43)–(R.45) into (R.36) we obtain:  !   κ f λ˜ 2n t n (t) = Rd sin(λ˜ n rw ) qn |2004 + μ0 f n |2004 exp − 2 Rd +Rd2

pn |2004 κ f λ˜ 2n

κ f λ˜ 2 t 1 − exp − 2n Rd

!! .

(R.46)

Having substituted (R.46) and (R.40) into (R.37) Eq. (16) of [2] is obtained, assuming that μ0 is constant during the timesteps and remembering that κ, introduced in [2], is equal to κ used in this appendix divided by Rd2 .

References 1. Nissar Z, Rybdylova O, Sazhin SS, Heikal M, Aziz ARBA, Ismael MA (2020) A model for puffing/microexplosions in water/fuel emulsion droplets, Int. J Heat Mass Transfer 149:119208. 2. Sazhin SS, Krutitskii PA, Abdelghaffar WA, Sazhina EM, Mikhalovsky SV, Meikle ST, Heikal MR (2004) Transient heating of diesel fuel droplets, Int. J Heat Mass Transfer 47:3327–3340. 3. Sazhin SS, Bar-Kohany T, Nissar Z, Antonov D, Strizhak PA, Rybdylova O (2020) A new approach to modelling micro-explosions in composite droplets. Int. J Heat Mass Transfer 161:120238.

Appendix S

Solution of Equation (5.40)

In this Appendix, the details of the solution of Eq. (5.40) (Yli (t, R)) for t ≥ 0 and Rw ≤ R ≤ Rd are presented following [4]. Let us rewrite Condition (5.41) as:

  αm ∂Yli − Yli  = −αm εi (t)Dl−1 . ∂R Dl R=Rd

(S.1)

We look for a solution to Eq. (5.40) as: Yli (t, R) = y(t, R) + ε(t),

(S.2)

where subscript i at y and ε is hereafter omitted. Having substituted (S.2) into Eq. (5.40) and using (S.1), (5.42) and (5.43) we can rewrite this equation and the corresponding boundary and initial conditions as: ∂y = Dl ∂t



∂2 y 2 ∂y + ∂ R2 R ∂R

 −

dε(t) , dt

 ∂y αm  − y = 0, ∂R Dl  R=Rd −0

(S.3)

(S.4)

 ∂ y  = 0, ∂ R  R=Rw +0

(S.5)

y(t = 0) = Yli0 (R) − ε(0) ≡ Yli0 (R) − ε0 .

(S.6)

Using new variable R = R − Rw

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

(S.7)

575

576

Appendix S: Solution of Equation (5.40)

and ignoring the contribution of the term dε(t) in Eq. (S.3)3 , we can rewrite (S.3)–(S.6) dt as:

2  ∂ y ∂y 2 ∂y = Dl + , (S.8) ∂t ∂R 2 R + Rw ∂R

 ∂y αm  y  = 0, − ∂R Dl R =Rd −Rw

(S.9)

 ∂ y  = 0, ∂R R =0

(S.10)

y|t=0 = Yli0 (R + Rw ) − ε(0) ≡ Yli0 (R + Rw ) − ε0 .

(S.11)

Using new variable u(t, R) = y(t, R)(R + Rw ) allows us to rewrite (S.8)–(S.11) as: ∂ 2u ∂u = Dl , ∂t ∂R 2

∂u − ∂R



αm 1 + Rd D

Rw

   u 

R =Rd −Rw

(S.12)

= 0,

  ∂u − u  = 0, ∂R R =0

u(R, 0) = (R + Rw ) (Yli0 (R + Rw ) − ε0 ) .

(S.13)

(S.14) (S.15)

We look for a solution to Eq. (S.12) as: u=

∞

n (t)vn (R),

(S.16)

n=0

where vn (R) is the full set of non-trivial solutions to the equation: ∂ 2v + pv = 0, ∂R 2 3

(S.17)

This assumption is consistent with other assumptions made in our analysis, remembering that the final solution will be used during short time steps in the numerical codes, including the assumption that Rd = const during individual time steps. The validity of this assumption was confirmed using the numerical analysis of the evaporation of multicomponent droplets without water sub-droplets (Rw = 0) taking and not taking into account this term in the analytical solution [3].

