Application of Control Volume Based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer [1 ed.] 0128141522, 9780128141526

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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER MOHSEN SHEIKHOLESLAMI

Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Islamic Republic of Iran

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright r 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-814152-6 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

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Contents Biography Preface

1. Detailed Explanation of Control Volume-based Finite Element Method

1

1.1 Introduction 1.2 The Discretization: Grid, Mesh, and Cloud 1.3 The Element and the Interpolation Shape Functions 1.4 Region of Support and Control Volume 1.5 Discretization and Solution References

1 1 3 4 5 9

2. Simulation of Vorticity Stream Function Formulation by Means of CVFEM 2.1 CVFEM Stream Function-Vorticity Solution for a Lid Driven Cavity Flow 2.2 CVFEM Stream Function-Vorticity Solution for Natural Convection References

20 30

33 33 37 63

5. Buongiorno Model for Nanofluid Treatment Using CVFEM 5.1 Introduction 5.2 Buongiorno Model for Nanofluid Flow and Heat Transfer Using Heatline Analysis 5.3 Two-phase Model for Nanofluid Natural Convection Heat Transfer

6.1 Introduction 6.2 Magnetic Nanofluid Mixed Convection Heat Transfer Treatment in the Presence of Variable Magnetic Field 6.3 Forced Convection of Nanofluid in a Porous Lid Driven Enclosure in the Presence of Lorentz Forces 6.4 Influence of Lorentz Forces on Nanofluid Flow Inside a Porous Enclosure With Moving Wall 6.5 Single-phase Model Simulation of Nanofluid Forced Convection Inside a Permeable Enclosure With Sinusoidal Wall References

15

3.1 Introduction 3.2 Simulation of Nanofluid Flow and Heat Transfer References

4.1 Introduction 4.2 Nanofluid Hydrothermal Analysis in a Complex Shaped Cavity 4.3 Natural Convection Heat Transfer in a Nanofluid Filled Enclosure With Elliptic Inner Cylinder 4.4 Nanofluid Free Convection Heat Transfer in a Tilted Cavity References

6. Nanofluid Forced and Mixed Convection Heat Transfer by Means of CVFEM

15

3. Various Application of Nanofluid for Heat Transfer Augmentation

4. Single-phase Model for Nanofluid Free Convection Heat Transfer by Means of CVFEM

5.4 MHD Natural Convection of Al2O3-water Nanofluid Considering Thermophoresis and Brownian Motion Effects References

ix xi

7. Effect of Uniform Lorentz Forces on Nanofluid Flow Using CVFEM 7.1 Introduction 7.2 Nanofluid Free Convection Heat Transfer in an Enclosure Between a Circular and a Sinusoidal Cylinder in the Presence of Magnetic Field 7.3 Influence of a Magnetic Field on Free Convection in an Inclined Half-annulus Enclosure Filled With Cu-water Nanofluid 7.4 MHD Nanofluid Convective Flow in an Inclined Enclosure With Sinusoidal Wall 7.5 MHD Nanofluid Flow in a Cavity With Heat Flux Boundary Condition References

73 73 73 77

8. Influence of Variable Lorentz Forces on Nanofluid Free Convection Using CVFEM

87 94

8.1 Introduction 8.2 Influence of External Variable Magnetic Field on Ferrofluid Flow and Convective Heat Transfer 8.3 Ferrofluid Flow and Heat Transfer in a Semiannulus Enclosure in the Presence of Thermal Radiation 8.4 Influence of Spatially Variable Magnetic Field on Ferrofluid Flow and Heat Transfer Considering Constant Heat Flux Boundary Condition 8.5 Effect of Space Dependent Magnetic Field on Free Convection of Fe3O4-Water Nanofluid

99 99 99 107

v

110 123

127 127

127 135 144

151 158

163 163

163

171 177 187 196

201 201 201 207

218 229

vi 8.6 Nonuniform Magnetic Field Effect on Nanofluid Hydrothermal Treatment Considering Brownian Motion and Thermophoresis Effects 8.7 External Magnetic Source Effect on Water Based Nanofluid Convective Heat Transfer 8.8 Nanofluid Transportation in a Curved Cavity in the Presence of Magnetic Source 8.9 Ferrofluid Convective Heat Transfer Under the Influence of External Magnetic Source 8.10 Nanofluid Hydrothermal Treatment in a Cavity With Variable Magnetic Field 8.11 Magnetic Source Impact on Magnetic Nanofluid Convective Heat Transfer References

9. Nanofluid Forced Convective Heat Transfer in Presence of Variable Magnetic Field Using CVFEM 9.1 Introduction 9.2 Effect of Nonuniform Magnetic Field on Forced Convection Heat Transfer of Fe3O4-Water Nanofluid 9.3 Magnetic Nanofluid Forced Convective Heat Transfer in the Presence of Variable Magnetic Field Using Two-Phase Model 9.4 Forced Convection Heat Transfer in a Semiannulus Under the Influence of a Variable Magnetic Field 9.5 Flow and Convective Heat Transfer of a Ferronanofluid in a Double-Sided Lid-Driven Cavity With a Wavy Wall in the Presence of a Variable Magnetic Field References

10. Influence of Shape Factor on Nanofluid Heat Transfer Improvement Using CVFEM 10.1 Introduction 10.2 Forced Convection of Nanofluid in the Presence of Constant Magnetic Field Considering Shape Effects of Nanoparticles 10.3 Effect of Shape Factor on Fe3O4-Water Nanofluid Forced Convection in the Presence of External Magnetic Field 10.4 Magnetic Source Effect on Nanofluid Flow in Porous Medium Considering Various Shape of Nanoparticles 10.5 Magnetohydrodynamic CuO-Water Transportation Inside a Porous Cavity Considering Shape Factor Effect 10.6 Magnetic Field Influence on CuO-H2O Nanofluid Convective Flow in a Permeable Cavity Considering Various Shapes for Nanoparticles References

11. Electrohydrodynamic Nanofluid Natural Convection Using CVFEM 11.1 Introduction 11.2 Electrohydrodynamic Free Convection Heat Transfer of a Nanofluid in a Semiannulus Enclosure With a Sinusoidal Wall

CONTENTS

236 241

11.3 Free Convection of Nanofluid Under the Effect of Electric Field in a Porous Enclosure 11.4 Nanofluid Natural Convection Under the Influence of Coulomb Force in a Porous Enclosure References

379 387 395

252 260 273 281 288

293 293 293

12. Forced Convection of Nanofluid in Existence of Electric Field Using CVFEM 12.1 Introduction 12.2 EHD Nanofluid Force Convective Heat Transfer Considering Electric Field Dependent Viscosity 12.3 Electrohydrodynamic Nanofluid Hydrothermal Treatment in an Enclosure With Sinusoidal Upper Wall 12.4 Effect of Electric Field on Hydrothermal Behavior of Nanofluid in a Complex Geometry 12.5 Effect of Coulomb Forces on Fe3O4-H2O Nanofluid Thermal Improvement 12.6 Active Method for Nanofluid Heat Transfer Enhancement by Means of EHD References

399 399

399

405 414 419 427 437

298 304

312 323

327 327

327

335 343

348

360 368

373 373

373

13. Darcy Model for Nanofluid Flow in a Porous Media by Means of CVFEM 13.1 Introduction 13.2 Magnetohydrodynamic CuO-Water Nanofluid in a Porous Complex Shaped Enclosure 13.3 Analysis of Water-Based Nanofluid Flow and Heat Transfer Due to Magnetic Field in a Porous Enclosure 13.4 Magnetohydrodynamic Nanofluid Convection in a Porous Enclosure Considering Heat Flux Boundary Condition 13.5 Effect of Lorentz Forces on Nanofluid Flow in a Porous Cylinder Considering Darcy Model References

14. Non-Darcy Model for Nanofluid Hydrothermal Treatment in a Porous Medium Using CVFEM 14.1 Introduction 14.2 MHD Nanofluid Free Convective Heat Transfer in a Porous Tilted Enclosure 14.3 Magnetic Nanofluid Flow in a Porous Cavity Using CuO Nanoparticles 14.4 Nanofluid Transportation in Porous Media Under the Influence of External Magnetic Source 14.5 Nanofluid Convective Heat Transfer Intensification in a Porous Circular Cylinder 14.6 Convective Flow of Nanofluid Inside a Lid-Driven Porous Cavity 14.7 Nanofluid Heat Transfer in a Permeable Enclosure in Presence of Variable Magnetic Field References

441 441 441

447

459 468 479

483 483 483 488 499 513 521 532 544

CONTENTS

15. Thermal Nonequilibrium Model for Nanofluid Flow in a Porous Enclosure by Means of CVFEM 15.1 Introduction 15.2 Simulation of Nanofluid Flow Inside a Porous Enclosure via Nonequilibrium Model 15.3 Nanofluid Free Convection in a Porous Cavity Considering the Two-Temperature Model 15.4 Nanofluid Flow in a Porous Sinusoidal Cavity Considering Thermal Nonequilibrium Model References

16. Nonuniform Magnetic Field Effect on Nanofluid Convective Flow in a Porous Cavity 16.1 Introduction 16.2 Effect of Variable Magnetic Field on Nanofluid Convective Heat Transfer in a Porous Curved Enclosure 16.3 Nanofluid Natural Convection in Porous Media in the Presence of a Magnetic Source 16.4 Heat Transfer of Fe3O4-Water Nanofluid in a Permeable Medium With Thermal Radiation 16.5 Effect of External Magnetic Source on Fe3O4-H2O Nanofluid Behavior in a Permeable Cavity Considering Shape Effect References

17. Thermal Radiation Influence on Nanofluid Flow in a Porous Medium in the Presence of Coulomb Forces Using CVFEM 17.1 Introduction 17.2 Combined Natural Convection and Radiation Heat Transfer of Nanofluid Under the Impact of Electric Field in a Porous Cavity 17.3 Nanofluid Free Convection Under the Influence of an Electric Field in a Porous Wavy Enclosure 17.4 EHD Nanofluid Flow in a Porous Medium Considering Radiation Parameter References

18. Influence of Electric Field on Forced Convection of Nanofluid in a Porous Medium by Means of CVFEM 18.1 Introduction

547 547 547 558 568 577

581 581

581 589

18.2 EHD Nanofluid Flow in a Permeable Enclosure With Sinusoidal Wall 18.3 Effect of Shape Factor on Electrohydrodynamic Nanofluid Flow in a Porous Medium 18.4 Effect of Elective Field on Nanofluid Flow in a Porous Lid Driven Cavity in Existence of Electric Field References

19. Nanofluid Heat Transfer Enhancement in Presence of Melting Surface Using CVFEM 19.1 Introduction 19.2 Melting Heat Transfer Influence on Nanofluid Flow Inside a Cavity in the Presence of a Magnetic Field 19.3 Simulation of CuO-Water Nanofluid Heat Transfer Enhancement in the Presence of a Melting Surface 19.4 CuO-Water Nanofluid Magnetohydrodynamic Natural Convection Inside a Sinusoidal Annulus in the Presence of Melting Heat Transfer 19.5 MHD Nanofluid Natural Convection Inside a Half Annulus With Melting Surface References

vii 649 655

662 670

675 675

675 682

691 697 704

595

606 620

623 623

623 629 637 644

649 649

20. Nanofluid Convective Heat Transfer Considering Magnetic Field Dependent (MFD) Viscosity by Means of CVFEM 20.1 Introduction 20.2 Natural Convection of Magnetic Nanofluid Considering MFD Viscosity Effect 20.3 Magnetic Field Influence on Nanofluid Thermal Radiation in a Cavity With Tilted Elliptic Inner Cylinder 20.4 Thermal Radiation of Ferrofluid in Existence of Lorentz Forces Considering Variable Viscosity 20.5 Magnetic Nanofluid Natural Convection in Presence of Thermal Radiation Considering Variable Viscosity 20.6 Numerical Study of the Effect of Magnetic Field on Fe3O4-Water Ferrofluid Convection With Thermal Radiation References

Nomenclature Appendix: A CVFEM Code for Lid Driven Cavity Index

707 707 707

713 723 733

737 746

751 753 761

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Biography

Mohsen Sheikholeslami Kandelousi (M. Sheikholeslami) Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Islamic Republic of Iran Dr. Mohsen Sheikholeslami works in the Department of Mechanical Engineering, Babol Noshirvani University of Technology, Iran. He is the head of renewable energy systems and nanofluid applications in the heat transfer laboratory, Babol Noshirvani University of Technology. His research interests include nanofluid, CFD, simulation, mesoscopic modeling, nonlinear science, magnetohydrodynamic, ferrohydrodynamic, electrohydrodynamic, and heat exchangers. He has written several papers and books in various fields of mechanical engineering. He is also the first author of Applications of Nanofluid for Heat Transfer Enhancement, Application of Semi-Analytical Methods for Nanofluid Flow and Heat Transfer, Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method, and External Magnetic Field Effects on Hydrothermal Treatment of Nanofluid, published by Elsevier. According to Thomson Reuters’ report (Clarivate Analytics), he was selected as the Web of Science Highly Cited Researcher (Top 0.01%) in 2016 and 2017.

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Preface

In this book, I provide readers the new powerful numerical method, namely Control Volume-based Finite Element Method (CVFEM). This method uses the advantages of both the finite volume and finite element methods. It combines the flexibility of the finite element methods to discretize complex geometry with the conservative formulation of the finite volume methods, in which variables can be easily interpreted physically in terms of fluxes, forces, and sources. Applications of CVFEM in nanofluid simulation in the presence of various external forces are presented. This text is suitable for senior undergraduate students, postgraduate students, engineers, and scientists. In the first and second chapters, the basic idea of CVFEM is presented and then Navier
Stokes equations in the form of vorticity stream function formulation are solved. The third chapter of this book deals with the necessary fundamentals of nanotechnology. The various models for simulation of nanofluid are discussed. Homogeneous and Buongiorno models in various geometries are investigated in Chapter 4, Single-phase Model for Nanofluid Free Convection Heat Transfer by Means of CVFEM, and Chapter 5, Buongiorno Model for Nanofluid Treatment Using CVFEM. Chapter 6, Nanofluid Forced and Mixed Convection Heat Transfer by Means of CVFEM, deals with nanofluid forced and mixed convection heat transfer. Chapter 7, Effect of Uniform Lorentz Forces on Nanofluid Flow Using CVFEM, and Chapter 8, Influence of Variable Lorentz Forces on Nanofluid Free Convection Using CVFEM, give a complete account of uniform and nonuniform magnetic field effects on nanofluid natural convection heat transfer. Nanofluid forced convective heat transfer in the presence of a variable magnetic field is demonstrated in Chapter 9, Nanofluid Forced Convective Heat Transfer in Presence of Variable Magnetic Field Using CVFEM. Chapter 10, Influence of Shape Factor on Nanofluid Heat Transfer Improvement Using CVFEM, deals with the effect of nanoparticles’ shape on nanofluid behavior. The influence of an electric field on natural and forced convection heat transfer of nanofluid is explained in Chapter 11, Electrohydrodynamic Nanofluid Natural Convection Using CVFEM, and Chapter 12, Forced Convection of Nanofluid in Existence of Electric Field Using CVFEM. There exist several models for the simulation of porous media: the Darcy model, the non-Darcy model, and the thermal nonequilibrium model. These various models are investigated in Chapters 13
15. Nonuniform magnetic field effect on nanofluid convective flow in a porous cavity is analyzed in Chapter 16, Nonuniform Magnetic Field Effect on Nanofluid Convective Flow in a Porous Cavity. The influence of thermal radiation on nanofluid flow in a porous medium in the presence of Coulomb forces is illustrated in Chapter 17, Thermal Radiation Influence on Nanofluid Flow in a Porous Medium in the Presence of Coulomb Forces Using CVFEM. Chapter 18, Influence of Electric Field on Forced Convection of Nanofluid in a Porous Medium by Means of CVFEM, deals with the influence of Coulomb forces on forced convection of nanofluid in a porous medium. Nanofluid heat transfer enhancement in the presence of a melting surface is analyzed in Chapter 19, Nanofluid Heat Transfer Enhancement in Presence of Melting Surface Using CVFEM. Chapter 20, Nanofluid Convective Heat Transfer Considering Magnetic Field Dependent (MFD) Viscosity by Means of CVFEM, gives the reader a full account of the theory and practice associated with nanofluid convective heat transfer considering magnetic field-dependent viscosity. A sample FORTRAN code is presented in the Appendix. The readers can extend this code for other complex geometries and physics. Mohsen Sheikholeslami Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Islamic, Republic of Iran

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C H A P T E R

1 Detailed Explanation of Control Volume-based Finite Element Method 1.1 INTRODUCTION Fluid flow has several applications in engineering and nature. Mathematically, real flows are governed by a set of nonlinear partial differential equations in complex geometry. So, suitable solutions can be obtained through numerical techniques such as the finite difference method (FDM), the finite volume method (FVM), and the finite element method (FEM). In last decade, the finite element method has been developed for use in the area of computational fluid dynamics (CFD), and now this method has become a powerful method to simulate complex geometry. However, the finite volume method is applied the most to calculate fluid flows. The control volume finite element method (CVFEM) combines interesting characteristics from both the finite volume and finite element methods. The CVFEM was presented by Baliga and Patankar [1,2] using linear triangular finite elements, and by Raw and Schneider [3] using linear quadrilateral elements. Several authors have improved CVFEM from that time to now. Raw et al. [4] applied the nine-noded element to solve heat conduction problems. Banaszek [5] made a comparison of the Galerkin and CVFEM methods in diffusion problems using six-noded and nine-noded elements. Campos Silva [6] developed a computational program using nine-noded finite elements based on a control volume formulation to simulate 2-D transient, incompressible, viscous fluid flows. Campos Silva and Moura [7] and Campos Silva et al. [8] presented results for fluid flow problems. CVFEM combines the flexibility of the finite element methods to discretize complex geometry with the conservative formulation of the finite volume methods in which the variables have easy physical interpretation in terms of fluxes, forces, and sources. Saabas and Baliga [9,10] referenced a list of several works in FVMs and CVFEMs. Voller [11] presented the application of CVFEM for Fluids and Solids. Sheikholeslami et al. [12] studied the problem of natural convection between a circular enclosure and a sinusoidal cylinder. They concluded that streamlines, isotherms, and the number, size, and formation of the cells inside the enclosure strongly depend on the Rayleigh number, values of amplitude, and the number of undulations of the enclosure. Since the mid-2010s, this method has been applied for various applications [13
125].

1.2 THE DISCRETIZATION: GRID, MESH, AND CLOUD Generally, three ways exist to place the node points into the domain [11].

1.2.1 Grid A basic approach assigns the location of nodes using a structured grid where, in a two-dimensional domain, the location of a node is uniquely specified by a row and a column index, see Fig. 1.1A. Although such a structured approach can lead to very convenient and efficient discrete equations it lacks flexibility in accommodating complex geometries or allowing for the local concentration of nodes in solution regions of particular interest.

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00001-1

1

© 2019 Elsevier Inc. All rights reserved.

2

1. DETAILED EXPLANATION OF CONTROL VOLUME-BASED FINITE ELEMENT METHOD

1.2.2 Mesh Geometric flexibility, usually at the expense of solution efficiency, can be added by using an unstructured mesh. Fig. 1.1B shows an unstructured mesh of triangular elements. In two-dimensional domains triangular meshes are good selections because they can tessellate any planar surface. Note however, other choices of elements can be used in place of or in addition to triangular elements. The mesh can be used to determine the placement of the nodes. A common choice is to place the nodes at the vertices of the elements. In the case of triangles, this will allow for the approximation of a dependent variable, over the element, by linear interpolation between the vertex nodes. Higher order approximations can be arrived by using more nodes (e.g., placing nodes at midpoints) or alternative elements (e.g., quadrilaterals). In considering an unstructured mesh it is important to recognize the following: 1. The quality of the numerical solution obtained is critically dependent on the mesh. For example, a key quality requirement for a mesh of triangular elements is to avoid highly acute angles. The generation of appropriate meshes for a given domain is a complex topic worthy of a monograph in its own right. Fortunately, for twodimensional problems in particular, there is a significant range of commercial and free software that can be used to generate quality meshes. 2. The term unstructured is used to indicate a lack of a global structure that relates the position of all the nodes in the domain. In practice, however, a local structure—the region of support—listing the nodes connected to a given node i, is required. Establishing, storing, and using this local data structure is one of the critical ingredients in using an unstructured mesh.

1.2.3 Cloud The most flexible discretization is to simply populate the domain with node points that have no formal background grid or mesh connecting the nodes. Solution approaches based on this mesh-less form of discretization create “cloud” of neighboring nodes (see Fig. 1.1C [11]).

FIGURE 1.1 Different forms of discretization.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

1.3 THE ELEMENT AND THE INTERPOLATION SHAPE FUNCTIONS

3

1.3 THE ELEMENT AND THE INTERPOLATION SHAPE FUNCTIONS The building block of the discretization is the triangular element, Fig. 1.2. For linear triangular elements the node points are placed at the vertices. In Fig. 1.2 the nodes, moving in a counterclockwise direction, are labeled 1, 2, and 3. Values of the dependent variable
φare calculated and stored at these node points. In this way, values at an arbitrary point x; y within the element can be approximated with linear interpolation φ  ax 1 by 1 c

ð1:1Þ

where the constant coefficients a, b, and c satisfy the nodal relationships φi 5 axi 1 byi 1 c;

i 5 1; 2; 3

Eq. (1.1) can be more conveniently written in terms of the shape function, N1 ; N2 , and N3 , where ( 1 At node i
 Ni x; y 5 0 At all points on side opposite node i 3 X


 Ni x; y 5 1 At every point in the element

ð1:2Þ

ð1:3Þ ð1:4Þ

i51

such that, over the element, the continuous unknown field can be expressed as the linear combination of the values at nodes i 5 1; 2; 3 3
 X
 φ x; y  Ni x; y φi

ð1:5Þ

i51

With linear triangular elements a straightforward geometric derivation for the shape functions can be obtained. With reference to Fig. 1.2, observe that the area of the element is given by    1 x1 y1   1   1  A123 5  1 x1 y1  5 ðx2 y3 2 x3 y2 Þ 2 x1 ðy3 2 y2 Þ 1 y1 ðx3 2 x2 Þ ð1:6Þ  2 2 1 x y  1 1


 and the area of the subelements with vertices at points p; 2; 3 ; p; 3; 1 and p; 1; 2 , where pis an arbitrary and variable point in the element, are given by Ap23 5 ½ðx2 y3 2 x3 y2 Þ 2 xp ðy3 2 y2 Þ 1 yp ðx3 2 x2 Þ Ap31 5 ½ðx3 y1 2 x1 y3 Þ 2 xp ðy1 2 y3 Þ 1 yp ðx1 2 x3 Þ A

p12

ð1:7Þ

5 ½ðx1 y2 2 x2 y1 Þ 2 xp ðy2 2 y1 Þ 1 yp ðx2 2 x1 Þ

FIGURE 1.2 An element indicating the areas used in shape function definitions.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4

1. DETAILED EXPLANATION OF CONTROL VOLUME-BASED FINITE ELEMENT METHOD

With these definitions it follows that the shape functions are given by N1 5 AP23 =A123 ; N2 5 AP31 =A123 ; N3 5 AP12 =A123

ð1:8Þ

Note that, when point p coincides with node i ð1; 2 or 3Þ, the shape functionNi 5 1, and when point p is anywhere on the element side opposite node i, the associated subelement area is zero, and through Eq. (1.8), the shape functionNi 5 0. Hence the shape functions defined by Eq. (1.8) satisfy the required condition in Eq. (1.3). Further, note that at any point p, the sum of the areas:




AP23 1 AP31 1 AP12 5 A123

ð1:9Þ

such that the shape functions at xp ; yp will sum to unity. Hence the shape functions defined by Eq. (1.8) also satisfy the condition Eq. (1.4). For future reference it is worthwhile to note that the derivatives of the shape functions in Eq. (1.8) over the element are the following constants N1x 5

@N1 ðy2 2 y3 Þ 5 ; 2A123 @x

N1y 5

@N1 ðx2 2 x3 Þ 5 2A123 @y

N2x 5

@N2 ðy3 2 y2 Þ 5 ; 2A123 @x

N2y 5

@N2 ðx1 2 x3 Þ 5 2A123 @y

N3x 5

@N3 ðy1 2 y2 Þ 5 ; 2A123 @x

N3y 5

@N3 ðx2 2 x1 Þ 5 2A123 @x

ð1:10Þ

1.4 REGION OF SUPPORT AND CONTROL VOLUME The local structure on the mesh in Fig. 1.1B is defined in terms of the region of support—the list of nodes that share a common element with a given node i, in Fig. 1.3 [11]. In this region of support, as illustrated in Fig. 1.3, a control volume is created by joining the center of each element in the support to the midpoints of the element sides that pass through node i. This creates a closed polygonal control volume with 2m sides (faces); where m is the number of elements in the support. Each element contributes 1/3 of its area to the control volume area and the volumes from all the nodes tessellate the domain without overlap.

FIGURE 1.3 Region of support and control volume for node i in an unstructured mesh of Linear triangular elements.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

1.5 DISCRETIZATION AND SOLUTION

5

1.5 DISCRETIZATION AND SOLUTION 1.5.1 Steady State Advection
Diffusion With Source Terms To illustrate the solution procedure using the CVFEM, one can consider the general form of advection
diffusion equation for node i in integral form: ð ð ð 2 Q dV 2 krφUn dA 1 ðvUnÞφ dA 5 0 ð1:11Þ V

A

A

or point form 2rUðkrφÞ 1 rUðvφÞ 2 Q 5 0

ð1:12Þ

which can be represented by the system of CVFEM discrete equations as: ½ai 1 Qci 1 Bci φi 5

ni X

ai;j φSi;j 1 QBi 1 BBi

ð1:13Þ

j51

In the above, the a s are the coefficients, the index (i, j) indicates the jth node in the support of node i, the index Si, j provides the node number of the jth node in the support, the Bs account for boundary conditions, and the Qs for source terms. For the selected triangular element which is shown in Fig. 1.4 this approximation without considering the source term leads to


 2 ak1 1 au1 φi 1 ak2 1 au2 φSi;3 1 ak2 1 au2 φSi;4 5 0

ð1:14Þ

Using upwinding the advective coefficients, identified with the superscripts ( )u, are given by     au1 5 max qf1 ; 0 1 max qf2 ; 0   au2 5 max 2qf1 ; 0   au3 5 max 2qf2 ; 0

ð1:15Þ

FIGURE 1.4 A sample triangular element and its corresponding control volume.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

6

1. DETAILED EXPLANATION OF CONTROL VOLUME-BASED FINITE ELEMENT METHOD

And the diffusion coefficients, identified with the superscripts ( )k, are given by y f1 1 kf1 N1y Δ~ x f1 2 kf2 N1x Δ~ y f2 1 kf2 N1y Δ~ x f2 ak1 5 2 kf1 N1x Δ~ ak2 5 2 kf1 N2x Δ~ y f1 1 kf1 N2y Δ~ x f1 2 kf2 N2x Δ~ y f2 1 kf2 N2y Δ~ x f2 ak2

ð1:16Þ

5 2 kf1 N3x Δ~ y f1 1 kf1 N3y Δ~ x f1 2 kf2 N3x Δ~ y f2 1 kf2 N3y Δ~ x f2

In Eq. (1.27) the volume flow across face 1 and 2 in the direction of the outward normal, is   f1 f1 qf1 5 vUn A 5 vx Δ~ y f1 2 vy Δ~ y f1 f1   f2 f2 qf2 5 vUn A 5 vx Δ~ y f2 2 vy Δ~ y f2

ð1:17Þ

f2

The value of the diffusivity at the mid-point of face 1 can be obtained as kf1 5 ½N1 k1 1N2 k2 1N3 k3 f1 5

5 5 2 k1 1 k2 1 k3 12 12 12

ð1:18Þ

kf2 5 ½N1 k1 1N2 k2 1N3 k3 f2 5

5 2 5 k1 1 k2 1 k3 12 12 12

ð1:19Þ

and at the midpoint of face 2

The velocity components at the midpoint of face 1 are: f1

vx 5 f1 vy

5 5 2 vx 1 vx 1 vx 12 1 12 2 12 3

5 5 2 vy 1 1 vy 2 1 vy 5 12 12 12 3

ð1:20Þ

And on face 2: f2

vx 5 f2

vy 5

5 2 5 vx 1 vx 1 vx 12 1 12 2 12 3 5 2 5 vy 1 vy 1 vy 12 1 12 2 12 3

ð1:21Þ

These values can be used to update the ith support coefficients through the following equation: ai 5 ai 1 ak1 ð1:22Þ

ai;3 5 ai;3 1 ak2 ai;4 5 ai;4 1 ak3 In Eq. (1.28), moving counterclockwise around node i, the signed distances are: x3 x2 x1 2 2 ; 3 6 6 y3 y2 y1 2 2 ; Δ~ y f1 5 3 6 6

Δ~ x f1 5

x2 x3 x1 1 1 3 6 6 y2 y3 y1 Δ~ y f2 5 2 1 1 3 6 6 Δ~ x f2 5 2

ð1:23Þ

the derivatives of the shape functions are:


 y2 2 y3 @N1 N1x 5 5 ; N1y 5 2V ele @x
 y3 2 y1 @N2 5 N2x 5 ; N2y 5 @x 2V ele
 y1 2 y2 @N2 5 N3x 5 ; N3y 5 @x 2V ele

@N1 ð x3 2 x2 Þ 5 2V ele @y @N1 ð x1 2 x3 Þ 5 @y 2V ele

ð1:24Þ

@N3 ð x2 2 x1 Þ 5 @y 2V ele

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

1.5 DISCRETIZATION AND SOLUTION

and the volume of the element is


 x2 y3 2 x3 y2 1 x1 y2 2 y3 1 y1 ðx3 2 x2 Þ V 5 2

7




ele

ð1:25Þ

The obtained algebraic equations from the discretization procedure using CVFEM are solved by the Gauss
Seidel Method.

1.5.2 Implementation of Source Terms and Boundary Conditions The boundary conditions for the present problem can be enforced using BBi and BCi as follows: Insulated boundary: BBi 5 0 and BCi 5 0

ð1:26Þ

Fixed Value Boundary: BBi 5 φvalue 3 1016 and BCi 5 1016

ð1:27Þ

Fixed Flux Boundary: BBi 5 Ak 3 qv and BCi 5 0

ð1:28Þ

Where φvalue is the prescribed value on the boundary and Ak is the length of the control volume surface on the boundary segment. To provide a general treatment for boundary conditions some preliminary calculation of the boundary face areas associated with each node j in a given boundary segment are required. Fig. 1.5 shows a schematic of the kth ðk 5 3Þ boundary, indicating the data structure. Assuming unit depth, the face area associated with any node j of the boundary segment highlighted in Fig 1.5 is given by 8 Upper1 > < ð1:29Þ Ak;j 5 Upperj 1 Lowerj ; 2 # j # nB:k 2 1 > : LowernB;k where Upperj 5

1 2

Lowerj 5

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxj 2xj11 Þ2 1 ðyj 2yj11 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxj 2xj21 Þ2 1 ðyj 2yj21 Þ2

ð1:30Þ

The first and last lines on the right-hand side of Eq. (1.29) account for the first and last node on the boundary segment. This treatment assumes that there are at least two boundary segments—i.e., at least two contiguous regions of the domain boundary where different boundary conditions are applied. In cases where only one boundary condition is applied (e.g., a constant value over the whole boundary) the condition of a two-segment boundary can be artificially imposed. The volume source terms can be applied to Eq. (1.13) as: elements X ð Q dV  Qi Vi ð1:31Þ j51

Vj

FIGURE 1.5 A domain boundary segment with three sequential points detailed.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

8

1. DETAILED EXPLANATION OF CONTROL VOLUME-BASED FINITE ELEMENT METHOD

or after linearizing as the source term Qi Vi 5 2 QCi φi 1 QBi

ð1:32Þ

1.5.3 Unsteady Advection
Diffusion With Source Terms From the above the net flow rate of quantity into the control volume around node i can be approximated as ð ð ni ð ni X X κrφ:ndA 2 ðν:nÞφdA  2 ½ai 1 QCi 1 BCi φi 1 ai;j φSi;j 1 QBi 1 BBi ð1:33Þ Neti 5 QdV 1 Vi

j51

Aj

j51

Aj

In a steady-state problem this flow rate will be identically zero. In a transient problem, however, it will result in a change in storage of the quantity in the control volume, i.e., ð d φdV 5 Neti ð1:34Þ dt Vi

Using nodal lumping for the volume integration and finite difference in time this equation can be used to evaluate the nodal field values at time t 1 Δtin terms of the nodal field values at time t, Vi φnew 5 Vi φi 1 Δt½ð1 2 θÞNeti 1 θNetnew i i 

ð1:35Þ

where Δtis a time step and the superscript ðÞnew indicates evaluation at time t 1 Δt. The parameter 0 # θ # 1is a user-defined weighting factor, used to approximate the net flow into control volume i during the time interval ½t; t 1 Δt, in terms of the net flows at the beginning and end of the time step. Neglecting, for now, contributions from the boundaries and sources, the resulting discrete equations for three choices of θ are: Fully Implicit θ 5 1: 0 1 ni X new A Vi φnew ð1:36Þ 5 Vi φi 1 Δt@ ai;j φnew i Si;j 2 ai φi j51

The advantage of this choice is that it is unconditionally stable—i.e., for any choice of time step errors (induced or inherent) will not grow. The downside is that a system of equations needs to be solved to obtain the field values at the new time step φnew i ði 5 1::nÞ. Crank
Nicolson θ 5 0:5: 0 1 0 1 ni ni X X Δt Δt new A @ @ Vi φnew 1 ð1:37Þ 5 Vi φi 1 ai;j φnew ai;j φSi;j 2 ai φi A i Si;j 2 ai φi 2 2 j51 j51 This unconditionally stable scheme also requires a system solution. Unlike the 1st order in time, fully implicit method, however, the Crank Nicolson is 2nd order in time (i.e., time errors scale withΔt2 ) Fully Explicit θ 5 0: 0 1 ni X Vi φnew ð1:38Þ 5 Vi φi 1 Δt@ ai;j φSi;j 2 ai φi A i j51

This choice can be used to directly update the new time level values from the current time values without solution of a system of equations. This, however, comes at the price of a restriction on the time step size to ensure stability. The solution of Eq. (1.38) is likely to become unstable (error grow as the solution advances) if the net coefficient for φi becomes negative. This requires that the time step is chosen such that  Vi Δt , min ; ði 5 1. . .nÞ ð1:39Þ ai If a fine grid is used this value could be prohibitively small, i.e., the advance of time could be too slow to reach a practical value within reasonable computational resources. In many cases, however, the drawback for using a

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

9

small time step is offset by the ability to solve Eq. (1.29) without iteration. Driven by its simplicity and flexibility of modeling complex nonlinear terms the explicit time integration approach is the preferred choice in this work. When sources and boundary condition treatments are added the explicit scheme Eq. (1.29) can be written as 0 1 ni X ðVi 1 BCi Þφnew 5 Vi φi 1 Δt@ ai;j φSi;j 2 ai φi 1 QBi 2 QCi φi A 1 BBi ð1:40Þ i j51

The addition of the source term in Eq. (1.40) could require a further reduction in the time step to retain positive coefficients and stability, i.e.,  Vi Δt , min ; ði 5 1. . .nÞ ð1:41Þ ai 1 QCi

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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

10

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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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374. [94] M. Sheikholeslami, H.B. Rokni, Influence of melting surface on MHD nanofluid flow by means of two phase model, Chin. J. Phys. 55 (2017) 1352
1360. [95] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid natural convection in the presence of thermal radiation considering variable viscosity, Europ. Phys. J. Plus 132 (2017) 238. Available from: https://doi.org/10.1140/epjp/i2017-11498-4. [96] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4- Ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, Journal of Molecular Liquids, https://doi.org/10.1016/j.molliq.2017.11.048. [97] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, International Journal of Heat and Mass Transfer, In press. [98] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman, Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosity, Int. J. Heat Mass Transfer 108 (2017) 2558
2565. [99] M. Sheikholeslami, Numerical simulation for solidification in a LHTESS by means of nano-enhanced PCM, J. Taiwan Inst. Chem. Eng. 86 (2018) 25
41. [100] M. Sheikholeslami, Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424
438. [101] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transf. 123 (2018) 418
431. [102] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transf. 122 (2018) 1264
1271. [103] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transf. 122 (2018) 643
650. [104] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
462. [105] M. Sheikholeslami, S.A. Shehzad, Zhixiong Li, Water based nanofluid free convection heat transfer in a three dimensional porous cavity with hot sphere obstacle in existence of Lorenz forces, International Journal of Heat and Mass Transfer 125 (2018) 375
386. [106] M. Sheikholeslami, M. Jafaryar, D.D. Ganji, Zhixiong Li, Exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators, Journal of Molecular Liquids 262 (2018) 104
110. [107] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transf. 124 (2018) 980
989. [108] M. Sheikholeslami, Z. Li, M. Shamlooei, Nanofluid, MHD natural convection through a porous complex shaped cavity considering thermal radiation, Phys. Lett. A 382 (2018) 1615
1632. [109] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids, 30 (1), https://doi.org/10.1063/1.5012517. [110] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model, Int. J. Heat Mass Transf 120 (2018) 1200
1212. [111] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [112] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transf. 120 (2018) 772
781. [113] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process.: Process Intensif. 124 (2018) 71
82. [114] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [115] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [116] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [117] M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transf 118 (2018) 823
831. [118] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
746. [119] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4-ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Mol. Liq. 249 (2018) 429
437. [120] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transf 118 (2018) 182
192.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

13

[121] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [122] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [123] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Phys. B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [124] M. Sheikholeslami, D.D. Ganji, Numerical investigation of nanofluid transportation in a curved cavity in existence of magnetic source, Chem. Phys. Lett. 667 (2017) 307
316. [125] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cylinder considering Darcy model, J. Mol. Liq. 225 (2017) 903
912.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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C H A P T E R

2 Simulation of Vorticity Stream Function Formulation by Means of CVFEM 2.1 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR A LID DRIVEN CAVITY FLOW 2.1.1 Definition of the Problem and Governing Equation A classic test problem in computational fluid mechanics is a lid driven cavity flow. In this two-dimensional problem, the flow in a square conduit is induced by sliding the upper surface (the lid) at a constant 3 velocity U, see Fig. 2.1 [1]. The nature of the flow in the cavity is controlled by a Reynolds Number defined as: Re 5

UL υ

ð2:1Þ

where L is the dimension of the cavity and υ is the kinematic viscosity. In terms of the stream function Ψ and vorticity ω the governing equations, in integral form ð ων:n 2 υrω:ndS S

ð

ð2:2Þ

ð

2 ωdA 5

rΨ:ndS

ð2:3Þ

S

A

In terms of the velocity field, v, the vorticity and stream function are defined by ω5 νx 5

@ν y @ν x 2 @x @y

@Ψ ; @y

νy 5 2

ð2:4Þ @Ψ @x

ð2:5Þ

The boundary condition on Eq. (2.3) is Ψ 5 0 on all solid boundaries

ð2:6Þ

There is not an explicit boundary condition for the vorticity equation Eq. (2.2). In setting up the numerical solution of Eqs. (2.2) and (2.3) a boundary condition for the solution of the discrete form of Eq. (2.2) is established by using the discrete form of Eq. (2.3) at boundary nodes coupled with the known velocity conditions, i.e., ux 5 U on lid; ux 5 0 on all other boundaries uy 5 0 on all boundaries

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00002-3

15

ð2:7Þ

© 2019 Elsevier Inc. All rights reserved.

16

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

FIGURE 2.1 Lid driven cavity flow with streamline indicating flow.

j = 1 or 5 k = Si,1 or Si,5

j=4 K = Si,4

j=3 k = Si,3

j=2 k = Si,2

3=j=4

f2 f1

1=i

2=j=3

FIGURE 2.2 Arrangement of control volumes and elements.

2.1.2 The CVFEM Discretization of the Stream Function Equation In keeping with the rest of this book the improvement of the discrete equations is based on linear triangular elements. The key components of this discretization, the support, control volume, the element, and the control volume faces (f1, f2) in an element are illustrated in Fig. 2.2 [1]. It is convenient to first consider the

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.1 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR A LID DRIVEN CAVITY FLOW

17

discretization of the stream function equation Eq. (2.3). The form of this equation can be identified as steadystate diffusion with a volume source. The general form for the discrete equation, for node i in the support shown in Fig. 2.2, is h

ni i X aΨi;j Ψ Si;j 1 QΨBi 1 BΨBi aΨi 1 QΨCi 1 BΨCi Ψ i 5

ð2:8Þ

j51



 where the as are the coefficients, the index i; j indicates the jth node j 5 1; 2; ::; ni in the support of node i, the index Si;j provides the global node number ði 5 1; 2; ::; nÞ of the jth node in the support, the Bs account for boundary conditions, and the Qs for source terms. 2.1.2.1 Diffusion Contributions Following the presentation in the previous chapter the contribution to the coefficients as in Eq. (2.8), obtained from consideration of the diffusion flux across the control volume faces (f1, f2) for the particular element in Fig. 2.2, are ,

,

,

,

a1 k 5 2 N1x Δ y f1 1 N1y Δ x f1 2 N1x Δ y f2 1 N1y Δ x f2 ,

,

,

,

,

,

,

,

a2 k 5 N2x Δ y f1 2 N2y Δ x f1 1 N2x Δ y f2 2 N2y Δ x f2

ð2:9Þ

a3 k 5 N3x Δ y f1 2 N3y Δ x f1 1 N3x Δ y f2 2 N3y Δ x f2 These values can be used to update the ith support coefficients through aΨi 5 aΨi 1 aκ1 aΨi;3 5 aΨi;3 1 aκ2 aΨi;4

5 aΨi;4

ð2:10Þ

1 aκ4

In Eq. (2.9), moving counterclockwise around node i, the signed distances ,

x3 x2 x1 2 2 ; 3 6 6

Δ x f2 5 2

,

y3 y2 y1 2 2 ; 3 6 6

Δ y f1 5 2

Δ x f1 5 Δ y f1 5

,

x2 x3 x1 1 1 3 6 6

,

y2 y3 y1 1 1 3 6 6

ð2:11Þ

the derivatives of the shape functions N1x 5

@N1 ðy2 2 y3 Þ 5 ; @x 2V ele

N1y 5

@N1 ðx3 2 x2 Þ 5 @y 2V ele

N2x 5

@N2 ðy3 2 y1 Þ 5 ; @x 2V ele

N2y 5

@N2 ðx1 2 x3 Þ 5 @y 2V ele

N3x 5

@N3 ðy1 2 y2 Þ 5 ; @x 2V ele

N3y 5

@N3 ðx2 2 x1 Þ 5 @y 2V ele

ð2:12Þ

and the volume of the element V ele 5

ðx2 y3 2 x3 y2 Þ 1 x1 ðy2 2 y3 Þ 1 y1 ðx3 2 x2 Þ 2

ð2:13Þ

2.1.2.2 Source Terms The left-hand side of Eq. (2.3) is identified as a volume source term which can be represented in Eq. (2.8) by the setting QΨBi 5 ωi Vi ;

QΨCi 5 0

ð2:14Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

where Vi is the volume of the control volume, made up of contributions of V ele =3 from each element in the support of node i. The coupling with the nodal vorticity field will require an iterative solution of Eq. (2.8). For any iteration, the source term in Eq. (2.14) will need to be reevaluated to reflect the updating of the iterative values of the nodal vorticity field. 2.1.2.3 Boundary Conditions The boundary conditions of Ψ 5 0 on all boundaries are set in Eq. (2.8) by setting, for each node i lying on the domain boundary BΨBi 5 0;

BΨCi 5 1015

ð2:15Þ

2.1.3 The CVFEM Discretization of the Vorticity Equation The vorticity equation Eq. (2.2) is seen to have the form of a steady state advection
diffusion equation for a scalar. The general form for this equation for node i in the support shown in Fig. 2.2 is ½aωi 1 BωCi ωi 5

ni X

aωi;j ωSi;j 1 BωBi

ð2:16Þ

j51

2.1.3.1 Diffusion Contributions The contributions to the coefficients a in Eq. (2.16), from consideration of the diffusion flux across the control volume faces (f1, f2) in Fig. 2.2, are ,

,

,

,

a1 κ 5 2νN1x Δ y f1 1 νN1y Δ x f1 2 νN1x Δ y f2 1 νN1y Δ x f2 ,

,

,

,

a2 κ 5 νN2x Δ y f1 2 νN2y Δ x f1 1 νN2x Δ y f2 2 νN2y Δ x f2 , a3 κ 5 νN3x Δ y f1

,

, 2 νN3y Δ x f1 1 νN3x Δ y f2

ð2:17Þ

, 2 νN3y Δ x f2

These values can be used to update the ith support coefficients through aωi 5 aωi 1 a1 κ aωi;3 5 aωi;3 1 a2 κ

ð2:18Þ

aωi;4 5 aωi;4 1 a4 κ 2.1.3.2 The Advection Coefficients The contributions to the coefficients a in Eq. (2.8), from consideration of the advection flux across the control volume faces (f1, f2) in Fig. 2.2, are, using upwinding, au1 5 max½qf1 ; 0 1 max½qf2 ; 0 au2 5 max½ 2qf1 ; 0 au3

ð2:19Þ

5 max½ 2qf2 ; 0

These values can be used to update the ith support coefficients through aωi 5 aωi 1 ak1 1 au1 aωi;3 5 aωi;j 1 ak2 1 au2 aωi;4

5 aωi;j

1 ak3

ð2:20Þ

1 au3

In Eq. (2.19) the volume flow rates out across the control volume faces are calculated by ,

,

,

,

qf1 5 ν:nAjf1 5 ν x e Δ y f1 2 ν y e Δ x f1 qf2 5 ν:nAjf2 5 ν x e Δ y f2 2 ν y e Δ x f2

ð2:21Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.1 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR A LID DRIVEN CAVITY FLOW

19

where the velocities at the midpoint of the faces are approximated in terms of the constant element velocity, ν e 5 ðν x e ; ν y e Þ. This value, in turn is calculated directly from the nodal stream function field through the approximation of Eq. (2.5), e.g., for the element in Fig. 2.2 ν x e  N1y Ψ i 1 N2y Ψ Si;3 1 N3y Ψ Si;4 ν y e  2 N1x Ψ i 2 N2x Ψ Si;3 2 N3x Ψ Si;4

ð2:22Þ

This coupling with the nodal stream-function field will require an iterative solution of Eq. (2.16). At each iteration, the advection coefficients in Eq. (2.19) will need to be evaluated to reflect the updating of the iterative values of the nodal stream-function field. 2.1.3.3 Boundary Conditions Before Eq. (2.16) can be solved boundary conditions need to be prescribed. Fixed value boundary conditions are used. At each node i on a domain boundary the discrete form of the stream function Eq. (2.3) can be used to prescribe a value for the nodal vorticity. Use of this equation is allowed since, due to the fact that the stream function takes the known fixed value Ψ 5 0 on all boundaries, it is not needed in the stream function solution. At a node i on the boundary the control volume finite element discretization of Eq. (2.3), informed by Eq. (2.8), can be written as 

ni X X 1 Ψ Ψ ai;j Ψ Si;j 2 ArΨ:n ð2:23Þ ωi 5 a Ψi 2 Vi i j51 boundary where the coefficients aΨ are given by Eq. (2.10) and the definition of the source term in Eq. (2.14) has been used to isolate ωi . The last term in Eq. (2.23) represents contributions from control volume faces that coincide with a boundary segment—see the double line in Fig 2.3 [1]. In the sliding lid problem of Fig. 1.1 the only nonzero contribution from this term will be for nodes on the sliding lid and will have the form 2 3 X 4 ArΨ:n5 5 ΔU ð2:24Þ boundary

where Δ is the length of the control volume surface on the boundary segment, see Fig. 2.3 [1]. Hence, with current iterative values of the stream function known, Eqs. (2.23) and (2.24) can be used to estimate fixed nodal values of vorticity for boundary nodes i 2  ni X 1 4 ψ ΔU iE lid ψ ωi 5 a ψ 2 ai;j ψSi;j 2 ð2:25Þ 0 otherwise Vi i i j51 These values are forced into the solution of Eq. (2.16) by setting BωBi 5 ωi 3 1015 ;

BωCi 5 1015

ð2:26Þ

FIGURE 2.3 Node point on sliding lid boundary.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

2.1.4 Calculating the Nodal Velocity Field Following the calculation of the converged Ψ field, the nodal velocity field v can be calculated. Integrating over the ith control volume results in ð ð ð ð @Ψ @Ψ dV; dV; ð2:27Þ ν x dV 5 ν y dV 5 2 @y @x Vi

or using the divergence theorem

Vi

ð

Vi

I ν x dV 5

Vi

Vi

ð Ψny dS;

I ν y dV 5 2

Vi

S

Ψnx

ð2:28Þ

S

where the surface integral is over the control volume faces, and n 5 ðnx :ny Þ is the outward pointing normal on the control volume face. Using midpoint integration approximations 1X 1X ν xi 5 ΨAny ; ν yi 5 2 ΨAnx ð2:29Þ Vi faces Vi faces where A is the area (length 3 unit depth) of a given control volume face, and, since boundary values of Ψ 5 0, the summations are restricted to control faces that do not coincide with the domain boundaries. On recalling that the outward normal area product on a volume face is ,

Anx 5 Δ y ; ,

,

Any 5 2 Δ x

ð2:30Þ

,

According to definitions of Δ y and Δ x , the contributions to control volume i from the right-hand side of Eq. (2.29), for the selected element from the control volume in Fig. 2.2, can be calculated as 0 1 0 1 X 5 5 2 5 2 5 , , Ψ Si;3 1 Ψ Si;4 A Δ x f1 2 @ Ψ i 1 Ψ Si;3 1 Ψ Si;4 AΔ x f2 ΨAny 5 2@ Ψ i 1 12 12 12 12 12 12 f1;f2 ð2:31Þ 0 1 0 1 X 5 5 2 5 2 5 , , ΨS 1 Ψ S A Δ y f1 2 @ Ψ i 1 Ψ Si;3 1 Ψ S AΔ y f2 ΨAnx 5 2@ Ψ i 1 12 12 i;3 12 i;4 12 12 12 i;4 f1;f2 With the nodal velocity field calculated from Eq. (2.28) the velocity at given point x 5 ðx; yÞcan be determine by determining the triangular element (local nodes 1, 2, and 3) which contains the point x and then using the shape function interpolations ν x ðxÞ 5 N1 ν x1 1 N2 ν x2 1 N3 ν x3 ν y ðxÞ 5 N1 ν y1 1 N2 ν y2 1 N3 ν y3

ð2:32Þ

where the shape functions Ni ði 5 1; 2; 3Þ are defined in Eqs. (2.7) and (2.8).

2.1.5 Results Fig. 2.4 shows the streamline for different Reynolds number. In Appendix a FORTRAN code for the CVFEM solution of a steady-state advection
diffusion equation is provided.

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION 2.2.1 Definition of the Problem and Governing Equation The physical model and the corresponding triangular elements used in the present CVFEM program are shown in Fig. 2.5 [2], where rout is the radius of the outer cylinder. The surfaces of the inner and outer cylinders

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION

21

FIGURE 2.4 Streamlines for different Reynolds number.

are maintained at constant temperatures Th and Tc , respectively, under the condition that Th . Tc . The profile of the inner cylinder follows the pattern:

 r 5 rin 1 A cos N ζ 2 ζ 0 ð2:33Þ where rin is the base circle radius (rin 5 1), A and N are amplitude and number of undulations, respectively, and ζis the rotation angle. Under the Boussinesq approximation, the continuity, momentum, and energy equations for the laminar and steady-state natural convection in the cylindrical enclosure can be written as follows: @u @v 1 50 @x @y

ð2:34Þ

 2 @u @u 1 @P @ u @2 u 1v 52 1υ u 1 @x @y ρ @x @ x2 @ y2  2 @v @v 1 @P @ v @2 v 1v 52 1υ 1 1 g β ðT 2 Tc Þ u @x @y ρ @y @ x2 @ y2  2 @T @T @T @2 T 1v 5α 1 2 u @x @y @x2 @y

ð2:35Þ ð2:36Þ ð2:37Þ

Where u and v are the velocity components along the axes x and y, Tis the fluid temperature, P is the pressure, and the physical meaning of the other quantities is mentioned in the Nomenclature. We introduce the following dimensionless variables defined as X5

x ; L

Y5

y ; L

Ω5

ωL2 ; α

Ψ5

ψ ; α

Θ5

T 2 Tc Th 2 Tc

ð2:38Þ

where L 5 rout 2 rin 5 rin 5 1. Using these variables, Eqs. (2.34)
(2.37) can be written in the dimensionless form as   2 @Ψ @Ω @Ψ @Ω @ Ω @2 Ω @Θ 2 5 Pr 1 ð2:39Þ 1 Ra Pr 2 2 @Y @X @X @Y @X @Y @X  2 @Ψ @Θ @Ψ @Θ @ Θ @2 Θ 2 5 1 ð2:40Þ @Y @X @X @Y @X2 @Y2 @2 Ψ @2 Ψ 1 5 2Ω 2 @X @Y2

ð2:41Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

22

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

FIGURE 2.5 (A) Geometry and boundary conditions. (B) Computational mesh.

where Ra 5 g β ðTh 2 Tc Þ L3 =ðαυÞ is the Rayleigh number and Pr 5 υ=α is the Prandtl number. The boundary conditions of Eqs. (2.39)
(2.41) are, as shown in Fig. 2.5, Θ 5 1:0 on the inner sinusoidal circular cylinder Θ 5 0:0

on the outer circular enclosure

ð2:42Þ

Ψ 5 0:0 on all solid boundaries The values of the vorticity Ω on the boundaries of the enclosure can be obtained using the definition of the stream function and the known velocity conditions during the iterative solution procedure. The local Nusselt number on the cold wall Nuloc can be expressed as Nuloc 5

@Θ @n

ð2:43Þ

where n is the distance normal to the surface of the outer cylinder. The average Nusselt number on the cold circular wall Nuave is evaluated as 1 Nuave 5 2π

360 ð3

Nuloc ðζ Þ dζ

ð2:44Þ

0

2.2.2 Effect of Active Parameters Calculations are performed for various values of Rayleigh numbers Ra 5 103 ; 104 ; 105 , and 106 , amplitude parameter A 5 0:1; 0:3 and 0:5, and the number of undulations N 5 2; 3; 5 and 6 at constant Prandtl number, Pr 5 0:71 (air). Streamlines and isotherm obtained by the present code are compared with those reported by Kim et al. [3] for different values of the Rayleigh numbers (Fig. 2.6). The shape size and location of vortexes are the same in the result obtained by Kim et al. [3] and the present code. These comparisons illustrate an excellent agreement between the present calculations and the previously published results. We are, therefore, confident that the present results are correct and very accurate using the CVFEM code. Recently, this code was developed for various applications [4
93].

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION

23

FIGURE 2.6 Comparison of the present results with those reported by Kim et al. [3] for different Rayleigh numbers when Pr 5 0:7.

Fig. 2.7 shows the isotherms and streamlines for different values of amplitude parameter and number of undulations at Ra 5 103 . For all cases at this Rayleigh number the isotherms follow the shape of inner and outer walls. Moreover for lower Rayleigh numbers the absolute values of stream function are nearly small, which indicates the domination of conduction heat transfer mechanism. For N 5 3 increasing the amplitude of undulation parameter A, enhances the absolute value of stream function jΨ max j. It is due to the fact that with the increase of A the gap between the hot and cold walls decreases which results in a higher heat transfer. Fig. 2.7 also shows that as the amplitude of undulation increases from A 5 0:1 to A 5 0:5 the single vortex inside the enclosure divides into two vortices with different strength located at the top and bottom of the enclosure. Moreover, the streamlines indicate that the upper vortex is stronger than the lower one. The reason could be the more available space for circulation of fluid at the top of the enclosure and existence of the hot surface beneath the cold outer wall in this region, which helps the flow circulation. As seen at the bottom of the enclosure a very small secondary vortex appears near the vertical centerline, which appears due to the existence of the crest in that region. A similar flow pattern is seen at this region for N 5 5. The values of streamlines show that for N 5 4 increasing A from 0.1 to 0.3 enhances the values of jΨ max j, but as A increases furthermore its value decreases. It could be related to the effect of less available space for fluid circulation in this area.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

24

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

FIGURE 2.7 Isotherms (left) and streamlines (right) for different values of amplitude A and number of undulations N at Ra 5 103 .

The streamlines also show that there are two main vortices between the inner and outer cylinders for all values of A; an upper stronger vortex and a lower weaker one. Since the available spaces for circulation are equal for both vortexes, it can be concluded that the existence of the hot surface beneath the cold one assists the flow circulation at the upper half of the enclosure, while the existence of the hot surface over the cold one in the lower half of the enclosure has a reverse effect on the flow circulation. At N 5 4, there is no small secondary vortex at the bottom half of the enclosure because the lower crest is parallel to the flow path. A similar pattern exists for N 5 6. Also the flow pattern, size, and formation of vortices are nearly similar for N 5 5 and 6. As seen, APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION

25

increasing A for these values of the undulation number divides the space between the inner and outer cylinders into three circulation regions. Increase of A decreases jΨ max j considerably at N 5 5 and 6. Although an increase of A expands the effective surface area of hot surface, the fluid flow will be damped, especially in the gaps between undulated surfaces so these areas are almost stagnant. The streamlines and isotherms are shown for different values of N and A at Ra 5 104 in Fig. 2.8. As seen, the isotherms are disturbed at this Ra and thermal plumes gradually appear on the hot surface of the inner cylinder, which indicates that the convective heat transfer became comparable with the conduction mode of heat transfer at this Rayleigh number. In addition, the thickness of the thermal boundary layer between the crests of inner the cylinder becomes thinner in comparison with that of Ra 5 103 , which indicates that the flow ventilates the region between each of the two crests; hence the thermal boundary layer is thinner at these areas.

FIGURE 2.8 Isotherms (left) and streamlines (right) for different values of the amplitude A and the number of undulations N at Ra 5 104 . APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

26

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

FIGURE 2.9 Isotherms (left) and streamlines (right) for different values of the amplitude A and number of undulations N at Ra 5 105 .

At Ra 5 104 , the variation of jΨ max j with N and A is similar to that of Ra 5 103 . For N 5 3 with the increase of A from 0.1 to 0.3, a thermal plume is formed over the upper crest at ς 5 90 degrees with reverse direction owing to the two secondary vortices which are generated newly at this area. With more increase in A these secondary vortices disappear because the available space between the inner hot wall and outer cold cylinder decreases. For N 5 4, the primary vortex divides into two smaller vortices; an upper strong and a lower weak one. The flow pattern and number of vortices for N 5 5 and 6 is similar to those of Ra 5 103 except the value of jΨ max j is higher at Ra 5 104 because the convective heat transfer mechanism is more pronounced. Figs. 2.9 and 2.10 show the effects of A and N on the streamlines and isotherms for Ra 5 105 and 106 , respectively. As a general observation the isotherms are crowed near the crests which demonstrate the diminution in

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION

FIGURE 2.10

27

Isotherms (left) and streamlines (right) for different values of the amplitude A and number of undulations N at Ra 5 106 .

thermal boundary layer thickness at the bottom of the enclosure. At the top of the inner cylinder ðς 5 90 degreesÞ, a strong plume appears which strongly impinges upon the outer wall of the enclosure for all values of A and N. The maximum value of the stream function has a direct relation to the value of A for N 5 6 in which it enhances up to the value A 5 0:3 and then decreases. For N 5 3 and A 5 0:1, a primary vortex exists at the top of the inner cylinder. When A increases, this main vortex divides into two vortices with small cores which are located at the upper half of the cylinder. Finally for A 5 0:5 these two vortices merge together and form a single

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

Ra = 103

Ra = 104

2

Ra = 105

4 A = 0.1 A = 0.3 A = 0.5

1.6

Ra = 106 12

8 A = 0.1 A = 0.3 A = 0.5

3.5

A = 0.1 A = 0.3 A = 0.5

7

3

6

2.5

5

A = 0.1 A = 0.3 A = 0.5

10

0 90

135

180 ζ

225

3

1

2

0.5

1

0 90

270

2

135

180 ζ

225

0.8

2 0 135

180 ζ

225

90

270

3

6

10

5

8

4

1.5

3

1

2

0.5

1

180 ζ

225

270

A = 0.1 A = 0.3 A = 0.5

12

2.5 2

135

14 A = 0.1 A = 0.3 A = 0.5

7

Nuloc

Nuloc

Nuloc

1.2

6 4

8 A = 0.1 A = 0.3 A = 0.5

3.5

1.6

N=4

0 90

270

4 A = 0.1 A = 0.3 A = 0.5

4

1.5

0.4

Nuloc

2

Nuloc

Nuloc

Nuloc

N=3

0.8

Nuloc

8 1.2

6 4

0.4

180 ζ

225

0 90

270

180 ζ

225

0 90

270

4

2 A = 0.1 A = 0.3 A = 0.5

1.6

Nuloc

Nuloc 0.8

0 135

180 ζ

225

90

270

8 A = 0.1 A = 0.3 A = 0.5

3.5

1.2 n=5

135

3

6

2.5

5

2

3

1

2

0.5

1

180 ζ

225

270

A = 0.1 A = 0.3 A = 0.5

12 10

4

1.5

135

14 A = 0.1 A = 0.3 A = 0.5

7

Nuloc

135

Nuloc

0 90

2

8 6 4

0.4

180 ζ

225

0 90

270

180 ζ

225

0 90

270

4

2 A = 0.1 A = 0.3 A = 0.5

Nuloc

Nuloc

1.2

0.8

0 135

180 ζ

225

270

90

A = 0.1 A = 0.3 A = 0.5

7

3

6

10

5

8

4

1.5

3

1

2

0.5

1

180 ζ

225

270

A = 0.1 A = 0.3 A = 0.5

12

2.5 2

135

14

8 A = 0.1 A = 0.3 A = 0.5

3.5

1.6

N=6

135

Nuloc

135

Nuloc

0 90

2

6 4

0.4

0 90

FIGURE 2.11

2

135

180 ζ

225

270

0 90

135

180 ζ

225

270

0 90

0 135

180 ζ

225

270

Effects of the amplitude A, number of undulations N , and Rayleigh number Ra on the local Nusselt number Nuloc .

90

135

180 ζ

225

270

29

2.2 CVFEM STREAM FUNCTION-VORTICITY SOLUTION FOR NATURAL CONVECTION

1.3

1.8 A = 0.1 A = 0.2 A = 0.3

1.2

A = 0.1 A = 0.2 A = 0.3

1.7 1.1

Nuave

Nuave

1.6 1

1.5 0.9 1.4

0.8

0.7

1.3 3

3.5

4

4.5 N

5

5.5

6

3

3.5

4

Ra = 105

5

5.5

6

5

5.5

6

Ra = 106 5

2.9 A = 0.1 A = 0.2 A = 0.3

2.8

A = 0.1 A = 0.2 A = 0.3

4.8

2.7

Nuave

Nuave

4.6

2.6

4.4

2.5

4.2

2.4

3

3.5

4

4.5 N

FIGURE 2.12

4.5 N

5

5.5

6

4

3

3.5

4

4.5 N

Effects of the amplitude A, number of undulations N , and Rayleigh number Ra on the average Nusselt number Nuave .

vortex which is stronger than the first one. For N 5 4, increasing A from 0.1 to 0.3 and then to 0.5 divides the main cell into two cells with different strength. For N 5 5; 6 and A 5 0:1 a single vortex exists inside the enclosure. The increase of A up to 0.3 and then to 0.5 pushes this single cell upwards and then divides it into two cells. The flow pattern and number of vortices at this Rayleigh number is similar to those of Ra 5 105 except the value of jΨ max j is greater at Ra 5 106 . It should be noted that all of the computations in the case of Ra 5 106 are obtained by using an unsteady solution and time averaging for a period of time. Variation of the local Nusselt number Nuloc over the cold wall and between ς 5 90 and 270 degrees is depicted in Fig. 2.11. In general, the effects of increasing A on Nuloc are less pronounced at higher values of the Rayleigh number. In addition, the variation of Nuloc is greater for higher values of the undulation amplitude, while this variation decreases when the gap between the hot inner and cold outer cylinders increases. The number of extremum in the local Nusselt number profiles corresponds to the number of undulations of the inner cylinder. For N 5 3 there exist two local maximum, three maximum for N 5 4 and 5, and four local maximum for N 5 6. Fig. 2.11 also shows that Nuloc decreases as the angle ς increases from ς 5 90 to 270 degrees; at the top of the inner cylinder (ς 5 90 degrees), the flow accompanying cold fluid impinges upon the upper cold surface, which results in a thicker thermal boundary layer on the enclosure’s wall. Whereas in the case of ς 5 270 degrees, the hot surface locates at the top of the cold wall and the conduction mechanism is more pronounced. Thus, Nuloc decreases noticeably. At Ra 5 104 ; 105 , and 106 , the Nuloc profile uniformly reaches from its maximum value at ς 5 90 degrees to its minimum value near ς 5 270 degrees when A 5 0:1. For A 5 0:3 and 0:5, the profiles of the Nuloc profiles have local extremes, which are related to the thermal plumes and crests over the inner cylinder. Fig. 2.12 shows the effects of A; N, and Ra on the average Nusselt number Nuave . As the Ra increases, the value of Nuave enhances due to the convective heat transfer mechanism at higher values of Ra. At Ra 5 103 , the APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

30

2. SIMULATION OF VORTICITY STREAM FUNCTION FORMULATION BY MEANS OF CVFEM

average Nusselt number is approximately constant at A 5 0:1 for various numbers of undulations, whereas it increases uniformly for A 5 0:3 and 0.5 due to decreasing the space between the hot and cold surfaces. At Ra 5 104 , the maximum value for Nuave is obtained for N 5 4 and A 5 0:1 meaning more space is available for fluid flow, while it is obtained for N 5 6 at A 5 0:3 and 0.5. The Nuave profiles for Ra 5 105 indicate that the minimum values of Nuave are obtained for N 5 3, whereas the maximum values of Nuave occur at N 5 4 for A 5 0:1; 0:3 and at N 5 6 for A 5 0:5. It is worth pointing out that for higher values of the Rayleigh number the profiles of the average Nusselt number have a local minimum at N 5 5 for all values ofA.

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110. [79] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transfer 124 (2018) 980
989. [80] M. Sheikholeslami, Z. Li, M. Shamlooei, Nanofluid MHD natural convection through a porous complex shaped cavity considering thermal radiation, Phys. Lett. A 382 (2018) 1615
1632. [81] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids, 30, 1, 10.1063/1.5012517, https://doi.org/10.1063/1.5012517. [82] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via Non-equilibrium model, Int. J. Heat Mass Transfer 120 (2018) 1200
1212. [83] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [84] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transfer 120 (2018) 772
781. [85] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process.: Process Intensif. 124 (2018) 71
82. [86] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [87] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [88] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [89] M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transfer 118 (2018) 823
831. [90] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
746. [91] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [92] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transfer 118 (2018) 182
192. [93] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4- Ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Mol. Liq. 249 (2018) 429
437.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

3 Various Application of Nanofluid for Heat Transfer Augmentation 3.1 INTRODUCTION Nanofluids are produced by dispersing the nanometer-scale solid particles into base liquids with low thermal conductivity such as water, ethylene glycol (EG), oils, etc. Control of heat transfer in many energy systems is crucial due to the increase in energy prices. In recent years, nanofluids technology has been proposed and studied by some researchers experimentally or numerically to control heat transfer in a process. The nanofluid can be applied to engineering problems, such as heat exchangers, cooling of electronic equipment, and chemical processes. There are two ways for simulation of a nanofluid: single-phase and two-phase. In the first method, researchers assumed that nanofluids were to be treated as the common pure fluid and conventional equations of mass, momentum, and energy are used and the only effect of nanofluid is its thermal conductivity and viscosity which are obtained from the theoretical models or experimental data. These researchers assumed that nanoparticles are in thermal equilibrium and there aren’t any slip velocities between the nanoparticles and fluid molecules, thus they have a uniform mixture of nanoparticles. In the second method, researchers assumed that there are slip velocities between nanoparticles and fluid molecules. So the volume fraction of nanofluids may not be uniform anymore and there would be a variable concentration of nanoparticles in a mixture. There are several numerical and semianalytical methods which have been used by several authors in order to simulate nanofluid flow and heat transfer.

3.1.1 Definition of Nanofluid Low thermal conductivity of conventional heat transfer fluids such as water, oil, and EG mixture is a serious limitation in improving the performance and compactness of much engineering equipment, such as heat exchangers and electronic devices. To overcome this disadvantage, there is a strong motivation to develop advanced heat transfer fluids with substantially higher conductivity. An innovative way of improving the thermal conductivities of fluids is to suspend small solid particles in the fluid. Various types of powders such as metallic, nonmetallic, and polymeric particles can be added into fluids to form slurries. The thermal conductivities of fluids with suspended particles are expected to be higher than that of common fluids. Nanofluids are a new kind of heat transfer fluid containing a small quantity of nanosized particles (usually less than 100 nm) that are uniformly and stably suspended in a liquid. The dispersion of a small amount of solid nanoparticles in conventional fluids changes their thermal conductivity remarkably. Compared to the existing techniques for enhancing heat transfer, the nanofluids show a superior potential for increasing heat transfer rates in a variety of cases [1].

3.1.2 Model Description In the literature, convective heat transfer with nanofluids can be modeled using mainly the two-phase or single approach. In the two-phase approach, the velocity between the fluid and particles might not be zero [2] due to several factors such as gravity, friction between the fluid and solid particles, Brownian forces, Brownian diffusion, sedimentation, and dispersion. In the second approach, the nanoparticles can be easily fluidized and therefore, one may assume that the motion slip between the phases, if any, would be considered negligible [3]. The latter approach is simpler and more computationally efficient. Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00003-5

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© 2019 Elsevier Inc. All rights reserved.

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3. VARIOUS APPLICATION OF NANOFLUID FOR HEAT TRANSFER AUGMENTATION

3.1.3 Conservation Equations 3.1.3.1 Single-phase Model Although nanofluids are solid
liquid mixtures, the approach conventionally used in most studies of natural convection handles the nanofluid as a single-phase (homogenous) fluid. In fact, due to the extreme size and low concentration of the suspended nanoparticles, the particles are assumed to move with the same velocity as the fluid. Also, by considering the local thermal equilibrium, the solid particle
liquid mixture may then be approximately considered to behave as a conventional single-phase fluid with properties that are to be evaluated as functions of those of the constituents. The governing equations for a homogenous analysis of natural convection are continuity, momentum, and energy equations with their density, specific heat, thermal conductivity, and viscosity modified for nanofluid application. The specific governing equations for various studied enclosures are not shown here and they can be found in different references [4]. It should be mentioned that sometimes this assumption is not correct. For example, Ding and Wen [5] found that this assumption may not always remain true for a nanofluid. They investigated the particle migration in a nanofluid for a pipe flow and stated that at Peclet numbers exceeding 10 the particle distribution is significantly nonuniform. Nevertheless, many studies have performed the numerical simulation using single-phase assumption and reported acceptable results for the heat transfer and hydrodynamic properties of the flow. 3.1.3.2 Two-phase Model Several authors have tried to establish convective transport models for nanofluids [6]. Nanofluid is a twophase mixture in which the solid phase consists of nanosized particles. In view of the nanoscale size of the particles, it may be questionable whether the theory of conventional two-phase flow can be applied in describing the flow characteristics of nanofluid. On the other hand, several factors such as gravity, friction between the fluid and solid particles and Brownian forces, the phenomena of Brownian diffusion, sedimentation, and dispersion may affect a nanofluid flow. Consequently, the slip velocity between the fluid and particles cannot be neglected for simulating nanofluid flows. Since the two-phase approach considers the movement between the solid and fluid molecule, it may have better prediction in nanofluid study. To fully describe and predict the flow and behavior of complex flows, different multiphase theories have been proposed and used. The large number of published articles concerning multiphase flows typically employed the Mixture Theory to predict the behavior of nanofluids [7]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [8], using a model in which Brownian motion and thermophoresis are accounted for. Buongiorno developed a two-component four-equation nonhomogeneous equilibrium model for mass, momentum, and heat transfer in nanofluids. The nanofluid is treated as a two-component mixture (base fluid and nanoparticles) with the following assumptions: No chemical reactions; Negligible external forces; Dilute mixture (φ 5 1); Negligible viscous dissipation; Negligible radiative heat transfer; Nanoparticle and base fluid locally in thermal equilibrium. Invoking the above assumptions, the following equations represent the mathematical formulation of the nonhomogenous single-phase model for the governing equations as formulated by Buongiorno [8]: 3.1.3.2.1 Continuity Equation r:v 5 0

ð3:1Þ

 @φ rT 1 v:rφ 5 r: DB rφ 1 DT @t T

ð3:2Þ

where v is the velocity 3.1.3.2.2 Nanoparticle Continuity Equation

Here φ is nanoparticle volume fraction, DB is the Brownian diffusion coefficient given by the Einstein
Stokes’s equation: DB 5

kB T 3πμdp

ð3:3Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

35

3.1 INTRODUCTION

where μ is the viscosity of the fluid, dp is the nanoparticle diameter, kB 5 1:385 3 10223 is Boltzmann constant and DT is the thermophoretic diffusion coefficient, which is defined as   μ k DT 5 0:26 ð3:4Þ ρ k 1 kp In Eq. (3.4), k and kp are the thermal conductivity of the fluid and particle materials, respectively. 3.1.3.2.3 Momentum Equation v:rv 5 2 where

1 rp 1 r:τ 1 g ρnf

ð3:5Þ


 τ 5 2μnf rv 1 ðrvÞt

ð3:6Þ

where the superscript “t” indicates the transpose of rv. Also p is pressure. 3.1.3.2.4 Energy Equation ρp cp
 v:rT 5 r αnf rT 1 ρnf cnf

 DB rφ:rT 1 DT

rT:rT T

ð3:7Þ

where φ and T are nanoparticle concentration and temperature of nanofluid, respectively. This nanofluid model can be characterized as a “two-fluid” (nanoparticles 1 base fluid), four-equation (mass, momentum, energy), nonhomogeneous (nanoparticle/fluid slip velocity allowed) equilibrium (nanoparticle/fluid temperature differences not allowed) model. Note that the conservation equations are strongly coupled. That is, v depends on φ via viscosity; φ depends on T mostly because of thermophoresis; T depends on φ via thermal conductivity and also via the Brownian and thermophoretic terms in the energy equation: φ and T obviously depends on v because of the convection terms in the nanoparticle continuity and energy equations, respectively. In a numerical study by Behzadmehr et al. [9] for the first time a two-phase mixture model was implemented to investigate the behavior of Cu
water nanofluid in a tube and the results were also compared with previous works using a single-phase approach. The authors claimed that the simulation done by assuming that base fluid and particles behave separately possessed results that are more precise compared to the previous computational modeling. They implemented the mixture theory for their work. It was suggested that the continuity, momentum, and energy equations be written for a mixture of fluid and a solid phase. Some assumptions were also stated for the model such as a strong coupling between two phases and the fluid being closely followed by the particles with each phase owning a different velocity leading to a term called slip velocity of nanoparticles as in Eq. (3.8):

 ( ρp 2 ρm ρp d2p 1 1 0:15Re0:687 p Vpf 5 VP 2 Vf 5 a; a 5 g 2 ðVm :rÞVm ; fdrag 5 ð3:8Þ ρp 18μf fdrag 0:0183Rep The conservation equations (continuity, momentum, and energy, respectively) will be written for the mixture as follows:
 ð3:9Þ r: ρm Vm 5 0 ! n X
 φk ρk Vdr;k Vdr;k r: ρm Vm Vm 5 2rPm 1 r:½τ 2 τ t  1 ρm g 1 r: ð3:10Þ k51



 r: φp Vk ρk hk 1 p 5 r: keff rT 2 Cp ρm vt where Vdr;p is the particle draft velocity that is related to the slip velocity and is defined as: n X φk ρk Vdr;p 5 VP 2 Vf 5 Vpf 2 Vfk ρm k51

ð3:11Þ

ð3:12Þ

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36

3. VARIOUS APPLICATION OF NANOFLUID FOR HEAT TRANSFER AUGMENTATION

3.1.4 Physical Properties of the Nanofluids for Single-phase Model Base nanofluid properties have been published over the past few years in literature. However, only recently have some data on temperature-dependent properties been provided, even though they are only for nanofluid effective thermal conductivity and effective absolute viscosity. 3.1.4.1 Density In the absence of experimental data for nanofluid densities, constant-value temperature independent values, based on nanoparticle volume fraction, are used: ρnf 5 ρf ð1 2 φÞ 1 ρp φ

ð3:13Þ

3.1.4.2 Specific Heat Capacity It has been suggested that the effective specific heat can be calculated using the following equation as reported in [10]:


 ð3:14Þ Cp nf 5 Cp f ð1 2 φÞ 1 Cp p φ Other authors suggest an alternative approach based on a heat capacity concept [11]:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp p φ

ð3:15Þ

These two formulations may of course lead to different results for specific heat. Due to the lack of experimental data, both formulations are considered equivalent in estimating nanofluid specific heat capacity [12]. 3.1.4.3 Thermal Expansion Coefficient Thermal expansion coefficient of nanofluid can be obtained as follows [1]: ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þp φ 3.1.4.4 The Electrical Conductivity The effective electrical conductivity of nanofluid was presented by Maxwell [13] as below:



  σnf =σf 5 1 1 3 σP =σf 2 1 φ= σP =σf 1 2 2 σP =σf 2 1 φ

ð3:16Þ

ð3:17Þ

3.1.4.5 Dynamic Viscosity Various models have been suggested to model the viscosity of a nanofluid mixture that take into account the percentage of nanoparticles suspended in the base fluid. The classic Brinkman model [14] seems to be a proper one which has been extensively used in the studies on numerical simulation concerning nanofluids. Eq. (3.1) shows the relation between the nanofluid viscosity, base fluid viscosity, as well as the nanoparticle concentration in this model. μnf 5 μf =ð12φÞ2:5

ð3:18Þ

However, in some recent computational studies, other models have been selected to be used in the numerical process, like the work done by Abu-Nada and Chamkha [15] to investigate the convection of CuO
EG
water nanofluid in an enclosure where Namburu correlation for viscosity [16] was applied:
 ð3:19Þ log μnf 5 Ae2BT where A 5 1:837φ2 2 29:643φ 1 165:65 B 5 4 3 1026 φ2 2 0:001φ 1 0:0186

ð3:20Þ

In their study, the results where compared to that of viscosity modeled by Brinkman. It was outlined that as far as a value for normalized average Nusselt number for the fluid is concerned, for various values of Rayleigh number, the Brinkman model owns a prediction of higher value compared to that for the Namburu model showing the notable role of viscosity model used in the calculations. The authors also state that a combination of different models might also be implemented that will show different dependence on volume concentration as well as the geometry aspect ratio, along with the limitation that the models include only the ones mentioned in the APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

3.2 SIMULATION OF NANOFLUID FLOW AND HEAT TRANSFER

TABLE 3.1

37

Different Models for Viscosity of Nanofluids Used in Simulation

Model

Equation

Einstein model [17]

μnf 5 ð2:5φ 1 1Þμf ; φ , 0:05

Pak and Cho’s Correlation [18] Jang et al. model [19] Koo and Kleinstreuer [20]


 μnf 5 μf 1 1 39:11φ 1 533:9φ2 h i
22ε 2=3 μnf 5 ð2:5φ 1 1Þμf 1 1 η dp =H φ ðε 1 1Þ μnf 5 5 3 104 βφρf

( sffiffiffiffiffiffiffiffiffi β 5 0:0137ð100φÞ20:8229 for φ , 1% kB T f ðT; φÞ; dp ρp β 5 0:0011ð100φÞ20:7272 for φ . 1%

Maiga model [21]


 μnf 5 μf 1 1 7:3φ 1 123φ2

Brownian model [22]


 μnf 5 μf 1 1 2:5φ 1 6:17φ2

Nguyen model [23]


 μnf 5 μf 1 1 0:025φ 1 0:015φ2

Masoumi et al. [24]

μnf 5 μf 1 ρp VB d2p =ð72CδÞ

Gherasim et al. [25]

μnf 5 μf 0:904e14:8φ

study. Other studies have also shown that different models might lead to different results; a number of suggested relations for viscosity models used in numerical studies are presented in Table 3.1. 3.1.4.6 Thermal Conductivity Different nanofluid models based on a combination of the different formulas for the thermal conductivity adopted in the studies of natural convection are summarized in Table 3.2. Also Table 3.3 demonstrates the values of thermo-physical properties for different materials used as suspended particles in nanofluids.

