Effect Of Variable Viscosity On Convective Heat And Mass .

WSEAS TRANSACTIONS on MATHEMATICSM. B. K. Moorthy, K. SenthilvadivuEffect of Variable Viscosity on Convective Heat and Mass Transfer byNatural Convection from Vertical Surface in Porous MediumM.B.K.MOORTHY1, K.SENTHILVADIVU21Department of Mathematics, Institute of Road and Transport Technology,Erode – 638316, Tamilnadu, India. E-mail: mbk.moorthy@yahoo.com2Department of Mathematics, K.S.Rangasamy College of Technology,Tiruchengode - 637215, Tamilnadu, India. E-mail: senthilveera47@rediffmail.comCorresponding author: K.Senthilvadivu, Ph no: 91 98650 24343,E-mail: senthilveera47@rediffmail.comAbstract: - The aim of this paper is to investigate the effect of variable viscosity on free convective heat andmass transfer from a vertical plate embedded in a saturated porous medium. The governing equations ofcontinuity, momentum, energy and concentration are transformed into non linear ordinary differential equationsusing similarity transformations and then solved by using Runge – Kutta – Gill method along with shootingtechnique. Governing parameters for the problem under study are the variable viscosity, the buoyancy ratio andthe Lewis number. The velocity, temperature and concentration distributions are presented and discussed. TheNusselt and Sherwood number are also derived. The numerical values of local Nusselt and local Sherwoodnumbers have also been computed for a wide range of governing parameters. The viscous and thermalboundary layer thicknesses are discussedKey-Words: - Free convection, Heat transfer, Mass transfer, Variable viscosity, Porous medium.[7] investigated the effect of steady free convectionflow with variable viscosity and thermal diffusivityalong a vertical plate. Yih [8] analyzed the coupledheat and mass transfer in mixed convection about awedge for variable wall temperature andconcentration. Rami.Y.Jumah et al. [9] studied thecoupled heat and mass transfer for non-Newtonianfluids. Kumari [10] analyzed the effect of variableviscosity on free and mixed convection boundarylayer flow from a horizontal surface in a saturatedporous medium. Postelnicu et al. [11] investigatedthe effect of variable viscosity on forced convectionover a horizontal flat plate in a porous medium withinternal heat generation. Seddeek [15, 16] studiedthe effects of chemical reaction, variable viscosity,and thermal diffusivity on mixed convection heatand mass transfer through porous media. MohamedE- Ali [17] studied the effect of variable viscosityon mixed convection along a vertical plate. Alam etal [18] analyzed the study of the combined free –forced convection and mass transfer flow past avertical porous plate in a porous medium with heatgeneration and thermal diffusion. Pantokratoras [19]analyzed the effect of variable viscosity withconstant wall temperature. Seddeek et al. [20]1 IntroductionIn recent years the combined heat and mass transferby natural convection in a fluid saturated porousmedium has its own role in many engineeringapplication problems such as nuclear reactor design,geothermal systems, petroleum engineeringapplications, evaporation at the surface of a waterbody, energy transfer in a wet cooling tower and theflow in a desert cooler. A comprehensive account ofthe available information in this field is provided inrecent books by Ingham et al [13] and Vafai[14]Kassoy [1] studied the effect of variable viscosity onthe onset of convection in porous medium. Chengand Minkowyz [2] studied the effect of freeconvection about a vertical plate embedded in aporous medium with application to heat transferfrom a dike. Bejan and Khair [3] studied thebuoyancy induced heat and mass transfer from avertical plate embedded in a saturated porousmedium. Lai and Kulacki [5] studied the coupledheat and mass transfer by natural convection fromvertical surface in a porous medium. The sameauthors [6] also studied the effect of variableviscosity on convection heat transfer along a verticalsurface in a saturated porous medium. ElbashbeshyE-ISSN: 2224-2880751Issue 9, Volume 11, September 2012