Appendix S: Solution of Equation (5.40)

577

with the boundary conditions:

∂v − ∂R



1 αm + Rd D

Rw

   v 

R =Rd −Rw

= 0,

(S.18)

  ∂v = 0. − v  ∂R R =0

(S.19)

Note that p in this approach is dimensional and has units of 1/m2 . Equation (S.17) with boundary conditions (S.18) and (S.19) is the Sturm-Liouville problem. The cases when p < 0, p = 0, and p > 0 are considered separately.

Sturm-Liouville problem for p < 0 Assuming that p = − (λ/Rd )2 , the general solution to Eq. (S.17) is presented as

v = A sinh

λR Rd



+ B cosh

λR Rd

 .

(S.20)

Note that λ is dimensionless. Having substituted (S.20) into (S.18) and (S.19) we obtain: 

λ A cosh Rd



λ Rd



− K sinh

λ Rd





λ +B sinh Rd



λ Rd



− K cosh

λ Rd

 = 0,

(S.21) A

λRw − B = 0, Rd

(S.22)

where  and K are defined by Formulae (5.47). System (S.21) and (S.22) has a non-trivial solution if, and only if, its determinant is equal to zero:

(λ) =

R w λ2 −K Rd2



sinh

λ Rd



λ + (1 − Rw K ) cosh Rd



λ Rd

 = 0. (S.23)

Equation (S.23) can be simplified to

tanh

λ Rd

 =

λ(1 − Rw K )  . 2 Rd K − RRw λ2 d

Equation (S.24) can be further rearranged to

(S.24)

578

Appendix S: Solution of Equation (5.40)

Rw λ˜ 2 F (λ˜ ) ≡ K − 

! tanh λ˜ = λ˜ (1 − Rw K ).

(S.25)

˜ where λ˜ = λ/Rd . The graphical solution to Eq. (S.25) is the intersection of F (λ) with the straight line described by the right-hand side of (S.25). The details of this solution are shown in Appendix U. As follows from the analysis presented in Appendix U, Eq. (S.25) has only one solution for the chosen values of parameters. In the limit Rw = 0 Eq. (S.25) reduces to Eq. (L.13). Solution λ˜ > 0 to Eq. (S.25) is referred to as λ˜ 0 which corresponds to λ0 . Assuming that A = 1 we have from Eq. (S.22) that B = λRw /Rd = λ˜ Rw /. For λ = λ0 , Solution (S.20) is presented as:

v0 = sinh

λ0 R Rd

 +



λ0 R w λ0 R , cosh Rd Rd

(S.26)

where λ0 = λ˜0 Rd / is obtained from (S.25). Note that the assumption A = 1 does not affect the generality of the solution since the normalisation of u is obtained when calculating . Formula (S.26) can be presented in an alternative form:

v0 = AY 0 sinh

 λ0 R + ηY 0 . Rd

(S.27)

To obtain explicit expressions for coefficients AY and ηY , (S.27) is rewritten as

λ0 R v0 = AY 0 sinh Rd





λ0 R cosh ηY 0 + AY 0 cosh Rd

 sinh ηY 0 .

(S.28)

λ0 R w . Rd

(S.29)

Comparing (S.26) and (S.28) we obtain: cosh ηY 0 =

1 λ0 R w ; sinh ηY 0 = ; AY 0 Rd A Y 0

tanh ηY 0 =

Remembering that

cosh2 ηY 0 − sinh2 ηY 0 =

1 AY 0

+

we obtain: AY 0 =

1−

2

−

λ0 R w Rd

λ0 R w Rd A Y 0

2 =1

2 .

(S.30)

Appendix S: Solution of Equation (5.40)

579

Sturm-Liouville Problem for p = 0 For p = 0 the general solution to Eq. (S.17) can be presented as v = AR + B.

(S.31)

Having substituted (S.31) into (S.18) and (S.19) we obtain: A(1 − K ) − BK = 0,

A Rw − B = 0.

(S.32)

The determinant of this system is equal to zero if, and only if, K ( + Rw ) = Rd K = 1,

(S.33)

which is not possible since K > 1/Rd .

Sturm-Liouville Problem for p > 0 Assuming that p = (λ/Rd )2 , the general solution to Eq. (S.17) is presented as

v = A sin

λR Rd



+ B cos

λR Rd

 .