3.2 SIMULATION OF NANOFLUID FLOW AND HEAT TRANSFER Several semianalytical and numerical methods have been applied successfully in order to simulate nanofluid flow and heat transfer. In the following sections we present these works.

3.2.1 Semianalytical Methods Forced convective heat transfer to Sisko nanofluid past a stretching cylinder in the presence of variable thermal conductivity was presented by Khan and Malik [34]. They used Homotopy Analysis Method (HAM) to solve the governing equations. They found that the curvature parameter assisted the temperature as well as concentration profiles. Momentum and heat transfer characteristics from heated spheroids in water-based nanofluids has been investigated by Sasmal and Nirmalkar [35]. They showed that the smaller the nanoparticles’ size the better the heat transfer at low Reynolds number and volume fraction. Hayat et al. [36] studied the effects of homogeneous
heterogeneous reactions in flow of magnetite-Fe3O4 nanoparticles by a rotating disk. They showed that the axial, radial, and azimuthal velocity profiles are a decreasing function of the Hartman number. Sheikholeslami et al. [37] utilized Least Square and Galerkin Methods to investigate MHD nanofluid flow in a semiporous channel. They indicate that velocity boundary layer thickness decreases with an increase of Reynolds number and it increases as the Hartmann number increases. Sheikholeslami et al. [38] studied the squeezing unsteady nanofluid flow using Adomian Decomposition Method (ADM). They showed that the Nusselt number increases with an increase of nanoparticle volume fraction and Eckert number. Sheikholeslami and Ganji [39] applied Homotopy perturbation method (HPM) to analysis heat transfer of Cu-water nanofluid flow between parallel plates. They indicated that the Nusselt number has a direct relationship with nanoparticle volume fraction, the squeeze number, and Eckert number when two plates are separated. Application of ADM for nanofluid Jeffery
Hamel flow with high magnetic field has been presented by Sheikholeslami et al. [40]. They proved that in greater angles or Reynolds numbers a high Hartmann number is needed to reduce backflow.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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3. VARIOUS APPLICATION OF NANOFLUID FOR HEAT TRANSFER AUGMENTATION

TABLE 3.2 Different Models for Thermal Conductivity of Nanofluids Used in Simulation Model

Equation




 1 5 3 10 βφρf Cp f

sffiffiffiffiffiffiffiffiffi kB T f ðT; φÞ dp ρp

Koo and Kleinstreuer [20]

k 1 2k 2 2φðk 2 k Þ knf 5 kf kp 1 2kf 1 φ k f2 k p ð f pÞ p f

Jang and Choi model [26]

knf 3df 5 ð1 2 φÞ 1 Bkp φ 1 18 3 106 kf Re2dp Pr φ kf dp pffiffiffiffiffiffiffi knf 5 0:25kf ð3φ 2 1Þkp =kf 1 ½3ð1 2 φÞ 2 1 1 ΔB  2 ΔB 5 ð3φ21Þkp =kf 1 ð3ð12φÞ21Þ 1 8kp =kf h
0:369
0:7476 0:9955 1:2321 i knf 5 kf 1 1 64:7φ0:7640 df =dp kf =kp PrT Re

Bruggeman model [27]

Chon et al. model [28]

4

"
#  knf kp 1 2kf 2 2φ kf 2 kp
 1 1 bφPem 5 p kf kp 1 2kf 1 φ kf 2 kp

Charuyakorn et al. [29]


 knf 5 kf 1 1 kp φdf = kf ð1 2 φÞdp " #
 knf kp 1 2kf 2 2φ kf 2 kp ð11ηÞ3 5
 kf kp 1 2kf 1 φ kf 2 kp ð11ηÞ3

Staionary model [30] Yu and Choi [31]

Patel et al. [32]

" # knf kp df φ 2kB Tdp 11c 511 kf dp ð1 2 φÞ kf παf μf d2p

Mintsa et al. [33]

knf 5 kf ð1:72φ 1 1:0Þ

TABLE 3.3 The Thermo-physical Properties of the Nanofluid

Pure water

ρ ðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

βðK 21 Þ

997.1

4179

0.613

21 3 1025

σðΩUmÞ21

25

0:05

Copper (Cu)

8933

385

401

1.67 3 10

5:96 3 107

Silver (Ag)

10 500

235

429

1.89 3 1025

3:60 3 107

Alumina (Al2 O3 )

3970

765

40

0.85 3 1025

1 3 10210

Titanium Oxide (TiO2 )

4250

686.2

8.9538

0.9 3 1025

1 3 10212

Flow and heat transfer of Cu-water nanofluid between a stretching sheet and a porous surface in a rotating system was studied by Sheikholeslami et al. [41]. They showed that for both suction and injection, the heat transfer rate at the surface increases with increasing nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with the power of rotation parameter. Sheikholeslami et al. [42] used HAM to describe nanofluid flow over a permeable stretching wall in a porous medium. They found that an increase in the nanoparticle volume fraction will decrease the momentum boundary layer thickness and entropy generation rate while this increases the thermal boundary layer thickness. Sheikholeslami and Ganji [43] utilized Galerkin’s optimal homotopy asymptotic method to investigate magnetohydrodynamic nanofluid flow in a permeable channel. They showed that velocity boundary layer thickness decreases with the increase of Reynolds number and nanoparticle volume fraction, and it increases as the Hartmann number increases. Sheikholeslami et al. [44] presented an application of HPM for simulation of two-phase unsteady nanofluid flow and heat transfer between parallel plates in the presence of a timedependent magnetic field. Nanofluid flow and heat transfer between parallel plates considering Brownian motion has been investigated by Sheikholeslami and Ganji [45]. They used Differential Transformation Method (DTM) to solve the governing equations. They showed that skin friction coefficient increases with the increase of the squeeze number and Hartmann number. Sheikholeslami et al. [46] studied the steady nanofluid flow between parallel plates. They indicated that the Nusselt number augments with an increase of viscosity parameters but it decreases with the augment of the magnetic parameter, thermophoretic parameter, and Brownian parameter. DTM has been applied by Domairry et al. [47] to solve the problem of free convection heat transfer of non-Newtonian nanofluid between two vertical flat plates. They showed that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases. Table 3.4 shows the summary of the semianalytical method studies on nanofluid. APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

TABLE 3.4 Summary of the Semianalytical Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami et al. [37]

Least Square and Galerkin Methods

Cu-water

1 # Re # 10 0 # φ # 0:06 0 # Ha # 10 Pr 5 6:8

Velocity boundary layer thickness decreases with increase of Reynolds number and it increases as Hartmann number increases

Sheikholeslami et al. [38]

Adomian Decomposition Method

Cu-water

21#S#1 0 # φ # 0:06 0:01 # Ec # 2 Pr 5 6:2

Nusselt number increases with increase of nanoparticle volume fraction and Eckert number but it decreases with increase of the squeeze number

Ag-water Al2O3-water TiO2-water

Sheikholeslami and Ganji [39]

Homotopy perturbation method

Cu-water

21#S#1 0 # φ # 0:06 0 # Ec # 1 0#δ#1 Pr 5 6:2

Nusselt number has direct relationship with nanoparticle volume fraction, δ, the squeeze number, and Eckert number when two plates are separated but it has reverse relationship with the squeeze number when two plates are squeezed

Sheikholeslami et al. [40]

Adomian decomposition method

Cu-water

100 # Re # 300 2:5 # α # 5 0 # φ # 0:06 0 # Ha # 2000 Pr 5 6:2

In greater angles or Reynolds numbers high Hartmann number is needed to reduce backflow

Sheikholeslami et al. [41]

Homotopy Analysis Method

Cu-water

5 # R # 20 0:5 # Kr # 6 21#λ#1 0 # φ # 0:2 0 # Ha # 2000 Pr 5 6:2

For both suction and injection, the heat transfer rate at the surface increases with increasing nanoparticle volume fraction, Reynolds number, and injection/suction parameter and it decreases with power of rotation parameter

Sheikholeslami et al. [42]

Homotopy Analysis Method

Cu-water

0:1 # Re # 1:5 2 0:1 # fw # 1 0#n#5 0 # φ # 0:2 0 # Ha # 2000 Pr 5 6:2

Increase in the nanoparticle volume fraction will decrease momentum boundary layer thickness and entropy generation rate while this increases the thermal boundary layer thickness. Such effects are found to be more noticeable in the Ag-water solution than in the other solutions

Ag-water Al2O3-water TiO2-water

(Continued)

TABLE 3.4 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami and Ganji [43]

Galerkin Optimal Homotopy Asymptotic Method

Cu-water

0 # Ha # 20 1 # Re # 20 0 # φ # 0:06 Pr 5 6:2

Velocity boundary layer thickness decreases with increase of Reynolds number and nanoparticle volume fraction and it increases as the Hartmann number increases

Sheikholeslami et al. [44]

Homotopy perturbation method

Al2O3-water

0 # Ha # 12 0 # S # 12 0:01 # Ec # 0:4 0:5 # Sc # 12 Pr 5 10

Nusselt number is an increasing function of the Hartmann number, Eckert number, and Schmidt number but it is a decreasing function of the squeeze number

Sheikholeslami and Ganji [45]

Differential Transformation Method

CuO-water, Al2O3water

0 # S # 10 0 # Ha # 20 2 1 # Hs # 2 10 0:01 # Ec # 0:4 0:5 # Sc # 12 Pr 5 10

Skin friction coefficient increases with an increase of the squeeze number and Hartmann number but it decreases with an increase of nanofluid volume fraction. Nusselt number increases with augment of nanoparticle volume fraction, and Hartmann number while it decreases with increase of the squeeze number

Sheikholeslami et al. [46]

Differential Transformation Method

Al2O3-water

0 # S # 10 0 # Ha # 20 2 1 # Hs # 2 10 0:01 # Ec # 0:4 0:5 # Sc # 12 Pr 5 10

Nusselt number augments with increase of viscosity parameters but it decreases with augment of Magnetic parameter, thermophoretic parameter, and Brownian parameter

Domairry et al. [47]

Differential Transformation Method

Cu-water

0 # Ha # 20 0 # δ # 100 0 # φ # 0:1 0 # Ec # 6 Pr 5 6:2

As the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases

3.2 SIMULATION OF NANOFLUID FLOW AND HEAT TRANSFER

41

3.2.2 Runge
Kutta Method Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet has been investigated by Malvandi et al. [48]. They showed that Cu-water nanofluids exhibit a better thermal performance than the other considered nanofluids. Malvandi [49] investigated the unsteady flow of a nanofluid in the stagnation point region of a time-dependent rotating sphere. Ashorynejad et al. [50] studied nanofluid flow and heat transfer due to a stretching cylinder in the presence of magnetic field. They showed that choosing copper (for a small magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem. Heating a permeable stretching surface in a porous medium was studied by Sheikholeslami and Ganji [51]. Three-dimensional nanofluid flow, heat and mass transfer in a rotating system has been presented by Sheikholeslami and Ganji [52]. They showed that the Nusselt number has a direct relationship with Reynolds number while it has a reverse relationship with rotation parameter and magnetic parameter. Sheikholeslami et al. [53] studied the nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field. Sheikholeslami and Ganji [54] studied two-phase modeling of nanofluid in a rotating system with a permeable sheet. Unsteady nanofluid flow and heat transfer in the presence of a magnetic field considering thermal radiation has been investigated by Sheikholeslami and Ganji [55]. Sheikholeslami et al. [56] studied MHD nanofluid flow and heat transfer considering viscous dissipation. They showed that the magnitude of the skin friction coefficient is an increasing function of the magnetic parameter, rotation parameter, and Reynolds number and it is a decreasing function of the nanoparticle volume fraction. Sheikholeslami et al. [57] studied the effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of a two-phase model. Sheikholeslami [58] used KKL model for simulating nanofluid flow and heat transfer in a permeable channel. The effect of uniform suction on nanofluid flow and heat transfer over a cylinder has been studied by Sheikholeslami [59]. Sheikholeslami and Abelman [60] studied two-phase simulation of nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field. Nanofluid spraying on an inclined rotating disk for cooling process has been investigated by Sheikholeslami et al. [61]. Sheikholeslami et al. [62] investigated nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation. They showed that skin friction coefficient increases with the increase of Reynolds number and suction parameter but it decreases with the increase of nanoparticle volume fraction. Table 3.5 shows the summary of the Runge
Kutta method studies on nanofluid. Chamkha and Aly [63] have studied the boundary layer flow of a nanofluid past a vertical flat plate. They have considered the Brownian motion and the thermophoresis effect. They have transformed the governing equations to a nonsimilar form and used numerical techniques to solve the same. They have reported that the local skin friction coefficient increased as any of the suction, injection parameter, thermophoresis parameter, Lewis number, or heat generation or absorption parameters increased, while it decreased as any of the buoyancy ratio, Brownian motion parameter, or the magnetic field parameter increased.

3.2.3 Finite Difference Method Chamkha and Rashad [64] have studied the flow of a nanofluid around a nonisothermal wedge. They have considered the Brownian movement and the thermophoresis effects. They have concluded that the local skin friction coefficient, local Nusselt number, and the local Sherwood number reduced as either of the magnetic parameter or the pressure gradient parameter was increased. The presence of the Brownian motion and the thermophoresis effects caused the local Nusselt number to decrease and the Sherwood number to increase. Sheremet and Pop [65] used Buongiorno’s mathematical model for conjugate natural convection in a square porous cavity filled with nanofluid. They showed that high thermophoresis parameter, low Brownian motion parameter, low Lewis and Rayleigh numbers, and high thermal conductivity ratio reflect essential nonhomogeneous distribution of the nanoparticles inside the porous cavity. Sheremet et al. [66] studied the threedimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno’s mathematical model. Sheremet et al. [67] investigated the effect of thermal stratification on free convection in a square porous cavity filled with a nanofluid using Tiwari and Das’ nanofluid model. Ghalambaz et al. [68] studied the free convection heat transfer in a porous cavity filled with a nanofluid using Tiwari and Das’ nanofluid model. Double-diffusive mixed convection in a porous open cavity filled with a nanofluid using Buongiorno’s model has been studied by Sheremet et al. [69]. Sheremet and Pop [70] studied nanofluid free convection in a triangular porous cavity porous. Natural convection in a horizontal cylindrical annulus filled with a porous medium saturated by a nanofluid has been investigated by Sheremet and Pop [71]. Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid has been studied by Sheremet et al. [72].

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

TABLE 3.5 Summary of the Runge
Kutta Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Malvandi et al. [48]

Runge
Kutta
Fehlberg method

Cu-water, Al2O3-water and TiO2-water nanofluids

1 # dp # 100 nm 0 # φ # 0:2 Pr 5 6:2

Cu-water nanofluids exhibit a better thermal performance than the other considered nanofluids. Slip velocity at the walls increase the heat transfer rate

Malvandi [49]

Runge
Kutta
Fehlberg method

Nonhomogeneous mixtures

1 # dp # 100 nm 0 # φ # 0:2

Increasing the thermophoresis is found to decrease heat transfer and concentration rates. This decrease is suppressed for higher thermophoresis numbers. In addition, it was observed that unlike the heat transfer rate, a rise in Brownian motion leads to an increase in concentration rate

Ashorynejad et al. [50]

Fourth order Runge
Kutta method

Cu-water Ag-water Al2O3-water TiO2-water

1 # Re # 5 0 # M # 10 0 # φ # 0:2 Pr 5 6:2

Choosing copper (for small of magnetic parameter) and alumina (for large values of magnetic parameter) leads to the highest cooling performance for this problem

Sheikholeslami and Ganji [51]

Fourth order Runge
Kutta method

Cu-water

1 # Re # 5 2 0:5 # γ # 1 0#ε#1 0 # φ # 0:2 Pr 5 6:2

Choosing titanium oxide as the nanoparticle and ethylene glycol as base fluid proved to have the highest cooling performance for this problem

1 # R # 10 0 # M # 12 2 0:5 # γ # 1 Kr 5 0:5; Nt 5 0:1 Nb 5 0:1; Sc 5 0:1 Pr 5 10

Nusselt number has a direct relationship with Reynolds number while it has a reverse relationship with rotation parameter, magnetic parameter, Schmidt number, thermophoretic parameter and Brownian parameter

Ag-water Al2O3-water TiO2-water Cu- Ethylene glycol

Sheikholeslami and Ganji [52]

Fourth order Runge
Kutta method

Al2O3-water

Sheikholeslami et al. [53]

Fourth order Runge
Kutta method

Cu-water Ag-water Al2O3-water TiO2-water

5 # R # 20 0 # M # 50 0:5 # Kr # 6 0 # φ # 0:2 2 0:5 # λ # 1 Pr 5 6:2

The highest values are obtained when titanium oxide is used as nanoparticle. Also it can be found that the Nusselt number decreases with an increase of magnetic parameter due to the presence of Lorentz forces

Sheikholeslami and Ganji [54]

Fourth order Runge
Kutta method

Al2O3-water

1 # R # 10 0:5 # Kr # 6 0#λ#3 Nt 5 Nb 5 0:1 Pr 5 10

The Nusselt number has a direct relationship with the Reynolds number and injection parameter, while it has a reverse relationship with rotation parameter, Schmidt number, thermophoretic parameter and Brownian parameter

Sheikholeslami and Ganji [55]

Fourth order Runge
Kutta method

Al2O3-water

0:5 # S # 12 0:01 # Ec # 0:4 0:01 # Nt # 0:4 0:5 # Sc # 20 0 # Rd # 16 Pr 5 10

Concentration boundary layer thickness increases with increase of radiation parameter

Sheikholeslami et al. [56]

Fourth order Runge
Kutta method

CuO-water, Al2O3-water

0:1 # R # 2 0#M#9 0 # φ # 0:04 1 # Kr # 6 0:01 # Ec # 0:04 Pr 5 10

The magnitude of the skin friction coefficient is an increasing function of the magnetic parameter, rotation parameter, and Reynolds number and it is a decreasing function of the nanoparticle volume fraction

Sheikholeslami et al. [57]

Fourth order Runge
Kutta method

Al2O3-water

0:1 # R # 20 0 # Rd # 12 0:01 # Sc # 6 0 # M # 16 0 # φ # 0:04 0:1 # Kr # 12 Pr 5 10

Nusselt number has a direct relationship with the radiation parameter and Reynolds number while it has a reverse relationship with other active parameters. Also it can be found that concentration boundary layer thickness decreases with the increase of the radiation parameter

Sheikholeslami [58]

Fourth order Runge
Kutta method

Al2O3-water

21#α#2 1#R#4 0 # φ # 0:04 0#m#3 Pr 5 6:8

Heat transfer enhancement has a direct relationship with the Reynolds number when the power law index is equal to zero but an opposite trend is observed for other values of the power law index (Continued)

TABLE 3.5 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami [59]

Fourth order Runge
Kutta method

CuO-water, Al2O3-water

0:1 # Re # 2:7 0#γ#2 0 # φ # 0:04 Pr 5 6:2

Adding nanoparticle into the base fluid of this problem is capable of changing the flow pattern. It is found that the Nusselt number is an increasing function of nanoparticle volume fraction, suction parameter, and Reynolds number

Sheikholeslami and Abelman [60]

Fourth order Runge
Kutta method

Al2O3-water

1#R#4 1#M#4 0:01 # Ec # 0:1 0:01 # Nt # 0:4 0:01 # Nt # 0:6 0:001 # Ec # 0:02 Pr 5 10

The Nusselt number has a direct relationship with the aspect ratio and Hartmann number but it has a reverse relationship with the Reynolds number, Schmidt number, Brownian parameter, thermophoresis parameter. and Eckert number

Sheikholeslami et al. [61]

Fourth order Runge
Kutta method

Al2O3-water

0:2 # δ # 1 0:1 # Nt 5 Nb # 0:4 0:1 # Sc # 8 Pr 5 10

The Nusselt number is an increasing function of each active parameter. Latent heat has a direct relationship with the Schmidt number, Brownian parameter, and thermophoretic parameter but it has a reverse relationship with normalized thickness.

Sheikholeslami et al. [62]

Fourth order Runge
Kutta method

Cu-water

0:5 # γ # 2 0:5 # Rd # 2 0 # φ # 0:04 Pr 5 6:8

Skin friction coefficient increases with increase of Reynolds number and suction parameter but it decreases with increase of nanoparticle volume fraction

3.2 SIMULATION OF NANOFLUID FLOW AND HEAT TRANSFER

45

Khan et al. [73] studied the three-dimensional flow of nanofluid induced by an exponentially stretching sheet. They showed that existence of interesting Sparrow
Gregg-type hills for temperature distribution corresponding to some ranges of parametric values. Nanofluid flow with multimedia physical features for conjugate mixed convection and radiation has been studied by Hsiao [74]. Table 3.6 shows the summary of the finite difference method studies on nanofluid.

3.2.4 Finite Volume Method Garoosi and Hoseininejad [75] investigated the natural and mixed convection heat transfer between differentially heated cylinders in an adiabatic enclosure filled with nanofluid. Garoosi et al. [76] applied Buongiorno model for mixed convection of the nanofluid in heat exchangers. Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating has been studied by Garoosi et al. [77]. Teamah et al. [78] studied the augmentation of natural convective heat transfer in a square cavity by utilizing nanofluids in the presence of magnetic field. They showed that for a weak magnetic field the addition of nanoparticles is necessary to enhance the heat transfer but for a strong magnetic field there is no need for nanoparticles because the heat transfer will decrease. Santra et al. [79] studied the heat transfer augmentation in a differentially heated square cavity using copper
water nanofluid. Das and Ohal [80] investigated Natural convection heat transfer augmentation in a partially heated and partially cooled square cavity utilizing nanofluids. Oztop et al. [81] analyzed the nonisothermal temperature distribution on natural convection in nanofluid filled enclosures. They showed that an enhancement in heat transfer rate was registered for the whole range of Rayleigh numbers. Table 3.7 shows the summary of the finite volume method studies on nanofluid.

3.2.5 Finite Element Method MHD mixed convection of nanofluid filled partially heated triangular enclosure with a rotating adiabatic has been investigated by Selimefendigil and Oztop [82]. They showed that local and average heat transfer and total entropy generation are enhanced as the solid volume fraction of nanoparticle and angular rotational speed of the cylinder increases and the Hartmann number decreases. Heat transfer enhancements around 30% are achieved for the highest volume fraction compared to base fluid. Selimefendigil and Oztop [83] studied the natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation. Selimefendigil and Oztop [84] studied pulsating nanofluids jet impingement cooling of a heated horizontal surface. They showed that the combined effect of pulsation and inclusion of nanoparticles is not favorable for the stagnation point heat transfer enhancement for some combinations of Reynolds number and nanoparticle volume fraction. Selimefendigil and Oztop [85] studied MHD mixed convection in a nanofluid filled lid driven square enclosure with a rotating cylinder. Selimefendigil and Oztop [86] investigated numerical investigation and reduced order model of mixed convection at a backward facing step with a rotating cylinder subjected to nanofluid. Effect of nanoparticle shape on mixed convection due to rotating cylinder in an internally heated and flexible walled cavity filled with SiO2-water nanofluids has been investigated by Selimefendigil et al. [87]. They indicated that Nusselt number enhances with external Rayleigh number and nanoparticle volume fraction while the opposite behavior is seen as the value of internal Rayleigh number and flexibility of the wall increase. Conjugate natural convection in a cavity with a conductive partition and filled with different nanofluids on different sides of the partition has been studied by Selimefendigil and Oztop [88]. They proved that as the value of the Grashof number, thermal conductivity ratio (Kr), and nanoparticle volume fraction increase, average Nusselt number increases. Table 3.8 shows the summary of the finite element method studies on nanofluid.

3.2.6 Control Volume-based Finite Element Method Heatline analysis has been used by Sheikholeslami et al. [89] to investigate two-phase simulation of nanofluid flow and heat transfer. They found that Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then starts increasing. As Lewis number increases, this minimum value occurs at higher buoyancy ratio number. Natural convection heat transfer in a cavity with a sinusoidal wall filled with CuO-water nanofluid in presence of magnetic field has been studied by Sheikholeslami et al. [90]. Effects of a magnetic field on natural convection in different enclosures filled with naofluids have been examined by Sheikholeslami et al. [91
93]. Soleimani et al. [94] studied the natural convection heat transfer in a nanofluid filled semiannulus enclosure. They found that there is an optimum angle of turn in which the average Nusselt APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

TABLE 3.6 Summary of the finite Difference Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheremet and Pop [65]

Finite difference method

Porous cavity with water-based nanofluid

Ra 5 100
500; Le 5 1
10; Nb 5 0.1
0.4; Nt 5 0.1
0.4; Nr 5 0.1
0.4; Kr 5 0.1
10.0; D 5 0.1
0.3.

High thermophoresis parameter, low Brownian motion parameter, low Lewis and Rayleigh numbers, and high thermal conductivity ratio reflect essential nonhomogeneous distribution of the nanoparticles inside the porous cavity

Sheremet et al. [66]

Finite difference method

Porous cavity with water-based nanofluid

Ra 5 30
500, Le 5 1
10, Nr 5 0.1
0.4, Nb 5 0.1
0.4, Nt 5 0.1
0.4, aspect ratio (A 5 0.2
5.0).

Average Nusselt number at the hot surface is an increasing function of the Rayleigh number and Brownian motion parameter, and a decreasing function of the Lewis number, buoyancy-ratio, and thermophoresis parameters. The average Sherwood number at the hot surface is an increasing function of the Rayleigh number and thermophoresis parameter, and a decreasing function of the Lewis number, buoyancy-ratio, and Brownian motion parameters.

Sheremet et al. [67]

Finite difference method

Cu-water

Ra 5 10
500, ϕ 5 0.0
0.05, porosity of porous medium (ε 5 0.1
0.8), thermal stratification parameter (b 5 0
1.0) solid matrix of the porous medium (aluminum foam and glass balls)

An increase in the thermal stratification parameter leads to an essential reduction of the convective core sizes, displacement of this core close to the lower left corner and rotation of this vortex along the horizontal axis. More intensive attenuation of convective flow with ε and b occurs for low thermal conductivity material for the solid matrix of the porous medium

Ghalambaz et al. [68]

Finite difference method

Cu-water


60 , α , 60 , aspect ratio 0.1 , A , 10, 10 , Ra , 1000, 0 , ϕ , 0.1, ε 5 0.3, 0.5, 0.7, ks 5 2kbf

Presence of nanoparticles deteriorates the heat transfer in all studied cases. The decrease of the porosity increases the porous matrix thermal conductivity while the decrease of the inclination angle and of the aspect ratio would boost the deterioration of heat transfer

Sheremet et al. [69]

Finite difference method

Porous cavity with water-based nanofluid

Re 5 10
100, Pr 5 0.2, Ra 5 50
500, usual Lewis number (Le 5 1
50), Dufour-contaminant Lewis number (Ld 5 1
50), nanofluid Lewis number (Ln 5 1
50), Nr 5 0.1
0.4, regular double-diffusive buoyancy ratio (Nc 5 0.1
0.4), Nb 5 0.1
0.4, Nt 5 0.1
0.4, modified Dufour parameter (Nd 5 0.1
0.4)

Average Nusselt number at hot vertical wall is an increasing function of the Rayleigh and Reynolds numbers, and a decreasing function of the usual Lewis number. While the average Sherwood number at this vertical wall is an increasing function of the usual Lewis. Effects   of the Rayleigh and Reynolds numbers on Sh  and the thermophoresis parameter on the average Nusselt and Sherwood numbers are nonmonotonic. It has been shown that in the present porous problem the Richardson number does not define the prevailing of the forced or natural convection modes

Sheremet and Pop [70]

Finite difference method

Porous cavity with water-based nanofluid

Ra 5 100
500; Le 5 1
10; Nb 5 0.1
0.4; Nt 5 0.1
0.4; Nr 5 0.1
0.4; A 5 H/L 5 1.0.

Average Nusselt number is an increasing function of Ra, Le, and a decreasing function of Nr, Nb, Nt. At the same time the average Sherwood number is an increasing function of Ra, Le, Nb, Nt and a decreasing function of Nr.

Sheremet and Pop [71]

Finite difference method

Cu-water

Ra 5 200
700, ε 5 0.1
0.8, ϕ 5 0.0
0.05, the annulus radius ratio (R 5 1.5, 2.0, 2.5), solid matrix of porous medium (aluminum foam and glass balls)

Average Nusselt number is an increasing function of the Rayleigh number and annulus radius ratio and a decreasing function of the porosity of porous medium regardless of the solid matrix material. A decrease in the thermal conductivity of the solid matrix material leads to an attenuation of the convective heat transfer inside the annulus for high values of the porosity and solid volume fraction of nanoparticles

Sheremet et al. [72]

Finite difference method

Water-based nanofluid

Ra 5 105, Le 5 10, Pr 5 6.26, Nr 5 0.1, Nb 5 0.1, Nt 5 0.1, A 5 1, Ha 5 0
100, undulation number (κ 5 1
3), wavy contraction ratio (b 5 0.1
0.3), ϕ 5 0
π, τ 5 0
0.13.

An increase in the Hartmann number leads to an attenuation of convective flow and heat transfer and a formation of a double-core convective cell for high values of Ha. An increase in the wavy contraction ratio leads to an increase in the wave amplitude and an attenuation of the convective flow with more intensive heating of the wavy troughs

(Continued)

TABLE 3.6 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Khan et al. [73]

Finite difference scheme known as Keller-box method

Water-based nanofluid

0#λ#1 0:1 # Nb # 0:9 0:3 # Nt # 0:5 5 # Pr # 10 21#A#3

The existence of interesting Sparrow
Greggtype hills for temperature distribution corresponding to some range of parametric values

Kai-Long Hsiao [74]

An improved finitedifference with box method

Cu-water

K 5 400 (W/mK): ρ 5 8940 ðkg=m3 Þ Cp 5 385 ðJ=kgKÞ 101 # Re # 106 0 # Pr # 1000

The results show that dimensionless heat transfer effects increase with increasing values of Pr, R0, Nb, or G but decrease with increasing M, Nt, or Ec. Those, parameters Pr, R0, Nb, or G are important factors in this study for increasing the heat transfer effects. The fourth, for mass transfer of the values Sc and Nb can be obtained a good mass diffusion effect, but the parameter Nt has not been processed for this kind of function

TABLE 3.7 Summary of the Finite Volume Method Studies on Nanofluid Authors Garoosi and Hoseininejad [75]

Geometry of study

Method Finite volume method

Type of nanofluid

Properties

Cu-water

10 # Ra # 10 0 # φ # 0:05 25 # dp # 145 Pr 5 6:8

Changing the location of the heat source/sink from bottom
top to top
bottom configuration, the heat transfer rate decreases significantly

TiO2-water Al2O3-water

4

Remarks 7

Garoosi et al. [76]

Finite volume method

Al2O3-water

104 # Ra # 107 0:01 # Ri # 1000 102 # Gr # 104 0 # φ # 0:05 25 # dp # 145

At low Rayleigh numbers and high Richardson numbers, the particle distribution is fairly nonuniform while at high Ra and low Ri values particle distribution remains almost uniform for free and mixed convection cases, respectively

Garoosi et al. [77]

Finite volume method

Cu-water

104 # Ra # 107 0:01 # Ri # 1000 0 # φ # 0:05 25 # dp # 145

Thermophoretic effects are negligible for nanoparticles with high thermal conductivity. As a result, in such conditions the use of homogeneous and single-phase models is valid at any Ra and Ri

TiO2-water Al2O3-water

Teamah et al. [78]

Finite volume method

Water-based nanofluid

104 # Ra # 107 0 # φ # 0:05 0 # Ha # 60

At a weak magnetic field the addition of nanoparticles is necessary to enhance the heat transfer but for a strong magnetic field there is no need for nanoparticles because the heat transfer will decrease

Santra et al. [79]

Finite volume method

Cu-water

104 # Ra # 107 0 # φ # 0:05 A51

The heat transfer rate decreases with increase in solid volume fraction for a particular Rayleigh number. However, it increases with Rayleigh number for a particular solid volume fraction

(Continued)

TABLE 3.7 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Das and Ohal [80]

Finite volume method

Cu-water

104 # Gr # 107 0 # φ # 0:02 A51

The rate of heat transfer increases with an increase in the nanoparticles volume fraction

Oztop et al. [81]

Finite volume method

Cu-water TiO2-water Al2O3-water

104 # Gr # 107 0 # φ # 0:01 0 # ϕ # 903 A51

An enhancement in heat transfer rate was registered for the whole range of Rayleigh numbers

TABLE 3.8 Summary of the Finite Element Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Selimefendigil and Oztop [82]

Finite element method

Cu-water

10 # Gr # 10 0 # φ # 0:05 0 # Ha # 50 2 20 # ω # 20 Pr 5 7:1

Local and average heat transfer and total entropy generation enhance as the solid volume fraction of nanoparticle and angular rotational speed of the cylinder increases and Hartmann number decreases. Heat transfer enhancements around 30% are achieved for the highest volume fraction compared to base fluid

Selimefendigil and Oztop [83]

Finite element method

Cu-water

104 # RaE # 106 104 # RaI # 106 0 # φ # 0:05 0 # Ha # 50 Pr 5 6:9

As the value of the external Rayleigh number decreases, internal Rayleigh number and Hartmann number increase, average heat transfer enhance. Average Nusselt number enhances by about 40%
60% at the highest solid volume fraction when compared to base fluid with different obstacles installed within the cavity.

Selimefendigil and Oztop [84]

Finite volume method

Al2O3-water

100 # Re # 400 1 # f # 10 0 # φ # 0:06 Pr 5 7:1

The combined effect of pulsation and inclusion of nanoparticles is not favorable for the stagnation point heat transfer enhancement for some combinations of Reynolds number and nanoparticle volume fraction

Selimefendigil and Oztop [85]

Finite element method

Cu-water

0:001 # Ri # 10 0 # Ha # 50 0 # φ # 0:05 2 10 # Ω # 10 Pr 5 6:2

Average heat transfer is enhanced with Richardson number, nanoparticle volume fraction, and cylinder rotation, while magnetic field reduces the convection. The combined effect of magnetic field and cylinder rotation acts in a way to enhance local heat transfer for some locations along the heated wall

4

6

(Continued)

TABLE 3.8 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Selimefendigil and Oztop [86]

Finite element method

Al2O3-water

50 # Re # 200 Gr 5 104 0 # φ # 0:05 24#Ω#4 Pr 5 6:2

Cylinder rotation affects the flow and thermal patterns behind the step. The addition of the nanoparticles enhances the averaged heat transfer along the bottom wall downstream of the step. There is almost a linear relation between heat transfer enhancement and nanoparticle volume fraction

Selimefendigil et al. [87]

Finite element method

SiO2-water

103 # RaE # 5 3 105 104 # RaI # 106 5 3 102 # E # 106 0 # φ # 0:05 2 2000 # Ω # 2000

Average Nusselt number enhances with external Rayleigh number and nanoparticle volume fraction while the opposite behavior is seen as the value of internal Rayleigh number and flexibility of the wall increases. Among different nanoparticle types cylindrical ones show the best performance in terms of heat transfer enhancement

Selimefendigil and Oztop [88]

Finite element method

Al2O3-water and CuO-water

103 # Gr # 106 0o # θ # 270o 0:01 # Kr # 10 0 # φ # 0:04

As the value of the Grashof number, thermal conductivity ratio (Kr), and nanoparticle volume fraction increase, the average Nusselt number increases. When nanoparticles with low thermal conductivity on the right cavity are added it is more effective for the heat transfer enhancement compared to adding nanoparticles with high thermal conductivity

3.2 SIMULATION OF NANOFLUID FLOW AND HEAT TRANSFER

53

number is maximum for each Rayleigh number. Moreover, the angle of turn has an important effect on the streamlines, isotherms, and maximum or minimum values of local Nusselt number. Effects of MHD on Cu-water nanofluid flow and heat transfer has been studied by Sheikholeslami et al. [95]. Constant temperature and heat flux boundary conditions for an Al2O3-water nanofluid filled enclosure have been examined by Sheikholeslami et al. [96
99]. Sheikholeslami et al. [100] studied free convection heat transfer in a nanofluid filled inclined Lshaped enclosure. Ferrohydrodynamic and Magnetohydrodynamic effects on ferrofluid flow and convective heat transfer has been investigated by Sheikholeslami and Ganji [101]. They found that Nusselt number increases with augment of Rayleigh number and nanoparticle volume fraction but it decreases with an increase of Hartmann number. Magnetic number has a different effect on Nusselt number corresponding to the Rayleigh number. Sheikholeslami et al. [102] considered the effect of thermal radiation on ferrofluid flow and heat transfer in a semiannulus enclosure in the presence of a magnetic source. Sheikholeslami Kandelousi [103] studied the effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Free convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder has been presented by Sheikholeslami et al. [104]. Sheikholeslami and Rashidi [105] studied the effect of space dependent magnetic field on free convection of Fe3O4-water nanofluid. Effect of nonuniform magnetic field on forced convection heat transfer of nanofluid has been studied by Sheikholeslami et al. [106]. Electrohydrodynamic nanofluid hydrothermal treatment in an enclosure with sinusoidal upper wall has been investigated by Sheikholeslami and Ellahi [107]. Sheikholeslami et al. [108] applied a two-phase model for magnetic nanofluid forced convective heat transfer in the existence of variable magnetic field. Sheikholeslami et al. [109] investigated forced convection heat transfer in a semiannulus under the influence of a variable magnetic field. Sheikholeslami et al. [110] studied the effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. They found that the effect of an electric field on heat transfer is more pronounced at low Reynolds number. Sheikholeslami et al. [111] investigated nonuniform magnetic field effect on nanofluid hydrothermal treatment considering Brownian motion and thermophoresis effects. Sheikholeslami and Rashidi [112] studied ferrofluid heat transfer treatment in the presence of variable magnetic field. They found that the Nusselt number has direct relationship with Richardson number, nanoparticle volume fraction while it has an inverse relationship with Hartmann number and magnetic number. Table 3.9 shows the summary of the control volume-based finite element method studies on nanofluid.

3.2.7 Lattice Boltzmann Method Investigation of nanofluid flow and heat transfer in presence of magnetic field using KKL model has been studied by Sheikholeslami et al. [113]. Nanofluid hydrothermal behaviors in an enclosure with curve boundaries have been studied by Sheikholeslami et al. [114
117]. Ashorynejad et al. [118] studied magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus. They found that flow oscillations can be suppressed effectively by imposing an external radial magnetic field. MHD effects on nanofluid flow and heat transfer in a semiannulus enclosure has been studied by Sheikholeslami et al. [119,120]. They showed that the enhancement in heat transfer increases as Hartmann number increases but it decreases with increase of Rayleigh number. Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field has been investigated by Sheikholeslami and Gorji [121]. They found that particles with a smaller size have better ability to dissipate heat, and a larger volume fraction would provide a stronger driving force which leads to an increase in temperature profile. Sheikholeslami et al. [122] studied MHD free convection in an eccentric semiannulus filled with nanofluid. Sheikholeslami et al. [123] simulated MHD CuO-water nanofluid flow and convective heat transfer considering Lorentz forces. Entropy generation of nanofluid in the presence of magnetic field was studied by Sheikholeslami and Ganji [124]. Sheikholeslami et al. [125] simulated magnetohydrodynamic natural convection heat transfer of Al2O3-water nanofluid in a horizontal cylindrical enclosure with an inner triangular cylinder. Sheikholeslami and Ellahi [126] studied ferrofluid flow for magnetic drug targeting. They showed that back flow occurs near the region where the magnetic source is located. Sheikholeslami et al. [127] simulated the magnetic field effect on hydrothermal behavior of nanofluid in a cubic cavity. Three-dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid has been studied by Sheikholeslami and Ellahi [128]. Nanofluid heat transfer enhancement and entropy generation has been studied by Sheikholeslami et al. [129]. Effect of a magnetic source on free convection in a cavity subjugated to nanofluid has been investigated [130,131]. Table 3.10 shows the summary of the lattice Boltzmann Method studies on nanofluid. Recently several authors have applied new methods for simulation of hydrothermal behavior [132
268].

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

TABLE 3.9 Summary of the Control Volume-based Finite Element Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami et al. [89]

Control volumebased finite element method

Al2O3-water

10 # Ra # 10 0:1 # Nr # 4 1 # Le # 8 Nb 5 Nt 5 0:5 Pr 5 10

Average Nusselt number decreases as buoyancy ratio number increases until it reaches a minimum value and then starts increasing. As Lewis number increases, this minimum value occurs at higher buoyancy ratio number

Sheikholeslami et al. [90]

Control volumebased finite element method

CuO-water

103 # Ra # 105 0:1 # a # 0:3 0 # Ha # 100 0 # φ # 0:04 Pr 5 6:2

Nusselt number is an increasing function of nanoparticles volume fraction, dimensionless amplitude of the sinusoidal wall, and Rayleigh number while it is a decreasing function of Hartmann number

Sheikholeslami et al. [91]

Control volumebased finite element method

Cu-water

103 # Ra # 105 03 # λ # 903 0 # φ # 0:06 0 # Ha # 100 Pr 5 6:2

Hartmann number and the inclination angle of the enclosure can be control parameters at different Rayleigh numbers. In the presence of magnetic field the velocity field is retarded and hence convection and Nusselt number decreases

Sheikholeslami et al. [92]

Control volumebased finite element method

Cu-water

103 # Ra # 105 03 # γ # 903 0 # Ha # 100 0 # φ # 0:06 Pr 5 6:2

At Ra 5 103 , maximum value of enhancement for low Hartmann number is obtained at γ 5 03 , but for higher values of Hartmann number, maximum values of E occurs at γ 5 903

Sheikholeslami et al. [93]

Control volumebased finite element method

Cu-water

103 # Ra # 105 3#N#6 0 # Ha # 100 0 # φ # 0:06 A 5 0:5; Pr 5 6:2

Nusselt number is an increasing function of nanoparticle volume fraction, the number of undulations, and Rayleigh numbers while it is a decreasing function of Hartmann number.

3

5

Soleimani et al. [94]

Control volumebased finite element method

Cu-water

103 # Ra # 105 453 # γ # 1803 0 # φ # 0:06 Pr 5 6:2

There is an optimum angle of turn in which the average Nusselt number is maximum for each Rayleigh number. Moreover, the angle of turn has an important effect on the streamlines, isotherms, and maximum or minimum values of local Nusselt number

Sheikholeslami et al. [95]

Control volumebased finite element method

Cu-water

103 # Ra # 105 03 # γ # 903 0 # φ # 0:06 ε 5 0:9; a 5 0:8L Pr 5 6:2

Increasing Rayleigh number leads to decreased heat transfer enhancement while the opposite trend is observed with augment of Hartmann number

Sheikholeslami et al. [96]

Control volumebased finite element method

Al2O3-water

103 # Ra # 105 03 # Ha # 100 0 # φ # 0:04 0:2 # rin =L # 0:4 Pr 5 6:2

Domination of conduction mechanism causes heat transfer enhancement to increase. So enhancement in heat transfer increases with increase of Hartmann number and aspect ratio while it decreases with augment of Rayleigh number

Sheikholeslami et al. [97]

Control volumebased finite element method

Al2O3-water

103 # Ra # 105 2 903 # γ # 03 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:2

The heat transfer between cold and hot regions of the enclosure cannot be well understood by using isotherm patterns so heatline visualization technique is used to find the direction and intensity of heat transfer in a domain

Sheikholeslami et al. [98]

Control volumebased finite element method

Al2O3-water

303 # γ # 903 0:1 # Nr # 4 2 # Le # 8 Ra 5 105 Nb 5 Nt 5 0:5 Pr 5 10

Lewis number has no significant effect on Nusselt number at low values of buoyancy ratio number

Sheikholeslami et al. [99]

Control volumebased finite element method

Al2O3-water

103 # Ra # 105 0 # Ha # 100 303 # γ # 903 0:1 # Nr # 4 2 # Le # 8 Nb 5 Nt 5 0:5 Pr 5 10

As buoyancy ratio number increases the effects of other active parameters are more pronounced

(Continued)

TABLE 3.9 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami et al. [100]

Control volumebased finite element method

Cu-water

10 # Ra # 10 2 453 # ζ # 453 0 # φ # 0:06 Pr 5 6:2

The results show that for Ra 5 104 the maximum and minimum  average Nusselt number are corresponding to ζ 5 2 45 and  45 respectively, whereas the opposite trend is observed for Ra 5 105

Sheikholeslami and Ganji [101]

Control volumebased finite element method

Fe3O4-water

103 # Ra # 105 0 # Ha # 5 0 # MnF # 500 0 # φ # 0:04 Pr 5 6:8 Ec 5 1025

Nusselt number increases with augment of Rayleigh number and nanoparticle volume fraction but it decreases with increase of Hartmann number. Magnetic number has different effect on Nusselt number corresponding to Rayleigh number

Sheikholeslami et al. [102]

Control volumebased finite element method

Fe3O4-water

103 # Ra # 105 0 # Ha # 10 0 # MnF # 100 0 # Nr # 0:027 0 # φ # 0:04 Pr 5 6:8 Ec 5 1025

Nusselt number is an increasing function of Rayleigh number, nanoparticle volume fraction, and magnetic number while it is a decreasing function of with Hartmann number and Radiation parameter

Sheikholeslami Kandelousi [103]

Control volumebased finite element method

Fe3O4- water

103 # Ra # 105 0 # MnF # 100 0 # Ha # 10 0 # φ # 0:04 Pr 5 6:2

Enhancement in heat transfer decreases with increase of Rayleigh number and magnetic number but it increases with increase of Hartmann number

Sheikholeslami et al. [104]

Control volumebased finite element method

CuO-water

103 # Ra # 105 03 # γ # 903 0 # φ # 0:04 Pr 5 6:2

Increasing Rayleigh number leads to the decrease in ratio of heat transfer enhancement. For high Rayleigh number the minimum heat transfer enhancement ratio occurs at γ 5 90

3

5

Sheikholeslami and Rashidi [105]

Control volumebased finite element method

Fe3O4- water

103 # Ra # 105 0 # MnF # 100 0 # Ha # 10 0 # φ # 0:04 Pr 5 6:2

Nusselt number is an increasing function of magnetic number, Rayleigh number, and nanoparticle volume fraction while it is a decreasing function of Hartmann number

Sheikholeslami et al. [106]

Control volumebased finite element method

Fe3O4- water

10 # Re # 103 0 # Ha # 20 0 # φ # 0:04 Pr 5 6:2

Nusselt number has a direct relationship with Reynolds number and nanoparticle volume fraction while it has reverse relationship with Hartmann number

Sheikholeslami and Ellahi [107]

Control volumebased finite element method

Fe3O4-Ethylene glycol

3000 # Re # 6000 0 # Δϕ # 10 0 # φ # 0:04 Pr 5 149:54

Heat transfer rises with augment of supplied voltage and Reynolds number

Sheikholeslami et al. [108]

Control volumebased finite element method

Fe3O4- water

10 # Re # 500 2 # Le # 8 0 # Ha # 20

Nusselt number has direct relationship with Reynolds number while it has reverse relationship with Hartmann number and Lewis number

Sheikholeslami et al. [109]

Control volumebased finite element method

Fe3O4- water

10 # Re # 600 0 # MnF # 10 0 # Ha # 10 0 # φ # 0:04 Pr 5 6:8

Effects of Kelvin forces are more pronounced for high Reynolds number. Heat transfer enhancement has direct relationship with the Reynolds number and the magnetic number; while it has inverse relationship with the Hartmann number

Sheikholeslami et al. [110]

Control volumebased finite element method

Fe3O4-Ethylene glycol

3000 # Re # 6000 0 # Δϕ # 10 0 # φ # 0:04 Pr 5 149:54

Effect of electric field on heat transfer is more pronounced at low Reynolds number

(Continued)

TABLE 3.9 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami et al. [111]

Control volumebased finite element method

Fe3O4- water

103 # Ra # 105 0:1 # Nr # 4 0 # Ha # 10 2 # Le # 4 Pr 5 6:85

Nusselt number has a direct relationship with Rayleigh number, buoyancy ratio number, and Lewis number while it has a reverse relationship with Hartmann number

Sheikholeslami, and Rashidi [112]

Control volumebased finite element method

Fe3O4- water

0:001 # Ri # 10 0 # δ # 0:6 0 # MnF # 10 0 # Ha # 10 Re 5 100

Nusselt number has a direct relationship with Richardson number and nanoparticle volume fraction while it has reverse relationship with Hartmann number and Magnetic number

TABLE 3.10 Summary of the Lattice Boltzmann Method Studies on Nanofluid Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami et al. [113]

Lattice Boltzmann Method

Al2O3-water

10 # Ra # 10 0 # φ # 0:04 0 # Ha # 60 R=L 5 0:5 Pr 5 6:8

Enhancement in heat transfer increases with increase of Hartmann number except for Ra 5 104 in which Ha 5 40 roles as a critical Hartmann number

Sheikholeslami et al. [114]

Lattice Boltzmann Method

Cu-water

103 # Ra # 106 0 # φ # 0:06 0:65 # ε # 0:95 a 5 0:4 Pr 5 6:8

The minimum value of enhancement of heat transfer occurs at ε 5 0:95 for Ra 5 105 but for other values of Rayleigh number it is obtained at ε 5 0:65

Sheikholeslami et al. [115]

Lattice Boltzmann Method

Cu-water

104 # Ra # 106 0 # φ # 0:06 0 # Ha # 60 Pr 5 6:8

As the nanoparticle volume fraction and Rayleigh number increase, average Nusselt number increases but opposite trends are observed when the Hartmann number increases

Sheikholeslami et al. [116]

Lattice Boltzmann Method

Cu-water

104 # Ra # 106 0 # φ # 0:04 2 603 # γ # 603 R=L 5 0:5 Pr 5 6:8

The change of inclination angle has a significant impact on the thermal and hydrodynamic flow field

Sheikholeslami et al. [117]

Lattice Boltzmann Method

Cu-water Ag-water Al2O3-water TiO2water

103 # Ra # 106 0 # φ # 0:04 1:5 # λ # 4:5 Pr 5 6:8

Choosing copper as the nanoparticle leads to obtaining the highest enhancement for this problem

4

6

(Continued)

TABLE 3.10 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Ashorynejad et al. [118]

Lattice Boltzmann Method

Ag-water

10 # Ra # 10 0 # φ # 0:04 0 # Ha # 60 Pr 5 6:8

Flow oscillations can be suppressed effectively by imposing an external radial magnetic field

Sheikholeslami et al. [119]

Lattice Boltzmann Method

Al2O3-water

104 # Ra # 106 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:8

The enhancement in heat transfer increases as Hartmann number increases but it decreases with increase of Rayleigh number

3

6

Lattice Boltzmann Method Sheikholeslami et al. [120]

Lattice Boltzmann Method

Cu-water

104 # Ra # 106 0 # φ # 0:06 0 # Ha # 40 1:5 # λ # 4:5 Pr 5 6:8

Enhancement ratio increases with decrease of Rayleigh number and it increases with augment of Hartmann number

Sheikholeslami and Gorji [121]

Lattice Boltzmann Method

Cobalt-kerosene

104 # Ra # 106 0 # φ # 0:04 0:675 # A # 1:325 0:2 # ε # 0:8 Pr 5 6:8

Particles with a smaller size have better ability to dissipate heat, and a larger volume fraction would provide a stronger driving force which leads to increase in temperature profile

Sheikholeslami et al. [122]

Lattice Boltzmann Method

Cu-water

104 # Ra # 106 0 # φ # 0:06 0 # Ha # 40 δ 0:2 # # 0:8 L

Nusselt number has direct relationship with nanoparticle volume fraction and Rayleigh number but it has inverse relationship with Hartmann number and position of inner cylinder at high Rayleigh number

λ 5 3:5 Pr 5 6:8

Sheikholeslami et al. [123]

Lattice Boltzmann Method

CuO-water

103 # Ra # 105 0 # φ # 0:04 0 # Ha # 100 0:4 # ε # 0:8 Pr 5 6:8

Enhancement in heat transfer increases as Hartmann number and heat source length increase but it decreases with increase of Rayleigh number. Also it can be found that effect of Hartmann number and heat source length is more pronounced at high Rayleigh number

Sheikholeslami and Ganji [124]

Lattice Boltzmann Method

CuO-water

103 # Ra # 105 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:8

Heat transfer rate and Dimensionless entropy generation number increase with increase of the Rayleigh number and nanoparticle volume fraction but it decreases with increase of the Hartmann number

Sheikholeslami et al. [125]

Lattice Boltzmann Method

Al2O3-water

103 # Ra # 105 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:8

Lattice Boltzmann method with doublepopulation is a powerful approach for the simulation of natural convection heat transfer in nanofluids in regions with curved boundaries

Sheikholeslami and Ellahi [126]

Lattice Boltzmann Method

Fe3O4-Plasma

50 # Re # 400 0 # MnF # 100 φ 5 0:04 Ha 5 20

Back flow occurs near the region where the magnetic source is located. Also it can be found that skin friction coefficient is a decreasing function of Reynolds number and magnetic number

Sheikholeslami et al. [127]

Lattice Boltzmann Method

Al2O3-water

103 # Ra # 105 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:8

Nusselt number increases with increase of nanoparticle volume fraction and Rayleigh number while it decreases with increase of Hartmann number

(Continued)

TABLE 3.10 (Continued) Authors

Geometry of study

Method

Type of nanofluid

Properties

Remarks

Sheikholeslami and Ellahi [128]

Lattice Boltzmann Method

Al2O3-water

10 # Ra # 10 0 # φ # 0:04 0 # Ha # 100 Pr 5 6:8

Applying magnetic field results in a force opposite to the flow direction that leads to drag on the flow and then reduces the convection currents by reducing the velocities

Sheikholeslami et al. [129]

Lattice Boltzmann Method

Cu-water Ag-water Al2O3-water TiO2water

103 # Ra # 105 0 # φ # 0:04 2 # H=t # 9 Pr 5 6:8

The effect of nanoparticle volume fraction is found to be more pronounced for low Rayleigh number as compared to high Rayleigh number

Kefayati [130]

lattice Boltzmann method

Cu-water

103 # Ra # 105 0 # φ # 0:06 0 # Ha # 90 0:5 # A # 2

The heat transfer decreases by increment of Hartmann number. Heat transfer decreases with growth of the aspect ratio but this growth causes the effect of nanoparticles to increase. Magnetic field augments the effect of nanoparticles at High Rayleigh numbers (Ra 5 105). The effect of nanoparticles rises for high Hartmann numbers when the aspect ratio increases

Mejri et al. [131]

lattice Boltzmann method

Al2O3 -water

103 # Ra # 106 0 # φ # 0:06 0 # Ha # 60 γ 5 0; π=4; π=2; 3π=4; π

The heat transfer rate decreases with the increase of Hartmann number and increases with the rise of Rayleigh number

3

5

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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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3. VARIOUS APPLICATION OF NANOFLUID FOR HEAT TRANSFER AUGMENTATION

[232] M. Sheikholeslami, H.B. Rokni, Influence of melting surface on MHD nanofluid flow by means of two phase model, Chin. J. Phys. 55 (2017) 1352
1360. [233] M. Sheikholeslami, H.B. Rokni, Effect of melting heat transfer on nanofluid flow in existence of magnetic field considering Buongiorno Model, Chin. J. Phys. 55 (2017) 1115
1126. [234] S.R. Hosseini, M. Sheikholeslami, M. Ghasemian, D.D. Ganji, Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field by means of KKL model, Powder Technol. 324 (2018) 36
47. [235] R. Kumar, S. Sood, M. Sheikholeslami, S.A. Shehzad, Nonlinear thermal radiation and cubic autocatalysis chemical reaction effects on the flow of stretched nanofluid under rotational oscillations, J. Colloid Interface Sci. 505 (2017) 253
265. [236] M. Sheikholeslami, D.D. Ganji, Analytical investigation for Lorentz forces effect on nanofluid Marangoni boundary layer hydrothermal behavior using HAM, Indian J. Phys. 91 (12) (2017) 1581
1587. [237] M. Sheikholeslami, Ali J. Chamkha, Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field, Numer. Heat Transfer, Part A 69 (10) (2016) 1186
1200. Available from: https:// doi.org/10.1080/10407782.2015.1125709. [238] M. Sheikholeslami, H. Oztop, MHD free convection of nanofluid in a cavity with sinusoidal walls by using CVFEM, Chin. J. Phys. 55 (6) (2017) 2291
2304. [239] R. Kumar, S. Sood, S.A. Shehzad, M. Sheikholeslami, Numerical modeling of time-dependent bio-convective stagnation flow of a nanofluid in slip regime, Results Phys. 7 (2017) 3325
3332. Available from: https://doi.org/10.1016/j.rinp.2017.08.059. [240] M. Sheikholeslami, T. Hayat, A. Alsaedi, On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylinders, Int. J. Heat Mass Transfer 115 (2017) 981
991. [241] M. Sheikholeslami, Numerical simulation for solidification in a LHTESS by means of nano-enhanced PCM, J. Taiwan Inst. Chem. Eng. 86 (2018) 25
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438. [243] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transf. 123 (2018) 418
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1271. [245] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transf. 122 (2018) 643
650. [246] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
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540. [257] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
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[264] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
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660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [267] M. Sheikholeslami, D.D. Ganji, Influence of electric field on Fe3O4-water nanofluid radiative and convective heat transfer in a permeable enclosure, J. Mol. Liq. 250 (2018) 404
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311.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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C H A P T E R

4 Single-phase Model for Nanofluid Free Convection Heat Transfer by Means of CVFEM 4.1 INTRODUCTION Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals thermal conductivities are up to three times higher than the fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal conductivity of a metal. The term “nanofluid” was first proposed by Choi [1] to indicate engineered colloids composed of nanoparticles dispersed in a base fluid. Khanafer et al. [2] firstly conducted a numerical investigation on the heat transfer enhancement due to adding nanoparticles in a differentially heated enclosure. They found that the suspended nanoparticles substantially increase the heat transfer rate at any given Grashof number. Influence of axial magnetic field on nanofluid thermal management has been analyzed by Sheikholeslami and Abelman [3]. Sheikholeslami and Rashidi [4] applied a single-phase model for Fe3O4-water nanofluid. They indicated that Lorentz forces reduce the velocity. The influence of single magnetic source on Fe3O4-H2O flow style has been reported by Sheikholeslami and Ganji [5]. Sheikholeslami and Shehzad [6] employed lattice Boltzmann method for nanofluid flow in a porous cavity in the presence of a magnetic field. Sheikholeslami [7] presented a mesoscopic approach for nanofluid convective flow in a porous lid driven cubic cavity. Sheikholeslami Kandelousi [8] presented application of a Koo
Kleinstreuer
Li (KKL) model for simulation of nanofluid in a semiporous channel. Influence of uniform suction on nanofluid flow and heat transfer over a cylinder was investigated by Sheikholeslami [9]. Sheikholeslami [10] studied nanofluid single-phase model flow in a porous enclosure in the presence of a constant magnetic field. Sheikholeslami [11] simulated nanofluid flow and natural convection heat transfer in an open porous cavity. Sheikholeslami and Chamkha [12] demonstrated the impact of Marangoni convection on nanofluid forced convection in the presence of Lorentz forces. Sheikholeslami et al. [13] presented a numerical simulation of nanofluid forced convection heat transfer augmentation in the presence of a magnetic field using lattice Boltzmann method. Several numerical simulations have been presented for the simulation of a nanofluid [14
93].

4.2 NANOFLUID HYDROTHERMAL ANALYSIS IN A COMPLEX SHAPED CAVITY 4.2.1 Problem Definition Fig. 4.1 depicts the geometry and boundary conditions. The inner cylinder is hot and follows the formula:

  ð4:1Þ r 5 cos ζ 2 ζ 0 N A 1 rin each parameter has been shown in Fig. 4.1.

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00004-7

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© 2019 Elsevier Inc. All rights reserved.

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

(A)

(B) Tc Th

A

g

Region of support

y

Control volume

1.5

ζ

x

1 0.5

rin

0 –0.5 –1

rout

–1.5 –2.5 –2 –1.5 –1 –0.5 0

0.5

1

1.5

2

2.5

Si,3 (C) i Si,4

j=4

f1

f2 i=1 j=3

FIGURE 4.1

(A) Geometry and the boundary conditions with (B) the mesh of Geometry considered in this work; (C) A sample triangular element and its corresponding control volume.

4.2.2 Governing Equation The governing equations for laminar CuO-water free convective heat transfer must be presented as: @v @u 1 50 @y @x  μnf @2 u @2 u 1 @P @u @u 5v 1u 1 2 2 @y ρnf @x @y @x ρnf @x2  μnf @2 v @2 v @P 1 @v @v 1 5v 1u 2 ðTc 2 TÞgβ nf 2 @y2 @y ρnf @y @x ρnf @x2   2
 @ T @2 T @T @T knf 1u 1 5 ρCp nf v 2 2 @x @y @y @x

ð4:2Þ ð4:3Þ ð4:4Þ ð4:5Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4.2 NANOFLUID HYDROTHERMAL ANALYSIS IN A COMPLEX SHAPED CAVITY


 (ρnf ), ρCp nf , and β nf are defined as:









ρnf 5 ρf ð1 2 φÞ 1 ρs φ


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð4:7Þ

β nf 5 β f ð1 2 φÞ 1 β s φ

ð4:8Þ



ð4:6Þ

knf and μnf are obtained according to KKL model : 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi kf κb T 1 0 1 1 5 3 104 g0 ðφ; T; dp Þφρf cp;f knf 5 1 1 0 ρ p dp @kp 1 2A 2 @kp 2 1Aφ kf kf





2  g0 φ; T; dp 5 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 LnðφÞln dp 1 a10 Ln dp




2  1 LnðT Þ a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp

ð4:9Þ

Rf 1 dp =kp 5 dp =kp;eff ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð4:10Þ

All needed coefficients and properties are illustrated in Tables 4.1 and 4.2. Vorticity and stream function should be defined as: ω1

TABLE 4.1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð4:11Þ

The Coefficient Values of CuO-H2 O Nanofluid CuO-H2 O

Coefficient values a1

2 26.593310846

a2

2 0.403818333

a3

2 33.3516805

a4

2 1.915825591

a5

6.4218584E-02

a6

48.40336955

a7

2 9.787756683

a8

190.245610009

a9

10.9285386565

a10

2 0.72009983664

TABLE 4.2 Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK21 Þ

dp ðnmÞ

σðΩUmÞ21

H2O

997.1

4179

0.613

21

-

0:05

CuO

6500

540

18

29

10210

6500

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

By removing pressure gradient source terms from Eqs. (4.3) and (4.4), the final form of equations can be obtained as:   2 @ω @2 ω @T @ω @ψ @ω @ψ 2 ð4:12Þ υnf 1 2 2 β nf g 5 @x2 @y @x @x @y @y @x  2   @ T @2 T @T @ψ @T @ψ
2 knf 1 2 5 ð4:13Þ ρCp nf 2 @y @x @x @y @y @x @2 ψ @2 ψ 1 2 1ω50 @y2 @x

ð4:14Þ

Dimensionless parameters are as follows: Ω5

ωL2 ψ T 2 Tc y x ;Ψ 5 ;Θ5 ;Y5 ; X5 αf L L αf Th 2 Tc

ð4:15Þ

Using the above formulae the governing equations change to: 0 12 1 3 2 0 13 0 2 2 @Θ β ρ @ Ω @ ΩA 1 Raf Prf @ A4φ s 1 ð1 2 φÞ5 1 Prf =4ð12φÞ2:5 @φ s 1 ð1 2 φÞA5@ 2 1 @X @Y @X2 βf ρf @Ψ @Ω @Ψ @Ω 1 52 @X @Y @Y @X
 ! # 2  " ρCp s knf @ Θ @2 Θ @Θ @Ψ @Θ @Ψ
 1 1 = ð1 2 φ Þ 1 ð1 2 φ Þ 1 φ 52 @X2 @Y2 @Y @X @X @Y kf ρCp f Ω1

@2 Ψ @2 Ψ 1 50 @X2 @Y2

ð4:16Þ

ð4:17Þ ð4:18Þ






Rayleigh and Prandtl numbers are introduced as: Raf 5 gβ f ðTh 2 Tc ÞL3 = υf αf , Prf 5 υf =αf , respectively. Nuloc and Nuave over the cold wall should be calculated as:  knf @Θ ð4:19Þ Nuloc 5 kf @r ðγ 1 Nuave 5 Nuloc ðζ Þ dζ γ

ð4:20Þ

0

4.2.3 Effects of Active Parameters CuO-water hydrothermal analysis in a cavity with hot inner sinusoidal cylinder is studied via Control Volume-based Finite Element Method (CVFEM). Different values for number of undulations (N 5 2,3,5, and 6), nanoparticles volume fraction (φ 5 0%; 2%, and 4%), and Rayleigh number (Ra 5 103, 104, and 105) has been examined. The influence of nanoparticles on the velocity and temperature contours is illustrated in Fig. 4.2B. The strength of flow is enhanced by adding nanoparticles in H2O. Furthermore, augmenting the volume fraction of nanofluid causes the temperature to enhance. Influences of Rayleigh number and N on hydrothermal behavior are depicted in Figs. 4.3 and 4.4. jψmax j enhances with rise of buoyancy force. At low Reynolds number, augmenting the number of undulations causes jψmax j to enhance but at high Reynolds number this trend is not observed. Conduction heat transfer mode is dominant at low Reynolds number. So the isotherms are parallel to each other in those cases. Temperature gradient at the bottom of the hot wall is higher than other places. For an odd number of undulations, very small vortexes generate near the vertical centerline. The lower eddies are weaker than the upper ones because of smaller space for flow motion. As buoyancy forces augment, thermal plumes are generated on the hot wall. These plumes make isotherms to be denser and in turn the temperature gradient enhances.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER

Ψmax

nf

= 54.88 , Ψmax

f

77

= 45.53

FIGURE 4.2 Comparison of the streamlines (left) and isotherms (right) contours between nanofluid (φ 5 0:04) (
) and pure fluid (φ 5 0) (- - -) for N 5 6; A 5 0:5 at Ra 5 105 and Pr 5 6:2. jΨ max jnf 5 54:88; jΨ max jf 5 45:53

Effects of φ; Ra and N on Nuloc and Nuave are depicted in Figs. 4.5 and 4.6, respectively. The average Nusselt number can be obtained according to the following formula: Nuave 5 4:56414 2 2:58664 logðRaÞ 1 0:19N 2 8:075φ 1 0:021 logðRaÞ N 1 3:791 logðRaÞ φ 1 0:287 N φ 1 0:417 logðRaÞ2 2 0:023N 2 2 0:3074φ2

ð4:21Þ

The numbers of extremum in Nuloc are dependent on the number of undulations and thermal plumes. As ζ augments the conduction mechanism becomes dominant. So Nuloc reduces with rise of ζ. The Nusselt number is enhanced with the rise of φ; Ra, and N.

4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER 4.3.1 Problem Definition The schematic diagram and the mesh of the semiannulus enclosure used in the CVFEM program are shown in Fig. 4.7. The system consists of a circular enclosure with radius of L, within which an inclined elliptic cylinder is located and rotates from γ 5 03 to 903 . Th and Tc are the constant temperatures of the inner and outer cylinders, respectively and Th . Tc . Setting a as the major axis and b as the minor axis of elliptic cylinder, the eccentricity (ε) for the inner cylinder is defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ε 5 a2 2 b2 =a or b 5 1 2 ε2 : a ð4:22Þ In this study, for the inner ellipse, the eccentricity and the major axis are 0.9 and 0:8L, respectively.

4.3.2 Governing Equation The flow is steady, two-dimensional, laminar, and incompressible. The radiation, viscous dissipation, induced electric current, and Joule heating are neglected. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Neglecting displacement currents, induced magnetic field and using the Boussinesq approximation, the governing equations of heat transfer and fluid flow for nanofluid can be obtained as follows:

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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Ra = 104

Ra = 105

N=6

N=5

N=4

N=3

Ra = 103

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIGURE 4.3 Comparison of the isotherms contours for different values of Rayleigh number ðRaÞ and number of undulations ðN Þ atA 5 0:5

and φ 5 0:04.

@u @v 1 50 @x @y

 2 @u @u 1 @P @ u @2 u 1v 52 1 υnf 1 @x @y ρnf @x @x2 @y2  2 @v @v 1 @P @ v @2 v 1 υnf 1 1 β nf gðT 2 Tc Þ u 1v 52 @x @y ρnf @y @x2 @y2  2 @T @T @ T @2 T 1v 5 αnf 1 2 u @x @y @x2 @y u

ð4:23Þ ð4:24Þ ð4:25Þ ð4:26Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER

N=6

N=5

N=4

N=3

Ra = 103

Ra = 104

4 3 2 1 0.2 0.012 – 0.012 – 0.2 –1 –2 –3 –4

3 2.5 2 1.5 1 0.5 0.15 – 0.15 – 0.5 –1 – 1.5 –2 – 2.5 –3

2 1.5 1 0.5 0.046 0.01 – 0.01 – 0.046 – 0.5 –1 – 1.5 –2

2 1.5 1.2 0.9 0.5 0.15 – 0.15 – 0.5 – 0.9 – 1.2 – 1.5 –2

Ra = 105

18.5 15 10 5 2.5 0.25 0.15 0.05 –0.05 –0.15 –0.25 –2.5 –5 –10 –15 –18.5

15 10 8 6 4 2 –2 –4 –6 –8 –10 –15

12.8 12 10 8 5 2.5 0.5 0.25 0.1 –0.1 –0.25 –0.5 –2.5 –5 –8 –10 –12 –12.8

12 10 8 5 3.5 2 0.8 –0.8 –2 –3.5 –5 –8 –10 –12

45 40 30 20 10 5 0.4 –0.4 –5 –10 –20 –30 –40 –45

49 45 40 35 30 20 10 –10 –20 –30 –35 –40 –45 –49

40 30 25 20 15 10 5 1.5 –1.5 –5 –10 –15 –20 –25 –30 –40

35 30 20 15 10 5 –5 –10 –15 –20 –30 –35

FIGURE 4.4 Comparison of the streamlines contours for different values of Rayleigh number ðRaÞ and number of undulations ðN Þ at A 5 0:5 and φ 5 0:04.


 where the effective density (ρnf ), the thermal expansion coefficient (β nf ), and heat capacitance ρCp nf of the nanofluid are defined as: ρnf 5 ρf ð1 2 φÞ 1 ρs φ β nf 5 β f ð1 2 φÞ 1 β s φ


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð4:27Þ

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

Ra = 104

Ra = 103

Ra = 105

4.2

2.5

10 φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

2

φ=0 φ = 0.02 φ = 0.04

8

1.5

1

Nuloc

Nuloc

Nuloc

N=3

2.8

4

1.4 2

0.5

0

0 90

135

180

225

270

90

135

ζ

180

225

90

270

135

ζ

225

270

10 φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

2

180

ζ

4.2

2.5

φ=0 φ = 0.02 φ = 0.04

8

2.8 1.5

1

Nuloc

Nuloc

Nuloc

N=4

6

6

4

1.4 2

0.5

0

0 90

135

180

225

270

90

135

ζ

180

225

90

270

135

ζ

270

10

φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

2

225

ζ

4.2

2.5

180

φ=0 φ = 0.02 φ = 0.04

8

1

Nuloc

Nuloc

Nuloc

N=5

2.8 1.5

6

4

1.4 2

0.5

0

0 90

135

180

225

270

90

135

ζ

180

225

90

270

135

ζ

270

10

φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

2

225

ζ

4.2

2.5

180

φ=0 φ = 0.02 φ = 0.04

8

1.5

1

Nuloc

Nuloc

Nuloc

N=6

2.8 6

4

1.4 2

0.5

0

0 90

135

180

225

270

ζ

90

135

180

225

270

90

135

ζ

180

225

270

ζ

FIGURE 4.5 Effects of the nanoparticle volume fraction, number of undulations and Rayleigh number on Local Nusselt number.

The Brownian motion has a significant impact on the effective thermal conductivity.

kstatic kf

keff 5 kstatic 1 kBrownian

 k 3 kpf 2 1 φ   5 1 1 k kp p 1 2 2 2 1 φ kf kf

ð4:28Þ ð4:29Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER

81

FIGURE 4.6 Effects of the nanoparticle volume fraction, number of undulations and Rayleigh number on Average Nusselt number.

where, kstatic is the static thermal conductivity based on Maxwell classical correlation. The enhanced thermal conductivity component generated by microscale convective heat transfer of a particle’s Brownian motion and affected by ambient fluid motion is obtained via simulating Stokes’ flow around a sphere (nanoparticle). sffiffiffiffiffiffiffiffiffi κb T 4 kBrownian 5 5 3 10 βφρf cp;f fðT; φÞ ð4:30Þ ρ p dp In recent years, there has been an increasing trend to emphasize the importance of the interfacial thermal resistance between nanoparticles and base fluids. The KKL model combined β and f functions to develop a new g0 function which captures the influences of particle diameter, temperature, and volume fraction. The empirical g0 -function depends on the type of nanofluid.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

(A)

(B) Tc

Y g a

r

γ

b

ζ X

Th

Si,3 i

(C)

Si,4 j=4

f1 f2 j=3

i=1

FIGURE 4.7 (A)Geometry and the boundary conditions with (B) the mesh of enclosure considered in this work; (C) A sample triangular element and its corresponding control volume.

