Advances in heat transfer [52] 9780128207383, 0128207388

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Academic Press is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States 525 B Street, Suite 1650, San Diego, CA 92101, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 125 London Wall, London, EC2Y 5AS, United Kingdom First edition 2020 Copyright © 2020 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. ISBN: 978-0-12-820737-6 ISSN: 0065-2717 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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List of contributors J.P. Abraham University of St. Thomas, School of Engineering, St. Paul, MN, United States A. Andreozzi Dipartimento di Ingegneria Industriale, Universita` degli Studi di Napoli Federico II, Napoli, Italy Filippo de Monte Department of Industrial and Information Engineering and Economics, University of L’Aquila, L’Aquila, AQ, Italy Zhipeng Duan School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China J.M. Gorman Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, United States P. Alex Greaney Department of Mechanical Engineering, University of California Riverside—Riverside, Riverside, CA, United States A. Haji-Sheikh Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, United States Jackson R. Harter Reactor Physics Analysis, Idaho National Laboratory, Idaho Falls, ID, United States M. Iasiello Dipartimento di Ingegneria Industriale, Universita` degli Studi di Napoli Federico II, Napoli, Italy Erfan Kosari Mechanical Engineering Department, University of California, Riverside, CA, United States Hao Ma School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China W.J. Minkowycz Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL, United States Todd S. Palmer School of Nuclear Science and Engineering, Oregon State University, Corvallis, OR, United States

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List of contributors

Matthew Regnier Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, United States C. Tucci Dipartimento di Medicina Traslazionale, Universita` degli Studi del Molise, Campobasso, Italy Kambiz Vafai Mechanical Engineering Department, University of California, Riverside, CA, United States Tie Wei Department of Mechanical Engineering, New Mexico Institute of Mining and Technology, Socorro, NM, United States

CHAPTER ONE

Analyses of buoyancy-driven convection Tie Wei* Department of Mechanical Engineering, New Mexico Institute of Mining and Technology, Socorro, NM, United States *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Prediction of Nusselt number 2.1 Lessons from forced convection 3. Governing equations 3.1 Reynolds averaged equations for the mean flow and heat transport 3.2 Laminar DHVC solution 4. Dimensional analysis of buoyancy-driven convection 4.1 Dimensional analysis of laminar DHVC 4.2 Dimensional analysis of turbulent DHVC 5. Review of scaling patch approach 5.1 Layer structure of turbulent channel flow 5.2 Steps in scaling patch approach 6. Scaling analysis of laminar DHVC 7. Scaling analysis of the mean momentum equation in turbulent DHVC 7.1 Layer structure of the mean momentum balance equation 7.2 Properties of the Reynolds shear stress 7.3 Outer scaling of the mean momentum equation 7.4 Inner scaling of the mean momentum equation 7.5 Meso scaling of the mean momentum equation 8. Scaling analysis of the mean heat equation 8.1 Layer structure of the mean heat equation 8.2 Properties of the turbulent temperature flux Rwθ 8.3 Outer scaling of the mean heat equation 8.4 Inner scaling of the mean heat equation 8.5 Scaling patches in the mean heat equation 9. New prediction of Nusselt number 10. Summary and conclusions Acknowledgments References

Advances in Heat Transfer, Volume 52 ISSN 0065-2717 https://doi.org/10.1016/bs.aiht.2020.09.002

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2020 Elsevier Inc. All rights reserved.

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Abstract This article investigates the multilayer structure of turbulent flow and heat transport in buoyancy-driven convection, and in particular, introduces a relatively new scaling patch approach. A differentially heated vertical channel (DHVC) is used as an example of buoyancy-driven convection, and its multilayer structure is first qualitatively investigated by dimensional analysis. In the near-wall region of turbulent DHVC, flow and heat transport is strongly influenced by the molecular diffusion, and the kinematic viscosity and thermal diffusivity are important control parameters in the dimensional analysis. Flow and heat transport in the inner layer is controlled by two nondimensional numbers: the Prandtl number of the fluid and an inner Richardson number. Away from the wall, flow and heat transport are dominated by eddy motions, largely independent of molecular diffusion. The controlling nondimensional parameter in the outer layer of turbulent DHVC is an outer Richardson number. The multilayer structure in turbulent DHVC is then elucidated quantitatively by the scaling patch analysis. Based on the characteristics of force balance, the mean momentum equation is divided into three layers: an inner layer, a meso layer, and an outer layer. The inner and outer Richardson numbers, derived from the dimensional analysis, appear naturally in the properly scaled mean momentum equation. Another nondimensional number that appears naturally from the scaling patch analysis is the friction Reynolds number. The characteristic length scale in the inner layer is directly influenced by the friction Reynolds number, distinctively different from that in forced convection. The characteristic length scale in the meso layer is an Obukhov-style length scale. The mean heat equation can also be divided into multiple layers. In fact, an inherent hierarchy of layer structure (scaling patches) is revealed through a simple transformation of the turbulent temperature flux. A new prediction of the Nusselt number is developed based on the insight gained from the dimensional analysis and scaling patch analysis. The new prediction is directly connected to the multilayer structure of heat transport in turbulent DHVC and is fundamentally different from the traditional power-law correlations.

1. Introduction A cornerstone in the study of turbulence is the recognition that the dynamics of turbulent flow and scalar transport operate on a great many space and time scales (see, e.g., Monin and Yaglom [1], Tennekes and Lumley [2]). A better understanding of the multiscale structure of turbulence is critical in improving our predictive capabilities of turbulence, i.e., prediction of skin friction and heat transport rate. Two powerful tools to uncover the multiscale nature of turbulence are dimensional analysis and scaling analysis. This article applies these tools to elucidate the multilayer structure of turbulent buoyancy-driven convection. In particular, one aim of this article is to introduce a relatively new scaling approach to buoyancy-driven convection. The new approach, based on the “search of scaling patches,” was

Analyses of buoyancy-driven convection

3

originally developed for forced wall-bounded turbulence such as sheardriven turbulent flow over a flat plate or pressure-driven turbulent pipe and channel flow, in a series of papers by Fife, Klewicki, McMurtry, Wei, and coworkers [3–17]. Some concepts and ideas in the scaling patch approach are similar to previous scaling approaches, however, the logical trains of thought in the new approach are distinctly different. Generally speaking, a fluid expands when being heated, resulting in a density decrease. Density inhomogeneity can also be generated from the mixing of fluids with different densities. In the presence of a gravitational field, there is a net force that pushes upward a light fluid surrounded by a heavier fluid, and this upward force is called the buoyancy force. The buoyancy force gives rise to the ascending of the lighter fluid and descending of the heavier fluid. The bulk fluid motion induced by the buoyancy force is commonly called buoyancy-driven convection. If the fluid motion is driven solely by buoyancy, the convection is also called natural convection or free convection. If the fluid motion is driven by both buoyancy and shear or pressure, the convection is called mixed convection. Buoyancy-driven convections are encountered in a variety of natural phenomena and industry applications. Examples include atmospheric and oceanic convection, space heating and cooling, smoke and fire spreading, nuclear reactor containment, and solar collectors. There have been an enormous amount of studies about the effects of buoyancy on flow and heat (or mass) transport. It is impossible to give in one article an exhaustive review of buoyancy-driven flow and heat transport, which can be found in books devoted to the subject, e.g. by Turner [18], Jaluria [19], Gebhart et al. [20], Kakac et al. [21], Martynenko and Khramtsov[22], and Verman [23]. How buoyancy affects the flow and heat or mass transport is of important practical interest, and is also of great theoretical interest. To this day, there are still no reliable tools for predicting the heat or mass transfer coefficient in buoyancy-driven turbulence [24]. One of the earliest studies of turbulent flow affected by buoyancy is about the convective atmospheric boundary layer (ABL) by Taylor [25] and Schmidt [26]. A key quantity in the understanding and prediction of stratified ABL is the vertical transport of momentum, water vapor, sensible heat, or heat in latent form [27]. One of the landmarks in the development of our understanding of atmospheric turbulence was the formation of the concept of the Austausch coefficient by Schmidt and Taylor. The idea was built on the knowledge of pressureor shear-driven wall turbulence developed by Prandtl et al. [27]. While reasonable for shear dominated turbulence, the concept of eddy viscosity based

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on local properties becomes meaningless when the buoyancy effect is strong, e.g., in convective atmospheric boundary layer [28]. A better understanding of the underlying physics in buoyancy-driven turbulence is critical in developing more robust models. To understand the essential physics of buoyancy-driven turbulence, convection with simple geometry is typically used in physical experiments or numerical simulations. Two buoyancy-driven convections with simple geometry are illustrated in Fig 1: differentially heated vertical channel (DHVC) and Rayleigh-B
enard convection (RBC). In both cases, flow and heat transport occurs between two parallel plates, which are maintained at different temperatures. In DHVC, the temperature gradient is perpendicular to the gravity, but in RBC, the temperature gradient is aligned with the gravity. In DHVC, hotter fluid ascends along the hot plate side, and at the same time, colder fluid descends along the cold plate side. In turbulent RBC, a prominent feature is the rising up of hot fluid as plumes and falling down of cold fluid as inverted plumes. RBC has been extensively studied in physical laboratories for more than one hundred years. During the past thirty years, numerical simulation, especially direct numerical simulation (DNS), has become an important tool in the study of turbulent flows, including buoyancy-driven convection, RBC

Fig. 1 Canonical configurations of buoyancy-driven convection. (A) Differentially heated vertical channel (DHVC). The hotter fluid rises up on the left side, and the colder fluid descends on the right side. (B) Rayleigh–B
enard convection. The mean flow is zero def

and the mean temperature is antisymmetric about the mid-plane. Θ ¼ T hot  T is the mean transformed temperature.

Analyses of buoyancy-driven convection

5

and DHVC. Analysis of turbulent quantities, especially those involved derivatives, are often only feasible with DNS data, due to the accuracy and resolution requirement. In the interests of simplicity and clarity, buoyancy-driven DHVC is chosen as an example to introduce the newly developed scaling patch approach, but the approach can be readily adapted to other buoyancy-driven convection. DHVC is an interesting buoyancy-driven convection to test tools and concepts developed originally for forced convection. In turbulent DHVC, flow is driven by buoyancy, but the turbulent kinetic energy is produced by two mechanisms: shear-generation and buoyancy-generation [29]. From a practical point of view, a better understanding of DHVC can help our prediction of heat transfer from a hot or cold vertical wall, as in the heating or cooling of building spaces. Last, DHVC is selected because there have been direct numerical simulation (DNS) data from three independent studies by Versteegh [30], Kisˇ [31], and Ng [32]. As in the analysis of forced wall turbulence [3], high-quality DNS data are essential in the application and evaluation of the scaling patch approach. More details on DHVC were given in dissertations by Versteegh [30], Kisˇ [31], and Ng [32]. Here, only a brief review of previous studies on turbulent DHVC is provided. One of the earliest studies of buoyancy-driven turbulence next to a vertical plate was by George and Capp [33]. They used classical scaling arguments and proposed a three-layer structure of buoyancydriven turbulence: an inner layer adjacent to the solid wall, an outer layer away from the wall, and a buoyant sublayer in between. Applying an asymptotic matching approach, they proposed a power-law variation of the mean velocity profile and mean temperature profile in the buoyant sublayer. Their scaling analysis also leads to an explicit relationship between the Nusselt number and the Rayleigh number. Some of the early numerical simulations of buoyancy-driven convection in a vertical slot were by Phillips [34] and Boudjemadi et al. [35]. Due to the limit of computing power, early DNS typically used a small computational domain. The first benchmark DNS study of DHVC was by Versteegh [30,36], who used a large domain of 24  12  2 (Lx  Ly  Lz). Using the simulation data, they investigated the scaling behavior of the mean temperature, the mean velocity profile, and the profiles of various turbulence statistics. Kisˇ [31,37] performed direct numerical simulations of DHVC over a wider range of Rayleigh number on a large computation domain of 24  12  2. The effect of domain size, spatial resolution, time-averaging period,

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and error analysis was examined in detail. Budgets for Reynolds stress and turbulent heat flux were documented, and entropy generation was also investigated. Ng and coworkers [32,38–41] have performed the highest Rayleigh number simulation of DHVC so far. They used the simulation data to explore the scaling of the flow and heat transport, the structures of the flow field, and the spectral properties of the turbulence. The rest of the article is organized as follows. In Section 2, the current strategy in the prediction of Nusselt number is reviewed, and its shortcoming is pointed out. In Section 3, governing equations for flow and heat transport are presented. In Section 4, dimensional analysis is applied to both the laminar DHVC and turbulent DHVC. In Section 5, the scaling patch approach is reviewed and demonstrated using a pressure-driven turbulent channel flow. In Section 6, scaling analysis is applied to laminar DHVC. In Section 7, the scaling patch analysis is applied to the mean momentum balance equation in turbulent DHVC. In Section 8, the scaling patch analysis is applied to the mean heat equation in turbulent DHVC. In Section 9, a new prediction of Nusselt number is proposed. Section 10 summarizes the article.

2. Prediction of Nusselt number A centerpiece in many studies of convection, forced or buoyancydriven, is the prediction of heat transport rate. The prediction of flow and heat transport often starts with a dimensional analysis [42]. The first step in dimensional analysis is to identify parameters that affect the quantity being predicted. In buoyancy-driven convection, DHVC, or RBC, the conventionally used control parameters are listed in Table 1. There are a total of nine control parameters, and three primary dimensions are entailed: length L, time t, and temperature T. Following the Buckingham-Pi theorem [43], flow and heat transport in DHVC or RBC are governed by six nondimensional numbers. For example, selecting ΔT, ν, and Lz (the domain size in the direction of temperature gradient) as the repeating variables, the following six nondimensional numbers are found accordingly ν ¼ Pr; α Π2 ¼ Πβ ¼ β ΔT ;

Π1 ¼ Πα ¼

(1a) (1b)

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Analyses of buoyancy-driven convection

Table 1 Parameters that affect flow and heat transport in buoyancy-driven convection.

Material properties

ν

Kinematic viscosity

[L2/t]

α

Thermal diffusivity

[L2/t]

β

Thermal expansion coefficient

[1/T]

Force

!

Gravitational acceleration

[L/t2]

g

Thermal BC ΔT Geometry

∡

Temperature difference

[T] !

Angle between the temperature gradient and g [1]

Lx, Ly, Lz Domain size

[L]

The last column lists the primary dimensions of the parameters: length L, time t, and temperature T.

g L 3z ; ν2 Π4 ¼ Π∡ ¼ ∡; L Π5 ¼ Πx ¼ x ¼ Γ x ; Lz Ly ¼ Γy , Π6 ¼ Πy ¼ Lz Π3 ¼ Πg ¼

(1c) (1d) (1e) (1f)

where Π1 is the ratio between the kinematic viscosity and thermal diffusivity, called the Prandtl number. For the ideal configurations of DHVC and RBC within infinite parallel plates, the aspect ratios are Γx ≫ 1 and Γy ≫ 1. The angle between the temperature gradient and the gravitational acceleration is ∡ ¼ π=2 for DHVC and ∡ ¼ 0 for RBC. Thus, three nondimensional control parameters remain for DHVC or RBC: Π1 ¼ Pr, Π2 ¼ β ΔT, and Π3 ¼ g L 3z =ν2 . In traditional analyses of buoyancy-driven convection, the nondimensional numbers Π2 and Π3 are combined to form a Grashof number (the justification is given in Section 4). In this article, the Grashof number is defined using the channel half-width δ and the temperature difference between the wall and channel mid-plane Θmp ¼ 0.5ΔT as def

Gr ¼

3 gβΘmp δ3 1 gβΔT ð2δÞ ¼ : 16 ν2 ν2

(2)

The reason for using δ and Θmp is that the multilayer structures of flow and heat transport in DHVC or RBC are symmetrical about the channel

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Tie Wei

mid-plane. Such a defined Grashof number is compatible with the friction Reynolds number used in pressure-driven wall-bounded turbulence def

through a pipe or channel Reτ ¼ δuτ =ν, where δ is the channel half-width or pipe radius and uτ is the friction velocity [3]. The traditional Grashof number is defined using the channel whole width Lz ¼ 2δ and the temperature difference across the whole channel ΔT ¼ 2Θmp. Thus, the presently defined Grashof number is 1/16 of the traditionally defined Grashof number. Traditionally, Grashof number is interpreted as a ratio between the buoyancy force and the viscous force [44]. However, Grashof number can also be related to the ratio of length scales by rearranging as follows Gr1=3 ¼

δ ðν2 =gβΘ

1=3 mp Þ

:

(3)

Thus, one-third power of Grashof number can be interpreted as the ratio of the channel half-width to a “viscous length scale” defined as (ν2/gβΘmp)1/3. In the study of buoyancy-driven convection, a popular nondimensional number is the Rayleigh number defined as def

Ra ¼

gβΘmp δ3 ¼ Gr Pr: να

(4)

One of the most important quantities in the study of convection is the heat transport rate. A commonly used nondimensional number for heat transport efficiency is the Nusselt number, which is defined as the ratio between the convective transport rate and conductive transport rate:
dΘ
def qt dz wall Nu ¼ Θ ¼ Θ , (5) mp mp α δ δ def

where qt ¼ Qwall =ρref c p ¼ αdΘ=dzjwall is the temperature flux at the wall, Qwall is the wall heat flux, ρref is the fluid density, and cp is the heat capacity. Mathematically, Nusselt number can be interpreted as a normalized second kind boundary condition [44], i.e., the temperature gradient at the wall normalized by a temperature scale Θmp and a length scale δ. Accurate prediction of the Nusselt number is of great importance for engineering applications. Based on the dimensional analysis, it is obvious that Nusselt number in DHVC or RBC is a function of the Prandtl number of the fluid and the Rayleigh number:

Analyses of buoyancy-driven convection

Nu ¼ f ðPr, RaÞ:

9

(6)

Historically, power-law is often used to correlate the Nusselt number with the Rayleigh number and Prandtl number as Nu  C Prn Ram :

(7)

While experimental measurements of Nusselt number for DHVC are scant, abundant experimental measurements of Nusselt number are available for RBC. Hence, we will use the Nusselt number prediction in RBC to illustrate the current strategy of predicting the Nusselt number for buoyancydriven convection. Fig. 2 presents the Nusselt number data for RBC over a wide range of Rayleigh numbers. Numerous correlations have been proposed for the Nusselt number in RBC, and Table 2 lists just a few of them. Nearly all the correlations used a power-law for the Rayleigh number dependence as Ram. However, a long-standing, still heated debated, issue in the study of RBC is the value of the exponent m. In the 1950s, Malkus [45] analyzed turbulent RBC in terms of marginal stability of the mean flow and derived an exponent of m ¼ 1/3. The work was further developed by Howard [46]. In the 1960s, Kraichnan [47] refined

Fig. 2 Nusselt number vs Rayleigh number in Rayleigh–B
enard convection. Inset shows the trend at ultra-high Rayleigh numbers. Data of helium (U) are from experimental measurement Urban et al. [50]. Data of helium (N) are from the experimental measurement Niemela and Sreenivasan [51]. Data of water and silicone oil AK3 are from experimental measurement of Silveston [52].

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Table 2 Examples of correlations for Nusselt number in RBC. Year Researchers Fluids Correlations

1959

Globe and Dropkin [53]

Any fluids

Nu ¼ 0.069 Ra0.33 Pr0.074

1969

Rossby [54]

Mercury

Nu ¼ 0.147 Ra0.257

1975

Threlfall [55]

Helium

Nu ¼ 0.173 Ra0.28

1989

Castaing et al. [48]

Helium

Nu ¼ 0.23 Ra0.282

1996

Takeshita et al. [56]

Mercury

Nu ¼ 0.155 Ra0.27

The applicable Rayleigh number ranges of the correlations are referred to the original papers. More discussions on the correlations can be found in Ahlers et al. [57].

the similarity theory to include a double boundary layer. One interesting result of his analysis is that at ultra-high Rayleigh number (say Ra ≳ 1018 ), the exponent m will be larger than 1/3, because of the interaction of the boundary layer with a horizontal fluctuating wind. Based on cryogenic helium gas data over a wide range of Rayleigh numbers, Castaing et al. [48] observed that the exponent is smaller than m ¼ 1/3. Moreover, they developed a new scaling theory and suggested an exponent of m ¼ 2/7. A more recent, and currently very popular model of RBC has been developed by Grossmann and Lohse [49]. The model is built on a theoretical analysis of the dissipation rate in both the boundary layer and the well-mixed core layer. Depending on the ratio of the dissipation rate, the model provides a set of scaling relations between the Nusselt number and Rayleigh number, Prandtl number, and aspect ratio of the RBC cell. The difference in the exponent, for example, m ¼ 1/3 vs m ¼ 2/7, can lead to significant difference in the prediction of Nusselt number, up to a factor of 10 at Ra  1025, as shown in the inset of Fig. 2. To better diagnose the existence of a power law, it is common to plot the so-called compensated Nusselt number as Nu Ram vs the Rayleigh number, as shown in Fig. 3. The compensated Nusselt number with an exponent m ¼ 2/7 is shown in Fig. 3A, and a constant region is observed for traditional Rayleigh numbers between 105 and 1010. However, this compensated Nusselt number monotonically increases for Rayleigh number larger than 1010. For the exponent of m ¼ 1/3, the constant region of the compensated Nusselt number is at higher Rayleigh numbers between 1010 and 1014, as shown in Fig. 3B. However, the trend at ultra-high Rayleigh number (> 1014) is not clear presently. Nevertheless, Fig. 3 shows that the dependence of Nusselt number on the Rayleigh number in RBC can be approximated by a power-law, but over

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Analyses of buoyancy-driven convection

A

B

Fig. 3 Compensated Nusselt number vs Rayleigh number in RBC. (A) Nu Ra2/7 vs Ra. (B) Nu Ra1/3 vs Ra.

A

B

Fig. 4 Nusselt number in DHVC. (A) Nusselt number vs Rayleigh number. (B) Compensated Nusselt number vs Rayleigh number.

a limited range of Rayleigh number, and the exponent varies with the Rayleigh number. The nonconstancy of the exponent m in the powerlaw prediction of Nusselt number is not just a nuisance in engineering application, but, more importantly, it reflects an inherent drawback of the power-law prediction: it lacks a connection to the underlying physics in buoyancy-driven convection, i.e., the multilayer structure of turbulent flow and heat transport. For DHVC, the Nusselt number data are mainly from direct numerical simulations. Fig. 4A presents the Nusselt number data vs Rayleigh number in the turbulent DHVC regime. Like other buoyancy-driven convection, power-law correlations are also popular in predicting the Nusselt number in DHVC, as listed in Table 3. Compared with the RBC case, the Rayleigh number range of DHVC data is much more limited. Curve-fitting within a limited range of Rayleigh numbers can almost always produce a power-law

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Table 3 Correlations for Nusselt number in DHVC. Year Researchers Correlations

1979

George and Capp

Nu ¼ 0.059(Pr Ra)1/3

1998

Versteegh and Nieuwstadt

Nu ¼ 0.071(Pr Ra)1/3

2007

Balaji, Holling and Herwig

Nu ¼

2012

Kisˇ and Herwig

Nu ¼ 0.186(Pr Ra)1/3.2

2014

Ng, Chuang and Ooi

Nu ¼ 0.046(Pr Ra)1/3

Ra1=3 ½0:101 lnðRaÞ+6:30634=3

correlation, as shown in Fig. 4A. The deviation between the data and the power-law function seems small in Fig. 4A. However, when the compensated Nusselt number is plotted, a systematic deviation is noticeable in Fig. 4B as the Rayleigh increases, similar to the RBC case shown in Fig. 3. One goal of this article is to advocate that it is time to move beyond the power-law prediction of Nusselt number in buoyancy-driven convection using Eq. (7). In forced convection, the prediction of Nusselt number with power-law correlation has long been superseded by more accurate equations that were built on the underlying physics of heat transfer.

2.1 Lessons from forced convection Power-law correlations with Reynolds number and Prandtl number were popular in predicting Nusselt number in early studies of forced convection. For example, for heat transfer (treated as passive scalar) through a turbulent pipe flow, one of the earliest correlations is the well-known Dittus–Boelter equation developed in the 1930s (see, e.g., [58–62]) Nu  0:023 Pr0:4 Re0:8 b ,

(8)

where Reb is the bulk Reynolds number defined by the diameter of the pipe and the bulk mean velocity. The Dittus–Boelter equation can be found in many textbooks on heat transfer, e.g., [44]. Now, it is generally agreed that such power-law prediction is not accurate over a wide range of Reynolds numbers, in particular at high Reynolds numbers. More accurate predictions that are commonly used nowadays are the Petukhov equation [63] and the Gnielinski equation [64]:

Analyses of buoyancy-driven convection

13

f ðReb  1000ÞPr 8 rffiffiffiffi Nu  ; Petukhov equation (9a) f 2=3 ðPr  1Þ 1 + 12:7 8 " # f  23  0:11 ðReb  1000Þ Pr d Pr rffiffiffi Nu  8 1+ , Gnielinski equation L Prw f 2=3 1 + 12:7 ðPr  1Þ 8 (9b) where f is the friction coefficient. The Reynolds number and Prandtl number ranges for the validity of these equations can be found in standard heat transfer textbooks, e.g., Ref. [44]. Recently, Wei [65] and Wei and Abraham [66] showed that the Kader– Yaglom style equation [67] gives an excellent prediction of Nusselt number, over a wide range of Prandtl number and Reynolds numbers. The Kader– Yaglom style equation is presented as (see [65,66]) Nu ≡

2Pr Reτ 2PrReτ  , + 1 Θmix lnðPr Reτ Þ + Bθ,m κθ

(10)

where Reτ is the friction Reynolds number defined using the channel halfwidth or pipe radius, Θ+mix is the mixed temperature scaled by the friction temperature. It is known that the mean temperature in forced wall-bounded turbulence can be robustly approximated by a logarithmic function, except in the nearwall region and a small, and bounded, deviation in the core of the pipe [1]. The thermal “log-law” is analogous to the well-known “log-law” for the mean velocity that was found in the 1930s (see, e.g., Monin and Yaglom [1]). More importantly, the log-law approximation for the mean velocity distribution has been reliably and robustly observed in the high Reynolds number experiments such as Princeton Superpipe experiments [68–70]. The logarithmic function in the Kader–Yaglom style Eq. (10) is directly related to the “log-layer” for the mean temperature distribution. The function Bθ,m represents the temperature increments in the thermal diffusion sublayer, and the deviation in the core (outer layer) [65,66]. The effect of the Prandtl number in the near-wall region is also accounted for in Bθ,m. Thus, the Kader–Yaglom style equation is directly built on the multilayer structure of mean

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In Section 9, a new prediction of the Nusselt number in buoyancydriven convection, similar to the Kader–Yaglom style equation, is developed. The new prediction equation is also directly built on the multilayer structure of mean temperature in buoyancy-driven turbulence.

3. Governing equations For most buoyancy-driven convection, the velocity of bulk motion is sufficiently small, and the governing equations can be approximated by the Boussinesq–Oberbeck equations [1]. In the Boussinesq–Oberbeck approximation, the fluid is assumed to be incompressible, and the density variation is negligible except in the buoyancy term [1]. The governing equations for the instantaneous flow and heat transfer in the Cartesian coordinate systems can be written as follows [1] ∂e uj ¼ 0; ∂xj

(11a) 

∂ðe uj uei Þ pe ∂e ui ∂2 uei ∂ + ¼ν  ∂xj ∂t ∂xj ∂x j ∂xi ρref e uj θÞ ∂θe ∂ðe ∂2 θe + ¼α : ∂xj ∂xj ∂xj ∂t

 + gi ;

(11b) (11c)

In this article, the instantaneous flow or heat variable is denoted by a tilde. For example, uei is the instantaneous velocity in the i direction, and θe is the instantaneous modified temperature defined as the difference between the e : θe def mean wall temperature Twall and the instantaneous temperature T ¼ e T wall  T . gi is the component of the body force in the i direction. Eq. (11a) is the continuity equation, Eqs. (11b) are the well-known Navier-Stokes equation for the momentums in three directions, and Eq. (11c) is the energy or heat equation. One fluid property, the kinematic viscosity ν, appears in the momentum equations (11b), and another fluid property, thermal diffusivity α appears in the heat equation (11c). In this article, the material properties are assumed constant, and the variability of viscosity, thermal diffusivity, or non-Boussinesq effects are not considered [71–74]. The coordinate system is set up that z is in the wall-normal direction. For wall-bounded flow and heat transport, no-slip boundary conditions are applied for the instantaneous velocity and temperature as uei jz¼0 ¼ 0 and e θj z¼0 ¼ 0 [1].

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Analyses of buoyancy-driven convection

3.1 Reynolds averaged equations for the mean flow and heat transport General solutions to the instantaneous governing Eq. (11) lie beyond the scope of existing methods. Moreover, in practical applications, the knowledge of the instantaneous flow field is too cumbersome and is often not necessary. Instead, we are mainly interested in the mean flow and scalar transport properties, for example, the mean skin friction, the mean heat transfer rate, and sometimes the average of the fluctuation level. Reynolds averaging provides an extremely common framework for studying the mean flow and scalar transport [1]. It is a shortcut and erases a wealth of details about the fluid motions being studied. Conceptually, Reynolds averaging is ensemble averaging over a large number of realizations of the flows under nominally the same boundary conditions and initial conditions [1]. The Reynolds averaging decomposes an instantaneous flow or heat variable into a mean component and a fluctuation component. For example, the instantaneous velocity is decomposed as uei ¼ U i + ui :

(12)

Here, an upper case letter denotes a mean flow or heat variable, and a lower case letter denotes its fluctuation. Similarly the instantaneous temperature is decomposed as θe ¼ Θ + θ, where Θ is the mean temperature and θ is the temperature fluctuation. In an experimental or numerical study of statistically steady turbulence, a more practical averaging is obtained by time averaging. Moreover, for turbulence possessing statistically spatial homogeneity, the averaging is also applied spatially over the homogeneous plane. For example, in turbulent DHVC, mean flow and temperature vary only in the wall-normal z direction, and are homogeneous in the vertical x–y plane. Therefore, the experimental or simulation data of turbulent DHVC are both time-averaged and spatial averaged in the x–y plane. The averaged governing equations for the mean heat transport and mean flow in turbulent DHVC are as follows [30,75] d2 Θ dRwθ ; + dz dz2 d2 U dRwu 0¼ν 2 + + gβðΘmp  ΘÞ, dz dz 0¼α

(13a) (13b)

def

where Rwθ ¼ hwθi is the turbulent temperature flux in the vertical direcdef

tion. Rwu ¼ hwui is the turbulent flux of the streamwise momentum in the vertical direction, commonly called the kinematic Reynolds shear stress.

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Angle brackets h i denote the conceptual ensemble averaging. β is the thermal expansion coefficient, and Θmp is the mean temperature at the channel mid-plane. The Reynolds shear stress Rwu arises from the averaging of the nonlinear inertial term in the Navier-Stokes equations, representing a force produced by turbulent fluctuations. The turbulent temperature flux Rwθ arises from the averaging of the nonlinear advective term in the instantaneous heat equation. There are four unknowns Θ, U, Rwu, and Rwθ in two Eqs. (13a) and (13b). This is the well-known “closure” problem in RANS modeling. Being under-determined, the problem can yield no unique solutions by themselves. This conflicts with the usual image of asymptotic methods, which provide a sequence of more and more accurate approximations to a unique exact solution [5]. The scaling patch approach tacked this basic ill-posed problem by adding reasonable and minimal assumptions about the qualitative nature of the solutions [5]. The mean temperature Θ in Eq. (13a) and the mean velocity U in Eq. (13b) are second-order, and Rwu and Rwθ are first-order. Thus, two boundary conditions are required for U and Θ, and one boundary condition is required for Rwu and Rwθ. The commonly used boundary conditions in DHVC are z¼0:

U ¼ 0, z¼δ:

Θ ¼ 0, Rwu ¼ 0, Rwθ ¼ 0: U ¼ 0, Θ ¼ Θmp :

(14a) (14b)

The boundary conditions in Eq. (14a) are the no-slip condition at the solid surface. Note that all these boundary conditions are the first kind boundary condition [44]. However, a second kind boundary condition can also be applied at the solid surface as
q dU
¼ u;
ν dz z¼0
qt dΘ
¼ :
dz z¼0 α

(15a) (15b)

def

Here qu ¼ τwall =ρref denotes the kinematic momentum flux at the wall, where def

τwall is the wall shear stress and ρref is the fluid density. qt ¼ Qwall =ρref c p denotes the temperature flux at the wall. In Sections 4 and 7, qu and qt are shown to play a key role in the multiscaling analysis of the mean equations using the dimensional analysis approach and the scaling patch approach.

Analyses of buoyancy-driven convection

17

In the study of forced wall-bounded turbulence, two extremely important wall variables are the friction velocity and friction temperature (see Monin and Yaglom [1] and Tennekes and Lumley [2]) defined by the wall shear stress and wall heat flux as rffiffiffiffiffi τw pffiffiffiffi uτ ¼ ¼ qu ; ρ q def Q w ¼ t: θτ ¼ ρref c p uτ uτ def

(16a) (16b)

Integrating the mean heat Eq. (13a) and the mean momentum Eq. (13b) in the wall-normal z direction and applying boundary conditions produce a relation for the total temperature flux and total momentum flux as [75] dΘ + Rwθ ¼ uτ θτ ; dz Z z dU 2 ν + Rwu ¼ uτ  gβ ðΘmp  ΘÞdz: dz 0 α

(17a) (17b)

The total temperature flux consists of the molecular diffusion (first term on the left of Eq. (17a)) and the turbulent transport (second term). The physical meaning of Eq. (17a) is that the total temperature flux (or heat flux when multiplied by ρref c p) in a fully developed turbulent DHVC is a constant across the channel, equal to the wall temperature (or heat) flux. This constancy of total temperature flux is analogous to the constancy of total momentum flux in turbulent plane Couette flow [76]. The total momentum flux also consists of two components: a viscous transport (first term on the left of Eq. (17b)) and a turbulent transport (second term). The physical meaning of Eq. (17b) is that the total momentum flux equals to the wall momentum flux minus a spatial integral related to the temperature deficit. The variation of total momentum flux in DHVC is distinctively different from that in forced convection [3]. For example, in turbulent plane Couette flow, the total momentum flux is a constant, equal to the wall momentum flux [76]. In turbulent DHVC, the total momentum flux is directly related to the buoyancy effect, which plays a fundamental role in the shape and distribution of Reynolds shear stress, as shown in Section 7. Next, we present the analytical solutions of laminar DHVC. The analytic solutions themselves are trivial, but they provide a simple example to evaluate dimensional analysis and scaling patch approach.

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3.2 Laminar DHVC solution The governing equation for laminar DHVC can be obtained by setting turbulence terms to zero in the mean heat Eq. (13a) and the mean momentum Eq. (13b). Integrating along the wall-normal z direction and applying boundary conditions produce the analytical solution for the temperature and velocity distribution in laminar DHVC as z Θ ¼ Θmp ; δ   gβΘmp δ2 1 z 1 z2 1 z3 U¼ : +  ν 3δ 2 δ 6 δ

(18a) (18b)

The analytical solutions for the temperature and velocity distributions in laminar DHVC are shown in Fig. 5A. The temperature Θ varies linearly in the wall-normal direction, like a pure conduction distribution. The vertical velocity exhibits a maximum value near the hotter plate, then decreases in the core of the channel, and exhibits a minimum near the colder plate. The vertical velocity is antisymmetric about the channel mid-plane. For comparison, the mean velocity and mean temperature distributions in A

B

Fig. 5 Temperature and vertical velocity profiles in a DHVC. (A) Laminar case. (B) Turbulent cases. Data are from the DNS of Kiš [31] at two Grashof numbers: Gr ¼ 4.75  104 (solid curves) and Gr ¼ 4.4  105 (dashed curves).

19

Analyses of buoyancy-driven convection

turbulent DHVC are presented in Fig. 5B at two Grashof numbers. In turbulent DHVC, the mean temperature varies sharply in the near-wall region, and the variation of temperature in the core is much smaller. On the flow side, the variation of the mean velocity is also much sharper in the near-wall region, and the peak location of the maximum velocity moves closer to the wall with increasing Grashof number. The maximum vertical velocity location in laminar DHVC can be found by setting the velocity gradient to zero. The maximum location and value are found as zU max ¼ ð1  31=2 Þδ  0:423 δ; U max  0:064

gβΘmp δ2 gβqt δ3 ¼ 0:064 : να ν

(19a) (19b)

In buoyancy-driven convection, a commonly used velocity scale is the so-called free-fall velocity defined as def

U ff ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gβΘmp δ:

(20)

Hence, the maximum vertical velocity in laminar DHVC is related the freefall velocity as U max  0:064

δ U ff U ff , ν

or

U max δ U ff :  0:064 U ff ν

(21)

Thus, the so-called “free-fall” velocity is not a proper or “natural” scale for the vertical velocity in laminar DHVC, because Umax/Uff is not a constant, but increases as δUff/ν. In Section 4, it is shown that this “free-fall” velocity is not a proper scale for the vertical velocity in turbulent DHVC either. Although free-fall velocity is easy to compute from the inputs of the buoyancy-driven convection and is commonly used to normalize equations or numerical simulation data, such scaled variables are not appropriate and should be avoided in the scaling of buoyancy-driven convection. Based on Eq. (19b), a better velocity scale for laminar DHVC is defined as def

U c,lam ¼

gβqt δ3 : να

(22)

As shown in Section 4, the wall temperature and momentum fluxes qt and qu are critical in understanding buoyancy-driven convection. For laminar DHVC, the wall temperature flux and the wall momentum flux can be obtained from the analytical solutions as

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Θmp dΘ
; qt ¼ α
¼α δ dz z¼0
gβqt δ2 dU
qu ¼ ν :
¼ 3α dz z¼0

(23a) (23b)

Eq. (23a) indicates that the Nusselt number in laminar DHVC is Nu ¼ 1. In other words, in laminar DHVC, there is bulk fluid motion in the vertical direction, but the flow direction is along the isothermal line, perpendicular to the temperature gradient. Thus the bulk fluid motion does not advect heat, and the heat transport from the hot plate to the cold plate is by pure molecular diffusion. From Eq. (23b), the friction velocity uτ in laminar DHVC is rffiffiffiffiffiffiffiffiffiffiffiffi gβqt δ2 uτ ¼ : (24) 3α The friction Reynolds number for laminar DHVC can be obtained as pffiffiffiffiffiffi Gr def δu Reτ ¼ τ ¼ pffiffiffi : (25) ν 3 The ratio between the Umax and uτ in laminar DHVC depends on the Grashof number and the friction Reynolds number as pffiffiffiffiffiffi U max  0:11 Gr  0:19Reτ : (26) uτ

4. Dimensional analysis of buoyancy-driven convection Dimensional analysis is a powerful tool in the study of fluid dynamics and heat transfer (see, e.g., Buckingham [43], Bridgman [77], Taylor [78], Townsend [79], Sedov [80], Barenblatt [81]). According to Churchill [60], the fundamental basis for dimensional analysis was established by Fourier in 1822. Rayleigh [82] demonstrated the power of dimensional analysis in a short paper published in the Nature magazine in 1915, giving examples from various fields. A fascinating example illustrating the power of dimensional analysis is the prediction of the first atomic explosion yield by G. I. Taylor [78], using a series of pictures published in a magazine. We will first present a dimensional analysis of laminar DHVC, followed by turbulent DHVC. In the dimensional analysis of laminar DHVC, we present two options of selecting control parameters and discuss the commonality and difference.

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Analyses of buoyancy-driven convection

Table 4 Dimensional analysis of laminar DHVC. Option 1

Option 2

Control parameters

gβ, ν, α, δ, Θmp

gβ, ν, α, δ, qt, qu

Repeating variables

gβ, ν, δ

gβ, ν, δ

Nondim. parameters

Π1 ¼ Πα ¼ αν ¼ Pr

Π1 ¼ Πα ¼ αν ¼ Pr

Π2 ¼ ΠΘmp ¼

gβΘmp δ3 ν2

¼ Gr

Π2 ¼ Πqt ¼

gβqt δ4 ν3

Π3 ¼ Πqu ¼

qu δ2 ν2

¼ Gr Pr

Length scale

lc ¼ δ Ψ(Pr, Gr)

lc ¼ δ Ψ(Pr, Gr)

Velocity scale

uc ¼ νδ ΨðPr, GrÞ

uc ¼ νδ ΨðPr, GrÞ

Temperature scale

θc ¼ ΘmpΨ(Pr, Gr)

θc ¼ ΘmpΨ(Pr, Gr)

The generic nondimensional functions Ψ in the last three rows are, in general, different from each other, and cannot be determined by the dimensional analysis itself.

4.1 Dimensional analysis of laminar DHVC Proper selection of control parameters is of utmost importance in a successful dimensional analysis of any flow or heat transport problem. For laminar DHVC, two sets of control parameters can be selected for dimensional analysis, as listed in Table 4. In both options, only three primary dimensions are entailed: length, time, and temperature. However, there are five control parameters in option 1, and six in option 2. Hence, based on the Buckingham Pi theorem, option 1 produces two nondimensional control parameters, and option 2 produces three. The common control parameters in option 1 and option 2 are gβ, ν, α, δ, which arise from the momentum and heat equation, and the domain size (part of the boundary condition). The parameter g and β are grouped together based on the observation of Eq. (13b): the effect of gravity on the buoyancy force is always coupled with the thermal expansion coefficient. There is nowhere in the governing equations that g and β appear separately. Therefore, gβ should be treated as a single parameter, instead of as two parameters as in Table 1. The difference between option 1 and option 2 in Table 4 lies in the parameter(s) related to heat. In option 1, Θmp is used. Note that Θmp can be interpreted as the temperature difference between the wall and the channel mid-plane. In other words, Θmp is the difference of the first kind thermal boundary conditions at z ¼ 0 and z ¼ δ. More importantly, Θmp in general does not reflect any local rate of temperature change, at the wall, or near the

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channel mid-plane. In contrast, the two control parameters qt and qu in option 2 represent the local rate of change at the wall (a second kind boundary condition). Applying the standard dimensional analysis procedure [43], the two control parameters from option 1 in Table 4 are the Prandtl number of the fluid and the Grashof number. For example, based on the dimensional analysis results, the maximum vertical velocity in laminar DHVC can be presented as U max ¼ ΨðPr, GrÞ, ν=δ

(27)

where Ψ denotes a generic nondimensional function that depends on Pr and Gr, which has to be determined from analytical solutions or experimental and numerical data. The three nondimensional parameters resulting from option 2 in Table 4 are Pr, Π2 ¼ gβqtδ4/ν3, and Π3 ¼ quδ2/ν2. However, in laminar DHVC, the boundary conditions of qu and qt are not independent, but directly related to each other as shown in Eq. (23b): qu δ2 Pr gβqt δ4 : ¼ 3 ν2 α ν2

(28)

Thus, the third nondimensional number Π3 from option 2 in Table 4 is redundant, and both options in Table 4 produce identical nondimensional control parameters. An advantage of option 2 in Table 4 is that it leads directly to the definition of a flux Grashof number. The flux Grashof number is defined using the wall temperature flux qt as (e.g., Bejan [42]) Gr

def f

¼

gβqt δ4 ¼ Gr Nu: ν2 α

(29)

Similarly, a flux Rayleigh number can be defined as def

Ra f ¼

gβqt δ4 ¼ Pr Gr f : να2

(30)

Note that this flux Rayleigh number is similar to the H number defined by George and Capp [33] for natural convection along a vertical plate under constant heat flux. For laminar DHVC, Nu ¼ 1 and the flux Grashof number equals the traditional Grashof number Gr ¼ Grf and Ra ¼ Raf. The flux Grashof number is related to the ratio of two length scales, as showing in the following form

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Analyses of buoyancy-driven convection

Gr

1=4 f

¼

δ ðν2 α=gβqt Þ1=4

(31)

Thus, one-fourth power of the flux Grashof number can be interpreted as the ratio between the channel half-width δ and a “viscous length scale” defined as (ν2α/gβqt)1/4. Such defined viscous length scale is similar to the inner length scale defined by Townsend [79] for buoyancy-driven convection over a single horizontal plane or Rayleigh–B
enard convection. To recap, laminar DHVC is controlled by two nondimensional numbers. Two options are valid for the selection of control parameters. The difference between the two options lies in the selection of parameter(s) with a local rate of temperature change or a global temperature difference. In laminar DHVC, the temperature varies linearly, hence, the global temperature difference Θmp is directly related to the local rate of temperature change qt. Thus, the two options in Table 4 are equivalent. However, in turbulent DHVC, the mean temperature variation is not linear, and the selection of Θmp becomes inappropriate.

4.2 Dimensional analysis of turbulent DHVC In pressure- or shear-driven wall-bounded turbulence, it is well known that the flow and heat transport can be divided into several layers, and the characteristic scales may vary with respect to given layers [1,2]. For example, turbulent flow over a flat plate is traditionally divided into four layers: a viscous sublayer, a buffer layer, a log-layer, and an outer layer (see, e.g., Tennekes and Lumley [2]). In the viscous sublayer, the characteristic length scale is the viscous length scale ν/uτ. However, in the outer layer, the characteristic length scale is the boundary layer thickness. In turbulent DHVC, flow and heat transport can also be divided into several layers, and the control parameters in each layer may be different. For example, adjacent to the wall, the velocity and temperature profiles have large gradients (see Fig. 5B). As a result, the molecular diffusion is important, and ν and α should be selected as the control parameters in the near-wall region. On the other hand, in the core of the channel, flow and heat transport are dominated by eddy motions, and molecular diffusion is negligible. For example, in the core of the channel in Fig. 5B, the mean temperature variation is small, meaning that the molecular temperature flux αdΘ/dz is small. Therefore, the kinematic viscosity ν and thermal diffusivity α should not be selected as control parameters for the flow and heat transport in the core of the channel. At a sufficiently high Grashof number, the separation

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between the inner layer and the outer layer becomes large, and a meso layer emerges between the inner and outer layers. Based on physical reasoning [83], a turbulent DHVC is divided into an inner layer, a meso layer, and an outer layer, as illustrated in Fig. 6. The control parameters for each layer are shown in Fig. 6 and listed in Table 5.

Fig. 6 Inner, meso, and outer layer structure of turbulent DHVC, and the control parameters in each layer [83].

Table 5 Summary of dimensional analysis for the inner layer, meso layer, and outer layers in turbulent DHVC. Inner layer Meso layer Outer layer

Control parameters

gβ, ν, α, qu, qt.

gβ, qu, qt.

gβ, qu, qt, δ.

Repeating variables

gβ, ν, qu.

gβ, qu, qt.

gβ, qu, δ.

Nondim. parameters

Πqt ¼

gβqt ν q2u

¼ Rii

Πqt ¼

gβqt δ 3=2 qu

¼ Rio

Πα ¼ αν ¼ Pr Length scale

l i ¼ uντ ΨðPr, Rii Þ

3=2

qu L ¼ gβq

lo ¼ δΨ(Rio)

t

Velocity scale

ui ¼ uτΨ(Pr, Rii)

uo ¼ uτΨ(Rio)

Temperature scale

θi ¼ θτΨ(Pr, Rii)

θo ¼ θτΨ(Rio)

The generic nondimensional functions Ψ in the last three rows are, in general, different from each other and must be determined from experimental or numerical data.

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Analyses of buoyancy-driven convection

It is obvious that Θmp cannot be an appropriate control parameter in the inner layer, because Θmp is not a local quantity in the near-wall region. In some previous studies of turbulent DHVC, only qt was selected as a control parameter, but not qu. Here, both qt and qu are used. There are five control parameters for the inner layer: gβ, qu, qt, ν, α, and the standard dimensional analysis following Buckingham’s Pi theorem produces two nondimensional controlling parameters: one is the Prandtl number of the fluid, and the other one is called the inner Richardson number defined as [83] def

Rii ¼

Gr f gβqt ν gβθτ ν Gr Nu ¼ 3 ¼ ¼ : 2 4 qu uτ Pr Reτ Pr Re4τ

(32)

In the outer layer of highly turbulent flow and highly turbulent heat transfer DHVC, flow and heat transport is dominated by eddy motions, and the effect of molecular diffusion is negligible. As the channel width is a geometric constraint on the size of the eddies, channel half-width δ is chosen as a control parameter in the outer layer. So there are four control parameters for the outer layer gβ, qu, qt, δ, and dimensional analysis produces one nondimensional number, called the outer Richardson number defined as [83] def

Rio ¼

gβqt δ 3=2

qu

¼

Gr f gβθτ δ Gr Nu ¼ ¼ ¼ Rii Reτ : 2 3 uτ Pr Reτ Pr Re3τ

(33)

For a turbulent DHVC at a sufficiently high Rayleigh number, there is a meso layer that is far away from the wall and the molecular diffusion does not affect the flow or heat transport. At the same time, the meso layer is also far away from the outer layer and is not affected by the channel half-width δ either. Thus, the meso layer is controlled by three parameters gβ, qu, qt, which leads to the definition of an Obukhov-style length scale [83,84] def

L ¼

q3=2 u2 u ¼ τ : gβqt gβθτ

(34)

Therefore, the inner Richardson number can be interpreted as the ratio between an inner length scale and the Obukhov-style length scale: Rii ¼

ν=uτ ν=uτ , ¼ L u2τ =gβθτ

(35)

and the outer Richardson number can also be interpreted as a length scale ratio: the channel half-width to the Obukhov length scale

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δ δ ¼ : u2τ =gβθτ L

Rio ¼

(36)

Fundamentally, flow and heat transport in turbulent DHVC is controlled by two nondimensional parameters. One is the Prandtl number of the fluid, and the other can be any one of the following: Gr, Grf, Ra, Raf, Rii, Rio, or Reτ. The nondimensional numbers that can be used in the study of buoyancy-driven convection in DHVC are summarized in Table 6. These nondimensional parameters are not independent of each other. For example, if Gr and Pr are known, then Rii or Rio or Reτ each has a unique value. If Rio and Pr are known, then each of the Gr, Rii or Reτ also has a unique value. Fig. 7 shows the relationship among these nondimensional numbers. Over the range of DNS data, these nondimensional numbers can be approximated by simple power-laws. The solid curve at low Grashof numbers in Fig. 7 represents the analytical pffiffiffiffiffiffi Eq. (26), U max =uτ  0:11 Gr, for laminar DHVC. The gray vertical band indicate the transitional Grashof numbers, based on the stability analysis of Versteegh [30]. Also included in the figure is the data of the maximum vertical velocity scaled by the friction velocity Umax/uτ, because it is a convenient parameter in the scaling analysis of the mean equations as shown in Sections 7 and 8.

Table 6 Nondimensional numbers that can be used in the study of DHVC. Definition

Grashof number

Gr ¼ gβΘmp δ3 =ν2

Flux Grashof number

Gr

Rayleigh number

Ra ¼ gβΘmp δ3 =ðναÞ

Flux Rayleigh number

Ra

Inner Richardson number

Rii ¼ gβqt ν=q2u ¼ gβθτ ν=u3τ ¼ Gr Nu=ðPr Re4τ Þ

Outer Richardson number

2 3 Rio ¼ gβqt δ=q3=2 u ¼ gβθ τ δ=uτ ¼ Gr Nu=ðPr Reτ Þ

Friction Reynolds number

Reτ ¼ δuτ =ν

Friction P
eclet number

Peτ ¼ δuτ =α ¼ Pr Reτ

def

def

¼ gβqt δ4 =ðν2 αÞ ¼ Gr Nu

f

def

def f

¼ gβΘmp δ4 =ðνα2 Þ ¼ Ra Nu

def

def

def

def

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Analyses of buoyancy-driven convection

A

B

Fig. 7 (A) Relationship among the nondimensional numbers: Gr, 1/Rii, Rio, and Reτ. (B) Nondimensional numbers are multiplied by different factors to demonstrate similar dependence on the Grashof number. The DNS data are from Versteegh [30] (red), Kiš [31] (green), and Ng [32] (blue). The gray vertical band indicates the critical Grashof number range based on the stability analysis of Versteegh [30].

5. Review of scaling patch approach One aim of this article is to introduce a relatively new approach, called scaling patch analysis, to buoyancy-driven turbulence. The scaling patch approach was originally developed for forced wall-bounded turbulence, pressure-driven or shear-driven. The new approach relies on a concept of “scaling patch,” which was implicit in many of previous asymptotic endeavors, but the scaling patch approach follows distinctively different trains of logic thought. The goal of the scaling patch analysis is to uncover the spatial multiscale structure of the mean profiles and gain insight into some basic properties of the mean flow and scalar transport. The formal and general description of the scaling patch approach was presented in three papers by Fife and coworkers [5,7,10]. More applications of the scaling patch approach can be found in Refs. [3,4,6,8,9,11–17]. For the reader’s convenience, Table 7 summarizes some of the key concepts and definitions in the scaling patch approach, with a focus on fluid dynamics problems, and we will use a pressure-driven turbulent channel flow to demonstrate the steps in the scaling patch approach. The geometry and coordinate system of the channel flow are sketched in Fig. 8. The upper and bottom plates are assumed to be infinite long in the x direction and infinite wide in the y direction. The channel half-height is denoted as δ. The fluid within the channel is driven by an imposed pressure gradient in the streamwise x direction. For a fully developed channel flow,

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Table 7 Key concepts and definitions in scaling patch approach, with a focus on fluid mechanics applications. A transformation of the turbulent momentum flux or turbulent + temperature flux: Rⓐ Adjusted wu ¼ Rwu  a z=δ or ⓐ + turbulent flux Rwθ ¼ Rwθ  a z=δ

Admissible scaling

A scaled equation that has at least two terms of nominal order of magnitude 1.

Differential scaling

a transformation of differential by multiplying a rescaling factor which is a function of the governing nondimensional number (e.g., Reynolds number or adjusted parameter in the adjusted turbulent flux).

Natural scaling

a scaling in which variation of the rescaled dependent variables (e.g., mean velocity U) with respect to the rescaled independent one (e.g., wall-normal location z) is neither too rapid nor trivially slow. The rescaled derivatives are  O(1). Some derivatives must be O(1), otherwise, all derivatives would be small, and the variation of the rescaled dependent variables would be unnaturally slow.

Nominal order of The “nominal” order of magnitude of a term in an equation is magnitude based solely on its prefactor, without considering the value of the derivative. Numerical order The “numerical” order of magnitude of a term considers the of magnitude values of both its prefactor and the derivative, which may take on a value that is not O(1). Scaling patch

An interval I (subdomain) of the flow in which a natural scaling exists and satisfies the following criteria • The size of the interval I is O(1) with respect to the rescaled independent variable (say wall-normal location z). • Derivatives of the rescaled dependent variable (e.g., the mean velocity U) up to some prescribed order are bounded in magnitude in I, independent of the governing nondimensional number of the flow (e.g., Reynolds number). Moreover, at least one of these derivatives is O(1). • If the scaling factor for the independent variable is replaced by one with a large magnitude, while the scaling factors for the dependent variables remain unchanged, some derivatives with respect to the newly scaled independent variable are not bounded independently of the governing nondimensional number.

Formal and general definitions are given in Refs. [5,7,10].

Analyses of buoyancy-driven convection

29

Fig. 8 Sketch of a pressure-driven channel flow.

Fig. 9 Distribution of the mean velocity (stars) and Reynolds shear stress (circles) in a turbulent channel flow. The mean streamwise velocity is normalized by its value at the channel mid-plane, and the Reynolds shear stress is normalized by the square of the friction velocity u2τ . Profiles at two Reynolds numbers are depicted: Reτ ¼ 180 (gray) and Reτ ¼ 5200 (black). The data are from the DNS of Lee and Moser [85]. To avoid clutter, some DNS data points are skipped.

the mean pressure gradient in the x direction is a constant, and the mean statistics, e.g., the mean velocity or Reynolds stresses, varies only in the wall-normal z direction and is homogeneous in the x and y (into the plane) direction. Prior to an analysis of any turbulent flow, it is important to scrutinize the distributions of some important flow quantities. Fig. 9 presents the mean velocity U and Reynolds shear stress Rwu profiles at two Reynolds numbers. The figure shows that the mean velocity U profile in a fully developed turbulent regime is symmetric about the channel mid-plane, and the Reynolds shear stress profile Rwu is antisymmetric about the channel mid-plane, where

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the Reynolds shear stress is zero. The Reynolds shear stress profile features a prominent peak in the near-wall region and then decreases almost linearly with respect to z in the core of the channel. As Reynolds number increases, Fig. 9 shows that the region in which the mean velocity varies sharply becomes a smaller fraction of the channel, and at the same time, the Reynolds shear stress peak location moves closer to the wall. As demonstrated in the following analysis of the mean momentum equation, the shape of the Reynolds shear stress profile plays a key role in the scaling patch analysis. The scaling patch approach is applied to the mean momentum balance equation. For a fully developed turbulent channel flow, the mean momentum balance equation in the streamwise direction can be written as [3,86] 0¼ν

d2 U dRwu 1 dP  + : dz ρ dx dz2

(37)

and the boundary conditions at the wall and the channel mid-plane are U ¼ 0, Rwu ¼ 0: dU z¼δ: ¼ 0: dz

z¼0:

(38a) (38b)

At the wall, the kinematic wall shear stress is used to define the well-known friction velocity as τwall =ρ ¼ νdU=dzj0 ¼ u2τ . Integrating the mean momentum balance Eq. (37) in the wall-normal z direction and applying boundary conditions in Eq. (38a) give the onceintegrated mean momentum equation as 0 ¼ ðν

dU 1 dP  u2τ Þ + ðRwu  0Þ  ðz  0Þ: dz ρ dx

(39)

Evaluating the once-integrated mean momentum Eq. (39) at the channel mid-plane produces a relation between the mean pressure gradient and the friction velocity as 

1 dP u2τ ¼ : δ ρ dz

(40)

Hence, the mean momentum balance Eq. (37) for a fully developed turbulent channel flow can be expressed as 0¼ν

u2 d2 U dRwu + τ: + 2 dz δ dz

(41)

Analyses of buoyancy-driven convection

31

Before applying scaling patch analysis on the mean momentum balance equation, it is important to develop an understanding of the layer structure of the turbulent flow.

5.1 Layer structure of turbulent channel flow Physically, the mean momentum balance Eq. (41) for turbulent channel flow represents the balance of three forces in the streamwise direction: a viscous force Fvisc ¼ ν d2U/dz2, a Reynolds force Fturb ¼ dRwu/dz that results from turbulence (will be called hereafter as turbulent force), and a pressure force F pres ¼ u2τ =δ. The three forces have to remain balanced across the whole channel. However, the contribution of the individual force may vary along the vertical direction. The typical profiles of the three forces across the channel are illustrated in Fig. 10. A prominent feature of Fig. 10 is that the force balance is established differently depending on the wall-normal location. For example, the viscous force and turbulent force have a sharp peak adjacent to the wall, and the magnitude of the peak is much larger than the pressure force. Away from the wall, the viscous force decreases rapidly toward zero and does not contribute to the force balance in the core of the channel.

Fig. 10 Typical force distributions in a turbulent channel flow. The magnitudes of the forces are not shown in the figure, because they depend on the normalization. The data are from the DNS at Reτ ¼ 180 [85], a relatively moderate Reynolds number to better show the near-wall region.

32

Tie Wei

Fig. 11 Ratio of the viscous force to turbulent force in turbulent channel flow. The vertical dash lines demarcate the four-layer structure based on the mean momentum equation.

The contribution of different forces to the balance of Eq. (41) can be ascertained directly by plotting the ratio between two of the three forces. In Fig. 11 the ratio of the viscous force to turbulent force Fvisc/Fturb is plotted vs the wall-normal distance scaled by the channel half-width. In a thin layer adjacent to the wall, the ratio goes to infinite. This is because the Reynolds shear stress Rwu increases with the wall-normal distance in a fashion of z3 adjacent to the wall (see, e.g., Monin and Yaglom [1]). Thus in a thin layer adjacent to the wall, dRwu/dz  0, and the force balance is between the viscous force and pressure force. This layer is called layer I. Adjoining layer I, there exists a region where Fvisc/Fturb 1, meaning that the force balance is between the viscous force (a drag) and the turbulent force (driving force), and the pressure force is negligible. This layer is called layer II or stress gradient balance layer [3]. Around the peak of the Reynolds shear stress profile, the turbulent force changes sign from being positive to zero, and then to negative. In this layer, all three forces contribute to the balance of the mean momentum equation and is called layer III or meso layer. In the core of the channel, the force ratio Fvisc/Fturb is essentially zero, meaning that the force balance is between the turbulent force (a drag force) and the pressure force (a driving force). This layer is called the outer layer or layer IV. In summary, based on the characteristics of force balance in the mean momentum Eq. (41), each half of a turbulent channel flow can be divided

33

Analyses of buoyancy-driven convection

A

B

Fig. 12 Reynolds number dependence of the layer thicknesses in a turbulent channel flow. (A) Wall-normal distance scaled by the channel half-width: z/δ. (B) Wall-normal distance scaled by the viscous length scale: z/(ν/uτ). The DNS data are from Lee and Moser [85].

into a four-layer structure, as sketched in Fig. 11. The layer structures in the lower and upper half are symmetric about the channel mid-plane. The data points in Fig. 11 represents the force ratio of Fvisc/Fturb. Details on the definitions of layer demarcation are given in Wei et al. [3]. The thickness of different layers is important in the scaling analysis of the mean momentum equation. Fig. 12A presents the force ratio Fvisc/Fturb vs z/δ at three Reynolds numbers. The figure is plotted on log-linear axes, to better show the near-wall region. As the Reynolds number increases, Fig. 12A clearly shows that layer II becomes a smaller fraction of the channel. The thickness of layer I and layer III also becomes a smaller fraction of the channel with increasing Reynolds number. In Fig. 12B, the wall-normal location is normalized by the so-called viscous length scale ν/uτ. Fig. 12B clearly shows that the thickness of layer II normalized by the viscous length scale increases with the Reynolds number. It is found that the thickness of layer II or layer III scales with the geometric pffiffiffiffiffiffiffiffiffiffiffi mean of the inner length scale and the outer layer scale: δν=uτ . This meso length scale has been reported by Long and Chen [87], Afzal [88], Sreenivasan and Sahay [89], and Wei et al. [3]. The characteristics of the four-layer structure are summarized in Table 8, and more details are given in Refs. [3,14].

5.2 Steps in scaling patch approach One motivation of the scaling patch approach is the observation that models involving small or large parameters are commonplace in natural science, and

34

Tie Wei

Table 8 Characteristics of the four-layer structure in a turbulent channel flow. Layer Force balance Layer thickness Mean velocity increment

I

jFviscj  jFpresj ≫ jFturbj

ΔU ! 0.5Ump

jFviscj  jFturbj  jFpresj

Δz ¼ O(ν/uτ) pffiffiffiffiffiffiffiffiffiffiffi Δz ¼ Oð δν=uτ Þ pffiffiffiffiffiffiffiffiffiffiffi Δz ¼ Oð δν=uτ Þ

II

jFviscj  jFturbj ≫ jFpresj

III IV

ΔU ¼ O(uτ)

jFpresj  jFturbj ≫ jFviscj

Δz ! δ

ΔU ! 0.5Ump

ΔU ¼ O(uτ)

More details are given in Refs. [3,14].

in general, the processes involve actions operating on more than one, often many, different space and time scales. Hence, the phenomenon being studied can then be most clearly and naturally represented, in certain subdomains, in terms of a function of “rescaled variables.” Rescaling means that new dependent and independent variables are defined as linear transformations of the original ones [5,7,10]. The scaling patch analysis starts with a generic rescaling of independent and dependent variables. The rescaling can be performed on the variables themselves as z ; lc def U ; U* ¼ uc def R wu : R*wu ¼ Rwu,c z* ¼ def

(42a) (42b) (42c)

Here the superscript ( )* denotes a generic normalized variable and will be changed to special notations for different layers. lc, uc, and Rwu,c are the characteristic length, velocity, and Reynolds shear stress scale, respectively, in a subdomain of the flow. In general, lc, uc, or Rwu,c may be different in a different layer of the flow. The rescaling may also be performed on the differential, referred to as the differential scaling in the scaling patch approach: dz ; lc def dU dU * ¼ ; uc def dR wu : dR*wu ¼ Rwu,c dz* ¼ def

(43a) (43b) (43c)

35

Analyses of buoyancy-driven convection

The differential scaling is more convenient in transforming between different scalings as shown in the scaling for the meso layer. Substituting the scaled variables in Eq. (42) or Eq. (43) into the mean momentum balance Eq. (41) yields 0¼

u2τ Rwu,c dR*wu νuc d 2 U * : + + lc dz* δ l 2c dðz* Þ2

(44)

The boundary conditions at the wall in the scaled variables are U * j0 ¼ 0;

(45a)

R*wu j0 ¼ 0;
dU *

lu u ¼ c τ τ:
* ν uc dz

(45b) (45c)

0

The scaling patch approach seeks an “admissible” scaling in a subdomain of the flow. An “admissible” scaling is defined as a scaled equation that must have at least two terms of nominal order of magnitude 1, and the rescaled boundary conditions should also be bounded. An admissible scaling for the inner layer requires that the viscous term in Eq. (44) must have a nominal order of magnitude 1. Multiplying l2c =νuc to Eq. (44) gives a nondimensional equation: 0¼

lc Rwu,c dR*wu d2 U * l u lu + + c τ c τ: 2 * * νu δ uc ν dz c dðz Þ

(46)

It is known that the friction velocity is a proper scale in the near-wall region: uc ¼ uτ. Then to set the boundary condition Eq. (45c) to 1 dictates a length scale lc ¼ ν/uτ. The prefactor to the turbulence term is then lc Rwu,c ν=uτ Rwu,c Rwu,c ¼ ¼ 2 : νuc νuτ uτ

(47)

A reasonable scale for the Reynolds shear stress is then Rwu,c ¼ u2τ , which makes the prefactor to the turbulence term as 1. In summary, based on physical reasoning, boundary condition requirement, and admissible scaling requirement, the characteristic scales for the inner layer of turbulent channel flow are set as uc ¼ uτ ;

lc ¼ ν=uτ ,

Rwu,c ¼ u2τ :

(48)

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Tie Wei

The scaled variables used for the inner scaling of the mean momentum equation are denoted as z z ¼ ¼ z+ ; lc ν=uτ U U ¼ ¼ U +; uc uτ Rwu R ¼ wu ¼ R+wu : Rwu,c u2τ

(49a) (49b) (49c)

The superscript of ()+ denotes variable scaled by the wall variables (friction velocity and kinematic viscosity), almost universally adopted in the study of wall-bounded turbulence, dating back to Prandtl (e.g., see Monin and Yaglom [1]). Using the inner-scaled variables, the inner scaling of the mean momentum balance equation can be presented as 0¼

dR+wu d2 U + 1 + + : + 2 + dz Reτ dðz Þ

(50)

Eq. (50) is the well-known inner-scaled mean momentum equation for wall-bounded turbulence, and can be found in many textbooks on fluid mechanics (e.g., [86]). Here, we show that the traditional inner-scaled equation is an admissible scaling in the scaling patch approach. Eq. (50) highlights the force balance in layer II, with the viscous force term and the turbulent force term as nominal order of magnitude 1. The pressure force in layer II is a high order term because 1/Reτ ≪ 1. Fig. 13A shows that the inner-scaled mean velocity profiles from different Reynolds number merge well onto a single curve in the near-wall A

B

Fig. 13 Inner scaling for a turbulent channel flow. (A) Inner-scaled mean velocity. (B) Inner-scaled Reynolds shear stress.

Analyses of buoyancy-driven convection

37

region, supporting the validity of the inner-scaled Eq. (50). Adjacent to the wall, it is well known that the growth of the inner-scaled mean velocity can be approximated by a linear function of the inner-scaled wall-normal distance: U+  z+, as represented by a dashed curve in Fig. 13A. The “loglaw,” U+  2.5ln(z+) + 5, is also plotted in Fig. 13A by a solid curve. Fig. 13B shows that the inner-scaled Reynolds shear stress profiles from different Reynolds numbers also merge well onto a single curve in the nearwall region. Adjacent to the wall, the growth of the Reynolds shear stress is proportional to the cube of the wall-normal distance: R+wu ∝ ðz+ Þ3 (see Monin and Yaglom [1]). Hence, the turbulence force dR+wu =dz+ is close to zero in layer I. Now, we will seek an admissible scaling for the outer layer, where the turbulent force is balanced by the pressure force. A nondimensional equation with the scaled pressure force as 1 is obtained by multiplying δ=u2τ to Eq. (44): 0¼

δ uc ν d 2 U * δ Rwu,c dR*wu + + 1: l c uτ lc uτ dz* l c u2τ dz*

(51)

In the core of the channel, flow is dominated by eddy motions, and the largest eddy size is constrained by the channel width. Thus, a natural length scale in the outer layer is the channel half-width lc ¼ δ. Hence, the prefactor to the turbulent term in Eq. (51) becomes δ Rwu,c Rwu,c ¼ 2 : lc u2τ uτ

(52)

For the turbulence term to have a nominal order of magnitude 1, the characteristic scale for the Reynolds shear stress should be set as Rwu,c ¼ u2τ . The characteristic scale is kept as uc ¼ uτ (this is not the only option. For example, Zagarola-Smits scale has also been used in the outer layer of wall-bounded turbulence (see [68,90])). The scaled variables used in the outer layer are denoted as z z ¼ ¼ z ; (53a) lc δ U U ¼ ¼ U +; (53b) uc uτ Rwu R ¼ wu ¼ R+wu : (53c) Rwu,c u2τ The wall-normal distance scaled by the channel half-width is denoted by a superscript (), to contrast the superscript ()+ for inner scaling. Substituting

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Tie Wei

the scaled variables in Eq. (53) into Eq. (51), the outer scaled mean momentum balance equation can be written as 0¼

dR+wu 1 d2 U + + + 1: dz Reτ dðz Þ2

(54)

Eq. (54) is an admissible scaling because it has two terms of nominal order of magnitude 1: the second and the third terms on the right. The first term has a nominal order of magnitude of 1/Reτ. In a fully developed channel turbulent flow, the Reynolds number is large Reτ ≫ 1. Hence, the first term or the viscous force is a high order term in the outer layer and does not contribute to the force balance, consistent with the force balance depicted in Fig. 8. Eq. (54) is the well-known outer scaling of the mean momentum equation for wall-bounded turbulence (see, e.g., Pope [86]). The mean velocity and Reynolds shear stress profiles consistent with the outer scaling Eq. (54) are presented in Fig. 14. Given the Galilean invariance of the equations of mechanics, the addition of any constant to all velocities do not change the momentum flux transmitted through the fluid [1]. In the outer layer, the mean velocity deficit is relevant in evaluating the outer scaling, not the mean velocity itself, because it is affected by the scaling from other layers in an additive fashion. Fig. 14A shows that the scaled mean velocity deficit profiles from different Grashof numbers merge well onto a single curve in the outer layer. The scaled Reynolds shear stress profiles from different Grashof numbers also merge onto a single curve in the outer layer, as shown in Fig. 14B. So far, the scaling patch analysis results for the inner layer and the outer layer are just a reproduction of the well-known traditional inner scaling and A

B

Fig. 14 Outer scaling for a turbulent channel flow. (A) Scaled mean velocity deficit. (B) Scaled Reynolds shear stress.

