Kuvshinski fluid through porous medium

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1 Effects of chemical reaction and mass transfer on radiation and MHD free convection flow of Kuvshinski fluid through porous medium

4.1 INTRODUCTION Unsteady free convection flows in a porous medium have wived much attantion in recent time due to their wide applications in geothermal and oil reservoir engineering as well as other

2 4.1 INTRODUCTION Unsteady free convection flows in a porous medium have wived much attantion in recent time due to their wide applications in geothermal and oil reservoir engineering as well as other geophysical and astrophysical studits. Apar~ from this a considerable interest has been shown in radiation with convection for heat and mass transfer in fluids. This is due to the signifi cant role of thermal radiation in the surface heat transfer when convection heat transfer is small, particularly in free convection problems involving absorbing-emitting fluids. In many transport processes in nature and in industrial applications in which heat and mass transfer with heat radiation is a consequence of buoyancy effects caused by diffusion of heat and chemical species. The study of such processes is useful for improving a number of chemical technologies, such as polymer production and f d processing. The present trend in the field of chemical reaction with heat radiation analysis is to give a mathematical model for the system to predict the reactor performance. A large amount of research work has been reported in this field. In particular, the study of chemical reaction, heat and mass transfer with radiation is of considerable importance in chemical and hydrometallurgical industries, Chemical reaction can be codified as either heterogeneous or homogeneous processes. This depends on whether they occur at an interface or as a single phase volume reaction. A reaction is said to be first order if the rate of reaction is directly proponional to its concentration. Radiative convective flows are encountered in countless industrial end environmental processes e.g. heating and cooling chambers, fossil fuel combustion energy processes, evaporation fiom large open water reservoirs astrophysical flows, solar power technology and space vehicle re-entry. Radiative heat and mass transfer play an important role in manufacturing industries for the design of reliable equipment, Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and s pw vehicles are examples of such engineering applications. If the temperature of the surrounding fluid is rather high, radiation effects play an important role and this situation does exist in space technology. In such cam, one has to take into BCOOWI~ the

effect of thermal radiation and mass diffusion. The unsteady flow past a moving plate in the presence of free convection and radiation was studied by Mansour [S].

3 effect of thermal radiation and mass diffusion. The unsteady flow past a moving plate in the presence of free convection and radiation was studied by Mansour [S]. Raptis [15] has analyzed the thermal radiation and free convection flow through porous medium by using perturbation technique. Jaypal Singh and Gupta [6] have discussed radiation and free convection flow of Kuvshinski fluid through porous medium. Effect of mass transfer on radiation and free convection flow of Kuvshinski fluid through a porous medium was studied by Harish Kumar Sharma and Kaanodiya [4]. Raji Reddy and Srihari [I 11 have analyzed the Numerical solution of unsteady flow of a radiating and chemically reacting fluid with time-dependent suction. The science of magneto hydrodynamics (MHD) was concerned with geophysical and astrophysical problems for a number of years. In recent years the possible use of MHD is to affect a flow stream of an electrically conducting fluid for the purpose of thermal protection, braking, propulsion and control. From the point of applications, model studies on the effect of magnetic field on free convection flows have been made by several investigators. Raptis [14] studied mathematically the case of time varying twodimensional natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous plate embedded in a porous medium. Alagoa et al. [I] studied radiative and free convection effects on MHD flow through porous medium between infinite parallel plates with time dependent suction. Radiation and mass transfer effects on an unsteady MHD free convection flow past a heated vertical plate in a porous medium with viscous dissipation was analyzed by Ramachandra Prasad and Bhaskar Reddy [13]. Raptis and Perdikis 1171 studied the effects of thermal radiation and free convective flow past moving plate. Das et al. [3] have analyzed the radiation effects on flow past an impulsively started infinite isothermal vertical plate. Chamka et a1 [2] studied the effect of radiation on free convective flow past a semi-infinite vertical plate with mass transfer. Transient radiative hydro magnetic free convection flow past an impulsively started vertical plate with uniform heat and mass flux was studied by Ramachandra Prasad et a1 [12]. Kim [7] studied the Unsteady MHD convective heat transfer past a semi-infinite vertical plate with variable suction Muthucumaraswamy and Ganesan [9] studied the problem of unsteady flow past an impulsively started isothermal vertical plate with mass transfer by an implicit finite