Also, by introducing a thermal interfacial resistance Rf 5 4 3 1028 km2 =W the original kp in Eq. (4.30) was replaced by a new kp;eff in the form: Rf 1

dp dp 5 kp kp;eff

ð4:31Þ

For different based fluids and different nanoparticles, the function should be different. Only water-based nanofluids are considered in the current study. For CuO-water nanofluids, this function follows the format:



2  g0 T; φ; dp 5 a1 1 a2 ln dp 1 a3 lnðφÞ 1 a4 lnðφÞ ln dp 1 a5 ln dp lnðTÞ

ð4:32Þ


2  1 a6 1 a7 ln dp 1 a8 lnðφÞ 1 a9 lnðφÞ ln dp 1 a10 ln dp where the coefficients ai (i 5 0.10) are based on the type of nanopartices, CuO-water nanofluids has an R2 of 96% and 98%, respectively (Table 4.1). Finally, the KKL correlation is written as: sffiffiffiffiffiffiffiffiffi κb T 0 4 kBrownian 5 5 3 10 φρf cp;f g ðT; φ; dp Þ ð4:33Þ ρp dp For the effective viscosity due to micromixing in suspensions, they proposed: μeff 5 μstatic 1 μBrownian 5 μstatic 1 where μstatic 5

μf ð12φÞ2:5

μf kBrownian 3 kf Prf

ð4:34Þ

is viscosity of the nanofluid, as given originally by Brinkman.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER

83

The stream function and vorticity are defined as: u5

@ψ ; @y

v52

@ψ ; @x

ω5

@v @u 2 @x @y

ð4:35Þ

The stream function satisfies the continuity Eq. (4.23). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e., by taking the y-derivative of Eq. (4.24) and subtracting from it the x-derivative of Eq. (4.25). This gives:   2 @ψ @ω @ψ @ω @ ω @2 ω @T 2 5 υnf 1 g ð4:36Þ 1 β nf 2 2 @y @x @x @y @x @y @x  2 @ψ @T @ψ @T @T @2 T 2 5 αnf 1 ð4:37Þ @y @x @x @y @x2 @y2 @ 2 ψ @2 ψ 1 2 52ω @x2 @y

ð4:38Þ

By introducing the following nondimensional variables: X5

x y ωL2 ψ T 2 Tc ; Y5 ; Ω5 ;Ψ 5 ;Θ5 L L αf αf Th 2 Tc

ð4:39Þ

where in Eq. (4.39) L 5 rout . Using the dimensionless parameters, the equations now become: 3 2 70 6 1 7 2 6 2 7 @ Ω Prf @Ψ @Ω @Ψ @Ω 6 @ Ω @ A 0 17 2 56 7 @X2 1 @Y2 @Y @X @X @Y 6 7 6 4ð12φÞ2:5 @ð1 2 φÞ 1 φ ρs A5 ρf

ð4:40Þ

30 1 β @Θ 1 Raf Prf 4ð1 2 φÞ 1 φ s 5@ A @X βf 2

2

3  7 @2 Θ @2 Θ knf =kf @Ψ @Θ @Ψ @Θ 6 7  2 56 1 @Y @X @X @Y 4 @Y2 ðρCpÞ 5 @X2 ð1 2 φÞ 1 φ ρCp s ð Þf

ð4:41Þ

@2 Ψ @2 Ψ 1 52Ω @X2 @Y2

ð4:42Þ
 where Raf 5 gβ f L3 ðTh 2 Tc Þ= αf υf is the Rayleigh number for the base fluid and Prf 5 υf =αf is the Prandtl number for the base fluid. The thermophysical properties of the nanofluid are given in Table 4.2. The boundary conditions as shown in Fig. 4.7 are: Θ 5 1:0

on the inner elliptic boundary

Θ 5 0:0

on the outer circular boundary

Ψ 5 0:0

on all solid boundaries

ð4:43Þ

The values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure. The local Nusselt number of the nanofluid along the cold wall can be expressed as:  knf @Θ Nuloc 5 ð4:44Þ kf @r

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

where r is the radial direction. The average Nusselt number on the cold circular wall is evaluated as: ð 1 2π Nuave 5 Nuloc ðζ Þ dζ 2π 0

ð4:45Þ

To estimate the enhancement of heat transfer between the case of φ 5 0:04 and the pure fluid (base fluid) case, the enhancement is defined as:
 Nuðφ 5 0:04Þ 2 Nu basefluid
 3 100 ð4:46Þ E5 Nu basefluid

4.3.3 Effects of Active Parameters In this section, natural convection heat transfer between a circular enclosure and an elliptic cylinder filled with nanofluid is investigated numerically using the control volume-based finite element method (CVFEM). The fluid in the enclosure is CuO-water nanofluid. Calculations are carried out for constant eccentricity (ε 5 0:9), major axis (a 5 0:8L), and Prandtl (Pr 5 6.2) at different values of Rayleigh numbers (Ra 5 103 ; 104 ; 105 ) and inclined angle of elliptic inner cylinder (γ 5 03 ; 303 ; 603 ; 903 ) and volume fraction of nanoparticles (φ 5 0%; 2%, and 4%). The effects of adding nanoparticles on the streamlines and isotherms are shown in Fig. 4.8. As can be seen, the absolute values of stream functions indicate that the strength of flow increases with an increase in the volume fraction of nanofluid. Also it is obvious to say that adding nanoparticle leads to an increase in Nusselt number due to increment in thermal conductivity. Since the reduction in temperature gradient due to the presence of nanoparticles is much smaller than the thermal conductivity ratio, therefore an enhancement in the Nusselt number occurs by adding nanoparticle. Fig. 4.9 shows the isotherms and streamlines for different values of Ra and γ at φ 5 0:04. At Ra 5 103 the isotherms are parallel to each other and take the form of the inner and outer wall and the stream function magnitude is relatively small which indicates the domination of the conduction heat transfer mechanism. Increasing the inclination angle leads to an increase in the absolute value of the maximum stream function (jΨ max j) at this Rayleigh number. At γ 5 0 degrees the streamlines and isotherms are symmetric with respect to the vertical centerline of the enclosure. Each pair of cells has two cells. The top vortex is stronger because at this area the hot surface is located beneath the cold one which helps the flow circulation, whereas the arrangement of the presence of the cold wall under the hot cylinder at the bottom of the enclosure resists the flow circulation. As γ 5 0 degree increases these two pairs of cells merge together and form two single cells at different locations inside the enclosure. At γ 5 90 degrees again the streamlines patterns become symmetric with respect to the vertical centerline of the enclosure. At Ra 5 104 the thermal plumes start to appear over the hot elliptic cylinder. Besides, the stream function values start to grow which shows that the convection heat transfer mechanism has become comparable with conduction. At λ 5 0 degree the temperature contour becomes stratified beneath the hot cylinder. With the increase of γ to 30 degrees a secondary vortex appears at the top of the enclosure and the thermal plume slants to the left because of more available space at this area. As the inclination angle of the inner cylinder increases further this secondary vortex disappears and the streamlines show two main vortices in the enclosure. Also it can be found that effect of increasing inclination angle on jΨ max j becomes less pronounced at γ . 30 degrees. When the Rayleigh number increases up to Ra 5 105 isotherms are totally distorted at the top of the enclosure while it is stratified at the bottom of the enclosure, which shows the heat transfer mechanism is dominated by convection. The thermal plume is completely formed which impinges the hot fluid to the cold wall of the enclosure. As seen, FIGURE 4.8 Comparison of the streamlines (left) and isotherms (right) contours between nanofluid (φ 5 0:04) (
) and pure fluid (φ 5 0) (- -) for ε 5 0:9; a 5 0:8L at Ra 5 105 and Pr 5 6:2.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4.3 NATURAL CONVECTION HEAT TRANSFER IN A NANOFLUID FILLED ENCLOSURE WITH ELLIPTIC INNER CYLINDER

85

the secondary vortex exists at γ . 30 and 60 degrees γ . 30 degrees and 603 which can slant the thermal plume to the left. Also, as seen in Fig. 4.9 the maximum value of jΨ max j occurs at γ . 60 degrees. Fig. 4.10 shows the distribution of local Nusselt numbers along the surface of the outer circular wall for different inclination angle and Rayleigh number. As the Rayleigh number increases the local Nusselt number increases

FIGURE 4.9 Isotherms (up) and streamlines (down) contours for different values of Ra and γ at φ 5 0:04 APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

86

4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

due to increasing convection effect. At Ra 5 103 the Nuloc profile is nearly symmetrical with respect to the horizontal centerline. As the Rayleigh number enhances (e.g., Ra 5 104 and Ra 5 105 ), the Nuloc profile is no longer symmetrical and the local Nusselt number is considerably small over the bottom wall of the enclosure. These local Nusselt number profiles are more complex due to the presence of the thermal plume at the vicinity of the top wall of the enclosure. Fig. 4.11 depicts the effects of the inclination angle and Rayleigh number on the average Nusselt number. For all values of γ, the average Nusselt number increases with the increase of Rayleigh number. As seen, the inclination angle has no significant effect on the average Nusselt number at low Rayleigh number. While for higher

Ra = 103 6

6

γ = 0º γ = 30º γ = 60º γ = 90º

5

inclination angle and Rayleigh number on Local Nusselt number when φ 5 0:04.

γ = 0º γ = 30º γ = 60º γ = 90º

5

4

4 Nuloc

Nuloc

FIGURE 4.10 Effects of the

Ra = 10 4

3

3

2

2

1

1

0

0

90

180 ζ

270

0

360

0

90

180 ζ

270

360

Ra = 10 5 12

γ = 0º γ = 30º γ = 60º γ = 90º

10

Nuloc

8 6 4 2 0 0

90

180 ζ

270

360

FIGURE 4.11

5 Ra = 103 Ra = 104 Ra = 105

4.5

Effects of the inclination angle and Rayleigh number on average Nusselt number when φ 5 0:04.

Nuave

4 3.5 3 2.5 2

0

30 Ra = 10 5

60

90

γ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4.4 NANOFLUID FREE CONVECTION HEAT TRANSFER IN A TILTED CAVITY

TABLE 4.3

Constant Coefficient for Using Eq. (4.47) i51

i52

i53

i54

i55

i56

j51

7.564525

2 3.38154

2 0.25235

0.506444

0.019843

2.008494

j52

0.484538

2 0.31956

0.711765

2 0.04265

0.023808

0.1894

aij

19.2 Ra = 103 Ra = 104 Ra = 105

19.1

FIGURE 4.12 Effects of the inclination angle and Rayleigh number on the ratio of heat transfer enhancement due to addition of nanoparticles when Pr 5 6:2.

19

E

18.9 18.8 18.7 18.6 18.5 0

30

60

90

γ

values of the Rayleigh number, an increase in inclination angle leads to an increase in the average Nusselt number. Also, the corresponding polynomial representation of such a model for the Nusselt number is as follows: Nu 5 a12 1 a22 γ 1 a32 Y1 1 a42 γ 2 1 a52 Y1 2 1 a62 γY1 Y1 5 a11 1 a21 Ra 1 a31 φ 1 a41 Ra2 1 a51 φ2 1 a61 φRa

ð4:47Þ

Also aij can be found in Table 4.3, e.g., a21 is equal to (21.52352). The heat transfer enhancement ratio due to the addition of nanoparticles for different values of inclination angle and Rayleigh number is shown in Fig. 4.12. It can be found that the effect of nanoparticles is more pronounced at low Rayleigh number than at high Rayleigh number because of the greater enhancement rate. This observation can be explained by noting that at low Rayleigh number the heat transfer is dominated by conduction. Therefore, the addition of high thermal conductivity nanoparticles will increase the conduction and make the enhancement more effective. It is an interesting observation that the enhancement in heat transfer for Ra 5 103 is the same for all inclination angles.
At Ra 5 104 the maximum heat transfer enhancement ratio is obtained at γ 5 30 . At high Rayleigh number Ra 5 105 , at first the percentage of the heat transfer enhancement decreases and then increases. At this Rayleigh number the minimum heat transfer enhancement ratio occurs at γ 5 90 .

4.4 NANOFLUID FREE CONVECTION HEAT TRANSFER IN A TILTED CAVITY 4.4.1 Problem Definition Fig. 4.13 illustrates the important geometric parameters of current geometry. Also a sample mesh is presented. Constant heat flux is applied on the inner surface and the other conditions are clear in Fig. 4.13.

4.4.2 Governing Equation The flow is laminar and steady and two-dimensional. Boussinesq approximation has been considered for momentum equations. Nanofluid is assume to be an homogenous fluid. According to these assumptions the governing equations can be presented as: @v @u 1 50 @y @x

ð4:48Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

(Α)

(Β)

Region of support

Control volume

2.5

g

Tc

2 1.5 q

′′

ζ

r in

1

r out

0.5

y

0

λ

–0.5 –2 –1.5 –1 –0.5 0

x

0.5

1

1.5

2

FIGURE 4.13 (A) Geometry and the boundary conditions with (B) the mesh of half-annulus enclosure considered in this work.

 @u @u 1 @P μnf @2 u @2 u 1u 1 5 v 1 2 @y @x ρnf @x @x ρnf @y2  μnf @2 v @2 v @v @v @P 1 5 1 2 ðTc 2 T Þgβ nf u 1v 1 @x @y @y ρnf @y2 ρnf @x2  2  knf @ T @2 T @T @T
 1 u 1 5 v @y2 @y @x ρCp nf @x2

ð4:49Þ ð4:50Þ ð4:51Þ


 β nf , ρCp nf , and ρnf are defined as:












β nf 5 β f ð1 2 φÞ 1 β s φ 

 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ ρnf 5 ρf ð1 2 φÞ 1 ρs φ



knf and μnf are obtained according to the KKL model: 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi kf κb T 4 0 1 0 1 1 5 3 10 g ðφ; T; dp Þφρf cp;f knf 5 1 1 0 ρ p dp @ k p 1 2 A 2 @ k p 2 1A φ kf kf





2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞ ln dp 1 a5 Ln dp LnðTÞ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð4:52Þ ð4:53Þ ð4:54Þ

ð4:55Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð4:56Þ

All necessary coefficients and properties are illustrated in Tables 4.1 and 4.2. Vorticity and stream function should be defined as: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð4:57Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

4.4 NANOFLUID FREE CONVECTION HEAT TRANSFER IN A TILTED CAVITY

89

By removing pressure gradient source terms from Eqs. (4.49) and (4.50), the final form of the equations can be obtained as:   2 @ ω @2 ω @T @ω @ψ @ω @ψ 2 ð4:58Þ υnf 1 2 2 β nf g 5 @x2 @y @x @x @y @y @x  2   @ T @2 T @T @ψ @T @ψ
2 knf 1 2 5 ð4:59Þ ρCp nf 2 @y @x @x @y @y @x @ 2 ψ @2 ψ 1 2 1ω50 @y2 @x

ð4:60Þ



 y; x ψ; ωL2 T 2 Tc  ; ðY; XÞ 5 ; ðΨ; ΩÞ 5 Θ5
L αf qvL=kf

ð4:61Þ

Dimensionless parameters are as follows:

Using the above formulae the governing equations change to: 0 12 1 3 2 0 130 2 2 @Θ β ρ @ Ω @ Ω s 2:5 s A 1 Raf Prf @ A4φ 1 ð1 2 φÞ5 1 Prf =4ð12φÞ @φ 1 ð1 2 φÞA5@ 2 1 @X @Y @X2 βf ρf @Ψ @Ω @Ψ @Ω 1 52 @X @Y @Y @X
 !#  2  " ρCp s @ Θ @2 Θ knf @Ψ @Θ @Ψ @Θ  2 1 = ð1 2 φÞ 1 ð1 2 φÞ 1 φ
5 @X2 @Y2 @Y @X @X @Y kf ρCp f 1

@2 Ψ @2 Ψ 1 52Ω @Y2 @X2

ð4:62Þ

ð4:63Þ ð4:64Þ


 Prandtl and Rayleigh numbers are introduced as: Prf 5 υf =αf ; Raf 5 gβ f L4 qv= kf αf υf , respectively. Nuloc and Nuave over the hot wall should be calculated as:  knf 1 ð4:65Þ Nuloc 5 kf Θ ðπ 1 Nuave 5 Nuloc ðζ Þ dζ π

ð4:66Þ

0

4.4.3 Effects of Active Parameters In this work, the influence of CuO nanoparticle on the hydrothermal behavior in an inclined half-annulus is studied using CVFEM. Simulations are made for different values of Rayleigh number (Ra 5 103 ; 104 , and 105 ), inclination angle (λ 5 0, 45, 90, 135, and 180 degrees), and volume fraction of CuO (φ 5 0% and 4%). Fig. 4.14 depicts the effect of CuO nanoparticles on hydrothermal characteristics. The nanofluid velocity augments due to enhancing the solid movements. The temperature gradient augments with the rise of the volume fraction of CuO. Figs. 4.15 and 4.16 illustrate the impact of inclination angle, Rayleigh number, and CuO volume fraction. Conduction mode dominates at low Rayleigh number. Two rotating vortices exist in streamlines. As buoyancy forces grow, the temperature gradient is enhanced and a thermal plume appears near the vertical centerline. The core of vortices moves upward. At λ 5 45 degrees, the counterclockwise vortex is stronger than the other one due to more space for circulation. At λ 5 90 degrees, the two main eddies merge into one counterclockwise eddy. The streamlines and isotherms at λ 5 135 and 180 degrees are depicted in Fig. 4.16. As seen, the impact of the Rayleigh number on the size of eddies for λ 5 135 degrees is opposite to that of λ 5 45 degrees. By increasing buoyancy forces, the size of the secondary eddy decreases and finally at Ra 5 105 a small vortex appears on the top of the secondary eddy, which turns in a clockwise direction.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

FIGURE 4.14 Comparison of the streamlines (left) and isotherms (right) contours between nanofluid (φ 5 0:04) (- - -) and pure fluid (φ 5 0)  (
)for different values of Ra at λ 5 0 and Pr 5 6:2.





FIGURE 4.15 Isotherms (down) and streamlines (up) contours for different values of Rayleigh number when λ 5 45 and 90 for CuOwater nanofluid (φ 5 0:04).

91

4.4 NANOFLUID FREE CONVECTION HEAT TRANSFER IN A TILTED CAVITY





FIGURE 4.16 Isotherms (down) and streamlines (up) contours for different values of Rayleigh number when λ 5 135 and 180 for CuOwater nanofluid (φ 5 0:04).

This behavior of the thermal boundary layer may be due to the stronger flow circulation at higher Rayleigh numbers at these points. The isotherms and streamlines for λ 5 180 degrees are almost symmetric to the vertical centerline. However, enhancing the buoyancy forces has no significant influence on the location of the two main eddies’ core. In fact, when the hot circular wall locates above the cold one, the impact of convection on the velocity and temperature is less pronounced. Similar to the inclination angle of 135 degrees, the temperature gradient becomes thinner near the inner wall except at the two inner corners. The influence of important parameters on Nuloc and Nuave are depicted in Figs. 4.17
4.19. The correlation for this parameter is as follows: Nuave 5 2:12 2 1:2 logðRaÞ 1 0:01λ 2 11:81φ 2 4:88 λ logðRaÞ 1 7:94 φ logðRaÞ 2 0:027λφ 1 0:38logðRaÞ2 1 2:19λ2 1 0:07φ2

ð4:67Þ

Response surface methodology (RSM) is utilized to find this correlation. This method presented polynomials formulation according to input data. As volume fraction of CuO augments, Nuloc augments. Nusselt number aug     ments with rise of Ra. Nuloc profiles are symmetric to ζ 5 90 when λ 5 0 and 180 . At λ 5 0 and 45 minimum     amount of Nuloc is located at ζ 5 90 and 55 respectively. Also Fig. 4.19 shows that for λ 5 135 and 180 maxi  mum amount of Nuloc is located at ζ 5 100 and 90 respectively. The Nusselt number decreases with

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

92

4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

Ra = 103

Ra = 104

Ra = 105 10

5.6

2.8 φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

4.8

φ=0 φ = 0.02 φ = 0.04

2.4

2

Nu loc

Nu loc

Nu loc

λ = 0º

7.5 4

3.2

5 1.6 2.4

1.6

1.2 0

45

90

135

2.5 0

180

45

90

135

180

0

45

90

ζ

ζ 5.6

2.8 φ=0 φ = 0.02 φ = 0.04

135

180

ζ 9 φ=0 φ = 0.02 φ = 0.04

4.8

φ=0 φ = 0.02 φ = 0.04

8

2.4

2

Nu loc

Nu loc

Nu loc

λ = 45º

7 4

3.2

6 5

1.6

2.4

4

1.6

1.2 0

45

90

135

3 0

180

45

135

180

0

2.7

4.4

φ=0 φ = 0.02 φ = 0.04

6.6

180

135

180

5.4

1.8

Nu loc

3.2

Nu loc

Nu loc

λ = 90º

135

φ=0 φ = 0.02 φ = 0.04

6

3.6

2.8 2.4

1.5

0.9 0

45

90

135

4.2

1.6

3

1.2

2.4 0

180

45

90

135

180

0

45

90

ζ

ζ

ζ 6

4

2.7 φ=0 φ = 0.02 φ = 0.04

φ=0 φ = 0.02 φ = 0.04

3.5

Nu loc

2.1

1.8

φ=0 φ = 0.02 φ = 0.04

5

3

Nu loc

2.4

4.8

3.6

2 1.2

Nu loc

90

ζ φ=0 φ = 0.02 φ = 0.04

4

2.1

λ = 135º

45

ζ

ζ

2.4

90

2.5

4

3 2

1.5

1.5

1.2 0

45

90

135

2 0

180

45

90

135

180

0

90

φ=0 φ = 0.02 φ = 0.04

135

180

135

180

ζ

4.5

2.7

2.4

45

ζ

ζ

7 φ=0 φ = 0.02 φ = 0.04

4

φ=0 φ = 0.02 φ = 0.04

6

1.8

Nu loc

Nu loc

Nu loc

λ = 180º

3.5 2.1

3

5

4

2.5 1.5

3

2

1.2

2

1.5 0

45

90

ζ

135

180

0

45

90

ζ

135

180

0

45

90

ζ

FIGURE 4.17 Effects of the nanoparticle volume fraction, Rayleigh number and inclination angle for CuO-water nanofluids on Local Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

FIGURE 4.18 Effects of the nanoparticle volume fraction, Rayleigh number and inclination angle for CuO-water nanofluids on average Nusselt number.

(A)

(B)

180.00

0.04

135.00

0.03

λ 90.00

φ 0.02

45.00

0.01

0.00

0.00 3.00

3.50

4.00 log (Ra)

4.50

3.00

5.00

3.50

4.00 log (Ra)

φ = 0.04

(C)

λ = 180º

0.04

0.03

φ 0.02

0.01

0.00 0.00

45.00

90.00 λ log (Ra) = 5

FIGURE 4.19

Contour plots of average Nusselt number.

135.00

180.00

4.50

5.00

94

4. SINGLE-PHASE MODEL FOR NANOFLUID FREE CONVECTION HEAT TRANSFER BY MEANS OF CVFEM

FIGURE 4.20 Effects of λ and Ra on the heat transfer enhancement due to addition of nanoparticles when Pr 5 6:2.

enhancement of the inclination angle. Fig. 4.20 shows the influence of λ and Ra on the heat transfer enhancement. This parameter can be calculated as:
 Nuðφ 5 0:04Þ 2 Nu basefluid
 3 100 ð4:68Þ E5 Nu basefluid According to this figure, heat transfer enhancement increases with the increase of the inclination angle. Higher Rayleigh number leads to lower values of E. At low Rayleigh number and λ 5 180 degrees, the dominant mechanism is conduction and in this way adding nanoparticles has a significant impact on thermal conductivity.

References [1] S. Choi, in: D.A. Siginer, H.P. Wang (Eds.), Enhancing Thermal Conductivity of Fluids With Nanoparticles in Developments and Applications of Non-Newtonian Flows, 66, ASME, 1995, pp. 99
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1632. [78] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids, Volume 30, Issue 1, 10.1063/1.5012517, https://doi.org/10.1063/1.5012517. [79] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via Non-equilibrium model, Int. J. Heat Mass Transfer 120 (2018) 1200
1212.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

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[80] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [81] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transfer 120 (2018) 772
781. [82] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process. Process Intensif. 124 (2018) 71
82. [83] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [84] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [85] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [86] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
746. [87] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4-ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Mol. Liq. 249 (2018) 429
437. [88] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [89] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [90] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Phys. B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [91] M. Sheikholeslami, A. Zeeshan, Numerical simulation of Fe3O4-water nanofluid flow in a non-Darcy porous media, Int. J. Numer. Methods Heat Fluid Flow 28 (3) (2018) 641
660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [92] M. Sheikholeslami, D.D. Ganji, Influence of electric field on Fe3O4-water nanofluid radiative and convective heat transfer in a permeable enclosure, J. Mol. Liq. 250 (2018) 404
412. [93] M. Sheikholeslami, M.B. Gerdroodbary, S.V. Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90o elbow channel by non-uniform magnetic field, J. Magnet. Magnet. Mater. 460 (2018) 302
311.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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C H A P T E R

5 Buongiorno Model for Nanofluid Treatment Using CVFEM 5.1 INTRODUCTION Control of heat transfer in many energy systems is crucial due to the increase in energy prices. In recent years, nanofluids technology has been proposed and studied by some researchers experimentally or numerically to control heat transfer in a process. The nanofluid can be applied to engineering problems, such as heat exchangers, cooling of electronic equipments, and chemical processes. Almost all of the researchers assumed that nanofluids are to be treated as the common pure fluid and conventional equations of mass, momentum, and energy are used and the only effect of the nanofluid is its thermal conductivity and viscosity, which are obtained from the theoretical models or experimental data. These researchers assumed that nanoparticles are in thermal equilibrium and there aren’t any slip velocities between the nanoparticles and fluid molecules, thus they have a uniform mixture of nanoparticles Single-phase models assumed that there aren’t any slip velocities between nanoparticles and fluid molecules and assumed that the nanoparticle concentration is uniform. It is believed that in natural convection of nanofluids, the nanoparticles could not accompany fluid molecules due to some slip mechanisms such as Brownian motion and thermophoresis, so the volume fraction of nanofluids may not be uniform anymore and there would be a variable concentration of nanoparticles in a mixture. Nield and Kuznetsov [1] studied the natural convection in a horizontal layer of a porous medium saturated by a nanofluid. Their analysis revealed that for a typical nanofluid (with large Lewis number) the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, the contribution of nanoparticles to the thermal energy equation being a second-order effect. Khan and Pop [2] published a paper on boundary-layer flow of a nanofluid past a stretching sheet. They indicated that the reduced Nusselt number is a decreasing function of each dimensionless number. Cheng [3] studied the natural convection boundary layer flow over a truncated cone embedded in a porous medium saturated by a nanofluid with constant wall temperature. He showed that an increase in the thermophoresis parameter or the Brownian parameter tends to decrease the local Nusselt number. Sheikholeslami et al. [4] presented the effects of Thermophoresis and Brownian motion on nanofluid treatment in the presence of a magnetic field. Sheikholeslami et al. [5] studied the two-phase modeling of nanofluid free convection heat transfer in an L-shaped enclosure. Sheikholeslami and Abelman [6] simulated nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field by means of two-phase model. Sheikholeslami and Rokni [7] reported a numerical simulation for melting heat transfer effect on nanofluid flow by means of Buongiorno model. Sheikholeslami and Rokni [8] utilized the two-phase model for effect of Lorentz forces on nanofluid flow in the existence of melting heat transfer. Recently, numerical simulations about the application of nanofluid have been presented in various papers [9
91].

5.2 BUONGIORNO MODEL FOR NANOFLUID FLOW AND HEAT TRANSFER USING HEATLINE ANALYSIS 5.2.1 Problem Definition The physical model and the corresponding triangular elements used in the present CVFEM program are shown in Fig. 5.1. The inner and outer walls are maintained at constant temperatures Th and Tc , respectively. The shape of the inner cylinder profile is assumed to mimic the following pattern Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00005-9

99

© 2019 Elsevier Inc. All rights reserved.

100

5. BUONGIORNO MODEL FOR NANOFLUID TREATMENT USING CVFEM

FIGURE 5.1 (A) Geometry and the boundary conditions; (B) the mesh of enclosure considered in this work.

r 5 rin 1 A cosðN ðζ ÞÞ

ð5:1Þ

in which rin is the base circle radius, rout is the radius of outer cylinder, A and N are amplitude and number of undulations, respectively. ζ is the rotation angle. In this study A and N are equal to 0.2 and 8, respectively.

5.2.2 Governing Equation The nanofluid’s density, ρ is ρ 5 φρp 1 ð1 2 φÞρf n o D φρp 1 ð1 2 φÞ ρf0 ð1 2 βðT 2 Tc Þ

ð5:2Þ

where ρf , is the base fluid’s density, Tc , is a reference temperature, ρf0 is the base fluid’s density at the reference temperature, andβ is the volumetric coefficient of expansion. Taking the density of base fluid as that of the nanofluid, the density ρ in Eq. (5.2), thus becomes   ð5:3Þ ρ D φρp 1 ð1 2 φÞ ρ0 ð1 2 βðT 2 Tc Þ ρ0 is the nanofluid’s density at the reference temperature. The continuity momentum under Boussinesq approximation and energy equations for the laminar and steady-state natural convection in a two-dimensional enclosure can be written in dimensional form as follows: @u @v 1 50 @x @y   2 @u @u @P @ u @2 u 1v 1μ ρf u 1 52 @x @y @x @x2 @y2   2 
 @v @v @P @ v @2 v 1μ 1 52 2 φ 2 φc ρp 2 ρf0 g 1 ð1 2 φc Þρf0 ðT 2 Tc Þg ρf u 1 v 2 2 @x @y @y @x @y

ð5:4Þ ð5:5Þ ð5:6Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

5.2 BUONGIORNO MODEL FOR NANOFLUID FLOW AND HEAT TRANSFER USING HEATLINE ANALYSIS

8 9 3

r:V 5 0 > > 0 1 > > >

-  > > @V > > 1 V :r V A 5 2 rp 1 μnf r2 V - 1 qE ρnf @ > > > @t > > > > 0 1 > > >

-  >
 -> @T < ρC @ 1 V :r T A 5 knf r2 T 1 J :E p nf @t > > > > > @q > > 50 r: J 1 > > > @t > > > > > > > > r:εE 5 q > > > > :E 5 2 rϕ
 ρnf ; ρCp nf ; αnf ; β nf ; μnf , and knf are defined as: ρnf 5 ρs φ 1 ρf ð1 2 φÞ


ρCp

 nf

ð12:7Þ

ð12:8Þ



 5 ρCp s φ 1 ρCp f ð1 2 φÞ

ð12:9Þ


 αnf 5 knf = ρCp nf

ð12:10Þ

β nf 5 β s φ 1 β f ð1 2 φÞ

ð12:11Þ

knf 5 kf

2 2φðkf 2 ks Þ 1 2kf 1 ks 1 φðkf 2ks Þs 1 2kf 1 k

ð12:12Þ

Table 12.1 illustrate the properties of the base fluid and nanoparticles. The effect of electric field on viscosity of nanofluid has been taken into account: μ 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3

ð12:13Þ

Table 12.2 shows the coefficient values of this equation. Nondimensional parameters are presented as follows: t5

tULid P y x v u ;p5 ;y 5 ;x 5 ;v 5 ;u 5 ; 2 L L ULid ULid L ρULid

T 2 T0 ϕ 2 ϕ0 q E θ5 ; rT 5 T1 2 T0 ; ϕ 5 ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 q0 E0 rT rϕ

TABLE 12.1

ð12:14Þ

Thermophysical Properties of Water and Nanoparticles ρ ðkg=m3 Þ

Cp ðj=kgkÞ

k ðW=m:kÞ

μ ðPa:sÞ

Ethylene glycol

1110

2400

0.26

0.0162

Fe3 O4

5200

670

6




APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.2 EHD NANOFLUID FORCE CONVECTIVE HEAT TRANSFER CONSIDERING ELECTRIC FIELD DEPENDENT VISCOSITY

TABLE 12.2

403

The Coefficient Values of Eq. (12.13) φ50

Coefficient values

φ 5 0:05

A1

1.0603E 1 001

9.5331

A2


2.698E-003


3.4119E-003

A3

2.9082E-006

5.5228E-006

A4


1.1876E-008


4.1344E-008


 where rT and rϕ are ðT1 2 T0 Þ and ϕ1 2 ϕ0 , respectively. By eliminating the over bar, the equations are: 8 - > > r: V 5 0 > > > > 0 1 > > >

-  > ρnf =ρf 1 2 @V SE > > @ > r V1 1 V :r V A 5 2 rp 1 qE > > Re @t μ =μ ρ =ρ > nf f nf f > > > > 0 1
> > @t ρCp nf = ρCp f ρCp nf = ρCp f RePr > > > > > > > @q > > 50 > r: J 1 > @t > > > > > > : r:εE 5 q; E 5 2 rϕ Stream function and vorticity can be defined as: v52

@ψ @ψ @v @u ωL ψ ;u5 ;ω5 2 ; Ω5 ;Ψ 5 @x @y @x @y ULid LULid

ð12:16Þ

Stream function can satisfy the continuity equation. Vorticity equation can be derived by eliminating pressure sources. Nuloc and Nuave along the lid wall can be obtained as: Nuloc 5

Nuave 5

1 L

 knf @Θ kf @Y

ð12:17Þ

rð out

Nuloc dX

ð12:18Þ

rin

12.2.3 Effects of Active Parameters Nanofluid flow in a semi lid-driven enclosure in existence of electric field has been investigated. The working fluid is ethylene glycol and Fe3O4. Viscosity of nanofluid relies on the strength of the electric field. The effects of volume fraction of nanoparticles (φ 5 0% and 5%), Reynolds number (Re 5 3000; 4500, and 6000), and supplied voltage (Δϕ 5 0; 4; 6, and 10kV) are examined. In all cases Pr and Ecare equal to 149.54 and 1026, respectively. Figs. 12.3
12.5 depict the impacts of supplied voltage and Reynolds number on isotherm and streamlines.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

FIGURE 12.3 Effect of supplied voltage on streamlines and isotherm when Re 5 3000; φ 5 0:05.

At low Reynolds number, streamline has one main eddy and two very small vortexes at upper corners of the cavity. By increasing the electric field the main eddy converts to two smaller eddies. So the thermal boundary layer thickness decreases near the moving wall and in turn the temperature gradient reduces. As the Reynolds number increases, the thermal boundary layer thickness decreases. So the rate of heat transfer enhances with the rise of the Reynolds number. The effect of Δϕ and Re on local and average Nusselt number are shown in Figs. 12.6 and 12.7. Enhancing supplied voltage and Reynolds number leads to more distortion of isotherms. So Nusselt number augments with rise of these parameters. By applying the electric field stronger mixing leads to the destruction of the thermal boundary layer and creates vortex flow, thinning the boundary layer, and therefore causes the rise in heat transfer rate. Existence of extremums in the local Nusselt number is due to the generation of thermal plumes. The effect of coulomb forces on rate of heat transfer is more sensible for lower Reynolds number. This is due to this fact that the boundary layer thickness at low Reynolds number is thicker than that at high Reynolds number. So the electric field can be more effective t low Reynolds number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

405

12.3 ELECTROHYDRODYNAMIC NANOFLUID HYDROTHERMAL TREATMENT IN AN ENCLOSURE WITH SINUSOIDAL UPPER WALL

FIGURE 12.4

Effect of supplied voltage on streamlines and isotherm when Re 5 4500; φ 5 0:05.

12.3 ELECTROHYDRODYNAMIC NANOFLUID HYDROTHERMAL TREATMENT IN AN ENCLOSURE WITH SINUSOIDAL UPPER WALL 12.3.1 Problem Definition Fig. 12.8 illustrates the physical geometry along with the important parameters and mesh of the enclosure. The lower wall has the velocity of ULid and others are stationary. The lower wall has constant temperature T1 and the temperature of other walls is T0 . Also the retained boundary conditions are depicted in Fig. 12.8A. The formula of the upper sinusoidal wall is: 

 Y 5 1 2 0:03 1 1 sin 2πx 2 π=2

ð12:19Þ

Fig. 12.9 depicts the distribution of electric density for different Reynolds number and supplied voltage.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

406

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

FIGURE 12.5 Effect of supplied voltage on streamlines and isotherm when Re 5 6000; φ 5 0:05.

12.3.2 Governing Equation In order to simulate nanofluid hydrothermal treatment in the presence of an electric field, we should combine the equations of electric fields with those of hydrothermal. The formulas of electric field are:

-  r: E ε 5 q -

ð 2rϕÞ 5 E -

r: J 1

@q 50 @t

ð12:20Þ

ð12:21Þ ð12:22Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.3 ELECTROHYDRODYNAMIC NANOFLUID HYDROTHERMAL TREATMENT IN AN ENCLOSURE WITH SINUSOIDAL UPPER WALL

407

45

45 Δϕ = 0 kV

Δϕ = 0 kV

Δϕ = 4 kV

Δϕ = 4 kV

Δϕ = 6 kV Δϕ = 10 kV

Δϕ = 6 kV Δϕ = 10 kV

30

Nuloc

Nuloc

30

15

0

15

1

1.2

1.4

1.6

0

2

1.8

1.2

1

1.4

X

1.6

1.8

2

X

(B) Re = 4500

(A) Re = 3000 45 Δϕ = 0 kV Δϕ = 4 kV Δϕ = 6 kV Δϕ = 10 kV

Nuloc

30

15

0

1

1.2

1.4

1.6

1.8

2

X

(C) Re = 6000

FIGURE 12.6 Effects of Reynolds number and supplied voltage on local Nusselt number when φ 5 0:05. (A) Re 5 3000, (B) Re 5 4500,

(C) Re 5 6000.

There exist two models for charge distribution: (1) conductivity model and (2) mobility model. In the first model, electroconvection relies on a temperature gradient. But in the second model, electroconvection is independent of the temperature gradient in the liquid. In the case of free charge origination, the second model is more acceptable according to experimental results. Electric current density can be defined as: -

-

-

J 5 σ E 2 Drq 1 q V

-

ð12:23Þ

-

where σ E is ionic mobility, Drq is diffusion, qV is convection.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

408

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

18

15

Nuave

12

9

Re = 3000 6

Re = 4500 Re = 6000

3

0

2

4

6

8

10

Δϕ (kV) FIGURE 12.7 Effects of Reynolds number and supplied voltage on average Nusselt number.

FIGURE 12.8 (A) Geometry of the problem and boundary conditions; (B) the mesh of enclosure considered in this work.

According to Eqs. (12.22) and (12.23), the equation for electric charge density can be obtained as follows: @q @q @q 1u 1v @y @x @t 2 0 1 3 1 4 @@Ey @Ex A @q @q 5 1 1 1 Ey 1 Ex q Re PrE @y @x @y @x 0 1 2 2 ρnf =ρf 1 @@ q 1 @ q A 5 μnf =μf Re De @y 2 @x 2

ð12:24Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.3 ELECTROHYDRODYNAMIC NANOFLUID HYDROTHERMAL TREATMENT IN AN ENCLOSURE WITH SINUSOIDAL UPPER WALL

FIGURE 12.9

409

Electric density distribution injected by the bottom electrode.

The diffusion term can be taken negligible. Also Drq in Eq. (12.23) can be taken as negligible and σ 5 bq. So Eq. (12.23) can be considered as: -

-

-

J 5 q V 1 qb E

ð12:25Þ

In presence of electric field Coulomb forces should be added to the momentum equation and the Joule heating effect should be added in the energy equation. So we have: 8 > 1 > > r:V05-0 >

-  > > @V > > 1 V :r V A 5 2 rp 1 μnf r2 V 1 qE ρnf @ > > > @t > > > 0 1 > > >

-  >  -@T

> > > @q > > > 50 r: J 1 > > @t > > > > > > > r:εE 5 q > > :E 5 2 rϕ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

410

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM




ρnf ; ρCp

 nf

; αnf ; μnf , and knf are defined as:


ρnf 5 ρs φ 1 ρf ð1 2 φÞ

 ρCp nf 5 ρCp s φ 1 ρCp f ð1 2 φÞ
 αnf 5 knf = ρCp nf 

μnf 5 knf 5 kf

μf ð12φÞ2:5

2 2φðkf 2 ks Þ 1 2kf 1 ks 1 φðkf 2ks Þs 1 2kf 1 k

ð12:27Þ ð12:28Þ ð12:29Þ ð12:30Þ ð12:31Þ

The thermophysical properties of the working fluid are given in Table 12.1. Nondimensional parameters are introduced as follow: t5

tULid P y x ;p 5 ;y 5 ;x 5 ; 2 L L L ρULid

v5

v u T 2 T0 ; rT 5 T1 2 T0 ; ;u 5 ;θ5 ULid ULid rT

ϕ5

ϕ 2 ϕ0 q E ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 q0 E0 rϕ

ð12:32Þ

In order to reach a clear formulation, the over bar will be deleted in the next equations. So, the governing equations can be considered as follows: 8 - > r: V 5 0 > > > > 0 1 > > >

 > ρ =ρ > > @@V 1 V :r V A 5 2 rp 1 nf f 1 r2 V 1 SE q E > > > > @t μnf =μf Re ρnf =ρf > > > > 0 1 > > >

- - > < @θ -  knf =kf 1 S Ec @ 1 V :r θA 5

 E
 J :E r2 θ 1
ð12:33Þ @t ρCp nf = ρCp f RePr ρCp nf = ρCp f > > > > > > @q > > > 50 r: J 1 > > @t > > > > > > > > r:εE 5 q > > > > :E 5 2 rϕ The formulas of vorticity and stream function are: ω5

@v @u 2 ; @x @y

v52 Ω5

@ψ @ψ ;u5 ; @x @y

ð12:34Þ

ωL ψ ;Ψ 5 ULid LULid

Continuity equation has been satisfied by the stream function. By eliminating pressure between x-momentum and y- momentum, the vorticity equation can be obtained.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.3 ELECTROHYDRODYNAMIC NANOFLUID HYDROTHERMAL TREATMENT IN AN ENCLOSURE WITH SINUSOIDAL UPPER WALL

Nuloc and Nuave along the lid wall can be obtained as: Nuloc 5

 knf @Θ kf @Y

ðL 1 Nuave 5 Nuloc dX L

411

ð12:35Þ

ð12:36Þ

0

12.3.3 Effects of Active Parameters The electric field effect on hydrothermal behavior of nanofluid in an enclosure with moving lower and sinusoidal upper walls is presented. Table 12.1 illustrates the properties of ethylene glycol and Fe3O4. Calculations are prepared for various values of supplied voltage (Δϕ 5 0; 4; 6, and 10kV), volume fraction of nanoparticles (φ 5 0% and 4%) and Reynolds number numbers (Re 5 3000; 4500, and 6000). In all calculations, the Prandtl number (Pr) and Eckert number (Ec) are set to 149.54, 0.0, and 10-6. The influence of Reynolds number and supplied voltage on streamlines and isotherms are shown in Figs. 12.10
12.12. At Re 5 3000, one main eddy and two very small eddies at the upper corners of the enclosure exist in streamline. As the electric field is applied the main eddy turns into two smaller ones. Also the isotherms become denser near the hot wall due to the existence of these eddies. As the Reynolds number increases up to 6000, the upper right eddy becomes stronger and isotherms become denser near the bottom wall. Since convective heat transfer is helped more successfully at a greater flow rate, the Nusselt number is improved at greater Reynolds numbers. So the isotherms become more distorted at greater values of Reynolds number. As the electric field increases, the main cell converts to two eddies which rotate in opposite directions. Figs. 12.13 and 12.14 depict the influences of Δϕ and Re on Nuloc and Nuave along the lid wall. As the Reynolds number increases, the Nusselt number increases due to the decrease in thermal boundary layer thickness. Increasing supplied voltage makes the isotherms more distorted. Local Nusselt number profiles have extremums at higher values of supplied voltage because of the presence of thermal plumes. The Nusselt number is an

FIGURE 12.10

Effect of supplied voltage on streamlines and isotherms when Re 5 3000; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

FIGURE 12.11

Effect of supplied voltage on streamlines and isotherms when Re 5 4500; φ 5 0:04.

FIGURE 12.12

Effect of supplied voltage on streamlines and isotherms when Re 5 6000; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

35

35

35

Δϕ = 0 kV Δϕ = 4 kV Δϕ = 6 kV Δϕ = 10 kV

20

Δϕ = 4

kV

Δϕ = 6

kV

Δϕ = 0 kV

20

20

15

10

10

10

5

5

5

0.5

1 X

(A) Re = 3000

FIGURE 12.13

1.5

2

0

0

0.5

Δϕ = 10 kV

25

15

0

kV

Δϕ = 6 kV

15

0

Δϕ = 4

30

Δϕ = 10 kV

25

Nuloc

Nuloc

25

kV

Nuloc

30

30

Δϕ = 0

1

1.5

2

0

X

1 X

(B) Re = 4500

(C) Re = 6000

0

0.5

Effects of Reynolds number and supplied voltage on local Nusselt number when φ 5 0:04. (A) Re 5 3000, (B) Re 5 4500, (C) Re 5 6000.

1.5

2

414

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

16

Re = 3000 Re = 4500

14

Re = 6000 12

Nuave

10

8

6

4

2

FIGURE 12.14

0

2

4

Δϕ (kV)

6

8

10

Effects of Reynolds number and supplied voltage on average Nusselt number.

increasing function of supplied voltage. In absence of electric field, the Nusselt number for Re 5 6000 is 1.117 times higher than that of Re 5 3000 while in the presence of electric field ðΔϕ 5 10Þ, the Nusselt number for Re 5 6000 is 2.298385 times lower than that of Re 5 3000. Also it can be concluded that the Nusselt number at Δϕ 5 10 for Re 5 3000, 4500, and 6000 are higher than those of obtained at Δϕ 5 0. This observation confirms that the impact of electric field is more marked for a lower Reynolds number.

12.4 EFFECT OF ELECTRIC FIELD ON HYDROTHERMAL BEHAVIOR OF NANOFLUID IN A COMPLEX GEOMETRY 12.4.1 Problem Definition Fig. 12.15 illustrates the physical geometry along with the important parameters and mesh of the enclosure. The lower wall has the velocity of ULid and others are stationary. The lower wall has constant temperature T1 and the temperature of other walls is T0 . Also the retained boundary conditions are depicted in Fig. 12.15A. The shape of the inner cylinder profile is assumed to mimic the following pattern r 5 rin 1 A cosðN ðζ ÞÞ

ð12:37Þ

in which rin is the base circle radius, rout is the radius of outer cylinder, A and N are amplitude and number of undulations, respectively. ζ is the rotation angle. In this study A and N are equal to 0.025 and 48, respectively. Fig. 12.16 depicts the distribution of electric density for different Reynolds number and supplied voltage.

12.4.2 Governing Equation In order to simulate nanofluid hydrothermal treatment in the presence of an electric field, we should combine the equations of electric fields with those of hydrothermal. The formulas of the electric field are:

-  ð12:38Þ r: E ε 5 q -

ð 2rϕÞ 5 E -

r: J 1

@q 50 @t

ð12:39Þ ð12:40Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.4 EFFECT OF ELECTRIC FIELD ON HYDROTHERMAL BEHAVIOR OF NANOFLUID IN A COMPLEX GEOMETRY

415

FIGURE 12.15

(A) Geometry of the problem and boundary conditions; (B) the mesh of enclosure considered in this work; (C) a sample triangular element and its corresponding control volume.

-

-

-

J 5 σ E 2 Drq 1 q V " ! # ρnf =ρf 1 @Ey @q @q @q 1 @Ex @q @q 1u 1 1 1 Ex 1v q 1 Ey 5 @y @x Re PrE @y @x @y @x μnf =μf Re De @t

@2 q @2 q 1 @y 2 @x 2

!

ð12:41Þ ð12:42Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

416

FIGURE 12.16

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Electric density distribution injected by the bottom electrode.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.4 EFFECT OF ELECTRIC FIELD ON HYDROTHERMAL BEHAVIOR OF NANOFLUID IN A COMPLEX GEOMETRY

417

Diffusion term can be taken as negligible. Also Drq in Eq. (12.41) can be taken as negligible and σ 5 bq [51]. So Eq. (12.5) can be considered as: -

-

-

J 5 q V 1 qb E

ð12:43Þ

In the presence of an electric field Coulomb forces should be added to the momentum equation and the Joule heating effect should be added in the energy equation. So we have: 8 > r:V05 0 1 > > >

 > > > > ρnf @@V 1 V :r V A 5 2 rp 1 μnf r2 V 1 qE > > > @t > > > 0 1 > > >

 >
 -@T < ρCp nf @ 1 V :r T A 5 knf r2 T 1 J :E ð12:44Þ @t > > > > @q > > > r: J 1 50 > > @t > > > > > > > r:εE 5q > > :E 5 2 rϕ
 ρnf ; ρCp nf ; αnf ; knf , and μnf are defined as: ρnf 5 ρs φ 1 ρf ð1 2 φÞ


 ρCp nf 5 ρCp s φ 1 ρCp f ð1 2 φÞ
 αnf 5 knf = ρCp nf knf 5 kf

2 2φðkf 2 ks Þ 1 2kf 1 ks 1 φðkf 2ks Þs 1 2kf 1 k μnf 5

μf ð12φÞ2:5

ð12:45Þ ð12:46Þ ð12:47Þ ð12:48Þ ð12:49Þ

Nondimensional parameters are introduced as follows: tULid P y x v u ;p 5 ;y 5 ;x 5 ;v 5 ;u5 ; L L ULid ULid L ρULid 2 ð12:50Þ T 2 T0 ϕ 2 ϕ0 q E ; rT 5 T1 2 T0 ; ϕ 5 ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 θ5 q0 E0 rT rϕ
 where rT and rϕ are ðT1 2 T0 Þ and ϕ1 2 ϕ0 , respectively. In order to reach a clear formulation, the over bar will be deleted in the next equations. So, the governing equations can be considered as follows: 8 - > r: V 5 0 > > 0 1 > > >

-  > ρnf =ρf 1 2 > @V SE >@ > r V1 1 V :r V A 5 2 rp 1 qE > > > @t μnf =μf Re ρnf =ρf > > > 0 1 > > >

- - < @θ -  knf =kf 1 S Ec @ 1 V :r θA 5

 E
 J :E r2 θ 1
ð12:51Þ @t ρCp nf = ρCp f ρCp nf = ρCp f RePr > > > > > @q > > r:> 50 J 1 > > @t > > > > > > > r:εE 5 q > > :E 5 2 rϕ t5

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

418

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

The formulas of vorticity and stream function are: ω5

@v @u 2 ; @x @y

v52 Ω5

@ψ @ψ ;u5 ; @x @y

ð12:52Þ

ωL ψ ;Ψ 5 ULid LULid

Continuity equation has been satisfied by the stream function. By eliminating pressure between x-momentum and y-momentum, the vorticity equation can be obtained. Nuloc and Nuave along the lid wall can be obtained as:  knf @Θ Nuloc 5 kf @Y

1 Nuave 5 L

ð12:53Þ

rð out

Nuloc dX

ð12:54Þ

rin

12.4.3 Effects of Active Parameters Effect of nonuniform electric field on nanofluid hydrothermal behavior in an enclosure with sinusoidal wall is presented. The working fluid is ethylene glycol and Fe3O4. Calculations are prepared for various values of supplied voltage (Δϕ 5 0; 4; 6, and 10kV), volume fraction of nanoparticles (φ 5 0% and 5%) and Reynolds number numbers (Re 5 3000; 4500, and 6000). In all calculations, the Prandtl number (Pr) and Eckert number (Ec) are set to 149.54 and 10
6. Influence of Reynolds number and supplied voltage on streamlines and isotherm are shown in Figs. 12.17
12.19. At Re 5 3000, one main eddy and two very small eddies at the upper corners of the enclosure exist in the streamlines. As the electric field is applied the main eddy turns into two smaller ones. Also the isotherms become denser near the hot wall due to the existence of these eddies. As the Reynolds number increases up to 6000, the upper right eddy becomes stronger and isotherms become denser near the bottom wall. Since convective heat transfer is helped more successfully at a greater flow rate, the Nusselt number is improved at greater Reynolds numbers. So the isotherms become more distorted at greater values of Reynolds number. As the electric field increases, the main cell converts to two eddies which rotate in opposite directions. Figs. 12.20 and 12.21 depict the influences of Δϕ and Re on Nuloc and Nuave along the lid wall. As the Reynolds number increases, the Nusselt number increases due to a decrease in thermal boundary layer thickness. Increasing the supplied voltage makes the isotherms more distorted. The local Nusselt number profiles have extremums at higher values of supplied voltage because of the existence of thermal plumes. The Nusselt number is an increasing function of supplied voltage. In the absence of an electric field, the Nusselt number for Re 5 6000 is 1.135921 times higher than that of Re 5 3000 while in the presence of electric field ðΔϕ 5 10Þ, the Nusselt number for Re 5 6000 is 1.768835 times lower than that of Re 5 3000. Also it can be concluded that Nusselt number at Δϕ 5 10 for Re 5 3000, 4500, and 6000 are 5.816335, 4.112412, and 2.894771 times higher than those of obtained at Δϕ 5 0. This observation confirms that the impact of an electric field is more marked for a lower Reynolds number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.5 EFFECT OF COULOMB FORCES ON FE3O4-H2O NANOFLUID THERMAL IMPROVEMENT

FIGURE 12.17

419

Effect of supplied voltage on streamlines and isotherms when Re 5 3000; φ 5 0:05.

12.5 EFFECT OF COULOMB FORCES ON FE3O4-H2O NANOFLUID THERMAL IMPROVEMENT 12.5.1 Problem Definition Fig. 12.22 demonstrates the schematic of this problem and its boundary conditions. The influence of an electric field on Fe3O4-H2O nanofluid is considered. Only the top wall can move. The contours of electric density for various values of active parameters is depicted in Fig. 12.23.

12.5.2 Governing Equation According to Gauss’s law and Maxwell’s relation, the electric field can be defined as: -

q 5 r:ε E

ð12:55Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

420

FIGURE 12.18

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Effect of supplied voltage on streamlines and isotherms when Re 5 4500; φ 5 0:05. -

E 5 2 rϕ -

r: J 1 -

-

ð12:56Þ

@q 50 @t

ð12:57Þ -

J 5 q V 2 Drq 1 σ E

The governing equations are as follows: 8 > r:V05 0 1 > > >

- - @V > > > A 5 qE 1 μnf r2 V 2 rp > > ρnf @ V :r V 1 > > @t > > > 0 1 > > -> > knf @T J :E < @ -   r2 T 1
 V :r T 1 A 5
@t ρC ρC p nf p nf > > > > @q > > > r: J 1 50 > > @t > > > > > > > r:εE 5 q > > :E 5 2 rϕ

ð12:58Þ

ð12:59Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.5 EFFECT OF COULOMB FORCES ON FE3O4-H2O NANOFLUID THERMAL IMPROVEMENT

FIGURE 12.19




ρnf , ρCp



nf

421

Effect of supplied voltage on streamlines and isotherms when Re 5 6000; φ 5 0:05.

; μnf , and knf can be obtained as: ρnf 5 ρf ð1 2 φÞ 1 ρs φ; μf ; μnf 5 ð12φÞ2:5




ρCp





 5 ρCp f ð1 2 φÞ 1 ρCp s φ; knf ks 1 2kf 2 2φðkf 2 ks Þ 5 ks 1 2kf 1 φðkf 2 ks Þ kf nf

Properties of Fe3O4 and H2O are illustrated in Table 12.3. So, the final dimensionless equations in the presence of an electric field are: 8 > r:V 5 0 0 1 > > >

- - @V > ρ =ρ > > @ V :r V 1 A 5 SE qE 1 nf f 1 r2 V 2 rp > > > > @t ρnf =ρf μnf =μf Re > > > 0 1 > > >

- - > knf =kf @θ 1 S Ec

> > > > > > E 5 2 rϕ > > > > q 5 r:εE > > > > @q > > > : r: J 5 2 @t

ð12:60Þ

ð12:61Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

422

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

30 Δϕ = 0 kV

Δϕ = 0 kV

Δϕ = 4 kV

Δϕ = 4 kV

Δϕ = 6 kV Δϕ = 10 kV

Δϕ = 6 kV Δϕ = 10 kV

15

Nuloc

Nuloc

30

0

15

0 1.05

1.2

1.4

1.6

1.8

2

1.05

1.2

1.4

X

1.6

1.8

2

X

(A) Re = 3000

(B) Re = 4500

30 Δϕ = 0 kV Δϕ = 4 kV

Nuloc

Δϕ = 6 kV Δϕ = 10 kV

15

0 1.05

1.2

1.4

1.6

1.8

2

X

(C) Re = 6000

FIGURE 12.20 (C) Re 5 6000.

Effects of Reynolds number and supplied voltage on the local Nusselt number when φ 5 0:05. (A) Re 5 3000, (B) Re 5 4500,

where u5

u v y x ;v 5 ;y 5 ;x 5 ; ULid ULid L L

t5

tULid P q E ;p 5 q5 ;E5 ; 2 q0 E0 L ρULid

rT 5 T1 2 T0 ; θ 5

ð12:62Þ

T 2 T0 ϕ 2 ϕ0 ; rϕ 5 ϕ1 2 ϕ0 ; ϕ 5 rT rϕ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.5 EFFECT OF COULOMB FORCES ON FE3O4-H2O NANOFLUID THERMAL IMPROVEMENT

423

12

Re = 3000 Re = 4500 Re = 6000

Nuave

9

6

3

0

FIGURE 12.21

0

2

4

Δϕ (kV)

6

8

10

Effects of Reynolds number and supplied voltage on the average Nusselt number.

By eliminating the pressure gradient, the vorticity and stream function can be introduced as: ω5

@v @u @ψ @ψ ψ ωL2 2 ;2 5 v; 5 u; Ψ 5 ; Ω 5 @x @y @x @y αf αf

ð12:63Þ

Nuloc and Nuave along the left wall are defined as: Nuloc 5

 @Θ knf @X kf

ðL 1 Nuloc dY Nuave 5 L

ð12:64Þ

ð12:65Þ

0

12.5.3 Effects of Active Parameters Coulomb forces impact on Fe3O4
H2O hydrothermal behavior in a cavity with moving wall is studied. Various values of active parameters are examined, such as Reynolds number (Re 5 3000; 4000; 5000, and 6000), volume fraction of solid particle (φ 5 0% and 4%), and supplied voltage (Δϕ 5 0; 5; 8, and 10kV). Impacts of Δϕ and Re on isotherms and streamlines are demonstrated in Figs. 12.24
12.27. At low Re, one main vortex exists in the streamlines. As the Coulomb forces augment the main vortex turns into three smaller vortexes and the temperature gradient near the hot walls is enhanced. As lid velocity increases, another eddy appears at the bottom right of the enclosure. As the voltage enhances, the previous cell alters to two vortexes which revolve in opposite directions. A further increase in the Reynolds number leads to the generation of another small eddy at the bottom left of the enclosure. Temperature gradient augments at high Reynolds numbers due to stronger convective heat transfer.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

424

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

FIGURE 12.22 (A) Geometry of the problem and boundary conditions; (B) the mesh of enclosure considered in this work; (C) a sample triangular element and its corresponding control volume.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.5 EFFECT OF COULOMB FORCES ON FE3O4-H2O NANOFLUID THERMAL IMPROVEMENT

FIGURE 12.23

425

Electric density distribution injected by the bottom electrode.

Impacts of Δϕ and Re on Nuloc and Nuave are demonstrated in Figs. 12.28 and 12.29. As the lid wall velocity augments, the Nusselt number enhances due to a rise in the temperature gradient. Augmenting the Coulomb forces causes the isotherms near the left wall to be denser. The presence of thermal plumes for high values of supplied voltage leads to the existence of extremums for Nuloc profiles. The Nusselt numbers in the presence of electric field with supplied voltage Δϕ 5 10kV of Re 5 3000, 4000, 5000, and 6000 are, respectively, 1.09, 2.15, 8.96, and 1.76 times higher than those in the absence of an electric field.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

426

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

FIGURE 12.23

TABLE 12.3

(Continued).

Thermophysical Properties of Water and Nanoparticles ρ ðkg=m3 Þ

Cp ðj=kgkÞ

k ðW=m:kÞ

H2O

997.1

4179

0.613

Fe3 O4

5200

670

6

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD

FIGURE 12.24

427

Effect of supplied voltage on streamlines and isotherms when Re 5 3000; φ 5 0:04.

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD 12.6.1 Problem Definition Fig. 12.30 illustrates the schematic of this problem and its boundary conditions. The influence of an electric field on Fe3O4-ethylene glycol nanofluid is considered. Only the bottom wall can move. Fig. 12.31 illustrates the contour of q for various values of rϕ and Re. As rϕ increases the distortion of isoelectric density lines become greater and one cell appears on the right side. The influence of rϕ on q is more sensible than Re.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

428

FIGURE 12.25

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Effect of supplied voltage on streamlines and isotherms when Re 5 4000; φ 5 0:04.

12.6.2 Governing Equation According to Gauss’s law and Maxwell’s relation, the electric field can be defined as: -

q 5 r:ε E -

E 5 2 rϕ

ð12:66Þ ð12:67Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD

FIGURE 12.26

429

Effect of supplied voltage on streamlines and isotherms when Re 5 5000; φ 5 0:04.

-

r: J 1 -

-

@q 50 @t

ð12:68Þ -

J 5 q V 2 Drq 1 σ E

ð12:69Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

430

FIGURE 12.27

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Effect of supplied voltage on streamlines and isotherms when Re 5 6000; φ 5 0:04.

The governing equations are:

8 > > > r:V 5 0 > -! >

- - @V > μnf 2 - rp qE > > > V :r V 1 5 1 r V2 > > ρnf @t ρnf ρnf > > > ! > ->

-  > > knf @T J :E <  r2 T 1
 5
V :r T 1 @t ρCp nf ρCp nf > > > > > rϕ 5 2 E > > > > @q > > 5 2 r: J > > > @t > > > > : q 5 r:εE

ð12:70Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

431

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD

0.6

0.6

Re = 3000

Δϕ = 0 kV

Re = 4000

Δϕ = 5 kV Δϕ = 8 kV Δϕ = 10 kV

Re = 5000

0.4

0.4

Nuloc

Nuloc

Re = 6000

0.2

0.2

0

0

FIGURE 12.28

20 0.25

40 0.5

60 0.75

0

80 1

0

20 0.25

40 0.5

Y

Y

(A) without EHD

(B) Re = 6000

60 0.75

80 1

Effects of Reynolds number and supplied voltage on the local Nusselt number when φ 5 0:04. (A) without EHD, (B) Re 5 6000.

20

Re = 3000 Re = 4000 Re = 5000

Nuave

16

Re = 6000

12

8

4

0

2

4

6

8

10

Δϕ (kV)

FIGURE 12.29

Effects of Reynolds number and supplied voltage on the average Nusselt number.


 knf , ρCp nf ; μnf , and ρnf can be obtained as: knf 2 2φðkf 2 ks Þ 1 ks 1 2kf 5 ; kf φðkf 2 ks Þ 1 ks 1 2kf


ρCp

 nf



 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ð12:71Þ

μ 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

432

FIGURE 12.30

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Geometry of the problem and boundary conditions.

Properties of Fe3O4 and ethylene glycol are illustrated in Table 12.1. So, the final dimensionless equations in the presence of an electric field are: 8 > > > r:V 5 0 > > > -  > > - > @V 1 ρnf =ρf 2 SE > > r V 2 rp 1 qE V :r V 1 5 > > > Re μnf =μf @t ρnf =ρf > > > > >  >

- - > -  > knf =kf @θ 1 1 > >

 r2 θ 1

 SE Ec J : E :r θ 1 5 V < @t PrRe ρCp nf = ρCp f ρCp nf = ρCp f ð12:72Þ > > > > > > > E 5 2 rϕ > > > > > > > q 5 r:εE > > > > > > @q > > > r: J 5 2 > : @t where
  y; x ðu; vÞ ϕ 2 ϕ0
T 2 T0 ; y; x 5 ; ðu; v Þ 5 ;ϕ5 ;θ5 ULid rϕ rT L t5

tULid P q E ;p 5 q5 ;E5 ; q0 E0 L ρULid 2

ð12:73Þ

rT 5 T1 2 T0 ; rϕ 5 ϕ1 2 ϕ0 :

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD

FIGURE 12.31

433

Electric density distribution injected by the bottom electrode.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

434

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

By eliminating pressure gradient, vorticity and stream function can be introduced as: v52

@ψ @ψ ψ ωL2 @v @u ; 5 u; Ψ 5 ; Ω 5 2 ;ω5 @x @y αf @x @y αf

ð12:74Þ

Nuloc and Nuave along the hot wall are calculated as: Nuloc 5 1 Nuave 5 L

FIGURE 12.32

 knf @Θ kf @X ðL Nuloc dY

ð12:75Þ

ð12:76Þ

0

Effect of supplied voltage on streamlines and isotherms when Re 5 3000; φ 5 0:05.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

12.6 ACTIVE METHOD FOR NANOFLUID HEAT TRANSFER ENHANCEMENT BY MEANS OF EHD

435

12.6.3 Effects of Active Parameters Fe3O4-ethylene glycol nanofluid forced convection in an enclosure is examined in the presence of an electric field. The bottom wall is the moving lid. The roles of supplied voltage (Δϕ 5 0 to 10kV), Reynolds number (Re 5 3000 to 6000), and volume fraction of Fe3O4 (φ 5 0% to 5%) are presented graphically. The effects of Δϕ and Re on streamlines and isotherms are illustrated in Figs. 12.32
12.34. Two eddies appear in streamlines which rotate in opposite directions. The counterclockwise eddy is stronger and located at the left side upper of the square obstacle. The presence of such eddies leads to the generation of two thermal plumes at the left and upper side of the square obstacle. As an electric field is applied, the clockwise eddy becomes stronger and the other one converts to two smaller counterclockwise eddies which are located at the left side of enclosure. The isotherms become more disturbed by augmenting Δϕ. By increasing Re, the strength of the rotating eddies is enhanced and the distortion of the isotherms becomes greater than before. The temperature gradient enhances at high Reynolds numbers due to stronger convective heat transfer. At high Reynolds

FIGURE 12.33

Effect of supplied voltage on streamlines and isotherms when Re 5 4500; φ 5 0:05.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

436

FIGURE 12.34

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

Effect of supplied voltage on streamlines and isotherms when Re 5 6000; φ 5 0:05.

number, the Coulomb force is weaker than the viscous force, so the impact of adding an electric field becomes weaker with the increase of Re. The influences of Re and Δϕ on Nuave are illustrated in Fig. 12.35. A correlation for average Nusselt number is presented as: Nuave 5 4:86 2 1:52Re 1 1:01Δϕ 2 0:12Re Δϕ 1 0:17ðRe Þ2 2 0:02ðΔϕÞ2

ð12:77Þ

where Re 5 Re 3 1023 and Δϕ is voltage supply in Kilovolt. In the absence of an electric field, as the velocity of the bottom wall augments, convective heat transfer becomes stronger and the temperature gradient enhances with rise of Re. Coulomb force helps the convention mode to enhance. So Nuave augments with the rise of Re and Δϕ. In the presence of a high electric field enhancing Re shows the reverse behavior and the Nusselt number decreases with the increase of Re. These outputs proved that using an electric field is more useful at low Reynolds number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES:

FIGURE 12.35

437

Effects of Reynolds number and supplied voltage on average Nusselt number when φ 5 0:05.

References: [1] H.S. Shu, F.C. Lai, Effect of Electrical Field on Buoyancy-Induced Flows in an Enclosure, Conference Record—IAS Ann. Meet. (IEEE Ind. Applicat. Soc.) vol. 2 (1995) 1465
1471. [2] N. Kasayapanand, J. Tiansuwan, W. Asvapoositkul, N. Vorayos, T. Kiatsiriroat, Effect of the electrode arrangements in tube bank on the characteristic of electrohydrodynamic heat transfer enhancement: low Reynolds number, J. Enhanced Heat Transf. 9 (2002) 229
242. [3] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman, EHD nanofluid force convective heat transfer considering electric field dependent viscosity, Int. J. Heat Mass Transfer 108 (2017). [4] M. Sheikholeslami, R. Ellahi, Electrohydrodynamic nanofluid hydrothermal treatment in an enclosure with sinusoidal upper wall, Appl. Sci. 5 (2015) 294
306. Available from: https://doi.org/10.3390/app5030294.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

438

12. FORCED CONVECTION OF NANOFLUID IN EXISTENCE OF ELECTRIC FIELD USING CVFEM

[5] M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry, J. Mol. Liq. 213 (2016) 153
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437. [34] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transfer 118 (2018) 182
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water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
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[70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
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1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
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386. [75] M. Sheikholeslami, M. Jafaryar, D.D. Ganji, Z. Li, Exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators, J. Mol. Liq. 262 (2018) 104
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660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Des. 120 (2017) 382
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25.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

13 Darcy Model for Nanofluid Flow in a Porous Media by Means of CVFEM 13.1 INTRODUCTION Convective flows in porous media have occupied the central stage in many fundamental heat transfer analyses and have received considerable attention over the last few decades. This interest is due to their wide range of applications in high performance insulation for buildings, chemical catalytic reactors, packed sphere beds, grain storage, and such geophysical problems as frost heave. Porous media are also of interest in relation to the underground spread of pollutants, solar power collectors, and to geothermal energy systems. The literature concerning convective flows in porous media is abundant and representative studies in this may be found in the books by Oosthuizen and Naylor [1]. Darcy’s law can be used for porous media. Because of the complexity of pore geometries in a porous medium, Darcy’s law has to be used to obtain any meaningful insights into the physics of flow in porous media. Darcy’s law has been verified by the results of many experiments. Theoretical backing for it has been obtained in various ways, with the aid of either deterministic or statistical models [2]. Vafai [3] have described that Darcy’s approximation is reasonable for the geothermal flows, except perhaps near boreholes [3]. Darcy’s law is valid only when the pore Reynolds number is of the order of 1, and for many practical applications Darcy’s law is not valid, and boundary and inertial effects need to be accounted for [3]. For example, modeling fluid flows in the fuel cells. This is because the fuel cell consists of a number of distinct layers of different porosities. Darcy’s law assumes no effect of boundaries and the fluid velocity in Darcy’s equation is determined by the permeability of the matrix. Therefore at the interface between the regions of different porosity in the fuel cell, particularly between the free fluid flow region, such as a gas flow channel, and a permeable medium, then a discontinuity in the fluid velocity and/or the shear stress could emerge [4]. Sheikholeslami and Rokni [5] studied the magnetic field effect on nanofluid flow in a porous complex shaped enclosure. Sheikholeslami and Zeeshan [6] analyzed flow and heat transfer of water-based nanofluid due to magnetic field in a porous enclosure with constant heat flux. Sheikholeslami and Shehzad [7] studied magnetohydrodynamic nanofluid convection in a porous enclosure considering the heat flux boundary condition. Sheikholeslami [8] employed Darcy’s law for porous enclosure filled with nanofluid. Sheikholeslami and Ganji [9] investigated the transportation of MHD nanofluid free convection in a porous semiannulus using numerical approach. Sheikholeslami [10] presented the CuO-water nanofluid free convection in a porous cavity considering Darcy’s law. Sheikholeslami et al. [11] simulated nanofluid flow in a porous media in the presence of Lorentz forces. Recently, several simulation works have been published in various journals [12
93].

13.2 MAGNETOHYDRODYNAMIC CUO-WATER NANOFLUID IN A POROUS COMPLEX SHAPED ENCLOSURE 13.2.1 Problem Definition Fig. 13.1 shows the geometry, boundary condition, and sample element. The formula of the inner cylinder is:

 ð13:1Þ r 5 rin 1 A cos N ζ 2 ζ 0 ; A 5 0:5 in which rin ; rout are radius of the base circle and outer cylinder. Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00013-8

441

© 2019 Elsevier Inc. All rights reserved.

442

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Region of support

1.5

Nanofluid

Tc Th

A

g

Control volume

1 0.5

B

y γ

0

ζ

o x

–0.5 –1

rin rout

–1.5 porous media –2.5

–2

–1.5 –1

(A)

–0.5

0

0.5

1

1.5

2

2.5

(B) Si , 3

i Si , 4

j=4

f1

f2 i=1

j=3 (C)

FIGURE 13.1 (A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

13.2.2 Governing Equation 2D steady convective flow of nanofluid in a porous media is considered in the presence of a constant horizontal magnetic field. The PDEs equations are: -

r: V 5 0

 μnf - V 5 rp 1 I 3 B 1 ρnf g K
 -  ρCp nf V :r T 5 knf r2 T

ð13:2Þ ð13:3Þ ð13:4Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.2 MAGNETOHYDRODYNAMIC CUO-WATER NANOFLUID IN A POROUS COMPLEX SHAPED ENCLOSURE

-

r: I 5 0

-  σnf V 3 B 2 rϕ 5 I

443 ð13:5Þ ð13:6Þ

In Eq. (13.2), Darcy’s model is used for a porous medium. Eqs. (13.5) and (13.6) reduce to r2 ϕ 5 0. So the electric field can be neglected. So the above equations turn into: @v @u 1 50 @y @x 2 2




ρCp

 nf

 K @p σnf KB20
2 2u sin2 γ 1 v sinγ cosγ 5 u μnf @x μnf

ðρβ Þnf K  K @p σnf KB20
2 2v cos2 γ 1 u sinγ cosγ 1 g ð T 2 Tc Þ 5 v μnf @y μnf μnf   2
 @T @T @ T @2 T 1v 1 2 ρCp nf u 5 knf @x @y @x2 @y

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð13:7Þ ð13:8Þ ð13:9Þ ð13:10Þ

ð13:11Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ

ð13:12Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð13:13Þ

kn f ; μn f are obtained according to the Koo
Kleinstreuer
Li (KKL) model:  kp 3 21 φ sffiffiffiffiffiffiffiffiffi kf κb T 4 0  1 5 3 10 g ðφ; T; dp Þφρf cp;f knf 5 1 1  kp kp ρp dp 12 2 21 φ kf kf



2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðT Þ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð13:14Þ

ð13:15Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð13:16Þ

All required coefficients and properties are illustrated in Tables 13.1 and 13.2. Introducing dimensionless quantities:
 T 2 Tc Ψ 5 ψ=αnf ; ðX; YÞ 5 x; y =L; θ 5 Th 2 Tc By discarding the pressure, the final equations are: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θ 2 Ra A4 A5 @X

ð13:17Þ

ð13:18Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

444

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

TABLE 13.1 The Coefficient values Of CuO 2 Water Nanofluid Coefficient values

CuO 2 Water

a1

2 26.593310846

a2

2 0.403818333

a3

2 33.3516805

a4

2 1.915825591

a5

6.42185846658E-02

a6

48.40336955

a7

2 9.787756683

a8

190.245610009

a9

10.9285386565

a10

2 0.72009983664

TABLE 13.2 Thermophysical Properties of Water and Nanoparticles

Water CuO

ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK21 Þ

dp ðnmÞ

σðΩUmÞ21

997.1

4179

0.613

21

-

0:05

6500

540

18

-10

29

10

@2 θ @2 θ @θ @Ψ @θ @Ψ 2 1 5 2 2 @Y @X @X @Y @Y @X

6500



where Ra 5

g K ðρβ Þf L ΔT μf αf

and Ha 5

σf K B20 μf

ð13:19Þ

are the Rayleigh and Hartmann numbers for the porous media. Also

Ai ði 5 1::6Þ are constants parameters which are obtained as: A1 5

ρnf ρf

A2 5

;

A3 5

ðρCP Þnf ðρCP Þf

;

ðρβ Þnf ðρβ Þf

A4 5

; A5 5

μnf μf

;

knf σnf ; A6 5 kf σf

ð13:20Þ

and boundary conditions are: Θ 5 1:0

on inner wall

Θ 5 0:0

on outer wall

Ψ 5 0:0

on all walls

ð13:21Þ

Local and average Nusselt numbers over the cold cylinder can be calculated as: Nuloc 5 A4 1 Nuave 5 π

@Θ @r

ð13:22Þ

3π=2 ð

Nuloc ðζ Þ dζ

ð13:23Þ

π=2

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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13.2 MAGNETOHYDRODYNAMIC CUO-WATER NANOFLUID IN A POROUS COMPLEX SHAPED ENCLOSURE

13.2.3 Effects of Active Parameters Lorentz forces’ influence on nanofluid flow in a porous enclosure with complex hot cylinder is investigated. Darcy’s model is taken into account to present the governing equations. Numerical simulations are examined for various Rayleigh numbers for porous medium (Ra 5 102 ; 250, and 103 ), number of undulations (N 5 3; 4; 5, and 6), volume fraction of CuO-water (φ 5 0 and 0.04), and Hartmann number for porous medium (Ha 5 0 to 20). Figs. 13.2
13.4 demonstrate the impacts of number of undulations, Rayleigh and Hartmann numbers for porous medium on streamlines and isotherms. As nanofluid temperature augments, the nanofluid begins moving from the warm surface to the outer one and dropping along the circular cylinder, afterwards rising again at the inner cylinder, generating a clockwise revolving vortex inside the cavity. Under the dominance of the conduction mode, isotherms follow the shape of cylinders. jΨ max j augments as buoyancy force augments and it reduces as Lorentz force enhances. For odd values of N, a very small eddy is generated around ζ 5 903 . This phenomenon is observed because of the opposition of the crest in contradiction of the flow movement between the hot and cold walls. But there is no secondary eddy near ζ 5 903 for even values of N because the crest is parallel to the gravity force. As buoyancy forces enhances, the eddies become stronger and the thermal plume is generated near the centerline. Increasing the Hartmann number causes the thermal plume to diminish and change the heat transfer mechanism from convection to conduction. So, Nu reduces with the rise of Lorentz forces. In the absence of Lorentz force when N 5 3, Ra 5 1000, two vortexes rotating in opposite directions appear near the vertical centerline. So, a strong thermal plume appears in this region. As Lorentz force augments, these two vortexes merge together and the thermal plume vanishes.

N=3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

N=4

Ha = 0

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha = 20

0.03 –0.05 –0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 –1.56

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –1 –1.1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.001 –0.002 –0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03

FIGURE 13.2 Isotherm (left) and streamline (right) contours for different values of number of undulations and Hartmann number for porous medium when φ 5 0:04; Ra 5 100.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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N=6

N=5

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.045 –0.2 –0.4 –0.6 –0.8 –0.9 –1 –1.2 –1.34

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.001 –0.005 –0.01 –0.015 –0.02 –0.023 –0.025 –0.03 –0.035

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.2 –0.4 –0.58 –0.6 –0.65 –0.7 –0.8 –1 –1.2

–0.005 –0.01 –0.015 –0.018 –0.019 –0.02 –0.025 –0.03 –0.035

FIGURE 13.2 (Continued).

Figs. 13.5 and 13.6 depict the influence of N; φ ; Ra , and Ha for porous media on Nuloc, Nuave. The correlation for Nuave corresponding active parameters is: Nuave 5 0:98 1 5:5 3 1024 Ra 2 0:016N 1 0:51φ 2 0:021Ha 2 3:2 3 1026 Ra N 2 1:71 3 1025 Ra φ 2 3:6Ra Ha 1 0:15 Nφ 1 3:15 3 1024 N Ha 1 0:016φ Ha 27

23

23

1 1:4 3 10 Ra 1 7:9 3 10 N 1 46:5φ 1 1:16 3 10 Ha 2

2

2

ð13:24Þ

2

The number of extremum in in Nuloc profile is matching to existence of thermal plume and number of undulation. As the buoyancy force increases, the temperature gradient increases and in turn Nuave enhances with the rise of buoyancy forces. As the nanofluid volume fraction augments, the temperature of the fluid is enhanced and Nuave increases with the enhancement of the thermal conductivity of the fluid. Enhancing the Lorentz force causes the nanofluid flow to retard and the Nusselt number reduces. The impacts of number of undulations, buoyancy, and Lorentz forces on heat transfer improvement are demonstrated in Table 13.3 and Fig. 13.7. Adding CuO nanoparticle into water can enhance the thermal conductivity of fluid. So, this passive method is more effective when a conduction mechanism is dominant. According to this fact, heat transfer improvement increases with the rise of Lorentz forces and it reduces with the enhancement of buoyancy forces. As the number of undulations increases, E is enhanced. The maximum value of E is 12.43, which is obtained at N 5 6, Ha 5 20, Ra 5 100.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

447

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

N = 4

N=3

Ha = 0

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha = 20

0.1 –0.1 –0.29 –0.5 –1 –1.5 –2 –2.5 –3 –3.5 –3.85

–0.5 –1 –1.5 –2 –2.4 –2.7 –3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.001 –0.003 –0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09 –0.1

–0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.078

FIGURE 13.3 Isotherm (left) and streamline (right) contours for different values of number of undulations and Hartmann number for porous medium when φ 5 0:04; Ra 5 250.

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE 13.3.1 Problem Definition Fig. 13.8 depicts the geometry, boundary condition, and sample element. The formula of the outer cylinder is: pffiffiffiffiffiffiffiffiffiffiffiffiffi b 5 1 2 ε2 : a ð13:25Þ where a; b; ε are the major, minor axis of elliptic cylinder, and eccentricity for the inner cylinder. The inner cylinder has constant heat flux condition.

13.3.2 Governing Equation 2D steady convective flow of nanofluid in a porous medium is considered in the presence of a constant magnetic field. The PDEs equations are: -

r: V 5 0

 μnf - V 5 2rp 1 I 3 B 1 ρnf g K

ð13:26Þ ð13:27Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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N=6

N=5

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.09 –0.5 –1 –1.5 –1.86 –2 –2.28 –2.5 –3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.5 –1 –1.5 –1.4 –1.5 –1.8 –2 –2.5 –3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.004 –0.01 –0.02 –0.03 –0.04 –0.05 –0.057 –0.06 –0.07 –0.08 –0.09

–0.01 –0.02 –0.03 –0.04 –0.046 –0.05 –0.06 –0.07 –0.08 –0.09

FIGURE 13.3 (Continued).