39

Analyses of buoyancy-driven convection

traditional outer scaling [2]. One contribution of the scaling patch approach is the development of a meso scaling for layer III in a turbulent channel flow. The scaling patch analysis for the meso layer III is best accomplished through the transformation of inner scaling Eq. (50) or outer scaling Eq. (54), using the rescaled differential scaling. Here, the transformation is performed on the outer scaled Eq. (54). The reason we transform from the outer scaled equation is that in buoyancy-driven convection, the outer length scale is obvious based on physical reasoning, but the inner length scale is often not obvious, as shown in Section 7.1. The meso scaled differentials for the wall-normal location and Reynolds shear stress are rescaled from the differential scaling for the outer layer as dzΛ ¼ ϕz dz ; dRΛwu

¼

ϕRwu dR+wu ,

(55a) (55b)

where the superscript ()Λ denotes the meso scaled variable. The generic symbol ϕ (in honor of Dr. Fife for his developments of the scaling patch approach) denotes the rescaling factor between the differentials. For example, ϕz is the rescaling factor between the meso scaled wall-normal location and the outer scaled wall-normal location. Substituting the meso scaled differentials in Eq. (55) into the outer scaled Eq. (54) yields 0¼

ϕ2z d 2 U + ϕz dRΛwu + + 1: Reτ dðzΛ Þ2 ϕRwu dzΛ

(56)

In layer III all three forces contribute to the balance of the equation. Therefore, an admissible scaling for layer III (the meso layer) requires all three terms have a nominal order of magnitude 1. The prefactors to the viscous term and turbulence term should be 1: ϕ2z ϕ ¼ z ¼ 1: Reτ ϕRwu The rescaling factors can then be obtained as pffiffiffiffiffiffiffiffi ϕz ¼ ϕRwu ¼ Reτ :

(57)

(58)

Substituting the meso scaled variables, the admissible scaling for the meso layer can be written as

40

Tie Wei

0¼

dRΛwu d2 U + + + 1: dzΛ dðzΛ Þ2

The rescaled distance in the wall-normal direction is pffiffiffiffiffiffiffiffi dzΛ ¼ Reτ dz ,

(59)

(60)

meaning that the meso scaled distance in the wall-normal direction is pffiffiffiffiffiffiffiffi stretched by a factor of Reτ with respect to the outer scaled wall-normal pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi distance. It is known that the thickness of layer III is Oð δν=uτ Þ or 1= Reτ when normalized by the channel half-width δ. Thus, stretched by a factor of pffiffiffiffiffiffiffiffi Reτ , the thickness of the meso layer is O(1) in the meso scaled distance. The variation of Reynolds shear stress around its peak is small, pffiffiffiffiffiffiffiffi ΔR+wu jIII ≪ 1. The meso scaled Reynolds shear stress dRΛwu ¼ Reτ dR+wu pffiffiffiffiffiffiffiffi is also stretched by a factor of Reτ , making the variation of the Reynolds shear stress in meso scaling across layer III be O(1) [91]. Differential scaling has the advantage of representing the local variation rate, instead of the local value, which is related to the scaling in the adjoining layer. The meso scaled wall-normal location and Reynolds shear stress are defined as pffiffiffiffiffiffiffiffi  z+  z+m pffiffiffiffiffiffiffiffi ; Reτ ðz  z Þ ¼ m Reτ p ffiffiffiffiffiffiffiffi def RΛwu ¼ Reτ ðR+wu  R+wu jzm Þ:

zΛ ¼ def

(61a) (61b)

where zm denotes the peak Reynolds shear stress location. Here, the origin of the meso scaled wall-normal location is shifted to the peak Reynolds shear stress location to offset the influence of the adjoining layer. The meso scaled Reynolds shear stress is also shifted with respect to the peak value. Under such definitions, the meso scaled Reynolds shear stress peaks at zΛ ¼ 0 with a peak value of RΛ wu ¼ 0, as shown in Fig. 15A. The variation of the meso scaled Reynolds shear stress is ΔΛ Rwu ¼ Oð1Þ over a region of ΔzΛ ¼ O(1), as shown in the figure inset. Fig. 15B presents the profiles of the mean velocity shifted by its value at zm: U +  U +zm vs the meso scaled wall-normal location. The figure inset shows that the variation of the mean velocity around the center of the meso layer is also ΔU+ ¼ O(1) over a range of ΔzΛ ¼ O(1). Therefore, the meso layer in a turbulent channel flow is a scaling patch. A more significant contribution of the scaling patch approach is the discovery that a hierarchy of layer structure (scaling patches) can be obtained by

41

Analyses of buoyancy-driven convection

A

B

Fig. 15 Meso scaling for a turbulent channel flow. (A) Reynolds shear stress. (B) Mean velocity.

a simple transformation of the Reynolds shear stress. The transformed Reynolds shear stress, called adjusted Reynolds shear stress, for a turbulent channel flow is defined as +  Rⓐ  az , wu ¼ R wu + z def

(62)

where a is called the adjusting parameter. Note that in the previous presentation of the scaling patch approach (e.g., [4] or [5]), the adjusted Reynolds shear stress is typically transformed using z+ and the adjusting parameter uses the symbol of β. Here the symbol is changed to a special character for two reasons: (1) in buoyancy-driven convection, β is reserved for the thermal expansion coefficient; (2) the special character emphasizes that it is not a power exponent, but a superscript denoting a normalized variable. The adjusted Reynolds shear stress profiles with five values of a are illustrated in Fig. 16. By increasing the value of parameter a continuously, the peak location of the adjusted Reynolds shear stress profile also continuously moves from the core of the channel toward the wall. The continuous scaling patch (or hierarchy of layer) is centered on the peak of the adjusted Reynolds shear stress profile, or the meso layer with the adjusted Reynolds shear stress. The scaling patch is best elucidated by a meso scaling of the mean momentum equation with the adjusted Reynolds shear stress. From the definition of the adjusted Reynolds shear stress, the gradient of the Reynolds shear stress is dR+wu dRⓐ ¼ wu + a  1: dz dz

(63)

42

Tie Wei

Fig. 16 Adjusted Reynolds shear stress in a turbulent channel flow, with five different adjusting parameters.

Substituting Eq. (63) into the outer scaling Eq. (54) produces an outer scaling of the mean momentum equation with the adjusted Reynolds shear stress as 0¼

dRⓐ 1 d2 U + wu + + a: dz Reτ dðz Þ2

(64)

To obtain an admissible scaling for the region around the peak of the adjusted Reynolds shear stress (the meso layer with the adjusted Reynolds shear stress), differential scaling is used to transform outer scaling to meso scaling as dzΛa ¼ ϕz dz ; dRΛwua

¼

ϕRwu dRⓐ wu :

(65a) (65b)

Substituting the meso scaled variables, Eq. (64) becomes 0¼

ϕ2z d2 U + ϕz dRΛwua + + a: Reτ dðzΛa Þ2 ϕRwu dzΛa

(66)

An admissible scaling for the meso layer requires all three terms to have a nominal order of magnitude 1, or ϕ2z ϕ ¼ z ¼ a: Reτ ϕRwu

(67)

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Analyses of buoyancy-driven convection

It follows that the rescaling factors should be set as pffiffiffiffiffiffiffiffiffiffi ϕz ¼ aReτ ; rffiffiffiffiffiffiffiffi Reτ ϕRwu ¼ : a

(68a) (68b)

Substituting the meso scaled variables, the meso scaling with the adjusted Reynolds shear stress becomes 0¼

dRΛwua d2 U + + + 1: dzΛa dðzΛa Þ2

(69)

For brevity, the results for the meso scaled mean momentum equation with adjusted Reynolds shear stress are presented only for two cases with a ¼ 0.25 in Fig. 17 and a ¼ 4 in Fig. 18. Like the meso scaling for the actual “meso layer,” the thickness of the meso layer of the adjusted Reynolds shear stress is O(1) in the rescaled wall-normal distance, and the variation of the adjusted Reynolds shear stress is also O(1) in the rescaled dRΛwua , as shown in the insets of Fig. 17B and Fig. 18B. Therefore, the meso layer with the adjusted Reynolds shear stress is a scaling patch. By varying a continuously, a hierarchy of scaling patches can be obtained in a turbulent channel flows. More details on the upper and lower bound on the scaling patch are given in Refs. [4,5,10]. A rigorous connection between the scaling hierarchy and the mean velocity profile has been established by Fife and coworkers [4,11,14]. The connection is through a certain function A(z+) defined in terms of the hierarchy, which remains O(1) for all z+. The mean velocity satisfies an exact logarithmic growth law in an interval of the hierarchy if and only if A is A

B

Fig. 17 (A) Adjusted Reynolds shear stress with a ¼ 0.25. (B) Meso scaled adjusted Reynolds shear stress.

44 A

Tie Wei

B

Fig. 18 (A) Adjusted Reynolds shear stress with a ¼ 4. (B) Meso scaled adjusted Reynolds shear stress. Inset highlights the scaling around the “meso layer.”

constant. In general, A is not an exact constant in any such interval, however, it is arguably almost constant under certain circumstances in some regions. This derivation of log-law is completely independent of classical inner/outer/overlap scaling arguments, for example, the Izakson–Millikan style argument [92,93]. The asymptotic value of the von Ka´rma´n constant in the log-law was determined by Klewicki et al. [14] by an emerging condition of dynamic self-similarity on an interior inertial domain. The scaling patch approach has been applied to other forced wallbounded turbulence, for example, turbulent Poiseuille-Couette flow [8], turbulent Taylor-Couette flow [17]. The approach has also been applied to passive scalar transport in turbulent channel flow [6,15,16]. The scaling patch approach has shed insights into the underlying physics that was not obtained from other approaches.

6. Scaling analysis of laminar DHVC We start the scaling patch analysis of buoyancy-driven convection with laminar DHVC. The analysis itself is trivial, but it provides a good example to compare the traditional scaling analysis and scaling patch approach. In traditional scaling analysis of buoyancy-driven convection (laminar or turbulent), the commonly used length scale is the domain size lc ¼ δ and the commonly used temperature scale is related to the temperature difference across the domain θc ¼ Θmp. Two velocity scales are commonly used: one is a diffusion velocity scale uc ¼ ν/δ, and the other is the free-fall pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity uc ¼ U ff ¼ gβΘmp δ.

45

Analyses of buoyancy-driven convection

If the characteristic scales are set as lc ¼ δ, θc ¼Θmp, and uc ¼ ν/δ, the momentum equation for laminar DHVC can be transformed into a nondimensional form as   d 2 ðU=ðν=δÞÞ Θ 0¼ : + Gr 1  (70) Θmp dðz=δÞ2 Eq. (70) is a good example to explain the concepts of nominal order of magnitude and numerical order of magnitude defined in Table 7. The nominal order of magnitude of the two terms in Eq. (70) are 1 and Gr. However, there are only two terms in Eq. (70), which have to be balanced. Thus, the numerical orders of magnitude of the two terms must be the same. The variation of (1  Θ=Θmp ) in the channel is O(1), but Gr can be very small (≪ O(1)) or reasonable large (on the order of 100–1000, not above the critical Grashof number to maintain a laminar state [30]). This means that the numerical magnitude of the first term can also vary as ≪ O(1) for small Gr or ≫ O(1) for large Gr, indicating that either the velocity scale or the length scale in the viscous term is not appropriate. In this case, the velocity scale uc ¼ ν/δ is not appropriate. As the two terms in Eq. (70) are not nominal order of magnitude 1, this scaling is not an admissible scaling. Another traditional scaling analysis of laminar DHVC sets lc ¼ δ, θc ¼Θmp, and uc ¼ Uff, and the scaled momentum equation becomes   d2 ðU=U ff Þ pffiffiffiffiffiffi Θ 0¼ + Gr 1  (71) : Θmp dðz=δÞ2 pffiffiffiffiffiffi The nominal order of magnitude of the two terms in Eq. (71) are 1 and Gr. Hence, Eq. (71) is not an admissible scaling either. Now we will present a scaling patch analysis of the momentum equation for laminar DHVC. As reviewed in Section 5, the first step in seeking an admissible scaling is to scale the independent variable z and the dependent variables U and Θ by their characteristic scales z ; lc def Θ Θ* ¼ ; θc def U U* ¼ : uc z* ¼ def

(72a) (72b) (72c)

46

Tie Wei

Substituting the scaled variables in Eq. (72) into the momentum and heat equations produce αθc d 2 Θ* ; l2c dðz* Þ2

(73a)

νuc d2 U * + gβθc ðΘ*mp  Θ* Þ: l2c dðz* Þ2

(73b)

0¼ 0¼

To transform these equation into nondimensional form, multiplying l2c =ðαθc Þ to Eq. (73a), and multiplying l2c =ðνuc Þ to Eq. (73b) yield d 2 Θ* 2; dðz* Þ

(74a)

gβθc l 2c * d2 U * + ðΘmp  Θ* Þ: 2 * νu c dðz Þ

(74b)

0¼ 0¼

The normalized boundary conditions at the wall and the channel midplane are z* ¼ 0 : U * ¼ 0; Θ* ¼ 0; Θmp δ z* ¼ : U * ¼ 0; Θ* ¼ : θc lc

(75a) (75b)

To make the normalized boundary conditions admissible, the length scale is set as lc ¼ δ, and the temperature scale as θc ¼ Θmp. For the scaled momentum Eq. (74b) to be admissible, the prefactor to the buoyancy term should be 1: gβΘmp δ2 gβθc l2c ¼ ¼ 1, νuc υuc

(76)

from which the characteristic velocity scale is obtained as uc ¼ gβΘmp δ2 =ν:

(77)

Thus, the properly scaled momentum and heat equations, and the boundary conditions for laminar DHVC become d 2 Θ* 2; dðz* Þ

(78a)

d2 U * * * 2 + ðΘmp  Θ Þ: * dðz Þ

(78b)

0¼ 0¼

47

Analyses of buoyancy-driven convection

and z* ¼ 0 :

U * ¼ 0;

Θ* ¼ 0;

(79a)

z* ¼ 1 :

U * ¼ 0;

Θ* ¼ 1:

(79b)

Eq. (78b) is an admissible scaling. In summary, laminar DHVC is a single scale problem, and the proper length, velocity, and temperature scales are lc ¼ δ; uc ¼

(80a)

gβΘmp δ gβqt δ ¼ ; ν να θc ¼ Θmp : 2

3

(80b) (80c)

Using variables scaled by the characteristic scales in Eq. (80), the laminar DHVC solutions (Eqs. 18a and 18b) becomes Θ* ¼ z * ; U* ¼

*

* 2

(81a) * 3

ðz Þ ðz Þ z  : + 3 2 6

(81b)

Hence, velocity and temperature distributions in laminar DHVC are “selfsimilar” i.e., properly scaled variables do not vary with the Prandtl number or the Grashof number. It has been shown that the traditionally used diffusion velocity scale or free-fall velocity scale is not a proper velocity scale in laminar DHVC. The traditional velocity scales are related to the proper velocity scale in laminar DHVC as follows uc ¼ Gr; (82a) ν=δ pffiffiffiffiffiffi uc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Gr: (82b) gβΘmp δ Thus, for small Grashof numbers Gr ≪ 1, the traditional velocity scales are too large to capture the characteristics of the fluid motion in laminar DHVC. For large Grashof numbers Gr ≫ 1 (but smaller than the critical Grashof number), the traditional velocity scales are too small to represent the fluid motion in laminar DHVC. The single scale nature of laminar DHVC is distinctively different from turbulent DHVC where multiple scales are essential. The multiscale nature of turbulent DHVC is elucidated next by the scaling patch analysis of the mean momentum and mean heat equations.

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Tie Wei

7. Scaling analysis of the mean momentum equation in turbulent DHVC In the dimensional analysis of turbulent DHVC in Section 4, physical intuition and experience are employed to conceive qualitatively an inner layer, a meso layer, and an outer layer. However, assisted with direct numerical simulation data, a multilayer structure can be distinguished directly from the mean momentum or heat equation. The mean momentum Eq. (13b) for turbulent DHVC represents the balance of three forces in the vertical direction (the mean flow direction): a viscous force Fvisc ¼ νd2U/dz2, a turbulent force Fturb ¼ dRwu/dz, and a buoyancy force Fbuoy ¼ gβ(Θmp Θ). The typical distribution of the three forces is illustrated in Fig. 19. Like the force balance in turbulent channel flow shown in Fig. 10, the force balance in turbulent DHVC is established differently depending on the wall-normal location.

7.1 Layer structure of the mean momentum balance equation One way to appraise the contribution of different forces to the balance of the mean momentum Eq. (13b) is to plot the ratio between two of the three

Fig. 19 Typical force distributions in turbulent DHVC. Magnitudes of the forces are not shown, for they depend on the normalization. The data are from the DNS of Kiš [31]. Turbulent DHVC is divided into three layers, based on the characteristic of the force balance. In Layer I, the force balance is between the viscous force and buoyancy force. In layer III, the force balance is between the turbulent force and the buoyancy force. In layer II all the forces contribute to the balance.

49

Analyses of buoyancy-driven convection

A

B

Fig. 20 Force ratio in turbulent DHVC. (A) Ratio of the viscous force to turbulent force. (B) Ratio of the viscous force to buoyancy force. The near-wall region is better shown in the inset of the figure.

forces. In Fig. 20A, the ratio between the viscous force and turbulent force is shown. In a thin layer adjacent to the wall, the viscous force is much larger than the turbulent force. As Grashof (or Reynolds) number increases, this layer becomes a smaller fraction of the channel. In this thin layer, the force balance is essentially between the viscous force and buoyancy force, and the ratio between the viscous force and buoyancy force is close to 1, as shown in the inset of Fig. 20B. This thin layer adjacent to the wall is called layer I of the mean momentum equation, in which the viscous force is important in the balance of the equation. In layer I, the molecular properties, such as the kinematics viscosity and thermal diffusivity, are important parameters that affect the flow and heat transport, supporting the selection of ν and α as the control parameters in the dimensional analysis of the inner layer in turbulent DHVC (see Section 4). Away from the wall, the viscous force decreases rapidly and becomes essentially zero in the core of the channel, as shown in Fig. 20. Hence, the force balance in the core of the channel is established by the turbulent force and the buoyancy force. The core of the channel is called the outer layer or layer III. As the viscous force plays a negligible role in the force balance of the outer layer, the molecular properties ν or α are not controlling parameters for the flow or heat transport. This is consistent with the physical intuition used in the dimensional analysis of the outer layer in Section 4. Between the inner layer and the outer layer, there exists a transitional layer and is called layer II, in which all three forces contribute to the balance of the equation. The three-layer structure is sketched by vertical dash lines in Fig. 19. The structures of the right and left halves of the channel are symmetric.

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A

B

Fig. 21 The thickness of layer I for the mean momentum equation in a turbulent DHVC. The thickness is normalized by ν/uτ. (A) Normalized thickness vs Gr. (B) Normalized thickness vs Reτ.

Pragmatically, the end of layer I is defined as the location where Fvisc/ Fturb ¼ 4 or Fvisc/Fbuoy ¼ 0.8, as indicated by the horizontal dash line in Fig. 20. If a factor larger than Fvisc/Fturb ¼ 4 is used for the demarcation, extremely fine resolution is required to determine the end of layer I (see Fig. 20). The layer I thickness normalized by ν/uτ is plotted in Fig. 21A vs the Grashof number, and in Fig. 21B vs the friction Reynolds number. The data are from three independent DNS studies by Versteegh [30], Kisˇ [31], and Ng [32]. The Prandtl number in all the simulations is for air at Pr  0.7. In forced wall-bounded turbulence such as turbulent pipe flow or turbulent channel flow, the thickness of the viscous sublayer scales with ν/uτ, largely independent of the Reynolds number [1,2]. Fig. 21 shows that the layer I thickness normalized by ν/uτ increases with the Grashof number or Reynolds number in turbulent DHVC. Also shown in Fig. 21 is the maximum mean vertical velocity normalized by the friction velocity Umax/uτ, which also increases with the Grashof or Reynolds number. As shown in the following analysis of the mean momentum equation, the maximum vertical velocity normalized by the friction velocity U +max ¼ U max =uτ plays a critical role in the scaling analysis. Based on the DNS data, the layer I thickness and the maximum mean vertical velocity display similar dependences on the Grashof or Reynolds number as zjF visc =F turb ¼4 U  1:25 max : uτ ν=uτ

(83)

As discussed in dimensional analysis for the inner layer of turbulent DHVC in Section 4, the inner length scale lν can be presented as lν ¼

ν Ψ ðPr, Rii Þ, uτ ν

(84)

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Analyses of buoyancy-driven convection

where the nondimensional function Ψν has to be determined from DNS data. Here, the viscous length scale lν is defined to be proportional to the layer I thickness shown in Fig. 21. Thus, the nondimensional function Ψν can be approximated as Ψν ðPr, Rii Þ 

U max ¼ U +max : uτ

(85)

This inner length scale lν ¼ ν=uτ Ψν ¼ ν=uτ U +max in the buoyancy-driven turbulent DHVC is distinctively different from that in pressure- or sheardriven turbulent wall-bounded flows, where the viscous length scale is ν/uτ and is largely independent of the Reynolds number.

7.2 Properties of the Reynolds shear stress As shown in Section 3, all mean equations for turbulent flows, forced or buoyancy-driven, are under-determined. It is practically impossible to determine all the characteristic scales theoretically. Hence, direct numerical simulation data are essential to gain more insight into the unclosed terms and the equations themselves. Before seeking an admissible scaling for each layer of turbulent DHVC, we will determine the properties of some key quantities in the mean momentum Eq. (13b). One of the most important quantities in the scaling of the mean momentum equation is the Reynolds shear stress Rwu [94]. Applying dimensional analysis for the outer layer of turbulent DHVC in Section 4, the Reynolds shear stress magnitude at the channel mid-plane can be found as jRwu jmp ¼ ΨðRio Þ, u2τ

(86)

where Ψ denotes a generic nondimensional function that has to be determined experimentally or numerically. Fig. 22A presents the magnitude of the Reynolds shear stress at the channel mid-plane normalized by the square of the friction velocity jRwu jmp =u2τ vs the Grashof number. In Fig. 22B, data of jRwu jmp =u2τ are plotted vs Rio, which is the controlling nondimensional number based on the dimensional analysis of outer layer in Section4. Also plotted in the figure is the data of the maximum vertical velocity normalized by the friction velocity (Umax/uτ), because this quantity plays a key role in the scaling analysis of the mean momentum equation [94].

52 A

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B

Fig. 22 Scaling of the magnitude of the Reynolds shear stress at the channel mid-plane. (A) jRwu jmp =u2τ vs Gr. (B) jRwu jmp =u2τ vs Rio.

Based on the presently available DNS data (at Pr  0.7) shown in Fig. 22, the nondimensional function Ψ in Eq. (86) can be approximated by a simple power-law of Rio with an exponent of 1: jRwu jmp ¼ ΨðRio Þ  Rio : u2τ

(87)

Substituting the definition of the outer Richardson number in Eq. (33) into Eq. (87) produces a relation between the magnitude of the Reynolds shear stress at the channel mid-plane and the buoyancy parameter as jRwu jmp  gβθτ δ:

(88)

Hence, the Reynolds shear stress at the channel mid-plane is directly affected by the buoyancy parameter. The DNS data also indicate that the maximum vertical velocity scaled by the friction velocity also increases linearly with the outer Richardson number: U max  2:0Rio : uτ

(89)

Dividing Eq. (87) by Eq. (89) produces a scaling for the Reynolds shear stress at the channel mid-plane: jRwu jmp  0:5uτ U max :

(90)

Therefore, the Reynolds shear stress in the outer layer of turbulent DHVC can be scaled by a mixed scale of uτUmax [94]. This mixed scale for the Reynolds shear stress in a turbulent DHVC is distinctively different from that in forced wall-bounded turbulence, which is scaled by u2τ . The mixed

Analyses of buoyancy-driven convection

53

scale uτUmax  2gβθτδ for the Reynolds shear stress in a turbulent DHVC reflects directly the effect of buoyancy on the Reynolds shear stress. The properties of the Reynolds shear stress and the maximum mean velocity obtained from DNS data, Eqs. (87), (88), (89), (90), are essential to determine the characteristic scales for an admissible scaling of the mean momentum balance equation. As presented in Section 5, the search for an admissible scaling starts with a generic scaling of the independent variables and dependent variables, with undetermined characteristic scales. The generic scalings of the variables in the mean momentum Eq. (13b) for a turbulent DHVC are denoted as z z* ¼ ; lc U U* ¼ ; uc Θ Θ* ¼ ; θc Rwu * , Rwu ¼ Rwu,c

(91a) (91b) (91c) (91d)

where lc, uc, θc are the characteristic length scale, velocity scale, and temperature scale, respectively, for a subdomain of the channel. Rwu,c is the characteristic scale for the Reynolds shear stress. In forced wall-bounded turbulence, the scale for the Reynolds shear stress is typically taken as the square of the characteristic velocity scale u2c . However, in buoyancy-driven turbulence, DNS data indicate that the Reynolds shear stress in the outer layer is not scaled by u2c (see Eq. (90)). Substituting the scaled variables in Eq. (91) into the mean momentum Eq. (13b) produces 0¼

Rwu,c dR*wu νuc d 2 U * + + gβθc ðΘ*mp  Θ* Þ: l c dz* l2c dðz* Þ2

(92)

This equation will be rearranged to obtain admissible scaling for each layer of turbulent DHVC. The characteristic scales will be determined by seeking admissible scaling for a subdomain (or layer) of the channel.

7.3 Outer scaling of the mean momentum equation For a scaling analysis of wall-bounded turbulence, shear-driven or buoyancy-driven, it is better to start with the outer layer because the natural

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length scale is the domain size, which constrains the size of the largest eddies. For turbulent DHVC, a natural length scale in the outer layer is the channel half-width lc ¼ δ. In contrast, the proper length scale for the inner layer of wall-bounded turbulence, especially in buoyancy-driven DHVC or RBC, is typically not known a priori. It is known that the force balance in the outer layer is essentially between the turbulent force and the buoyancy force, as shown in Fig. 19. Thus, in the admissible scaling for the outer layer, the turbulence term and buoyancy term should have a nominal order of magnitude 1. A nondimensional equation, in which the turbulence term has a nominal order of magnitude 1, can be obtained by multiplying lc/Rwu,c to Eq. (92): 0¼

dR*wu gβθc lc * νuc d2 U * + + ðΘ  Θ* Þ: * l c Rwu,c dðz* Þ2 Rwu,c mp dz

(93)

For Eq. (93) to be an admissible scaling in the outer layer, the buoyancy term must also have a nominal order of magnitude 1. Thus, the prefactor to the buoyancy term must be O(1), providing a constraint on the selection of the characteristic scale in the outer layer: gβθc lc ¼ Oð1Þ: Rwu,c

(94)

The length scale has been set as lc ¼ δ based on physical reasoning. There are still two scales θc and Rwu,c to be determined with one constraint. An additional constraint is obtained from the relevant boundary condition. At the channel mid-plane (edge of the outer layer), the scaled boundary conditions are U * jz¼δ ¼ 0; Θjmp ; Θ* jz¼δ ¼ θc Rwu jmp : R*wu jz¼δ ¼ Rwu,c

(95a) (95b) (95c)

From the boundary condition, one reasonable choice for the characteristic Reynolds shear stress is Rwu,c ¼ jRwujmp, which makes the scaled boundary condition as R*wu ¼ 1. Substituting lc ¼ δ and Rwu,c ¼ jRwujmp, the prefactor to the buoyancy term becomes

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Analyses of buoyancy-driven convection

gβθc lc gβθc δ gβθτ δ θc ¼ ¼ : Rwu,c jRwu jmp jRwu jmp θτ

(96)

Eq. (88) indicates that gβθτδ/jRwujmp  1. Hence, the prefactor Eq. (96) will be O(1) if the characteristic temperature scale is set as θc ¼ θτ. Based on physical intuition assisted with the mean velocity profile shown in Fig. 5B, it is reasonable to set the characteristic velocity scale in the outer layer as uc ¼ Umax. In summary, based on physical reasoning, admissible scaling requirement, boundary condition requirement, and empirical observation, the characteristic scales for the outer layer in a turbulent DHVC are set as lc ¼ δ;

uc ¼ U max ;

θc ¼ θτ ;

Rwu,c ¼ jRwu jmp :

(97)

The scaled variables for the outer layer are denoted as z z ¼ ¼ z ; lc δ U U ¼ ¼ U ; uc U max Θ Θ ¼ ¼ Θ+ ; θc θτ Rwu Rwu ¼ ¼ R wu : Rwu,c jRwu jmp

(98a) (98b) (98c) (98d)

The superscript ( )+ follows the so-called inner scaling convention almost universally used in wall-bounded turbulence [2]. To contrast the notation for the inner scaling with wall variables, a superscript ( ) denotes variable scaled by an outer scale (δ, Umax or jRwujmp) Substituting the outer scaled variables in Eq. (98) into the mean momentum balance Eq. (93), the nondimensional mean momentum balance equation can be written as 0¼

 dR gβθτ δ  + νU max d 2 U  + wu + + Θ  Θ : mp dz δjRwu jmp dðz Þ2 jRwu jmp

(99)

Note that the prefactor of the buoyancy term can be presented as gβθτ δ gβθ δ u2τ Rio ¼ 2τ ¼ : uτ jRwu jmp jRwu jmp =u2τ jRwu jmp

(100)

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Empirically it is found that jRwu jmp =u2τ  Rio (see Eq. 87). Thus, the prefactor to the buoyancy term has a nominal order of magnitude 1. The prefactor to the viscous term can be rearranged as νU max ν uτ U max 1 uτ U max ¼ ¼ : δjRwu jmp δuτ jRwu jmp Reτ jRwu jmp

(101)

Empirically it is found that jRwujmp  0.5uτUmax (see Eq. (90)). Thus, the viscous term has a nominal order of magnitude 1/Reτ, a high order term. The outer scaling equation for the mean momentum equation can then be written as 0¼

  dR 1 uτ U max d2 U  Rio + + wu + + Θ  Θ : (102) mp dz Reτ jRwu jmp dðz Þ2 jRwu jmp =u2τ

This equation has two terms with nominal order of magnitude 1: the turbulence term and the buoyancy term, and a high order viscous term, so it is an admissible scaling for the outer layer. Two nondimensional numbers, Reτ and Rio, appear “naturally” in the outer scaled mean momentum balance Eq. (102), in particular, the friction Reynolds number gives a direct measure of the contribution of the viscous term. Therefore, we advocate using the friction Reynolds number Reτ in the study of turbulent DHVC, instead of the Grashof number Gr. The friction Reynolds number, inner Richardson number, and outer Richardson number for six simulations of Kisˇ [31] are listed in Table 9. As discussed in Section 4, the inner and outer Richardson numbers can be interpreted as the ratio of length scales. For example, at the highest Grashof number simulation Gr ¼ 1.5  106, the outer Richardson number Table 9 Friction Reynolds number, inner Richardson number, and outer Richardson number for six simulations of Kiš [31]. Gr Reτ Rii Rio

4.75  104

61.2

0.0255

1.560

1.19  105

86.5

0.0216

1.868

2.86  10

5

120.9

0.0179

2.169

5

160.8

0.0150

2.426

1.0  106

195.0

0.0136

2.657

1.5  10

227.0

0.0126

2.866

6.0  10

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Analyses of buoyancy-driven convection

57

is Rio ¼ 2.866, meaning that the channel half-width is 2.866 times of the Obukhov-style length scale or δ ¼ 2.866L. For the same Grashof number, the inner Richardson number is Rii ¼ 0.0126, meaning that the Obukhovstyle length scale is 1/0.0126 ¼ 79.4 times of the length scale of ν/uτ. The friction Reynolds number represents the ratio between the channel halfwidth and ν/uτ, which is 227 for this Grashof number. If the mixed scale uτUmax is used to scale the Reynolds shear stress, an alternative outer scaling equation can be presented as 0¼

dðRwu =ðuτ U max ÞÞ gβθτ δ 1 d2 U  + + ðΘ+  Θ+ Þ: (103)  2  Reτ dðz Þ uτ U max mp dz

Note that the prefactor to the buoyancy term is gβθτδ/(uτUmax)  0.5. In the outer layer 0 < z < 1, U is bounded between 0 and 1. Moreover, U is a regular function with derivatives (say first and second-order) bounded < O(1). At high Grashof numbers, the prefactor to the viscous term 1/Reτ is much smaller than 1. Thus, it is clear that the first term, the viscous force, is a high order term and is negligible in the force balance of the outer layer, consistent with the force distributions depicted in Fig. 19. If scaling is appropriate for a subdomain of the flow, then the scaled dependent variables from different Grashof (or Reynolds) numbers should merge onto a single curve in the subdomain when plotted vs the scaled independent variable. Fig. 23 shows that the outer scaled Reynolds shear stress profiles from different Grashof numbers merge well onto a single

Fig. 23 Outer scaled Reynolds shear stress in a turbulent DHVC. The data are from the DNS of Kiš [31].

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curve around the channel mid-plane, supporting the validity of the outer scaled Eq. (102). Fig. 23 shows that the Reynolds shear stress is a regular function in the outer layer. Moreover, the Reynolds shear stress is an even function around the channel mid-plane. A Taylor series expansion for the Reynolds shear stress in the neighborhood of the channel mid-plane can be presented as 2 4   R wu ¼ a1 + a2 ðz  1Þ + a3 ðz  1Þ + ⋯

(104)

By definition, the outer scaled Reynolds shear stress is  1 at the channel mid-plane R wu jz ¼1 ¼ 1. Thus, the coefficient a1 is  1. Dropping high order terms, the Reynolds shear stress in the neighborhood of the channel mid-plane can be approximated as 2  R wu  1 + 0:92ðz  1Þ ,

(105)

where a2  0.92 comes from the curve-fitting of the DNS data. Fig. 23 shows that the approximation Eq. (105) fits the DNS data excellently in the core of the channel. Empirically it is known that the Reynolds shear stress can also be scaled by a mixed scale of uτUmax (see Eq. 90). Fig. 24 presents the Reynolds shear stress profiles normalized by the mixed scale uτUmax. The deviation is noticeable at low Grashof numbers (or low Reynolds numbers) in Fig. 24. This

Fig. 24 Reynolds shear stress scaled by uτUmax in a turbulent DHVC. The data are from the DNS of Kiš [31].

Analyses of buoyancy-driven convection

59

Fig. 25 The outer scaled mean vertical velocity in turbulent DHVC. Data are from the DNS of Kiš [31].

low Reynolds number effect is not surprising in wall-bounded turbulence, indicating that the turbulence may not be fully established yet. At a lower Reynolds number, especially in the transitional Reynolds number regime, the deviation in Fig. 24 will be more pronounced. The outer scaled mean vertical velocity profiles from different Grashof numbers also merge onto a single curve in the core of the channel, as shown in Fig. 25. The mean vertical velocity is a regular odd function about the channel mid-plane, and an approximation function from Taylor series expansion can be obtained as (dropping higher-order terms) U ¼

U  1:15ðz  1Þ: U max

(106)

The mean temperature profiles for the outer layer will be presented in the scaling analysis of the mean heat equation in Section 8.

7.4 Inner scaling of the mean momentum equation In layer I of the mean momentum equation, the force balance is between the viscous force and the buoyancy force, and the contribution of the turbulent force is negligible, as shown in Figs. 19 and 20. Hence, we will seek an admissible scaling in the inner layer by making the viscous term and buoyancy term nominal order of magnitude 1. The viscous term in the mean

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momentum equation can be made nominal order of magnitude 1 by multiplying l 2c =ðνuc Þ to Eq. (92) as 0¼

dR* gβθc l2c * lR d2 U * + c wu,c wu + ðΘmp  Θ* Þ: 2 * * νu νu dz c c dðz Þ

(107)

For an admissible scaling in the inner layer, the prefactor to the buoyancy term gβθc l2c =ðνuc Þshould be 1 by setting lc, uc, and θc properly. There is only one constraint, but three undetermined characteristic scales. Additional constraints come from boundary conditions, physical reasoning, and empirical observation. The boundary conditions at the wall (one edge of layer I) are U * j0 ¼ 0;

(108a)

Θ* j0 ¼ 0;
dU *

lu u ¼ c τ τ:
* ν uc dz

(108b) (108c)

0

As discussed in Section 7.1 (see Eqs. (84) and (85)), the inner length scale lν for turbulent DHVC is lν ¼ ν/uτΨν(Pr, Rii). The characteristic velocity scale uc can be determined by setting the boundary condition in Eq. (108c) to be 1: uc ¼

ν=uτ Ψν uτ l c uτ u ¼ uτ ¼ uτ Ψν : ν τ ν

(109)

A proper temperature scale in the near-wall region is found as (see Eq. (144) in Section 8.4) θc ¼ θτ Ψα ðPr, Rii Þ:

(110)

The inner-scaled variables for the mean momentum balance equation are denoted as z z z+ ¼ ¼ ; (111a) lc ν=uτ Ψv Ψν U U U+ ¼ ¼ ; (111b) uc Ψν uτ Ψν Θ Θ Θ+ ¼ ¼ , (111c) θc Ψα θτ Ψα where the notations z+ ¼ z/(ν/uτ), U+ ¼ U/uτ, and Θ+ ¼ Θ/θτ follow the inner scaling convention used in forced wall-bounded turbulence [2]. Substituting the inner-scaled variables in Eq. (111) into Eq. (107), the inner-scaled mean momentum equation can be rewritten as

Analyses of buoyancy-driven convection

 d 2 ðU + =Ψν Þ Rwu,c dðRwu =Rwu,c Þ 0¼ + u2τ dðz+ =Ψν Þ dðz+ =Ψν Þ2 !  + Θmp Θ+ gβθτ ν + : Ψν Ψα  Ψα Ψα u3τ

61

(112)

For Eq. (112) to be an admissible scaling in the inner layer, the buoyancy term must have a nominal order of magnitude 1, and the turbulence term must have a nominal order magnitude  O(1). The characteristic scale for the Reynolds shear stress must be equal to or smaller than u2τ , otherwise, the prefactor to the turbulence term will be > O(1). A reasonable choice for the Reynolds shear stress in the inner layer is Rwu,c ¼ u2τ , which makes the prefactor of the turbulence term as 1. The inner-scaled Reynolds shear stress is denoted as R+wu ¼ Rwu =u2τ . In summary, assisted with physical reasoning, boundary condition requirement, and admissible scaling requirement, the characteristic scales for the inner layer are set as ν Ψ; uτ ν uc ¼ Ψν uτ ; θc ¼ Ψα θτ ;

(113b) (113c)

Rwu,c ¼ u2τ :

(113d)

lc ¼ lν ¼

(113a)

With the inner scaling notations, the inner scaling for the mean momentum balance equation can be presented as 0¼

Θ+mp Θ+ dR+wu d2 ðU + =Ψν Þ + Ri + ½ Ψ Ψ ð  Þ: i ν α Ψα Ψα dðz+ =Ψν Þ dðz+ =Ψν Þ2

(114)

For this inner scaling Eq. (114) to be admissible, the prefactor of the buoyancy term must have nominal order of magnitude of O(1), independent of the Reynolds number: gβθc l2c gβθ ν ¼ 3τ Ψν Ψα ¼ Rii Ψν Ψα ¼ Oð1Þ: νuc uτ

(115)

Fig. 26 shows that the inner-scaled mean velocity profiles from different Grashof numbers merge well onto a single curve in the near-wall region, an indication of proper scaling. The figure shows that the inner scaling is valid to z/lν  1  2. Thus, the thickness of the inner layer is O(1) with

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A

B

Fig. 26 Inner scaling of the mean vertical velocity. (A) On linear-linear axes. (B) On loglinear axes to better shown the near-wall region. The nondimensional function Ψν is approximated as U+max base on the DNS data at Pr ¼ 0.7 (see Eq. (85)).

A

B

Fig. 27 Inner scaling of the Reynolds shear stress. (A) On linear-linear axes. (B) On loglinear axes to better shown the near-wall region.

the inner-scaled wall-normal location, meaning that the inner layer is a scaling patch. Small deviation at low Grashof numbers is attributed to low Reynolds number effect. Fig. 27 shows that inner-scaled Reynolds shear stress profiles from different Grashof numbers also merge well onto a single curve in the near-wall region. The figure shows that for the region z+/Ψν < 1, the inner-scaled Reynolds shear stress has very small magnitude, resulting in negligible turbulent force in layer I.

7.5 Meso scaling of the mean momentum equation As shown in Fig. 19, all three forces contribute to the balance of the equation in layer II. An admissible scaling for layer II can be conveniently transformed from the outer scaling equation using the rescaled differential scalings

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Analyses of buoyancy-driven convection

dzΛ ¼ ϕz dz ; dRΛwu

¼

ϕRwu dR wu ,

(116a) (116b)

where dzΛ is the differential scaling for the meso scaled wall-normal distance, and dRΛ Rwu is the differential scaling for the meso scaled Reynolds shear stress. The rescaling factors ϕz and ϕRwu will be determined from the requirement of an admissible scaling. Substituting the meso scaled variables in Eq. (116) into the outer scaled mean momentum Eq. (102) yields 0¼

Λ uτ Umax ϕ2z d2 U  ϕz dRwu Rio + ðΘ +  Θ + Þ: (117) + 2 Λ jRwu jmp Reτ dðzΛ Þ ϕRwu dðz Þ jRwu jmp =u2τ mp

It is known that uτUmax/jRwujmp ¼ O(1) (see Eq. 90) and Rio =ðjRwu jmp =u2τ Þ ¼ Oð1Þ (see Eq. 87). Thus, the buoyancy term has a nominal order of magnitude 1. For an admissible scaling in layer II, the viscous term and the turbulence term should also have a nominal order of magnitude 1, which requires that the prefactors to the viscous term and turbulence term scale as O(1): ϕ2z ϕ ¼ z ¼ Oð1Þ: Reτ ϕRwu

(118)

It follows that the rescaling factors should be pffiffiffiffiffiffiffiffi Reτ ; pffiffiffiffiffiffiffiffi ¼ Reτ :

ϕz ¼ ϕRwu

(119a) (119b)

The meso scaled wall-normal distance and the meso scaled Reynolds shear stress are then pffiffiffiffiffiffiffiffi dzΛ ¼ Reτ dz ; (120a) pffiffiffiffiffiffiffiffi  Λ dRwu ¼ Reτ dRwu : (120b) Thus, both the meso scaled wall-normal distance and the meso scaled pffiffiffiffiffiffiffiffi Reynolds shear stress are stretched by a factor of Reτ with respect to the outer scaled values. Substituting the meso scaled variables in Eq. (120) to Eq. (117), the meso scaled mean momentum equation for layer II can be presented as

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0¼

dRΛwu uτ U max d2 U  Rio + ðΘ+  Θ+ Þ: + 2 Λ Λ jRwu jmp dðz Þ jRwu jmp =u2τ mp dðz Þ

(121)

This meso scaled equation is admissible, because all three terms have a nominal order of magnitude 1. Note that the meso scaled wall-normal distance can be presented as dzΛ ¼

pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi dz dz Reτ dz ¼ Reτ ¼ pffiffiffiffiffiffiffiffiffiffiffi : δ δν=uτ

(122)

Thus the characteristic length scale in layer II is the geometric mean of the pffiffiffiffiffiffiffiffiffiffiffi channel half-width δ and ν/uτ: δν=uτ . Pragmatically, the end of layer II can be defined as the location where Fvisc/Fturb ¼ 0.5 (see Fig. 20). Then at this location, the contribution of the forces Fvisc : Fturb : Fbuoy scales as  1 : 2 : 3, all of the same order of magnitude. The location of end of layer II, zII, obtained from the DNS data is presented in Fig. 28. There is noticeable deviation between pffiffiffiffiffiffi the scaling of zII/δ and 1= Reτ . The causes of the deviation may include: (1) low Reynolds number effect. The Reynolds numbers of the simulation are only moderate, meaning that the separation between the inner and outer scales is not wide. (2) The definition of zII is somewhat arbitrary.

Fig. 28 The location of Fvisc/Fturb ¼ 0.5 (filled color symbols), the Obukhov-style length scale (open color symbols), and the location of maximum vertical velocity (filled black symbols). The locations are normalized by the channel half-width. The horizontal dotted line at low Reynolds numbers represents the location of the maximum vertical velocity in laminar DHVC at z 0.423 (see Eq. 19a).

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Nevertheless, the DNS data in Fig. 28 indicate that the mixed length scaled pffiffiffiffiffiffiffiffiffiffiffi δν=uτ is a reasonable length scale in layer II. The definition of the end of layer II with a force ratio requires data of forces in turbulent DHVC, which involves derivatives and can be obtained only with high resolution DNS data. Fig. 28 presents the location of the maximum vertical velocity zjU max , which is easier to be obtained from experimental measurements or numerical simulation. In laminar DHVC, the maximum vertical velocity occurs at z  0.423, independent of the Grashof or Reynolds number (see Eq. 19a). The maximum mean vertical velocity location moves closer toward the wall with the Grashof or Reynolds number in turbulent DHVC, as shown in Fig. 28. The figure shows that at a sufficiently high Reynolds number, Reτ ≳ 100, the scaled location zjU max =δ shows similar Reynolds number dependence as that of zII/δ. The trend needs to be verified with simulations at higher Reynolds numbers. If the trend holds, then, at sufficiently high Reynolds numbers, the maximum mean vertical velocity location can be used to represent the location of layer II. In the dimensional analysis of turbulent DHVC in Section 4, an Obukhov-style length scale is defined as L ¼ u2τ =ðgβθτ Þ (see Eq. (34)). The Obukhov-style length normalized by the channel half-width is shown in Fig. 28 vs the friction Reynolds number. The presently available DNS pffiffiffiffiffiffiffiffiffiffiffi data indicate that L also scales as δν=uτ : L 5  pffiffiffiffiffiffiffiffi , δ Reτ

or

rffiffiffiffiffiffiffi ν L5 δ : uτ

(123)

Therefore, the characteristic length scale in layer II is the Obukhov-style length scale. The meso scaled Reynolds shear stress profiles from different Grashof numbers are presented in Fig. 29. The meso layer occupies the region around the maximum vertical velocity location, represented by a vertical dotted line in Fig. 29, and the width of the meso layer is O(1) in the meso scaled wall-normal distance: ΔzΛ ¼ O(1). The figure shows that the variation of the meso scaled Reynolds shear stress is indeed O(1) in the meso layer. An approximate function for the meso scaled Reynolds shear stress in the meso layer is RΛwu  2:4  2:4ðzΛ  1:5Þ,

(124)

where 1.5 is the meso scaled location of the maximum mean vertical velocity, and the coefficient  2.4 is from the curve-fitting of the DNS data.

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Fig. 29 Meso scaled Reynolds shear stress vs meso scaled wall-normal location.

Fig. 30 Mean temperature deficit vs meso scaled wall-normal location.

The mean temperature deficit Θ+mp  Θ+ profiles are presented vs the meso scaled wall-normal distance in Fig. 30, showing also an O(1) variation in the meso layer. Around the meso layer, an approximation function for the mean temperature deficit is obtained by curve-fitting as Θ+mp  Θ+ 

3:8 : ðzΛ Þ0:6

(125)

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Fig. 31 Mean vertical velocity vs meso scaled wall-normal distance.

The mean vertical velocity profiles are presented vs the meso scaled wallnormal distance in Fig. 31. Around the maximum vertical velocity location, the mean vertical velocity can be approximated as 2

U   1  cðzΛ  1:5Þ ,

(126)

where c is the curvature of the profile and estimated as c  0.04 by curvefitting the DNS data. The small curvature means that the variation of the mean vertical velocity is smooth around its peak in the meso scaled wallnormal distance.

8. Scaling analysis of the mean heat equation Like the analysis of the mean momentum equation in Section 7, it is important to scrutinize the distribution of the terms in the mean heat Eq. (13a) before attempting scaling analysis. Since the mean heat Eq. (13a) has only two terms: a diffusion term and a turbulence term, the two terms have to remain balanced across the whole channel, as illustrated in Fig. 32. The distributions of the molecule diffusion temperature flux αdΘ/dz and the turbulent temperature flux Rwθ are illustrated in the inset of Fig. 32, which shows that the molecular diffusion temperature flux peaks at the wall, and decreases rapidly away from the wall. In contrast, turbulent temperature flux is zero at the wall and increases rapidly away from the wall. In the core of

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Fig. 32 Balance of the mean heat equation. Inset shows the molecular temperature flux αdΘ/dz and turbulent heat flux Rwθ, normalized by the wall temperature flux qt.

the channel, the molecular diffusion temperature flux is much smaller than the turbulent temperature flux.

8.1 Layer structure of the mean heat equation A reasonable delimiter for the inner layer of the mean heat equation is the peak location of the gradient of the turbulent temperature flux zdRwθ =dzjmax . Another choice is the crosspoint of molecular temperature flux αdΘ/dz and the turbulent temperature flux Rwθ. The thickness of the inner layer of the mean heat equation normalized by ν/uτ is plotted vs the Grashof number in Fig. 33A, and vs the inverse of the inner Richardson number in Fig. 33B. As Grashof number increases, the thickness normalized by ν/uτ also increases. Based on the DNS data shown in Fig. 33, the thickness of the thermal inner layer can be approximated by a power-law of the maximum mean vertical velocity as 0:33

z+Iθ  ð0:6  1:4ÞðU +max Þ

:

(127)

Note that in Ref. [75], the exponent was reported as 0.5. However, the DNS data seems to agree better with an exponent of 0.33. From dimensional analysis in Section 4, the characteristic length scale for the thermal diffusion sublayer, denoted as lα, can be presented as

A

B

Fig. 33 Thickness of inner layer in the mean momentum equation normalized by ν/uτ. (A) vs the Grashof number. (B) vs the inverse of the inner Richardson number.

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lα ¼

α ν Ψ ðPr, Rii Þ ¼ Pr Ψα ðPr, Rii Þ, uτ α uτ

(128)

where Ψα(Pr, Reτ) is a nondimension function that may depend on the Prandtl number of the fluid and the inner Richardson number of the flow, which has to be determined experimentally or numerically. We will set the thermal diffusion length scale lα proportional to the thickness of the inner layer shown in Fig. 33. Therefore, the nondimensional function Ψα in lα can be approximated as Ψα  ðU +max Þ

0:33

:

(129)

8.2 Properties of the turbulent temperature flux Rwθ From the integrated mean heat Eq. (17a), the turbulent temperature flux at the channel mid-plane is
dΘ
Rwθ jmp ¼ uτ θτ  α
: (130) dz mp The mean temperature gradient in the core of the channel is small (see Fig. 5B), meaning that the molecular diffusion is very small in the core of the channel (see inset of Fig. 32). Therefore, the turbulent temperature flux at the channel mid-plane jRwθjmp is very close to uτθτ. Fig. 34 presents

Fig. 34 Difference between the turbulent temperature flux at the channel mid-plane Rwθjmp and the wall temperature flux qt ¼ uτθτ.

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the difference between jRwθjmp and uτθτ. Based on the DNS data at Pr  0.7, the difference can be approximated as uτ θτ  Rwθ jmp 1:8 ¼ 1  R+wθ jmp  , uτ θ τ Peτ

(131)

where Peτ ¼ δuτ/α is the P
eclet number. Thus, the difference between jRwθjmp and uτθτ decreases with the P
eclet number, proportional to 1/Peτ. At an infinite P
eclet number, the turbulent temperature flux at the channel mid-plane equals to uτθτ. Like the scaling analysis of the mean momentum equation in Section 7, the scaling analysis of the mean heat equation starts with a generic scaling of the independent and dependent variables, which are denoted as z z* ¼ ; (132a) lc Θ Θ* ¼ ; (132b) θc R (132c) R*wθ ¼ wθ , Rwθ,c where lc is the characteristic length scale for a subdomain of the channel, θc is the characteristic temperature scale, and Rwθ,c is the characteristic scale for the turbulent temperature flux. Substituting the scaled variables in Eq. (132) into the mean heat Eq. (13a) yields 0¼

Rwθ,c dR*wθ αθc d2 Θ* + : lc dz* l 2c dðz* Þ2

(133)

A generic nondimensional mean heat equation can be obtained by multiplying lc/Rwθ,c to Eq. (133) as 0¼

dR*wθ αθc d2 Θ* + : 2 lc Rwθ,c dðz* Þ dz*

(134)

Next, we will identify proper scales different subdomains, starting with the outer layer.

8.3 Outer scaling of the mean heat equation In the outer layer, a natural length scale is the channel half-width, lc ¼ δ. The relevant boundary conditions at the channel mid-plane are

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Θmp ; θc Rwθ jmp ¼ : Rwθ,c

Θ* jz¼δ ¼ R*wθ jz¼δ

(135a) (135b)

The characteristic scale for the turbulent temperature flux can obtained by setting the boundary condition in Eq. (135b) to be 1, which leads to Rwθ,c ¼ jRwθjmp. The temperature scale is set as θc ¼ θτ, which comes from the requirement of admissible scaling for the mean momentum balance equation in the outer layer in Section 7 (see Eq. (96)). The scaled variables used in the outer scaling of the mean heat equation are denoted as z z ¼ ¼ z ; lc δ

(136a)

Θ Θ ¼ ¼ Θ+ ; θc θτ Rwθ R ¼ wθ ¼ R wθ : jRwθ jmp Rwθ,c

(136b) (136c)

Substituting the scaled variables in Eq. (136a) into Eq. (134), the outer scaled mean heat equation can be presented as 0¼

dR αθτ d 2 Θ+ wθ + , 2 dz δjRwθ jmp dðz Þ

MHB : Outer Scaling (137)

Fig. 35 shows that the scaled mean temperature deficit Θ+mp  Θ+ from different Grashof numbers or Reynolds numbers merge onto a single curve in the outer layer, supporting the outer scaling Eq. (137). Fig. 36 shows that the outer scaled turbulent temperature flux R wθ from different Rayleigh numbers also collapse well around the channel mid-plane, which is better shown in the inset of the figure. The shape of the turbulent temperature flux Rwθ around the channel mid-plane can be estimated as follows. The turbulent temperature flux R wθ is a regular even function [95] in the neighborhood around the channel mid-plane, and a Taylor series expansion around the channel mid-plane can be presented as 2 4   R wθ ¼ b1 + b2 ðz  1Þ + b3 ðz  1Þ + ⋯

(138)

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Fig. 35 Outer scaling of the mean temperature deficit.

Fig. 36 Outer scaling of the turbulent temperature flux.

By definition, R wθ jz ¼1 ¼ 1 and b1 ¼ 1. The coefficient b2  1.8/Peτ is based on the curve-fitting of the DNS data. Thus, the turbulent temperature flux in the neighborhood of channel mid-plane can be approximated as R wθ  1 

1:8  ðz  1Þ2 : Peτ

(139)

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Fig. 37 Turbulent heat flux in the core of the channel, and the approximate Eq. (139).

The turbulent temperature flux profiles in the core of the channel are presented in Fig. 37 for three Grashof numbers, showing good agreement with the approximate Eq. (139). The curvature of the profile is 1.8/Peτ, meaning that the turbulent temperature flux profile near the channel mid-plane becomes flatter as the Reynolds number increases. 8.3.1 Alternative scaling of the turbulent temperature flux A disadvantage of using the turbulent temperature flux at the channel midplane Rwθjmp as a characteristic scale is that it has to be obtained by measurements at the channel mid-plane. In the study of wall-bounded turbulence, measurements at the wall are abundant. Here, an alternative scale is presented for the turbulent temperature flux as the wall temperature flux qt ¼ uτθτ. The turbulent heat flux scaled by uτθτ is denoted as def

R+wθ ¼

Rwθ Rwθ ¼ : qt uτ θτ

(140)

and an alternative outer scaled mean heat equation can be presented as 0¼

dR+wθ 1 d 2 Θ+ + : dz Peτ dðz Þ2

(141)

This form of the outer scaled mean heat equation is commonly used in the study of passive scalar transport in a turbulent pipe or channel flow [6,16].

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Fig. 38 Turbulent temperature flux scaled by uτθτ.

Fig. 38 presents the turbulent temperature flux profiles scaled by uτθτ, and the inset depicts the core of the channel. At the lowest Grashof number (Gr ¼ 4.75  104 or Reτ ¼ 61), the deviation of the peak value from 1 is apparent. As Grashof number increases, the difference between R+wθ jmp and 1 monotonically decreases, consistent with the trend shown in Fig. 34. Taylor series can also be used to obtain an approximation function in the neighborhood of the channel mid-plane as R+wθ  R+wθ jmp 

1:8  ðz  1Þ2 , Peτ

(142)

The approximation Eq. (142) is presented in Fig. 39 for three Grashof numbers, showing good agreement with the scaled turbulent temperature flux in the core of the channel.

8.4 Inner scaling of the mean heat equation For the inner layer of the mean heat equation, the relevant boundary conditions are those at the solid surface: Θ* j0 ¼ 0; R*wθ j0

¼ 0;

(143a) (143b)

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Fig. 39 Turbulent temperature flux scaled by uτθτ around the channel mid-plane, and the approximation Eq. (142).


dΘ*

lu θ ¼ c τ τ:
* α θc dz 0

(143c)

As discussed in Section 8.1, the proper length scale in the inner layer of the mean heat equation is the thermal diffusional length scale: lc ¼ lα ¼ α/uτΨα. Substituting lc ¼ lα to the boundary condition Eq. (143c) yields α
Ψ u dΘ*

uτ α τ θτ Ψα θτ ¼ ¼ : α θc θc dz*
0

(144)

To set the boundary condition Eq. (143c) as dΘ*/dz*j0 ¼ 1, the characteristic temperature scale is determined as θc ¼Ψαθτ. Substituting lc ¼ α/uτΨα and θc ¼Ψαθτ to Eq. (134), the prefactor to the diffusion term becomes αθc αΨ θ uθ ¼ α α τ ¼ τ τ : lα Rwθ,c R wθ,c Ψ R uτ α wθ,c

(145)

A simple choice to make the diffusion term a nominal order of magnitude 1 is by setting Rwθ,c ¼ uτθτ. In summary, the characteristic scales for the inner layer of the mean heat equation are determined from physical reasoning, boundary condition and admissible scaling requirement as

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lc ¼ lα ¼

α Ψ ; uτ α

θ c ¼ Ψα θ τ ;

Rwθ,c ¼ uτ θτ :

(146)

The scaled variables for the inner layer of the mean heat equation are denoted as z z Prz+ ¼ ¼ ; Ψα lc α=uτ Ψα

(147a)

Θ Θ Θ+ ¼ ¼ ; θc Ψα θτ Ψα Rwθ R ¼ wθ ¼ R+wθ : Rwθ,c uτ θτ

(147b) (147c)

The inner-scaled mean heat equation can then be presented as 0¼

dR+wθ d2 ðΘ+ =Ψα Þ + : 2 dðPrz+ =Ψα Þ dðPrz+ =Ψα Þ

(148)

The inner-scaled mean temperature profiles from different Grashof numbers are presented in Fig. 40. To better shown the near-wall region, the profiles are plotted on log-linear axes in Fig. 40B, clearly showing that the scaled mean temperature profiles from different Grashof numbers merge onto a single curve in the near-wall region. Like in forced convection, the mean temperature profile adjacent to the wall can also be approximated by a linear growth function: Θ+ Prz+  , Ψα Ψα

or

Θ+  Prz+ :

(149)

The thickness of the linear growth region is about Prz+/Ψα ¼ 1.5 or in dimensional form as 1.5Ψαα/uτ. In comparison, the thermal diffusion A

B

Fig. 40 Inner-scaled mean temperature. (A) On linear-linear axes. (B) On log-linear axes. The dashed curve represents the linear growth equation (149).

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sublayer thickness in forced convection scales as Prmα/uτ where m is a constant of about 0.5 for low Prandtl numbers and 0.66 for large Prandtl numbers [16]. At this time, it is not clear how the Prandtl number affects the layer I thickness of the mean heat equation in turbulent DHVC. More DNS simulations over a wide range of Prandtl numbers are required to clarify the Prandtl number effect in buoyancy-driven turbulent DHVC. The inner-scaled turbulent temperature flux profiles are presented in Fig. 41. The near-wall region is better shown in Fig. 41B on log-linear axes. The scaled turbulent temperature flux profiles from different Grashof numbers also merge onto a single curve in the near-wall region. Adjacent to the wall, it can be shown that the turbulent temperature flux grows as the third power of the wall-normal distance [1]:  + 3 Prz + : (150) Rwθ jz!0  Ψα

8.5 Scaling patches in the mean heat equation The mean heat equation has only two terms, similar to the mean momentum equation for a turbulent plane Couette flow. Applying a simple transformation, Fife et al. [4] discovered that the turbulent plane Couette flow consists of an intrinsic hierarchy of “scaling patches.” Applying a similar transformation, an adjusted turbulent temperature flux is defined as +  Rⓐ wθ ¼ Rwθ  a z ,

A

(151)

B

Fig. 41 Inner-scaled turbulent temperature flux. (A) On linear-linear axes. (B) On loglinear axes. The inset depicts the near-wall region and the cube growth with respect to the wall-normal distance.

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Fig. 42 Adjusted turbulent temperature flux in turbulent DHVC, with six different adjusted parameters. The data are based on the DNS of Kiš [31] at Gr ¼ 1.0  106.

where a is a constant. Examples of adjusted turbulent temperature flux profiles are illustrated in Fig. 42 for six values of a. For a ¼ 0 (the actual turbulent temperature flux), the peak of the profile occurs at the channel mid-plane. With increasing a value, the peak location of the adjusted turbulent temperature flux profile continuously moves closer to the wall. The scaling patches will be built around the peaks of the adjusted turbulent temperature flux profiles. Substituting the adjusted turbulent temperature flux defined in Eq. (151) into the outer scaled mean heat Eq. (141) gives  2 + ⓐ 1 d Θ dRwθ (152) 0¼ + + a: Peτ dðz Þ2 dz The scaling patch for the mean heat equation is obtained by seeking an admissible scaling for the meso layer of the adjusted turbulent temperature flux profile. The rescaling of the wall-normal distance and the adjusted turbulent temperature flux is best performed using the following differential scaling: dzΛa ¼ ϕz dz ;

(153a)

dRΛwθa ¼ ϕRwθ dRⓐ wθ :

(153b)

These rescaled variables are called meso scaled, and denoted by a superscript ðÞΛa . The rescaling factors ϕz and ϕRwθ are determined by admissible scaling

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requirement. Substituting the meso scaled variables in Eq. (153) into Eq. (152) gives 0¼

Λ ϕ2z d2 Θ+ ϕz dRwθa + + a: Peτ dðzΛa Þ2 ϕRwθ dzΛa

(154)

The goal of meso scaling is to make all the three terms nominal order of magnitude 1, which can be achieved by setting the rescaling factors as follows: pffiffiffiffiffiffiffiffiffi ϕz ¼ aPeτ ; (155a) rffiffiffiffiffiffiffi Peτ ϕRwθ ¼ : (155b) a The meso scaling of the mean heat equation with the adjusted turbulent temperature flux can be written as 0¼

dRΛwθa d 2 Θ+ + + 1: dzΛa dðzΛa Þ2

(156)

Fig. 43A presents the adjusted turbulent temperature flux for the case of a ¼ 1. As the Grashof (or Reynolds) number increases, the peak location of the adjusted turbulent temperature flux moves closer to the wall, and the peak value increases toward to 1. Overall, the shape of the adjusted turbulent temperature flux with a ¼ 1 is similar to the Reynolds shear stress profile in turbulent channel flow (see Fig. 9). The “meso scaled” adjusted turbulent temperature flux profiles for a ¼ 1 are presented in Fig. 43B. The inset shows clearly that the rescaled “meso layer” A

B

Fig. 43 Adjusted turbulent temperature flux with a ¼ 1. (A) Adjusted turbulent temperature flux vs outer scaled wall-normal location. (B) Meso scaled turbulent temperature flux vs meso scaled wall-normal distance.

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A

B

Fig. 44 Adjusted turbulent temperature flux with a ¼ 0.25. (A) Adjusted turbulent temperature flux vs outer scaled wall-normal location. (B) Meso scaled turbulent temperature flux vs meso scaled wall-normal distance. A

B

Fig. 45 Adjusted turbulent temperature flux with a ¼ 4. (A) Adjusted turbulent temperature flux vs outer scaled wall-normal location. (B) Meso scaled turbulent temperature flux vs meso scaled wall-normal distance.

has a thickness of O(1), and the variation of the rescaled adjusted turbulent temperature flux is also O(1). Therefore, this meso layer is a scaling patch. Results for a ¼ 0.25 and a ¼ 4 are shown in Figs. 44 and 45. For the case of a ¼ 0.25, the thickness of the “meso layer” is wider in terms of z than that with a ¼ 1 case (see Figs. 44A and 43A). The rescaling factor ϕz ¼ pffiffiffiffiffiffiffiffiffiffi aReτ for a ¼ 0.25 is half of that for a ¼ 1 case. On the other hand, for the case of a ¼ 4, the thickness of the “meso layer” is narrower in terms pffiffiffiffiffiffiffiffiffiffi of z (see Fig. 45A). The rescaling factor ϕz ¼ aReτ for a ¼ 4 is twice of that for a ¼ 1 case. The rescaled wall-normal location dzΛa ¼ ϕz dz makes the meso layer have a thickness of O(1) for any a, as shown in the insets of Figs. 43B, 44B, and 45B. Hence, the meso layer with different a is also a scaling patch, results a hierarchy of scaling patches for the mean heat equation.

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9. New prediction of Nusselt number By definition, the Nusselt number for buoyancy-driven DHVC can be presented as
dΘ
uθ Pr Reτ def α dz 0 Nu ¼ : (157) ¼ τ τ ¼ Θmp Θmp Θ+mp α α δ δ Hence, the Nusselt number in DHVC is directly related to the friction Reynolds number and the mean temperature at the channel mid-plane normalized by the friction temperature Θ+mp ¼ Θmp =θτ . Therefore, a better understanding of the mean temperature growth in DHVC is critical in improving the prediction of the Nusselt number. Based on the dimensional analysis and scaling patch analysis in the preceding sections, it is clear that heat transport in turbulent DHVC consists of a multilayer structure, and the characteristic of the temperature growth rate is different in each layer. In the inner layer, the temperature growth can be approximated by a linear function (see Fig. 40B). At a sufficiently high Grashof (or Reynolds) number, Fig. 40B shows that there exists a region, in which the mean temperature growth can be approximated by a logarithmic function. The “log-layer” is better shown in Fig. 46. For comparison, the mean temperature profiles in turbulent channel flow are also plotted in Fig. 46.

Fig. 46 Mean temperature profiles in a turbulent DHVC. For comparison, mean temperature profiles in a turbulent channel (or plane-Poiseuille flow, PPF) are also plotted. The DNS data of PPF at Reτ ¼ 180 and Reτ ¼ 1020 are from Kawamura and coworkers [96–98].

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In the DNS of heat transfer through turbulent channel flow (or planePoiseuille flow) [96–98], temperature is treated as a passive scalar. Fig. 46 shows that, at Reτ ¼ 1020, the mean temperature exhibits a region that fits a logarithmic function reasonable well. The thermal “log-law” is analogous to the “log-law” for the mean velocity profile in turbulent flow over a flat plate or turbulent pipe or channel flow. According to Kader and Yaglom [67], the thermal “log-law” was first proposed by Landau and Lifshitz in 1944 (in the first Russian edition of the book [99]). The robustness of the thermal “log-law” motivated the Kader–Yaglom style prediction of the Nusselt number discussed in Section 2.1 (see Eq. (10)). The Reynolds (or P
eclet) number range of the presently available DNS of turbulent DHVC is still relatively moderate. For example, the highest P
eclet number for turbulent DHVC in Fig. 46 is about Peτ  227. In classical arguments for forced wall-bounded turbulence, this P
eclet number is not high enough for the emergence of a log-layer. However, Fig. 46 shows that a log-layer starts to emerge in DHVC. The slope (related to the von Ka´rma´n constant in the thermal log-law) in turbulent DHVC is about half of that in the forced convection. Following the Kader–Yaglom style equation, a new prediction of Nusselt number for turbulent DHVC is proposed as Nu ¼

Pr Reτ Pr Reτ  : ð1=κ θ ÞlnðPr Reτ Þ+BðPr, Reτ Þ Θ+mp

(158)

Based on DNS data at Pr  0.7 (see Figs. 40 and 46), the function B is approximated as B  1:5Ψα + 1:5  1:19ðReτ Þ0:15 + 1:5,

(159)

where 1.5Ψα represents the temperature increment in the linear growth region (layer I), and the additive 1.5 represents the temperature increment across a transition layer between the linear growth region and the log-layer, and also the deviation from the log-law in the core of the channel. The new Nusselt number prediction Eq. (158) for turbulent DHVC is plotted in Fig. 47, agreeing well with the DNS data. However, the Reynolds number range of the presently available data is still limited. More DNS data over a wider range of Grashof number is required to evaluate the validity of Eq. (158), and determine more precisely the coefficients in Eq. (159). Fig. 47 shows that the new Nusselt number prediction Eq. (158) is similar to the Kader–Yaglom style equation for passive scalar transport in a turbulent pipe or channel flow [65, 66]. However, there is a major difference

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due to the temperature increment in the thermal diffusion layer. For passive scalar transport in turbulent channel flow, the temperature increment in the thermal diffusional layer is proportional to Prm, but largely independent of the Reynolds number [1,16]. However, the temperature increment in layer I of turbulent DHVC is proportional to 1.5Ψα  1.19(Reτ)0.15, directly affected by the Reynolds number. Currently, Grashof or Rayleigh number is almost universally used in presenting Nusselt number data for buoyancy-driven convection. Like in forced convection [16, 65, 66], an approximate relation is established between the Reynolds number and the Grashof number. Based on the DNS data at Pr  0.7, the relationship between the two nondimensional number for turbulent DHVC can be approximated as Reτ  1:0Gr0:38 :

(160)

Substituting the approximate Eq. (160) for Reτ to Eq. (158), a new Nusselt number prediction equation using the Grashof number is obtained as Nu 

Pr Gr0:38 : 1:1 lnðPr Gr0:38 Þ + 1:19 Gr0:056 + 1:5

(161)

Fig. 48 presents the Nusselt number vs Grashof number with new prediction Eq. (161). The new prediction is similar to a power-law with an exponent of m ¼ 1/3, even to ultra-high Grashof numbers Gr ¼ 1020. However, the difference between the new prediction Eq. (161) and a power-law prediction

Fig. 47 Prediction of Nusselt number for DHVC. DNS of the turbulent PPF are from Kawamura and coworkers [96–98,100] and Pirizzoli et al. [101].

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Fig. 48 New prediction of Nusselt number in turbulent DHVC using Eq. (161). The inset depicts the prediction over an extended range of Grashof numbers.

