Thermal analysis of a constructal T-shaped porous fin with simultaneous heat and mass transfer

Saheera Azmi Hazarika,Tuhin Deshamukhya,Dipankar Bhanja*,Sujit Nath

Department of Mechanical Engineering,National Institute of Technology Silchar,Assam 788 010,India

1.Introduction

Fins are an integralpart of any equipment which requires transfer of heat between solid surface and the ambient atmosphere.There has been a lotofresearch in this in the pastseveralyears.In an experimental analysis,Baby and Balaji[1]studied the performance of finned heat sinks made of aluminium used in portable electronic devices.The results showed that fins in heat sinks filled with PCM improve the performance of these devices.The heat transfer characteristics of PCM based pin fin heat sinks were studied experimentally by Baby and Balaji[2]by changing the volume fraction, fin numbers and power density.A hybrid optimization algorithm based on artificial neural network and Genetic Algorithm had been developed for maximizing the operating time for the n-eicosane based heatsink.Transientin-house experiments on a fin losing heat by natural convection have been conducted by Gnanasekaran and Balaji[3].The fin parameters“m”and the thermal diffusivity “α”were estimated separately by Bayesian approach while MCMC technique is used for sampling the parameters“m”and “α”.In every fin analysis problem,the determination of the optimum fin geometry is of paramount importance in order to obtain maximum bene fit from using fins and for implementing the fin in practical applications.In a numericalanalysis Sasikumar and Balaji[4]performed a numerical analysis of natural convection heat transfer and entropy generation from an array of vertical fins on a horizontal duct with turbulent fluid flow inside.Also,optimal dimensions of the fin system for maximum heat transfer rate per unit mass and for minimum total entropy generation rate were determined by using Genetic Algorithm.In another work[5]both the authors performed a holistic optimization study of convecting–radiating fin array,which stands vertically outside a rectangular duct by taking in to account,a fully developed turbulent flow inside the duct.The results showed convection to be significant compared to radiation.Also the arrays having shorter height showed better performance.

There are many applications of fins where they are employed to undergo simultaneous heat and mass transfer.Examples of such kind can be easily seen in a number of industrial applications such as airconditioning,refrigeration,chemical industries and so on.Working under moist condition results in deposition of moisture from the air to the fin surface due to the formation of humidity gradient.As the surface temperature of the fins in such cases is lower than the dew point temperature of the air,a thin deposition layer of moisture is observed at the fin surface.Based on this, fins can be classified under three categories:(1)Dry fin,(2)Partially wet fin,and(3)Fully wet fin.The first case is obtained when the fin maintains a temperature which is greater than the dew point temperature of the air being cooled.If the tip of the fin is greater than the dew point temperature while the base is at lower temperature we observe a case of partially wet fin.However in this work the authors are concerned with the third case,where the entire length of the fin remains at a temperature less than the dew point temperature of the air being cooled.There always exists a relationship between humidity ratio of saturated air adjacent to the fin surface and the corresponding fin surface temperature and thus it becomes necessary to analyse this relationship.Generally a linear relationship is considered between these two parameters by researchers to avoid or reduce the non-linearity in the equations.Kundu and Barman[6]presented an analysis for the performance and optimization of an annular fin assembly with a trapezoidal pro file under dehumidifying conditions by assuming that the humidity ratio varies linearly with fin surface temperature.Sharqawy and Zubair[7,8]in two separate works studied the efficiency of annular and straight fins ofdifferentpro files respectively undergoing heatand masstransferconsidering the fully wet fin condition.They reported a direct in fluence of atmospheric pressure on the efficiency of the fin.But a linear model can never be observed in real life situations,which is why the need for an accurate psychometric relation is required.Considering a cubic polynomial relationship between humidity ratio and fin temperature,Kundu and Lee[9]analysed the performance of wet fins of various geometries.Temperature dependent thermal conductivity and variable heat transfer co-efficient have been considered for this analysis.Kundu[10]in his work,in order to determine the optimum fin pro file of both fully wet and partially wet longitudinal fins,considered a nonlinear relationship between these two parameters.In another attempt to study the performance of wet fins analytically,Kundu[11]considered a cubic polynomial relationship between specific humidity and dry bulb temperature and analysed the performance of wet fins.In a comparative study Kundu and Miyara[12]established an analytical method for determining the performance of a fully wet fin by considering a cubic polynomialrelationship between the humidity ratio and temperature.The resultobtained in this work showed appreciable difference from that obtained by Sharqawy and Zubair[7].Kundu and Lee[13]forwarded an exactanalyticaltechnique to determine the minimumprofile shape of wet longitudinal fins for effectively transferring energy.A mathematical analysis of semi-spherical fin with simultaneous heat and mass transfer was conducted by Sabbaghiet al.[14]where they considered a closed form analytical solution to obtain the efficiency of the fin.In a work by Xuetal.[15]the efficiency ofwet fins has been evaluated by modifying the McQuiston model.The effect of condensate film moving over the fin surface on the heat transfer has been studied and this effect was included in the governing equation of the fin.