ρCp

 -  V :r T 5 knf r2 T nf -

r: I 5 0

-  σnf V 3 B 2 rϕ 5 I

ð13:28Þ ð13:29Þ ð13:30Þ

In Eq. (13.26), Darcy’s model is used for a porous medium. Eqs. (13.29) and (13.30) reduce to r2 ϕ 5 0. So the electric field can be neglected. So the above equations turn into: @v @u 1 50 @y @x 2 2

 K @p σnf KB20
2 2u sin2 γ 1 v sinγ cosγ 5 u μnf @x μnf

ðρβ Þnf K  K @p σnf KB20
2 2v cos2 γ 1 u sinγ cosγ 1 g ð T 2 Tc Þ 5 v μnf @y μnf μnf   2
 @T @T @ T @2 T 1v 1 2 ρCp nf u 5 knf @x @y @x2 @y

ð13:31Þ ð13:32Þ ð13:33Þ ð13:34Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

449

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

Ha = 0

Ha = 20

N=3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.25 –0.25 –1 –2 –4 –6 –8 –10 –12 –12.7

N=4

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –10.8

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.008 0.006 0.004 –0.015 –0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3

FIGURE 13.4 Isotherm (left) and streamline (right) contours for different values of number of undulations and Hartmann number for porous medium when φ 5 0:04; Ra 5 1000.




ρCp

 nf

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð13:35Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ

ð13:36Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ

ð13:37Þ

 σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð13:38Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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N=5

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

N=6

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.5 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.4 0.3 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10

0.012 –0.05 –0.1 –0.15 –0.2 –0.23 –0.25 –0.3 –0.35

–0.05 –0.1 –0.15 –0.18 –0.19 –0.2 –0.25 –0.3 –0.35

FIGURE 13.4 (Continued).

knf ; μnf are obtained according to the Koo
Kleinstreuer
Li (KKL) model:  kp 3 21 φ sffiffiffiffiffiffiffiffiffi kf κb T 4 0  1 5 3 10 g ðφ; T; dp Þφρf cp;f knf 5 1 1  kp kp ρ p dp 12 2 21 φ kf kf





2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðTÞ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð13:39Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð13:40Þ

All required coefficients and properties are illustrated in Tables 13.1 and 13.2. Introducing dimensionless quantities:
 T 2 Tc qvL ; ΔT 5 Ψ 5 ψ=αnf ; ðX; YÞ 5 x; y =L; θ 5 kf ΔT

ð13:41Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

451

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

Ra = 102 Ha = 0 Ha = 5 Ha = 10 Ha = 20

Ha = 0 Ha = 5 Ha = 10 Ha = 20

4

2

1.5 1 0.5 90º

180º

225º

0.5 90º

270º

1 135º

ζ

225º

270º

Ha = 0 Ha = 5 Ha = 10 Ha = 20

6 Nuloc

Nuloc

5

1.5

2

4 3

1.5

2 1

135º

180º

225º

0.5 90º

270º

1 135º

180º

225º

0 90º

270º

135º

3.5 Ha = 0 Ha = 5 Ha = 10 Ha = 20

2

3

270º

Ha = 0 Ha = 5 Ha = 10 Ha = 20

6 5 Nuloc

Nuloc

225º

7 Ha = 0 Ha = 5 Ha = 10 Ha = 20

2.5 1.5

180º

ζ

ζ

2.5

Nuloc

180º

7 Ha = 0 Ha = 5 Ha = 10 Ha = 20

ζ

2

4 3

1.5

2

1 1 0.5 90º

135º

180º

225º

270º

0.5 90º

1 135º

ζ

180º

225º

0 90º

270º

3

6

ζ FIGURE 13.5

225º

270º

4

2

1 180º

5 3

1.5 1

135º

270º

Ha = 0 Ha = 5 Ha = 10 Ha = 20

7

Nuloc

Nuloc

Nuloc

2

0.5 90º

225º

8

Ha = 0 Ha = 5 Ha = 10 Ha = 20

2.5 1.5

180º

ζ

3.5 Ha = 0 Ha = 5 Ha = 10 Ha = 20

2

0.5 90º

135º

ζ

2.5

N=6

135º

ζ

2.5

1

N=5

0 90º

270º

225º

3 Ha = 0 Ha = 5 Ha = 10 Ha = 20

2 Nuloc

180º

ζ

2.5

0.5 90º

3 2

1

135º

Ha = 0 Ha = 5 Ha = 10 Ha = 20

5 Nuloc

1.5

6

2.5 Nuloc

Nuloc

N=3

2

N=4

Ra = 103

Ra = 250 3

2.5

1 135º

180º

ζ

225º

270º

0 90º

135º

180º

225º

270º

ζ

Effects of the number of undulations, Rayleigh number, and Hartmann number for porous medium on local Nusselt number

at φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

2.02 1.8275 2.02 1.635 1.8025

Nueve

1.4425

1.585

Nueve

1.25

6.00

1.3675

5.25

1.15 20.00

1000.00 15.00

4.50

775.00 550.00

10.00

Ha

20.00

325.00

5.00

15.00

Ra

0.00 100.00

N

3.75

10.00

5.00

Ha

0.00 3.00

Ra = 103, φ = 0.04

φ = 0.04, N = 6

1.39 1.325 2.02 1.26 1.8175

Nueve

1.195

Nueve

1.615 0.04

1.4125

1.13

0.03

1.21 6.00

0.02

1000.00 5.25

20.00

775.00 4.50

15.00

550.00 3.75

N

325.00 3.00 100.00

Ra

φ

0.01 5.00

0.00 0.00

Ha

Ha = 20, φ = 0.04

Ra = 103, N = 6

1.38

1.39

1.2925

1.3375

1.205

Nueve

10.00

1.285

Nueve 1.1175

1.2325

1.03

1.18

0.04

6.00 5.25

0.03

1000.00 775.00

0.03

4.50

0.02

φ

0.04

3.75

0.01 0.00 3.00

Ra = 103, Ha = 20

N

φ

0.02

550.00 325.00

0.01 0.00 100.00

Ra

Ha = 20, N = 6

FIGURE 13.6 Effects of the number of undulations, nanoparticle volume fraction, Rayleigh number, and Hartmann number for porous medium on average Nusselt number.

By discarding the pressure, the final equations are: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θ Ra 2 A4 A5 @X

ð13:42Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

Ha

20.00

20.00

15.00

15.00

Ha

10.00

10.00

5.00

5.00

0.00 0.00

0.00 3.75

3.00

4.50

5.25

6.00

0.01

0.02

Ra = 103, φ = 0.04

0.04

775.00

1000.00

Ra = 103, N = 6

20.00

6.00

15.00

5.25

N

10.00

4.50

3.75

5.00

0.00 100.00

3.00 325.00

550.00

775.00

100.00

1000.00

325.00

Ra

550.00

Ra Ha = 20, φ = 0.04

φ = 0.04, N = 6

φ

0.03

φ

N

Ha

453

0.04

0.04

0.03

0.03

0.02

φ 0.02

0.01

0.01

0.00

0.00 100.00

325.00

550.00

Ra Ha = 20, N = 6 FIGURE 13.6

775.00

1000.00

3.00

3.75

4.50

5.25

6.00

N Ra = 103, Ha = 20

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

TABLE 13.3 Effects of the Number of Undulations and Hartmann Number on the Ratio of Heat Transfer Enhancement When Ra 5 102 N Ha

3

4

6

0

11.49639

11.62185

11.70252

5

12.43029

12.43048

12.42186

10

12.43759

12.4375

12.43239

20

12.43968

12.43959

12.43681

18 Ra = 100 Ra = 250 Ra = 1000

16 14 12

E 10 8 6 4 2 0

5

10

15

20

Ha FIGURE 13.7

Effects of the Hartmann number and Rayleigh number for porous medium on the ratio of heat transfer enhancement due to addition of nanoparticles when N 5 6.



@2 θ @2 θ @θ @Ψ @Ψ @θ 1 1 52 2 2 @Y @X @Y @X @Y @X

where Ra 5

g K ðρβ Þf L ΔT μf αf

and Ha 5

σf K B20 μf

ð13:43Þ

are the Rayleigh and Hartmann numbers for the porous media. Also

Ai ði 5 1::6Þ are constants parameters which are obtained as: A1 5

ρnf ρf

A2 5

;

A3 5

ðρCP Þnf ðρCP Þf

;

ðρβ Þnf ðρβ Þf A4 5

; A5 5

μnf μf

;

knf σnf ; A6 5 kf σf

ð13:44Þ

and boundary conditions are: @θ 5 1:0 @n

on inner wall

θ 5 0:0

on outer wall

Ψ 5 0:0

on all walls

ð13:45Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

Region of support g

455

Control volume

1.5 1

B

A y

γ

o

B

q″

ζ

0.5 S

x

0

C

–0.5 Nanofluid

–1 –1.5 –2

–1.5

–1

–0.5

0

(A)

0.5

1

1.5

2

(B) Si,3

Si,4

j=4

f1

f2 i=1

j=3 (C)

FIGURE 13.8 (A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

Local and average Nusselt numbers over the inner cylinder can be obtained as:  1 knf Nuloc 5 θ kf ðs 1 Nuave 5 Nuloc ds S

ð13:46Þ

ð13:47Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

456

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

FIGURE 13.9 Impact of nanofluid volume fraction on streamline (right) and isotherm (left) contours (nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (- - -)) when Ra 5 105 .

13.3.3 Effects of Active Parameters CuO-water nanofluid motion in permeable media in the presence of Lorentz forces is demonstrated. KKL and Darcy models are utilized for nanofluid and permeable media, respectively. CVFEM is utilized to present the effects of active parameters such as Rayleigh number for porous medium (Ra 5 102 ; 250 and 103 ), inclination angle (ξ 5 03 to 903 ), Hartmann number for porous medium (Ha 5 0 to 20), and volume fraction of CuO-water (φ 5 0 and 0.04). Fig. 13.9 illustrates the impact of φ on hydrothermal behavior. As φ increases, nanoparticles interaction increases and in turn nanofluid motion increases. Also nanofluid velocity enhances with the rise of φ. The impact of φ on isotherms is more sensible in the absence of Lorentz forces. Figs. 13.10 and 13.11 show the influences of inclination angle, Hartmann and Rayleigh numbers for porous medium on streamlines and isotherms. A single clockwise eddy exists in streamlines. At low Rayleigh number, isotherms follow the shape of cylinders. jΨ max j enhances with the increase of Ra. As Ra increases, the strength of the eddy enhances and the center of this eddy moves upward. Also a thermal plume is generated near the vertical centerline. As Ha increases the thermal plume vanishes and jΨ max j reduces. As the outer cylinder rotates, the main eddy is stretched vertically and by increasing Ra, the center of the eddy moves to the top middle. In the presence of Lorentz forces, the distribution of isotherms reduces and the eddy moves downward. Figs. 13.12 and 13.13 depict the influence of ξ; Ra and Ha on Nuloc ; Nuave . The correlation for Nuave corresponding to active parameters is: Nuave 5 2:19 2 0:03ξ 1 0:81 3 1024 Ra 2 0:05Ha 2 7:55 3 1025 ξ Ra 1 3:9 3 1023 ξ Ha 2 5:58 3 1025 Ra Ha 23

ð13:48Þ

27

2 7:6 3 10 ξ 2 1 2:96 3 10 Ra2 1 7:6Ha2 The root mean squared error of this formula is equal to 0.98. The presence of the thermal plume makes the Nuloc have extremum points.Nuave increases with the increase of the Rayleigh number and nanofluid volume fraction. Increasing ξ; Ha leads to changing the convection mechanism to conduction mode, so Nuave detracts with the enhancement of ξ; Ha . Table
13.4 illustrates the influences of inclination angle, Lorentz and buoyancy forces  on heat transfer augmentation E 5 100 3 ðNuφ50:04 2 Nuφ50 Þ=Nuφ50 . Adding CuO nanoparticle into H2O can augment the thermal conductivity of fluid. So, this technique is more efficient when the conduction mode is dominant. Therefore E increases with the rise of the inclination angle and Lorentz forces but it decreases with the rise of buoyancy forces.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

457

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4

Ha = 0

Ra = 250

Ha = 20

Ra = 100

Ha = 0

13.3 ANALYSIS OF WATER-BASED NANOFLUID FLOW AND HEAT TRANSFER DUE TO MAGNETIC FIELD IN A POROUS ENCLOSURE

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.003 –0.005 –0.008 –0.01

–0.2 –0.4 –0.6 –0.8 –0.9

FIGURE 13.10 Isotherm (left) and streamline (right) contours for different values Rayleigh and Hartmann numbers for porous medium when φ 5 0:04; ξ 5 03 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

458

Ha = 20

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

0.4 0.35 0.3 0.25 0.15 0.1

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.005 –0.01 –0.015 –0.02 –0.025

–1 –2 –3

Ha = 20

Ra = 1000

Ha = 0

0.05 0.03

FIGURE 13.10

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.05 –0.07 –0.1 –0.11

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION

Ra = 250

Ra = 100

Ha = 0

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

459

Ha = 20

–0.05 –0.1 –0.15 –0.2 –0.25

–0.2 –0.4 –0.6

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.002 –0.004 –0.006 –0.006

–0.005 –0.01 –0.015 –0.016 –0.017

FIGURE 13.11 Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers for porous medium when φ 5 0:04; ξ 5 903 .

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION 13.4.1 Problem Definition Fig. 13.14 shows the geometry, boundary condition, and sample element. The formula of the inner cylinder is:  2n^  2n^ X Y 1 51 ð13:49Þ a b where n^ . 1 geometry is square when a 5 b.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

460

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Ra = 1000

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

FIGURE 13.11

–0.3 –1 –2 –3

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.008 –0.02 –0.04 –0.06

(Continued).

13.4.2 Governing Equation Two-dimensional steady convective flow of nanofluid in a porous media is considered in the presence of a constant horizontal magnetic field. The PDEs equations are: -

r: V 5 0

 μnf - V 5 rp 1 I 3 B 1 ρnf g K
 -  ρCp nf V :r T 5 knf r2 T -

r: I 5 0

-  σnf V 3 B 2 rϕ 5 I

ð13:50Þ ð13:51Þ ð13:52Þ ð13:53Þ ð13:54Þ

In Eq. (13.51), Darcy’s model is utilized for porous medium. Eqs. (13.53) and (13.54) reduce to r2 ϕ 5 0. So the electric field can be neglected. Therefore the governing equations are: @v @u 1 50 @y @x 2 2




ρCp

 nf

 K @p σnf KB20
2 2u sin2 γ 1 v sinγ cosγ 5 u μnf @x μnf

ðρβ Þnf K  K @p σnf KB20
2 2v cos2 γ 1 u sinγ cosγ 1 g ð T 2 Tc Þ 5 v μnf @y μnf μnf   2
 @T @T @ T @2 T 1v 1 ρCp nf u 5 knf @x @y @x2 @y2

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð13:55Þ ð13:56Þ ð13:57Þ ð13:58Þ

ð13:59Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION

ξ = 0º

ξ = 90º

3

3.2 Ha = 0 Ha = 20

2.8

2.8

Nuloc

Nuloc

Ha = 0 Ha = 20

3

2.6

Ra = 100

461

2.4

2.6 2.4

2.2 2.2 2 1.8

30 A

2 60

1.8 30 A

90 C

B S

3.6 Ha = 0 Ha = 20

3.2

3.2

3

3

2.8

2.8

2.6

2.6

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6 A 30

60 B

1.6 30 A

90 C

S

90 C

B

S Ha = 0 Ha = 20

4.5

Ha = 0 Ha = 20

4.5

4

4

3.5

3.5

Nuloc

Nuloc

60

5

5

Ra = 1000

Ha = 0 Ha = 20

3.4

Nuloc

Nuloc

Ra = 250

3.4

3

3

2.5

2.5

2

2 A 30

60 B

S

FIGURE 13.12

90 C

B

S

3.6

1.5

60

90 C

A 30

60 B

90 C

S

Effects of the inclination angle, Rayleigh number, and Hartmann number for porous medium on local Nusselt number at

φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

462

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Nuave

Nuave 2.68

2.74

2.54

2.58

2.4

2.42

2.26

2.26

2.12 2.1 0.00

1.57

1000.00 775.00

Ra

1.18 550.00

0.79 325.00

0.39

0.00

ζ

0.39 5.00

10.00

0.79 15.00

Ha

100.00 0.00

1.18 20.00 1.57

ζ

Ra = 550

Ha = 10 Nuave 3.3 3 2.7 2.4 2.1 20.00

1000.00 15.00

775.00 550.00

10.00 325.00

5.00

Ha

0.00 100.00

Ra

ξ = 0.785

FIGURE 13.13

Effects of the inclination angle, Rayleigh number, and Hartmann number for porous medium on average Nusselt number.

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ

ð13:60Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð13:61Þ

knf ; μnf are obtained according to the Koo
Kleinstreuer
Li (KKL) model:  kp 3 21 φ sffiffiffiffiffiffiffiffiffi kf κb T 4 0  1 5 3 10 g ðφ; T; dp Þφρf cp;f knf 5 1 1  kp kp ρ p dp 12 2 21 φ kf kf



2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðTÞ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð13:62Þ

ð13:63Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð13:64Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION

1000.00

20.00

775.00

15.00

Ra 550.00

Ha 10.00

325.00

5.00

100.00 0.00

0.39

0.79

1.18

0.00 0.00

1.57

0.39

0.79

1.18

463

1.57

ζ

ζ

Ra = 550

Ha = 10 20.00

15.00

Ha 10.00

5.00

0.00 100.00

325.00

775.00

550.00

1000.00

Ra ξ = 0.785

FIGURE 13.13

(Continued).

TABLE 13.4 Effects of ξ; Ha and Ra on Heat Transfer Enhancement Ra

ξ

Ha

E

100

0

0

11.44969

100

0

20

12.43305

250

0

0

8.840375

250

0

20

12.41954

1000

0

0

3.717265

1000

0

20

12.31083

100

1.57

0

11.73324

100

1.57

20

12.43413

250

1.57

0

9.087008

250

1.57

20

12.42324

1000

1.57

0

4.142252

1000

1.57

20

12.34446

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

464

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Tc Region of support

Control volume

0.5

Nanofluid

0.4

g

0.3 B γ

Tc

0.2

ζ

y

q″

Tc

x

0.1 0

rin

–0.1 –0.2

porous medium

–0.3 –0.4 –0.5 –0.6

Tc

–0.4

–0.2

0

0.2

0.4

0.6

(B)

(A) Si,3

i

Si,4

j=4

f1

f2 i=1

j=3 (C)

FIGURE 13.14

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

All required coefficients and properties are illustrated in Tables 13.1 and 13.2. Introducing dimensionless quantities:
 T 2 Tc Ψ 5 ψ=αnf ; ðX; YÞ 5 x; y =L; θ 5 Th 2 Tc By discarding the pressure, the final equations are: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θ 2 Ra A4 A5 @X

ð13:65Þ

ð13:66Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION

@2 θ @2 θ @θ @Ψ @θ @Ψ 2 1 5 @Y2 @X2 @X @Y @Y @X

465



where Ra 5

g K ðρβ Þf L ΔT μf αf

and Ha 5

σf K B20 μf

ð13:67Þ

are the Rayleigh and Hartmann numbers for the porous media. Also

Ai ði 5 1::6Þ are constants parameters which are obtained as: A1 5

ρnf ρf

;

A3 5

ðρβ Þnf ðρβ Þf

; A5 5

μnf μf

;

ðρCP Þnf

knf σnf A2 5 ; A4 5 ; A6 5 ðρCP Þf kf σf

ð13:68Þ

and boundary conditions are: @θ 5 1:0 @n

on inner wall

θ 5 0:0

on other wall

Ψ 5 0:0

on all walls

ð13:69Þ

Nulocal ; Nuave over the inner cylinder and E can be calculated as:  1 Nuloc 5 A4  θ inner wall 2ðπ 1 Nuave 5 Nuloc dζ 2π

ð13:70Þ

0


 Nuðφ 5 0:04Þ 2 Nu basefluid
 3 100 E5 Nu basefluid

ð13:71Þ

13.4.3 Effects of Active Parameters Effect of Lorentz forces on CuO-water nanofluid flow in a porous semiannulus with constant heat flux is examined using Control Volume-based Finite Element Method. KKL and Darcy’s models are utilized to obtain the governing equations. Numerical simulations are presented for various Rayleigh numbers for porous medium (Ra 5 102 ; 250 and 103 ), radius of inner cylinder (rin 5 0:2 to 0:4), CuO-water volume fraction (φ 5 0 to 0.04), and Hartmann number for porous medium (Ha 5 0 to 20). Figs. 13.15 and 13.16 illustrate the influence of rin ; Ra, and Ha for porous medium on isotherms and streamlines. At low Rayleigh number, isotherms follow the shape of cylinders and one main vortex can be observed in the streamlines. As the Rayleigh number increases, jΨ max j increases and the center of the vortex moves upward. Also a thermal plume is generated near ζ 5 903 . By applying magnetic forces, the center of the vortex returns to the centerline and the thermal plume vanishes. At a high value of rin , there is not enough space for the nanofluid to rotate, so the conduction mode becomes dominant. Figs. 13.17 and 13.18 illustrate the influence of φ; rin ; Ra, and Ha on Nuloc ; Nuave . The correlation for Nuave corresponding effective parameters is: Nuave 5 2:17 1 3:7 3 1024 Ra 1 12:85rin 1 13:42φ 2 0:018Ha 2 4:5 3 1024 Ra rin 2 3:82 3 1024 Ra φ 2 1:45 3 1025 Ra Ha 1 3:45 rin φ 1 0:04rin Ha 1 0:036φ Ha

ð13:72Þ

1 5:6 3 1028 Ra2 1 9:44rin 2 1 27:9φ2 1 3:471024 Ha2 The number of extremum in the Nuloc profile matches the presence of thermal plumes. The influence of adding a magnetic field is more pronounced at lower values of rin . As buoyancy force increases, the temperature

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

466

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Ra = 1000

Ra = 250

Ra = 100

Ha = 0

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

Ha = 20

–0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09

–0.02 –0.04 –0.06 –0.08 –0.1 –0.12 –0.14 –0.16 –0.18 –0.2 –0.22 –0.23

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

–0.0005 –0.001 –0.0015 –0.002 –0.0024

–0.001 –0.002 –0.003 –0.004 –0.005 –0.006

–0.005 –0.01 –0.015 –0.02 –0.022 –0.024

FIGURE 13.15 Isotherm (left) and streamline (right) contours for different values of Rayleigh number and Hartmann number for porous medium when φ 5 0:04; rin 5 0:2.

gradient augments and in turn Nuave is enhanced with the increase of buoyancy forces. A similar effect is observed for φ; rin . Enhancing the Lorentz force causes the nanofluid flow to retard and the Nusselt number reduces. The impacts of Ra and Ha on heat transfer improvement are depicted in Fig. 13.19. Thermal conductivity has an important role in conduction mode and, in turn, the addition of nanoparticles is more sensible in this condition. According to this fact, heat transfer improvement augments with rise of Lorentz forces and it reduces with the enhancement of buoyancy forces.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.4 MAGNETOHYDRODYNAMIC NANOFLUID CONVECTION IN A POROUS ENCLOSURE CONSIDERING HEAT FLUX BOUNDARY CONDITION

Ra = 1000

Ra = 250

Ra = 100

Ha = 0

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

467

Ha = 20

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03

–0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

–0.0002 –0.0004 –0.0006 –0.0008

–0.0005 –0.001 –0.0015 –0.002 –0.0023

–0.002 –0.004 –0.006 –0.008 –0.0088

FIGURE 13.16 Isotherm (left) and streamline (right) contours for different values of Rayleigh number and Hartmann number for porous medium when φ 5 0:04; rin 5 0:4.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

468

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Ra = 102 5.9

Ha = 0 Ha = 5 Ha = 10 Ha = 20

7

Ha = 0 Ha = 5 Ha = 10 Ha = 20

6.1 6

Nuloc

Nuloc

5.7

Ha = 0 Ha = 5 Ha = 10 Ha = 20

6.5

5.9

Nuloc

rin = 0.2

5.8

Ra = 103

Ra = 250 6.2

5.8 5.7

6 5.5

5.6 5.6

5.5

5

5.4 5.5 90º

135º

180º

225º

5.3

270º

90º

135º

180º

135º

180º

225º

270º

13 12.5 12 11.5 11 10.5 10 9.5 9 8.5 8

Ha = 0 Ha = 5 Ha = 10 Ha = 20

90º

135º

180º

225º

270º

13 12.5 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5

ζ

ζ FIGURE 13.17

4.5 90º

135º

180º

225º

270º

225º

270º

ζ

Nuloc

Nuloc

Nuloc

rin = 0.4

Ha = 0 Ha = 5 Ha = 10 Ha = 20

90º

270º

ζ

ζ 13 12.5 12 11.5 11 10.5 10 9.5 9 8.5 8

225º

Ha = 0 Ha = 5 Ha = 10 Ha = 20

90º

135º

180º

ζ

Effects of rin ; Ra and Ha for porous medium on local Nusselt number at φ 5 0:04.

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL 13.5.1 Problem Definition Fig. 13.20 shows the geometry, boundary condition, and sample element. The formula of the inner cylinder is:  2n^  2n^ X Y 1 51 ð13:73Þ a b where n^ . 1 geometry is square when a 5 b.

13.5.2 Governing Equation 2D steady convective flow of nanofluid in a porous media is considered in the presence of a constant horizontal magnetic field. The PDEs equations are: -

r: V 5 0

 μnf - V 5 rp 1 I 3 B 1 ρnf g K
 -  ρCp nf V :r T 5 knf r2 T -

r: I 5 0

-  σnf V 3 B 2 rϕ 5 I

ð13:74Þ ð13:75Þ ð13:76Þ ð13:77Þ ð13:78Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

469

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL

9.7

5.73

8.675

5.56

Nuave

5.39

7.65

Nuave 5.22

6.625 5.6

5.05

0.40 0.35 0.30

rin

0.25

1000.00 775.00 550.00 325.00 Ra 0.20 100.00

0.04

1000.00 775.00

0.03

φ

0.02 0.01

φ = 0.04, Ha = 20

rin = 0.2, Ha = 20

6.11

9.7

6.0025

8.525

Nuave

5.895

Nuave

7.35

5.7875

6.175

5.68

5

20.00 15.00 10.00 5.00

Ha

1000.00 775.00 550.00 325.00 Ra 0.00 100.00

0.04

0.40 0.03

0.35 0.02

φ

9.7

9.71

8.7

9.505

7.7

0.25 0.00 0.20

0.30 rin

9.3

Nuave

6.7

9.095

5.7

8.89

20.00

0.40 15.00

20.00

0.35 0.25

5.00 0.00 0.20

0.04 0.03

15.00

0.30

10.00

Ha

0.02

10.00

rin

0.00

5.00

Ha

φ = 0.04, Ra = 1000 FIGURE 13.18

0.01

Ra = 1000 , Ha = 20

φ = 0.04, rin = 0.2

Nuave

550.00 325.00 Ra 0.00 100.00

0.00 0.00

φ

rin = 0.4, Ra = 1000

Effects of φ; rin ; Ra, and Ha for porous medium on average Nusselt number.

Electric field can be neglected. So the above equations turn into: @v @u 1 50 @y @x  K @p σnf KB20
2 2u sin2 γ 1 v sinγ cosγ 5 u μnf @x μnf

ð13:80Þ

ðρβ Þnf K  K @p σnf KB20
2 2v cos2 γ 1 u sinγ cosγ 1 g ð T 2 Tc Þ 5 v μnf @y μnf μnf

ð13:81Þ

2 2

ð13:79Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

470

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

rin

0.40

0.04

0.35

0.03

φ

0.30

0.25

0.01

0.20 100.00

550.00

325.00

775.00

0.00 100.00

1000.00

Ra φ = 0.04, Ha = 20

Ha

0.02

20.00

0.04

15.00

0.03

φ

10.00

325.00

0.00 100.00

325.00

550.00

775.00

1000.00

0.00 0.20

0.25

20.00

15.00

15.00

10.00

Ha

0.40

10.00

5.00

0.00

0.00 0.20

0.25

0.30

0.35

0.40

0.00

φ = 0.04, Ra = 1000 FIGURE 13.18

0.01

0.02

0.03

0.04

φ rin = 0.4 , Ra = 1000

rin

(Continued).




nf

0.35

Ra = 1000 , Ha = 20

20.00



0.30

rin

5.00

ρCp

1000.00

0.02

Ra φ = 0.04, rin = 0.4




775.00

Ra rin = 0.4 , Ha = 20

0.01

5.00

Ha

550.00

  2 @T @T @ T @2 T 1v ρCp nf u 1 2 5 knf @x @y @x2 @y 

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð13:82Þ

ð13:83Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL

471

13

12

11 1

Ra =1000 Ra = 2550

E 10 0

Ra =1000

9

8

0

5

10

15

20

Ha FIGURE 13.19 Effects of the Ra and Ha for porous medium on the ratio of heat transfer enhancement due to addition of nanoparticles when rin 5 0:2.

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ

ð13:84Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð13:85Þ

ð13:86Þ

knf ; μnf are obtained according to the Koo
Kleinstreuer
Li (KKL) model: 

kp 21 φ sffiffiffiffiffiffiffiffiffi kf κb T 4 0  1 5 3 10 g ðφ; T; dp Þφρf cp;f knf 5 1 1  kp kp ρp dp 12 2 21 φ kf kf



2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðT Þ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp 3

ð13:87Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð13:88Þ

All required coefficients and properties are illustrated in Tables 13.1 and 13.2. Introducing dimensionless quantities:
 T 2 Tc Ψ 5 ψ=αnf ; ðX; YÞ 5 x; y =L; θ 5 Th 2 Tc

ð13:89Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

472

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Tc g Nano fluid rin

B

Th γ

ζ

o

rout Porous medium

(A)

Si,3

Si,4 Region of support 0.5

Control volume

j=4

0.4 0.3 0.2 0.1 0 –0.1

f1

–0.2 –0.3

f2

–0.4 –0.5 –0.6

–0.4

–0.2

0

0.2

0.4

i=1

0.6

(B)

j=3 (C)

FIGURE 13.20

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

By discarding the pressure, the final equations are: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θ Ra 2 A4 A5 @X @2 θ @2 θ @θ @Ψ @θ @Ψ 2 1 5 2 2 @Y @X @X @Y @Y @X

ð13:90Þ



ð13:91Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL

1

473

-2

2

-3

3

-4

0.8

-4

4

0.9

rin = 0.1

0.95

0.1 0.05

0.5

-1

1

0.7

rin = 0.25

2

0.9

-2

3

-3

-1

1

0.4 0.1

FIGURE 13.21 Comparison of the streamline (left) and isotherm (right) contours between nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (- - -) when Ra 5 103 ; Ha 5 0

where Ra 5

g K ðρβ Þf L ΔT μf αf

and Ha 5

σf K B20 μf

are the Rayleigh and Hartmann numbers for the porous media. Also

Ai ði 5 1::6Þ are constants parameters which are obtained as: A1 5

ρnf ρf

A2 5

;

ðρCP Þnf ðρCP Þf

A3 5

ðρβ Þnf ðρβ Þf

; A4 5

; A5 5

μnf μf

knf σnf ; A6 5 kf σf

; ð13:92Þ

and boundary conditions are: θ 5 1:0

on inner wall

θ 5 0:0

on outer wall

Ψ 5 0:0

on both walls

ð13:93Þ

Nulocal ; Nuave over the inner cylinder and E can be calculated as: Nuloc 5 A4

@θ @r

2ðπ 1 Nuave 5 Nuloc dζ 2π

ð13:94Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

474

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

Ha = 20

Ra = 1000

Ra = 250

Ra = 100

Ha = 0

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4 –0.45 –0.48

–0.2 –0.4 –0.6 –0.8 –1 –1.2

–0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.002 –0.004 –0.006 –0.008 –0.01 –0.012 –0.013

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03

–0.02 –0.04 –0.06 –0.08 –0.1 –0.12 –0.13

FIGURE 13.22 Isotherms (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers for porous medium when φ 5 0:04; rin 5 0:1.


 Nuðφ 5 0:04Þ 2 Nu basefluid
 E5 3 100 Nu basefluid

ð13:95Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

475

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL

Ha = 20

Ra = 1000

Ra = 250

Ra = 100

Ha = 0

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9

–0.2 –0.3 –0.5 –1 –1.5 –2 –2.5 –3 –3.5

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.001 –0.002 –0.003 –0.004 –0.005 –0.006 –0.007 –0.008 –0.009

–0.002 –0.004 –0.006 –0.008 –0.01 –0.012 –0.014 –0.016 –0.018 –0.02 –0.022 –0.024

–0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09

FIGURE 13.23 Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers for porous medium when φ 5 0:04; rin 5 0:25.

13.5.3 Effects of Active Parameters The effect of magnetic field on nanofluid natural convection in a circular permeable cylinder with hot square cylinder is studied. The names of the models which are applied for porous media and nanofluid are Darcy and KKL, respectively. The roles of Rayleigh number for porous medium (Ra 5 102 ; 250, and 103 ), length of inner cylinder (rin 5 0:1 to 0:25), CuO-water volume fraction (φ 5 0 to 0.04) and Hartmann number for porous medium (Ha 5 0 to 20) are illustrated as figures.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

476

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

4.5

2.4 Ha = 0 Ha = 5 Ha = 20

Ha = 0 Ha = 5 Ha = 20

4 Nuloc

Nuloc

Ra = 100

1.8 3.5

1.2 3

2.5

0.6 90º

135º

180º

225º

270º

3

1.8

Nuloc

Nuloc

Ra = 250

180º

225º

270º

Ha = 0 Ha = 5 Ha = 20

4

1.2

3.5

3

90º

135º

180º

225º

2.5

270º

9

90º

135º

180º

225º

270º

6.5 Ha = 0 Ha = 5 Ha = 20

7.5

Ha = 0 Ha = 5 Ha = 20

6 5.5 5

6 Nuloc

4.5 Nuloc

Ra = 1000

135º

4.5 Ha = 0 Ha = 5 Ha = 20

2.4

0.6

90º

4.5 3

4 3.5 3 2.5

1.5

2 0

FIGURE 13.24

90º

135º

180º

225º

270º

1.5

90º

135º

180º

225º

270º

Effects of Ha; rin , and Ra on local Nusselt number along the cold wall.

Fig. 13.21 depicts the effect of adding nanoparticles on hydrothermal treatment. Isotherms move away from the hot cylinder on the addition of nanoparticles. jΨ max j increases with the increase of φ. The impacts of rin ; Ra, and Ha for porous medium on streamlines and isotherms are shown in Figs. 13.22 and 13.23. At low Ra, one main vortex appears in the streamlines and isotherms follow the shape of cylinders. As Ra augments, a thermal plume generates at ζ 5 903 and the main vortex moves upward. As the Hartmann number augments, the thermal plume diminishes and the strength of the vortex reduces. As rin augments, the main vortex stretches vertically and isotherms become denser near the outer wall. So the temperature gradient is enhanced with increasing rin .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

477

13.5 EFFECT OF LORENTZ FORCES ON NANOFLUID FLOW IN A POROUS CYLINDER CONSIDERING DARCY MODEL

1000.00

0.04

775.00

0.03

Ra 550.00

φ 0.02

325.00

0.01

100.00 0.10

0.14

0.17

0.21

0.00 0.10

0.25

rin

0.03

Ha 10.00

φ 0.02

5.00

0.01

0.00 100.00

0.00

0.14

0.17

0.21

0.25

325.00

550.00

φ = 0.04 , ra = 1000

Ha = 10, rin = 1000

20.00

20.00

15.00

15.00

Ha 10.00

Ha 10.00

5.00

5.00

0.00

325.00

550.00

775.00

1000.00

0.00 0.00

0.01

0.02

0.03

0.04

φ

Ra

φ = 0.04 , rin = 0.1 FIGURE 13.25

1000.00

Ra

rin

100.00

775.00

Ha = 20, Ra = 1000

0.04

15.00

0.10

0.25

rin

φ = 0.04 , Ha = 0

20.00

0.21

0.17

0.14

rin = 0.25 , Ra = 1000

Effects of φ; Ha; rin , and Ra on average Nusselt number along the cold wall.

Figs. 13.24 and 13.25 depict the impact of φ; rin ; Ra, and Ha on Nuloc ; Nuave . Table 13.5 shows the heat transfer improvement for various cases. The correlation for Nuave corresponding to effective parameters is: Nuave 51 0:52 1 8:18rin 1 1:4 3 1023 Ra 2 3:02 φ 2 0:04Ha 2 1:9 3 1023 rin Ra 1 38:4 rin φ 1 0:14rin Ha 2 1:01 3 1023 Ra φ 2 0:45 3 1023 Ra Ha 27

ð13:96Þ

24

1 0:078φ Ha 1 9:4rin 2 1 1:45 3 10 Ra2 1 70:85φ 2 8:6 3 10 Ha2 2

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

478

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

4.3

3.8

3.6

3.175

2.9

Nuave

Nuave

2.55

2.2

1.925

1.5

1.3

1000.00 775.00 550.00 0.17 Ra 325.00100.00 0.10 0.14 rin

0.25

0.04

0.25

0.21

0.21

0.03 0.02

φ = 0.04 , Ha = 0

Nuave

3.85

3.575

3.6925

2.85

Nuave 3.535

2.125

3.3775

1.4

3.22

20.00

0.14 0.00 0.10

1000.00 775.00

0.03

0.21 0.17 5.00

rin

0.14 0.00 0.10

0.04

0.25 15.00

Ha

0.01

Ha = 20, Ra = 1000

4.3

10.00

0.17

φ

φ

rin

0.02

550.00

0.01

325.00

Ra

0.00 100.00

φ = 0.04 , Ra = 1000

Ha = 10, rin = 1000

2.7 4.21

2.375

3.9675 2.05

Nuave

Nuave

3.725

1.725

3.4825

1.4

3.24 20.00

20.00 15.00 10.00

Ha

5.00 0.00 100.00

1000.00 775.00 550.00 325.00 Ra

φ = 0.04 , rin = 0.1 FIGURE 13.25

0.04 0.03

15.00 0.02

10.00

Ha

5.00

0.01

φ

0.00 0.00

rin = 0.25 , Ra = 1000

(Continued).

The number of extremum in the Nuloc profile matches the presence of thermal plumes. As the buoyancy force increases, the temperature gradient augments and in turn Nuave increases with the increase of buoyancy forces. A similar influence is seen in φ; rin . Enhancing the Lorentz force causes the nanofluid flow to retard and the Nusselt number decreases. Heat transfer improvement increase with the rise of rin and Lorentz forces, and it reduces with enhancement of buoyancy forces due to more variations in thermal conductivity.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

479

REFERENCES

TABLE 13.5 Effects of Ha; rin and Ra on Heat Transfer Enhancement rin

Ra

Ha

E

0.1

100

0

11.5852

0.1

100

5

12.42115

0.1

100

20

12.43677

0.1

250

0

8.189697

0.1

250

5

12.36346

0.1

250

20

12.4291

0.1

1000

0

4.420415

0.1

1000

5

11.58128

0.1

1000

20

12.35301

0.25

100

0

12.3712

0.25

100

5

12.43895

0.25

100

20

12.44018

0.25

250

0

12.05567

0.25

250

5

12.43441

0.25

250

20

12.43958

0.25

1000

0

8.280751

0.25

1000

5

12.37197

0.25

1000

20

12.4336

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248. Available from: https://doi.org/10.1140/epjp/i2014-14248-2.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

480

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

[19] M. Sheikholeslami, Numerical simulation for external magnetic field influence on Fe3O4-water nanofluid forced convection, Engineering Computations, Inpress. [20] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cavity by means of Non- Darcy model, Eng. Comput. 34 (8) (2017) 2651
2667. Available from: https://doi.org/10.1108/EC-01-2017-0008. [21] M. Sheikholeslami, H.B. Rokni, Nanofluid convective heat transfer intensification in a porous circular cylinder, Chem. Eng. Proc. Process Intensif. 120 (2017) 93
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H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transfer 118 (2018) 182
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92. [32] M. Sheikholeslami, H.B. Rokni, Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force, Comput. Methods Appl. Mech. Eng. 317 (2017) 419
430. [33] M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Appl. Math. Comput. 298 (2017) 272
282. [34] M. Sheikholeslami, H.B. Rokni, Nanofluid two phase model analysis in existence of induced magnetic field, Int. J. Heat Mass Transfer 107 (2017) 288
299. [35] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman, Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosity, Int. J. Heat Mass Transfer 108 (2017) 2558
2565. [36] M. Sheikholeslami, Magnetic field influence on CuO-H2O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles, Int. J. Hydrogen Energy 42 (2017) 19611
19621. [37] M. Sheikholeslami, M.M. Bhatti, Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles, Int. J. Heat Mass Transfer 111 (2017) 1039
1049. [38] M. Sheikholeslami, M. Shamlooei, Fe3O4-H2O nanofluid natural convection in presence of thermal radiation, Int. J. Hydrogen Energy 42 (9) (2017) 5708
5718. [39] M. Sheikholeslami, M.M. Bhatti, Active method for nanofluid heat transfer enhancement by means of EHD, Int. J. Heat Mass Transfer 109 (2017) 115
122. [40] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transfer 106 (2017) 745
755. [41] M. Sheikholeslami, A.J. Chamkha, Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field, Numer. Heat Transfer, Part A 69 (10) (2016) 1186
1200. Available from: https:// doi.org/10.1080/10407782.2015.1125709. [42] M. Sheikholeslami, A.J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Numer. Heat Transfer, Part A 69 (7) (2016) 781
793. https://doi.org/10.1080/10407782.2015.1090819. [43] M. Sheikholeslami, M.K. Sadoughi, Simulation of CuO- water nanofluid heat transfer enhancement in presence of melting surface, Int. J. Heat Mass Transfer 116 (2018) 909
919. [44] M. Sheikholeslami, H.B. Rokni, Simulation of nanofluid heat transfer in presence of magnetic field: a review, Int. J. Heat Mass Transfer 115 (2017) 1203
1233. [45] M. Sheikholeslami, H.B. Rokni, Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic field, Int. J. Heat Mass Transfer 114 (2017) 517
526. [46] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM, Int. J. Heat Mass Transfer 113 (2017) 796
805. [47] M. Sheikholeslami, Magnetohydrodynamic nanofluid forced convection in a porous lid driven cubic cavity using Lattice Boltzmann Method, J. Mol. Liq. 231 (2017) 555
565. [48] M. Sheikholeslami Kandelousi, KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel, Phys. Lett. A 378 (45) (2014) 3331
3339. [49] M. Sheikholeslami, Effect of uniform suction on nanofluid flow and heat transfer over a cylinder, J. Braz. Soc. Mech. Sci. Eng. 37 (2015) 1623
1633. [50] M. Sheikholeslami, Lattice Boltzmann Method simulation of MHD non-Darcy nanofluid free convection, Phys. B 516 (2017) 55
71.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

481

[51] M. Sheikholeslami, Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann Method, J. Mol. Liq. 234 (2017) 364
374. [52] M. Sheikholeslami, A.J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, J. Mol. Liq. 225 (2017) 750
757. [53] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using Lattice Boltzmann Method, Int. J. Heat Mass Transfer 108 (2017) 1870
1883. [54] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, Int. J. Heat Mass Transfer 89 (2015) 799
808. [55] M. Sheikholeslami, S. Abelman, Two phase simulation of nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field, IEEE Trans. Nanotechnol. 14 (3) (2015) 561
569. [56] M. Sheikholeslami, R. Ellahi, Simulation of ferrofluid flow for magnetic drug targeting using Lattice Boltzmann method, J. Zeitschrift Fur Naturforschung A 70 (2) (2015) 115
124. [57] M. Sheikholeslami, M. Seyednezhad, Lattice Boltzmann Method simulation for CuO-water nanofluid flow in a porous enclosure with hot obstacle, J. Mol. Liq. 243 (2017) 249
256. [58] M. Sheikholeslami, M. Sadoughi, Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles, Int. J. Heat Mass Transfer 113 (2017) 106
114. [59] M. Sheikholeslami, A. Zeeshan, Mesoscopic simulation of CuO-H2O nanofluid in a porous enclosure with elliptic heat source, Int. J. Hydrogen Energy 42 (22) (2017) 15393
15402. [60] M. Sheikholeslami, S.A. Shehzad, CVFEM for influence of external magnetic source on Fe3O4
H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
191. [61] M. Sheikholeslami, M. Seyednezhad, Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEM, Int. J. Heat Mass Transfer 114 (2017) 1169
1180. [62] M. Sheikholeslami, H.B. Rokni, Free convection of CuO-H2O nanofluid in a curved porous enclosure using mesoscopic approach, Int. J. Hydrogen Energy 42 (22) (2017) 14942
14949. [63] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
746. [64] M. Sheikholeslami, H.B. Rokni, Influence of melting surface on MHD nanofluid flow by means of two phase model, Chin. J. Phys. 55 (2017) 1352
1360. [65] M. Sheikholeslami, H.B. Rokni, Effect of melting heat transfer on nanofluid flow in existence of magnetic field considering Buongiorno Model, Chin. J. Phys. 55 (2017) 1115
1126. [66] M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of Al2O3
water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
524. [67] M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field, Int. J. Heat Mass Transfer 92 (2016) 339
348. [68] M. Sheikholeslami, Numerical simulation for solidification in a LHTESS by means of nano-enhanced PCM, J. Taiwan Inst. Chem. Eng. 86 (2018) 25
41. [69] M. Sheikholeslami, Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424
438. [70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
431. [71] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transfer 122 (2018) 1264
1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
462. [74] M. Sheikholeslami, S.A. Shehzad, Z. Li, Water based nanofluid free convection heat transfer in a three dimensional porous cavity with hot sphere obstacle in existence of Lorenz forces, Int. J. Heat Mass Transfer 125 (2018) 375
386. [75] M. Sheikholeslami, M. Jafaryar, D.D. Ganji, Z. Li, Exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators, J. Mol. Liq. 262 (2018) 104
110. [76] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transfer 124 (2018) 980
989. [77] M. Sheikholeslami, Z. Li, M. Shamlooei, Nanofluid MHD natural convection through a porous complex shaped cavity considering thermal radiation, Phys. Lett. A 382 (2018) 1615
1632. [78] M. Sheikholeslami, H. B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids, 30, 1, https://doi.org/10.1063/1.5012517. [79] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model, Int. J. Heat Mass Transfer 120 (2018) 1200
1212. [80] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [81] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transfer 120 (2018) 772
781. [82] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process. Process Intensif. 124 (2018) 71
82.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

482

13. DARCY MODEL FOR NANOFLUID FLOW IN A POROUS MEDIA BY MEANS OF CVFEM

[83] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [84] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [85] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [86] M. Sheikholeslami, D.D. Ganji, Influence of electric field on Fe3O4-water nanofluid radiative and convective heat transfer in a permeable enclosure, J. Mol. Liq. 250 (2018) 404
412. [87] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Physica B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [88] M. Sheikholeslami, A. Zeeshan, Numerical simulation of Fe3O4-water nanofluid flow in a non-Darcy porous media, Int. J. Num. Methods Heat Fluid Flow 28 (3) (2018) 641
660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Des. 120 (2017) 382
393. [90] M. Sheikholeslami, D.D. Ganji, Numerical modeling of magnetohydrodynamic CuO-water transportation inside a porous cavity considering shape factor effect, Colloids Surf. A 529 (2017) 705
714. [91] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [92] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [93] M. Sheikholeslami, M. Barzegar Gerdroodbary, S. Valiallah Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field, J. Magn. Magn. Mater. 460 (2018) 302
311.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

14 Non-Darcy Model for Nanofluid Hydrothermal Treatment in a Porous Medium Using CVFEM 14.1 INTRODUCTION The knowledge of free or forced convection heat transfer inside geometries of irregular shape (e.g., wavy channel and pipe bend) for porous media has many significant engineering applications; e.g., geothermal engineering, solar-collectors, performance of cold storage, and thermal insulation of buildings. A considerable number of published articles are available that deal with flow characteristics, heat transfer, flow and heat transfer instability, transition to turbulence, design aspects, etc. For a non-Darcy porous medium, Kumar and Gupta [1] reported the flow and thermal fields’ characteristics in wavy cavities. Sheikholeslami [2] investigated MHD nanofluid free convective heat transfer in a porous tilted enclosure by means of a non-Darcy porous medium. Sheikholeslami and Ganji [3] studied the magnetic nanofluid flow in a porous cavity using CuO nanoparticles. Sheikholeslami and Ganji [4] investigated the nanofluid transportation in porous media under the influence of an external magnetic source. Sheikholeslami and Rokni [5] reported nanofluid convective heat transfer intensification in a porous circular cylinder. Sheikholeslami and Shamlooei [6] utilized CVFEM for convective flow of nanofluid inside a lid-driven porous cavity. Sheikholeslami and Seyednezhad [7] simulated the nanofluid heat transfer in a permeable enclosure in the presence of a variable magnetic field. Sheikholeslami [8] demonstrated the influence of Lorentz forces on nanofluid flow in a porous cavity by means of a non-Darcy model. Sheikholeslami and Zeeshan [9] presented the numerical simulation of Fe3O4-water nanofluid flow in a non-Darcy porous media. Nanofluid flows in various mediums were studied in recent years [10
92].

14.2 MHD NANOFLUID FREE CONVECTIVE HEAT TRANSFER IN A POROUS TILTED ENCLOSURE 14.2.1 Problem Definition Fig. 14.1 illustrates the important geometric parameters of current geometry. Also a sample mesh is presented. The inner and outer cylinders are considered as hot and cold walls, respectively. A horizontal magnetic field has been considered.

14.2.2 Governing Equation 2D steady convective flow of nanofluid in a porous media is considered in the presence of a constant magnetic field. The PDEs equations are: @v @u 1 50 @y @x

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00014-X

483

ð14:1Þ

© 2019 Elsevier Inc. All rights reserved.

484

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

Tc

(A) uid um i ed sm u o r ofl

n Na Po

ζ

Th

ri

x

o

ro

λ

ut

g

B

n

y

γ

Si,3 (B)

i

(C) Region of support

Si,4

Control volume

2.5

j=4

2 1.5 1

f1 0.5

f2

0 –0.5 –2

i=1 –1.5

–1

–0.5

0

0.5

1

1.5

j=3

2

FIGURE 14.1

(A) Geometry and the boundary conditions with (B) the mesh of half-annulus enclosure considered in this work; (C) a sample triangular element and its corresponding control volume.

1 2 2 @ u @ u 1 @P 1 μnf @ 2 u 2 ðTc 2 TÞβ nf gsinγ 1 2A 2 2 @y ρnf @x ρnf K ρnf @x   @u @u 1u 1 σnf B20 2uðsinλÞ2 1 vðsinλÞðcosλÞ 5 v @y @x

μnf

0

1 2 2 @ v @ v @P 1 1 μnf @ v 1 2 A 2 ðTc 2 TÞβ nf gcosγ 2 2 2 @y @y ρnf ρnf K ρnf @x   @v @v 1 σnf B20 2vðcosλÞ2 1 uðsinλÞðcosλÞ 5 v 1 u @y @x

μnf

0





ρCp

 nf

ð14:2Þ

  2 @T @T @ T @2 T 1u ρCp nf v 1 2 5 knf @y @x @x2 @y 

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð14:3Þ

ð14:4Þ

ð14:5Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ;

ð14:6Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ

 σs 3 2 1 φ σnf σ f  511 σs σs σf 1 2 2 2 1 φ σf σf

ð14:7Þ ð14:8Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

485

14.2 MHD NANOFLUID FREE CONVECTIVE HEAT TRANSFER IN A POROUS TILTED ENCLOSURE

knf ; μnf are obtained according to the Koo
Kleinstreuer
Li (KKL) model: knf 5 kstatic 1 kBrownian 0

1 k p 3@ 2 1A φ kf kstatic 1 0 1 511 0 kf k k @ p 1 2A 2 @ p 2 1A φ kf kf sffiffiffiffiffiffiffiffiffi kBrownian κb T 4 0 5 5 3 10 g ðφ; T; dp Þφρf cp;f ρ p dp kf



2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðT Þ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð14:9Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð14:10Þ

All required coefficients and properties are illustrated in Tables 14.1 and 14.2. Vorticity and stream function should be used to eliminate pressure source terms: ω1

TABLE 14.1

TABLE 14.2

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð14:11Þ

The Coefficient Values of CuO 2 Water Nanofluid

Coefficient values

CuO 2 Water

a1

2 26.5933108

a2

2 0.403818333

a3

2 33.3516805

a4

2 1.915825591

a5

6.421858E-02

a6

48.40336955

a7

2 9.787756683

a8

190.245610009

a9

10.9285386565

a10

2 0.72009983664

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK21 Þ

dp ðnmÞ

σðΩUmÞ21

Water

997.1

4179

0.613

21

-

0:05

CuO

6500

540

18

29

45

10210

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

Introducing dimensionless quantities:


 x; y uL vL T 2 Tc ; U5 ; ΔT 5 Th 2 Tc ; ðX; YÞ 5 ;V5 ; θ5 L αnf αnf ΔT Ψ5

ψ ωL2 ;Ω5 αnf αnf

ð14:12Þ

The final formulae are: @2 Ψ @2 Ψ 1 5 2 Ω; @Y2 @X2 0

ð14:13Þ

1

@Ω @Ω A5 A2 @@ Ω @ ΩA 1 V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @@U @V @U @V cosλsinλ 2 ðcosλÞ2 1 ðsinλÞ2 2 cosλsinλA 1 PrHa2 @X @Y @Y A1 A4 @X 0 1 2 A3 A2 @ @θ @θ Pr A5 A2 cosγ 2 sinγ A 2 1 Pr Ra Ω; @Y Da A1 A4 A1 A24 @X

ð14:14Þ

 2 @θ @θ @ θ @2 θ U1 V5 1 2 @X @Y @X2 @Y

ð14:15Þ

2

2

U

where dimensionless and constants parameters are defined as:

 qffiffiffiffiffiffiffiffiffiffiffiffi Pr 5 υf =αf ; Ra 5 gðρβ Þf ΔTL3 = μf αf ; Ha 5 LB0 σf =μf
 ρCp nf ρnf ðρβ Þnf  ; A3 5 A1 5 ; A2 5
; ρf ðρβ Þf ρCp f μnf knf σnf A4 5 ; A5 5 ; A6 5 kf μf σf

ð14:16Þ

and boundary conditions are: θ 5 1:0 θ 5 0:0 @θ 5 0:0 @n

on inner wall on outer wall on other walls

Ψ 5 0:0

on all walls

ð14:17Þ

Local and average Nusselt numbers over the hot wall can be calculated as: Nuloc 5 A4 1 Nuave 5 0:5π

0:5π ð

@θ @r

Nuloc ðζ Þ dζ

ð14:18Þ ð14:19Þ

0

14.2.3 Effects of Active Parameters In this chapter, magnetohydrodynamic nanofluid flow and convective heat transfer in a porous tilted annulus is investigated. CVFEM is utilized to obtain the outputs for various values of Hartmann number (Ha 5 0 to 40),  Rayleigh number (Ra 5 103 ; 104 , and 105 ), tilted angle (γ 5 0 to 90 ), Darcy number (Da 5 0:01 to 100), and volume fraction of CuO (φ 5 0% and 4%). APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.2 MHD NANOFLUID FREE CONVECTIVE HEAT TRANSFER IN A POROUS TILTED ENCLOSURE

487

Figs. 14.2 and 14.3 demonstrate the influences of Hartmann, Rayleigh, Darcy numbers, and tilted angle on hydrothermal behavior. At γ 5 03 , in conduction mode, there are two vortexes in streamlines which rotate in opposite directions. As buoyancy forces increase, the vortexes become stronger and their centers move upward. Then the thermal plume appears at ξ 5 903 . Appling a magnetic field reduces the strength of the vortexes and the thermal plume. As the tilted angle increases, the convective mode becomes less than before. At γ 5 453 , the main vortexes convert to new two vortexes in which the upper one rotates clockwise. As the Rayleigh number increases, the primary vortexes become stronger and the thermal plume appears in the region between the two vortexes. At γ 5 903 , only one vortex exists in the absence of a magnetic field. As the buoyancy forces increase, the main vortex converts to two vortexes and the thermal plume is generated. Increasing the Darcy number makes the convective heat transfer increase. It is an interesting observation that in the presence of a magnetic field at high values of Ra and Da, two thermal plumes are generated at the upper region due to the existence of three rotating vortexes. Impacts of significant parameters on Nuloc and Nuave are illustrated in Figs. 14.4 and 14.5. The correlation for Nuave is as follows: Nuave 5 5:54 1 0:29γ 2 0:47Da 2 3:18 logðRaÞ 1 0:02Ha 2 0:02γDa 2 0:086γHa 1 0:014γlogðRaÞ 1 0:2Da Ha 2 0:13logðRaÞHa
2 2 0:039γ 2 2 0:015ðDa Þ2 1 0:53 logðRaÞ 1 0:097ðHa Þ2

ð14:20Þ

where Ha 5 0:1Ha; Da 5 0:01Da. Due to symmetric geometry and boundary conditions, Nuloc profiles are sym  metric respect to ζ 5 90 when γ 5 0 . The Nusselt number increases with the increase of Darcy and Rayleigh numbers. The rate of heat transfer reduces with the increase of the tilted angle. Lorenz forces have a reverse effect on the Nusselt number due to the increase in thermal boundary layer thickness with the increase of the Hartmann number.

Ha = 0

Ra = 103

Ra = 104

0.4 0.3 0.2 0.1 –0.1 –0.2 –0.3 –0.4

Ha = 40

γ = 0º

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.06 0.04 0.02 –0.02 –0.04 –0.06

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Ra = 105

4 3 2 1 –1 –2 –3 –4

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.6 0.4 0.2 –0.2 –0.4 –0.6

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

15 10 5 –5 –10 –15

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

6 4 2 –2 –4 –6

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.2 Isotherm (bottom) and streamline (top) contours for different values of Rayleigh number and Hartmann number when Da 5 0:01.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

488

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

4 3 2 1 –1 –2 –3

20 15 10 5 –5 –10 –15

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Ha = 0

0.5 0.4 0.3 0.2 0.1 –0.1

γ = 45º

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Ha = 40

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 –0.02

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 –0.2

12 10 8 6 4 2 –2 –4

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.2 (Continued)

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES 14.3.1 Problem Definition Fig. 14.6 demonstrates the important geometric parameters of current geometry. Also a sample mesh is presented. Constant heat flux is introduced from the inner wall. The outer wall is cold and the other walls are adiabatic. A horizontal magnetic field is taken into account. The radiation effect is considered in a porous medium.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

489

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

20 15 10 5 –5 – 10

Ha = 40

γ = 90º

Ha = 0

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.2

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

12 10 8 6 4 2 –2 –4

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

(Continued)

14.3.2 Governing Equation Free convective MHD nanofluid flow in a porous media is considered. A non-Darcy model is used for porous media. The PDEs equations are: @v @u 1 50 @y @x

ð14:21Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

490

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

Ra = 103

Ra = 104

Ra = 105

Ha = 0

1.5 1 0.5 –0.5 –1 –1.5

γ = 0º

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

30 20 10 –10 –20 –30

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.6 0.4 0.2 –0.2 –0.4 –0.6

6 4 2 –2 –4 –6

Ha = 40

0.06 0.04 0.02 –0.02 –0.04 –0.06

8 6 4 2 –2 –4 –6 –8

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.3 Isotherm (bottom) and streamline (top) contours for different values of Rayleigh number and Hartmann number when Da 5 100.

0

1 2 2 @ u @ u @P μnf 2 u μnf @ 2 1 2 A 2 @x @y @x K

1 @u @u 1u A 1 σnf B0 2uðsinλÞ2 1 vðsinλÞðcosλÞ 5 ρnf @v @y @x  2



0

0

1

@ v @ vA @P μnf 2 v 1 2 2 ðTc 2 T Þðρβ Þnf g 2 2 @x @y @y K 0 1   @v @v 1 σnf B20 2vðcosλÞ2 1 uðsinλÞðcosλÞ 5 ρnf @v 1 u A @y @x

μnf @




ð14:22Þ

2

2



0

1

0

1

@v @T 1 u @T A 5 knf @@ T 1 @ TA 2 @qr ; @y @x @x2 @y2 @y 2 3 4 4qr 5 2 4σe @T ; T 4 D4T 3 T 2 3T 4 5 c c 3β R @y

ρCp

ð14:23Þ

2

2

nf

ð14:24Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

491

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES

Ha = 0

1.5 1 0.5 –0.5

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

30 20 10 –10 –20

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

γ = 45º

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

10 8 6 4 2 –2 –4 –6 –8

2.5 2 1.5 1 0.5 –0.3

16 14 12 10 8 6 4 2 –2 –4

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Ha = 40

0.25 0.2 0.15 0.1 0.05 –0.02 –0.03

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.3




ρCp

 nf

(Continued)

; ðρβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð14:25Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ;

ð14:26Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð14:27Þ

ð14:28Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

492

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

12 10 8 6 4 2

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

3 2.5 2 1.5 1 0.5

16 14 12 10 8 6 4 2 –2 –4

30 20 10 –10 –20

Ha = 40

γ = 90º

Ha = 0

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

FIGURE 14.3 (Continued)

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

493

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES

γ = 0º

γ = 45º

8

8

Ra = 103 Ra = 104 Ra = 105

7

6

5

4 3

3

2

2

2

1

1

1

0

0 45º

90º

135º

180º

0 0º

45º

5

Nuloc

6

5 4 3

3

2

2

2

1

1

1

0

0

135º

180º

0 0º

45º

ζ

135º

180º

0º

5

Nuloc

6

5

Nuloc

6

4 3

3

2

2

2

1

1

1

0

0 90º

135º

0 0º

180º

45º

90º

135º

180º

0º

5

Nuloc

6

5

Nuloc

6

4

4

3

3

3

2

2

2

1

1

1

0

0 45º

90º

ζ

135º

180º

180º

Ra = 103 Ra = 104 Ra = 105

7

5

0º

135º

8

Ra = 103 Ra = 104 Ra = 105

7

6

4

90º

ζ

8

Ra = 103 Ra = 104 Ra = 105

7

45º

ζ

ζ 8

180º

4

3

45º

135º

Ra = 103 Ra = 104 Ra = 105

7

5 4

90º

8

Ra = 103 Ra = 104 Ra = 105

7

6

0º

45º

ζ

8

Ra = 103 Ra = 104 Ra = 105

7

Nuloc

90º

ζ

8

180º

4

3

90º

135º

Ra = 103 Ra = 104 Ra = 105

7

6

45º

90º

8

5

0º

45º

ζ

6

4

Nuloc

0º

180º

Ra = 103 Ra = 104 Ra = 105

7

Nuloc

Ha = 40

135º

8

Ra = 103 Ra = 104 Ra = 105

7

Ha = 0

90º

ζ

8

Ha = 40

4

3

ζ

Da = 100

6

Nuloc

Nuloc

4

Ra = 103 Ra = 104 Ra = 105

7

5

0º

Da = 0.001

8

Ra = 103 Ra = 104 Ra = 105

7

5

Nuloc

Ha = 0

6

FIGURE 14.4

γ = 90º

0 0º

45º

90º

135º

180º

0º

45º

ζ

90º

135º

180º

ζ

Effects of the Hartmann number, Rayleigh number, and tilted angle on Local Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

494

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.5 Effects of the Hartmann number, Rayleigh number, and tilted angle on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES

FIGURE 14.5

495

(Continued)

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

496

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

(A)

Tc

Sun

g B

Nanofluid

C

λ

B q′′ Porous medium y S

A

x

O

(B)

Si,3

(C) i Region of support

Control volume

Si,4

1 0.9 0.8

j=4

0.7 0.6 0.5 0.4

f1

0.3 0.2

f2

0.1 0

0

0.2

0.4

0.6

0.8

i=1

1

j=3

FIGURE 14.6 (A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

knf ; μnf are obtained according to the Koo
Kleinstreuer
Li (KKL) model: knf 5 kstatic 1 kBrownian 0

1 k p 3@ 2 1A φ kf kstatic 1 0 1 511 0 kf k k p p @ 1 2 A 2 @ 2 1A φ kf kf sffiffiffiffiffiffiffiffiffi kBrownian κb T 4 0 5 5 3 10 g ðφ; T; dp Þφρf cp;f ρ p dp kf



2  g0 φ; T; dp 5 a1 1 a2 Ln dp 1 a3 LnðφÞ 1 a4 LnðφÞln dp 1 a5 Ln dp LnðTÞ




2  1 a6 1 a7 Ln dp 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð14:29Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð14:30Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.3 MAGNETIC NANOFLUID FLOW IN A POROUS CAVITY USING CUO NANOPARTICLES

497

All required coefficients and properties are illustrated in Tables 14.1 and 14.2. Vorticity and stream function should be used to eliminate pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð14:31Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tc U5 ; ; ΔT 5 qvL=kf ; ðX; YÞ 5 ;V5 ; θ5 L αnf αnf ΔT

Ψ5

ψ ωL2 ;Ω5 αnf αnf

ð14:32Þ

The final formulae are: @2 Ψ @2 Ψ 1 5 2 Ω; 2 @Y @X2 0 1 2 2 @Ω @Ω A5 A2 @@ Ω @ ΩA 1 V 5 Pr 1 U @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @@U @V @U @V cosλsinλ 2 ðcosλÞ2 1 ðsinλÞ2 2 cosλsinλA 1 PrHa2 @X @Y @Y A1 A4 @X 0 1 2 A3 A2 @ @θ A Pr A5 A2 2 1 Pr Ra Ω; Da A1 A4 A1 A24 @X  2 @θ @θ @ θ @2 θ 4 1 @2 θ U1 V5 1 Rd 1 @X @Y @X2 @Y2 3 A4 @Y2 where dimensionless and constants parameters are defined as:

 qffiffiffiffiffiffiffiffiffiffiffiffi
 Pr 5 υf =αf ; Ra 5 gðρβ Þf ΔTL3 = μf αf ; Ha 5 LB0 σf =μf ; Rd 5 4σe Tc3 = β R kf
 ρCp nf ρnf ðρβ Þnf  ; A3 5 A1 5 ; A2 5
; ρf ðρβ Þf ρCp f μnf knf σnf A4 5 ; A5 5 ; A6 5 kf μf σf

ð14:33Þ

ð14:34Þ

ð14:35Þ

ð14:36Þ

and boundary conditions are: @θ 5 1:0 @n

on inner wall

θ 5 0:0 @θ 5 0:0 @n

on outer wall on other walls

Ψ 5 0:0

on all walls

Local and average Nusselt numbers over the hot wall can be calculated as:   21 ! knf 1 knf 4 Nuloc 5 1 1 Rd θ kf 3 kf Nuave 5

ðs 1 Nuloc ds S

ð14:37Þ

ð14:38Þ

ð14:39Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

498

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

14.3.3 Effects of Active Parameters A non-Darcy model is applied for the natural convection of nanofluid in a porous enclosure. The influence of thermal radiation and magnetic field are taken into account. The KKL model is utilized for estimating viscosity and thermal conductivity of CuO-water nanofluid. The numerical procedure is conducted by means of CVFEM. Effects of Darcy number (Da 5 0:01 to 100), radiation parameter (Rd 5 0 to 0:8), Rayleigh number (Ra 5 103 to 105 ), Hartmann number (Ha 5 0 to 40), and volume fraction of nanofluid (φ 5 0 to 0:04) are examined. Fig. 14.7 demonstrates the effect of adding nanoparticle into the base fluid on hydrothermal behavior. The thermal boundary layer thickness reduces with the addition of nanoparticles. So, the rate of heat transfer enhances with the increase of the volume fraction of nanofluid. Also the nanofluid velocity is greater than the base fluid velocity due to the increase in nanoparticles’ motion. Besides, the influence of adding nanoparticles is more sensible in the presence of a magnetic field. The influence of the radiation parameter on nanofluid flow and heat transfer is shown in Fig. 14.8. As the radiation parameter increases, the thermal boundary layer thickness is enhanced. jΨ max j augments with the increase of the radiation parameter. The impacts of Darcy, Hartmann, and Rayleigh numbers on hydrothermal behavior of nanofluid are reported in Figs. 14.9
14.11. In low Darcy and Rayleigh numbers, the conduction mechanism can be seen, so the isotherms follow the shape of the cylinders. As the Rayleigh number increases, the buoyancy forces enhance the convection heat transfer. So the isotherms become more disturbed with the increase of Ra. Also jΨ max j increases with the rise of Ra. As the Darcy number increases the permeability of the medium increases and the convective mechanism is enhanced. So, the rate of heat transfer and absolute values of stream function are enhanced with the rise of Da. As the magnetic field increases, the Lorentz forces are generated and these forces reduce the velocity of the nanofluid. Also the rate of heat transfer reduces with the rise of the Hartmann number.

FIGURE 14.7 Influence of nanofluid volume fraction on streamlines (left) and isotherms (right) contours (nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (- - -)) when Ha 5 40; Da 5 100; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

499

FIGURE 14.8 Influence of radiation parameter on streamline (left) and isotherm (right) contours (nanofluid (Rd 5 0:8)(

) and pure fluid (Rd 5 0) (2U 2 )) when Ra 5 105 ; Ha 5 0.

Figs. 14.12 and 14.13 demonstrate the influence of Rd; Da; Ra , and Ha on Nuloc ; Nuave . The formula for Nuave is: Nuave 5 4:15 2 0:816Rd 2 1:45 logðRaÞ 2 0:67Da 1 0:79Ha 1 0:67RdlogðRaÞ 1 0:14Rd Da 2 0:16Rd Ha 1 0:039 logðRaÞDa
2 2 0:28logðRaÞHa 2 0:009Da Ha 1 0:29Rd2 1 0:29 logðRaÞ 1 0:66ðDa Þ2 1 0:046ðHa Þ2

ð14:40Þ

where Ha 5 0:1Ha; Da 5 0:01Da. The root mean squared error of this formula is equal to 0.98. The existence of extremum points on the local Nusselt number is relevant to the presence of the undulation of the inner wall and the thermal plume. The Nusselt number increases with the increase of Rayligh and Darcy numbers due to the increase of the convective heat transfer mechanism. Also Fig. 14.13 indicates that rate of heat transfer increases with the increase of the radiation parameter. Futhermore, Lorentz forces reduce the convective heat transfer mode. So, the Nusselt number decreases with the enhancement of the Hartmann number.

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE 14.4.1 Problem Definition Boundary conditions are depicted in Fig. 14.14. The inner elliptic wall has constant temperature and is considered as a hot wall. The outer circular wall is a cold wall, the others are adiabatic. A magnetic source has been considered as shown in Fig. 14.15. Hx ; Hy ; H can be calculated as follows: h
i21 γ 2 Hy 5 b2y 1 ða2xÞ2 ða 2 xÞ; ð14:41Þ 2π

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

500

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

Streamlines

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05 –0.055

Ha = 40

Da = 100

Ha = 0

Ha = 40

Da = 0.01

Ha = 0

Isotherms

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.001 – 0.002 – 0.003 – 0.004 – 0.005 – 0.006 – 0.007 – 0.008 – 0.009 – 0.01 – 0.011 – 0.012 – 0.013

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.02

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11

0.55 0.45 0.4 0.3 0.25 0.2 0.15 0.1 0.05

– 0.002 – 0.004 – 0.006 – 0.008 – 0.01 – 0.012 – 0.014

FIGURE 14.9

Isotherm (left) and streamline (right) contours for different values of Darcy and Hartmann numbers when Ra 5 103 ; φ 5 0:04; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

Streamlines

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4 –0.45 –0.5 –0.55

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11 – 0.12 – 0.13

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –1 –1.1

Ha = 0 Ha = 40

Da = 100

Ha = 40

Da = 0.01

Ha = 0

Isotherms

501

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.01

– 0.02 – 0.04 – 0.06 – 0.08 – 0.1 – 0.12 – 0.14 – 0.16

FIGURE 14.10 Isotherm (left) and streamline (right) contours for different values of Darcy and Hartmann numbers when Ra 5 104 ; φ 5 0:04; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

502

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

Streamlines

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

Ha = 0 Ha = 40

Da = 100

Ha = 40

Da = 0.01

Ha = 0

Isotherms

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4

–0.5 –1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5 –5 –5.5 –6 –6.5

–0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 –1.6 –1.8

FIGURE 14.11 Isotherm (left) and streamline (right) contours for different values of Darcy and Hartmann numbers when Ra 5 105 ; φ 5 0:04; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE 10

10 Ra = 103 Ra = 104

8

Ra = 103 Ra = 104

8

Ra = 105

Ra = 105

Nuloc

6

Nuloc

6

4

4

2

2

0

A

B

0

C

A

B

C

S

S

Rd = 0 , Da = 0.01, Ha = 0

Rd = 0 , Da = 0.01, Ha = 40 12

12 Ra = 103 Ra = 104

10

Ra = 103 Ra = 104

10

Ra = 105

Ra = 105 8 Nuloc

Nuloc

8 6

6

4

4

2

2

0

A

B

0

C

A

S

B

Rd = 0 , Da = 100, Ha = 0

Rd = 0 , Da = 100, Ha = 0 20

Ra = 103 Ra = 104 16

Ra = 103 Ra = 104 16

Ra = 105

12

Ra = 105

Nuloc

Nuloc

12

8

8

4

4

0

0 A

B

C

A

S

B

C

S

Rd = 0.8, Da = 0.01, Ha = 0

Rd = 0.8, Da = 0.01, Ha = 40

24

24 Ra = 103 Ra = 104

20

Ra = 103 Ra = 104

20

Ra = 105

16

Ra = 105

16 Nuloc

Nuloc

C

S

20

12

12

8

8

4

4

0

0 A

FIGURE 14.12

503

B

C

A

B

S

S

Rd = 0.8, Da = 100, Ha = 0

Rd = 0.8, Da = 100, Ha = 0

C

Effects of radiation parameter, Darcy, Rayleigh, and Hartmann numbers on local Nusselt number at φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

504

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.13

Effects of radiation parameter, Darcy, Rayleigh, and Hartmann numbers on local Nusselt number at φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

FIGURE 14.13

505

(Continued)

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

(A) Nanofluid

Tc

g Magnetic source

Th

Porous medium

y ζ O x

Si,3

(C)

(B)

i Si,4 2.5

Region of support

Control volume

2

j=4

1.5 1 0.5

f1

0 –0.5 –2

f2 –1.5

–1

–0.5

0

0.5

1

1.5

2

i=1

j=3

FIGURE 14.14

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

FIGURE 14.15

Contours of the (A) magnetic field intensity component in x direction Hx; (C) magnetic field
field strength
 H; (B) magnetic
 intensity component in y direction Hy. (A) H x; y , (B) Hx x; y , (C) Hy x; y .