(S.34)

Having substituted (S.34) into (S.18) and (S.19) we obtain: 

λ A cos Rd



λ Rd



− K sin

λ Rd







 λ λ λ +B − − K cos = 0, sin Rd Rd Rd (S.35) λRw − B = 0. (S.36) A Rd

System (S.35) and (S.36) has a non-trivial solution if, and only if, its determinant is equal to zero:

(λ) = −

R w λ2 +K Rd2



sin

λ Rd

 +

λ (1 − Rw K ) cos Rd



λ Rd

 = 0. (S.37)

Equation (S.37) is simplified to

tan

λ Rd

 =

λ(1 − Rw K )  . 2 Rd K + RRw λ2 d

(S.38)

580

Appendix S: Solution of Equation (5.40)

Equation (S.38) can be further rearranged to R λ˜ 2 ˜ ≡ K + w G (λ) 

! ˜ − Rw K ). tan λ˜ = λ(1

(S.39)

The graphical solutions to this equation are the intersections of G (λ˜ ) with the straight line described by the right-hand side of (S.39). The details of this solution are shown in Appendix U. For Rw = 0 Eq. (S.39) reduces to Eq. (L.15). Equation (S.39) has a countable number of solutions (eigenvalues) 0 < λ 1 < λ2 < · · · < λn < · · ·

(S.40)

For each of these eigenvalues, Solution (S.34) is presented as:

vn = An sin

λn R Rd



+ Bn cos



λn R Rd

,

n ≥ 1.

(S.41)

As in the case of p < 0 we assume that An = 1, which implies that Bn = Rw λn /Rd , and rewrite (S.41) as:

λn R vn = sin Rd





R w λn λn R + . cos Rd Rd

(S.42)

Formula (S.42) is presented in an alternative form:  λn R vn = AY n sin + ηY n . Rd

(S.43)

To obtain explicit expressions for coefficients AY n and ηY n (S.43) is rewritten as

vn = AY n sin

λn R Rd



cos ηY n + AY n cos

λn R Rd

 sin ηY n .

(S.44)

R w λn . Rd

(S.45)

Comparing (S.42) and (S.44) we obtain: cos ηY n =

1 λn R w , sin ηY n = , AY n A Y n Rd

tan ηY n =

Remembering that

cos ηY n + sin ηY n = 2

2

1 AY n

2

+

R w λn A Y n Rd

2 = 1,

Appendix S: Solution of Equation (5.40)

581

+

we obtain AY n =

1+

R w λn Rd

2 .

(S.46)

For Rw = 0 Eqs. (S.42) and (S.43) reduce to Eq. (L.14).

Orthogonality and Norms of vn (n ≥ 0) The orthogonality of vn for R in the range 0 to Rd − Rw was proven earlier for the case when Rw = 0. Also, it follows from the general theory of the Sturm-Liouville boundary value problem [1]. Completeness of these functions follows from the theory of the Sturm-Liouville boundary value problem [1] remembering that we have found all solutions to this problem. These are the norms of v0 and vn (n ≥ 1):

   2 2 2 2 λ0 R v0 (R) = v0 (R) dR = AY 0 sinh + ηY 0 dR Rd 0 0

A2 = Y0 2

=

=

1−



λ0 R w Rd

2



2 1−



λ0 R w Rd

    λ R  Rd Rd 0 sinh 2 + ηY 0 − sinh 2ηY 0 −  2λ0 Rd 2λ0

2 ⎡

  ⎤ tanh λR0d + ηY 0 R tanh η R d Y0 ⎣ d  − − ⎦ λ0 1 − tanh2 λ0  + η λ0 1 − tanh2 ηY 0 Y 0 Rd

2

=

   λ R  Rd 0 sinh 2 + ηY 0 − R  2λ0 Rd 0

1−



λ0 R w Rd

2

vn (R) = 2



0

A2 = Yn 2

2 ⎛ ⎜ ⎝

⎞ K Rw ⎟ 2 − ⎠ ,  2 −  K 2 − Rλ0d 1 − λ0RRd w

vn2 (R) dR

=

A2Y n





sin 0

2

λn R + ηY n Rd



 λ R  Rd n  R− sin 2 + ηY n  2λn Rd 0

 dR

(S.47)

582

Appendix S: Solution of Equation (5.40)

=

=

=

1+



λn R w Rd

2



2 1+



λn R w Rd

2 ⎡

  ⎤ tan λRnd + ηY n tan η R R d Yn ⎦ ⎣ − d  + λn 1 + tan2 λn  + η λn 1 + tan2 ηY n Yn Rd

2 1+



   λ R  Rd Rd n + − sin 2 + ηY n sin 2ηY n 2λn Rd 2λn

λn R w Rd

2

2

 +

 Rw K . − 1 + (λn Rw /Rd )2 K 2 + (λn /Rd )2

(S.48)

In the limit when Rw = 0 Expressions (S.47) and (S.48) reduce to Expressions (K.17) and (K.22), respectively.