Fig. 49 Compensated Nusselt number vs Grashof number in a turbulent DHVC. The curve represents Eq. (161) multiplied by Gr1/3.

can be discerned from the compensated Nusselt number shown in Fig. 49. The figures shows that the subtle decrease of the compensated Nusselt number over the range of 104 < Gr < 107 is captured well by the new prediction Eq. (161). More DNS data over a wider range of Prandtl number and Grashof number are required to check the new prediction and better determine the relevant coefficients.

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10. Summary and conclusions Improving the accuracy of Nusselt number prediction is at the core of many research on convection, forced or buoyancy-driven. A better prediction of the Nusselt number depends on our understanding of the underlying physics and structure of the flow and heat transport. In this work, the multilayer structure of buoyancy-driven convection, i.e., differentially heated vertical channel, is elucidated using two tools: dimensional analysis and scaling patch approach. Dimensional analysis is a powerful tool in the study of fluid dynamics and heat transfer. Proper selection of control parameters is of utmost importance in the success of dimensional analysis. However, governing equations are not directly employed in dimensional analysis, and the proper selection of control parameters relies heavily on physical intuition and experience. For heat transport in wall-bounded turbulence, forced or buoyancy-driven, the mean velocity and mean temperature vary sharply adjacent to the solid surface. As a result, molecular transport of momentum and heat is always important in the near-wall region. Therefore, the kinematic viscosity ν and thermal diffusivity α are control parameters for flow and heat transport in the layer adjacent to the solid surface. On the other hand, flow and heat transport in the region away from the wall is dominated by eddy motions, and molecular diffusion is negligible. In a dimensional analysis of turbulent DHVC, flow and heat transport are divided into three layers based on physical intuition. In the inner layer adjacent to the wall, five control parameters are selected including the buoyancy parameter gβ, wall temperature flux qt, wall momentum flux qu, kinematic viscosity ν, and thermal diffusivity α. Hence, flow and heat transport in the inner layer are controlled by two nondimensional numbers: one is the Prandtl number of the fluid, and the other is called the inner Richardson number. In the outer layer away from the solid surface, flow and heat transport are controlled by four parameters including gβ, qt, qu, and the channel half-width δ, which results in one nondimensional control parameter called the outer Richardson number. Between the inner layer and the outer layer, there exists a meso layer. At a sufficiently high Grashof number, flow and heat transport in the meso layer is not strongly influenced by the molecular diffusion. At the same time, flow and heat transport in the meso layer is also not strongly affected by the large eddy of size δ. Thus, the control parameters in the meso layer are gβ, qt, and qu, which leads to an Obukhov-style length scale.

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A relatively new scaling approach, called the scaling patch approach, is introduced in this article to the buoyancy-driven turbulence. The analysis uncovers quantitatively a multilayer structure for the mean momentum equation and mean heat equation. The inner and outer Richardson numbers derived from dimensional analysis appear naturally from the properly scaled mean momentum equation. Moreover, a friction Reynolds number also appears naturally in the outer scaled mean momentum equation and is directly related to the multilayer structure of flow and heat transport in turbulent DHVC. On the contrary, the Grashof number or Rayleigh number commonly used in buoyancy-driven convection is not directly related to any sublayer in turbulent DHVC, because the global temperature increment ΔT used in the definition of Grashof number or Rayleigh number is not a characteristic temperature scale in any layer of turbulent DHVC (in RBC as well). Therefore, the friction Reynolds number is more appropriate than the Grashof number or Rayleigh number in the investigation of the underlying physics in turbulent DHVC. For the reader’s convenience, the inner and outer scaling for the mean momentum equation and the mean heat equation in turbulent DHVC are summarized in Table 10, and compared with those for passive scalar transport in turbulent channel flow. There are striking similarities between the buoyancy-driven turbulence and pressure-driven turbulence, and also distinctive differences. The forced wall-bounded turbulence (e.g., turbulent channel flow) and the buoyancy-driven turbulence (e.g., turbulent DHVC) both consist of a multilayer structure and the characteristics of the force balance is different in each layer. For example, in the inner layer of turbulent channel flow, the force balance is between the viscous force and pressure force (layer I) or between the viscous force and turbulence force (layer II). In comparison, in the inner layer of turbulent DHVC, the force balance is between the viscous force and buoyancy force (its layer I). In both turbulent channel flow and turbulent DHVC, there exists a “meso layer,” in which all three forces contribute to the balance of the equation. In the outer layer of turbulent channel flow or turbulent DHVC, flow and heat transport are dominated by eddy motions, and molecular diffusion is negligible. Based on the insight gained from the dimensional analysis and scaling patch analysis, a new prediction of Nusselt number is proposed for turbulent DHVC. The new prediction is built on a multilayer structure of the mean heat equation. In specific, the new prediction is directly related to the characteristic of the mean temperature growth rate in different layers of the flow:

Table 10 Summary of the inner and outer scaling of the mean momentum equation and the mean heat equation in a turbulent DHVC. DHVC Passive scalar transport in turbulent channel 2

2

+ gβðΘmp  ΘÞ

0 ¼ ν ddzU2 +

 dxd ðPρ Þ

MMB

0 ¼ ν ddzU2 +

Inner

l c ¼ uντ Ψν , uc ¼ uτ Ψν , Rwu,c ¼ u2τ , θc ¼ θτ Ψα :

lc ¼ uντ , uc ¼ uτ , Rwu,c ¼ u2τ :

+ z , U + ¼ uUτ , R+wu ¼ Ruwu ¼ θΘτ . z+ ¼ ðν=u 2 , Θ τÞ

z z+ ¼ ðν=u , U + ¼ uUτ , R+wu ¼ Ruwu 2 : τÞ

dRwu dz

τ

0 ¼ ddðzðU+ =Ψ=ΨÞν2Þ + 2

+

ν

Outer

dR+wu dðz+ =Ψ

νÞ

Θ+mp

dRwu dz

τ

2

U+ + 2

d 0 ¼ dðz

+ Rii Ψν Ψα ð Ψα  ΘΨα Þ +

dR+wu dðz+ Þ

+

Þ

1 Reτ

+

lc ¼ δ, uc ¼ Umax, Rwu,c ¼ jRwujmp, θc ¼ θτ.

lc ¼ δ, uc ¼ uτ , Rwu,c ¼ u2τ :

Rwu + Θ z ¼ zδ , U  ¼ UUmax , R wu ¼ jRwu j , Θ ¼ θτ .

z ¼ zδ , U + ¼ uUτ , R+wu ¼ Ruwu 2 . τ

mp

0¼

uτ U max 1 d 2 U  jRwu jmp Reτ dðz Þ2

+

dR wu dðz Þ

+

Rio jRwu jmp =u2τ

ðΘ+mp

Θ Þ +

2

U+  2

d 0 ¼ Re1 τ dðz

+

Þ

+1

uτ θ τ U δ Ub

MHB

0 ¼ α ddzΘ2 +

Inner

l c ¼ uατ Ψα , θc ¼ θτ Ψα , Rwθ,c ¼ uτ θτ :

lc ¼ uατ Ψα , θc ¼ θτ Ψα , Rwθ,c ¼ uτ θτ , uc ¼ uτ .

z z+ ¼ ðν=u , Θ+ ¼ θΘτ , R+wθ ¼ uRτwθ θτ : τÞ

z z+ ¼ ðν=u , Θ+ ¼ θΘτ , U + ¼ uUτ , R+wθ ¼ uRτwθ θτ : τÞ

d ðΘ =Ψα Þ + 0 ¼ dðPr z+ =Ψ Þ2

d ðΘ =Ψα Þ 0 ¼ dðPr + z+ =Ψ Þ2

2

2

2

dRwθ dz

+

α

Outer

0 ¼ α ddzΘ2 +

dR+wu dðz Þ

dR+wθ dðPr z+ =Ψα Þ

2

dRwθ dz

+

α

+

dR+wθ dðPr z+ =Ψα Þ

+

1 U+ Pr Reτ =Ψα U +b

lc ¼ δ, θc ¼ θτ, Rwθ,c ¼ uτθτ.

lc ¼ δ, θc ¼ θτΨα, Rwθ,c ¼ uτθτ, uc ¼ uτ.

z ¼ zδ , Θ+ ¼ θΘτ , R+wθ ¼ uRτwθ θτ :

z ¼ zδ , Θ+ ¼ θΘτ , U + ¼ uUτ , R+wθ ¼ uRτwθ θτ :

d Θ + 0 ¼ Pr 1Reτ dðz  Þ2 2

+

dR+wθ dðz Þ

d Θ 0 ¼ Pr 1Reτ dðz +  Þ2 2

+

dR+wθ dðz Þ

+

U+ U +b

Based on the DNS data at Pr  0.7, the nondimensional functions for the inner layer are approximated as Ψν  U +max and Ψα  ðU +max Þ0:33. For the outer layer, it is found that jRwujmp  Rio  0.5uτUmax. Scalings for passive scalar transport in turbulent channel flow are also presented to demonstrate similarity and difference. The nondimensional Ψα for the thermal diffusion sublayer in turbulent channel flow is empirically determined as Ψα Prm where m  1/2 for Pr ≪ 1 and m  2/3 for Pr ≳ 1 [16].

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a linear growth region adjacent to the wall, a transitional layer, a log-layer, and the core region of the channel with small deviation from the log-law. The new Nusselt prediction equation is analogous to the Kader–Yaglom style equation for forced convection. However, there is an important difference between the forced convection and the buoyancy-driven convection: the temperature increment in the thermal diffusion sublayer of turbulent DHVC is influenced by the Reynolds number. The main messages of the present article can be condensed as follows • The multilayer structure of buoyancy-driven turbulence can be qualitatively discerned from dimensional analysis. • The scaling patch approach is a powerful tool to quantitatively elucidate the multilayer structure of buoyancy-driven turbulence. • The Grashof number and Rayleigh number do not reflect directly the multilayer structure of buoyancy-driven turbulence, because the global temperature difference used in their definitions is not appropriate in any layer of the buoyancy-driven turbulence. For turbulent DHVC, the friction Reynolds number is more meaningful, directly related to the multilayer structure of the turbulent flow. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi • The “free-fall” velocity defined as gβΘmp δ should be avoided in the scaling of velocity in buoyancy-driven convection, also due to the inappropriateness of the global temperature difference in any layer of the buoyancy-driven turbulence. For turbulent DHVC, more appropriate velocity scales can be obtained from the wall momentum flux and wall temperature flux. • The power-law correlation prediction of the Nusselt number does not reflect the multilayer structure of heat transport in buoyancy-driven turbulence and should be superseded by new predictions.

Acknowledgments The author is very grateful to Dr. T. A. M. Versteegh, Dr. P. Kisˇ, and Dr. C. S. Ng for generously sharing their DNS data.

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CHAPTER TWO

Convective heat transfer in different porous passages A. Haji-Sheikha,*, Filippo de Monteb, and W.J. Minkowyczc a

Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, United States Department of Industrial and Information Engineering and Economics, University of L’Aquila, L’Aquila, AQ, Italy c Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, IL, United States *Corresponding author: e-mail address: [email protected] b

Contents 1. Introduction 2. The governing equations 2.1 Physical interpretation of relaxation times 2.2 Energy equations with dimensionless coordinates and time 3. Micro-scale bio-heat diffusion 3.1 Boundary and initial conditions 3.2 Uncoupling-based solution method 3.3 Dual-phase lag bio-heat diffusion equation 3.4 Transformations of the dependent variable: The DPL equation 3.5 Temperature solution in finite regular tissues 3.6 Temperature solution in a laser-irradiated biological tissue 4. Transient thermal diffusion in porous devices 4.1 Transient temperature field with a moving fluid 4.2 Transient temperature field with stationary fluid 5. Steady state heat transfer to flow in porous passages 5.1 Formulation of heat transfer in porous ducts 5.2 Porous ducts with no-axial conduction 5.3 Porous ducts with axial conduction 5.4 Orthogonality condition and the determination of coefficients 5.5 Frictional heating effects 6. Rapidly switched heat regenerators in counterflow 6.1 Mathematical formulation for cyclic steady operation 6.2 Slowly- and rapidly-switched heat regenerators 6.3 General solution of the gas energy equation 6.4 “Cold” particles 6.5 “Hot” particles 6.6 “Internal” particles

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6.7 Fluid temperature solution 6.8 Effectiveness and heat stored in the regenerator 7. Concluding remarks References

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Abstract This presentation concerns with the reported development of the exact series solutions for the computation of temperature in porous passages. Preliminary consideration is given to rapid heating within biological systems. Then, parallel plate and circular porous passages received significant considerations. These porous passages are filled with solids having relatively different thermal conductivity from that of the fluid materials. Therefore, this study includes the contribution of axial conduction for selected cases. This is to demonstrate the mathematical procedure leading to exact series solutions for selected cases. This leads to the exact solutions for a set of modified Graetz type problems for parallel plate channels and circular pipes. In general, when solid material has higher thermal conductivity, the numerical procedure yields a Nusselt number that changes significantly when the Peclet number changes. To show this phenomenon, the Nusselt number is computed for selected values of the Peclet number. Another application deals with rapidly switched heat regenerators in counterflow where the phenomenon of the so-called ‘flush’ phase plays an important role and affects the regenerator effectiveness.

1. Introduction This presentation describes heat transfer, via forced convection, to fluid flowing inside porous passages. These phenomena exist within biological applications, heat regenerators, and in rapidly developing modern technologies where the porous materials include metal foams. In biological applications, this phenomenon exists during the transport of energy between blood and biological tissues. Others exist in Stirling cycle machines (prime movers, coolers, heat pumps and cry-coolers), petroleum industries, high power batteries, compact heat exchangers, power electronics, and in various other applications. A porous bed could have various compositions including a packed bed of spherical-shaped materials, stacks of cylindrical rods, woven wire screens or random fibers, metal foams constructed from wire-type ligaments, etc. It is common knowledge that there is an induced drag force as flow passes over, below, and around these ligaments. Therefore, it is appropriate to re-examine the basic formulation of the momentum equation. The energy equation may have different features depending on the rate that energy

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transfer to or from the fluid. Accordingly, there are two different views of the energy equation: one view with local thermal equilibrium condition or another with local thermal non-equilibrium condition. The Local Thermal Non-Equilibrium (LTNE) hypothesis emerges when studying a rapid transport of heat in porous media. The nonequilibrium phenomenon is an interesting issue during a rapid heating or a cooling process. This phenomenon occurs in various engineering applications such as nuclear devices [1], fuel cells [2], electronic systems [3], micro devices [4], Stirling heat regenerators [5], and others. Also, it occurs in living biological tissues when tumor cells are subject to a heating therapy [6–8]. Generally, a classical approach is used to determine the materials temperature in the presence of Local Thermal Equilibrium (LTE). In the absence of LTE, the single energy equation needs to be replaced with two energy equations, one for the solid and another for the fluid. Earlier, Vick and Scott [9] determined the heat transfer in a matrix with embedded particles using a finite-difference approach. They reported a thermal lag that depends on the property ratio and contact conductance while the temperature response due to an applied heat flux differs from that predicted using the classical heat conduction technique. Also, Amiri and Vafai [10] used a two-equation model and performed an analytical determination of the solid and fluid temperature differential in porous media within a forced convective flow. The studies reported in [11] show that the thermal non-equilibrium condition in a fluidized porous bed depends on the structure of solid materials, mean pore size, interstitial heat transfer coefficient, and thermophysical properties. For a porous medium subject to rapid transient heating or cooling, the studies in [12] established conditions for departure from local thermal equilibrium. Therefore, the occurrence of the LTNE in the presence of a rapidly changing heat source depends on the magnitude of a dimensionless quantity called the Sparrow number in [11]. Accordingly, it determines the existence of the local thermal equilibrium condition and it controls the transport of energy through the porous media.

2. The governing equations The basic equations for convective energy transport in porous media within a differential element ΔV such as the one depicted in Fig. 1, as recommended in [10], are

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Fig. 1 Schematic of a differential element in a porous region.

εC f


 ∂T f ðr, t Þ + C f V  rT f ¼ r  qf ðr, t Þ + hap T s  T f ∂t
 ∂T ðr, tÞ ð1  εÞC s s ¼ r  qs ðr, tÞ  hap T s  T f ∂t

(1a) (1b)

where ε ¼ ΔVf /ΔV is the porosity that stands for the fraction of a volume occupied by the fluid. In a differential element ΔV, the parameter Tf stands for the mean fluid temperature while Ts stands the mean temperature of the porous ligaments. Also, qf and qs are the heat flux vectors in the fluid and solid materials, respectively; while V is the fluid velocity vector. Other parameters are the volumetric heat capacitances Cf and Cs for fluid and solid materials, interstitial heat transfer coefficient h, the parameter of great concern, and the contact area parameter ap ¼ ΔAp/ΔV between fluid and solid materials within a differential element ΔV while ΔAp is the surface contact area. Adding Eqs. (1a) to (1b) and after replacing qf + qs with q, the resulting relation is. r  qðr, t Þ ¼ εC f

∂T f ðr, t Þ ∂T ðr, tÞ + C f V  rT f + ð1  εÞC s s (2) ∂t ∂t

In phase change application, Tf becomes the temperature of the phase change materials as the pores may contain vapor, liquid, solid, or a combination. For the sake of generality, it is appropriate to add a volumetric heat source function S(r, t) to Eq. (2) to become

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Convective heat transfer in different porous passages

∂Tf ðr, tÞ + C f V  rTf ∂t ∂Ts ðr, t Þ + ð1  εÞCs ∂t

r  qðr, tÞ + Sðr, t Þ ¼ εCf

(3)

Under local thermal equilibrium condition, the solid and the adjacent fluid are at the same temperature, Ts ¼ Tf. However, during a rapid heating or cooling, the fluid and solid are not at the same temperature, locally. Therefore, before the onset of equilibrium, there is an energy exchange between the solid phase and the fluid phase within the pores, and temperature undergoes a transient process, as defined by the equation Cf ΔVp


 ∂Tf ¼ hΔAp Ts  Tf ∂t

(4)

where ΔVp is the mean pore volume, ΔAp is its contact surface area with the solid phase, and h is the interstitial heat transfer coefficient. As a shorthand notation, rh ¼ ΔVp/ΔAp is considered to be a pore hydraulic radius in Eq. (4) and it becomes Ts ðr, tÞ ¼ Tf ðr, t Þ + τt

∂Tf ðr, t Þ ∂t

(5)

where τt ¼ rhCf/h has units of time. In this formulation, the solid matrix plays the primary role of energy transport and reversing it would change this formulation. Eq. (3) contains the heat flux vector q that depends on temperature. Under the local thermal equilibrium condition, when Ts ¼ Tf, the Fourier equation applies, qðr, tÞ ¼ ke rT f ðr, t Þ

(6)

where ke ¼ εkf + (1  ε)ks is the effective thermal conductivity of the porous medium. It is reported in Fournier and Boccara [12] that Eq. (6) does not hold under a rapid heating process. In the absence of local thermal equilibrium, when the departures of q and rT from local equilibrium are relatively small, it is suggested in Tzou [13,14] that the Fourier equation, as given in Eq. (6), needs to be modified to become
 q r, t + τq ¼ ke rTf ðr, t + τx Þ (7) as time changes. In detail, τq is the so-called relaxation time that describes the time delay as heat travels from pore to pore; therefore, it depends on the

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direction of energy transport between solid and fluid. When the solid structure has a higher local temperature than the materials in a pore, as an approximation, τq  CsRcΔVs/ΔAs; the parameter ΔVs is the differential volume within the solid structure while Rc and ΔAs are the contact resistance and contact area between individual solid structures, respectively. The relaxation time τx describes the time delay as temperature changes in the fluid and solid material. The physical descriptions of these parameters are presented in the net subsection. Then, by applying a Taylor series expansion to both sides of Eq. (7) and performing a first-order approximation, Eq. (7) can be written as qðr, tÞ + τq

n o ∂ qðr, tÞ ∂ rTf ðr, tÞ  ke rTf ðr, tÞ + τx ∂t ∂t

(8)

Eqs. (3), (5), and (8) contain three unknowns, Tf, Ts, and the heat flux vector q. One can eliminate Ts and retain Tf using Eq. (5) and eliminate q using Eq. (8). Then, following some simplifications, Eq. (3) takes the form 
  
  ∂ r  ke rTf ∂ 

∂S + S + τq L Tf + τx  τq L Tf + τq ∂t ∂t ∂t  (9) 2
 ∂T ∂ T ∂ f f + τe τq 2 ¼ Ce Tf + τe + τq ∂t ∂t ∂t where Ce ¼ εCf + (1  ε)Cs is the effective thermal capacitance of the porous medium; while τe ¼ (1  ε) CsCf rh/(hCe). Also, Tf ¼ Tf (r, t), S ¼ S(r, t), and L(Tf) ¼ r  (ke r Tf)  CfV  r Tf. An alternative form of τe, that is, τe ¼ (1  ε) (Cs/Ce) τt indicates that τe < τt that appears in Eq. (5). The terms within the square brackets on the right side of Eq. (9) include the first two terms of the Taylor series expansion and a third term. This third term is smaller than the third term of this Taylor series (τe + τq)2(∂2 Tf/∂ t2)/2! since τeτq < (τe + τq)2/2!. As stated earlier, the higher order terms of the Taylor series expansion in Eq. (8) are eliminated; therefore, for consistency of these formulations, the third term within the square brackets on the right-hand-side of Eq. (9) is small and should be neglected. Then, the reduced form of Eq. (9) is 
  
  ∂ r  ke rTf ∂ 

∂S + S + τq L Tf + τx  τq L Tf + τq ∂t ∂t ∂t  (10) 2
 ∂Tf ∂ Tf ¼ Ce + τe + τq ∂t ∂t2

Convective heat transfer in different porous passages

101

that represents the dual-phase lag (DPL) heat equation. For a few special cases, Eq. (10) has exact analytical solutions; however, in general, the solution of Eq. (10) requires a numerical procedure. In the presence of a moving fluid, the solution of Eq. (10) depends on the functional form of the velocity vector V.

2.1 Physical interpretation of relaxation times The lag time τt that appears in Eq. (10) through τe is a parameter that influences the state of local thermal non-equilibrium phenomena in porous media. The physical nature of the lag time τt ¼ rhCf/h, formerly in Eq. (5), is well defined and its value is obtainable once the interstitial heat transfer coefficient h is known. The other parameters, rh and Cf in the definition of τt, are usually available for a well-defined porous system. A relatively large value of τt indicates that there will be detectable temperature variations in the presence of a rapidly changing energy input. Large differences between Tf and Ts can be realized when the condition of local thermal equilibrium fails to exist depending on the values of the thermophysical properties including the time delay parameters. For a system with a specified characteristic length L, Minkowycz et al. [11] introduced a dimensionless quantity called the Sparrow number that can indicate the presence of local thermal non-equilibrium condition. It is defined as

 2 kf hL 2 L Sp≜ ¼ Nur h (11) ke r h ke rh Alternatively, the parameter τt controls the size of the Sparrow number, that can also be written as Sp ¼ (Cf/Ce)/(αeτt/L2), where αe ¼ ke/Ce is the effective thermal diffusivity of the porous medium. For stationary materials in the pores, the interstitial Nusselt number Nurh ¼ hrh/kf has a value between 1 and 2 depending on the geometry of the pores. Therefore, according to Eq. (11), in addition to the thermal conductivity ratio, the ratio of a characteristic length L to the hydraulic radius rh plays a significant role in the determination of the Sparrow number, Sp. For spherical bodies, cylindrical prisms, and square prisms, the values of Nurh ¼ 1.09, 1.45, and 1.23 are reported in Minkowycz et al. [11], respectively. For a thermally fully developed laminar flow in circular passages, the value of Nurh ¼ hrh/kf  0.92. Other useful correlations may be found still in [11], and then in [15], where a correlation for a packed bed of spherical particles is given.

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Another parameter that enters the analysis is τq. Experimental studies are needed to ascertain the proper value of τq in Eq. (10). Often, it is possible to provide an estimate of τq values using existing information in the literature. According to the Fourier equation, the heat flux across a control volume is related to rT. However, prior to onset of LTE, the actual heat flux is larger because additional thermal energy must be supplied to the individual structures across contact surfaces and through constrictions within the differential element. To account for the change in heat flux, one can set ΔAsΔq  ΔVsCs ∂(ΔTs)/∂ t, where ΔTs is the temperature difference across a constriction and/or a contact surface. Substituting for ΔTs ¼ qRc results in the relation Δq 

∂q ΔV s CR ΔAs s c ∂ t

(12)

where Rc is the contact resistance. A comparison of Eq. (12) with Eqs. (7) and (8) suggests that τq  CsRcΔVs/ΔAs. The value of Rc depends on various factors including geometry, applied pressure, and the constriction effect; hence, it is difficult to develop an accurate prediction in the absence of experimental data. The study of physical parameters that control the value of τx is the next topic for discussion. Although the experimental determination of a reasonably accurate value of τx is desirable, different theories are cited for its determination. They include taking τx ¼ τt which is a reasonable choice when τq is much smaller than τt. A theoretical reasoning for determination of τx by viewing the temperature of solid materials Ts and the temperature of pore materials Tp to be different under LTNE condition leads to [11] τx ¼ τt +

ð1  εÞC s τq : C

(13)

2.2 Energy equations with dimensionless coordinates and time The time and spatial coordinates can be made dimensionless by multiplying both sides of Eqs. (1a) and (1b) by L2/ke, where L is the characteristic length, to get ε

h i
 Cf ∂Tf ðer, et Þ e  rT e f ¼ r e  Lqf ðer, et Þ=ke + εSp Ts  Tf + Pe V e Ce ∂t
 Cs ∂Ts ðer, teÞ e  ½Lqs ðer, et Þ=ke   εSp Ts  Tf ð1  εÞ ¼ r Ce ∂et

(14a) (14b)

Convective heat transfer in different porous passages

103

where et ¼ αe t=L 2, er ¼ r=L, and re ¼ L r. Another referenced parameter is e ¼ V=U r . As shown in Eq. (14a), when rp is the the velocity Ur to get V hydraulic radius of the pore, the Sparrow number defined by Eq. (11) controls the difference between the solid and the fluid temperatures [11]. The standard Peclet number Pe ¼ CfLUr/ke appears in the presence of a moving fluid. Similarly, Eq. (10) can be rewritten in a dimensionless form as h  i e f e  ke rT ∂ r
 
 
 ∂ e 1 eτx  eτq Le Tf + eτq L Tf + k e ∂et ∂t e (15)  ∂T 2
 ∂2 Tf L ∂S f + S + eτq + eτe + eτq ¼ 2 ke ∂et ∂et ∂et where  

 L2 1 e f  PeVe  rT e f ¼ re  ke rT Le Tf ¼ L Tf ke ke

 Cs 1  ε CsC f eτe ¼ ð1  εÞ eτt ¼ Ce Sp C 2e

(16a) (16b)

Eq. (15), along with Eqs. (16a) and (16b), states that the transient temperature distribution of a moving fluid in a porous medium depends on both Sparrow and Peclet numbers. However, for a stationary fluid, only the Sparrow number affects its thermal field. These effects are analyzed in [11], where a parametric study describes the role of the Sparrow number in applications where there is a rapid change in the surface heat flux. Further related information is presented in the selected numerical examples that appear in the following sections.

3. Micro-scale bio-heat diffusion The two-step model defined through Eqs. (1a) and (1b) can be modified and extended to biological applications as follows [7,8]:   ∂T b + V  rT b ¼ εkb r2 T b + U ðT s  T b Þ + εSh (17a) ερb c b ∂t ∂T ð1  εÞρs c s s ¼ ð1  εÞks r2 T s + U ðT b  T s Þ + ð1  εÞSm ∂t + ð1  εÞSh (17b)

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Fig. 2 Biological domain as an assembly of repeated hexagonal units with blood vessels at the center [16].

where the blood phase (liquid, denoted by the subscript “b”) and biological tissue phase (solid matrix, indicated by the subscript “s”) are here assumed to be isotropic and homogeneous. Also, Sm is the metabolic heating, while Sh is an external heat source applied locally to the tissue, for example during hyperthermia therapy (e.g., laser, microwave, magnetic fluid). A schematic of the biological domain is shown in Fig. 2, where ds is the diameter of an equivalent circular tissue, db is the inner diameter of vessels and L is the vessel distance or pitch between two adjacent vessels that may be pffiffiffi taken as L ¼ 3ds =2. Under this ideal representation, the tissue porosity pffiffi  2 may be taken as ε ¼ 2π9 3 ddbs . In Eqs. (17a) and (17b) the thermal conductivities of blood and biological tissue have been assumed temperature-independent and U is the coupling factor between solid (tissue) and fluid (blood) phases accounting for both convection and perfusion phenomena. It may be taken as U¼

4εhb 4εk 4εk + c b ρb ω ¼ Nu 2 b + c b ρb ω ¼ 4:93 2 b + c b ωb db |{z} db db

(18)

ωb

where the numerical value of 4.93 assigned to the Nusselt number comes from Ref. [17]. Some numerical values for U are listed in Table 1 for five different configurations of the biological region. The contribution of convective heat transfer (4εhb/db, with db inner diameter of the vascular tube)

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Table 1 Parameters characterizing the geometry of the biological domain and the energy exchanges between solid and fluid phases for five different cases [8]. Case ds (mm) db (mm) ε hb (W/(m2 K))

1

19.82

2.28

0.0160

1080

2

14.42

2.28

0.0302

1080

3

12.06

2.28

0.0432

1080

4

10.48

2.28

0.0572

1080

5

9.92

2.28

0.0639

1080

Case

4εhb/db (W/(m3 K))

ωb (kg/(m3 s))

cbωb (W/(m3 K))

U (W/(m3 K))

1

30,348

1

3770

34,118

2

57,281

2

7540

64,821

3

81,939

3

11,310

93,249

4

108,493

4

15,080

123,573

5

121,202

5

18,850

140,052

and blood perfusion (cbωb) to the coupling coefficient is also given in the same table. For that purpose, the blood specific heat is taken as cb ¼ 3770 J/(kg K) and its thermal conductivity as kb ¼ 0.5 W/(m K). Eqs. (17a) and (17b) are two coupled linear, non-homogeneous, parabolic partial differential equations of the second-order in two unknowns, i.e. Ts and Tb, whose solution requires to assign boundary and initial conditions, as shown in next subsection.

3.1 Boundary and initial conditions Boundary conditions for biological tissue and blood can take on various forms depending on the type of boundary surface. If the ith boundary is an impermeable one (for example, when the tissue is in contact with the wall of an artery), generalized non-homogeneous boundary conditions for biological tissue and blood may be taken, respectively, as

 ∂T s ks ¼ hw,i ½T ∞,i  T s ðri , tÞ + qi,s (19a) ∂ni ri

 ∂T b kb ¼ hw,i ½T ∞,i  T b ðri , tÞ + qi,b (19b) ∂ni ri

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where T∞,i ¼ Tb,in ¼ 37 ° C; ri is the location of the ith boundary in a specific coordinate system and ni is the direction of the outward pointing unit vector ni normal to the surface at the same boundary. While the terms qi,s and qi,b are unknown, their combination qw,i ¼ qi ¼ ð1  εÞqi,s + εqi,b

(19c)

is known. Note that hw,i is the heat transfer coefficient of the porous medium with the surrounding fluid (blood within the artery, for example) at the ith impermeable boundary surface; and qw,i ¼ qi is the overall surface heat flux applied to the same boundary of the porous medium that can in general be a function of both space and time. If the ith boundary of the biological domain is a permeable one, that is, a surface crossed by a fluid supplied from (or received by) a reservoir at temperature T∞,i and having v ∞ (ni) ¼ v∞  ni as undisturbed velocity normal to that boundary, the generalized non-homogeneous boundary conditions for solid and fluid phases are, respectively,

 ∂T s ks ¼ hw,i ½T ∞,i  T s ðri , tÞ + qi,s (20a) ∂ni ri

 ∂T b ¼ vð∞ni Þ ρb c b ½T ∞,i  T b ðri , tÞ + qi,b (20b) kb ∂ni ri where T∞,i ¼ Tb,in ¼ 37 ° C; while qi,s and qi,b are unknown but their combination qw,i is known and given by Eq. (19c). Note that in Eq. (20a) hw,i  v ∞ (ni)ρbcb under local thermal equilibrium conditions between the two phases; and the thermal energy exchange in Eq. (20b) is due to a mass transfer process. As regards the initial conditions for tissue and blood phases, they can in general be arbitrary functions of space as T s ðr, 0Þ ¼ T s,in ðrÞ,

T b ðr, 0Þ ¼ T b,in ðrÞ:

(21)

However, it is here assumed that Tb(r, 0) ¼ Tb,in ¼ 37 ° C.

3.2 Uncoupling-based solution method The energy balance Eqs. (17a) and (17b) along with the boundary and initial conditions (19a)-(22) were solved in Ref. [8] by applying an uncoupling procedure. This procedure is based on an approximate relation between Ts and Tb coming from Eq. (17a), that is,

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Ts  Tb +

ερb c b ∂T b εkb 2  r Tb U ∂t U |ffl{zffl} |fflfflfflfflffl{zfflfflfflfflffl} τS

(22)

harmonic term

where the term accounting for hyperthermia therapy and the convective term have been neglected according to de Monte and Haji-Sheikh’s approach [7]. Note that Eq. (22) is an extension of the approximate relation Eq. (5) suggested by Minkowicz et al. [11] and, then, by Zhang [16], where the fluid (here blood) is approximated to behave using a lumped capacitance approach. Also, Eq. (22) exhibits a time lag between tissue and blood temperatures when neglecting the harmonic term. This time lag (or relaxation time) is given by τS ¼ ερbcb/U as its RHS may be seen as a first-order approximation of Tb(r, t + τS) when using a Taylor series expansion. In addition, Eq. (22) is in accordance with Vadasz’s work [18], where the harmonic term is taken into account. By some algebra and simplifying assumptions (the reader can refer to Ref. [8]), the above uncoupling approach applied to Eqs. (17a) and (17b) leads to a DPL equation similar to the one expressed by Eq. (10). In detail,  

 ∂2 Ts ∂Ts ερb cb ∂V ∂
2  2 τq 2 + ¼ α r Ts + τT r Ts + rTs  V + τS ∂t ∂t ∂t ∂t ρc  ð1  εÞSm + Sh ∂ ð1  εÞSm + Sh + τS + ∂t ρc ρc (23) where the tissue temperature Ts is the sole unknown. Also, τq and τT may be seen as phase-lag times for heat flux and temperature gradient, respectively. They are related to the thermal properties of both tissue and blood as follows C s,eff C b,eff ½ð1  εÞρs c s   ðερb c b Þ  ¼
U ½ð1  εÞρs c s + ερb c b  U C s,eff + C b,eff

1 1 1 1 ¼ + U C s,eff C b,eff

τq ¼

½ð1  εÞks   ðερb c b Þ + ðεkb Þ  ½ð1  εÞρs c s  U ½ð1  εÞks + εkb  1 ks,eff C b,eff + kb,eff C s,eff ¼ ks,eff + kb,eff U

(24a)

τT ¼

(24b)

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A. Haji-Sheikh et al.

Note that ks,eff and kb,eff are the effective thermal conductivities of the solid and blood phases, respectively, as well as Cs,eff and Cb,eff are the effective volumetric heat capacities of the corresponding phases. In addition, other selected parameters appearing in Eq. (23) k ¼ ð1  εÞks + εkb ¼ ks,eff + kb,eff

(25a)

ρc ¼ ð1  εÞρs c s + ερb c b ¼ C s,eff + C b,eff

(25b)

α¼

k ρc

(25c)

are the overall thermal conductivity, the overall volumetric heat capacity, and the overall thermal diffusivity of the porous medium (biological tissue filled by blood). Similarly, by applying the same uncoupling procedure to the boundary conditions defined in the previous subsection, it results in [7,8]. "  # "  # ∂T s ∂ ∂T s k + P i T s ðri , t Þ + τS k + P i T s ðri , tÞ ∂ni ri ∂ni ri ∂t (26)
  ∂
¼ P i T b,in + qw,i + τS P T + qw,i ∂t i b,in where Pi ¼ hw,i for non-permeable boundaries, while Pi ¼ v ∞ (ni)ρbcb for permeable ones. Also, it has been set Tb ¼ Tb,in ¼ 37 ° C and τS ¼ ð1  εÞτS .

3.3 Dual-phase lag bio-heat diffusion equation The contribution of the blood flow to the tissue temperature distribution is represented by the third term on the LHS of Eq. (23). If the blood velocity is assumed to be time-independent, i.e., ∂V/∂t  0, it is reasonable to assume that ερb c b V  rT s  U ðT s  T b Þ

(27)

as suggested first by Khaled and Vafai [19] and, then, by Zhang [16]. Substituting Eq. (27) into Eq. (23) yields h i ∂2 T ∂T s ∂
2  1 ¼ α r2 T s + τ T r T s  U ðT s  T b Þ τq 2 s + ∂t ρc ∂t ∂t |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} bioheat term   1 ∂S S + τS + (28) ρc ∂t where S ¼ (1  ε)Sm + Sh.

Convective heat transfer in different porous passages

109

Eq. (28) is similar to the well-established dual-phase lag (DPL) bio-heat equation. However, contrary to this equation, (1) the phase lags τq and τS are not the same in Eq. (28) and (2) the blood temperature appearing in the bioheat term of Eq. (28) is space- and time-dependent. As regards the former, it is shown in Ref. [8] that τS  τq. As far as the latter is concerned, the blood temperature in Eq. (28) is considered uniform and constant according to Pennes’ equation [20], say Tb ¼ Tb,in ¼ 37 ° C, that is the blood temperature in the body’s core. This simplifying assumption is limited only to Eq. (28), as the blood temperature is actually space and time dependent, and can be calculated as shown at the end of the next subsection.

3.4 Transformations of the dependent variable: The DPL equation The “bio-heat” term appearing in Eq. (28) may be canceled out by suitably defining a new dependent variable Ψ(r, t) related to θs(r, t) ¼ Ts(r, t)  Tb,in by θs ðr, tÞ ¼ Ψðr, tÞeσt where σ is a constant defined as (see Ref. [7, app. A]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1  1  4 τq U=ρc σ¼ 2τq

(29)

(30)

Note that the argument of the above square root is always positive or equal to zero as τq U=ρc  1=4 [7]. Now, substituting Eq. (29) into Eq. (28), bearing in mind that α ¼ k=ρc and, then, multiplying the resulting equation by exp(σt) yield

 ∂
2  1 ∂Sσ ∂Ψ ∂2 Ψ 2 ∗ ∗ e ατT (31) αr Ψ + e rΨ + Sσ + τ S + τ∗q 2 ¼ e ∂t ρc ∂t ∂t ∂t where 1  τT σ ke ¼α 1  2τq σ e ρc 1  τT σ ke ¼ k 1  τS σ 1  2τq σ e ¼ ρc ρc 1  τS σ τT ∗ τT ¼ 1  τT σ

e α¼

(32a) (32b) (32c) (32d)

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A. Haji-Sheikh et al.

τq 1  2τq σ τS τ∗S ¼ 1  τS σ Sσ ¼ Seσt

τ∗q ¼

(32e) (32f) (32g)

Eq. (31) is the dual-phase lag (DPL) diffusion equation where the term τS∗(∂ Sσ /∂t) is the so-called ‘apparent’ heating term introduced by Tzou [13,14]. Numerical values for the constant σ defined by Eq. (30) and the transformed phase-lag times appearing in the DPL Eq. (31) are listed in Table 2 for the same five cases of Table 1. Once the Ψ(r, t) function is computed, the tissue temperature may be taken as T s ðr, tÞ ¼ T b,in + θs ðr, tÞ ¼ T b,in + Ψðr, t Þeσt

(33)

Then, the blood temperature Tb(r, t) can be derived by integrating Eq. (22) where the harmonic term can be neglected. Following a standard integration procedure (see Ref. [21], p. 1096, #16.316), the result is Z et=τS t t0 =τS T b ðr, tÞ ¼ T b,in e + e T s ðr, t 0 Þdt0 τS t0 ¼0 Z 0 et=τS t ¼ T b,in + Ψðr, t0 Þeðσ1=τS Þt dt0 τS t0 ¼0 t=τS

(34)

where t0 is a dummy time variable. As regards the boundary conditions; similarly, substituting Eq. (29) into Eq. (26) yields

Table 2 Transformed phase-lag times. Case σ (s21) τ∗q (s)

τ∗ T (s)

τ∗ S (s)

1

0.009

1.890

3.780

1.889

2

0.018

1.910

3.820

1.907

3

0.026

1.930

3.860

1.925

4

0.035

1.964

3.928

1.957

5

0.040

1.954

3.908

1.946

Convective heat transfer in different porous passages

111

"  # "  # ∂Ψ ∂ ∂Ψ + Pei Ψðri , tÞ + τ∗T + Pei Ψðri , tÞ ke ke ∂ni ri ∂t ∂ni ri ðσ Þ

¼ qw,i + τ∗S

ðσ Þ

∂qw,i ∂t

(35)

where keand τ∗ S are defined by Eqs. (32b) and (32f ), respectively. Also, other selected parameters are: 1  τT σ P Pei ¼ 1  τS σ i 1  τT σ σt ðσ Þ e qw,i ¼ qw,i 1  τS σ τS 1  τS σ τ∗T ¼ 1  στS 1  τT σ

(36a) (36b) (36c)

where τS ¼ ð1  εÞτS . For qw,i ¼ 0, Eq. (35) reduces to a homogeneous boundary condition of the third kind "  # "  # ∂Ψ ∂ ∂Ψ ke + Pei Ψðri , tÞ + τ∗T + Pei Ψðri , t Þ ¼ 0 (37) ke ∂ni ri ∂t ∂ni ri Then, for Pei ! ∞ and Pei ! 0, Eq. (37) reduces, respectively, to



 ∂ ∂Ψ ∂ ∂Ψ + τ∗T ¼0 Ψðri , tÞ + τ∗T Ψðri , tÞ ¼ 0 ∂t ∂ni ri ∂t ∂ni ri

(38)

It is interesting to observe that, when applying the modified separation of variables (SOV) method proposed by Hays-Stang and Haji-Sheikh [22] for solving the governing equations, the homogeneous boundary conditions of the Sturm-Liouville eigenvalue problem associate to the DPL diffusion equation reduce the ones of the conventional parabolic diffusion, that is,



 ∂F ∂F ke + Pei F ðri Þ ¼ 0 F ðri Þ ¼ 0 ¼0 (39) ∂ni ri ∂ni ri where the space-variable function F(r) is the eigenfunction of the SturmLiouville problem. As regards the initial condition for the tissue phase appearing in Eq. (28), by using Eq. (29) in the first of the two Eqs. (21), its transformation gives

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Ψðr, 0Þ ¼ Ψin ðrÞ ¼ T s,in ðrÞ  T b,in ,

(40)

where Tb,in ¼ 37 ° C. Another initial condition is needed for the dual-phase lag diffusion _ in ðrÞ ¼ ð∂Ψ=∂tÞ . This can be derived from Eq. (29) Eq. (28), that is, Ψ t¼0 by taking its time partial derivative for t ¼ 0. The result is   ∂Ψðr, tÞ ∂Ts ðr, t Þ _ ¼ Ψ in ðrÞ ¼ σΨðr, 0Þ + ¼ σΨin ðrÞ (41) ∂t ∂t t¼0 t¼0 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} ¼0

3.5 Temperature solution in finite regular tissues The temperature solution may be obtained in an integral form by using the Dual-Phase Lag (DPL) “alternative” Green’s function solution equation (AGFSE) derived by de Monte and Haji-Sheikh [7]. This equation leads to a much faster convergence of the series-solution due to non-homogeneous boundary conditions than the DPL GFSE (here not derived). It is Ψðr, tÞ ¼ ΨI ðr, t Þ + ΨS ðr, t Þ + ΨB ðr, tÞ

(42)

where the function ΨI(r, t) represents a solution using the initial conditions, ΨS(r, t) is a solution that defines the contribution of volumetric heat source, while ΨB(r, t) accounts for non-homogeneous boundary conditions. In detail, they are given by ! Z ∗    τ q   e  ΨI ðr, tÞ ¼ Ψin ðr0 ÞdV 0 GDPL + ∗ GDPL ατ∗T r20 GDPL 1 2 1 τ¼0 τ¼0 τ¼0 τ V S Z   Ψ _ ðr0 ÞdV 0 + τ∗q GDPL 1 τ¼0 in V

ΨS ðr, tÞ ¼

1 e ρc

Z

t τ¼0

(43a)  0 ∂S ðr , τÞ dV 0 dτ G1DPL ðr, tj r0 , τÞ Sσ ðr0 , τÞ + τ∗S σ ∂τ V

Z

(43b)

∗  2 ∗ ∂ΨB B ∂Ψ G1DPL ðr, tj r0 , τÞ + τ∗q 2 B dV 0 dτ ΨB ðr, tÞ ¼ Ψ∗B ðr, tÞ  ∂t ∂t τ¼0 V  Z

∗   τq DPL  2 DPL  DPL  ∗   G1 τ¼0 + ∗ G2 τ¼0  e ατT r0 G1 τ¼0 Ψ∗B, in ðr0 ÞdV 0 τ V S Z  ∗ 0 0 _  τ∗q G1DPL τ¼0 Ψ B, in ðr ÞdV Z

t

Z

V

(43c)

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where Ψ∗(r, t) satisfies the original non-homogeneous boundary conditions B but does not need to satisfy the two initial conditions (see example 5.3 given by Cole et al. [23] for parabolic heat diffusion). Therefore, the function Ψ∗(r, t) satisfies the following governing equations B r2 Ψ∗B ¼ 0 "

# "

# ∗ ∗ ∂Ψ ∂Ψ ∂ B B ke ke + Pei Ψ∗B ðri , tÞ + τ∗T + Pei Ψ∗B ðri , t Þ ∂t ∂ni ri ∂ni ri ðσ Þ

¼ qw,i + τ∗S

(44a)

ðσ Þ

∂qw,i ∂t

(44b)

The dual-phase lag (DPL) based Green’s functions (GFs) appearing in Eqs. (43a)–(43c) are given in appendix of Ref. [8], and related corrigendum Ref. [24].

3.6 Temperature solution in a laser-irradiated biological tissue For effective laser treatment processes, such as laser hyperthermia, it is important to understand the evolution of temperature in the biological tissues. As the laser light can be highly absorbed or strongly scattered [25,26], two cases can in general be considered according to these situations. In both cases, a broad laser beam with a uniform incident irradiance (qIR) is applied at time t ¼ 0 up to tIR normally to a slab of perfused biological tissue with a thickness of Ls and a uniform initial temperature of Ts,in ¼ Tb,in ¼ 37 ° C. As the spot size of the broad laser beam is much larger than the thickness of the thermally affected zone for the time interval of interest, a one-dimensional model with the x ¼ Ls boundary considered thermally insulated would be sufficient for analyzing the thermal response of the heated porous medium [25,26], as shown in Fig. 3. According to Fig. 2, the blood vessels of the tissue depicted in Fig. 3 are considered as a bundle of staggered tubes placed along the direction normal to the incident laser radiation. However, in the current subsection, only the case of a laser irradiation highly absorbed by a tissue (e.g., for some UV and IR wavelengths) is analyzed. In such a case, as the laser light is absorbed within a very small depth of the biological region, a few μm, its consequent heating can reasonably be approximated as a surface heat flux at x ¼ 0. Alternatively, the irradiated tissue surface x ¼ 0 can be considered thermally insulated, and the laser heating can be modeled as a local (plane) heat source located at x ¼ 0, that is, Sh ðx, tÞ ¼ ð1  Rd ÞqIR ðtÞδðxÞ

(45a)

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Fig. 3 Schematic of a biological tissue subject to a laser irradiation at the boundary x ¼ 0 for a finite time period of tIR.

where Rd is the diffuse reflectance of light at the irradiated surface, δ(.) is the Dirac delta function and qIR(t) is given by ð0Þ

qIR ðtÞ ¼ qð0, tÞ ¼ qIR ½H ðt Þ  H ðt  tIR Þ

(45b)

In the above equation, q(0) IR is the time-independent incident laser heat flux, tIR is the laser exposure time and H(.) is the so-called Heaviside or unit step function, as depicted in Fig. 3, where a single-layer finite 1D rectangular body of biological tissue is considered. As regards the initial condition, some authors such as Baish et al. [27] and, then, Okajima et al. [28], assume as initial temperature for the biological tissue its steady-state temperature that may be taken as Ts,in ¼ Tb,in + Sm/(cbωb), where Sm is the metabolic heat generation (Sm ¼ 4.2  103 W/m3 [28]) and cbωb represents the blood heat sink term whose numerical values are given in Table 1. This equilibrium temperature derives from a steady-state balance of the heat generation term with the heat sink, in the absence of diffusion, starting with Pennes’ bio-heat equation. If the steady-state balance is applied to the current Eq. (28), it results in Ts,in ¼ Tb,in + (1  ε)Sm/U, where U is the coupling factor defined by Eq. (18). When dealing with this temperature as initial condition, for the sake of consistency the metabolic term has to be removed from the BHDPL Eq. (28). Therefore, S ¼ (1  ε)Sm + Sh appearing in Eq. (32g) related to the following DPL Eq. (31) reduces to S ¼ Sh that is defined by Eq. (45a). In the current case, the initial tissue temperature is considered uniform and equal to Tb,in ¼ 37 ° C. Therefore, Eqs. (40) and (41) reduce to Ψin(r) ¼ 0 _ in ðrÞ ¼ 0, respectively. However, as the metabolic heating term Sm and Ψ

Convective heat transfer in different porous passages

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is usually very small compared to the external heating power Sh [29], it is negligible and so S ¼ Sh. The one-dimensional transient governing equations for this heating problem can be derived from the more general ones defined in the previous subsections. As the initial and boundary conditions are homogeneous, the only driving term is due to laser heating through Sσ ¼ Seσt. The solution to the current 1D rectangular problem is given by Eq. (43b) readapted in one-dimensional form (the reader can refer to [8]). Plots of tissue and blood temperature as a function of both space and time are shown in Figs. 4 and 5 and refer to cases “1” and “5” of Tables 1 and 2, respectively. In particular, these two cases consider blood vessels having a diameter of 2.28 mm but with different porosity and perfusion rate. The plots were drawn for a tissue of Ls ¼ 5 cm as a thickness, a laser 2 power density of q(0) IR ¼ 2 W/cm , a diffuse reflectance of light at the irradiated surface Rd ¼ 0.05 and a laser exposure time tIR ¼ 5 s [25,26]. Both Figs. 4 and 5 have two parts. In the former (a), where the location is assumed as a parameter, the phase lag time between the tissue and blood temperature is evident enough and in accordance with the time lag τS defined by Eq. (22). In detail, for case “1” τS ¼ 1.856 s; while for case “5” τS ¼ 1.806 s. In the latter (b and c), where the time is considered as a parameter, the heating (laser “on”) and cooling (laser “off”) processes of the perfused tissue are put into evidence separately. It may be noted that the irradiated surface reaches a peak value of about 67 ° C for the tissue and of about 60 ° C for the blood in the “1st” case (Fig. 4A), characterized by ε ¼ 0.016, hb ¼ 1080 W/(m2 K) and ωb ¼ 1 kg/(m3s). These values of temperature are lower in the “5th” case (Fig. 5A), characterized by the same heat transfer coefficient hb ¼ 1080 W/(m2 K), but with a higher void fraction ε  0.064 and a higher perfusion rate ωb ¼ 5 kg/(m3s). In fact, the temperature at the heated surface reaches a maximum value of about 64 ° C for the tissue and of about 58 ° C for the blood. This is due to a major cooling effect of the blood within the vessels for higher porosity and perfusion rate. Then, Figs. 4B and C and 5B and C show that only a few mm of tissue is affected by the laser therapy. In fact, at a distance of 3 mm from the irradiated surface, the tissue heating is practically negligible. Once the temperature distribution is known, the thermal damage of the tissue can be estimated by using the well-established damage integral of Henriques as described in [30], where a careful review of quantitative models of thermal damage is provided. A critical temperature of this damage

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tissue and blood temperatures [°C]

A

location, x [mm], as a parameter

70 65 x = 0 mm

60 55

0.5 mm

50

1 mm

45

1.5 mm 40 35

2 mm 0

2

4

8

6

10

12

14

16

18

20

time, t [s]

tissue and blood temperatures [°C]

B

time, t [s], as a parameter

70 65 60 55 50

5s 45

3s

40 t=0 s 35

0

1s

0.4

C

0.8 1.2 space, x [mm]

1.6

2

1.6

2

time, t [s], as a parameter

tissue and blood temperatures [°C]

70 65 60 55 t=6 s

10 s

50

15 s 45

20 s

40 35

0

0.4

0.8 1.2 space, x [mm]

Fig. 4 Plots of tissue (solid line) and blood (dashed line) temperatures for a laser power 2 density of q(0) IR ¼ 2 W/cm : (A) as a function of time; (B) vs space with laser “on” (heating); and (C) vs space with laser “off” (cooling). Case “1” of Tables 1 and 2.

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tissue and blood temperatures [°C]

A

65 60

tissue and blood temperatures [°C]

x = 0 mm

55 50

0.5 mm

45

1 mm 1.5 mm

40 35

B

location, x [mm], as a parameter

70

2 mm 0

2

4

8

10 12 time, t [s]

14

16

18

20

time, t [s], as a parameter

70 65 60 55

5s

50 3s 45

2s 1s

40 35

t=0 s 0.4

0

C

0.8 1.2 space, x [mm]

1.6

2

1.6

2

time, t [s], as a parameter

70 tissue and blood temperatures [°C]

6

65 60 55 50

t=6 s 8s 10 s

45

15 s

40

20 s

35 0

0.4

0.8 1.2 space, x [mm]

Fig. 5 Plots of tissue (solid line) and blood (dashed line) temperatures for a laser power 2 density of q(0) IR ¼ 2 W/cm : (A) as a function of time; (B) vs space with laser “on” (heating); and (C) vs space with laser “off” (cooling). Case “5” of Tables 1 and 2.

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is calculated to be 74.8 °C, that is always greater than the maximum temperature observed in Figs. 4 and 5. In general, higher values of porosity and perfusion rate of the biological tissue allow the risk of its thermal damage to be reduced.

4. Transient thermal diffusion in porous devices The objective of this section is to show two classical solutions for selected geometries containing homogeneous porous materials. Two different solutions of Eq. (10) are presented, one for a semi-infinite body and the second for a thin plate. In the presence of a moving fluid, the solution of Eq. (10) depends on the functional form of the velocity vector V. Among various possible cases, two different and simple functional forms are examined in the next two sections. The velocity is finite in the first special case while the second example is for a stationary fluid with V ¼ 0, neglecting the CfV  r Tf in Eq. (10).

4.1 Transient temperature field with a moving fluid Various studies related to the effects of local thermal non-equilibrium in porous media are in the archival literature. As an illustration, the studies related to the developing forced convection is reported by Khashan et al. [31], Hao and Tao [32], Part I and [33], Part II, Nield et al. [34], and others. Certainly, in the presence of phase change in the flow field, the condition of LTNE exists and this phenomenon is discussed in Duval et al. [35] and Cao et al. [36]. For a fluid moving with a finite velocity, the computation of temperature, from Eq. (10), is possible by numerical means. Also, analytical solutions are possible for special cases, and these solutions can serve as valuable tools for verification objectives. One special case is when the velocity vector in Eq. (10) has a constant value. Further simplification can be realized when the two parameters τx and τq have nearly the same value; this latter condition modifies Eq. (10) to become   

 ∂2 Tf ∂Tf ∂ 
 ∂S + τe + τq L Tf + τq (46a) L Tf + S + τq ¼C ∂t ∂t ∂t ∂ t2 Using a dimensionless temperature that satisfies the non-homogeneous boundary conditions leads to the equation

Convective heat transfer in different porous passages

119




  ∂2 θ f ∂θf
∂ 
 ∂S + τe + τq L θf + τq L θf + S + τq + Sb ¼ C ∂t ∂t ∂t ∂t2 (46b) where Sb depends on the functional form of the transformed temperature. For this specific case, when the velocity has a constant value, it is often possible to use a series solution and obtain a solution for a regular geometry. However, for this case, the function Fn(r) in θf ðr, tÞ ¼

∞ X

ψ n ðtÞFn ðrÞeγ n t

(47a)

n¼1

can be the solution of equation L ½F n ðrÞ ¼ γ n CF n ðrÞθ

(47b)

Therefore, one could provide the final temperature solution once the function θf (r, t), the solution for Eq. (46b), is known. It should be stated that it is often possible to find a transformation that produces a partial differential equation without the convection terms for all values of r. To demonstrate the behavior of this solution methodology, consideration is given to a relatively thick porous plate initially at temperature Ti. A fluid is entering or leaving this plate in x-direction and, in the absence of other walls, the velocity u has a constant value. In the presence of heat flux at x ¼ 0, this example represents an interesting application of transpiration cooling for surface protection in the presence a suddenly applied large heat flux. The case of a sudden temperature change to Tο at x ¼ 0 is included in this example, mainly for comparing the two solutions. When θ ¼ (T  Ti)/ Tref while there is no volumetric heat source, the governing energy equation takes the form

2 

  ∂ θf ∂3 θ f ∂θ f ∂2 θ f ∂θ f ∂2 θ f ke + τx 2 + τq + τa 2 Cfu ¼C ∂x ∂ x∂t ∂t ∂ x2 ∂x ∂t ∂t (48) where τa, τq, and τx are as defined earlier. Using the dimensionless variables pffiffiffiffiffiffiffi ξ ¼ x= ατa , η ¼ t/τa, r ¼ τx/τa, rq ¼ τq/τa, and rt ¼ τt/τa, Eq. (48) becomes ∂2 θ f ∂3 θ f H + r ∂ ξ2 ∂ ξ2 ∂η



 ∂θ f ∂2 θ f ∂θ f ∂2 θ f + rq ¼ + ∂ξ ∂ ξ∂η ∂η ∂ η2

(49a)

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∂θ f (49b) ∂t
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wherein H ¼ ðu=jujÞ C f =C C juj2 τa =ke . The quantity (u/j uj) ¼ 1 when fluid is moving in the positive x-direction and (u/j u j) ¼  1 when fluid is moving in the negative x-direction. Using the definition of θ, the initial conditions are θf (ξ, 0) ¼ 0 and ∂ θf (ξ, t)/∂ t jt¼0 ¼ 0, while the boundary conditions are to be specified later. Also, the reference temperature Tref depends on the specified boundary condition at x ¼ 0 and it will be determined later. By defining the interstitial Peclet number as Pe ¼ Cu rh/ke, it
3=2 pffiffiffiffiffiffiffiffiffi makes the controlling parameter H ¼  C f =C Pe= Nurh. This shows that the value of H strongly depends on Pe, since Cf/C < 1 and Nurh is of the order of 1. It is known that the effect of axial conduction becomes negligible when Pe is larger than 1. Therefore, in the presence of axial conduction in Eq. (49a), the parameter H is generally smaller than 1. A simple method of finding the solution of Eq. (49a) is by using the Laplace transform method. When θðξ, sÞ is the Laplace transform of θ (ξ, η), the Laplace transform of Eq. (49a) with the specified initial conditions takes the following form: θs ¼ θ f + r t

ð 1 + r sÞ


dθf
 d2 θ f ¼ s + s2 θ f 1 + r qs 2 H dξ dξ

(50)

This is an ordinary second-order differential equation with constant coefficients whose solution is 2 H ð1+r q sÞξ 4

θ f ðξ, sÞ ¼ e

+ D2 e

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 H 2 ð1+r q sÞ +4ð1+r sÞðs+s2 Þξ



D1 e

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 H 2 ð1+r q sÞ +4ð1+r sÞðs+s2 Þξ 5

(51)

When H is negative, the problem under consideration simulates flow leaving a porous device with insulated sidewalls. Therefore, for a sufficiently thick device, the condition of finite θ f ðξ, sÞ as ξ ! ∞ makes D2 ¼ 0. The first set of data in Fig. 6A is for suddenly applied heat flux qw at x ¼ 0, see the inset of pffiffiffiffiffiffiffiffiffi Fig. 6A for qw direction. For the data in this figure, T ref ¼ qw ατa =ke, and

pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi this makes θ f ¼ ke Tf  T i = qw ατa and θs ¼ ke ðT s  T i Þ= qw ατa .

Convective heat transfer in different porous passages

121

A

B

Fig. 6 The temperature variation with time at selected locations within solid and pore materials in the presence of a flow field: (A) boundary condition of the second kind and (B) boundary condition of the first kind.

The application of the boundary condition of the second kind at x ¼ 0 requires the use of Eq. (8) with a constant qw, that is n o ∂ qw ¼ ke rTf ð0, t Þ + τx rTf ð0, t Þ (52) ∂t In dimensionless space, this equation provides the value of D1 in Eq. (51) and it becomes  
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2 exp H 1 + rq s  H 1 + rq s + 4ð1 + r sÞðs + s ξ  θf ðξ, sÞ ¼
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2 sð1 + rsÞ H 1 + rq s  H 1 + rq s + 4ð1 + r sÞðs + s (53)

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The second set of data in Fig. 6B is for the boundary condition of the first kind at x ¼ 0. Assuming the reference temperature is Tref ¼ Tο  Ti, the porous materials temperature θs (0, η) ¼ 1 readily provides the value of D1 ¼ 1/[s(1 + rts)], using the boundary condition at ξ ¼ 0. This makes Eq. (51) to become qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 H ð1+r q sÞ ξ  H 2 ð1+r q sÞ +4ð1+r sÞðs+s2 ξÞ e e θ f ðξ, sÞ ¼ (54) sð1 + r t sÞ The inverse Laplace transforms for these two cases, Eqs. (53) and (54), are acquired numerically, see a symbolic programming in Mathematica. Fig. 6A and B is prepared to show this trend when H ¼  1/2 and τq/τt ¼ (1  ε)Cs/C ¼ 1/2; therefore, the input parameters in Eq. (54) become r ¼ 1.25, rq ¼ 0.5, and rt ¼ 1. The data plotted in both figures show the condition of LTNE that becomes LTE at larger times. Of course, as before, LTNE condition exists when r is relatively large. The data in Fig. 6A show a gradual increase in the solid materials temperature θs(0, η) at ξ ¼ 0 location, while θs(0, η) undergoes its rapid jump in Fig. 6B due to specified boundary condition. At larger values, e.g., ξ ¼ 2, the functional variations of temperature within the solid materials and the fluid in the pores become similar.

4.2 Transient temperature field with stationary fluid In the next example, consideration is given to determination of transient temperature field in a plate with stationary fluid in the pores; wherein, the governing equation is  ∂2 θ f ∂3 θ f ∂2 θ f 1 ∂θ f + τa 2 + τx ¼ (55a) ∂x2 ∂t∂x2 α ∂t ∂t This is an interesting special case when τx ¼ τa. The solution of this equation in a plate with a thickness of b and with the boundary conditions 8 > < θ f ¼ θs ¼ 0 when t ¼ 0 θs ¼ 1 at x ¼ 0 and t > 0 (55b) > : ∂θ f =∂x ¼ 0 at x ¼ b is of interest. The second condition provides the surface temperature for the pore materials; it is θ f ð0, tÞ ¼ 1  exp ðt=τt Þ

(56a)

Convective heat transfer in different porous passages

123

This suggests using a transformation θ f ðx, tÞ ¼ ψðx, tÞ + 1  exp ðt=τt Þ

(56b)

in order to have a homogeneous boundary condition at x ¼ 0. Following the use of this transformation, the initial condition and the boundary conditions for ψ are 8 when t ¼ 0 >

: ∂ψ=∂x ¼ 0 at x ¼ b Note that all the boundary conditions in Eq. (57) are homogeneous. Moreover, in one-dimensional space, the transformation in Eq. (56b) makes Eq. (55a) to take the following form, " # ∂2 ψ ∂3 ψ 1 ∂ ψ 1 t=τt ∂2 ψ τa t=τt + τx ¼ + τa 2 + e (58)  e ∂ x2 ∂ t ∂ x2 α ∂ t τ t ∂t ðτt Þ2 When τt ¼ τa, the two exponential terms in this equation have the same magnitudes but with opposite signs; they would drop out and this equation also becomes homogeneous. Therefore, the solution for this equation with the specified boundary conditions for ψ, in Eq. (57), is a trivial one; that is ψ 5 0. Certainly, this is unacceptable because it makes θf (x, t) ¼ exp( t/τt), independent of the axial coordinate x. It is possible to eliminate this phenomenon when τx ¼ τt ¼ τa. This is expected when using the τx ¼ τt model and the relation τq ¼ τt(1  ε)Cf /C. Under these conditions, Eq. (55a) takes the form whose solution is satisfied if the equation α

∂2 θ f ∂θ f ¼0  ∂t ∂x2

(59a)

is satisfied; which is the classical diffusion equation. The above transformation, Eq. (56b), makes Eq. (59a) to become α

∂2 ψ ∂ψ 1  + et=τt ¼ 0 ∂t τt ∂x2

(59b)

subject to the boundary conditions as given by Eq. (57). pffiffiffiffiffiffi In dimensionless space, when η ¼ t/τt and ξ ¼ x= ατt , the solution of the function ψ(ξ, η), as given in [23] is

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Fig. 7 The variation of temperature with time at selected locations within solid and pore materials in a plate, when r ¼ τt/τa ¼ 1.

ψ ðξ, ηÞ ¼

2 ∞ 2 X eη  e½ðm1Þπ=ξb  η sin ½ðn  1=2Þπξ=ξb  ξb m¼1 ½ðm  1Þπ=ξb 2  1

(60)

pffiffiffiffiffiffi where ξb 5b= ατt . Following the computation of ψ(ξ, η), Eqs. (56a) and (56b) provide the values of θf (ξ, η), and the values of θs(ξ, η) are obtainable using Eq. (49b). These quantities are computed and the data are plotted in Fig. 7 for x/b ¼ ξ/ξb ¼ 0.1, 0.2, 0.3, 0.5 and 1; the data plotted in Fig. 7 are for ξb ¼ 1. The solid lines represent the temperatures of the solid matrix and the dash lines are for the materials in pores. It is essential to use an alternative method of analysis for the purpose of verifying the accuracy of this solution. Accordingly, Eq. (50), when u ¼ 0 leads to Eq. (55a) with initial and boundary conditions as given in Eq. (55b). Then, the solution of Eq. (55a) with specified conditions in Eq. (55b) is in the Laplace transformed domain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  cosh sðs + 1Þ=ð1 + rsÞðξb  ξÞ 1 θ f ðξ, sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  sð1 + rsÞ cosh sðs + 1Þ=ð1 + rsÞ ξb

(61)

where θ f ðξ, sÞis the Laplace transform of θf (ξ, η), with dimensionless parameters. The inverse Laplace transform of Eq. (61) is acquired numerically, using the symbolic methodology in Mathematica. The circular symbols in Fig. 7 depict these numerically computed data. The data equally agree well

Convective heat transfer in different porous passages

125

with both the solid lines and dash lines in Fig. 6. In fact, these numerical computed data from Eqs. (60) and (61) are nearly identical to all six significant figures over the entire range of acquired data plotted in Fig. 7.

5. Steady state heat transfer to flow in porous passages The studies related to heat transfer to flow in porous ducts are presented in this section. Two types of mathematical analysis are presented: one with no axial conduction and one with axial conduction. It is to be noted that the contribution of axial conduction becomes significant [37] when the passages filled with porous materials with relatively high thermal conductivity values. An extensive study of flow in porous media is available in Nield and Bejan [38], Kaviany [39], and Vafai [40]. Also, acquired Green’s function solutions related to heat conduction in porous passage are in [41,42]. A series solution requires the computation of a set of eigenvalues and the numerical computation for determination of certain eigenvalues can become a formidable task. To improve this situation, the weighted residual is used in [42,43]. The study of heat transfer with axial conduction, with specified wall temperature, is in [43,44] while with specified wall heat flux is in [45]. In this work, the first section describes a modified Graetz problem in parallel plate channels and circular tubes with no axial conduction of thermal energy, while consideration is given to axial conduction in the second section. The exact series solution for parallel plate and circular ducts are presented using the Brinkman’s model. Then, the results are compared to those acquired from a numerical study based on the method of weighted residuals. Also, the method of weighted residuals provided flow and heat transfer data in elliptical passages. The results include the computation of heat transfer to fluid flowing through elliptical passages with different aspect ratios. Two different solutions are presented. The first one considers that there is no axial conduction in the fluid passage. However, the second case includes the contribution of axial conduction.

5.1 Formulation of heat transfer in porous ducts The process begins by separating the variable and getting an exact solution. To verify the accuracy of this solution, an alternative numerical solution is selected. This alternative solution uses the method of weighted residuals. These two solutions provide comparable accuracy over an extended range

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y

A

H

x

u (y)

z

B

r

rο x

u (r)

y

Fig. 8 Schematic of a flow (A) in a parallel plate channel and (B) in a circular pipe.

of variables. For a finite number of eigenvalues, the method of weighted residuals provides results with comparable accuracy at larger values of the axial coordinate. Since this is based on the minimization principle in variational calculus, it yields higher accuracy near the thermal entrance region. In addition, standard-computing packages can produce all eigenvalues with ease instead of getting them one at a time in the exact solution. In summary, the presentation in this section begins by providing the exact mathematical solution for heat transfer in the entrance region of a parallel plate channel, in Fig. 8A. The procedure is then extended to compute similar parameters for a circular duct, in Fig. 8B. The study of heat transfer to a fully developed flow though selected porous media, confined within impermeable walls, is of interest. This work considers heat transfer in parallel plate channels, circular pipes, and elliptical passages in the absence of any volumetric heat source and axial heat conduction. The Brinkman momentum equation for a fully developed flow is μe r2 u 

∂p μ u ¼0 ∂x K

(62)

Here, fluid flows in the direction of x and the Laplace operator takes different forms depending on the shape of the passages. The classical energy equation in its reduced form is u

∂T k 2 1 rT+ S ¼ ∂x ρ c p ρ cp

(63)

Convective heat transfer in different porous passages

127

where S is the classical volumetric heat source that includes the contribution of frictional heating and T is the mean temperature. The contributions of volumetric heat source and frictional heating in porous media, using the Green’s function, are in [41,42] while various other related theories are in Nield et al. [46,47]. For a few cases presented in Sections 5.2–5.4, it is assumed that there is no frictional heating, S ¼ 0. Eqs. (62) and (63) in conjunction with the given wall and entrance conditions provide velocity and temperature fields. Using the energy balance on a differential fluid element would produce the relation hCd(Tw  Tb) ¼ ρUA (dTb/dx) where U is the average velocity, Cd is the contour of the duct, A is the passage area, Tw is the constant wall temperature, and Tb ¼ Tb(x) is the local bulk or mean temperature of the fluid defined as Z   1 u Tb ¼ TdA (64) A A U It is appropriate to have Lc as the characteristic length selected depending on the shape of a passage while U to designate the average fluid velocity in the passage. As an example, for a circular pipe, Lc ¼ ro where ro is the pipe radius. This produces the working relation for computing the dimensionless local heat transfer coefficient, hLc/k, as



 dT b ðxÞ=dx dθb ðxÞ=dx hL c A A ¼ (65a) ¼ k θb ðxÞ C d T w  T b ðxÞ Cd where x ¼ x=ðPe L c Þ with Pe ¼ ρ cpLcU/k, A ¼ A=L 2c , C d ¼ C d =L c , and assuming Tb(0) ¼ Ti to be a constant, it makes θb ðxÞ ¼ ½Tb ðxÞ  Tw = ðTi  Tw Þ. This equation yields the standard definition of the Nusselt number NuD ¼ hDh/k, based on the hydraulic diameter Dh ¼ 4A/C, as

2  Dh dθb ðxÞ=dx hDh (65b) ¼ k θb ðxÞ 4L 2c Using the definition of the average heat transfer coefficient, Z 1 x H¼ HDX, X 0 in Eqs. (65a) and (65b), it produces the relations

 hL c A ln θb ðxÞ ¼ k x C

(66a)

(66b)

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A. Haji-Sheikh et al.

and hDh ¼ k



D2h 4L 2c



ln θb ðxÞ x

(66c)

The subsequent analyses report the computed results from the use of these four definitions of the dimensionless heat transfer coefficients.