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

Hx 5

h


i21 γ
2  y2b ; b2y 1 ða2xÞ2 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 H 5 H x 1 H y:

507 ð14:42Þ ð14:43Þ

14.4.2 Governing Equation 2D laminar nanofluid flow and free convective heat transfer is taken into account. The governing PDEs are: @v @u 1 5 0; @y @x 0 1 0 1

 2 2 @u @u @ u @ u @P 1 vA 5 @ 2 1 2 Aμnf 2 ρnf @u @x @y @y @x @x μ nf 2 μ20 σnf Hy2 u 1 σnf μ20 Hx Hy v 2 u; K 1 0 1 2 2 @v @v @ v @ v @P ρnf @ u 1 vA 51 μnf @ 2 1 2 A 2 @x @y @x @y @y μ nf 1 μ20 Hy σnf Hx u 2 μ20 Hx σnf Hx v 2 v K

ð14:44Þ

ð14:45Þ

0

1 ðT 2 Tc Þβ nf gρnf ; 1 0 1 2 2

 @T @T @ T @ T 2 1 u A 5 σnf μ20 Hx v2Hy u 1 knf @ 2 1 2 A ρCp nf @v @y @x @x @y 8 0 12 0 12 0 12 9 < @u @v @u @v = 1 μnf 2@ A 1 2@ A 1 @ 1 A ; : @x @y @y @x ;

ð14:46Þ

0

ð14:47Þ


 ρnf ; ρCp nf ; β nf ; knf , and σnf are calculated as ρnf 5 ρf ð1 2 φÞ 1 ρs φ;


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ð14:48Þ

β nf 5 β f ð1 2 φÞ 1 β s φ;  ks 2 2φðkf 2 ks Þ 1 2kf knf 5 kf ; ks 1 φðkf 2 ks Þ 1 2kf

 3ðσ1 2 1Þφ 11 ; σ1 5 σs =σf : σnf 5 σf ðσ1 1 2Þ 2 ðσ1 2 1Þφ

ð14:50Þ

ð14:49Þ

ð14:51Þ ð14:52Þ

μnf is obtained as follows:


 μnf 5 0:035μ20 H 2 1 3:1μ0 H 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

Dimensionless parameters are defined as:

  b; a
Hy ; Hx ; H p ðb; aÞ 5 ;P5
; Hy ; Hx ; H 5 2 L H0 ρf αf =L
 x; y uL vL T 2 Tc ; ðX; YÞ 5 U5 ;V5 ; Θ5 : αf αf ð Th 2 Tc Þ L

ð14:53Þ

ð14:54Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

508

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

So equations change to: @V @U 1 5 0; @Y @X

ð14:55Þ

2 30 1 2 2 μnf =μf @U @U @ U @ U 5@ A U 1 V 5 Pr4 1 @X @Y @Y2 @X2 ρnf =ρf 2 3 2 3

 @P μ =μ σ =σ Pr nf f nf f 5 H 2 U 2 Hx Hy V 2 4 5U; 2 2 Ha2 Pr4 y @X Da ρnf =ρf ρnf =ρf

ð14:56Þ

0 12 3 2 2 μnf =μf @V @V @ V @ V A4 5 1U 5 Pr@ 2 1 V @Y @X @Y @X2 ρnf =ρf 2 3
 σ =σ nf f 5 H 2 V 2 Hx Hy U 2 Ha2 Pr4 x ρnf =ρf 2 3 2 3 μnf =μf β nf @P Pr 4 5V; 1 RaPr4 5Θ 2 2 @Y Da ρnf =ρf βf

ð14:57Þ

2 30 1 @Θ @Θ 4knf ðρCP Þf 5@@2 Θ @2 ΘA 1U 5 V 1 @Y @X @Y2 @X2 kf ðρCP Þnf 0 1 ð ρC Þ   σ P f nf A V Hx 2U Hy 2 1 Ha2 Ec@ ðρCP Þnf σf 2 3 μnf 8 0 12 0 12 0 12 9 6 μ 7 < = f 6 7 @@U A @@VA @@U @V A 7 16 6ðρCP Þnf 7Ec:2 @X 1 2 @Y 1 @Y 1 @X ; 4 5 ðρCP Þf

ð14:58Þ

and dimensionless parameters are qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

 h i Ec 5 μf αf = ðρCP Þf ΔT L2 ; Da 5 K= L2 :

ð14:59Þ

The thermophysical properties of Fe3O4 and water are presented in Table 14.3. The pressure gradient source terms are discarded by the vorticity stream function. Ω5

TABLE 14.3

 ωL2 ψ @u @v @ψ @ψ 1 ; ðu; vÞ 5 ;2 ;Ψ 5 ;ω52 : αf @y @x @y @x αf

ð14:60Þ

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

dp ðnmÞ

σðΩUmÞ21

Pure water

997.1

4179

0.613

-

0:05

Fe3 O4

5200

670

6

47

25000

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

509

According to Fig. 14.14, boundary conditions are: on inner wall on outer wall on other walls

θ 5 1:0 θ 5 0:0 @θ 50 @y

on all walls

Ψ 5 0:0

Nuloc ; Nuave along the cold wall are :    knf @Θ ; Nuloc 5  kf @r  1 Nuave 5 0:5π

ð14:61Þ

ð14:62Þ

0:5π ð

Nuloc ðζ Þ dζ;

ð14:63Þ

0

14.4.3 Effects of Active Parameters In this section, the influence of magnetic field on Fe3O4-water nanofluid in a porous enclosure is reported. The governing equations have been solved via CVFEM and the outputs are depicted in several plots for the influence of various parameters on the flow and heat transfer. These parameters are Darcy number (Da), Rayleigh number (Ra), Hartmann number (Ha), and volume fraction of Fe3O4 (φ). Pr and Ec are 6.8 and 1025, respectively. Fig. 14.16 demonstrates the influence of adding Fe3O4 into water on the hydrothermal characteristics. This figure depicts that an increase in nanoparticle volume fraction results in an increase in nanofluid velocity. It is also found that the thermal boundary layer thickness of water-based nanofluid is higher than pure fluid. Figs. 14.17
14.19 illustrate the impact of Darcy, Hartmann, and Rayleigh numbers on isotherms and streamlines. In dominant conduction modes, one main clockwise cell appears in half of the enclosure. An increase in magnetic field results in the generation of a secondary cell near the vertical centerline. As the permeability of porous media enhances, the convective heat transfer improves and a thermal plume appears. An increase in buoyancy forces result in an increase in strength of the main eddy and a thermal plume is generated near the vertical centerline. As Lorentz forces increase, the position of the thermal plume becomes far from the

FIGURE 14.16 Impact of nanofluid volume fraction on streamline (top) and isotherm (bottom) contours (nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (- - -)) when Ra 5 105 ; Da 5 100; Ha 5 0.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.17

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 103 .

vertical centerline. It is a fantastic observation that in the case of Da 5 100, Ra 5 105, jΨ max j reaches to its maximum value and the main eddy stretches horizontally. Also one powerful thermal plume generates near the ζ 5 903 . Applying a magnetic field for such a case converts the main eddy to three smaller ones. The middle one rotates counterclockwise. The existence of such eddies generates two thermal plumes over the hot elliptic wall.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.4 NANOFLUID TRANSPORTATION IN POROUS MEDIA UNDER THE INFLUENCE OF EXTERNAL MAGNETIC SOURCE

FIGURE 14.18

511

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 104 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.19

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.5 NANOFLUID CONVECTIVE HEAT TRANSFER INTENSIFICATION IN A POROUS CIRCULAR CYLINDER

513

The rate of heat transfer is depicted in Fig. 14.20. The formula of Nuave corresponding to important parameters is: Nuave 5 4:37 1 0:15Da 2 2:77logðRaÞ 1 0:03Ha 1 0:03Da logðRaÞ 2 0:21Da Ha 2 0:39logðRaÞHa
2 1 0:07Da2 1 0:51 logðRaÞ 1 0:61Ha2

ð14:64Þ

where Da 5 0:01Da; Ha 5 0:1Ha. Increasing the permeability of the porous media results in an increase in the rate of heat transfer. Enhancing the Rayleigh number makes the convective heat transfer increase. So this nondimension parameter has a similar effect on the Nusselt number to that of obtained for Darcy number. As the Lorentz force increases, Nuave reduces due to the domination of the conduction mode. Adding Fe3O4 nanoparticles into base fluid enhances the Nusselt number. Table 14.4 demonstrates the influence Ha, and Ra on    of Da; 
heat transfer improvement. This output is defined as E 5 100  Nuave φ50:04 2 Nuave φ50 =Nuave φ50 . In conduction mode, the influence of adding nanoparticles has more benefit because of more changes in thermal conductivity. Therefore, heat transfer improvement enhances with enhance of Hartmann number but it decreases with the rise of Darcy and Rayleigh numbers.

14.5 NANOFLUID CONVECTIVE HEAT TRANSFER INTENSIFICATION IN A POROUS CIRCULAR CYLINDER 14.5.1 Problem Definition Fig. 14.21 depicts the geometry, boundary condition, and sample element. The inner cylinder has a constant heat flux condition and the outer cylinder is cold. A horizontal magnetic field has been applied in this porous media. A non-Darcy model is utilized for the porous media.

14.5.2 Governing Equation Steady convective nanofluid flow in a porous enclosure is considered in the presence of a uniform magnetic field. The PDEs equations are: @v @u 1 50 @y @x  

 2 @u @u @ u @2 u @P μnf 21 1u 2 v u ρnf 1 2 5 σnf Bx By v 2 σnf B2y 1 μ @y @x @y2 @x2 nf @x K 0 1 0 1 2 2 μnf @v @v @ v @ v @P 2 Bx σnf Bx v 1 By σnf Bx u 2 ρnf @ u 1 vA 5 μnf @ 2 1 2 A 2 v @x @y @x @y @y K 1 ðT 2 Tc Þβ nf gρnf ;




ρCp

 nf

Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @ T @2 T 1u 1 ρCp nf v 5 knf @y @x @x2 @y2 ; ðρβ Þnf , ρnf ; knf , and σnf are defined as:


 ρCp nf 5 φ ρCp s 1 ð1 2 φÞ ρCp f ðρβ Þnf ðρβ Þf

5φ

ðρβ Þs 1 ð1 2 φÞ ðρβ Þf

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  ks 1 2kf 1 2φðks 2 kf Þ knf 5 kf ks 2 φðks 2 kf Þ 1 2kf

ð14:65Þ ð14:66Þ

ð14:67Þ

ð14:68Þ

ð14:69Þ ð14:70Þ ð14:71Þ ð14:72Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

514

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.20

Effects of Da; Ha, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

515

14.5 NANOFLUID CONVECTIVE HEAT TRANSFER INTENSIFICATION IN A POROUS CIRCULAR CYLINDER

TABLE 14.4 Effects of Da; Ha, and Ra on Heat Transfer Enhancement Ra

Da

Ha

E

10

3

0.01

0

11.09238

10

3

0.01

20

11.42401

10

4

0.01

0

4.20269

10

4

0.01

20

7.652818

10

5

0.01

0

2.613391

10

5

0.01

20

3.245394

10

3

100

0

5.874633

10

3

100

20

7.615688

10

4

100

0

3.787017

10

4

100

20

6.464888

10

5

100

0

3.757416

10

5

100

20

1.560289

(A)

(B)

Nanofluid

Tc

A

Nanofluid A

q″ q″

g B

g B

y γ

y γ

S

o

S

o

B

x

x

B

C C Porous medium

Porous medium

ζ=0

ζ = 45 Si,3

(C)

i

(D) Region of support

Si,4

Control volume

2

j=4

1.5 1 0.5 0

f1

–0.5 –1

f2

–1.5 –2 –2.5

–2–2 –1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

i=1

j=3

FIGURE 14.21 (A, B) Geometry and the boundary conditions with (C) the mesh of geometry considered in this work; (D) a sample triangular element and its corresponding control volume.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

516

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

2
3  !21 
21σs =σf 2 σs =σf 21 φ
 σnf 5 4 1 15σf 3φ 211σs =σf

ð14:73Þ

μnf is obtained as follows:


 μnf 5 0:035B2 1 3:1B 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

ð14:74Þ

The properties of nanofluid are provided in Table 14.1. Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð14:75Þ

Introducing dimensionless quantities:
 ðY; XÞ 5 y; x =L; P 5




p

2 ; U 5

ρnf αnf =L

uL vL T 2 Tc ; ΔT 5 qvL=kf ;V5 ; θ5 αnf αnf ΔT

ð14:76Þ

The final formulae are: @2 Ψ @2 Ψ 1 Ω 1 5 0; @X2 @Y2 0 1 2 2 @Ω @Ω A 5 A 2 @@ Ω @ ΩA U1 V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @@U @V 2 @U 2 @V Bx By 2 Bx 1 B 2 Bx B y A 1 PrHa2 @X @Y y @Y A1 A4 @X 0 1 A3 A22 @θ Pr @A5 A2 A 2 Ω; 1 Pr Ra Da A1 A4 A1 A24 @X V

 2 @θ @θ @θ @2 θ 1 U5 1 : @Y @X @Y2 @X2

ρnf ρf

;

A3 5

ðρβ Þnf ðρβ Þf

; A5 5

μnf μf

; A2 5

ðρCP Þnf ðρCP Þf

ð14:78Þ

ð14:79Þ

where dimensionless and constants parameters are illustrated as: qffiffiffiffiffiffiffiffiffiffiffiffi
 K Pr 5 υf =αf ; Ra 5 gβ f qvL4 = kf υf αf ; Ha 5 LB0 σf =μf ; Da 5 2 ; L A1 5

ð14:77Þ

; A4 5

knf σnf ; A6 5 kf σf

ð14:80Þ

and boundary conditions are: @θ 5 1:0 @n

on inner wall

θ 5 0:0 Ψ 5 0:0

on outer wall on all walls

Local and average Nusselt numbers over the inner cylinder can be calculated as:  1 knf Nuloc 5 θ kf Nuave 5

ðs 1 Nuloc ds S

ð14:81Þ

ð14:82Þ

ð14:83Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.5 NANOFLUID CONVECTIVE HEAT TRANSFER INTENSIFICATION IN A POROUS CIRCULAR CYLINDER

517

14.5.3 Effects of Active Parameters The influence of magnetic field on nanofluid transportation in a porous cylinder with inner inclined square obstacle is presented. The working fluid is considered as Fe3O4-water and its viscosity is a function of φ and Ha. Results are demonstrated for several values of volume fraction of Fe3O4-water (φ 5 0 to 0.04), Darcy number (Da 5 0:001 to 100), Hartmann number (Ha 5 0 to 40), Rayleigh number (Ra 5 103 to 105 ), and inclination angle (ξ 5 0 and 45 ). Fig. 14.22 demonstrates the impact of φ on isotherms and streamlines. Augmenting the nanofluid volume fraction leads to an increase in the temperature boundary layer thickness. The nanofluid velocity increase because of the increase of φ. The impacts of Ha; ξ; Da, and Ra on hydrothermal behavior are demonstrated in Figs. 14.23
14.25. As nanofluid temperature increases, the nanofluid initiates, moving from the inner cylinder to the cold one and dropping along the outer cylinder. Conduction mode is dominant at low Rayleigh and Darcy numbers. So isotherms follow the shape of the enclosure. At ξ 5 03 , one main eddy exists and when the inner cylinder is inclined ðξ 5 453 Þ the main eddy converts to two similar ones. The strength of this main eddy is enhanced with the rise of convective heat transfer. So jΨ max j rises with the increase of Da; Ra . Also a thermal plume appears near the vertical centerline when convection mode is dominating. As ξ increases, the distortion of the isotherms is enhanced. As the magnetic field increases, jΨ max j reduces and the center of the main eddy moves upward. Also the Lorentz force makes the thermal plume diminish.

FIGURE 14.22 Da 5 100; Ra 5 105 .

Effect of volume fraction of nanofluid on isotherm (left) and streamline (right) contours (φ 5 0 (22 ), φ 5 0:04 (2)) when

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

518

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.23

Effect of Darcy and Hartmann numbers on isotherm (left) and streamline (right) contours when φ 5 0:04; Ra 5 103 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.5 NANOFLUID CONVECTIVE HEAT TRANSFER INTENSIFICATION IN A POROUS CIRCULAR CYLINDER

FIGURE 14.24

519

Effect of Darcy and Hartmann numbers on isotherm (left) and streamline (right) contours when φ 5 0:04; Ra 5 104 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

520

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.25

Effect of Darcy and Hartmann numbers on isotherm (left) and streamline (right) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

521

Figs. 14.26 and 14.27 illustrate the impact of ξ; Da; Ra , and Ha on Nuave ; Nuloc . With respect to the active parameters, the following equation can be obtained: Nuave 5 4:58 2 0:66ξ 2 2logðRaÞ 2 0:58Da 1 0:18Ha 1 0:19ξ logðRaÞ 1 0:09ξDa 2 0:04ξHa 1 0:18logðRaÞDa
2 2 0:15logðRaÞ Ha 2 0:41Da Ha 1 0:17ξ2 1 0:33 logðRaÞ 1 0:1Rd2 1 0:08Ha2

ð14:84Þ

where Ha 5 0:1Ha; Da 5 0:01Da. The number of extremum in Nuloc matches the presence of the corners of the square cylinder and thermal plume. Nuave increases with the increase of Da; Ra; ξ but it decreases with the rise of Ha.

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY 14.6.1 Problem Definition Sample element, boundary condition, and geometry are depicted in Fig. 14.28. The south wall is hot and the others are cold. Also the south wall can move horizontally. A porous cavity is filled with nanofluid and affected by a horizontal magnetic field.

14.6.2 Governing Equation Nanofluid forced convective non-Darcy flow is taken into account in the presence of a uniform magnetic field. The equations are: @v @u 1 50 @y @x 

  2 μnf @ u @2 u @P @u 1 By σnf Bx v 2 B2y σnf u 1 u 5 ρ 1 2 2 μ nf nf v 2 2 @y @x @x @y K 0 0 1 2 2 μnf @ v @ v @P @v μnf @ 2 1 2 A 2 1 By σnf uBx 2 Bx σnf Bx v 2 v 5 ρnf @ u 1 @y @x @y @x K Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @ T @2 T @T u 1 v ; knf 1 5 ρC p nf @x @x2 @y2 @y
 σnf ; ρCp nf and ρnf are :  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf


 ρCp nf 5 φ ρCp s 1 ð1 2 φÞ ρCp f

ð14:85Þ @u u @x 1 @v A v ; @y

ð14:86Þ

ð14:87Þ

ð14:88Þ

ð14:89Þ

ð14:90Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ;

ð14:91Þ

μf kBrownian 1 μf ð12φÞ22:5 3 kf Pr

ð14:92Þ

μnf ; knf can be presented as: μnf 5

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

3

3 Ha = 0 Ha = 40

2.6

2.6

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4 A

Ha = 0 Ha = 40

2.8

Nuloc

Nuloc

2.8

1.4 B S

A

C

Da = 0.01, Ra = 103, ζ = 0º

B S

Da = 0.01, Ra = 103, ζ = 45º 3.2

3.2 Ha = 0 Ha = 40

Ha = 0 Ha = 40

2.4

2.4

Nuloc

2.8

Nuloc

2.8

2

2

1.6

1.6

1.2

A

1.2 B S

C

A

Da = 0.01, Ra = 104, ζ = 0º

B S

4.8 Ha = 0 Ha = 40

Ha = 0 Ha = 40

4.2

3.6

3.6

Nuloc

4.2

3

3

2.4

2.4

1.8

1.8

1.2 A

B S

Da = 0.01, Ra = 103, ζ = 0º

FIGURE 14.26

C

Da = 0.01, Ra = 104, ζ = 45º

4.8

Nuloc

C

C

1.2 A

B S

C

Da = 0.01, Ra = 105, ζ = 45º

Effects of inclination angle, Darcy, Rayleigh, and Hartmann numbers on local Nusselt number when φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

523

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

3

3 Ha = 0 Ha = 40

2.6

2.6

2.4

2.4

2.2

2.2

2

2

1.8

1.8

1.6

1.6

1.4 A

B S

Ha = 0 Ha = 40

2.8

Nuloc

Nuloc

2.8

1.4 A

C

Da = 100, Ra = 103, ζ = 0º

B S

C

Da = 100, Ra = 103, ζ = 45º 4

3.6 Ha = 0 Ha = 40

Ha = 0 Ha = 40

3.6

3.2

3.2

Nuloc

Nuloc

2.8 2.4 2

2.8 2.4 2

1.6

1.6 1.2

1.2 A

B S

A

C

Da = 100, Ra = 104, ζ = 0º

B S

C

Da = 100, Ra = 104, ζ = 45º

5.4

8

Ha = 0 Ha = 40

4.8

Ha = 0 Ha = 40

7

4.2

Nuloc

Nuloc

6 3.6 3

4

2.4

3

1.8 1.2

1.2

A

B S

Da = 100, Ra = 105, ζ = 0º

FIGURE 14.26

5

C

A

B S

C

Da = 100, Ra = 105, ζ = 45º

(Continued)

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

524

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.27

Effects of inclination angle, Darcy, Rayleigh, and Hartmann numbers on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

FIGURE 14.27

525

Effects of inclination angle, Darcy, Rayleigh, and Hartmann numbers on average Nusselt number.

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

v = 0,u = 0,T = Tc

(A)

B λ

v = 0, u = 0, T = Tc

v = 0, u = 0, T = Tc

Nanofluid Porous media

y x v = 0,u = ULid ,T = Th Si,3 (C)

(B)

i Si,4

Control volume

Region of support

1

j=4 0.8

0.6

0.4

0.2

f1

0

f2

0.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

i=1

j=3

FIGURE 14.28

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

knf 5 kBrownian 1 kstatic
 sffiffiffiffiffiffiffiffiffi 3 21 1 kp =kf φ kBrownian κb T kstatic 4 0 1 0 1 ; ρ cp;f ; 5 5 3 10 g ðdp ; T; φÞφ 511 0 ρ p dp f kf kf k k p p @ 1 2A 2 @ 2 1A φ kf kf




 
 2 g0 dp ; T; φ 5 LnðT Þ Ln dp a2 1 a5 Ln dp 1 a1 1 a3 LnðφÞ 1 LnðφÞLn dp a4




2  1 a7 Ln dp 1 a6 1 Ln dp a9 LnðφÞ 1 a8 LnðφÞ 1 a10 Ln dp ; Rf 5 4 3 1028 km2 =W;   Rf 5 2kp 21 1 kp;eff 21 dp ;

ð14:93Þ

Properties and needed parameters are provided in Tables 14.1 and 14.2. ψ; ω can be defined as: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð14:94Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

527

Introducing dimensionless quantities:


 x; y T 2 Tc ψL ;Ψ 5 θ5 ; ΔT 5 Th 2 Tc ; ðX; YÞ 5 ; L ULid ΔT v ω u ;Ω5 ;U5 V5 ULid LULid ULid

ð14:95Þ

The final formulae are: @2 Ψ @2 Ψ 1Ω1 5 0; 2 @X @Y2 0 1 @Ω @Ω 1 A5 @@2 Ω @2 Ω A V1 U5 1 @Y @X Re A1 @Y2 @X2 0 1 Ha2 A6 @@U 2 @V 2 @U @V A B 2 B x 1 By B x 2 Bx By 1 @X @Y Re A1 @Y y @X 2

ð14:96Þ

ð14:97Þ

1 A5 Ω; Re Da A1   A4 @2 θ @2 θ @θ @θ 1 U 1 2 5 PrRe V @Y @Y @X A2 @X2

ð14:98Þ

where dimensionless and constants parameters are: Re 5 A1 5

ρf ULid L ρnf

μf

; Ha 5 LB0

; A5 5

qffiffiffiffiffiffiffiffiffiffiffiffi σf =μf ;

μnf

; ρf μf
 ρCp nf knf σnf  ; A4 5 A2 5
; A6 5 k σf ρCp f f

ð14:99Þ

and boundary conditions are: Ψ 5 0:0 θ 5 1:0 θ 5 0:0 Nuloc and Nuave over the south wall are:

on all walls on south wall on other walls

 knf @θ Nuloc 5 kf @y 1 Nuave 5 L

ð14:100Þ

ð14:101Þ

rð out

Nuloc dx

ð14:102Þ

rin

14.6.3 Effects of Active Parameters Nanofluid MHD forced convection in a porous sinusoidal enclosure is examined numerically in this section. Numerical outputs are presented for various Darcy numbers (Da 5 0:01 to 100), Hartmann numbers (Ha 5 0 to 20), Reynolds numbers (Re 5 100 to 600), and volume fraction of CuO (φ 5 0% to 4%). Fig. 14.29 illustrates the influence of φ on isotherms and streamlines. Increasing φ leads to an increase in thermal boundary layer thickness. The nanofluid velocity is enhanced by adding nanoparticles.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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FIGURE 14.29 Influence of nanofluid volume fraction on streamlines (left) and isotherm (right) contours (nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (- - -)) when Da 5 100.

Figs. 14.30 and 14.31 illustrate the effect of the Darcy, Hartmann, and Reynolds numbers on hydrothermal behavior. In the absence of a magnetic field, when Darcy and Reynolds number are low, only one eddy exists in streamlines and isotherms are parallel to each other. As the Reynolds number increases, the isotherms become denser at the lid wall due to the increase of convective mode. Also jΨ max j augments with the rise of the Reynolds number. As the Darcy number increases, the convective mode becomes stronger due to an increase in the permeability of the medium. So the temperature gradient over the hot wall increases with the increase of the Darcy number. Increasing the Lorentz forces makes the isotherms becomes less dense. Also velocity reduces with the rise of the Hartmann number. The influence of important parameters on Nuave is depicted in Fig. 14.32. The correlation for average Nusselt number is as follows: Nuave 5 7:56 1 4:19Re 1 0:45Da 2 0:1Ha 1 25:6φ 1 0:017Re Da 1 0:048Re Ha 1 0:45φRe 2 0:0018Da Ha 1 0:6Da φ 2 0:048Ha Da 2 0:38Re2 1 0:52Da2 2 0:22Ha2 2 1:35φ2

ð14:103Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

529

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

Ha = 20

Re = 600

Ha = 0

Ha = 20

Re = 300

Ha = 0

Ha = 20

Re = 100

Ha = 0

Isotherms

FIGURE 14.30

Streamlines 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05 –0.055 –0.06 –0.065

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05 –0.055 –0.06

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.005 – 0.01 – 0.015 – 0.02 – 0.025 – 0.03 – 0.035 – 0.04 – 0.045 – 0.05 – 0.055 – 0.06 – 0.065

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.005 – 0.01 – 0.015 – 0.02 – 0.025 – 0.03 – 0.035 – 0.04 – 0.045 – 0.05 – 0.055

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05 –0.055 –0.06

–0.0001 –0.003 –0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05 –0.055

Isotherm (left) and streamline (right) contours for different values of Reynolds and Hartmann numbers when

Da 5 0:01; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.31 Isotherm (left) and streamline (right) contours for different values of Reynolds and Hartmann numbers when Da 5 100; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.6 CONVECTIVE FLOW OF NANOFLUID INSIDE A LID-DRIVEN POROUS CAVITY

531

FIGURE 14.32 Influences of the volume fraction of nanofluid and Darcy, Reynolds, and Hartmann numbers on average Nusselt number.

where Re 5 0:01Re; Ha 5 0:1Ha. Adding nanoparticles makes the Nusselt number increase due to the increase in knf . Nuave improves with the increase of Darcy and Reynolds numbers because of the increase in convective heat transfer. As Ha improves, the temperature gradient decreases and in turn the Reynolds number is reduced with the increase of Lorentz forces.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

532

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.32

(Continued)

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD 14.7.1 Problem Definition Geometry, boundary condition, and sample element are demonstrated in Fig. 14.33. External magnetic source is applied (see Fig. 14.34). H; Hx ; Hy are: i21 2 γ h
Hy 5 ða 2 xÞ b2y 1 ða2xÞ2 ; ð14:104Þ 2π

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

533

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

Tc

(A)

Nanofluid g Magnetic source

q′′

y

Porous medium

O x

Si,3

(B) i

Si,4 j=4

f1 f2

i=1

FIGURE 14.33

j=3

(A) Geometry and the boundary conditions with; (B) a sample triangular element and its corresponding control volume.

(A)

(B) 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9

(C) –1 –2 –3 –4 –5 –6 –7 –8 –9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17 – 18 – 19

FIGURE 14.34 Contours of the (A) magnetic field intensity component in x direction Hx; (C) magnetic field
field strength
 H; (B) magnetic
 intensity component in y direction Hy. (A) H x; y , (B) Hx x; y , (C) Hy x; y .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

i21
 γ h
2 Hx 5 y 2 b b2y 1 ða2xÞ2 ; 2π

2  2 0:5 H 5 H y 1H x :

ð14:105Þ ð14:106Þ

14.7.2 Governing Equation Two-dimensional convective non-Darcy flow of nanofluid is considered in the presence of an external magnetic source. The governing equations are:

0

@u @v 52 ; @x @y

1 2 2 @ u @ u @P 2 @ 2 μ20 σnf H y u 1 σnf μ20 Hx H y v 1 2 Aμnf 2 2 @y @x @x 0 1

 @u μnf @u vA ; u 5 ρnf @ u 1 2 @x @y K 1 2 2 μnf @ v @ v @P μnf @ 2 1 2 A 2 1 μ20 H y σnf H x u 2 μ20 H x σnf H x v 2 v @x @y @y K 0 1 @v @v 1 ðT 2 Tc Þβ nf gρnf 5 ρnf @ u 1 vA; @x @y

ð14:107Þ

ð14:108Þ

0

0 1 1 2 2
 @ T @ T @q @T @T r knf @ 2 1 2 A 2 1 u A ρCp nf ; 5 @v @y @x @y @x @y 2 3 4 4qr 5 2 4σe @T ; T 4 D4T 3 T 2 3T 4 5: c c 3β R @y

ð14:109Þ

0




ρCp

 nf

; ρnf , ðρβ Þnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð14:110Þ

ð14:111Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ

ð14:112Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ;

 3φðσ1 2 1Þ 11 ; σ1 5 σs =σf : σnf 5 σf ð1 2 σ1Þφ 1 ð2 1 σ1Þ

ð14:113Þ

μnf is calculated as follows:

 2 μnf 5 0:035μ20 H 1 3:1μ0 H 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

ð14:114Þ

ð14:115Þ

knf can be calculated as:

 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

ð14:116Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

535

TABLE 14.5 The Values of Shape Factor of Different Shapes of Nanoparticles m

Spherical

3

5.7 Platelet 4.8 Cylinder

Brick

3.7

The properties of the nanofluid are depicted in Table 14.3. Different values of shape factors for various shapes of nanoparticles are illustrated in Table 14.5. Vorticity and stream function should be used to eliminate pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5 u: @y @x @x @y

Dimensionless parameters are defined as:


 Hy ; Hx ; H b; a Hy ; Hx ; H 5 ; ðb; aÞ 5 ; L H0
 x; y uL vL T 2 Tc ; ΔT 5 qvL=kf ; ;V5 ; ðX; YÞ 5 ;θ5 U5 αnf αnf ΔT L Ψ5

ð14:117Þ

ð14:118Þ

ψ ωL2 ;Ω5 : αnf αnf

So equations change to: @2 Ψ @2 Ψ 1 Ω 1 5 0; @Y2 @X2 0 1 2 2 @Ω @Ω A5 A2 @@ Ω @ ΩA U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @@U @V 2 @U 2 @V Hy Hx 2 Hx 1 H 2 Hy Hx A 1 PrHa2 @X @Y y @Y A1 A4 @X 1 Pr Ra

ð14:119Þ

ð14:120Þ

A3 A22 @θ Pr A5 A2 2 ; 2 Da A1 A4 A1 A4 @X

 2 @θ @θ @ θ @2 θ 4 1 @2 θ U1V 5 1 Rd : 1 @X @Y @X2 @Y2 3 A4 @Y2

ð14:121Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

and dimensionless parameters are:

qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

 h i Da 5 K= L2 ; Ec 5 μf αf = ðρCP Þf ΔT L2 ;
 ρCp nf ρnf ðρβ Þnf  ; A3 5 A1 5 ; A2 5
; ρf ðρβ Þf ρCp f μnf knf σnf A4 5 ; A5 5 ; A6 5 ; kf μf σf
 Rd 5 4σe Tc3 = β R kf

ð14:122Þ

and boundary conditions are: @θ 5 1:0 @n

on inner wall

on other walls

θ 5 0:0 @θ 50 @n

on all walls

Ψ 5 0:0

on outer wall

ð14:123Þ

Nuloc ; Nuave over the hot wall can be calculated as: !   21 knf knf 4Rd 1 ; Nuloc 5 11 3 θ kf kf

ð14:124Þ

ðS 1 Nuave 5 Nuloc ds: S

ð14:125Þ

0

14.7.3 Effects of Active Parameters The impact of nonuniform magnetic field on Fe3O4-water flow in a permeable enclosure is simulated. Nanofluid viscosity is estimated according to previous experimental data. The shape effect of nanoparticles on knf is taken into consideration. CVFEM is utilized to find the effects of radiation parameter ðRd 5 0 to 0:8Þ, Darcy number (Da 5 0:01 to 100), Rayleigh number (Ra 5 103 ; 104 ; 105 ), volume fraction of Fe3O4-water (φ 5 0% to 4%), shape of nanoparticle, and Hartmann number (Ha 5 0 to 10). The impacts of shape of the nanoparticles on Nuave are presented in Table 14.6. The maximum Nuave is obtained for Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been selected to complete this section. Fig. 14.35 shows the impact of adding nanoparticles to water on the hydrothermal treatment. The temperature gradient reduces with the increase of φ. Velocity increases with the addition of nanoparticles because of an increase in the solid movements. Fig. 14.36 demonstrates the effect of the radiation parameter on streamline and TABLE 14.6 Effect of Shape of Nanoparticles on Nusselt Number When Da 5 100; Ra 5 105 ; Rd 5 0:8; φ 5 0:04 Ha 0

10

Spherical

12.32892

10.1131

Brick

12.40613

10.15818

Cylinder

12.52868

10.23001

Platelet

12.62989

10.28956

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

537

FIGURE 14.35 Impact of nanofluid volume fraction on streamline (top) and isotherm (bottom) contours (nanofluid (φ 5 0:04)(
) and pure fluid (φ 5 0) (2U 2 )) when Ra 5 105 ; Da 5 100; Rd 5 0:8.

isotherm contours. The thermal boundary layer thickness increases with the increase of Rd. By adding a magnetic field, the impact of the radiation parameter on streamlines becomes not significant. Figs. 14.37
14.39 illustrate the effects of Da; Ra; Ha on isotherms and streamlines. Only one eddy appears in streamlines. By increasing the Hartmann number, the main eddy moves downward and the distortion of isotherms becomes less than before. As the buoyancy forces increase, a thermal plume is generated near the vertical symmetric line. Increasing the Lorentz forces shifts the thermal plume to left and reduces jΨ max j. As the permeability of the media increases, the convective mode becomes stronger and the shape of isotherms becomes more complicated.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.36 Impact of radiation parameter on streamline (top) and isotherm (bottom) contours (Rd 5 0:8 (
), Rd 5 0 (- - -)) when Ra 5 105 ; Da 5 100; φ 5 0:04.

Effects of significant parameters on Nuave are depicted in Fig. 14.40. The correlation for Nuave is: Nuave 5 31:9 2 5:2Rd 2 15:3logðRaÞ 2 0:6Da 1 1:25Ha 1 2:93RdlogðRaÞ 1 0:44Rd Da 2 0:57Rd Ha 1 0:28Da logðRaÞ 2 0:37Ha logðRaÞ 2 0:69Da Ha
2 2 4:13Rd2 1 2:02 logðRaÞ 1 0:47ðDa Þ2 2 0:27ðHa Þ2

ð14:126Þ

where Da 5 0:01Da; Ha 5 0:1Ha. Heat transfer rate increases with the increase in permeability of the porous media. A similar treatment is reported for the Rayleigh number. The temperature gradient is reduced with the increase of the Hartmann number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

FIGURE 14.37

539

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 103 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.38

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 104 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

FIGURE 14.39

541

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

542

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

FIGURE 14.40

Effects of Da; Ha; Rd, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

14.7 NANOFLUID HEAT TRANSFER IN A PERMEABLE ENCLOSURE IN PRESENCE OF VARIABLE MAGNETIC FIELD

FIGURE 14.40

543

(Continued)

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

544

14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

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1848. [2] M. Sheikholeslami, Numerical investigation of MHD nanofluid free convective heat transfer in a porous tilted enclosure, Eng. Computat. 34 (2017) 1939
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248. Available from: https://doi.org/10.1140/epjp/i2014-14248-2. [18] M. Sheikholeslami, Numerical simulation for external magnetic field influence on Fe3O4-water nanofluid forced convection, Engineering Computations, Inpress. [19] M. Sheikholeslami, M.K. Sadoughi, Numerical modeling for Fe3O4 -water nanofluid flow in porous medium considering MFD viscosity, J. Mol. Liq. 242 (2017) 255
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3078. [21] M. Sheikholeslami, K. Vajravelu, Forced convection heat transfer in Fe3O4- ethylene glycol nanofluid under the influence of Coulomb force, J. Mol. Liq. 233 (2017) 203
210. [22] M. Sheikholeslami, H.B. Rokni, Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall, J. Mol. Liq. 232 (2017) 390
395. [23] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid natural convection in the presence of thermal radiation considering variable viscosity, Europ. Phys. J. Plus 132 (2017) 238. Available from: https://doi.org/10.1140/epjp/i2017-11498-4. [24] M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transfer 118 (2018) 823
831. [25] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4- Ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Mol. Liq. 249 (2018) 429
437. [26] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transfer 118 (2018) 182
192. [27] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition, Int. J. Heat Mass Transfer 106 (2017) 1261
1269. [28] M. Sheikholeslami, S.A. Shehzad, Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity, Int. J. Heat Mass Transfer 109 (2017) 82
92. [29] M. Sheikholeslami, H.B. Rokni, Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force, Comput. Methods Appl. Mech. Eng. 317 (2017) 419
430. [30] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transfer 106 (2017) 745
755. [31] M. Sheikholeslami, A.J. Chamkha, Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field, Numer. Heat Transfer, Part A 69 (2016) 1186
1200. Available from: https://doi. org/10.1080/10407782.2015.1125709. [32] M. Sheikholeslami, A.J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Numer. Heat Transfer, Part A 69 (2016) 781
793. Available from: http://dx.doi.org/10.1080/10407782.2015.1090819.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

545

[33] M. Sheikholeslami, M.K. Sadoughi, Simulation of CuO- water nanofluid heat transfer enhancement in presence of melting surface, Int. J. Heat Mass Transfer 116 (2018) 909
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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
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299. [60] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman, Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosity, Int. J. Heat Mass Transfer 108 (2017) 2558
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14. NON-DARCY MODEL FOR NANOFLUID HYDROTHERMAL TREATMENT IN A POROUS MEDIUM USING CVFEM

[66] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [67] M. Sheikholeslami, M. Barzegar Gerdroodbary, S. Valiallah Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field, J. Magn. Magn. Mater. 460 (2018) 302
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438. [70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
431. [71] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transfer 122 (2018) 1264
1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
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412. [87] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Physica B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [88] M. Sheikholeslami, A. Zeeshan, Numerical simulation of Fe3O4-water nanofluid flow in a non-Darcy porous media, Int. J, Num. Methods Heat Fluid Flow Vol. 28 (Issue: 3) (2018) 641
660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Des. 120 (2017) 382
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161.

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C H A P T E R

15 Thermal Nonequilibrium Model for Nanofluid Flow in a Porous Enclosure by Means of CVFEM 15.1 INTRODUCTION The study of fluid flow and heat transfer in a porous medium has been a subject of continuous interest for the past several decades because of the wide range of applications, such as geothermal systems, drying technologies, production of thermal isolators, the control of pollutant spread in groundwater, the insulation of buildings, solar power collectors, design of nuclear reactors, and compact heat exchangers, etc. A comprehensive literature survey on this subject can be found in the books by Nield and Bejan [1]. In porous media applications, such as the environmental impact of buried nuclear heat generating waste, chemical reactors, thermal energy transport/storage systems, the cooling of electronic devices, etc., a temperature discrepancy between the solid matrix and the saturating fluid has been observed and recognized. Therefore an analysis of separate energy equations for the two phases, i.e., a local thermal nonequilibrium (LTNE) model, has led to the increasing importance of such approach. Problems on boundary layer flows and heat transfer past a vertical flat plate and a horizontal circular cylinder embedded in a porous medium using LTNE have been studied by several authors and references on this topic may be found in the review papers by Kuznetsov [2]. Convective flow in porous enclosures using the LTE model is a topic studied very intensively during the last several decades and reviews can be found in the books mentioned above and in the review papers by Bejan [3], and Nield [4]. However, it appears that Baytas and Pop [5] were the first to investigate LTNE for the problem of steady free convection in a square porous cavity in order to modify the flow behavior and heat transfer rates. Unsteady laminar natural convection flow in a rectangular enclosure formed by nonisothermal walls, filled with a fluid saturated porous medium in the case of LTNE condition was considered by Hossain and Wilson [6]. In recent years, nanofluid flow in a porous medium has been developed [7
101].

15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL 15.2.1 Problem definition Fig. 15.1 demonstrates the boundary condition and geometry. The hot wall formulation is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 5 1 2 ε1 2 :a

ð15:1Þ

where a; b; ε1 are the major, minor axis of elliptic cylinder, and eccentricity for the inner cylinder. A horizontal magnetic field is employed. CuO-water nanofluid is a working fluid. The enclosure has porous media.

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00015-1

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© 2019 Elsevier Inc. All rights reserved.

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15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

FIGURE 15.1 Geometry and the boundary conditions.

15.2.2 Governing equation The thermal nonequilibrium model and Boussinesq
Darcy law for flow are employed. So, the twotemperature model is utilized. Considering these conditions, the governing PDEs are: -

r: V 5 0

2

μnf K





ρCp

 nf

- -
2 ðρβ Þnf Tnf 2 Tc g 2 σnf V 3 B 2 rp 5 0
 hnfs ks  r 2 Ts 1
 Tnf 2 Ts 5 0 ρCp s ð1 2 εÞ ρCp s

; ðρβ Þnf , ρnf , and σnf can be defined as:


 ρCp nf 5 φ ρCp p 1 ð1 2 φÞ ρCp f

ð15:2Þ ð15:3Þ ð15:4Þ

ð15:5Þ

ðρβ Þnf 5 ð1 2 φÞðρβ Þf 1 φðρβ Þp

ð15:6Þ

ρnf 5 ρf ð1 2 φÞ 1 ρp φ

ð15:7Þ

σnf σp ðMM 2 1Þφ 1 1; MM 5 53 φð1 2 MMÞ 1 ðMM 1 2Þ σf σf

ð15:8Þ

knf ; μnf can be estimated via the Koo
Kleinstreuer
Li (KKL) model: 0 1 k p @1 2 A φ sffiffiffiffiffiffiffiffiffi kf knf κb T 4 0 1 0 1 1 5 3 10 g ðdp ; T; φÞρf φ cp;f 51230 ρ kf p dp k k @ 1 2 p A φ 1 @ p 1 2A kf kf



2

 g0 dp ; T; φ 5 LnðTÞ a2 Ln dp 1 a5 Ln dp 1 a1 1 a3 LnðφÞ 1 a4 Ln dp LnðφÞ




2  1 a7 Ln dp 1 a6 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð15:9Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W

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15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL

μnf 5

μf ð12φÞ

2:5

1

μf
 kBrownian =Pr kf

ð15:10Þ

Properties of CuO-water and related coefficients are depicted in Tables 15.1 and 15.2. Nondimensional quantities are: v52

@ψ @ψ ;u5 ; @x @y

ð15:11Þ


 θ s 5 ð Ts 2
Tc Þ=  ðTh 2 Tc Þ; θnf 5 Tnf 2 Tc =ðTh 2 Tc Þ; ðX; YÞ 5 x; y =L; Ψ 5 ψ=αnf According to the above definitions, we have: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θnf 2 Ra A4 A5 @X @2 θnf @2 θnf ε 1 @Y2 @X2

!


 @θnf @Ψ @Ψ @θnf 1 1 Nhs θs 2 θnf 5 2 @Y @X @Y @X

ð15:13Þ

 2
 @ θs @2 θs ε 1 1 Nhs δs θnf 2 θs 5 0 2 2 @Y @X

TABLE 15.1

ð15:14Þ

The Coefficient Values of CuO 2 Water Nanofluid CuO 2 Water

Coefficient values

TABLE 15.2

ð15:12Þ

a1

2 26.593310846

a2

2 0.403818333

a3

2 33.3516805

a4

2 1.915825591

a5

6.42185846658E-02

a6

48.40336955

a7

2 9.787756683

a8

190.245610009

a9

10.9285386565

a10

-0.72009983664

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK21 Þ

dp ðnmÞ

σ ðΩUmÞ21

Water

997.1

4179

0.613

21

-

0:05

CuO

6500

540

18

29

29

10-10

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15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

where the constant and dimensionless parameters are: A1 5

ρnf ρf

A2 5 Ra 5

;

A3 5

ðρCP Þnf ðρCP Þf

ðρβ Þnf ðρβ Þf

; A6 5

g K ðρβ Þf L ΔT μf αf

Nhs 5 hnfs L2 =knf ;

; A5 5

μnf μf

;

σnf knf ; A4 5 ; σf kf ; Ha 5

σf K μf

B20

ð15:15Þ

;

δs 5 knf =½ks ð1 2 εÞ

The inner wall is hot. Ψ 5 0:0 θnf 5 θs 5 0:0 θnf 5 θs 5 1:0 Nuloc ; Nuave along the cold wall are:

on all walls on outer wall on inner wall

knf @θnf kf @r ð 2π 1 Nuave 5 Nuloc dr 2π 0

ð15:16Þ



Nuloc 5

ð15:17Þ ð15:18Þ

15.2.3 Effects of active parameters As in the application of CVFEM, nanofluid natural convection in porous media is analyzed by means of the Nonequilibrium model. Simulations are depicted for the impacts of Rayleigh number (Ra 5 100; 500, and 103 ), the porosity (ε 5 0:3 to 0:9), the solid
nanofluid interface heat transfer parameter (Nhs 5 10 to 1000), and Hartmann number (Ha 5 0 to 20) on nanofluid hydrothermal behavior. Figs. 15.2
15.7 depict the impacts of Ra; Nhs; ε, and Ha on streamlines, isotherms for solid and nanofluid.
In conduction mode, the influences of other parameters are not significant. So, isotherms for the nanofluid θnf the thermal plume can be and the solid ðθs Þ have the same shapes. As buoyancy forces are enhanced,
 observed near the vertical centerline and isotherms for the nanofluid θnf have a complex shape while isotherms for the solid ðθs Þ are parallel to each other. Augmenting the Lorentz forces reduces the nanofluid velocity and the temperature fields are stratified. As Nhs enhances, convective flow become stronger and ðΨ max Þ augments. But the temperature gradient of the nanofluid reduces and in turn Nuave decreases. As porosity of porous medium augments, the pore volume inside the cavity increases. Therefore, improvement in the convective flow can be observed. Also, the effect of ε on isotherms is similar to that of Nhs. Fig. 15.8 depicts the influence of Ra; Ha; ε; Nhs on average Nusselt number. According to obtained data Nuave can be expressed as: Nuave 5 2:63 1 1:42Ra 2 0:16Ha 2 0:12ε 2 0:11Nhs 2 0:35Ra Ha 2 0:56Ra ε 2 0:63Ra Nhs 1 0:12Ha ε 1 0:14Ha Nhs 1 0:29εNhs 2 0:05ðRa Þ2 1 0:018ðHa Þ2 2 0:29ðεÞ2 2 0:15ðNhs Þ2

ð15:19Þ

where Ra 5 0:001Ra; Ha 5 0:1Ha; Nhs 5 0:001Nhs. Increasing the porous medium porosity leads to a decrease in Nusselt number and a similar impact is shown for the Hartmann number. The Nhs has a direct relationship with the rate of heat transfer because the thermal boundary layer thickness reduces with the rise of Nhs. Nuave is enhanced with the increase of buoyancy forces.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL

FIGURE 15.2

551


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 0; φ 5 0:04.

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15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM






FIGURE 15.3 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL

FIGURE 15.4

553


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM






FIGURE 15.5 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL

FIGURE 15.6

555


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

556

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM






FIGURE 15.7 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.2 SIMULATION OF NANOFLUID FLOW INSIDE A POROUS ENCLOSURE VIA NONEQUILIBRIUM MODEL

FIGURE 15.8

557

Effects of Ra; Ha; ε; Nhs on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

558

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

FIGURE 15.8 (Continued).

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL 15.3.1 Problem definition Fig. 15.9 demonstrates the boundary conditions and geometry. A horizontal magnetic field is employed. CuOwater nanofluid is a working fluid. The enclosure has porous media.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL

FIGURE 15.9

559

Geometry and the boundary conditions.

15.3.2 Governing equation The thermal nonequilibrium model and Boussinesq
Darcy law for flow are employed. So, the twotemperature model is utilized. Considering these conditions, the governing PDEs are: -

r: V 5 0

2




ρCp

 nf

μnf K

- -
2 ðρβ Þnf Tnf 2 Tc g 2 σnf V 3 B 2 rp 5 0


 hnfs k
s  r 2 Ts 1
 Tnf 2 Ts 5 0 ρCp s ð1 2 εÞ ρCp s ; ðρβ Þnf , ρnf , and σnf can defined as:


 ρCp nf 5 φ ρCp p 1 ð1 2 φÞ ρCp f

ð15:20Þ ð15:21Þ ð15:22Þ

ð15:23Þ

ðρβ Þnf 5 ð1 2 φÞðρβ Þf 1 φðρβ Þp

ð15:24Þ

ρnf 5 ρf ð1 2 φÞ 1 ρp φ

ð15:25Þ

σnf σp ðMM 2 1Þφ 1 1; MM 5 53 φð1 2 MMÞ 1 ðMM 1 2Þ σf σf

ð15:26Þ

knf ; μnf can be estimated via the KKL model: 0 1 k p @1 2 Aφ sffiffiffiffiffiffiffiffiffi kf knf κb T 4 0 0 1 0 1 cp;f 5123 1 5 3 10 g ðdp ; T; φÞρf φ ρ kf p dp k k p p @ 1 2 A φ 1 @ 1 2A kf kf



2

 g0 dp ; T; φ 5 LnðT Þ a2 Ln dp 1 a5 Ln dp 1 a1 1 a3 LnðφÞ 1 a4 Ln dp LnðφÞ




2  1 a7 Ln dp 1 a6 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð15:27Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

1

μf
 kBrownian =Pr kf

ð15:28Þ

Properties of CuO-water and related coefficients are depicted in Tables 15.1 and 15.2.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

560

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

Nondimensional quantities are: v52

@ψ @ψ ;u5 ; @x @y


 θs 5 ðTs 2
Tc Þ=  ðTh 2 Tc Þ; θnf 5 Tnf 2 Tc =ðTh 2 Tc Þ; ðX; YÞ 5 x; y =L; Ψ 5 ψ=αnf According to the above definitions, we have: 2 3 @2 Ψ @2 Ψ A6 4@2 Ψ
2  @2 Ψ
2  @2 Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha sin γ 1 cos γ 1 2 @X2 @Y2 @Y2 @X2 @X @Y A5 A3 A2 @θnf 2 Ra A4 A5 @X @2 θnf @2 θnf ε 1 @Y2 @X2

!


 @θnf @Ψ @Ψ @θnf 1 1 Nhs θs 2 θnf 5 2 @Y @X @Y @X

 2
 @ θs @2 θs ε 1 1 Nhs δs θnf 2 θs 5 0 2 2 @Y @X

ð15:29Þ

ð15:30Þ

ð15:31Þ ð15:32Þ

where the constant and dimensionless parameters are: A1 5

ρnf ρf

A2 5 Ra 5

;

A3 5

ðρCP Þnf ðρCP Þf

ðρβ Þnf ðρβ Þf

; A6 5

g K ðρβ Þf L ΔT μf αf

Nhs 5 hnfs L2 =knf ;

; A5 5

μnf μf

;

σnf knf ; A4 5 ; σf kf ; Ha 5

σf K μf

B20

ð15:33Þ

;

δs 5 knf =½ks ð1 2 εÞ

The inner wall is hot. Ψ 5 0:0 θnf 5 θs 5 0:0 θnf 5 θs 5 1:0 Nuloc ; Nuave along the cold wall are:

on all walls on outer wall on inner wall

knf @θnf kf @r ð 2π 1 Nuave 5 Nuloc dr 2π 0

ð15:34Þ



Nuloc 5

ð15:35Þ ð15:36Þ

15.3.3 Effects of active parameters The nonequilibrium model is employed for the porous media to analyze nanofluid flow and natural convection. This section is an application of CVFEM. The KKL model is selected for nanofluid. Simulations are shown for the influences of Rayleigh number (Ra 5 100; 500 and 103 ), the porosity (ε 5 0:3 to 0:9), the solid
nanofluid interface heat transfer parameter (Nhs 5 10 to 1000), and Hartmann number (Ha 5 0 to 20) on nanofluid
hydro thermal behavior. Effects of Ra; Nhs; ε, and Ha on streamlines ðΨ Þ, isotherms for solid ðθs Þ and nanofluid θnf are are not visible because of conduction mode shown in Figs. 15.10
15.15. In low Ra, effects of other
parameters  dominance. In this case, isotherms for the nanofluid θnf and the solid ðθs Þ have the same shapes. As the buoyancy forces increase, the thermal plume can be observed near the vertical centerline and isotherms for the

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL

FIGURE 15.10

561


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

562

FIGURE 15.11

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL

FIGURE 15.12

563


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

564

FIGURE 15.13

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL

FIGURE 15.14

565


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

566

FIGURE 15.15

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 20; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.3 NANOFLUID FREE CONVECTION IN A POROUS CAVITY CONSIDERING THE TWO-TEMPERATURE MODEL




567



nanofluid θnf have complex shapes while isotherms for the solid ðθs Þ are parallel to each other. Increasing the Lorentz forces reduces the nanofluid velocity and the temperature fields are stratified. As Nhs enhances, convective flow becomes stronger and ðΨ max Þ augments. But the temperature gradient of nanofluid reduces and in turn Nuave decreases. As the porous medium porosity enhances, the pore volume inside the cavity increases. Therefore, an improvement in the convective flow can be observed. Also, the effect of ε on isotherms is similar to that of Nhs. Fig. 15.16 depicts the impact of Ra; Ha; ε; Nhs on average Nusselt number. According to obtained data, Nuave can be expressed as:

FIGURE 15.16

Effects of Ra; Ha; ε; Nhs on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

568

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

Nuave 5 1:48 1 3:24Ra 2 0:57Ha 2 0:04ε 2 0:49Nhs 2 0:81Ra Ha 2 0:76Ra ε 2 1:15Ra Nhs 1 0:4Ha ε 1 0:62Ha Nhs 1 0:8εNhs 2 0:46ðRa Þ2 1 0:007ðHa Þ2 2 0:66ðεÞ2 2 0:63ðNhs Þ2

ð19Þ

where Ra 5 0:001Ra; Ha 5 0:1Ha; Nhs 5 0:001Nhs. The Nhs has a direct relationship with the rate of heat transfer because the thermal boundary layer thickness reduces with the rise of Nhs. Increasing the porous medium porosity leads to a decrease in Nusselt number and a similar impact is shown for the Hartmann number. Nuave is enhanced with the increase of buoyancy forces.

15.4 NANOFLUID FLOW IN A POROUS SINUSOIDAL CAVITY CONSIDERING THERMAL NONEQUILIBRIUM MODEL 15.4.1 Problem definition Fig. 15.17 demonstrates the boundary conditions and geometry. The sinusoidal wall formulation is:

 rout 5 rin 1 A cos N ζ 2 ζ 0

ð15:37Þ

A horizontal magnetic field is employed. CuO-water nanofluid is a working fluid. The enclosure has porous media.

15.4.2 Governing equation The thermal nonequilibrium model and Boussinesq
Darcy law for flow are employed. So, the twotemperature model is utilized. Considering these conditions, the governing PDEs are: -

r: V 5 0

2

μnf K





ρCp

 nf

- -
2 ðρβ Þnf Tnf 2 Tc g 2 σnf V 3 B 2 rp 5 0
 hnfs ks  r 2 Ts 1
 Tnf 2 Ts 5 0 ρCp s ð1 2 εÞ ρCp s

; ðρβ Þnf , ρnf , and σnf can defined as:


 ρCp nf 5 φ ρCp p 1 ð1 2 φÞ ρCp f ðρβ Þnf 5 ð1 2 φÞðρβ Þf 1 φðρβ Þp

FIGURE 15.17

ð15:38Þ ð15:39Þ ð15:40Þ

ð15:41Þ ð15:42Þ

Geometry and the boundary.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.4 NANOFLUID FLOW IN A POROUS SINUSOIDAL CAVITY CONSIDERING THERMAL NONEQUILIBRIUM MODEL

569

ρnf 5 ρf ð1 2 φÞ 1 ρp φ

ð15:43Þ

σnf σp ðMM 2 1Þφ 1 1; MM 5 53 φð1 2 MMÞ 1 ðMM 1 2Þ σf σf

ð15:44Þ

knf ; μnf can be estimated via the KKL model: 0 1 k @1 2 p Aφ sffiffiffiffiffiffiffiffiffi kf knf κb T 4 0 1 0 1 1 5 3 10 g ðdp ; T; φÞρf φ cp;f 51230 ρ p dp kf k k p p @ 1 2 A φ 1 @ 1 2A kf kf



2

 g0 dp ; T; φ 5 LnðT Þ a2 Ln dp 1 a5 Ln dp 1 a1 1 a3 LnðφÞ 1 a4 Ln dp LnðφÞ




2  1 a7 Ln dp 1 a6 1 a8 LnðφÞ 1 a9 ln dp LnðφÞ 1 a10 Ln dp

ð15:45Þ

Rf 5 dp =kp;eff 2 dp =kp ; Rf 5 4 3 1028 km2 =W μnf 5

μf ð12φÞ

2:5

μf
 kBrownian =Pr kf

1

ð15:46Þ

Properties of CuO-water and related coefficients are depicted in Tables 15.1 and 15.2. Nondimensional quantities are: v52

@ψ @ψ ;u5 ; @x @y


 θ s 5 ð Ts 2
Tc Þ=  ðTh 2 Tc Þ; θnf 5 Tnf 2 Tc =ðTh 2 Tc Þ; ðX; YÞ 5 x; y =L; Ψ 5 ψ=αnf According to the above definitions, we have: 2 3 2 2 2
2
2   @ Ψ @ Ψ A6 @ Ψ @ Ψ @ Ψ ðsinγ Þ ðcosγ Þ5 1 5 2 Ha4 2 sin2 γ 1 cos2 γ 1 2 @X2 @Y2 @Y @X2 @X @Y A5 A3 A2 @θnf 2 Ra A4 A5 @X @2 θnf @2 θnf ε 1 @Y2 @X2

!


 @θnf @Ψ @Ψ @θnf 1 1 Nhs θs 2 θnf 5 2 @Y @X @Y @X

 2
 @ θs @2 θs ε 1 1 Nhs δs θnf 2 θs 5 0 @Y2 @X2

ð15:47Þ

ð15:48Þ

ð15:49Þ ð15:50Þ

where the constant and dimensionless parameters are: A1 5

ρnf ρf

A2 5 Ra 5

;

A3 5

ðρCP Þnf ðρCP Þf

ðρβ Þnf ðρβ Þf

; A6 5

g K ðρβ Þf L ΔT μf αf

Nhs 5 hnfs L2 =knf ;

; A5 5

μnf μf

;

σnf knf ; A4 5 ; σf kf ; Ha 5

σf K μf

B20

ð15:51Þ

;

δs 5 knf =½ks ð1 2 εÞ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

570

FIGURE 15.18

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 0; φ 5 0:04.

The inner wall is hot. Ψ 5 0:0 θnf 5 θs 5 0:0 θnf 5 θs 5 1:0

on all walls on outer wall on inner wall

ð15:52Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.4 NANOFLUID FLOW IN A POROUS SINUSOIDAL CAVITY CONSIDERING THERMAL NONEQUILIBRIUM MODEL

FIGURE 15.19

571


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 100; Ha 5 20; φ 5 0:04.

Nuloc ; Nuave along the cold wall are:

 knf @θnf Nuloc 5 kf @r ð 1 2π Nuave 5 Nuloc dr 2π 0

ð15:53Þ ð15:54Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

572

FIGURE 15.20

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 0; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.4 NANOFLUID FLOW IN A POROUS SINUSOIDAL CAVITY CONSIDERING THERMAL NONEQUILIBRIUM MODEL

FIGURE 15.21

573


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 500; Ha 5 20; φ 5 0:04.

15.4.3 Effects of active parameters The nonequilibrium model is employed for porous media to analyze nanofluid flow and natural convection. This section is an application of CVFEM. The KKL model is selected for nanofluid. Simulations are shown for the influences of Rayleigh number (Ra 5 100; 500 and 103 ), the porosity (ε 5 0:3 to 0:9), the solid
nanofluid interface heat transfer parameter (Nhs 5 10 to 1000), and Hartmann number (Ha 5 0 to 20) on nanofluid hydrothermal

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

574

FIGURE 15.22

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 0; φ 5 0:04.


 behavior. Effects of Ra; Nhs; ε, and Ha on streamlines ðΨ Þ, isotherms for solid ðθs Þ and nanofluid θnf are shown in Figs. 15.18
15.23. In low Ra, the effects of other
parameters are not visible because of conduction mode dominance. In this case, isotherms for the nanofluid θnf and the solid ðθs Þ have the same shapes. As buoyancy
forces increase, a thermal plume can be observed near the vertical centerline and isotherms for the nanofluid θnf have

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

15.4 NANOFLUID FLOW IN A POROUS SINUSOIDAL CAVITY CONSIDERING THERMAL NONEQUILIBRIUM MODEL

FIGURE 15.23

575


 Streamlines ðΨ Þ, isotherms for the nanofluid θnf and the solid ðθs Þ at Ra 5 1000; Ha 5 20; φ 5 0:04.

complex shapes while isotherms for the solid ðθs Þ are parallel to each other. Increasing the Lorentz forces reduces the nanofluid velocity and the temperature fields are stratified. As Nhs enhances, convective flow become stronger and ðΨ max Þ augments. But the temperature gradient of the nanofluid is reduced and in turn Nuave decreases. As the porous medium porosity enhances, the pore volume inside the cavity increases. Therefore, improvement

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

576

FIGURE 15.24

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

Effects of Ra; Ha; ε; Nhs on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

577

in the convective flow can be observed. Also, the effect of ε on isotherms is similar to that of Nhs. Fig. 15.24 depicts the impact of Ra; Ha; ε; Nhs on average Nusselt number. According to obtained data, Nuave can be expressed as: Nuave 5 5:04 1 4:95Ra 2 0:07Ha 1 0:24ε 2 2:04Nhs 2 1:13Ra Ha 2 1:38Ra ε 2 1:88Ra Nhs 1 0:69Ha ε 1 1:11Ha Nhs 1 1:36εNhs 2 0:69ðRa Þ2 2 0:44ðHa Þ2 2 1:33ðεÞ2 2 0:12ðNhs Þ2

ð15:55Þ

where Ra 5 0:001Ra; Ha 5 0:1Ha; Nhs 5 0:001Nhs. The Nhs has direct relationship with rate of heat transfer because the thermal boundary layer thickness reduces with the increase of Nhs. Increasing the porous medium porosity leads to a decrease in Nusselt number and a similar impact is shown for the Hartmann number. Nuave is enhanced with the increase of buoyancy forces.

References [1] D.A. Nield, A. Bejan, Convection in Porous Media, fourth ed., Springer, New York, USA, 2013. [2] A.V. Kuznetsov, Thermal nonequilibrium forced convection in porous media, in: D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Vol. 1, Pergamon Press, Oxford, UK, 2005, pp. 103
129. [3] A. Bejan, Convective heat transfer in porous mediaChapter 16 in: S. Kakac¸, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987. [4] D.A. Nield, Modelling fluid flow in saturated porous media and at interfaces, in: D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media., Vol. 2, Pergamon Press, Oxford, UK, 2002, pp. 1
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248. Available from: https://doi.org/10.1140/epjp/i2014-14248-2. [16] M. Sheikholeslami, Numerical simulation for external magnetic field influence on Fe3O4-water nanofluid forced convection, Eng. Comput., Inpress. [17] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cavity by means of Non- Darcy model, Eng. Comput. 34 (8) (2017) 2651
2667. Available from: https://doi.org/10.1108/EC-01-2017-0008. [18] M. Sheikholeslami, H.B. Rokni, Nanofluid convective heat transfer intensification in a porous circular cylinder, Chem. Eng. Proc. Process Intensif. 120 (2017) 93
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250. [22] M. Sheikholeslami, K. Vajravelu, Forced convection heat transfer in Fe3O4- ethylene glycol nanofluid under the influence of Coulomb force, J. Mol. Liq. 233 (2017) 203
210. [23] M. Sheikholeslami, H.B. Rokni, Influence of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall, J. Mol. Liq. 232 (2017) 390
395. [24] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid natural convection in the presence of thermal radiation considering variable viscosity, Europ. Phys. J. Plus 132 (2017) 238. Available from: https://doi.org/10.1140/epjp/i2017-11498-4. [25] M. Sheikholeslami, H.B. Rokni, Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation, Int. J. Heat Mass Transfer (2018). Available from: https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.041.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

578

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

[26] M. Sheikholeslami, M. Shamlooei, R. Moradi, Fe3O4- Ethylene glycol nanofluid forced convection inside a porous enclosure in existence of Coulomb force, J. Mol. Liq. 249 (2018) 429
437. [27] M. Sheikholeslami, S.A. Shehzad, Numerical analysis of Fe3O4
H2O nanofluid flow in permeable media under the effect of external magnetic source, Int. J. Heat Mass Transfer, In press. [28] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition, Int. J. Heat Mass Transfer 106 (2017) 1261
1269. [29] M. Sheikholeslami, S.A. Shehzad, Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity, Int. J. Heat Mass Transfer 109 (2017) 82
92. [30] M. Sheikholeslami, H.B. Rokni, Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force, Comput. Methods Appl. Mech. Eng. 317 (2017) 419
430. [31] M. Sheikholeslami, K. Vajravelu, Nanofluid flow and heat transfer in a cavity with variable magnetic field, Appl. Math. Comput. 298 (2017) 272
282. [32] M. Sheikholeslami, H.B. Rokni, Magnetohydrodynamic CuO-water nanofluid in a porous complex shaped enclosure, ASME, J. Thermal Sci. Eng. Applicat. 9 (4) (2017) 041007. Available from: https://doi.org/10.1115/1.4035973. [33] M. Sheikholeslami, H.B. Rokni, Nanofluid two phase model analysis in existence of induced magnetic field, Int. J. Heat Mass Transfer 107 (2017) 288
299. [34] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman, Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosity, Int. J. Heat Mass Transfer 108 (2017) 2558
2565. [35] M. Sheikholeslami, Magnetic field influence on CuO -H2O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles, Int. J. Hydrogen Energy 42 (2017) 19611
19621. [36] M. Sheikholeslami, M.M. Bhatti, Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles, Int. J. Heat Mass Transfer 111 (2017) 1039
1049. [37] M. Sheikholeslami, A. Zeeshan, Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM, Comput. Methods Appl. Mech. Eng. 320 (2017) 68
81. [38] M. Sheikholeslami, M. Shamlooei, Fe3O4- H2O nanofluid natural convection in presence of thermal radiation, Int. J. Hydrogen Energy 42 (9) (2017) 5708
5718. [39] M. Sheikholeslami, M.M. Bhatti, Active method for nanofluid heat transfer enhancement by means of EHD, Int. J. Heat Mass Transfer 109 (2017) 115
122. [40] M. Sheikholeslami, A.J. Chamkha, Flow and convective heat transfer of a ferro-nanofluid in a double-sided lid-driven cavity with a wavy wall in the presence of a variable magnetic field, Numer. Heat Transfer, Part A 69 (10) (2016) 1186
1200. Available from: https:// doi.org/10.1080/10407782.2015.1125709. [41] M. Sheikholeslami, A.J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Numer. Heat Transfer, Part A 69 (7) (2016) 781
793. Available from: http://dx.doi.org/10.1080/10407782.2015.1090819. [42] M. Sheikholeslami, M.K. Sadoughi, Simulation of CuO- water nanofluid heat transfer enhancement in presence of melting surface, Int. J. Heat Mass Transfer 116 (2018) 909
919. [43] M. Sheikholeslami, H.B. Rokni, Simulation of nanofluid heat transfer in presence of magnetic field: a review, Int. J. Heat Mass Transfer 115 (2017) 1203
1233. [44] M. Sheikholeslami, H.B. Rokni, Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic field, Int. J. Heat Mass Transfer 114 (2017) 517
526. [45] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM, Int. J. Heat Mass Transfer 113 (2017) 796
805. [46] M. Sheikholeslami, Magnetohydrodynamic nanofluid forced convection in a porous lid driven cubic cavity using Lattice Boltzmann Method, J. Mol. Liq. 231 (2017) 555
565. [47] M.S. Kandelousi, KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel, Phys. Lett. A 378 (45) (2014) 3331
3339. [48] M. Sheikholeslami, Effect of uniform suction on nanofluid flow and heat transfer over a cylinder, J. Braz. Soc. Mech. Sci. Eng. 37 (2015) 1623
1633. [49] M. Sheikholeslami, Lattice Boltzmann Method simulation of MHD non-Darcy nanofluid free convection, Phys. B 516 (2017) 55
71. [50] M. Sheikholeslami, Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann Method, J. Mol. Liq. 234 (2017) 364
374. [51] M. Sheikholeslami, Ali J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, J. Mol. Liq. 225 (2017) 750
757. [52] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using Lattice Boltzmann Method, Int. J. Heat Mass Transfer 108 (2017) 1870
1883. [53] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, Int. J. Heat Mass Transfer 89 (2015) 799
808. [54] M. Sheikholeslami, S. Abelman, Two phase simulation of nanofluid flow and heat transfer in an annulus in the presence of an axial magnetic field, IEEE Trans. Nanotechnol. 14 (3) (2015) 561
569. [55] M. Sheikholeslami, R. Ellahi, Simulation of ferrofluid flow for magnetic drug targeting using Lattice Boltzmann method, J. Zeitschrift Fur Naturforschung A 70 (2) (2015) 115
124. [56] M. Sheikholeslami, M. Seyednezhad, Lattice Boltzmann Method simulation for CuO-water nanofluid flow in a porous enclosure with hot obstacle, J. Mol. Liq. 243 (2017) 249
256. [57] M. Sheikholeslami, M. Sadoughi, Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles, Int. J. Heat Mass Transfer 113 (2017) 106
114.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

REFERENCES

579

[58] M. Sheikholeslami, A. Zeeshan, Mesoscopic simulation of CuO-H2O nanofluid in a porous enclosure with elliptic heat source, Int. J. Hydrogen Energy 42 (22) (2017) 15393
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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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1180. [61] M. Sheikholeslami, H.B. Rokni, Free convection of CuO-H2O nanofluid in a curved porous enclosure using mesoscopic approach, Int. J. Hydrogen Energy 42 (22) (2017) 14942
14949. [62] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq., In press. [63] M. Sheikholeslami, H.B. Rokni, Influence of melting surface on MHD nanofluid flow by means of two phase model, Chin. J. Phys. 55 (2017) 1352
1360. [64] M. Sheikholeslami, H.B. Rokni, Effect of melting heat transfer on nanofluid flow in existence of magnetic field considering Buongiorno Model, Chin. J. Phys. 55 (2017) 1115
1126. [65] M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of Al2O3
water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
524. [66] M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field, Int. J. Heat Mass Transfer 92 (2016) 339
348. [67] M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry, J. Mol. Liq. 213 (2016) 153
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438. [70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
431. [71] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J.Heat Mass Transfer 122 (2018) 1264
1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of Non-equilibrium model, J. Mol. Liq. 254 (2018) 446
462. [74] M. Sheikholeslami, S.A. Shehzad, Z. Li, Water based nanofluid free convection heat transfer in a three dimensional porous cavity with hot sphere obstacle in existence of Lorenz forces, Int. J. Heat Mass Transfer 125 (2018) 375
386. [75] M. Sheikholeslami, M. Jafaryar, D.D. Ganji, Z. Li, Exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators, J. Mol. Liq. 262 (2018) 104
110. [76] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transfer 124 (2018) 980
989. [77] M. Sheikholeslami, Z. Li, M. Shamlooei, Nanofluid MHD natural convection through a porous complex shaped cavity considering thermal radiation, Phys. Lett. A 382 (2018) 1615
1632. [78] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids, Volume 30, Issue 1. Available from: https://doi.org/10.1063/1.5012517. [79] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model, Int. J. Heat Mass Transfer 120 (2018) 1200
1212. [80] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [81] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transfer 120 (2018) 772
781. [82] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng, Process. : Process Intensif. 124 (2018) 71
82. [83] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [84] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [85] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [86] M. Sheikholeslami, D.D. Ganji, Influence of electric field on Fe3O4-water nanofluid radiative and convective heat transfer in a permeable enclosure, J. Mol. Liq. 250 (2018) 404
412. [87] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Physica B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [88] M. Sheikholeslami, A. Zeeshan, Numerical simulation of Fe3O4-water nanofluid flow in a non-Darcy porous media, Int. J. Num. Methods Heat Fluid Flow Vol. 28 (Issue: 3) (2018) 641
660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Des. 120 (2017) 382
393.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

580

15. THERMAL NONEQUILIBRIUM MODEL FOR NANOFLUID FLOW IN A POROUS ENCLOSURE BY MEANS OF CVFEM

[90] M. Sheikholeslami, D.D. Ganji, Numerical modeling of magnetohydrodynamic CuO-water transportation inside a porous cavity considering shape factor effect, Colloids Surf. A 529 (2017) 705
714. [91] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [92] M. Sheikholeslami, H.R. Kataria, Akhil, S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [93] M. Sheikholeslami, M. Barzegar Gerdroodbary, S. Valiallah Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field, J. Magn. Magn. Mater. 460 (2018) 302
311. [94] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transfer 106 (2017) 745
755. [95] M. Sheikholeslami, M. Jafaryar, Ahmad Shafee, Zhixiong Li, Investigation of second law and hydrothermal behavior of nanofluid through a tube using passive methods, J. Mol. Liq. 269 (2018) 407
416. [96] Mohsen Sheikholeslami, Ahmad Zeeshan, Aaqib Majeed, Control volume based finite element simulation of magnetic nanofluid flow and heat transport in non-Darcy medium, J. Mol. Liq. 268 (2018) 354
364. [97] M. Sheikholeslami, S.A. Shehzad, Zhixiong Li, Ahmad Shafee, Numerical modeling for Alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law, Int. J. Heat Mass Transferr 127 (2018) 614
622. [98] M. Sheikholeslami, Application of Darcy law for nanofluid flow in a porous cavity under the impact of Lorentz forces, J. Mol. Liq. 266 (2018) 495
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355. [101] M. Sheikholeslami, M. Jafaryar, S. Saleem, Zhixiong Li, Ahmad Shafee, Yu Jiang, Nanofluid heat transfer augmentation and exergy loss inside a pipe equipped with innovative turbulators, Int. J. Heat Mass Transfer 126 (2018) 156
163.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

16 Nonuniform Magnetic Field Effect on Nanofluid Convective Flow in a Porous Cavity 16.1 INTRODUCTION Free convection in porous media occurs in many systems and in nature, including in geophysical, environmental, and technological problems. Such problems are of great interest, e.g., in the geothermal, high-performance building insulation, postaccident heat removal from pebble-bed nuclear reactors, multishield structures used in the insulation of nuclear reactors, pollutant dispersion in aquifers, solar power collector, etc. The state-of the-art has been very well summarized in the recent books by Pop and Ingham [1]. The problem of free convection flow in differentially heated cavities, with top and bottom walls insulated, and filled with Darcian or non-Darcian fluid-saturated porous media, is of fundamental interest to many technological applications in the modern industry. Sheikholeslami [2] utilized CVFEM for magnetic nanofluid convective heat transfer in a porous curved enclosure. Magnetic nanofluid natural convection in porous media has been investigated by Sheikholeslami [3]. Sheikholeslami et al. [4] presented the heat transfer of Fe3O4-water nanofluid in a permeable medium with thermal radiation in the presence of constant heat flux. Sheikholeslami and Shehzad [5] investigated the influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect. Natural convection of nanofluid was analyzed in several articles [6
90].

16.2 EFFECT OF VARIABLE MAGNETIC FIELD ON NANOFLUID CONVECTIVE HEAT TRANSFER IN A POROUS CURVED ENCLOSURE 16.2.1 Problem Definition A curved cavity with hot left wall is considered. Boundary conditions are depicted in Fig. 16.1. A magnetic source exists near the left bottom corner. Magnetic source has been considered as shown in Fig. 16.2. Hx ; Hy ; H can be calculated as follow: h
i21 γ 2 Hy 5 b2y 1 ða2xÞ2 ða 2 xÞ; ð16:1Þ 2π h
i21 γ
2  y2b ; Hx 5 b2y 1 ða2xÞ2 ð16:2Þ 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð16:3Þ H 5 Hx 1 Hy

16.2.2 Governing Equation 2D laminar nanofluid flow and forced convective heat transfer is taken into account. The governing PDEs are: @v @u 1 5 0; @y @x

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00016-3

581

ð16:4Þ

© 2019 Elsevier Inc. All rights reserved.

582

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

(A)

(B)

1.4

Region of support

Control volume

1.2

Tc

1

g 0.8

C

q″

0.6

Fe3O4 –Water

gn Ma

B

0.4

etic rce

sou

S

0.2

porous medium

0

A

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

FIGURE 16.1 (A) Geometry and the boundary conditions (B) sample mesh.

(A)

(B)

g Ma

r ce ou

r ce ou

cs

cs

i net

i net

g Ma

19 17 15 14 12 10 9 8 7 6 5 4 3 2 1.7 1.45

9 7 6 5 4 3 2 1.2 1 –1 – 1.2 –2 –3 –4 –5 –6 –7 –9

(C)

g Ma i net cs r ce ou

– 0.6 – 1.18 –2 –3 –4 –5 –6 –7 –8 –9 – 10 – 11 – 12 – 14 – 17 – 19

FIGURE 16.2

Contours of the (A) magnetic strength field intensity component in x direction Hx; (C) magnetic field
field 
H; (B) magnetic
 intensity component in y direction Hy. (A) H x; y , (B) Hx x; y , (C) Hy x; y .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.2 EFFECT OF VARIABLE MAGNETIC FIELD ON NANOFLUID CONVECTIVE HEAT TRANSFER IN A POROUS CURVED ENCLOSURE

  2  21 μnf @u @u @ u @2 u @P 2 2 2 1 v 5 ρnf 2 μ0 σnf Hy u 1 σnf μ0 Hx Hy v 2 u u 1 2 μnf 2 @x @y @y2 @x @x K 0 1 0 1 2 2 μnf @v @v @ v @ v @P ρnf @ u 1 vA 51 μnf @ 2 1 2 A 2 1 μ20 Hy σnf Hx u 2 μ20 Hx σnf Hx v 2 v @x @y @x @y @y K

583



ð16:5Þ

ð16:6Þ

1 ðT 2 Tc Þβ nf gρnf 1 0 1 2 2

 @T @T @ T @ T 2 1 u A 5 σnf μ20 Hx v2Hy u 1 knf @ 2 1 2 A ρCp nf @v @y @x @x @y 0

8 0 12 0 12 0 12 9 < @u @v @u @v = 1 μnf 2@ A 1 2@ A 1 @ 1 A ; : @x @y @y @x ;

ð16:7Þ


 ρnf ; ρCp nf ; αnf ; β nf ; μnf ; knf , and σnf are calculated as


ρnf 5 ρf ð1 2 φÞ 1 ρs φ;

 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ; 

ð16:8Þ ð16:9Þ

knf  ; αnf 5
ρCp nf

ð16:10Þ

β nf 5 β f ð1 2 φÞ 1 β s φ:

ð16:11Þ

μnf 5

μf ð12φÞ2:5

:

 ks 2 2φðkf 2 ks Þ 1 2kf knf 5 kf ; ks 1 φðkf 2 ks Þ 1 2kf

 3ðσ1 2 1Þφ 11 ; σ1 5 σs =σf σnf 5 σf ðσ1 1 2Þ 2 ðσ1 2 1Þφ Dimensionless parameters are defined as:

  b; a
Hy ; Hx ; H p ; Hy ; Hx ; H 5 ðb; aÞ 5 ;P5
2 L H0 ρf αf =L
 x; y uL vL T 2 Tc ; ΔT 5 qvL=kf ; ðX; YÞ 5 U5 ;V5 ; Θ5 αf αf ΔT L

ð16:12Þ ð16:13Þ ð16:14Þ

ð16:15Þ

So equations change to: @V @U 1 5 0; @Y @X 2 30 1 2 2 μ =μ @U @U @ UA nf f 5 @@ U 1 V 5 Pr4 U 1 @X @Y @Y2 @X2 ρnf =ρf 2

3 2 3

 @P μ =μ σ =σ Pr 4 nf f 5 nf f5 2 Hy2 U 2 Hx Hy V 2 U; 2 Ha2 Pr4 @X Da ρnf =ρf ρnf =ρf

ð16:16Þ

ð16:17Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

584

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

0 12 3 2 2 μnf =μf @V @V @ V @ V A4 5 1U 5 Pr@ 2 1 V @Y @X @Y @X2 ρnf =ρf 2

3
 σ =σ nf f 5 H 2 V 2 Hx Hy U 2 Ha2 Pr4 x ρnf =ρf

ð16:18Þ

2 3 2 3 μnf =μf β nf @P Pr 4 5V; 1 RaPr4 5Θ 2 2 @Y Da ρnf =ρf βf 2 30 1 @Θ @Θ 4knf ðρCP Þf 5@@2 Θ @2 ΘA 1U 5 1 V @Y @X @Y2 @X2 kf ðρCP Þnf 0

1 ð ρC Þ   σ P f nf A V Hx 2U Hy 2 1 Ha2 Ec@ ðρCP Þnf σf

ð16:19Þ

3

2

μnf 8 0 12 0 12 0 12 9 6 μ 7 < = f 7 6 7 @@U A @@VA @@U @V A 16 6ðρCP Þnf 7Ec:2 @X 1 2 @Y 1 @Y 1 @X ; 5 4 ðρCP Þf and dimensionless parameters are

qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

ð16:20Þ

 h i Ec 5 μf αf = ðρCP Þf ΔT L2 ; Da 5 K= L2

The thermophysical properties of Fe3O4 and water are presented in Table 16.1. Pressure gradient source terms are discarded by the vorticity stream function.  ωL2 ψ @u @v @ψ @ψ 1 ; ðu; vÞ 5 ;2 Ω5 ;Ψ 5 ;ω52 ð16:21Þ αf @y @x @y @x αf According to Fig. 16.1, boundary conditions are

TABLE 16.1

on left wall

@Θ 5 1:0 @n

on all walls

Ψ 5 0:0

on right wall

Θ 5 0:0

on other walls

@Θ 5 0:0 @n

ð16:22Þ

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

dp ðnmÞ


σ ΩUmÞ21

Pure water

997.1

4179

0.613

-

0:05

Fe3 O4

5200

670

6

47

25,000

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.2 EFFECT OF VARIABLE MAGNETIC FIELD ON NANOFLUID CONVECTIVE HEAT TRANSFER IN A POROUS CURVED ENCLOSURE

Nulocal ; Nuave along curve wall are:

 knf 1 Nuloc 5 kf θ Nuave 5

585

ð16:23Þ

ðs 1 Nuloc ds S

ð16:24Þ

0

16.2.3 Effects of Active Parameters Influences of magnetic field on nanofluid free convection are simulated. Influences of Darcy number (Da), Hartmann number (Ha), Rayleigh number (Ra), and volume fraction of Fe3O4 (φ) are presented. Pr and Ec are 6.8 and 1025, respectively. The impact of Fe3O4 nanoparticles on hydrothermal treatment is depicted in Fig. 16.3. Temperature gradient enhances with the rise of volume fraction Fe3O4. The nanofluid motion augments due to enhancement of the solid movements.

