International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 3, Aug 2013, 13-22 ÂŠ TJPRC Pvt. Ltd.

CHEMICAL REACTION EFFECTS ON MHD FREE CONVECTION FLOW THROUGH A POROUS MEDIUM BOUNDED BY AN INCLINED SURFACE S. MASTHANRAO1, K. S. BALAMURUGAN2 & S. V. K. VARMA3 1,2

Department of Mathematics, RVR & JC College of Engineering, Guntur, Andhra Pradesh, India 3

Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India

ABSTRACT In this analysis the effect of chemical reaction have been discussed on steady two-dimensional free convection flow of a viscous incompressible electrically conducting fluid through a porous medium bounded by an inclined surface with constant suction velocity, constant heat and mass flux in the presence of uniform magnetic field. The non-linear partial differential equations governing the fluid flow are solved using perturbation method and the expressions are obtained for velocity, temperature and concentration fields. The skin friction coefficient and the rate of heat transfer in terms of Nusselt number at the surface are also derived. The numerical results are presented graphically for different values of the parameters entering into the problem. All numerical calculations are done with respect to air (Pr = 0.71 at 20 0C).

KEYWORDS: MHD, Free Convection, Porous Medium, Chemical Reaction, Inclined Surface INTRODUCTION The problem of free convection and mass transfer flow of an electrically conducting fluid past an inclined surface under the influence of a magnetic field has attracted interest in view of its application to geophysics, astrophysics and many engineering problems. Such as cooling of nuclear reactors, the boundary layer control in aerodynamics and cooling towers. In light of these applications, Umemura and Law [1] developed a generalized formulation for the natural convection boundary layer flow over a flat plate with arbitrary inclination. They found that the flow characteristics depend not only on the extent of inclination but also on the distance from the leading edge. Hossain et al. [2] studied the free convection flow from an isothermal plate inclined at a small angle to the horizontal. Anghel et al. [3] presented a numerical solution of free convection flow past an inclined surface. Chen [4] performed an analysis to study the natural convection flow over a permeable inclined surface with variable wall temperature and concentration He observed that increasing the angle of inclination decreases the effect of buoyancy force. Ganesan and Palani [5] have dealt with the natural convection past an inclined plate where in they have studied the effects of magnetic field under the conditions of variable surface heat and mass flux. Sivasankaran et al. [6] presented a Lie group analysis of natural convection heat and mass transfer in an inclined surface. Alam and Rahman [7] studied the MHD free convection and mass transfer flow past an inclined semi infinite surface in the presence of heat generation. Bhuvaneswari et al. [8] studied exact analysis of radiation convective flow heat and mass transfer over an inclined plate in a porous medium. Diffusion rates can be altered tremendously by chemical reactions. The Effect of a chemical reaction depends whether the reaction is homogeneous or heterogeneous. This depends on whether they occur in an interface or as a single phase volume reaction. In a well mixed system, the reaction is heterogeneous if the reactants are in multiple phase, and homogeneous if the reactants are in the same phase. In most cases of chemical reactions, the reaction rate depends on the concentration of the species itself. Kandasamy and Devi [9] studied the effects of chemical reaction, heat and mass transfer on non-linear laminar boundary-layer flow over a wedge with suction or injection. Kandasamy et al. [10] studied the

14

S. Masthanrao, K. S. Balamurugan & S. V. K. Varma

effects of chemical reaction heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection. The problem of unsteady two-dimensional flow of a radiating and chemically reacting fluid with time dependent suction was studied by Prakash and Ogulu [11]. Kandasamy and Hashim [12] investigated the effects of variable viscosity, heat and mass transfer on nonlinear mixed convection flow over a porous wedge with heat radiation in the presence of chemical reaction. Kandasamy et al. [13] studied thermophoresis and variable viscosity effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of chemical reaction. Jyothi Bala and Vijaya Kumar [14] analyzed the problem of unsteady MHD heat and mass transfer flow past a semi infinite vertical porous moving plate with variable suction in the presence of heat generation and homogeneous chemical reaction. Hitesh Kumar [15] presented an analytical solution to the problem of radiative heat and mass transfer over an inclined plate at prescribed heat flux in the presence of chemical reaction. The objective of the present study is to investigate the effect of chemical reaction on steady two-dimensional free convection flow of a viscous incompressible electrically conducting fluid through a porous medium bounded by an inclined surface with constant suction velocity, constant heat and mass flux under the influence of uniform magnetic field applied normal to the direction of flow.