5.2 Porous ducts with no-axial conduction The first flow model considers a steady and hydrodynamically fully developed flow between two impermeable parallel plates (i.e., parallel plates ducts), 2H apart, see Fig. 8A. The computation begins by considering the Brinkman momentum equation μe

∂p ∂2 u μ  u ¼0 ∂x ∂y2 K

(67a)

Assuming constant pressure gradient Φ ¼  ∂ p/∂x and using the effective viscosity μe, the fluid viscosity μ, the permeability K, and Lc ¼ H, Eq. (67a), in dimensionless form, reduces to an ordinary differential equation M

d2 u 1  u+1¼0 2 Da dy

(67b)

where y ¼ y=H , M ¼ μe/μ, u ¼ μu=ðΦH 2 Þ, and Da ¼ K/H2 is the Darcy number. The solution of Eq. (67b) using the boundary conditions u ¼ 0 at y ¼ 1 and the symmetry condition ∂ u=∂ y ¼ 0 at y ¼ 0 becomes  cosh ðωyÞ u ¼ Da 1  (68) cosh ðωÞ where ω ¼ (MDa)1/2. Then, using the mean velocity defined by the relation Z 1 H udy (69) U¼ H 0 and Eq. (68) leads to a relation for the reduced mean velocity U as,

and therefore

U ¼ Da ½1  tanh ðωÞ=ω

(70a)

 cosh ðωyÞ u u ω ¼ ¼ 1 U U ω  tanh ðωÞ cosh ðωÞ

(70b)

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Convective heat transfer in different porous passages

The definition of the average velocity U ¼ μU=ðΦH 2 Þ leads to a relation for the friction factor  2 ð∂ p=∂ xÞDh 2 1 Dh F f ¼ ¼ ¼ (71) 2 Re D ρU =2 U Re D H where F ¼ ð2=U ÞðDh =H Þ2 . Moreover, Eq. (69) provides the value of U once the function u replaces u in the integrand. The temperature distribution assuming local thermal equilibrium is obtainable from the energy equation u

∂T k ∂2 T ¼ ∂x ρ c p ∂y2

(72a)

Introducing the Peclet number Pe ¼ ρ cpHU/k where U is the mean velocity in the duct, and dimensionless x ¼ x=ðPe H Þ, the energy equation, Eq. (72a), reduces to u ∂ θ d2 θ ¼ U ∂ x d y2

(72b)

where θ is the dimensionless temperature θ ¼ (T  Tw)/(Ti  Tw). In this formulation, it is assumed that the inlet temperature Ti and the wall temperature Tw have constant values. The solution of the partial differential Eq. (72b) is obtainable using the method of separation of variables; that is, let θðx, yÞ ¼ X ðxÞY ðyÞ. The substitution of this functional for θ in Eq. (72b) yields u X 0 Y 00 ¼ Y UX

(73)

leading to two ordinary differential equations X 0 ðxÞ + λ2 X ðxÞ ¼ 0 and 0

Y 0 ðyÞ + λ2

  u Y ðyÞ ¼ 0 U

(74a)

(74b)

The parameter λ is the eigenvalue in this eigenvalue problem that produces the temperature solution θ¼

∞ X m¼1

Bm Y m ðyÞ eλm x 2

(75)

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where the term eλm x in Eq. (75) is the solution of Eq. (72a) once the parameter λ is replaced by λm. However, the solution of Eq. (74b) remains as the major task for this investigation and it is presented in the following sections. Following substitution for u/U, Eq. (74b) takes the following form    cosh ðωyÞ ω 00 2 Y ðyÞ + λ (76) 1 Y ðyÞ ¼ 0 ω  tanh ðωÞ cosh ðωÞ 2

Next, consideration is given to the exact and numerical solutions of this ordinary differential equation. For simplicity of this presentation, let ψ¼

λ2 =ω ω  tanh ðωÞ

(77a)

1 cosh ðωÞ

(77b)

and β¼ then Eq. (76) reduces to become Y 00 ðyÞ + ω2 ψ ½1  β cosh ðω yÞ Y ðyÞ ¼ 0

(78)

This differential equation has the form of a modified Mathieu differential equation whose solution is associated with the modified Mathieu Function McLachlan [48]. Various classical solutions of the Mathieu differential equation are in [48]. However, the plus sign instead of a minus sign in Eq. (78) makes its solution more demanding than the classical modified Mathieu Function. Since this is a special Mathieu differential equation, it is best to describe the method of solution for this special case. Among a few possible solutions, two different solutions are presented here. 5.2.1 First solution For convenience of algebra, the abbreviated form of this ordinary differential equation is Y 00 + ω2 ψ ½1  β cosh ðω yÞ Y ¼ 0

(79)

where ψ ¼ (λ2/ω)/[ω  tanh(ω)] and β ¼ 1/ cosh(ω). The boundary conditions require having a symmetric condition at y ¼ 0 and Y ¼ 0 at y ¼ H or y ¼ 1; they are Y0 (0) ¼ Y(1) ¼ 0. For convenience of algebra, using a new independent variable η ¼ cosh ðωyÞ  1, one obtains

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Convective heat transfer in different porous passages

and

dY dY dη dY ¼ ¼ ½ω sinh ðωyÞ dy dη dy dη

(80)

 d2 Y d2 Y dY  2 ¼ 2 ½ω sinh ðωyÞ2 + ω cosh ðωyÞ 2 dη dη dy 2 d Y dY ¼ ω2 sinh 2 ðωyÞ 2 + ω2 cosh ðωyÞ dη dη 2   d Y dY ¼ ω2 ðη + 1Þ2  1 + ω2 ð1 + ηÞ dη dη2

(81)

Following appropriate substitution, the resulting differential equation is
2  d2 Y dY + ðη + 1Þ + ψ ð1  β  βηÞY ¼ 0 η + 2η 2 dη dη

(82)

It is interesting to note that the boundary conditions dY/dy ¼ 0 when y ¼ 0 or η ¼ 0 is automatically satisfied regardless of the value of dY/dη. The solution of this differential equation is a hypergeometric function and two different solutions are in the following sections. To obtain this solution by a standard technique, let ∞ X Y ðηÞ ¼ c n ηn (83a) n¼0

and then differentiate to get ∞ dY ðηÞ X c n nηn1 , ¼ dη n¼0

for n > 0

(83b)

and ∞ d2 Y ðηÞ X ¼ c n nðn  1Þηn2 , dη2 n¼0

for n > 1

(83c)

The substitution for Y(η) and its derivatives in the differential equation, Eq. (82), produces ∞ X

∞ X

 c n nðn  1Þ ηn + 2ηn1 + c n n ηn + ηn1

n¼2

+

∞ X n¼0

n¼1



c n ψ ð1  βÞη  βη n

n+1



¼0

(84)

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The examination of Eq. (83a) shows that c0 is a constant since Y(0) 6¼ 0. This leads to the condition that dY/dη 6¼ 0 when η ¼ 0 or c1 6¼ 0. Therefore, in the above equation, the terms that contain the parameter η0 take the following form, c 1  1  η0 + c 0 ψ ð1  βÞη0 ¼ 0

(85a)

which makes c1 + c0ψ (1  β) ¼ 0 or c1 ¼  c0ψ (1  β). Next, the examination of the coefficients that multiply by η1 shows that ½4c 2 + c 1 + 2c 2 + c 1 ψ ð1  βÞ  c 0 ψ β η1 ¼ 0

(85b)

This procedure is to be repeated for η values with exponents 2, 3, … and so on. In general, the examination of the coefficients that multiply ηn1 leads to the relation ½ðn  1Þðn  2Þcn1 + 2nðn  1Þcn + ðn  1Þcn1 + nc n + ψ ð1  βÞcn1  ψβcn2 ηn1 ¼ 0

(85c)

Since η 6¼ 0, then the solution for cn is n2  2n  n + 2 + n  1 + ψ ð1  βÞ ψ β c n1 + 2 c 2n2  2n  n 2n  2n  n n2 ðn  1Þ2 + ψ ð1  βÞ ψ β c n1 + ¼ c nð2n  1Þ nð2n  1Þ n2 (85d) In summary, the solution can be rewritten as cn ¼ 

Y ðyÞ ¼ 1 +

∞ X

c n ½ cosh ðω yÞ  1n

(86)

n¼1

In this formulation, c0 ¼ 1 is selected arbitrarily, then c1 ¼  c0ψ (1  β) and the recursive relation (85d) provide the remaining coefficients cn for n 2. It should be noted that there are other possible variations of this solution, e.g., η ¼ β cosh(ωy), η ¼ ½1  cosh ðωyÞ= cosh ðωÞ with 0  η  1. 5.2.2 Second solution As an alternative solution of this ordinary differential equation, Eq. (79), let Y ðηÞ ¼

∞ X n¼0

c n ηn

(87)

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Convective heat transfer in different porous passages

using a different independent variable η ¼ y . Moreover, Eqs. (83b) and (83c) provide the first and second derivatives for Y(η) in Eq. (87). Following substitution in the differential equation, Eq. (79), and, after removing the zero terms, one obtains ∞ X

c n nðn  1Þηn2 + ω2 ψ  ½1  β cosh ðωηÞ

∞ X

n¼2

c n ηn ¼ 0

(88)

n¼0

After substituting for β cosh ðωηÞ ¼

∞ X

ai ηi

(89a)

i¼0

where ( ai ¼

β ðωÞi i! 0

when i is even

(89b)

when i is even

in Eq. (88), it can be written as ∞ X

c n nðn  1Þηn2 + ω2 ψ 

n¼2

∞ X

c n ηn  ω2 ψ 

n¼0

∞ X

d n ηn ¼ 0

(90a)

n¼0

where dn ¼

n X

c j anj

(90b)

j¼0

The term that includes η0 suggests c0 ¼ constant ¼ 1 whereas the terms that include η1 require c1 ¼ 0 because of symmetry at y ¼ 0. Accordingly, all the terms with odd power vanish in this solution. The values of other constants are obtainable from the recursive relation c n+2 ¼ 

ω2 ψ ðc n  d n Þ ðn + 2Þ2  ðn + 2Þ

(91)

Since the terms with odd power vanish, the working relations are rearranged as Y ðηÞ ¼

∞ X n¼0

c n η2n

(92a)

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c 0 ¼ 1 and c n ¼ 

ω2 ψ ðc n1  dn1 Þ for n 1 4n2  2n

(92b)

Also, Eq. (89a) is rearranged to take the form ∞ X β β cosh ðωηÞ ¼ ðωηÞ2i ð 2i Þ! i¼0

(93a)

This causes the parameter ai in Eq. (89b) to take the form ai ¼

β ðωÞ2i ð2iÞ!

(93b)

while the constant dn remains as given by Eq. (90b). Once the solution for Y ðyÞ is known, the next step is the computation of the eigenvalues. The condition Y ðyÞ ¼ 0 at y ¼ η ¼ 1 leads toward accomplishing the task of finding the eigenvalues. Once the eigenvalues are known, the function Y m ðyÞ describes the function Y ðyÞ for the mth eigenvalue; that is, when λ ¼ λm in Eq. (76). A sample of eigenvalues, λ2m, for selected MDa coefficients is in [49] where the first eigenvalue yields the heat transfer coefficient for thermally fully developed flow using the Eq. (86a) to get hH/K ¼ λ21Dh/(4H). Following the computation of the eigenvalues, the thermal condition at the entrance location provides the constants Bm in Eq. (75). The use of a constant temperature at x ¼ 0; that is, θð0, yÞ ¼ 1, greatly simplifies the determination of Bm. As an intermediate step, it is necessary to utilize the orthogonality condition  Z 1  0 u Y m ðyÞY n ðyÞdy ¼ U Nm 0

when n 6¼ m when n ¼ m

(94a)

and a Graetz-solution type of analysis provides the norm Z 1  u ½Y m ðyÞ2 dy Nm ¼ U 0   ∂Y m ðyÞ 1 ∂Y m ðyÞ  ¼ 2λm ∂ y y¼1 ∂ λm y¼1

(94b)

Also, the utilization of the orthogonality condition leads toward the determination of the integral that reduces to

Convective heat transfer in different porous passages

Z 1  u Am ¼ Y ðyÞd y U m 0  1 ∂Y m ðyÞ ¼ 2 ∂ y y¼1 λm

135

(94c)

when integrating this equation over y. Finally, the use of the initial condition and orthogonality condition yields the coefficient Bm as Am Nm

  ∂Y m ðyÞ 2 ¼ = λm ∂ λm y¼1

Bm ¼

(95)

Brief samples of the norm Nm and the coefficient Am for different MDa values are shown in [49]. Additionally, the computed temperature values are also in [49]. The main objective of this study is to present the computed values of local and average heat transfer coefficients. For parallel plate channels, Lc is selected as H and then Eq. (65a) provides the local dimensionless heat transfer coefficient. Fig. 9A shows the discrete computed local dimensionless heat transfer coefficient hH/k data plotted vs x ¼ x=ðPe H Þ using 40 eigenvalues depending on the value of MDa. The data in a log–log plot show near linear behavior as x goes toward zero. For verification, the discrete data are compared with the acquired values using the weighted residual method. A mixed symbolic and numerical computation was used to accomplish this task. The computation of eigenvalues when m is large becomes demanding. Using Mathematica, a mixed symbolic and numerical procedure with a high degree of precision was written to perform the task of finding the eigenvalues. Next, the dimensionless average heat transfer coefficient hH=k , see Eq. (66b), is plotted vs x in Fig. 9B. 5.2.3 Circular pipes Consideration is given to heat transfer to a fluid passing through a porous medium bounded by an impermeable circular wall, see Fig. 8B. The procedure to obtain a temperature solution is similar to that described for the parallel plate channel. The second method used earlier is modified here. In cylindrical coordinates, the momentum equation is

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A

8 7 6 5 4

MDa=0 MDa=1/10000

3

hH/k

2

MDa=1/1000 MDa=1/100

101 8 7 6 5

MDa=1/10 MDa=∞ MDa=1

4

Nu=2.467

3 2

Nu=1.885 100 2

3 4 5 6 7 10-3

2

3 4 5 6 7 10-2

2

3 4 5 6 7 10-1

2

3 4 5 6 7 100

x = x/(HPe) B 102

MDa=0 MDa=1/10000

8 7 6 5 4

MDa=1/1000

hH/k

3

MDa=1/100

2

MDa=1/10 101 8 7 6 5 4

MDa=∞ MDa=1

Nu=2.467

3 2

Nu=1.885 100 10-4 2 3 4 56 10-3 2 3 4 56 10-2 2 3 4 56 10-1 2 3 4 567100

2 3 4 567 9

x = x/(HPe) Fig. 9 The dimensionless heat transfer coefficient in parallel plate ducts for different MDa values: (A) local values and (B) average values.

2  ∂p ∂u 1 ∂u μ  u ¼0 μe + ∂x r ∂r K ∂r 2

(96)

where r is the local radial coordinate and x is the axial coordinate, Fig. 8B. If the pipe radius is designated by rο and Lc ¼ ro, the dimensionless quantities

Convective heat transfer in different porous passages

137

defined for flow in parallel plate channels are repeated after some modifica
 tions; the modified quantities are: Lc ¼ ro, r ¼ r=r ο , u ¼ μ u= Φ r 2ο , Da ¼ K/r2ο, and ω ¼ (M Da)1/2. Then, the momentum equation reduces to

2  d u 1 du u  + +1¼0 (97) M r dr Da dr 2 Using the boundary condition u ¼ 0 at r ¼ 1 and the condition ∂u=∂r ¼ 0 at r ¼ 0, the solution becomes  I0 ðωr Þ (98) u ¼ Da 1  I0 ðωÞ where I0 ðωr Þ is the Bessel function. Here, as the mean velocity is defined by the relation Z 2 rο U¼ 2 urdr (99) rο 0 the velocity profile takes the form

 ω I0 ðωÞ I0 ðωr Þ u u 1 ¼ ¼ U U ω I0 ðωÞ  2I1 ðωÞ I0 ðωÞ

(100)

Using the computed values of U ¼ Da ½1  ð2=ωÞI1 ðωÞ=I0 ðωÞ, the relation

2 ð∂ p=∂ xÞDh 2 1 Dh f ¼ ¼ (101) ρU 2 =2 U Re D r o provides the pipe pressure drop. The steady-state form of the energy equation in cylindrical coordinates is

 ∂T k ∂2 T 1 ∂T (102) + ¼ u ∂x ρ c p ∂ r 2 r ∂r Defining the dimensionless temperature θ ¼ (T  Tw)/(Ti  Tw) where Ti is the inlet temperature and Tw is the wall temperature, one obtains u ∂θ ∂2 θ 1 ∂θ ¼ + r ∂r U ∂ x ∂ r2

(103)

where x ¼ x=ðPe r ο Þ and Pe ¼ ρ cprοU/k. As before, separating the variables, θðx, r Þ ¼ X ðxÞRðr Þ leads to solution of two ordinary differential equations, X 0 ðxÞ + λ2 X ðxÞ ¼ 0

(104a)

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A. Haji-Sheikh et al.

and R00 ðr Þ +

  1 0 u R ðr Þ + λ2 R ðr Þ ¼ 0 r U

(104b)

The major task remaining is to find the exact value of R(r ) and its verification by comparing it with the numerically obtained data. The parameter λ in Eq. (104a) serves as the eigenvalue. The final temperature solution, after the computation of eigenvalues, is θ¼

∞ X

Bm Rm ðr Þ eλm x 2

(105)

m¼1

Eq. (104b) following substitution for u/U takes the following form    ω I0 ðωÞ I ðωr Þ 1 1 0 Rðr Þ ¼ 0 (106) R00 ðr Þ + R0 ðr Þ + λ2 r ω I0 ðωÞ  2I1 ðωÞ I0 ðωÞ Using the abbreviations β ¼ 1=I 0 ðωÞ

(107a)

and ψ¼

I0 ðωÞ λ2 , ω I0 ðωÞ  2I1 ðωÞ ω

(107b)

in the following analysis, Eq. (106) is rewritten as R00 ðr Þ +

1 0 R ðr Þ + ω2 ψ ½1  β I0 ðωr Þ Rðr Þ ¼ 0 r

(107c)

Eq. (107c) is subject to the boundary conditions R0 (0) ¼ R(1) ¼ 0. Like the previous case, one can select η ¼ I0(ωr)  1; however, this selection did not produce sufficient simplification to warrant its implementation. Therefore, a direct derivation of a series solution will follow. To obtain this solution, let η ¼ r and then set RðηÞ ¼

∞ X

c n ηn

(108)

n¼0

then ∞ dRðηÞ X c n nηn1 , ¼ dη n¼0

for n > 0

(109a)

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Convective heat transfer in different porous passages

and ∞ d 2 RðηÞ X ¼ c n nðn  1Þηn2 , dη2 n¼0

for n > 1

(109b)

After removing the zero terms in Eqs. (109a) and (109b), the substitution R(η) and its derivatives in Eq. (107c) yields ∞ X

c n n ðn  1Þηn +

n¼2

∞ X

c n n ηn + ðωηÞ2 ψ  ½1  β I0 ðωηÞ

n¼1

∞ X

c n ηn ¼ 0

n¼0

(110) After substituting for ∞ X

β I0 ðωηÞ ¼

ai ηi

(111)

i¼0

Eq. (110) can be written as ∞ ∞ ∞ ∞ X X X X cn nðn  1Þηn + cn nηn + ω2 ψ  cn ηn + 2  ω2 ψ  dn ηn + 2 ¼ 0 n¼2

n¼1

n¼0

n¼0

(112a) where dn ¼

n X

c j anj

(112b)

j¼0

The term that includes η0 suggests c0 ¼ constant ¼ 1 whereas the terms that include η1 require c1 ¼ 0. Accordingly, all the terms with odd power vanish in the solution. The values of other constants are obtainable from the recursive relation c n+2 ¼ 

ω2 ψ ðc n + dn Þ ðn + 2Þ2

(113)

Since the terms with odd power vanish, the working equation may be rewritten as ∞ X RðηÞ ¼ c n η2n (114a) n¼0

c 0 ¼ 1,

cn ¼ 

ω ψ ðc n1  dn1 Þ 4n2 2

for n 1

(114b)

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A. Haji-Sheikh et al.

Moreover, the relation ∞   X β ωη 2i 2 2 i¼0 ði!Þ

β I0 ðωηÞ ¼

(115a)

makes ai ¼

  β ω 2i ði!Þ2 2

(115b)

and the constant dn remains as d2n ¼

n X

c j anj

(115c)

j¼0

for insertion in this recursive relation. As for the previous case, the solution for Rðr Þ must satisfy the condition Rðr Þ ¼ 0 when r ¼ η ¼ 1. This condition leads toward the computation of eigenvalues. A sample of computed eigenvalues is in [3.49]. Accordingly, the function Rm ðr Þ, in Eq. (105), describes the function Rðr Þ for the mth eigenvalue; that is, when λ ¼ λm also in Eq. (105). To compute Bm for inclusion in Eq. (105), one needs to use the thermal condition at the entrance location; that is θð0, yÞ ¼ 1 at x ¼ 0 in this study. The analysis leads to the following orthogonality condition  Z 1  0 when n 6¼ m u rRm ðr ÞRn ðr Þdr ¼ (116a) U N m when n ¼ m 0 where the norm Nm assumes the following form Z 1  u Nm ¼ r ½Rm ðr Þ2 dr U 0   ∂Rm ðr Þ 1 ∂Rm ðr Þ ¼  ∂r ∂λm r¼1 2λm r¼1

(116b)

Also, the orthogonality condition leads to the determination of the second integral Z 1  u Am ¼ rRm ðr Þdr U 0  (116c) 1 ∂Rm ðr Þ ; ¼ 2 ∂r λm r¼1 obtained by integrating Eq. (104b) after replacing λ with λm and R with Rm.

141

Convective heat transfer in different porous passages

Finally, the coefficients Bm for inclusion in Eq. (105) is Am Nm

  ∂Rm ðr Þ 2 = ¼ ∂λm r¼1 λm

Bm ¼

(117)

To facilitate the computation of Bm from Eq. (117) for inclusion in Eq. (105), the coefficients Am and Nm were computed for a selected number of eigenvalues. Following the determination of temperature, the local and average heat transfer coefficients presented in Fig. 10. For circular pipes, ro replaces the characteristic length Lc and the compute fluid temperature using Eq. (84). Then, the computed bulk temperature leads toward the determination of heat transfer coefficients. Fig. 10A shows the computed discrete dimensionless local heat transfer coefficient hro/k data plotted vs x using 40 eigenvalues. As in the previous case, the data, in a log–log plot, show near linear behavior as x goes toward zero. Fig. 10B shows the dimensionless average heat transfer coefficient hr o =k plotted vs x ¼ x=ðPe r ο Þ. 5.2.4 An alternative method of solution There are ducts for which the temperature solution does not accept the separation of variables technique. For these ducts, it is possible to get a temperature solution similar to that for transient heat conduction in Cole et al. [23, chapter 10] when, the axial coordinate, x replaces time t; the mathematical process is in [42]. This type of analysis leads to a Green’s function solution method that uses the weighted residuals technique. This method is tested for verification of exact series solution data for parallel and circular ducts. The method of solution presented here equally applies to ducts described earlier and it can be extended to accommodate the ducts having various cross section shapes. As significant advantages, the computation of eigenvalues is automatic, and the computation of other coefficients is also automatic. In this technique, for preselected M eigenvalues, the proposed solution is a modification of Eq. (75), that is, θ¼

∞ X m¼1

Bm Ψm ðyÞeλm x 2

(118a)

142

A. Haji-Sheikh et al.

A 8 7 6 5 4

MDa=0 MDa=1/10000

3

MDa=1/1000

2

hro / k

MDa=1/100 10

1

MDa=1/10

8 7 6 5 4

MDa=∞ MDa=1

Nu=2.892

3 2

Nu=1.828 100

2

3 4 5 67 10-3

2

3 4 5 67 10-2

2

3 4 5 67 10-1

2

3 4 5 67 100

x/(roPe) B

Fig. 10 The dimensionless heat transfer coefficient in circular pipes for different MDa values: (A) local values and (B) average values.

where Ψm ðηÞ ¼

N X

dmj f j ðηÞ

(118b)

j¼1

and the function fj(η) are selected so that they satisfy the homogeneous boundary conditions along the surface of the ducts; that is,

143

Convective heat transfer in different porous passages


 f j ðηÞ ¼ 1  η2 η2ð j1Þ

(118c)

For a parallel plate channel, Ψm(η) stands for Ym(η) and, for a circular pipe, Ψm(η) stands for Rm(η). The remaining steps apply equally to both cases with minor modifications to be identified later. As described in Cole et al. [23, chapter 10], the next task is the computation of eigenvalues λ2m and the coefficients dmj in Eq. (118b). The computation begins by finding the elements of two matrices, A and B, Z 1 aij ¼ fi ðηÞr2 fj ðηÞJdη (119a) 0

and Z bij ¼

1

ρcp

u

0

U

fi ðηÞfj ðηÞJdη

(119b)

The parameter J in Eqs. (119a) and (119b) is the Jakobian; J ¼ 1 for parallel plate channels and J ¼ η is for circular ducts. This analysis leads to an eigenvalue problem [49].
 A + λ2 B d ¼ 0 (120a) that can be rewritten as.
1  B A + λ2 I d ¼ 0

(120b)

The symbolic software Mathematica was used to produce the elements of matrices A and B and subsequently the eigenvalues λ2m and the corresponding coefficients dmj embedded within the eigenvectors d. These eigenvectors become the rows of a matrix designated as D. The basic Mathematica statements to accomplish this task, when using Eqs. (120a) and (120b) are in [49]. Accordingly, as discussed in [49], the coefficients Bm for inclusion in Eq. (118a) are obtainable from the relation [23, eq. (10.53)], Bm ¼

N X i¼1

Z

1

pmi 0

ρcp

u U

fi ðηÞdη

The parameters pmi in Eq. (121) are the elements of a matrix  1 P ¼ ðD  BÞT ,

(121)

(122)

that is, the matrices D multiplied by matrix B and the resulting matrix is transposed and then inverted. Following computation of D and P, the

144

A. Haji-Sheikh et al.

Green’s function is readily available to include the effect of frictional heating, see the Green’s function solution in [42]. This task can be performed conveniently using Mathematica. This numerical computation of temperature can be extended to passages having various shapes, e.g., triangular ducts. The needed modifications are described in [23, chapter 10]. Following the computation of temperature, the next task is to compute local and average heat transfer coefficients for fluid flowing through parallel plate ducts and circular pipes within the range of 106  (x/Dh)/ (ReDPr) < ∞ where Dh ¼ 4A/C, ReD ¼ ρUDh/μ, and Pr ¼ μ cp/k. As many as 50 eigenvalues did not provide results with sufficient accuracy using the exact analysis; however, 40 eigenvalues did yield relatively accurate results using this alternative solution. The data in Table 3 show excellent agreement between computed local heat transfer coefficients within a small range of the dimensionless axial coordinate, see columns 3 and 4. Also, these two completely different methods of solution yield the average heat transfer coefficients in columns 5 and 6 that are in excellent agreement.

5.3 Porous ducts with axial conduction The porous materials with relatively high thermal conductivity have small Peclet numbers and the contribution of axial conduction is significant. Nield et al. [46] numerically studied the contribution of axial conduction for flow through a porous medium located between two parallel plates. A similar study for flow through circular porous passages is in [47]. The study in this section is devoted to the presentation of the exact series solutions for selected flow passage by using a Graetz-type analysis that includes the effect of axial conduction. The mathematical procedures using series solutions are presented in [37,44]; however, some numerical modifications become necessary. 5.3.1 Parallel plate ducts This model considers a steady and hydro-dynamically fully developed flow between two impermeable parallel plates, 2H apart, as depicted in Fig. 7A. The classical Brinkman momentum equation described earlier by Eqs. (67a) and (67b), applies for determination of steady state and fully developed velocity in these channels. The velocity solution as given by Eq. (70b) is  cosh ðωyÞ u u ω ¼ ¼ 1 (123) U U ω  tanh ðωÞ cosh ðωÞ where ω 5 (M Da)1/2 and y5y=H.

145

Convective heat transfer in different porous passages

Table 3 Comparison of computed data for parallel plate channels and for cylindrical pipes.     ðx=Dh Þ MDa [NuD]Exa [NuD]WRb NuD Ex a NuD WR b [θb]Exact Re D Pr

Comparison of computed data for parallel plate channels 102

104 5  10 10

4

3

5  10

3

102 5  10 10

2

1

0.5 103

104 5  10

4

103 5  10 10

2

5  10 10

3

2

1

0.5 104

104 5  10

4

103 5  10 10

3

2

5  102 10

1

0.5

38.074

38.074

58.383

58.383

0.97692

21.634

21.634

33.275

33.275

0.93562

17.053

17.053

26.127

26.127

0.90077

10.516

10.516

15.249

15.249

0.73714

9.3025

9.3025

12.502

12.502

0.60648

8.9626

8.9626

9.6932

9.6932

0.14390

8.9626

8.9626

9.3279

9.3279

0.2397  101

8.9626

8.9626

9.0356

9.0356

0.1418  107

48.638

48.637

77.477

77.479

0.96948

25.806

25.806

42.049

42.048

0.91934

19.672

19.672

32.143

32.143

0.87935

11.377

11.377

17.529

17.529

0.70427

9.9424

9.9424

14.001

14.001

0.57118

9.5605

9.5605

10.474

10.474

0.12311

9.5605

9.5605

10.017

10.017

0.1819  101

9.5605

9.5605

9.6518

9.6518

0.4136  10–8

56.280

56.271

96.297

96.319

0.96221

27.624

27.623

48.570

48.572

0.90743

20.587

20.587

36.036

36.037

0.86576

11.625

11.625

18.630

18.630

0.68894

10.149

10.149

14.661

14.661

0.55630

9.7710

9.7710

10.773

10.773

0.11594

9.7710

9.7710

10.272

10.272

0.1643  101

9.7710

9.7710

9.8712

9.8712

0.2667  108 Continued

146

A. Haji-Sheikh et al.

Table 3 Comparison of computed data for parallel plate channels and for cylindrical pipes.—cont’d     ðx=Dh Þ MDa [NuD]Exa [NuD]WRb NuD Ex a NuD WR b [θb]Exact Re D Pr

Comparison of computed data for cylindrical pipes 102

5  104

17.447

17.447

26.938

26.938

0.94755

13.645

13.645

21.093

21.093

0.91909

7.8710

7.8710

11.999

11.999

0.78665

6.3810

6.3810

9.4985

9.4985

0.68390

4.8349

4.8349

6.0584

6.0584

0.29770

4.7905

4.7905

5.4293

5.4293

0.11398

0.5

4.7899

4.7899

4.9178

4.9178

0.5351  104

1

4.7899

4.7899

4.8538

4.8538

0.3699  108

22.272

22.251

35.482

35.541

0.93149

17.010

17.002

27.354

27.377

0.89636

9.2264

9.2255

14.828

14.830

0.74338

7.3122

7.3119

11.464

11.465

0.63221

5.4677

5.4677

6.9970

6.9972

0.24674

5.4277

5.4277

6.2165

6.2166

0.8319  101

0.5

5.4273

5.4273

5.5852

5.5852

0.1409  104

1

5.4273

5.4273

5.5062

5.5062

0.2721  10–9

25.650

25.647

43.943

43.951

0.91587

18.942

18.941

32.842

32.845

0.87690

9.7254

9.7253

16.628

16.628

0.71709

7.6266

7.6265

12.556

12.556

0.60518

5.7043

5.7043

7.4093

7.4093

0.22721

101

5.6685

5.6685

6.5425

6.5425

0.7302  101

0.5

5.6682

5.6682

5.8431

5.8431

0.8409  105

1

5.6682

5.6682

5.7557

5.7557

0.1003  109

10

3

5  10 10

3

2

5  102 10

103

1

5  104 10

3

5  103 10

2

5  10 10

104

2

1

5  104 10

3

5  103 10

2

5  10

a

2

Nusselt number from exact analysis. Nusselt number using method of weighted residuals.

b

147

Convective heat transfer in different porous passages

The temperature distribution assuming obtainable from the energy equation within

∂T k ∂2 T + ¼ u ∂x ρc p ∂x2

local thermal equilibrium is a differential element,  ∂2 T (124) ∂y2

Therefore, the temperature and thermophysical properties have effective values. Introducing the Peclet number Pe ¼ ρcpHU/k where U is the mean velocity in the duct, and dimensionless x ¼ x=ðPeH Þ, the energy equation, Eq. (124), becomes d2 T u ∂T 1 d2 T  ¼ U ∂x Pe2 dx2 dy2

(125)

It is hypothesized that there is an unheated section in a passage, wherein the wall temperature is Tw ¼ T1, before the heated section. As shown in Fig. 11, the wall temperature of the heated section is designated as Tw ¼ T2. When T1 and T2 in Fig. 11 are constants, one can select two different dimensionless temperatures, θ1 ¼ (T  T1)/(T1  T2) when x < 0 and θ2 ¼ (T  T2)/ (T1  T2) when x > 0. Then, the partial differential equation, Eq. (125) for these two regions takes the form d2 θi u ∂θi 1 d 2 θi  ¼ U ∂x Pe2 dx2 dy2

For i ¼ 1 or 2

(126)

The solution of the partial differential Eq. (126) is obtainable using the method of separation of variables by setting θðx, yÞ ¼ X ðxÞY ðyÞ. The substitution of this functional form, for θ, in Eq. (126) yields

Fig. 11 Schematic of a duct with prescribed surface temperatures.

148

A. Haji-Sheikh et al.

Y 00 u X0 1 X 00 ¼  2 Y U X Pe X

(127)

Because u ¼ u(y) on the right side of Eq. (127), it is possible to separate the variables if one considers the relation X 0 ðxÞ ¼ λ2 X ðxÞ and then differentiate it to get X 00 ðxÞ ¼ λ2 X ðxÞ ¼ λ4 X ðxÞ. This leads to the following differential equation for computation of Y ðyÞ, Y 00 ðyÞ  λ2

  u λ4 Y ðyÞ + 2 Y ðyÞ ¼ 0 U Pe

(128)

The parameter λ is the eigenvalue in this equation and it is the solution
 of the differential equation X 00 ðxÞ + λ4 X ðxÞ ¼ 0; that is, exp λ2 x and
 exp λ2 x . This methodology leads to the following temperature solutions, θi ¼

∞ h i X 2 2 Am eλm x + Bm eλm x Y m ðyÞ

for i ¼ 1 or 2:

(129)

m¼1

In this equation θ1 is finite as x !  ∞ and θ2 is finite as x ! + ∞. Therefore, for x < 0, the eigenvalue λm is real and Bm ¼ 0. However, when x > 0 the coefficient Am ¼ 0 and λm in Eq. (128) becomes imaginary. This leads to the following two relations in which λ2m is real for both θ1 and θ2, θ1 ¼

∞ X

Am Y m ðyÞeλm x

when x < 0

(130a)

Bm Y m ðyÞeλm x

when x > 0

(130b)

2

m¼1

and θ2 ¼

∞ X

2

m¼1

Because the computation of θ2 is the main objective of this study, Eq. (128) is utilized with imaginary λm. Once the value of u/U from Eq. (123) is placed in Eq. (128), it becomes    cosh ðωyÞ ω λ4 00 2 Y ðyÞ + λ 1 Y ðyÞ + 2 Y ðyÞ ¼ 0 ω  tanh ðωÞ cosh ðωÞ Pe (131) Next, consideration is given to the exact and numerical solutions of this ordinary differential equation using a methodology similar to that in [42,49]. For simplicity of this presentation, let ψ¼

1 ω½ω  tanh ðωÞ

(132a)

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Convective heat transfer in different porous passages

and β¼

1 cosh ðωÞ

(132b)

then Eq. (131) reduces to   Y 00 ðyÞ + λ4 =Pe2 + ω2 λ2 ψ½1  β cosh ðωyÞY ðyÞ ¼ 0

(133)

This differential equation has the form of a modified Mathieu differential equation and the solution for this special Mathieu differential equation is described below. 5.3.2 Solution The solution of Eq. (133) has a hypergeometric form, Y ðηÞ ¼

∞ X

c n ηn

(134)

n¼0

wherein the independent variable η ¼ y ¼ y=H . Following substitution Y(y) ¼ Y(η) from Eq. (134) in Eq. (133) and after removing the zero terms, the results is ∞ X c n nðn  1Þηn2 + ω2 λ2 ψ½1  βcoshðωηÞ n¼2



∞ X

∞
X c n η + λ4 =Pe2 c n ηn ¼ 0

(135)

n

n¼0

n¼0

Next, using the relation β cosh ðωηÞ ¼

∞ X

ai ηi

(136)

i¼0

where ( ai ¼

β ðωÞi i! 0

when i is even

(137)

when i is odd

further reduces Eq. (135) and it can be written as ∞ X


 c n nðn  1Þηn2 + λ4 =Pe2 + ω2 λ4 ψ

n¼2



∞ X n¼0

cnη  ω ψ n

2

∞ X n¼0

(138) dn η ¼ 0 n

150

A. Haji-Sheikh et al.

where dn ¼

n X

c j anj

(139)

j¼0

The term that includes η0 suggests c0 ¼ constant ¼ 1 whereas the terms that include η1 require c1 ¼ 0 because of symmetry at y ¼ 0. Accordingly, all the terms with odd power vanish in this solution. When n > 1, the constants are obtainable from the recursive relation  λ4 c n =Pe2 + ω2 λ2 ψ ðc n  d n Þ c n+2 ¼  (140) ðn + 2Þ2  ðn + 2Þ When Pe ! ∞, this relation reduces to the form presented in [49,50]. Following the determination of cn, Eq. (134) yields the solution for Y ðyÞ and the next step is the computation of the eigenvalues. The eigencondition Y(1) ¼ 0 when y ¼ η ¼ 1 leads toward accomplishing the task of finding the eigenvalues. Once the mth eigenvalue is known, the function Y m ðyÞ for the mth eigenvalue replaces Y ðyÞ in Eq. (134) for subsequent insertion in Eqs. (130a) and (130b). Following the computation of the eigenvalues, the thermal compatibility conditions at x ¼ 0 location would provide the constants Am and Bm also for inclusion in Eqs. (130a) and (130b). According to the definitions of θ1 ¼ (T  T1)/(T1  T2) and θ2 ¼ (T  T2)/(T1  T2), the first compatibility condition at x ¼ 0 is θ2  θ1 ¼ ðT  T 1 Þ=ðT 1  T 2 Þ  ðT  T 2 Þ=ðT 1  T 2 Þ ¼ 1

(141)

and the next compatibility condition is ∂θ1 =∂xjx¼0 ¼ ∂θ2 =∂xjx¼0

(142)

The method of determination of Am and Bm using these compatibility conditions is combined with that for circular pipes and it is presented next. 5.3.3 Circular ducts The next presentation concerns the determination of temperature field in a fluid passing through a porous medium bounded by an impermeable circular wall. The mathematical procedure is similar to that described for the parallel plate channel. In cylindrical coordinates, the momentum equation is given by Eq. (97) and the velocity profile by Eq. (100) as

Convective heat transfer in different porous passages

 ωI0 ðωÞ I0 ðωr Þ u u 1 ¼ ¼ U U ωI0 ðωÞ  2I1 ðωÞ I0 ðωÞ

151

(143)

where ω ¼ (M Da)1/2, r ¼ r=r O and I0 ðωr Þ is the Bessel function. Because the velocity is fully developed, the steady-state form of the energy equation, in cylindrical coordinates, is

 ∂T k ∂2 T 1 ∂T ∂2 T u + (144) ¼ + ∂x ρc p ∂r 2 r ∂r ∂x2 Defining the dimensionless temperature θi ¼ (T  Ti)/(T1  T2) where i ¼ 1 or 2, one obtains u ∂θi ∂2 θi 1 ∂θi 1 ∂2 θ ¼ 2 + + 2 2i r ∂r U ∂x ∂r Pe ∂x

(145)

where x ¼ x=ðPe r O Þ and Pe ¼ ρcprOU/k. To use the method of separation of
 variables, let θ1 ðx, r Þ ¼ Rðr Þ exp λ2 x and following its insertion in Eq. (145), one obtains

4   1 0 λ 00 2 u R ðr Þ ¼ 0 (146) R ðr Þ + R ðr Þ  λ Rðr Þ + r U Pe2 and λ in Eq. (146) is imaginary for θ2. The next task is the determination of the function R(r ) with emphasis on θ2. The parameter λ in Eq. (146) serves as the eigenvalues. Eq. (146) following substitution for u/U takes the following form

4    1 0 ωI0 ðωÞ I0 ðωr Þ λ 00 2 R ðr Þ+ R ðr Þ+ λ 1 Rðr Þ ¼ 0 Rðr Þ+ ωI0 ðωÞ  2I1 ðωÞ I0 ðωÞ r Pe2 (147) Using the abbreviations ψ¼

I0 ðωÞ 1 ωI0 ðωÞ  2I1 ðωÞ ω

(148a)

β ¼ 1=I 0 ðωÞ,

(148b)

and

Eq. (147) becomes R00 ðr Þ +

1 0 R ðr Þ + ω2 λ2 ψ½1  βI0 ðωr ÞRðr Þ + λ4 Rðr Þ=Pe2 ¼ 0 r

(148c)

152

A. Haji-Sheikh et al.

The solution to Eq. (148c) should satisfy the boundary conditions R0 (0) ¼ R(1) ¼ 0. The derivation of an exact series solution for Eq. (148c) is presented below and the procedure is similar to that in [49] or [50]. 5.3.4 Solution To obtain the solution, let η ¼ r ¼ r=r O and then set ∞ X

RðηÞ ¼

c n ηn

(149)

n¼0

then ∞ dRðηÞ X c n nηn1 , ¼ dη n¼0

for n > 0

(150a)

and ∞ d2 RðηÞ X ¼ c n nðn  1Þηn2 , dη2 n¼0

for n > 1

(150b)

After removing the zero terms in Eqs. (150a) and (150b), the substitution R(η) and its derivatives in Eq. (148c) yields ∞ X n¼2

c n nðn  1Þηn +

∞ X

c n nηn + ðωηÞ2 λ2 ψ

n¼1

∞ X

∞ η2 λ4 X n ½1  βI0 ðωηÞ cnη + cnη ¼ 0 Pe2 n¼0 n¼0

(151)

n

Following substitution for βI0 ðωηÞ ¼

∞ X

ai ηi

(152)

i¼0

Eq. (151) can be written as ∞ X

c n nðn  1Þηn +

n¼2

∞ X

c n nηn + ω2 λ2 ψ 

n¼1

ω λ ψ  2 2

∞ X n¼0

∞ X n¼0

dn η

n+2

c n ηn+2

∞ λ X n+2 + 2 cnη ¼ 0 Pe n¼0 4

(153a)

153

Convective heat transfer in different porous passages

where dn ¼

n X

c j anj

(153b)

  β ω 2i 2 2 ði!Þ

(153c)

j¼0

and ai ¼

The term that includes η0 suggests c0 ¼ constant ¼ 1 whereas the terms that include η1 require c1 ¼ 0. Accordingly, all the terms with odd power vanish in the solution. The values of other constants are obtainable from the recursive relation c n+2 ¼ 

λ4 c n =Pe2 + ω2 λ2 ψðc n + dn Þ ðn + 2Þ2

(154)

As Pe ! ∞, this equation reduces to the form presented in [49,50]. As for the previous case, the solution for Rm ðr Þ must satisfy the eigencondition Rm(1) ¼ 0 when r ¼ η ¼ 1 . This condition leads toward the computation of eigenvalues and the function Rm ðr Þ describes the function Rðr Þ for the mth eigenvalue; that is, when λ ¼ λm. The final temperature solutions, after the computation of eigenvalues, has the form θ1 ¼

∞ X

Am Rm ðr Þeλm x

when x < 0

(155a)

Bm Rm ðr Þeλm x

when x > 0

(155b)

2

m¼1

and θ2 ¼

∞ X

2

m¼1

To compute Am or Bm for inclusion in Eq. (155a) and (155b), one needs to use the compatibility thermal conditions as given by Eqs. (141) and (142). The method of determination of coefficients Am or Bm for circular pipes is similar to that for parallel plate channels; therefore, they are combined in the following subsection.

154

A. Haji-Sheikh et al.

5.4 Orthogonality condition and the determination of coefficients The methodology presented in this section is similar to that in [51,52]. The modified formulations, presented here, are to assure the completeness of this presentation. Using the eigenvalues λ2m and λ2n in Eq. (128) or (146), the differential equation for both parallel plate and circular passages may be written as   2 uðηÞ 1 d e dψm ðηÞ 2 λm + λ  (156a) η ψm ðηÞ ¼ 0, m ηe dη dη U Pe2  2  uðηÞ 1 d e dψn ðηÞ 2 λn + λ  (156b) ψn ðηÞ ¼ 0: η n U ηe dη dη Pe2 In these two relations ψ stands for Y in Eq. (128) for parallel plate channels if the exponent e assumes a value of 0 and ψ stands for R in Eq. (146) for circular pipes if e ¼ 1. Eqs. (156a) and (156b) may be written as   2 2 1 d e dψm ðηÞ uðηÞ λ2m λ2n 2 λm + λn + λ  ð η Þ  ψ ðηÞ ¼ 0, (157a) η ψ m m ηe dη dη U Pe2 Pe2 m   2 2 λ2m λ2n uðηÞ 1 d e dψn ðηÞ 2 λm + λn + λ  ð η Þ  ψ ðηÞ ¼ 0 (157b) η ψ n n ηe dη dη U Pe2 Pe2 n by adding and subtracting the last terms in these equations. Multiply both sides of Eq. (157a) by ηeψn(η) and both sides of Eq. (157b) by ηeψm(η) and, then, subtracting the resulting relations would leads to   d e dψm ðηÞ d e dψn ðηÞ η η ψn ðηÞ  ψm ðηÞ dη dη dη dη (158)  2 2
2  uðηÞ e 2 λm + λn + λm  λn  η ψn ðηÞψm ðηÞ ¼ 0: U Pe2 The integration of this equation from 0 to 1 and then integrating the first two terms by parts yields, Z 
2  1 λ2m + λ2n uðηÞ e 2 λm  λn  (159a) r ψn ðηÞψm ðηÞdη ¼ 0 U Pe2 0 Therefore, the orthogonality condition is  Z 1 2 0 λm + λ2n uðηÞ e  ψn ðηÞψm ðηÞη dη ¼ 0 2 U Pe Nm 0

when n 6¼ m when n ¼ m

(159b)

155

Convective heat transfer in different porous passages

for parallel plate channels when e ¼ 0 and for circular pipes when e ¼ 1, as stated earlier. This orthogonality condition is to be used to compute the coefficients Am and Bm from the compatibility condition given by Eqs. (141) and (142). Because the eigenfunctions and the eigenvalues in x + side and x  side appear in different summations, it is appropriate to identify θ1 and θ2 by using different symbols, e.g., when x < 0, ∞ X

θ1 ¼

Am f m ðηÞeβm x

with β2m > 0

(160a)

Bm gm ðηÞeλm x

with λ2m > 0

(160b)

2

m¼1

and when x > 0 θ2 ¼

∞ X

2

m¼1

The eigenfunctions fm(η) and gm(η) in Eqs. (160a) and (160b) are solutions of the following differential equations,  2  uðηÞ d e df m ðηÞ 2 βm + βm 2  f m ðηÞηe ¼ 0, η U dη dη Pe  2  uðηÞ d e dgn ðηÞ 2 λn + λn 2  η g ðηÞηe ¼ 0: dη U n dη Pe

(161a) (161b)

The substitution of θ1 and θ2 from Eqs. (160a) and (160b) into Eqs. (141) and (142) leads to the relations for the determination of the coefficients Am and Bm. The condition θ1 ¼ θ2 at x ¼ 0 yields ∞ X

Bm gm ðηÞ ¼ 1 +

m¼1

∞ X

Am f m ðηÞ

(162a)

m¼1

and the relation ∂θ1 =∂xjx¼0 ¼ ∂θ2 =∂xjx¼0 yields ∞ X m¼1

λ2m Bm gm ðηÞ ¼

∞ X

β2m Am f m ðηÞ

(162b)

m¼1

Following the application of the orthogonality condition and other algebraic manipulations in [51,52], the coefficients are

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Z 1h

+

uðηÞ U

2β2n Pe2

+

uðηÞ U

An ¼ Z 0 1 h 0

i

β2n Pe2

i

f n ðηÞηe dη (163a) f 2n ðηÞηe dη

that becomes An ¼ and

2 1 βn ½df n ðηÞ=dβn η¼1

Z 1h Bn ¼ Z 0 1 h 0

λ2n Pe2

2λ2n Pe2

(163b)

i gn ðηÞηe dη

+

uðηÞ U

+

uðηÞ U

i

(164a) g2n ðηÞηe dη

that becomes Bn ¼ 

2 1 λn ½dgn ðηÞ=dλn η¼1

(164b)

It is remarkable that An, Eq. (163b), depends only on the parameters within the θ1 solution and Bn, Eq. (164b), depends only on the parameters within the θ2 solution. The temperature solution, using the aforementioned exact analysis provides accurate results when x is relatively large. However, the solution may require many eigenvalues at small values of x. Once the temperature solution is known, the heat transfer coefficient h ¼ qw/(T2  Tb) is obtainable following the determination of the wall heat flux qw ¼ (∂T/∂r)jr¼r and the bulk temperature Tb. It can be written in dimensionless form as Nu ¼ hL c =ke  ∞  1 X dψðηÞ ¼ Bm  θb m¼1 dη 

eλm x 2

(165)

η¼1

where

Tb  T2 T1  T2 Z 1  ∞ X u λ2m ¼ ð1 + eÞ Bm e ψm ðηÞηe dη U 0 m¼1 "  # ∞ 2 Z 1 X  2 λ ð Þ dψ η 1  Bm eλm  m2 ψm ηe dη  2 ¼ ð1 + eÞ  dη Pe λ 0 m m¼1 η¼1

θb ¼

(166)

Convective heat transfer in different porous passages

157

As stated earlier, for parallel plate channels, the parameter e ¼ 0 and ψm(η) stands for Ym ðyÞ in Eq. (130b) and θb stands for θ2,b. For circular pipes, the parameter e ¼ 1, ψm(η) stands for Rm ðyÞ in Eq. (155b), and θb stands for θ2,b. For parallel plat ducts and circular pipes, this numerical study evaluates the exact series solution in the presence of axial conduction. Three different M Da values are selected, 1, 103, and 105, for each case. This will permit one to observe the influence of Pe at small, intermediate, and large values ofM Da. For each c value, the data for Pe values of 1, 2, 5, 10, and ∞ are presented. 5.4.1 Parallel plate channels For parallel plate channels, data are acquired for the Nusselt number Nu ¼ hH/ke and the bulk temperature θb ¼ (Tb  T2)(T1  T2) for three selected M Da values. It is customary to show the variation of Nusselt number as a function of (x/H)/Pe while having Pe as a parameter. Fig. 12A shows the Nusselt number and Fig. 12B provides the bulk temperature as a function of (x/H)/Pe for M Da ¼ 105. Similarly, Figs. 13 and 14 display the variation of the Nusselt number and the bulk temperature when M Da ¼ 103, and 1. It is to be noted that both values of M Da and θb are needed in order to determine the heat flux values at the wall; that is qw H ¼ Nu  θb ke ðT 2  T 1 Þ

(167a)

5.4.2 Circular pipes The data for circular pipes are also acquired for Nu ¼ hro/ke and θb when M Da ¼ 105, 103, and 1. Fig. 15A displays the variation of Nusselt number as a function of (x/H)/Pe when Pe ¼1, 2, 5, 10, and ∞ while Fig. 15B provides the corresponding values of the bulk temperature θb. Moreover, Figs. 16 and 17 demonstrate the influence of M Da values of 103 and 1, respectively. Again, the values of Nu and θb are needed in order to determine the heat flux values at the wall; that is qw r o ¼ Nu  θb ke ðT 2  T 1 Þ

(167b)

It is to be noted that for small values of Pe numbers, the spacing between the adjacent eigenvalues will reduce significantly. This indicates that a larger number of eigenvalues should be computed in order to achieve a desired accuracy. Accordingly, as Pe decreases, it is necessary to compute a larger

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A

B

Fig. 12 Local Nusselt number and bulk temperature for parallel plate channels when M Da ¼ 105.

number of eigenvalues for a comparable accuracy. Accurate computation of eigenvalues is an essential part of this study. All computations were performed symbolically using Mathematica.

5.5 Frictional heating effects In previous subsections, it was assumed that the effect of viscous dissipation was negligible. The method of including the effects of frictional heating and

Convective heat transfer in different porous passages

159

A

B

Fig. 13 Local Nusselt number and bulk temperature for parallel plate channels when M Da ¼ 103.

a few mathematical details are the objects of this subsection. The energy equation with frictional heating for parallel plate channels is "

2 # ∂2 T i μu2 ∂u ke 2 + + μe K ∂y ∂y (168a) ∂T i ∂2 T i  ke 2 , for i ¼ 1 and 2 ¼ Cu ∂x ∂x

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A

B

Fig. 14 Local Nusselt number and bulk temperature for parallel plate channels.

that in dimensionless form becomes d 2 θi + Br dy2

"

2 #  2 ∂ðu=U Þ u + M Da ∂y U

u ∂θi 1 d2 θ  2 2i , ¼ U ∂x Pe dx

for i ¼ 1 and 2

(168b)

Convective heat transfer in different porous passages

161

A

B

Fig. 15 Local Nusselt number and bulk temperature for circular pipes when M Da ¼ 105.

where Br ¼ μU2/[Da ke(T1  T2)] is the Brinkman number. All the boundary conditions remain the same as those described earlier. The transformation to convert Eq. (168b) to Eq. (126) is readily available in [41, eq. (32)]; it is θi, s ðyÞ ¼

 2
 ω2 2 2 2ω 1  y  8½1  cosh ðωyÞ= 4½ω  tanh ðωÞ cosh ðωÞ + ½ cosh ð2ωÞ  cosh ð2ωyÞ=cosh 2 ðωÞg

(169)

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A

B

Fig. 16 Local Nusselt number and bulk temperature for circular pipes when M Da ¼ 103.

Therefore, adding Br θi,s ðy, Þ to θi ðy, xÞ is required in order to include the effect of viscous dissipation S. Subsequently, the application of the Fourier heat conduction at the wall leads to the relation  ∂θi,s ðyÞ ω  ¼ ∂ y y¼1 ω  tanh ðωÞ

(170)

that determines the contribution of frictional heating to the wall heat flux. As an example, for M Da¼ 105, 103, and 1, Eq. (170) yields θi,s0 (1) ¼

Convective heat transfer in different porous passages

163

A

B

Fig. 17 Local Nusselt number and bulk temperature for circular pipes when M Da ¼ 1.

1.00317, 1.03266, and 4.19453. The contribution of the viscous dissipation to the bulk temperature is readily available in [41, eq. (30a)],  3 ω2 1 θb,s ðωÞ ¼ 48 ωcoshðωÞ  sinh ðωÞ 
 
12ω3  150ω cosh ðωÞ + 4ω3  30ω  cosh ð3ωÞ + 57 sinh ðωÞ + 41 sinh ð3ωÞ

(171)

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A. Haji-Sheikh et al.

Therefore, the dimensionless bulk temperature θb should be augmented by the quantity Br θb,s. Similar equations, but with different form are also available in [46]. For M Da ¼ 105, 103, and 1 used in Figs. 12 and 14, Eq. (171) provides θb,s ¼0.33649, 0.36443, and 1.26162. The process is similar for circular pipes. The governing diffusion equation in cylindrical coordinates is   μ u2  2 1 ∂ ∂T i ∂u e ke + μe + r ∂r K r ∂r ∂r ∂T ∂2 T ¼ Cu i  ke 2 i , ∂x ∂x

(172a)

for i ¼ 1 and 2

that in dimensionless form becomes "



2 #  2 ∂ðu=U Þ 1∂ ∂θi u r + Br + MDa r ∂r ∂r U ∂r u ∂θi 1 d2 θ  2 2i , ¼ U ∂x Pe dx

(172b)

for i ¼ 1 and 2

The transformation to covert this equation to the form of Eq. (145) without any changes in the boundary condition is readily available in [50, eq. (43)]; it is Br θi,s where ( )  
 1 I0 ðωr Þ I0 ðωr Þ 2 2 2 68 +2 ω 1r θi, s ðr Þ ¼ I0 ðωÞ I0 ðωÞ 4½ω  2I1 ðωÞ=I0 ðωÞ2 (173) As in the previous case, Br∂θi,s ðr Þ=∂rjr¼1 yields the contribution of frictional heating to the wall heat flux where ∂θi,s ðr Þ=∂rjr¼1 ¼

ω2  2I1 ðωÞ=I0 ðωÞ 2½ω  2I1 ðωÞ=I0 ðωÞ2

(174)

For MDa ¼ 105, 103, and 1 used in Figs. 15–17, Eq. (174) produces θi,s0 (1) ¼ 0.50318, 0.53318, and 4.66331. Finally, the dimensionless bulk temperature in circular pipes θi,b must be augmented by the influence of frictional heating Br θb,s where [50, eq. (38)].

Convective heat transfer in different porous passages

θb,s

  
2 ω2 2 2 ¼ 3 ω ω  32 + 20I1 ðωÞ =I0 ðωÞ 8½ω  2I1 ðωÞ=I0 ðωÞ 
  + 4 14  ω2 I1 ðωÞ + 2ωI2 ðωÞ + ω2 I3 ðωÞ =I0 ðωÞ  Z 1 + 8ω ½I0 ðωr Þ=I0 ðωÞ3 rdr

165

(175)

0 5

For M Da ¼ 10 , 103, and 1, Eq. (175) yields θb,s ¼ 0.12737, 0.14880, and 1.13609. The influence of frictional heating for different Br values is well documented in [46,47] with graphical presentation of data for parallel plate channels and circular pipes. For the data presented in Figs. 12–17, one can compute the Nusselt that includes the contribution of frictional heating. Using the product of Nu  θb as given in Figs. 12–17, the relation Nu  θb  Br  θ01,s ð1Þ Nutotal θb + Br  θb,s

(176)

can provide the total effects due to wall temperature change and frictional heating. Test results indicate that they agree well with the reported data in [46,47]. However, there are some small differences mainly due to expected numerical errors from numerical computations. As an illustration, in the absence of axial conduction, the accuracy of the fully developed Nusselt number is directly related to the first eigenvalue, and the reported Nusselt numbers [46] for MDa ¼ 105 and 1 are 2hH/ke ¼ 4.920 and 3.806. These agree well with 2  4.60 ¼ 4.920 and 2  1.901 ¼ 3.802 taken from Figs. 12A and 14A, respectively. This limiting value changes in the presence of frictional heating. For the same MDa values, [46] reports 2hH/ ke ¼ 6.641 for MDa ¼ 1 when using viscous dissipation in [50]. For this case, Eq. (176) yields 2hK/ke ¼  2(θ1,s0 (1)/θ1,b) ¼ 2(4.19453/1.26162) ¼ 6.649. Similarly, in the absence of frictional heating, for MDa ¼ 105 and 1, [47] reports 2hrO/ke ¼ 5.750 and 3.958. These are in reasonably good agreement with 2  2.873 ¼ 5.746 while there is a noticeable difference with 2  1.847 ¼ 3.694 when using the data in Figs. 15A and 17A, respectively. In the presence of frictional heating, Ref. [47] reports 2hrO/ke ¼ 8.206 for MDa ¼ 1 and the viscosity dissipation model in [53]. Using Eq. (125), and the procedure described earlier, this study yields 2hrO/ ke ¼ 2(4.66331/1.13609) ¼ 8.209. In summary, the exact series solution permits one to compute temperature and heat flux with a high degree of accuracy. Therefore, such solutions

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A. Haji-Sheikh et al.

are valuable for the purpose of verification of the numerical solutions. The data reported here indeed shows that the numerically acquired data in Nield et al. [46], for entrance flow problems in the presence of frictional heating and axial conduction for parallel plate channels, are sufficiently accurate.

6. Rapidly switched heat regenerators in counterflow Rapidly-switched thermal regenerators in countercurrent [5] used in Stirling cycle machines [54–58] represent another application of the twostep model for porous media defined in Section 2 of this chapter through Eqs. (1a) and (1b). In general, a heat regenerator is a thermal energy storage device where the processes of heat storage and heat retrieval are cyclically repeated and the hot and cold fluids usually flow in opposite directions (counter-flow operation), as depicted in Fig. 18. The performance of a heat regenerator during the cyclic steady operation depends on three dimensionless variables, namely NTU, U and α. The wellestablished number of transfer units NTU and the utilization factor U were first introduced by Hausen ([59,60], chapter 35). The former is defined as the A

B

J(x + = 0,t +b) = 0

Cold particles – J k

J(x + = 1,t r+ ) = 1 HOT SPACE

Internal particles – J i COLD SPACE

Hot particles – J h

0

1

x+

Fig. 18 Schematic representation for the analysis of the regenerator. (A) general scheme; (B) “Cold,” “hot” and “internal” particles of gas.

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Convective heat transfer in different porous passages

heat transfer coefficient—surface area of matrix product to the heat capacity flow rate of the gas (as a matter of fact, it was termed as “reduced length” by Hausen and denoted by the symbol Λ). The latter however specifies the ratio of the thermal capacity of the gas per pass to that of the matrix. (Actually, Hausen used the “reduced period” Π, but since U ¼ Π/Λ, this is only a minor change suggested by Baclic et al. [61,62] with the advantage to not include the heat transfer coefficient.) As regards the flush ratio α, it was first introduced by Organ [63,64] and, then, used by de Monte [5,65] for properly simulating the Stirling regenerator performance. It is defined as the ratio of the time required for a gas particle to complete a regenerator traverse to the duration of a period, namely the “blow” or “reverse” periods. (As a matter of fact, it was formerly denoted NFL and defined as NFL ¼ α1.) It accounts for the flush phase, i.e. the fact that some of the working fluid might not pass all the way through the regenerator but could remain (or be “held up”) inside the regenerator. Thus, the utilization factor may be written as U ¼ (αβ)1, where β is the thermal capacity ratio specifying the heat capacity of a length unit of the matrix, ρwcw(A  Af), to the heat capacity of a length unit of the gas, ρcpAf.

6.1 Mathematical formulation for cyclic steady operation The regenerator defining equations represent the transfer of heat to/from the working fluid and from/to the matrix on the passage of fluid through the porous regenerator. When dealing with a one-dimensional transient treatment, they may be derived from mass, momentum (Darcy) and energy balances concerning an element dx at a location x of the working fluid and matrix using an Euler formulation (Fig. 1A). Thus, for both transient and cyclic operation it results in ∂ρ ∂ ðρ uÞ ¼ 0 + ∂t ∂x ∂p μ  u ¼0 K ∂x  ∂ ∂
∂2 T h ðρc v T Þ + ρ uc p T ¼ k 2  ðT  T w Þ ∂t ∂x rh ∂x 2 ∂T ∂ Tw h  ε ðT w  T Þ ð1  εÞρw c w w ¼ ð1  εÞkw ∂t rh ∂x2

(177a) (177b) (177c) (177d)

where K is the regenerator permeability, μ and k are the dynamic viscosity and thermal conductivity of the fluid, rh ¼ Af/P is the hydraulic radius, with Af cross-sectional area for flow and P matrix wetted perimeter, u is the gas

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velocity, p its pressure, and ε ¼ Af/A is the matrix porosity with A crosssectional area of the regenerator. Also, T and Tw are the working gas and matrix temperatures, respectively; cp and cp are the specific heats of the gas at constant pressure and volume, respectively; ρwcw and kw are the volumetric heat capacity and thermal conductivity of the matrix; and h is the convective heat transfer coefficient that can be evaluated by using the empirical correlations as defined in [54]. Eqs. (177a)–(177d) are four partial differential equations in the unknowns u, p, T and Tw. As far as the gas density ρ is concerned, in fact, it can be related to T and p by means of the fluid equation-of-state (EOS), for example through the ideal gas law ρ ¼ p/(RT), with R specific gas constant. It is interesting to note that, substituting Eq. (177a) in Eq. (177c) and applying the ideal gas law, by some algebra the gas energy balance Eq. (177c) may be rewritten as ρc p

∂p ∂T ∂T ∂2 T h + uρ c p ¼ k 2  ðT  T w Þ + ∂t ∂x rh ∂t ∂x

(178)

where the gas pressure time-variation appears explicitly. The solution of the equations listed before requires to assign boundary and initial conditions. Before proceeding to solve these equations, some simplifying assumptions are however made: (1) the longitudinal heat conduction within both gas and matrix is ignored (see Section 5.2); (2) the gas pressure and velocity time- and space-variations are neglected as well; (3) the gas-matrix heat transfer coefficient h is assumed to be the same everywhere in the regenerator; and (4) the gas density is considered constant and evaluated at the average temperature (Th + Tk)/2. Behaving like this, the governing equations reduce to only the energy balance equations for the gas and matrix that simplify notably. In dimensionless form and for only cyclic steady operation, they may be taken as     + + + + ∂ϑ x , t ∂ϑ x , t
 j j α + sign u j + + ∂x 2 ∂t j h    i + + ¼ NTU ϑw x , t j  ϑ x+ , t+j (179a) ð j ¼ b, r Þ   + + h    i αβ ∂ϑw x , t j NTU ϑ x+ , t +j  ϑw x+ , t+j ð j ¼ b, r Þ (179b) ¼ 2 ∂t+j where the dimensionless quantities are defined as follows

Convective heat transfer in different porous passages

169

t+j ¼ t/τ0 is the dimensionless time during the jth period referred to the duration τ0 of an entire operation cycle. In detail, j ¼ b denotes the “blow” period of duration τ0/2; while j ¼ r indicates the “reverse” period having the same duration. It is assumed, in fact, that the gas flow switches in equal time intervals. Therefore, t+j  [t+in,j, t+in,j + 1/2] where t+in,j is the dimensionless initial time for the jth period. In particular, for the nth cycle (with n ¼ 1, 2, …), t+b  [n, n + 1/2] during the blow period and t+r  [n + 1/2, n + 1] during the reveres period; • “sign(uj)” appearing in Eq. (179a) takes account of the possibility of positive and negative uj as the flow direction alternates. For uj > 0, sign(uj) ¼ 1; while for uj < 0, sign(uj) ¼  1. Also, sign(0) ¼ 0; where the “sign(.)” function is defined by Oldham et al. ([66], p. 69). As the space coordinate system labeled in Fig. 18 considers as positive the gas flow from the cold end of the regenerator to the hot one, i.e. ub ¼  ur ¼ u > 0, it results in sign(ub) ¼ 1 and sign(ur) ¼  1; • x+ ¼ x/L is the dimensionless space coordinate referred to the regenerator length L. In particular, x+  [0, 1]; T k • ϑ ¼ TT h T is the non-dimensional temperature referred to the temperk ature jump between the hot and cold spaces; • NTU ¼ hL/(ρcpurh) is the number of transfer units; • α ¼ (L/u)/(τ0/2) is the flush ratio. It is defined as the ratio of the time L/u required for a gas particle having u as velocity to complete a regenerator traverse of length L to the duration τ0/2 of a period, namely the “blow” or “reverse” periods. However, it may also be defined as α ¼ L/[u(τ0/2)], that is, the ratio of the regenerator length L to the space u(τ0/2) traveled by a fluid particle having u as velocity during the “blow” or “reverse” periods of duration τ0/2. In both cases, it results in α ¼ 2L/(τ0u). In addition, if a dimensionless gas velocity is defined as u+ ¼ u/uref, with uref ¼ L/ (τ0/2), the flush ratio may be seen as the inverse of this velocity, that is, α ¼ 1/u+; • β ¼ cwρw(1  ε)/(cpρε) is the thermal capacity ratio between gas and matrix. Eqs. (179a) and (179b) are two coupled, linear, partial differential equations of the first-order in the unknowns ϑ(x+, t+j ) and ϑw(x+, t+j ), with j ¼ b, r, whose solution requires to assign boundary conditions for the gas and matrix. As the partial derivative along x appearing in the energy balance Eq. (179a) for the gas is of the first-order, only one boundary condition for its temperature is needed. However, as j ¼ b or r, two boundary conditions have to be assigned. In detail, according to Fig. 18, they are •

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  ϑ x+ ¼ 0, t+j ¼   ϑ x+ ¼ 1, t+j ¼

( (

for t+j ¼ t+b ½n, n + 1=2

0

unknown for t+j ¼ t+r ½n + 1=2, n + 1 unknown for t+j ¼ t+b ½n, n + 1=2 for t+j ¼ t+r ½n + 1=2, n + 1

1

(180a)

(180b)

with n ¼ 1, 2, … On the contrary, no boundary condition is necessary for the matrix temperature as partial derivatives along x do not appear in the related energy balance Eq. (179b). As far as the initial conditions for both gas and matrix are concerned, they do not play any role in the current treatment as it deals only with the cyclic steady operation of the regenerator. During this operation, the heat released from the matrix during the flow of the cold gas stream (blow period) is equal at any location to the heat transferred to the matrix during the flow of the hot gas stream (reverse period). This relation may be written symbolically as Z

n+1=2 

Z

n

¼



 ϑw x+ , t+b  ϑ x+ , t +b dt +b

n+1

n+1=2


+ +
 ϑ x , tr  ϑw x+ , t+r dt +r ,

(181)

and the two unknown temperatures ϑ(x+, t+j ) and ϑw(x+, t+j ), with j ¼ b, r, have to satisfy the above constraint.

