Entropy generation in thermal systems with solid structures – A concise review

8/17/2019 Entropy generation in thermal systems with solid structures – A concise review

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Review

Entropy generation in thermal systems with solid structures – A concise

review

Mohsen Torabi a,b,⇑, Kaili Zhang b, Nader Karimi c, G.P. Peterson a

a The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USAb Department of Mechanical and Biomedical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong c School of Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom

a r t i c l e i n f o

Article history:

Received 24 October 2015Received in revised form 21 February 2016Accepted 3 March 2016Available online 15 March 2016

Keywords:

Entropy generationHeat transferIrreversibilitiesSolid material

a b s t r a c t

Analysis of thermal systems on the basis of the second law of thermodynamics has recently gained con-siderable attention. This is, in part, due to the fact that this approach along with the powerful tools of entropy generation and exergy destruction provides a unique method for the analysis of a variety of sys-tems encountered in science and engineering. Further, in recent years therehas been a surge of interest inthe thermal analysis of conductive media which include solid structures. In this work, the recentadvances in the second law analyses of these systems are reviewed with an emphasis on the theoreticaland modeling aspects. The effects of including solid components on the entropy generation within differ-ent thermal systems are first discussed. The mathematical methods used in this branch of thermodynam-ics are, then, reviewed. This is followed by the conclusions regarding the existing challenges andopportunities for further research.

2016 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9172. Entropy generation in thermal systems with conductive parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919

2.1. Pure conductive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9202.2. Conjugate heat transfer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9212.3. Porous media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9222.4. Thermoelectric systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924

3. Analysis methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9253.1. Exact analytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9263.2. Approximate analytical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9273.3. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9283.4. Combined analytical–numerical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

1. Introduction

Multidisciplinary efforts to achieve better control of the heattransfer rates in thermal systems are currently of significant

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.03.007

0017-9310/ 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: The George W. Woodruff School of MechanicalEngineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.

E-mail addresses: [email protected] (M. Torabi), [email protected] (K. Zhang), [email protected] (N. Karimi), [email protected] (G.P. Peterson).

International Journal of Heat and Mass Transfer 97 (2016) 917–931

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academic and industrial interest [1]. This mainly stems from therapidly growing concerns about energy efficiency in a wide rangeapplications spanning from industrial to domestic appliances [2].Further, the introduction of micro and bio-heat transfer and energytechnologies has set new challenges for the thermal energy opti-mization [3–5]. Fundamentally, heat transfer has close ties withthe first law of thermodynamics and this is, effectively, the onlythermodynamic principle used in conventional heat transfer anal-ysis. The second law of thermodynamics provides a measure of entropy generation rate, or irreversibility, within a system or pro-cess, and as a result, impacts the efficiency of the heat transfer pro-cess. Over the last few decades, there has been an increasingawareness about the influence of irreversibility on energy interac-tions [6]. This has led to the formulation of exergy analysis and the

definition of exergetic efficiencies as an addition to the conven-tional energy efficiency approach [2,6,7]. Although initially devel-oped for thermo-mechanical processes, the concept of exergywas extended to chemical systems as well [8], and it is now beingapplied to the problems in ecology, environment and economy[9,7]. Central to the conduction of exergy analysis is the calculationof entropy generation, which then determines the rate of exergydestruction [10,6]. This evaluates the level of degradation of energyand therefore introduces the concept of energy quality [11]. Thefirst law analysis, however, does not recognize the variations inenergy quality and only conducts an energy accounting [12]. Thisdifference is the principal reason for the veracity of second lawanalyses when compared to those, which are solely on the basisof the first law of thermodynamics.

Application of second law analysis in thermal engineering, pro-vides the possibility of optimizing a given system or process on thebasis of the energy quality, which is very different from a first lawanalysis [13]. For example, it is well documented that the heattransfer in heat exchangers can be enhanced using various profilesof extended surfaces [14,15], or by optimizing the volume flow rateor the heat transfer coefficient [16]. However, design of a specificfin on the basis of the minimum entropy generation [17] resultsin different configurations compared to those obtained by the clas-sical optimization methods [16]. This is also true when one isapplying the concept of entropy generation to optimize the tem-perature field in electronic devices [18]. These types of thermody-namic analyses, i.e., the entropy generation and exergy analyses of a system, have been interestingly further extended to exergetic

analysis of the human body [19,20]. Even for the human body,the obtained results from the energetic and the exergetic analyses

were quite different to each other [19], which leads researchers tore-consider thermal systems by using the perspective of the secondlaw of thermodynamics. This has been extended further to the con-sideration of the second law of thermodynamics over the first lawfor mechanical analysis in solid structures [21,22]. For instance,Baneshi et al. [21] used maximum entropy generation criterionto calculate the worst case scenario for mechanical systems underthermomechanical loads.

In reality, all thermal processes include some level of irre-versibility, primarily due to the existence of temperature gradients.This renders an exergetic efficiency loss and results in reducing inthe energy quality. In heat transfer processes, entropy generationhas been reported in conduction [23–26], convection [27–29], radi-ation [30,31], and/or any combination of these modes of heat

transfer [32,33]. Further, there are other sources of irreversibilitiessuch as viscous dissipation [27–29] and magnetic fields [34]. Initi-ated by the seminal works of Bejan [35,36], several studies havebeen conducted on the minimization of entropy generation [37–41,10,12,42]. To date, there has been a significant focus on exer-getic processes in forced and free convection [37–41,10,12,42].However, much less attention has been paid to the entropy gener-ation in heat conductive media [43–45]. Given that solid structuresare found in, nearly all thermal systems, this field has major poten-tials to grow, and is still far from maturity. Perhaps more impor-tantly, after the introduction of complex solid components intothermo-mechanical systems, such as composite and multilayerstructures, the first and second law analyses of conductive systemshas gained considerable attention [46–48]. Hence, obtaining

Nomenclature

asf interfacial area per unit volume of porous media, m1

c p specific heat at constant pressure, J kg1

K1

hsf fluid-to-solid heat transfer coefficient, W m2 K1

J current density, amp m2

k thermal conductivity of solid material, W m1 K1

kef effective thermal conductivity of the fluid ðek f Þ,W m1 K1

kes effective thermal conductivity of the solid ðð1 eÞksÞ,W m1 K1

N 000s dimensionless local entropy generation rateQ Dimensionless volumetric internal heat generation rate_q Volumetric internal heat generation rate, W m3

_S 000 f local entropy generation rate within the fluid phase of the porous medium, W m3 K1

_S 000s local entropy generation rate within the solid phase of the porous medium, W m3 K1

T temperature, K

T f temperature of the fluid phase of the porous medium, KT s temperature of the solid phase of the porous medium, Kt time, Su f velocity of the fluid in the porous medium, m s

1

X dimensionless axial distance

x axial distance, m y vertical distance, m

Greek symbolsr electric conductivity, X1 m1

e porosityj permeability, m2

l f fluid viscosity, kg m1 s1

leff effective viscosity of porous medium, kg m1 s1

h dimensionless temperatureq fluid density, kg m3

Fig. 1. Configuration of a solid wall with different modes of heat losses.