The last decade has seen a lot of advancement in the world of heat transfer and one such revolutionary concept is the introduction of porous fins[16,17],which by the virtue of their greater surface area facilitates higher heat transfer rate.Kundu and Bhanja[18]performed an analyticalstudy to understand the performance and optimumdesign analysis of porous fins.They did a comparative study of three models to understand the feasibility of each model.Hatami and Ganji[19]used least square method to study the temperature distribution of circular convective–radiative porous fins.The analytical results obtained are verified by numerical values.In another work Hatamiet al.[20]investigated the heat transfer through porous fins by three different analytical techniques for predicting the temperature distribution in a porous fin with temperature dependent internal heat generation.This study showed a direct dependence of temperature distribution on Darcy number and Rayleigh number.Bhanjaet al.[21]investigated the temperature distribution,performance parameters and heat transfer rate through a porous pin fin in natural convection condition.Kunduet al.[22]analysed straight fins of four different pro files.They found a significant increase in heat transfer rate through porous fins as compared to that of solid fins for a low porosity and high flow parameter.Das and Ooi[23]predicted five important parameters to meeta de finite temperature field through inverse analysis.Saedodin and Olank[24]made a comparison between the temperature distribution of porous fins and solid fins under natural convection.In an experimental study conducted by Kimet al.[25]the impact of porous fins on the heat transfer characteristics and pressure drop has been studied.They made a comparative study between the conventional louvered fins and porous fins on the basis of heat transfer and friction characteristics.Turkyilmazoglu[26]studied a case of exponential porous fins undergoing simultaneous heat and mass transfer by the virtue of temperature and humidity ratio differences.The results showed that the exponential wetporous fins have better efficiencies than the straightones.An investigation of circular porous fin of variable cross section undergoing simultaneous heat and mass transfer was done by Hatami and Ganji[27]where they used least square method to find an exact analytical solution.By employing the same method,Hatamiet al.[28]and Vahabzadehet al.[29]analysed fully wet semi-spherical porous fins and pin fins respectively.

The thirst for higher heat transfer through fins has prompted the researchers to take help of constructal designs.Since its inception,constructal theory[30],due to its versatility has been used in various fields of science and technology.This theory works on the basic principle ofproviding branching to the low resistive links,thereby increasing orimproving the performance of flow.Bejan and Almogbel[31]and Lorenziniet al.[32]analysed T-shaped assembly of fins.Hazarikaet al.[33]performed an analytical study on a fully wet constructal T-shaped fin to predict its performance and optimum design parameters.Kundu and Bhanja[34]considered a case ofvariable thermalconductivity ofa constructal fin to study its thermalperformance.Bhanja and Kundu[35]in an attemptto study the temperature distribution of constructal T-shaped porous fin used the Adomian decomposition method to solve the non linear governing equation.Chen[36]made a detailed study of constructal theory and its various applications.In another work,Chen and Feng[37]discussed about the multi-objective optimizations for fluid flow.A numberofnote worthy works have been published on the constructaloptimization of heat conduction problems[38–41]and convective fins[42–49].

The literature survey described above points out that a very few works have been done to analyse porous fins under dehumidifying conditions and till date,no researcher has taken the effort to analyse constructal porous fins under simultaneous heat and mass transfer.What makes this work different from the previously published works in this area is that the presentwork is devoted to determine analytically the temperature distribution and fin efficiency of a fully wet T-shaped porous fin.Also,keeping in mind the role played by various important parameters,the authors have performed a multivariable geometric optimization study which will help in proper selection of parameters for designing constructal fins under the stated conditions.There are some cases where different authors have solved non-linear fin equations by exact analysis[50–52].However,in the present work the nonlinearity and complexity of the problem is in such a magnitude that the traditional techniques may not be applicable to obtain an exact solution.Hence,the authors have employed a semi-analytical technique called Differential Transform Method(DTM)to solve the highly nonlinear governing equations.Moreover,due to unavailability of any experimental work on the present problem,the results of present analytical model are validated using a numerical model,based on Finite Difference scheme.

2.Mathematical Formulations

A steady state analytical study has been carried out on a T-shaped porous fin having uniform rectangular cross-section,attached to an isothermal surface at temperatureTBand exposed to moving moist air at constant dry bulb temperatureTA.The various geometric parameters involved along with the coordinate system taken are shown in Fig.1.The fin material being porous,the fluid in filtrates through it.Because of the presence of the pores,the effective thermal conductivity of the fin is reduced but this effect is counteracted by the simultaneous increase in the effective surface area for convection.Fully wet fin surface is considered for the analysis.For simplifying the analysis,the certain assumptions[16,33]are considered which are listed below.

