International Scholarly Research NetworkISRN Chemical EngineeringVolume 2012, Article ID 240297, 6 pagesdoi:10.5402/2012/240297

Research Article

Analysis of Entropy Generation Minimization inCircular Porous Fins

Seyfolah Saedodin,1 Siamak Sadeghi,2 and Majid Shahbabaei2

1 Science and Research Branch, Department of Mechanical Engineering, Islamic Azad University, P.O. Box 35131-3711, Semnan, Iran2 Department of Mechanical Engineering, Islamic Azad university, Semnan Branch, P.O. Box 35131-3711, Semnan, Iran

Correspondence should be addressed to Siamak Sadeghi, siamak.sadeghi1986@gmail.com

Received 27 July 2012; Accepted 5 September 2012

Academic Editors: A. Gil and A. M. Seayad

Copyright © 2012 Seyfolah Saedodin et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This work introduces a simple method of exergy analysis in a typical circular porous fin. The entropy generation of anythermodynamic system provides a useful measure of the extent of irreversibility. The irreversibility causes the loss of useful work(exergy) in the system and hence the loss of exergy has to be minimized. Entropy generation is a parameter that quantifies theloss of exergy. Circular fins are relatively good heat transfer augmentation features with superior aerodynamic performance andas a result find application in some solar air heaters. In this paper, the entropy generation in a circular porous fin is calculatedand its performance is compared with respect to entropy generation. Also shown in porous fins, with increase of porosity. Theentropy generation number Ns will increase; also states with porosity have higher entropy generation number than Ns states withnonporosity (ϕ = 0). Also at higher Reynolds number the effect of the ReL on NsT is negligible, but at lower Reynolds number thevariation of the NsP is negligible. Also we can see that with increased porosity (ϕ), the entropy generation (Sgen) will decrease. Theentropy generation is calculated for circular porous fins with mass constraint.

1. Introduction

The term extended surfaceis used to describe a system inwhich the area of a surface is increased by the attachmentof fins. A fin accommodates energy transfer by conductionwithin its boundaries, while its exposed surfaces transferenergy to the surroundings by convection or radiation orboth. Fins are commonly used to augment heat transferfrom electronic components, automobile radiators, engineand compressor cylinders, control devices, and a host ofother applications. Fins are used to enhance convective heattransfer in a wide range of engineering applications and offera practical means for achieving a large total heat transfersurface area without the use of an excessive amount ofprimary surface area. Fins are commonly applied for heatmanagement in electrical appliances such as computer powersupplies or substation transformers. Other applicationsinclude IC engine cooling, such as fins in a car radiator.Heat and mass transfer through saturated porous media is animportant development and an area of very rapid growth in

contemporary heat transfer research. Although the mechan-ics of fluid flow through porous media has preoccupiedengineers and physicists for more than a century, the studyof heat transfer has reached the status of a separate field ofresearch during the last three decades [1]. The descriptionof heat and fluid flow through a porous medium saturatedwith fluid (liquid or gas) is based on a series of specialconcepts that are not found in the pure-fluid heat transfer.Examples are the porosity and the permeability of the porousmedium and the volume-averaged properties of the fluidflowing through the porous medium. The porosity of theporous medium is defined as

ϕ = void volume contained in porous medium sampletotal volume of porous medium sample

.

(1)

The increased awareness that the world’s energy resourcesare limited has caused many countries to reexamine theirenergy policies and take drastic measures in eliminatingwaste. It has also sparked interest in the scientific community