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further familiarity with the process of entropy generation in solidcomponents and their modeling is clearly of interest.

The current investigation presents a survey of the literature onthe progress and challenges of the evaluation of entropy genera-tion in primarily conductive systems. These include purelyconductive problems and also those convective-conductive prob-lems in which conductive process have dominating roles. Thereview covers the physical aspects of entropy generation in theseproblems and also explains the mathematical tools that can beutilized for the analysis and modeling. It ultimately identifiesthe ongoing challenges and the areas of interest for futureresearch.

2. Entropy generation in thermal systems with conductive parts

The energy equation for a three-dimensional object that experi-ences internal heat generation/consumption is governed by the fol-lowing formula.

rðkrT Þ þ _q ¼ 0 ð1Þ

Calculation of the temperature field through this structure isthe first step in obtaining the entropy generation rate. Dependingon the nature of the thermal conductivity, the internal heat sourceand the boundary conditions, the temperature field can be found

analytically or through numerical simulations. Regardless of themethod of heat transfer analysis, the second law analysis can beconducted on this system to provide a ready-to-use relationshipfor determining the rate of entropy generation. Using theClassius-Duhem statement of the second law, the entropy balanceequation may be written as [49,6]:

d

dt

Z V

sdV ¼

I A

1

T ~q ~nd A þ

Z V

1

T _qdV þ

Z V

_S 000dV : ð2Þ

Considering steady-state and applying the divergence theorem,the surface integral on the right hand side of the integral can bewritten in terms of the volume integral. That is,

I A

1

T ~q ~nd A ¼

Z V r

~q

T d

V : ð3

Þ

Using Eq. (3) to replace the surface integral with the right-handside of Eq. (2), we obtain

Z V

r ~q

T

dV þ

Z V

1

T _qdV þ

Z V

_S 000dV ¼ 0: ð4Þ

Since each integral in Eq. (4) is a volume integral, it follows that

_

S 000 ¼ r ~q

T

_q

T : ð5Þ

By consideration of a one-dimensional solid wall of thickness L,made of a material with thermal conductivity k(T) as illustrated inFig. 1, the energy Eq. (1) can be simplified to [50,49]

d

d x kðT Þ

dT

d x

þ _q ¼ 0: ð6Þ

Assuming one-dimensional heat transfer, which holds in thecurrent configuration, and incorporating energy, Eq. (6), into Eq.(5), the following equation is obtained for the local entropy gener-ation rate.

_S 000 ¼ kðT Þ

T 2

dT

d x

2

: ð7Þ

Eq. (7) is one of the fundamental equations used in entropyrelated analyses in solid media. It is imperative to note that asdemonstrated by Aziz and Khan [49], the internal heat generationin solid media does not explicitly appear in the entropy generationequation.

In the pioneering work of Ibanez et al. [51] it was shown thatentropy generation in a cooling process could be optimized by con-trolling the convective cooling parameters. The entropy generationin a process of heat conduction is a function of the temperaturefield [23,50]. It, therefore, depends upon the thermo-physical prop-erties of the medium, internal energy generation, and the thermalconditions imposed on the boundaries of the medium. FollowingRef. [51], a number of studies calculated the classical entropy gen-

eration in pure conduction processes or minimized that in the pro-cesses within the pure conductive environments. Strub et al. [52]analyzed entropy generation in a wall with a purely sinusoidaltemperature boundary condition on one side and a constant tem-perature condition on the other side. Later, Al-Qahtani and Yilbas[53] used the concept of entropy generation to analyze the one-dimensional transient heat conduction during laser pulse heating.Recently, Ali et al. [54] extended this work to the pure analyticaltreatment of energetic and entropic analyses of a cylinder due tolaser short-pulse irradiation. It was shown that the general behav-ior of the entropy generation was similar to the temperature field.However, when a parameter varies, the variation on the tempera-ture and entropy generation fields may not be similar [54]. Azizand Khan [17] studied heat transfer and entropy generation

analyses of convective pin fins with convective heating at the baseand convective heat loss at the tip. The fluid friction at the base,lateral surface and tip of the fin were also included in thecalculations [17].

Subsequent to the introduction of entropy generation in solidmedia [51] a number of studies contributed to the advancementof this field, see for example [55,52,53]. These investigationsfocused on various conductive media such as homogeneous struc-tures, functionally graded materials and composite geometries[23,50]. The next step was to include solid systems in convectivemedia which resulted in the investigation of conductive–convective media. This group of problems is different from theconventional convection problems by the fact that the overallheat transfer in a conductive–convective system is strongly depen-

dent upon the conduction of heat in the solid components. Entropygeneration analysis, then, found its way into porous thermal

Fig. 2. Total entropy generation rate versus heat transfer rate in a hollow cylinder[23].

M. Torabi et al. / International Journal of Heat and Mass Transfer 97 (2016) 917–931 919

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systems [56–59], as an important class of conductive–convectivesystems.

In the following, the literature on these sub-categories is

discussed.

2.1. Pure conductive media

One of the first articles on entropy generation in solid mediawas authored by Bisio in 1988 [60]. He analyzed entropy genera-tion in one-dimensional transient conduction in a system withtime-dependent thermal boundary conditions. Here, the authorsassumed that the thermal conductivity of the slab varies withthe local temperature as well as the coordinate along the directionof the heat conduction [60]. Many investigators followed Bisio’spath and opted in favor of a thermodynamic analyses of conductivesystems. The investigations of Ibanez et al. [51] and Kolenda et al.

[55] were among these works. Ibanez et al. [51] considered entropygeneration in a slab with uniform internal heat generation and

different convective cooling conditions on the opposing faces of the slab. In this work, it was shown that the total entropy genera-tion in the system can be minimized with the proper choice of theconvective cooling parameters [51].

The work of Kolenda et al. [55] demonstrated that entropy gen-eration in steady state conduction (one-dimensional or multidi-mensional) could be always minimized by placing heat sources

in the conducting medium. Given this interesting conclusion, otherresearchers applied the second law of thermodynamics to conduc-tive systems to analyze the entropy production within these struc-tures [49,61,23]. These attempts led to the development of somecontentions and debates. As an example, Aziz and Khan [49] con-ducted a theoretical analysis of entropy generation in heat gener-ating solids and questioned the validity of some other entropygeneration analyses of this problem. These authors mathematicallydemonstrated that the thermal energy sources do not explicitlyappear in the entropy generation equation [49]. They therefore,concluded that the previous analyses, which considered internalheat generation as an extra source of irreversibility were incorrect[62,63]. In a subsequent study, Aziz and Khan [61] examined thisnew formulation of entropy generation [49] on all the three funda-mental configurations, namely slabs, cylinders and spheres withconstant boundary temperatures.

Later, Torabi and Aziz [23] investigated the generation of entropy in an asymmetrically cooled hollow cylinder withtemperature-dependent thermal conductivity, internal heat gener-ation and radiation effects on the outside surface. They calculatedthe local and total entropy generation rates for this geometry.These authors [23] demonstrated that with a proper choice of theconvection parameters, the total entropy generation could be min-imized (see Fig. 2). Torabi and Zhang [50] addressed entropy gen-eration rate problems in the regular and functionally gradedmaterial slabs with temperature-dependent internal heat genera-tion and convective-radiative boundary conditions. In addition tothe effects of convection, this study included those due to radia-tion. Fig. 3 shows that radiation could have similar effects to con-

vection as in the previous studies [23]. This figure also indicatesthat the total entropy generation rate can be minimized by opti-mizing the radiation.