WSEAS TRANSACTIONS on MATHEMATICSM. B. K. Moorthy, K. Senthilvadivustudied the effects of chemical reaction, variableviscosity, on hydro magnetic mixed convection heatand mass transfer through porous media. Singh et al.[21] used integral treatment to obtain theexpressions for Nusselt number and Sherwoodnumber. Recently the authors analysed the effect ofvariable viscosity on convective heat and masstransfer by natural from horizontal surface in porousmedium.The purpose of the present work is to studythe effect of variable viscosity on heat and masstransfer along a vertical surface embedded in asaturated porous medium.3. Method of solution2. Analysis kgβ Tx Ra x is the Rayleigh number να Introducing the stream function Ψ (x, y) such that ψ ψ,v u (10) y xwhereu v uuκ p ρg µ x κµ p y c c 2c v D x y y 2{}T T Tw T (13)φ c c cw c (14)(1)andN (2)Substitution of these transformations (10) to (15) toequations (2) to (5) along with the equations (6) and(7), there results the non linear ordinary differentialequations,f ′′ µ 1µ {1 γ (T T )}E-ISSN: 2224-2880(15)(16)1fθ ′2Le 1φ ′′ (5)(17)1fφ ′2(18)together with the boundary conditions(6)η 0, f 0,θ 1,φ 1(19)η , f ′ 0,θ 0,φ 0(20)where(7)Le αistheLewisnumberandDθ r 1 γ (T T )w which is reasonable for liquids such as water and oil.Here γ is a constant.The boundary conditions are y 0, v 0, T Tw , c cwy , u 0, T T , c c β (cw c )β (Tw T ) θ θr f ′θ ′ (θ ′ φ ′N ) θ θ r θ r θ ′′ The viscosity of the fluid is assumed to be aninverse linear function of temperature and can beexpressed as1(12)θ (4)ρ ρ 1 β (T T ) β (c c )y(Ra x ) 1 2xDefine(3) 2T T T α v y x y 2(11)η Consider a vertical surface embedded in a saturatedporous medium. The properties of the fluid andporous medium are isotropic and the viscosity of thefluid is assumed to be an inverse linear function oftemperature. Using Boussinesq and boundary layerapproximations, the governing equations ofcontinuity, momentum, energy and concentrationare given by u v 0 x y1ψ αf ( Ra x ) 2is the parameter characterizing theinfluence of viscosity. For a given temperaturedifference, large values of θ r implies either γ or(Tw -T ) are small. In this case the effect of variableviscosity can be neglected.The effect of variable viscosity is important ifθ r is small. Since the viscosity of liquids decreaseswith increasing temperature while it increases forgases, θ r is negative for liquids and positive for(8)(9)752Issue 9, Volume 11, September 2012

WSEAS TRANSACTIONS on MATHEMATICSM. B. K. Moorthy, K. Senthilvadivugases. The concept of this parameter θ r was firstintroduced by Ling and Dybbs [4] in their study offorced convection flow in porous media. Theparameter N measures the relative importance ofmass and thermal diffusion in the buoyancy – drivenflow. It is clear that N is zero for thermal- drivenflow, infinite for mass driven flow, positive foraiding flow and negative for opposing flow. It maynote that in the absence of mass transfer, theequations (16) and (17) together with the boundaryconditions (19) and (20) reduce to that obtained byLai and Kulacki [5].The heat transfer coefficient in terms of theNusselt number is given byNu x θ ′(0 )(Ra x ) 1 2plate as N 0 for opposing flows (N 0) for givenvalues of θ r and Le. From Fig.4 it is realized thatthe temperature and concentration decreases asN 0 for opposing flows (N 0) and increases asN 0 for aiding flows (N 0) for given values ofθ r and Le.(21)The mass transfer coefficient in terms of theSherwood number is given bySh x(Ra x ) 1 2 φ ′(0 )(22)Results and discussionFig.1.Velocity profiles for different values of θ r forN 1 and Le 1The equations (16), (17) and (18) are integratednumerically by using Runge – Kutta – Gill methodalong with shooting technique. The parametersinvolved in this problem are: θ r – the variableviscosity, Le- Lewis number and N-the buoyancyparameter. To observe the effect of variableviscosity on heat and mass transfer we have plottedthe velocity functionf ', temperature function θand concentration function Φ against η for variousvalues of θ r , Le and N.The effect of variable viscosity θ r on velocity,temperature and concentration are shown in figures1 and 2. It is realized that the velocity increases nearthe plate and decreases away from the plate asθ r 0 in the case of liquids ( θ r 0) and decreasesnear the plate and increases away from the plate asθ r 0 in the case of gases ( θ r 0 ) for given valuesof N and Le. From Fig.2 it is evident that thetemperature and concentration increases as θ r 0for θ r 0(i.e. for gases) and decreases as θ r 0 forθ r 0(i.e. for liquids) for other values are fixed.The effect of buoyancy ratio N on velocity,temperature and concentration are shown in figures3 and 4. From Fig.3 it is observed that the velocitydecreases near the plate and increases away from theplate as N 0 for aiding flows (N 0) andincreases near the plate and decreases away from theE-ISSN: 2224-2880Fig.2.Temperature and Concentration profiles fordifferent values of θ r for N 1 and Le 1753Issue 9, Volume 11, September 2012