2 ISRN Chemical Engineering

to take a closer look at the energy conversion devices and todevelop new techniques to better utilize the existing limitedresources. The first law of thermodynamics deals with thequantity of energy and asserts that energy cannot be createdor destroyed. This law merely serves as a necessary toolfor the bookkeeping of energy during a process and offersno challenges to the engineer. The second law, however,deals with the quality of energy. More specifically, it isconcerned with the degradation of energy during a process,the entropy generation, and the lost opportunities to dowork. The second law of thermodynamics has proved tobe a very powerful tool in the optimization of complexthermodynamic systems. The second law of thermodynamicsuses an exergy balance for the analysis and the design ofthermal systems. The importance of this thermal designtechnique in the general area of heat transfer augmentationand energy conservation is fully recognized by the heattransfer community [2, 3]. The traditional approach to theoptimization of fins consists of minimizing the consumptionof the fin material for the execution of a specified heattransfer task. More than a half-century ago, Schmidt statedintuitively that a two-dimensional fin must have a parabolic-law based cross-sectional profile if it is to require the leastmaterial (volume) for a certain heat transfer rate, and thisdesign principle was later proved by [4] who worked on theformulation of variational calculus [5]. This design principlehas been steadily brought closer to the realities of finmanufacturing and heat exchanger operation by a numberof contributors who have analyzed the role of radiation, two-directional heat transfer (curvature) temperature-dependentthermal conductivity and variable heat transfer coefficient[6–10]. Entropy generation in a thermodynamics system isdefined as the difference between the entropy change of thesystem and the net entropy transport into the system and isa path function of the thermodynamics system. Accordingto the second law of thermodynamics, the entropy changeof a system is always greater than or equal to the netentropy transport into the system. The first and second lawsof thermodynamics, taken together, state that the entropygenerated by any engineering system is proportional to thework lost (destroyed) irreversibly by the system. Truth isexpressed concisely as the Gouy-Stodola theorem [11]:

Wlost = T0

∑

total

Sgen, (2)

where Wlost is the lost available work (lost exergy) [12], T0

is the absolute temperature of the environment, and Sgen isthe entropy generated in each compartment of the system.Equation (2) implies the thermodynamics irreversibility(entropy generation) of each system component contributesto the aggregate loss of available work in the system (Wlost).Circular fins serve as components in a wide range ofheat transfer applications. Therefore, in order to conserveavailable work (exergy), it is necessary to approach the designof such fins from the point of view of entropy generationminimization. The trade-off between heat transfer and fluidfriction is a classical dilemma in heat exchanger design[3, 5]. The irreversibility minimization philosophy placestrade-off on a solid foundation, as heat transfer and fluid

qb

Tb

D

L

U∞

Figure 1: Porous fin nomenclature.

drag are both mechanisms for entropy generation. Using thebulk level of entropy production, [13] found the optimaldesign of radial fin geometry with respect to Reynoldsnumber in both laminar and turbulent convection. [14]showed a numerical code in order to evaluate the effects ofdifferent parameters of combustion in porous media, thatthe governing equations including Navier-stokes, the solidand gas energy and the chemical species transport equationsare solved using a multistep reduced kinetic mechanism.In [15], EGM (entropy generation minimization) approachis implemented for the analysis of optimal thermal mixingand temperature uniformity due to natural convection insquare cavities filled with porous medium for the materialprocessing application.

2. Materials and Methods

2.1. Analysis Using the First and Second Laws of Thermo-dynamics. The entropy generated by a single porous fin incross-flow can be evaluated based on the general modelpresented in Figure 1. A circular porous fin of arbitrary cross-section is suspended in a uniform stream with velocity U∞,temperature T∞, and solar radiation qr . The heat transferqb is driven by the temperature difference between the finbase Tb and the free stream, T∞. In addition, the cross-flowarrangement is responsible for a net drag force FD, which istransmitted through the porous fin. At steady state, by usingfirst and second laws of thermodynamics we have:

qb + qr − qfluid + FDU∞ = 0, (3)

Sgen = qfluid

T∞− qb

Tb− qr

T∞≥ 0. (4)

The following assumptions are made to solve this problem.

(i) The porous medium is isotropic and homogenous.

(ii) The porous medium is saturated with single-phasefluid.

(iii) Physical properties of both fluid and solid matrix areconstant.

(iv) The temperature inside fin is only function of x andvelocity inside fin is very low.

(v) There is no temperature variation across the finthickness.

(vi) The solid matrix and fluid are assumed to be at localthermal equilibrium with each other.

ISRN Chemical Engineering 3

(vii) The interactions between the porous medium and theclear fluid can be simulated by the Darcy formula-tion.

(viii) All conditions are the same as flow around a solid finand only variable parameter is porosity.