■The porous medium is homogenous,isotropic and saturated with a single phase liquid.

■The solid matrix and the fluid are considered to be in local thermal equilibrium with each other.

■Both the fluid and the solid matrix have invariable physicalpropertiesexceptthe fluid density,which may vary and thisaffects the buoyancy term where Boussinesq approximation is applied.

■The interaction between the fluid and the porous media is expressed mathematically using the Darcy model.

■The temperature distribution in the fin is considered to be one dimensional since the fin thickness is small compared to the other dimensions.

■The heat transfer at the fin tip is considered to be negligibly small due to small cross section at fin tip.

■The thermal resistance associated with the thin film of condensed water on the fin surface is neglected.

■The convective heat transfer coefficient,the mass transfer coefficient and latent heat of condensation of water vapour are constants.

■There is no heat production within the fin and radiation heat exchange between the fins and the ambient are neglected.

■Contact resistances at the point of attachment of the stem with the primary wall and at the junction point between the stem and the flange are neglected.

Fig.1.Schematic of constructal T-shaped porous fin.

2.1.Governing differential equations

The heat transfer due to convection constitutes of two parts,one is the surface convection at the fin surface due to the moving air and the other part is due to the motion of the air through the pores.Applying the energy conservation principle to in finitely small elements perpendicular to the direction of heat flow in both the flange and stem parts of the fin(Fig.1),

where,Qcis the rate of heat transfer by convection from the ambient air to the solid part of the fin surface,Qmis the rate of heat transfer to the fin surface due to condensation ofmoisture on the fin surface andQc′is the rate of heat transferfrom the air thatpasses through the pores of the fin.The subscripts F and S stand for the flange and stem part respectively.

Thus,from Eqs.(1)and(2)the governing differential equations for the flange and the stem part of the porous T-shaped fin under simultaneous heat and mass transfer condition are

wherekEffis the effective thermal conductivity and it is de fined as[27]

Eq.(3)cannot be solved until a relationship is considered between the humidity ratio and the fin temperature.Though consideration of a linear relation simplifies the analysis,but this linear model may give impractical results.Hence for obtaining more practical results,the humidity ratio of saturated air adjacent to the fin surface and the corresponding fin surface temperature is related by the same cubic polynomial relationship,as considered in[9–12,33].

whereA0,A1,A2andA3are constant parameters whose values are 3.7444×10?3,0.3078×10?3/°C?1,0.46×10?5/°C?2and 0.4×10?6/°C?3respectively for the temperature range of 0°C≤T≤30°C.

Substituting Eqs.(4)–(8)in Eq.(3)yields the following equation

The dimensionless parameters involved in the above equation are listed below

2.2.Boundary conditions

The boundary conditions in non-dimensional from for Eq.(10)are

2.3.Temperature distribution

Eq.(10)is second order,non-homogenous and highly non-linear.Hence its solution using usual linearization techniques becomes cumbersome.So the present analysis adopts a semi-analytical transformation technique called Differential Transform Method(DTM),based on the Taylor's series of expansion.This method gives the solution as an in finite power series and the accuracy can be improved by taking more number of terms in the solution.The details of the DTM solution procedure can be found in the work of Zhou[54].A brief introduction to this method has been provided below.

The one-dimensional differential transform of a functionf(Z)at the pointZ=Z0is de fined as

Some of the mathematical operations,involved in the present analysis performed by Differential Transform Method are listed in Table 1.

Table 1The fundamental operations of the Differential Transform Method[54]

The differential transform of governing Eq.(10)can be expressed in the following way

Boundary condition(16a)is transformed as

AssumingF(0)=P,S(0)=QandS(1)=Rand puttingi,j=0,1,2,3,..…in Eq.(19),the following terms are obtained For the flange,

Similarly for the stem part,the terms obtained are as below

The inverse differential transforms ofF(i)andS(i)can be written as

Since the present method yields a solution that converges faster,a small number of terms(i,j=12)give sufficiently accurate results.Incorporating the values ofF(i)andS(j)(i,j=1,2,3,.…12)from Eqs.(22)–(40)in Eq.(41),(42)respectively,θF(X)and θS(Y)can be expressed in terms ofthree unknown parameters,i.e.,P,QandR.The values of these unknown parameters have been obtained by using the boundary conditions(16b–16d)and adopting the Newton–Raphson iterative procedure,satisfying a desired accuracy of 10?6.After getting the values ofP,QandR,the values of allF(i)andS(j)(i,j=1,2,3,.…12)are obtained and subsequently the temperature distribution in the flange and the stem has been determined using Eqs.(41)and(42)respectively.