Given the increasing importance of the thermal analysis of com-posite multilayer structures this approach has recently gainedincreased significance. These types of structures are used exten-sively in manufacturing applications, largely due to their excellentmechanical and thermo-physical properties, when compared tothose of conventional materials. Kaska and Yumrutas [64] pre-sented experimental and theoretical studies on the transient tem-perature and heat flow in multilayer walls and flat roofs. Inaddition, Amini-Manesh et al. [65] numerically demonstrated theimportance of using composite materials as a substrate, which sig-nificantly enhances the flame stabilization in a reactive nano-film.

Fig. 3. Total entropy generation rate versus conduction-radiation parameter for (a)homogenous slab and (b) FGM slab [50].

Fig. 4. Configuration of double-layer composite hollow cylinder studied by Torabiand Zhang [47].



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Conduction heat transfer is also of significance in composite media,due to widespread applications of composite structures in the elec-tronics industry [66–68]. It follows that predictions of the temper-ature and entropy generation within composite structures areimportant for many industrial applications. As a result, in recentyears there have been a significant number of attempts to examinecomposite and multilayer media, from the second law perspective.

Torabi and Zhang [47] were among the first who investigatedthe generation of entropy in composite solid media. They consid-

ered one-dimensional steady conduction in a two-layer composite,hollow cylinder with temperature-dependent thermal conductiv-ity and constant temperature (case one) or pure convective coolingon the internal and external surfaces (case two). Fig. 4 shows theconfiguration of the double-layer hollow cylinder in their study[47]. These authors [47] further assumed perfect thermal contactbetween the two layers and varied the internal heat generationin each layer of the material. Substantial differences between thetotal entropy generation rate for the two cases were observed[47]. This implies that, in general, consideration of a constant ther-mal boundary condition in place of the actual convective boundarycondition, cannot be adequately justified. Further, the currentinvestigation indicates that in addition to the different values forthe total entropy generation rate in the two investigated cases,

the trend could sometimes be different. For example, by increasingthe interface radius, the total entropy generation rate is increasedin the first case. However, through a similar variation of the inter-face radius, the total entropy generation rate may decrease orincrease in the second case [47]. More recently, by consideringimperfect thermal contact resistance (TCR), Torabi et al. [48]extended the analyses of Ref. [47] to entropy generation analysisfor the three fundamental double-layer solid structures previouslyinvestigated in Ref. [61]. It was observed that the existence of TCR could lead to an interesting phenomenon within the compositemedia, in which increasing the TCR reduces the total entropy gen-eration rate. Hence, neglecting TCR in entropy generation simula-tion in multilayer materials can cause overestimation of the totalentropy generation rate [48].

The investigations presented and discussed here, consideredonly a fraction of the general problem and there exists a handful

of structures and conditions, which still need to be analyzed. Forinstance, the second law analyses of multilayer composite struc-tures with more than two layers have yet to be conducted. Becausethe TCR has been shown to have a substantial influence upon theentropy generation in composite structures [48], the temperature jumps are speculated to play a similar role within the interfaceof conductive–convective systems. Further, velocity slip isexpected to have a strong impact on the entropy generation ratewithin the system. Rigorous investigations of these effects remain

as future research tasks.The above discussion mainly has considered the steady-state

processes. The number of transient analyses of entropy generationin pure conductive configurations is very limited. This is mainlydue to the inherent difficulty of these analyses, which transformthe ordinary differential equations into partial differential ones. Astudy of transient entropy generation analysis has been recentlyperformed by Ali et al. [54]. This investigation opted in favor of transient entropy generation rate in a cylinder under laser-pulseheating [54]. Here, as mentioned in the introduction, a purely ana-lytical solution was developed to obtain the temperature field. Thecalculated temperature field was then incorporated into theentropy generation formulation. However, due to the complexityof the derived mathematical formulation, the local entropy gener-

ation is difficult to be integrated over the entire geometry, andhence the total entropy generation has not been reported.A summary of the recent investigated geometries and the

employed solution methods, which will be discussed in Section 3,has been provided in Table 1.

2.2. Conjugate heat transfer systems

The thermal aspects of the problem of conjugate heat transferhave been already reviewed [69,70]. As stated previously,entropy generation in convective systems has been the subjectof intensive research for a relatively long time [42,71]. However,less attention has been paid to the media that contain both theconductive and convective constituencies. This is also true for

conjugate conduction-radiation heat problems. Recently, thisapproach has been investigated by a number of researchers

Table 1

Summary of the recent investigated pure conductive geometries.

Authors Geometry Boundary condition Solution method Main contribution

Kolendaet al.[55]

One-dimensional ormulti-dimensionalwalls

Constant temperature Numerical One of the pioneering works

Aziz andKhan

[49]

Wall Asymmetric convection Exact analytical Incorporation of internal heat source

Aziz andKhan[61]

Wall, hollowcylinder and sphere

Inner and outer constant temperatures Exact analytical+ numerical forsome cases

Incorporation of internal heat source + newfundamental geometries

Torabi andAziz[23]

Hollow cylinder Inner convective, outer convective-radiative Analytical DTM Analytical solution with radiation boundarycondition

Torabi andZhang[50]

Walls Convective-radiative at both sides Analytical DTM Incorporation of walls materials such astemperature-dependent or spatial-dependentthermal conductivities

Torabi andZhang[47]

Double-layer hollowcylinder

Constant temperature for both inner and outer layers+ convective cooling for both inner and outer layers(Two cases)

Combinedanalytical–numerical

Composite cylindrical structure

Torabi andZhang[128]

Double-layer wall Convective-radiative at both sides Analytical DTM Composite walls with radiation heat loss

Torabi

et al.[48]

Double-layer wall,

hollow cylinder andsphere

Inner convective, outer convective-radiative Combined

analytical–numerical

Incorporation of imperfect thermal contact for the

interface boundary condition

Ali et al.[54]

Cylinder Constant temperature Combinedanalytical–numerical

Transient heat transfer and entropy generationanalyses of a cylinder


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through the application of the second law to a number of convective-conductive or conductive–radiative media. Here, theprocess of entropy generation in these systems is considered,

and a particular attention is paid on the role of the solid phasein this process.Ibáñez and Cuevas [72] considered the fluid flow, heat transfer

and entropy generation rates within a parallel wall microchannel,under a uniform transverse magnetic field. The magnetic field pro-duced the Lorentz force and this was assumed to generate a fullydeveloped flow [72]. The conjugate heat transfer problem in anMHD flow between two parallel solid walls of finite thicknesses,was analyzed from an optimization perspective. The principalobjective in this investigation was to better understand the influ-ences of the thermal conductivity of the fluid and channel wall,on the entropy generation [72]. This investigation demonstratedthe existence of optimal values of these quantities, which resultin the minimum entropy generation rates [72]. Later, these analy-ses were repeated for a viscous flow between parallel solid walls of finite thickness [45]. It was concluded that by the incorporation of wall dissipation in the evaluation of entropy generation in naturalor forced convection, the channel wall thickness can be optimized[72,45]. This conclusion was also true for the optimization of thewall to fluid thermal conductivity ratio [72,45]. These findingsare therefore of importance in the design of heat transferringdevices [45,72].

