Entropy Generation in an Asymmetrically Cooled Slab withTemperature-Dependent Internal Heat Generation

A. Aziz1 and Waqar A. Khan2

1Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA, USA

2Department of Engineering Sciences, PN Engineering College, National University of Sciencesand Technology, Karachi, Pakistan

The paper presents an entropy generation analysis for steady conduction in aslab with temperature-dependent volumetric internal heat generation. The slab expe-riences asymmetric convective cooling on its two faces. The exact analytical solutionfor the temperature distribution is used to compute dimensionless local and totalentropy generation rates in the slab. The total entropy generation rate depends on fivedimensionless parameters: reference heat generation temperature Q, the heat genera-tion–temperature variation parameter a, the temperature asymmetry parameter λ, andBiot numbers Bi1 and Bi2. Graphs illustrating the effect of these five parameters on thelocal and total entropy generation rates are presented and discussed. It is found thatthe total entropy generation in the slab can be minimized with a suitable choice of thecooling parameters. The paper corrects the flawed entropy results published recently.The present results for the special case of uniform internal heat generation confirm theresults presented in a 2003 paper. © 2012 Wiley Periodicals, Inc. Heat Trans AsianRes, 41(3), 260–271, 2012; Published online 10 March 2012 in Wiley Online Library(wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20404

Key words: local and total entropy generation rates, slab, temperature-depend-ent internal heat generation, asymmetric cooling

1. Introduction

The elementary analysis for one-dimensional steady state conduction in a slab with constantand uniform internal heat generation appears in every undergraduate heat transfer textbook, forexample, Bergman et al. [1]. The temperature distribution in the slab for the more general case oftemperature-dependent internal heat generation has been presented by Gebhart [2]. He also discussesthe criterion that must be met to avoid thermal instability which may lead to thermal destruction ofthe slab due to excessively high temperatures. Aziz and El-Ariny [3] refined the basic solution givenin Gebhart [2] by considering exponential and power law type of heat generation dependence ontemperature. Additionally, they included the effect of temperature-dependent thermal conductivity ofthe slab. Their approach was to use a two-parameter perturbation method to deal with the twononlinearities, one associated with the heat generation term and the other appearing in the thermal

© 2012 Wiley Periodicals, Inc.

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conductivity expression. The perturbation solutions were compared with the numerical solutionsgenerated by converting the boundary value problem into an initial value using the group transfor-mation method [4] and subsequently solving the initial value problem using the standard Runge–Kuttanumerical scheme. The perturbation solutions were found to be accurate for a range of valuesparameters characterizing the heat conduction process. The same problem was solved by Aziz andNa [5] using a non-iterative procedure called general parameter mapping.

The studies cited so far had one common feature, namely, the temperatures at the exposedsurfaces of the slab were assumed constant and identical. The use of this restrictive boundary conditionlimits the applicability of the solution for practical engineering situations where the two surfaces ofthe slab may be exposed to two different convective environments and in that case the surfacetemperatures on the two sides would be different. Zivic et al. [6] analyzed in detail the local and totalentropy generation rates in one-dimensional steady conduction in a plane with no internal heatgeneration and fixed boundary temperatures. Ibanez et al. [7] and Bautista et al. [8] considered entropygeneration in a slab with uniform internal heat generation and asymmetric convective cooling on thetwo faces. They concluded that the total entropy generation in a slab can be minimized for certaincombinations of the cooling parameters. A similar conclusion was reached by Ibanez et al. [7] forviscous flow between parallel plates with asymmetric convective cooling at the plate and by Aziz [9]in the study of a pressure gradient assisted Couette flow with the asymmetrical convective boundaryconditions at the plates. More recently, Aziz and Makinde [10] studied one-dimensional steadyconduction in a slab with temperature-dependent internal heat generation and asymmetric surfacecooling conditions. They obtained analytical results for the temperature distribution in the slab,established the thermal stability criterion, and evaluated the entropy generation results. While theresults for the temperature distribution and the thermal stability criterion are correct, their results forthe entropy generation are flawed as pointed out in a private communication to the first author byDelhaye [11]. The purpose of the present paper is to present the results for the local and the totalentropy generation based on the correct expression for the volumetric rate of entropy generation in aheat generating slab. Because the temperature distribution and stability criterion are discussed by Azizand Makinde [10], only the entropy generation results will be presented and discussed here. However,for the sake of completeness, the analytical results for the temperature distribution and thermalstability criterion will be included. Based on the entropy generation results, it is concluded that thedimensionless local and the total entropy generation rates in the slab are minimized for certaincombinations of the cooling parameters.