MATHEMATICAL FORMULATION We consider steady two dimensional motion of viscous incompressible electrically conducting fluid through a porous medium occupying semi-infinite region of space bounded by an inclined surface under the action of uniform magnetic field applied normal to the direction of flow. The flow is assumed to be in the x-direction, which is taken along the semi-infinite inclined plate and y-axis is taken normal to it. The magnetic Reynolds number is assumed to be small. The effect of induced magnetic field is neglected. The fluid properties are assumed constant except for the influence of density in the body force term. As the boundary surface is infinite in length, all the variables are functions of y only. Hence, by the usual boundary layer approximation the basic equations for steady flow through highly porous medium are:

v 0 y

v

B02 u 2u u u v 2 g cos (T T ) g cos * (C C ) y k y

T 2T u v y C p y 2 C p y

v

(1)

(2)

2

(3)

C 2C D 2 D1 (C C ) y y

(4)

Where u and v are the corresponding velocity components along and perpendicular to the surface, the kinematic viscosity, g the acceleration due to gravity,

* is

the coefficient of volume expansion for the heat transfer and

the volumetric coefficient of expansion with species concentration, T the fluid temperature, T ∞ the far field

temperature, λ the thermal conductivity, ρ the density of the fluid, Cp specific heat at constant pressure, C the species concentration, C∞ the far field concentration, D the chemical molecular diffusivity,

D1 the rate of chemical reaction and k

Chemical Reaction Effects on MHD Free Convection Flow through a Porous Medium Bounded by an Inclined Surface

the permeability of porous medium,

15

is the angle of inclination, is the fluid electrical conductivity, B0 is the magnetic

field component along y axis.

METHOD OF SOLUTION The equation of continuity (1) gives v = constant = - v 0

(5)

where v 0 > 0 corresponds to steady suction velocity (normal) at the surface. In view of equation (5), equations (2), (3) and (4) are reduced to

v0

B02 u 2u u u v 2 g cos (T T ) g cos * (C C ) y k y

T 2T u v0 y C p y 2 C p y v0

(6)

2

(7)

C 2C D 2 D1 (C C ) y y

(8)

The relevant boundary conditions are

u =0,

T q , y

C m y D

u =0,

T = T∞,

C = C∞

y=0

(9)

y→∞

Where q is the heat flux per unit area and m is the mass flux per unit area. The dimensionless parameters introduced in the above equations are defined as follows:

f ( )

Cp v y u , 0 , Pr , v0

Sc

D

gq 2 g * m 2 2 , Gr 4 , Gm , v0 v 04 D v 02 k

Where

,

(T T )v 0 (C C )v 0 D , C* , q m

B02 v 03 D M , E , Kr 2 1 2 q C p v0 v0

(10)

is the distance, f ( ) is the velocity, Pr is the Prandtl number, Sc is the Schmidt number, Gr is the

Grashof number, Gm is the solutal Grashof number, E is the Eckert number, M is the magnetic field parameter, Kr is the Chemical reaction parameter,

k

is the Permeability of porous medium, α is the permeability parameter, θ is the

dimensionless temperature, C is the dimensionless concentration. Now substituting equation (10) in equations (6) – (8) and dropping asterisk, we obtain:

f '' f ' ( M 1 ) Gr1 Gm1C

(11)

'' Pr ' E Pr( f ' ) 2

(12)

C '' ScC ' KrScC 0

(13)

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S. Masthanrao, K. S. Balamurugan & S. V. K. Varma

Where

Gr1 Gr cos

and Gm1 Gm cos and primes denote differentiation with respect to η.