In a recent study, Ibáñez et al. [73] extended the investigation of Refs. [45,72] to entropy generation analysis in a microchannel byconsidering the uniform heat flux boundary conditions and hydro-dynamic slip between the channel walls and the fluid. This studyillustrated the possibility of finding an optimum value of the heatflux, slip length, wall to fluid thermal conductivity ratio and Pecletnumber to minimize the global entropy generation rate. Further, as

shown in Fig. 5, Ibáñez et al. [73] concluded that by fixing all otherparameters and increasing the wall to fluid thermal conductivityratio, the optimum rate of the global entropy generation increases.

Several other recent articles investigated the entropy genera-tion in conjugate convective-conductive systems [74,75]. Torabiet al. [74] analyzed the first and second laws of thermodynamicswithin a cylindrical system. Nanofluid flow was considered

between the inner and outer walls and these two solid structureswere further included in the model. Temperature-dependent ther-mal conductivities, for solid materials, were assumed. Constant,but uneven, values of internal heat generations were incorporatedinto the energy equation of the solid parts. In a separate study,Torabi and Zhang [75] modeled temperature and entropy genera-tion within a cylindrical system. Their model considers the innersolid cylinder, the MHD flow between the inner and outer cylindersand the outer hollow cylinder [75]. Thermal conductivities of thesolid materials were assumed to be temperature-dependent whilethat of the fluid was considered constant [75]. These studies[74,75] demonstrated that due to the almost constant temperaturewithin the inner solid layer of the rotating cylinder, this layermight not participate in the entropy generation. Nonetheless, theouter layer greatly affects the local and total entropy generationrates.

Currently, there is a shortage of information on the conjugateradiation-conduction heat transfer relationship and the associatedentropy generation. To date, the only investigation in this area isthe work of Makhanlall and Liu [32]. This investigation, includeda second law analysis of a solid–liquid phase change systemsinvolving both conduction and radiation [32], and assumed thatboth phases were semi-transparent and that the liquid motiondoes not occur within the system, as the material properties areidentical for both phases. By calculating the entropy generation,the exergy destruction due to the conduction and radiation modesof heat transfer could be calculated [32]. Makhanlall and Liu [32]concluded that, although often neglected, radiation heat transferplays an important role in the exergy destruction. Future research

on the second law analysis of conductive–convective–radiativesystems is expected to provide further insight into the influencesof these combined modes of heat transfer upon the generation of entropy.

For the sake of completeness the above review has been tabu-lated in Table 2.

2.3. Porous media

It is well understood that porous solids can greatly facilitateheat transfer rates [76,77]. This is due to the dual influences of the solid medium on increasing the effective thermal conductivityof the composite solid–fluid medium, and also providing anextensive heat transfer surface area [77]. This has led to the wide

Fig. 5. Total entropy generation rate versusslip lengthin a microchannel studied byIbáñez et al. [73].

Table 2

Summary of the recent investigations in conjugate thermal systems.

Authors Geometry Modes of heat transfer Solution method Main contribution

Ibáñez and Cuevas[72]

Microchannel Conduction andconvection

Analytical A pioneering work about conductive–convective systems

Ibáñez et al. [45] Microchannel Conduction andconvection

Analytical Optimization of walls’ thickness

Ibáñez et al. [73] Microchannel Conduction andconvection

Analytical Incorporation of slip boundary

Torabi et al. [74] Cylindricalsystem

Conduction andconvection

Combined analytical–numerical

Consideration of copper–water nanofluid

Torabi and Zhang[75]

Cylindricalsystem

Conduction andconvection

Combined analytical–numerical

Consideration of magnetohydrodynamic flow

Makhanlall and Liu[32]

Square cavity Conduction andradiation

Numerical A pioneering work about entropy generation in conductive–radiative systems



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application of porous materials in a variety of processes spanningfrom cooling technologies to bio–heat transfer [78]. However, theexcellent heat transfer characteristics of porous media are associ-ated with significant impedance of the flow. As a result, there havebeen considerable efforts on thermo–hydraulic optimisation of heat transfer processes in porous media, see for example Refs.[79–82]. A challenge before this approach is the multiplicity of pos-

sible optimisation criteria. For instance, optimisation can be on thebasis of Nusselt numbers (defined at different locations across thesystem), temperature profiles or pressure drop. It is not immedi-ately clear which of these is the most suitable criterion. Further,these features of the system are strongly interconnected and vary-ing one would change the others [79]. This may highly obscure theoptimisation process and even cause confusions. One possible rem-edy for this situation is the introduction of a single optimisationcriterion, which includes all the pertinent thermal and hydrody-namic effects. The total entropy generation clearly provides suchcriterion. The irreversibilities encountered in heat transfer pro-cesses within porous media are, generally, due to the transfer of thermal energy across finite temperature difference and the vis-cous dissipation of the flow kinetic energy. The former, usually,occurs by conduction of heat within the solid matrix or convectiveand radiative heat exchanges between the solid and fluid phases[77]. The latter is due to the interactions occurring between theviscous flow and the solid matrix, and the extended contact surfacearea, between the moving fluid and the solid surface [76,83,84].

To model this situation, the method of representative elemen-tary volume is often employed [76] and a system of equations thatdescribe the transport of momentum and thermal energy aresolved [76,77]. Depending upon the pore scale Reynolds numbereither of Darcy, Darcy-Brinkman or Darcy-Brinkman-Forchheimerequations are employed [76]. Consideration of local thermody-namic equilibrium divides the thermal energy transport into twomajor sub-categories. Local thermal equilibrium (LTE) greatly sim-plifies the problem and renders a single differential energy equa-tion for the composite medium [77]. However, this approach

ignores the generation of entropy as a result of heat exchangesbetween the solid and fluid phases. A more accurate approachreleases the assumption of local equilibrium and establishes thelocal thermal non-equilibrium (LTNE) condition, and hence consid-ers the irreversibility of local heat exchanges within the medium[78,85]. The generation of entropy due to the flow of fluid and heatin porous media has been investigated by a number of authors. Inthese works, usually, the problem of heat transfer is solved firstand the resultant temperature and hydrodynamic fields are usedto calculate the rates of entropy generation. The relative signifi-cance of heat transfer irreversibility (HHI) and fluid flow irre-versibility (FFI) are measured by the so-called Bejan numberdefined as Be = HHI/(HHI + FFI) [10]. Generation of entropy in por-ous media under natural convection has been studied most exten-