2.4.Actual heat transfer rate

By applying Fourier's law of heat conduction at the fin base,the non-dimensional form of actual heat transfer rate per unit width is obtained as

2.5.Fin ef ficiency

The ideal heat transfer rate is the maximum possible heat transfer rate which would be transferred if the entire fin is maintained at the base temperature or,in other words,the fin is made of a material of in finite thermal conductivity.The ideal heat transfer rate per unit width is expressed in dimensionless form as

2.6.Optimization analysis

Fins augmentthe heattransferrate and this bene fitfrom fins should outweigh the added costand complexity thatare associated with fins.Hence to ensure that maximum bene fit is obtained from fins and for using the fin in real life applications,the determination of the optimum dimensions of the fin is of paramount importance in every fin analysis problem.Optimization of fins is concerned with determining the fin geometry which either transfers the maximum heat transfer rate for a given fin volume,or conversely which results in the minimum fin volume for a given heat duty.

The volume of a T-shaped fin per unit width is expressed in non-dimensional form as

Both the dimensionless heat transfer rate and the fin volume depends four parameters,Bi,τ ,γTand γL,other parameters being considered constant.Lagrange multiplier technique has been adopted for the present optimisation analysis.Elimination of the Lagrange multiplier from the Euler system of equations generates the following optimality criteria

Depending upon the design requirements,the constraint may be either the heat transfer rate or the fin volume and it can be written as

Using Eq.(49),Eq.(48)can be written as

Now,either the fin volume or the heat transfer rate can be taken as the constant and the constraint equation along with Eq.(50)can be solved simultaneously to get the optimum values ofBi,γL,γTand τ.For the solution,initially guess values of the four parameters are taken and the iterative Newton–Raphson method is used to get the optimum values of the parameters(Bi,γL,γTand τ),satisfying the necessary and sufficient convergence criteria[34].Once the optimum design parameters are obtained,the maximum heat transfer rate or the minimum fin volume,whichever is applicable,can be obtained.

3.Results and Discussions

The results of the above mathematical formulations can be best analysed by generating the results for the temperature distribution,actual heat transfer rate and fin efficiency in graphical form for a wide range of the various parameters involved.Before proceeding any further, first it is important to validate the solutions obtained using the present analytical technique.Unfortunately validation of the present result is not possible directly because no work has been performed so far on constructal T-shaped porous fins under simultaneous heat and mass transfer.However,few works are available on longitudinal porous fin subjected to combined heat and mass transfer[21–26]and a work is done on constructal T-shaped solid fin under same condition[33].Thus the validation has been done by comparing the present analytical results with that from a numerical analysis,as seen in Fig.2(a).For generating the numerical results, first the discretized equations are obtained from Eq.(10)using the Taylor's series central difference scheme and subsequently the difference equations are solved using the Gauss Seidel iterative procedure,satisfying a desired convergence criterion of 10?6.Both the results are found to be in excellent agreement with each other.Fig.2(b)is drawn not only to compare the present analytical results and the results obtained from the published model[33].but also to show the effects of porous fin under dehumidifying condition.In the published model,as already mentioned in the literature,temperature distribution and fin efficiency have been obtained for a fully wet solid T-shaped fin using the Adomian decomposition method(ADM),which can be obtained fromthe presentmodelby setting ?=0,Ra=0,Da=0.The dimensionless temperature for the wet solid fin is found to be higher than that for the wet porous fin and hence the dimensional fin surface temperature is lower for the solid fin than that for the porous fin.The reason is that for porous fin,due to the presence of pores the effective thermal conductivity decreases and simultaneously more hot fluid comes in contactwith the fin surface,hence the fin surface temperature increases.Further,it has been seen from Fig.2(a)that as the porosity is increased,the dimensionless fin surface temperature decreases,i.e.,the dimensional fin surface temperature increases.This trend is expected because increasing the porosity of the fin causes increased contact with the hot fluid and decreased effective thermal conductivity.

Fig.2.Comparison of temperature distribution between(a)present analytical and published[33]results(b)present analytical and numerical results.

Fig.3 suggests that the dimensional temperature for wet porous fins is higher than that of dry porous fins.Unlike that of dry fins,in wet fins condensation ofmoisture on the fin surface occurs with the evolution of latent heat,which causes the temperature of wet fins to be higher than thatofdry fins.Also,asseen in Fig.3,the dry porous fin exhibits a higher fin surface temperature than the corresponding dry solid fin,due to the same reason,as explained for wet fins.When the relative humidity of the incoming air stream is increased at constant ambient temperature and pressure,the moisture content of the air also increases,which ultimately causes the release of more latent heat due to increased condensation on the fin surface.This results in the rise of the fin surface temperature with the increase in relative humidity ofthe ambientair,as seen in the same figure.