Calculation of Coefficients n in Expansion (S.16) Having substituted (S.16) into (S.12) we obtain: ∞

n (t)vn (R) = Dl

n=0

∞

n (t)vn (R),

(S.49)

n=0

where n =

d n ; dt

vn (R) =

d2 vn . dR 2

Remembering that ∂ 2 v0 = ∂R 2



λ0 Rd

2

∂ 2 vn =− ∂R 2

v0 ,



λn Rd

2 vn ,

n ≥ 1,

(S.50)

and the uniqueness of the Fourier expansion (Eq. (S.49) is satisfied only when it is satisfied for each term), the following equations for n are obtained: 0 (t) n (t)

= Dl

= −Dl

λn Rd

λ0 Rd

2 0 (t),

(S.51)

2 n (t),

n ≥ 1.

(S.52)

To solve Eqs. (S.51) and (S.52) we need to find the initial conditions n (0) for n ≥ 0. To find n (0) we substitute (S.16) into (S.15) to obtain

Appendix S: Solution of Equation (5.40) ∞

583

n (0)vn (R) = (R + Rw ) (Yli0 (R + Rw ) − 0 ) .

(S.53)

n=0

Expanding the right-hand side of (S.53) into a Fourier series with respect to functions vn the latter equation can be rewritten as: ∞

∞

n (0)vn (R) =

n=0

where qin =

(qin − 0 Q n ) vn (R),

(S.54)

n=0

1 ||vn ||2





(R + Rw )Yli0 (R + Rw )vn (R)dR,

(S.55)

0



1 Qn = ||vn ||2



(R + Rw )vn (R)dR.

(S.56)

0

The details of the derivation of the explicit expressions for Q n are presented in Appendix T. Equation (S.54) is satisfied if, and only if, it is satisfied for each term in the Fourier series. Hence, (S.57) n (0) = qin − 0 Q n , n ≥ 0. Remembering (S.57), solutions to Eqs. (S.51) and (S.52) are presented as: 



0 (t) = exp Dl 



n (t) = exp −Dl

λn Rd

λ0 Rd

2  t [qi0 − 0 Q 0 ] ,

2  t [qin − 0 Q n ] , n ≥ 1.

(S.58)

(S.59)

Having substituted (S.58) and (S.59) into (S.16), using the definitions of u and y the final solution to Eq. (5.40), subject to boundary conditions (5.41)–(5.42) and initial condition (5.43), is obtained as Expression (5.44). In the limit when Rw = 0, R = R and Expression (5.44) reduces to Expression (4.21). Solution (5.44) could also be obtained from the general solution to Eq. (S.12) subject to boundary and initial conditions (S.13)–(S.15) in the form [2]: u(R, t) = 0

where



(R, ξ, t)g(ξ ) dξ,

(S.60)

584

Appendix S: Solution of Equation (5.40)

(R, ξ, t) =

∞ n=0

1 vn (R)vn (ξ ) exp (Dpn t) , vn 2

(S.61)

  g(R) = (R + Rw ) Yli0 (R + Rw ) − 0 , vn are the eigenfunctions corresponding to eigenvalues pn of the Sturm–Liouville problem (see Eq. (S.17)) with the corresponding boundary conditions.

References 1. Boyce WE, Di Prima RC. (2017) Elementary Differential Equations and Boundary Value Problems. 11th Edition. Wiley and Sons Inc. 2. Polyanin A. (2002) Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton. 3. Rybdylova O, Poulton L, Al Qubeissi M, Elwardany AE, Crua C, Khan T, Sazhin SS. (2018) A model for multi-component droplet heating and evaporation and its implementation into ANSYS Fluent. Int. Commun. Heat Mass Transfer 90:29–33. 4. Sazhin SS, Shchepakina E, Sobolev VA, Antonov D, Strizhak PA. (2022) Puffing/microexplosion in composite multi-component droplets. Int. J Heat and Mass Transfer 184: 122210.