sively. However, investigations of entropy generation under forcedconvection are less frequent. For conciseness reasons, the followingdiscussion and review mostly concern the forced convection of fluid through a porous solid. The reader is referred to Ref. [71]for the survey of literature on entropy generation in free and mixedconvection within porous media. Further, some very recent exam-ples of investigations in this area could be found in [86–89]. Hoo-man and Ejlali [56] numerically solved the Darcy-Brinkman modeland the LTE energy model for a fully filled porous pipe under devel-oping condition. They calculated the rate of entropy generation andBejan number for this configuration [56]. The effects of viscous dis-sipation on the temperature fields and entropy generation in a fullyfilled channel were investigated by Hooman and Haji Sheikh [90].They conducted an extensive numerical study and demonstrated

that viscous dissipation decreases the Nusselt number in the devel-oping and developed regions of the flow conduit under isothermal

wall heating [90]. In an attempt to provide optimization guides forthe fully filled porous channels, Hooman et al. [57] analyticallysolved the Darcy-Brinkman model and the LTE energy equation.They considered three different boundary conditions on theexternal surface of the channel, and provided expressions for theNusselt and Bejan numbers and rates of entropy generation. Thisanalysis was later extended to the developing flows by considering

a simple Darcian flow with the LTE condition and under constanttemperature boundary condition [91]. A parametric study was con-ducted which demonstrated that entropy generation rate inverselycorrelates with the Peclet number and is directly related to Brink-man number [91]. The problem of entropy generation in a fullyfilled porous channel under fully developed condition was alsoconsidered by Mahmud and Fraser [92]. They considered constantbut dissimilar wall temperatures and conducted analytical andnumerical studies on the heat transfer and entropy generationcharacteristics of the system [92]. This investigation was laterextended to unsteady cases by Kamish [93]. Further, through lin-earization of the governing equations, the unsteady heat transfer,fluid flow and entropy generation through a metal foam was inves-tigated analytically by Mahmud et al. [94]. In a separate work,Mahmud and Fraser [95] calculated the rate of entropy generationduring the natural convection of heat in a porous cavity undermagneto–hydrodynamic effects. Hydro-magnetic effects were,also, considered in the problem of entropy generation in a channelfully filled with porous media and under mixed convection [96].Entropy generation rates in partially filled conduits were calcu-lated by Morosuk [97], who demonstrated that the maximum rateof entropy generation occurs on the porous-fluid interface. In thiswork, the behavior of entropy generation rate was mainly attribu-ted to the hydrodynamic characteristics of the problem [97].Entropy generation minimization has been used as a design crite-rion in porous heaters by Shakouhmand et al. [98]. They calculatedthe optimal porosity for a fully filled porous conduit and demon-strated that the optimum matrix porosity increases with theReynolds number [98]. More recently, Mahdavi et al. [99] conducted

a numerical study on the partially filled conduits and calculatedthe flow field, Nusselt number and entropy generation rates. Theyconsidered two configurations in which porous insert is eitherplaced at the core of the pipe or is attached to the inner wall

Fig. 6. Total entropy generation rate versus porous thickness in a partially porouschannel using LTNE model [59].



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[99]. Their results highlighted the effects of the placement of por-ous material in the conduit on the heat transfer and entropy gen-eration rate [99]. They also demonstrated that the thermalconductivity ratio can dominate the level of heat transfer enhance-ment [99].

Assumption of LTE is the common feature of all the precedingstudies of entropy generation in porous media. Deviation from

thermodynamic equilibrium implies higher levels of irreversibility.In such cases, entropy generation is not limited to those due to vis-cous dissipation and external heat transfer sources and, the inter-nal heat exchange processes introduce an important mechanism of entropy generation. When LTNE model is employed for thermalanalysis of a porous system, an extra energy equation is used inthe analysis due to the heat exchanges between the fluid and solidphases of the porous medium [100]. This new equation, which iscoupled with the energy equation of the fluid phase, providesinformation on the temperature of the porous solid structure of the system. The difference between the solid and fluid temperatureand the resultant heat exchange between the two is a major sourceof entropy generation in the solid porous structure. Given this, it issurprising that there is currently a small number of studies onentropy generation in porous systems under the LTNE conditions.In a recent work, Buonomo et al. [58] conducted a study on porousfilled micro channels through using LTNE model. They analyticallyinvestigated the hydrodynamic and thermal processes betweentwo parallel plates filled with a porous medium [58]. Due to themicroscale size of the channel and rarefaction effects of the gasflow under consideration, the first order velocity slip and temper-ature jump conditions at the fluid–solid interface were used [58].This study provided analytical expressions for the velocity andtemperature fields as well as the local and total entropy generationrates [58]. Torabi et al. [59] took an LTNE approach and investi-gated temperature distribution, Nusselt number, and local andtotal entropy generation rates within a channel partially filled witha porous material. The lower wall of the channel was exposed to aconstant heat flux and the upper wall was assumed to be adiabatic.

Viscous dissipation effects were incorporated into the energyequations and analytical solutions were developed for the velocityand temperature fields [59]. Further, Nusselt number and local andtotal entropy generation rates were evaluated and a temperatureand Nusselt number bifurcation phenomena were observed [59].Furthermore, for the first time, a bifurcation phenomenon regard-ing the entropy generation rate was reported by this work (Fig. 6)[59]. The bifurcation in some other heat transfer characteristics,such as heat flux or Nusselt number, is defined as a sudden jumpfrom a positive value to a negative value or vice versa[59,101,102]. However, since the entropy production is a positivequantity, by entropy bifurcation we mean that the value of entropy

generation features a sudden jump [59]. Torabi et al. [100] consid-ered the exothermic or endothermic feature of physical or physic-ochemical processes for entropy generation analysis of a channelpartially filled with porous materials. Here, both local and totalentropy generation rates have been investigated and illustrated.It was shown that, for a specific set of parameters, a certain thick-ness for the porous medium can be found such that the Nusselt

number or the total entropy generation of the system can beoptimized. Trevizoli and Barbosa [103] used the second law of thermodynamics to optimize a regenerator. One-dimensionalBrinkman–Forchheimer equation was used to describe the fluidflow in the porous matrix and LTNE equation model was employedto determine the temperatures in the fluid and solid phases [103].By using finite volume method, cycle-average entropy generationdue to axial heat conduction, fluid friction and interstitial heattransfer were calculated. Variation of the total entropy generationversus particle diameter and aspect ratio showed the second lawoptimization is dependent upon other parameters of the system[103]. In a series of recent works, Ting et al. [104,105] analyticallyexamined the thermal characteristics and entropic behavior of afully-filled porous channel saturated with nanofluid. Their investi-gated configuration was subject to fully developed flow and asym-metric thermal boundary conditions [104], and could also includeinternal heat generation within the solid phase [105]. Theseauthors applied LTNE between the nanofluid and the solid struc-ture and demonstrated that the addition of nanoparticles reducesthe local temperature difference between these two phases [105].It was shown that the existence of internal heat sources in the solidphase majorly affects the system irreversibility. In particular, thisstudy showed that the internal heat generation in the solid phasecould have destructive influences upon the second law perfor-mance of the system [105]. Table 3 provides a summary of therecent investigation on LTNE analysis of the entropy generationin solid porous structures experiencing forced convection of heat.

In general, the analysis of entropy generation under non-localequilibrium in porous solids is still in its early stages. The previous

heat transfer investigations [106–109] can be extended to the sec-ond law analyses. In particular, the bifurcation of entropy genera-tion rate, deserves further attention as currently there is only asmall number of studies of this phenomenon [58,59,100].