2.2. Circular Porous Fin. Circular fin is one of the simplestgeometries, since the heat transfer and drag force depend ononly two dimensions, the length L and the diameter of thecircular cross-section D. In (3) and (4), qb is the amountof heat which is transferred from the base wall to the fin,qfluid is the amount of heat, which is transferred from porousfin to the rest of the fluid, and qr is the amount of heat,which is transferred from solar to porous fin. The free streamtemperature and free stream velocity are taken as T∞ and U∞,respectively. The entropy generation in the heat transfer andfluid flow arrangement is Sgen. By combining (3) and (4),qb and qr can be eliminated and the following equation isobtained:

Sgen = qbT∞[1 + T∞/θb]

+FDU∞T∞

. (5)

The relationship between the fin base heat flux and the steamtemperature difference is (Kern 1980)

θb = qb(π/4)Kf D2m tanh(ml)

. (6)

With the specification of Sgen and qb relations essentially, theentropy generation number Ns associated with the fin fordesign purposes can be determined by [16]:

Ns =Sgen(

q2bU∞/kνT2∞

) . (7)

From the Darcy’s model for porous media, we have

U∞ = gkβ

ν(Tb − T∞). (8)

Also the effective thermal conductivity of the porous fin isgiven by

Keff = ϕk f +(1− ϕ

)ks. (9)

Substitution of (5) and (6) into (7) gives

Ns = 1

(Kr)(π/2)(ReL)(λ)(T/T∞) + (π/2)(ReL)(Nu)0.5(Kr)(Kf

)2tanh

[2(Nu)0.5(1/Kr)

0.5] + B(ReL)2

(1Kr

)2(1λ

)2

(CD). (10)

Or

Ns = NsT + NsP, (11)

whereKr = Keff/K f , the dimensionless number, is the ratio ofporous fin and fluid thermal conductivities. The slendernessratio (λ = 1/L) is defined as the ratio between the lengthand characteristic dimension of the circular fins and B is aconstant value. The equation for entropy generation numberfor circular porous fin is given by (10) respectively. In thisequation, the first item indicates the entropy generationnumber due to temperature difference, NsT , but the seconditem indicates NsP. The Nusselt number and drag coefficientcan be evaluated from the results developed for a singlecircular cylinder in cross flow and are given by (13a) and(13b). From [17] one has:

B = ρν3T∞πD3(Tb − T∞)2 (12)

1 ≤ ReL ≤ 40, Nu = 0.657ReL0.4,

40 ≤ ReL ≤ 1∗ 103, Nu = 0.447ReL0.5,

1∗ 103 ≤ ReL ≤ 2∗ 105, Nu = 0.228ReL0.6,

2∗ 105 ≤ ReL ≤ 1∗ 106, Nu = 0.067ReL0.7.

(13a)

From [18], one has

1 ≤ ReL ≤ 4, CD = 10ReL−0.6,

4 ≤ ReL ≤ 4∗ 103, CD = 5.484ReL−0.246,

4∗ 103 ≤ ReL ≤ 2∗ 105, CD = 1.1.

(13b)

2.3. Optimization. The first step to optimize a thermody-namic system is to determine a proper objective function.The next step is to minimize this objective function whichcan be done using a number of different numerical, analyticaland graphical methods. Using the equation presented in,Section 2.2, the objective function is optimized. Here, theslenderness ratio is chosen as objective function. Minimiza-tion of Ns with respect to λ is achieved in a straightforwardmanner by solving (12) as follows:

∂Ns

∂λ= 0. (14)

3. Results

The engineering significance of (10) is that the optimumfin length can be calculated immediately, provided ReL isspecified. Substituting (10) into (8), it is possible to obtainthe minimum Ns for circular porous fins, corresponding tooptimum pin length, Ns(λopt, ReL) as shown in Figure 2.The optimum slenderness ratio of the circular porous finto produce minimum entropy generation number variousvalues of ReL.

4 ISRN Chemical Engineering

0

50

100

150

200

250

300

2000 4000 6000 8000 10000 12000

φ = 0.9, Kr = 80, B =

λ opt

ReL

7E − 09

Figure 2: Optimum slenderness ratio of the circular porous fin forminimum entropy.