The boundary conditions (9) then turn into:

0

f 0 f 0

' 1 0

C ' 1 C 0

(14)

Now, equations (11) – (13) are coupled nonlinear differential equations, which can be obtained by expanding the velocity, temperature and concentration in powers of E. As in the case of incompressible fluids, E is always very small. Hence, we can write

f ( ) f 0 ( ) E f1 ( ) O( E 2 )

( ) 0 ( ) E 1 ( ) O( E 2 )

(15)

C ( ) C0 ( ) E C1 ( ) O( E 2 ) Substituting (15) in equations (11) - (13) and equating harmonic and non-harmonic terms, and neglecting higher 2

order terms of O( E ) , we obtain

f 0'' f 0' ( M 1 ) f 0 Gr1 0 Gm1C 0

(16)

f1'' f1' ( M 1 ) f1 Gr11 Gm1C1

(17)

0'' Pr 0' 0

(18)

1'' Pr 1' Pr( f 0' ) 2

(19)

C0'' Sc C0' KrSc C0 0

(20)

C1'' Sc C1' KrSc C1 0

(21)

With the corresponding boundary conditions

f 0 0, f 0 0,

0' 1, 1' 0, C 0' 1, C1' 0 at 0 f1 0, 0 0, 1 0, C 0 0, C1 0 at f1 0,

(22)

Solution of equation (16) - (21) with boundary condition (22) is as follows:

0

e Pr Pr

C 0 k1e t1

f 0 k 3 e Pr k 4 e t1 k 5 e t2

(23)

(24)

(25)

17

Chemical Reaction Effects on MHD Free Convection Flow through a Porous Medium Bounded by an Inclined Surface

1 k6 e 2t k7 e 2 Pr k8e 2t k9 e (t Pr) k10 e (t Pr) k11e (t t ) 2

1

2

1

1

2

(26)

k12 e Pr

f1 k13e k3 k14 e Pr k15 e 2 k3 k16 e 2 Pr k17 e 2 Sc k18e ( k3 Sc ) k19 e ( pr Sc ) k 20 e ( k3 pr )

(27)

SKIN FRICTION From velocity field, now we study the skin friction coefficient at the surface. It is given in the non dimensional form as

xy / v02 y 0 Pr k3 t1k 4 t 2 k5 0 f

Pr k13 2 prk14 2t1k15 t 2 k16 2t 2 k17 (t1 Pr)k18 E ( pr t ) k ( t t ) k 2 19 1 2 20

(28)

NUSSELT NUMBER From temperature field, now we study Nusselt number (rate of change of heat transfer). It is given in non dimensional form as

prk12 2t 2 k 6 2 prk 7 2t1k 8 (t 2 pr )k 9 1 E Nu 0 (t1 Pr)k10 (t1 t 2 )k11

(29)

RESULTS AND DISCUSSIONS In order to get a clear insight of the physical problem the velocity, temperature and concentration have been discussed by assigning numerical values to the parameters like Chemical reaction parameter ( Kr), Magnetic parameter (M), Schmidt parameter (Sc), angle of inclination ( ), Eckert number (E), Permeability parameter ( ), thermal Grashof number (Gr) and mass Grashof number (Gm) from figures 1-8 for the cases of cooling (Gr > 0) and heating (Gr <0) of plate. The heating and cooling takes place by setting up free convection currents due to temperature and concentration gradient. Figure 1 depicts the effect of magnetic field parameter on the fluid velocity and we observed that an increase in magnetic field parameter the velocity decreases in case of cooling of the plate, it is due to the presence of magnetic field normal to the flow in an electrically conducting fluid introduces a Lorentz force which acts against to the flow while it increases in case of heating. The velocity profiles for different values of permeability parameter

are shown in Figure 2.