2.4. Thermoelectric systems

The need for electrical power is constantly growing across theworld. The economic and environmental concerns associated withthe conventional energy sources for conversion to electric powergeneration has led scientist to newconcepts such as thermoelectricsystems [110]. An example of such devices is a thermoelectric

Table 3

Summary of the recent investigations in forced convection in porous systems using LTNE model.

Authors Geometry Boundary condition Fully orpartiallyfilled

Solution method Main contribution

Buonomo et al.[58]

Channel Constant heat flux Fully filled Analytical First entropy generation investigation for LTNEmodel

Torabi et al. [59] Channel Constant heat flux lower wall andadiabatic upper wall

Partiallyfilled


Entropy generation study for partially filledchannel

Torabi et al.[100]

Channel Constant heat flux Partiallyfilled


Incorporation of internal heat sources

Trevizoli andBarbosa [103]

A simplified passiveregenerator

Constant temperature andsymmetry boundaries

Fully filled Numerical Incorporation of oscillatory flow using a time-dependent pressure function

Ting et al. [104] Channel Constant heat flux Fully filled Analytical Addition of nanoparticles to the fluid phaseTing et al. [105] Channel Constant heat flux Fully filled Analytical Addition of nanoparticles to the fluid phase and

heat generation in solid phase



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cooler which works as a reversed heat engine operating betweenthe two heat reservoirs [25]. These devices which are in a solid

state have better efficiency compared with the conventional sys-tems for low capacity power applications [111]. Importantly, theirsolid structure offers advantages such as noiseless operation andlow-cost maintenance [24,110]. Due to the solid structure of thesesystems, Fourier heat transfer model is considered for the thermalanalyses [112–114]. In general, the unsteady energy balance inthermoelectric devices is written as [113]:

qc @ T

@ t ¼ rðkrT Þ þ

J 2

r b J

! rT ð8Þ

The first, second, and third terms of the right hand side of Eq. (8)describe the heat conduction, Joule heating, and Thomson effect,respectively. However, when it comes to micro and nanoscales,more appropriate models such as Cattaneo-Vernotte are required

[26]. Therefore, Eq. (8) may change to the following energy equa-tion for the small scale devices under electrothermal effects [26].

s@ 2T

@ t 2 þ qc

@ T

@ t ¼ rðkrT Þ þ

J 2

r b J

! rT ð9Þ

where the first term on the left hand side of Eq. (9) accounts for thestructure and size of the solid structure on the heat transfer equa-tion. As mentioned above, another two main terms, which are Jouleheating and Thomson effect, appear in the energy equations[112–114]. In a recently published article [115], yet another heatgeneration term called Peltier effect has been discussed throughthe formulations. The thermal analysis of these systems has beenwell investigated throughout the literature; see Refs. [116,114] fora review of this topic. However, thermodynamic analysis of thesesolid systems is rather limited [24–26,114]. In all of these analyses[24–26,114] the considered geometry has been treated one-dimensional fin-type [117], to eliminate multi-dimensional partialdifferential equations. These analyses can be performed for steadystate [24,25] or transient [26] processes. Similar to that discussedin the earlier subsections, when entropy generation in solidthermoelectric systems is considered, first the temperature field isobtained and consequently the thermal field is digested into theentropy formulation. Expectedly, that the entropy generationformulation consists of new terms which account for entropygeneration due to Joule heating and Thomson effects [114]. Kaushikand Manikandan [24] provided some information regarding thegeometry of the system for exergy analysis. With the exception of the cited articles there is no further thermodynamic optimizationof thermoelectric solid systems. Hence, this topic needs furtherinvestigation and attention. Moreover, the transient energy analy-ses and consequently entropic analyses of nanoscales thermoelec-tric systems can be further analysed by using dual-phase-lag heattransfer models [118,119]. The most important contributionsabout thermodynamic analyses of thermoelectric systems havebeen tabulated in Table 4.

3. Analysis methods

In general, theoretical heat transfer modeling encounters one ormore challenging partial differential equations, which could becoupled with each other. To solve these set of equations, the first

step is to make reasonable simplifying assumptions. For instance,in solving energy equations within solid media unidirectional heattransfer is often considered to convert partial differentialequation (PDE) into an ordinary differential equation (ODE)[120]. Let us consider a three dimensional wall (Fig. 7). The steadystate energy equation without internal energy sources for this wallis given by [121]

d

d x k

dT

d x

þ

d

d y k

dT

d y

þ

d

d z k

dT

d z

¼ 0 ð10Þ

If the four surfaces which are perpendicular to y and z axes areassumed to be well insulated, the heat merely transfers through xdirection. Then Eq. (7) reduces to the following ODE [122]:

d

d x k dT

d x

¼ 0 ð11Þ

Although due to some nonlinearities involved within the prob-lem (say k = k(T )), the obtained ordinary differential equation maynot be readily solvable, they take less expenses to be tackled.

Following this step, some mathematical manipulations may benecessary to decouple the equations. For example, the conductionenergy equation in solid phase of porous media can be coupledwith fluid phase energy equation in through convection terms. Thisoccurs during the thermal analyses of porous systems under LTNEmodel (so-called two-energy equation model). Therefore, the fol-lowing coupled equations may be considered [83,59]:

qc pu f @ T f @ x ¼ kef

@ 2T f @ y2 þ hsf asf ðT s T f Þ þ

l f j u

2

f þ leff @ u f @ y

2ð12aÞ

Table 4

Summary of the recent entropy generation investigations in thermoelectric solid

systems.

Authors Geometry Steady ortransient

Solutionmethod

Maincontribution

Chakrabortyet al. [114]

Rectangularthermoelectricleg

Steady Analytical A pioneerwork on theentropy

analysisKaushik and

Manikandan[24],Manikandanand Kaushik[25]

Annularthermoelectricleg

Steady Analytical Influence of Thomsoneffect andexergyanalysis

Figueroa andVázquez [26]

Rectangularthermoelectricleg

Transient Numerical Hyperbolicheat transferanalysis forsmall scaledevices

Fig. 7. Configuration of a three dimensional cooled wall.