6.2 Slowly- and rapidly-switched heat regenerators In the regenerators used in blast and glass melting furnaces, in metallurgical and chemical processing industries (of fixed-bed type) as well as in electrical power generating stations for air preheating, and in gas turbine power plants (of rotary type) [67], the time required for an element of gas to pass through the regenerator (L/u) is very short compared to the time of either period (τ0/2). These regenerators are in fact usually very large heat exchangers, some having spatial dimensions of up to 40 m and having unidirectional flow periods of many hours. This indicates that they are “slowly switched” (α ! 0) and their performance may be well described by only two dimensionless variables, namely NTU and U, as done by Hausen ([59,60], chapter 35). Notice that the utilization factor U ¼ (αβ)1 is finite as α ! 0 but β ! ∞. The latter is due to the fact that the matrices employed in the engineering applications stated before are usually metallic. For such regenerators, as α ! 0, the partial

171

Convective heat transfer in different porous passages

derivative ∂ ϑj/∂ t+j on the left-hand side of Eq. (178a) (which represents the heat storage in the gas) vanishes according to Hausen’s theory. In the regenerators used in Stirling machines [54,64] (of fixed-bed type), the time required for a gas particle to pass through the regenerator is however approximately equal to the blow time (or reverse time). These regenerators are in fact considered large if their diameter exceeds 6 cm and unidirectional flow periods are more likely to be in the millisecond range. This indicates that they are “rapidly switched” (α ¼ finite) and the effects of the flush phase have to be considered. Their performance may hence be well described by only two dimensionless variables, namely NTU and α, in that the utilization factor U ¼ (αβ)1 approaches zero as α ¼ finite but β ! ∞, unless the matrix used in the heat regenerators is not metallic (β ¼ finite). In such a case, in fact, only using all of the three dimensionless variables, namely NTU, U (or β) and α, ensures an appropriate description of their operation. For rapidly-switched heat regenerators (α ¼ finite), when employing a metallic matrix (β ! ∞), which are here of interest, Eq. (179b) reduces to ∂ ϑw/∂t+j ¼ 0 as αβ ! ∞. This indicates that the matrix temperature at each location x+ is essentially constant with time and, hence, ϑw ¼ ϑw(x+). Therefore, the defining Eqs. (179a) and (181) may be taken as     + + + + ∂ϑ x , t j
 ∂ϑ x , t j 2 + sign u j + ∂x+ α ∂t j h  i 2NTU ¼ (182) ϑw ðx+ Þ  ϑ x+ , t+j ð j ¼ b, r Þ α Z n+1=2 Z n+1

 ϑw ðx+ Þ ¼ ϑ x+ , t+b dt +b + ϑ x+ , t+r dt +r (183) n

n+1=2

with the boundary conditions for the fluid still defined by Eqs. (180a) and (180b). The unknown temperatures are ϑ(x+, t+j ) ( j ¼ b or r) and ϑw(x+).

6.3 General solution of the gas energy equation According to Refs. [5, 63–65], the general solution of the gas energy balance Eq. (182) will be derived by using a Lagrange frame of reference. Concerning this, the law of motion x+(t+j ) of the gas particles in a dimensionless form may be taken as   
 2 x+ t +j ¼ ξ+ + sign u j t+j  τ+j (184) ð j ¼ b, r Þ α

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where the ξ+ denotes a generic location of an element of gas within the regenerator (i.e., ξ+  [0, 1]) to which corresponds a generic time τ+j . At a generic time instant of t+j > τ+j , the gas particle is located at the position x+(t+j ) defined by Eq. (184). In particular, x+(t+j ) > ξ+ for j ¼ b; while x+(t+j ) < ξ+ for j ¼ r. Also, when t+j ¼ τ+j , the particle is located within the regenerator at the position x+(t+j ¼ τ+j ) ¼ ξ+. Taking the time derivative of Eq. (184) yields dx+/dt+j ¼ 2 sign(uj)/α that is the dimensionless gas velocity during the jth period (blow or reverse) Therefore, the left-hand side of Eq. (182) is recognized to be the substantial derivative, d/dt+j , of the fluid temperature. Thus, this equation may be rewritten as. h   i h   i d 2NTU + + + ϑ x t + ϑ x+ tj+ , tj+ , t j j α dtj+ h  i 2NTU ϑw x+ tj+ ¼ ð j ¼ b, r Þ α

(185)

Now, the integration of this linear, first-order, differential equation along the path Eq. (184) followed by a selected gas particle may be performed analytically, as shown in Ref. ([21], p. 1096, #16.316). Its solution gives the temperature at time τ+j when the element of gas is positioned at the location ξ+, that is,   2NTU + + + + ϑ ξ , τ j ¼ ϑin,j e α ðτ j tin,j Þ Z + h  i 2NTU + + 2NTU τ j + ϑw x+ t+j e α ðt j τ j Þ dt+j ð j ¼ b, r Þ (186) α t+ in,j

+ + where ϑin,j ¼ ϑ(xin, j, tin, j) is the “initial” temperature of the gas particle. In other words, it is the temperature of the gas particle at its “initial” location + + + + xin, j ¼ x (tin, j) at the time tin, j to which corresponds the beginning of the jth period (blow period or reverse period) of the nth operation cycle (t+in,b ¼ n and t+in,r ¼ 1/2 + n). The knowledge of the initial temperature will be discussed afterwards, while the initial location of the gas particle may be related + to (ξ+,τ+j ) through Eq. (184) simply setting t+j ¼ tin, j

  
 2 ð j ¼ b, r Þ x+in,j ¼ x+ t+in,j ¼ ξ+ + sign u j t +in,j  τ+j α

(187)

Now, by means of an algebraic substitution of the dummy variable in the integral on the RHS of Eq. (186), that is, time variable t+j ! space variable

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Convective heat transfer in different porous passages

x+ according to the law of motion (184), after some algebraic steps making use of Eq. (187), Eq. (186) becomes " # Z ξ+   + NTU NTU + NTU +  ξ x x NTU in,j
 ϑw ðx+ Þe signðu j Þ dx+ ϑ ξ+ , τ+j ¼ e signðu j Þ ϑin,j e signðu j Þ + sign u j x+in,j (188) where the independent variable τ+j appears implicitly through x+in, j ¼ x+(t+in, j) defined by Eq. (187). Eq. (188) has the advantage to provide us with the fluid temperature in a much more straightforward manner than Eq. (186). When using Eq. (188), the computation of the fluid temperature requires. • the knowledge of the matrix temperature ϑw(x+). It may be evaluated through the cyclic steady operation condition Eq. (183) that, according to the Lagrange system (ξ, τ+j ) here assumed, has to be rewritten as Z

n+1=2

ϑw ðξ Þ ¼ +

n


 ϑ ξ+ , τ+b dτ+b +

Z

n+1

n+1=2


 ϑ ξ+ , τ+r dτ+r

ðξ+ ½0, 1Þ (189)

As ϑ(ξ+, τ+b ) and ϑ(ξ+, τ+r ) may be taken through Eq. (188) as a function of ϑw(ξ+), Eq. (189) becomes an integral equation in the unknown ϑw. However, according to Refs. [5, 63, 64], a uniform gradient for the matrix temperature has been assumed. In particular, the well-known expression deriving from the Nusselt classical theory of the regenerators (α ! 0) has been used, i.e., NTU 1 ξ+ + (190) NTU + 2 NTU + 2 + + • the knowledge of the ‘initial’ temperature ϑin,j ¼ ϑ(xin, j, tin, j) of the gas particles, that is, the temperature of the gas particles when the blow and reverse periods start. It depends on the initial location of the element of gas. When the blow period starts, for example, the element of gas may be located inside or outside the regenerator. Similarly, when the reverse period starts. For this reason, the boundary conditions (180a) and (180b) are relevant only for the fluid particles located outside the regenerative matrix. When the flush ratio α  1, the initial temperatures of the gas particles and, hence, the blow and reverse fluid temperatures can be derived as shown by de Monte in Ref. [65], where an integral equation was solved to calculate the matrix temperature. In this work, it was shown that the matrix ϑw ðξ+ Þ ¼

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temperature is nearly linear-in-space according to Eq. (190). For α > 1 (which is of interest in the current chapter), the mathematical treatment is much more complex because a slug of fluid oscillates within the matrix without exiting either end [5]. In such a case, to determine the initial temperatures of the gas particles, it is convenient to classify them as cold (k), hot (h) and internal (i) particles according to their initial locations x+in,j, as shown in Fig. 18B. In particular, these locations may be related to ξ+ and τ+j by using Eq. (187), where the ξ+ range depends on the value of α and type of particle (i.e., cold, hot and internal), as shown in the following sections, namely Sections 6.4–6.6.

6.4 “Cold” particles The “cold” gas particles are defined as those particles that, at the beginning of the blow period, are located inside the cold space and can enter the regenerator; while, when the reverse period starts, they are positioned inside the regenerator. Therefore, during the cyclic steady operation of the regenerator, for α 1 they fluctuate between the cold space and regenerator without entering the hot space, as shown in Fig. 18. In particular, during the reverse period it is assumed that the dimensionless temperature of the particles becomes instantaneously equal to ϑk ¼ 0, that is, a step change in temperature occurs at ξ+ ¼ 0 when exiting the matrix and entering the cold space. Also, for α ¼ 1 and during the blow period, the ‘cold’ particles can only reach the right end ξ+ ¼ 1 of the regenerator without entering the hot space. However, for α < 1, they can exit the matrix at the right-hand side ξ+ ¼ 1 and enter the hot space during the blow period. In such a case, these particles become of ‘hot’ type (see Section 6.5) and it is assumed that their dimensionless temperature becomes suddenly equal to ϑh ¼ 1, that is, a step change in temperature at ξ+ ¼ 1 occurs. Similarly, they can exit the matrix at the left-hand side ξ+ ¼ 0 and enter the cold space during the reverse period. In such a case, it is still assumed that their dimensionless temperature becomes instantaneously equal to ϑk ¼ 0, that is, a step change in temperature occurs at ξ+ ¼ 0. The blow and reverse periods will be considered separately in the next paragraphs. 6.4.1 Blow period At the beginning of the blow period (t+in,b ¼ n), the cold particles are located inside the cold space, i.e., x+in,b  [(1/α  ξ+), 0], with ξ+  [0, 1] for α  1 and ξ+  [0, 1/α] for α > 1. (Note that when x+in,b <  (1/α  ξ+) with ξ+ ¼ 0, i.e., x+in,b <  1/α, the particle will never reach the matrix and, hence, it does

Convective heat transfer in different porous passages

175

not play any role.) In particular, if x+in,b ¼  (1/α  ξ+), that is, the particle is located at the lower bound of this spatial range, using Eq. (187) for j ¼ b yields τ+b ¼ n + 1/2. This indicates that, at the end of the blow period, the fluid particle can reach any position ξ+  [0, 1] if α  1; while it can reach only some of the locations within the regenerator if α > 1, in detail only the locations ξ+  [0, 1/α], which define the so-called “cold” zone of the + matrix, as shown in Fig. 19. However, if xin, b ¼ 0, that is, the particle is located at the entrance of the regenerator when the blow period starts, the use of Eq. (187) for j ¼ b yields τ+b ¼ n + ξ+ α/2. This indicates that the time (τ+b  n) ¼ ξ+ α/2 that it takes for a fluid particle to reach the position ξ+ depends on both the position itself and the value of the flush ratio (i.e., less, equal or greater than 1), unless ξ+ ¼ 0. In fact, when ξ+ ¼ 0, it results in (τ+b  n) ¼ 0. For ξ+ other than zero, the situation depends on α. When α  1, the time (τ+b  n) ¼ ξ+ α/2 that it takes to reach ξ+ is less than or equal to ½ for any ξ+  [0, 1]. Therefore, when α < 1, a gas particle reaches the exit of the regenerator ξ+ ¼ 1 before the end of the blow period. As an example, if α ¼ 1/2, the gas element reaches ξ+ ¼ 1 at the time τ+b (ξ+ ¼ 1, α ¼ 1/2) ¼ n + 1/4 and, hence, at the end of the blow period this element is located within the hot space at the dimensionless temperature of ϑh ¼ 1. If α ¼ 1, the gas particle reaches the exit of the regenerator ξ+ ¼ 1 at the end of the blow period. When α > 1, however, the time (τ+b  n) ¼ ξ+ α/2 that it takes to reach ξ+ is less than or equal to ½ only for ξ+  [0, 1/α]. In such a case, a fluid particle reaches the location ξ+ ¼ 1/α only at the end of the blow period. In addition, as the cold particles are located inside the cold space at the beginning of the blow period (t+in, b ¼ n) and, hence, are outside the regenerator, there is no heat exchanged between the cold elements of gas and the matrix in the first part of the blow period, say in the range t+b  [n, t(k)+ b ]. The (k)+ time tb may be evaluated through Eq. (184) for the blow period (j ¼ b) (k)+ simply setting x+(t(k)+ ¼ τ+b  ξ+ α/2  [n, n + 1/2]. b ) ¼ 0. It follows that tb + This would indicate that the initial time tin,b ¼ n appearing in Eq. (186) for j ¼ b has to be replaced with the ‘heat exchange’ initial time t(k)+ b . Similarly, the initial location x+in,b which appears in Eq. (188) for j ¼ b has to be replaced by the position x+(t(k)+ b ) ¼ 0. Also, the initial temperature + + ϑin,b ¼ ϑ(xin,b, tin,b) appearing in both Eqs. (186) and (188) has to be replaced with ϑ(0, t(k)+ b ), which is equal to zero according to the first of the two Eqs. (180a). Therefore, Eq. (188) for j ¼ b giving the temperature of a cold element of gas at the location ξ+ and time τ+b reduces to

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A

COLD SPACE

HOT SPACE

COLD ZONE ≡ HOT ZONE

0

1

x+

B

HOT SPACE

COLD SPACE

COLD ZONE HOT ZONE SUPERPOSITION ZONE

0

1–1/α

1/α

1

x+

C

COLD SPACE

HOT SPACE

COLD ZONE

0

1/α

HOT ZONE

1–1/α

1

x+

Fig. 19 “Cold” and “hot” zones of the matrix. (A) α  1 (complete superposition with no internal particles of gas); (B) 1 < α  2 (partial superposition containing cold, hot and internal elements of gas); and (C) α > 2 (zone ξ+  [1/α, 1  1/α] containing only the internal particles).

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ðkÞ

ϑb ðξ+ Þ ¼ NTUeNTUξ

+

Z

ξ+

+

ϑw ðx+ ÞeNTUx dx+

(191)

0

where ξ+  [0, 1] for α  1; while ξ+  [0, 1/α] for α > 1. Also, ϑw(x+) is defined by Eq. (190). Notice that the above temperature is independent of τ+b . 6.4.2 Reverse period At the beginning of the reverse period (t+in,r ¼ n + 1/2), the cold particles are located inside the regenerator, i.e., x+in,r  [ξ+, 1] with ξ+  [0, 1] for α  1, and x+in,r  [ξ+, 1/α] with ξ+  [0, 1/α] for α > 1. In the former case (α  1), if x+in,r ¼ ξ+, that is, the element of fluid is located at the lower bound of the spatial interval, using Eq. (187) for j ¼ r yields τ+r ¼ n + 1/2. However, if x+in,r ¼ 1, that is, the particle is positioned at the right end of the regenerator when the reverse period starts, the use of Eq. (187) for j ¼ r gives τ+r ¼ (n + 1/2) + (1  ξ+)α/2. This indicates that the time [τ+r  (n + 1/2)] ¼ (1  ξ+)α/2 that it takes for a fluid particle to reach the position ξ+ depends on both the position itself and the value of the flush ratio (less than or equal to 1). In particular, this time is always less than or equal to ½ for any ξ+  [0, 1]. Hence, a gas particle reaches the left end of the regenerator ξ+ ¼ 0 before the end of the reverse period. As an example, if α ¼ 1/2, the gas element reaches ξ+ ¼ 0 at the time τ+r (ξ+ ¼ 0, α ¼ 1/2) ¼ (n +1/2) + 1/4 and, hence, at the end of the reverse period this element is located within the cold space at the dimensionless temperature of ϑk ¼ 0. If α ¼ 1, the gas particle reaches the left end of the regenerator ξ+ ¼ 0 only at the end of the reverse period. In conclusion, when α  1, the gas particle reaches the position ξ+  [0, 1] at the time τ+r  [(n + 1/2), (n + 1/2) + (1  ξ+)α/2]. In the latter case (α > 1), the initial location of the cold particles is x+in,r  [ξ+, 1/α] with ξ+  [0, 1/α], where the ξ+ range defines the “cold” zone of the matrix mentioned in the previous paragraph and depicted in Fig. 19. If ξ+ > 1/α, the cold particles would not enter the cold space and, hence, could not be termed as cold particles. If x+in,r ¼ ξ+ with ξ+  [0, 1/α), that is, the element of gas is located at the lower bound of the spatial interval, using Eq. (187) for j ¼ r yields τ+r ¼ n + 1/2. In such a case, at the end of the reverse period the particles will be located within the cold space at the dimensionless temperature of ϑk ¼ 0. However, if x+in,r ¼ 1/α, that is, the particle is positioned at the right end of the “cold” zone when the reverse period starts, the use of Eq. (187) for j ¼ r gives τ+r ¼ (n + 1/2) + (1  ξ+ α)/2. Therefore, the time [τ+r  (n + 1/2)] ¼ (1  ξ+ α)/2 that it

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A. Haji-Sheikh et al.

takes for a fluid particle to reach the position ξ+ depends not only on the position itself but also on the value of the flush ratio (here greater than 1). In particular, this time is always less than or equal to ½ for any ξ+  [0, 1/α]. Hence, a gas particle starting from x+in,r ¼ 1/α can reach the left end of the regenerator ξ+ ¼ 0 only at the end of the reverse period. As an example, if α ¼ 2, the gas element reaches ξ+ ¼ 1/4 at the time τ+r (ξ+ ¼ 1/4, α ¼ 2) ¼ (n + 1/2) + 1/4 and, hence, at the end of the reverse period this element will be located at left end of the regenerator ξ+ ¼ 0. In conclusion, when α > 1, the gas particle reaches the position ξ+  [0, 1/α] at the time τ+r  [(n + 1/2), (n + 1/2) + (1  ξ+ α)/2]. Therefore, by using Eq. (188) with j ¼ r, the temperature of a cold element of gas at the location ξ+ and time τ+r may be evaluated as
 ϑðr kÞ ξ + , τr+ ¼ " # Z ξ+
 + + + eNTUξ ϑðr kÞ xin+, r , n + 1=2 eNTUxin, r  NTU ϑw ðx + ÞeNTUx dx + xin+, r

(192) (k) + where the initial temperature ϑ(k) in,r ¼ ϑr (xin,r, n + 1/2) depends on the thermal history of the same element during the preceding blow period + (k) + (τ+b  [n, n + 1/2]), that is, ϑ(k) r (xin,r, n + 1/2) ¼ ϑb (xin,r, n + 1/2). Now, the (k) + temperature ϑb (xin,r, n + 1/2) may be obtained by means of Eq. (191) simply setting ξ+ ¼ x+in,r. Thus, Eq. (192) becomes " Z xin+, r
+ + + ðkÞ NTUξ + 2NTUxin+, r ϑr ξ , τr ¼ NTUe e ϑw ðx + ÞeNTUx dx +

0

Z

ϑw ðx + ÞeNTUx dx +  +

ξ + xin+, r

(193)

where ξ+  [0, 1] for α  1; while ξ+  [0, 1/α] for α > 1. Also, ϑw(x+) is defined by Eq. (190). Notice that the above temperature depends on τ+r by means of x+in,r defined through Eq. (187) for j ¼ r.

6.5 “Hot” particles The “hot” gas particles are defined as those particles that, at the beginning of the reverse period, are located inside the hot space and can enter the regenerator; while, when the blow period starts, they are positioned inside the regenerator. Therefore, when the regenerator works in cyclic operation,

Convective heat transfer in different porous passages

179

for α 1 they fluctuate between the regenerator and hot space without entering the cold space, as shown in Fig. 18. In particular, during the blow period it is assumed that the dimensionless temperature of the particles becomes instantaneously equal to ϑh ¼ 1, that is, a step change in temperature occurs at ξ+ ¼ 1 when exiting the matrix and entering the hot space. Also, for α ¼ 1 and during the reverse period, the “hot” particles can only reach the left end ξ+ ¼ 0 of the regenerator without entering the cold space. However, for α < 1, they can exit the matrix at the left-hand side ξ+ ¼ 0 and enter the cold space during the reverse period. In such a case, these particles become of ‘cold’ type (see Section 6.4) and it is assumed that their dimensionless temperature becomes suddenly equal to ϑk ¼ 0, that is, a step change in temperature at ξ+ ¼ 0 occurs. Similarly, they can exit the matrix at the right-hand side ξ+ ¼ 1 and enter the hot space during the blow period. In such a case, it is still assumed that their dimensionless temperature becomes instantaneously equal to ϑh ¼ 1, that is, a step change in temperature occurs at ξ+ ¼ 1. The blow and reverse periods will be considered separately in the next paragraphs. 6.5.1 Reverse period At the beginning of the reverse period (t+in,r ¼ n + 1/2), the hot particles are located inside the hot space, i.e., x+in,r  [1, 1/α + ξ+] with ξ+  [0, 1] for α  1 and ξ+  [1  1/α, 1] for α > 1. (Note that when x+in,b > (1/α + ξ+) with ξ+ ¼ 1, i.e., x+in,b > (1/α + 1), the particle will never reach the matrix and, hence, it does not play any role.) For α  1, the hot particles can reach any position ξ+ within the regenerator, i.e., ξ+  [0, 1]. For α > 1, instead, these particles of gas can reach only those positions having ξ+  [1  1/ α, 1] which define the so-called “hot” zone of the matrix, as shown in Fig. 19. In both cases, however, a gas particle starting with the location x+in,r  [1, 1/α + ξ+] reaches the position ξ+ at τ+r  [(n + 1/2) + (1  ξ+)α/2, n + 1]. In fact, setting x+in,r ¼ 1 in Eq. (187) for j ¼ r, it results in the lower limit of the above time interval. Similarly, substituting x+in,r ¼ 1/α + ξ+ in the same equation, we obtain τ+r ¼ n + 1. In addition, as the hot particles are located inside the hot space at the beginning of the reverse period (t+in,r ¼ n + 1/2) and, hence, are outside the regenerator, there is no heat exchanged between the hot elements of gas and the matrix in the first part of the reverse period, say in the range t+r  [(n + 1/2), t(h)+ ]. The time t(h)+ may be evaluated through Eq. (184) r r for the reverse period (j ¼ r) simply setting x+(t(h)+ ) ¼ 1. It follows that r

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A. Haji-Sheikh et al.

t(h)+ ¼ τ+r + (ξ+  1)α/2  [(n + 1/2), (n + 1)]. This would indicate that the r initial time t+in,r ¼ n + 1/2 appearing in Eq. (186) for j ¼ r has to be replaced with the above “heat exchange” initial time t(h)+ . Similarly, the initial locar tion x+in,r which appears in Eq. (188) for j ¼ r has to be replaced by the position x+(t(h)+ ) ¼ 1. Also, the initial temperature ϑin,r ¼ ϑ(x+in,r, t+in,r) apper aring in both Eqs. (186) and (188) has to be replaced with ϑ(0, t(h)+ ), which r is equal to the unit according to the second of the two Eq. (180b). Therefore, Eq. (188) giving the temperature of a hot element of gas for j ¼ r at the location ξ+ and time τ+r reduces to " # Z + ϑðr hÞ ðξ+ Þ ¼ eNTUξ eNTU  NTU

ξ

+

ϑw ðx+ ÞeNTUx dx+ +

(194)

1

where ξ+  [0, 1] for α  1; while ξ+  [1  1/α, 1] for α > 1. Also, ϑw(x+) is defined by Eq. (190). Notice that the above temperature is independent of τ+r . 6.5.2 Blow period At the beginning of the blow period (t+in,b ¼ n), the hot particles are located inside the regenerator. For α  1, x+in,b  [0, ξ+] where ξ+  [0, 1]. In this case, the gas particle reaches the position ξ+ at the time τ+b  [n, n + ξ+ α/2]. In fact, setting x+in,b ¼ 0 in Eq. (187) for j ¼ b, it results in τ+b ¼ n + ξ+ α/2. Similarly, substituting x+in,b ¼ ξ+ in the same equation, we get τ+b ¼ n. For α > 1, however, x+in,b  [1  1/α, ξ+] with ξ+  [1  1/α, 1], where the ξ+range defines the “hot” zone of the matrix mentioned in the previous paragraph and depicted in Fig. 19. If ξ+ < 1  1/α, the hot particles would not enter the hot space and, hence, they could not be termed as hot particles. In such a case, the element of gas reaches the position ξ+ at the time τ+b  [n, n + 1/2  (1  ξ+)α/2]. In fact, setting x+in,b ¼ 1  1/α in Eq. (187) for j ¼ b, it is found the upper limit of the above time interval. Therefore, by using Eq. (188) with j ¼ b, the temperature of a cold element of gas at the location ξ+ and time τ+b may be evaluated as " # Z ξ+

 + + + ðhÞ ðhÞ ϑb ξ+ , τ+b ¼ eNTUξ ϑb x+in,b , n eNTUxin,b + NTU ϑw ðx+ ÞeNTUx dx+ x+in,b

(195) (h) + ϑ(h) in,b ¼ ϑb (xin,b, n)

depends on the thermal hiswhere the initial temperature tory of the same element during the reverse period at the (n  1)th operation + (h) + cycle (τ+r  [n  1/2, n]), that is, ϑ(h) b (xin,b, n) ¼ ϑr (xin,b, n). Now, the

181

Convective heat transfer in different porous passages

+ temperature ϑ(h) r (xin,b, n) may be obtained by means of Eq. (194) simply setting ξ+ ¼ x+in,b. Thus, Eq. (195) becomes

 ðhÞ
ϑb ξ+ , τ+ b

" NTUξ+

¼e





NTU 2x+in,b 1

e

Z

ξ+

+ NTU x+in,b

+

Z

x+in,b

NTUe2NTUxin,b


 + ϑw x+ eNTUx dx+

#


 + ϑw x+ eNTUx dx+

1

(196)

where ξ  [0, 1] for α  1; while ξ  [1  1/α, 1] for α > 1. Also, ϑw(x+) is defined by Eq. (190). Notice that the above temperature depends on τ+b by means of x+in,b defined through Eq. (187) for j ¼ b. +

+

6.6 “Internal” particles During the cyclic steady operation of the regenerator, the “internal” particles of gas oscillate within the matrix without exiting either end, as shown in Fig. 18. Of course, they vanish for α  1 and, in this case, only the cold and hot particles treated in the previous two sections take place within the regenerator. 6.6.1 Blow period At the beginning of the blow period (t+in,b ¼ n), the internal particles are located at x+in,b  [0, 1  1/α]. In fact, if x+in,b  [1  1/α, 0], the internal particles would enter the hot space and, hence, they could not be termed as internal particles. Two possible cases can occur: • 1 < α  2. In this case, the internal particles, starting from x+in,b  [0, 1  1/α], can reach both the domains ξ+  [0, 1  1/α] and ξ+  [1  1/α, 1/α], where the latter is given by the superposition of the cold ([0, 1/α]) and hot ([1  1/α, 1]) zones of the matrix, as shown in Fig. 19B. Notice that an internal particle starting from x+in,b  [ξ+  1/α, 1  1/α] can also reach the domain ξ+  [1/α, 1]; • α > 2. In the current case, the internal particles, starting from x+in,b  [0, 1  1/α], can reach both the ξ+  [0, 1/α] and ξ+  [1/α, 1  1/α] domains, where the former is the cold zone of the matrix depicted in Fig. 19C. Also, an internal particle starting from the position x+in,b  [ξ  1/α, 1  1/α] can reach the domain ξ+  [1  1/α, 1] (hot zone of the matrix). Applying Eq. (188) for j ¼ b, the temperature of an internal element of fluid at the location ξ+ and time τ+b may be evaluated as

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A. Haji-Sheikh et al.

  + + ðiÞ
ðiÞ
ϑb ξ+ , τ+b ¼ ϑb x+in,b , n eNTU ðxin,b ξ Þ Z ξ+ + + + NTU ϑw ðx+ ÞeNTU ðx ξ Þ dx+

(197)

x+in,b

where the initial temperature ϑ(iin,) b ¼ ϑ(ib )(x+in, b, n) may be calculated making use of the cyclic operation of the regenerator, that is,   ðiÞ
ðiÞ
ϑb x+in,b , n ¼ ϑb x+in,b , n  1 (198) Now, the temperature on the LHS of the above equation depends on the thermal history of the same particle during the reverse period of the (n  1)th operation cycle (τ+r  [n  1/2, n]), i.e. ϑ(ib )(x+in,b, n) ¼ ϑ(ir )(ξ+ ¼ x+in,b, τ+r ¼ n). In particular, the temperature ϑ(ir )(ξ+ ¼ x+in,b, τ+r ¼ n) may be derived by applying Eq. (188) for j ¼ r, with x+in,r ¼ x+(t+in,r ¼ n  1/2). Also, it depends on ϑ(ir )(x+in,r, n  1/2) which is the temperature of the internal particles at the beginning of the (n  1)th reverse period with x+in,r ¼ x+in,b + 1/α. However, since this temperature may also be seen as the temperature at the end of the (n  1)th blow period (τ+b  [n  1,n  1/2]), it results in ϑ(ir )(x+in,r, n  1/2)¼ ϑ(ib )(ξ+ ¼ x+in,r, τ+b ¼ n  1/2), where the second temperature may be determined by means of Eq. (188) for j ¼ b, with x+in,b ¼ x+(t+in,b ¼ n  1). Notice also that it depends on ϑ(ib )(x+in,b, n  1), which is the temperature of the internal particles at the beginning of the (n  1)th blow period, that is, the temperature on the RHS of Eq. (198). Thus, solving Eq. (198) for ϑ(ib )(x+in,b, n), it is found  ðiÞ
ϑb xin+, b , n ¼

NTU NTU

2 α (1Z e+ x + 1=α in, b

xin+, b

 )
 2 + + + + ϑw ðx + Þ eNTU xin, b + αx +eNTU ðxin, b x Þ dx + (199)

where x+in,b depends on the time τ+b in the form of Eq. (187) for j ¼ b. The time and space intervals where Eq. (197), along with the companion Eq. (199), may be used depend strictly on the value of α, as it will be illustrated in Section 6.7. 6.6.2 Reverse period At the beginning of the reverse period (t+in,r ¼ n + 1/2), the internal particles are located at x+in,r  [1/α, 1]. In fact, if x+in,r < 1/α, the internal particles would

183

Convective heat transfer in different porous passages

enter the cold space and, hence, they could not be termed as internal particles. Two possible cases can occur: • 1 < α  2. In this case, the internal particles starting from x+in,r  [1/α, 1] can reach both the domains ξ+  [1/α, 1] and ξ+  [1  1/α, 1/α], where the latter is given by the superposition of the cold ([0, 1/α]) and hot ([1  1/α, 1]) zones of the matrix shown in Fig. 19B. Notice that an internal particle starting from x+in,r  [1/α + ξ+, 1] can also reach the domain ξ+  [0, 1  1/α]; • α > 2. In the present case, the internal particles are still located initially at x+in,r  [1/α, 1] and they can reach both the domains ξ+  [1  1/α, 1] and ξ+  [1/α, 1  1/α], where the former is the hot zone of the matrix depicted in Fig. 19C. Also, an internal particle starting from x+in,r  [1/ α + ξ+, 1] can reach the domain ξ+  [0, 1/α] (the cold zone of the matrix). Applying Eq. (188) for j ¼ r, the temperature of an internal element of fluid at the location ξ+ and time τ+r may be evaluated as

 + + ϑðr iÞ ξ+ , τ+r ¼ ϑðr iÞ x+in,r , n + 1=2 eNTU ðxin,r ξ Þ Z ξ+ + +  NTU ϑw ðx+ ÞeNTU ðξ x Þ dx+

(200)

x+in,r

where the initial temperature ϑ(ir )(x+in,r, n + 1/2) may be calculated making use of the cyclic steady operation of the regenerator, that is, ϑ(ir )(x+in,r, n + 1/2) ¼ ϑ(ir )(x+in,r, n  1/2). Then, following a procedure similar to the one applied to the blow period in the previous paragraph and solving the above equation for ϑ(ir )(x+in,r, n + 1/2), it results in
 ϑðr iÞ xin+, r , n + 1=2 ¼

NTU NTU 1  e2 α

(Z

xin+, r

xin+, r 1=α

ϑw ðx + Þ

) 
 2 NTU xin+, r  αx + NTU ðx + xin+, r Þ + e +e dx

(201)

where x+in,r depends on the time τ+r in the form of Eq. (187) for j ¼ r. The time and space intervals where Eq. (200), along with the companion Eq. (201), may be used depend strictly on the value of α, as it will be described in the next subsection.

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6.7 Fluid temperature solution As the matrix temperature ϑw(ξ+) may be evaluated approximately through Eq. (190), Eqs. (191), (196) and (197), along with Eq. (199), allow the fluid (h) + + (i ) + + + temperatures ϑ(k) b (ξ ), ϑb (ξ , τb ) and ϑb (ξ , τb ) to be evaluated during the blow period. Similarly, Eqs. (193), (194) and (200), along with Eq. (201), (h) + (i ) + + + + allow the gas temperature distributions ϑ(k) r (ξ , τr ), ϑr (ξ ) and ϑr (ξ , τr ) to be calculated during the reverse period as well. Thus, solving the integrals related to the equations stated before, it is found.
   1  sign uj NTU ðpÞ + + + ϑj ξ , τj ¼ ξ + NTU + 2 NTU + 2
 NTU ðpÞ 2λj sign uj signðuj Þðxin+, j ξ + Þ + ð j ¼ b, r andp ¼ h, i, kÞ e NTU + 2 (202) (h) (h) (k) (i ) (i ) where λ(k) b ¼ λr ¼ 0, λb ¼ λr ¼ 1 and λb ¼ λr ¼ 1/[1 + exp(NTU/α)]. Eq. (202) states that the fluid temperature depends exponentially on the time τ+j by means of x+in,j defined through Eq. (187). As regards the time and space intervals (including the values of α) where this equation may be used, they are summarized in Tables 4 and 5. Fig. 20 shows the fluid temperature for NTU ¼ 1 and α ¼ 5/3 as a function of ξ+ with τ+j (j ¼ b, r) as a parameter for both the blow a) and reverse b) periods. The matrix temperature (time-independent) is plotted in the same diagrams and its cold and hot zones depending on α are pointed out. When the gas flows from the cold space to the hot one (blow period, Fig. 20A), the fluid temperature is not continuous at the points (ξ+, τ+b ) which satisfy the times τ+b,1 and τ+b,2 given in Table 4. For example, when α ¼ 5/3 and τ+b ¼ n + 1/6 (second plot in Fig. 20A), the gas temperature is discontinuous at ξ+ ¼ 0.2 and ξ+ ¼ 0.6, and so on. This discontinuity is due to the proposed analytical treatment which deals with the boundary condition Eq. (180b). In fact, in order that this boundary condition can really occur, the dimensionless temperature of the gas exiting the matrix at its right-hand side (x+ ¼ 1) during the blow period (t+b  [n, n + 1/2]) and entering the hot space has to approach suddenly the unity. The sudden variation of temperature causes the above discontinuity. Similarly, during the reverse period (Fig. 20B), the fluid temperature is not continuous at the points (ξ+, τ+r ) which satisfy the times τ+r,1 and τ+r,2 given in Table 5. As an example, when α ¼ 5/3 and τ+r ¼ (n + 1/2) + 1/3 ¼ n + 5/6 (third plot in Fig. 20B), the gas temperature is discontinuous

185

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Table 4 Time and space intervals for the fluid temperatures as a function of α during the blow period. Also, τ+b,1 ¼ (n + 1/2)  (1  ξ+)α/2 and τ+b,2 ¼ n + ξ+ α/2. Blow period (τ+b  [n, n + 1/2]) α

α1 12

[0, 1/α]

at ξ+ ¼ 0.2 and ξ+ ¼ 0.6, and so on. This discontinuity is due to the boundary condition Eq. (180a). The above discontinuities are in agreement with Organ’s findings [63,64]. In fact, using the simplification of incompressibility, Organ demonstrated that one-dimensional, cyclically reversing flow in a duct is inevitably accompanied by discontinuities in the lengthwise distribution of temperature. Fig. 20A also shows that, at a prescribed time τ+b during the blow period, the matrix does in general not give up heat to the gas at all the locations ξ+ within the regenerator. As an example, when α ¼ 5/3 and τ+b ¼ n + 1/6 (second plot in Fig. 20A), the matrix temperature is higher than the fluid temperature for only ξ+  [0, 0.2]. This indicates that the heat is transferred from the fluid to the matrix for ξ+  [0.2, 1] at the time τ+b ¼ n + 1/6 even though the gas at that time is flowing from the cold space to the hot one. All this is due to the phenomenon of the flush phase occurring only in rapidly switched heat regenerators. Similar considerations may be done for the reverse period.

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Table 5 Time and space intervals for the fluid temperatures as a function of α during the reverse period. Also, τ+r, 1 ¼ n + 1  ξ+ α/2 and τ+r, 2 ¼ n + 1/2 + (1  ξ+)α/2. Reverse period (τ+r  [n + 1/2, n + 1]) α

α1 12

[0, 1/α]

6.8 Effectiveness and heat stored in the regenerator The regenerator effectiveness ε may be defined as the ratio of the heat actually exchanged to an ideal amount of heat which would be exchanged if the temperature of the cold gas could be increased to the entrance temperature of the hot gas [60, chapter 35]. Therefore, ε¼

T b ðξ ¼ L Þ  T k ¼ ϑb ðξ+ ¼ 1Þ Th  Tk

(203)

here ϑb ðξ+ ¼ 1Þ is the time-average dimensionless temperature of the fluid over the blow period (whose dimensionless duration is of 1/2) at the hot end of the regenerator (ξ+ ¼ 1). Bearing in mind the time and space intervals of Tables 4 and 5, Eq. (203) has to be split into two cases. Thus, it results in

187

Convective heat transfer in different porous passages

A

B

Fig. 20 Dimensionless temperatures (fluid and matrix) as a function of ξ+ with τ+j (j ¼ b, r) as a parameter for NTU ¼ 1 and α ¼ 5/3. (A) blow period and (B) reverse period.

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ε¼

A. Haji-Sheikh et al.

1 ð1=2 8ÞZ n+α=2 Z n+1=2  + > ðhÞ
+ ðkÞ + > > ϑ ξ ¼ 1, τ + ϑb ðξ+ ¼ 1Þdτ+b dτ b b b < n+α=2 n  Z n+1=2 >  > ð hÞ
> : ϑ ξ+ ¼ 1, τ+ dτ+ n

b

b

b

α1 α>1 (204)

Substituting Eq. (202) in Eq. (204), the effectiveness in an exact closed-form is obtained as a function of the flush ratio α and the Number of Transfer Units NTU, that is, εðNTU, αÞ ¼

NTU 2α + + 2 NTU NTU ð NTU + 2Þ |fflfflfflfflfflffl{zfflfflfflfflfflffl} ε (α¼0 ð1  eNTU Þ α  1  
1  eN TU =α α > 1

(205)

The first term on the RHS of Eq. (205) is the well-known effectiveness εα¼0 deriving from the classical regenerator theory (Nusselt) and applicable only to slowly switched heat regenerators in countercurrent. Eq. (205) says that ε increases linearly with α for α  1. Also, it states that, for α > 1, the effectiveness increases with α but less rapidly than a linear trend. In fact, for α ! ∞ it approaches the unity whichever value is given to NTU. In such a special case, however, only internal gas particles are present within the regenerator and, consequently, it does not work. Fig. 21 shows the regenerator effectiveness as a function of both α and NTU. The effect of flush ratio on ε is considerable when the Number of Transfer Units is low. This effect however decreases when NTU increases. It may also be noted that the effectiveness is a monotonically increasing function with NTU for low values of α. On the contrary, for values of α approximately greater than 0.7, the effectiveness presents a trade-off in NTU (in particular, a minimum value). Once the efficiency is known, the so-called “regenerator losses” due to imperfect heat transfer from gas to matrix and vice-versa may be taken as (1  ε)Q+r , where Q+r is the dimensionless heat stored in the regenerator during each blow. It is given by

Convective heat transfer in different porous passages

189

A

B

Fig. 21 Regenerator effectiveness (A) as a function of NTU with α as a parameter and (B) vs α with NTU as a parameter.

Z Q+r

Z

ðn+1=2Þτ0

ξ¼0 τb ¼nτ0

¼ Z ¼

L

1

Z

hP ðT w  T b Þdτb dξ

hðPL ÞðT h  T k Þτ0 n+1=2
 ϑw  ϑb dτ+b dξ+

ξ+ ¼0 τ+b ¼n

(206)

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where PL ¼ V/rh is the area of the matrix wetted surface. Solving the above integral and using Eq. (205) give 8 ε α α1  > > < 2NTU NTU ðNTU + 2Þ +

 NTU Qr ¼ ε 1 1 4  NTU 2 e α > >   α>1
: 2 NTU NTU NTU  1 + e α 2NTU ðNTU + 2Þ 1 + e α (207) In Stirling regenerator applications NTU is usually greater than 20. For example, in the Philips MP1002CA air engine (250 W), with Tk ¼ 60 ° C, Th ¼ 700 ° C and PL ¼ 0.455 m2 ([64], p. 565), it is found that NTU ¼ 37 and α ¼ 0.3. Therefore, by using Eqs. (205) and (207), ε ¼ 0.949 and Q+r ¼ 0.0126 ()a regenerator loss of about 37 W with h ¼ 100 W/(m2K)). In the United Stirling P-40 “U” engine (4.5 kW), where Tk ¼ 60 ° C, Th ¼ 750 ° C and PL ¼ 4.1 m2 ([64], p. 569), it results in NTU ¼ 130 and α ¼ 1.45. Substituting these values in Eqs. (205) and (207) provides an efficiency of about 0.985 and Q+r ¼ 0.0037 ()a regenerator loss of 310 W), respectively. Eqs. (205) and (207) may in every respect be used in an initial stage of the regenerator design, without the need of numerical computations. On the other hand, it must be pointed out that high temperature recovery cannot be the sole design criterion, as its increase with increasing NTU goes always with a corresponding increase in pumping losses along the regenerator. The optimum NTU value shouldn’t therefore be as high as possible (to maximize ε), but the best compromise between thermal performance and pumping losses, as indicated by Organ [63,64—chapters. 16 and 17].

7. Concluding remarks In the mathematical formulation of heat transfer in porous passages presented in Section 2, it was noted that the Sparrow number, Eq. (11), can define the appearance of the local thermal non-equilibrium conditions. This leads to different sets of mathematical formulations. As given in [11], the condition of local thermal non-equilibrium is expected when the Sparrow number is less than 100. This phenomenon has a direct application in bio-heat diffusion, as presented in the next section. The dual-phase lag bio-heat diffusion equation under local thermal nonequilibrium conditions was derived in Section 3 starting from the two-step

Convective heat transfer in different porous passages

191

model for porous media. By using an appropriate dependent-variable transformation, the “bio-heat” term was canceled out and the DPL diffusion equation was analytically solved by using a Dual-phase lag-based Green’s functions approach. An application to two types of biological tissues subject to thermal therapy when the laser light is highly absorbed from the same tissues was given. A computer code using MATLAB software allowed the temperature distribution of both tissue and blood phases to be obtained as a function of both space and time. In particular, it was noted that the blood temperature is delayed because of the relaxation time between the two phases. Also, higher values of porosity and perfusion rate of the biological tissue allow the cooling effect of the blood within the vessels to be enhanced, so reducing the risk of thermal damage during the heating therapy. The mathematical solutions for heat transfer in porous passages with no flow condition is presented in Section 4. It is shown in Section 5 that both free flow and slip flow conditions provided reasonable limiting values for the Nusselt numbers. Figs. 9 and 10, with no axial conduction, show that the Nusselt numbers for two different geometries have limiting values for large and small MDa values. They approach the Nusselt numbers for free flow when MDa ¼ 1 before the exact limiting value of infinity. However, the Nusselt number when MDa ¼ 1/100,000 is below that, at x ¼ 0, for slug flow condition. Additionally, as shown in Figs. 12–17, neglecting the axial conduction can produce a relatively large error if the axial thermal conduction is neglected. Clearly, even for Pe ¼ 10, this error would be large. However, the effect of the axial conduction becomes negligible when the Peclet number is well above 10. Section 6 outlines the methodology for analytically solving the 1D governing equations of rapidly switched heat regenerators in counter-flow with a uniform gradient of the metallic matrix temperature. It has been shown that, when a Lagrange system is used, it is convenient to classify the gas particles flowing within the regenerator as “cold,” “hot” and “internal” particles according to Organ’s concept of independent flow regimes. Also, the definition of the so-called “cold” and “hot” zones of the matrix gives insight when the gas energy equation is integrated along the path of a selected gas particle. It was found that the phenomenon of the flush phase is able to affect positively and considerably the heat transfer performance of a rapidly switched regenerator, especially when the number of transfer units is low and the flush ratio is high. In fact, when the blow period (i.e., the heating period of the fluid) starts, the fluid gives up heat to the matrix, and only subsequently

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absorbs heat from it. Similarly, when the reverse period (i.e., the cooling period of the fluid) starts, the fluid absorbs heat from the matrix, and only subsequently supplies heat to it. Also, it was found that the one-dimensional, cyclically reversing flow in a regenerator is inevitably accompanied by discontinuities in the lengthwise distribution of temperature. Finally, an easy-to-handle expression for the regenerator effectiveness as a function of NTU and flush ratio has been provided. It indicates that in rapid cyclic flow situations (typical of Stirling regenerators) the effectiveness is underestimated by the conventional regenerator theory as it fully neglects the flush phase. The investigation of the pressure fluctuations remains a subject for future research.

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CHAPTER THREE

Heat exchange between the human body and the environment: A comprehensive, multi-scale numerical simulation J.M. Gormana, Matthew Regniera, and J.P. Abrahamb a

Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, United States University of St. Thomas, School of Engineering, St. Paul, MN, United States

b

Contents 1. Introduction 1.1 Overall solution domain geometry 2. Numerical methods 3. Bioheat modeling 3.1 Pennes’ solution geometry 3.2 Numerical methods for the investigation into Pennes’ model 3.3 Pennes’ model boundary conditions 3.4 Pennes’ model computational grid and convergence 3.5 Pennes’ model thermophysical properties 4. Pennes’ model results 4.1 Heat transfer from the skin surface 4.2 Quantitative Pennes’ model temperature comparisons 5. Computational grid and convergence of the wind chill model 5.1 Boundary conditions 5.2 Thermophysical properties 5.3 Initial conditions 6. Results 6.1 Comparison of published Nusselt number correlations 6.2 Simulation validation using cylinder Nusselt number correlations 6.3 New facial Nusselt number correlation 6.4 Comparison of predicted facial temperatures with measured values 6.5 Temperature contour diagrams 6.6 Discussion of wind chill 6.7 Predicted facial temperatures 7. Concluding remarks References

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Abstract This paper presents a new whole-body numerical simulation model to study wind chill and predict actual facial temperatures. This model is based on a physically realistic healthy human male and includes biothermal heat transfer processes. The human model was exposed to a parametric variation in wind speed (using an atmospheric wind velocity profile), wind temperature, and two different wind directions. For the range of wind speeds under investigation (U10 from 1 to 20 m/s) the corresponding Reynolds number range based on UFace was between 8000 and 150,000, and for the Reynolds number based on U10, the range was 10,000–220,000. The practical results of the present work include new Nusselt number correlations for the human face for both a forward wind and a side wind direction. The new Nusselt number correlation for the forward wind direction ranged between 3% and 34% lower than a previously reported correlation in the literature. In addition, the predicted facial temperatures, as a function of time, are reported for the range of wind speeds and the temperatures investigated. The present facial temperatures compared favorably to the limited available experimental data reported in the literature. A parametric study of Pennes’ equilibrium constant was performed. The current model successfully predicted most of the measured data to within 5%. The success of the model encourages its application to other parts of the body and other thermal processes.

Abbreviations A cp F1, F2 h k kturb Pk Pr Prturb Q Re S T T∞ u ui xi

SST model constant specific heat at constant pressure blending functions in the SST model local heat transfer coefficient thermal conductivity turbulent thermal conductivity production term for the turbulent kinetic energy Prandtl number turbulent Prandtl number rate of heat transfer Reynolds number absolute value of the shear strain rate temperature inlet free stream air temperature free stream velocity component in the x-direction free stream velocity component in the i-direction tensor coordinate direction

Greek symbols α β1, β2 ε

SST model constant SST model constants turbulence dissipation

Heat exchange between the human body and the environment

κ μ μturb ν ρ σ ω

199

turbulent kinetic energy molecular or dynamic viscosity turbulent viscosity kinematic viscosity fluid density diffusion coefficient specific rate of turbulence dissipation

1. Introduction In cold climates, the concept of wind chill, more specifically the Wind Chill Index (WCI) or the Wind Chill Equivalent Temperature (WCET), has been of interest for both human safety and human comfort. Unfortunately, creating a universal metric is made complicated by many factors. From a biological standpoint, individuals exposed to a cold environment will respond differently based on factors such as age, size, health, activity level, clothing, acclimated to a local climate, time spent in a warm environment prior to a cold environment, etc. From an environmental standpoint, factors such as buildings/structures, snow/moisture, sky conditions (clouds, sun), time of day, etc., in addition to wind speed and air temperature values, also play an important role. In order to develop the best possible wind chill metric (or model) it is necessary to examine the important critical factors and accurately incorporate them into the analysis. Multiple models and investigations exist in the published literature and will be examined in the forthcoming paragraphs. These investigations can be divided into three different categories (experimental, analytical, and numerical), although there is often overlap. Experimental studies range from inanimate objects used as a surrogate human to actual human experiments in cold environments [1–7]. Analytical studies are limited to one-dimensional or resistive network-based models that make use of heat transfer coefficients taken from the published literature [8–18]. The third category, numerical, is comprised of three-dimensional humanoid models that are simulated with computational fluid dynamics (CFD) or finite element analysis (FEA) to obtain more realistic conditions. Currently only one paper in the published literature focused on wind chill falls into this latter category, and it studied frostbite of a single human finger [19]. Each of these three approaches has associated advantages and disadvantages that will become apparent.

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Credit is given to Siple and Passel for pioneering the use of the wind chill index to provide guidance on the increased severity of surrogate skin cooling due to a combination of cold air and wind [1]. In 1945, they reported their dry atmospheric cooling measurements for various subfreezing temperatures in Antarctica. Measurements were conducted using 250 g of water in a cylinder of Pyralin, a cellulose nitrate. An empirical formula for the cooling rate of the water was used to summarize their results. The original wind chill index was created with a dimensionless scale of numbers and descriptors based on wind velocity and air temperature. Later, in 1995, Osczevski [2] reviewed the wind chill index created by Siple and Passel and performed experiments in a wind tunnel using a thermal model of a head. The thermal model had heating circuits and thermally conducting skin put onto a polyurethane foam head shape to facilitate the wind tunnel experiments. Osczevski criticized Siple and Passel for having too thin of a plastic wall to model human tissue. His facial heat loss experiments showed that the plastic wall used by Siple and Passel overestimated the amount of heat transfer from a human’s face. Osczevski created very simple empirical relationships for the heat transfer from the “face” in the form of a heat transfer coefficient and thermal resistance of a cheek. This model agrees with the trend created by Siple and Passel; however, it increased the wind chill equivalent temperature by 20 °C because of the larger Osczevski cheek thermal resistance than the Pyralin cylinder used by Siple and Passel. In general, the experiments by Osczevski were limited and relied on many assumptions. The physical model was also confined to the face and head. The currently used wind chill temperature index was implemented in 2001 by the US National Weather Service [4,20,21]. This “new” index was developed to include human factors such as blood perfusion, which had not been considered previously. An experimental study was paired with a numerical method for calculating the new index. A total of six male and six female subjects had thermal transducers attached to their faces while walking on a treadmill in a chilled wind tunnel. The skin (cheek) thermal resistance, measured as the heat loss rate divided by the temperature difference between the internal body and the cheek, of the most susceptible of the 12 subjects was used to model the 95th percentile of the population as the worst case scenario. This was then used as the basis of the new wind chill temperature (WCT) index. The supporting theoretical calculations for the study were accomplished by analytically modeling the human face as a cylinder in cross flow (a rather incorrect model). A simple numerical solution was used to find the heat loss rate for a base wind speed of 3 mph. The physical measurements were compared to the numerical solution to find the wind speed and

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temperature that corresponded to a “still” ambient condition temperature, or the wind chill temperature. In 2002 Osczevski and Tikuisis improved upon their previous experiment by analyzing the effect of a varying thermal resistance and cheek dimensions [8]. It was concluded that experimentally decreasing the thermal resistance of the cheek increases the blood perfusion and therefore the heat gain (loss) to the skin. Other thermal effects, such as radiation from the sun or a clear night sky were ignored. Again, this model used a simplistic hollow cylinder with a fixed internal boundary temperature. Another prominent author on the topic of wind chill is Schitzer [10–12,17,18,22] who uses analytical methods to quantify the heat exchange from the human head. The head is modeled as a hollow cylinder in cross flow with several layers (annuli) to represent the blood circulation. The results show that an increased blood perfusion to the face decreases the wind chill effective temperature, while a lower blood perfusion should intuitively cause frostbite to occur quicker [11]. Schitzer also studied the effect of varying convective heat transfer coefficients on the Wind Chill Effective Temperature (WCET) [10]. Several assumptions were made for modeling the cylinder: thermal steady state, homogenous material thermal properties, no metabolic effects, a constant internal body temperature, and no evaporation or solar radiation effects. In addition, the models often relied on heat transfer coefficients created for cylinders in cross flow from the literature. Fiala also developed a model [15,16,23,24] of human thermal regulation and comfort for the Universal Thermal Climate Index, which considers the effect of clothing. The model is made of 12 spherical and cylindrical components, all built in concentric tissue layers. Fiala used Pennes’ bio-heat equation to model the body’s heat and mass transport. The Fiala model had 187 tissue nodes with unique tissue material properties such as thermal conductivity, density, heat capacity, basal heat generation rations, and basal blood perfusion rates. The results are limited by the crude nature of a spherical and cylindrical model and use of heat transfer coefficients from the literature specific for cylinders in cross flow. Detailed reviews of the wind chill literature and the shortcomings of the present models can be found in Refs. [18,22,25]. More generic literature reviews focused on heat transfer in human body modeling can be found in Refs. [26,27]. The underlying concept behind wind chill is to take account of weather conditions, such as temperature and wind speed, and related its effect on a biological (human) system. These metrics are then used to give people a more informed idea how to prepare themselves before entering cold/winter

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environments to avoid frostbite/hypothermia [20], for rescue personnel to predict survival times in extreme circumstances [25], or for human thermal comfort [28]. The plan of the present investigation is to start by examining a human male model, approximately 25–30 years old, for the basis of an anatomically correct numerical wind chill simulation to compare with existing wind chill related studies and models. The numerical approach, aided and abetted by powerful computing resources, permitted the use of a realistic model that avoids the simplifying assumptions of the published literature. In addition to developing a predictive model that does not rely on literature heat transfer coefficients, this new model will provide detailed results not obtainable by experimental or analytical methods. The present results will be focused on the human face, and parameters such as wind speed and wind direction will be varied. However, the entire human body including appropriate clothing will be included in the overall model. The present human model will include thermal-biological processes such as metabolic heat generation and blood perfusion.

1.1 Overall solution domain geometry In order to create the present human model, geometry files for the human anatomy and clothing were obtained from both CGTrader (www.cgtrader. com) and TurboSquid (www.turbosquid.com), respectively. By the use of the acquired geometry files as a template, the present authors edited, modified, and prepared the files for use. The resulting human and clothing geometry to be used in the present work are shown in Fig. 1. The solution

1.73 m

0.514 m

Fig. 1 Schematic diagram of the human body used in the numerical simulations showing dimensions.

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1.83 m

2.39 m

Y

4.33 m

X Z

Fig. 2 Isometric view of the modeled solution domain around the human model.

domain surrounding the human model, shown in Fig. 2, was created to give ample distance between the human model and the boundary conditions. Specific details about the boundary conditions and the human model will be discussed in the forthcoming sections.

2. Numerical methods The fluid flow to be investigated here will encompass both the laminar and turbulent flow regimes. The simulations for the turbulent case will make use of the Reynolds-Averaged Navier-Stokes (RANS) equations, in conjunction with an appropriate turbulence model. Here, it was chosen to use the Shear Stress Transport (SST) κ-ω turbulence model [29,30] since it has been repeatedly shown to give good agreement with a wide range of experimental comparisons [31–39]. The SST model is a blending of two different but well-known turbulence models, respectively, the κ-ω and κ-ε models. The former has been shown to provide very good results in the near-wall region whereas the latter model gives good results far from a wall boundary. In Cartesian tensor notation, the RANS and continuity equations for unsteady, three-dimensional, constant property turbulent flow are, respectively; ρ



 ∂u j ∂u j ∂u j ∂p ∂ ¼ i ¼ 1, 2, 3 j ¼ 1, 2, 3 (1) + ρ ui + ðμ + μturb Þ ∂t ∂xi ∂xi ∂x j ∂xi ∂u j ∂ρ + ¼0 (2) ∂xi ∂t

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In Eqs. (1) and (2), ρ is the fluid density, u is a velocity component, t is time, x is a direction coordinate, p is pressure, and μ is the dynamic viscosity. The term μturb denotes the turbulent or eddy viscosity. In reality, this quantity was created to model the Reynolds stresses, given by μturb

∂u j ¼ ρu0i u0j ∂xi

(3)

where the ui0 represent the fluctuating component of ui. To calculate the numerical values of μturb, it is necessary to use a turbulence model. The application of the SST κ-ω turbulence model involves solving a pair of simultaneous partial differential equations that are linked to the RANS and mass conservation equations. The κ and ω equations, which are discussed in detail in Refs. [29,30], are   ∂ðρκ Þ ∂ðρui κÞ ∂ ∂κ ∗ + ¼ P κ  β ρκω + ðμ + σ κ μturb Þ (4) ∂xi ∂xi ∂t ∂xi and   ∂ðρωÞ ∂ðρui ωÞ ρ ∂ ∂ω + ¼α P κ  βρω2 + ðμ + σ ω μturb Þ ∂t ∂xi μturb ∂xi ∂xi 1 ∂κ ∂ω + 2ρð1  F 1 Þσ ω2 ω ∂xi ∂xi

(5)

where the turbulent viscosity μturb is obtained from μturb ¼

α1 ρκ max ðα1 ω, SF 2 Þ

and F1 is a blending function given by
pffiffiffi  
4 κ 500μ 4ρσ ω2 κ F 1 ¼ tanh min max ∗ , 2 , β ωy ρy ω CDκω y2 where y is the distance from a wall boundary, and
 1 ∂κ ∂ω 10 CDκω ¼ max 2ρσ ω2 , 10 ω ∂xi ∂xi

(6)

(7)

(8)

In addition, 
pffiffiffi  2 2 κ 500μ F 2 ¼ tanh max ∗ , 2 β ωy ρy ω

(9)

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The term α in Eq. (5) is equal to α1F1 + α2(1  F1), where α1 and α2 are known model constants. Similarly, the term β is equal to β1F1 + β2(1  F1), where β1 and β2 are also model constants; the same is also true for the terms σ κ and σ ω. A complete listing of the constants and coefficients that were used in the preceding equations for the SST κ-ω model are given in Table 1. For laminar flow, the quantity μturb that appears in Eq. (1) is removed. The remaining terms in Eq. (1) constitute the Navier–Stokes equations which are appropriate for laminar flow. The energy equation for the fluid domain is   ∂ðui T Þ ∂T ∂ ∂T ρc p + ρc p ¼ ðk + kturb Þ (10) ∂xi ∂t ∂xi ∂xi In this equation, T is the fluid temperature, cp is the specific heat at constant pressure, ρ is the fluid density, and k is the thermal conductivity. The quantity kturb is commonly referred to as the turbulent thermal conductivity. Its value is closely connected to that of the turbulent viscosity by means of the turbulent Prandtl number, Pr turb ¼

c p μturb kturb

(11)

There is extensive research that has shown that a constant value of Prturb ¼ 0.85 [40,41] gives rise to highly accurate heat transfer results. The energy equation for the solid domains are
2  ∂T ∂T ρc + S_ met + S_ perfusion i ¼ 1, 2, 3 (12) ¼k ∂t ∂x2i where S_ met is the metabolic heat generation inside of the human tissue and S_ perfusion represents the contribution of blood perfusion to energy transfer within the tissue. For the clothing, these two source terms are omitted from Eq. (12). The numerical values of S_ met are obtained based on location in the human body whereas S_ perfusion comes from the well-known approach by Pennes [42]. The perfusion source term takes the form   (13) ρc p b ψ ð1  ζÞðT b  T t Þ Table 1 Dimensionless constants for the SST κ-ω turbulence model. Constant α1 α2 β∗ β1 β2 σ ω1 σ ω2

σ κ1

σ κ2

Value

0.85

1

5/9

0.44

0.09

0.075

0.0828

0.5

0.856

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where the subscripts t and b refer to local tissue and to blood, respectively, ψ is the tissue-specific perfusion rate, and ζ is an equilibrium constant between capillary blood and the tissue, which is assumed to be uniform throughout the tissue where 0  ζ  1. In the present model, the temperature Tb is the mass-weighted average temperature of the blood within a specific region of the body, and Tt is the local tissue temperature anywhere within the body. Based on the work presented in Ref. [43], ζ is believed to be 0.6. In the present work, a separate investigation to determine the appropriate value of ζ will be performed and is presented in the following sections.

3. Bioheat modeling Modeling bioheat transfer in the human body is an area of decadeslong interest, and one of the most influential papers in that regard was by Pennes in 1948 [42]. In that paper, Pennes used experimental data obtained for the human arm to deduce a model for bioheat transfer. That model was represented by a simple one-dimension bioheat transfer equation that is still widely used today [44–46]. Pennes’ original paper received multiple criticisms relating to his analysis of the experimental data, the simplicity of the model, and the use of isotropic perfusion in tissue. In 1998, Wissler [47] re-analyzed Pennes’ data to address the concerns of some investigators and showed that the Pennes bioheat equation was supported by Pennes’ measured temperature profiles. Despite the criticism, the Pennes bioheat equation has been widely used in a variety of numerical simulations where the whole or part of a human body have been modeled as a collection of cylindrical elements [43] or with a homogenous lumpedtissue approach [48]. Alternatives to the Pennes’ model include using effective material properties, such as an effective thermal conductivity, or modeling the tissues as porous media [49]. These numerical approaches have provided useful information and results but have thus far been limited to the restrictions of one-dimensional equations and lumped analyses. For instance, limitations of these approaches include: (1) relying on heat transfer coefficients for an external cylinder, (2) use of physiologically incorrect geometry, (3) inability to obtain threedimensional temperature distributions within individual tissues, and (4) not directly modeling the circulatory system (blood flow) in contact with the tissue. The one-dimensional/lumped analyses have the advantage of being solved easily and quickly, but lack the detailed information needed for certain biomedical research and applications, such as implantable medical devices or medication transport.

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A great deal of research is currently being done to create highly detailed and accurate biomedical human circulatory models [50] (which eliminates the need for Pennes’ model), but the computational resources needed for these models are still beyond those of most practitioners. As a result, numerical approaches that make use of Pennes’ model are expected to continue into the near future. To bridge the gap between the one-dimensional/ lumped approaches and the highly detailed approaches (modeling every capillary), the present authors are investigating whether a hybrid model using Pennes’ bioheat equation for blood perfusion in tissue can be combined with a more detailed physically realistic three-dimensional model. Here, Pennes’ model will be used in perfused tissues, and major arteries or veins will be directly modeled. Since Pennes’ bioheat equation was created from data from a section of the human arm, it is appropriate to start with the same physical situation. This will be done by numerically modeling a realistic non-homogenous section of the human arm, using Pennes’ equation when appropriate, and comparing the results with available temperature information.

3.1 Pennes’ solution geometry In order to examine the application of Pennes’ bioheat equation in a threedimensional model, a section of a human arm was chosen in the region that Pennes originally used for his experimental and theoretical work in Ref. [42]. Fig. 3 shows the general region of the human arm under investigation. As seen in the figure, the segment to be modeled is 3.0cm long and located

3.0 cm

36 cm

Fig. 3 Three-dimensional arm segment chosen for the investigation on Pennes’ bioheat model.

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approximately 36 cm from the end of the hand on the right arm of a 30-year-old male. Using the STL CAD model dimensions as a template, the present authors formed the relevant arm components into a useable geometry for simulation. In some instances, such as with the skin layers, the geometry had to be idealized (uniform thicknesses and smoothed surfaces). Fig. 4 provides a detailed breakdown of the various components to be included in the present model. These regions include the skin layers (dermis and epidermis), bones, muscles, subcutaneous fat, interstitial tissues, and the artery and vein walls including the blood. The circumference of the arm at this location was approximately 25.2 cm and has an effective diameter of 8.02 cm (matching the arm diameter of Ref. [47]). The epidermis layer was 0.08 mm thick; the dermis layer was 2 mm thick; and the subcutaneous fat layer was ranged from 0.5 to 10 mm thick depending on the location. The skin layers, subcutaneous fat, and interstitial A

Subcutaneous fat Arteries

Bones

Interstitial tissue

Skin layers Muscles

Veins

B

C

D

Fig. 4 Detailed cross-sections of the simulated arm segment. (A) Front view showing components of the model. (B) Isometric view showing muscle tissue regions within the arm segment. (C) Isometric view showing bone regions within the arm segment. (D) Isometric view showing artery and veins within the arm segment.

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Table 2 Volume of each component in the arm segment. Component Volume (cm3)

Arterial blood

0.42

Artery tissue

0.31

Bones

8.83

Dermis

7.78

Epidermis

1.53

Interstitial tissue

36.18

Muscles

46.53

Subcutaneous fat

19.18

Vein blood

0.58

Vein tissue

0.37

Total volume

121.71

tissue were assumed to be continuous (uninterrupted) regions. As seen in Fig. 4B–C, there are 13 unique muscle regions, 2 separate bone regions, 2 main arteries, and 7 veins. The modeled arteries and veins were between 0.4 and 3.75 mm in diameter; any artery or vein less than 0.4 mm was not directly modeled. In addition to the aforementioned physical dimensions, Table 2 lists the physical volumes occupied by each of the modeled components in the arm segment. In addition to the segment of the human arm it was necessary to model the air region surrounding the arm. This allows for natural convection and radiation from the arm to the environment to be modeled directly instead of relying on heat transfer coefficients. Fig. 5A shows the wireframe of the complete model with the surrounding air region and Fig. 5B shows the coordinate system, which will be referenced later. The air solution domain is 30 cm tall, 30 cm wide, and 3 cm deep.

3.2 Numerical methods for the investigation into Pennes’ model The simulations encompass two fluid domains including the air surrounding the arm and the blood flowing in the arteries and veins. Any fluid motion external to the arm is caused by natural convection between the skin and the ambient air and is expected to vary with time. The internal blood flow,

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A

B

y 30 cm x z

30 cm

Fig. 5 Wireframe of the numerical model showing the arm segment with the surrounding air solution domain. (A) Front view of the complete solution domain. (B) Isometric view of the solution domain showing the origin located in the center of the model.

pumped by the heart, is specified as a time varying mass flow boundary condition. In both fluid domains, the fluid flow will be in the laminar flow regime and the simulations will make use of the unsteady Navier-Stokes equations. In Cartesian tensor notation, the equations of fluid motion and continuity for unsteady, three-dimensional, constant property flow are, respectively ∂ρu j + ∂t



 ∂ρui u j ∂u j ∂p ∂ ¼ i ¼ 1, 2, 3 j ¼ 1, 2, 3 (14) + ρg j + μ ∂xi ∂xi ∂xi ∂x j ∂ρu j ∂ρ + ¼ 0: (15) ∂t ∂xi

In Eqs. (14) and (15), ρ is the fluid density, u is a velocity component, t is time, x is a direction coordinate, p is pressure, g is gravity, and μ is the dynamic viscosity. Compared to Eq. (1), Eq. (14) includes gravity to account for natural convection around the arm and does not contain the eddy viscosity term because the fluid flow is expected to be laminar (to match Pennes’ conditions). Heat transfer will occur both in the fluid domains and in the solid domains. The energy equation for the fluid domain is
2  ∂ðρui T Þ ∂ρT ∂T cp + cp ¼k (16) ∂t ∂xi ∂x2i In this equation, T is the fluid temperature, cp is the specific heat at constant pressure, ρ is the fluid density, and k is the thermal conductivity.

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As a reminder, the energy equation for the solid tissue domains are ρc p


2  ∂T ∂T + S_ met + S_ perfusion i ¼ 1, 2, 3 ¼k ∂t ∂x2i

(17)

where S_ met is the rate of metabolic heat generation inside of the human tissue and S_ perfusion represents the contribution of perfusion inside some of the tissues. The numerical values of S_ met are prescribed based on location (tissue type) and assumed to be spatially uniform and temporally constant, whereas S_ perfusion comes from the approach by Pennes [42] and varies with local temperature. The perfusion source term is found from   S_ perfusion ¼ ρc p b ψ ð1  ζÞðT a  T t Þ

(18)

where the subscripts t and a refer to local tissue and arterial blood, respectively, ψ is the tissue-specific perfusion rate, and ζ is an equilibrium constant between capillary blood and the tissue, which is assumed to be uniform throughout the tissue where 0  ζ  1. For the present numerical calculations, the temperature Tb is the mass-based average temperature of the blood within the arteries and Tt is a local tissue temperature anywhere within the tissues with perfusion. Pennes, as stated in his original paper, believed the physical nature of capillary circulation favored thermal equilibrium; meaning ζ approaches 1.0. However, if ζ ¼ 1.0, there would be no perfusion. Determining the proper values for ζ is difficult and limited work has been done in the literature to obtain appropriate values. Pennes [42] and other investigators [48] have set ζ ¼ 0, whereas some researchers believe ζ values between 0.3 and 0.9 are appropriate depending on the physical situation [51,52]. Other investigators conclude ζ is an immeasurable quantity and chose an arbitrary value of 0.6 that is in the middle of expected values [43]. Either ψ or ζ could be implemented as a spatially varying quantity to improve results based on human physiology (such as in Ref. [51]) or either quantity could be varied in conjunction with mass transport of medication that directly impacts perfusion. Here, ζ will be varied parametrically between 0, 0.6, 0.9, and 1 to see the influence on the results. Changing the value of ζ within the tissue is expected to have a similar result to directly varying the perfusion values. Since the present model is a hybrid (using the Pennes model in perfused tissue and modeling major arteries and veins larger than 0.4 mm), it is difficult to know the proper usage of ζ a priori. Ideally, with the present modeling goals, setting ζ ¼ 0 would be the best scenario to avoid specifying somewhat arbitrary values.

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Therefore, ζ ¼ 0 will be considered the base case with which other simulations will be compared. Radiation heat transfer from the skin surface to the surrounding environment was modeled using a Monte Carlo method, assuming a gray body model (emissivity ¼ absorptivity). For each iteration, 100,000 photon packets were emitted from a randomly determined location on the external surface of the solution domain with a randomly determined trajectory and tracked until being absorbed or exiting the solution domain.

