Chemical Reaction Effects on a Three Dimensional MHD Mass Transfer Flow past a Vertical Plate
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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 3 Ver. IV (Ma - Jun. 5), PP -3.iosrjournals.org Chemical Reaction Effects on a Three Dimensional MHD Mass Transfer Flo past a Vertical Plate D. Sarma and P.K.Mahanta Department of Mathematics, Cotton College,Guahati-,India. Department of Mathematics, Jalukbari H.S. School, Guahati-4. Abstract: The objective of the present paper is to stud the effects of chemical reaction on a three dimensional MHD mass transfer flo past a vertical plate in presence of heat source. The governing non dimensional equations relevant to the problem are solved b regular perturbation technique. Under certain assumptions, the solutions for velocit field, temperature distribution and species concentration are obtained. The expressions for skin-friction, Nusselt number and Sherood number are performed. The influences of various parameters on the velocit field, temperature field, concentration field, skin-friction, Nusselt number and Sherood number are studied graphicall. The obtained results have shon that the chemical reaction effect has a great influence in the stud of flo and heat transfer process in the presence of magnetic field of some tpes of fluids considered. Keords: MHD, chemical reaction, mass transfer, perturbation. I. Introduction: Combined heat and mass transfer in fluid finds applications in a variet of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical cataltic reactors and processes, geothermal and geophsical engineering. Double diffusive flo is driven b buoanc due to temperature and concentration gradients. The investigation of MHD convection problems have attracted the attention of a number of scholars because of its ide application in meteorolog, solar phsics, cosmic fluid dnamics, astrophsics, geophsics and in the motion of earth s cone. From technological point of vie, MHD convection flo problems are also ver significant in the fields of stellar and planetar magnetospheres, aeronautics, chemical engineering and electronics. The application of MHD principles in medicines and biolog are of paramount interest oing to their significance in bio-medical engineering in general and in the treatment of various pathological states in particular. The bio-medical engineering includes cardiac MRI, ECG etc. The problems of above phenomena of MHD convection have been studied b man authors. Raptis and Soundalgekar [3], Ferraro and Plumpton [6], Cramer and Pai [5], Sanal and Bhattachara [4], and Soundalgekar et al. [8] are some of them. Ahmed and Sarma [], singh et al. [7], and Choudhar and Chand [ 4] have investigated the effect of three dimensional flo caused b the periodic motion perpendicular to the main flo hen the difference beteen the all temperature and free stream temperature gives rise to buoanc force in the direction of the free stream on heat transfer characteristics. In man transport processes in nature and in industrial applications, the heat and mass transfer ith variable viscosit is a consequence of buoanc effects caused b the diffusion of heat and chemical species. The stud of such processes is useful for improving a number of chemical technologies such as polmer production and food processing. In nature the presence of pure air or ater is impossible, because some foreign mass ma be presented either naturall or mixed ith air or ater. The stud of heat and mass transfer ith chemical reaction is of considerable importance in the chemical and hdrometallurgical industries. Chemical reaction can be codified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniforml through a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or ithin the boundar of a phase. A reaction is said to be first order, if the rate of reaction is directl proportional to the concentration itself. Chemical reactions usuall accompan a large amount of exothermic and endothermic reactions. These characteristics can be easil seen in a lot of industrial processes such as the polmer production, the manufacturing of ceramics or glassare, the food processing and so on. It has been realied that