Appendix T

Calculations of Q n based on (S.56)

Having substituted (S.26) into (S.56) we obtain, following [1]: 1 Q0 = v0 2 −1 = v0 2



 



  λ0 R w λ0 R λ0 R  + sinh cosh (R + Rw ) dR Rd Rd Rd 0

λ  Rd (Rd − Rw ) 0 cosh (Rd − Rw ) + − λ0 Rd

Rd2 λ20

! − Rw Rd sinh

λ

0

Rd

#  (Rd − Rw ) .

(T.1) Having substituted (S.42) into (S.56) we obtain, following [1]: Qn = 1 = vn 2



1 vn 2

 



 λn R w λn R λn R  + R + Rw dR sin cos Rd Rd Rd 0

λ  Rd (Rd − Rw ) n − cos (Rd − Rw ) + λn Rd

! # λ  Rd2 n + Rw Rd sin (Rd − Rw ) , λ2n Rd

(T.2) where n ≥ 1. In the limit Rw = 0, Expressions (T.1) and (T.2) reduce to Expression (K.28). When reducing (T.1) and (T.2) to the latter expression we took into account that for Rw = 0: λ0 cosh λ0 = K sinh λ0 , Rd λn cos λn = K sin λn Rd

n ≥ 1.

These formulae follow from boundary condition (S.13).

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

585

586

Appendix T: Calculations of Q n Based on (S.56)

Reference 1. Sazhin SS, Shchepakina E, Sobolev VA, Antonov D, Strizhak PA. (2022) Puffing/microexplosion in composite multi-component droplets. Int. J Heat and Mass Transfer 184: 122210.

Appendix U

Graphical Solutions of (5.45) and (5.46)

The aim of this Appendix is to present graphical illustrations of the solutions of Eqs. (5.45) and (5.46) for typical values of parameters for kerosene droplet heating and evaporation at atmospheric pressure and ambient temperature 473 K, following [1]. The initial droplet and water sub-droplet radii were taken equal to 0.85 and 0.395 mm, which corresponds to a water volume fraction equal to 10%. Thus  = 0.455 × 10−3 m. As in Sect. 5.7 it was assumed that Dl = 1.1043 × 10−9 m2 /s. Focusing on droplet evaporation at t = 0 it was assumed that αm = 8.2254 × 10−8 m/s, which leads to K = 1250.9562 1/m. For these values of input parameters, the plots of the left (F ) and right-hand sides of (5.45) versus λ are shown by dotted and solid curves in Fig. U.1a, respectively. Note that the right-hand side of this equation is a linear function of λ. As can be seen in this figure, the curves intersect at one point; at λ = 0.4617. Plots similar to those shown in Fig. U.1a, but ignoring the contribution of water (Rw = 0) are shown in Fig. U.1b. Comparing Figs. U.1a, b, it can be seen that the curves are strongly affected by the value of Rw , but the values of λ at which they intersect are rather close. They intersect at λ = 0.4387 for the case shown in Fig. U.1b. Using the same values of input parameters, the plots of the left (G ) and righthand sides of (5.46) versus λ are shown by dotted and solid curves in Figure U.2a, respectively. As in the case of Fig. U.1a, the right-hand side of (5.46) is a linear function of λ. As can be seen in this figure, in contrast to Fig. U.1a, the curves intersect at an infinite number of points: λ1 = 6.1712, λ2 = 11.8939 etc. Plots similar to those shown in Fig. U.2a, but ignoring the contribution of water (Rw = 0) are shown in Fig. U.2b. In contrast to the cases shown in Fig. U.1, the values of λ at which they intersect are strongly affected by the presence of water. In the case shown in Fig. U.2b the curves intersect at λ1 = 4.4793, λ2 = 7.7172 etc.

Reference 1. Sazhin SS, Shchepakina E, Sobolev VA, Antonov D, Strizhak PA. (2022) Puffing/microexplosion in composite multi-component droplets. Int. J Heat and Mass Transfer 184: 122210.