WSEAS TRANSACTIONS on MATHEMATICSM. B. K. Moorthy, K. SenthilvadivuFrom Fig.7 it is realized that as the Lewisnumber Le decreases, the concentration decreasesfor fixed values of θ r and N.Fig.8 illustrates the effect of variable viscosity θ ron the slip velocity f'(0) for different values of thebuoyancy ratio N for given value of Le. It isobserved that the slip velocity decreases as θ r 0 inthe case of gases (for θ r 0) but in the cases ofliquids ( θ r 0) the slip velocity increases as θ r 0for aiding (N 0), thermal-driven (N 0) andopposing (N 0) flows. These observations areobviously same as drawn in Table 1.Fig. 3.Velocity profiles for different values of Nfor θ r 5 and Le 1Fig. 5.Velocity profiles for different values of Lefor θ r 5 and N 1Fig.4.Temperature and Concentration profiles fordifferent values of N for θ r 5 and Le 1The effect of Lewis number Le on velocity,temperature and concentration are shown in figures5, 6 and 7. From Fig.5 it is evident that as the Lewisnumber Le decreases, the velocity decreases forgiven values of θ r and N. From Fig.6 it is observedthat as Le increases, the temperature decreases forfixed values of θ r and N.E-ISSN: 2224-2880Fig. 6.Temperature profiles for different values ofLe for θ r 5 and N 1754Issue 9, Volume 11, September 2012

WSEAS TRANSACTIONS on MATHEMATICSM. B. K. Moorthy, K. 0.3540.3340.318-0.1Slip velocity1250.637 1.1415 2.1235 4.2370.7666 1.7036 2.5554 5.11080.8730 1.9401 2.9102 5.82041.1752 2.6115 3.9173 7.83471.0363 2.3033.4546 6.9090.9271 2.0601 3.0901 6.1803Nusselt and Sherwood 11.1016Table1. Numerical values of slip velocity, Nusseltand Sherwood numbers for different values of N andθ r for Le 1.Fig.7.Concentration profiles for different values ofLe for θ r 5 and N 1Fig.9.Effect of variable viscosity θ r and buoyancyratio N on heat and mass transfer rates for Le 1Fig.9 displays the variation of local Nusselt andSherwood numbers at the surface with variableviscosity θ r covering aiding flows (N 0)andopposing flows (N 0) for fixed value of Le. It isevident that the heat transfer and mass transfer ratesdecreases as θ r 0 for θ r 0 and increases asθ r 0 for θ r 0 for both aiding and opposing flows.This is also evident from Table 1. Similar behaviourFig. 8.Effect of variable viscosity θ r and buoyancyratio N on the slip velocity for Le 1E-ISSN: 2224-2880755Issue 9, Volume 11, September 2012