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0 ¼ kes@ 2T s@ y2

hsf asf ðT s T f Þ ð12bÞ

where Eq. (12a) represents the energy equation in fluid phase andEq. (12b) governs the heat flow through the solid phase of the por-ous system. As can be clearly seen in Eqs. (12a) and (12b), are cou-pled with each other through the term hsf asf ðT s T f Þ which countsfor the convective heat transfer between the two phases. Byincreasing the order of derivatives, these coupled equations aredecoupled to yield a new set of differential equations. Interestedreaders may refer to the recently published papers on heat transferand entropy generation in porous media using LTNE model[123,124,102,59]. In some cases this step helps providing exact ana-lytical solution for the problem [83,59]. However, under moreinvolved situations, application of numerical methods is unavoid-able [32,125,48]. Numerical techniques, such as Newton–Raphson,bisection, secant or fixed-point iteration methods, can be appliedto find the coefficients of the solution [50,48]. They can be, further,applied to the original energy equation together with boundaryconditions [32,125]. In the latter case, finite volume or finite differ-ence methods could be employed. Consequently, the obtained tem-perature distributions are digested into the entropy formulations. If

the system is a purely conductive one, the provided entropy equa-tion in Section 2 is used. Regarding systems with other heat transfermodes, the appropriate formulation for the entropy generation hasto be considered. For example, if a porous system under LTNE modelis investigated, the entropy formulas may be written as:

_S 000 f ¼ kef

T 2 f

@ T f @ x

2þ

@ T f @ y

2" #þ

hsf asf ðT s T f Þ2

T sT f þ

l f jT f

u2 f

þleff T f

@ u f @ y

2ð13aÞ

_S 000s ¼ kes

T 2s

@ T s@ x

2

þ @ T s@ y

2" #

þ hsf asf ðT s T f Þ

2

T sT f ð13bÞ

where Eqs. (13a) and (13b) give local entropy generation rate insolid and fluid phases of the porous medium, respectively. A morecomprehensive discussion can be found in Refs. [59,83,104,100,58].

Here, we briefly discuss the mathematical method used for theanalysis of entropy generation in solid systems.

3.1. Exact analytical methods

It is often possible to find exact analytical solutions for thelinear ordinary differential equations [14,121]. By takingthis approach, along with the associated simplifying assumptions,

the model may lack some realistic features. However, exact analyt-ical solutions help developing a deep understanding of the behav-ior of the system under investigation. Importantly, the analyticalresults can be, further, used for validation of numerical tools. Ana-lytical approach has been taken in a number of investigations of entropy generation in solid media [126,49,61], entropy generationin conjugate convection-conduction systems [72,45], and even for

heat transfer and entropy generation within convective fins [17]. Itshould be noted that some energy equations for very fundamentalgeometries, such as the following discussed cooled homogeneousslab, which contain temperature-dependent parameters may bealso solvable with exact analytical methods. An example of thiscan be seen in the interesting work of Aziz and Khan [61]. How-ever, when dealing with the systems containing radiation effects[32,48], numerical methods are necessary to obtain the tempera-ture distribution of the system and predict the entropy generationrate within the system.

To make the given descriptions clearer, an exemplary problemis solved analytically in this section. We consider a homogeneousslab made of a material with temperature-dependent thermal con-ductivity, which is cooled asymmetrically. The slab generates heatwith a constant rate. Fig. 8 shows the configuration of the problem.Considering these assumptions, the one-dimensional non-linearenergy differential equation within the slab can be expressed as:

k d

d x 1 þ a T T 0ð Þð Þ

dT

d x

þ _q ¼ 0 0 < x 6 L; ð14Þ

with the following boundary conditions

dT

d x

x¼0

¼ 0 ð15aÞ

T j x¼L ¼ T c ð15bÞ

These equations can be easily non-dimensionalized using thedimensionless parameters described in the literature [50,47].Hence, the nondimensional form of the equations are written as

d

d X 1 þ aðh 1Þð Þ

dh

d X

þ Q ¼ 0 0 < X 6 1 ð16Þ

Fig. 8. Configuration of a cooled wall.Fig. 9. Temperature distribution and local entropy generation rate for the cooledwall.



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dh

d X

X ¼0

¼ 0 ð17aÞ

hj X ¼1 ¼ hc ð17bÞ

Although the main energy equation for this problem is nonlin-ear, it can be still solved using exact analytical methods. In thiscase, the energy Eq. (16) is a nonlinear second order ODE, which

can be easily solved through a change of variable (v ¼ dhd X ) and,therefore, transforming the original equation into a first orderODE, see Ref. [127]. Employing Maple 14, using the availableboundary conditions and after some manipulation, the followingrelation can be used to describe the temperature of the slab:

hð X Þ ¼

a 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 2a þ 1 X 2aQ þ 2a Q

2 ahc þ hc þ

ah2c 2

r a

ð18Þ

This formula can be then incorporated into the entropy genera-tion relationship for the solid structure [50,47]. For the presentone-dimensional problem, the volumetric entropy generation Eq.(5) can be written as:

_S 000 ¼ d

dx

q00 xT

1

T _q ð19Þ

Using Fourier’s law, for a slab with regular material Eq. (19)becomes

_S 000 ¼ d

dx

1

T kðT Þ

dT

dx

1

T _q ð20Þ

By incorporating the energy equation, i.e., Eq. (14) in Eq. (20),the latter reduces to

_S 000 ¼ kðT Þ

T 2dT

dx

2ð21Þ

With the introduction of dimensionless variables, the dimen-sionless local volumetric entropy generation rate N s for the slabwith homogeneous material is given by

N s ¼ ð1 þ bðh jÞÞ dh

dX

2h

2 ð22Þ

Here, it should be noted that, as discussed by Aziz and Khan[49], the internal heat generation will not appear in the entropygeneration formulation. In fact, the effects of internal heat genera-tion on the temperature formulation influence the entropy gener-ation rate in an indirect way. Due to the analytic nature of thismethod, appropriate parametric values can be used to determinethe temperature and local entropy generation rate as depicted inFig. 9.

This analysis can be further extended to double-layer compositewalls by incorporating another energy equation for the second walland related boundary conditions at the interface of the two walls.The two walls, depending on their size and surface condition, mayor may not be in perfect thermal contact with each other. This hasbeen discussed thoroughly in a series of recent publication byTorabi and co-workers [128,47,48]. For example, if the imperfectthermal contact is valid for the interface of the two walls, thefollowing boundary conditions, in addition to the outer layersboundary conditions should be employed [48]:

T 1ð x ¼ x2Þ T 2ð x ¼ x2Þ ¼ Rc k1 1 þ a1ðT 1 T 0Þð ÞdT 1d x

x¼ x2

ð23aÞ

k1

ð1

þ a1

ðT 1

T 0

ÞÞ

dT 1

d x x¼ x2 ¼ k

2 1þ a

2

ðT 2

T 0

Þð Þ

dT 2

d x x¼ x2 ð

23bÞ

where the right hand side of Eq. (23a) introduces a temperature jump into the system. For more information regarding the TCR and its effects on the temperature field of conductive systems, read-ers are encouraged to consult with the recently published articles inthis regard [129–131].

3.2. Approximate analytical techniques

In conduction problems the governing equation becomesnonlinear if the thermal conductivity of the material istemperature-dependent [23]. Similarly, the convection termbecomes nonlinear due to the power law dependence of heattransfer coefficient on the temperature difference. The radiationterm contains two nonlinearities, one due to the Stefan–Boltz-mann law of radiation, and the second due to the fact that thesurface emissivity is a function of temperature. Other nonlinearterms arise in the differential equation of the material by intro-ducing temperature-dependent internal heat generation[132,128]. Under most situations, even with one nonlinear term,the ODE describing the energy balance does not admit an exactanalytical solution. The following energy equations have some of

the above-mentioned characteristics and cannot be treatedanalytically [50,23].

d

d x k 1 þ bðT T 0Þ½

dT

d x

þ _qð1 þ aT Þ ¼ 0 ð24Þ

1

r

d

dr k 1 þ bðT T 0Þ½

dT

dr

þ _qð1 þ aT Þ ¼ 0 ð25Þ

Consequently, a number of non-exact solution procedures canbe used to address these types of problems. These methods whichdeliver a relation for the temperature distribution and, accordinglyfor the entropy generation within the material, are regarded as theapproximate analytical techniques. They transfer the original non-

linear differential equation into an infinite number of simpler, ana-lytically solvable differential equations. Since these tend to sum upa very large numbers of solutions, they are regarded as approxi-mate solutions. When a reasonable number of transformed differ-ential equations are solved and their results are added together,further differential equations are neglected. This imposes a trun-cated error in the results, which is an inherent characteristic of these techniques [133].