4 difference method. Soundalgekar and Takhar [I81 have considered the radiative free convective flow of an optically thin gray-eas past a semi-infinite plate. Kadiation eff'ect on mixed convection along a isothermal vertical plate ujere studied by tfossuin and Takhar [5]. Muthucumaraswamy et al 1101 analyzed the studg ol' MtiIl and radiation effects on moving vertical plate with variable temptraturc and mass diftusion. Magnetohydrodynamic free convection flow an mass transfer trough porous medium bounded by an infinite vertical plate with constant heat flus was studied hy Kupris and Kafoussias [I 61 Since no attempt has been made to analyze the chr~nical rsuction and muss transfer effects on radiation and MHD frcc convection llow of Kusshinski fluid tlirough porous medium, we have investigated it in tliis chapter. The equations of continuity, linear momentum, energy and diffusion, which govern the flow licld, nre soliled hy using perturbation technique and for difl'erent values of dimcnsionicss parameters of \lie problem under consideration for the purpose of illustrating the results grapliicnlly. Examination of such flow models rcveals the influence of chemical renction and magnetic parameters on velocity, temperature and concentration profiles. The results obtained show that thc fluid is influenced appreciably by the presence of chemical reaction and magnetic parameters. It is hoped that tlic rssults ohtilined will riot only provide useful information for applications, but also serve as a conil~lc~~~en~ 10 11ic previous studies, 4.2 FORMULATION OF THE PROBLEM An unsteady of two-dimensional free convectiori llaw of a viscaua, incompressible, radiating and chemically reacting Kuvshinski fluid through a porous medium occupying a semi-infinite region of the spacc bounded by an infinite vcrlical plate is considered. The x -axis is taken along the vcnical plate in thc upward direction and y' - axis is taken normal to it. Initially, it is assumed that the plate and the fluid are at the same temperature Ti and concentration level (', everywhere in the fluid, Thc level of foreign mass is assumed to be low, so that Soret and Uufour effects are

5 negligible. The radiative heat flux in the x' - direction is considered negligible in comparison to that in the y' - direction. The fluid assumed to be gray emitting and absorbing radiation but non-scattering medium. All the fluid properties are considered constant except the influence of the density variation with temperature is considered only in the body-force term. Now, under the above assumptions, the flow field is governed by the following equations. Continuity equation Momentum equation Energy equation Diffusion equation The initial and boundary conditions are: Where u' and v' are the parallel and perpendicular components of velocity to the plate, t'is the time, pis the fluid density, vis the kinematic viscosity, C, is the specific heat at constant pressure, g is the acceleration due to gravity, Pis the volumetric coefficient of thermal expansion, P is the volumetric coefficient of concentration expansion, T' is the dimensional temperature, Ti is the wall temperature, Ti is the free stream temperature far away from the plate, C is the species concentration, C', is the

6 concentration at the plate, Ca is the fw man concentration in fluid far away from the plate, u is the thermal conduction, D is the mass difisivity, q, is the radiation heat flux, K' is the permeability of the porous medium, C, is the specific heat at constant pressure, Kr' is the chemical reaction parameter. The third term on right hand side of equation (4.3) represents radiative heat flux term. The equation of continuity (4,l) gives v =-Yo Where v, is the constant suction velocity normal to the plate. By using the Rosseland approximation, the radiative heat flux 9, is given by 40' ata q, =--- 3k' dy' (4.6) whete a'is the Stefan-Boltzmann constant and k' is the mean absorption coeficient. It should be noted that by using the Rosseland approximation the present analysis is limited to optically thick fluids. Assuming that the differences in temperature within flow are such that T" can be expressed as a linear combination of the temperature, we expand T" in a Taylor's series about T: as follows T" =T: +~T:(T-T~)+~T:(T-T*Y t... and neglecting higher order terms beyond the first degree in (T' -T*). we get 4 4 T z~t, T'-~T. (4.8) Substituting equation (4.8) in equation (4.3) we obtain Using equation (5.8) in equation (5.3) we obtain

