An Integral Treatment For Heat And Mass Transfer Along A .
Computational Methods in Multiphase Flow IV143An integral treatment for heat and masstransfer along a vertical wall by naturalconvection in a porous mediaB. B. SinghDepartment of Mathematics,Dr. Babasaheb Ambedkar Technological University, Lonere-402103,Dist. Raigad (M.S.), IndiaAbstractThis paper deals with the free convective heat and mass transfer along a verticalwall embedded in a fluid saturated porous medium by using an integral methodof the Von-Karman type in the presence of temperature and concentrationgradients. Mathematical expressions for the local Nusselt number and localSherwood number have been derived in terms of boundary layer thickness ratio.The governing parameters for the flow-field are buoyancy ratio (N) and Lewisnumber (Le). The numerical values of the local Nusselt number and localSherwood number have been computed for a wide range of values of N and Le.The variations of local Nusselt number and local Sherwood number with N havealso been studied with the help of graphs for the different values of Le.Similarly, the variations of local Nusselt number and local Sherwood numberwith Le have been studied for different values of N with the help of graphs.It has been found that the local Nusselt number increases as N increases for thedecreasing value of Le, whereas the local Sherwood number increases as Nincreases for the increasing values of Le. The local Nusselt number and the localSherwood number increase as Le increases for increasing values of N. Thenumerical values of the thermal boundary layer and concentration boundary layerthicknesses have also been computed for the flow-field. It has been found thatthe results obtained by the integral method are in good agreement with thoseobtained by Bejan and Khair [Heat and Mass Transfer by Natural Convection ina Porous medium, Int. J. Heat Mass Transfer, 28, pp. 909-918, 1985].Keywords: natural convection, porous media, heat and mass transfer.WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)doi:10.2495/MPF070141
144 Computational Methods in Multiphase Flow IV1IntroductionThe natural convection flows arising out of the combined buoyancies due tothermal and mass diffusion in a porous medium are of importance because of thefundamental nature of the problem and broad range of their applicationspertaining to manufacturing and process industries such as geothermal systems,fibre and granular insulation, storage of nuclear waste materials, usage of porousconical bearings in lubrication technology, chemical catalytic reactors, dispersionof chemical contaminant through water saturated soil , natural gas storage tanks ,etc.On account of the afore-mentioned applications, Bejan and Khair [1] used theDarcy’s law to study the features of the free convection boundary layer flowdriven by temperature and concentration gradients. Recently, Lai and Kulacki [2]have re-examined the free convection boundary layer along a vertical wall withconstant heat and mass flux including the effect of wall injection. The heat andmass transfer by natural convection near a vertical wall in a porous mediumunder boundary layer approximations has been studied by Nakayama andHossain [3] and Singh and Queeny [4]. A further review of coupled heat andmass transfer by natural convection in porous medium is given by Nield andBejan [5].The objective of the present paper is to apply integral method to analyze freeconvection problem along a vertical wall in the presence of temperature andconcentration gradients. A comparison of the numerical values of the localNusselt and local Sherwood numbers obtained by the integral method has beendone with those obtained by Bejan and Khair [1] for different values of thebuoyancy ratio.It has been found that the results obtained by the present method are in goodagreement with those obtained by Bejan and Khair.2 Mathematical analysisWe consider a two-dimensional laminar flow over a vertical flat plate in a porousmedium embedded in a Darcian fluid. The co-ordinate system and the physicalmodel are shown in figure 1. In the mathematical formulation of the problem, wenote the following conventional assumptions:i)the physical properties are considered to be constant, except for thedensity term that is associated with the body force;ii)flow is sufficiently slow so that the convecting fluid and the porousmatrix are in local thermodynamic equilibrium;iii)Darcy’s law, the Boussinesq and boundary layer approximations havebeen employed.With these assumptions, the governing equations are given byWIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