-1

-2

Ha = 0

-5 -7

-9

0.06

0.08

0.1 0.14

0.04

-7

0.02

-4 -2 -1

-0.2 -0.6 -1.2

0 .0

Ha = 20

-1.6 0 .0

-2

2

4

0.06 0 .0

-2.2

8

0.12 0 .1

6

-2.2

-1 .8 -0

.2

FIGURE 16.3 Influence of nanofluid volume fraction on streamline (left) and isotherm (right) contours (nanofluid (φ 5 0:04)(—) and pure fluid (φ 5 0) (- - -)) when Ra 5 105 ; Da 5 100.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

586

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

– 0.001 – 0.002 – 0.003 – 0.004 – 0.005 – 0.006 – 0.007 – 0.008 – 0.009 – 0.01 – 0.011 – 0.012 – 0.013 – 0.014 – 0.015

0.32 0.26 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.01 0.005

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11 – 0.12 – 0.13 – 0.14 – 0.15

0.32 0.28 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.01 0.007 0.003

– 0.001 – 0.002 – 0.003 – 0.004 – 0.005 – 0.006 – 0.007 – 0.008 – 0.009 – 0.01 – 0.011 – 0.012 – 0.013 – 0.014 – 0.015 – 0.016 – 0.017 – 0.018 – 0.019

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Ha = 20

Da = 100

Ha = 20

– 0.005 – 0.01 – 0.015 – 0.02 – 0.025 – 0.03 – 0.035 – 0.04 – 0.045 – 0.05 – 0.055 – 0.06

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Ha = 0

Da = 0.01

Ha = 0

Figs. 16.4
16.6 illustrate the impacts of Da; Ha, and Ra on isotherms and streamlines. One clockwise vortex can be seen in streamline. At low Rayleigh number, isotherms are parallel to each other. As Ha increases the primary eddy stretches vertically and moves downward. Increasing the Darcy number makes the isotherms become

FIGURE 16.4 Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 103 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

Ha = 20

Da = 100

Ha = 0

Ha = 20

Da = 0.01

Ha = 0

16.2 EFFECT OF VARIABLE MAGNETIC FIELD ON NANOFLUID CONVECTIVE HEAT TRANSFER IN A POROUS CURVED ENCLOSURE

FIGURE 16.5

– 0.015 – 0.05 – 0.1 – 0.15 – 0.2 – 0.25 – 0.3 – 0.35 – 0.4 – 0.45 – 0.5 – 0.55 – 0.6 – 0.63

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11 – 0.12 – 0.13 – 0.14 – 0.15

0.32 0.28 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.036 0.02 0.01 0.006

– 0.1 – 0.2 – 0.3 – 0.4 – 0.5 – 0.6 – 0.7 – 0.8 – 0.9 –1 – 1.1 – 1.2 – 1.3 – 1.4 – 1.5

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11 – 0.12 – 0.13 – 0.14 – 0.15 – 0.16 – 0.17 – 0.18 – 0.19

587

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 104 .

denser and nanofluid flow is enhanced. Also jΨ max j increases with the increase of Da. Convection mode becomes stronger with the rise of Ra; Da and isotherms become more distorted. As the Lorentz force is enhanced, the distortion of isotherms reduces and jΨ max j reduces.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

588

Ha = 20

Da = 100

Ha = 0

Ha = 20

Da = 0.01

Ha = 0

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

– 0.5 –1 – 1.5 –2 – 2.5 –3 – 3.5 –4 – 4.5 –5 – 5.4

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

– 0.1 – 0.2 – 0.3 – 0.4 – 0.5 – 0.6 – 0.7 – 0.8 – 0.9 –1 – 1.1 – 1.2 – 1.3 – 1.4 – 1.5 – 1.6 – 1.7

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

– 0.5 –1 – 1.5 –2 – 2.5 –3 – 3.5 –4 – 4.5 –5 – 5.5 –6 – 6.5 –7 – 7.5 –8 – 8.5 –9

–0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 –1.6 –1.8 –2 –2.2

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02

FIGURE 16.6 Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

589

16.3 NANOFLUID NATURAL CONVECTION IN POROUS MEDIA IN THE PRESENCE OF A MAGNETIC SOURCE

Impacts of φ; Ha; Rd, and Ra on Nuave ; Nuloc are depicted in Figs. 16.7 and 16.8. The formula for Nuave is: Nuave 5 7:68 2 0:22Da 2 1:6logðRaÞ 1 0:96Ha 1 0:17Da logðRaÞ 2 0:1Da Ha 2 0:23logðRaÞHa
2 2 0:28Da2 1 0:25 logðRaÞ 2 0:093Ha2

ð16:25Þ

where Da 5 0:01Da; Ha 5 0:1Ha. Nusselt number rises with the increase of Darcy number due to the domination of convection. As the buoyancy force rises, the temperature gradient near the hot wall increases and in turn Nuave is enhanced with the increase of Rayleigh number. Augmenting the Lorentz forces leads to a reduction of Nuave due to domination of conduction mode. Adding Fe3O4 nanoparticles into water makes Nuave increase. Table 16.2 the impacts    of Ha;  Rd, and Ra on heat transfer improvement which is defined as:
depicts E 5 100  Nuave φ50:04 2 Nuave φ50 =Nuave φ50 . As thermal conductivity changes, this parameter changes. Therefore, the impact of adding nanoparticles is greater in the conduction mechanism. E is enhanced with the rise of Ha while it is reduced with the increase of Da; Ra.

16.3 NANOFLUID NATURAL CONVECTION IN POROUS MEDIA IN THE PRESENCE OF A MAGNETIC SOURCE 16.3.1 Problem Definition A curved cavity with a hot left wall is considered. Boundary conditions are depicted in Fig. 16.9. A magnetic source exists near the left bottom corner. Magnetic source has been considered as shown in Fig. 16.10. Hx ; Hy ; H can be calculated as follows: h
i21 γ 2 Hy 5 b2y 1 ða2xÞ2 ða 2 xÞ; ð16:26Þ 2π h
i21 γ
2  y2b ; Hx 5 b2y 1 ða2xÞ2 ð16:27Þ 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð16:28Þ H 5 H x 1 H y:

16.3.2 Governing Equation 2D laminar nanofluid flow and forced convective heat transfer are taken into account. The governing PDEs are: @v @u 1 5 0; @y @x    2  21 μnf @u @u @ u @2 u @P 2 2 2 1 v 5 ρnf 2 μ0 σnf Hy u 1 σnf μ0 Hx Hy v 2 u u ; 1 2 μnf 2 @x @y @y2 @x @x K 0 1 0 1 2 2 μnf @v @v @ v @ v @P ρnf @ u 1 vA 51 μnf @ 2 1 2 A 2 1 μ20 Hy σnf Hx u 2 μ20 Hx σnf Hx v 2 v @x @y @x @y @y K 1 ðT 2 Tc Þβ nf gρnf ; 1 0 1 2 2

 @T @T @ T @ T 2 1 u A 5 σnf μ20 Hx v2Hy u 1 knf @ 2 1 2 A ρCp nf @v @y @x @x @y (   2  2 ) 2 @v 1 μnf 2 @u 1 2 @v 1 @u ; @x @y @y 1 @x

ð16:29Þ ð16:30Þ

ð16:31Þ

0

ð16:32Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

590

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

Da = 0.01

Da = 100

16

16

Ha = 0

Ha = 0

Ha = 20

Ha = 20 12

Nuloc

Nuloc

Ra = 103

12

8

4

0

8

4

A

0

C

B

A

S 16

16

Ha = 0

Ha = 0

Ha = 20

Ha = 20 12

Nuloc

Nuloc

Ra = 104

12

8

4

0

8

4

A

0

C

B

A

S 16

Ha = 0

Ha = 0

Ha = 20

Ha = 20 12

Nuloc

Nuloc

12

Ra = 105

C

B

S

16

8

4

0

C

B

S

8

4

A

B

S

C

0

A

B

C

S

FIGURE 16.7 Effects of Ha; Ra, and Da on local Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.3 NANOFLUID NATURAL CONVECTION IN POROUS MEDIA IN THE PRESENCE OF A MAGNETIC SOURCE

FIGURE 16.8

591

Effects of Da; Ha, and Ra on average Nusselt number when φ 5 0:04.

TABLE 16.2 Effects of Da; Ha, and Ra on Heat Transfer Enhancement Ra

Da

Ha

E

3

0.01

0

19.5528

3

0.01

20

19.55272

5

0.01

0

16.3129

5

0.01

20

18.64209

3

100

0

19.55295

3

100

20

19.5528

5

100

0

15.36061

5

100

20

18.13567

10 10 10 10 10 10 10 10


 ρnf ; ρCp nf ; αnf ; β nf ; μnf ; knf , and σnf are calculated as ρnf 5 ρf ð1 2 φÞ 1 ρs φ;


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ð16:33Þ ð16:34Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

592

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

FIGURE 16.9 Geometry and the boundary conditions.

(A)

(B)

9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9

18 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

(C)

–1 –2 –3 –4 –5 –6 –7 –8 –9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17 – 18 – 19

FIGURE 16.10

Contours of the (A) magnetic field intensity component in x direction Hx; (C) magnetic field
field strength
 H; (B) magnetic
 intensity component in y direction Hy. (A) H x; y , (B) Hx x; y , (C) Hy x; y .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.3 NANOFLUID NATURAL CONVECTION IN POROUS MEDIA IN THE PRESENCE OF A MAGNETIC SOURCE

593

knf  ; αnf 5
ρCp nf

ð16:35Þ

β nf 5 β f ð1 2 φÞ 1 β s φ;

ð16:36Þ

μnf 5

μf ð12φÞ2:5

;

 ks 2 2φðkf 2 ks Þ 1 2kf knf 5 kf ; ks 1 φðkf 2 ks Þ 1 2kf

 3ðσ1 2 1Þφ 11 ; σ1 5 σs =σf : σnf 5 σf ðσ1 1 2Þ 2 ðσ1 2 1Þφ Dimensionless parameters are defined as:

  b; a
Hy ; Hx ; H p ; Hy ; Hx ; H 5 ðb; aÞ 5 ;P5
2 L H0 ρf αf =L
 x; y uL vL T 2 Tc ; ðX; YÞ 5 U5 ;V5 ; Θ5 : αf αf ð Th 2 Tc Þ L

ð16:37Þ ð16:38Þ ð16:39Þ

ð16:40Þ

So equations change to: @V @U 1 5 0; @Y @X 2 30 1 2 2 μ =μ @U @U @ UA nf f 5 @@ U 1 V 5 Pr4 U 1 @X @Y @Y2 @X2 ρnf =ρf 2 3 2 3

 @P μ =μ σ =σ Pr 4 nf f 5 nf f5 2 2 Ha2 Pr4 Hy2 U 2 Hx Hy V 2 U; @X Da ρnf =ρf ρnf =ρf 0 12 3 2 2 μnf =μf @V @V @ V @ V A4 5 1U 5 Pr@ 2 1 V @Y @X @Y @X2 ρnf =ρf 2 3
 σ =σ nf f 5 H 2 V 2 Hx Hy U 2 Ha2 Pr4 x ρnf =ρf 2 3 2 3 μnf =μf β nf @P Pr 4 5V; 2 1 RaPr4 5Θ 2 @Y Da ρnf =ρf βf 2 30 1 @Θ @Θ 4knf ðρCP Þf 5@@2 Θ @2 ΘA 1U 5 V 1 @Y @X @Y2 @X2 kf ðρCP Þnf 0 1 ð ρC Þ 2 σ Pf nf A V Hx 2U Hy 1 Ha2 Ec@ ðρCP Þnf σf 2 3 μnf 8 0 12 0 12 0 12 9 6 μ 7 < = f 6 7 @U @V @U @V A 7Ec 2@ A 1 2@ A 1 @ 1 16 6ðρCP Þnf 7 : @X @Y @Y @X ; 4 5 ðρCP Þf

ð16:41Þ

ð16:42Þ

ð16:43Þ

ð16:44Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

594

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

and dimensionless parameters are

qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

 h i Ec 5 μf αf = ðρCP Þf ΔT L2 ; Da 5 K= L2 :

ð16:45Þ

The thermophysical properties of Fe3O4 and water are presented in Table 6.11. Pressure gradient source terms are discarded by the vorticity stream function.  ωL2 ψ @u @v @ψ @ψ 1 ; ðu; vÞ 5 ;2 Ω5 ;Ψ 5 ;ω52 : ð16:46Þ αf @y @x @y @x αf According to Fig. 16.9, boundary conditions are: on left wall

θ 5 1:0

on all walls

Ψ 5 0:0

on other walls

θ 5 0:0

Nulocal ; Nuave along the curved wall are: Nuloc 5

Nuave 5

 knf @Θ ; kf @r

1 0:5π

ð16:47Þ

ð16:48Þ

0:5π ð

Nuloc ðζ Þ dζ;

ð16:49Þ

0

16.3.3 Effects of Active Parameters Fe3O4-water ferrofluid convective heat transfer in permeable medium is investigated. Impacts of Darcy number (Da), Rayleigh number (Ra), Hartmann number (Ha), and volume fraction of Fe3O4 (φ) are presented. Ec and Pr are 1025 and 6.8, respectively. Fig. 16.11 depicts the impact of Fe3O4 nanoparticles on hydrothermal characteristic. The nanofluid motion is enhanced due to the increase in solid particles interaction. The thermal boundary layer thickness increases with the rise of the volume fraction of Fe3O4. Impacts of Da; Ha, and Ra on isotherms and streamlines are illustrated in Figs. 16.12
16.14. In all conditions, one main clockwise vortex is generated in the cavity. At low Darcy and Rayleigh number the main heat transfer mechanism is conduction. Isotherms are parallel together in this mode. As Ha enhances the main vortex moves upward. As the Rayleigh number increases, the convective heat transfer becomes stronger. So isotherms become more disturbed and jΨ max j increases. As Darcy number is enhanced the nanofluid flow becomes easier. The impact of Da on hydrothermal behavior is similar to that of Ra. Therefore the temperature boundary layer thickness near the curved wall is reduced and in this way the temperature gradient is enhanced. As Lorentz forces increase, jΨ max j is reduced and the distribution of isotherms decreases. Figs. 16.15 and 16.16 demonstrate the impact of Da; Ha, and Ra on Nuave ; Nuloc . The correlation for Nuave corresponding to important parameters is: Nuave 5 5:96 2 0:35Da 2 2:9logðRaÞ 1 0:77Ha 1 0:23Da logðRaÞ 2 0:05Da Ha 2 0:17logðRaÞHa
2 2 0:22Da2 1 0:5 logðRaÞ 2 0:06Ha2 :

ð16:50Þ

where Da 5 0:01Da; Ha 5 0:1Ha. Nuave is enhanced with the increase of Da due to an increase in the temperature gradient near the curved wall. A similar trend is seen for buoyancy forces, so Nuave increases with the rise of Rayleigh number. As Lorentz forces increase, Nuave is reduced due to the domination of the conduction mode. Adding Fe3O4 nanoparticles into water makes Nuave be enhanced.  16.3 depicts theinfluence  of Da; Ha, and Ra on heat transfer augmentation, which
Table is defined as: E 5 100  Nuave φ50:04 2 Nuave φ50 =Nuave φ50 . E is relevant to the variation of thermal conductivity and k has more changes when the temperature boundary layer thickness is larger. So, heat transfer augmentation is enhanced with the rise of the Hartmann number while it decreases with the enhancement of Ra; Da.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

0.9

-4

0 .7

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

5 0.45

-1

1

-8

0 .5

595

-1

4

0.4 -15

Ha = 0

0.35 0.3 -15

0.25

-13

0.2 -10

0.15 0.1

-4

0.05

-1

2

-2

-1.

-3.2

0.85

0 .4

-4

0.5

0.6 -4.4

0.25

-4

0.2

-3.8

0.15

-3 .4

Ha = 40

0.3

-3.2 - 2 .8

-2.4

0.1

-0.8 -0

.4

0.05

-0.2

FIGURE 16.11 Impact of nanofluid volume fraction on streamline (left) and isotherm (right) contours (nanofluid (φ 5 0:04) (—) and pure fluid (φ 5 0) (- - -)) when Ra 5 105 ; Da 5 100.

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION 16.4.1 Problem Definition Sample element, boundary condition, and geometry are depicted in Fig. 16.17. Fig. 16.18 shows the external magnetic source. H; Hx ; Hy are: i21 2 γ h
Hy 5 ða 2 xÞ b2y 1 ða2xÞ2 ; ð16:51Þ 2π i21
 γ h
2 Hx 5 y 2 b b2y 1 ða2xÞ2 ; ð16:52Þ 2π

2  2 0:5 H 5 H y 1H x : ð16:53Þ

16.4.2 Governing Equation Two-dimensional steady convective non-Darcy flow of nanofluid is considered in the presence of a constant magnetic field. The PDEs are:

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

596

Ha = 40

Da = 100

Ha = 0

Ha = 40

Da = 0.01

Ha = 0

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

FIGURE 16.12

– 0.02 – 0.04 – 0.06 – 0.08 – 0.1 – 0.12 – 0.14 – 0.16 – 0.18 – 0.2 – 0.22 – 0.24 – 0.26 – 0.28 – 0.3

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.002 – 0.004 – 0.006 – 0.008 – 0.01 – 0.012 – 0.014 – 0.016 – 0.018 – 0.02 – 0.022 – 0.024 – – 0.026 – 0.028 – 0.03 – 0.032 – 0.034 – 0.036 – 0.038 – 0.04 – 0.042 – 0.044 – 0.046 – 0.048

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.05 – 0.1 – 0.15 – 0.2 – 0.25 – 0.3 – 0.35 – 0.4 – 0.45 – 0.5 – 0.55 – 0.6

– 0.002 – 0.004 – 0.008 – 0.01 – 0.012 – 0.014 – 0.016 – 0.018 – 0.022 – 0.026 – 0.032 – 0.034 – 0.036 – 0.04 – 0.042 – 0.044 – 0.046 – 0.05 – 0.052

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 103 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

597

Ha = 40

Da = 100

Ha = 0

Ha = 40

Da = 0.01

Ha = 0

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

FIGURE 16.13

– 0.2 – 0.4 – 0.6 – 0.8 –1 – 1.2 – 1.4 – 1.6 – 1.8 –2 – 2.2 – 2.4 – 2.6

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.02 – 0.06 – 0.1 – 0.14 – 0.16 – 0.18 – 0.22 – 0.26 – 0.3 – 0.34 – 0.36 – 0.38 – 0.4 – 0.42 – 0.44 – 0.46

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.2 – 0.4 – 0.6 – 0.8 –1 – 1.2 – 1.4 – 1.8 – 2.2 – 2.6 –3 – 3.4 – 3.6 –4 – 4.4 – 4.6

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.02 – 0.06 – 0.1 – 0.12 – 0.16 – 0.18 – 0.22 – 0.24 – 0.28 – 0.32 – 0.36 – 0.38 – 0.4 – 0.42 – 0.44 – 0.46 – 0.48 – 0.5 – 0.52

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 104 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

598

Ha = 40

Da = 100

Ha = 0

Ha = 40

Da = 0.01

Ha = 0

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

FIGURE 16.14

– 0.5 –1 – 1.5 – 2.5 – 3.5 – 4.5 – 5.5 – 6.5 – 7.5 – 8.5 – 9.5 – 10.5 – 11.5

–0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 –1.6 –1.8 –2 –2.2 –2.4 –2.6 –2.8 –3 –3.2 –3.4 –3.6 –3.8 –4

–1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15

–0.2 –0.6 –0.8 –1.2 –1.4 –2 –2.2 –2.4 –2.6 –2.8 –3 –3.2 –3.4 –3.6 –3.8 –4 –4.2 –4.4

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Influence of Da; Ha on streamline (left) and isotherm (right) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

599

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

Da = 0.01

Da = 100 30

30

Ha = 0

Ha = 0 25

20

Nuloc

Nuloc

20

Ra = 103

Ha = 40

25

Ha = 40

15

15

10

10

5

5

0

0 0°

30°

ζ

0°

90°

60°

30

30°

ζ

60°

30

Ha = 0

Ha = 0

Ha = 40

25

20

Nuloc

Nuloc

Ra = 104

Ha = 40

25

20

15

15

10

10

5

5

0

0

0°

30°

ζ

60°

90°

0°

30°

ζ

60°

Ha = 0

Ha = 0

20

Nuloc

20

Nuloc

Ha = 40

25

Ha = 40

25

Ra = 105

90°

30

30

15

15

10

10

5

5

0

0 0°

FIGURE 16.15

90°

30°

ζ

60°

90°

0°

30°

ζ

60°

90°

Influence of Ha; Ra, and Da on local Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

600

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

0

@u @v 52 ; @x @y

1 2 2 @ u @ u @P 2 @ 2 μ20 σnf H y u 1 σnf μ20 Hx H y v 1 2 Aμnf 2 2 @y @x @x 0 1

 @u μnf @u vA ; u 5 ρnf @ u 1 2 @x @y K 1 2 2 μnf @ v @ v @P μnf @ 2 1 2 A 2 1 μ20 H y σnf H x u 2 μ20 H x σnf H x v 2 v @x @y @y K 0 1 @v @v 1 ðT 2 Tc Þβ nf gρnf 5 ρnf @ u 1 vA; @x @y

ð16:54Þ

ð16:55Þ

0

FIGURE 16.16

ð16:56Þ

Effects of Da; Ha, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

FIGURE 16.16

601

(Continued).

TABLE 16.3 Effects of Da; Ha, and Ra on Heat Transfer Enhancement Ra

Da

Ha

E

3

0.01

0

9.113388

3

0.01

40

9.180014

5

0.01

0

9.396332

5

0.01

40

8.235231

3

100

0

8.706959

3

100

40

9.170092

5

100

0

9.150151

5

100

40

8.044214

10 10 10 10 10 10 10 10

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

602

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

FIGURE 16.17

Geometry and the boundary conditions.

(A)

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

(B)

9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9

–1 –2 –3 –4 –5 –6 –7 –8 –9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17 – 18 – 19

(C)

FIGURE 16.18

Contours of the (A) magnetic field intensity component in x direction Hx; (C) magnetic field
field strength
 H; (B) magnetic
 intensity component in y direction Hy. (A) H x; y , (B) Hx x; y , (C) Hy x; y .

0 1 1 2 2
 @ T @ T @q @T @T r 1 u A ρCp nf ; 5 @v knf @ 2 1 2 A 2 @y @x @y @x @y 2 3 4 4qr 5 2 4σe @T ; T4 D4T3 T 2 3T 4 5: c c 3β R @y 0


 ρCp nf ; ρnf ; ðρβ Þnf ; σnf are:




ρCp

 nf



 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð16:57Þ

ð16:58Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

ρnf 5 ρf ð1 2 φÞ 1 ρs φ

ð16:59Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ;  3φðσ1 2 1Þ 11 ; σ1 5 σs =σf : σnf 5 σf ð1 2 σ1Þφ 1 ð2 1 σ1Þ

ð16:60Þ



μnf is obtained as follows:

 2 μnf 5 0:035μ20 H 1 3:1μ0 H 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T knf can be obtained as:

603



 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

ð16:61Þ

ð16:62Þ

ð16:63Þ

The properties of nanofluid are depicted in Table 16.1. Different values of shape factors for various shapes of nanoparticles are illustrated in Table 16.4. Vorticity and stream function should be used to eliminate pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5 u: @y @x @x @y

Dimensionless parameters are defined as:


 Hy ; Hx ; H b; a ; Hy ; Hx ; H 5 ; ðb; aÞ 5 L H0
 x; y uL vL T 2 Tc U5 ; ΔT 5 qvL=kf ; ;V5 ; ðX; YÞ 5 ;θ5 αnf αnf ΔT L Ψ5

ð16:64Þ

ð16:65Þ

ψ ωL2 ;Ω5 : αnf αnf

So equations change to: @2 Ψ @2 Ψ 1 Ω 1 5 0; @Y2 @X2

TABLE 16.4

ð16:66Þ

The Values of the Shape Factor of Different Shapes of Nanoparticles

Spherical

3

Platelet

5.7

Cylinder

4.8

Brick

3.7

m

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

604

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

0 1 @Ω @Ω A5 A2 @@2 Ω @2 Ω A U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A A @U @V @U @V 6 2 @ Hy Hx 2 Hx 2 1 H2 2 Hy Hx A 1 PrHa2 @X @Y y @Y A1 A4 @X 1 Pr Ra

ð16:67Þ

A3 A22 @θ Pr A5 A2 2 Ω; Da A1 A4 A1 A24 @X

 2 @θ @θ @ θ @2 θ 4 1 @2 θ U1V 5 1 Rd : 1 @X @Y @X2 @Y2 3 A4 @Y2

ð16:68Þ

and dimensionless parameters are

qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

 h i Da 5 K= L2 ; Ec 5 μf αf = ðρCP Þf ΔT L2 ;
 ρCp nf ρnf ðρβ Þnf  ; A3 5 ; A2 5
; A1 5 ρf ðρβ Þf ρCp f

ð16:69Þ

μnf knf σnf ; A5 5 ; A6 5 ; kf μf σf
 Rd 5 4σe Tc3 = β R kf :

A4 5

and boundary conditions are: on inner wall on outer wall on other walls

@θ 5 1:0 @n θ 5 0:0 @θ 50 @n

ð16:70Þ

Ψ 5 0:0

on all walls

Nuloc ; Nuave over the hot wall can be calculated as: !   21 knf knf 4Rd 1 ; Nuloc 5 11 3 θ kf kf ðS 1 Nuave 5 Nuloc ds: S

ð16:71Þ

ð16:72Þ

0

16.4.3 Effects of Active Parameters The impact of nonuniform Lorentz forces on Fe3O4-water nanofluid treatment in a permeable medium is modeled. The shape effect of nanoparticles on knf is taken into consideration. μnf of Fe3O4-water are estimated according to previous experimental data. CVFEM is utilized to find the influences of radiation parameter ðRd 5 0 to 0:8Þ, Darcy number (Da 5 0:01 to 100), Rayleigh number (Ra 5 103 ; 104 ; 105 ), volume fraction of Fe3O4 (φ 5 0% to 4%), shape of nanoparticle, and Hartmann number (Ha 5 0 to 10). Impacts of shape of the nanoparticles on Nuave are reported in Table 16.5. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. Therefore, Platelet nanoparticle has been selected for further investigation. Fig. 16.19 demonstrates the effect of adding nanoparticles in the water on velocity and temperature contours. The temperature gradient is reduced

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.4 HEAT TRANSFER OF FE3O4-WATER NANOFLUID IN A PERMEABLE MEDIUM WITH THERMAL RADIATION

605

TABLE 16.5 Effect of Shape of Nanoparticles on Nusselt Number When Da 5 100; Ra 5 105 ; Rd 5 0:8; φ 5 0:04 Ha 0

10

Spherical

8.754093

6.893915

Brick

8.802073

6.930307

Cylinder

8.87772

6.987786

Platelet

8.940132

7.034996

5

5

-5 -10

15

-10

5

Ha = 0

-5

0.25 0.15

0.15

0.1 0.05

0.1

0.05

FIGURE 16.19 Impact of nanofluid volume fraction on streamlines (top) and isotherms (bottom) contours (nanofluid (φ 5 0:04)(—) and pure fluid (φ 5 0) (2U 2 )) when Ra 5 105 ; Da 5 100; Rd 5 0:8.

with the increase of φ. jΨ max j increases with the addition of nanoparticles because of the increase in the solid movements. Fig. 16.20 shows the impact of the radiation parameter on velocity and temperature contours. The thermal boundary layer thickness increases with the increase of Rd. As Hartmann number increases, the impact of the radiation parameter on streamlines becomes not significant. Figs. 16.21
16.23 depict the effects of Da; Ra; Ha on isotherms and streamlines. One eddy can be seen in streamlines. By applying a magnetic field, the main eddy moves downward. The distortion of isotherms becomes less than before. As Ra increases, a thermal plume is generated near the vertical symmetric line. Augmenting Ha shifts the thermal plume to the left and decreases jΨ max j. As the Darcy number increases, the convective mode becomes stronger and the primary eddy converts into two eddies. Fig. 16.24 demonstrates the influence of significant parameters on Nuave . The correlation for Nuave is: Nuave 5 22:3 2 3:13Rd 2 10:5ogðRaÞ 2 0:43Da 1 1:03Ha 1 1:88RdlogðRaÞ 1 0:36Rd Da 2 0:54Rd Ha 1 0:19Da logðRaÞ 2 0:28Ha logðRaÞ 2 0:52Da Ha
2 2 2:5Rd2 1 1:37 logðRaÞ 1 0:29ðDa Þ2 2 0:17ðHa Þ2

ð16:73Þ

where Da 5 0:01Da; Ha 5 0:1Ha. Heat transfer rate augments with the rise of permeability of porous media. A similar treatment is observed for buoyancy forces. The temperature gradient is reduced with the increase of the Hartmann number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

606

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

5

-5 -11

Ha = 0

11

0.2 0.15 0.1

0 .1 5 0.1

0.05

0 .0

5

Impact of radiation parameter on streamlines (top) and isotherms (bottom) contours (Rd 5 0:8(—), Rd 5 0 (- - -)) when Ra 5 105 ; Da 5 100; φ 5 0:04.

FIGURE 16.20

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR IN A PERMEABLE CAVITY CONSIDERING SHAPE EFFECT 16.5.1 Problem Definition Fig. 16.25 shows the sample element and boundary conditions of current geometry. As shown in Fig. 16.26, the external magnetic source affects the nanofluid flow. H; Hx ; Hy are: i21 2 γ h
Hy 5 ða 2 xÞ b2y 1 ða2xÞ2 ; ð16:74Þ 2π i21
 γ h
2 Hx 5 y 2 b b2y 1 ða2xÞ2 ; ð16:75Þ 2π

2  2 0:5 H 5 H y 1H x : ð16:76Þ

16.5.2 Governing Equation Convective nanofluid flow in existence of nonuniform magnetic field is considered using a non-Darcy model. The PDEs are:

0

@u @v 52 ; @x @y

1 2 2 @ u @ u @P 2 @ 2 μ20 σnf H y u 1 σnf μ20 Hx H y v 1 2 Aμnf 2 @y2 @x @x 0 1

 @u μnf @u vA ; 2 u 5 ρnf @ u 1 @x @y K

ð16:77Þ

ð16:78Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

FIGURE 16.21

607

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 103 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

608

FIGURE 16.22

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 104 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

FIGURE 16.23

609

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

610

FIGURE 16.24

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

Effects of Da; Ha; Rd, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

FIGURE 16.25 (A) Geometry and the boundary conditions with; (B) a sample triangular element and its corresponding control volume.

(A)

FIGURE 16.26 Contours of the (A) magnetic field strength H; (B) magnetic field intensity component in x direction Hx; (C) magnetic field
inten sity
component in y direction Hy. (A) H x; y , (B) 
 Hx x; y , (C) Hy x; y .

(B)

9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

(C)

–1 –2 –3 –4 –5 –6 –7 –8 –9 – 10 – 11 – 12 – 13 – 14 – 15 – 16 – 17 – 18 – 19

612

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

0

1 2 2 μnf @ v @ v @P 1 μ20 H y σnf H x u 2 μ20 H x σnf H x v 2 v μnf @ 2 1 2 A 2 @x @y @y K 0 1 @v @v 1 ðT 2 Tc Þβ nf gρnf 5 ρnf @ u 1 vA; @x @y 0 1 1 2 2
 @ T @ T @q @T @T r knf @ 2 1 2 A 2 1 u A ρCp nf ; 5 @v @y @x @y @x @y 2 3 4 4σ @T e 4qr 5 2 ; T4 D4Tc3 T 2 3Tc4 5: 3β R @y

ð16:79Þ

0

ð16:80Þ


 ðρβ Þnf ; ρCp nf , ρnf , and σnf are calculated as:


ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ; 

 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  3φðσ1 2 1Þ 11 ; σnf 5 σf ð1 2 σ1Þφ 1 ð2 1 σ1Þ

σ1 5 σs =σf :

 2 μnf 5 0:035μ20 H 1 3:1μ0 H 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

knf can be obtained as:

ð16:82Þ ð16:83Þ



μnf is calculated as:

ð16:81Þ



 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

ð16:84Þ

ð16:85Þ

ð16:86Þ

Tables 16.1 and 16.4 show the properties of nanofluid and shape factor values. Vorticity and stream function can be employed to eliminate pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5 u: @y @x @x @y

Dimensionless parameters are defined as:


 Hy ; Hx ; H b; a ; Hy ; Hx ; H 5 ; ðb; aÞ 5 L H0
 x; y uL vL T 2 Tc U5 ; ΔT 5 qvL=kf ; ;V5 ; ðX; YÞ 5 ;θ5 αnf αnf ΔT L Ψ5

ð16:87Þ

ð16:88Þ

ψ ωL2 ;Ω5 : αnf αnf

So equations change to: @2 Ψ @2 Ψ 1Ω1 5 0; 2 @Y @X2

ð16:89Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

0 1 @Ω @Ω A5 A2 @@2 Ω @2 Ω A U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A A @U @V @U @V 6 2 @ Hy Hx 2 Hx 2 1 H2 2 Hy Hx A 1 PrHa2 @X @Y y @Y A1 A4 @X 1 Pr Ra

613

ð16:90Þ

A3 A22 @θ Pr A5 A2 2 ; Da A1 A4 A1 A24 @X

 2 @θ @θ @ θ @2 θ 4 1 @2 θ U1V 5 1 Rd : 1 @X @Y @X2 @Y2 3 A4 @Y2

ð16:91Þ

and dimensionless parameters are:

qffiffiffiffiffiffiffiffiffiffiffiffi
 Raf 5 gβ f L3 ΔT= αf υf ; Prf 5 υf =αf ; Ha 5 Lμ0 H0 σf =μf ;

 h i Da 5 K= L2 ; Ec 5 μf αf = ðρCP Þf ΔT L2 ;
 ρCp nf ρnf ðρβ Þnf  ; A3 5 ; A2 5
; A1 5 ρf ðρβ Þf ρCp f

ð16:92Þ

μnf knf σnf ; A5 5 ; A6 5 ; kf μf σf
 Rd 5 4σe Tc3 = β R kf

A4 5

and boundary conditions are: on inner wall on outer wall on other walls

@θ 5 1:0 @n θ 5 0:0 @θ 50 @n Ψ 5 0:0

on all walls

Nuloc ; Nuave over the hot wall can be calculated as: !   21 knf knf 4Rd 1 ; Nuloc 5 11 3 θ kf kf Nuave 5

1 S

ð16:93Þ

ðS Nuloc ds:

ð16:94Þ ð16:95Þ

0

16.5.3 Effects of Active Parameters The effect of an external magnetic source on Fe3O4-water hydrothermal treatment in a permeable enclosure is demonstrated. Magnetic field-dependent viscosity of nanofluid is considered. The shape effect of nanoparticles on knf is taken into consideration. CVFEM is employed to find the influences of Darcy number (Da 5 0:01 to 100), radiation parameter ðRd 5 0 to 0:8Þ, Rayleigh number (Ra 5 103 ; 104 ; 105 ), volume fraction of Fe3O4 (φ 5 0% to 4%), shape factor, and Hartmann number (Ha 5 0 to 10). The effect of shape factor on Nuave is discussed in Table 16.6. The highest Nuave is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet shape was selected for further investigation.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

614

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

TABLE 16.6 Effect of Shape of Nanoparticles on Nusselt Number When Da 5 100; Ra 5 105 ; Rd 5 0:8; φ 5 0:04 Ha 0

10

Spherical

7.072375

5.865269

Brick

7.120273

5.901963

Cylinder

7.196075

5.960649

Platelet

7.258503

6.009695

FIGURE 16.27 Impact of nanofluid volume fraction on streamline (right) and isotherm (left) contours (nanofluid (φ 5 0:04)(—) and pure fluid (φ 5 0) (2U 2 )) when Ra 5 105 ; Da 5 100; Rd 5 0:8; Ha 5 0.

-2

-8 -14 0.15 -18

0.2

0.1

0.25

0.05

FIGURE 16.28

Impact of radiation parameter on streamline (right) and isotherm (left) contours (Rd 5 0:8(—), Rd 5 0 (- - -)) when Ra 5 105 ; Da 5 100; φ 5 0:04; Ha 5 0.

-4

-1 0

0.2

-18

0.15

-18 0.1 0.25 0. 35

0.05

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

615

Fig. 16.27 illustrates the influence of adding nanoparticles in the base fluid on velocity and temperature contours. The temperature gradient is reduced with the increase of φ. jΨ max j increases with the addition of nanoparticles into the base fluid. Fig. 16.28 shows the impact of the radiation parameter on isotherms and streamlines.

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.15 0.05

0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha = 10

Da = 100

Ha = 0

Ha = 10

Da = 0.01

Ha = 0

FIGURE 16.29 Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 103 .

0.65 0.6 0.5 0.4 0.3 0.25 0.2 0.15 0.1 0.05

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1

– 0.005 – 0.01 – 0.015 – 0.02 – 0.025 – 0.03 – 0.035 – 0.04 – 0.045 – 0.05 – 0.055 – 0.06

– 0.05 – 0.1 – 0.15 – 0.2 – 0.25 – 0.3 – 0.35 – 0.4

– 0.01 – 0.02 – 0.03 – 0.04 – 0.05 – 0.06 – 0.07 – 0.08 – 0.09 – 0.1 – 0.11 – 0.12

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

616

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

Increasing the radiation parameter leads to an increase in the thermal boundary layer thickness. In the presence of Lorentz forces, the impact of radiation parameter on flow style becomes not sensible. Figs. 16.29
16.31 depicts the influences of Da; Ra; Ha; Rd, on isotherms and streamlines. In conduction mode (low Reynolds number), FIGURE 16.30

Ha = 10

Da = 100

Ha = 0

Ha = 10

Da = 0.01

Ha = 0

Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 104 .

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.1 – 0.2 – 0.3 – 0.4 – 0.5 – 0.6 – 0.7 – 0.8 – 0.9 –1 – 1.1

– 0.05 – 0.1 – 0.15 – 0.2 – 0.25 – 0.3 – 0.35 – 0.4 – 0.45 – 0.5 – 0.55 – 0.6 – 0.65

– 0.5 –1 – 1.5 –2 – 2.5 –3 – 3.5 –4 – 4.5 –5 – 5.5

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –1 –1.1 –1.2 –1.3 –1.4 –1.5

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

617

streamlines show one rotating eddy. In the presence of a magnetic field, the main eddy moves downward. The distortion of the isotherms becomes less than before. As buoyancy force is enhanced, a thermal plume is generated. As the Darcy number is enhanced, the convective mode becomes stronger and the rate of heat transfer is enhanced.

Ha = 10

Da = 100

Ha = 0

Ha = 10

Da = 0.01

Ha = 0

FIGURE 16.31 Influence of Da; Ha on streamline (right) and isotherm (left) contours when φ 5 0:04; Ra 5 105 .

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–1 –2 –3 –4 –5 –6 –7 –8 –9 –10

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

– 0.5 –1 – 1.5 –2 – 2.5 –3 – 3.5 –4 – 4.5 –5 – 5.5 –6 – 6.5 –7 – 7.5 –8

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–2 –4 –6 –8 –10 –12 –14 –16 –18

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

618

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

The effect of significant parameters on Nuave is demonstrated in Fig. 16.32. The formula for Nuave is: Nuave 5 15:28 2 1:12Rd 2 6:6logðRaÞ 2 0:18Da 1 0:47Ha 1 1:07RdlogðRaÞ 1 0:32Rd Da 2 0:44Rd Ha 1 0:08Da logðRaÞ 2 0:11Ha logðRaÞ 2 0:3Da Ha
2 2 0:9Rd2 1 0:85 logðRaÞ 1 0:15ðDa Þ2 2 0:07ðHa Þ2

FIGURE 16.32

ð16:96Þ

Effects of Da; Ha, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

16.5 EFFECT OF EXTERNAL MAGNETIC SOURCE ON FE3O4-H2O NANOFLUID BEHAVIOR

FIGURE 16.32

619

(Continued).

where Da 5 0:01Da; Ha 5 0:1Ha. Nusselt number is enhanced with the increase of buoyancy forces and the permeability of porous media. The temperature gradient reduces with the increase of the Hartmann number. Also, the Nusselt number is enhanced with the increase of radiation number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

620

16. NONUNIFORM MAGNETIC FIELD EFFECT ON NANOFLUID CONVECTIVE FLOW IN A POROUS CAVITY

References [1] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [2] M. Sheikholeslami, CVFEM for magnetic nanofluid convective heat transfer in a porous curved enclosure, Eur. Phys. J. Plus 131 (2016) 413. Available from: https://doi.org/10.1140/epjp/i2016-16413-y. [3] M. Sheikholeslami, Numerical simulation of magnetic nanofluid natural convection in porous media, Phys. Lett. A 381 (2017) 494
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1200. Available from: https:// doi.org/10.1080/10407782.2015.1125709. [30] M. Sheikholeslami, A.J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Numer. Heat Transfer, Part A 69 (7) (2016) 781
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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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248. Available from: https://doi.org/10.1140/epjp/i2014-14248-2. [41] M. Sheikholeslami, Numerical simulation for external magnetic field influence on Fe3O4-water nanofluid forced convection, Engineering Computations, In press. [42] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cavity by means of Non- Darcy model, Eng. Comput. 34 (8) (2017) 2651
2667. Available from: https://doi.org/10.1108/EC-01-2017-0008. [43] M. Sheikholeslami, R. Ellahi, Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid, Int. J. Heat Mass Transfer 89 (2015) 799
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water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
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1883. [64] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [65] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25.

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[66] M. Sheikholeslami, M. Barzegar Gerdroodbary, S. Valiallah Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field, J. Magn. Magn. Mater. 460 (2018) 302
311. [67] M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry, J. Mol. Liq. 213 (2016) 153
161. [68] M. Sheikholeslami, Numerical simulation for solidification in a LHTESS by means of nano-enhanced PCM, J. Taiwan Inst. Chem. Eng. 86 (2018) 25
41. [69] M. Sheikholeslami, Numerical modeling of Nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424
438. [70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
431. [71] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transfer 122 (2018) 1264
1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid: an experimental procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model, J. Mol. Liq. 254 (2018) 446
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1212. [80] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys 56 (2018) 270
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660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Des. 120 (2017) 382
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714.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

17 Thermal Radiation Influence on Nanofluid Flow in a Porous Medium in the Presence of Coulomb Forces Using CVFEM 17.1 INTRODUCTION One of the effective active techniques for heat transfer augmentation is Electrohydrodynamics. Nanofluids are widely used in coolants for computers and nuclear reactors, cancer therapy, safer surgery by cooling, lubricants, heat exchangers, microchannel heat sinks; cooling of a new class of super powerful and small computers, and other electronic devices for use in military systems; vehicle cooling, and transformer cooling; in designing the waste heat removal equipment; major process industries including materials and chemicals, oil and gas, food and drink, paper and printing etc. The thermal radiation effects are of vital importance at high absolute temperature due to basic differences between radiation and convection and conduction energy-exchange mechanisms. For space applications, some devices are designed to operate at high temperature levels in order to obtain high thermal efficiency. That is why the radiation effects are significant while determining thermal effects in the processes with high temperatures. Akber et al. [1] investigated the effect of thermal radiation on MHD convective flow of nanofluid past a shrinking surface. Natural convection and radiation heat transfer has been investigated [2
90].

17.2 COMBINED NATURAL CONVECTION AND RADIATION HEAT TRANSFER OF NANOFLUID UNDER THE IMPACT OF ELECTRIC FIELD IN A POROUS CAVITY 17.2.1 Problem Definition Fig. 17.1 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for the bottom wall. The influence of Darcy and Reynolds numbers on the contour

T =T0, T =T0, ϕ = ϕ0,

∂q = 0, u = 0, v = 0 ∂y

Nanofluid y O

FIGURE 17.1

∂ϕ ∂q T =T0, = 0, = 0, u = 0, v = 0 ∂n ∂n γ

x

∂ϕ ∂q = 0, = 0, u = 0, v = 0 ∂n ∂n

g

Porous media

T =T1, ϕ = ϕ1, q = q0, u = 0, v = 0

Geometry and the boundary conditions.

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00017-5

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© 2019 Elsevier Inc. All rights reserved.

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17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

of q is demonstrated in Fig. 17.2. The effect of Re on q is less sensible than Da. As the Darcy number augments, the distortion of the isoelectric density lines become greater.

17.2.2 Governing Equation The definition of the electric field is: -

E 5 2 rϕ

ð17:1Þ

-

ð17:2Þ

q 5 r:ε E -

-

-

J 5 q V 2 Drq 1 σ E -

r: J 1

Ra = 500

Ra = 50

Da = 0.01

ð17:3Þ

@q 50 @t

ð17:4Þ

Da = 100

0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 0.989 0.988 0.987 0.986 0.985 0.984 0.983 0.982

0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 0.989 0.988 0.987 0.986 0.985 0.984 0.983 0.982 0.981 0.98

0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 0.989 0.988 0.987 0.986 0.985 0.984 0.983 0.982 0.98 0.97

0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.99 0.989 0.988 0.987 0.986 0.985 0.984 0.983 0.982

FIGURE 17.2 Electric density distribution injected by the bottom electrode when Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

17.2 COMBINED NATURAL CONVECTION AND RADIATION HEAT TRANSFER OF NANOFLUID UNDER THE IMPACT OF ELECTRIC FIELD

The governing formulae are: 8 > > r:V05 0; > 1 > > >

-  > μnf @V > > > 1 V :r V A 5 2 rp 1 μnf r2 V 1 qE 2 ðρβ Þnf g ðT 2 T0 Þ 2 V; ρnf @ > > @t K > > > 2 3 1 > > 0 -> 4 > knf

> > > > > > rϕ 5 2 E ; > > > > @q > 5 2 r: J ; > > > @t > > > > : q 5 r:εE
 ρCp nf ; ðβ Þnf ; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ; ðβ Þnf 5 ðβ Þf ð1 2 φÞ 1 ðβ Þs φ; μnf 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ

625

ð17:5Þ

ð17:6Þ

Properties of Fe3O4 and ethylene glycol are illustrated in Table 17.1. Table 17.2 illustrates the coefficient values of this formula. knf can be expressed as:

 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 ð17:7Þ kf mkf 1 kf 2 kp φ 1 kf 1 kp Different values of shape factors for various shapes of nanoparticles are illustrated in Table 17.3.

TABLE 17.1

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

β 3 1025 ð1=kÞ

kðW=m:kÞ

Ethylene glycol

1110

2400

65

0.26

Fe3 O4

5200

670

1.3

6

TABLE 17.2

The Coefficient Values of Eq. (17.6)

Coefficient values

φ50

φ 5 0:05

A1

1.0603E 1 001

9.5331

A2

22.698E-003

23.4119E-003

A3

2.9082E-006

5.5228E-006

A4

21.1876E-008

24.1344E-008

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

TABLE 17.3

The Values of Shape Factor of Different Shapes of Nanoparticles

Spherical

3

Platelet

5.7

Cylinder

4.8

Brick

3.7

m

So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V 5 0; > 0 1 2 3 > > >

-  > μnf =μf β nf @V S Pr μnf =μf > E > @ > 1 V :r V A 5 2 rp 1 Prr2 V 1 q E 2 RaPr4 5θ 2 V > > Da ρnf =ρf @t ρnf =ρf ρnf =ρf βf > > > 0 1 > >  21 >

- - 4 > knf =kf knf =kf

> > > > > E 5 2 rϕ > > > > > > q 5 r:εE > > > @q > > > r: J 5 2 > : @t

ð17:8Þ

where tαf P y x vL uL ;p5
;u 5 ; 2 ; y 5 ; x 5 ; v 5 L L α αf L2 f ρ αf =L T 2 T0 ϕ 2 ϕ0 q E θ5 ; rT 5 T1 2 T0 ; ϕ 5 ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 q0 E0 rT rϕ

t5

ð17:9Þ

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

@ψ @ψ ψ ωL2 @v @u ; 5 u; Ψ 5 ; Ω 5 2 ;ω5 @x @y αf @x @y αf

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @θ Nuloc 5 1 1 Rd 3 @X kf kf 1 Nuave 5 L

ðL Nuloc dY

ð17:10Þ

ð17:11Þ ð17:12Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

627

17.2 COMBINED NATURAL CONVECTION AND RADIATION HEAT TRANSFER OF NANOFLUID UNDER THE IMPACT OF ELECTRIC FIELD

17.2.3 Effects of Active Parameters The effect of electric field on nanofluid free convection heat transfer is investigated in the presence of thermal radiation. Electric field dependent viscosity is considered for nanofluid. The porous enclosure is filled with Fe3O4Ethylene glycol and has one lid wall. Roles of Darcy number (Da 5 1022 to 102 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 6kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Rayleigh number (Ra 5 50 to 500) are illustrated graphically. The influence of shape factor on rate of heat transfer is reported in Table 17.4. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. Impacts of Da; Δϕ, and Ra on isotherms and streamlines are demonstrated in Figs. 17.3
17.6. At low Rayleigh number, there are two eddies which rotate in opposite direction. In the presence of an electric field, the eddies are stretched and a thermal plume is generated. Isotherms become more disturbed in the presence of an electric field. As buoyancy TABLE 17.4

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Ra 5 500; Δϕ 5 6; φ 5 0:05 Da 1022

102

Spherical

18.15263

21.44078

Brick

18.94481

21.67239

Cylinder

13.05472

22.01274

Platelet

19.70303

22.27031

Da = 102

Da = 10–2

Streamline

FIGURE 17.3

Isotherm

0.0045 0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 –0.0005 –0.001 –0.0015 –0.002 –0.0025 –0.003 –0.0035 –0.004

0.008 0.006 0.004 0.002 –0.002 –0.004 –0.006

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

Da = 102

Da = 10–2

Streamline

Isotherm

16 14 12 10 6 2 –2 –6 –10 –14 –18 –20 –22 –24 –26 –28 –30

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

60 50 40 30 20 10 – 20 – 30 – 40 – 50 – 60 – 70 – 80 – 90 – 100 – 110 – 120 – 130 – 140 – 150 – 160

0.95 0.9 0.85 0.8 0.75 0.7 0.6 0.55 0.5 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

FIGURE 17.4 Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

Da = 102

Da = 10–2

Streamline

Isotherm

0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 –0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.45 0.35 0.25 0.15 0.05

0.09 0.07 0.05 0.03 0.02 0.01 –0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07

0.95 0.85 0.75 0.65 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

FIGURE 17.5 Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

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17.3 NANOFLUID FREE CONVECTION UNDER THE INFLUENCE OF AN ELECTRIC FIELD IN A POROUS WAVY ENCLOSURE

Streamline

Da = 10–2

20 15 10 5 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 –60 –65

–10 –20 –30 –40 –50 –60 –70 –80 –90 –100 –110 –120 –130 –140 –150 –160 –170 –180 –190 –200

Da = 102 FIGURE 17.6

Isotherm

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

forces increase, another eddy is generated. At high values of the Rayleigh number, the increasing Darcy number leads to the conversion of all eddies to one clockwise eddy and the thermal plume shifts to the left side. Nuave versus Ra; Da; Rd, and Δϕ is depicted in Fig. 17.7. The related formula is: Nuave 5 4:07 2 0:53Δϕ 1 2:39Ra 1 0:35Da 1 0:92Rd 1 0:06ΔϕRa 1 0:49ΔϕDa 2 0:98ΔϕRd 2 0:61Ra Da 1 0:41Ra Rd 1 3:2Da Rd 1 0:17Δϕ2 2 0:39ðRa Þ2 1 0:82ðDa Þ2 1 2:5Rd2

ð17:13Þ

where Ra 5 0:01Ra; Da 5 0:01Da, and Δϕ is voltage supply in Kilovolt. Rate of heat transfer is enhanced with the rise of Rayleigh number. The electric field helps the convective mode to enhance. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of Darcy number is the same as the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

17.3 NANOFLUID FREE CONVECTION UNDER THE INFLUENCE OF AN ELECTRIC FIELD IN A POROUS WAVY ENCLOSURE 17.3.1 Problem Definition Fig. 17.8 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for the bottom wall. The influence of the Darcy and Reynolds numbers on the contour of q is demonstrated in Fig. 17.9. The effect of Re on q is less sensible than Da. As the Darcy number increases the distortion of the isoelectric density lines become greater.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

FIGURE 17.7 Effects of Da; Δϕ; Rd, and Ra on average Nusselt number.

T =T0, ϕ = ϕ0,

T =T0,

∂ϕ ∂q = 0, = 0, u = 0, v = 0 ∂n ∂n

∂q = 0, u = 0, v = 0 ∂y

Nanofluid g

T =T0,

∂ϕ ∂q = 0, = 0, u = 0, v = 0 ∂n ∂n

Porous media T =T1, ϕ = ϕ1, q = q0, u = 0, v = 0

FIGURE 17.8 Geometry and the boundary conditions.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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17.3 NANOFLUID FREE CONVECTION UNDER THE INFLUENCE OF AN ELECTRIC FIELD IN A POROUS WAVY ENCLOSURE

Da = 0.01

Da = 100

Ra = 50

0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9 0.895 0.89 0.885 0.883

0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9 0.895 0.891 0.89 0.885 0.883

Ra = 500 FIGURE 17.9

0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.83 0.82 0.815 0.81 0.803

0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.81 0.807 0.805

Electric density distribution injected by the bottom electrode when Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

17.3.2 Governing Equation The definition of the electric field is: -

E 5 2 rϕ

ð17:14Þ

-

ð17:15Þ

q 5 r:ε E -

-

-

J 5 q V 2 Drq 1 σ E -

r: J 1

@q 50 @t

The governing formulae are: 8 > > r:V05 0; > 1 > > >

-  > μnf @V > @ > > ρnf 1 V :r V A 5 2 rp 1 μnf r2 V 1 q E 2 ðρβ Þnf g ðT 2 T0 Þ 2 V; > > @t K > > > 0 1 2 3 > > - > 4 > k < @ -  @T J :E 1 @qr 4 4σe @T nf  r2 T 1
 2
 ; qr 5 2 ; T4 D4Tc3 T 2 3Tc4 5 ; V :r T 1 A 5
@t @y 3β @y ρC ρC ρC p p p R > nf nf nf > > > > > rϕ 5 2 E ; > > > > @q > > 5 2 r: J ; > > > @t > > > > : q 5 r:εE
 ρCp nf ; ðβ Þnf ; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ; ðβ Þnf 5 ðβ Þf ð1 2 φÞ 1 ðβ Þs φ; μnf 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ

ð17:16Þ ð17:17Þ

ð17:18Þ

ð17:19Þ

Properties of Fe3O4 and ethylene glycol are illustrated in Table 17.1. Table 17.2 illustrates the coefficient values of this formula.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

knf can be expressed as:



 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

Different values of shape factors for various shapes of nanoparticles are illustrated in Table 17.3. So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V 5 0; >0 1 2 3 > > >

-  > μnf =μf β nf @V S Pr μnf =μf > E > @ > 1 V :r V A 5 2 rp 1 Prr2 V 1 q E 2 RaPr4 5θ 2 V > > Da ρnf =ρf @t ρnf =ρf ρnf =ρf βf > > > 0 1 > >  21 >

- - 4 > knf =kf knf =kf

> > > > > E 5 2 rϕ > > > > > q 5 r:εE > > > > @q > > > r: J 5 2 > : @t

ð17:20Þ

ð17:21Þ

where tαf P y x vL uL ;p5
;u 5 ; 2 ; y 5 ; x 5 ; v 5 L L α αf L2 f ρ αf =L T 2 T0 ϕ 2 ϕ0 q E θ5 ; rT 5 T1 2 T0 ; ϕ 5 ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 q0 E0 rT rϕ

t5

ð17:22Þ

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

@ψ @ψ ψ ωL2 @v @u ; 5 u; Ψ 5 ; Ω 5 2 ;ω5 @x @y αf @x @y αf

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @θ Nuloc 5 1 1 Rd 3 @X kf kf 1 Nuave 5 L

ðL Nuloc dY

ð17:23Þ

ð17:24Þ ð17:25Þ

0

17.3.3 Effects of Active Parameters Electrohydrodynamic nanofluid free convection heat transfer through the porous enclosure is investigated in the presence of thermal radiation. Electric field dependent viscosity is considered for nanofluid. The porous enclosure is filled with Fe3O4-Ethylene glycol and has one lid wall. The roles of Darcy number (Da 5 1022 to 102 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 6kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Rayleigh number (Ra 5 50 to 500) are illustrated graphically. The influence of shape factor on the rate of heat transfer is reported in Table 17.5. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. Impacts of Da; Δϕ, and Ra on isotherms and streamlines are demonstrated in Figs. 17.10
17.13. At low Rayleigh number, there are two eddies which rotate in opposite directions. In the presence of an electric field, smaller eddies are generated and a thermal plume is generated. Isotherms become more disturbed in the presence of an electric field. Increasing the Darcy number makes the isotherms denser near the positive electrode. As

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

633

17.3 NANOFLUID FREE CONVECTION UNDER THE INFLUENCE OF AN ELECTRIC FIELD IN A POROUS WAVY ENCLOSURE

TABLE 17.5

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Ra 5 500; Δϕ 5 6; φ 5 0:05 Da 22

10

102

Spherical

31.22357

48.90455

Brick

33.84407

49.91443

Cylinder

37.50945

50.10212

Platelet

40.45061

51.85191

Streamline

Da = 10–2 Da = 10–2 Da = 102

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.025 0.02 0.015 0.01 0.005 –0.005 –0.01 –0.015 –0.02 –0.025

Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

Streamline

FIGURE 17.11

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.01 0.008 0.006 0.004 0.002 –0.002 –0.004 –0.006 –0.008 –0.01

Da = 102 FIGURE 17.10

Isotherm

Isotherm 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.3 0.25 0.2 0.15 0.1 0.05

360 340 300 260 220 180 140 100 80 60 40 20 11 –11 –40

900 800 700 600 500 400 300 200 100 –100 –200 –300

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

Da = 10–2

Streamline

Da = 10–2 Da = 102

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

Streamline

FIGURE 17.13

Isotherm

0.3 0.25 0.2 0.15 0.1 0.05 –0.05 –0.1 –0.15 –0.2 –0.25 –0.3

Da = 102 FIGURE 17.12

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 –0.01 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.08 –0.09 –0.1

Isotherm 180 160 120 80 60 40 20 –20 –60 –100 –140 –180 –200 –220 –240 –260 –280

150 100 50 –50 –100 –150 –200 –250 –300 –350 –400 –450 –500 –550 –600 –650 –700 –750

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.25 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

buoyancy forces increase, the shapes of the streamlines become more complex. At high values of Rayleigh number, increasing the Darcy number leads to a shift of the thermal plume to the middle of the enclosure. Nuave versus Ra; Da; Rd, and Δϕ is depicted in Fig. 17.14. The related formula is: Nuave 5 2:04 2 0:75Δϕ 1 3:12Ra 1 1:25Da 1 0:12Rd 1 0:12ΔϕRa 1 0:55ΔϕDa 2 1:04 3 10215 ΔϕRd 2 0:066Ra Da 1 1:57 3 10215 Ra Rd 1 6:39 3 10215 Da Rd 1 0:56Δϕ2 2 0:55ðRa Þ2 2 1:37ðDa Þ2 2 2:15Rd2

ð17:26Þ

where Ra 5 0:01Ra; Da 5 0:01Da, and Δϕ is voltage supply in Kilovolt. The rate of heat transfer enhances with the rise of the Rayleigh number. The electric field helps to enhance the convective mode. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of the Darcy number is the same as the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

17.3 NANOFLUID FREE CONVECTION UNDER THE INFLUENCE OF AN ELECTRIC FIELD IN A POROUS WAVY ENCLOSURE

FIGURE 17.14

635

Effects of Da; Δϕ; Rd, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

636

17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

FIGURE 17.14

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

637

17.4 EHD NANOFLUID FLOW IN A POROUS MEDIUM CONSIDERING RADIATION PARAMETER

17.4 EHD NANOFLUID FLOW IN A POROUS MEDIUM CONSIDERING RADIATION PARAMETER 17.4.1 Problem Definition Fig. 17.15 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for the bottom wall. The influence of Darcy and Reynolds numbers on contour of q is demonstrated in Fig. 17.16. The effect of Re on q is less sensible than Da. As the Darcy number increases, the distortion of the isoelectric density lines become greater. T =T0, ϕ = ϕ0,

∂q = 0, u = 0, v = 0 ∂y

T =T0,

∂ϕ ∂q = 0, = 0, u = 0, v = 0 ∂n ∂n

g

T =T0,

∂ϕ ∂q = 0, = 0, u = 0, v = 0 ∂n ∂n

Nanofluid

Porous media

y x O

FIGURE 17.15

T =T1, ϕ = ϕ1, q = q0, u = 0, v = 0

Geometry and the boundary conditions.

Ra = 500

Ra = 50

Da = 0.01

FIGURE 17.16

Da = 100

0.9995 0.999 0.9985 0.998 0.9975 0.997 0.9965 0.996 0.9955 0.995 0.9945 0.994 0.9935 0.993

0.9995 0.999 0.998 0.997 0.9965 0.996 0.9955 0.995 0.994 0.9935 0.993

0.9995 0.999 0.9985 0.998 0.9975 0.997 0.9965 0.996 0.9955 0.995 0.9945 0.994 0.9935 0.993

–2 –6 –10 –14 –18 –22 –24 –26 –28 –30 –32 –34 –36

Electric density distribution injected by the bottom electrode when Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

638

17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

17.4.2 Governing Equation The definition of the electric field is: -

E 5 2 rϕ

ð17:27Þ

-

ð17:28Þ

q 5 r:ε E -

-

-

J 5 q V 2 Drq 1 σ E

TABLE 17.6

ð17:29Þ

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Ra 5 500; Δϕ 5 6; φ 5 0:05 Da 1022

102

Spherical

17.64276

18.65795

Brick

17.83949

18.84945

Cylinder

18.12832

19.13087

Platelet

19.34793

20.34507

Da = 102

Da = 10–2

Streamline

FIGURE 17.17

Isotherm

0.0022 0.002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 5E-06 –0.0002 –0.0004 –0.0006 –0.0008 –0.001 –0.0012 –0.0014

0.003 0.0025 0.002 0.0015 0.001 0.0005 –0.0005 –0.001 –0.0015 –0.002

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.01

Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

639

17.4 EHD NANOFLUID FLOW IN A POROUS MEDIUM CONSIDERING RADIATION PARAMETER

-

r: J 1

@q 50 @t

ð17:30Þ

The governing formulae are: 8 > > r:V05 0; > 1 > > >

-  > μnf @V > > > ρnf @ 1 V :r V A 5 2 rp 1 μnf r2 V 1 q E 2 ðρβ Þnf g ðT 2 T0 Þ 2 V; > > @t K > > > 0 1 2 > > -> > k < @ -  @T J :E 1 @qr 4 4σe @T 4 nf  r2 T 1
 2
 ; qr 5 2 ; V :r T 1 A 5
@t 3β R @y ρCp nf ρCp nf ρCp nf @y > > > > > > rϕ 5 2 E ; > > > > @q > > 5 2 r: J ; > > > @t > > > > : q 5 r:εE

Da = 102

Da = 10–2

Streamline

FIGURE 17.18

3 T4 D4Tc3 T 2 3Tc4 5 ;

ð17:31Þ

Isotherm

–2 –4 –6 –8 –10 –12 –14 –16 –18 –20 –22 –24 –26 –28 –30 –32 –34 –36

–2 –4 –6 –8 –10 –14 –18 –22 –26 –30 –34 –38 –40 –44 –46

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 50; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

640


ρCp

17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

 nf

; ðβ Þnf ; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ðβ Þnf 5 ðβ Þf ð1 2 φÞ 1 ðβ Þs φ; ð17:32Þ

μnf 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ 2

Properties of Fe3O4 and ethylene glycol are illustrated in Table 17.1. Table 17.2 illustrates the coefficient values of this formula. knf can be expressed as:

 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 ð17:33Þ kf mkf 1 kf 2 kp φ 1 kf 1 kp Different values of shape factors for various shapes of nanoparticles are illustrated in Table 17.3.

Isotherm

0.022 0.02 0.016 0.012 0.01 0.006 0.004 0.002 0.0001 1E-05 –0.002 –0.004 –0.006 –0.01 –0.014 –0.016

0.95 0.85 0.8 0.7 0.65 0.55 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.01

0.03 0.025 0.02 0.015 0.01 0.005 –0.005 –0.01 –0.015 –0.02

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.02 0.01

Da = 102

Da = 10–2

Streamline

FIGURE 17.19

Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

641

17.4 EHD NANOFLUID FLOW IN A POROUS MEDIUM CONSIDERING RADIATION PARAMETER

So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V 5 0; > 0 1 2 3 > > >

-  > μnf =μf β nf @V S Pr μnf =μf > E > @ > 1 V :r V A 5 2 rp 1 Prr2 V 1 q E 2 RaPr4 5θ 2 V > > Da ρnf =ρf @t ρnf =ρf ρnf =ρf βf > > > 0 1 > >  21 >

- - 4 > knf =kf knf =kf < @ -  @θA SE Ec @2 θ knf 2





 V :r θ 1 r θ1 J :E 1 Rd 5 k @t 3 ρCp nf = ρCp f f @Y2 ρCp nf = ρCp f ρCp nf = ρCp f > > > > > > E 5 2 rϕ > > > > > > q 5 r:εE > > > @q > > > r: J 5 2 > : @t

ð17:34Þ

where tαf P y x vL uL ;p 5
;u5 ; 2 ; y 5 ; x 5 ; v 5 L L α αf L2 f ρ αf =L T 2 T0 ϕ 2 ϕ0 q E θ5 ; rT 5 T1 2 T0 ; ϕ 5 ; rϕ 5 ϕ1 2 ϕ0 ; q 5 ; E 5 q0 E0 rT rϕ

t5

Da = 102

Da = 10–2

Streamline

FIGURE 17.20

ð17:35Þ

Isotherm

–2 –6 –10 –14 –18 –22 –24 –26 –28 –30 –32 –34 –36

0.95 0.9 0.85 0.8 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

1.25 0.21 –2 –4 –6 –8 –12 –14 –18 –20 –24 –26 –30 –32 –36 –38 –40 –44 –46

0.9 0.8 0.75 0.65 0.6 0.5 0.4 0.35 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Ra 5 500; Δϕ 5 6kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

642

17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

FIGURE 17.21

@ψ @ψ ψ ωL2 @v @u ; 5 u; Ψ 5 ; Ω 5 2 ;ω5 @x @y αf @x @y αf

ð17:36Þ

Effects of Da; Δϕ; Rd, and Ra on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

17.4 EHD NANOFLUID FLOW IN A POROUS MEDIUM CONSIDERING RADIATION PARAMETER

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @θ 1 1 Rd Nuloc 5 3 @X kf kf

FIGURE 17.21

643

ð17:37Þ

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

644

17. THERMAL RADIATION INFLUENCE ON NANOFLUID FLOW IN A POROUS MEDIUM IN THE PRESENCE OF COULOMB FORCES

1 Nuave 5 L

ðL Nuloc dY

ð17:38Þ

0

17.4.3 Effects of Active Parameters Electrohydrodynamic nanofluid free convection heat transfer through the porous enclosure is investigated in the presence of thermal radiation. Electric field dependent viscosity is considered for nanofluid. The porous enclosure is filled with Fe3O4-Ethylene glycol and has one lid wall. The roles of Darcy number (Da 5 1022 to 102 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 6kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Rayleigh number (Ra 5 50 to 500) are illustrated graphically. The influence of shape factor on the rate of heat transfer is reported in Table 17.6. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. The impacts of Da; Δϕ, and Ra on isotherms and streamlines are demonstrated in Figs. 17.17
17.20. At low Rayleigh number, there are two eddies which rotate in opposite directions. In the presence of an electric field, the two eddies convert to one clockwise eddy. The isotherms become more disturbed in the presence of an electric field and a thermal plume is generated. Increasing the Darcy number makes the isotherms denser near the positive electrode. As buoyancy forces increase, the shapes of the streamlines become more complex. At high values of Rayleigh number, increasing the Darcy number generates a secondary eddy. Nuave versus Ra; Da; Rd, and Δϕ is depicted in Fig. 17.21. The related formula is: Nuave 5 7:36 1 0:04Δϕ 1 3:86Ra 1 0:2Da 1 5:9Rd 2 6:23 3 1023 ΔϕRa 1 0:07ΔϕDa 2 0:046ΔϕRd 2 9:84 3 1023 Ra Da 2 0:15Ra Rd 2 0:073Da Rd 1 0:05Δϕ2 2 0:64ðRa Þ2 1 0:16ðDa Þ2 1 4:64Rd2

ð17:39Þ

where Ra 5 0:01Ra; Da 5 0:01Da, and Δϕ is voltage supply in Kilovolt. The rate of heat transfer enhances with the increase of the Rayleigh number. The electric field helps to enhance the convective mode. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of the Darcy number is the same as the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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[45] M. Sheikholeslami, M. Seyednezhad, Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEM, Int. J. Heat Mass Transfer 114 (2017) 1169
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1126. [50] M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of Al2O3
water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
524. [51] M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field, Int. J. Heat Mass Transfer 92 (2016) 339
348. [52] M.S. Kandelousi, Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition, Eur. Phys. J. Plus (2014) 129
248. Available from: https://doi.org/10.1140/epjp/i2014-14248-2. [53] M. Sheikholeslami, CVFEM for magnetic nanofluid convective heat transfer in a porous curved enclosure, Eur. Phys. J. Plus 131 (2016) 413. Available from: https://doi.org/10.1140/epjp/i2016-16413-y. [54] M. Sheikholeslami, Magnetic source impact on nanofluid heat transfer using CVFEM, Neural Computing and Applications (2016). Available from: https://doi.org/10.1007/s00521-016-2740-7. [55] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cylinder considering Darcy model, J. Mol. Liq. 225 (2017) 903
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2667. Available from: https://doi.org/10.1108/EC-01-2017-0008. [62] M. Sheikholeslami, M. Darzi, Z. Li, Experimental investigation for entropy generation and energy loss of nano-refrigerant condensation process, Int. J. Heat Mass Transfer 125 (2018) 1087
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781. [82] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process. Process Intensif. 124 (2018) 71
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C H A P T E R

18 Influence of Electric Field on Forced Convection of Nanofluid in a Porous Medium by Means of CVFEM 18.1 INTRODUCTION Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. It is the study of the motions of ionized particles or molecules and their interactions with electric fields and the surrounding fluid. This active method can be combined with useful passive techniques such as nanofluid. Sheikholeslami [1] investigated Fe3O4-H2O nanofluid thermal improvement in the presence of Coulomb forces. Sheikholeslami and Ganji [2] considered variable viscosity for simulation of nanofluid forced convection in the presence of an electric field. Sheikholeslami, and Bhatti [3] presented an active method for nanofluid heat transfer enhancement by means of EHD. Sheikholeslami and Rokni [4] presented the impact of EFD viscosity on nanofluid forced convection in a cavity with sinusoidal wall. Active and passive techniques for heat transfer enhancement can be combined [5
90].

18.2 EHD NANOFLUID FLOW IN A PERMEABLE ENCLOSURE WITH SINUSOIDAL WALL 18.2.1 Problem Definition Fig. 18.1 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for the bottom wall. The influence of Darcy and Reynolds numbers on the contours of q is demonstrated in Fig. 18.2. The effect of Re on q is less sensible than Da. As the Darcy number augments, the distortion of the isoelectric density lines become greater.

FIGURE 18.1

Geometry and the boundary conditions.

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00018-7

649

© 2019 Elsevier Inc. All rights reserved.

650

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

Da = 105

Re = 6000

Re = 3000

Da = 102

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.35 0.25 0.15 0.05

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.03

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05

FIGURE 18.2 Electric density distribution injected by the bottom electrode when Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

18.2.2 Governing Equation The definition of the electric field is: -

E 5 2 rϕ

ð18:1Þ

-

ð18:2Þ

q 5 r:ε E -

-

-

J 5 q V 2 Drq 1 σ E -

r: J 1

ð18:3Þ

@q 50 @t

The governing formulae are: 8 > > r:V 5 0; > 0 1 > > >

- - @V > μnf 2 - rp μnf qE >@ > A5 > V :r V 1 1 r V2 2 V; > > ρnf @t ρnf ρnf Kρnf > > > 0 1 2 > > -> > knf < @ -  @TA J :E 1 @q 4σe @T 4 r  r2 T 1
 2
 5
; 4qr 5 2 ; V :r T 1 @t 3β R @y ρCp nf @y ρCp nf ρCp nf > > > > > > rϕ 5 2 E ; > > > > @q > > 5 2 r: J ; > > > @t > > > > : q 5 r:εE

ð18:4Þ

3 T 4 D4Tc3 T 2 3Tc4 5 ;

ð18:5Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

651

18.2 EHD NANOFLUID FLOW IN A PERMEABLE ENCLOSURE WITH SINUSOIDAL WALL




ρCp

 nf

; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ð18:6Þ

μ 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ

Properties of Fe3O4 and ethylene glycol are illustrated in Table 18.1. Table 18.2 illustrates the coefficient values of this formula. knf can be expressed as:

 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 ð18:7Þ kf mkf 1 kf 2 kp φ 1 kf 1 kp Different values of shape factors for various shapes of nanoparticles are illustrated in Table 18.3.

TABLE 18.1

Thermophysical Properties of Ethylene Glycol and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

Ethylene glycol

1110

2400

0.26

Fe3 O4

5200

670

6

TABLE 18.2

The Coefficient Values of Eq. (18.6) φ50

Coefficient Values

φ 5 0:05

A1

1.0603E 1 001

9.5331

A2

2 2.698E-003

2 3.4119E-003

A3

2.9082E-006

5.5228E-006

A4

2 1.1876E-008

2 4.1344E-008

TABLE 18.3

m

The Values of Shape Factor of Different Shapes of Nanoparticles Spherical

3

Platelet

5.7

Cylinder

4.8

Brick

3.7

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

652

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V > 0 5 0; 1 > >  >

- - @V > 1 ρnf =ρf 2 SE 1 μnf ρnf 21 > > @ A > 5 V :r V 1 r V 2 rp 1 qE 2 V > > Re μnf =μf Re Da μf ρf @t ρnf =ρf > > > 0 1 > ! > >

- - 4 k 21 @2 θ > knf =kf < @ -  @θA 1 1 nf

 r2 θ 1 SE

 Ec J : E 1 V :r θ 1 Rd 2 5 @t PrRe ρCp nf = ρCp f 3 kf @Y ρCp nf = ρCp f > > > > > > E 5 2 rϕ > > > > > > q 5 r:εE > > > @q > > > r: J 5 2 > : @t where


  y; x ðu; vÞ ϕ 2 ϕ0
T 2 T0 ;θ5 ; y; x 5 ; ðu; v Þ 5 ;ϕ5 L ULid rϕ rT tULid P q E ;p 5 t5 q5 ;E5 ; 2 q0 E0 L ρULid rT 5 T1 2 T0 ; rϕ 5 ϕ1 2 ϕ0 ;

ð18:8Þ

ð18:9Þ

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

@ψ @v @u @ψ ψL ω ;ω5 2 ; 5 u; Ψ 5 ;Ω5 @x @x @y @y ULid LULid

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @Θ Nuloc 5 1 1 Rd 3 @X kf kf 1 Nuave 5 L

ðL Nuloc dY

ð18:10Þ

ð18:11Þ ð18:12Þ

0

18.2.3 Effects of Active Parameters Forced convection of nanofluid in the presence of Coulomb forces is presented. The radiation term is added to the energy equation. The porous enclosure is filled with Fe3O4-Ethylene glycol and has one lid wall. The roles of Darcy number (Da 5 102 to 105 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 10kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Reynolds number (Re 5 3000 to 6000) are illustrated graphically. The impact of shape factor on the rate of heat transfer is reported in Table 18.4. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. Impacts of Da; Δϕ, and Re on isotherms and streamlines are demonstrated in Figs. 18.3
18.6. At low Reynolds number, there is one clockwise eddy in the streamlines. The center of the main eddy is near the lid wall. Augmenting the Darcy number leads to the generation of the second eddy which rotates counterclockwise and the center of the main eddy shifts to upwards. Applying an electric field causes the strength of the main eddy to enhance and shifts the center of the eddy upwards. The isotherms become more disturbed in the presence of an electric field. A thermal plume appears when the Reynolds number is increased. Also, by increasing Re, the strengths of the rotating eddies are augmented. As the Coulomb force increases, the secondary eddy diminishes and the strength of main eddy is enhanced.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

653

18.2 EHD NANOFLUID FLOW IN A PERMEABLE ENCLOSURE WITH SINUSOIDAL WALL

TABLE 18.4

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Re 5 6000; Δϕ 5 10; φ 5 0:05 Da 2

10

105

Spherical

2.338308

2.884048

Brick

2.359831

2.913817

Cylinder

2.391644

2.957562

Platelet

2.415995

2.990852

Da = 105

Da = 102

Streamline

FIGURE 18.3

Isotherm

–1E–07 –5E–07 –2E–06 –2E–05 –4E–05 –6E–05 –8E–05 –0.0001 –0.00012 –0.00016 –0.0002 –0.00024 –0.00028 –0.00032 –0.00036

2.8E–06 1.5E–06 3E–07 2E–08 –2E–05 –5E–05 –0.0001 –0.0002 –0.0003 –0.0004 –0.0005 –0.0006 –0.0007 –0.0008 –0.0009 –0.001 –0.0011 –0.0012

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.01

Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

Nuave versus Re; Da; Rd, and Δϕ is depicted in Fig. 18.7. The related formula is: Nuave 5 2 2:08 2 2:65Δϕ 1 1:3Re 1 1:27logðDaÞ 1 0:8Rd 2 0:3 3 1025 ΔϕRe 1 0:04ΔϕlogðDaÞ 1 0:3 3 1024 ΔϕRd 2 0:017Re logðDaÞ 2 0:16Re Rd 1 0:16logðDaÞRd
2 1 0:02Δϕ2 2 0:15ðRe Þ2 2 0:175 logðDaÞ 1 1:76Rd2

ð18:13Þ

where Re 5 0:001Re and Δϕ is voltage supply in Kilovolt. In the absence of Coulomb force, the Nusselt number augments with the rise of the Reynolds number but the opposite behavior is reported in the presence of such forces. The electric field helps the convective mode to enhance. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of the Darcy number is the same as that for the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

Da = 105

Da = 102

Streamline

Isotherm

–2E–05 –3.4E–05 –4E–05 –6E–05 –8E–05 –0.0001 –0.00012 –0.00014 –0.00016 –0.00018 –0.0002 –0.00022 –0.00024 –0.00026 –0.00028 –0.0003 –0.00032 –0.00034 –0.00036 –0.00038

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–0.0001 –0.0002 –0.0003 –0.0004 –0.0005 –0.0006 –0.0007 –0.0008 –0.0009 –0.001 –0.0011 –0.0012 –0.0013

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

FIGURE 18.4 Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

Da = 105

Da = 102

Streamline

Isotherm

–3E–07 –2E–06 –1E–05 –5E–05 –0.00015 –0.00025 –0.00035 –0.00045 –0.00055 –0.00065

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.02

4E–05 2E–05 7E–06 2E–06 –9E–05 –0.0004 –0.0008 –0.0012 –0.0016 –0.002 –0.0024 –0.0028

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

FIGURE 18.5 Effect of Darcy number on streamlines and isotherms when Re 5 6000; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

18.3 EFFECT OF SHAPE FACTOR ON ELECTROHYDRODYNAMIC NANOFLUID FLOW IN A POROUS MEDIUM

Isotherm

Da = 105

Da = 102

Streamline

FIGURE 18.6

655

–2E–06 –1E–05 –2E–05 –5E–05 –0.0001 –0.00015 –0.0002 –0.00025 –0.0003 –0.00035 –0.0004 –0.00045 –0.0005 –0.00055 –0.0006 –0.00065

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

–8E–06 –7E–05 –0.0002 –0.0003 –0.0004 –0.0006 –0.0008 –0.001 –0.0012 –0.0014 –0.0016 –0.0018 –0.002 –0.0022 –0.0024 –0.0026 –0.0028 –0.003

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Re 5 6000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

18.3 EFFECT OF SHAPE FACTOR ON ELECTROHYDRODYNAMIC NANOFLUID FLOW IN A POROUS MEDIUM 18.3.1 Problem Definition Fig. 18.8 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for the bottom wall. The influence of Darcy and Reynolds numbers on the contours of q is demonstrated in Fig. 18.9. The effect of Re on q is less sensible than forDa. As the Darcy number increases the distortion of the isoelectric density lines become greater.