Fig.4(a)and(b)demonstrates the role played by Darcy number and Rayleigh number on the temperature distribution in the fin.Flow of fluid through the pores depends on not only the number and size of the pores,but also on how well connected are the pores,or in other words on the permeability of the fin.With the increase in Darcy number,the permeability of the fin increases,causing more fluid to pass through the fin and thus increases the fin temperature,which results in the decrease of dimensionless fin temperature,as obvious through Fig.4(a).The same trend has been observed in case of Rayleigh number in Fig.4(b).A higher value of Rayleigh number means a higher value of the effective convective heat transfer coefficient between the fin and the ambient air,thus enhancing the rate of heat transfer to the fin from the surrounding air and so the fin surface temperature increases orthe dimensionless temperature declines.The dimensionless temperature distribution in the fin is found to exhibit an increasing tendency with the increase in the thermal conductivity ratiokR,as clearly seen in Fig.4(c).This is an expected variation since as the value ofkRincreases,the thermal conductivity of the fin increases,thereby lowering the fin surface temperature.The same figure also shows that as the ambient temperature increases,the dimensionless temperature decreases and thus dimensional fin surface temperature increases.This is due to the evolution of more latent heat of condensation on the fin surface because of the increased moisture content in the air.

Fig.5(a)represents the variation of dimensionless fin surface temperature as a function of Biot number and thickness ratio.A higher value ofBiotnumberis achieved by increasing the conductive resistance in the fin and/or decreasing the convective resistance between the fin and the ambient air,which results in a higher fin surface temperature and subsequently a lower dimensionless temperature.Again,at a certain Biot number,a higher value of thickness ratio translates into a lower stem thickness and/or larger flange thickness.Thus a low dimensionless tip temperature is obtained for higher thickness ratio.Fig.5(b)clearly demonstrates that the dimensionless temperature is an incremental function of τ.With the increase in τ,the stem thickness increases or the stem length decreases.For a constantstem length,with the increase inτresults in the decrease of fin surface temperature due to decrease in conduction resistance.Fig.5(b)also shows that increasing the length ratio decreases the dimensionless temperature.At higher length ratio with constant stem length,the flange length becomes more causing an increase in the conductive resistance in the flange part,which amplifies the dimensional fin tip temperature.

Fig.3.Comparative study of temperature distribution between dry and wet porous fin(γT=0.5, γL=0.5, τ=0.15,T A=30°C,T B=5°C,Bi=0.0005,k R=8000,Ra=105,Da=0.0001).

Fig.4.Variation of dimensionless fin surface temperature with variation of:(a)Darcy number,(b)Rayleigh number,(c)thermal conductivity ratio and ambient temperature.

Next section has been devoted to analysing the role played by the various parameters on the fin efficiency.Compared to that of solid fin,porous fins have more effective surface area for convection but at the same time,they have reduced effective fin thermal conductivity.Because of these two con flicting behaviour of the porous fins,the heat transfer rate may either increase or decrease than the corresponding solid fin.From Fig.6(a),it has been noticed that at low values of porosities,the actual heat transfer rate for porous fin is higher than the solid fin.But as the value of the porosity is increased,the actual heat transfer rate decreases and beyond a certain value of ?,the actual heat transfer rate for porous fins becomes lower than the solid fin.This particular value of ? may vary depending upon the values of the other parameters involved.Therefore,for better heat transfer rate than solid fin,selection of various related parameters likeRa,Daand ? is very important.It is also evident that for some values of porosity,whether heat transfer is more or less than that of the solid fin depends upon the geometric parameterofthe porous fin.Forsome porosity,ifcross overpointexists,then decrement of heat transfer due to decreased effective thermal conductivity is not significant compare to the solid fin,but increment of heat transfer due to increment of convective heat transfer through the porous medium is significant.And this enhancement occurs with the increase of contact flow area through the porous medium or thickness ofthe porous fin.Fig.6(a)shows,for porosity 0.4 and for a constant fin length of porous fin,if τ increases more than a critical value,heat transfer more than solid fin can be obtained,because increase in thickness of the fin, fluid has to pass more contact surface area through the porous medium.This is a very important finding for the design perspective of any system where both increase in heat transfer and lightweight are major concern.So that low weight can be maintained keeping high porosity value with higher heat transfer rate compare to the solid fin.Fig.6(b)demonstrates thatwhatever be the value ofporosity,efficiency of porous fins is always lower than that of the corresponding solid fin.Again,in the same figure it is seen that increasing porosity decreases the efficiency,which is an expected observation because of decreasing effective thermalconductivity.Fin efficiency compares the heattransfer rate in fins to an idealised situation and hence for high efficiency fins should be shorter.But for shorter fins,heat transferred to the fin reduces.This reason clears the fact that increasing the value of τ(which reduces the stem length with constant stem thickness)increases the fin efficiency,as envisioned in Fig.6(b).