3.3 Pennes’ model boundary conditions

Mass flow rate (kg/s)

As seen in Fig. 5, the arm segment is surrounded by an air domain. The boundary condition at the perimeter air domain (the region external to the arm segment) was fixed at an ambient gauge pressure of 0 Pa and a temperature of 26.6 °C to match the conditions used in Pennes’ experiments. For the radiation boundary condition, radiation entering from the far surface was set to be from blackbody emitting surfaces at 26.6 °C. The skin surface had a specified emissivity of 0.95 [43]. The positive and negative z-faces of the air domain are treated as symmetry boundaries, and the positive and negative z-faces of the solids/tissues were assumed to be adiabatic. The adiabatic condition coincides with the assumption that tissue temperatures just upstream or downstream from the solution domain are virtually identical. The blood flow through the arteries and veins was prescribed as a repeating time-varying mass flow rate, as shown in Fig. 6, from the results presented in Ref. [53]. The blood entering the solution domain through the arteries was set at a temperature of 36.8 °C to correspond with the analysis done 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 -0.001 0.0

0.2

0.4

0.6

0.8

1.0

Time (s)

Fig. 6 Inlet blood mass flow rate variation with time determined from Ref. [53].

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by Wissler of Pennes’ work in Ref. [47]. The blood entering the solution domain from the veins traveling in the opposite direction of the artery blood flow was an unknown value. To avoid prescribing a fixed value each vein inlet, the temperatures were specified as an average temperature of the surrounding tissue and was subsequently reevaluated every time step.

3.4 Pennes’ model computational grid and convergence The grid or mesh for each simulation was generated using ANSYS Meshing, and the numerical solutions were generated using ANSYS CFX 19.1 software. A mesh independency study using several different computational meshes were employed by systematically varying the node count (a node is the location where the calculations are performed). For the mesh independence study, it was found that increasing the total node number from 1.3 million (5.3 million elements) to 19.2 million nodes (94.9 million elements) gave little to no noticeable changes in the results, i.e., changes were less than 0.1% (quantities, such as area-average skin temperature changed from 33.54 to 33.56 °C with the mesh size increase). A representative image of the mesh used for the presented results is shown in Fig. 7. The solutions were considered to be converged to sufficient accuracy when the root-mean-square (RMS) residuals for all of the governing

Fig. 7 Representative image of the mesh used for the presented results. (A) Image of the mesh in the air region away from the arm. (B) Typical mesh at the skin-air (fluid-solid) interface. (C) Representative mesh at a tissue–tissue interface.

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equations were 106 or less. The simulations were performed as a transient, using a time step of 0.001 s and were run for of 5 min of simulated time. The RSM currant number for the simulations was 0.03 or less. The CPU time was 150 h for 1 s of simulated time.

3.5 Pennes’ model thermophysical properties In order to perform the numerical simulations, it is necessary to know the material properties of each fluid and solid material. Table 3 has been prepared to list all of the material properties used in the present simulations. The biological/tissue material values listed in the table represent average values of what is reported in the literature [43,48,54–68]. For air, to account for natural convection surrounding the arm, the Ideal Gas Law is used to determine the fluid density. The ambient environment was at atmospheric pressure and 26.6 °C to match Pennes’ original experimental measurements. For the blood, in recognition of the blood’s viscosity sensitivity with temperature, a temperature-dependent expression (in Celsius) from Ref. [43] was used where 0.00307 Pa-s is the viscosity at 37 °C. In addition to the material properties, the metabolic heat generation and volumetric perfusion values are needed for several components, which are listed in Table 4.

Table 3 Thermophysical material properties obtained from averaging values from Refs. [43,48,54–68]. Thermal Density Specific heat conductivity Viscosity (W/m °C) (Pa s) Material (kg/m3) (J/kg °C)

Air

P/RT

1004

0.0261

1.831 105

Blood

1055

3575

0.605

0.00307e0.015(37-T)

Artery/vein

1068

3700

0.42

–

Muscle

1068

3700

0.42

–

Interstitial tissue

850

2405

0.16

–

Skin

1042

3725

0.34

–

Subcutaneous fat

850

2500

0.20

–

Bone

1528

1835

1.51

–

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Table 4 Metabolic heat generation and perfusion values from Refs. [47,48,55,57,58,69]. Volumetric perfusion Metabolic heat generation (mL/mL s) Material (W/m3)

Artery/vein

0

0.0005

Muscle

694

0.000538

Interstitial tissue

3.4

0.0003146

Subcutaneous fat

3.4

0.00000422

Skin (dermis)

500

0.001477

4. Pennes’ model results 4.1 Heat transfer from the skin surface Both natural convection and radiation between the skin surface and surrounding environment were modeled. The resulting spatially and temporally average skin temperature was 33.56 °C when the surrounding environmental temperature was fixed at 26.6 °C; the skin had an emissivity of 0.95, and ζ ¼ 0. The temporal and spatial average convective heat flux was 22.15 W/m2 and the radiative heat flux was 41.8 W/m2. Using the standard definition of a heat transfer coefficient h ¼ q/(Tskin  Tambient), the corresponding convective heat transfer coefficient would be 3.18W/m2 °C whereas the radiative heat transfer coefficient would be 6.0W/m2 °C. The combined heat transfer coefficient from these results (9.18W/m2 °C) is remarkably similar to the value used by Wissler in Ref. [47] for a horizontal cylinder, 8.37 W/m2 °C. A representative image of the results for ζ ¼ 0, at a typical moment of time, is shown in Fig. 8. Fig. 8A shows a color contour temperature diagram and Fig. 8B shows a velocity vector diagram in the air region through the center cross section (z ¼ 0 cm) of the solution domain. The natural convection plume is clearly visible in both images.

4.2 Quantitative Pennes’ model temperature comparisons The first comparison of the temperature results will be with Pennes’ experimental data and theoretical results. Several arbitrarily chosen experimental datasets were extracted from Ref. [47] in addition to Pennes’ theoretical model and has been reproduced in Fig. 9. The present simulation results corresponding to ζ ¼ 0 have also been plotted on Fig. 9 at a position along

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Fig. 8 Representative results at z ¼ 0 cm for a typical moment of time. (A) Color contour diagram of temperature. (B) Vector diagram for velocity in the air region surrounding the arm. 1.20

Temperature

1.00 0.80 Present simulation ζ = 0 Pennes theory Subject 1 Subject 2 Subject 4 Subject 5 Subject 7 Subject 8

0.60 0.40 0.20 0.00 -1.20

-0.80

-0.40

0.00

0.40

0.80

1.20

r / Rarm

Fig. 9 Dimensionless temperature profiles from the present simulation where ζ ¼ 0 and from data extracted from Ref. [47].

the x-axis from 1  x/Rarm  1 at z ¼ 0 cm (Rarm ¼ 4.01 cm). The temperatures were made dimensionless using (Tx  Ts)/(Ta  Ts) where Tx is the temperature along the x-axis, Ts is the area-averaged skin temperature (33.56 °C), and Ta is the artery temperature (36.8 °C). As can be seen in Fig. 9, there is some scatter in the experimentally measured data points, and Pennes’ theoretical curve is symmetric about

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Heat exchange between the human body and the environment

the center. As expected, the present simulations do not provide a perfectly symmetric curve since the temperature results are being extracted along a line crossing multiple dissimilar tissues. Overall, there appears to be good agreement between the present simulation and the data extracted from Ref. [47], which suggests ζ ¼ 0 was appropriate. Since the exact location and geometry of each measurement Pennes made in Ref. [42] are not precisely known, it is worthwhile to examine the present temperature results at several positions along the arm. Fig. 10 has been created to display temperature profiles along the x-axis from 1  x/Rarm  1 at z ¼ 1, 0, and 1 cm. As can be seen in the figure, there is an observable, however small, temperature change at different positions in z. The largest temperature differences between these three locations are on the order of 3%–6%. This result indicates that temperature data extracted along the x-axis anywhere in this region would provide similar temperature profiles. The next set of temperature results to be compared is for the parametric variation the equilibrium constant ζ for values of 0, 0.6, 0.9, and 1, and is displayed in Fig. 9. The absolute range of ζ is between 0 and 1; Pennes set ζ ¼ 0 in his original work [42] and ζ ¼ 1 is a special case where there is no thermal contribution from perfusion. The incoming arterial blood temperature in all cases was maintained at 36.8 °C; however, the resulting average skin temperature varied with changing values of ζ. Values of ζ ¼ 0, 0.6, 0.9, and 1 corresponded to skin temperatures of 33.56, 32.43, 31.05, and 30.15 °C, respectively. 1.20

Temperature

1.00 0.80 0.60 Centerline at z = +1 cm 0.40

Centerline at z = 0 cm Centerline at z = -1 cm

0.20 0.00 -1.20

-0.80

-0.40

0.00

0.40

0.80

1.20

r / Rarm

Fig. 10 Dimensionless temperature profiles from the present simulation where ζ ¼ 0 along the x-axis from 1  x/Rarm  1 at z ¼ 1, 0, and 1 cm.

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1.20

Temperature

1.00 0.80 0.60

ζ=0 ζ = 0.6 ζ = 0.9 ζ=1 Pennes data

0.40 0.20 0.00 -1.20

-0.80

-0.40

0.00

0.40

0.80

1.20

r / Rarm

Fig. 11 Dimensionless temperature profiles from the present simulation where ζ ¼ 0, 0.6, 0.9 and 1 along the x-axis from 1  x/Rarm  1 at z ¼ 0 cm and Pennes’ temperature data extracted from Ref. [47].

Examination of Fig. 11 reveals that the results are very sensitive to ζ, and the temperature profiles become less symmetric with increasing values of ζ. From the figure, based on the data points, it is clear that a value of ζ ¼ 0 leads to dimensions temperature profiles near the upper bound and ζ ¼ 0.6 provides a lower bound. Quantitative deviations of Pennes’ data from the curves obtained by numerical simulation were calculated. It was found that 85% of the data points were within 5% of the ζ ¼ 0 curve. Correspondingly, for the ζ ¼ 0.6 curve, 73% of the data fell within 5% deviation. The outcome of this comparison suggests that the ζ ¼ 0 representation is more favorable.

5. Computational grid and convergence of the wind chill model The grid or mesh for each simulation was generated using ANSYS Meshing, and the numerical solutions were carried out using ANSYS CFX 19.1 software. The non-dimensional near-wall mesh-quality metric known as y + was used to ensure the computational mesh near fluid-solid boundary surfaces was sufficiently resolved. For the SST κ-ω turbulence model, y+ values on the order of 1.0 or less are preferred to achieve the most accurate results. In addition to obtaining appropriate y + values, a mesh independency study using several different computational meshes was employed by

Heat exchange between the human body and the environment

219

systematically varying the node count (a node is the location where the calculations are performed). For the mesh independence study, it was found that increasing the node number from 24 million to 57 million nodes gave rise to variations of the Nusselt number no greater than 0.2%. A representative image of the mesh is shown in Fig. 12.

Fig. 12 Representative image of the mesh used in the simulations.

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The solutions were considered to be converged to sufficient accuracy when the root-mean-square (RMS) residuals for all of the governing equations were 106 or less. The simulations were performed as a transient using time steps between 0.1 and 0.0001 s depending on wind speed to keep the RSM Courant number for all of the simulations near 1.0.

5.1 Boundary conditions The inlet boundary condition for velocity was chosen to be the atmospheric boundary layer (ABL). There are multiple profiles and equations in the published literature that purport to provide an algebraic representation of the ABL. The selected algebraic equation describing the ABL velocity profile was taken from Ref. [70] and is stated in Eq. (19).
 y + y0 u∗ (19) UðyÞ ¼ ln K y0 In addition to the velocity profile, the turbulence characteristics of the ABL are needed and are given in Eqs. (20) and (21). u2 κ ¼ p∗ffiffiffiffiffiffi Cμ εðyÞ ¼

u∗ 3 K ðy + y0 Þ

(20) (21)

In the foregoing, U(y) is the streamwise velocity at the inlet to the solution domain, κ is the turbulence kinetic energy, ε(y) is the eddy dissipation rate, K is the von Karman constant (0.42), y is the height above ground, y0 is the surface roughness, and Cμ is a model constant (0.09). The friction velocity u∗ is defined in Eq. (22). u∗ ¼

K U y0 0

ln y y+y0

(22)

0

In this equation, Uy0 is a specified reference velocity at a height y0 . The value of y0 was taken to be that of grassland (y0 ¼ 0.03 m), as given in Ref. [71]. The calculated velocity profiles in this ABL model are based on reference velocities that occur 10 m from the ground (Uy0 ¼ U10). The aforementioned ABL equations have been shown to have good agreement with experimental measurements [72].

Heat exchange between the human body and the environment

221

For the present study, U10 was varied for values between 1 and 20 m/s. This corresponds to velocities in the range of 0.7–13 m/s at the person’s face (between 1.4 and 1.7 m from the ground). The wind directions were limited to head-on (0°) and from the side (90°). In addition to the inlet cross section and the ground (no slip) boundary conditions, all of the remaining boundaries were specified as openings (entrainment) at an ambient gauge pressure of 0 Pa. At the opening boundaries, the turbulence was specified as a zero gradient. These “weak” boundary conditions allowed the air to enter or leave the solution domain, at any velocity or direction, without prescribing or imposing values of turbulence at the opening’s surfaces.

5.2 Thermophysical properties In order to simulate the present model, the thermophysical properties are needed for all of the components, such as the air, the biological materials (skin, bone, human hair, etc.) and the clothing. For the biological tissues, information about metabolic heat generation and blood perfusion are also required. The human hair, which is to be modeled as a porous medium, requires additional information about porosity and pressure drop. The human model was divided into several convenient segments; head, hair, torso, arms, hands, legs, feet, and clothing. Within each of these segments the model was further subdivided into categories of biological tissue; core, muscle, fat, skin, and bone. The term core in this case represents various organs such as the brain, liver, stomach, etc. Detailed information about the volume and mass of each of the biological components in the current model is listed in Table 5 for reference. The clothing inherently had non-uniform contact with the external human skin creating air gaps between the clothes and the skin. The present human-body model, as shown in Fig. 1, is approximately 70% biological tissues by volume, and the remaining 30% is split between the clothing and the air gaps formed between the skin and clothes. To simplify the analysis, the clothing was assumed not to deform. The clothing (a combination of the shirt, tie, coat, pants, shoes, etc.) had a volume of 10,912 cm3 and a mass of 16.8 kg. The air had a density of 1.185 kg/m3, a specific heat of 1000 J/kg-° C, a thermal conductivity of 0.0261 W/m-°C, and a dynamic viscosity of 1.835  105 Pa-s. The blood properties for Eq. (13) were assigned a density of 1204 kg/m3 and a specific heat of 2672 J/kg °C.

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Table 5 Volume and mass information for the human model. Component Core Muscle Fat Skin

Bone

Arms Volume composition (%)

0.124 1

0

0.671 0.124 0.08

Volume (cm )

0

4684.6 867.2 558.2 867.2 6977.1

Mass (kg)

0

4.919 0.737 0.558 1.127 7.341

Volume composition (%)

0

0.209 0.418 0.128 0.245 1

3

Volume (cm )

0

408.7 817.3 249.2 478.4 1953.6

Mass (kg)

0

0.429 0.695 0.249 0.622 1.995

0

0.27

Volume (cm3)

0

269.1 348.9 179.4 199.3 996.7

Mass (kg)

0

0.283 0.297 0.179 0.259 1.018

3

Feet

Hands Volume composition (%)

Head Volume composition (%)

Legs

Total

0.35

0.18

0.2

1

0.538 0.174 0.138 0.077 0.073 1

Volume (cm3)

2026.1 655.04 517.94 289.44 274.2 3762.7

Mass (kg)

2.755 0.688 0.44

Volume composition (%)

0

0.487 0.268 0.069 0.175 1

Volume (cm3)

0

6977.1 3837.4 986.8 2511.8 14,313

Mass (kg)

0

7.326 3.262 0.987 3.265 14.84

Torso Volume composition (%)

0.289 0.356 4.528

0.393 0.263 0.285 0.027 0.032 1

Volume (cm3)

10,565 7086.7 7674.8 717.6 867.2 26,912

Mass (kg)

11.622 7.441 6.524 0.718 1.127 27.431

The specific material properties (density, specific heat, thermal conductivity) for all of the biological tissues were obtained from Ref. [48] with the exception of the hair’s thermophysical properties which were obtained from Refs. [73,74]. The thermophysical properties of each of the major segments of the human model (arms, feet, hands, head, legs, torso) are listed by component (core, muscle, fat, skin, and bone) in Table 6. For simplification of the model, the various material properties in Table 6 were averaged using a mass-weighted-average based on the mass values listed in Table 5. These resulting effective material properties are presented in Table 7 along with the relevant clothing and hair properties. As seen in Table 7, the human head has been treated separately from the rest of the human body since it

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Table 6 Thermophysical properties used for the human tissues. Density Specific heat Thermal conductivity (kg/m3) (J/kg °C) (W/m °C)

Arms

Feet

Hands

Head

Legs

Torso

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.16

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.16

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.16

Core

1360

3350

0.53

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.2

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.16

Core

1100

3350

0.55

Muscle

1050

3770

0.5

Fat

850

2500

0.2

Skin

1000

3770

0.21

Bone

1300

1590

1.16

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J.M. Gorman et al.

Table 7 Effective thermophysical properties for the human model. Specific Mass Density heat Volume (kg) (kg/m3) (J/kg °C) (cm3)

Thermal conductivity (W/m °C)

Clothing

10,912

16.8

1540

1400

0.04

Hair

607.61

0.79

1300

1353

0.704

Head

3763

4.53

1236

2672

0.526

Human bodya

51,152

52.63

1029

3134

0.511

a

Effective human body properties exclude the head, hair, and clothing.

is the most relevant component to the present wind chill analysis. Similarly, the hair on the human head was treated separately because of the additional unique modeling aspects which will now be discussed. In order to numerically model the hair, it was necessary to model the volume occupied by the hair as a porous media since modeling each individual hair was not practical in the current approach. After an extensive literature search, none of the relevant information required for modeling hair as a porous medium was found. It was then necessary to create a separate set of simulations of human hair in order to develop a Darcy-Forchheimer pressure-drop relationship which could be used in conjunction with the present simulation. This relationship, accredited to Forchheimer [75], results in an equation of the form dp ¼ AU  BU 2 : dx

(23)

in which U is the superficial air velocity, A is the linear resistance coefficient, and B is the quadratic resistance coefficient. In order to determine the coefficients A and B, a series of simulations were created in which a segment of the human scalp was modeled. Focus was limited to one hair type, based on Ref. [76], where each hair had a 0.065 mm diameter and a distribution of 170 hairs per cm2 on the human scalp. A simple code was written in MATLAB® to randomly distribute 170 hairs per cm2 for a total of 60 different geometric configurations. For each geometric configuration, the air velocity was parametrically varied to obtain a velocity-pressure drop relationship. The velocity-pressure drop data for all 60 geometric cases were combined and fitted with the Darcy-Forchheimer equation shown in Eq. (23)

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Heat exchange between the human body and the environment

to obtain A and B. The hair simulations resulted in values of 213.54 kg/m3 s and 121.62 kg/m4 for A and B, respectively, and a volume porosity of 95%. Over 300 independent simulations of the hair on a representative human scalp were necessary to obtain the needed information to create a porous media model of the hair. In addition to the thermophysical properties, it is necessary to specify the volume-averaged values for the metabolic heat generation and the blood perfusion for the relevant biological tissues needed for Eqs. (12) and (13). Values used in the present model are listed in Table 8 and were obtained from the literature [48,77–79]. For the entire human head, the effective metabolic heat generation was 6245 W/m3, and the effective perfusion value was 1.91 105 m3/s. The remaining sections of the human body had an effective metabolic heat generation of 1422 W/m3 and an effective perfusion value of 2.94105 m3/s.

5.3 Initial conditions Since the simulations were time dependent, it was necessary to start the human model in a reasonably comfortable environment before being exposed to cold wind. All of the wind chill simulations were started with the person in a 25 °C calm environment for an hour before walking outside Table 8 Metabolic heat generation and perfusion values used for the human tissues. Component Core Muscle Fat Skin Bone

Arms

Feet

Metabolic generation (W/m3) 0

706.6

3.5

1003.2 0

Perfusion (cm3/s)

2.52

0.003

0.78

709.6

3.7

1003.2 0

0.22

0.003

0.53

706.1

2.9

1003.3 0

0.14

0.0012 0.59

0

Metabolic generation (W/m ) 11,352 458.0

2.3

656.4

0

Perfusion (cm3/s)

0.23

0

1.37

0

705.2

3.4

1013.4 0

3.75

0.013

1.19

3.4

1003.3 0

0.026

1

0 3

Metabolic generation (W/m ) 0 3

Perfusion (cm /s)

0

Hands Metabolic generation (W/m3) 0 3

Perfusion (cm /s) Head

Legs

0 3

17.5 3

Metabolic generation (W/m ) 0 3

Perfusion (cm /s)

0

Torso Metabolic generation (W/m3) 4230.8 705.5 3

Perfusion (cm /s)

48

3.81

0

0

0

0

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o

Simulation: 34.5 C Measured: 34.4 oC Simulation: 33.5 oC Measured: 32.2 oC o

Simulation: 33.75 C Measured: 33.3oC

Simulation: 34.25 oC Measured: 34.3 oC Simulation: 33.0 oC Measured: 32.3 oC Simulation: 33.37 oC Measured: 33.3 oC

Fig. 13 Color contour diagram of temperatures on the simulated human face in a calm 25 °C environment compared with measured values from Ref. [80].

into a windy environment. This approach allowed the simulated person to develop an appropriate steady-state temperature distribution with a core temperature around 37 °C. The area average face temperature (all areas of the skin that are exposed and not covered by hair or clothing) was 33.85 °C just before entering the windy environment. The resulting facial temperatures from the simulation of a person within a calm environment at a temperature of 25 °C are shown in Fig. 13. These facial temperatures are compared with measured facial temperatures reported in Ref. [80]. The authors of Ref. [80] investigated the facial temperatures of 30 healthy individuals in an ambient environment with a mean temperature of 22 °C and reported the mean of the measured temperatures for the 30 participants after 10 min. The asserted accuracy of their temperature measurement system was stated at 0.5 °C. These measured temperatures from Ref. [80] are also shown in Fig. 13 along with the simulation-based temperature results at the same locations. Overall, there is very good agreement (within 4%) between the simulation results and the measured values.

6. Results 6.1 Comparison of published Nusselt number correlations Based on the review of the literature regarding wind chill, the head of a person has been a primary focus. The majority of the papers based on analytical models treat the human head as a cylinder in cross flow. The most commonly used Nusselt number correlations, which are still used in modern publications, have been presented in Table 9. In the published literature,

Table 9 Commonly used convective correlations from the literature that have been used to model the heat transfer from a human head. Ref. # Year Geometry Correlation Range Notes

[81]

Before 1958

Cylinder

 1 N uθ ¼ 1:14Re2 Pr 0:4 1  ðθ=90Þ3

0 < θ < 80

[82]

1974

Cylinder

Nu ¼ ð0:43 + 0:5Re0:5 ÞPr 0:38

1 < Re  1000

Nu ¼ 0:25Re0:6 Pr 0:38

103 < Re < 2  105

h 1  Re 58 i45 1 D 0:62 Re 2D Pr 3 1 + 282000 Nu ¼ 0:3 + h  2=3 i1=4 1 + 0:4 Pr

RePr > 0.2

b

a

[83]

1977

Cylinder

[84]

2004

Cylinder

14  1 2 Nu ¼ 0:25 + 0:4 Re 2 + 0:06Re3 Pr 0:37 μ μ

1 < Re < 105

c

Human Face

Nu ¼ 0.869Re

θ ¼ 50

d

wall

[6]

2012

0.5765

Pr [1  (50/90) ] 0.4

3

a The Reynolds number range was not given in Ref. [81]; however, when compared to experimental data from Ref. [85], this equation performed better at lower Reynolds numbers. The correlation was within 10% of the expected experimental results from Ref. [85] at Re ¼ 20,000. b The authors of Ref. [83] implied that this correlation represents Nu values for low free-stream turbulence for an isothermal cylinder with neglectable end effects and natural convection. c The correlation is best for gas flows with a Prandtl number near 1 and for liquids at higher Prandtl numbers. d Reynolds number range was not given in Ref. [6]; however, the formula is very similar to that from Ref. [81].

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there are human-specific Nusselt number correlations, but they are primarily focused on specific cases that are not directly relevant to wind chill. These include correlations specific for natural convection of a seated person or forced convection of a person on an operating table. With the exception of Ref. [6], all of the other correlations listed in Table 9 were created from cylinder-based experiments. The authors of Ref. [6] modified the correlation from Ref. [81] using estimated values from limited human studies. In the present work, the commonly used correlations, as seen in Table 9, will be compared with the current human-specific results. This comparison is very important since heat transfer coefficients have been a critical element of all previously published wind chill studies and models. The major different is that in the present work, heat transfer coefficients are a result of numerical simulations whereas for all other non-experimental studies, a heat transfer coefficient from the literature is used as an input to the model.

6.2 Simulation validation using cylinder Nusselt number correlations In order to more thoroughly validate the existing numerical method and approach (beyond the comparison of experimentally measured facial temperatures in Section 5.3), a simple cylinder in cross flow was simulated to compare with appropriate Nusselt number correlations from Table 9. This was done in lieu of using suitable human-specific data available for comparison. The cylinder was modeled two-dimensionally with a 1 cm diameter. The solution domain around the simulated cylinder extended 3 cm in front of the leading edge, 3 cm to either side, and 15 cm downstream of the cylinder. The cylinder wall was given a uniform wall temperature of 50 °C, and the oncoming air flow had a temperature of 25 °C. The upstream velocity was specified as a uniform value that was parametrically varied, and the side and end boundaries were specified as entrainment openings at ambient gauge pressure (0 Pa). For the mesh, it was found that 1 million nodes were sufficient for high accuracy, and the results did not change with further mesh refinement. Fig. 14 has been created to compare the Nusselt number results for the simulated cylinder with several well-accepted cylinder-in-cross flow Nusselt number correlations presented in Table 6. As seen in the figure, there is some variation among the literature Nusselt number correlations; however, the present numerical results show very good agreement, within 2%, of the expected values from Ref. [84] and 3%–5% greater than the values from Ref. [82].

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200

Nu

150

[82] [83] [84] Cylinder Simulation

100

50 20,000

30,000

40,000

50,000

60,000

70,000

80,000

Re

Fig. 14 Comparison of simulated Nusselt numbers for a simulated cylinder in cross flow to several correlations from literature as listed in Table 6.

6.3 New facial Nusselt number correlation With regard to the present human-model results, the average facial Nusselt number NuFace will be defined as NuFace ¼

hDhyd k

(24)

where the heat transfer coefficient h is based on the oncoming wind temperature and velocity, the area-averaged facial skin surface temperature, and the area average facial heat flux. The facial area (610 cm2) was considered to be any skin region exposed to the air and not covered by hair or clothing (i.e., Fig. 13). Not unexpectedly, both the facial temperatures and heat flux from the human face given by the simulations varied with time. In order to get timeindependent heat transfer coefficients, the values of facial temperatures and heat fluxes had to be time-averaged. Since the current model mimics an average human, certain combinations of skin temperature and time durations would be considered fatal and could not be used (for core temperatures dropping below 28 °C, the modeled person is expected to be unconscious). Therefore, the values of facial temperatures and heat fluxes had to be time-averaged and restricted to moderate wind temperatures, at or above 0 °C. After the initial “shock” of entering a colder windy environment, the facial heat transfer coefficient oscillated around an average value (by 0.5 W/m2 °C). The temporally averaged values of the present

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heat transfer coefficients were used to create the current Nusselt number correlation. The hydraulic diameter Dhyd of the head was 17.6 cm and the air’s thermal conductivity k was given previously in Section 5.2. With regard to the Reynolds number definition for the human face, there are two logical choices for the velocity. The first option is to use the local oncoming velocity at face height UFace which is useful for comparing with other correlations or when local wind velocities are known. The second option is to make use of the U10 value since it can be related directly to weather data. Both choices are defined in Eq. (25) and will be used for discussing the results. Re Face ¼

ρU Face Dhyd ρU 10 Dhyd or μ μ

(25)

Attention will now be turned to the graphical display of the facial Nusselt number results shown in Fig. 15, where they are plotted as a function of the Reynolds based on the face-height velocity. As seen in the figure, the Nusselt numbers from the present work for both wind directions (front and side) are plotted as a function of Reynolds number where the velocity 500 400

Nu

300 200 100 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 Re Simulation - Forward wind

Simulation - Side wind

[6]

[81] (theta = 50)

[83]

[84]

Fig. 15 Comparison of Nusselt number correlations. The curves Simulation—Forward wind and Simulation—Side wind represent the present numerical results for a forward wind direction and a side wind direction, respectively. The Reynolds number for the present numerical results is based on the local wind velocity at face height. The other Nusselt number correlations are listed in Table 9.

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is based on UFace. Also shown in the figure, for comparison, are several of the aforementioned commonly used correlations. As seen in Fig. 15, the present results for the Nusselt numbers were much higher compared with the correlations for cylinders in cross flow [81,83,84]. This is not surprising since the Nusselt number correlation from Ref. [6], for a human face, was also much higher than the correlations for cylinders. The present results for the forward wind Nusselt numbers ranged between 3% and 34% lower than the correlation from Ref. [6] where the closest agreement was around Re ¼ 40,000. With regard to the effect of wind direction, the Nusselt numbers corresponding to the forward wind were slightly higher. For Reynolds numbers less than 40,000, the forward wind direction corresponded to Nusselt numbers that were 9%–14% higher compared to the wind coming from the side. As the Reynolds numbers increased beyond Re ¼ 70,000, the percent difference in Nusselt numbers for the two wind directions were less than 6%. The resulting Nusselt numbers from the simulations were formulated into a correlation of the form N uFace ¼ C 1 Re nFace Pr 0:4 

C2 Re mFace

(26)

where Pr is the Prandtl number equal to 0.71, and C1, C2, n, and m are constants used for curve fitting. The equation constants, listed in Table 10, were adjusted until the correlation fit the values to within 3% for the investigated range of Reynolds numbers. For the range of wind speeds under investigation (U10 from 1 to 20 m/s) the corresponding Reynolds number range based on UFace was between 8000 and 150,000, and for the Reynolds number based on U10 the range was 10,000–220,000.

Table 10 Equation constants for Eq. (26). Wind direction Reference velocity

Forward

Side

C1

C2

n

m

U10

1.38

3700

0.5

1.0

UFace

1.64

3700

0.5

1.0

U10

1.27

180

0.51

0.26

UFace

1.50

142

0.512

0.2

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6.4 Comparison of predicted facial temperatures with measured values In addition to developing Nusselt number correlations, the present model provides detailed information such as the change of facial temperatures with time in a cold environment. For comparison purposes, several measured facial temperature data points were extracted from Ref. [6]. This data, reproduced in Fig. 16, corresponds to the average measured facial temperatures of four males in a 10 °C environment experiencing a 2 m/s wind speed over a period of time. The error bars on the data from Ref. [6] shown in the figure represent the upper and lower bounds of the measured temperatures on the four male subjects. Fig. 16 also displays the results from the present human simulation corresponding to a U10 ¼ 3 m/s (approximately 2 m/s at the face height of 1.5 m) and an oncoming wind temperature of 10 °C. As seen in Fig. 16, the facial temperatures predicted by the current simulation fall within the bounds of the measured values present in Ref. [6]. At times less than 5 min, the simulated facial temperatures are less than 1% different from the average measured values. After 5 min, the simulated facial temperatures approach the upper bounds of the measured temperatures, for a percent difference of 6% from the average measured values. Considering the small sample size (four males) and the variations in physiology from person-to-person, the agreement between the present simulation and the measured temperature data is remarkably good.

Temperature (oC)

36 34

Human simulation

32

Average measured temperature

30 28 26 24 22

0

5

10

15

20

25

Time (minutes)

Fig. 16 Comparison of facial temperatures. The data points, from Ref. [6], represent average measured values for four males in a 10 °C environment experiencing a 2 m/s wind speed over a period of time. The solid line represents the current simulation results corresponds to a U10 ¼ 3 m/s (which is approximately 2 m/s at the face height) and a wind temperature of 10 °C.

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6.5 Temperature contour diagrams With the present human model, temperature values at any location or moment of time can be obtained. The first set of temperature results, shown in Fig. 17 as temperature color contour diagrams, is positioned at a plane bisecting the human model in the vertical direction. The separate regions (hair, head, and clothing/body) are outlined with a black line for reference. At time equal to zero minutes, the human model leaves the 25 °C environment with an average facial temperature of 33.85 °C and experiences a wind in the forward direction. The temperature contour results shown in Fig. 17 correspond to a wind with a temperature of 10 °C and a velocity corresponding to U10 ¼ 1 m/s. As seen in Fig. 17A, after 1 min the human model is starting to experience a drop in temperature at the face surface and within the hair region. The lowest temperature on the face at this moment of time is at the nose, 30 °C. After 10 min, Fig. 17B shows that the nose has dropped to 26 °C and other areas of the face are in the 30 °C 0.5 °C temperature range. In addition, most of the hair has cooled down to 23 °C or lower. Fig. 17C and D, corresponding to 20 and 30 min of exposure, show only slight temperature A

B

time = 1 minute C

time = 10 minutes D

time = 20 minutes

time = 30 minutes

Fig. 17 Temperature contours, as a function of time, at the center cross section of the human model. The results correspond to a forward wind direction at a temperature of 10 °C and a velocity corresponding to U10 ¼ 1 m/s.

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changes compared to the results at 10 min, with the nose still being the coldest location on the face (23.5 °C) shown in the cross section. As expected, the temperature contours in Fig. 18 show the hair is acting as a form of thermal insulation to reduce the heat loss from the head surface. The parts of the head covered by hair are seen to be warmer by an average of 4 °C compared to uncovered areas of the head. For instance, the forehead

Fig. 18 Temperature contours after 10 min in a forward direction wind for (A) a wind temperature of 10 °C at a velocity corresponding to U10 ¼ 1 m/s, and (B) a wind temperature of 10 °C at a velocity corresponding to U10 ¼ 1 m/s.

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at 20 min is 30 °C whereas the region covered by hair just above the forehead is 34 °C. In addition to the comparison of temperature changes with time, comparisons can also be made between different wind temperatures. Fig. 18 has been created to present the facial temperatures for two contrasting wind temperatures, 10 °C and 10 °C. In both cases, the wind velocity corresponds to U10 ¼ 1 m/s, and the wind was in the forward direction. The temperature values seen in the figure correspond to being exposed to these wind conditions for 10 min. As can be expected, Fig. 18B shows the human model’s facial temperatures are much lower at the 10 °C wind temperature. For instance, Fig. 18A shows the coldest temperatures on nose and ears at 26 °C while Fig. 18B shows the temperature at these same locations at 6 °C. At 10 °C the average facial temperature is 30°C, whereas at 10 °C the average facial temperature is closer to 13 °C. In both cases, the facial temperatures shown in the figure are nonuniform and vary considerably with location.

6.6 Discussion of wind chill Defining wind chill or the wind-chill temperature is difficult because it is not an exact temperature but instead an index used to quantify cold and windy conditions. The best description of wind-chill temperature is the equivalent temperature felt at an average human face that would result in the same rate of heat loss as in calm wind conditions [4,6]. In this definition, a calm wind speed is defined as 1.3 m/s to be compatible with the average walking speed of a person [4]. Currently, the wind-chill equivalent temperature is calculated from simple algebraic algorithms relating only wind speed (at 10 m from the ground) and wind temperature as discussed in Ref. [4]. This equivalent temperature is then often used to estimate the time until someone may experience frostbite. A difficulty with the description of wind chill is the concept of the equivalent temperature “felt” at an average human face. This is interpreted as an equivalent air temperature which could presumably be used in conjunction with a heat transfer coefficient associated with calm winds conditions to determine the heat loss. Therefore, the calculated wind-chill equivalent temperature used by the National Weather Service would not be the same as the temperature directly on a human’s face (skin temperature). Since Williamson [4] does not discuss the heat transfer coefficients and the heat loss from a human face under calm wind conditions, it is not possible to

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use the same algebraic algorithms to calculate an average face temperature. This would be a useful comparison since the method promulgated by National Weather Service, as described in Ref. [4], is still in use today. Determination of the wind-chill equivalent temperatures could be done using a convective thermal energy balance at a person’s face,   qface ¼ hface T face  T wind chill

(27)

where qface is the heat flux leaving a person’s face, hface is the heat transfer coefficient for the human face, Tface is the surface temperature of the face, and Twind chill is the resulting wind chill equivalent temperature. In order to use the aforementioned definition of a wind-chill temperature, qface would represent the same rate of heat loss as in calm wind conditions, hface would be a known constant for each wind speed, and Tface would be determined for various combinations of wind speed and air temperature. However, in reality, the actual values of qface, hface, and Tface vary with time in addition to wind speed and air temperature. For a comparison of predicted facial temperatures, attention can be turn to Ref. [17], which is a more recent publication related to the study of wind chill. The approach used in Ref. [17] relied on the use of heat transfer coefficients and reported steady-state facial temperatures (450 min). This is in contrast to the present work since the current human model’s facial temperatures (at lower wind temperatures and higher wind velocities) were found to vary with time. If the skin temperatures, or temperatures of other tissues, dropped too low, the tissue would be damaged or killed. In Ref. [17], facial temperatures are predicted under similar conditions (temperatures and wind speeds) as presented in Ref. [4] and used by the National Weather Service. Since the present human model changes with time, it is not appropriate to report and compare steady-state facial temperatures. However, for discussion purposes, at 30 km/h (U10 value) and a 20 °C wind temperature, Shabat et al. [17] predicts a steady-state facial (cheek) temperature of 12 °C (after 450 min). In the current model, under similar conditions, the average facial temperature drops by 10 degrees within the first 3 min. After 18min, the current human model predicts a facial (cheek) temperature of 14.8 °C (20% difference compared to Ref. [17]) with a nose and ear temperature around 17 °C. For a sufficient duration of time, the present model predicts that the facial temperatures will be colder than the values reported in Ref. [17] since the present human model cannot sustain its heat generation under certain conditions.

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In general, tissue damage in extreme conditions is a function of time and air temperature along with several other factors. Within the literature on cold-related injuries, there is not a precise definition of the time and temperatures necessary to cause tissue damage in the form of frostbite. Some of the literature suggests a slow freezing of tissue and tissue temperatures around 0.55 to 2 °C are necessary for frostbite [86–88]. However, as mentioned in Ref. [89], it is possible for tissue exposed to temperatures in the 1.4–4.4 °C range for prolong periods of time to have injuries similar to frostbite. For the human finger, the author of Ref. [90] implies there is a 5% chance of frostbite if the finger’s skin tissue is 4.8 °C and a 95% change of frostbite if the skin tissue is 7.8 °C. As mentioned in Ref. [90], biological processes such as cold-induced vasodilatation can protect the tissue from freezing at certain low temperatures (>4.8 °C). Another observation, as discussed in Ref. [90], indicates some investigators found that people who were acclimated to cold environments had a lower risk of frostbite. Considering the number of factors which influence the risk of frostbite, it is clear why there is not a consistent consensus within the literature about the time and temperatures necessary to cause frostbite-related tissue damage. Within the cryosurgery field of research [91–94], it is clear the rate of cooling and the tissue type are important factors in determining cell injury in addition to the time duration and temperature. The authors of Ref. [92] reported the results on 243 studies of the freezing and thawing rates on dog skin in the temperatures ranges of 15 to 50 °C. The results presented in Ref. [92] indicate that a slow freezing of tissue (for 3 min) with a rapid thaw (less than 2 min) did the least damage to the tissue (freezing dog skin tissues to 15 to 24 °C in 3 min with an active heating for 2 min resulted in no damage). However, Gage et al. [92] also showed faster rates of freezing (in about 1 min) or holding the tissues at temperatures between 15 and 50 °C for 3 min did more damage (freezing the skin tissues to 15 to 24 °C in 1 min and holding and waiting 3 min before actively heating for 2 min resulted in 50% of the cells to be damaged). For cryosurgery, which is typically performed on internal organs, it is suggested that 50–60s at 20 °C will completely damage the tissue or cooling the tissue to a temperature of 40 °C (or lower) will damage 100% of the tissue cells immediately [93,94]. If the tissues/cells are killed, the affected regions would no longer able to deliver heat (no perfusion, metabolic generation, shivering, etc.). In the present model, if the human model’s core temperature dropped too low, the human model would eventually no longer function (at a core temperature of 28 °C the human would be unconscious). Eventually, a long exposure

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to cold wind would cause the heat transfer coefficient to approach zero as the skin temperature approaches the wind temperature. Therefore, the present model is not able to provide steady-state results when the conditions are extreme enough to pose a serious health risk. In addition to some of the earlier concerns and difficulties raised about the determination of wind chill, the aforementioned description did not explicitly include the effects of radiation. The worst-case scenario for radiation was mentioned in Ref. [4] but it is not clear how (or if ) this was incorporated into the calculation. For instance, if the effect of the worst-case scenario for radiation was included, the wind chill equivalent temperature under calm winds conditions should be lower due to the combined convective and radiative heat losses. However, the text of Ref. [4] suggests that the wind chill equivalent temperature from the algebraic algorithms should be equal to the air temperature under calm wind speeds. Therefore, it is not clear if radiation was considered to be part of the description/calculation of the wind-chill temperature or if the wind-chill temperature is solely based on convective losses. If frostbite warnings were the primarily goal of wind chill temperatures, then radiation effects should be included. In the model presented in this paper, the authors decided to omit radiation heat losses to obtain convective Nusselt numbers which were directly comparable with the published literature. Radiation heat losses could be expected to reduce the facial temperatures further, especially if the human model was exposed to a clear night sky and was away from surrounding structures. In order to quantify the potential impact of radiation on facial cooling, two separate simulations were performed, assuming a gray model, in which radiation heat losses were investigated using a Monte Carlo model. In the first case, the human model was located in a large enclosed room with still air. The enclosure walls were maintained at a constant temperature of 10 °C. The human skin was assigned an emissivity of 0.96 and the surrounding enclosure surfaces had an emissivity of 0.95. Based on the heat flux at the cheek, the linearized radiative heat transfer coefficient hrad was found to be 5.22 W/m2-°C after 5 min. For comparison, the convective heat transfer coefficient hface for U10 ¼ 1 m/s was 12.11 W/m2-°C, whereas when U10 ¼ 20 m/s, hface was 84.7 W/m2-°C. At the lowest wind speed, the convective heat transfer coefficient was slightly more than double the linearized radiative heat transfer coefficient, and at the highest wind speed the convective heat transfer coefficient was roughly 16 times larger. In the second case involving radiation heat loss, the human was located in a large outdoor environment away from any buildings or structures.

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The ground was treated as an adiabatic surface and the human model was exposed to a clear night sky (without clouds). The air was stationary at a temperature of 10 °C with a dew point of 13 °C. Based on these conditions, Martin and Berdahl and Parsons and Sharp [95,96] estimate the night sky temperature to be 21.5 °C with a sky emissivity of 0.85. Under these conditions, the linearized radiative heat transfer coefficient hrad at the cheek was found to be 9.38 W/m2-°C after 5 min. This value of the radiative heat transfer coefficient is 80% larger than the previous case where the person was in an enclosed room.

6.7 Predicted facial temperatures Reporting the predicted facial temperatures on a person’s face has more of a practical significance compared to reporting a wind-chill equivalent temperature. A facial temperature is experienced directly by a person whereas the wind-chill equivalent temperature is an artificially constructed quantity. A wind-chill equivalent temperature has historically been an index relating wind speed and air temperature based on heat transfer coefficients, but it lacks a physical meaning, and does not directly relate to a person’s exposure time. Tables 11–14 have been created to display the predicted facial temperatures at the cheek for exposures times of 3, 5, 7, and 9 min, respectively, over a range of temperatures and wind speeds in the forward direction. Without a clear and precise definition of when skin tissue damage (frostbite) occurs, it is difficult to estimate safe exposure times. However, skin temperatures at prolonged freezing temperatures would certainly be a safety concern. Cheek temperatures listed in the tables below 0 °C have been highlighted. Table 11 Predicted facial temperatures (at the cheek) as a function of wind speed (U10 value) and air temperature after 3 min of exposure. Air temperature (°C) 10

5

0

25

210 215 220 225 230

Wind speed U10 (km/h) 10 26.4 24.5 22.6 20.8 18.9 17.0 15.1 13.2 11.3 20 24.7 22.5 20.3 18.1 15.9 13.7 11.5 9.2

7.0

30 23.4 20.9 18.4 16.0 13.5 11.0 8.5

6.1

3.6

40 22.5 19.8 17.2 14.5 11.9 9.2

6.6

3.9

1.3

50 21.6 18.7 15.9 13.1 10.3 7.5

4.7

1.8

1.0

60 20.8 17.9 14.9 12.0 9.0

3.1

0.1

2.8

6.1

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Table 12 Predicted facial temperatures (at the cheek) as a function of wind speed (U10 value) and air temperature after 5 min of exposure. Air temperature (°C) 10

5

0

25

210 215 220 225

Wind speed U10 10 25.5 23.4 21.3 19.2 17.0 14.9 12.8 10.7 (km/h) 20 23.8 21.3 18.8 16.4 13.9 11.5 9.0 6.6

230

8.5 4.1

30 22.3 19.6 16.9 14.2 11.5 8.7

6.0

3.3

0.6

40 21.4 18.5 15.6 12.8 9.9

7.0

4.1

1.2

1.7

50 20.6 17.5 14.4 11.4 8.3

5.3

2.2

0.8 3.9

60 19.9 16.7 13.5 10.3 7.1

3.9

0.7

2.5 5.6

Table 13 Predicted facial temperatures (at the cheek) as a function of wind speed (U10 value) and air temperature after 7 min of exposure. Air temperature (°C) 10

5

0

25

210 215 220

Wind speed U10 10 25.1 22.8 20.6 18.3 16.0 13.8 11.5 (km/h) 20 23.2 20.7 18.1 15.5 13.0 10.4 7.9

225

230

9.3

7.0

5.3

2.8 1.0

30 21.8 19.0 16.1 13.3 10.4 7.6

4.7

1.8

40 20.9 17.9 14.9 11.9 8.9

5.9

2.9

0.2 3.2

50 20.1 16.9 13.7 10.6 7.4

4.3

1.1

2.1 5.2

60 19.4 16.1 12.8 9.5

2.9

0.4 3.7 7.0

6.2

Table 14 Predicted facial temperatures (at the cheek) as a function of wind speed (U10 value) and air temperature after 9 min of exposure. Air temperature (°C) 10

5

0

25

210 215 220

Wind speed U10 10 24.8 22.5 20.2 17.8 15.5 13.2 10.8 (km/h) 20 23.0 20.3 17.6 15.0 12.3 9.6 7.0

225

230

8.5

6.2

4.3

1.6 1.8

30 21.6 18.6 15.7 12.8 9.9

7.0

4.1

1.2

40 20.7 17.6 14.5 11.4 8.4

5.3

2.2

0.9 3.9

50 19.8 16.6 13.3 10.1 6.9

3.6

0.4

2.8 6.1

60 19.1 15.8 12.4 9.1

2.4

1.0 4.3 7.7

5.7

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As seen in Table 12, after 5 min, the cheek reaches freezing temperatures at wind speeds from 50 to 60 km/h and air temperatures from 25 to 30°C. In the present model the face has exposed ears and nose, and these extremities would reach freezing temperatures sooner than the cheek (typically, the exposed ears and nose are 2–3 °C colder than the cheek temperatures). After 9 min, as seen in Table 14, the cheek reaches freezing temperatures at wind speeds from 30 to 60 km/h and air temperatures from 20 to 30 °C.

7. Concluding remarks This paper presents a new modeling approach of the human body to better study wind chill and predict facial temperatures. This model, employing time-dependent numerical simulation, uses a physically-realistic healthy human male subject and physically realistic thermophysical properties. This model also includes the biothermal heat transfer processes in the form of metabolic heat generation and blood perfusion. The human model was exposed to a parametric variation in wind speed (using an atmospheric wind velocity profile), wind temperature, and two wind directions. The present human model was compared to data reported in the literature; however, there is limited data available for comparison. With regard to ambient facial temperatures exposed to air temperature of 25 °C, the present model was within 4% of expected measured values at the forehead, cheek, and chin. With regard to facial cooling with time, the present model was within 6% of the average facial temperatures measured on four males for a period of time in a 10 °C environment experiencing a 2 m/s wind speed. However, over long periods of time in cold and windy conditions, the present model predicts colder facial temperatures than results reported in Ref. [17]. The practical results of the present work include new Nusselt number correlations for the human face for both a forward wind and a side wind direction. For the range of wind speeds under investigation (U10 from 1 to 20 m/s) the corresponding Reynolds number range based on UFace was between 8000 and 150,000, and for the Reynolds number based on U10, the range was 10,000–220,000. The new Nusselt number correlation for the forward wind direction ranged between 3 and 34% lower than a previously reported correlation in the literature. The authors were unable to find a directly comparable Nusselt number correlation in the literature for the side wind results. With regard to wind chill, the present model is more encompassing than is previously published research. Previous models and studies relied on

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Nusselt numbers (heat transfer coefficients) from the literature purported to be applicable for the human face whereas in the present work, Nusselt numbers were found as a result. However, more significantly, the current model predicts the actual facial temperatures based on wind speed, wind direction, wind temperature, and biological processes. Since the present human model is time dependent, the model can then be used to study the time at which tissue damage (frostbite) occurs. Because of this, predicting the actual facial temperatures would have more direct benefit than creating a fictitious index such as wind chill equivalent temperatures. The present results show that it takes 3 min of exposure in 50–60 km/h winds at 30 °C to reach cheek temperatures below 0 °C. After 5 min, winds in the 40–60 km/h range at 25 to 30 °C reach temperatures below 0 °C. Then, after 9 min, winds in the 30–60 km/h range at 20 to 30 °C reach temperatures below 0 °C. The ears and nose, which were uncovered and exposed to the wind, were routinely 2–3 °C colder than the cheek temperatures. More information is needed in order to determine the extent of tissue damage due to these predicted facial temperatures over these amounts of time. Further work needs to be done to: (a) study the sensitivity of geometric facial features on the predicted Nusselt number results, (b) incorporate a quantifiable tissue damage metric for cold conditions, (c) examine the effects of protective gear such (as scarf, hat, etc.), (d) parametrically vary biological responses (blood perfusion, metabolic heat generation, shivering, etc.) to represent different age and/or health conditions, (e) include the effects of thermal radiation, (f ) investigate effects of unsteady wind gusts, (g) include blowing snow/ice carried by the wind, and (h) investigate the effects that surrounding structures, such as trees and buildings, might have on the resulting facial temperatures.

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CHAPTER FOUR

Pressure drop and heat transfer in the entrance region of microchannels Zhipeng Duan* and Hao Ma School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Entrance length 2.1 Hydrodynamic entrance length 2.2 Thermal entrance length 3. Circular microchannels 4. Parallel plate microchannels 5. Rectangular microchannels 5.1 Continuous flow in the entrance region 5.2 Slip flow in the entrance region 5.3 Hydrodynamic development length 6. Elliptical microchannels 6.1 Fully developed flow and heat transfer 6.2 Laminar flow in the entrance region 6.3 Heat Transfer in the entrance region 6.4 Entrance length 7. Microchannel plate fin heat sinks 7.1 Liquid slip entrance flow 7.2 Nanofluid entrance flow Acknowledgments References

251 252 252 254 256 261 266 266 273 279 283 284 285 293 311 316 316 321 329 329

Abstract The entrance region constitutes a considerable fraction of the channel length in microsized devices. When the hydrodynamic and thermal development lengths are the same magnitude as the microchannel length, entrance effects have to be taken into account, especially in relatively short channels. This chapter deals with issues of hydrodynamic and thermal development in circular and noncircular microchannels, with particular emphasis on the microchannels of rectangular and elliptical cross sections that are the most useful and common channel shapes in practical applications. Slip flow in

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2020 Elsevier Inc. All rights reserved.

249

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Zhipeng Duan and Hao Ma

the entrance region of circular and parallel plate microchannels is first considered by solving a linearized momentum equation. General correlations are developed for slip flow and continuum flow in the hydrodynamic entrance of circular, parallel plate and rectangular microchannels. Three-dimensional laminar flow and convective heat transfer for various thermal boundary conditions at the entrance region in elliptical microchannels are discussed in detail. Models and corresponding correlations are proposed and developments of the velocity distribution are presented for entrance flows in microchannels of circular and noncircular cross sections. In addition, slip flow and nanofluid flow in short microchannel plate fin heat sinks are demonstrated.

Nomenclature Roman symbols A a b cp Dh e f h K Kn Kn∗ L Lhy Lhy+ Lth Lth∗ Nu P p p* Po Pr q00 R S_ gen T U u u v w x x+

cross section area (m2) semi-minor axis of the rectangle or ellipse (m) semi-major axis of the rectangle or ellipse (m) specific heat (J kg
1 K
1) hydraulic diameter (m) eccentricity of the ellipse Fanning friction factor convective heat transfer coefficient (W m
2 K
1) incremental pressure drop factor Knudsen number, ¼ λ/Dh modified Knudsen number, ¼ Kn(2
σ)/σ channel length (m) hydrodynamic entrance length (m) dimensionless hydrodynamic entrance length, ¼ Lhy/(DhRe) thermal entrance length (m) dimensionless thermal entrance length, ¼ Lth/(DhRePr) Nusselt number, ¼ hDh/λ wetted perimeter (m) pressure (N m
2) dimensionless pressure, ¼ p/ρum2 Poiseuille number Prandtl number, ¼ μcp/λ wall heat flux (W m
2) correlation coefficient entropy generation rate (W K
1) temperature (K) dimensionless velocity, ¼u=u axial velocity (m s
1) average velocity (m s
1) velocity component (m s
1) velocity component (m s
1) x-coordinate (m) dimensionless axial distance, ¼ x/(DhRe)

Heat transfer in the entrance region of microchannels

x* y z

251

reciprocal Graetz number, ¼ x/(DhRePr) y-coordinate (m) z-coordinate (m)

Greek symbols α σ ε η θ λ λ μ ξ ρ τ ϕ

eigenvalue tangential momentum accommodation coefficient aspect ratio, ¼ a/b dimensionless channel coordinate dimensionless temperature molecular mean free path (m) thermal conductivity (W m
1 K
1) fluid dynamic viscosity (kg m
1 s
1) dimensionless axial distance, ¼ x/(DhRe) fluid density (kg m
3) wall shear stress (N m
2) nanoparticle volume fraction (%)

Subscripts app en f fd hy m th w

apparent entrance fluid fully developed hydrodynamic entrance region mean thermal entrance region wall

1. Introduction As one of the hotspots in current fluid mechanics research, the study on flow characteristics in microchannels has important theoretical and practical significance, especially for the rapid progress in micro-electro-mechanical system (MEMS) which is promoting the research boom. With the advantages of heat transfer enhancement, increased heat flux dissipation and cooling capacity of microscale devices, microchannels have shown great commercial value and potential applications. Microchannels are the fundamental parts of microfluidic systems. Understanding the flow characteristics of microchannel flows is very important in determining the pressure drop, heat transfer, and transport properties

252

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of the flow for minimizing the effort and expense of experiments [1]. Microchannel heat sinks have received considerable attention owing to their high surface-area-to-volume ratio, large convective heat transfer coefficient, and small mass and volume. For the effective design and optimization of microchannel heat sinks, it is significant to understand the fundamental characteristics of fluid flow and heat transfer in microchannels [2,3]. The recent development of microscale fluid systems has attracted much academic research of fluid flow and heat transfer in microchannels with different cross sections [4–29]. Although the characteristics of flow and heat transfer in microchannels are widely studied, there are some problems that could easily be ignored, such as the research of entrance region. It is noted that some studies focused on the fully developed flow or specified flow as fully developed and paid less attention to the entrance region of developing flow in microchannels. For a short duct, the length of channel in developing region may be a major portion of the whole microchannel length, and the flow is not fully developed. The entrance region in a microchannel is particularly of interest due to the presence of comparatively large pressure drop and heat transfer. Given that the convective heat transfer behavior in the developing region differs from that in the fully developed region, and given that many microchannel heat exchangers are short, this effect of entrance region is significant. If the entrance region is not treated properly when dealing with relevant data, it will undeniably cause remarkable errors.

2. Entrance length 2.1 Hydrodynamic entrance length When a viscous fluid enters a duct with the uniform velocity distribution at the entrance, boundary layers develop along the walls and the velocity is gradually redistributed due to the viscosity. Eventually the fluid will reach a location where the velocity is independent of the axial direction, and under such conditions the flow is termed the hydrodynamically fully developed. It is of paramount importance that the velocity profiles undergo rapid transformations from essentially uniform inlet profiles to fully-developed parabolic profiles. The hydrodynamic entrance length Lhy is defined as the duct length required to achieve a maximum velocity of 99% of that for fully developed flow [30]. Since the pressure gradients found in small diameter channels are quite high, the friction factor is an important parameter of fluid flow, which can be

Heat transfer in the entrance region of microchannels

253

used to evaluate pressure drop characteristics in microchannels and heat transfer performance of devices. For fully developed laminar flow, the result of Fanning friction factor ƒ is presented in the following form f ¼

Po Re

(1)

where Po is the Poiseuille number, which relates to the flow channel geometry. And Reynolds number is defined as Re ¼ ρumDh/μ, μ is the fluid viscosity, Dh is the hydraulic diameter of a channel, ρ is the fluid density, and um is the mean longitudinal velocity. The Fanning friction factor ƒ in terms of the wall shear stress is also expressed as f ¼

τw ð1=2Þρu2m

(2)

The frictional pressure drop Δp of a length x can be given as Δp ¼

2f ρu2m x Dh

(3)

where x is local position in the flow direction. The friction factor Reynolds number product in developing flow should contain the entrance region. Obviously, due to the greater velocity and pressure gradient in the entrance region, the pressure drop of developing flow is greater than that of the fully developed flow. So, the pressure drops of fully developed flow and developing flow are definitely different. For the developing region, the pressure drop equations are presented in related to an apparent friction factor ƒapp, which represents a true value of the friction factor from the entrance region to the local position along the flow direction under calculation. Thus, the pressure drop of Eq. (3) involved the apparent friction factor ƒapp can be represented into Δp ¼

2 f app ρu2m x Dh

(4)

The apparent friction factor fapp is devised for operational convenience and defined as f app ¼

ΔpDh 2ρu2m x

(5)

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and the apparent friction factor Reynolds number product fappRe can be derived as f app Re ¼

ΔpDh Re Δp ¼ 2ρu2m x 2ρu2m x+

(6)

The relationship between the apparent friction factor ƒapp and fully developed friction factor ƒ is presented in Ref. [31]. For ease of understanding, pressure drop commonly contains two parts at the entrance region of laminar flow, the pressure drop of Eq. (4) can be given as
 2 2 f app Re μum x 2ð fReÞμu x m + ρum Δp ¼ ¼ + K ð x Þ (7) 2 D2h D2h Here, K(x+) is the incremental pressure defect. It can be expressed as

 4x K ðx+ Þ ¼ f app
f ¼ 4x+ f app Re
fRe (8) Dh A definition of a non-dimension flow distance is: x+ ¼

x Dh  Re

(9)

where x+ is the non-dimension flow distance. Similarly, the dimensionless hydrodynamic entrance length is defined as L +hy ¼

L hy Dh  Re

(10)

2.2 Thermal entrance length The thermal entrance region of the duct is that region where the temperature boundary layer is developing. For this region, the dimensionless temperature profile (T-Tin)/(Tw-Tin) of the fluid changes from the initial profile at a point where the heating is started to an invariant form downstream. The flow in this region is designated as thermally developing flow [30]. The velocity profile in this region could be either developed or developing. Thermally developing flow with a developing velocity profile is referred to as simultaneously developing flow. The thermal entry length Lth is defined as the axial distance needed to achieve a value of the local Nusselt number Nu(x), which is 1.05 times the fully developed Nusselt number, when the entering fluid temperature profile is uniform [30,32].

255

Heat transfer in the entrance region of microchannels

In order to study the effect of thermal boundary on heat transfer performance at the entrance region in details, three thermal boundaries are all taken into consideration in this chapter, that is, T boundary, H1 boundary and H2 boundary. For T boundary condition, the first kind thermal boundary condition is set at the wall, namely, constant wall temperature condition. While H1 and H2 boundary conditions are the second kind thermal boundary conditions, namely, constant heat flux conditions. The distinction between H1 and H2 is the wall thermal conductivity around the channel [30]. The H1 boundary condition is applied to a heat exchanger with highly conductive materials. Based on the obtained temperature distribution of fluid and wall, the local surface heat flux q00 (x, y, z) is defined as. q00 ðx, y, zÞ ¼ λ

∂T ∂n

(11)

The direction of the local heat flux is the normal direction of the corresponding wall. Then the local convective heat transfer coefficient h(x) can be determined by the following equation. hðxÞ ¼

q00 ðxÞ T w ðxÞ
T m ðxÞ

(12)

where the Tw(x) is the temperature of the wall and Tm(x) is the massweighted average temperature of the fluid in the cross section. Moreover, q00 (x, y, z) is a known parameter for H1 and H2 boundary conditions. For T boundary condition, the known parameter is Tw(x). The local Nusselt number Nu(x) is obtained as. NuðxÞ ¼

hðxÞ  Dh λ

(13)

A definition of a nondimensional axial coordinate for the thermal entrance region is: x∗ ¼

x Dh RePr

(14)

where Pr is the Prandtl number. Similarly, the dimensionless thermal entrance length L∗th is defined as ∗ ¼ Lth

Lth Dh RePr

(15)

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Zhipeng Duan and Hao Ma

3. Circular microchannels Rarefaction effects, usually characterized by the Knudsen number, which is defined as the ratio of the molecular mean free path of gas to a characteristic dimension of flow domain. According to the value of Kn, the flow can be classified into four flow regimes, the continuum flow (Kn  0.001), the slip flow (0.001 0.001), the effects of geometry become more pronounced and the solutions for circular tubes and parallel plates Eq. (21) or Eq. (28) are no longer valid for non-circular ducts.

5. Rectangular microchannels For a specified cross-sectional shape, different aspect ratios of cross section have an impact on the friction and pressure drop characteristics [41,42]. Researches have shown that due to the stability and extremely high thermal performance of rectangular cross section, rectangular microchannels are good choices which have been used in a vast amount of applications. In this section, the friction characteristics of developing laminar flow in the entrance region of three-dimensional rectangular microchannels with aspect ratios from 0.1 to 1 are given [43,44].

5.1 Continuous flow in the entrance region Duan et al. [43] numerically investigated the flow characteristics and extended the data of friction factor and Reynolds number product of hydrodynamically developing laminar flow in three-dimensional rectangular microchannels with different aspect ratios. Using a finite-volume approach, the friction factor characteristics of Newtonian fluid in three-dimensional rectangular ducts with aspect ratios from 0.1 to 1 were conducted numerically under no slip boundary conditions. Currently, the available analytical solutions of the friction factor for laminar fluid flow and forced convection heat transfer in noncircular pipes had been summarized in Ref. [30]. Considering only the entrance region results of ƒappRe at ε ¼ 1, 0.5, 0.2 for developing flow in rectangular ducts were presented, it is necessary to calculate the data of ƒappRe for other aspect ratios in rectangular microchannels. Furthermore, if the values of ƒappRe for arbitrary aspect ratios of rectangular cross section can be obtained, it will be more convenient and efficient for engineering practice and application.

267

Heat transfer in the entrance region of microchannels

Therefore, it is of great importance that using a model to approximately predict flow characteristics of microchannels with different cross sections, especially in the developing laminar flow. For fully developed flow, the analytical solutions such as circular tubes and parallel plates with cross-section defined by a single coordinate can be easily obtained. The investigation of flow characteristics in rectangular ducts required a two-dimensional analysis can also be solved for fully developed flow. However, it is fairly complex for developing laminar flow, which is a three-dimensional analysis in rectangular ducts, and there is no analytical solution. Duan and Muzychka [33,45] examined slip flow and continuum flow in non-circular microchannels and developed a simple model to predict the friction factor and Reynolds number product ƒRe for fully developed flow and even the developing flow in the noncircular microchannels. Using the model, the friction factor and Reynolds number product ƒRe for both slip flow and continuum flow of noncircular microchannels at arbitrary cross sections and arbitrary axial distances can be approximately calculated. For a rectangular channel with aspect ratio defined as αc ¼ a/b (a < b), Shah and London [30] provided the equation of the friction factor Reynolds number product for fully developed laminar flow   fRe ¼ 24 1
1:3553αc + 1:9467α2c
1:7012α3c + 0:9564α4c
0:2537α5c (31) Most researchers use Eq. (31) to calculate the friction factor Reynolds number product, which is a fitting formula. In fact, for fully developed flow in rectangular microchannels, there exists a more accurate theoretical solution, which can be obtained from Duan and Muzychka [33,45]. The expression is as follows: fReDh ¼

24  π ð1 + εÞ2 1
192ε tanh π5 2ε +

1 243

tanh

3π

(32)

2ε

Here, ε is the aspect ratio of rectangular cross section, which is the ratio of a to b (a < b). A comparison between Eq. (31) and Eq. (32) is presented in Table 2. It has been found that the fitting formula has reached a certain precision and the maximum error is less than 0.11%. For the calculation of the high accuracy requirements, the theoretical solution can also be utilized more conveniently and accurately for calculating friction factor Reynolds number product ƒRe.

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Table 2 The comparison of ƒRe between Eq. (31) and Eq. (32). ε

1

Eq. (31) 14.23

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

14.27

14.38

14.61

14.99

15.56

16.38

17.52 19.07

0.1

21.18

Eq. (32) 14.218 14.254 14.372 14.601 14.976 15.545 16.366 17.51 19.069 21.168 (%)

0.8

0.11

0.07

0.05

0.06

0.08

0.07

0.03

0.01

0.04

1

u/umax

0.8 0.6

0.2 0

x+=0.03 x+=0.05 x+=0.07 x+=0.1

x+=0.001 x+=0.003 x+=0.005 x+=0.007 x+=0.01

0.4

0

0.2

0.4

0.6

0.8

1

z/W Fig. 9 Developing dimensionless velocity profiles at ε ¼ 1.

We focus on the non-dimensional flow distance of x+ from 0.001 to 0.1. Staying with the ε ¼ 1 as an example, the developing dimensionless velocity profiles at different x+ in the entrance region of the central axis are shown in Fig. 9, in which the spanwise direction (z) with the width (W) of the microchannel, and the axial velocity magnitude in different positions of x for the developing flow are plotted. The gradual process of change for dimensionless velocity profile is observed in the entrance region of rectangular cross section until velocity reaches a parabolic profile curve in the upper section which indicates the flow has been in the fully developed regime and the axial velocity gradient ∂ u/∂ x ¼ 0. In the fully developed flow, it is a two-dimensional analysis including the transverse velocity components v and w. For the flow boundary layer and the wall surface, the presence of the significant pressure drop has an evident effect on the dimensionless velocity profile and the skin friction coefficient. Therefore, the investigation of friction characteristics and pressure drop in the entrance region requires a three-dimensional analysis and gets even more complicated.

Heat transfer in the entrance region of microchannels

269

The apparent friction factor and Reynolds number product ƒappRe had been presented in the available literature by Shah and London [30]. They reviewed the available information and made a figure for the apparent friction factor. The data of ƒappRe in a tabular form with the aspect ratios of ε ¼ 1, 0.5, 0.2 were presented, which are shown intuitively in Table 3 as the contrast and comparisons. The obtained results show a good agreement and the maximum relative error is less than 3.9%, which confirm the efficiency and accuracy of ƒappRe in developing flow for this numerical method. On the basis of the accurate and reliable simulated results, the data of ƒappRe for other aspect ratios of rectangular microchannels are significantly extended. The numerical results of the apparent friction factor and Reynolds number product ƒappRe of 10 different aspect ratios in rectangular microchannels are presented in Table 3. It has been shown that ƒappRe is a function of axial dimensionless distance and the aspect ratio. As expected, ƒappRe is very large at the beginning of entrance region, but rapidly decreases with axial position and then changes slowly until reaching the full developed values. In addition, a larger aspect ratio corresponds to a smaller value of ƒappRe. For x+ > 1, the incremental pressure defect reaches a constant value K (∞), known as Hagenbach’s factor. Steinke and Kandlikar [46] provided the following curve-fit equation for the Hagenbach’s factor of rectangular channels: K ð∞Þ ¼ 0:6796 + 1:2197ε + 3:3089ε2
9:5921ε3 + 8:9089ε4
2:9959ε5

(33)

Now a new curve-fit equation for the Hagenbach’s factor of rectangular channels can be obtained from the accurate and comprehensive data of Table 3. According to Eqs. (8) and (9), for x+ > 1, the results of K(0.04), K(0.05), K(0.06), K(0.07), K(0.08), K(0.09), K(0.1), K(0.2) and Eq. (33) at ε of 0.1–1 for rectangular channels are calculated respectively, as is shown in Table 4. On account of the Hagenbach’s factor, which means the incremental pressure defect attains a constant for x+ up to the hydrodynamically developing entrance region. Combined with Table 3, the new curve-fit equation for the Hagenbach’s factor is fitted for x+ ¼ 0.2. Through the calculation and comparison, when the formula reaches the four-order, the accuracy has been improved. The expression is as follows: K ð∞Þ ¼ 0:7208
0:4489ε + 8:0598ε2
12:7107ε3 + 5:7791ε4

(34)

Table 3 ƒappRe for developing laminar flow at different aspect ratios.