18.3.2 Governing Equation The definition of the electric field is: -

E 5 2 rϕ -

q 5 r:ε E

ð18:14Þ ð18:15Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

656

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

FIGURE 18.7 Effects of Da; Δϕ; Rd, and Re on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18.3 EFFECT OF SHAPE FACTOR ON ELECTROHYDRODYNAMIC NANOFLUID FLOW IN A POROUS MEDIUM

FIGURE 18.7

657

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

658

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

FIGURE 18.8 Geometry and the boundary conditions.

Da = 105

Re = 6000

Re = 3000

Da = 102 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1

FIGURE 18.9 Electric density distribution injected by the bottom electrode when Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

-

-

-

J 5 q V 2 Drq 1 σ E -

r: J 1

@q 50 @t

ð18:16Þ ð18:17Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

659

18.3 EFFECT OF SHAPE FACTOR ON ELECTROHYDRODYNAMIC NANOFLUID FLOW IN A POROUS MEDIUM

The governing formulae are: 8 > > r:V 5 0; > 0 1 > > >

- - @V > μnf 2 - rp μnf qE > > @ A > 5 V :r V 1 1 r V2 2 V; > > ρnf @t ρnf ρnf Kρnf > > > 2 0 1 > > -> > knf < @ -  @TA J :E 1 @q 4σe @T 4 r  r2 T 1
 2
 V :r T 1 5
; 4qr 5 2 ; @t 3β R @y ρCp nf ρCp nf ρCp nf @y > > > > > > rϕ 5 2 E ; > > > > @q > > > > @t 5 2 r: J ; > > > > > : q 5 r:εE
 ρCp nf ; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

3 T4 D4Tc3 T 2 3Tc4 5 ;

μ 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ knf can be expressed as:



 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V 5 0; >0 1 > >  >

- - @V > 1 ρnf =ρf 2 SE 1 μnf ρnf 21 > > @ A > V :r V 1 5 r V 2 rp 1 qE 2 V > > Re μnf =μf Re Da μf ρf @t ρnf =ρf > > > 0 1 > ! > >

- - 4 k 21 @2 θ > knf =kf < @ -  @θA 1 1 nf

 r2 θ 1 SE

 Ec J : E 1 5 V :r θ 1 Rd 2 @t PrRe ρCp nf = ρCp f 3 kf @Y ρCp nf = ρCp f > > > > > > > E 5 2 rϕ > > > > > q 5 r:εE > > > @q > > > r: J 5 2 > : @t where


  y; x ðu; vÞ ϕ 2 ϕ0
T 2 T0 ;θ5 ; y; x 5 ; ðu; v Þ 5 ;ϕ 5 L ULid rϕ rT tULid P q E ;p 5 t5 q 5 ;E 5 ; 2 q0 E0 L ρULid rT 5 T1 2 T0 ; rϕ 5 ϕ1 2 ϕ0 ;

ð18:18Þ

ð18:19Þ

ð18:20Þ

ð18:21Þ

ð18:22Þ

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

@ψ @v @u @ψ ψL ω ;ω5 2 ; 5 u; Ψ 5 ;Ω5 @x @x @y @y ULid LULid

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @Θ Nuloc 5 1 1 Rd 3 @X kf kf

ð18:23Þ

ð18:24Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

660

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

1 Nuave 5 L

ðL

ð18:25Þ

Nuloc dY 0

18.3.3 Effects of Active Parameters EHD nanofluid forced convection in the presence of thermal radiation is reported. The porous enclosure is filled with Fe3O4-Ethylene glycol and has one lid wall. The roles of Darcy number (Da 5 102 to 105 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 10kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Reynolds number (Re 5 3000 to 6000) are illustrated graphically. The impact of shape factor on the rate of heat transfer is reported in Table 18.5. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. Impacts of Da; Δϕ, and Re on isotherms and streamlines are demonstrated in Figs. 18.10
18.13. At low Reynolds number, there is one clockwise eddy in the streamlines. The center of the

TABLE 18.5

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Re 5 6000; Δϕ 5 10; φ 5 0:05 Da 2

10

105

Spherical

3.697271

6.084579

Brick

3.740571

6.147401

Cylinder

3.805097

6.239713

Platelet

3.855149

6.318966

Da = 105

Da = 102

Streamline

FIGURE 18.10

Isotherm

0.001 –6.09E–05 –0.0001 –0.00018 –0.00025 –0.00038 –0.001 –0.002 –0.003 –0.004

0.001 –0.0003 –0.001 –0.002 –0.0023 –0.003 –0.004

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

661

18.3 EFFECT OF SHAPE FACTOR ON ELECTROHYDRODYNAMIC NANOFLUID FLOW IN A POROUS MEDIUM

Streamline

Da = 102 Da = 102 Da = 105

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

–0.0024 –0.005 –0.01 –0.015 –0.02 –0.025 –0.03

Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

Streamline

FIGURE 18.12

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.001 –0.001 –0.002 –0.003 –0.0035 –0.004

Da = 105

FIGURE 18.11

Isotherm

Isotherm 0.0005 3.7E–05 –3E–05 –6E–05 –0.0001 –0.00017 –0.00024 –0.0005 –0.001 –0.0015 –0.002

–0.0003 –0.0005 –0.001 –0.002 –0.003 –0.004 –0.005

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Effect of Darcy number on streamlines and isotherms when Re 5 6000; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

662

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

Streamline

Da = 102

0.0006 0.0004 0.0002 –0.0002 –0.0004 –0.0006 –0.0008 –0.001 –0.0012 –0.0014 –0.0016 –0.0018 –0.002

–0.002 –0.0025 –0.004 –0.006 –0.008 –0.01 –0.012 –0.014 –0.016

Da = 105

FIGURE 18.13

Isotherm

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Effect of Darcy number on streamlines and isotherms when Re 5 6000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

main eddy is near the lid wall. Augmenting the Darcy number leads to the generation of the second eddy which rotates counterclockwise and the center of the main eddy shifts upward. Applying an electric field causes the strength of the main eddy to enhance and shifts the center of the eddy upward. The isotherms become more disturbed in the presence of an electric field. A thermal plume appears when the Reynolds number is increased. Also, by increasing Re, the strengths of the rotating eddies are augmented. As the Coulomb force increases, the secondary eddy diminishes and the strength of main eddy is enhanced. Nuave versus Re; Da; Rd, and Δϕ is depicted in Fig. 18.14. The related formula is: Nuave 5 2 2:26 1 0:08Δϕ 1 2:01Re 1 1:05logðDaÞ 1 3:25Rd 2 0:036ΔϕRe 1 0:023ΔϕlogðDaÞ 1 0:15ΔϕRd 2 0:31Re logðDaÞ 2 0:64Re Rd 1 0:41logðDaÞRd
2 1 0:015Δϕ2 2 0:2ðRe Þ2 2 0:14 logðDaÞ 1 1:46Rd2

ð18:26Þ

where Re 5 0:001Re and Δϕ is voltage supply in Kilovolt. In the absence of the Coulomb force, the Nusselt number increases with the rise of Reynolds number but the opposite behavior is reported in the presence of such forces. An electric field helps to enhance the convective mode. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of Darcy number is the same as that for the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

18.4 EFFECT OF ELECTIVE FIELD ON NANOFLUID FLOW IN A POROUS LID DRIVEN CAVITY IN EXISTENCE OF ELECTRIC FIELD 18.4.1 Problem Definition Fig. 18.15 depicts the porous enclosure and its boundary conditions. Ethylene glycol-Fe3O4 nanofluid is utilized. All walls are stationary except for bottom wall. The influence of Darcy and Reynolds numbers on the

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18.4 EFFECT OF ELECTIVE FIELD ON NANOFLUID FLOW IN A POROUS LID DRIVEN CAVITY IN EXISTENCE OF ELECTRIC FIELD

FIGURE 18.14

663

Effects of Da; Δϕ; Rd, and Re on average Nusselt number.

contours of q is demonstrated in Fig. 18.16. The effect of Re on q is less sensible than for Da. As the Darcy number augments the distortion of the isoelectric density lines become greater.

18.4.2 Governing Equation The definition of electric field is: -

E 5 2 rϕ -

q 5 r:ε E

ð18:27Þ ð18:28Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

664

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

FIGURE 18.14

(Continued)

-

-

-

J 5 q V 2 Drq 1 σ E -

r: J 1

@q 50 @t

ð18:29Þ ð18:30Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18.4 EFFECT OF ELECTIVE FIELD ON NANOFLUID FLOW IN A POROUS LID DRIVEN CAVITY IN EXISTENCE OF ELECTRIC FIELD

FIGURE 18.15

Geometry and the boundary conditions.

Re = 6000

Re = 3000

Da = 102

FIGURE 18.16

665

Da = 105 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.85 0.8 0.75 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.22 0.15 0.1 0.05

0.95 0.8 0.65 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.12 0.1

0.6 0.5 0.4 0.35 0.3 0.25 0.2 0.15

Electric density distribution injected by the bottom electrode when Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

The governing formulae are: 8 > > r:V 5 0; > 0 1 > > >

- - @V > μ μ > > @ V :r V 1 A 5 qE 1 nf r2 V 2 rp 2 nf V ; > > > ρnf @t ρnf ρnf Kρnf > > > 1 2 >0 > -> > knf < @ -  @TA J :E 1 @qr 4 4σe @T 4 2  r T1
 2
 5
; qr 5 2 ; V :r T 1 @t 3β R @y ρCp nf ρCp nf ρCp nf @y > > > > > > rϕ 5 2 E ; > > > > @q > > > > @t 5 2 r: J ; > > > > > : q 5 r:εE

3 T

4

D4Tc3 T 2 3Tc4 5 ;

ð18:31Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

666


ρCp

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

 nf

; μnf , and ρnf can be obtained as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ; μ 5 A1 1 A2 ðΔϕÞ 1 A3 ðΔϕÞ2 1 A4 ðΔϕÞ3 ; ρnf 5 ρf ð1 2 φÞ 1 ρs φ

knf can be expressed as:



 knf 2 m kf 2 kp φ 1 kp 2 kf φ 1 mkf 1 kp 1 kf
 5 kf mkf 1 kf 2 kp φ 1 kf 1 kp

So, the final PDE in the presence of thermal radiation and an electric field in a porous medium are: 8 > > r:V 5 0; > 0 1 > >  >

- - @V > 1 ρnf =ρf 2 SE 1 μnf ρnf 21 > > @ A > 5 V :r V 1 r V 2 rp 1 qE 2 V > > Re μnf =μf Re Da μf ρf @t ρnf =ρf > > > 0 1 > ! > >

- - 4 k 21 @2 θ > knf =kf < @ -  @θA 1 1 nf

 r2 θ 1 SE

 Ec J : E 1 5 V :r θ 1 Rd 2 @t PrRe ρCp nf = ρCp f 3 kf @Y ρCp nf = ρCp f > > > > > > E 5 2 rϕ > > > > > > q 5 r:εE > > > @q > > > r: J 5 2 > : @t where


  y; x ðu; vÞ ϕ 2 ϕ0
T 2 T0 ;θ5 ; y; x 5 ; ðu; v Þ 5 ;ϕ5 L ULid rϕ rT tULid P q E ;p 5 t5 q5 ;E5 ; 2 q0 E0 L ρULid rT 5 T1 2 T0 ; rϕ 5 ϕ1 2 ϕ0 ;

ð18:32Þ

ð18:33Þ

ð18:34Þ

ð18:35Þ

Vorticity and stream function should be employed in order to diminish the pressure gradient: v52

@ψ @v @u @ψ ψL ω ;ω5 2 ; 5 u; Ψ 5 ;Ω5 @x @x @y @y ULid LULid

Nuloc and Nuave along the bottom wall are calculated as:   21 ! knf knf 4 @Θ Nuloc 5 1 1 Rd 3 @X kf kf Nuave 5

1 L

ðL Nuloc dY

ð18:36Þ

ð18:37Þ ð18:38Þ

0

18.4.3 Effects of Active Parameters Forced convection of nanofluid in the presence of Coulomb forces is presented. The radiation term is added to the energy equation. The porous enclosure is filled with Fe3O4-Ethylene glycol and has one lid wall. The roles of Darcy number (Da 5 102 to 105 ), Radiation parameter (Rd 5 0 to 0:8), supplied voltage (Δϕ 5 0 to 10kV), volume fraction of Fe3O4 (φ 5 0% to 5%), and Reynolds number (Re 5 3000 to 6000) are illustrated graphically. The impact

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18.4 EFFECT OF ELECTIVE FIELD ON NANOFLUID FLOW IN A POROUS LID DRIVEN CAVITY IN EXISTENCE OF ELECTRIC FIELD

667

of shape factor on the rate of heat transfer is reported in Table 18.6. In this table, various shapes of nanoparticles are utilized. The maximum Nu is caused by Platelet, followed by Cylinder, Brick, and Spherical. So, Platelet nanoparticle has been utilized for further investigation. Impacts of Da; Δϕ, and Re on isotherms and streamlines are demonstrated in Figs. 18.17
18.19. At low Reynolds number, there is one clockwise eddy in the streamlines. The center of the main eddy is near the lid wall. As the Darcy number increases, the center of the main eddy shifts upward and jΨ max j increases. Applying an electric field causes the strength of the main eddy to be enhanced and shifts the center of the eddy upward. The isotherms become more disturbed in the presence of an electric field. A thermal plume appears when the Reynolds number is increased. Also, by increasing Re, the strength of the rotating eddy is enhanced. As the Coulomb force increases, the strength of main eddy is enhanced and a stronger thermal plume is generated. Nuave versus Re; Da; Rd, and Δϕ is depicted in Fig. 18.20. The related formula is: Nuave 5 2 0:99 1 0:04Δϕ 2 0:18Re 1 1:7logðDaÞ 1 1:04Rd 2 0:038ΔϕRe 1 0:03ΔϕlogðDaÞ 1 0:12ΔϕRd 2 0:07Re logðDaÞ 2 0:32Re Rd 1 0:25logðDaÞRd
2 1 0:013Δϕ2 2 0:05ðRe Þ2 2 0:203 logðDaÞ 1 0:36Rd2

TABLE 18.6

ð18:39Þ

Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Re 5 6000; Δϕ 5 10; φ 5 0:05 Da 2

10

105

Spherical

3.225443

3.741018

Brick

3.267538

3.780537

Cylinder

3.32923

3.838521

Platelet

3.375827

3.882675

Da = 105

Da = 102

Streamline

FIGURE 18.17

Isotherm

0.0004 0.0002 –5E–05 –9E–05 –0.0002 –0.0004 –0.0006 –0.00067 –0.0008 –0.001 –0.0012 –0.0014 –0.0016 –0.0018

–0.0001 –0.0005 –0.001 –0.0015 –0.002 –0.0025 –0.003 –0.0035 –0.004 –0.0045 –0.005 –0.0055 –0.006 –0.0065 –0.007 –0.0075 –0.008

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 0kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

668

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

where Re 5 0:001Re and Δϕ is voltage supply in Kilovolt. In the presence of the Coulomb force, the Nusselt number decreases with the rise of Reynolds number. The electric field helps to enhance the convective mode. So, Nuave augments with the increase of Δϕ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of the Darcy number is the same as that for the radiation parameter. Therefore, Nuave is an increasing function of Rd; Da.

Streamline

Isotherm

Da = 102

–0.002 –0.004 –0.006 –0.008 –0.01 –0.012 –0.014 –0.016 –0.018 –0.02 –0.022 –0.024 –0.026 –0.028 –0.03

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Da = 105

–0.0033 –0.005 –0.01 –0.015 –0.02 –0.025 –0.03 –0.035 –0.04 –0.045 –0.05

FIGURE 18.18

Effect of Darcy number on streamlines and isotherms when Re 5 3000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

Da = 105

Da = 102

Streamline

FIGURE 18.19

Isotherm –0.002 –0.004 –0.006 –0.008 –0.01 –0.012

–0.002 –0.004 –0.006 –0.008 –0.01 –0.012 –0.014 –0.016 –0.018 –0.02 –0.022 –0.024 –0.026 –0.028

0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Effect of Darcy number on streamlines and isotherms when Re 5 6000; Δϕ 5 10kV; φ 5 0:05; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

18.4 EFFECT OF ELECTIVE FIELD ON NANOFLUID FLOW IN A POROUS LID DRIVEN CAVITY IN EXISTENCE OF ELECTRIC FIELD

FIGURE 18.20

669

Effects of Da; Δϕ; Rd, and Re on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

670

18. INFLUENCE OF ELECTRIC FIELD ON FORCED CONVECTION OF NANOFLUID IN A POROUS MEDIUM BY MEANS OF CVFEM

FIGURE 18.20

(Continued)

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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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C H A P T E R

19 Nanofluid Heat Transfer Enhancement in Presence of Melting Surface Using CVFEM 19.1 INTRODUCTION Heat transfer generated during the melting process of a stretching sheet has various industrial applications including preparation of semiconductor materials, magma solidification, melting of permafrost, and thawing of frozen ground, etc. Epstein and Cho [1] examined the characteristics of melting heat transfer in the laminar flow over a flat plate. The work of Epstein and Cho [1] has been extended by the various researchers. Cheng and Lin [2] investigated the melting effect in mixed convective heat transfer with aiding and opposing external flows from the vertical plate in a liquid-saturated porous medium. Ishak et al. [3] studied melting heat transfer in steady laminar flow over a moving surface. Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet in a micropolar fluid has been analyzed by Yacob et al. [4]. Recently, Ahmad and Pop [5] have studied the effect of melting parameter on the dual solutions for the problem of mixed convection boundary-layer flow past a vertical flat surface embedded in a fluid-saturated porous medium. They considered only opposing flow which shows that the effect of surface melting reduces the heat transfer rate and expedites the boundary-layer separation at the solid
liquid interface. Hayat et al. [6] analyzed the stagnation-point flow of coupled stress fluid with melting heat transfer. Sheikholeslami and Rokni [7] studied the melting heat transfer influence on nanofluid flow inside a cavity in the presence of a magnetic field. Sheikholeslami and Sadoughi [8] simulated the CuO-water nanofluid heat transfer enhancement in the presence of a melting surface. Sheikholeslami et al. [9] investigated CuO-water nanofluid magnetohydrodynamic natural convection inside a sinusoidal annulus in presence of melting heat transfer. Also, nanotechnology is utilized for other applications [10
96].

19.2 MELTING HEAT TRANSFER INFLUENCE ON NANOFLUID FLOW INSIDE A CAVITY IN THE PRESENCE OF A MAGNETIC FIELD 19.2.1 Problem Definition Fig. 19.1 depicts the geometry, boundary condition, and sample element. The bottom wall is the hot wall ðT 5 Th Þ and the top one is the melting surface ðT 5 Tm Þ. Other walls are adiabatic. A horizontal magnetic field has been applied. The enclosure is filled with a nanofluid.

19.2.2 Governing Equation 2D steady convective nanofluid flow is considered in the presence of a constant magnetic field. The PDEs are: @v @u 1 50 @y @x

ρnf



  2  @u @u @ u @2 u @P 1u 1 2 5 By σnf vBx 2 B2y σnf u 1 v μ @y @x @y2 @x2 nf @x

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00019-9

675

ð19:1Þ ð19:2Þ

© 2019 Elsevier Inc. All rights reserved.

676

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

L

(A)

Melting surface Tm →

g

→

B0

CuO – Water

λ

L

Th

(B)

(C) Si,3 i Control volume

Region of support 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Si,4 j=4

f1 f2 0

0.2

0.4

0.6

0.8

i=1

1

j=3

FIGURE 19.1 (A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work (C) a sample triangular element and its corresponding control volume.

0

1 @v @v ρnf @v 1 uA 5 uBy σnf Bx 1 ðT 2 Tc Þβ nf gρnf 2 vBx σnf Bx @y @x 0 1 2 2 @P @ v @ v 1 μnf @ 2 1 2 A; 2 @y @y @x




ρCp

 nf

Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @ T @2 T 1u 1 ρCp nf v 5 knf @y @x @x2 @y2 , ρnf ; ðρβ Þnf , and σnf are calculated as:


 ρCp nf 5 φ ρCp s 1 ð1 2 φÞ ρCp f ρnf 5 φρs 1 ð1 2 φÞρf

ð19:3Þ

ð19:4Þ

ð19:5Þ ð19:6Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.2 MELTING HEAT TRANSFER INFLUENCE ON NANOFLUID FLOW INSIDE A CAVITY IN THE PRESENCE OF A MAGNETIC FIELD

ðρβ Þnf 5 φðρβ Þs 1 ð1 2 φÞðρβ Þf

σnf 1 21σs =σf 2φ 211σs =σf
 5 σf 3φ 211σs =σf

677 ð19:7Þ

!21 11

ð19:8Þ

kn f ; μn f are calculated via the Koo
Kleinstreuer
Li (KKL) model: 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi k f knf κb T 4 0 1 0 1 1 5 3 10 g ðφ; T; dp Þφρf cp;f ; 511 0 dp ρp kf k k p p @ 1 2A 2 @ 2 1A φ kf kf ð19:9Þ

Rf 5 4 3 1028 km2 =W; Rf 5 2 dp ð1=kp 2 1=kp;eff Þ;



2

 g0 φ; dp ; T 5 LnðT Þ a1 1 a3 LnðφÞ 1 a2 Ln dp 1 a5 Ln dp 1 a4 ln dp LnðφÞ



2
 1 a6 1 a8 LnðφÞ 1 a7 Ln dp 1 a10 Ln dp 1 a9 LnðφÞln dp μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð19:10Þ

Properties and needed parameters are provided in Tables 19.1 and 19.2. Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð19:11Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tm P5 ;V5 ; θ5 ; ðX; YÞ 5 ;
2 ; U 5 αnf αnf Th 2 Tm L ρnf αnf =L p

Ψ5

ψ ωL2 ;Ω5 αnf αnf

TABLE 19.1

ð19:12Þ

The Coefficient Values of CuO 2 Water Nanofluid

Coefficient values

CuO 2 Water

a1

2 26.5933108

a2

2 0.403818333

a3

2 33.3516805

a4

2 1.915825591

a5

6.421858E-02

a6

48.40336955

a7

2 9.787756683

a8

190.245610009

a9

10.9285386565

a10

2 0.72009983664

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

678 TABLE 19.2

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK 21 Þ

dp ðnmÞ

σðΩUmÞ21

Water

997.1

4179

0.613

21




0:05

CuO

6500

540

18

29

45

10210

The final formulae are: @2 Ψ @2 Ψ 1 5 0; @Y2 @X2 0 1 2 2 @Ω @Ω A 5 A 2 @@ Ω @ ΩA U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 Ω1

0 1 A A @V @U @V @U 6 2@ Bx By 2 B y Bx 1 B y By A 1 PrHa2 2 Bx Bx 1 @X @X @Y @Y A1 A4 1 Pr Ra

ð19:13Þ

ð19:14Þ

A3 A22 @θ ; A1 A24 @X  2 @θ @θ @ θ @2 θ U1 V5 1 2 @X @Y @X2 @Y

ð19:15Þ

Boundary conditions are: θ50

on top wall

θ 5 1:0 @θ 5 0:0 @n Ψ 5 0:0 and on the melting surface, we have: 

on bottom wall on other walls

ð19:16Þ

Right; left and bottom walls

  @θ  A2 1 @Ψ  5 ðL 1 cs ðTm 2 T0 ÞÞ @n Y51 A1 δ @n Y51

ð19:17Þ

where dimensionless and constants parameters are illustrated as:


 qffiffiffiffiffiffiffiffiffiffiffiffi ρCp f ðTh 2 Tm Þ  Pr 5 υf =αf ; Ra 5 gβ f ΔTL3 = υf αf ; Ha 5 LB0 σf =μf ; δ 5 ρ f 1 1 c s ð Tm 2 T0 Þ


ρnf

μnf

ðρCP Þnf

ðρβ Þnf knf σnf A1 5 ; A5 5 ; A2 5 ; A4 5 ; A6 5 ; A3 5 ρf μf ðρCP Þf kf σf ðρβ Þf It should be mentioned that δ is related to Stefan numbers. Local and average Nusselt over the hot wall can be calculated as:  knf @θ Nuloc 5 kf @y ðL 1 Nuave 5 Nuloc dx L

ð19:18Þ

ð19:19Þ

ð19:20Þ

0

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.2 MELTING HEAT TRANSFER INFLUENCE ON NANOFLUID FLOW INSIDE A CAVITY IN THE PRESENCE OF A MAGNETIC FIELD

679

19.2.3 Effects of Active Parameters The impacts of melting heat transfer on nanofluid magnetohydrodynamic flow in a cavity is examined. μnf ; knf of CuO-water nanofluid are estimated by means of KKL model. Graphs and tables are depicted for different amounts of CuO-water volume fraction (φ 5 0 to φ 5 0:04), melting parameter (δ 5 0 to 0:8), Hartmann number (Ha 5 0 to 40), and Rayleigh number (Ra 5 103 to 104 ). The impact of adding nanoparticles in the base fluid on velocity and temperature contours is depicted in Fig. 19.2. Temperature gradient decreases with the increase of φ. jΨ max j augments with the addition of nanoparticles because of the increase in the solid movements. In the presence of melting heat transfer and magnetic field, the effect of adding nanoparticles on isotherms becomes negligible. Fig. 19.3 depicts the impact of Rayleigh and Hartmann numbers in absence of melting heat transfer. There is only one eddy in the streamlines. At low Rayleigh number, the conduction mechanism is dominant. As Ra increases the distortion of the isotherms is enhanced close to the hot wall. Adding a magnetic field makes isotherms become parallel. Figs. 19.4 and 19.5 demonstrate the influence of Ra and Ha on streamlines and isotherms in the presence of the melting process. In the presence of melting heat transfer the primary eddy diminishes and the increasing Lorentz forces generate three layers for streamlines. Increasing the melting parameter increases the strength of the bottom eddy. Streamlines

Isotherms

0.1 –1

–0. 5

0 .2

–1.5

Ha = 0, δ = 0

–2

0.3

–2.5

0.4

0.5

–3

0.6 0.7 0.8

0.9

0.2

0.5

0.

6

0.4

–6

0.

0.3

Ha = 0, δ = 0.8

0.1

–8

7

–4

0 .8 –2

0.9

–0.4

–0.25

0.1

–0.1 0.2

Ha = 40, δ = 0.8

–0.05

0.3

0.00955 115 0.4

0.5

0.6 0.7

–0.000330904 2.3E–05

0.8 0.9

Influence of nanofluid volume fraction on streamline (left) and isotherm (right) contours (nanofluid (φ 5 0:04)(

) and pure fluid (φ 5 0) (- - -)) when Ra 5 104 .

FIGURE 19.2

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

680

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.3 Streamline (left) and isotherm (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0; φ 5 0:04.

Fig. 19.6 illustrates the impact of δ; Ra, and Ha on Nuave . The formula for Nuave corresponding to active parameters is: Nuave 5 0:425 1 0:122δ 2 1:63logðRaÞ 1 0:35Ha 1 1:5logðRaÞδ 2 0:58δHa 2 0:66logðRaÞHa
2 1 0:45Ha2 1 0:53 logðRaÞ 2 4:04δ2

ð19:21Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.2 MELTING HEAT TRANSFER INFLUENCE ON NANOFLUID FLOW INSIDE A CAVITY IN THE PRESENCE OF A MAGNETIC FIELD

FIGURE 19.4

681

Streamline (left) and isotherm (right) contours for different values of Rayleigh and Hartmann numbers when

δ 5 0:2; φ 5 0:04.

where Ha 5 0:1Ha. As the melting parameter augments, the temperature gradient is enhanced, and in turn the Nusselt number increases. Increasing the bouyancy forces leads the thermal boundary layer thicknesss to reduce. So the Nusselt number increases with the enhancement of Ra. As the Hartmann number augments, the isotherms become parallel to each other. Therefore, the Nusselt number has an inverse relationship with Ha.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

682

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.5 Streamline (left) and isotherm (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0:8; φ 5 0:04.

19.3 SIMULATION OF CUO-WATER NANOFLUID HEAT TRANSFER ENHANCEMENT IN THE PRESENCE OF A MELTING SURFACE 19.3.1 Problem Definition Fig. 19.7 depicts the geometry, boundary condition, and sample element. The inner wall is the hot wall ðT 5 Th Þ and the outer one is the melting surface ðT 5 Tm Þ. The other walls are adiabatic. A horizontal magnetic field has been applied. The enclosure is filled with nanofluid.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

683

19.3 SIMULATION OF CUO-WATER NANOFLUID HEAT TRANSFER ENHANCEMENT IN THE PRESENCE OF A MELTING SURFACE

FIGURE 19.6

Influences of the melting parameter, Reynolds number, and Hartmann numbers on average Nusselt number.

19.3.2 Governing Equation Nanofluid steady convective flow is considered in the presence of a constant magnetic field. The PDEs are: @v @u 1 50 @y @x

ð19:22Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

684

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.7 Geometry and the boundary conditions.



  2

 @u @u @ u @2 u @P 1u ρnf v 1 5 By σnf vBx 2 B2y σnf u 1 μnf 2 @y @x @y2 @x2 @x

ð19:23Þ

0

1 @v @v ρnf @ v 1 uA 5 uBy σnf Bx 1 ðT 2 Tc Þβ nf gρnf 2 vBx σnf Bx @y @x 0

1 2 2 @ v @ v @P 1 @ 2 1 2 Aμnf 2 ; @x @y @y

ð19:24Þ

Bx 5 Bo cosλ; By 5 Bo sinλ



ρCp

 nf

ρCp

  2 @T @T @ T @2 T 1 u v 1 5 k nf nf @y @x @x2 @y2



ð19:25Þ

, ρnf ; ðρβ Þnf , and σnf are calculated as:


ρCp

 nf



 5 φ ρCp s 1 ð1 2 φÞ ρCp f

ρnf 5 ρf ð1 2 φÞ 1 φρs ðρβ Þnf ðρβ Þf

5φ

ðρβ Þs 1 ð1 2 φÞ ðρβ Þf

2
3 
 !21 21σs =σf 2 σs =σf 21 φ
 σnf 5 4 1 15σf 3φ 211σs =σf

ð19:26Þ ð19:27Þ ð19:28Þ

ð19:29Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.3 SIMULATION OF CUO-WATER NANOFLUID HEAT TRANSFER ENHANCEMENT IN THE PRESENCE OF A MELTING SURFACE

knf ; μnf are calculated via the KKL model: 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi kf knf κb T 4 0 0 1 0 1 ; 511 1 5 3 10 g ðφ; T; dp Þφρf cp;f d kf p ρp k k p p @ 1 2A 2 @ 2 1A φ kf kf Rf 5 4 3 1028 km2 =W;

685

ð19:30Þ

Rf 5 2 dp ð1=kp 2 1=kp;eff Þ;




2
 g0 φ; dp ; T 5 LnðT Þ a1 1 a3 LnðφÞ 1 a2 Ln dp 1 a5 Ln dp 1 a4 ln dp LnðφÞ



2
 1 a6 1 a8 LnðφÞ 1 a7 Ln dp 1 a10 Ln dp 1 a9 LnðφÞln dp μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð19:31Þ

Properties and needed parameters are provided in Tables 19.1 and 19.2. Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð19:32Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tm P5 ;V5 ; θ5 ; ðX; YÞ 5 ;
2 ; U 5 α α T 2 T L m nf nf h ρnf αnf =L p

ψ ωL2 ;Ω5 Ψ5 αnf αnf

ð19:33Þ

The final formulae are: @2 Ψ @2 Ψ 1 Ω 1 5 0; @Y2 @X2 0 1 @Ω @Ω A5 A2 @@2 Ω @2 Ω A U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @ @V @U @V @U A Bx 1 By Bx 2 By B x 1 B y By 1 PrHa2 2Bx @X @X @Y @Y A1 A4 1 Pr Ra

ð19:34Þ

ð19:35Þ

A3 A22 @θ ; A1 A24 @X  2 @θ @θ @ θ @2 θ U1 V5 1 @X @Y @X2 @Y2

ð19:36Þ

Boundary conditions are: @θ 5 0:0 @n θ50 θ 5 1:0 Ψ 5 0:0

on other walls on outer wall on inner wall on all walls except melting surface

ð19:37Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

686

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

and on the melting surface, we have: @θ A2 1 @Ψ 5 ðL 1 cs ðTm 2 T0 ÞÞ @n A1 δ @n

ð19:38Þ

where dimensionless and constants parameters are illustrated as:


 qffiffiffiffiffiffiffiffiffiffiffiffi ρCp f ðTh 2 Tm Þ  Pr 5 υf =αf ; Ra 5 gβ f ΔTL = υf αf ; Ha 5 LB0 σf =μf ; δ 5 ρ f 1 1 c s ð Tm 2 T0 Þ 3

ρnf

μnf




ðρCP Þnf

σnf knf ; A5 5 ; A2 5 ; A6 5 ; A4 5 ; A1 5 ρf μf ðρCP Þf σf kf

A3 5

ðρβ Þnf ðρβ Þf

It should be mentioned that δ is related to Stefan numbers. Nuloc and Nuave over the cold wall can be calculated as:  knf @θ Nuloc 5 kf @r Nuave 5

1 0:5π

ð19:39Þ

ð19:40Þ

0:5π ð

Nuloc dξ

ð19:41Þ

0

19.3.3 Effects of Active Parameters Nanofluid in the presence of melting heat transfer is presented under the effect of Lorentz forces. The KKL model is selected to calculate properties of the nanofluid. Outputs are demonstrated for various number of undulation (N 5 3 to 5), volume fraction of nanofluid (φ 5 0 to 0:04), Hartmann number (Ha 5 0 to 40), melting parameter (δ 5 0 to 0:2), and Rayleigh number (Ra 5 500 to 5000). Fig. 19.8 presents the influence of using nanofluid instead of base fluid. Velocity increases with the use of nanofluid due to the increase in the solid movements. In the presence of a magnetic field and melting heat transfer, the impact of adding nanoparticles on isotherms becomes insignificant. The temperature gradient reduces by the use of the nanofluid. Influences of Hartmann and Rayleigh numbers without melting heat transfer are illustrated in Figs. 19.9
19.11. The streamlines have just one rotating eddy. Conduction mode is prominent in low Raleigh numbers. As the buoyancy force is enhanced, the distortion of isotherms increases close to the hot wall. Increasing the Hartmann number causes the isotherms to become parallel. In the presence of melting heat transfer the primary eddy diminishes and enhancing Lorentz forces generate three layers for streamlines. The bottom eddy becomes stronger with the increase of the melting parameter. As the number of undulations increases, the temperature gradient near the hot wall is enhanced. The effects of δ; Ra; N, and Ha on average Nusselt number are depicted in Fig. 19.12. The following correlation can be offered: Nuave 5 0:068 1 0:505N 2 0:15δ 1 0:08Ra 1 0:11Ha 0:13Nδ 1 0:002N Ra 2 0:021NHa 1 0:056Ra δ 2 0:14δHa 2 0:008Ra Ha 2 0:0485N 2

ð19:42Þ

1 0:49δ2 2 0:007Ra2 2 0:011Ha2 where Ha 5 0:1Ha; Ra 5 0:001Ra. The temperature gradient increases with the increase of melting parameter. So, the Nusselt number has a direct relationship with δ. Augmenting the Rayleigh number leads to a reduction in thermal boudary layer thicknesss. Therefore, the Nusselt number is enhanced with the increase of Ra. As the Lorentz forces increase, the conduction mechanism become stronger. Thus, Nuave reduces with the increase of the Hartmann number. In addition, the rate of heat transfer increases with the rise of N.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.3 SIMULATION OF CUO-WATER NANOFLUID HEAT TRANSFER ENHANCEMENT IN THE PRESENCE OF A MELTING SURFACE

687

FIGURE 19.8 Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0; φ 5 0:04; N 5 3.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

688

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.9 Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0:2; φ 5 0:04; N 5 3.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.3 SIMULATION OF CUO-WATER NANOFLUID HEAT TRANSFER ENHANCEMENT IN THE PRESENCE OF A MELTING SURFACE

689

FIGURE 19.10

Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0; φ 5 0:04; N 5 5.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

690

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.11

Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0:2; φ 5 0:04; N 5 5.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.4 CUO-WATER NANOFLUID MAGNETOHYDRODYNAMIC NATURAL CONVECTION INSIDE A SINUSOIDAL ANNULUS

FIGURE 19.12

691

Influences of the number of undulation, Darcy, Reynolds, and Hartmann numbers on average Nusselt number.

19.4 CUO-WATER NANOFLUID MAGNETOHYDRODYNAMIC NATURAL CONVECTION INSIDE A SINUSOIDAL ANNULUS IN THE PRESENCE OF MELTING HEAT TRANSFER 19.4.1 Problem Definition Fig. 19.13 depicts the geometry, boundary condition, and sample element. The inner wall is the hot wall ðT 5 Th Þ and the outer one is the melting surface ðT 5 Tm Þ. The other walls are adiabatic. A horizontal magnetic field has been applied. The enclosure is filled with nanofluid.

19.4.2 Governing Equation 2D steady convective nanofluid flow is considered in the presence of a constant magnetic field. The PDEs are: @v @u 1 50 @y @x

ð19:43Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

692

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.13

(A) Geometry (B) Sample element.



  2

 @u @u @ u @2 u @P 1u ρnf v 1 5 By σnf vBx 2 B2y σnf u 1 μnf 2 @y @x @y2 @x2 @x 0 1 @v @v ρnf @ v 1 uA 5 uBy σnf Bx 1 ðT 2 Tc Þβ nf gρnf 2 vBx σnf Bx @y @x 0 1 2 2 @ v @ v @P 1 @ 2 1 2 Aμnf 2 ; @x @y @y




ρCp

 nf

Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @ T @2 T 1u 1 2 ρCp nf v 5 knf @y @x @x2 @y

ð19:47Þ

ρnf 5 ρf ð1 2 φÞ 1 φρs

σnf 5 σf 4

ðρβ Þnf 5 φðρβ Þs 1 ð1 2 φÞðρβ Þf

ð19:45Þ

ð19:46Þ

, ρnf ; ðρβ Þnf , and σnf are calculated as:


 ρCp nf 5 φ ρCp s 1 ð1 2 φÞ ρCp f

2

ð19:44Þ

ð19:48Þ 3



!21 1 21σs =σf 2φ 211σs =σf
 1 15 3φ 211σs =σf

knf ; μnf are calculated via the KKL model: 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi k f knf κb T 4 0 1 0 1 1 5 3 10 g ðφ; T; dp Þφρf cp;f ; 511 0 dp ρ p kf k k p p @ 1 2A 2 @ 2 1A φ kf kf Rf 5 4 3 1028 km2 =W; Rf 5 2 dp ð1=kp 2 1=kp;eff Þ;



2

 g0 φ; dp ; T 5 LnðT Þ a1 1 a3 LnðφÞ 1 a2 Ln dp 1 a5 Ln dp 1 a4 ln dp LnðφÞ



2
 1 a6 1 a8 LnðφÞ 1 a7 Ln dp 1 a10 Ln dp 1 a9 LnðφÞln dp

ð19:49Þ ð19:50Þ

ð19:51Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.4 CUO-WATER NANOFLUID MAGNETOHYDRODYNAMIC NATURAL CONVECTION INSIDE A SINUSOIDAL ANNULUS

μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

693 ð19:52Þ

Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð19:53Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tm P5 ; ;V5 ; θ5 ; ðX; YÞ 5
2 ; U 5 L α α T 2 T m nf nf h ρnf αnf =L p

ψ ωL2 Ψ5 ;Ω5 αnf αnf

ð19:54Þ

The final formulae are: @2 Ψ @2 Ψ 1 5 0; 2 @Y @X2 0 1 @Ω @Ω A5 A2 @@2 Ω @2 Ω A U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 Ω1

ð19:55Þ

0 1 A A @V @U @V @U 6 2 @ 2 B x Bx 1 By Bx 2 By B x 1 B y By A 1 PrHa2 @X @X @Y @Y A1 A4 1 Pr Ra

ð19:56Þ

A3 A22 @θ ; A1 A24 @X  2 @θ @θ @ θ @2 θ U1 V5 1 @X @Y @X2 @Y2

ð19:57Þ

Boundary conditions are: @θ 5 0:0 @n

on other walls

θ50

on outer wall

θ 5 1:0

on inner wall

Ψ 5 0:0

on all walls except melting surface

ð19:58Þ

and on the melting surface, we have: @θ A2 1 @Ψ 5 ðL 1 cs ðTm 2 T0 ÞÞ @n A1 δ @n

ð19:59Þ

where dimensionless and constants parameters are illustrated as:


 qffiffiffiffiffiffiffiffiffiffiffiffi ρCp f ðTh 2 Tm Þ
 Pr 5 υf =αf ; Ra 5 gβ f ΔTL3 = υf αf ; Ha 5 LB0 σf =μf ; δ 5 ρf 1 1 cs ðTm 2 T0 Þ A1 5

ρnf ρf

; A5 5

μnf μf

; A2 5

ðρCP Þnf ðρCP Þf

; A6 5

σnf knf ; A4 5 ; σf kf

A3 5

ðρβ Þnf

ð19:60Þ

ðρβ Þf

It should be mentioned that δ is related to Stefan numbers.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

694

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

Local and average Nusselt over the cold wall can be calculated as:  knf @θ Nuloc 5 kf @r 1 Nuave 5 0:5π

ð19:61Þ

0:5π ð

ð19:62Þ

Nuloc dξ 0

19.4.3 Effects of Active Parameters Nanofluid flow in a half sinusoidal annulus due to magnetic field in presence of melting surface is examined. μnf ; knf of CuO-water nanofluid are estimated by means of the KKL model. Graphs and tables are depicted for different amounts of CuO-H2O volume fraction (φ 5 0 to 0:04), melting parameter (δ 5 0 to 0:2), Rayleigh number (Ra 5 500 to 5000), and Hartmann number (Ha 5 0 to 40). The impact of adding CuO nanoparticles in water on velocity and temperature contours is depicted in Fig. 19.14. The temperature gradient decreases with the increase of φ. jΨ max j augments with the addition of nanoparticles because of the increase in the solid movements. In the presence of melting heat transfer and magnetic field, the effect of the addition of nanoparticles on the isotherms becomes negligible. Figs. 19.15 and 19.16 depict the impact of Rayleigh and Hartmann numbers in the absence of melting heat transfer. There is only one eddy in the streamlines. At low Rayleigh number, the conduction mechanism is dominant. As Ra increases, the distortion of isotherms is enhanced close to the hot wall. Adding a magnetic field makes the isotherms become parallel. In the presence of melting heat transfer the primary eddy diminishes and increasing Lorentz forces generate three layers of the streamlines. Increasing the melting Isotherms

00.6

–16

0.4

0.2

00.33

Streamlines

0..1

–112

0.7

Ha = 0

–8

0..9 –4 –2

0.1

0.2 –0.2 –0.2

0.3

–0.6

0 .4

–0.8 –1

0.5

Ha = 40

–1

0.

7

0.9

FIGURE 19.14 Impact of adding CuO in water on streamlines and isotherms (nanofluid (φ 5 0:04)(

) and pure fluid (φ 5 0) (- - -)) when Ra 5 5000; δ 5 0:2.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.4 CUO-WATER NANOFLUID MAGNETOHYDRODYNAMIC NATURAL CONVECTION INSIDE A SINUSOIDAL ANNULUS

FIGURE 19.15

695

Isotherms and streamlines for various Ha; Ra when δ 5 0; φ 5 0:04.

parameter enhances the bottom eddy. Fig. 19.17 illustrates the impact of δ; Ra , and Ha on Nuave . The formula for Nuave corresponding to active parameters is: Nuave 5 0:799 1 0:305δ 1 0:19Ra 2 0:16Ha 1 0:003Ra δ 2 0:0039δHa 2 0:028Ra Ha 1 0:058δ2 2 0:012Ra2 1 0:039Ha2

ð19:63Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

696

FIGURE 19.16

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

Isotherms and streamlines for various Ha; Ra when δ 5 0:2; φ 5 0:04.

where Ha 5 0:1Ha; Ra 5 0:001Ra. As the melting parameter increases, the temperature gradient is enhanced and in turn the Nusselt number is enhanced. Increasing the bouyancy forces leads to a reduction in the thickness of the thermal boundary layer . So the Nusselt number increases with the enhancement of Ra. As the Hartmann number augments, the isotherms become parallel. Therefore the Nusselt number has an inverse relationship with Ha.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.5 MHD NANOFLUID NATURAL CONVECTION INSIDE A HALF ANNULUS WITH MELTING SURFACE

FIGURE 19.17

697

Impacts of the δ; Ha; Ra on Nuave .

19.5 MHD NANOFLUID NATURAL CONVECTION INSIDE A HALF ANNULUS WITH MELTING SURFACE 19.5.1 Problem Definition Fig. 19.18 depicts the geometry, boundary condition, and sample element. The inner wall is the hot wall ðT 5 Th Þ and the outer one is the melting surface ðT 5 Tm Þ. The other walls are adiabatic. A horizontal magnetic field has been applied. The enclosure is filled with nanofluid.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

698

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

FIGURE 19.18

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

19.5.2 Governing Equation 2D steady convective nanofluid flow is considered in the presence of a constant magnetic field. The PDEs are: @v @u 1 50 @y @x



  2

 @u @u @ u @2 u @P 2 1u ρnf v 1 2 μnf 2 5 By σnf vBx 2 By σnf u 1 @y @x @y2 @x @x 0 1 @v @v ρnf @v 1 uA 5 uBy σnf Bx 1 ðT 2 Tc Þβ nf gρnf 2 vBx σnf Bx @y @x 0 1 2 2 @P @ v @ v 1 μnf @ 2 1 2 A; 2 @y @y @x Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @ T @2 T 1u ρCp nf v 1 5 knf @y @x @x2 @y2

ð19:64Þ ð19:65Þ

ð19:66Þ

ð19:67Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.5 MHD NANOFLUID NATURAL CONVECTION INSIDE A HALF ANNULUS WITH MELTING SURFACE




ρCp

 nf

, ρnf ; ðρβ Þnf , and σnf are calculated as:


 ρCp nf 5 φ ρCp s 1 ð1 2 φÞ ρCp f

699

ð19:68Þ

ρnf 5 φρs 1 ð1 2 φÞρf

ð19:69Þ

ðρβ Þnf 5 φðρβ Þs 1 ð1 2 φÞðρβ Þf

ð19:70Þ



!21 σnf 1 21σs =σf 2φ 211σs =σf
 5 11 σf 3φ 211σs =σf

ð19:71Þ

knf ; μnf are calculated via the KKL model: 0 1 k p 3@ 2 1A φ sffiffiffiffiffiffiffiffiffi kf knf κb T 4 0 0 1 0 1 ; 511 1 5 3 10 g ðφ; T; dp Þφρf cp;f d kf p ρp k k p p @ 1 2A 2 @ 2 1A φ kf kf Rf 5 4 3 1028 km2 =W;

ð19:72Þ

Rf 5 2 dp ð1=kp 2 1=kp;eff Þ;



2

 g0 φ; dp ; T 5 LnðT Þ a1 1 a3 LnðφÞ 1 a2 Ln dp 1 a5 Ln dp 1 a4 ln dp LnðφÞ



2
 1 a6 1 a8 LnðφÞ 1 a7 Ln dp 1 a10 Ln dp 1 a9 LnðφÞln dp μnf 5

μf ð12φÞ

2:5

1

μf kBrownian 3 kf Pr

ð19:73Þ

Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

ð19:74Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tm P5 ;V5 ; θ5 ; ðX; YÞ 5 ;
2 ; U 5 α α T 2 T L m nf nf h ρnf αnf =L p

ψ ωL2 ;Ω5 Ψ5 αnf αnf

ð19:75Þ

The final formulae are: @2 Ψ @2 Ψ 1 5 0; @Y2 @X2 0 1 @Ω @Ω A5 A2 @@2 Ω @2 Ω A U1V 5 Pr 1 @X @Y @X2 A1 A4 @Y2 0 1 A6 A2 @ @V @U @V @U B x By 2 By B x 1 By B y A 1 PrHa2 2 B x Bx 1 @X @X @Y @Y A1 A4 Ω1

1 Pr Ra

ð19:76Þ

ð19:77Þ

A3 A22 @θ ; A1 A24 @X  2 @θ @θ @ θ @2 θ U1 V5 1 @X @Y @X2 @Y2

ð19:78Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

700

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

Boundary conditions are: @θ 5 0:0 @n θ50 θ 5 1:0 Ψ 5 0:0

on other walls ð19:79Þ

on outer wall on inner wall on all walls except melting surface

and on the melting surface, we have: @θ A2 1 @Ψ 5 ðL 1 cs ðTm 2 T0 ÞÞ @n A1 δ @n

ð19:80Þ

Isotherms

–3

–1

–2

Streamlines

0.7 0.8

–5

Ha = 0, δ = 0

–6

0.6

0.9

0.5 0.4

0.3 0.2 0.1

0.1

0.6

0.2

0.5

0.3

0.7

Ha = 0, δ = 0.2

–16

0 .8 –14

0.9 –12

–8

–6 –4

0.8 0.9

0.7

0.6 0 .5

–1

0 .4

Ha = 40, δ = 0.2

–2

0.3 – 0.8

0.2 –0

–0

0.1 .6

.4

– 0.2

FIGURE 19.19 Influence of nanofluid volume fraction on streamlines (left) and isotherms (right) contours (nanofluid (φ 5 0:04)(

) and pure fluid (φ 5 0) (- - -)) when Ra 5 5000

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.5 MHD NANOFLUID NATURAL CONVECTION INSIDE A HALF ANNULUS WITH MELTING SURFACE

FIGURE 19.20

701

Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when δ 5 0; φ 5 0:04.

where dimensionless and constants parameters are illustrated as:


 qffiffiffiffiffiffiffiffiffiffiffiffi ρCp f ðTh 2 Tm Þ
 Pr 5 υf =αf ; Ra 5 gβ f ΔTL = υf αf ; Ha 5 LB0 σf =μf ; δ 5 ρf 1 1 cs ðTm 2 T0 Þ 3

ρnf

μnf

ðρCP Þnf

ðρβ Þnf knf σnf A1 5 ; A5 5 ; A2 5 ; A4 5 ; A6 5 ; A3 5 ρf μf ðρCP Þf kf σf ðρβ Þf

ð19:81Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

702

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

It should be mentioned that δ is related to Stefan numbers. Local and average Nusselt over the hot wall can be calculated as:  knf @θ Nuloc 5 kf @r

FIGURE 19.21

ð19:82Þ

Isotherm (left) and streamline (right) contours for different values of Rayleigh and Hartmann numbers when

δ 5 0:2; φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

19.5 MHD NANOFLUID NATURAL CONVECTION INSIDE A HALF ANNULUS WITH MELTING SURFACE

1 Nuave 5 0:5π

703

0:5π ð

Nuloc dξ

ð19:83Þ

0

19.5.3 Effects of Active Parameters The effect of melting heat transfer on nanofluid MHD flow in a half annulus is examined. μnf ; knf of CuOwater nanofluid are estimated by means of the KKL model. Graphs and tables depict the different amounts of CuO-H2O volume fraction (φ 5 0 to φ 5 0:04), Melting parameter (δ 5 0 to 0:8), Hartmann number (Ha 5 0 to 40), and Rayleigh number (Ra 5 103 to 104 ). The impact of the addition of nanoparticles in the base fluid on velocity and temperature contours is depicted in Fig. 19.19. The temperature gradient decreases with the increase of φ. jΨ max j augments with the addition of the nanoparticles because of the increase in the solid movements. In the presence of melting heat transfer and a magnetic field, the effect of adding nanoparticles on the isotherms becomes negligible. Figs. 19.20 and 19.21 depict the impacts of Rayleigh and Hartmann numbers in the absence of melting heat transfer. There is only one eddy in the streamlines. At low Rayleigh number, the conduction mechanism is dominant. As Ra increases the distortion of the isotherms is enhanced close to the hot wall. Adding a magnetic field makes isotherms become parallel. In the presence of melting heat transfer the primary eddy diminishes and increasing Lorentz forces generate three layers of the streamlines. Increasing the melting parameter enhances the bottom eddy. Fig. 19.22 illustrates the impact of δ; Ra , and Ha on Nuave . The formula for Nuave corresponding to active parameters is: Nuave 5 0:425 1 0:122δ 2 1:63logðRaÞ 1 0:35Ha 1 1:5logðRaÞδ 2 0:58δHa 2 0:66logðRaÞHa
2 1 0:45Ha2 1 0:53 logðRaÞ 2 4:04δ2

ð19:84Þ

where Ha 5 0:1Ha. As the melting parameter increases, the temperature gradient is enhanced and in turn the Nusselt number is enhanced. Increasing the bouyancy forces leads to the reduction in thickness of the thermal boudary layer. So the Nusselt number increases with the enhancement of Ra. As the Hartmann number augments, the isotherms become parallel. Therefore the Nusselt number has an inverse relationship with Ha.

FIGURE 19.22

Influences of the Darcy, Reynolds, and Hartmann numbers on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

704

19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

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124. [58] M. Sheikholeslami, M. Seyednezhad, Lattice Boltzmann Method simulation for CuO-water nanofluid flow in a porous enclosure with hot obstacle, J. Mol. Liq. 243 (2017) 249
256. [59] M. Sheikholeslami, M. Sadoughi, Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles, Int. J. Heat Mass Transfer 113 (2017) 106
114. [60] M. Sheikholeslami, A. Zeeshan, Mesoscopic simulation of CuO-H2O nanofluid in a porous enclosure with elliptic heat source, Int. J. Hydrogen Energy 42 (22) (2017) 15393
15402. [61] M. Sheikholeslami, S.A. Shehzad, CVFEM for influence of external magnetic source on Fe3O4
H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
191. [62] M. Sheikholeslami, M. Seyednezhad, Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEM, Int. J. Heat Mass Transfer 114 (2017) 1169
1180. [63] M. Sheikholeslami, H.B. Rokni, Free convection of CuO-H2O nanofluid in a curved porous enclosure using mesoscopic approach, Int. J. Hydrogen Energy 42 (22) (2017) 14942
14949. [64] M. Sheikholeslami, Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method, J. Mol. Liq. 249 (2018) 739
746.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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19. NANOFLUID HEAT TRANSFER ENHANCEMENT IN PRESENCE OF MELTING SURFACE USING CVFEM

[65] M. Sheikholeslami, H.B. Rokni, Influence of melting surface on MHD nanofluid flow by means of two phase model, Chin. J. Phys. 55 (2017) 1352
1360. [66] M. Sheikholeslami, H.B. Rokni, Effect of melting heat transfer on nanofluid flow in existence of magnetic field considering Buongiorno Model, Chin. J. Phys. 55 (2017) 1115
1126. [67] M. Sheikholeslami, T. Hayat, A. Alsaedi, MHD free convection of Al2O3
water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
524. [68] M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Forced convection heat transfer in a semi annulus under the influence of a variable magnetic field, Int. J. Heat Mass Transfer 92 (2016) 339
348. [69] M. Sheikholeslami, Numerical modeling of Nano enhanced PCM solidification in an enclosure with metallic fin, J. Mol. Liq. 259 (2018) 424
438. [70] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer 123 (2018) 418
431. [71] M. Sheikholeslami, S.A. Shehzad, CVFEM simulation for nanofluid migration in a porous medium using Darcy model, Int. J. Heat Mass Transfer 122 (2018) 1264
1271. [72] M. Sheikholeslami, M. Darzi, M.K. Sadoughi, Heat transfer improvement and Pressure Drop during condensation of refrigerant-based Nanofluid; An Experimental Procedure, Int. J. Heat Mass Transfer 122 (2018) 643
650. [73] M. Sheikholeslami, H.B. Rokni, CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of Non-equilibrium model, J. Mol. Liq. 254 (2018) 446
462. [74] M. Sheikholeslami, S.A. Shehzad, Z. Li, Water based nanofluid free convection heat transfer in a three dimensional porous cavity with hot sphere obstacle in existence of Lorenz forces, Int. J. Heat Mass Transfer 125 (2018) 375
386. [75] M. Sheikholeslami, M. Jafaryar, D.D. Ganji, Z. Li, Exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators, J. Mol. Liq. 262 (2018) 104
110. [76] M. Sheikholeslami, M. Jafaryar, Z. Li, Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles, Int. J. Heat Mass Transfer 124 (2018) 980
989. [77] M. Sheikholeslami, Z. Li, M. Shamlooei, Nanofluid MHD natural convection through a porous complex shaped cavity considering thermal radiation, Phys. Lett. A 382 (2018) 1615
1632. [78] M. Sheikholeslami, H.B. Rokni, Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects, Phys. Fluids 30 (1). Available from: https://doi.org/10.1063/1.5012517. [79] M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via Non-equilibrium model, Int. J. Heat Mass Transfer 120 (2018) 1200
1212. [80] M. Sheikholeslami, S.A. Shehzad, Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force, Chin. J. Phys. 56 (2018) 270
281. [81] M. Sheikholeslami, M. Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, Int. J. Heat Mass Transfer 120 (2018) 772
781. [82] M. Sheikholeslami, M. Shamlooei, R. Moradi, Numerical simulation for heat transfer intensification of nanofluid in a permeable curved enclosure considering shape effect of Fe3O4 nanoparticles, Chem. Eng. Process.: Process Intensif. 124 (2018) 71
82. [83] M. Sheikholeslami, T. Hayat, T. Muhammad, A. Alsaedi, MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method, Int. J. Mech. Sci. 135 (2018) 532
540. [84] M. Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, J. Mol. Liq. 249 (2018) 1212
1221. [85] M. Sheikholeslami, CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion, J. Mol. Liq. 249 (2018) 921
929. [86] M. Sheikholeslami, D.D. Ganji, Influence of electric field on Fe3O4- water nanofluid radiative and convective heat transfer in a permeable enclosure, J. Mol. Liq. 250 (2018) 404
412. [87] M. Sheikholeslami, S.A. Shehzad, Z. Li, Nanofluid heat transfer intensification in a permeable channel due to magnetic field using Lattice Boltzmann method, Phys. B (2018). Available from: https://doi.org/10.1016/j.physb.2018.03.036. [88] M. Sheikholeslami, A. Zeeshan, Numerical simulation of Fe3O4 -water nanofluid flow in a non-Darcy porous media, Int. J. Numer. Methods Heat Fluid Flow 28 (3) (2018) 641
660. Available from: https://doi.org/10.1108/HFF-04-2017-0160. [89] M. Sheikholeslami, D.D. Ganji, Numerical approach for magnetic nanofluid flow in a porous cavity using CuO nanoparticles, Mater. Design 120 (2017) 382
393. [90] M. Sheikholeslami, D.D. Ganji, Numerical modeling of magnetohydrodynamic CuO-water transportation inside a porous cavity considering shape factor effect, Colloids Surf., A 529 (2017) 705
714. [91] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation for forced convection flow of MHD CuO-H2O nanofluid inside a cavity by means of LBM, J. Mol. Liq. 249 (2018) 941
948. [92] M. Sheikholeslami, H.R. Kataria, A.S. Mittal, Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium, J. Mol. Liq. 257 (2018) 12
25. [93] M. Sheikholeslami, M. Barzegar Gerdroodbary, S. Valiallah Mousavi, D.D. Ganji, R. Moradi, Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field, J. Magn. Magn. Mater. 460 (2018) 302
311. [94] M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry, J. Mol. Liq. 213 (2016) 153
161. [95] M. Sheikholeslami, Numerical simulation for solidification in a LHTESS by means of Nano-enhanced PCM, J. Taiwan Inst. Chem. Eng. 86 (2018) 25
41. [96] M. Sheikholeslami, M. Darzi, Z. Li, Experimental investigation for entropy generation and exergy loss of nano-refrigerant condensation process, Int. J. Heat Mass Transfer 125 (2018) 1087
1095.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

C H A P T E R

20 Nanofluid Convective Heat Transfer Considering Magnetic Field Dependent (MFD) Viscosity by Means of CVFEM 20.1 INTRODUCTION Ferrohydrodynamics deals with the study of ferrofluids in the presence of a magnetic field and it is a special branch of magnetohydrodynamics. During the last decades, extensive research work has been done on these fluids since the effect of magnetization has yielded interesting information leading to diverse fascinating technological applications [1]. Ferrofluids consist of colloidal suspensions of single domain magnetic nanoparticles and it has been recognized that they have promising potential for heat transfer applications in electronics, engines, micro- and nanoelectromechanical systems (MEMS and NEMS), air-conditioning and ventilation systems [2]. Under the circumstances, the study of thermal convection in ferrofluids is gaining much importance in recent years. Moreover, many physical properties of these fluids can be tuned by varying the magnetic field. One of the well-known phenomena generated by the influence of magnetic fields on ferrofluids is the change of their viscous behavior. Realizing the importance of magnetic field dependent (MFD) viscosity on ferrofluid flows, several studies have been undertaken in the past. The effect of a homogeneous magnetic field on the viscosity of a fluid with solid particles possessing intrinsic magnetic moments has been investigated by Shliomis [3]. The effect of MFD viscosity on the onset of ferroconvection in a rotating ferrofluid layer is discussed by Vaidyanathan et al. [4]. Nanjundappa et al. [5] have investigated the effect of MFD viscosity on the onset of convection in a ferromagnetic fluid layer in the presence of a vertical magnetic field by considering the bounding surfaces are either rigidferromagnetic or stress- free with constant heat flux conditions. Sheikholeslami et al. [6] considered MFD viscosity effect on natural convection of ferrofluid. Sheikholeslami [7] studied the magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder. Sheikholeslami and Shehzad [8] studied the thermal radiation of ferrofluid in the presence of Lorentz forces considering variable viscosity. Sheikholeslami and Rokni [9] reported the magnetic nanofluid natural convection in the presence of thermal radiation. Sheikholeslami and Abelman [10] presented the numerical analysis of the effect of magnetic field on Fe3O4-water ferrofluid convection with thermal radiation. Sheikholeslami et al. [11] investigated the nanofluid radiation and natural convection in an enclosure with elliptical cylinders. Sheikholeslami and Sadoughi [12] studied the Fe3O4water nanofluid flow in porous medium considering MFD viscosity. The magnetic field effect on nanofluid treatment was investigated in recent years [13
98].

20.2 NATURAL CONVECTION OF MAGNETIC NANOFLUID CONSIDERING MFD VISCOSITY EFFECT 20.2.1 Problem Definition The geometry of this problem is shown in Fig. 20.1A. The heat source is centrally located on the bottom surface and its length L/3. The cooling is achieved by the two vertical walls. The heat source has constant heat flux qv

Application of Control Volume based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer. DOI: https://doi.org/10.1016/B978-0-12-814152-6.00020-5

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© 2019 Elsevier Inc. All rights reserved.

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20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.1 (A) Geometry and the boundary conditions; (B) a sample triangular element and its corresponding control volume.

while the cooling walls have a constant temperature Tc ; all the other surfaces are adiabatic. Also, it is also qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi assumed that the uniform magnetic field (B 5 Bx ex 1 By ey ) of constant magnitude B 5 B2x 1 B2y is applied, where ex

-

and ey are unit vectors in the Cartesian coordinate system. The orientation of the magnetic field forms an angle
 θM with horizontal axis such that θM 5 cot21 Bx =By . The electric current J and the electromagnetic force F are

- -

- - defined by J 5 σ V 3 B and F 5 σ V 3 B 3 B , respectively.

20.2.2 Governing Equation The flow is steady, two-dimensional, laminar, and incompressible. The induced electric current and Joule heating are neglected. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected compared to the applied magnetic field. Neglecting displacement currents, induced magnetic field, and using the Boussinesq approximation, the governing equations of heat transfer and fluid flow for nanofluid can be obtained as follows: @u @v 1 50 @x @y  2  @u @u @P @ u @2 u ρnf u 1v 1η 1 2 52 @x @y @x @x2 @y  1 σnf B0 2 v sinθM cosθM 2 u sin2 θM  2  @v @v @P @ v @2 v 1η ρnf u 1 v 1 52 1 ρnf β nf gðT 2 Tc Þ @x @y @y @x2 @y2
 1 σnf B0 2 u sinθM cosθM 2 v cos2 θM u

 2 @T @T @ T @2 T 1v 5 αnf 1 @x @y @x2 @y2

ð20:1Þ

ð20:2Þ

ð20:3Þ

ð20:4Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.2 NATURAL CONVECTION OF MAGNETIC NANOFLUID CONSIDERING MFD VISCOSITY EFFECT

709

--

where η 5 ð1 1 δ :B Þμnf , the variation of MFD viscosity ðδÞ has been taken to be isotropic, δ1 5 δ2 5 δ3 5 δ. The
 effective density (ρnf ), the thermal expansion coefficient (β nf ), heat capacitance ρCp nf , and electrical conductivity
 of nanofluid σnf of the nanofluid are defined as: ρnf 5 ρf ð1 2 φÞ 1 ρs φ β nf 5 β f ð1 2 φÞ 1 β s φ


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ
 σnf 3 σs =σf 2 1 φ 
 511
σf σs =σf 1 2 2 σs =σf 2 1 φ

ð20:5Þ

ð20:6Þ

The Koo
Kleinstreuer
Li (KKL) model is used to simulate thermal conductivity of nanofluid: keff 5 kstatic 1 kBrownian
 3 kp =kf 2 1 φ kstatic 
 511
kf kp =kf 1 2 2 kp =kf 2 1 φ sffiffiffiffiffiffiffiffiffi κb T 0 kBrownian 5 5 3 104 φρf cp;f g ðT; φ; dp Þ ρ p dp dp dp 5 ; Rf 5 4 3 1028 km2 =W kp kp;eff



2  g0 T; φ; dp 5 a1 1 a2 ln dp 1 a3 lnðφÞ 1 a4 lnðφÞln dp 1 a5 ln dp lnðT Þ




2  1 a6 1 a7 ln dp 1 a8 lnðφÞ 1 a9 lnðφÞln dp 1 a10 ln dp Rf 1

ð20:7Þ ð20:8Þ ð20:9Þ ð20:10Þ

ð20:11Þ

with the coefficients ai (i 5 0.10) based on the type of nanopartices, Al2O3-water nanofluids has an R2 of 98%, respectively (Table 20.1). The effective viscosity due to micro
mixing in suspensions can be obtained as follows: μeff 5 μstatic 1 μBrownian 5 μstatic 1 where μstatic 5

μf ð12φÞ2:5

μf kBrownian 3 kf Prf

ð20:12Þ

is viscosity of the nanofluid, as given originally by Brinkman.

TABLE 20.1

The Coefficient Values of Al2 O3 2 Water Nanofluid

Coefficient values

Al2 O3 2 Water

a1

52.813488759

a2

6.115637295

a3

0.6955745084

a4

4.17455552786E-02

a5

0.176919300241

a6


298.19819084

a7


34.532716906

a8


3.9225289283

a9


0.2354329626

a10


0.999063481

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The stream function and vorticity are defined as: u5

@ψ @ψ @v @u ; v52 ; ω5 2 @y @x @x @y

ð20:13Þ

The stream function satisfies the continuity Eq. (20.1). The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e., by taking the y-derivative of Eq. (20.2) and subtracting from it the x-derivative of Eq. (20.3). This gives:  2  @ψ @ω @ψ @ω @ω @2 ω @T 2 5 υnf ð1 1 δB0 ðcosθM 1 sinθM ÞÞ 1 g 1 β nf 2 2 @y @x @x @y @x @y @x 0 1 δv δu sinθM cosθM 1 sin2 θM 2 ð20:14Þ 2B C δy δy σnf B0 B C 1 B C ρnf @ 1 δu sinθ cosθ 2 δv cos2 θ A M M M δx δx  2 @ψ @T @ψ @T @ T @2 T 2 5 αnf 1 2 @y @x @x @y @x2 @y

ð20:15Þ

@2 ψ @2 ψ 1 2 52ω @x2 @y

ð20:16Þ

By introducing the following nondimensional variables: X5

x y ωL2 ψ T 2 Tc uL vL ; U5 ; Y5 ; Ω5 ; Ψ 5 ; Θ5
; V5 L L αf α αf αf qvL=kf f

Using the dimensionless parameters, the equations now become:
  2   μnf ρf kf ρCp nf β nf @Θ @Ψ @Ω @Ψ @Ω @Ω @2 Ω 
 2 5 Prf ð1 1 δ ðcosθM 1 sinθM ÞÞ 1 1 Raf Prf @Y @X @X @Y @X2 @Y2 μf ρnf knf ρCp f β f @X  σnf ρf δV δU δU δV 1 Ha2 Prf tan2 θM 1 tanθM 2 2 tanθM 1 δY δY δX δX σf ρnf

ð20:17Þ

ð20:18Þ


  @Ψ @Θ @Ψ @Θ knf ρCp f @2 Θ
 2 5 @Y @X @X @Y kf ρCp nf @X2

ð20:19Þ

@2 Ψ @2 Ψ 1 52Ω @X2 @Y2

ð20:20Þ

qffiffiffiffiffiffiffiffiffiffiffiffi
 where Raf 5 gβ f L4 qv= kf αf υf is the Rayleigh number for the base fluid, Ha 5 LBx σf =μf is the Hartmann number, and Prf 5 υf =αf is the Prandtl number for the base fluid. Also δ 5 δB0 is the viscosity parameter. The thermophysical properties of the nanofluid are given in Table 20.2. The boundary conditions as shown in Fig. 20.1 are: @Θ=@n 5 2 1:0 Θ 5 0:0 @Θ=@n 5 0:0 Ψ 5 0:0 TABLE 20.2

on the heat source on the left and right

ð20:21Þ

on all the other adiabatic surfaces on all solid boundaries

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

β 3 105 ðK21 Þ

dp ðnmÞ

σðΩUmÞ21

water

997.1

4179

0.613

21

-

0:05

Al2 O3

3970

765

25

0.85

47

1 3 10210

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.2 NATURAL CONVECTION OF MAGNETIC NANOFLUID CONSIDERING MFD VISCOSITY EFFECT

711

The values of vorticity on the boundary of the enclosure can be obtained using the stream function formulation and the known velocity conditions during the iterative solution procedure. The local Nusselt number of the nanofluid along the heat source can be expressed as:   knf 1  Nulocal 5 ð20:22Þ kf θ L=3 , X , 2L=3;Y50 The average Nusselt number is evaluated as: 1 Nuave 5 L=3

2L=3 ð

Nuloc ðXÞ dX

ð20:23Þ

L=3

The Nusselt ratio is defined as follows:

 Nuave δ 51  Nuratio 5 Nuave  

ð20:24Þ

δ 50

20.2.3 Effects of Active Parameters In this study magnetohydrodynamic natural convective heat transfer in a cooling system of electronic components is investigated. MFD viscosity effect is taken into account. Brownian motion effect is considered for simulating effective thermal conductivity and viscosity. The influence of active parameters such as Rayleigh number (Ra 5 103 ; 104 and 105 ), Hartmann number (Ha 5 0; 15; 30 and 60 ), and viscosity parameter (δ 5 0, 1 ) on flow and heat transfer are examined when φ 5 0:04; Pr 5 6:2. The effects of viscosity parameter, Hartmann number, and Rayleigh number on isotherms and streamlines are shown in Figs. 20.2
20.4. At low Rayleigh number, the streamlines take the enclosure geometry. By increasing the Rayleigh number the prominent heat transfer mechanism is switched from conduction to convection. When the magnetic field is imposed on the enclosure, the velocity field is suppressed owing to the retarding effect of the Lorenz force. So the intensity of convection weakens significantly. The decelerating effect of the magnetic field is observed from the maximum stream function value. The core vortex is shifted downwards vertically as the Hartmann number increases. Also, imposing a magnetic field leads to the omission of the thermal plume over the bottom wall. At high Hartmann number the conduction heat transfer mechanism is more pronounced.

FIGUE 20.2 Effects of viscosity parameter (δ 5 1 (222 ) and δ 5 0 (
)) and Hartmann number on isotherms (right) and streamlines (left) when φ 5 0:04; Ra 5 103 .

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20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

Effects of viscosity parameter (δ 5 1 (


) and δ 5 0 (
)) and Hartmann number on isotherms (right) and streamlines (left) when φ 5 0:04; Ra 5 104 .

FIGURE 20.3

Effects of viscosity parameter (δ 5 1 (


) and δ 5 0 (
)) and Hartmann number on isotherms (right) and streamlines (left) when φ 5 0:04; Ra 5 105 .

FIGURE 20.4

For this reason the isotherms are parallel to each other. Considering MFD viscosity effect increases the thermal boundary layer thickness and in turn the Nusselt number decreases with the increase of the viscosity parameter. Also it can be seen that flow circulation increases with the increase of this parameter. The effects of Hartmann number and Rayleigh number on local and average Nusselt number are shown in Figs. 20.5 and 20.6. Increasing the Rayleigh number leads to the enhancement of the Nusselt number. Increasing the Lorentz forces makes the thermal boundary layer thickness increase and in turn the Nusselt number decreases with the enhancement of the Hartmann number. Fig. 20.7 depicts the effect of the Hartmann number and the Rayleigh number on the Nusselt number ratio. As the viscosity parameter increases, the rate of heat transfer decreases. This reduction is more pronounced for higher values of Rayleigh number and lower values of Hartmann number. It is an interesting observation that increasing the Lorentz forces increases the effect of MFD viscosity on the Nusselt number. At low Rayleigh number, the MFD viscosity has no significant effect on Nusselt number.

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20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER 12

7.5

Ra = 10 10

Ra = 10 Ra = 10

3

Ra = 10

7

4

Ra = 10

5

Ra = 10

3 4 5

Nuloc

Nuloc

6.5

8

6

6 5.5

4 L/3

5L/12

5 L/3

L/2

5L/12

x

L/2

x

Ha = 0

Ha = 60

12

Ha = 0 Ha = 15

10

Ha = 30

Nuloc

Ha = 60

8

6

4 L/3

5L/12

L/2

x Ra = 105

FIGURE 20.5

Effects of Hartmann number and Rayleigh number on local Nusselt number when φ 5 0:04; δ 5 1; Pr 5 6:2.

20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER 20.3.1 Problem Definition Fig. 20.8 depicts the geometry, boundary condition, and sample element. The formula of the inner cylinder is: b5

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ε2 : a

ð20:25Þ

where a; b; ε are the major, minor axis of elliptic cylinder, and eccentricity for the inner cylinder. The inner cylinder has constant heat flux condition.

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20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

3.2

Ra = 103 Ra = 104 Ra = 105

Nuave

2.8

2.4

2

1.6

0

15

30

45

60

Ha

FIGURE 20.6 Effects of Hartmann number and Rayleigh number on average Nusselt number when φ 5 0:04; δ 5 1; Pr 5 6:2. 1.2

Ra = 103 Ra = 104 Ra = 105

Nuratio

1.1

1

0.9

0.8

0

15

30

45

60

Ha

FIGURE 20.7 Effects of Hartmann number and Rayleigh number on Nusselt number ratio when φ 5 0:04; Pr 5 6:2.