From this it is seen that the velocity increases with increasing values of permeability parameter in the case of cooling of the plate but decreases in the case of heating of the plate. Figure 3 shows the effect of angle of inclination The velocity decreases as the

on the velocity.

increases. The fluid has higher velocity when the surface is vertical than, when inclined

because of the fact that the buoyancy effect decreases due to gravity components (g cos ) as the plate is inclined. In the case of heating of the plate, it is observed that

increases the velocity. This is because the negative Gr makes the rate of

18

S. Masthanrao, K. S. Balamurugan & S. V. K. Varma

heat transfer also negative and at wall, heat flux become positive and that accelerate the convection effect at the wall; Stream function for lower values of thermal Grashof number become more thinner (diluted) due to stronger convection, hence the velocity reduces. Figure 4 reveals the influence of chemical reaction parameter Kr on the velocity profile. It is observed that the increase in chemical reaction parameter leads to fall in velocity in the both cases of cooling and heating of the plate.

Figure 1: Velocity Profiles When Pr = 0.71, Sc = 0.60, Gm = 2.0, E = 0.02, α = 1.0,

= π/6, Kr = 0.5

Figure 2: Velocity Profiles When Pr = 0.71, Sc = 0.60, Gm = 2.0, E = 0.02, M = 1.0,

= π/6, Kr =0.5

Figure 3: Velocity Profiles When Pr = 0.71, Gm = 2.0, E = 0.02, α = 1.0, M = 1.0, Sc= 0.60, Kr = 0.5

19

Chemical Reaction Effects on MHD Free Convection Flow through a Porous Medium Bounded by an Inclined Surface

Figure 4: Velocity Profiles When Pr = 0.71, Sc = 0.60, Gm = 2.0, E = 0.02, M = 1.0,

= π/6, α = 1.0

Figure 5: Temperature Profiles When Pr = 0.71, Gm = 2.0, M = 1, E = 0.02, α = 1.0, Sc = 0.60, Kr = 0.5

Figure 6: Temperature Profiles When Pr = 0.71, Gm = 2.0, M = 1, E = 0.02, α = 1.0, Sc = 0.60,

= π/6

20

S. Masthanrao, K. S. Balamurugan & S. V. K. Varma

Figure 7: Concentration Profiles When Kr = 0.1

Figure 8: Concentration Profiles When Sc = 0.60 The influence of angle of inclination

ď Ś and chemical reaction parameter Kr on the fluid temperature are

illustrated in figures 5 and 6. Figure 5 shows the effect of

ď Ś on temperature of the air (Pr = 0.71) boundary layer. It is seen

that the thermal boundary layer decreases with the increase of angle of inclination in both cases of cooling and heating of the plate. The temperature profile for different values of chemical reaction parameter is shown in Figure 6. From this it is seen that the temperature decreases with increasing values of Kr in the case of cooling of the plate but increases in the case of heating of the plate. The concentration profiles for different values of Sc (Schmidt number), chemical reaction parameter Kr are presented through figures 7 and 8. From these figures it is seen that the concentration decreases with an increase in Sc while it increases with Kr.

CONCLUSIONS In this paper, chemical reaction effect on steady two-dimensional free convection flow of a viscous incompressible electrically conducting fluid through a porous medium bounded by an inclined surface with constant suction velocity, constant heat and mass flux in the presence of uniform magnetic field has been studied. The

21

dimensionless governing equations are solved using perturbation technique. From this investigation it is found that velocity decreases with increasing inclined surface ď Ś , chemical reaction Kr in the case of cooling of the plate. The temperature of the fluid decreases as the angle of inclination

ď Ś increases in both cases of cooling and heating of the plate. The

temperature decreases as chemical reaction Kr increases in the case of cooling of the plate and the trend is reversed in the heating of the plate. The concentration profiles decreases with an increase in Schmidt number Sc while it increases with chemical reaction Kr.