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

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Fig. U.1 The plots of the left (dashed) and right (solid) hand sides of Eq. (5.45) versus λ for typical kerosene droplets for the cases when Rw = 0.395 mm (a) and Rw = 0 (b). Reprinted from [1]. Copyright Elsevier (2022)

Appendix U: Graphical Solutions of (5.45) and (5.46)

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Fig. U.2 The same as Fig. U.1 but for Eq. (5.46). Reprinted from [1]. Copyright Elsevier (2022)

Appendix V

Verification of the Numerical Code

Analytical solution (5.44) in the limit when Rw = 0 was implemented into ANSYS Fluent via User Defined Functions (UDF) and used at each time step of the calculations performed by this software [1]. In what follows, the results of these calculations are compared with those predicted by the numerical code described in Sect. 5.7 in the limit when Rw = 0 for heating/cooling and evaporation of acetone/ethanol mixture droplets, following [2]. The focus is on the case when the mass fractions of ethanol and acetone were 25 and 75%, respectively. The following input parameters were used in calculations: initial droplet temperature 305.65 K, initial droplet diameter 133.8 µm, ambient gas temperature 294.25 K, and droplet velocity (assumed to be constant) 12.75 m/s [1]. The time evolution of the surface, average and central temperatures of ethanol/ acetone droplets predicted by the customised version of ANSYS Fluent and the newly developed numerical code is presented in Fig. V.1. As follows from this figure the values of temperatures predicted by both codes are reasonably close. The deviation between the predicted temperatures does not exceed 0.2218%. This is comparable with the deviation between the predictions of the customised version of ANSYS Fluent and the in-house code in which the analytical solution for Rw = 0 was originally used (0.1636%) [1]. Possible reasons for this deviation are discussed in [1]. The results of calculation of the time evolution of mass fractions of acetone and ethanol for the same droplets as in Fig. V.1 are presented in Fig. V.2. As follows from the latter figure, the results predicted by both codes are reasonably close although the deviation between them is larger than in the case shown in Fig. V.1 (0.3226%). This deviation is comparable with the deviation between the predictions of the customised version of ANSYS Fluent and the in-house code in which the analytical solution for Rw = 0 was originally used (0.7933%) [1]. This allows us to conclude that our new numerical code is verified in the limiting case of Rw = 0.

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

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Fig. V.1 Time evolution of a 25% ethanol/75% acetone droplet surface (bottom curves), average (middle curves) and central (top curves) temperatures (Ts , Tav and Tc ) as predicted by the customised version of ANSYS Fluent (dashed curves) and the new numerical code (solid curves). Reprinted from [2]. Copyright Elsevier (2022)

Appendix V: Verification of the Numerical Code

593

Fig. V.2 Time evolution of surface mass fractions of ethanol (Yls,e ) and acetone (Yls,a ) for the same droplet as in Fig. V.1, predicted by the customised version of ANSYS Fluent (dashed curves) and the new numerical code (solid curves). Reprinted from [2]. Copyright Elsevier (2022)

References 1. Rybdylova O, Poulton L, Al Qubeissi M, Elwardany AE, Crua C, Khan T, Sazhin SS. (2018) A model for multi-component droplet heating and evaporation and its implementation into ANSYS Fluent. Int. Commun. Heat Mass Transfer 90:29–33. 2. Sazhin SS, Shchepakina E, Sobolev VA, Antonov D, Strizhak PA. (2022) Puffing/microexplosion in composite multi-component droplets. Int. J Heat and Mass Transfer 184: 122210.

Appendix W

Tikhonov Theorem

Tikhonov’s theorem [2, 3] allows the system order to be reduced based on traditional tools in the theory of singularly perturbed systems. In this section, this theorem is formulated, following [1], for a simplified case of an autonomous system with one small parameter: dx dτ dy ε dτ

= f (x, y)

(W.1)

= g(x, y),

(W.2)

where x(0) = x0 , y(0) = y0 , f : D → R m and g : D → R n are continuous vector functions, D is an open set in R n+m , (x0 , y0 ) ∈ D, and ε is a small positive parameter. Definitions. The vector differential equation dy = g(x, y) , d τ¯