Nomenclature

A: slab area normal to heat flow or surface area of the solid, m2

a: internal heat generation–temperature dependence parameter, dimensionlessBi1: Biot number on the left face of the slab, dimensionlessBi2: Biot number on the right face of the slab, dimensionlessh1: convection heat transfer coefficient on the left face of the slab, W/m2Kh2: convection heat transfer coefficient on the right face of the slab, W/m2Kk: thermal conductivity of the slab material, W/mKL: slab thickness, mNs: local entropy generation rate, dimensionlessNt: total entropy generation rate, dimensionless

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n: unit vector normal to the surfaceQ: reference internal heat generation parameter, dimensionlessq.: volumetric internal heat generation rate, W/m3

q.

0: reference volumetric heat generation rate, W/m3

q: heat flux vector, W/m2

qxgg: heat flux along the x direction, W/m2

s: specific entropy, J/kg⋅KS.ggg: volumetric entropy generation rate, W/m3 K

S.: entropy generation rate, W/K

T: temperature, KT1: coolant temperature on the left face of the slab, KT2: coolant temperature on the right face of the slab, KV: volume of the solid, m3

X: dimensionless coordinatex: coordinate measured from the left face of the slab, m

Greek Symbols

α: parameter associated with internal heat generation, K–1

θ: dimensionless temperatureλ: coolant temperature ratio, T2/T1

2. Temperature Distribution and Thermal Stability Criterion

Consider a one-dimensional slab of thickness L made of a material with uniform and constantthermal conductivity k as illustrated in Fig. 1. The slab is experiencing internal heat generation whichis a linear function of temperature, that is,

where q.

0 and α are constants. The left face (x = 0) is cooled by a convection process characterized bythe coolant temperature T1 and heat transfer coefficient h1. The right face of the slab is cooled by aconvection process characterized by the coolant temperature T2 and heat transfer coefficient h2. Theone-dimensional steady heat conduction equation is

(1)

(2)

Fig. 1. Asymmetrically cooled slab with temperature-dependent internal heat generation.

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The boundary conditions are

With the introduction of the following dimensionless variables

Equations (2)–(4) now become

Equation (6) subject to the boundary conditions (7) and (8) has been solved analytically byAziz and Makinde [10]. Their final result is

where

and

It is evident that θ is a function of X, Q, a, Bi1, λ, and Bi2. It would be particularly interestingto study how the location of maximum temperature in the slab is affected by parameters Q, a, Bi1, λ,and Bi2. This point is discussed in detail in Aziz and Makinde [10].

For the temperature in the slab to remain finite, the following thermal stability criterion wasderived by Aziz and Makinde [10]:

(5)

(3)

(4)

(7)

(8)

(9)

(10)

(11)

(13)

(12)

(6)

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The reader may refer to Aziz and Makinde [10] for further discussion on the thermal stability criterion.

3. Entropy Generation

Since a formal derivation of the expression for the volumetric entropy generation in a heatgenerating solid is not available elsewhere, we derive this expression by applying the Classius–Duhemstatement of the second law of thermodynamics. Consider a solid of volume V and surface area A withunit vector n pointing normally outward as shown in Fig. 2. Let q denote the heat flux vector, s denotethe entropy per unit volume, and S

.ggg the volumetric rate of entropy generation. Using the Classius–

Duhem statement of the second law, the entropy balance equation may be written as

For steady state, the time-dependent term on the left-hand side of Eq. (14) vanishes. Further, applyingGauss’ divergence theorem, the surface integral on the right hand of the integral may be written interms of the volume integral, that is,

Using Eq. (15) to replace the surface integral on the right-hand side of Eq. (14), we obtain

Because each integral in Eq. (16) is a volume integral, it follows that

For the present one-dimensional problem, Eq. (17) may be written as

Using Fourier’s law, Eq. (18) becomes

If the thermal conductivity is assumed to be a constant, then Eq. (19) takes the following form.