Fig.5.Effect of Biot number and geometric parameters on the dimensionless temperature distribution:(a)variation with Bi and γT,(b)variation with γL and τ.

Fig.6.Effect of porosity ? and geometric parameter τ on:(a)actual heat transfer rate,(b) fin efficiency.

Fig.7 illustrates that the efficiency of wet porous fins is lower than that of dry porous fins.Contrary to that of dry fins,due to moisture condensation in case of wet fins the actual heat transfer rate though becomes more but at the same time ideal heat transfer rate also increases due to dehumidification parameter ξ as evident in Eq.(45),which declines the fin efficiency.The same reason stands for the effect of relative humidity.The same figure demonstrates that the efficiency of dry solid fin is higher than the dry porous fin and the reason is the higher thermal conductivity of dry fins.Further,the effect of Biot number on the fin efficiency is shown in the same figure.One-dimensional heat conduction in fins can be assumed only ifBi?0.1 in the thickness direction[55].Typically,the fin materialhas a high thermalconductivity and fin thickness is very smallcompared to the other dimensions.Hence in the present analysis,Biis considered in the range of(0.0005–0.004).A high value of Biot number results in a high value of conductive resistance in the fin,which declines the efficiency,as exhibited in the same figure.

Fig.7.Comparative study of dry and wet fin efficiency as a function of Biot number.

The role played by Rayleigh number and Darcy number on the fin efficiency has been portrayed through Fig.8(a)and(b)respectively.Increasing the value of Rayleigh number leads to more effect of the buoyancy force and consequently higher value of heat transfer rate due to convection.But the fin efficiency reduces due to the simultaneous increase of the ideal heat transfer rate as expressed in Eq.(45).The same trend has been noticed in case of Darcy number also.As already explained previously,a high value of Darcy number can be achieved by using a more permeable fin,which results in more fluid flow through the fin and hence enhanced rates of both the actual and ideal heat transfer rate.But the increase in ideal heat transfer rate is more than that of actual heat transfer rate.This reduces the fin efficiency.The fin efficiency is a incrementalfunction of the thermal conductivity ratio,kR.This is an expected observation due to the increase in the thermal conductivity of the fin with the increase inkR.Increasing the thickness ratio increases the gradient of dimensionless temperature in the stem,which enhances the actual heat transfer rate.But thickness ratio has no effect on the ideal heat transfer rate and hence the fin efficiency enhances,as observed in Fig.8(c).A reverse effect can be seen in case of length ratio in Fig.8(c),which is because increasing the length ratio,increases the actual heat transfer rate due to the increase of gradient of dimensionless temperature in the stem,but simultaneously the ideal heat transfer rate also increases,which declines the fin efficiency.When the base temperature is increased,the temperature difference between the ambient and the fin base reduces,which declines the actual as well as the ideal heat transfer rates and therefore the fin efficiency.

Next,the results of the optimization analysis have been presented for a constant fin volume condition.First a complete multivariable geometric optimisation is done,where all the geometric parameters τ,γL,γTare varied simultaneously to obtain the optimum condition.The optimum condition is determined by initially taking guess values of these three parameters τ ,γL,γTand thereby adopting the iterative Newton–Raphson method for multiple variables.The results are presented for three different values of ? and fin volume.At constant ?,increasing the fin volume enhanced the maximum heat transfer remarkably.Also at constant fin volume,actual heat transfer rate showed a decreasing trend with increasing porosity.This is due to the removal of solid material,which lowers the conductive heat transfer through the fins.

Though the optimum values τ ,γL,γTobtained in Table 2 from the multivariable optimization analysis are better and more accurate,but from manufacturing point of view these optimum values may be difficult or costly to realize practically.Hence the optimization results are generated by taking the objective function and the constraint equation as functions of onlyBiand τ.The values of the other two geometric parameters along with the fin volume are selected priorto the optimization analysis,depending upon the design requirements.In each plot,the heat transfer rate initially increases with the increase in τ,reaches a maximum value and thereafter declines.Hence,for each operating condition,there is an optimum value of τ at which the heat transfer rate becomes maximum for a constraint fin volume.In porous fin,porosity parameter plays a significant role in transferring heat to the fin body by facilitating convection heat transfer.But Fig.9(a)shows a decreasing trend of the actual heat transfers with increasing porosity.This happens because of the removal of solid material which lowers the conductive heat transfer through the fins.However for a de finite value of the porosity,porous fin transfers more heat than its solid counterpart.Thus the porosity of the fin should be selected judiciously.On the other hand, fin efficiency of the porous fin is lower than that of solid fin.Although in a porous fin,up to a particular porosity,actual heat transfer rate is higher than solid fin but ideal heat transfer is also far higher than solid fin as de fined in Eq.(45).Thus fin efficiency is less as envisioned in Fig.9(b).