(W.3)

where τ¯ = τ/ε, and x is a parameter, is called a boundary layer equation. The mdimensional surface S g(x, y) = 0 (W.4) is called a slow surface. It describes all steady states of the boundary layer Eq. (W.3). This surface is given as a graph of a function. Thus there exists a continuous mapping φ : X → R n from some compact set X in R m such that (x, φ(x)) ∈ D for all x ∈ X and S = {(x, y) : y = φ(x), x ∈ X }. Following Liapunov’s stability theory, referring to systems depending on parameters, the steady state y = φ(x) of (W.3) is called 1. Stable, if for any μ > 0 there exists such η that any solution y(τ¯ ) to (W.3) with y(0) − φ(x) < η may be extended on τ¯ > 0 and it satisfies y(τ¯ ) − φ(x) < μ. The steady state is asymptotically stable if it is stable and lim y(τ¯ ) = φ(x) for all solutions satisfying the condition y(0) − φ(x) < η.

τ¯ →∞

© The Editor(s) (if applicable) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. S. Sazhin, Droplets and Sprays: Simple Models of Complex Processes, Mathematical Engineering, https://doi.org/10.1007/978-3-030-99746-5

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2. Attractive, if a domain of attractivity exists i.e. some neighbourhood V of S such that any solution y(τ¯ ) to (W.3) with y(0) ∈ V may be extended for all τ¯ > 0 and lim y(τ ) = φ(x). τ¯ →∞

The domain of attractivity of the steady state y = φ(x) is uniform in X if there exists such a > 0 that for all x ∈ X , the ball B = {x ∈ R m : y − φ(x) ≤ a} with the centre in φ(x) with radius a, is the domain of attractivity for φ(x). The stable and attractive steady state is asymptotically stable. The following assumptions are made: (1). For all x ∈ X , the boundary layer Eq. (W.3) has a unique solution for a given initial value. (2). For all x ∈ X , y = φ(x) is an isolated root of Eq. (W.4), i.e. g(x, φ(x)) = 0, and there exists such positive number δ > 0 that the conditions x ∈ X , y − φ(x) < δ and y = φ(x) imply g(x, y) = 0. This does not mean that Eq. (W.4) has no other roots except φ(x). (3). For any x ∈ X , the point y = φ(x) is an asymptotically stable steady state of Eq. (W.3) and the domain of attractivity y = φ(x) is uniform with respect to X . (4). The system dx = f (x, φ(x)) (W.5) dτ with a given initial condition has a unique solution. (5). For any interior point x0 in X , the point y0 belongs to the domain of attractivity of the steady state y = φ(x). Remarks. Let us consider the initial value problem for the boundary layer equation: dy = g(x0 , y) y(0) = y0 , (W.6) d τ¯ where x0 is an arbitrary interior point in X . Let y¯ (τ¯ ) be the solution to this problem, function y¯ (τ¯ ) is defined for all τ¯ ≥ 0 and lim y¯ (τ¯ ) = φ(x0 ). τ¯ →∞

The reduction of Eq. (W.1) leads to the initial value problem dx = f (x, φ(x)) x(0) = x0 . dτ

(W.7)

Let x(τ ¯ ) be the solution to the reduced problem for τ ∈ I = [0, T ), where T ∈ (0, ∞] is the maximal interval where this solution is defined. The Tikhonov theorem is formulated as: If assumptions (1)–(5) are valid then the solution (x(τ, ε), y(τ, ε)) to the initial value problem (W.1)–(W.2)) exists in [0, T ] and the following conditions take place

Appendix W: Tikhonov Theorem

597

lim x(τ, ε) = x(τ ¯ ), 0 ≤ τ ≤ T ;

(W.8)

¯ )), 0 < τ ≤ T. lim y(τ, ε) = φ(x(τ

(W.9)

ε→0

ε→0

The convergence in (W.8) and (W.9) is uniform in the interval 0 ≤ τ ≤ T for x(τ, ε) and in any interval 0 < τ1 ≤ τ ≤ T for y(τ, ε).

References 1. Sazhin SS, Shchepakina EA, Sobolev VA (2010) Order reduction of a non-Lipschitzian model of monodisperse spray ignition. Mathematical and Computer Modelling 52:529–537. 2. Tikhonov AN (1952) Systems of differential equations with small parameters multiplying the derivatives. Matematicheskii Zbornik (in Russian) 31:575–586. 3. Vasil’eva AB, Butuzov VF, Kalachev LV (1985) The Boundary Function Method for Singular Perturbation Problems, SIAM Studies in Appl. Math. vol. 14.