Computational Methods in Multiphase Flow IV145 u v 0 x yu (1)gK(β T (T T β C (C C )v(2) T T 2T v α 2 x y y(3) C C 2C v D 2 x y Figure 1:Physical model and coordinate system.Figure 2:Heat transfer coefficientas a function of buoyancyratio.The symbols have got their meanings as mentioned in the Nomenclature.The boundary conditions at the wall arey 0 : v 0 , T Tw , C Cw(5)and at infinity arey ; u 0 , T T , C C 3(6)Integral methodThe boundary layer equations (2)–(4) along with boundary conditions (5) and (6)have been solved by using integral method. The partial differential equations getconverted into the ordinary differential equations by making use of the followingtransformations:WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
146 Computational Methods in Multiphase Flow IVη y(Ra x )1 / 2xψ α ( Ra x )1 / 2 f (η )where Ra x (7)(8)θ T TwTW T (9)φ C CwC w C (10)gβ T Kx(Tw T )is the modified local Rayleigh number, ψ isαvthe stream function.After transformation the resulting equations becomef″(η) - θ′(η) – N φ′(η) 0(11)θ″(η) ½ f(η) θ′( η) 0(12)φ″(η) ( Le/2) f(η) φ′( η) 0(13)f(0) 0 , θ (0) φ(0) 1(14)f'( ) θ( ) φ( ) 0(15)with boundary conditionswhere primes denote the differentiation with respect to ‘η’, η [ 0, ). Here,f'(η) is non-dimensional velocity related to the stream function ψ(x,y).In the above equations (11) – (13), N is the buoyancy ratio defined byN β C (C w C )β T (Tw T )(16)and Le is the Lewis number defined byLe α / D(17)From (12) and (13), we get θ ' (0) 1f 'φdη2 0 θ ' (0) Lef 'φdη2 0(18) WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)(19)
Computational Methods in Multiphase Flow IV147The infinity is boundary layer thickness for temperature and concentration. Wenow assume the exponential temperature and concentration profiles as follows:θ(η) exp (- η/δT)(20)φ (η) exp (- ξ η/δT)(21)Here δT is arbitrary scale for the thermal boundary layer thickness whereas ξ isits ratio to the concentration boundary thickness δC. With the help of aboveprofiles, and using equation (11), the equations (18) and (19) can be obtained intwo distinct expressions as1δT1δT2 2ξ 1 2N4(ξ 1)(22) 2ξ N (ξ 1) Le 2 4ξ (ξ 1)(23)The above two equations (22) and (23) can be combined together to give thefollowing cubic equation for determining the boundary layer thickness ratio ξ asξ3 ( 1 2N) ξ2 – [ (2 N) Le ] ξ – N Le 0(24)As ξ is determined by using the computer programming like MATLAB from theequation (24), the local Nusselt and Sherwood numbers which are of our maininterest in terms of heat and mass transfer respectively, are given as ξ 1 2N Nu 0.5( Ra x )1 / 2 ξ 1 1/ 2(25)and ξ 1 2N Sh 0.5ξ 1/ 2( Ra x ) ξ 1 1/ 2(26)The accuracy acquired in the above approximations may be examined bycomparing the heat and mass transfer results against those obtained by Bejan andKhair [1]. It is not unusual to have an error of 5 % or more, depending on theassumed profile. However, the situation can be remedied by adjusting themultiplicative constant, namely, replacing 0.5 by 0.444. Thus, the followingapproximate formulae are proposed:WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
148 Computational Methods in Multiphase Flow IV ξ 1 2N Nu 0.444 1/ 2( Ra x ) ξ 1 ξ 1 2N Sh 0.5ξ 1/ 2( Ra x ) ξ 1 1/ 2(27)1/ 2(28)4 Results and discussionsThe formulae (27) and (28) give the values of the local Nusselt number (Nu) andSherwood number (Sh) as 0.444 for N 0 and Le 1. These values are the sameas obtained by Bejan and Khair [1]. The above assertion is clear from table 1.We have done calculations for a wide range of the parameters N (buoyancy ratio)and Le (Lewis number) in order to understand their influence on the combinedheat and mass transfer along a vertical wall due to free convection. These valueshave been given in table 1. From the table, it is evident that the values of localNu and Sh obtained by the integral method for different values of Le are inexcellent agreement with those obtained by Bejan and Khair who obtained thecorresponding values by the similarity solution technique.From the table, it is clear that the thermal boundary layer thickness δT showsan increasing trend for N 1, 4 for the increasing values of the Lewis numberLe. On the contrary, the concentration boundary layer thickness δC shows adecreasing trend for N 0, 1, 4 for the increasing values of Le. From the table, itis obvious that the Lewis number has more pronounced effect on theconcentration field than it has on temperature field. From the table, it is furtherevident that the magnitudes of the thermal boundary layer and concentrationboundary layer thicknesses are equal for N 0, Le 1; N 1, Le 1 and N 4,Le 1.The local Nusselt