Fig.10(a)shows that the actual heat transfer rate of porous wet fins is higher than that of the dry porous fins.This is due to the release of latent heat,because of moisture condensation in case of wet fins.Also,for dry fins,the maximum heat transfer rate is obtained at lower value of τ.Fig.10(a)also demonstrates thatactual heat transferrate increases with the increase in relative humidity and the optimum value of τ increases slightly.The same observation can be noticed for ambient temperature,in Fig.10(b).When the relative humidity is increased at constant ambient temperature,or conversely when the ambient temperature is increased at constant relative humidity,the moisture content of the air increases,hence the actual heat transfer rate enhances.

Fig.11(a)suggests that fins with higher length ratio transfer more heat and also the τ value needs to be increased for maintaining the optimum condition.But a higher length ratio with a constant stem length increases the envelop size of the fin,which is not desirable when space restriction is there in the direction of the flange.Hence length ratio should be adopted as the constraint for the optimisation study,when space is limited in the flange direction.A reverse trend has been noticed in case of thickness ratio in Fig.11(b).At lower thickness ratio,though the heat transfer rate increases and optimum τ value also increases,but at lower thickness ratio the flange thickness may become too small,which may cause manufacturing problem.In that case thickness ratio may also be taken as the constraint for the optimisation analysis.

Fig.8.Variation of fin efficiency with(a)Ra and Bi,(b)Da and k R,(c)γL,γT and T B.

Table 2Optimum geometric parameters and maximum heat transfer rate conditions(T A=30°C,T B=5°C,RH=70%,Bi=0.001,Ra=105,Da=0.0001,k R=8000)

Fig.9.Comparison of optimum design parameters between solid and porous fins as a function of τ:(a)actual heat transfer rate,(b) fin efficiency.

Fig.10.Variation of optimum heat transfer rate with:(a)relative humidity,RH(b)ambient temperature,T A.

Fig.11.Variation of optimum heat transfer rate with:(a)length ratio,γL,(b)thickness ratio,γT.

Rayleigh number plays a vital role in the heat transfer through porous fin as it boosts the convective heat transfer by increasing the temperature difference between the surface and the surrounding.Fig.12(a)justifies that this logic as a marked increase in the actual heat transfer rate is observed for higher values of Rayleigh numbers.The graph shows that the pointof maximum heattransfer shifts toward the right side,i.e.,toward higher τ values with increasing Rayleigh number.It is an obvious fact that for heat transfer to improve through porous fins the working fluid must penetrate well so as to facilitate the convective heat transfer.So the fins with higher permeability transfer heat better than those with lower permeability values.The parameter Darcy number which is directly proportional to the permeability of the fin is thus a deciding factor in this respect.In Fig.12(b)the heat transfer rate is seen to increase when the Darcy number is increased.Fig.12(c)shows the role played by thermal conductivity ratio,kRon the optimum heat transfer rate.With the increase inkR,the gradient of dimensionless temperature in the stem decreases as noticed in Fig.4(c)and thereby the actual heat transfer decreases as clearly seen in the figure.Also,the optimum value of τ decreases slightly with the increase inkR.Increase in fin volume amplifies the actual heat transfer rate,as envisioned in Fig.12(d).Also from the same figure,it is clear thata largerτvalue is required for transferring the optimumheattransfer rate.

Fig.12.Variation of optimum heat transfer rate with:(a)Rayleigh number,(b)Darcy number,(c)thermal conductivity ratio,(d) fin volume.

4.Conclusions

In the present study, fin efficiency and optimum design parameter analyses have been carried out on a constructal T-shaped porous fin under fully wet condition by considering a cubic polynomial variation of the humidity ratio with the corresponding fin surface temperature.The following conclusions can be arrived upon from the above discussion:

?The results of the present analyticalmodel based on DTMare found to be in excellentagreementwith thatofa numericalmodeland hence it can be concluded that this transformation technique can serve as a simple and accurate tool for analysis of problems involving high degree of non-linearity.

?Heat transfer rate for porous fins may be higher or lower than that of the corresponding solid fin and this depends upon the porosity of the fin.At low values of porosity,porous fin transfers more heat than the corresponding solid fin.As the fins become more and more porous,heat transfer rate decreases and beyond particular value porosity,heat transferred from the porous fins becomes lower than the solid if n.Hence the value of the porosity should be carefully selected while using porous fins.