it is not alas permissible to neglect the convection effects in porous constructed chemical reactors. Das et al. [9] considered the effects of first order chemical reaction on the flo past an impulsivel started infinite vertical plate ith constant heat flux and mass transfer. Muthucumarsam and Ganesan[9] and Muthucumarsam [] studied first order homogeneous chemical reaction on the flo past infinite vertical plate. In vie of the importance of the chemical reaction effects, several authors have carried out their research orks to investigate the effects of it on some mass transfer flo problems. Some of them are Poornima and Redd [], Sharma et al. [5], Shivaiah and Rao[6], Ibrahim et al.[7], Rajeshari et al. [], Kandasam et al.[ 8 ] and Chamkha [3]. DOI:.979/ iosrjournals.org Page
2 The object of the present ork is to investigate the chemical reaction effects on a three dimensional MHD mass transfer flo past a vertical plate. This ork is an extension of the problem studied b Ahmed and Gosami [ ]. II. Mathematical analsis: A stead, three dimensional free and forced convection flo of an incompressible and electricall conducting viscous fluid past a vertical porous plate in presence of chemical reaction is considered b making the folloing assumptions. (i) All the fluid properties except the densit in the buoanc force term are constant. (ii) A magnetic field of uniform strength B is applied transversel to the direction of the main flo. (iii) The magnetic Renolds number is so small that the induced magnetic field can be neglected. (iv) The viscous dissipation and magnetic dissipations of energ are negligible. (v) T T and C C Let us consider a co-ordinate sstem x,, ith X - axis verticall upards along the plate, Y - axis perpendicular to it directed into the fluid region and Z - axis along the idth of the plate. Let q iu ˆ ˆjv k ˆ be the fluid velocit at the point x,, and B B ˆ j be the applied magnetic field. The suction velocit distribution is taken as follos: v( ) V [ cos ] L hich consists of a basic stead distribution - V ith superimposed eak distribution V cos. Since the L plate is infinite in length in X - direction, therefore all the quantities except possibl the pressure are assumed to be independent of x. With these assumptions and under usual boundar laer approximations, the governing equations of the problem are: Equation of continuit: v Momentum equations: u u u u B v g T T g CC Uu v v p v v v () () (3) p B v (4) Energ equation: T T k T T Q v T T Cp Cp (5) Species concentration equation: C C C C T T v D D K CC m T The relevant boundar conditions are: At : u, v v,, T T, C C (7) At : u U, v V,, T T, C C, p p (8) DOI:.979/ iosrjournals.org Page (6)
3 To normalie the flo model, e introduce the folloing non-dimensional quantities: u v U TT CC,, u, v,, U,,, L L V V V V T T C C DT( TT ) Sr ( CC ) Lg ( T T ) Lg ( C C ) Gr, Gm V C p B VL, Pr, Sc, M, Re V k Dm V p p p, p, L L K QL K, Q V Vk The non-dimensional forms of the equations () to (6) are: v u u u u v Gr Gm M Re U u Re v v p v v v Re Re p v M Re Re Re Q v RePr Pr Sr v K Re Sc Re Re (9) () () () (3) (4) With relevant boundar conditions: : u, v cos,,, (5) : u U, v,,,, p p (6) Method of Solution: Let us suppose the solutions of the equations from (9) to (4) be of the form: u u u, o, o, o, o, o, o vv v p p p ith p p, = Substituting these in equations (9) to (4) and b equating the coefficients of the similar terms and neglecting, e get the folloing eroth-order and first-order equations. Zeroth-order equations: dv d (7) DOI:.979/ iosrjournals.org 3 Page
4 du d d u (8) Re d v Gr Gm M Re U u d d Q (9) Pr v d Pr Re d v d d Sr d K Re () d Sc Re d Re d First-order equations: v u du u u v v Gr Gm M Re u d Re v p v v v Re Re p v M Re Re Re d Q v v d Pr Re Pr d Sr v v K Re d Sc Re Re ith the boundar conditions: o: u, u, v, v cos,,,,,, : u U, u, v, v,,,,,,, p The solutions of the equations (7) to () under the boundar condition (7) are: v e A A A A e A e 3 3 A A A u A e A e A e U Cross flo solution: We shall first consider the equations (), (3) and (4) for v (, ), (, ) and p (, ) hich are independent of the main flo component u, temperature field and concentration field We assume v, and p as: v v cos v sin p Re p cos On substitution of the above, the equation () is satisfied and the equations (3) and (4) reduce to the ordinar differential equations as: Re v Rev v p (8) v Re v ( M Re ) v Re p (9) The corresponding boundar conditions are:. () () (3) (4) (5) (6) (7) DOI:.979/ iosrjournals.org 4 Page