0

1

2

3

4

2000 4000 6000 8000 10000 12000

λ = 10, B = 7E − 09

= 0.8

= 0.5

= 0.3

ϕ

ϕ

ϕ

Kr = 550

Kr = 400

Kr = 160

Ns

(E−

06)

ReL

Figure 3: Variation of entropy generation number with Reynoldsnumber of circular porous fin with variation of values ϕ.

Figure 3 shows the entropy generation number Ns forcircular porous fin with the variation of values of ϕ and ReL.The values of ϕ were varied between 0.3, 0.5, and 0.8, whilekeeping λ = 10. From Figure 3, we can see that by increasingporosity ϕ, in low Reynolds number, the entropy generationnumber Ns will increase. But in high Reynolds number, theentropy generation due to the fluid friction is approximatelyconstant.

Figure 4 shows comparison between fins with porosityand without porosity. It is observed that in fins with porosity(ϕ = 0.9), the entropy generation number is more thanwithout porosity (ϕ = 0). Also in the state of ϕ = 0.9, in highReynolds due to the fluid friction, Ns will increase.

In Figures 5 and 6, the variation of NsT and NsP withrespect to the characteristic Reynolds number for the circularporous fins is presented. At high Reynolds number the effectof the ReL on NsT is negligible and the values of the NsP atlower Reynolds number are negligible.

Figure 7 shows variation entropy generation (Sgen) withReynolds number and variable values of ϕ, Kr . From Figure 7we can see that with increased that porosity (ϕ), the entropygeneration will decrease. Also with the increase of Reynolds

λ = 10, B = 7E − 09

2000 4000 6000 8000 10000 12000

0.2

0.4

0.6

0.8

1

= 0.9

= 0

Ns

(E−

06)

ϕ

ϕ

Kr = 80

Kr = 800

ReL

Figure 4: Comparison between fins with porosity and withoutporosity.

0.5

1

1.5

2

2000 4000 6000 8000 10000 12000

λ = 100

= 0

= 0.9

ϕ

ϕ

Kr = 80Kr = 800

ReL

NsT

( E−

06)

Figure 5: Variation of NsT with Reynolds number and ϕ, Kr .

number, the values of entropy generation (Sgen) at all them,will increase.

4. Discussion and Conclusion

A quiet revolution is taking place in thermodynamics and itamounts to the closing of the gap between thermodynamicsand heat transfer. The method and field that unite theseclassical disciplines is entropy generation minimization.Today EGM is an established method in both fundamentaland applied heat transfer. This work introduces a simplemethod of analysis to study exergy in porous fin. Byusing the first and second laws of thermodynamics we willcontribute to entropy generation. The entropy generation ofany thermodynamics system provides a useful measure of theextent of irreversibility. Entropy generation is one parameterthat quantifies the loss of exergy. In this paper, the entropygeneration of circular porous fin in cross-flow of air andradiation is calculated and their performances are comparedwith respect to entropy generation. This paper shows thatthe entropy generation number Ns has a clear minimumwith respect to Reynolds number ReL. In porous fins, with

ISRN Chemical Engineering 5

2000 4000 6000 8000 10000 12000

= 0

= 0.9

0

50

100

150

200

250λ = 100, B = 7E − 09

ϕ

ϕ

Kr = 80Kr = 800

NsP

(E−

06)

ReL

Figure 6: Variation of NsP with Reynolds number and ϕ, Kr .

2

4

6

8

10

12

S gen

2000 4000 6000 8000 10000 12000

λ = 100, B = 7E − 09

= 0, = 800= 0.3, = 550

= 0.5, = 400= 0.9, = 80

ϕ

ϕ

ϕϕ

Kr Kr

KrKr

ReL

Figure 7: Variation of Sgen with Reynolds number and ϕ, Kr .

increase of porosity, the entropy generation number Ns

will increase, also states of porosity have higher entropygeneration number Ns than states of non-porosity. Also athigher Reynolds number, the effect of the ReL on NsT isnegligible, but at lower Reynolds number the variation of theNsP is negligible. Also we can see that with increased porosity(ϕ), the entropy generation (Sgen) will decrease.