20.3.2 Governing Equation 2D steady convective flow of nanofluid in a porous media is considered in the presence of a constant magnetic field. The PDEs equations are: @u @v 1 5 0; @x @y 

  2 @u @u @ u @2 u @P
21 2 1 u 5 2σnf By u 1 σnf Bx By v 1 ρnf ; v 1 2 μnf 2 @y @x @y2 @x @x   2 @v @v @ v @2 v @P u 1 v 51 μnf 1 By σnf Bx u 2 Bx σnf Bx v 1 ðT 2 Tc Þβ nf gρnf ; 1 ρnf 2 2 2 @x @y @x @y @y Bx 5 Bo cosλ; By 5 Bo sinλ;

ð20:26Þ ð20:27Þ

ð20:28Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER

715

(A)

Tc A B

y

g γ

ζ

S

O

B

q″

x

C Nanofluid

(B)

(C) Si,3 i Si,4

Region of support Control volume

2.5 2

j=4

1.5 1 0.5 0

f1

–0.5 –1

f2

–1.5

i=1

–2

j=3

–2.5 –3

–2

–1

0

1

2

3

FIGURE 20.8 (A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

  2
 @T @T @ T @2 T @qr 1u ; ρCp nf v 1 2 2 5 knf 2 @y @x @x @y @y " # 4σe @T 4 4 3 4 ; T D4Tc T 2 3Tc : qr 5 2 3β R @y


ρCp

 nf

; ðρβ Þnf , ρnf ; knf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ;

ð20:29Þ

ð20:30Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ;

ð20:31Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ;

ð20:32Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

716

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

TABLE 20.3

Thermophysical Properties of Water and Nanoparticles ρðkg=m3 Þ

Cp ðj=kgkÞ

kðW=m:kÞ

dp ðnmÞ

σðΩUmÞ21

Pure water

997.1

4179

0.613

-

0:05

Fe3 O4

5200

670

6

47

25000



ks 1 2kf 1 2φðks 2 kf Þ ; ks 2 φðks 2 kf Þ 1 2kf  σs 3 21 φ σnf σf  : 511  σs σs σf 12 2 21 φ σf σf knf 5 kf

ð20:33Þ

ð20:34Þ

μnf is obtained as follows:


 μnf 5 0:035B2 1 3:1B 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

ð20:35Þ

Properties of nanofluid are shown in Table 20.3. Vorticity and stream function should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5 u: @y @x @x @y

ð20:36Þ

Introducing the dimensionless quantities: uL vL T 2 Tc ; ΔT 5 qvL=kf ; ;V5 ; θ5 αnf αnf ΔT
 x; y ψ ωL2 ðX; YÞ 5 ;Ω5 : ;Ψ 5 αnf αnf L U5

ð20:37Þ

The final formulae are: @2 Ψ @2 Ψ 1 5 2 Ω; 2 @Y @X2  @Ω @Ω A5 A2 @2 Ω @2 Ω 1 V 5 Pr U 1 @X @Y @X2 A1 A4 @Y2  @U @V 2 @U 2 @V A3 A22 @θ 2 A6 A2 Bx B y 2 Bx 1 By 2 Bx By 1 Pr Ra ; 1 PrHa @X @Y @Y A1 A4 @X A1 A24 @X  2 @θ @θ @ θ @2 θ 4 1 @2 θ U1 V5 1 Rd : 1 @X @Y @X2 @Y2 3 A4 @Y2 where dimensionless and constants parameters are illustrated as:

 qffiffiffiffiffiffiffiffiffiffiffiffi Pr 5 υf =αf ; Ra 5 gðρβ Þf ΔT L3 = μf αf ; Ha 5 LB0 σf =μf ;
 ρCp nf ρnf ðρβ Þnf  ; A3 5 ; A2 5
; A1 5 ρf ðρβ Þf ρCp f μnf knf σnf A4 5 ; A5 5 ; A6 5 ; kf μf σf
 Rd 5 4σe Tc3 = β R kf :

ð20:38Þ

ð20:39Þ

ð20:40Þ

ð20:41Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER

717

and boundary conditions are: @θ 5 1:0 @n

on inner wall

θ 5 0:0

on outer wall

Ψ 5 0:0

on all walls

Local and average Nusselt over the hot cylinder can be calculated as:  21 !  knf 4 1 knf Nuloc 5 1 1 Rd ; 3 θ kf kf ðS 1 Nuave 5 Nuloc ds: S

ð20:42Þ

ð20:43Þ

ð20:44Þ

0

20.3.3 Effects of Active Parameters CVFEM is utilized to simulate the magnetic field impact on nanofluid free convection. The radiation source term is taken into account. MFD viscosity is considered. Results are reported for different levels of Radiation parameter (Rd 5 0 to 0:8), volume fraction of Fe3O4-water (φ 5 0 to 0.04), Rayleigh number (Ra 5 103 ; 104 and 105 ), inclination angle (ξ 5 03 and 903 ), and Hartmann number (Ha 5 0 to 40). Fig. 20.9 illustrates the impact of radiation parameter on nanofluid hydrothermal treatment. Radiation parameter can enhance the rate of heat transfer. Also augmenting Rd increases the nanofluid motion. This impact is more obvious in the presence of a magnetic field. Influences of ξ; Ha and Ra on hydrothermal characteristics are demonstrated in Figs. 20.10 and 20.11. As the nanofluid temperature increases, the nanofluid begins moving from the warm surface to the outer one and dropping along the outer cylinder. At low Rayleigh number, conduction is more signification than convection.jΨ max j augments as buoyancy force increases and it reduces as the Lorentz force increases. When ξ 5 03 ; Ha 5 0, three vortexes appear in the streamlines and these vortexes convert to two smaller ones in the presence of Lorentz forces. As ξ increases, the distortion of isotherms is enhanced. A thermal plume is generated near the vertical centerline. As the Lorentz force increases, the two upper vortexes merge together and the thermal plume vanishes.

FIGURE 20.9 Effect of Hartmann number and radiation parameter on isotherm (left) and streamline (right) contours (Rd 5 0 (2UU 2 ), Rd 5 0:8(2)) when ξ 5 903 ; φ 5 0:04; Ra 5 105 .

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

718

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.10

Effect of Rayleigh and Hartmann numbers on isotherm (left) and streamline (right) contours when ξ 5 0; φ 5 0:04; Rd 5 0:8.

Figs. 20.12
20.14 illustrate the impact of ξ; Rd; Ra , and Ha on Nuloc ; Nuave . The formula for Nuave corresponding to active parameters is: Nuave 5 9:94 2 0:22ξ 2 4:44Rd 2 4:76logðRaÞ 1 0:98Ha 1 0:4ξ Rd 2 0:06ξlogðRaÞ 2 0:06ξHa 1 1:69logðRaÞRd
2 2 0:351logðRaÞ Ha 2 0:25Rd Ha 2 0:23ξ2 1 0:72 logðRaÞ 1 1:49Rd2

ð20:45Þ

2 0:06Ha2

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER

FIGURE 20.11 Effect ξ 5 903 ; φ 5 0:04; Rd 5 0:8.

of

Rayleigh

and

Hartmann

numbers

on

isotherm

(left)

and

streamline

(right)

contours

719

when

Where Ha 5 0:1Ha. The number of extremum in Nuloc profile matches the presence of the thermal plumes and the shape of the outer cylinder. Nuave increases with the rise of Rd. As the buoyancy forces augment, the temperature decreases and in turn Nuave increases with the rise of Ra. The impact of inclination angle on Nuave is similar to that obtained for Ra. Besides, this figure depicts that the Nusselt number has a reverse relationship with the Hartmann number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

720

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

Rd = 0

Rd = 0.8 6

2.5

Ha = 0 Ha = 40

4.5

Nuloc

2

Nuloc

Ra = 103

Ha = 0 Ha = 40

3

1.5

1

A

1.5

B S

C

A

B S

C

7.5

3.5

Ha = 0 Ha = 40

Ha = 0 Ha = 40 3

Nuloc

Nuloc

Ra = 104

6

2.5

4.5

2

3 1.5

1.5

1

A

B S

A

C

6

B S

C

12

Ha = 0 Ha = 40

Ha = 0 Ha = 40

4.5

Nuloc

9

Nuloc

Ra = 105

10.5

7.5

6

3

4.5

1.5

FIGURE 20.12 φ 5 0:04; ξ 5 03

3

A

B S

C

A

B S

C

Effects of the radiation parameter, Hartmann number, and Rayleigh numbers on local Nusselt number when

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.3 MAGNETIC FIELD INFLUENCE ON NANOFLUID THERMAL RADIATION IN A CAVITY WITH TILTED ELLIPTIC INNER CYLINDER

Rd = 0

Rd = 0.8

3

4.5

Ha = 0 Ha = 40

Ha = 0 Ha = 40

2.5

Nuloc

4

Nuloc

Ra = 103

721

2

1.5

3.5

A

B S

3

C

A

B S

C

7.5

4.5

Ha = 0 Ha = 40

Ha = 0 Ha = 40

7

4

6

Nuloc

3.5

Nuloc

Ra = 104

6.5

3

5.5 5

2.5

4.5 4

2

3.5 1.5

A

B S

3

C

A

B S

C

15

8

Ha = 0 Ha = 40

Ha = 0 Ha = 40

7

12

Nuloc

Nuloc

Ra = 105

6

5

9 4

3

6

2

1

A

FIGURE 20.13 φ 5 0:04; ξ 5 903 .

B S

C

3

A

B S

C

Effects of the radiation parameter, Hartmann number, and Rayleigh numbers on local Nusselt number when

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

722

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.14

Effects of inclination angle, radiation parameter, Rayleigh number, and Hartmann numbers on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY

FIGURE 20.14

723

(Continued).

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY 20.4.1 Problem Definition Fig. 20.15 depicts the geometry, boundary condition, and sample element. The formula of the inner cylinder is: pffiffiffiffiffiffiffiffiffiffiffiffiffi b 5 1 2 ε2 : a ð20:46Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

724

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

(A) Tc

g A q″ y

B γ

B

ζ S

O x

C

Nanofluid

(B)

(C) Si,3 i Region of support

Control volume

Si,4

2

j=4

1.5 1 0.5 0

f1

–0.5

f2

–1

i=1 –1.5 –2 –2.5 –2 –1.5

–1

–0.5

0

0.5

1

1.5

2

j=3

2.5

FIGURE 20.15

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

where a; b; ε are the major, minor axis of elliptic cylinder, and eccentricity for the inner cylinder. The inner cylinder has constant heat flux condition.

20.4.2 Governing Equation 2D steady convective flow of nanofluid is considered in the presence of a constant magnetic field. The PDEs equations are: @u @v 1 50 @x @y 

  2 @u @u @ u @2 u @P
21 2 1 u 5 2σnf By u 1 σnf Bx By v v 1 2 μnf 2 ρnf @y @x @y2 @x @x

ð20:47Þ ð20:48Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY

 ρnf

 2 @v @v @ v @2 v @P u 1 v 51 μnf 1 By σnf Bx u 2 Bx σnf Bx v 1 ðT 2 Tc Þβ nf gρnf ; 1 2 2 @x @y @x2 @y @y

Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @ T @2 T @qr 1u ; ρCp nf v 1 2 2 5 knf 2 @y @x @x @y @y

 4σe @T 4 qr 5 2 ; T 4 D4Tc3 T 2 3Tc4 3β R @y


ρCp

 nf

; ðβ Þnf , ρnf ; knf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

725

ð20:49Þ

ð20:50Þ

ð20:51Þ

ðβ Þnf 5 ðβ Þf ð1 2 φÞ 1 ðβ Þs φ

ð20:52Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  ks 1 2kf 1 2φðks 2 kf Þ knf 5 kf ks 2 φðks 2 kf Þ 1 2kf  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð20:53Þ ð20:54Þ

ð20:55Þ

μnf is obtained as follows:


 μnf 5 0:035B2 1 3:1B 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

Vorticity and stream function should be used to eliminate pressure source terms: @u @v @ψ @ψ 2 5 0; 5 2 v; 5u ω1 @y @x @x @y

ð20:56Þ

ð20:57Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tc P5
; ΔT 5 qvL=kf ; ðX; YÞ 5 ; V5 ; Θ5 2 ; U 5 L αf αf ΔT ρf αf =L p

ð20:58Þ

The final formulae are: @V @U 1 5 0; @Y @X

 2 μnf =μf @U @U @ U @2 U U 1 V 5 Pr 1 @X @Y @Y2 @X2 ρnf =ρf

  @P σnf =σf 2 2 ; 2 Ha Pr By U 2 Bx By V 2 @X ρnf =ρf  2

 μnf =μf @V @V @ V @2 V 1U 5 Pr 1 @Y @X @Y2 @X2 ρnf =ρf



   @P β nf σnf =σf
2 2 1 RaPr 2 Ha Pr Bx V 2 B x By U 2 Θ; @Y ρnf =ρf βf

ð20:59Þ

ð20:60Þ

V

ð20:61Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

726

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

" #   21

 ðρCP Þf knf @2 Θ ðρCP Þf knf @2 Θ @Θ @Θ 4 knf U1 V5 11 Rd 1 @X @Y 3 kf @X2 ðρCP Þnf kf @Y2 ðρCP Þnf kf where dimensionless and constants parameters are illustrated as: qffiffiffiffiffiffiffiffiffiffiffiffi

 Pr 5 υf =αf ; Ra 5 gβ f qvL4 = kf υf αf ; Ha 5 LB0 σf =μf ; Rd 5 4σe Tc3 = β R kf ;

ð20:62Þ

ð20:63Þ

and boundary conditions are: @Θ 5 1:0 @n

on inner wall

Θ 5 0:0

on outer wall

Ψ 5 0:0

on all walls

Local and average Nusselt over the hot cylinder can be calculated as:   21  knf 4 1 knf Nuloc 5 1 1 Rd 3 θ kf kf ðs 1 Nuave 5 Nuloc ds S

ð20:64Þ

ð20:65Þ

ð20:66Þ

0

20.4.3 Effects of Active Parameters Fe3O4-H2O nanofluid free convection in the presence of radiation source term is examined considering the magnetic field effect. MFD viscosity is taken into account. Numerical simulations are examined for various amounts of Radiation parameter (Rd 5 0 to 0:8), volume fraction of Fe3O4-water (φ 5 0 to 0.04), Rayleigh number (Ra 5 103 ; 104 and 105 ), inclination angle (ξ 5 03 and 903 ), and Hartmann number (Ha 5 0 to 40). The impact of radiation parameter on nanofluid hydrothermal behavior is illustrated in Fig. 20.16. Thermal boundary layer thickness increases with the increase of Rd. Also increasing Rd enhances the nanofluid motion. This influence is more evident in the presence of a magnetic field.

Effect of Hartmann number and radiation parameter on isotherm (left) and streamline (right) contours (Rd 5 0 (2UU 2 ), Rd 5 0:8(2)) when ξ 5 903 ; φ 5 0:04; Ra 5 105 .

FIGURE 20.16

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY

727

Influence of ξ; Ha and Ra on streamlines and isotherms are depicted Figs. 20.17 and 20.18. As nanofluid temperature augments, the nanofluid begins moving from the warm surface to the outer one and dropping along the outer cylinder. At low Rayleigh number, conduction is more signification than convection.jΨ max j augments as the buoyancy force augments and it reduces as the Lorentz force increases. When ξ 5 03 ; Ha 5 0, as the Rayleigh number increases, the lower vortex diminishes and the upper one becomes stronger. As the Hartmann number increases, the centers of the vortexes move to x 5 0. jΨ max j and the temperature gradient is enhanced with the rise of Ra. As ξ increases, the distortion of the isotherms increases. Increasing the Hartmann number causes the thermal plume to diminish and changes the heat transfer mechanism from convection to conduction. As the Lorentz force augments, the two upper vortexes merge together and the thermal plume vanishes.

FIGURE 20.17

Effect of Rayleigh and Hartmann numbers on isotherm (left) and streamline (right) contours when ξ 5 0; φ 5 0:04; Rd 5 0:8.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

728

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.18 Effect ξ 5 903 ; φ 5 0:04; Rd 5 0:8.

of

Rayleigh

and

Hartmann

numbers

on

isotherm

(left)

and

streamline

(right)

contours

when

Figs. 20.19
20.21 illustrate the impact of ξ; Rd; Ra , and Ha on Nuloc ; Nuave . The formula for Nuave corresponding to active parameters is: Nuave 5 6:91 2 0:12ξ 2 4:3Rd 2 3:5logðRaÞ 1 1:26Ha 1 0:36ξ Rd 1 0:2ξlogðRaÞ 2 0:07ξHa 1 1:61logðRaÞRd
2 2 0:31logðRaÞ Ha 2 0:45Rd Ha 2 0:2ξ2 1 0:62 logðRaÞ 1 1:5Rd2

ð20:67Þ

2 0:08Ha2

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY

Rd = 0

6

2.5

729

Rd = 0.8 Ha = 0 Ha = 40

Ha = 0 Ha = 40

Nuloc

2

Nuloc

Ra = 103

5

4

3 1.5

2

1

1

A

A

C

B S

Ha = 0 Ha = 40

3.5

6

3

5

Nuloc

Nuloc

Ha = 0 Ha = 40

2.5

4

2

3

1.5

2

1

1

A

A

C

B S

B S

Ha = 0 Ha = 40

5

10

4

8

Nuloc

Nuloc

Ra = 105

Ha = 0 Ha = 40

3

2

1

φ 5 0:04; ξ 5 03 .

6

4

2

A

C

12

6

FIGURE 20.19

C

7

4

Ra = 104

B S

B S

C

A

B S

C

Effects of the radiation parameter, Hartmann number, and Rayleigh numbers on local Nusselt number when

Where Ha 5 0:1Ha. The number of extremum in Nuloc profile matches the presence of the thermal plumes. Nuave is enhanced with the rise of Rd. As the buoyancy forces increase, the temperature decreases and in turn Nuave increases with the rise of Ra. Also this figure shows that the Nusselt number has a direct relationship with inclination angle.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

730

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

Rd = 0

Rd = 0.8

3

5

Ha = 0 Ha = 40

2.5

Nuloc

4

Nuloc

Ra = 103

Ha = 0 Ha = 40

2

3

1.5

1

A

2

C

B S

Ha = 0 Ha = 20

4.5

C

Ha = 0 Ha = 40

8

4

7

3.5

Nuloc

Nuloc

B S

9

5

Ra = 104

A

3

6 5

2.5

4

2

3

1.5 1

A

2 3A

C

B S

8

Ha = 0 Ha = 40

12

6 10

Nuloc

Nuloc

C

14

Ha = 0 Ha = 40

7

Ra = 105

B S

5

8

4 6

3

4

2 1

A

FIGURE 20.20 φ 5 0:04; ξ 5 903 .

B S

C

2

A

B S

C

Effects of the radiation parameter, Hartmann number, and Rayleigh numbers on local Nusselt number when

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.4 THERMAL RADIATION OF FERROFLUID IN EXISTENCE OF LORENTZ FORCES CONSIDERING VARIABLE VISCOSITY

FIGURE 20.21

731

Effects of inclination angle, radiation parameter, Rayleigh number, and Hartmann numbers on average Nusselt number.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

732

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.21

(Continued).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.5 MAGNETIC NANOFLUID NATURAL CONVECTION IN PRESENCE OF THERMAL RADIATION CONSIDERING VARIABLE VISCOSITY

733

20.5 MAGNETIC NANOFLUID NATURAL CONVECTION IN PRESENCE OF THERMAL RADIATION CONSIDERING VARIABLE VISCOSITY 20.5.1 Problem Definition Fig. 20.22 shows the geometry, boundary condition, and sample element. The formula of the outer cylinder is:

 r 5 rin 1 A cos N ζ 2 ζ 0 ð20:68Þ in which rin ; rout are the radii of the base circle and outer cylinder, respectively.

20.5.2 Governing Equation 2D steady convective flow of nanofluid is considered in the presence of a constant magnetic field. The PDEs equations are: @u @v 1 50 @x @y 

  2 @u @u @ u @2 u @P
21 2 1 u 5 2σnf By u 1 σnf Bx By v ρnf v 1 2 μnf 2 @y @x @y2 @x @x

ð20:69Þ ð20:70Þ

(A) A

Tc

Nanofluid g

q″ y

B γ

ζ

O x rin rout

(B)

(C)

Si,3

2.5 2

i Region of support

1.5

Control volume

Si,4

1

j=4

0.5 0 –0.5

f1

–1

f2

–1.5

i=1

j=3

–2 –2.5 –2 –1.5 –1 –0.5

0

0.5

1

1.5

2

2.5

FIGURE 20.22

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

734

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

 ρnf

 2 @v @v @ v @2 v @P u 1 v 51 μnf 1 By σnf Bx u 2 Bx σnf Bx v 1 ðT 2 Tc Þβ nf gρnf ; 1 2 2 @x @y @x2 @y @y

Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @T @2 T @qr 1u ; ρCp nf v 1 2 2 5 knf 2 @y @x @x @y @y

 4σe @T4 ; T4 D4Tc3 T 2 3Tc4 qr 5 2 3β R @y


ρCp

 nf

ð20:71Þ

ð20:72Þ

; ðβ Þnf , ρnf , and σnf are defined as:


 ρCp nf 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð20:73Þ

ðβ Þnf 5 ðβ Þf ð1 2 φÞ 1 ðβ Þs φ

ð20:74Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ  σs 3 21 φ σnf σf  511  σs σs σf 12 2 21 φ σf σf

ð20:75Þ

ð20:76Þ

Properties of Fe3O4-water nanofluid are presented in Table 20.3. The Maxwell model and Hamilton
Crosser model for irregular particle geometries by introducing a shape factor can be expressed as
 knf kp 1 ðm 1 1Þkf 2 ðm 1 1Þφ kf 2 kp
 5 ð20:77Þ kf kp 1 ðm 1 1Þkf 1 φ kf 2 kp in which kp and kf are the conductivities of the particle material and the base fluid. In this equation “m” is shaper factor. Table 20.4 shows the different values of shape factors for various shapes of nanoparticles.μn f is obtained as follows [41]:
 ð20:78Þ μnf 5 0:035B2 1 3:1B 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T Vorticity and stream function should be used to eliminate pressure source terms:

TABLE 20.4 The Values of Shape Factor of Different Shapes of Nanoparticles

Spherical

3

Platelet

5.7

Cylinder

4.8

Brick

3.7

m

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.5 MAGNETIC NANOFLUID NATURAL CONVECTION IN PRESENCE OF THERMAL RADIATION CONSIDERING VARIABLE VISCOSITY

ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5u @y @x @x @y

735 ð20:79Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tc P5
; ΔT 5 qvL=kf ; ðX; YÞ 5 ; V5 ; Θ5 2 ; U 5 αf αf ΔT L ρf αf =L p

ð20:80Þ

The final formulae are: @V @U 1 5 0; @Y @X

 2 μnf =μf @U @U @ U @2 U U 1 V 5 Pr 1 @X @Y @Y2 @X2 ρnf =ρf

  @P σnf =σf 2 ; 2 Ha2 Pr By U 2 Bx By V 2 @X ρnf =ρf

ð20:81Þ

ð20:82Þ

 2

 μnf =μf @V @V @ V @2 V 1U 5 Pr V 1 @Y @X @Y2 @X2 ρnf =ρf



   @P β nf σnf =σf
2 1 RaPr 2 Ha2 Pr Bx V 2 B x By U 2 Θ; @Y ρnf =ρf βf

ð20:83Þ

# " #  21 !" ðρCP Þf knf @2 Θ ðρCP Þf knf @2 Θ @Θ @Θ 4 knf U1 V5 11 Rd 1 @X @Y 3 kf @X2 ðρCP Þnf kf @Y2 ðρCP Þnf kf

ð20:84Þ

where dimensionless and constants parameters are illustrated as: qffiffiffiffiffiffiffiffiffiffiffiffi

 Pr 5 υf =αf ; Ra 5 gβ f qvL4 = kf υf αf ; Ha 5 LB0 σf =μf ; Rd 5 4σe Tc3 = β R kf ;

ð20:85Þ

and boundary conditions are: @Θ 5 1:0 @n

on inner wall

Θ 5 0:0

on outer wall

Ψ 5 0:0

on all walls

Local and average Nusselt over the hot cylinder can be calculated as:  21 !  knf 4 1 knf Nuloc 5 1 1 Rd 3 θ kf kf 1 Nuave 5 ξ

ð20:86Þ

ð20:87Þ

1:5π ð

Nuloc dξ

ð20:88Þ

0:5π

20.5.3 Effects of Active Parameters Fe3O4-water nanofluid magnetohydrodynamic natural convection heat transfer in the presence of thermal radiation is studied. MFD viscosity has been considered. Control volume-based finite element method is selected to solve the governing equations. Numerical simulation are examined for various values of radiation parameter (Rd 5 0 to 0:8), Rayleigh number (Ra 5 103 ; 104 and 105 ), volume fraction of nanofluid (φ 5 0 to 0.04), shaper of nanoparticles, and Hartmann number (Ha 5 0 to 20). Table 20.5 shows the shape of nanoparticle effect on Nusselt number. As illustrated in this table, Platelet shape has highest rate of heat transfer. So, Platelet shape is selected for further investigation. APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

736

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

TABLE 20.5 Effect of Shape of Nanoparticles on Nusselt Number When Rd 5 0:8; Ra 5 105 ; φ 5 0:04 Ha 0

20

Spherical

8.269555

6.119096

Brick

8.336758

6.170139

Cylinder

8.442229

6.250367

Platelet

8.52838

6.316004

FIGURE 20.23 Ra 5 103 ; Rd 5 0:8.

Effect of Hartmann number on isotherm (left) and streamline (right) contours when (A) Ha 5 0, (B) Ha 5 20 and

Figs. 20.23
20.25 illustrate the effect of Ha and Ra on the streamlines and isotherms. As nanofluid temperature increase, the nanofluid begins moving from the warm surface to the outer one and dropping along the outer cylinder, afterwards climbing again at the inner cylinder, generating a revolving vortex inside the cavity. At low Rayleigh number, conduction is more signification than convection.jΨ max j augments as the buoyancy force augments and it reduces as the Lorentz force is enhanced. As buoyancy forces increase, the eddies become stronger and a thermal plume is generated near the centerline. Increasing the Hartmann number causes the thermal plume to diminish and changes the heat transfer mechanism from convection to conduction.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION

FIGURE 20.24 Ra 5 104 ; Rd 5 0:8.

737

Effect of Hartmann number on isotherm (left) and streamline (right) contours when (A) Ha 5 0, (B) Ha 5 20 and

Figs. 20.26 and 20.27 demonstrate the influence of Rd; Ra , and Ha on Nuloc, Nuave. The correlation for Nuave corresponding active parameters is: Nuave 5 18:8 2 5:4Rd 2 8:1logðRaÞ 2 1:2Ha 1 1:68RdlogðRaÞ 2 0:2Rd Ha 2 0:32Ha logðRaÞ
2 1 1:6Rd2 1 1:07 logðRaÞ 2 0:15ðHa Þ2

ð20:89Þ

where Da 5 0:01Da; Ha 5 0:1Ha. The number of extremum in in Nuloc profile matches the presence of the thermal plumes and the number of undulations. As the buoyancy force and thermal radiation increase, Nuave is enhanced. Enhancing the Hartmann number causes the nanofluid flow to retard and the Nusselt number reduces.

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION 20.6.1 Problem Definition Fig. 20.28 depicts the geometry, boundary condition, and sample element. The formula for the inner cylinder is:

 ð20:90Þ r 5 rin 1 A cos N ζ 2 ζ 0 in which rin is radius of the base circle. The inner cylinder has constant heat flux condition.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

738

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.25 Ra 5 105 ; Rd 5 0:8.

Effect of Hartmann number on isotherm (left) and streamline (right) contours when (A) Ha 5 0, (B) Ha 5 20 and

20.6.2 Governing Equation 2D steady convective flow of nanofluid in a porous medium is considered with constant magnetic field. The PDEs are: @u @v 1 50 @x @y 

  2 @u @u @ u @2 u @P
21 1 u 5 2σnf B2y u 1 σnf Bx By v 1 ρnf v 1 2 μ @y @x @y2 @x2 nf @x   2 @v @v @ v @2 v @P u 1 v 51 μnf 1 By σnf Bx u 2 Bx σnf Bx v 1 ðT 2 Tc Þβ nf gρnf ; 1 2 2 ρnf @x @y @x2 @y @y Bx 5 Bo cosλ; By 5 Bo sinλ   2
 @T @T @T @2 T @qr 1u ; 1 2 2 ρCp nf v 5 knf @y @x @x2 @y @y

 4σe @T4 ; T4 D4Tc3 T 2 3Tc4 qr 5 2 3β R @y

ð20:91Þ ð20:92Þ

ð20:93Þ

ð20:94Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION

(A)

(B)

6

Ha = 0 Ha = 20

739

7

Ha = 0 Ha = 20

6.5

5.5

Nuloc

Nuloc

6

5

5.5

5

4.5

4.5

4 4

3.5 90°

180°

3.5 90°

270°

180°

ζ

270°

ζ

(C)

(D) 10

11

Ha = 0 Ha = 20

Rd = 0 Rd = 0.8

10

9 9

8

Nuloc

Nuloc

8

7

7 6

6 5

5 4

4 90°

180°

270°

3

90°

180°

ζ

270°

ζ (E)

9

Rd = 0 Rd = 0.8

8

Nuloc

7

6

5

4

3

2

90°

180°

270°

ζ FIGURE 20.26 Effects of radiation parameter, Hartmann number, and Rayleigh number on local Nusselt number when (A) Ra 5 103 ; Rd 5 0:8, (B) Ra 5 104 ; Rd 5 0:8, (C) Ra 5 105 ; Rd 5 0:8, (D) Ra 5 105 ; Ha 5 0, (E) Ra 5 105 ; Ha 5 20 and φ 5 0:04.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

740

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.27 Effects of radiation parameter, Hartmann number, and Rayleigh number on average Nusselt number when (A) Ha 5 10; φ 5 0:04, (B) Ra 5 104 ; φ 5 0:04, (C) Rd 5 0:4; φ 5 0:04, (D) Ha 5 10; φ 5 0:04, (E) Ra 5 104 ; φ 5 0:04, (F) Rd 5 0:4; φ 5 0:04.




ρCp

 nf

; ðρβ Þnf , ρnf ; knf , and σnf are defined as:


ρCp

 nf



 5 ρCp f ð1 2 φÞ 1 ρCp s φ

ð20:95Þ

ðρβ Þnf 5 ðρβ Þf ð1 2 φÞ 1 ðρβ Þs φ

ð20:96Þ

ρnf 5 ρf ð1 2 φÞ 1 ρs φ

ð20:97Þ



ks 1 2kf 1 2φðks 2 kf Þ ks 2 φðks 2 kf Þ 1 2kf  σs 3 21 φ σnf σf  : 511  σs σs σf 12 2 21 φ σf σf knf 5 kf

ð20:98Þ

ð20:99Þ

μnf is obtained as follows:


 μnf 5 0:035B2 1 3:1B 2 27886:4807φ2 1 4263:02φ 1 316:0629 e20:01T

ð20:100Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION

FIGURE 20.27

741

(Continued).

Vorticity and stream functions should be used to eliminate the pressure source terms: ω1

@u @v @ψ @ψ 2 5 0; 5 2 v; 5 u: @y @x @x @y

ð20:101Þ

Introducing dimensionless quantities:


 x; y uL vL T 2 Tc P5
; ΔT 5 qvL=kf ; ðX; YÞ 5 ; V5 ; Θ5 2 ; U 5 αf αf ΔT L ρf αf =L p

ð20:102Þ

then the final equations are: @V @U 1 5 0; @Y @X

U

 2 μnf =μf @U @U @ U @2 U 1 V 5 Pr 1 @X @Y @Y2 @X2 ρnf =ρf

  @P σnf =σf 2 2 ; 2 Ha Pr By U 2 Bx By V 2 @X ρnf =ρf

ð20:103Þ

ð20:104Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

742

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

(A) A a

g

y

B γ

B

O

S x

C

Nanofluid

(C)

(B)

Si,3 i

Region of support Control volume 2

Si,4

1.5

j=4

1 0.5

f1

0 –0.5

f2 i=1

–1

j=3

–1.5 –2 –2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

FIGURE 20.28

(A) Geometry and the boundary conditions with (B) the mesh of geometry considered in this work; (C) a sample triangular element and its corresponding control volume.

 2

 μnf =μf @V @V @ V @2 V 1U 5 Pr V 1 @Y @X @Y2 @X2 ρnf =ρf



   @P β nf σnf =σf
2 1 RaPr 2 Ha2 Pr Bx V 2 Bx By U 2 Θ; @Y ρnf =ρf βf

ð20:105Þ

" #  21 ! 2 ðρCP Þf knf @2 Θ @Θ @Θ 4 knf @Θ U1 V5 11 Rd 1 @X @Y 3 kf @Y2 @X2 ðρCP Þnf kf

ð20:106Þ

where dimensionless and constants parameters are: qffiffiffiffiffiffiffiffiffiffiffiffi

 Pr 5 υf =αf ; Ra 5 gβ f qvL4 = kf υf αf ; Ha 5 LB0 σf =μf ; Rd 5 4σe Tc3 = β R kf ;

ð20:107Þ

and boundary conditions are: @Θ 5 1:0 @n

on inner wall

Θ 5 0:0 ψ 5 0:0

on outer wall on all walls

ð20:108Þ

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION

743

FIGURE 20.29

Effect of Rayleigh number, Hartmann number, and radiation parameter on Isotherms (left) and streamlines (right) contours (Rd 5 0 (2UU 2 ), Rd 5 0:8(2)) when N 5 3; φ 5 0:04.

Local and average Nusselt over the hot cylinder are determined from:  21 !  knf 4 1 knf Nuloc 5 1 1 Rd 3 θ kf kf ðs 1 Nuloc ds Nuave 5 S

ð20:109Þ

ð20:110Þ

0

20.6.3 Effects of Active Parameters MHD Fe3O4-water nanofluid natural convection with radiation source term is examined. MFD viscosity is taken into consideration. Numerical simulations are examined for various values of the radiation parameter (Rd 5 0 to 0:8), volume fraction of Fe3O4-water (φ 5 0 to 0.04), Rayleigh number (Ra 5 103, 104 and 105), number of undulations (N 5 3,4 and 5), and Hartmann number (Ha 5 0 to 40).

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

744

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.30

Effect of Rayleigh and Hartmann numbers on isotherm (left) and streamline (right) contours when N 5 3; φ 5 0:04; Rd 5 0:8.

The effect of radiation parameter on nanofluid hydrothermal behavior is illustrated in Fig. 20.29. The inner wall temperature increases with increasing Rd. Also increasing Rd results in increasing nanofluid motion. This influence is more evident when convection flow is dominant. The influence of N; Ha , and Ra on streamlines and isotherms are depicted in Figs. 20.30
20.32. As nanofluid temperature increases, the nanofluid begins moving from the warm surface to the outer one and dropping along the circular cylinder. At low Rayleigh number, conduction is more significant than convection.jΨ max j increases as

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

20.6 NUMERICAL STUDY OF THE EFFECT OF MAGNETIC FIELD ON FE3O4-WATER FERROFLUID CONVECTION WITH THERMAL RADIATION

FIGURE 20.31

745

Effect of Rayleigh and Hartmann numbers on isotherm (left) and streamline (right) contours when N 5 4; φ 5 0:04; Rd 5 0:8.

buoyancy force increases and it decreases as the Lorentz force increases. As the Hartmann number increases, the center of the vortices moves to x 5 0. As Ra increases, convection becomes stronger. Thus jΨ max j and the temperature gradient increase with increasing Ra. As N increases, the distortion of isotherms reduces and vortices become weaker. Increasing the Hartmann number leads to a reduction in the thermal plume and the heat transfer mechanism changes from convection to conduction. So, Nu decreases with increasing Lorentz forces. At N 5 5, increasing the Lorentz force causes the main vortices to convert into three smaller ones. As the Lorentz force increases, the two upper vortices merge together and the thermal plume vanishes.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

746

20. NANOFLUID CONVECTIVE HEAT TRANSFER CONSIDERING MAGNETIC FIELD DEPENDENT (MFD) VISCOSITY BY MEANS OF CVFEM

FIGURE 20.32

Effect of Rayleigh and Hartmann numbers on isotherm (left) and streamline (right) contours when N 5 5; φ 5 0:04; Rd 5 0:8.

References [1] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, London, 1985. [2] R. Ganguly, S. Sen, I.K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater. 271 (2004) 63
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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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1200. Available from: https:// doi.org/10.1080/10407782.2015.1125709. [45] M. Sheikholeslami, A.J. Chamkha, Electrohydrodynamic free convection heat transfer of a nanofluid in a semi-annulus enclosure with a sinusoidal wall, Numer. Heat Transfer, Part A 69 (7) (2016) 781
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569. [59] M. Sheikholeslami, R. Ellahi, Simulation of ferrofluid flow for magnetic drug targeting using Lattice Boltzmann method, J. Zeitschrift Fur Naturforschung A 70 (2) (2015) 115
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256. [61] M. Sheikholeslami, M. Sadoughi, Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles, Int. J. Heat Mass Transfer 113 (2017) 106
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H2O nanofluid behavior in a permeable cavity considering shape effect, Int. J. Heat Mass Transfer 115 (2017) 180
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water nanofluid considering thermal radiation: a numerical study, Int. J. Heat Mass Transfer 96 (2016) 513
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APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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431.

APPLICATION OF CONTROL VOLUME BASED FINITE ELEMENT METHOD (CVFEM) FOR NANOFLUID FLOW AND HEAT TRANSFER

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Nomenclature

A B Cp DB DT Ec En ~ g Hx ; Hy H Ha k L Le MnF M N Nb Nt Nu Pr r Ra Nr T 0 Tc u; v U; V ~ g qv x; y

amplitude magnetic induction specific heat at constant pressure Brownian diffusion coefficient thermophoretic diffusion coefficient Eckert number heat transfer enhancement gravitational acceleration vector components of the magnetic field intensity the magnetic field strength Hartmann number thermal conductivity gap between inner and outer boundary of the enclosure Lewis number magnetic number arising from FHD magnetization number of undulations Brownian motion parameter thermophoretic parameter Nusselt number Prandtl number nondimensional radial distance thermal Rayleigh number Buoyancy ratio number Fluid temperature curie temperature velocity components in the x-direction and y-direction dimensionless velocity components in the X-direction and Y-direction gravitational acceleration vector heat flux space coordinates

GREEK SYMBOLS ζ α φ μ υ ψ&Ψ

angle measured from the isulated right plane thermal diffusivity volume fraction dynamic viscosity kinematic viscosity stream function and dimensionless stream function

751

752 Θ ρ β μ0

NOMENCLATURE

dimensionless temperature fluid density thermal expansion coefficient magnetic permeability of vacuum

SUBSCRIPTS c h loc ave nf f p

cold hot local average nanofluid base fluid solid particles

Appendix: A CVFEM Code for Lid Driven Cavity

753

754

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

755

756

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

757

758

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

759

760

APPENDIX: A CVFEM CODE FOR LID DRIVEN CAVITY

Index

Note: Page numbers followed by “f” and “t” refer to figures an d tables, respectively.

A Active parameter effects Buongiorno model for nanofluid flow and heat transfer, 102
107 combined natural convection and radiation heat transfer, 627
629, 630f CuO-water nanofluid heat transfer enhancement simulation, 686
690 magnetohydrodynamic natural convection, 694
696 CVFEM stream function-vorticity solution for natural convection, 22
30 EHD free convection heat transfer of nanofluid, 376
379 EHD nanofluid flow in permeable enclosure, 652
654 in porous medium considering radiation parameter, 643f, 644 elective field effect on nanofluid flow, 666
669 external magnetic source effect on Fe3O4-H2O nanofluid behavior, 613
619 on water based nanofluid convective heat transfer, 250
251 external variable magnetic field flow, 206
207 Fe3O4-water ferrofluid convection, 743
745 ferrofluid convective heat transfer, 268
272 ferrofluid flow and heat transfer in semiannulus enclosure, 215
217 forced convection of nanofluid in porous lid driven enclosure, 138
143 free convection of nanofluid, 383
386 heat transfer of Fe3O4-water nanofluid in permeable medium, 604
605 Lorentz forces on nanofluid flow inside porous enclosure, 146
150 magnetic field on free convection, 174
176, 178f, 179f influence on nanofluid thermal radiation, 717
722 magnetic nanofluid mixed convection heat transfer treatment, 131
134 magnetic source impact on magnetic nanofluid convective heat transfer, 285
287

melting heat transfer influence on nanofluid flow, 679
681 MFD viscosity effect, 711
712 MHD nanofluid convective flow, 183
187, 184f, 185f flow in cavity with heat flux boundary condition, 192
195 natural convection inside half annulus, 703 MHD natural convection of AL2O3-water nanofluid, 116
123 nanofluid flow in porous sinusoidal cavity, 573
577 simulation via nonequilibrium model, 550
557 nanofluid free convection, 560
568 under influence of electric field, 632
636, 636f nanofluid free convection heat transfer, 89
94, 167
171 nanofluid hydrothermal analysis in complex shaped cavity, 76
77 treatment in cavity, 278
281 nanofluid natural convection, 393
395 in porous media, 594 nanofluid transportation in curved cavity, 258
260 natural convection, 735
737 natural convection heat transfer in nanofluid filled enclosure, 84
87 nonuniform magnetic field effect, 241 shape factor effect on EHD nanofluid flow, 660
662 single-phase model simulation, 158 space dependent magnetic field effect, 233
235 spatially variable magnetic field on ferrofluid flow and heat transfer, 226
228 thermal radiation of ferrofluid, 726
732 two-phase model for nanofluid natural convection heat transfer, 110 variable magnetic field effect on nanofluid convective heat transfer, 585
589 Adomian Decomposition Method (ADM), 37 Advection coefficients, 18
19 Advection
diffusion with source terms, steady state, 5
7 Aeronautics, 127 Al2O3-water nanofluids, 189
190, 190t

761

Average Nusselt number, 250, 711
712, 714f on cold circular wall, 22, 83
84, 174, 311, 320 contour plots of, 93f correlation for, 83, 332
333 on hot circular wall, 131, 166
167, 183, 191, 206, 215, 241, 278, 297, 304

B Boundary conditions, 7
8, 18
19, 101, 581 Boussinesq approximation, 21, 87
88, 294, 300
303, 306
308, 316
317, 708
709 Brownian motion, 34, 38, 99 effect, 711 MHD natural convection of Al2O3-water nanofluid, 110
123 nonuniform magnetic field effect, 236
241 impact on effective thermal conductivity, 80
81 parameter, 303
304 Buongiorno’s mathematical model, 41 Buoyancy ratio number, 102, 104f

C Cartesian coordinate system, 188 CFD. See Computational fluid dynamics (CFD) Charge distribution, models for, 374, 399
400, 407 Circular cylinder nanofluid convective heat transfer intensification, 513
521 in presence of magnetic field, 163
171 Classic Brinkman model, 36 Cloud, 2 Computational fluid dynamics (CFD), 1 Conductivity models, 374, 399
400, 407 Conservation equations, 34
35 single-phase model, 34 two-phase model, 34
35 Constant heat flux boundary condition, 218
228 Continuity equation, 34, 410, 418 Control volume-based finite element method (CVFEM), 1, 45
53, 54t, 76, 84, 102, 127, 138, 192, 218, 226, 293
294, 297, 338, 347, 355
356, 363
364, 483, 486, 536, 550, 604
605 discrete equations, 5

762 Control volume-based finite element method (CVFEM) (Continued) discretization, 1
2, 5
9 element and interpolation shape functions, 3
4 region of support and control volume, 4, 4f stream function-vorticity solution for LID driven cavity flow, 15
20 Convective flow of nanofluid inside lid-driven porous cavity effects of active parameters, 527
531 governing equation, 521
527 problem definition, 521 in porous media, 441 Convective heat transfer, 201
207 Conventional heat transfer fluids, 201, 327 Coulomb forces, 375 effect on FE3O4-H2O nanofluid thermal improvement, 419
426 in porous enclosure, 387
395 CuO nanoparticles, 483 magnetic nanofluid flow in porous cavity using, 488
499 CuO-H2O nanofluid convective flow, magnetic field influence on, 360
367 CuO-water hydrothermal analysis, 76 CuO-water nanofluid, 82, 547, 558 coefficient values, 549t heat transfer enhancement simulation effects of active parameters, 686
690 governing equation, 683
686 problem definition, 682 inclined half-annulus enclosure filled with, 171
176 magnetohydrodynamic natural convection effects of active parameters, 694
696 governing equation, 691
694 problem definition, 691 CVFEM. See Control volume-based finite element method (CVFEM)

D Darcy model for nanofluid flow, 441 Lorentz force effect on nanofluid flow considering Darcy model, 468
478 MHD CuO-water nanofluid, 441
446 MHD nanofluid convection considering heat flux boundary condition, 459
467 water-based nanofluid flow and heat transfer analysis, 447
458 Darcy number, 138
142, 158, 330
331, 339, 379, 509, 594, 653, 654f, 655f, 660f, 661f, 662f Darcy’s law, 441 Density, 36 Differential Transformation Method (DTM), 38 Diffusion contributions, 17
18 Dimensionless parameters, 76, 89, 296
297 Dimensionless quantities, 136
137, 145, 153, 329
330, 338, 355, 362, 497, 516, 527, 677
678, 685, 693, 699, 716, 735, 741
743

INDEX

Dimensionless variables, 21
22, 277 Discretization, 1
2, 2f cloud, 2 grid, 1 implementation of source terms and boundary conditions, 7
8 mesh, 1
2 steady state advection
diffusion with source terms, 5
7 unsteady advection
diffusion with source terms, 8
9 Double-sided lid-driven cavity, 312
323 DTM. See Differential Transformation Method (DTM) Dynamic viscosity, 36
37 of nanofluids, 129
130, 165, 212
213, 295
296

E Eckert number (Ec), 37, 130
131, 226, 297, 303
304, 318
319, 376
379, 408
409 Effective electrical conductivity of nanofluid, 130 Effective thermal conductivity, 130, 165, 213
214 Effective viscosity, 190, 709 EG. See Ethylene glycol (EG) EHD. See Electrohydrodynamics (EHD) Einstein-Stokes’s equation, 34
35 Electric charge density, 374, 408
409 Electric current density, 407 Electric field, 375, 389
390, 399, 624
625, 631, 638
639, 650, 655
659, 663
665 effect on hydrothermal behavior of nanofluid, 414
418 effects of active parameters, 418 governing equation, 414
418 problem definition, 414 equations, 399 free convection of nanofluid, 379
386 porous lid driven cavity in electric field existence, 662
669 in porous wavy enclosure, 629
636 Electric field dependent viscosity, 393 EHD nanofluid force convective heat transfer, 399
404 coefficient values, 403t effects of active parameters, 403
404 governing equation, 399
403 problem definition, 399 thermophysical properties of water and nanoparticles, 402t Electrical conductivity, 36 Electro-fluid-dynamics (EFD). See Electrohydrodynamics (EHD) Electroconvection, 374 Electrohydrodynamics (EHD), 623, 649. See also Magnetohydrodynamics (MHD) effect, 373 free convection heat transfer of nanofluid effects of active parameters, 376
379 governing equation, 373
376 problem definition, 373 nanofluid, 632, 644

nanofluid flow in permeable enclosure with sinusoidal wall, 649
654 active parameters effects, 652
654 governing equation, 650
652 problem definition, 649, 649f nanofluid flow in porous medium considering radiation parameter, 637
644 nanofluid hydrothermal treatment in enclosure with sinusoidal upperwall, 405
414 effects of active parameters, 411
414 governing equation, 406
411 problem definition, 405 nanofluid natural convection free convection of nanofluid under effect of electric field, 379
386 under influence of Coulomb force in porous enclosure, 387
395 shape factor effect on EHD nanofluid flow in porous medium, 655
662 active parameters effects, 660
662 governing equation, 655
660 problem definition, 655 Electrokinetics. See Electrohydrodynamics (EHD) Electromagnetic force, 177
178 Element and interpolation shape functions, 3
4, 3f Elliptic inner cylinder, nanofluid filled enclosure with, 77
87 Energy efficiency, 127 Energy equation, 35 Ethylene glycol (EG), 33, 327 thermophysical properties of, 651t External magnetic source, 260
272 effect on Fe3O4-H2O nanofluid behavior effects of active parameters, 613
619 governing equation, 606
613 problem definition, 606, 611f effects of active parameters, 250
251 governing equation, 247
250 nanofluid transportation in porous media, 499
513 problem definition, 241
247, 247f External variable magnetic field flow effects of active parameters, 206
207 governing equation, 203
206 problem definition, 201
203, 202f

F FDM. See Finite difference method (FDM) Fe3O4 nanoparticles, 585 on hydrothermal behavior, 259, 259f Fe3O4-water ferrofluid convective heat transfer, 594 flow, 226 hydrothermal treatment, 613 magnetic field effect on Fe3O4-water ferrofluid convection active parameter effects, 743
745 governing equation, 738
743 problem definition, 737
745 Fe3O4-water nanofluid, 293
298, 581 behavior, 606
619

INDEX

Coulomb forces effect on Fe3O4-H2O nanofluid thermal improvement, 419
426 effects of active parameters, 423
426 governing equation, 419
423 problem definition, 419 free convection, 229
235 magnetohydrodynamic natural convection heat transfer, 735 in permeable medium with thermal radiation, 595
605 shape factor effect on Fe3O4-water nanofluid forced convection effects of active parameters, 338
342 governing equation, 335
338 problem definition, 335 FEM. See Finite element method (FEM) Ferrofluid(s), 707. See also Nanofluids convective heat transfer effects of active parameters, 268
272 governing equation, 261
268 problem definition, 260
261, 266f flow, 201
207, 218
228 flow and heat transfer effects of active parameters, 215
217 governing equation, 208
215 problem definition, 207, 212f, 213f natural convection, 285 Ferrohydrodynamics (FHD), 129, 206, 308
310, 312, 707. See also Electrohydrodynamics (EHD); Magnetohydrodynamics (MHD) FHD. See Ferrohydrodynamics (FHD) Finite difference method (FDM), 1, 41
45, 46t Finite element method (FEM), 1, 45, 51t Finite volume method (FVM), 1, 45, 49t Fixed flux boundary, 7 Fixed value boundary, 7 Flow and convective heat transfer of ferronanofluid effects of active parameters, 321
323 governing equation, 316
320 problem definition, 312
316 Fluid flow, 1 Fluid in enclosure, 174 Forced convection of Fe3O4-water nanofluid, 229
235 heat transfer, 41
53 of nanofluid, 327 in constant magnetic field, 327
334 effects of active parameters, 138
143 governing equation, 135
138 problem definition, 135, 135f of nanofluid in existence of electric field, 399 active method for nanofluid heat transfer enhancement of EHD, 427
436 Coulomb forces effect on FE3O4-H2O nanofluid thermal improvement, 419
426 EHD hydrothermal treatment in enclosure with sinusoidal upperwall, 405
414

EHD nanofluid force convective heat transfer considering electric field dependent viscosity, 399
404 electric field effect on hydrothermal behavior of nanofluid, 414
418 in porous media, 581 Forced convection heat transfer magnetic nanofluid, 298
304 nonuniform magnetic field effect on, 293
298 in semiannulus effects of active parameters, 312 governing equation, 306
311 problem definition, 304
306 FVM. See Finite volume method (FVM)

G Galerkin’s optimal homotopy asymptotic method, 38 Gauss
Seidel method, 7 Geometric flexibility, 2 Governing equation active method for nanofluid heat transfer enhancement of EHD, 428
434 Buongiorno model for nanofluid flow and heat transfer, 100
102 combined natural convection and radiation heat transfer of nanofluid, 624
626 Coulomb forces effect on FE3O4-H2O nanofluid thermal improvement, 419
426 CuO-water nanofluid magnetohydrodynamic natural convection, 691
694 EHD free convection heat transfer of nanofluid, 373
376 EHD nanofluid flow, 650
652 in porous medium considering radiation parameter, 638
644 EHD nanofluid force convective heat transfer considering electric field dependent viscosity, 399
403 EHD nanofluid hydrothermal treatment in enclosure, 406
411 electric field effect on hydrothermal behavior of nanofluid, 414
418 effect on nanofluid flow, 663
666 external magnetic source effect, 247
250 external variable magnetic field flow, 203
206 Fe3O4-water ferrofluid convection, 738
743 ferrofluid convective heat transfer under influence of external magnetic source, 261
268 ferrofluid flow and heat transfer in semiannulus enclosure, 208
215 forced convection of nanofluid in porous lid driven enclosure, 135
138 free convection of nanofluid, 379
383 heat transfer of Fe3O4-water nanofluid in permeable medium, 595
604 Lorentz forces

763 effect on nanofluid flow, 468
474 on nanofluid flow inside porous enclosure, 144
146 magnetic field on free convection, 172
174 influence on nanofluid thermal radiation, 714
717 magnetic nanofluid natural convection, 733
735 magnetic source impact on magnetic nanofluid convective heat transfer, 282
284 melting heat transfer influence on nanofluid flow inside cavity, 675
678 MFD viscosity effect, 708
711 MHD CuO-water nanofluid, 442
444 MHD nanofluid convection, 460
465 convective flow in inclined enclosure, 179
183 flow in cavity with heat flux boundary condition, 188
192 natural convection inside half annulus, 698
703 MHD natural convection of Al2O3-water nanofluid, 113
116 nanofluid flow in porous sinusoidal cavity, 568
572 simulation, 547
550 nanofluid free convection heat transfer, 164
166 thermophysical properties of water and nanoparticles, 166t in tilted cavity, 87
89 nanofluid free convection under influence of electric field, 631
632 in porous cavity, 559
560 nanofluid hydrothermal analysis in complex shaped cavity, 73
76 treatment in cavity, 275
278 nanofluid natural convection, 389
392 in porous media, 589
594 nanofluid transportation in curved cavity in, 254
258 natural convection heat transfer in nanofluid filled enclosure, 77
84 nonuniform magnetic field effect, 238
241 shape factor effect on EHD nanofluid flow, 655
660 simulation of CuO-water nanofluid heat transfer enhancement, 683
686 single-phase model simulation, 151
157 space dependent magnetic field effect, 230
233 spatially variable magnetic field on ferrofluid flow and heat transfer, 220
226 thermal radiation of ferrofluid, 724
726 two-phase model for nanofluid natural convection heat transfer, 108
109 variable magnetic field effect on nanofluid convective heat transfer, 581
585

764 Governing equation (Continued) water-based nanofluid flow and heat transfer analysis, 447
455 Grid, 1

H HAM. See Homotopy Analysis Method (HAM) Hamilton
Crosser model, 329 Hartmann number (Ha), 37, 163, 167, 167f, 168f, 206
207, 227
228, 229f, 235, 258, 260f, 261f, 262f, 263f, 297, 303
304, 312, 330, 339, 509, 594, 710
712, 713f, 714f Heat flux boundary condition, 187
195 Heat function (H), 192, 304 Heat transfer, 129, 163, 180f, 226, 589, 675. See also Melting heat transfer; Nanofluid heat transfer Buongiorno model for, 99
107 characteristics, 127 considering constant heat flux boundary condition, 218
228 enhancement, 226, 228, 266t, 297, 311
312 ratio, 87 of Fe3O4-water nanofluid in permeable medium effects of active parameters, 604
605 governing equation, 595
604 problem definition, 595, 602f fluids, 73 nanofluid application for heat transfer augmentation, 33 conservation equations, 34
35 model description, 33 nanofluid, 33 nanofluid flow and heat transfer simulation, 37
62 physical properties of nanofluids for single-phase model, 36
37 in semiannulus enclosure in presence of thermal radiation, 207
217 Heatline analysis Buongiorno model for nanofluid flow and heat transfer using, 99
107 active parameters effects, 102
107 governing equation, 100
102 problem definition, 99
100 Heatline analysis, 45
53 Homotopy Analysis Method (HAM), 37 Homotopy perturbation method (HPM), 37

I Inclined half-annulus enclosure filled with Cu-water nanofluid, 171
176 Insulated boundary, 7 Interpolation shape functions, element and, 3
4

J Jeffery
Hamel flow, 37 Joule heating, 204, 248, 294
295 effect, 375, 409
410, 417

INDEX

K Kelvin force, 129, 204, 308
310 Koo
Kleinstreuer
Li model (KKL model), 53, 73, 75, 81
82, 88, 136, 354, 362, 443, 450, 471, 484
485, 491
497, 548
549, 559, 569, 677, 684
685, 692
693, 699 correlation, 82, 189
190, 328
329 to simulate thermal conductivity of nanofluid, 709

L Laminar CuO-water free convective heat transfer, 74
75 Lattice Boltzmann method, 53
62, 59t, 73 Lewis number (Le), 45
53, 102, 105f, 303
304 Lid driven cavity, CVFEM code for, 753 Lid driven cavity flow, CVFEM stream function-vorticity solution for, 15
20, 16f arrangement of control volumes and elements, 16f CVFEM discretization of stream function equation, 16
18 of vorticity equation, 18
19 nodal velocity field calculation, 20 problem and governing equation, 15 results, 20 streamlines for Reynolds number, 21f Lid velocity, 127 Lid-driven porous cavity, convective flow of nanofluid inside, 521
531 Linear triangular elements, 3 Local Nusselt number, 22, 29, 41, 83
84, 102, 116, 131, 183, 191, 206, 215, 226, 240
241, 250, 278, 297, 304, 311, 320, 376, 418, 711
712, 713f Local thermal nonequilibrium model (LTNE model), 547 Lorentz forces, 135
143, 227
228, 250, 268, 297, 308
310, 338, 347, 355
356, 445, 498, 586
587, 594 effect on nanofluid flow in porous cylinder considering Darcy model, 468
478 effects of active parameters, 475
478 governing equation, 468
474 problem definition, 468 on nanofluid flow inside porous enclosure with moving wall effects of active parameters, 146
150 governing equation, 144
146 problem definition, 144, 144f thermal radiation of ferrofluid in existence of, 723
732 LTNE model. See Local thermal nonequilibrium model (LTNE model)

M Magnetic field, 37, 41, 163, 167
169 circular and sinusoidal cylinder in presence of, 163
171 effect on Fe3O4-water ferrofluid convection, 737
745

on free convection effects of active parameters, 174
176 governing equation, 172
174 problem definition, 171
172, 171f influence on CuO-H2O nanofluid convective flow active parameter effects, 355
359 governing equation, 360
363 problem definition, 360 influence on nanofluid thermal radiation, 713
722 intensity, 127, 129, 201
203, 207, 230, 231f, 239f nanofluid flow inside cavity in presence of, 675
681 shape factor effect on Fe3O4-water nanofluid forced convection, 335
342 strength, 127, 128f, 201
203, 203f, 213f, 230, 231f, 239f, 247, 248f, 257f, 267f Magnetic field dependent viscosity (MFD viscosity), 131, 707 magnetic field effect on Fe3O4-water ferrofluid convection, 737
745 magnetic field influence on nanofluid thermal radiation, 713
722 magnetic nanofluid natural convection, 733
737 natural convection of magnetic nanofluid, 707
712 thermal radiation of ferrofluid, 723
732 Magnetic induction, 248
249 Magnetic nanofluid. See also Nanofluids convective heat transfer, 281
287 flow in porous cavity effects of active parameters, 498
499 governing equation, 489
497 problem definition, 488 forced convective heat transfer effects of active parameters, 304 governing equation, 300
304 problem definition, 298
299 mixed convection heat transfer treatment effects of active parameters, 131
134 geometry and boundary conditions, 128f governing equation, 129
131 problem definition, 127 thermophysical properties of water and nanoparticles, 131t natural convection considering MFD viscosity effect, 707
712 natural convection active parameter effects, 735
737 governing equation, 733
735 in porous media, 581 problem definition, 733 values of shape factor, 734t Magnetic number, 206
207, 227
228, 229f, 235, 236f, 237f Magnetic Reynolds number, 77
80, 129, 164
165, 172, 220
222, 294, 708
709 Magnetic source, 581 effect on nanofluid flow effects of active parameters, 347 governing equation, 343
347 problem definition, 343

INDEX

impact on magnetic nanofluid convective heat transfer effects of active parameters, 285
287 governing equation, 282
284 problem definition, 281
282, 281f, 282f nanofluid natural convection in porous media in presence, 589
594 nanofluid transportation in curved cavity in presence, 252
260 Magnetization, 129, 210, 248 Magneto caloric effect, 222
223, 294
295 Magnetohydrodynamic CuO-water nanofluid in porous complex shaped enclosure, 441
446 effects of active parameters, 445
446 governing equation, 442
444 problem definition, 441 transportation in porous cavity effects of active parameters, 355
359 governing equation, 353
355 problem definition, 348
352 Magnetohydrodynamics (MHD), 129, 206, 308
310, 312, 623, 703. See also Electrohydrodynamics (EHD); Ferrohydrodynamics (FHD) effects, 53, 116 Fe3O4-water ferrofluid natural convection, 743 nanofluid convection, 459
467 effects of active parameters, 465
467 governing equation, 460
465 problem definition, 459 nanofluid convective flow in inclined enclosure effects of active parameters, 183
187 governing equation, 179
183 problem definition, 177
178, 181f nanofluid flow in cavity with heat flux boundary condition effects of active parameters, 192
195 governing equation, 188
192 problem definition, 187
188, 188f nanofluid free convective heat transfer, 483 effects of active parameters, 486
487 governing equation, 483
486 problem definition, 483 nanofluid natural convection inside half annulus effects of active parameters, 703 governing equation, 698
703 problem definition, 697 natural convection of AL2O3-water nanofluid, 110
123 Mass flow, 110, 122 Maxwell classical correlation, 80
81 Maxwell model, 329 Maxwell
Garnetts model (MG model), 130, 172
173, 296 Mean absorption coefficient, 208
210 Melting heat transfer, 675. See also Heat transfer influence on nanofluid flow inside cavity effects of active parameters, 679
681 governing equation, 675
678 problem definition, 675 sinusoidal annulus in presence, 691
696

Melting surface MHD nanofluid natural convection inside half annulus with, 697
703 simulation of CuO-water nanofluid heat transfer enhancement, 682
690 MEMS. See Microelectromechanical systems (MEMS) Mesh, 1
2 MFD viscosity. See Magnetic field dependent viscosity (MFD viscosity) MG model. See Maxwell
Garnetts model (MG model) MHD. See Magnetohydrodynamics (MHD) Microelectromechanical systems (MEMS), 707 Mixed convection, 127 Mixed convective flows, 127 Mobility model, 374, 399
400, 407 Momentum equations, 35, 206

N Namburu correlation for viscosity, 36 Nanoelectromechanical systems (NEMS), 707 Nanofluid convective heat transfer intensification, 513
521 effects of active parameters, 517
521 governing equation, 513
516 problem definition, 513 Nanofluid flow Buongiorno model for, 99
107 inside cavity in presence of magnetic field, 675
681 EHD nanofluid flow in porous medium considering radiation parameter effects of active parameters, 644 governing equation, 638
644 problem definition, 637 elective field effect, 662
669 and heat transfer simulation, 37
62 control volume-based finite element method, 45
53, 54t finite difference method, 41
45, 46t finite element method, 45, 51t finite volume method, 45, 49t lattice Boltzmann method, 53
62, 59t Runge
Kutta method, 41, 42t semianalytical methods, 37
40, 39t magnetic source effect, 343
347 in porous sinusoidal cavity, 568
577 thermal nonequilibrium model nanofluid free convection in porous cavity, 558
568 simulation of nanofluid flow inside porous enclosure, 547
557 Nanofluid forced convection in constant magnetic field effects of active parameters, 330
334 governing equation, 327
330 problem definition, 327 influence of electric field on EHD nanofluid flow in permeable enclosure, 649
654 elective field effect on nanofluid flow, 662
669

765 shape factor effect on EHD nanofluid flow in porous medium, 655
662 inside permeable enclosure, 151
158 Nanofluid forced convective heat transfer flow and convective heat transfer of ferronanofluid, 312
323 forced convection heat transfer in semiannulus, 304
312 magnetic nanofluid forced convective heat transfer, 298
304 nonuniform magnetic field effect on forced convection heat transfer, 293
298 Nanofluid free convection. See also Nanofluid natural convection heat transfer effects of active parameters, 89
94, 167
171 governing equation, 87
89, 164
166 governing equation problem definition, 87, 163
164, 164f under influence of electric field effects of active parameters, 632
636 governing equation, 631
632 problem definition, 629
630 Nanofluid heat transfer. See also Heat transfer; Melting heat transfer; Natural convection heat transfer enhancement in presence of melting surface CuO-water nanofluid MHD natural convection, 691
696 melting heat transfer influence on nanofluid flow, 675
681 MHD nanofluid natural convection inside half annulus, 697
703 simulation of CuO-water, 682
690 enhancement method of EHD, 427
436 effects of active parameters, 435
436 governing equation, 428
434 problem definition, 427 in permeable enclosure effects of active parameters, 536
543 governing equation, 534
536 problem definition, 532
534 Nanofluid hydrothermal analysis in complex shaped cavity effects of active parameters, 76
77 governing equation, 73
76 problem definition, 73 treatment, 236
241 effects of active parameters, 278
281 governing equation, 275
278 problem definition, 273
275, 275f, 276f Nanofluid natural convection. See also Nanofluid free convection under influence of Coulomb force in porous enclosure, 387
395 effects of active parameters, 393
395 electric density distribution injected by bottom electrode, 388f geometry and boundary conditions, 388f governing equation, 389
392 problem definition, 387
388 MHD nanofluid natural convection inside half annulus, 697
703 in porous media effects of active parameters, 594

766 Nanofluid natural convection (Continued) governing equation, 589
594 problem definition, 589, 592f two-phase model for nanofluid natural convection heat transfer, 107
110 effects of active parameters, 110 governing equation, 108
109 problem definition, 107 Nanofluid thermal radiation, magnetic field influence on active parameter effects, 717
722 governing equation, 714
717 problem definition, 713 thermophysical properties of water and nanoparticles, 716t Nanofluid transportation in curved cavity in presence of magnetic source effects of active parameters, 258
260 governing equation, 254
258 problem definition, 252
254, 256f in porous media effects of active parameters, 509
513 governing equation, 507
509 problem definition, 499
507 Nanofluid treatment, Buongiorno model for Buongiorno model for nanofluid flow and heat transfer, 99
107 MHD natural convection of AL2O3-water nanofluid, 110
123 two-phase model for nanofluid natural convection heat transfer, 107
110 Nanofluids, 33, 73, 163, 201, 207f, 293, 623. See also Ferrofluid(s); Magnetic nanofluid convective heat transfer, 581
589 density, 238 effective electrical conductivity, 130 effective thermal conductivity, 130 EHD free convection heat transfer of, 373
379 filled enclosure with elliptic inner cylinder, 77
87 free convection of, 379
386 effects of active parameters, 383
386 governing equation, 379
383 problem definition, 379 under impact of electric field in porous cavity, 623
629 magnetohydrodynamic flow, 163 MHD forced convection, 527 in porous lid driven enclosure, 135
143 steady convective flow, 683
684 volume fraction, 138, 138f, 331 Nanomaterials, 201 Nanoparticles, 145, 201, 250 continuity equation, 34
35 nanoparticles shape on Nusselt number, 653t, 660t, 667t thermophysical properties, 549t, 651t of water, 205t values of shape, 145t values of shape factor, 651t Nanotechnology, 201

INDEX

Natural convection CVFEM stream function-vorticity solution for, 20
30 effect of active parameters, 22
30 geometry and boundary conditions and computational mesh, 22f problem and governing equation, 20
22 streamlines and isotherms, 24f, 25f, 26f, 27f of magnetic nanofluid considering MFD viscosity effect active parameter effects, 711
712 coefficient values of Al2O3
water nanofluid, 709t governing equation, 708
711 problem definition, 707
708 thermophysical properties of water and nanoparticles, 710t Natural convection heat transfer, 110, 163, 241. See also Nanofluid heat transfer in nanofluid filled enclosure effects of active parameters, 84
87 governing equation, 77
84 problem definition, 77 NEMS. See Nanoelectromechanical systems (NEMS) Nodal velocity field calculation, 20 Non-Darcy model, 498, 606
612 Non-Darcy model for nanofluid hydrothermal treatment. See also Darcy model for nanofluid flow convective flow of nanofluid inside liddriven porous cavity, 521
531 magnetic nanofluid flow in porous cavity using CuO nanoparticles, 488
499 MHD nanofluid free convective heat transfer, 483
487 nanofluid convective heat transfer intensification, 513
521 nanofluid heat transfer in permeable enclosure, 532
543 nanofluid transportation in porous media, 499
513 Non-Darcy porous medium, 483 Nonequilibrium model, 560
568 simulation of nanofluid flow inside porous enclosure via, 547
557 active parameters effects, 550
557 governing equation, 547
550 problem definition, 547 thermal, 568
577 Nonlinearity of fluid magnetization, 201 Nonuniform magnetic field, 606
612 Nonuniform magnetic field effect effects of active parameters, 241 on forced convection heat transfer of FE3O4-water nanofluid, 293
298 effects of active parameters, 297
298 governing equation, 294
297 problem definition, 293
294 governing equation, 238
241 on nanofluid convective flow external magnetic source effect on Fe3O4-H2O nanofluid behavior, 606
619

heat transfer of Fe3O4-water nanofluid in permeable medium, 595
605 nanofluid natural convection in porous media, 589
594 variable magnetic field effect on nanofluid convective heat transfer, 581
589 problem definition, 236
238, 238f Nusselt number, 22, 37, 41, 110, 114f, 193
195, 347, 376
379, 411
414 Nusselt ratio, 711, 714f

P PDEs, 327
328, 626, 641
642, 675
676, 683
684, 691
692, 698
699, 714
715 Permeable cavity considering shape effect, 606
619 magnetic field influence on CuO-H2O nanofluid convective flow, 360
367 Permeable enclosure, EHD nanofluid flow in, 649
654 Platelet nanoparticle, 363
364 Porous cavity magnetic nanofluid flow, 488
499 magnetohydrodynamic CuO-water transportation, 348
359 nanofluid free convection, 558
568 nanofluid under impact of electric field, 623
629 Porous circular cylinder, nanofluid convective heat transfer intensification in, 513
521 Porous curved enclosure, 581
589 Porous enclosure, 627
629 free convection of nanofluid under effect of electric field in, 379
386 nanofluid natural convection under influence of Coulomb force in, 387
395 Porous lid driven cavity, elective field effect on nanofluid flow in, 662
669 active parameters effects, 666
669 governing equation, 663
666 problem definition, 662
663 Porous lid driven enclosure, nanofluid in, 135
143 Porous media/medium, 135
136, 581 applications, 547 EHD nanofluid flow in porous medium considering radiation parameter, 637
644 magnetic source effect on nanofluid flow in, 343
347 nanofluid natural convection in, 589
594 nanofluid transportation in, 499
513 shape factor effect on EHD nanofluid flow in, 655
662 active parameters effects, 660
662 governing equation, 655
660 problem definition, 655 Porous sinusoidal cavity, nanofluid flow in, 568
577 Porous tilted enclosure, MHD nanofluid free convective heat transfer in, 483

767

INDEX

Porous wavy enclosure, electric field in, 629
636 Prandtl number (Pr), 21
22, 76, 89, 226, 297, 303
304, 710
711 Pressure gradient source, 594

R Radiation heat transfer, 285 combined natural convection and effects of active parameters, 627
629 governing equation, 624
626 problem definition, 623
624 Radiation parameter (Rd), 258, 260f, 261f, 262f, 263f Rayleigh number (Ra), 21
22, 23f, 76, 89, 163, 169f, 170f, 206
207, 227
228, 229f, 235, 258, 262f, 263f, 376
379, 509, 594, 710
712, 713f, 714f Response surface methodology (RSM), 91
94 Reynolds number (Re), 15, 131, 152f, 297, 303
304, 312, 330 Richardson number, 134 RSM. See Response surface methodology (RSM) Runge
Kutta method, 41, 42t

S Semianalytical methods, 37
40, 39t Semiannulus enclosure in presence of thermal radiation, 207
217 Shape factors, 145 influence on nanofluid heat transfer forced convection of nanofluid in constant magnetic field, 327
334 magnetic field influence on CuO-H2O nanofluid convective flow, 360
367 magnetic source effect on nanofluid flow in porous medium, 343
347 magnetohydrodynamic CuO-water transportation in porous cavity, 348
359 shape factor effect on Fe3O4-water nanofluid forced convection, 335
342 values of different shapes of nanoparticles, 145t Shapes of nanoparticles, 625 magnetic field influence on CuO-H2O nanofluid convective flow, 360
367 magnetic source effect on nanofluid flow in porous medium, 343
347 Single-phase model, 34, 99. See also Twophase model nanofluid free convection heat transfer in tilted cavity, 87
94 nanofluid hydrothermal analysis in complex shaped cavity, 73
77 natural convection heat transfer in nanofluid filled enclosure, 77
87 physical properties of nanofluids, 36
37 density, 36 dynamic viscosity, 36
37 electrical conductivity, 36 models for viscosity of nanofluids in simulation, 37t

specific heat capacity, 36 thermal conductivity, 37, 38t thermal expansion coefficient, 36 thermo-physical properties of nanofluid, 38t simulation effects of active parameters, 158 governing equation, 151
157 problem definition, 151, 152f Single-phase nanofluid simulation, 33 Sinusoidal annulus in presence of melting heat transfer, 691
696 Sinusoidal cylinder in presence of magnetic field, 163
171 Sinusoidal wall EHD free convection heat transfer of nanofluid, 373
379 EHD nanofluid flow in permeable enclosure with, 649
654 active parameters effects, 652
654 governing equation, 650
652 problem definition, 649, 649f formulation, 568 inclined enclosure with, 177
187 nanofluid forced convection inside permeable enclosure with, 151
158 Sisko nanofluid, 37 Slip velocity of nanoparticles, 35 Source terms, 17
18 Space dependent magnetic field effect effects of active parameters, 233
235 governing equation, 230
233 problem definition, 229
230, 230f Spatially variable magnetic field effects of active parameters, 226
228 governing equation, 220
226 problem definition, 218
220 Specific heat capacity, 36 Steady state advection
diffusion, 5
7 Stefan numbers, 678 Stream function, 15, 75
76, 83, 88
89, 101, 109, 115
116, 131, 136, 145, 296
297, 303, 311, 318
319, 329, 337
338, 346, 354
355, 362, 403, 418, 485
486, 497, 626, 632, 642
643, 677, 685, 693, 699 CVFEM discretization, 16
18 boundary conditions, 18 diffusion contributions, 17 source terms, 17
18 values, 84
85

T Temperature gradient, 679, 694
696, 703 Thermal conductivity, 37, 38t, 399 Thermal diffusivity, 164
165 Thermal expansion coefficient, 36 Thermal nonequilibrium model nanofluid flow in porous sinusoidal cavity considering, 568
577 active parameters effects, 573
577 governing equation, 568
572 problem definition, 568 nanofluid free convection in porous cavity, 558
568

simulation of nanofluid flow inside porous enclosure, 547
557 Thermal plumes, 76, 84
85, 376
379 Thermal radiation, 41, 53, 595
605, 653 effects, 623 of ferrofluid active parameter effects, 726
732 governing equation, 724
726 problem definition, 723
724 influence on nanofluid flow combined natural convection and radiation heat transfer of nanofluid, 623
629 EHD nanofluid flow in porous medium, 637
644 nanofluid free convection under influence of electric field, 629
636 magnetic field effect on Fe3O4-water ferrofluid convection, 737
745 magnetic nanofluid natural convection in presence of, 733
737 semiannulus enclosure in, 207
217 Thermal Rayleigh number, 102, 103f Thermophoresis, 99 effects, 236
241 MHD natural convection of AL2O3-water nanofluid, 110
123 Thermophoretic diffusion coefficient, 34
35 Thermophoretic parameter (Nt), 303
304 Thermophysical properties of Fe3O4 and water, 258, 584, 584t of nanofluid, 191, 191t, 250 Tiwari and Das’ nanofluid model, 41
45 Two-dimension (2D) convective non-Darcy flow of nanofluid, 534 laminar nanofluid flow, 581
583, 589
591 steady convective flow of nanofluid, 135
136, 327
328, 442
443, 460 nanofluid flow, 675
676, 691
692, 698
699 non-Darcy flow of nanofluid, 144, 151
153, 335
336, 343
345, 353
354, 360
362 Two-phase model, 34
35, 304. See also Single-phase model continuity equation, 34 energy equation, 35 magnetic nanofluid forced convective heat transfer, 298
304 momentum equation, 35 nanoparticle continuity equation, 34
35 Two-phase nanofluid simulation, 33 Two-temperature model, nanofluid free convection considering, 558
568 active parameters effects, 560
568 governing equation, 559
560 problem definition, 558

U Uniform Lorentz force effect on nanofluid flow magnetic field on free convection, 171
176 MHD nanofluid

768 Uniform Lorentz force effect on nanofluid flow (Continued) convective flow in inclined enclosure, 177
187 flow in cavity with heat flux boundary condition, 187
195 nanofluid free convection heat transfer, 163
171 Unsteady advection
diffusion with source terms, 8
9 Unsteady two-dimensional natural convection, 163

V Variable Lorentz forces on nanofluid free convection external magnetic source effect, 241
251 external variable magnetic field, 201
207 ferrofluid convective heat transfer, 260
272 ferrofluid flow and heat transfer, 207
217 magnetic source impact on magnetic nanofluid convective heat transfer, 281
287 nanofluid hydrothermal treatment in cavity, 273
281

INDEX

nanofluid transportation in curved cavity in presence of magnetic source, 252
260 nonuniform magnetic field effect, 236
241 space dependent magnetic field effect, 229
235 spatially variable magnetic field on ferrofluid flow and heat transfer, 218
228 Variable magnetic field, 127
134, 273
281 effect on nanofluid convective heat transfer effects of active parameters, 585
589 governing equation, 581
585 problem definition, 581, 582f flow and convective heat transfer of ferronanofluid, 312
323 forced convection heat transfer in semiannulus, 304
312 magnetic nanofluid forced convective heat transfer in, 298
304 nanofluid heat transfer in permeable enclosure, 532
543 Velocity components, 6, 131
134, 233 Viscosity parameter, 711
712, 712f Vorticity, 15, 75
76, 83, 88
89, 101, 109, 115
116, 131, 329, 337
338, 346,

354
355, 362, 403, 418, 485
486, 497, 626, 632, 642
643, 677, 685, 693, 699 Vorticity equation, 83, 136, 145, 296
297, 303, 311, 318
319 CVFEM discretization, 18
19 advection coefficients, 18
19 boundary conditions, 19 diffusion contributions, 18 node point on sliding lid boundary, 19f Vorticity stream function simulation formulation CVFEM stream function-vorticity solution for LID driven cavity flow, 15
20 for natural convection, 20
30

W Water, 327 based nanofluid convective heat transfer, 241
251 thermophysical properties of, 549t Water-based nanofluid flow and heat transfer analysis, 447
458 effects of active parameters, 456
458 governing equation, 447
455 problem definition, 447