7 In view of (4.6) and (4.9) equations (4.2), (4.3) and (4.4) now reduce to We introduce now the following non-dimensional variables:, a >,=-, u y = ~ ex-, T-T, #=-, c0-c; Gr= V@(T;-T;) YO v T,-T:' c,-c, vi V~J* (c; 4:) e 8; Gm =, M=- 3 E=-j, v,' v2 K=$K' v,' k'k Kr v vi t vo2 Pr=%,N=-, k Kr=-, a,=(i+$), a2=i-, I=- 4u T, V: v v In view of (4.12) the equations (4.9), (4.10) and (4.1 1) are reduced to the following non-dimensional form ~(4.12) Where Gr is the thermal Grashof number, Gm is the modified Grashof number, Pr is the fluid Prandtl number, Sc is the Schmidt number and Kr is the chemical reaction parameter. The corresponding boundary conditions become

8 43 METHOD OF SOLUTION To solve the coupled non-linear equations (4.13), (4.14) and (4.15) subject to the boundary conditions (4.16), we may represent the velocity (u), temperature (8) and concentration 0) of the fluid in powers of Echen number E, assuming that it is very small. Hence we can write substituting equations (4.17) into equations (4.13), (4.14) and (4.15) and equating harmonic and non-harmonic terms, neglecting terms in E2 and higher order. We get, here the primes denote differentiation with respect toy The corresponding boundary conditions now become

9 Solving equations (4.18)-(4.23) under the boundary conditions (4.24) and substituting the solutions into equations (4.17), we take 4.4 RESULTS AND DISCUSSION The problem of an unsteady free convection flow of a viscous, incompressible, radiating and chemically reacting Kuvshinski fluid flow past was formulated and solved by means of a perturbation method. The expressions for the velocity, temperature and concentration were obtained. The behavior of these physical quantities, with respect to the variations in the governing parameters viz., the thermal Grashof number Gr, the solutal Grashof number Gc, Prandtl number Pr, Schmidt number Sc, the radiaiion parameter N, the chemitx$@eaction parameter K and the permeability parameter k is studies - The velocity andlidncentration profiles for different values of the chemical reaction parameter Kr are displays in the Figures l(a) and I(b). It is notice that an increase in Kt leads to decrease in both the values of velocity and concentration. The velocity profiles for different values of the thermal Grashof number Gr are described in Figure 2(a). It is observed that an increase in Gr leads to rise in the values of velocity. For the case of different values of the modified Grashof number Gm, the! velocity profiles are show~'in.the Figure 2(b). It is observed that an increase in Gr leads to a rise in the values of velocity.

10 Figures 3(a) and 3(b) illustrate the behavior of the velocity and temperatures for different values of the Prandtl number Pr. The numerical results show that the effect of increasing values of Prandtl number results in a deceasing velocity. From Figure 3(b), is observed that an increase in the Prandtl number results in a decreasing in temperature. The effects of Schmidt number Sc on the velocity and concentration an shown in Figures 4(a) and 4(b). As the Schmidt number increases, the concentration dacnases. This causes the concentration buoyancy effects to decrease the fluid velocity. Figures S(a) and 5(b) represent the velocity and temperature prof les for different values of the thermal radiation parameter N. It is noticed that an increase in the thermal radiation parameter results a decrease in the velocity and temperature. The velocity and temperature profiles are shown in Figures 6(a) and 6(b) for different values of Eckert number E. An increase in Eckert number E leads to increase in both velocity and temperature. The effect of different values of Hanmann number (magnetic parameter M) for velocity profile is shown in Figure 7. decrease in velocity. It is observed that an increase in M leads to Figure 8 illustrate that the velocity profile for different values of permeable parameter k. From this figure it is clear that the velocity increases with increase of k. it 4.5 APPENDIX

11 -Pra:B: B, = b(20,)' t2pr0, -(n2'pra, -npr] - 2a6 B, =, - ~r Za, ( r)b, B, B, = b[az +Pr(a2 -F)-(~~ Pra, -npr)' - Pr a2a6b, B, B, =, B,,=I-(B,+B,+B,+B, +B,+B,) b(a, ta6)'t~r(a2 tab)-(n2 Pra, -npr) -Gr B,, -Gr B4