REFERENCES 1.

Umemura, A. and Law, C. K. (1990), Natural convection boundary layer flow over a heated plate with arbitrary inclination, J. Fluid Mech., Vol. 219, pp. 571-584.

2.

Hossain, M. A. Pop, I. and Ahamad, M. (1996), MHD free convection flow from an isothermal plate inclined at a small angle to the horizontal, J. Theo. Appl. Fluid Mech., Vol.1, pp. 194-207.

3.

Anghel, M. Hossain, M. A. Zeb, S. and Pop, I. (2001), Combined heat and mass transfer by free convection past an inclined flat plate, Int. J. Appl. Mech. and Engng., Vol. 2, pp.473-497.

4.

Chen, C. H. (2004), Heat and mass transfer in MHD flow by natural convection from a permeable inclined surface with variable wall temperature and concentration, Acta Mechanica., Vol. 172, pp. 219- 235.

5.

Ganesan, P. and Palani, G. (2004), Finite difference analysis of unsteady natural convection MHD flow past an inclined plate with variable surface heat and mass flux, Int. J. Heat Mass Transfer., Vol. 47, pp. 4449-4457.

6.

Sivasankaran, S. Bhuvaneswari, M. Kandaswamy, P. and Ramasami, E. K. (2006), Lie Group Analysis of Natural Convection Heat and Mass Transfer in an Inclined Surface, Non-linear Analysis; Modeling and Control., Vol. II (l), pp. 201- 212.

7.

Alam, M.S. and Rahman, M. M (2006), MHD Free Convective Heat and Mass Transfer Flow Past an Inclined Surface with Heat Generation, Thammasat Int. J. Sc. Tech., Vol. 1l, No. 4, PP. 1-8.

8.

Bhuvaneswari, M. Sivasankaran, S. and Kim, Y J. (2010), Exact analysis of radiation convective flow heat and mass transfer over an inclined plate in a porous medium, World Applied Sciences Journal. Vol. 10, pp. 774-778.

9.

Kandasamy, R. and Anjali Devi, S P. (2004), Effects of chemical reaction, heat and mass transfer on non-linear laminar boundary-layer flow over a wedge with suction or injection, Journal of Computational and Applied Mechanics, Vol. 5, pp. 21-31.

10. Kandasamy, R. Periasamy, K. and Sivagnana Prabhu, K K. (2005), Effects of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection, Int. J. Heat Mass Transfer, Vol. 48, pp. 1388- 1396. 11. Prakash, J and Ogulu. A. (2006), Unsteady two-dimensional flow of a radiating and chemically reacating MHD fluid with time dependent suction, Ind. J. pure & Appl. Phys., Vol. 44, pp. 805-810. 12. Kandasamy, R. and Hashim, I. (2009), Effects of variable viscosity, heat and mass transfer on nonlinear mixed convection flow over a porous wedge with heat radiation in the presence of homogeneous chemical reaction, ARPN J. Engg. Applied Sci., Vol. 2, pp. 44-53.

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S. Masthanrao, K. S. Balamurugan & S. V. K. Varma

13. Kandasamy, R. Muhaimin, I. and Khamis, Azme B. (2009), Thermophoresis and variable viscosity effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of chemical reaction, Heat and Mass Transfer, Vol. 45 pp. 703-712. 14. Jyothi Bala, A. and Vijaya Kumar Varma, S. (2011), unsteady MHD heat and mass transfer flow past a semi infinite vertical porous moving plate with variable suction in the presence of heat generation and homogeneous chemical reaction, Int. J. of Appl. Math and Mech. Vol. 7 (7) pp. 20-44. 15. Hitesh Kumar. (2012), An analytical solution to the problem of radiative heat and mass transfer over an inclined plate at prescribed heat flux with chemical reaction, J. Serb. Chem. Soc. Vol. 77, pp. 1-14.

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