One of the powerful approximate methods, which has beenused in heat transfer problems [134] and entropy generation anal-yses of solid media [23,135,128,50], is the differential transforma-tion method (DTM). DTM is based upon Taylor series expansion,and is able to deliver highly accurate results in various challengingsituations such as problems with convective-radiative boundary

heat loss [23,135,50] and highly nonlinear internal heat generation[128]. Using such approximate methods can provide relations forthe temperature and entropy generation along with a physicalinsight into the effective parameters. This is typically superior tothe traditional numerical methods under many situations. Thismethod transfers each term of the differential equation into anew term which is used in the series [23]. Afterwards, using thedifferential inverse operator, the final solution for the energy equa-tion of the structure can be obtained. For example, consider theenergy equation for the homogenous cooled wall in Ref. [50]. Theenergy equation is as follows:

d

d X 1 þ bðh cÞð Þ

dh

d X

þ Q ð1 þ ahÞ ¼ 0 ð26Þ

Using the DTM technique and taking the differential transformof Eq. (26), the above equation is transformed to:


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ð1 b cÞðk þ 2Þðk þ 1ÞHðk þ 2Þ

þ bXkl¼0

HðlÞðk þ 2 lÞðk þ 1 lÞHðk þ 2 lÞ

!

þ bXkl¼0

ðl þ 1ÞHðl þ 1Þðk þ 1 lÞHðk þ 1 lÞ

!

þ Q a HðkÞ þ Q dðkÞ ¼ 0 ð27Þ

Now by varying parameter k form zero, the values of HðkÞ aredetermined. Then, by the following equation which is called differ-ential inverse transform function is used to obtain the final energyequation.

hð X Þ ¼Xnk¼0

X kHðkÞ ð28Þ

A more detailed discussion of this technique can be found inRefs. [136,137].

When the convective aspects are incorporated into the system,the momentum and energy equations can be coupled [138,74,75].In some cases, the velocity field can be directly incorporated intothe energy equation [74]. More frequently, however, direct incor-

poration of the velocity into the energy equation result in an ana-lytically unsolvable problem and therefore necessitates the use of numerical techniques [75,138]. To circumvent this barrier, a Tylorseries of the velocity field is used. This makes the obtained temper-ature approximate, yet it has the advantage of producing an analyt-ical relation for the temperature field. Recently, Mahian et al. [138]and Torabi and Zhang [75] used this procedure to obtain tempera-ture distribution in convective media and accordingly incorporatedthese temperatures into entropy equations. The work of Mahianet al. [138] was performed in connection with purely convectivesystems, while Torabi and Zhang [75] investigated a conductive–convective system.

3.3. Numerical simulations

If the geometry is complicated and/or the energy equations,together with the boundary conditions involve strong nonlineari-ties, the use of numerical solution is mostly unavoidable. Further,numerical solutions are often used when energy and momentumequations are strongly coupled [99]. In these situations there aretwo main approaches to numerical solution of the equations. Onemethod is to make use of commercial modeling software, cus-tomized for heat transfer and thermal systems. Two well-knownof these are ANSYS Fluent and COMSOL Multiphysics. The othermethod is to develop a code in programing languages such as FOR-TRAN. One of the early investigation for the numerical entropygeneration analyses of thermal systems, which involves heat con-duction, has been done by Orhan et al. [139]. Here, finite control

volume approach was used to predict the entropy generation ratefor a phase-change process in a parallel plate channel. In a recentlypublished work by Makhanlall and Liu [32] FLUENT 6.3 has beenused to tackle the energy equations in a conductive–radiative envi-ronment. Under this setting, special treatment is needed to calcu-late the entropy generation rate within the system. This can bedone by employing a feature of this software, called user definedfunctions. By adopting this approach, Makhanlall and Liu were ableto provide local and total entropy generation and exergy destruc-tion [32].

3.4. Combined analytical–numerical techniques

The last major approach for the solution of conduction domi-

nated problems is to employ analytical and numerical techniquessimultaneously. In some situations, the general solution of the

problem can be found analytically. However, due to complexboundary conditions, such as radiation, in many cases, a particularsolution cannot be developed analytically. Hence, it is necessary toapply a numerical method to calculate the constant coefficients forthe particular solution. This approach has received considerableattention recently, as it presents a solution approach for a largenumber of problems [74,48,140]. To determine the coefficients a

system of equations must be solved. Commercial mathematicalpackages such as Maple or Wolfram Mathematica may be usedfor this purpose. In a series of work [74,48,140], Maple’s built-infsolve command, which numerically approximates the roots of analgebraic function, was used in the second law analysis of variousproblems. This command employs numerical techniques, such asNewton–Raphson, bisection, secant and fixed-point iterationmethods. Upon calculation of the coefficients, by incorporatingthem into the analytical formulation for the general solution, thetemperature field and consequently the entropy generation ratescan be readily calculated.

4. Conclusion

In recent years, entropy generation in solids has attracted con-siderable attention from the research community. The literature onentropy generation in thermal systems with solid structures wasreviewed in the current work. The importance of the second lawof thermodynamics in the analysis of these solid systems was pre-sented and discussed. Following a concise mathematical discussionof the entropy generation in solid media, the literature pertainingto the entropy generation in pure conductive media, thermal sys-tems with conjugate heat transfer and porous media, werereviewed. A relatively new technological manifestation of conduc-tion dominated processes was detected in thermoelectric devicesand the recent energy and exergy analysis in this field werereviewed. Subsequently, the review was extended to the solutionmethod of the differential equations, required to address the most

significant problems in this field. It was observed that there exists,significant potential for further research on the second law analy-ses of solid systems. Combined analytical–numerical and thenumerical methods were identified as the techniques that offerthe most attractive features for the foreseeable future of entropygeneration modeling.

Moreover, authors would like to emphasize that as mentionedthorough the review, the entropy generation studies for purelyradiative systems are very rare [30,31]. Since the primary heattransport modes in electronic systems are conduction and radia-tion [141], there is an urgent need for the consideration of entropygeneration in radiative environments. As it has been stated inBright and Zhang investigation [31], the general entropy genera-tion equation for a radiative environment is similar to a conductive

system. However, the intrinsic feature of radiative environmentespecially at micro and nanoscales, involves phonon transportequations which may hinders the solution of energy equations.This challenging and yet exciting task remains before the researchcommunity to be tackled.

Acknowledgements

This work was supported by the Hong Kong Research GrantsCouncil (project no. CityU 11216815) and NSAF (Grant no.U1330132).

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Entropy generation in thermal systems with solid structures – A concise review