5 : v, v : v, v Under these boundar conditions, the solutions of the equations (8) and (9) are A9 A 9 A A9 v A e A e AA p A e A e 9 A9 A Re A A9 Hence the solutions for the velocit component v, and pressure p are as follos. A v A e A e cos A A A AA 9 A9 A e e sin A9 A Re AA p A e A e cos 9 A9 A A A9 DOI:.979/ iosrjournals.org 5 Page Solution for first order flo, concentration and temperature field: e no consider the equations (), (5) and (6).To reduce these partial differential equations into ordinar differential equations, e consider the folloing assumptions for u, and.,, u u cos cos cos Substituting these expressions in equations (),(5) and (6), e get the folloing ordinar differential equations. u Reu M Re u Rev u Gr Re Gm Re (3) Q v Pr Re Re Pr Re (3) Sc Re KSc Re Sc Rev SrSc (3) With boundar conditions: : u,, : u,, The solution of the equations (3), (3) and (3) subject to the above boundar conditions are as follos: A3 A A4 A e A e A e A6 A8 A A A4 A3 A3 e A7e A8e A9e A3e A3e u Le A L A A A A A A e Le L3e L4e L5e L6e A3 A6 L e L e 7 8 Hence the solutions for the velocit component u, temperature and the concentration are as follos: A56 A A8 A36 A4 A A4 u [ L e L e L e L e L e L e L e A3 A6 L7e L8 e ]cos [ A e A e A e ]cos A3 A A A A A A A A [ A e A e A e A e A e A e ]cos
6 Thus the complete solutions of the equations (9) to (4) are: A A A A A A A u A e A e A e U L e L e L e L e [ 3 L e A L e A L e A L e A L e A ]cos A v A e 9 A e A 9 cos A9 A AA 9 e A9 e A sin A A 9 Re AA p p A A A e A e cos A 9 9 A9 A A A A e [ A e A e A e ]cos A A A A A A A A A e A e [ A e A e A e A e A e A e ]cos Skin-friction at the plate: The non-dimensional skin-friction at the plate in the direction of the free stream is given b u u v Re u u cos Re Q cos here Re u Re A A A A A A Q u Re Re L A L A L A L A L A L A L A L A L A The coefficient of rate of heat transfer: The heat flux from the plate to the fluid in terms of Nusselt number Nu is given b Nu k T Pr Re vc p T T here Nu Q cos Nu Pr Re A Pr Re Q Pr Re Pr Re A A A A A A DOI:.979/ iosrjournals.org 6 Page
7 The coefficient of rate of mass transfer: the mass transfer at the all in terms of Sherood number Sh is given b Sc Re = cos Sc Re = Sh Q cos 3 Sh= Sh 3 3 Sc Re Sc Re A A A A Q3 Sc Re A A A A A A A A A A A A Sc Re here III. Results And Discussion: In order to get a phsical insight of the problem, the numerical calculations are carried to illustrate the influence of various phsical parameters vi, chemical reaction parameter K, heat source parameter Q, Soret number Sr, and magnetic parameter on the velocit, concentration, skin-friction and Sherood number. The effect of heat source parameter Q on temperature profile is also presented. Throughout the calculations, Pr (Prandtl number) is considered to be equal.7 hich corresponds to air. Since the ater vapour is used as a diffusing chemical species of common interest in air therefore the value of Sc (Schmidt number) is taken to be.6 (ater vapour). The value of the Grashof number Gr for heat transfer has chosen as (externall cooled plate), The value of Grashof number Gm for mass transfer is considered to be 5, the free stream velocit U is selected to be, the value of (frequenc of oscillation) is considered as. and the remaining parameters are chosen arbitraril. The variation of velocit profiles u against for different values of chemical reaction parameter K, heat source parameter Q, Soret number Sr, and Hartmann number M are depicted in figures -4 respectivel. It is seen from these figures that the velocit quickl increases up to some thin laer of the liquid adjacent to the plate and after this liquid laer the fluid velocit decreases asmptoticall toards as. DOI:.979/ iosrjournals.org 7 Page