?efficiency of porous fins is always lower than the corresponding solid fin.The efficiency decreases with the increase in the value of porosity.

?With the increase in thermal conductivity ratio, fin efficiency increases.But a reverse trend is seen in case of Rayleigh number and Darcy number.

?Fin efficiency decreases with the increase in Biot number.The same effect can be seen in case of relative humidity of the ambient air.But at very low values of Biot number,effect of variation of relative humidity on the fin efficiency becomes insignificant.

?Increment of the aspectratio of the stemτenhances the fin efficiency.The same effect is observed in case of thickness ratioγTwhereas increase in length ratioγLdegrades the fin efficiency.Out of these three geometric parameters,effect ofτon the fin efficiency is found to be more than the other two parameters.A slight variation inτ changed the fin efficiency considerably.

?For the multivariable optimization analysis at constantBi,the optimum heat transfer rate increased with the increase in fi n volume but a reverse trend is noticed in case?.

?The optimum design curves are generated as a function ofτfor a constraint fi n volume condition and by maximizing the heat transfer rate.In each curve,there is an optimum value ofτat which the heat transfer rate to the fin becomes maximum.

?The optimum value ofτdoes not vary signi f cantly with the variation of fin porosity.

?With the increase in relative humidity or ambient temperature,the actual heat transfer rate enhances and the optimumτvalue also increases.

?A larger value of length ratio results in higher actualheattransfer rate,as well as the larger optimum value ofτ.But at larger length ratio,the envelop size of the fi n becomes more.Hence the length ratio should be taken as a constraint for the optimisation study,when space is restricted in the flange direction.

?Actual heat transfer rate becomes more at lower values of thickness ratioγTand results in higher value of optimumτ.But lower value of thickness ratio may restrict manufacturing the flange part.

?Actual heat transfer rate is an incremental function of the fi n volume.Also,fin efficiency at the optimum condition increases with the increase in fin volume.

Nomenclature

Across-sectional area perpendicular to the direction of heat flow

BiBiot number

Cp,aspecific heat of the ambient air at constant pressure,J·kg?1·K?1

DaDarcy number of the porous fin

Ggradient of dimensionless temperature at the junction

gacceleration due to gravity,m·s?2

hconvective heattransfer coefficientover the surface of the fin,W·m?2·K?1

hfglatent heat of condensation of moisture,J·kg?1

hmmass transfer coefficient,kg·m?2·s?1

Kpermeability of the porous fin,m2

kEffeffective thermal conductivity of fi n material,W·m?1·K?1

kFlthermal conductivity of the fluid,W·m?1·K?1

kRthermal conductivity ratio,kSl/kFl

kSlthermal conductivity of the solid,W·m?1·K?1

LeLewis number

LFhalf the flange length,m

LSstem length,m

m˙ local mass flow rate of the fluid passing through the pores,kg·s?1

Qadimensionless actual heat transfer rate through the fin per unit width

QIdimensionless ideal heat transfer rate per unit width

qaactual heat transfer rate through the fin per unit width,W·m?1

qIideal heat transfer rate through the fin per unit width,W·m?1

RHrelative humidity of surrounding air

RaRayleigh number of the fluid

Tlocal fin surface temperature,°C

TAambient temperature,°C

TBif n base temperature,°C

tFlf ange thickness,m

tSstem thickness,m

Udimensionlessfin volume per unit width

Vif n volume per unit width,m2

vlocal fluid velocity,ms?1

wif n width,m

Xdimensionless distance,x/LF

xaxial length measured from flange tip as shown in Fig.1,m

Ydimensionless length,y/LS

yaxial length measured from the junction along the stem as shown in Fig.1,m

βco-efficient of thermal expansion,K?1

γLlength ratio,LF/LS

γTthickness ratio,tF/tS

ηfin efficiency

θFdimensionless temperature for the flange part of the fin,(TA?TF)/(TA–TB)

θSdimensionless temperature for the stem part of the fin,(TA?TS)/(TA–TB)

ξdehumidification parameter,hfg/(Cp,aLe2/3),°C

ρdensity of the fl uid,kg·m?3

τthickness to length ratio of stem,tS/LS

τFthickness to length ratio of flange,tF/LF

υkinematic viscosity,m2·s?1

?porosity

ψratio ofeffective thermalconductivity to thermalconductivity of fluid,kEff/kFl

ωlocal humidity ratio of the air on the fi n surface,kg of water vapour per kg of dry air

ωAhumidity ratio of surrounding air,kg of water vapour per kg of dry air

Subscripts

A ambient condition

B if n base

F flange

S stem

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