12 4.6 REFERENCES 1. Alagoa K. D., Tay G. and Abbey T. M., "Radiative and free convection effects of a MHD flow through porous medium between infinite parallel plates with time dependent suction", Astrophysics. Space. Sci., Vol. 260, pp , Chamkha, A. J. Takhar, H.S. and Soundalgekar, V.M., "Radiation effects on free convection flow past a sem-infinite vertical plate with mass transferu, Chem. Engg. J, 84, pp , Das, U.N., Deka, R, K and Soundalgekar, V.M., "Radiation effects on flow past an impulsively started vertical plate-an exact solution", J. Thto, Appl, Fluid Mech, 1(2), pp , Harish Kumar Sharma and Kaanodiya, K.K., "Effect of mass transfer on radiation and free convection flow of Kuvshinski fluid through a porous medium", Indian Journal of Mathematics and Mathematical Sciences, Val, 3, No. 2, pp , Dec, 2007, 5. Houssin M. A, and Takhar H. S.,."Radiation effecr on mixed convection dong a vertical plate with uniform surface temperature", Heat and Mass Transfer, Vol. 3 I. pp , 1996.

13 Jaypal Singh and C. 0. Gupta., "Radiation and free convection flow of Kuvshinski fluid through a porous medium", Journal of MANIT, 36, pp Kim, Y. J., "Unsteady MHD convective heat transfer past a semi-infinite vertical plate with variable suction", International Journal of Engineering Science, Vol. 38, pp 833,2000. Mansour, M.H., "Radiative and free convection effects on the oscillatory flow past a vertical plate", Astrophysics and space science 166, pp Muthucumaraswamy R, and Ganesan P., "Unsteady flow past an impulsively started vertical plate with heat and mass transfer", Heat Mass Transfer, Vol. 38, pp ,1998. Muthucumaraswamy, R, Maheswari, J. and Pandurangan, J., "Study of MHD and Radiation Effects on Moving Vertical Plate with Variable Temperature and Mass Diffusion", International Review of Pure and Applied Mathematics, Vol. 3, No. I, pp Raji Reddy, S, and Srihari, K., "Numerical solution of unsteady flow of a radiating and chemically reacting fluid with time-dependent suction", Indian Journal of Pure & Applied Physics, Vol. 47. pp 7-1 1, January Rarnachandra Prasad, V, Bhaskar Reddy, N and Muthucumaraswarny, R., "Transient radiative hydromagnetic free convection flow past an impulsively started vertical plate with uniform heat and mass flux", Theoret, Appl. Mehc., Vol. 33, No.], pp , Belgrade Ramachandra Prasad, R and Bhaskar Reddy, N., "Radiation and mass transfer effects on an unsteady MHD free convection flow past a heated vertical plate in a porous medium with viscous dissipation", Theoret. Appl. Mech., Vol. 34, No.2, pp , Belgrade, Raptis A,, "Flow through a porous medium in the presence of magnetic field", Int. J. Energy Res., Vol. 10, pp , Raptis, A,, "Radiation and free convection flow through a porous medium", Int. Cornm. Heat Mass Transfer 25 (2) pp , 1998.

14 16. Raptis, A. and Kafoussias, N. G., "Magnetohydrodynamic fne convection flow and mass transfer through porous medium bounded by an infinite vertical porous plate with constant heat flux", Can. J, Phys. Vol. 60, pp , Raptis, A, Perdikis, C., "Radiation and free convection flow past a moving. plate", Appl. Mech. Eng. 4. pp , Soundalgekar, V.M, and Takhar, H.S., "Radiation emcts on free convection flow past a semi-infinite vertical plate", Modeling Measurement and Control, Vol. R 51, pp. 3140,1993.

15 Y Figure - I (b) Concentration profile for different values of fi

16 Figure - 2(a) Velocity profile for different values of Gr Figure - 2(b) Velocity profile for different values of Gm

17 - Y Figure - 3(a) Velocitypmfile for different values ofpr Y Figure - 3(b) Temperature profile for different values ofpr

18 h=0.71,n=2 Figure - 4(a) Velocity profile for different values of Sc Y Figure - 4@) Concentmiion profile for different values of Sc 74

19 Figure - S(a)Velocity profile for different values ofn Figure - S(a)Temperahwe profile for different values ofn

20 Kr=0.2,Sc=0.6. E=O. A "-L --- i Y Figure - b(a)velocity profile for different values of E I '.-. M=5,k=3,t=OGI E= ~ -.- d - a. Oo Y Figure. 6(a)Temperature profile for different values of E 76

21 Figure - 7Velocitypmfile for different values ofm Y Figure - 8 Velocity profile for different values of k

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