Nomenclature

A : Cross-sectional area of the fins in mD: Diameter of the circular cross-sectionFD: Drag force exerted by the cross-flow over the

fin in Nk: Thermal conductivity of the working fluid in

W/m·KL: Characteristic dimension of the fins, D for

circular finsl: Length of the fins in mKr : Thermal conductivity ratio, (keff/k f )Ns: Entropy generation numberθb: Base temperature difference, (Tb − T∞)

qb: Heat transfer through the base of the fin in wq: Heat transfer to the free stream in wSgen: Entropy generation in W/KT : Free stream temperature in KTb: Base fin temperature in KU∞: Free stream velocity in m/secWlost: Power lost due to the irreversibility in Wν: Kinematic viscosityρ: DensityΦ: Porosityβ: Coefficient of volumetric thermal expansiong: Gravity constantReL: Reynolds number in Length of finNsT : Entropy generation number by Temperature

differenceNsP: Entropy generation number by Drag force.

Subscripts

s: Solid propertiesf : Fluid propertieseff: Porous properties.

References

[1] D. A. Nield and A. Bejan, Convection in Porous Media,Springer, New York, NY, USA, 2nd edition, 1999.

[2] A. E. Bergles, R. L. Webb, G. H. Junkhan, and Jensen,“Bibliography on augmentation of convective heat and masstransfer,” Report HTL-19 ISU–ERI–AMES–79206, lowa StateUniversity, 1979.

[3] G. H. Junkhan, A. E. Bergles, and R. L. Webb, “Researchworkshop on energy conservation through enhanced heattransfer,” Report HTL-21 ISU-ERI-AMES-80063, lowa StateUniversity, 1979.

[4] R. J. Duffin, “A variational problem relating to cooling fins,”Journal of Mathematics and Mechanics, vol. 8, pp. 47–56, 1959.

[5] D. Poulikakos and A. Bejan, “Fin geometry for minimumentropy generation in forced convection,” Transactions of theASME, vol. 104, no. 4, pp. 616–623, 1982.

[6] C. J. Maday, “The minimum weight one-dimensional straightfin,” ASME Journal of Engineering for Industry, vol. 96, no. 1,pp. 161–165, 1974.

[7] S. Guceri and C. J. Maday, “A least weight circular cooling fin,”ASME Journal of Engineering for Industry, vol. 97, no. 4, pp.1190–1193, 1975.

[8] P. Razeols and K. Imre, “The optimum dimensions of circularfins with variable thermal parameters,” ASME Journal of HeatTransfer, vol. 102, pp. 420–425, 1980.

[9] D. Q. Kern and A. D. Kraus, Extended Surface Heat Transfer,McGraw-Hill, New York, NY, USA, 1980.

[10] A. D. Kraus and A. D. Snider, “New parameterizations for heattransfer in fins and spines,” ASME Journal of Heat Transfer, vol.102, no. 3, pp. 415–419, 1980.

[11] J. Szargut, “International progress in second law analysis,”Energy, vol. 5, no. 8-9, pp. 709–718, 1980.

[12] J. Kestin, “Availability: the concept and associated terminol-ogy,” Energy, vol. 5, no. 8-9, pp. 679–692, 1980.

[13] B. N. Taufiq, H. H. Masjuki, T. M. I. Mahlia, R. Saidur, M.S. Faizul, and E. Niza Mohamad, “Second law analysis foroptimal thermal design of radial fin geometry by convection,”

6 ISRN Chemical Engineering

Applied Thermal Engineering, vol. 27, no. 8-9, pp. 1363–1370,2007.

[14] M. Bidi, M. R. H. Nobari, and M. S. Avval, “A numericalevaluation of combustion in porous media by EGM (EntropyGeneration Minimization),” Energy, vol. 35, no. 8, pp. 3483–3500, 2010.

[15] R. S. Kaluri and T. Basak, “Entropy generation due to naturalconvection in discretely heated porous square cavities,” Energy,vol. 36, no. 8, pp. 5065–5080, 2011.

[16] A. Bejan, Entropy Generation Minimization, CRC Press, NewYork, NY, USA, 1996.

[17] A. Zukauskas, “Heat transfer from tubes in crossflow,”Advances in Heat Transfer, vol. 8, pp. 93–160, 1972.

[18] B. Gebhart, Heat Transfer, McGraw-Hill, New York, NY, USA,1971.

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