8 It is observed from figures, and 4 that chemical reaction parameter K, heat source parameter Q and Hartmann number M lead the fluid motion to retard hereas the motion is accelerated due to the Soret number Sr as seen from figure 3. The Soret number Sr defines the effect of temperature gradients inducing significant mass diffusion effects. Hence mass diffusion leads to increase the fluid motion. The application of the transverse magnetic field ill result in a resistive tpe force (Lorent force) similar to drag force hich leads to resist the fluid flos and thus reduces its velocit. Figure 5 shos the effect of the heat source parameter Q on the temperature. It is observed from this figure that ith an increase in the heat source parameter, the temperature decreases ithin the boundar laer. DOI:.979/ iosrjournals.org 8 Page
9 The concentration profiles for different values of the chemical reaction parameter, heat source parameter and Soret number are presented in figures 6-8. Figure 6 exhibits that, as the chemical reaction parameter K increases, the concentration decreases. Figures 7 and 8 sho that the concentration increases ith the increase of heat source parameter Q and Soret number Sr. Variation of Nusselt number Nu (rate of heat transfer) against Renolds number Re for different values of the heat source parameter Q is presented in figure 9. It is inferred from this figure that Nu increases ith the increasing values of Q hereas it decreases ith increasing Re. Figures, and demonstrate ho the skin-friction is effected b the Renolds number Re under the influence of chemical reaction parameter K, heat source parameter Q and Soret number Sr. It is noticed from these figures that an increase in the value of Re causes the magnitude of to decrease. It is clear from figures and that there is stead fall in the magnitude of due to chemical reaction parameter and the generating heat source hereas increases ith Soret number. DOI:.979/ iosrjournals.org 9 Page
10 The variations of Sherood number (rate of mass transfer) Sh against Renolds number Re for different values of K,Q and Sr are displaed in figures 3, 4 and 5. It is observed from these figures that an increase in the value of Re causes Sh to decrease. Figures3 and 5 sho that Sh increases ith the increasing values of K and Sr hereas figure 4 exhibits that Sh decreases ith the increasing values of Q. DOI:.979/ iosrjournals.org Page
11 IV. Conclusion: From the results and discussions of the present stud, the folloing conclusions can be dran: The velocit as ell as concentration of the fluid decreases ith an increase in the chemical reaction parameter. The heat source effect leads the fluid motion to retard hereas this motion is accelerated due to the thermal diffusion effect. An increase in value of magnetic parameter leads to fall in the velocit hich is consistent ith the la of phsics. Magnitude of skin friction decreases due to the application of generating heat source as ell as chemical reaction parameter. Magnitude of Sherood number increases due to the chemical reaction parameter and it falls due to the generating heat source. It is interesting to note that the velocit increases near the plate and then decreases smoothl aa from the plate in all the cases. NOMENCLATURE: x,, is the coordinate sstem u, v, are the components of the fluid velocit q u,, v are the non dimensional components of the fluid velocit V is the mean suction velocit U is the free stream velocit U is the non dimensional free stream velocit B is the strength of applied magnetic force C is the species concentration C is the species concentration in the free stream C is the species concentration at the plate C is the specific heat at constant pressure p D is the co-efficient of chemical molecular diffusivit m D is the co-efficient of chemical thermal diffusivit T g is the acceleration due to gravit Gr is the Grashof number for heat transfer Gm is the Grashof number for mass transfer DOI:.979/ iosrjournals.org Page