(14)

(15)

(16)

(17)

(18)

(19)

Fig. 2. Solid of volume V and surface area A with unit vector n outwardly normal to the surface.

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Using the energy equation, i.e., Eq. (2) in Eq. (20), the latter reduces to

which is the well-known result derived by Bejan [12] for one-dimensional steady conduction with nointernal heat generation. Thus we have formally proved that Eq. (21) is valid irrespective of whetherthere is internal generation in the solid or not.

A comparison of Eq. (21) with Eq. (16) in Aziz and Makinde [10] shows that their result forthe volumetric entropy generation rate is incorrect.

With the introduction of dimensionless variables defined by Eq. (5), the dimensionless localvolumetric entropy generation rate Ns is given by

and the total dimensionless entropy generation rate Nt is given by

4. Results and Discussion

Local entropy generation

Figure 3 illustrates how the local entropy generation rate in the slab responds to the changesin the coolant temperature ratio λ and the parameter a, the latter characterizing the dependence ofinternal heat generation on temperature. Because Bi1 = Bi2, the convective heat transfer coefficientson the two faces of the slab are identical, i.e., h1 = h2. Each curve manifests the same pattern ofvariation. The local entropy generation rate decreases as X increases, reaches a zero value at someintermediate value of X, and then increases again as X = 1 is approached. This is understandablebecause as X increases, the local temperature increases and the local temperature gradient decreases,the combined effect causing a decrease in the local entropy generation rate in accordance with theentropy generation, i.e., Eq. (22). The point of zero entropy generation coincides with the location ofmaximum temperature where the temperature gradient is zero. Although the convective heat transfercoefficients on the two faces of the slab are the same, the coolant temperatures are identical only whenλ = 1 or T1 = T2. In this case, the local entropy generation is symmetrical about the centerline (X =0.5) of the slab. However, for λ = 0.5, T2 = 0.5 T1. The lower coolant temperature on the right face ofthe slab extracts a larger amount of internally generated heat through the right face of the slab. Thisinduces larger temperature gradients accompanied by lower temperatures. This situation leads tohigher entropy generation rates in the right half of the slab compared with those in the left half. Fora given coolant temperatures ratio λ, the point of zero entropy generation rate shifts slightly towardthe left face of the slab as the internal heat generation becomes more strongly dependent on the localtemperature, i.e., as the parameter a increases.

(20)

(21)

(22)

(23)

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Figure 4 highlights the effect of internal heat generation parameters Q on the local entropygeneration rate when the Biot numbers on the two faces of the slab are identical. Because thetemperature of the coolant on the right face of the slab is lower than on the left face (T2 = 0.5 T1), onewould expect, despite identical Biot numbers, larger heat flow through the right face of the slab whichis accomplished with larger temperature gradients and lower temperatures in the right half of the slab.This combination of larger temperature gradients and lower temperatures gives rise to larger entropygeneration rates in the right half of the slab compared with those in the left hand of the slab, as wasthe case in Fig. 3. As the internal heat generation parameter Q increases, a larger proportion of the

Fig. 3. Local entropy generation rate in a slab: effects of parameters λ and a. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

Fig. 4. Local entropy generation rate in a slab: effects of parameters Q and a. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

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additional heat generated in the slab migrates toward the right face of the slab because the coolant onthe right face of the slab is half the temperature of the coolant on the left face of the slab. As aconsequence, one notices higher local entropy generation in the right half of the slab than in the lefthalf of the slab. As the internal heat generation parameter Q increases, the entropy generation ratesalso increase correspondingly as the temperatures decrease and the temperature gradients increase toaccommodate the removal of additional heat generated in the slab.

In Fig. 5, we illustrate the effect of unequal Biot numbers on the local entropy generation ratein the slab. While keeping the parameters Q, λ, Bi1 fixed, we have shown the local entropy generationrates for two values of Bi2, in other words, for two values of the convective heat transfer coefficientson the right face of the slab. For Bi2 = 0.8, the convective heat transfer coefficient h2 is larger thanthe convective heat transfer coefficient h1 (Bi1 = 0.5). The lower coolant temperature and the higherheat transfer coefficient on the right face of the slab gives rise to larger heat dissipation from the rightface of the slab compared with that from its left face. Thus the entropy generation rates in the righthalf of the slab exceed those in the left half of the slab. The zero entropy generation rate (zerotemperature gradient) and the maximum temperature occurs at X ≈ 0.35. For Bi2 = 0.3, the convectiveheat transfer coefficient h1 (Bi1 = 0.5) is larger than the convective heat transfer coefficient h2, a largerportion of the heat generated flows through the left face of the slab inducing larger entropy generationrates in the left half of the slab compared with those occurring in the right half of the slab. The pointof zero entropy generation rate occurs at X ≈ 0.60.