ƒappRe

ε51

ε 5 0.5

ε 5 0.2

Shah/London x+ ε51 0.001 111.05 111.0

ε 5 0.9 ε 5 0.8 ε 5 0.7 ε 5 0.6 111.09 111.07 111.14 111.09

Shah/London ε 5 0.5 111.01 111.0

ε 5 0.4 ε 5 0.3 111.04 111.15

Shah/London ε 5 0.2 111.15 111.0

ε 5 0.1 112.01

0.002 81.39

80.2

81.42

81.42

81.47

81.48

81.47

80.2

81.54

81.70

81.83

80.2

82.48

0.003 67.68

66.0

67.70

67.71

67.76

67.78

67.80

66.0

67.90

68.08

68.26

66.1

68.86

0.004 59.28

57.6

59.30

59.31

59.35

59.39

59.43

57.6

59.54

59.73

59.94

57.9

60.51

0.005 52.98

51.8

52.98

53.00

53.04

53.08

53.13

51.8

53.25

53.44

53.67

52.5

54.22

0.006 48.75

47.6

48.76

48.78

48.82

48.86

48.92

47.6

49.05

49.24

49.49

48.4

50.03

0.007 45.97

44.6

45.98

45.99

46.04

46.08

46.15

44.6

46.28

46.49

46.75

45.3

47.31

0.008 43.08

41.8

43.09

43.11

43.15

43.20

43.27

41.8

43.40

43.61

43.88

42.7

44.43

0.009 41.07

39.9

41.07

41.09

41.13

41.19

41.27

40.0

41.40

41.62

41.91

40.6

42.48

0.010 39.35

38.0

39.36

39.38

39.42

39.47

39.56

38.2

39.70

39.93

40.23

38.9

40.81

0.015 33.03

32.1

33.03

33.06

33.10

33.17

33.27

32.5

33.44

33.70

34.07

33.3

34.75

0.020 29.72

28.6

29.73

29.75

29.81

29.89

30.02

29.1

30.21

30.53

31.00

30.2

31.84

0.030 25.29

24.6

25.30

25.34

25.40

25.51

25.69

25.3

25.95

26.38

27.03

26.7

28.13

0.040 22.93

22.4

22.94

22.98

23.07

23.20

23.43

23.2

23.77

24.30

25.10

24.9

26.39

0.050 21.54

21.0

21.56

21.61

21.71

21.88

22.16

21.8

22.56

23.19

24.12

23.7

25.56

0.060 20.31

20.0

20.33

20.39

20.50

20.70

21.00

20.8

21.45

22.13

23.13

22.9

24.65

0.070 19.63

19.3

19.64

19.71

19.85

20.06

20.41

20.1

20.91

21.65

22.74

22.4

24.35

0.080 18.84

18.7

18.86

18.93

19.07

19.30

19.67

19.6

20.19

20.97

22.09

22.0

23.74

0.090 18.45

18.2

18.48

18.55

18.71

18.96

19.35

19.1

19.92

20.75

21.93

21.7

23.65

0.100 18.02

17.8

18.04

18.12

18.29

18.55

18.96

18.8

19.56

20.41

21.63

21.4

23.39

0.200 15.96

15.8

16.01

16.03

16.34

16.73

17.12

17.0

17.81

18.77

20.17

20.1

22.09

>1.0 14.22

14.2

14.25

14.37

14.60

14.98

15.55

15.5

16.37

17.51

19.07

19.1

21.17

Table 4 The K(∞) for different x+. x+ 5 0.05 ε x+ 5 0.04

x+ 5 0.06

x+ 5 0.07

x+ 5 0.08

x+ 5 0.09

x+ 5 0.1

x+ 5 0.2

Steinke/Kindlikar

1

1.3931

1.4645

1.4617

1.5039

1.4777

1.5248

1.5198

1.4029

1.5291

0.9

1.3897

1.4606

1.4576

1.4996

1.4732

1.5204

1.5155

1.3642

1.5410

0.8

1.3771

1.4470

1.4429

1.4849

1.4579

1.5054

1.5003

1.3797

1.5293

0.7

1.3540

1.4220

1.4162

1.4581

1.4302

1.4785

1.4736

1.3888

1.5002

0.6

1.3169

1.3820

1.3733

1.4151

1.3856

1.4352

1.4308

1.3593

1.4524

0.5

1.2615

1.3224

1.3094

1.3510

1.3188

1.3703

1.3664

1.2808

1.3808

0.4

1.1846

1.2402

1.2212

1.2623

1.2261

1.2798

1.2764

1.1601

1.2804

0.3

1.0851

1.1350

1.1083

1.1491

1.1074

1.1641

1.1607

1.0147

1.1492

0.2

0.9658

1.0107

0.9754

1.0171

0.9685

1.0295

1.0260

0.8676

0.9925

0.1

0.8346

0.8780

0.8343

0.8796

0.8235

0.8915

0.8885

0.7415

0.8259

Heat transfer in the entrance region of microchannels

273

The fourth-order fitting equation has higher precision and is more effective for different aspect ratios. The form of Eq. (34) becomes concise and the accuracy has been significantly improved.

5.2 Slip flow in the entrance region Ma et al. [44] carried out a study on a developing three-dimensional laminar slip flow in the entrance region of rectangular microchannels. When the hydrodynamic development length is the same magnitude as the microchannel length, entrance effects have to be taken into account, especially in relatively short ducts. Simultaneously, there are a variety of noncontinuum or rarefaction effects, such as velocity slip and temperature jump. The available data in the literature appearing on this issue is quite limited, the available study is the semi-theoretical approximate model (see Ref. [33]) to predict pressure drop of developing slip flow in rectangular microchannels with different aspect ratios. We applied the lattice Boltzmann equation method (LBE) to investigate the developing slip flow through a rectangular microchannel. The effects of the Reynolds number (1 < Re > ζ < c , Λg Eg ¼ (61) > h ch > : , 0 Ω

^ m  n0  


 
 Ω ^ , b r, Ω ^ ^  na r, Ω ^ b r, Ω ^ dΩdr, ¼ a r, Ω Γ ^  n0 Ω

^ m  n0

Z 
+ 
+   ^ ^ ψ rint , Ω nα!δ  Ω dΩ ¼ ϕdiffuse,δ rint

^ nα!δ  Ω>0

Z ¼ T α!δ

^ nα!δ  Ω>0


  ^ ψ r int , Ω nα!δ  Ω dΩ

Z

+ Rδ!δ

  nα!δ  Ω ^ dΩ

^ nα!δ  Ω0  Z 
 ^  ^ ψ r  Tα!δ int , Ω nα!δ  Ω dΩ ^ n  Ω>0  Z α!δ 
+  ^ nα!δ  Ω ^ dΩ , + Rδ!δ ψ rint ,Ω

(154)

^ nα!δ  Ω
> =


 
 ħωjvj 1 1   Gm,g ϕ0 T ¼ G ϕ0m,g T ¼ , wm w m m,g > > ħω > > : exp  1; kB T  0

 dϕ T 1 0,0 G , ϕm,g T ¼ w m m,g dT  2 0

 d ϕ T 1 0,00 : G ϕm,g T ¼ w m m,g dT 2 In the discretized PTE (Eq. 45) the only material properties that appear are the groups’ effective mean free path and radiance; however, each iteration of the transport simulation requires computing the local temperature and so we require a set of weighting terms to obtain the local temperature from the sum over fluxes. To compute the energy in a group from the angular flux, we define the group velocity as the averaged velocity weight by mode energy

406

Jackson R. Harter et al.


 Gm , g ϕ 0 T v m, g ¼  0
 : ϕ T Gm, g jvj For the local temperature calculation we need to know the total of the product of frequency, density of states, and velocity ωvm,g ¼ Gm,g fωjvjg, and the average frequency ωm,g ¼

Gm,g fωg : Gm,g fg

The approach for defining the various transport group properties that is laid out above is quite general; the choice of spectral parameterization has not been defined, and the method is applicable to anisotropic crystals provided if the set of transport ordinates is chosen to have the appropriate point symmetry. However, for the transport simulations of Si demonstrated in this chapter a series of further approximations were made to further simplify the transport groups. First, the exact phonon dispersion is replaced by an approximated dispersion that is spherically symmetric enabling us to use a standard set of nonsymmetric transport ordinates. In this approach we used just the dispersion for Si along the (100) crystal direction, and we collapsed the three optical branches and the two transverse acoustic branches into Table 2 Silicon material properties. G Λ [nm] ω  1013[s1] v [m  s21]   1027[m23] τ [ps] ωv [m22  s22]

1 (LA) 3120

1.49

8079

0.0736

386

12.9  1042

2 (LA) 155

4.25

6926

0.515

22.4

156  1042

3 (LA) 28.7

6.46

5152

1.4

5.6

458  1042

4 (TA) 898

0.99

5076

0.147

177

9.4  1042

5 (TA) 79.5

2.3

2063

1.03

38.5

42.2  1042

6 (TA) 11.1

2.75

721

2.8

15.4

51.8  1042

7 (O)

3.15

9.43

786

0.22

4

24.9  1042

8 (O)

3.8

8.92

1651

1.55

2.3

219  1042

9 (O)

4.58

8.43

1126

4.2

4.1

397  1042

407

Predicting mesoscale spectral thermal conductivity

16

14

12

10

8

6

4

2

0 0.1

0.2

0.3

0.4

0.5

0.6

Fig. 13 Dispersion relation in silicon [77].

a single branch with averaged frequency as shown in Fig. 13. The dispersion was parameterized in wave number, k, and this condensed (100) dispersion was used as the dispersion along all directions. The density of states was defined as
 3dp k2 ^ k, p ¼ ,  Ω, m,g πk3max a3Si where dp is the degeneracy that accounts for the merged branches and aSi is the lattice parameter of the Si unit cell. The integral of density of states accounts for all the phonon modes of Si 3 Z X p¼1 4π

Z

kmax

dΩ

dk 0

3dp k2 38 ¼ 3Nv , 3 ¼ 3 πkmax aSi a3Si

To test the performance of the model with different spectral resolution the approximated dispersion was split into between three and 20 equally sized k groups; we used a discretization of 9 groups in our simulation, split up among the polarizations (see Table 2, Fig. 14).

408

Jackson R. Harter et al.

9000

8000

7000

6000 2000 5000 1500 4000 1000

3000

2000

500

1000

0 0

0.005

0.01

0.015

0.02

0 0

10

20

30

40

50

60

Fig. 14 Mean free paths for dispersion branches vs group velocity. Units are consistent for inset figure [77].

3. Results In this section, we discuss results from various phonon transport simulations using different methods described in Section 2. We present results using the spectral transport approach and the frequency-independent, gray BTE.

3.1 Spectral phonon transport 3.1.1 Spectral phonon transport in silicon We performed spectrally resolved phonon transport simulations in grids of silicon with the geometric domain ranging from 10 nm to 10 μm. We report heat flux, thermal conductivity, and the equilibrium temperature distribution. We used the full phonon dispersion and density of states computed at room temperature from ab initio DFT simulations. Spatial domain sizes varied from 10 nm to 10 μm, and were spatially discretized using coarse (C) and fine (F) meshes constructed of triangular finite elements (Fig. 15). We used S4, S8, and S16 Gauss–Legendre angular quadrature for all C and F cases. We simulate a temperature gradient of 1 K along the x-axis, with boundary temperatures of TL ¼ 301 K, TR ¼ 300 K. Reflecting conditions

Predicting mesoscale spectral thermal conductivity

409

Fig. 15 Coarse (left, 926 elements) and fine (right, 5770 elements) spatial meshes [77].

are placed on remaining boundaries. We use a preconditioned GMRES [66, 110, 111] solver to tackle the linear system of equations, with convergence criteria set to E ¼ 108. The selected cases we discuss in this section were all simulated using S8 quadrature with the fine spatial mesh [77]. In centerline temperature profiles (Fig. 16), the condition of nonequilibrium arises when the spatial domain is small, this occurs because radiative equilibrium cannot be established. The incident phonon radiance from either side encounters interference due to the proximity; the distance between hot and cold sources is less than of the mean free path of the majority of phonons. This factor is more remarkable in smaller geometric domain sizes, exacerbated by the presence of ballistic phonons, which undergo very few collisions before reaching the opposite side of the domain. The temperature distribution has an equilibrium solution upon numerical convergence of the simulation, as the temperature distribution in each phonon mode is identical. If an equivalent transient simulation was conducted, the modal temperature distributions might not be identical. The temperature profiles produced from this work can provide a basis for the benchmarking of temperature distributions in MD simulations, which rely on the careful selection of fitting parameters; the temperature profiles computed by this method may aid in the accurate description of these fitting parameters [77, 112]. The closure term β (Fig. 17) provides a glimpse into the nonequilibrium behavior of the solution at various spatial domain sizes; β tightly couples the phonon radiance (ϕ0) and flux (ϕT). We have shown β multiplied by the average relaxation time τ in Fig. 17, in order to understand the fraction of total energy exchanged into the bath in one scattering event.

410

Jackson R. Harter et al.

301 300.9 300.8 300.7 300.6 300.5 300.4 300.3 300.2 300.1 300 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 16 Centerline temperature distribution for four cases. Temperature slips at the boundaries are observed, and their magnitude decreases in proportion to increasing domain size, as phonon boundary sources become further separated [77].

The total amount of energy exchanged in β between ϕT and ϕ00 is minuscule relative to the total energy of the system, but without the presence of β conservation is broken [77]. The total fluxes ΦT and Φ00 are not equivalent at small domain sizes; this occurs due to the difference between the boundary emission source and the local solution ϕT; which is where much of the “character” in the shape of β near the boundaries of Fig. 17 exists. This artifact is a consequence of the geometric system size, where the heated boundaries are in competition with each other. β exhibits a sign change, beginning negative at TL and turning positive at TR, with a sign change at the midpoint. In cases where β < 0, ϕT > ϕ0, and ϕT is strongly influenced by the boundary conditions where the incident angular intensity ψ is specified by the phonon radiance relation in Eq. (51). As phonons flow from hot to cold sides, the sign of β tends positive when ϕ0 > ϕT, after the spatial midpoint. This is because the local radiance ϕ0 is now stronger than local transport flux ϕT—the emission of phonon from the cold boundary is unable to overcome the flow of phonons from the hot side, which results in a positive β, but also elevates the temperature away from the cold boundary shown on the profiles in Fig. 16.

411

Predicting mesoscale spectral thermal conductivity

10–3

8

6

4

2

0

–2

–4

–6

–8 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 17 β  τ distributions along the x-axis for four domain sizes. As domain size increases, the behavior of β in the bulk tends toward zero as the total radiance (Φ00) and total flux (ΦT) come into balance and the influence of the heated boundaries subsides. The left boundary emits phonons at a higher temperature than the right boundary and so locally, ϕT > ϕ0. The opposite is true on the right side of the figure, in proximity of the colder phonon source. The influence of β is directly affected by the size of the domain; with smaller domain sizes, the emissive sources are in greater competition with each other and radiative equilibrium cannot be established [77].

Black-body phonon emission is very strong in proportion to the transport flux, and although convergence occurs the localized nonequilibrium effect is very apparent, especially in smaller domain sizes [77]. Centerline total heat flux profiles for the S8,F case are shown in Fig. 18. As system size increases, heat flux decreases. Due to the flat heat flux profiles observed in all cases, it is clear: including β to close the transport equation is a necessity. Without β, energy leaks out of the system and we observe a nonconserved heat flux. However, inspecting group heat flux depicts a clear image showing which groups carry the most heat in the system. It is, of course, the groups with the largest Λ, and those which remain the most ballistic over the entire geometric domain range—group velocity plays a key role in carrier transport. Fig. 19 shows group heat flux for the S8,F case for all domain sizes.

412

Jackson R. Harter et al.

0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 18 Centerline total heat flux for the S8, F case. Heat flux is approximately constant across the domain, with very minor fluctuations occurring at the hot and cold emitting boundaries. With increasing domain size, heat flux decreases [77].

It is clear that larger values of Λg are responsible for a higher heat flux; ballistic phonons carry energy further between collisions. The LA phonons are the dominant carriers due to their higher velocities and mean free path; TA and O phonons contribute lesser quantities. Acoustic phonons are generated by atomic displacements moving in phase, which is responsible for their higher velocity. Conversely, optic phonon motion is out of phase and are shorter range carriers. This phenomenon can be traced to the large magnitudes of the derivatives of the dispersion curves, which equate to higher wave propagation speed. Heat flux becomes flat as system size increases, shown by Fig. 19; in this way we can analyze the degree to which the heat-carrying groups are affected by the change in domain size. If phonon groups were decoupled, the relation between heat flux and domain length would remain fixed in each group, as individual groups are not influenced by the average energy of the domain. However, in a coupled simulation the equilibrium distribution in each phonon channel is impacted by the average material temperature. Results for heat flux and thermal conductivity vary by approximately 3% between all angular and spatial resolutions. For a homogeneous material, a coarse-to-moderate angular and spatial resolution is sufficient for accurate

413

Predicting mesoscale spectral thermal conductivity

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10,000

Fig. 19 Group heat flux for each geometric domain size for the S8, F case. Groups 1–3 are LA, 4–6 are TA, and 7–9 are O. Diffuse groups (7, 8, 9) always carry low amounts of heat and remain relatively flat independent of domain size [77].

results. This will likely change in heterogeneous environments such as porous materials, bulk material with dopants or inclusions (e.g., UO2 with Xe bubbles), where ray effects have been observed in gray simulations [77, 82]. Normalized spectral heat flux is shown in Fig. 20, giving insight into how rapidly each of the groups approach the ballistic limit. It is clear that q1 (with Λ1 ¼ 3120 nm) carries the dominant portion of energy throughout the entire range of domain sizes, as it barely begins to approach the asymptotic limit at a length of 10 microns. However, it is overtaken early on by groups 2–5, until the domain sizes reaches about 1000 nm. As expected, the diffuse groups do not contribute appreciably to the overall heat flux [77]. Thermal conductivity for all cases is shown in Fig 21. The increasing angular refinements reduce heat flux and thermal conductivity by negligible amounts, caused by additional information present due to a higher number of streaming directions. The domain sizes span 10 nm to 10 μm, and κeff is proportional to them. As domain size increases, κ eff asymptotically approaches a bulk value of about 165 W  m1  K1, and many researchers have reported values for κ eff with variation on the order of 100% [113]. As spatial and angular

414

Jackson R. Harter et al.

7

6

5

4

3

2

1

0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10,000

Fig. 20 Normalized group heat flux (S8, F case); the ballistic limit is achieved rapidly by most of the phonon groups, with the exception of group 1 (largest Λ in the system) [77].

discretizations change, so do the values of spectral κ eff. On smaller spatial meshes, Λg tends to be ballistic per cell, even if they would be diffuse on the global domain. The coarse spatial discretization is acoustically thicker where the fine spatial discretization is acoustically thinner. It is only as global domain increases in size that these groups become rather diffuse, and their contribution to total heat flux becomes quite negligible; this trend is shown in Figs. 19 and 20. Table 3 contains the contributions from optic phonons to κeff, and they have a stronger effect in smaller global domains. For simulations with large (greater than 1 μm) domain sizes, omitting optic phonons would cause a dramatic decrease in the amount of iterations required to converge the solution, with negligible loss of accuracy (Table 3). Overall, it is more important to finely resolve spatial cell (finite element) size in comparison to angular resolution [77]. We compare Φ00 and ΦT in Fig. 22. The offset between Φ00 and ΦT are related to the magnitude of β, and intuitively, as spatial domain size increases and the magnitude of β decreases, Φ00 and ΦT should converge on each other. However, for the 100 nm domain this is not the case; the two fluxes are

415

Predicting mesoscale spectral thermal conductivity

200 180 160 140 120 100 80 60 40 20 0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000 10,000

Fig. 21 Thermal conductivity for various angular and spatial (F ¼ fine, C ¼ coarse) discretizations up to 10 μm. The overall difference between the extreme cases is about 3% [77].

Table 3 Variation in κ eff with and without optic phonons for S8,F simulation. L [nm] κLA,TA,O κLA,TA Δκð%Þ

10

6.1

5.2

14.8

100

30.3

28.2

6.9

1000

90.8

88.4

2.6

10,000

168.6

166.7

1.1

κ is in units of ½W  m1  K1  [77].

further apart. Looking back to Fig. 19, there is a slight jump for groups 1 and 4, where heat flux experiences a slight rises rather than a monotonic decrease. The modal dominance of heat flux affects the magnitude between Φ00 and ΦT at 100 nm, where the system is still sufficiently far from equilibrium. The same effect is present in domains up to 600 nm, after which the heat flux contribution from group 4 (at Lx ¼ 700 nm) finally becomes less than the heat flux at 50 nm. This lag is caused by phonons in group 1 finally overtaking

416

Jackson R. Harter et al.

45.64

45.7

45.62

45.65

45.6 45.6 45.58 45.55

45.56 45.54 0

0.2

0.4

0.6

0.8

1

45.5 0

45.75

45.75

45.7

45.7

45.65

45.65

45.6

45.6

45.55

45.55

45.5

45.5

45.45 0

45.45 0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Fig. 22 Total radiance (Φ00) and transport flux (ΦT) comparisons domain sizes between 10 nm and 10 μm using an S8 quadrature. As the domain size increases, the systems relax to a more equilibrate state [77].

those from group 4; the transition happens in between 600 and 700 nm, observed in Fig. 20. As the geometric domain increases to the micrometer scale, Φ00 and ΦT become increasingly convergent and their respective difference reduces to 43 °C and RCEM ¼ 0.25 for T < 43 °C. Threshold values for t43 are available in literature depending on tissue examined, and they have been obtained from experimental data referred to tissue injuries at 43 °C [26–28]. An example of t43 calculation is shown in Fig. 4. About 44.5 °C are continuously applied for 10 min, resulting in equivalently 80 min at 43 °C. This result can be roughly achieved with Eq. (3) with t2 ¼ tf, T1 ¼ 43 °C and T2 ¼ 44.5 °C. Based on threshold values available for CEM43 from experiments [26–28],

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tf (min)

t43 (min)

1

1000

t43 = tf RCEM[43(°C)–T]

T (°C)

10

100

46 45 44 43

42

100

10 41

1000

1

Fig. 4 Monogram for t43 method prediction proposed by Sapareto and Dewey [24].

one can therefore establish if necrosis is achieved. For example, a tissue with a value of C120 means that necrosis would be achieved when the tissue is exposed at 43 °C for 120 min. Tissue temperature variations with time T(t) can be introduced in the CEM43 criterion as follow: CEM 43 ¼ t43 ¼

ð tf

RCEM ½43T ðtÞ dt 

0

¼ tf RCEM

ð43T Þ

tf X

RCEM ð43T i Þ Δti

t¼0

(3)

In this equation, a summation approximation can be employed with sufficiently small Δti and average temperatures T i evaluated for each Δti; if one assumes that the temperature is uniform (or, if an average value T is employed), then one could derive t43 in Eq. (3) without invoking any integral or summation, as for Eq. (1). An alternative method to establish tissue damage is the Arrhenius damage integral criterion. With this criterion, damage is obtained from an exponential relationship between tissue exposure temperature, time and parameters

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generally from experimental studies on cells survivability. In particular, these parameters are available for many tissues, and they have been obtained from fitting of known exposure times and temperatures with cell surviving probabilities. However, it is noticed that the Arrhenius damage integral criterion cannot be used a priori since temperature evolution with time needs to be known. This approach is based on Arrhenius equation with a first-order kinetic reaction, reported in the following together with the thermal damage probability Pt that is between 0 and 1 [29].   ðt ΔH  c ð0Þ (4a) ¼ Ae Rg T dt Ωt ðtÞ ¼ ln c ðtÞ 0 P t ¼ 1  eΩt ðtÞ

(4b)

In this expression, Ωt(t) is the tissue injury degree, c represents living cells concentration that depends on time, Rg is the universal gas constant, A is a kinetic energy frequency factor, and ΔH irreversible damage reaction activation energy [26]. Parameters A and ΔH depend on tissue types, for example for the liver they are A ¼ 7.39  1039 1/s and ΔH ¼ 2.577  105 J/mol [28], while for scald burns A ¼ 3.1  1098 1/s and ΔH ¼ 2.577  105 J/mol [30]. These two parameters A and ΔH can vary through a wide range of values, and they can be also temperature-dependent. In particular, A could vary of several order of magnitudes, while ΔH remains of an order of magnitude of about 105 J/mol [31,32]. Other values for the two parameters are available in Pearce [29]. From Eq. (4b), one can observe that for Ωt(t) ¼ 1 cell death probability is 63% (Pt ¼ 0.63), while if Ωt(t) ¼ 4.6, then cell death probability becomes 99% (Pt ¼ 0.99). This tissue injury value can be therefore assumed to be the value for which completed necrosis is achieved. Other two methods that can be used to quantify thermal dose, and then cells damage, are the Area Under the Curve criterion (AUC) [33] and the isotemperature contours criterion [34]. With the AUC criterion, temperature minus baseline is integrated over time to establish thermal damage, while with the isotemperature contours criterion one assumed that all the tissue above a prefixed temperature is necrotic. This threshold can vary between 43°C [35,36] and 59°C [37], depending on many variables like tissues considered. Various papers through the years compared various methods for thermal injury computation. Vallez et al. [38] suggested that CEM43 would be preferable for lower hyperthermia temperatures, while the Arrhenius

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thermal damage method is suggested for higher temperatures. An inverseproportionality relationship between CEM43 and Arrhenius thermal damage criteria has been analyzed and discussed by Viglianti et al. [39]. Pearce [29] compared CEM43 and Arrhenius thermal damage criterions for laser-induced heating, concluding that the latter would be preferable since it allows to separately study various thermodynamically-independent processes. Mertyna et al. [33] generated different ablation measures by using radiofrequency ablation, microwave ablation and laser diffusing fibers, and they compared area under the curve, CEM43 and Arrhenius thermal damage criteria. The authors conclude that thermal dose should not be established based on temperature at the end of coagulation zone since this is not constant but it depends on distance. Chang and Nguyen [28] simulated thermal and injury profiles for a radiofrequency ablation of 15 min, and comparisons between isotemperature contours, CEM43 and Arrhenius damage criteria have been shown. The authors conclude that choosing the appropriate criterion is important, since isothermal and CEM43 might cause significant errors in the estimation of lesion size.

1.4 Ablation procedures As already mentioned, various methodologies of thermal ablation depending on the application and on the employed applicator can be distinguished. Thermal ablation is used to treat tumors, cardiac arrhythmias, benign prostate hyperplasia, and so on. With references to tumor ablation, this is used mainly for liver tumors and other organs like kidney, lung or esophagus. Thermal ablation is a minimally-invasive technique useful for patients that cannot be go under surgery that presents good long-term survival rates and acceptable local tumor control [40]. On the other hand, it has to be mentioned that with thermal ablation quite-limited ablation volumes are achieved, that this technique is not applicable to all tumors, and a heat sink from surrounding vessels might occur, causing the risk of a local tumor progression during follow-up [40]. Thermal ablation can be done as Radio Frequency Ablation (RFA), MicroWave Ablation (MWA), High-Intensity Focused Ultrasound (HIFU), Laser ablation (LA) or Cryo Ablation (CA). Among these, the first two, and especially the first one, are the most used for tumor treatments. In all the cases, imaging techniques are used to place the applicator in the tumor that has to be treated. The first two techniques are based on electromagnetic waves that generate heat by Joule and dielectric heating effects. The HIFU ablation is

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based on the conversion of acoustic energy into dissipated heat, the LA consists into conversion of light to thermal energy, and finally with CA cold energy is used to achieve necrosis. As previously mentioned in Section 1, RFA is not used only for tumor ablation, but also for other treatments like atrial fibrillation, endovenous and pain. This technique has been introduced for tumor ablation by Rosenthal et al. [41] for osteoid osteomas in the early 90s. With reference to hepatocellular carcinoma, it is the most widely used ablation technique for earlystage and unresectable hepatic cancers [42]. In this treatment, a needle is placed in the tissue that needs to be treated. A generator provides alternate RF currents in a range between about 350 and 500 kHz. This causes ionic current to pass through the conductive tissue, thus a temperature increase occurs due to frictional agitation of ions that causes resistive heating (Joule heating). Since this phenomenon occurs only near the needle, temperature increases in the rest of tissue occur because of thermal conduction. In RFA, a ground pad is also necessary to close the body electric circuit body. This technique is useful especially for small and medium hepatocarcinoma [43]. Indeed, for larger tumors, incomplete ablation might happen because heating due to Joule effect occurs only close to the needle. For example, it has been reported by [44] an ablation rate of 62% for tumors between 5 and 7 cm. This ablation rate has been improved through the years by employing multiple bipolar electrodes or internally cooled electrodes, reaching, respectively, 81% and 90% ablation rates [45,46]. It is mentioned that one of the limits of RFA is the roll-off phenomenon. Indeed, during the ablation procedure, electrical issues in the needle might rise up because of heating process that causes fast impedance increases. Microwave ablation (MWA) has been introduced in 1994 [47], and nowadays it has an increasing interest. As for RFA, a needle is placed in the tissue that has to be treated. Oscillating electromagnetic waves usually between 900 and 2500 MHz are applied. Contrary to RFA, conductive currents are lower than displacement currents, thus oscillations of polar molecules are generated. These oscillations are converted into thermal energy because of frictional forces between molecules (dielectric heating). Such waves can propagate through tissues even if these present high impedances. In MWA, differently from RFA no ground pad is needed, and this is an advantage since there is no current circulation through the patient body. Higher wave penetration permits to have all related advantages like larger area of ablation, reduced heat sink effects that cause heat loss from surrounding tissues, reduced ablation times, and so on [48,49]. Many studies have

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been presented in literature showing comparisons of this technique with other ones [43]. For example, in the hepatocarcinoma treatment, tumor ablation rates were similar for both MWA and RFA, while recurrence rate was smaller for MWA than for RF ablation [50]. Glassberg et al. [51] showed that for hepatocarcinoma MWA is at least as safe and effective as RFA for liver cancer, with significantly reduced local tumor progression. For the same carcinoma, Poulou et al. [52] reported that MWA permits to treat multiple lesions, and larger coagulation zones are achieved with less time. On the other hand, it has to be mentioned here that therapeutic effect on hepatocarcinoma in terms of survival rates, complications and so on, is generally comparable to RFA [52,53]. For MWA, it is mentioned that too much high temperatures could make complex permittivity too low, thus antenna performances and delivered power become lower. High-Intensity Focused Ultrasounds (HIFU) have been first proposed in 1942 [54]. An ultrasound beam is focused on the ablation zone by employing a piezoelectric ultrasound transducer. The beam passes through the overlying skin, reaching the ablation zone, thus acoustic energy is converted in tissue heating due to particles agitation. This phenomenon could be increased because of acoustic cavitation. The employed beam presents high frequencies usually between about 0.5 and 10 MHz. Such temperatures can increase up to 85 °C, while higher temperatures are avoided because they can cause liquid evaporation [43]. The main advantage of this technique is that it is not invasive [55], and it can be coupled with Magnetic Resonance (MR) or Ultra Sonography (US) guidance. In particular, the former makes this technique very promising since imaging is crucial to achieve an accurate targeting of the focal volume and an accurate monitoring of the whole process [56]. Besides, MR-guidance is preferred since it presents a higher targeting compatibility [56]. The HIFU technique can be employed in various organs like for example prostate [57]. For this technique, if temperatures are too high then ultrasound transmission might be reduced. In Laser Ablation (LA) tissue necrosis is induced from a light beam that generates heat. This laser beam is generated with bare-tip quartz fibers in small-caliber needles. The monochromatic light interacts with tissue by means of absorption, reflection and scattering, and the light absorbed by the tissue becomes thermal energy. The ablation zone can be assumed to be a sphere with a diameter between 12 and 16 mm [58]. This technique has been proposed for phototherapy in tumors in 1983 [59]. Its main advantage is the optimal penetration depth in tumor tissues, making also temperature gradients smaller with lower risks of carbonization and tissue

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vaporization. Comparisons with other techniques like RFA have been presented by Orlacchio et al. [60] for hepatocarcinomas smaller than 4 cm in patients with cirrhotic liver. It has been shown that response rates were similar, while laser ablation presents higher recurrence for carcinomas higher than 2 cm. Authors conclude that generally RFA is more effective, but laser ablation can be considered a good option for smaller lesions. However, too high temperature might cause laser penetration difficult. Cryo ablation (CA) is based on Joule-Thomson effect, that causes cryoprobe cooling in relatively short times [61]. With this effect, a liquid or a real gas temperature changes due to fast pressure variations in devices like insulated valves. Except for gas likes hydrogen, helium or neon, flow temperature rapidly drops in this phenomenon. In CA, a metallic probe is placed in the tissue to be treated, and a gas that is not hydrogen, helium or neon (i.e., nitrogen or argon are often employed) is pumped in the probe and then throttled in order to achieve lower pressures very fast. Because of JouleThompson effect, heat from the tissue is absorbed, then cooling energy starts to diffuse in the tissue and temperature rapidly drops. Reached temperatures are about 160 °C, and various freeze-thaw cycles are performed until desired ablation is achieved. Ice crystal formation causes cell death because of physical damage to both plasma and cytoplasmic organelle membranes [62,63]. On the other hand, damage can also happen with fast freezing due to ice formation [63]. This technique is used for both primary and metastatic liver tumors. In some studies, comparisons with other ablation techniques are presented. For noncolorectal metastases, survival rates have been shown to be high [64], while for hepatic tumors recurrence rates compared to RF ablation have been shown to be higher [65,66].

1.5 Scope of the present chapter From the literature, it has been widely reported that thermal ablation technique is useful because it is minimally invasive [67] and because it allows to destroy only targeted tissue [68]. Generic hyperthermia coupled with chemotherapy has various advantages like improved drug delivery due to blood flow increase, or reduction of tumor oxygen demand due to heating. All these are advantages with respect to radiotherapy or chemotherapy for tumors like breast cancer or melanoma [67]. Besides, thermal ablation is widely used for the liver since up to 80% liver cancer patients are inoperable and techniques like radiotherapy or chemotherapy present some issues [69], and it is also widely used for kidney tumors [70] and others.

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However, it has been also reported that in the near future hyperthermia performances will be improved with new heating delivery technologies and monitoring strategies, useful to optimize delivered thermal dose [67]. Brace [69] reports that many thermal ablation solutions can be presented depending on the clinic situation, and that it is expected an improvement in thermal ablation techniques due to energy delivering technologies improvements. Besides, since multiple treatments are required to remove large tumors due to their sizes, improving strategies is of primary importance [71]. In this context, mathematical modeling also has a primary role because of small scales involved in such problems. What has been here introduced remarks the importance of new heating protocols and accurate predictive models. In this chapter, bioheat models for thermal ablation proposed through the years are described together with approaches to model heat source due to thermal ablation, and finally modulating-heat protocols are reviewed. In the introductory part, that is the first section, techniques employed for thermal ablation together with injury prediction techniques have been resumed. In the second section, various mathematical models for heat transfer in biological tissues developed through the years are presented together with various techniques used to predict heat source terms for such models. The latter aspect is very important because these are the cause of high temperatures for ablation. In the third section, state-of-art of various modulated-heat protocols presented through the years is shown in order to appreciate how these new approaches can lead to new frontiers in thermal ablation fields. The final scope of this chapter is to provide insights and sensitize the scientific community about the potential of modulating-heat procedures to improve tumor thermal ablation techniques.

2. Modeling In the following, bioheat transfer models proposed during the years are presented. Other reviews can be found in [72–75], while a review with an emphasis on how bioheat models can be applied to various hyperthermia applications can be found in Andreozzi et al. [76]. In this section, bioheat transfer models are presented without considering heat source term due to thermal ablation since this aspect has a separate subsection later.

2.1 Thermal problem The complex nature of human tissues makes bioheat transfer modeling quite challenging, and many models have been proposed through the years

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to overcome this issue. The first model to describe bioheat transfer has been proposed by Pennes [77]. In his work, experiments have been performed on human forearm temperature distributions, obtaining average temperature distributions for each section along the depth under different volumetric flow rates conditions. He assumed that this flow rate is uniform, and that there are two heat source terms. The first is a uniform heat source term due to tissue metabolism, while the second represents a heat sink term due to heat delivered from tissue to blood. Between these two, the latter could be a higher order of magnitude higher than the former. This heat sink term is described with a first-law balance between arterial and venous streams. Q_ p ¼ V_ C b ðTa  T v Þ

(5)

In Eq. (5), V_ is the blood mass flow rate per tissue unit volume, and Cb is the blood specific heat capacity. In this equation, it is observed that there is heat transfer between arteries and veins due to transcapillary heat exchange that occurs in the tissue. Eq. (5) has to be now expressed in terms of tissue temperature Tt, that is the main unknown of bioheat problems [77]. At some points, thermal equilibration will occur between capillary blood and surrounding tissue. Based on this, an equilibrium constant kT between 0 and 1 is introduced kT ¼

T v  Tt Ta  Tt

(6)

This constant kT is equal to zero when tissue temperature is equal to venous temperature, while it is equal to 1 if venous and arterial temperatures are in thermal equilibrium. By substituting Eq. (6) in Eq. (5), one obtains the following Q_ p ¼ V_ C b ðkT  1ÞðTt  Ta Þ

(7)

in which the heat rate per volume is proportional to the temperature difference between tissue and artery. The problem is now to characterize the term V_ C b ðkT  1Þ. If one assumes that arterial temperature is uniform through the tissue, and that equilibration between capillary blood and tissue is reached (kT ¼ 0), Eq. (7) can be expressed as Q_ p ¼ V_ C b ðTa  Tt Þ

(8)

that represents the heat source term previously introduced. In particular, Pennes [77] reports that its model matches pretty well experimental curves

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when V_ is between 0.2 and 0.3 kg per cubic meter per second, that are values close to what is reported in literature [78,79]. Besides, Pennes [77] also reports that for incomplete equilibration (kT ¼ 0.25–0.50) values of V_ become higher, say 0.3–0.5 kg per cubic meter per second. The parameter V_ can be expressed as the product of blood density and perfusion rate, i.e., V_ ¼ ρb ωp . The parameter ωp, that is perfusion rate, is a value that can be easily found in literature [80,81]. It is expressed in blood volume per tissue volume per second, thus it is 1/s if blood and tissue share the same volume, and it can be intended as the frequency with which the tissue is perfused by capillaries. For example, if one assumes that blood density is roughly equal to 1000 kg/m3, then Pennes’ found its model to match well experimental results with perfusion rates of ωp ¼ 0.0002 or 0.0003 m3/m3 s. Based on this, one can write Pennes’ bioheat equation [77]. ðρC Þt

∂Tt ¼ kt r2 Tt  ðρC Þb ωp ðTt  Ta Þ + Q_ m ∂t

(9)

Wulff [82] points out three issues related to Pennes bioheat equation. The first is that Eq. (9) reports at the same time local and global control systems, respectively, stored energy and thermal conduction, and perfusion term. The second one is that in Eqs. (5) and (9) three media with three different temperatures occupy the same space at the same time, which is not possible since two more equations might be required for arteries and venous spaces. Finally, the third issue is that perfusion term does not take into account directionality. In other words, heat is transferred in every direction by following local temperature gradients. Because of this, a model in which these effects are considered is proposed. After the definition of an average enthalpy flux that takes into account the angle-averaged expression of local velocity, it is possible to write the following heat equation, under the assumptions of no blood accumulation or depletion and blood temperature equal to tissue temperature, i.e., Tb ¼ Tt ðρC Þt

∂Tt ¼ kt r2 Tt  ðρC Þb juj  rTt + Q_ m ∂t

(10)

In this equation, the assumption of Tb ¼ Tt, i.e., local thermal equilibrium between tissue and blood inside the tissue, is still made as in Pennes [77]. Generally speaking, the main difference between the models consists in the advective term that replaces the perfusion term. In such equation, the velocity modulus j uj is computed as the solid angle-averaged velocity, that is harder to

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estimate compared to the blood perfusion rate from Pennes [77], except for simplified geometries [82]. Chen and Holmes [83] developed a bioheat transfer model that considers microvasculature effects by considering properties dependence on temperature field. Their model focuses on small-scale bioheat transfer, say length scales of the order of 10 mm temperature variations, thus large vessel contribution should be considered separately. The authors describe net mass exchange between vessel and solid by assuming that part of the lymph fluid volume is included in the vascular volume, while the other part is assumed to be in the tissue with the same temperature. In a generic control volume δV, two different subvolumes for solid tissue and blood, respectively, δVs and δVb, are distinguished. Their dimensions are small with respect to macroscopic temperature variation in the overall tissue, but large when references to temperature variations in a microscopic scale are made. A sketch of the control volumes employed is presented in Fig. 5. The ratio between two volumes can be defined as εb ¼

δV b δV b  ≪1 δV δV s

(11)

Volume-averaged temperatures for both solid tissue and blood are defined as follow

Fig. 5 A sketch of control volumes employed in Chen and Holmes [83].

Thermal ablation: Mathematical models and modulated-heat protocols

1 Ts ¼ δV s Tb ¼

1 δV b

507

ð δV s

TdV

(12a)

TdV

(12b)

ð

δV b

An energy balance can be written by considering volume-averaged properties where if Vb/Vs ≪ 1 then Tt  Ts, obtaining the following expression ðρC Þt

∂Tt ¼ r  ðkt rTt Þ + Q_ m + Q_ ∂t

(13)

with properties and local average tissue temperature that can be expressed as volumetric weighted-averages   ð1  εb ÞðρC Þs + εb ðρC Þb ðρC Þt ¼ ð1  εb Þρs + εb ρb   1 ð1  εb ÞðρC Þs T s + εb ðρC Þb T b Tt ¼ ðρC Þt

(14a) (14b)

In Eq. (13), the first term on the right side refers to thermal conduction through the tissue, i.e., stagnant conduction, without any perfusion. The term Q_ m is the metabolic heat per unit volume, while the term Q_ is the surface integral of an advective term per unit volume 1 Q_ ¼ δV

ð S

ðρC Þb T u  dS

(15)

The term under the integral of Eq. (15) cannot be described as usual for free flows since there are many contributions in different directions from blood flow in vessels across the surface S that should be considered. Chen and Holmes [83] split the problem of Eq. (15) in three different subproblems, thus the total term is expressed as Q_ ¼ Q_ 1 + Q_ 2 + Q_ 3. Each subproblem refers to a specific situation. The first subproblem refers to the blood flow and solid tissue temperatures thermal equilibrium problem, thus to the heat transfer between the two phases. The second subproblem refers to the heat transfer in the blood under the thermal equilibrium condition between blood and solid tissue everywhere, while the third subproblem expresses the heat transfer due to space-sinusoidal components of solid tissue temperature. Each term is independently analyzed as follows.

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The first term Q_ 1, is prone to a conventional blood perfusion term as in Pennes [77], and it can be evaluated as follows by making references to baseline arterial temperature for the blood   Q_ 1 ¼ ωp,j ðρC Þb T a,j  T s

(16)

with ωp, j the perfusion rate delivered through the jth generation of vessels, and T a,j is the blood flow-weighted average temperature referred to the jth generation of vessels. Indeed, there are j ¼ 12 generations of vessels from aorta to vena cava. Chen and Holmes [83] report that for a meaningfully small differential control volume large vessels are too few to be significantly considered in Eq. (15), thus the first two or three arteries generation should be considered separately and not included in the perfusion rate term computation (Eqs. 15 and 16). Because of this, in Eq. (16) the blood flow temperature is referred to the vessel at the jth generation and not to larger vessels temperatures since these had been already accounted for in the model. Chen and Holmes [83] also reported that perfusion heat generation in Eq. (16) is not the same of Pennes [77] since both perfusion rate ωp, j and average temperature are accounted from the jth generation vessel, thus T a,j is not referred to the major arteries temperature. In Chen and Holmes [83], it is also shown that most of the heat exchange between blood and solid tissue happens after arterial branches (j ¼ 3) but before arterioles (j ¼ 6), thus not in the capillary bed. This means that the Pennes’ equilibration constant kT [77] should be equal to 1 if one refers to the capillary bed. The second term Q_ 2 is derived from the assumption that blood temperature is equal to the solid tissue temperature. This term refers to a standard convective term that collects all the vessels across the surface S in Eq. (15), and the volumetric heat rate is then expressed as Q_ 2 ¼ ðρC Þb u  rT s

(17)

The third term Q_ 3 refers to the problem of blood temperature equilibration with space-sinusoidal varying components of tissue temperature, obtained via Fourier integral decomposition of solid phase temperature. One can assume that microvasculature presents isotropic properties, and if one collects all these vessels then it would be possible to evaluate this effect as an overall conductive heat transfer problem. All the contributions of all vessels and wavenumbers lead to the following combined perfusion thermal conductivity kp,

Thermal ablation: Mathematical models and modulated-heat protocols

Q_ 3 ¼ r  kp rT s

509

(18)

The perfusion thermal conductivity kp is a term that is not easy to establish [83], and it could have a role from vessels between terminal arterial branches and terminal veins. Finally, the following bioheat equation can be obtained from Eqs. (13) to (18) by reminding that Ts  Tt ðρC Þt

  ∂Tt ¼ r  kt rTt + ωp, j ðρC Þb T a, j  Tt  ðρC Þb u  rTt ∂t + r  kp rTt + Q_ m

(19)

Xuan and Roetzel [84] derived their model by assuming that the human body can be treated as a deformable multiphase porous medium. In other words, they observed that human structures looks like a porous medium saturated with blood [85]. Based on this assumption, two phases that are a solid phase with muscle tissue, vascular tissue and other solid parts, and a fluid phase in which blood flows with perfusion and advection, can be distinguished. Variables like temperature are volumetrically-averaged over a Representative Elementary Volume (REV) as shown in Fig. 6. The REV is the smallest differential volume that permits to obtain statistically meaningful average local properties that are representative of the whole domain. Each volumetric-averaged variable is then decomposed in a spatially-averaged value and in a spatially-fluctuating term, in which spatial average of this fluctuating term is assumed to be zero. Under the assumptions of local thermal nonequilibrium between the two phases, one can write the following expressions for blood and solid phases

Fig. 6 A sketch of representative elementary volume employed in Xuan and Roeztel [84].

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"

# h i ∂hTb ib b b εðρC Þb + hui  rhTb i ¼ r  keq, b rhTb ib ∂t  + hvol, b!s hTs is  hTb ib    ∂hTs is ð1  εÞðρC Þs ¼ r  keq, s rhTs is + hvol, b!s hTb ib  hTs is ∂t + Q_ m ð1  εÞ

(20a)

(20b)

In these equations, subscripts s and b refer to phase-intrinsic volumetricaverages of the variables over the REV. It is reminded that the subscript s is used here for the solid phase because it is made up of tissue and other components, but one could roughly assume that most of the solid phase is tissue. The term hvol is the volumetric heat transfer coefficient, derived from the volumetric average form of Newton’s law of cooling [86]. The last term on the right side of Eq. (20a), that is the opposite of the second term on the right side of Eq. (20b), is a coupling term that takes into account convective heat transfer between the two volumetrically-averaged phases. The two thermal conductivities keq,b and keq,s are equivalent thermal conductivities, that consider both stagnant thermal conductivity and a term that depends on tissue temperature fluctuation that is treated with a diffusive-approximation, as happens for turbulence modeling [84,87]. In the same study [84] and also in a follow-up study [88], with a similar approach the same authors present a three-equation model that can be applied to a limb [88]. Three equations are now established, respectively, for arterial, venous blood flows and solid phase, that is roughly mainly made up by tissue. With this model, counter-current effect between arterial and venous blood flow are considered. In cylindrical coordinates (r, z), under the assumption of steady state and uniform properties, one can write εa ðρC Þa juja

 s  ∂hTa ia d 2 hTa ia ¼ keq,a + hvol,a!s T s  hTa ia 2 ∂z dz

(21a)

 s  ∂hT v iv d 2 hT v iv ¼ keq,v + hvol,v!s T s  hT v iv (21b) 2 ∂z dz



∂hT s is ∂hT s is 1∂ ∂ + hvol,s!a ðhTa ia  hT v is Þ + k r k r ∂r eq,s ∂r ∂z eq,s ∂z (21c) v s _ + hvol,s!v ðhT v i  hT s i Þ + Qm ð1  εÞ ¼ 0 εv ðρC Þv jujv

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In such equations, porosities εv and εa are separately defined for each phase, while the total porosity in Eq. (21c) is ε ¼ εv +εa. Counter-currents effect between arterial and venous blood flow streams are considered in the two terms in the left sides of Eqs. (21a) and (21b) with different signs. A 1D model along the x coordinate is considered for blood flows because it has been shown that blood 2D effects can be neglected due to the transverse thermal dispersion of blood through tortuous capillary beds [88]. All the equations are mutually coupled with interfacial convective heat transfer heat exchange. Terms that refer to this heat transfer mode are the last terms of Eqs. (21a) and (21b) and the third and the fourth terms of Eq. (21c). The variable T s in Eq. (21b) refers to the tissue temperature averaged over limb transverse section at any axial position. A sketch of the counter-current effect in the three-equation model is shown in Fig. 7. A porous media model that accounts for flow field too has been proposed by Nakayama and Kuwahara [89]. As already done in other similar models [83,84,88], they use a volume-averaging approach, in which each variable is defined as the volumetric average over a representative elementary volume. The porous medium is divided in a vascular region (fluid), denoted here as blood region, and an extravascular region (solid), that includes cells and interstitium (Fig. 8). Again, temperature of the latter region can be assumed to be equal to tissue temperature. In the mass equation, the authors assume that there is no notable filtration of fluid between the two phases due to lymphatic system, and the following continuity and momentum equations can be derived as follows

Fig. 7 Sketch of the three-equation counter-current effect.

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Blood vessels

Vascular region

Cells

Extravascular region

Fig. 8 Different regions in the control volume from Nakayama and Kuwahara [89].

r  hui ¼ 0  ∂hui 1 ν b b + r  hui hui ¼  rhpib + νb r2 hui f  b εhuib ρ  ∂t K  b b 2  Fε hui hui

(22a)

b

(22b)

Contrary to Nakayama and Kuwahara [89], the index notation (i, j, k) is not used in the present paper. Eq. (22b) is the typical momentum equation for porous medium. The last two terms on the right side are, respectively, the Darcy and Forchheimer terms, that take into account microstructure viscous and inertial effects on flow field, respectively. In this equation, permeability K and inertial Forchheimer term F are expressed as tensors, and they can be determined from anatomic data if these are available [90]. Momentum equation (Eq. 22b) can be reduced to Darcy law if reference is made to small diameter vessels, thus this assumption is useful for regions in which there are not any large arteries or veins. Energy equations are written under the assumption of local thermal nonequilibrium between the two phases, as in Roetzel and Xuan [84]. Compared to the models previously described, blood perfusion and interfacial convective heat transfer are here separately considered εðρC Þb

!  ∂hTb ib b b + r  hui hTb i ¼ r  εkeq, b rhTb ib ∂t  Ss!b (23a) hs!b hTb ib  hTs is  Vs + Vb   ðρC Þb ωp hTb ib  hTs is

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ð1  εÞðρC Þs

  ∂hT s is Ss!b ¼ r  ðð1  εÞks rhT s is Þ + hs!b hT b ib  hT s is ∂t Vs +Vb   b s + ðρC Þb ωp hT b i  hT s i + Q_ m ð1  εÞ (23b)

In Eq. (23a), the effective thermal conductivity is the sum of the stagnant thermal conductivity and the dispersion thermal conductivity. In particular, the latter comes from a gradient diffusion hypothesis, as already done by Xuan and Roetzel [84]. Depending on radius sizes and Peclet number, by following Nakayama et al. [91] it is possible to describe the effective conductive term for blood as follow 8  2 b 1 > ð ρC Þ u R h i t b > > kb ; PeR < 1 < 48 kb keq,b ¼ εkb + kdis ¼ εkb + (24)  7=8 b > > ð ρC Þ u R h i t > 2:55 b : Pr1=8 k ; Pe > 1 kb

b

R

where the first expression refers to capillaries since they present small Peclet number, while the latter refers to larger vessels due to higher Peclet number. The Peclet number is computed with the tissue radius Rt as the characteristic length [89]. The last two terms of Eq. (23a) also appear in Eq. (23b) but with opposite signs. The first of these two terms is the interfacial convective heat transfer heat exchange term, obtained from the volume-averaged form of Newton’s law of cooling and previously shown in the model from Xuan and Roetzel [84]. However, differently from Xuan and Roetzel [84], the volumetric heat transfer coefficient hvol is here divided in specific surface area S/(Vs + Vb) and heat transfer coefficient hs ! b. In this model, the whole term in the equation in Watts per cubic meters represents the heat exchange between solid and fluid phases without considering mass flow penetration between the phases. On the other hand, mass flow penetration, that is the perfusion term, is taken into account in the last term of Eq. (23a) and in the third term of the right side of Eq. (23b). This term has been written by assuming lymphatic fluid at the same temperature of the tissue, and it represents transcapillary fluid exchange. It is important to notice that perfusion rates in Nakayama and Kuwahara are not the same of Pennes [77]. Indeed, Nakayama and Kuwahara compared their model with the one from Pennes [77], obtaining the following equation ωp,Pennes  ωp +

½Ss!b =ðV s + V b Þhs!b ðρC Þb

(25)

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with which it is possible to state that Pennes’ perfusion rate can be interpreted as an equivalent perfusion rate that considers both perfusion rate, that is as flow transcapillary exchange, and interfacial convective heat transfer between the two phases. Besides, the authors also remark that Pennes assumed a constant arterial temperature in their perfusion term, that is a reliable assumption only for small Peclet numbers. Besides, by comparing their model with the one from Xuan and Roetzel [84], the authors show that in Xuan and Roetzel [84] an overall interfacial heat transfer coefficient replaces perfusion and interfacial heat transfer terms hvol ¼ ½Ss!b =ðV s + V b Þhs!b + ðρC Þb ωp

(26)

In the right side of Eq. (26), convective term is usually higher than perfusion term, but perfusion heat rates could be significant for cases like extremities bioheat transfer, as shown by Chato [92]. It is mentioned that Nakayama and Kuwahara [89] developed a three-equation model similarly to [84,88], with the counter-current effect previously emphasized in Fig. 7. More recently, Dual-Phase Lag (DPL) based models have been applied to describe bioheat transfer. DPL models are based on the assumption that Fourier’s law is not valid for problems in which thermal effects are not propagating at infinite speed. In this case, such law might also take into account heat flux variation with time via a constant named thermal relaxation time, that has a 20–30 s order of magnitude for biological systems [93]. An extended DPL model has been reported in Tzou [94]. He assumed that temperature gradient causes heat flux vector, or vice versa; in other words, depending on the ratio between two time constants τq and τt, it can occur that heat flux is the result of temperature gradient at the same location but a previous time step (for τq/τt > 1), or that temperature gradients occurs because of heat flux at a previous time (for τq/τt < 1). Based on this, Fourier’s law of conduction is written as q + τq

h i ∂q ∂ ¼ k rT + τT ðrT Þ ∂t ∂t

(27)

The same concept has been applied by Antaki [95] to describe temperature evolution in processed meat. Starting from Pennes equation, Zhou et al. [96] derived a DPL bioheat model. They combined Pennes equation (Eq. 9) with DPL non-Fourier model (Eq. 27) in order to replace heat flux in Pennes equation, obtaining the following for the tissue temperature (the subscript b here refers to the artery)

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τq ðρC Þt

h i  ∂Tt ∂2 Tt  ∂ 2 2 ð Þ + 1 + ω ρC τ k r T + τ T r ¼ p q t t T t b ∂t ∂t ∂t 2 + ωp ðρC Þb ðTa  Tt Þ + ωp ðρC Þb τq

∂Ta ∂Q_ + Q_ m + τq m ∂t ∂t (28)

In this expression, differently from Zhou et al. [96] heat generation due to hyperthermia is not considered in order to be consistent with the other methods here shown, and it is reminded again that details on heat generation terms due to hyperthermia will be provided later in this chapter. The same DPL bioheat equation for the heat flux has been also derived in [96]. A DPL formulation based on a porous media local thermal nonequilibrium bioheat model has been proposed by Zhang [97], while the DPL bioheat model has been also applied to magnetic hyperthermia [98]. It is important to underline that such model is very useful in applications in which time scales are very short like laser ablation, for which a good agreement has been found [96]. Finally, one can also mention that all the aforementioned bioheat models can be modified by including various terms that consider other physical aspect For example, if temperature becomes higher than 100 °C then water evaporation might occur. This aspect can be modeled in two methods. The first has been proposed by Yang et al. [99] and arises from Pennes’ bioheat equation (Eq. 9). They include a heat source term in the bioheat transfer equation that represents the water content change depending on time dW Q_ ev ¼ λ dt

(29)

with λ the water latent heat of vaporization, and W the tissue water density in kilograms per cubic meter, that depends on temperature. After some manipulation, they obtained the following expression for Pennes’ bioheat equation ðρC Þeq

∂Tt ¼ kr2 Tt  ðρC Þb ωp ðTt  Ta Þ ∂t

(30)

where the equivalent volumetric heat capacity is defined as ðρC Þeq ¼ ðρC Þt  λ

dW dTt

(31)

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A. Andreozzi et al.

that makes Ceq > Ct during phase change due to water content increase together with temperature due to evaporation. The second method is similar and consists in the apparent heat capacity method [100]. With this method, one assumes that heat capacity during phase change has a very high finite value that depends on a small melting temperature range ΔT. With this assumption, one can use state equation with an apparent heat capacity applied to the bioheat equation transient term. The apparent volumetric heat capacity can be defined as [101] 8 ðρC Þl , if 0°C < Tt  99:5°C > > > < 0:75λ ðρC Þapp ¼ (32) , if 99:5°C < Tt  100:5°C ΔT > > > : ðρC Þg , if Tt > 100°C with ΔT ¼ 1 °C. The term 0.75 refers to the water tissue content that can be assumed equal to this value. In this case, for the apparent volumetric heat capacity during phase change the contribution caused by heat capacities from single components is neglected, differently from [100]. This approach has been also used in Abraham and Jeske [102]. The authors used this method to take into account water evaporation in boiling tissue due to burns. Specific heat is assumed to have an order of magnitude of 105 when tissue reaches temperatures in between 95 and 105 °C. Tissue water concentration and frozen tissue dependence of thermophysical properties has been also taken into account in cryoablation applications by Gonza´lez-Sua´rez and Berjano [103]. Finally, it is mentioned that in some cases perfusion rate can be multiplied with a 0–1 binary term that considers the perfusion to be zero if necrosis is achieved, or with a function that dumps these aspects. Schutt and Hammerich [104] compare a step binary term with a first-order kinetic Arrhenius model that considers a linear reduction of perfusion with the stasis degree, with a model that also includes increase of relative perfusion for low stasis, and finally with a model that considers the third model with four linear functions. The authors conclude that including the relative perfusion for low stasis improves the accuracy of the model [104]. A resume of variable-dependence of bioheat models here introduced (one or two equations) is presented in Table 1 with references to energy equations. This might be useful to understand how many input parameters are required to predict tissue temperature depending on the bioheat model. Porous-media models are here reported without brackets and with the

Thermal ablation: Mathematical models and modulated-heat protocols

517

Table 1 Resume of bioheat models energy equations for tissue and blood. References Equation Expression

Pennes [77] Wulff [82] Chen and Holmes [83]

  Tt ¼ f t, ðρC Þt , kt , ðρC Þb , ωp , Ta , Q_ m   Eq. (10) Tt ¼ f t, ðρC Þt , kt , ðρC Þb , juj, Q_ m   Eq. (19) Tt ¼ f t, ðρC Þt , kt , ðρC Þb , ωp,j , T a,j , u, kp , Q_ m Eq. (9)

  Eq. (20) Tt ¼ f t, ðρC Þt , keq,t , ε, T b , hvol,b!t , Q_ m Tb ¼ f [t, (ρC)b, keq, b, ε, Tt, hvol, b!t, u] h i Nakayama and Eq. (23) T ¼ f t, ðρC Þ , k , ε, ðρC Þ , ω , h , St!b , T , Q_ t t p t!b b m t b Vt + Vb Kuwahara [89] h i St!b T b ¼ f t, ðρC Þb , keq,b , ε, u, ωp , ht!b , V t + V b , Tt   Zhou et al. [96] Eq. (28) Tt ¼ f t, ðρC Þt , kt , ðρC Þb , ωp , Ta , τq , τt , Q_ m Xuan and Roetzel [84]

reasonable assumption of Ts ¼ Tt for the sake of comparison. It is clear that old models required less variables due to the higher number of assumptions that have been done.

2.2 Heat source term modeling As already mentioned, for all the models introduced hyperthermia can be modeled as a heat source term (W/m3) in the energy equations. In the two- and three-equation models (Eqs. 20a, 20b, 21a, 21b, 21c, 23a, and 23b), one usually might assume that the heat source is split in the various phases by employing porosity, that is a volumetric property, as the weighting term [105]. With references to RFA and MWA, the heat source is derived from electromagnetism theories. The antenna used for ablation emits electromagnetic waves that propagate through the tissue, and the electric field generated that causes heat generation can be obtained from Maxwell equations. Depending on the values of electric conductivity σ, angular frequency ω and permittivity εel, one can have different limiting conditions that generally refer to quasi-static lossy (σ ≫ ωεel), lossy (σ  ωεel) and wave equation low-lossy (σ ≪ ωεel) cases, that of course are dependent on the involved frequencies. Frequency-domain (∂/∂t ¼ jω and ∂/∂t2 ¼  ω2) Maxwell equations are here resumed rE¼

ρel εel

(33a)

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rB¼0 r  E ¼ μjωH r  H ¼ ðσ + jωεel ÞE

(33b) (33c) (33d)

The solution of these equations allows to obtain the electric field E. For RFA, since high wavelengths are involved, a quasi-static assumption with which negligible magnetic field variations with time can be done. The following relationship for the magnetic field B can be introduced B¼rA

(34)

with A the magnetic vector potential. By reminding that B ¼ μH, we substitute Eq. (34) in Eq. (33c), in order to obtaining the following r  E ¼ jωðr  AÞ

(35)

We can use curl properties to achieve r  ðE + jωAÞ ¼ 0 E + jωA ¼ rV el

(36a) (36b)

where the term in brackets of Eq. (36a) is expressed in Eq. (36b) as the gradient of a scalar field, namely r Vel. Under the quasi-static assumption, it is possible to establish the following relationship that is then valid for radiofrequencies E ¼ rV el

(37)

One can now perform divergence of Eq. (33d), that makes the term on the left side equal to zero, i.e., r  (r  H) ¼ 0; by reminding quasi-static assumption, one obtains the following expression r  ðσEÞ ¼ r  ðσrV el Þ ¼ 0

(38)

We recall that heat generated per unit volume depends on current density J and electric field E Q_ hyp ¼ J  E

(39)

with J current density. The term on the right side of Eq. (33d) can be also expressed as a total current density that considers both conductive current σE and displacement current jωεelE J ¼ ðσ + jωεel ÞE

(40)

Thermal ablation: Mathematical models and modulated-heat protocols

519

Under the quasi-static assumption, one can therefore substitute Eq. (40) in Eq. (39) in order to obtain the heat generation per unit volume Q_ hyp ¼ σ jEj2 ¼ σE  E∗ =2

(41)

where the asterisk refers to the complex conjugate of the electric field. It is important to underline that electrical conductivity varies with temperature, thus the electrical and thermal problems need to be coupled, and that this dependence could have an effect on results [106]. For MWA, since shorter wave lengths are involved (i.e., higher frequencies), the quasi-static assumption cannot be done, thus is not possible to obtain Eq. (37) from Eq. (36b). Starting from frequency-domain Maxwell equations (Eqs. 33a–33d), one can perform curl of Eq. (33c), r  ðr  EÞ ¼ r  ðμjωHÞ

(42)

By reminding that for the curl one can write r  (r  E) ¼ r (r  E)  r2E and that with no charge density the divergence of electric field approaches to zero (Eq. 33a), by finally using Eq. (33d) and with complex number properties one can obtain the following  σ r2 E  ω2 μ εel  j E ¼ 0 (43) ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the wave number k ¼ ω μðεel  jσ=ωÞ can be used in this expression. In the wave regime (σ ≪ ωεel) the resistive term in Eq. (40) might be neglected pffiffiffiffiffiffiffi and the wave number becomes k ¼ ω μεel . By solving Eq. (43) under the appropriate boundary conditions, one finally obtain electric field E. We now want to derive an expression valid for hyperthermia heat generation per unit volume. One can express Eq. (40) with an equivalent permittivity εel,eq that includes both electric conduction and permittivity J ¼ jωεel,eq E jωεel,eq ¼ σ + jωεel

(44a) (44b)

The equivalent permittivity εeq in Eq. (44b) can be expressed as follows  σ εel,eq ¼ εel  j εMWA + (45) ¼ εel  jε00el ω In such expression, the complex equivalent permittivity is divided into three parts. The real part εel refers to the lossless electromagnetic wave, while the imaginary part is the sum of dielectric and ionic conduction heating.

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This imaginary part is the heat generation cause, regrouped as εel00 and also named dielectric loss factor. This part is the only one that contributes to heat dissipation, thus the relationship between current density J and electric field E in Eq. (44a) can be expressed as follows   J ¼ jωεel + ωε00el E (46) Since only the dielectric loss factor part contributes to heat dissipation, one can finally obtain the following expression for the heat generation per unit volume Q_ hyp ¼ ωε00el jEj2 ¼ ωε00el E  E∗ =2

(47)

Finally, it is reminded that this term can be represented with an overall electric conductivity σ ¼ ωεel00 . This means that Eq. (47) can be expressed as for RFA in Eq. (41). High-Intensity Focused Ultrasound makes mechanical waves causing matter oscillations [107]. If ultrasounds are applied at high frequencies, the small wavelengths allow waves to penetrate tissues. The tissues compression and expansion cause absorption of mechanical energy that generates temperature gradients due to frictional heating. This means that acoustic and thermal energy problems have to be coupled. In order to analyze viscous losses, a constitutive equation that relates pressure with density, sound speed c0 and other parameters is employed [107]. Under the assumptions of homogenous medium, Westervelt wave equation can be written to describe nonlinear wave propagation in the tissue [107,108]. r2 p 

1 ∂2 p δ ∂3 p β ∂2 p2 + 4 3 + 4 2 ¼0 2 2 c 0 ∂t c 0 ∂t ρc 0 ∂t

(48)

with β nonlinearity coefficient available from literature [107,108] and δ acoustic diffusivity, that takes into account thermal and viscous dissipation in a fluid. Acoustic diffusivity δ can be expressed as follows δ¼

2c 30 αac ω2

(49)

with αac acoustic absorption coefficient. The hyperthermia heat generation term can be derived from an exponential model of ultrasound attenuation, in which it is assumed that ultrasounds exponentially decay with traveled distance depending on an acoustic absorption coefficient

Thermal ablation: Mathematical models and modulated-heat protocols

Q_ hyp ¼ 2αac I ac with Iac local acoustic density defined as follows [109]. 2 ∂p 1 I ac ¼ 2 ω ρc 0 ∂t

521

(50)

(51)

For laser ablation, since biological tissues can be assumed to be uniform, light propagation can be studied with references to optics theory based on transport theory. This approach is simpler than an electromagnetism-based analysis based on Maxwell equations. In the present chapter, light absorption and scattering are considered since the light travels through a participating media. Generally speaking, one has to solve the Radiative Transfer Equation (RTE) in order to derive radiation intensity [110], expressed under the quasi-steady state and negligible emitted energy assumptions in the following ð dI ðr, sÞ σs + ðαrt + σ s ÞI ðr, sÞ ¼ ϕðs, s’ÞI ðr, sÞdΩ (52) 4π 4π ds Reference is here made to the point at the coordinate r in the generic direction s, with s’ the direction after light scattering from the direction s. Because of the collimated nature of lasers [110], radiative intensity can be split in a diffusive and a collimated part, i.e., I ¼ Id + Ic. In particular, the diffusive part is the result of emission from the boundaries and the medium and scattering from the collimated beam; on the other hand, the collimated part is subject to progressive extinction due to scattering and absorption during beam travel across the medium. Under this assumption, it is possible to replace the intensity I with the diffused intensity Id, and the collimated heat source Ic is modeled as a heat source. The collimated heat source can be derived from Eq. (52) by assuming no scattering. After integration between the coordinates 0 and s and between the incoming radiative intensity I0 and the generic radiative intensity I, the following Beer-Lambert law is obtained I c ðr, sÞ ¼ I 0 eðαrt +σ s Þs f ðr Þ

(53)

with the generic function f(r) that could take into account radial variation of the collimated beam. By combining Eq. (52) with the radiative intensity equation I ¼ Id + Ic, one can finally derive the following expression, that also takes into account the solid angle dependence of the collimated component

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dI d ðr, sÞ σ + ðαrt + σ s ÞI d ðr, sÞ ¼ s 4π ds +

ð ϕðs, s’ÞI d ðr, sÞdΩ

4πð σs

4π

ϕðs, s’ÞI c ðr, sÞdΩ

(54)

4π

After obtaining the intensity function I ¼ Id + Ic, one can pass from the radiation intensity to the local fluence rate by integration over the solid angle ð ð ½I c ðr, sÞ + I d ðr, sÞdΩ ψ ðr Þ ¼ I ðr, sÞdΩ ¼ (55) 4π

4π

Besides, the heat source term can be computed by reminding that laser photo energy is converted into thermal energy Q_ hyp ðr Þ ¼ αrt ψ ðr Þ

(56)

Many methods are available to solve Eq. (52), like two-flux, Monte-Carlo, discrete ordinates, P1 methods and so on. In bioheat problems, the discrete ordinate method has been used in [111]. Another approach is based on the assumption of negligible diffusive radiative intensity with respect to collimated contribution, as in [112]. In this case, Beer-Lambert law (Eq. 53) can be directly coupled with Eqs. (55) and (56) in order to obtain the heat source. A function (i.e., a Gaussian) to consider collimated beam radial dependence has been employed too [112]. Another approach is the diffusion approximation, also known as the P1 approach (or spherical harmonics approach). This approach is common in bioheat transfer because it is accurate for near-isotropic radiative intensity media like tissues [110]. With this approach, radiative intensity can be expressed as a two-dimensional (r, s) Fourier series with spherical harmonics that include associated Legendre polynomials. The series is truncated with four terms, that are one for the isotropic component and three that account for anisotropy, respectively. Based on this decomposition, after obtaining Legrende polynomials one obtains the following expression for the radiative intensity as follows [110,113]. I ðr, sÞ ¼

ψ ðr Þ 3 + qðr Þ  s 4π 4π

(57)

One can express the radiative heat flux q by following Fick’s law for diffusion, with the current proportional to the fluence rate gradient, that is the light driving force

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qðr Þ ¼ 

rψ ðr Þ 3½αrt + ð1  gÞσ s 

(58)

where the coefficient g is the anisotropic coefficient equal to 0 for an isotropic medium. By reminding that we are separately considering diffusive and collimate contributions (in this case, ψ ¼ ψ d + ψ c), one can substitute Eqs. (57) and (58) in Eq. (54), and after some manipulation one can obtain the following differential equation r  ½Drψ d ðr Þ  αrt ψ d ðr Þ ¼ ψ c ðr Þ

(59)

with the radiative diffusion coefficient expressed as D¼

1 3½αrt + ð1  gÞσ s 

(60)

The collimated fluence rate ψ c can be obtained by combining Eq. (53) with Eq. (55) without considering Id. From the fluence rate, one can computes the heat source by following Eq. (56). This approach has been used in literature without Beer-Lambert term source in various papers on bioheat transfer [113,114]. Finally, a resume of all techniques used to calculate heat generation due to hyperthermia is presented in Table 2. Power rates for RFA and MWA can be expressed in terms of complex conjugates or equivalent electric conductivity with references to what has been expressed in Eqs. (41) and (47). For the laser ablation, the diffusion approximation to obtain fluence rate for Eq. (56) has been also presented in Eqs. (57)–(60). Table 2 A resume of hyperthermia heat generation term depending on the technique employed. Technique Heat generation Expressions Equation

Q_ hyp ¼ σ jEj2

r  (σ r Vel) ¼ 0

Eq. (38) Eq. (37)

MWA

Q_ hyp ¼ ωε00el jEj2

E ¼  r Vel  σ r2 E  ω2 μ εel  jω E¼0

HIFU

Q_ hyp ¼ 2αac I ac

RFA

LA

Q_ hyp ¼ αrt ψ

 2 ∂p I ac ¼ ω21ρc ∂t 0

Eq. (43) Eq. (50)

p ¼ f(c0, t, δ, β, ρ) Ð ψ ¼ 4π(Ic + Id)dΩ

Eq. (55)

Id ¼ f (r, s, αrt, σ s, ϕ, Ic)

Eq. (54)

Ic ¼ I0e(αrt+σ s)sf(r)

Eq. (53)

Eqs. (48) and (49)

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3. Modulated-heating protocols In this section, modulated-heating protocols for tumor ablation will be reviewed. Generally speaking, such type of protocols have been proposed through the years for problems related to pain treatment [115–117], drug delivery [118] and so on. In the first part of the section, protocols applied to RFA will be shown, while protocols referred to MWA will be introduced later. As it was previously mentioned, during RFA the roll-off issue might occur due to the very high reached impedance. This means that modulatedheating protocols have been proposed either to optimize ablation procedure to achieve better coagulation zones, or to avoid roll-off phenomenon even if higher powers are applied, or also for both aims.