12 k is the thermal conductivit L is the ave length of the periodic suction M is the Hartmann number K is the chemical reaction parameter K is the non dimensional chemical reaction parameter p is the pressure p is the pressure in the free stream p is the non dimensional pressure p is the non dimensional pressure in the free stream Q is the first order heat source Q is the non dimensional first order heat source Re is the Renolds number Sr is the Soret number Pr is the Prandtl number Sc is the Schmidt number T is the temperature in the boundar laer T is the temperature at the plate T is the fluid temperature in the free stream Greek smbols: is the co-efficient of volume expansion for heat transfer is the co-efficient of volume expansion for mass transfer is the kinematic viscosit is the electrical conductivit is the densit of the fluid is a small reference parameter is the non dimensional temperature is the non dimensional concentration is the coefficient of viscosit. References []. Ahmed, N. and Sarma, D. Three dimensional free convective flo and heat transfer through a porous medium. Indian J. Pure and Appl. Maths, 6 (), 345 (997) []. Ahmed, N. and Gosami, J.K. Mixed convection on a three dimensional MHD mass transfer flo past a vertical plate ith thermal diffusion in presence of a heat source. Bull. Cal. Math. Soc., 3(), 7-9 (). [3]. Chamkha, A J. MHD flo of a numerical of uniforml stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. International Communication in heat mass transfer, 3, (43-4), (3). [4]. Choudhur,R.C. and Chand, T. Three dimensional flo and heat transfer through porous medium. Int. J. Appl. Mech. Engineering, 7(4), 4. [5]. Cramer, K.P. and Pai, S.I. Magneto Fluid dnamics for Engineers and applied phsics. Mc Gra-Hill Book Co, Ne York (973). [6]. Ferraro, V.C.A. and Plumpton, C. An introduction to Magneto Fluid Mechanics. Clarendon Press, Oxford.(996). [7]. Ibrahim F.S,Elai A.M. and Bakr A.A. Effect of the chemical reaction and radiation absorption on the unstead MHD free convection flo past a semi infinite vertical permeable moving plate ith heat source and suction. Communications Nonlinear Science Numerical Simulation, 3, pp (8). [8]. Kandasam R., Periasam K. and Sivagnanaprabhu K.K. Effects of chemical reaction, heat and mass transfer along a edge ith heat source an concentration in the presence of suction/injection. International Journal of Heat and Mass Transfer, Vol. 48, pp (5). [9]. Muthucumarsam,R. and Ganesan, P. First order chemical reaction on flo past an Impulsivel started vertical plate ith uniform Heat and Mass Flux. Acta Mechanica, Vol. 47, No. -4, pp []. Muthucumarsam,R. Effects of a chemical reaction on moving isothermal vertical surface ith suction. Acta Mechanica, 55 (-), 65-7, (). []. Poornima,T and Redd N. bhaskar, Effects of Thermal radiation and Chemical reaction on MHD free Convective flo past a Semi-infinite vertical porous moving plate. Int. J. of Appl. Math and Mach, 9(7), 3-46 (3). []. Rajesari R, Jothiram B and Nelson VK, Chemical reaction, heat and mass transfer on linear MHD boundar laer flo through a vertical porous surface in the presence of suction. Applied Mathematical Sciences, 3, ,(9). [3]. Raptis A and VM Soundalgekar, MHD flo past a steadil moving infinite vertical plate ith mass transfer and constant heat flux. ZAMM,64, 7-3, (984). DOI:.979/ iosrjournals.org Page
13 [4]. Sanal, D.C.and Bhattachara,S, Similarit solutions of an unstead incompressible thermal MHD boundar flo b group theoretic approach. Int. J. Engg. Sci, 3, 56, (99). [5]. Sharma, Bhupendra K, Yadav K, Mishra, Nidish K and Chaudhur R.C, Soret and Dufour effects on unstead MHD mixed convection flo past a radiative vertical porous plate embedded in a porous medium ith chemical reaction. Applied Mathematics, 3, (() [6]. Shivaiah, S and Anand Rao J, Effects of Soret and Dufour and thermal radiation on unstead MHD free convective flo past an infinite vertical porous plate in the presence of chemical reaction. Int. J. ofappl. Math and Mech, 7(3), 58-76, () [7]. Sing P, Misra J. K, and Naraan K.A, Three dimensional convective flo and heat transfer in a porous medium. Indian J. Pure and Appl. Math, 9, 3, (988) [8]. Soundalgekar V. M, S. N. Ra and U.N. Das, MHD flo past an infinite vertical oscillating plate ith mass transfer and constant heat flux. Proc. Math. Soc., (995). [9]. U.N. Das, R.K. Deka and V.M. Soundalgekar, Effects of Mass Transfer on Flo past a Impulsivel Started Infinite Vertical Plate ith Constant Heat Flux and Chemical Reaction. Forschung in Ingenieuresen,Vol. 6, No., (994) DOI:.979/ iosrjournals.org 3 Page
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