Figure 6 also shows the effect of unequal Biot numbers on the local entropy generation ratein the slab except that now Bi2 is kept fixed and two values of Bi1 are chosen for study. Consideringthe curves for Bi1 = 0.5, we note that the heat transfer coefficients h1 and h2 are equal but the coolanttemperature T2 on the right face of the slab is one-half of the coolant temperature T1 on the left faceof the slab, and this drives more heat flow through the right half of the slab producing a larger entropygeneration rate in it. Examining the curves for Bi1 = 1.0, we note that h1 = 2h2 and a larger portion of

Fig. 5. Local entropy generation rate in a slab: effects of parameters Bi2 and a. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

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energy generated in the slab flows through the left face of the slab, producing in turn higher entropygeneration rates in the left half of the slab compared with those in the right half of the slab.

Total entropy generation

The effect of various parameters on the total entropy generation rate is illustrated in Figs. 7–9.In Fig. 7, we keep the parameters Q and λ fixed and allow Bi1, Bi2, and a to vary. For given valuesof Bi1 and Bi2, the entropy generation rate increases as the parameter a increases from 0.1 to 0.5. This

Fig. 6. Local entropy generation rate in a slab: effects of parameters Bi1 and a. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

Fig. 7. Total entropy generation rate in a slab: effects of parameters Bi1, Bi2, and a. [Color figurecan be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

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is to be expected because the larger value of a gives rise to higher internal energy generation andconsequently higher heat dissipation from the two faces of the slab. Each of the six curves in Fig. 7shows that there is a unique combination of Biot numbers Bi1 and Bi2 that minimizes the total entropygeneration rate in the slab. In other words, asymmetric cooling can be designed so that the systemoperates under a minimum entropy generation condition.

When the value of parameter Q is increased from 1.0 to 1.5, as shown in Fig. 8, the totalentropy generation rate in the slab increases. Again, each curve in Fig. 8 exhibits a minimum indicating

Fig. 8. Total entropy generation rate in a slab: effects of parameters Bi1, Bi2, and Q. [Color figurecan be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

Fig. 9. Total entropy generation rate in a slab: effects of parameters Bi1, Bi2, and λ. [Color figurecan be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]

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that the total entropy generation rate can be minimized by a proper choice of the coolants Biot numbersBi1 and Bi2. In Fig. 9, we have shown the effect of changing the coolant temperature ratio λ on theentropy generation rate. Because the smaller value of λ leads to higher combined heat flows throughthe faces of the slab, the entropy generation rates are higher for λ = 0.3 than for λ = 0.5. Each curvein Fig. 9 also manifests a point of minimum entropy generation for a unique combination of Biotnumbers Bi1 and Bi2.

In order to confirm the accuracy of our numerical results, we plot in Fig. 10 our results for thespecial case of a slab with uniform (temperature-independent) internal heat generation and comparethem with those of Ibanez et al. [7]. The fact that their results, shown by different symbols, fall onour curves confirms the validity and accuracy of our numerical computations.

5. Conclusions

This study has shown that the total entropy generation in a heat-generating slab can beminimized by providing asymmetric cooling on the two faces of the slab. This is true whether theinternal heat generation is uniform (independent of temperature) or a linear function of temperature.The optimum combination of Biot numbers, Bi1, Bi2 for minimum entropy generation depends on theratio of the coolant temperatures, λ, the dimensionless slope of the internal heat generation versustemperature line, a, and the dimensionless internal heat generation rate at the reference temperature,Q. Although we have identified the existence of the minimum entropy generation points for the caseof internal heat generation varying linearly with temperature, it is plausible to conjecture that suchminima also occur for other forms of temperature dependence such as the power law and exponentialforms.

Fig. 10. Comparison of present results with Ibanez et al. [6] for a slab with uniform internal heatgeneration and asymmetric cooling. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com/journal/htj.]

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