number has been plotted in figure 2 as a function ofbuoyancy ratio for various values of Lewis number (Le 0.1, 1, 10, 100). It isfound that the rate of heat transfer decreases with increasing Lewis number for N 0. Similarly the local Sherwood number has been plotted in figure 3 against thebuoyancy ratio N for various values of the Lewis number (Le 1, 10, 50, 100). Itis found that the rate of mass transfer increases with increasing Lewis number forall N.The local Nusselt number has been plotted in figure 4 as a function of Lewisnumber for various values of buoyancy ratio N 0, 2 and 4. It is found that thelocal Nusselt number decreases with increasing Lewis number for N 0.Similarly the local Sherwood number is plotted in figure 5 as a function of Lewisnumber for various values of buoyancy ratio N 0, 1 and 4. It is found that thelocal Sherwood number increases with increasing Lewis number for all N. Fromfigures 4 and 5, also it is evident that the values of local Nusselt and localSherwood numbers in the present case are in excellent agreement with thoseobtained by Bejan and Khair.WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
Computational Methods in Multiphase Flow IVTable 1:14Comparison of Local Nusselt and Sherwood numbers.Nu/(Ra x ) 1/2Le 52.5332.9363.29010.521Sh/(Ra x ) 03010.792 102100246NFigure 3:149Mass transfer coefficient as a function of buoyancy ratio.WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
150 Computational Methods in Multiphase Flow IV1.2N 0 (Present)10.8N 0 (Numerical)N 4N 2N 00.60.4N 2 (Present)N 2 (Numerical)0.20N 4 (Present)020406080100N 4(Numerical)120LeFigure 4:Heat transfer results.4N 0 (Present)3.5Sh/Ra x1/23N 0 (Numerical)N 42.5N 1(Present)2N 1N 1 (Numerical)N 01.51N 4 (Present)0.5N 4 (Numerical)0036LeFigure 5:5912Mass transfer results.Concluding remarksThis paper deals with the free convective heat and mass transfer along a verticalwall embedded in a fluid saturated porous medium. The heat and mass transfercoefficients obtained in the present study by the integral method agree very wellWIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
Computational Methods in Multiphase Flow IV151with those obtained by Bejan and Khair. In the present analysis, the results havebeen presented in such a way that any practicing engineer can easily obtain thephysical characteristic of the problem for arbitrary values of the buoyancy ratioand Lewis number. The advantage of this method is that it also provides withgreat freedom the approximate solutions to non-linear problems. The furtheradvantage of this method is that the results are obtained with more ease ascompared to Bejan and Khair.NomenclatureNTCDfghkKLeNuShRaxuvx, ybuoyancy ratiotemperatureconcentrationmass diffusivity of porousmediumdimensionless stream functiongravitational accelerationlocal heat transfer coefficientthermal conductivitypermeabilityLewis numberlocal Nusselt numberlocal Sherwood numbermodified Rayleigh numberDarcy’s velocity in x- directionDarcy’s velocity in y- directioncartesian co-ordinateGreek Symbolsα thermal diffusivity of porousmediumη similarity variableβT coefficient of thermal expansionofconcentrationβC coefficientexpansionδT arbitrary length scale for thermalboundary layerδC arbitrarylengthscaleforconcentration boundary layerψ stream functionφdimensionless concentrationξboundary layer thickness ratioνkinematic viscositySubscripts condition at the infinityw condition at the wallReferences[1][2][3][4][5]Bejan, A. and Khair, K.R., Heat and Mass Transfer by Natural Convectionin a Porous medium, Int. J. Heat Mass Transfer, 28, pp. 909-918, 1985.Lai, F.C. and Kulacki, Coupled Heat and Mass Transfer by NaturalConvection from Vertical Surface in Porous Media, Int. J. Heat MassTransfer, 34, pp. 1189-1194, 1991.Nakayama, A. and Hossain, M. A., An Integral Treatment for CombinedHeat and Mass Transfer by Natural Convection in a Porous Media, Int. J.Heat Mass Transfer, 38, pp. 761-765, 1995.Singh, P. and Queeny, Free convection Heat and Mass Transfer along aVertical Surface in a Porous Media; Acta Mechanica, 123, pp.69-73,1997.Nield, D. A. and Bejan, A., Convection in Porous Media, second edition,Springer-Verlag, New York, 1999.WIT Transactions on Engineering Sciences, Vol 56, 2007 WIT Presswww.witpress.com, ISSN 1743-3533 (on-line)
The heat and mass transfer by natural convection near a vertical wall in a porous medium under boundary layer approximations has been studied by Nakayama and Hossain [3] and Singh and Queeny [4]. A further review of coupled heat and mass transfer by natural convection in porous
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