3.1 Radiofrequency ablation An experimental necrotic tissue optimization of a RF pulsed protocol has been presented by Goldberg et al. [119]. Tissue current is monitored by using a computer algorithm. In particular, a 500 kHz monopole generator with a maximum of 2150 mA output (150 W) is employed. The pulsatilecurrent algorithm detects electrode and tissue impendences, and generator output is corrected based on this. Experiments were ran on ex vivo calf liver and in vivo porcine liver and muscle. Two separates pulsating-heat strategies were analyzed by the authors. The single trial lasts for 12-min, and it has been made also at a continuous current of 750 mA, that is the maximum applicable without too much high impendence. The first strategy consists in a constant-peak current but with variableapplication time, in which radiofrequency is applied at an established current with a minimum of 900 mA and a maximum of 1800 mA. Peak duration depends on tissue impedance. When it becomes higher than 20 Ω over baseline, the current becomes 100 mA for 15 s, then after it goes to the peak again. The scope of this strategy is to see if heat delivery can be increased with current peaks instead of a continuous lower current. The second strategy developed by Goldberg et al. [119] is a variable-peak current strategy with constantapplication time. As for the first strategy, when peak impedance is reached the current becomes 100mA for 15 s. Differently from the first strategy, the current remains at peak value for a constant time. However, peak current was also successively reduced in 100-mA decrements if the preceding cycle of high current could not be maintained for the specified minimum duration. A resume of these two strategies is presented in Fig. 9.

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2000

Current (mA)

1600 1200 800 400 0 0

40

80

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200

240

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360

400

440

480

520

560

600

640

680

720

440

480

520

560

600

640

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720

Duration (sec.) 2000

Current (mA)

1600 1200 800 400 0 0

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320

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400

Duration (sec.)

Fig. 9 Current-variable strategies employed in Goldberg et al. [119]: constant-peak current (upper) and constant-duration time (lower).

For the ex vivo experiments, results for the baseline-constant current of 750 mA provide a coagulation diameter of 2.9 cm  0.2. For the first strategy, the authors presented results for different peak currents ranging from 900 to 1800 mA. For all these cases, coagulation diameter was higher than baseline case, without any significant variation between 900 and 1800 mA. In particular, by averaging coagulation diameters obtained from different peak currents, it is possible to achieve a 3.5 cm  0.2 value, higher than the value of the baseline-constant current (say, 2.9 cm  0.2). For the second analyzed strategy, the authors found that coagulation diameter is strongly affected by the peak current duration. With a peak observed for a minimum duration of 10 s, coagulation diameter has been found to be 4.5 cm  0.2, while it becomes 3.7 cm  0.2 for 15 s peaks due to maximum current rapid decrease. The same peak current algorithm provides 3.7 cm  0.6 and 6.5 cm  0.9 coagulation diameters, respectively, for in vivo liver and muscle. For the sake of comparisons, such coagulation diameters were found to be 2.4 cm  0.2 and 5.1 cm  0.4 for the baseline case at continuous 1100 and 1500 mA delivered current, respectively. The authors also present temperature evolution with time under pulsating protocol for liver. In Fig. 10, temperature profiles are presented for three

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100 95 90 Tissue Temperature (°C)

85 80 75 70 65 60 55 50

Temperature @ 10mm Temperature @ 15mm Temperature @ 20mm

45 40 35 0

2

4

6

8

10

12

14

16

18

20

Duration (min.)

Fig. 10 Temperature evolution during pulsating protocol for different locations in the tissue from Goldberg et al. [119].

different locations from the heat source (10, 15, and 20 mm) during a 20-min pulsating-protocol application for the in vivo liver. It is possible to appreciate oscillations caused by pulsating-source that are more evident for points closer to the heat source. Goldberg et al. [119] also reported that temperatures of 107–110 °C were found for locations at 5 mm from the heat source. In conclusion, it is possible to establish that pulsed radiofrequency protocols can enhance coagulation necrosis, in particular referring to variable-peak current algorithms. Ahmed et al. [120] employed the same pulsating algorithm of Goldberg et al. [119] to analyze if NaCl solution injection can improve RFA coagulation. For this case, better heating has been found when the injected NaCl solution percentage increases. In particular, entire ablation of tumor (say, 5.2 cm  0.8 coagulation diameter) has been reached when a 36% NaCl percentage in a 6 mL solution injection is employed. For the sake of comparison, tumors without injection provided 3.1 cm  0.2 coagulation diameter. This means that the use of highly concentrated NaCl coupled with a pulsating-heat protocol can be advantageous for larger tumors. Lee et al. [121] used a saline infusion with an internally cooled perfusion for in vivo experiments for RFA. Three different groups of experiments were carried out on a porcine liver. In the first group, an internally cooled electrode with a 200 W generator for 12 min is used with a pulsatile protocol activated for more than 20 times. In the second group, the same power and time are used for an internally cooled perfusion electrode with hypertonic

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saline infusion, while the pulsed protocol is activated for a mean of 5.8 times. Finally, in the third group 40 W for 20 min were delivered with a perfusion electrode, and the impendence gradually increases to a target value of 85 Ω. A resume of pulsating procedures for the first two groups is shown in Fig. 11. Energy delivered in the three analyzed groups were, respectively, 52.3 kJ  10.3, 115.4 kJ  10.5, and 38.5 kJ  11.5. The authors conclude that the second group (internally cooled electrode) showed the best performance since it provides the largest ablation zone. Lopez Molina et al. [122] analytically analyzed temperature evolution in tissues during pulsed radiofrequency ablation. The geometry is modeled as a spherical active electrode embedded in a tissue, and they assume 1D unsteady temperature field. The pulsing protocol is applied by using a waveform for the applied power (Fig. 12), with a period ton of pulsed power application and toff as the rest period, making the frequency equal to 1/(ton + toff). The electrical problem is solved by using Laplace equation, similarly to what has been here shown in Eqs. (34)–(37). In the electric voltage equation, Dirichlet boundary conditions are assumed for the active electrode-tissue interface (constant value) and for the dispersive electrode (zero voltage). After solving the equation, the following Joule heat distribution is derived by assuming a constant electrical conductivity P ðt ÞR Q_ hyp ðr Þ ¼ 4πr 4

(61)

with the applied electrical power P is expressed as a function of time based on Fig. 6. Hyperbolic heat transfer and Fourier heat transfer equations were analytically solved in order to obtain temperature distribution and compared them. With only one pulse, authors found three typical waveforms of temperature behavior. In the hyperbolic heat transfer equation, authors also observe that temperature at any location is obtained by the overlap of various heat sources delay under different durations. Finally, it is relevant to notice that temperature peaks can be found also during the switch-off period, making thermal lag relevant for such applications. Fukushima et al. [123] performed RFA on 15 patients with hypervascular hepatocarcinoma smaller than 20 mm in diameter. Two methods are employed. The first one is a temperature-control method, in which 90 W are applied in order to make temperature reaching 105 °C; after this value is reached, temperature is controlled for 7 min. The second method is an impedance-control method, in which 10 W are initially applied, and this power level is increased of 10 W with intervals of 1 min for two times. This pulsed procedure was repeated until roll-off was evident.

Fig. 11 First, (A), (B) and (C), and second, (D), (E) and (F), groups from Lee et al. [121]: power, (A) and (D), current, (B) and (E), and impedance, (C) and (F), referred to each of the two groups.

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Fig. 12 Pulsing-functions used in Lopez Molina et al. [122] with j-cycles.

In both methods, a pulsatile procedure is required in order to control the power. The main diameter for each method is 32 mm, while the other is 25 mm for the first method and 31 mm for the other. This means that impedance-control method is preferred since it causes more uniform and bigger coagulation zones. Ablation time was 18.8 min for the first method, and 13.4 min for the second, while the delivered energy was more or less the same. This makes impedance-control method to be preferred also because of shorter ablation times. Trujillo et al. [124] numerically analyzed impendence-controlled pulsing protocols with a cooled electrode for RFA. Governing equations for heat transfer are based on Pennes bioheat equation by including evaporation of water with an apparent capacity method, while perfusion term considers tissue necrosis with a step function. In their model, either in vivo or ex vivo conditions are considered; for the in vivo condition, both perfused and nonperfused cases with a reference temperature of 37 °C are considered, while for the ex vivo case a non-perfused case with a reference temperature of 21 ° C is modeled. The authors consider also pulsed current or voltage as the boundary condition, making references to Goldberg et al. [119] for the applied protocol. They found that a current of 750 mA can be continuously applied without any roll-off for 12 min, and also 45 V for 12 min can be applied with references to constant-voltage boundary condition. Starting from this, the authors propose pulsating protocols based on roll-off. They use higher maximum current and voltage pulses, respectively, until rolloff is reached. When roll-off is reached, 15 s of zero-applied power are waited until reapplying the new current-voltage. With these techniques, authors found that if pulsatile-current protocol is applied coagulation zones are higher. For example, when 2000 mA current peaks are used,

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diameters can be about 2.3 times the diameter obtained from the reference case at 750 mA. For the pulsatile-voltage case, increase in terms of coagulation diameter can be about 1.3 times for the ex vivo case, if 70 V peaks are used. Finally, the authors also provide results for an optimized protocol, in which current or voltage peaks during periodic cycles are not constant through time, similarly to Goldberg et al. [119] peak-variable protocols. With this technique, coagulation zones are further increased. For example coagulation zone for 80 V has been found to be 2.68 cm for optimized case and 2.40 cm for non-optimized case. Hammerich et al. [125] simulated a 12 min impedance-controlled ablation in devices with internally cooling systems by using three different coolant temperatures, say 5, 15, and 25 °C. Pennes bioheat equation is used to predict temperatures, while Joule effect heat source is computed from Laplace equation for voltage. At the conductor, a 50 V value is assumed, and when impedance becomes higher than 20 Ω then constant voltage becomes 0 V for 20 s, following the scheme reported in Fig. 13. After simulations, authors also present ex vivo fresh bovine liver temperature profiles obtained from experiments. The authors concluded that coolant temperature does not have any relevant effect on lesion diameter. Zhang et al. [126] numerically analyzed the relationship between necrotic tissue area and target tissue size in liver tumors by employing pulsed radiofrequency ablation. Pennes equation coupled with Laplace equation is used to solve temperature field, while thermal conductivity and electrical conductivity are assumed to be variable with temperature, and blood perfusion rate is assumed to be zero for temperatures higher than 60 °C. Voltage is applied at the RF electrode, and two pulsed functions with a half-square (Fig. 14A) and a half-sine (Fig. 14B) wave form are analyzed. In these waveforms the peak voltage is 30 V, the reduced cooling voltage has to be as small as to make heat

Fig. 13 Applied voltage time-dependent protocol used in Hammerich et al. [125].

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Fig. 14 Half-square (left) and half-sine (right) voltage pulsed functions used in Zhang et al. [126].

Fig. 15 Half-sine (Sin.), half-square (Square), half-exponential (Exp.) and damped-sine (Damp.) voltage pulsed functions used in Lim et al. [127].

generation negligible (say, 4 V), and on and off times were, respectively, 10 and 15 s. Numerical simulations are also validated with experimental results. From the results, authors showed that the half-square procedure makes ablation area higher than with half-sine procedure. Lim et al. [127] analyzed the effect of input wave form and large blood vessel during radiofrequency ablation. They numerically analyzed various types of waveforms, say half-sine, half-square, half-exponential and damped-sine, reported in Fig. 15. In order to perform comparisons, the authors assume that the root-mean squared voltage V el,RMS of each wave form matches 25 V by means of the following equation integrated over a cycle length sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð ω V el,RMS ¼ (62) V 2el ðtÞdt 2π

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with ω/2π the cycle length and Vel(t) the waveform used as input for voltage. Based on this constraint, voltage vs time expression are derived (Fig. 15) in terms of numerical series [127]. In this paper, ex vivo experiments are presented to validate the numerical code too. The authors found that the damped-sine waveform provides smaller ablation zones, while the half-sine waveform provides the larger zones, in all cases with and without blood vessel modeling. Solazzo et al. [128] performed experiments with current-variable pulsatile protocols in ex vivo bovine and in vivo porcine. As for many papers previously introduced, experiments of 12min were carried out for various antennas, while simplex-optimization is used to establish the best pulsatile protocol to be used. For example (Fig. 16), starting from 2000 mA, consecutive 100 mA decrements are programmed. Thanks to the optimization, minimum on and absolute off times are established. For the case shown in Fig. 16, the on time is 10 s; since after the third pulse impedance becomes too high, 15 s of off time are required, and after these 15 s then power is reapplied but with 1900 mA delivered. From the results, the authors conclude that markedly larger coagulation zones can be achieved with optimized RFA procedures, and that longer pulsation intervals might be required compared to the one that they show.

3.2 Microwave ablation For MWA, it is important to observe that ablation can be performed either by controlling power or temperature, and that these two techniques implicitly cause a modulating-heat procedure [129]. For example, if one would like to

Fig. 16 Pulsatile-current protocol used in Solazzo et al. [128].

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control temperatures, then power should be varied and vice versa. Besides, no roll-off happens for microwave ablation, thus apart from controlling procedures the modulating-heat protocols might only refer to coagulation zones optimization. However, at this time not so many papers are available from literature about modulating-heat procedures for microwave ablation, perhaps because this is a relatively recent technique compared to RFA. Andreano et al. [130] compared RFA and MWA ablation in ex vivo liver and lung at equal delivered energy. RFA ablation is applied by using an impedance-based protocol and microwave ablation is performed at equal power levels and energy. The authors conclude that larger ablations can be achieved with MWA at equal delivered energy, especially for lung because RFA is limited by the high impedance baseline. Sommer et al. [131] analyzed the influence of delivered energy on MWA lesion shape. They performed 5-min ablation on ex vivo porcine livers by applying different power outputs, ranging from 20 to 105 W. The authors found that extension, but not shape, of the lesion depends on delivered energy, and very different volumes can be achieved if applied power is increased at equal ablation time. Bedoya et al. [132] compare continuous and pulsed protocols for microwave ablation procedures in ex vivo and in vivo liver models. The same amount of energy, i.e., 15 kJ at 2.45 GHz, is delivered for the ex vivo case, while 30 kJ are delivered for the in vivo case. For the latter, more energy is delivered since heat loss due to perfusion needs to be balanced during ablation. Step-functions protocols employed are resumed in Fig. 17 for the ex vivo case. The same pulsatile procedure is applied for the in vivo case,

Fig. 17 Power-pulsating protocols used in Bedoya et al. [132].

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Fig. 18 Temperature profiles for pulsating-heat protocols used in Bedoya et al. [132].

but with two times the total ablation time. The final temperature profiles present periodical trends due to pulsating procedures. For example, in Fig. 18 temperature profiles at 5 mm from the antenna shaft are shown. It is shown that pulsed protocols provide higher temperature peaks due to heat rates peaks. The authors compared their results in terms of ablation sizes. For the ex vivo cases, no significant differences in terms of ablation sizes have been found; on the other hand, the in vivo model presents significantly larger ablation zones for the pulsating-heat case, especially for smaller average applied powers. Besides, pulsing becomes most effective for a duty cycle less than 50%, that refers to a protocol in which the on time is equal to the off time.

4. Conclusions An overview on thermal ablation techniques has been presented in this chapter. Radiofrequency, microwave, high-intensity focused ultrasound, laser and cryo ablation procedures have been described, together with techniques used to estimate thermal damage like CEM43 and Arrhenius-based approach. Starting from the first model proposed in 1949 by Pennes’ [77], various mathematical models for heat transfer in tissues have been here presented, together with techniques to calculate the heat source generation depending on the employed ablation techniques. Finally, modulated-heat procedures to optimized ablation zones have been reviewed. In this technique, supplied heat is varied through time in order to appreciate if this procedure might improve necrotic zone and so on. It has been shown that this approach could be promising to enhance thermal ablation techniques efficacy, and this chapter can be a boost to propose further modulated-heat protocols in the next future.

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CHAPTER SEVEN

Thermal stimulation of targeted neural circuits via remotely controlled nano-transducers: A therapy for neurodegenerative disorders Erfan Kosari and Kambiz Vafai* Mechanical Engineering Department, University of California, Riverside, CA, United States *Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 1.1 Neurodegenerative disorders 1.2 Magnetothermal stimulation 2. Thermal field characterization 2.1 Blood flow analysis 2.2 Brain capillary analysis 2.3 Average thermophysical properties 3. Thermal models for nano transducers injected into brain capillaries 3.1 Single phase approach; homogenous method 3.2 Two-phase approach; Eulerian-Lagrangian method 3.3 Nanoparticle dynamics 4. Neuro-signaling model 5. Concluding remarks Acknowledgment References

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Abstract Neurodegenerative disorders, e.g., Alzheimer’s and Parkinson’s are signal irregularities in neuron cells. Deep brain stimulation (DBS) is a conventional method for Parkinson’s treatment. This method manages Parkinson’s symptoms, e.g., tremor, by surgically implanting an electrode to deliver constant stimulation in deep brain levels. The fatal impacts of DBS motivated scientists to propose a minimally invasive treatment, termed as “Magnetothermal Neuromodulation.” This technique employs an alternating magnetic field (AMF) to excite injected transducers, i.e., magnetic nanoparticles (MNP). The nanoparticles exposed to the magnetic field dissipate heat via thermal hysteresis. The local elevated temperature activates a type of heat-sensitive ion-channel, Advances in Heat Transfer, Volume 52 ISSN 0065-2717 https://doi.org/10.1016/bs.aiht.2020.09.001

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2020 Elsevier Inc. All rights reserved.

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which leads to calcium cation (Ca2+) influx. The Ca2+ influx eventually ameliorates the symptoms. In the magnetothermal modulation, the primary aim is to reach the minimum temperature required for stimulation (43°C) and maintain the tissue temperature below 50°C to avoid the thermal cytotoxic impacts. Hence, neurosurgical operations seek the Thermo engineering theories to alleviate the risk levels and effectively administer the remote therapy. The success and safety of the treatment strongly relies on thermal interactions of the brain capillary wall with blood flow carrying heat dissipative nano-transducers. Our emphasis is on microscale modeling of nanoparticles undergone magnetothermal stimulation and the consequences on the targeted neural circuits. As such, for medical advancements, the present contribution provides a mathematical model that extensively elaborates the physical attributes such as the blood flow dynamics, nanoparticle interactions with capillary flow, heat transfer, and the subsequent regulated neural signaling.

Nomenclature c cp d F F f h Ho k KB L Ms P q_ gen R R r Se Sm T T0 t u um 8 !

V V vBr

mean molecular speed ð8RT=π Þ0:5 specific heat capacity [J/ ° K] diameter [μm] force [N] Faraday constant [C/mol] magnetic field frequency [kHz] heat transfer coefficient [W/m2. ° K] electromagnetic amplitude [kA/m] thermal conductivity [W/m. ° K] Boltzmann constant [kg. m2/s2. ° K] characteristic length [μm] saturation magnetization [emu/g] pressure [Pa] uniform volumetric heat source [W/m3] specific gas constant [J/kg. ° K] capillary radius [μm] radius [μm] energy source term [W/m3] momentum source term [N/m3] temperature [°C] mean average velocity in vicinity of particle [°K] time [s] axial velocity [mm/s] average axial velocity [mm/s] volume [m3] velocity vector [mm/s] membrane potential [mV] 0:5
Brownian velocity 18KBT/πρp dp 3

A therapy for neurodegenerative disorders

Greek symbols γ_ λ μ ρ τ φ ω

shear rate [s1] mean free path [nm] dynamic viscosity [cP] density [kg/m3] relaxation time, ρp dp 2 =18μbf volume fraction [%] pulsation frequency [Hz]

Dimensionless numbers α Kn Nu Pr Re Re γ_ St

Womersley number, L(ωρ/μ)0.5 Knudsen number, λ/L Nusselt number, hd/k Prandtl number, cpμ/k Reynolds number, ρud/μ Shear Reynolds number, ρd2 γ_ =μ Stokes number, τu/d

Subscripts bf nf p

base fluid nano fluid particle

Acronyms AD AMF BBB DBS DPM LB MNP NCX NS PD RBC SEM SLP TRPV VTA

Alzheimer’s disease alternating magnetic field blood brain barrier deep brain stimulation discrete-phase model Lattice Boltzmann magnetic nano particle sodium/calcium exchanger Navier-Stokes Parkinson’s disease red blood cell scanning electron microscope specific loss power [W/g] transient receptor potential vanilloid ventral tegmental area

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1. Introduction During the past decades, mathematical modeling has assisted physicians in biological systems and the associated applications. Medical surgeons have sought the help of engineers to reduce the risk levels and to avoid the potential adverse side effects of operations. Recently, due to the cytotoxic impacts reported for the conventional treatment of neurodegenerative disorders, an achievement to find a noninvasive procedure is appealing. As such, it is essential to examine the newly established therapies and particularly “Magnetothermal Stimulation” in terms of heat transfer perspectives.

1.1 Neurodegenerative disorders The worldwide incidence of neurodegenerative disorder primarily affects patients that are mid to late life and it is anticipated to soar owing to the growth in the elderly population. The major associated financial burden is allocated to first, Alzheimer’s disease (AD) and second, Parkinson’s disease (PD). For instance, in the US, the treatment expenses incurred by PD patients is approximately USD 14.4 billion per year. As the population grows, it is expected that the expenses double by 2040 [1]. A failure to achieve a promising therapy to these kinds of disorders without noninvasive impacts might pose a huge threat to the public health. The incentive to propose an effective treatment motivated the bioengineering institutions in collaboration with pharmaceutical companies to develop new therapeutic strategies. Parkinson’s disease (PD) is characterized by a pathological condition which is mainly manifested by the formation of Lewy bodies, i.e., the abnormal aggregations of a protein type (α-synuclein), in the ventral tegmental area (VTA) situated in the midbrain [2]. In such a pathological condition, intercellular calcium cation (Ca2+) concentration plays a significant role in nervous system [3]. Hence, the main challenge for the benign therapeutic approach is to activate ion channels across the targeted region in order to regulate Ca2+ influx while avoiding any serious adverse effects. Recent endeavors offer the utilization of nano particles for ion channel activation and regarding the achievements in drug delivery field, we will soon be able to direct the nano particles to the VTA via bloodstream [4]. Thus, a profound understanding of anatomical view of brain capillaries and the processes involved in ion transport is crucial. Brain blood capillaries comprise a highly thick as well as selective semipermeable border known as blood brain barrier (BBB). BBB is composed of

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a layer of endothelial cells that shape the brain microvascular walls. Endothelial cells function as a permeable interface between vascular system and brain cells. The cell-cell junctions (i.e., tight junctions) throughout the endothelial layer maintain the microvasculature integrity and provides a severely selective transport of organic anions, ions, water, glucose and substantial nutrients to neuron cells (Fig. 1) [5]. Pericytes wrap around the capillaries and have contractile texture enabling them to control the capillary diameter for blood flow regulation [6]. Astrocytes are star-shaped glial cells that form contacts with capillaries and synapses. They are responsible for molecular transport from microvessels to neurons. Experimental results demonstrate that astrocytes propagate calcium cations in response to blood flow stimulations which due to an elevation of intracellular Ca2+ concentration in astrocyte end-feet includes impacts on neural signaling [7].

1.2 Magnetothermal stimulation The prevalent neurosurgical procedure for treatment of neurodegenerative disorders is deep brain stimulation (DBS). This technique is an approved therapy, though invasive, which utilizes permanently implanted electrodes to regulate neural signaling. The implanted electrodes in the long term might increase the risk of infection and bleeding [1]. Therefore, owing to the ineffectiveness and side effects of conventional stimulation, numerous studies have been conducted new stimuli technologies relying on acoustic [8,9], electromagnetic [10] and optic [11,12] signals. Among these stimuli, alternating magnetic field (AMF) enables us to deliver signals into the deep levels of brain tissue without attenuation. In magnetothermal stimulation, magnetic particles (MNP) are exposed to the AMF and via thermal hysteresis, heat generates within the targeted region. Ultimately, the locally elevated temperature activates a heatsensitive ion channel termed as transient receptor potential vanilloid (TRPV1) resulting in calcium influx (Fig. 2) [13]. In order to impose a controllable temperature elevation and to avoid adverse impacts of magnetic field, it is strongly recommended to restrict the magnetic power in which the product of electromagnetic frequency and amplitude (f  Ho) must be less than 5  109 [A/(m. s)] [14]. Further investigations exhibited that the induced temperature in excess of 43 ° C leads to the excitation of TRPV1 [14,15]. While, the temperatures higher than 50 ° C cause destructive impacts on the living tissue [16,17]. Thereby, the local temperature must be maintained within the safe domain.

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Fig. 1 Schematic of blood capillaries in brain tissue and Blood Brain Barrier (BBB). Endothelial cells are aligned in the axial direction that forms the tight junctions and by means of biomechanisms, BBB is capable of severely controlling solute crossings.

Fig. 2 Schematic of magnetothermal stimulation. MNP is exposed to an AMF which dissipates heat within the brain tissue. The elevated temperature activates a heat-sensitive ion channel (TRPV1) leading to Ca2+ influx.

The main goal of this chapter is to present a comprehensive mathematical model that incorporates blood flow dynamics, bioheat transfer and neural signaling. This chapter is generally divided into three sections; the first section focuses on the corresponding thermophysical properties of brain

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capillaries (Section 2), the second section introduces the governing equations from heat transfer and fluid dynamic perspectives (Section 3) and the last section provides a well-known neural signaling model (Section 4). By coupling the equations, the corresponding impacts of brain tissue exposure to the AMF on the neural signaling can be evaluated.

2. Thermal field characterization 2.1 Blood flow analysis 2.1.1 Blood flow homogeneity Human blood is not inherently a homogenous substance. Blood is a body fluid composed of numerous kinds of components. Four of the most important components are plasma, red blood cells (RBC/erythrocytes), white blood cells (leukocytes), and platelets. Plasma is responsible to transport blood cells [18]. RBCs are the most abundant cells in the blood, occupying 40 – 50% of the total blood volume [19]. White cells make up a very small portion of blood volume, normally around 1% and they protect the body from infections. Platelets are rather cell fragments without nuclei that help the blood clotting process or coagulation at the site of wounds. Due to larger volume concentrations, RBC and particularly plasma are of major importance for blood rheology [20]. Fig. 3 illustrates the 3D ultrastructural image of RBC, white blood cells and platelets.

Fig. 3 A three-dimensional image of a white blood cell or leukocytes (right), a platelet (center), and a red blood cell or erythrocytes (left). This image is in the public domain in the US and it is captured at NCI-Frederick using a Hitachi S-570 scanning electron microscope (SEM) equipped with a GW Backscatter Detector [21].

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A key dimensionless number for evaluation of microflow homogeneity is Knudsen number (Kn) which is defined as the ratio of mean free path (λ) over characteristic length (L). The mean free path is an average distance traversed by a moving atom or molecule between two collisions. The characteristic length scale corresponds to the gap length of which thermal or mass transfer occurs through the microfluid. For Knudsen numbers close or greater than one, the mean free path and the length scale are comparable which means the continuum assumption of fluid mechanics is no longer valid. Alternatively, small Knudsen numbers imply the compactness of moving particles along the length scale of the macroscopic flow. In this case, the flow behaves as a continuum phase [22]. Kn ¼

λ L

(1)

To estimate the degree of rarefaction impacts, Knudsen number can be used to classify the flow regimes. For Kn < 103, the flow is considered as a homogeneous medium. In this range, Navier-Stokes equations are sufficiently accurate to predict the flow behavior. As Knudsen number approaches 103, the first evidences of discontinuity appear near the wall known as the Knudsen layer. As Knudsen number increases to the range 103 < Kn < 0.1, the Knudsen layer grows and this type of flow is called the slip regime. In the range of 0.1 < Kn < 10, the Knudsen layer includes a large portion of the flow, and the flow regime shifts into the transition regime. At high Knudsen numbers (Kn > 10), the bulk homogeneous medium is eliminated and the continuum assumption is no longer valid [22,23]. Fig. 4 demonstrates that Navier-Stokes is still reliable for flow regimes with Knudsen number less than 0.1. Since plasma, as the primary constituent of blood, is mainly composed of water molecules, the mean free path (λ ¼ 2μ=ρc, c ¼ ð8RT =π Þ0:5; in order ˚ [24]) is of several orders of magnitude below capillary size (4  10 μm of 30 A [25]). In addition, to minimize particulate impacts, it can be assumed that the blood flow benefits uniform hematocrit and isotropic flow. Hence, Knudsen number is subsequently small and the blood flowing inside the brain capillaries is homogeneous [5,19,26]. 2.1.2 Blood rheology The blood rheological behavior highly depends on the features corresponded to the particles suspended in the blood flow. In small capillaries,

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102

Q

101

100

10-2

10-1

100

101

Kn

Fig. 4 Normalized mass flow rate in Poiseuille flow as a function of Knudsen number based on various methods. DSMC represents direct simulation Monte Carlo (●). D2Q9 refers to Lattice Boltzmann (LB) quadrature (■) and NS is Navier-Stokes solution (solid line). As indicated, Navier-Stokes approach is well qualified for predicting the flow behavior at Knudsen numbers below 0.1. Reproduced with permission from Ref. [85].

the particles tend to migrate toward the centerline leading to a dilute layer in vicinity of the vessel walls that has viscosity values close to water. This phenomenon is known as Fahraeus-Lindqvist effect [5,27,28]. The blood viscosity variation resulted from the aggregation of the blood particles, relies on shear rate and temperature [29]. Experimental observations on the independent impacts of temperature on blood viscosity demonstrate a nonlinear relation between blood viscosity and temperature [29,30]. The viscosity sharply rises with decreasing temperatures ranging from 15 ° C to 0 ° C. Depending on the shear rate, the decreasing temperature alone potentially can increase the viscosity nearly 450% compared to the values at the body temperature. For temperatures beyond 25° C, the impacts of temperature are less important, and the viscosity is no longer temperature dependent (Fig. 5). Therefore, in thermal neuromodulation that operates with the temperatures higher than body temperature, the blood viscosity is not a function of temperature [30]. From a rheological view, fluids can be categorized mainly into two groups based on the relation of viscosity with shear rate: 1. Newtonian fluids

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25

Viscosity [cP]

20

15

10

5

0

0

5

10

15

20

25

30

35

40

Temperature [∞C]

Fig. 5 Blood viscosity vs temperature for fixed shear rate (45 s1) and hematocrit (22.5%). Hemodilution was with 0.9% NaCl (), 5% albumin (●), autologous plasma (□), or 6% hydroxyethyl starch (■). Reproduced with permission from Ref. [30].

that the viscosity is independent of shear rate and 2. non-Newtonian fluids that the viscosity rather is a function of shear rate magnitude. Human blood behaves as a shear-thinning non-Newtonian fluid, i.e., the viscosity decreases as the shear rate increases. The rheologic properties measured over a wide range of shear rates exhibit an exponential increase in viscosity as shear rates decreasing. Also, it shows almost shear rate independence at shear rates above 45 s1 [30,31]. The particle aggregation is found to appear for small blood vessels approximately smaller than 800 μm [27]. For smaller arterial sizes in scale of capillaries, due to low shear rates which subsequently causes Fahraeus-Lindqvist impact, the blood viscosities are significantly high (Fig. 6). Consequently, an analytical solution in micro modeling must include non-Newtonian models proposed for blood viscosity. To consider rheologic properties of blood in analytical models, several non-Newtonian models have been introduced over decades. Among the proposed models, only few models accurately enough represent the blood viscosity behavior at various shear rates [31–33]. The main advantage of simple models such as the Power-Law is the availability of exact solutions in some geometries for numerical modeling [32]. Table 1 indicates a list of prominent non-Newtonian models proposed for blood flow.

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102

Viscosity [cP]

Aggregation

101

10-2

10-1

100

101

102

-1

Shear Rate [s ]

Fig. 6 Contribution of aggregation on blood viscosity for RBC at.%45 of hematocrit and at body temperature: For whole blood (●), the RBCs aggregate at low shear rates and conversely, for blood without plasma molecules (■) RBCs do not aggregate at low shear rates and viscosity is reduced compared to whole blood. The main idea and experimental data of this figure are reproduced with permission from Ref. [86].

Table 1 Proposed non-Newtonian models for blood viscosity including a list of model coefficients. Model Non-Newtonian equation Coefficient values for blood

Power-Law [31]

μðγ_ Þ ¼ κðγ_ Þn1

κ ¼ 0.42 n ¼ 0.61

Powel-Eyring [32]

μðγ_ Þ  μ∞ sinh 1 ðλ_γ Þ ¼ λ_γ μ0  μ∞ μ0  μ∞ μðγ_ Þ ¼ μ∞ + 1 + ðλ_γ Þm

μ∞ ¼ 3.45 cP, μ0 ¼ 56 cP λ ¼ 5.383 s

Cross [31,32]

μ∞ ¼ 3.45 cP, μ0 ¼ 56 cP λ ¼ 1.007 s, m ¼ 1.028

μ0  μ∞ μ∞ ¼ 3.45 cP, μ0 ¼ 56 cP ð1 + ðλ_γ Þm Þa λ ¼ 3.736 s, m ¼ 2.406, a ¼ 0.254

Modified Cross [31,32]

μðγ_ Þ ¼ μ∞ +

Carreau [32]

n1 μ ¼ 3.45 cP, μ0 ¼ 56 cP μðγ_ Þ  μ∞  ¼ 1 + ðλ_γ Þ2 2 ∞ λ ¼ 3.313 s, n ¼ 0.3568 μ0  μ∞

n1 μ∞ ¼ 3.45 cP, μ0 ¼ 56 cP Carreau-Yasuda [32] μðγ_ Þ  μ∞ ¼ ½1 + ðλ_γ Þa  a λ ¼ 1.902 s, n ¼ 0.22, a ¼ 1.25 μ0  μ∞

Excluding Power-Law model, all models are seen to agree well with the empirical observations at low and high shear rates.

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2.1.3 Pulsatile blood flow The pulsatility of blood in microvascular scale has been debated for many years [19,26,34,35]. Blood pulsatile flow is generated by the pumping action of the heart. In microcirculations, the flow is observed to be damped out. In blood vessels, a dimensionless quantity, i.e., Womersley number, is utilized to characterize whether the pulsatility assumptions are required for the vascular geometry. Womersley number correlates with the ratio of inertial forces of a pulsatile flow and viscous forces [36]. rffiffiffiffiffiffi ωρ α¼L μ

(2)

L stands for the characteristic length of the vascular geometry. ω represents the pulsation frequency. ρ and μ are the corresponding density and dynamic viscosity of blood flow, respectively. Throughout the circulatory system, μ, ρ and ω are almost constant. Whereas, for microvascular scale with distancing from the heart, the characteristic length strongly decreases [19,26]. Hence, in micro vessel modeling, the impacts associated with pulsatile flow are negligible (α < 0.1) [26].

2.2 Brain capillary analysis The quasi-fractal hierarchy of blood vessels contributes to the transportation and exchange of respiratory gases and nutrients throughout human body. The arterial system consists of branching patterns that the vessel diameters decrease at bifurcations resulting in minimizing the time required for providing resources [25,37,38]. The arterial system leads into the capillary network which is a three-dimensional interconnected grid and it ensures that the consuming tissues are nourished by vital substances. The mesh structure gives the capillaries the ability of which they benefit large surface area facilitating the exchange process [39]. In humans, the distribution of capillary diameter and length has been reported to be in range of 4  10 μm and 25  75 μm, respectively [37,40]. Blood vessels are also characterized by elastic walls and high curvature. Due to various roles assigned to the vessels with different scales and depending on the function of the organ, the vessel structure and the wall constituents are quite distinct [41]. Artery walls are composed of a layered

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structure consisting of intima, media and adventitia. In the literature, the comparison of elastic versus rigid walls indicates that for aorta, femoral, and carotid arteries there are 25%, 17%, 4  15% deviations, respectively [42]. Capillaries, in the micro scale, proportionally have a thinner and less elastic walls mainly constructed of endothelial cells, i.e., bEnd3 cells. In case of brain tissues, the capillaries are equipped with BBB which provides even thicker layers and subsequently, more rigidity for the wall [41,42]. The transvascular exchange between blood capillaries and the brain tissue occurs through channels or aqueous pores lying across the endothelial cells. The aqueous pores appear in different sizes that are sensitive to the solute size. Brain capillaries are reported to have tight junctions involving aqueous pores with less than 1 nm diameter [43]. Consistently, the experiments applied on a layer of bEnd3 cells demonstrate that the solutes with Stokes radii less than 1 nm have comparable permeability to larger particles across the endothelial barrier (Fig. 7) [44]. This implies the inability of MNPs in order of 22 nm or larger to traverse through the pores.

Permeability × 10-6 [cm/s]

102

101

100

10-1

10-2

0

0.5

1

1.5

2

2.5

3

3.5

4

Stokes radius [nm]

Fig. 7 Permeability as a function of Stokes radius of solutes. The experiments were carried out for the bEnd3 monoculture (●), coculture (), coculture with collagen mixture (■), and in vivo data from rat pial micro vessels (□). Stokes radius represents the radius of hard spheres with diffusion at the same rate as that of solute. Reproduced with permission from Ref. [44].

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The solute diffusivity of the endothelial layer has been investigated by means of a bubble tracker system (Fig. 8) [46,47]. In this BBB experimental model, the Transwell filter contains the endothelial monolayer which is sealed inside the chamber. The transport chamber is equipped with a laser excitation source and a detector. The laser source excites the solute fluorescent tags and the emissions are counted by the detector. The chamber is also connected to the water reservoir. A hydrostatic pressure (10 cm H2O; the hydrostatic pressure required to drive the fluid from the capillary into the tissue) is exerted across the filter. Eventually, the water flux through the endothelial layer is measured by using the bubble tracker [44,45]. Thus, according to the experimental results, the superficial velocity from the capillary to the brain tissue across the monolayer is measured to be 1.79  106 cm/s (SD 1.24  106) [45]. Blood typically flows in brain capillaries with the flow rate varying from 6 to 12 nL/min for capillaries in order of 10 μm in diameter. Subsequently, the corresponding average axial velocity is 25 mm/s [5,26]. Plasma pressure

Fig. 8 Schematic of the bubble tracker system used to measure the permeability of endothelial barrier to water and solutes. The endothelial monolayer is sealed in the chamber. By tracking the bubble displacement, water flux can be measured. Hydrostatic pressure is set to be 10 cm H2O. Adopted with permission by experimental setups provided in Refs. [44,45].

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along the trajectory from arterioles to capillaries varies from 40 to 20 mm Hg [48,49]. To take into account the parabolic and fully developed flow aspects, the inlet flow to the capillary is prescribed as a function of radius (r) and average velocity (um) [33,50,51]. u ¼ 2um ð1  ðr=RÞÞ2

(3)

To achieve a desirable therapeutic impact on the targeted tissue and to maintain the aqueous MNP solution at low concentrations, specific loss power (SLP) needs to be maximized under the benign magnetic field range (f  Ho < 5  109A/(m. s)). When the alternating magnetic field (AMF) is applied, the atomic dipoles of nanoparticles continuously align themselves with the field resulting in magnetic hysteresis. Hysteresis losses leading to SLP that mainly rely on saturation magnetization (Ms) and the effective anisotropy energy barrier [52,53]. Magnetic nanoparticles (MNP) with diameters within 18 nm to 22 nm significantly exhibit a larger hysteresis loop at the chosen magnetic field (Fig. 9) [52]. Among chemical compositions characterized for MNPs, the iron-based particles demonstrate the highest rate of power loss [53,54]. The experimental

15 nm 18 nm 22 nm 24 nm

1

M/Ms

0.5

0

-0.5

-1 -15

-10

-5

0

5

10

15

H [kA/m]

Fig. 9 Magnetization curves based on numerical simulations for Fe3O4 particles with diameters ranging from 18 to 24 nm. The simulations were carried out at f ¼ 500 kHz and Ho ¼ 15 kAm1. The SLP can be calculated by integrating the hysteresis loop area. Reproduced with permission from the investigation of Ref. [52].

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Table 2 Experimental SLP measurements based on MNP chemical composition and size obtained at f ¼ 500 kHz and Ho ¼ 15 kAm1. Sample MNP size [nm] SLP [W/g]

Fe3O4

MnFe2O4 Acac

MnFe2O4 Oleate

CoFe2O4 Acac

CoFe2O4 Oleate

10

19

15

176

18

654

22

715

24

310

11

32

16

62

26

301

11

13

16

40

19

30

28

42

9

10

12

45

14

20

13

19

20

4

The experimental results are taken from Ref. [52].

results at the given AMF amplitude and frequency indicate that Fe3O4 particles with 22 nm diameter generate the highest SLP values (Table 2) [52].

2.3 Average thermophysical properties In a high-volume fraction of particles submerged in the blood, the interaction between MNPs is considerable. Although there are biocompatible protocols to lower the risk of coagulation in brain capillaries, a criterion is required to establish an evaluation of interparticle forces. The criterion ensures that the nanofluid is dilute enough in which the interactive forces are negligible [55]. Volume fraction, as the criterion, classifies the nanofluids into two main groups; dilute and dense nanofluids [56,57]. As the ratio of

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nanoparticles volume over the total volume of nanofluid, the nanoparticle volume fraction can be written as [55,58]:


 =ρp 
  100 φ¼
mp =ρp + mbf =ρbf mp

(4)

A dilute mixture corresponds to a mixture with a solid volume fraction less than 0.1% [56,57]. In such a mixture, collisions or even particles interactions hardly occur. As more nanoparticles are mixed in nanofluid, the volume fraction exceeds 0.1%. Subsequently, the likelihood to have interparticle forces increases [55,56]. Other than particle concentration, the size of MNPs plays a significant role in the validity of governing equations based on single-phase assumption across the nanofluid. Large particles might cause a particle/fluid relative velocity that requires individual equations for the dispersed phase. Therefore, regarding the flow characteristics, the particle size must be sufficiently small to follow the flow streamlines that no-slip condition between fluid and particles is guaranteed. Thereby, the evaluation of a group of dimensionless numbers is essential. To ensure that the flow regime is within Stokes flow regime, it is needed to calculate Reynolds number. A very small value of Reynolds number compared to unity implies that the viscous forces are strongly dominant (Eq. 5). The average axial velocity is denoted by um and ρbf represents the base fluid density. μbf is base fluid dynamic viscosity. Re ¼

ρbf um d ≪1 μbf

(5)

Stokes number corresponds to the behavior of suspended particles. Stokes numbers less than unity specifies the dominancy of particle’s inertia in the nanofluid (Eq. 6). Consequently, it unveils that the MNPs only move along streamlines resulting in scarce sedimentation or coagulation. Hence, the dilute mixture with very fine particles can safely be assumed as a single-phase fluid with average thermophysical properties. τ is the relaxation time of the particle, ρp denotes the particle density, and dp represents the particle diameter. St ¼

τum ≪1 d

(6)

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Erfan Kosari and Kambiz Vafai

ρp d p 2 τ¼ 18μbf

(7)

In the case of single-phase assumption for nanofluids, the properties of submerged nanoparticles are taken into account through average thermophysical coefficients, namely density, specific heat, thermal conductivity, and dynamic viscosity. Empirical results [55,58,59] demonstrate that density and specific heat capacity are a linear function of volume fraction (φ) as defined by: ρnf ¼ ð1  φÞρbf + φρp

(8)

c p,nf ¼ ð1  φÞc p,bf + φc p,p

(9)

The other two properties, dynamic viscosity and thermal conductivity, other than volume fraction, are dependent on nanoparticle characteristics such as particle size, particle shape, and chemical compositions. Numerous studies strived to provide average coefficients that can effectively predict the nanofluid thermal conductivity. Maxwell model [60] as a prominent model for thermal conductivity has been widely used in numerical simulations [61–63]. Afterward, as a follow-up, researchers proposed accurate as well as generic models including Hamilton and Crosser [64], Wasp [59], and Yu and Choi [65] models, which are quite similar to Maxwell’s. Most of these traditional models are restricted to only consider the impacts of nanoparticle concentration leading to a failure to precisely predict the thermal conductivity [59,66]. Recently, many correlations have been proposed based on empirical data. Among all correlations, Hassani et al. provided a highly accurate correlation with a 3% deviation from experiments [67]. Additionally, this model covers a large variety of nanofluid characteristics, and it is able to account for the impacts of Brownian motion and nanofluid temperature [67]. The most correlations used for the average dynamic viscosity of nanofluids in convective heat transfer problems are the models developed several years ago by Brinkman [68] and Einstein [69]. According to empirical observations, these models normally predict an underestimation of nanofluid viscosity [59]. In an investigation carried out by Esfe et al. [70], an empiricalbased correlation has been proposed that exhibits a very good agreement with experiments. A summary of important models for conductivity and viscosity which are applicable for blood with submerged MNPs flowing inside brain capillaries, are listed in Tables 3 and 4, respectively.

Table 3 Summary of important thermal conductivity correlations proposed over a century. Model Thermal conductivity correlation

  Maxwell [60] ðkp + 2kbf Þ2φðkbf kp Þ knf=k ¼ bf ðkp + 2kbf Þ + φðkbf kp Þ   Hamilton kp + ðn1Þkbf Þðn1Þφðkbf kp Þ ð knf=k ¼ bf and Crosser ðkp + ðn1Þkbf Þ + φðkbf kp Þ [64]   Wasp [59] ðkp + 2kbf Þ2φðkbf kp Þ knf=k ¼ bf ðkp + 2kbf Þ + φðkbf kp Þ   Yu and Choi kp + 2kbf Þ2φðkbf kp Þð1 + ηÞ3 ð knf=k ¼ bf [65] ðkp + 2kbf Þ + φðkbf kp Þð1 + ηÞ3

Hassani et al. [67]

knf=k

bf

¼ 1:04 + φ1:11 φ0:99288 2 6 Pr 1:7 4


0:33

1 262 
0:33 kp Pr 1:7 kbf

vBr ¼

rffiffiffiffiffiffiffiffiffiffiffiffi

18K B T , πρp d p 3

kp kbf

dref ¼ 2.9A˚

Presumptions

Maxwell model is only a function of thermal conductivity and volume fraction This model is valid for spherical and cylindrical nanoparticles and n is a ratio of particle sphericity (ψ); n ¼ 3/ψ Wasp model is inspired by Hamilton and Crosser model with n ¼ 3 Yu and Choi model takes into account the interfacial layer and η refers to the ratio of nanolayer thickness over the nanoparticle radius

This model is valid for a specific range of 3 volume fraction (0.0313 %  φ  1%). Tbf 0:23

0:82

0:1 7 ! represents the base fluid boiling d ref vbf T bf Cp 7 temperature. v is Brownian velocity. d Br ref + 135 5 1 2 kp d p vBr kp T vBr refers to the molecular diameter of hydrogen and KB denotes Boltzmann constant

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Erfan Kosari and Kambiz Vafai

Table 4 Summary of important dynamic viscosity correlations. Model Dynamic viscosity correlation Presumptions

Brinkman [68]

μnf=μ

bf

¼ ð1φ1 Þ2:5

Brinkman model is only valid for spherical particles

Einstein [69]

μnf=μ

bf

¼ 1 + 2:5φ

Einstein model is valid for highly dilute nanofluids (φ < 0.05)

Esfe et al. [70]

μnf=μ

bf

 ¼ 1 + 0:1  φ0:69574  dp 0:44708 This model covers nanofluids containing Fe/water and volume fractions in the range of 0.0313 %  φ  1%

3. Thermal models for nano transducers injected into brain capillaries The incentive to raise the tissue temperature targeting TRPV1 channels in the ventral tegmental area (VTA), motivated researchers to employ the heat dissipation technique relying on activation of superparamagnetic ferrite nanoparticles in remotely applied alternating magnetic field (AMF). The main challenge in thermal modulation is to provide a controllable as well as uniform temperature distribution that can reach the minimum temperature requirement for TRPV1 activation (Ttissue > 43 ° C) [14,15]. Besides, there is a concern of thermal cytotoxicity (Ttissue < 50 ° C) [16,17]. To ensure that the tissue comes up with a temperature within the benign domain and to predict the heat distribution across the different layers of brain tissue, the computational approaches are found appealing. The progresses made in nanotechnology, spanning over a decade, resulted in a new generation of working fluid termed as nanofluid, i.e., a mixture comprising a base fluid and nanoparticles [71]. In microscale modeling of brain capillaries, by having a glimpse at nanofluid modeling, the advanced heat transfer methods of which the working fluid contains submerged nanoparticles, generally offer two approaches; single phase approaches and two-phase approaches. Single phase approaches seek to the velocity and temperature within the capillary undergone a magnetic stimulation. Blood flow with injected nanoparticles is inherently a two-phase flow. However, under some circumstances, it can be reasonably assumed as a single phase. Such assumption is

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valid until (1) the nanoparticles are fine enough to follow the streamlines, (2) the interphase forces are negligible, (3) there is no particle-particle interaction which indicates the diluteness, and (4) both continuum and dispersed phases are in thermal equilibrium. Regarding the flow characteristics analyzed in Section 2.3, homogenous method well suites the capillary modeling for low concentration of aqueous solution with very small MNPs. In this method, the energy and momentum equations are solved for a single-phase fluid with the average thermophysical properties. Eulerian-Lagrangian method, one of the subsets of two-phase approaches, attempts to model the mixture as two individual phases. The particles move through the blood flow with velocities and temperatures quite different from the continuum phase. The interactions between both phases are carefully monitored through the governing equations. This method involves two reference frames; Eulerian and Lagrangian. In Eulerian reference frame, the momentum and energy equations are examined for the base fluid. The energy and momentum exchanges of particles with fluid are traced through the Lagrangian frame. All governing equations from Eulerian and Lagrangian aspects are coupled by adding two source terms in corresponding equations which accounts for thermal and dynamic interactions. Eventually, the velocity field and temperature distribution are achieved. Subsequently, the trajectory of particles can be determined. Eulerian-Lagrangian method confronts a huge amount of computations leading to longer time taken compared to solving a given problem in single phase perspective. Despite the complexity, this model provides results with high compliance with experimental observations [55].

3.1 Single phase approach; homogenous method Due to the simplicity of the homogenous method, it has been used in various nanofluid flow simulations. Despite the ease of implementation, in a case of ultra-fine and uniformly dispersed particles in the continuum medium, the homogenous method is reasonably accurate. For small scale particles submerged in laminar flow (Re ≪ 1), Stokes numbers much less than unity (St ≪ 1) unveil that the particles follow the streamlines and there is no slip between fluid and particles. Therefore, together with dilute assumption, it implies that the interactive forces are negligible, and the mixture can be assumed as a single-phase flow with average thermophysical properties. The governing equations for the homogenous model are defined by:

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Erfan Kosari and Kambiz Vafai !

ρnf r:V ¼ 0

(10)

!
! ! ! ∂V + ρnf V :r V ¼ rp  μnf r2 V ρnf ∂t

(11)

Conservation of mass Conservation of momentum

Conservation of energy ρnf c p,nf


!   ∂T + ρnf c p,nf r: V T ¼ r: knf rT + q_ gen ∂t

(12)

Subscript nf refers to the corresponding average properties of the mixture. In homogenous model, q_ gen accounts for the constant uniform volumetric heat source within the capillary caused by the nanoparticle exposure to the AMF. As discussed earlier, the major challenge in this model is the specification of correlations for thermophysical properties which are consistent with the experimental observations. Depending on the physical scheme, the properties can be considered as a function of temperature or otherwise. For instance, the empirical observations demonstrate that the average viscosity is a function of temperature and the concentration of particles (see Section 2.3) [55].

3.2 Two-phase approach; Eulerian-Lagrangian method The Eulerian-Lagrangian method, i.e., discrete-phase model (DPM) is a precise and reliable model to simulate nanofluidic dynamics. This model is able to predict the particle distribution across the domain. It treats the base fluid as a continuum medium while the dispersed phase is modeled by tracking the individual spherical particles in the Lagrangian frame. The thermal and dynamic impacts of particles on the continuum phase is defined by two source terms in momentum and energy equations. The fundamental prerequisite for utilizing the Eulerian-Lagrangian approach is the nanofluid diluteness, i.e., the dispersed phase must occupy a small volume fraction such that the interaction between the particles is negligible. The major limitation pinpointed for this method is the high computational costs [72,73]. The following governing equations represents the Eulerian-Lagrangian formulation for both continuum and dispersed phases: Conservation of mass

! ρbf r:V bf ¼ 0 (13) Conservation of momentum

!
 ∂V bf

!

!

! ρbf + ρbf V bf :r V bf ¼ rp  μbf r2 V bf + Sm ∂t

(14)

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Conservation of energy ρbf c p,bf


  ∂T bf

! + ρbf c p,bf r: V bf T bf ¼ r: kbf rT bf + Se ∂t

(15)

Sm is the source term in the momentum equation which represents the integrated momentum exchange between the dispersed phase and the continuum medium. In the energy equation, Se accounts for the integrated heat transfer between the base fluid and particles. The source terms can be calculated by averaging the thermal and dynamic impacts of particles moving through a Eulerian element with a volume of δ8 (Fig. 10). The source/sink terms are defined as: Sm ¼

! n 1 X d Vp mp dt δ8 p¼1

(16)

Fig. 10 Schematic of Eulerian and Lagrangian frames in two-phase approach. Eulerian frame establishes the governing equations for continuum phase. Whereas, Lagrangian frame only monitors the nanoparticles. Both frames are coupled via source terms defined in the equations which the particle/fluid interactions for an individual cell are considered.

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Erfan Kosari and Kambiz Vafai

Se ¼

n dT p 1 X mp c p,p dt δ8 p¼1

(17)

In Eqs. 16 and 17, the subscript p refers to the corresponding properties of a single particle. δ8 represents the grid cell volume and n in the upper bound of summation notations is the number of nanoparticles that occupy the cell volume. The trajectory of particles is computed by the Lagrangian reference frame that represents a force balance between the particle inertia and interphase forces acting on the particle. The hydrodynamic forces exerted on the single-particle includes drag (FD), gravity (Fg), lift (FL), Brownian motion (FBr), thermophoretic (FT), pressure gradient (FP) and virtual mass forces (Fv) which are elaborately discussed in Section 3.3. ! ! ! ! ! d Vp ! ! ! ¼FD + Fg + FL + FBr + FT + FP + Fv mp dt

(18)

Regarding the small scale of nanoparticles, the interior temperature remains essentially uniform and the particles behave as a lumped system. Therefore, the particle-fluid convection and heat generated inside the particle (q_ gen) are in balance with the transient term as expressed by: mp c p,p

 dT p ¼ hAp T p  T + 8p q_ gen dt

(19)

q_ gen represents the constant heat source that is excited by AMF. 8p denotes the particle volume and the heat transfer coefficient (h) is identified by [59,72]: Nu ¼

hd p ¼ 2 + 0:6 Re p 0:5 Pr bf 1=3 kbf

(20)

where Rep and Prbf corresponds with particle Reynolds number and the base fluid Prandtl number, respectively. In numerical solution, after finalizing the initial/boundary conditions and applying the grid dependency assessments, firstly, the Eulerian equations are solved. Secondly, regarding the primary solution for continuum medium, the Lagrangian equations are employed. The interactive impacts of Eulerian-Lagrangian phases appear in the momentum and energy source terms. In a loop of calculations, the Eulerian and Lagrangian parts are repeatedly computed until the convergence criterion is satisfied. Fig. 11 illustrates the numerical algorithm for nanofluid simulation.

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Fig. 11 Numerical algorithm to solve a heat convective problem by means of EulerianLagrangian approach.

3.3 Nanoparticle dynamics A profound understanding of the force balance that exists upon continuum and dispersed phases, is essential to express the momentum exchanges. There are many forces that might apply on an individual particle but due to nanoscale size of particles, only a few forces are significant. In the following, the

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definition of acting forces is described and based on the model, it is meticulously expressed which forces are important of taking into account. 3.3.1 Drag force

! A nanoparticle suspended in the base fluid with velocity of Vp and diameter of dp, experiences an opposing force from the base fluid called drag force which is originally caused by the relative velocity between the particle ! ! and the fluid (Vbf  Vp ). As analyzed earlier, the capillary flow is within Stokes regime (Re ≪ 1) and the particle size is comparable with base fluid ˚ ). Hence, drag force can be derived by using modmean free path (λ ¼ 30 A ified Stokes principles as given by [55,74]:
! ! 3πμbf dp Vbf  Vp (21) FD ¼ Cc Cc represents Cunningham correction factor as defined by [72,74]: h i 1:1dp 2λ Cc ¼ 1 + 1:257 + 0:4 e 2λ dp

(22)

3.3.2 Gravity force The imposed impacts of gravity on the blood flow can be expressed by:
 g ρp  ρbf πd p 3 (23) Fg ¼ 6 In nanofluid problems, due to ultra-small size of the particles, the impacts of gravity are negligible [55]. 3.3.3 Lift force Nanoparticles in blood capillary confront a shear field which imposes a lift force upon the particle. Basically, lift force is caused by the inertia impacts in the viscous flow around the nanoparticle. For the first time, Saffman [75] introduced the lift equation as follows: !0:5 2
dp ! ! γ_ ρbf Vbf  Vp (24) FL ¼ 6:46μbf 2 μbf In a scale analysis applied on lift and drag forces, the ratio of Saffman lift over drag force is in direct relation to Re γ_ 0:5, where shear Reynolds number is [55]:

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Re γ_ ¼

ρbf dp 2 γ_ μbf

(25)

Due to nanoscale particle sizes, Re γ_ is much less than unity. Thus, the imposed lift force is negligible compared to drag force. 3.3.4 Brownian force The random motion of nanoparticles inside the continuum phase is known as Brownian motion, which is caused by particle/fluid molecules collisions. The particle motions generally boost the energy and momentum exchanges between the phases. To consider the impacts of random motions, Brownian force is given by [76]: rffiffiffiffiffiffiffiffi πS0 (26) F Br ¼ ξ Δt ξ is the independent Gaussian random number with zero mean and unit variance. Δt represents the time step used in the numerical modeling. S0 refers to spectral intensity function which is expressed by: S0 ¼

216υbf K B T
2 ρ π 2 ρbf d p 5 ρ p C c

(27)

bf

KB represents Boltzmann constant (KB ¼ 1.38  1023 m2kg/Ks2). Cc and υbf are Cunningham coefficient and base fluid kinematic viscosity, respectively. The magnitude of Brownian force is considerable for small nanoparticles and particularly, there are significant impacts on nanofluids with elevated temperature [55]. 3.3.5 Thermophoretic force In solid-liquid phase problems, an imposed temperature gradient across the field leads to migration of particles from high temperature zone to low temperature zone, which this phenomenon is known as thermo-migration and the corresponding force applied on the particle is termed as thermophoretic force. The equation introduced for thermophoretic force is basically defined for solid-gas mixtures [24]. However, in the literature, it is demonstrated that the relation is also in reasonable agreement with experiments for solid-liquid problems [24,72].

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Erfan Kosari and Kambiz Vafai

FT ¼

6πd p μbf 2 C s




kbf kp

 + C t Kn rT T0


 k ρbf ð1 + 3C m KnÞ 1 + 2 kbfp + 2C t Kn

(28)

T0 is the mean fluid temperature in the vicinity of the particle and Kn is Knudsen number. In the equation, momentum exchange coefficient (Cm), temperature jump coefficient (Ct) and thermal slip coefficient (Cs) are equal to 1.14, 2.18 and 0.75, respectively [55]. 3.3.6 Pressure gradient force The pressure gradient along the capillary induces the nanoparticles to move forward. The resultant force can be obtained by integration of pressure gradient over the nanoparticle volume [77]. ! ρbf πdp 3 D Vbf FP ¼ 6 Dt

(29)

3.3.7 Virtual mass force Virtual mass or added mass force is the inertia added to the system mainly due to accelerating nanoparticle to move the surrounding fluid. In other words, virtual mass is the additional force exerted on the particle to accelerate or decelerate in existence of opposing fluid. It is defined by [77]: ! !! ρbf πdp 3 d Vp D Vbf Fv ¼ Cvm (30)  6 dt Dt For spherical nanoparticles the virtual mass coefficient (Cvm) is 0.5. Eq. 30 is defined for inviscid flows. However, it has been demonstrated that it is sufficiently accurate for viscous forces in wide range of Reynolds numbers [77].

4. Neuro-signaling model Brain capillaries are surrounded by star-shaped glial cells known as astrocytes which they are responsible for many functions such as multiple ion exchange and provision of vital substances to neural system. Astrocytes are in possession of sodium/calcium exchangers (NCX) [78]. Temperature elevation across the capillary wall entails impacts on Ca2+ dynamics leading to ion exchange between astrocytes and neurons. Eventually, it results in propagation of action potentials along the axon.

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Fig. 12 Schematic of astrocytes which is a connection of brain capillaries to neuron system. A temperature elevation across the capillary wall triggers Ca2+ intercellular waves through astrocytes, eventually, leading to action potential propagation.

A Profound understanding of the impacts of thermal stimulation on neural signaling is crucial to achieve non-invasive therapies to neurodegenerative disorders (Fig. 12). The signaling model must render mathematical formulas taking into account the ionic fluxes through biomechanisms. Preceding articles suggest a circuit model well known as Hodgkin-Hexley modela [79–81] which represents an electrical behavior through a neuron. This model contains ionic currents that is primarily divided into three components; sodium ionic current, potassium ionic current and leakage current. Each ionic current is the product of electrical potential difference and permeability coefficient which can be assumed as a conductance. For instance, potassium current (iK) equals the multiplication of the potential difference (E  EK) and potassium conductance (gK). The electrical network is illustrated in Fig. 13 [82,83]. Total membrane current splits up into ionic currents and capacity current. In Eq. (31), the total membrane current is denoted by I, Iionic is the ionic a

A.L. Hodgkin and A.F. Huxley, together, carried out experiments on squid axon for which they were rewarded the Nobel prize in physiology and medicine 1963.

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Fig. 13 Schematic of electrical circuit proposed by Hodgkin and Huxley [82]. ENa, EK and El are the ionic potentials or Nernst potentials for individual ions. Based on this electrical network, the resulting potential (V) determines how much charge can be stored in the membrane capacitor (Cm).

current, V represents the membrane displacement potential from the rest value, Cm refers to the membrane capacity per unit area which is assumed constant and t is time. I ¼ Cm

dV + I ionic dt

(31)

The most important group of ion channels responsible for action potential propagation along axon is sodium and potassium channels [84]. Na+/K+ ions are carried through voltage-gated ion channels in parallel to the membrane capacity which leads to ionic current. To model sodium/potassium exchanges in neural system, in the electrical circuit, the ionic current must take into account three individual ion currents across the membrane; sodium ions (iNa), potassium ions (iK) and leakage current (il) that accounts for the currents made up by other ions. I ionic ¼ iNa + iK + il

(32)

The individual ionic currents are in direct relation with the equilibrium ion potentials and membrane permeability. In the electrical circuit, the membrane permeability is satisfactorily defined as ionic conductances (gNa, gK and gl ) [82]. The ionic currents can be expressed by: iNa ¼ gNa ðE  ENa Þ iK ¼ gK ðE  EK Þ il ¼ gl ðE  El Þ

(33) (34) (35)

where ENa and EK represents the equilibrium potentials for Na+ and K+. El refers to the equilibrium potential associated with other ions. The displacement potential is the potential difference from the membrane resting

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potential (Er) and it can be directly measured, e.g., V ¼ E  Er and V Na ¼ ENa  E r . As a matter of convenience, the current equations are rearranged based on the displacement potentials. iNa ¼ gNa ðV  V Na Þ iK ¼ gK ðV  V K Þ il ¼ gl ðV  V l Þ

(36) (37) (38)

The displacement potentials are simply proportional to the corresponding ion concentration across the membrane. Thus, the displacement potentials of V Na and V K according to Nernst equation, can be calculated using the internal and external ion concentrations, absolute temperature, thermodynamic gas constant (R ¼ 8:314 J=mol:°K ) and the Faraday constant (F ¼ 96, 485:3 C=mol) [84]. The associated displacement potential of other ions (V l) is set to be 10.61 mV in compliance with experimentally reported potentials [81,83].  RT ln ½Nae =½Nai F  RT 3 V K ¼ 10  ln ½K e =½K i F

V Na ¼ 103 

(39) (40)

Eq. (41) expresses the potassium conductance relation. gK is the maximal potassium conductance, n represents dimensionless potassium inactivation number which is in range of 0 to 1. αn and βn are rate coefficients and they are only voltage dependent. αn specifies the rate of inward potassium transfer while, βn determines the rate of potassium exchange in opposite direction [80,81]. g K ¼ g K n4

(41)

dn ¼ αn ð1  nÞ  βn n dt

(42)

To model transient changes in sodium conductance, two individual firstorder differential equations are defined (Eqs. (44) and (45)). By solving these equations, the sodium activation number (m) and sodium inactivation number (h) is attainable which both are dimensionless numbers varying between 0 and 1. Eq. (43) describes the sodium conductance as a function of m and h. gNa refers to the maximal sodium conductance through the sodium current component and it is assumed constant [82]. gNa ¼ m3 hgNa

(43)

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Erfan Kosari and Kambiz Vafai

dm ¼ αm ð1  mÞ  βm m dt dh ¼ αh ð1  hÞ  βh h dt

(44) (45)

Analogous to potassium conductance, α’s and β’s in sodium conductance are rate coefficients which are only a function of voltage. All rate-coefficients αn, βn, αm, βm, αh and βh, in previous equations can be calculated through the relations as given below [80,82]: 0:01ðV + 10Þ exp ½0:1ðV + 10Þ  1 βn ðV Þ ¼ 0:125 exp ðV=80Þ 0:1ðV + 25Þ αm ðV Þ ¼ exp ½0:1ðV + 25Þ  1 βm ðV Þ ¼ 4 exp ðV=18Þ αh ðV Þ ¼ 0:07 exp ðV=20Þ

(49) (50)

βh ðV Þ ¼ ½ exp ððV + 30Þ=10Þ + 11

(51)

αn ðV Þ ¼

(46) (47) (48)

At the initial state (t ¼ 0), the activation and inactivation numbers, i.e., n, m and h are stated based on rate coefficients when the voltage is zero mV. The resting values are given by: αn ð0Þ αn ð0Þ + βn ð0Þ αm ð0Þ m0 ¼ αm ð0Þ + βm ð0Þ αh ð0Þ h0 ¼ αh ð0Þ + βh ð0Þ n0 ¼

(52) (53) (54)

To consider the impacts of Ca2+ dynamics on signal modeling, in the literature, a potential shift is introduced. The potential shift (△V) comprises the impacts of external calcium concentration and it can be applied by adding this value to the voltage in rate coefficients. Hence, the rate coefficients must be calculated through Eqs. (46)–(51) as a function of V + △V rather than V, i.e., αðV + △V Þ and βðV + △V Þ [81,83].  (55) △V ¼ 9:32  ln ½Cae =41:8 In action potential calculations, the constant parameters in the equations given to this point are acquired based on a reference temperature 6.3 ° C. In order to make the signal modeling applicable for other temperatures and particularly

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for capillary walls which is induced to be slightly above body temperature, a scaling factor is proposed (ϕ). In this method, the rate coefficients are multiplied by the scaling factor; ϕ ¼ Q ðT T ref Þ=10 [82] or according to another 10

study ϕ ¼ A[1 + B(T  Tref)] [80]. By applying the above modifications, the neural signaling equations yield as follows: Cm

dV ¼ I  gK n4 ðV  V Na Þ  m3 hgNa ðV  V K Þ  gl ðV  V l Þ dt dn ¼ ϕ½αn ðV + △V Þð1  nÞ  βn ðV + △V Þn dt dm ¼ ϕ½αm ðV + △V Þð1  mÞ  βm ðV + △V Þm dt dh ¼ ϕ½αh ðV + △V Þð1  hÞ  βh ðV + △V Þh dt

(56) (57) (58) (59)

All constants, parameters and the initial condition for signal modeling are listed in Table 5. Table 5 A list of constant parameters and the initial condition required for neural signal modeling Parameter Value Units Description

Cm [80,81,83] gK [80,81,83] gNa [80,81,83]

1 36 120

mF/cm2

Membrane capacity

mS/cm

2

Maximal potassium conductance

mS/cm

2

Maximal sodium conductance

2

Maximal other ionic conductance

gl [80,81,83]

0.3

mS/cm

V initial [80]

0

mV

Initial membrane potential

V l [80]

10.61

mV

Displacement potential for other ions

[Na]i [81,83]

50

mM/kg

Internal sodium concentration

[Na]e [81,83]

440

mM/kg

External sodium concentration

[K]i [81,83]

400

mM/kg

Internal potassium concentration

[K]e [81,83]

20

mM/kg

External potassium concentration

[Ca]e [81,83]

2.22

mM/kg

External calcium concentration

Q10 [80,82]

3



Special constant

Tref [80,82]

6.3

°C

Reference temperature

A [80]

1.14



Rate of change of conductance with temperature

B [80]

0.06

°C1

Constant ratio between ionic conductances

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5. Concluding remarks Achieving new techniques for treatment of neurodegenerative disorders have been receiving significant attention. However, neurosurgeons are still facing the challenge to ensure the effectiveness and noninvasiveness of operations. An accurate computational solution based on engineering fundamentals is capable of providing a comprehensive model to lower the operation risks and guarantee a successful therapy. This chapter proposed a mathematical modeling in which the thermal and dynamic aspects of blood capillaries are taken into account. Subsequently, by using a well-known neuro-signaling model the corresponding impact of the locally raised temperature resonates across nervous system. Therefore, the proposed model is applicable in various biomedical applications of which require to investigate bioelectrical behavior of tissues undergone thermal stimulations.

Acknowledgment The authors’ research presented above has been jointly supported by UC Riverside graduate grant. E.K. would like to express a very great appreciation to Dr. A. Rahnama and Dr. N. Mirkhani for the information acquired through fruitful discussions.

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