Effect of Volumetric Heat Generation / Absorption on Convective Heat and Mass Transfer in Porous Medium Between Two Vertical Porous Plates

Effect of Volumetric Heat Generation / Absorption on Convective Heat and Mass Transfer in Porous Medium Between Two Vertical Porous Plates

P. R. Sharma

Professor, Department of Mathematics, University of Rajasthan, Jaipur-302004, India.

I. K. Dadheech

Assistant Professor, Birla Institute of Technology, Mesra (Ranchi)

Ext. Center Jaipur, India.

Abstract

Aim of the paper is to investigate the effect of

Q Q(T T ) ,

(1)

volumetric heat generation, which is a function of local species concentration on steady free convective mass transfer flow of a viscous fluid through a porous medium bounded by two stationary infinite vertical porous plates. The analytical solution to the problem is obtained and presented for different values of physical parameters through graphs. The skin friction coefficient, the heat and the mass transfer coefficients are derived, discussed numerically and presented through tables for different values of parameters.

1. Introduction

A large number of physical phenomena involve free/forced convection [Jaluria, (1)], which are enhanced and driven by internal heat generation. In such flows the buoyancy force is incremented due to heat generation resulting in modification of heat/mass transfer characteristics. Convection in the presence of internal heat generation/absorption has numerous

applications in the fields of geophysical science, fire

Q Q(x, y, z) . (2)

The equation (1) represents volumetric heat generation/absorption [Vajravelu and Nayfeh (3) and Vajravelu and Hadjinicolaou (4)], which depends on local fluid temperature and is considered for exothermic/endothermic chemical reactions. The temperature distribution is not known a priori. The equation (2) represents the volumetric heat generation [Crepeau and Clarksean (5), Chamkha and Khaled (6)], which has priori known space distribution not controlled by flow pattern and local temperature distribution directly.

To find a solution the governing equations are made dimensionless and the volumetric rate of heat generation/ absorption discussed above would take the form S , where S is heat source/sink parameter. The local concentration of specie should also be a major factor affecting the volumetric rate of heat generation/ absorption. Singh et al. [2] proposed a new kind of volumetric heat generation/absorption

and safety engineering, nuclear science, chemical engineering etc. The volumetric rate of heat

Q Q(C C )

(3)

generation Q in the boundary layer flows generally has been presented in the literatures. Singh et al. [2] investigated the effect of volumetric heat generation/absorption on mixed convection stagnation point flow on an iso- thermal vertical plate in porous media taking the following form

In the dimensionless form it would take the form S

where is the shape function i.e. the local concentration distribution.

Samad and Mohebujjaman [7] investigated MHD

heat and mass transfer free convection flow along a vertical stretching sheet in presence of magnetic field with heat generation. Jha and Ajibade [8] discussed free convective flow of a heat generating/absorbing fluid between vertical porous plates with periodic

heat input. Abdallah [9] derived analytic solution of heat and mass transfer over a permeable stretching plate affected by chemical reaction, internal heating, Dufour-Soret effect and Hall effect. Ahmed et al.

Makinde [11] discussed heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in highly porous medium with radiation and internal heat generation.

The boundary conditions are

y 0:u 0:T T0 :C C0 ;

y h:u 0:T T1 :C C1 . (8)

3. Method of Solution

Introducing the following non-dimensional quantities

In the present paper the effect of volumetric heat

y u T T

C C

generation, which is a function of local species

, f , ,

concentration on steady free convective mass transfer flow of a viscous fluid through a porous medium

y v0

hg T T

T0 T

hg C C

C0 C

~

bounded by two stationary infinite vertical porous plates is investigated.

Gr

0 , Gc c 0

v

v

2

2

0 0

, K K ,

p

2. Formulation of the Problem

The governing equations of continuity, momentum energy and specie are given by

Q S T0 Tv0

PrReh

into the equations (5) to (7), we get

f

dv 0

v v

(Constant), (4)

f //

Re f /

Re ( Gr Gc) , (9)

K

dy 0

//

Pr Re / S 0 , (10)

d u

v0

d 2 u

g T T

//

Sc Re /

0 , (11)

g c

d y

C C

d y 2

u

~

K

(5)

where Pr is the Prandtl number, Sc is Schmidt number, Re is the cross flow Reynolds number, Gr is the Grashof number for heat transfer, Gc is the Grashof number for mass transfer, Ec is the Eckert

v d T

p

0 d y C

d 2T d y 2

Q , (6)

number, K is the permeability parameter and Q is volumetric heat generation/absorption.

The corresponding boundary conditions are reduced in non-dimensional form as given below

dc d 2C

• v0 dy D dy 2

  , (7)

  0 : f 0 , 1, 1;

  1 : f

  0, m , n.

  (12)

  where is the density, is the kinematic viscosity,

  g is acceleration due to gravity, v0 is suction/injection velocity, is the coefficient of

  Equations (9) to (11) are ordinary second order differential equations and solved under the boundary conditions (12). Through straightforward calculations, the expressions of f ( ), () and

  volume expansion for heat transfer,

  c is the

  () are obtained.

  coefficient of volume expansion for mass transfer, D

  is the chemical molecular diffusivity,

  ~

  K is the

  f ( )

  f1 ( ) f 2

  ( ) f3

  ( ) , (13)

  permeability of the porous medium,

  C p is the

  ( ) A

  A exp{( Pr Re)} B

  specific heat at constant pressure, T is the 4 3 1 (14)

  temperature,

  T is the fluid temperature far away

  B2 B3 exp{( Re Sc)}

  from the plate, C is species concentration, C is the species concentration far away from the plate and

  () A2 A1 exp{( Re Sc)} , (15)

  u, v are velocity components along x, y -directions,

  respectively.

  1. Skin-friction coefficient

     df

     df

     ( C f ) 0

     d

     and

     (C f )1 d

     0

     1

     (16)

  2. Nusselt number

     (Nu)

     d

     d

     0 d and ( Nu )1 d

     0

     1

     (17)

  3. Sherwood number

     Figure 1. Velocity distribution versus when

     ( Sh ) 0

     d

     d 0

     and ( Sh )1

     d

     d 1

     (18)

     Gr = 5.0,Gc = 2.0,Pr = 1.0, Re = 1.0, m = 2.0, n = 2.0, K = 2.0, S = 0.0.

     1.6 Pr = 1, 5, 7

  4. Results and Discussion

     To get a physical view into the problem, the numerical value of velocity distribution, temperature distribution, species concentration distribution have been obtained and shown graphically.

     The variation in the velocity field under the influence of Schmidt number Sc or Prandtl number Pr or Reynolds number Re or permeability parameter K are presented in figures 1, 2, 3 and 4 respectively. The fluid velocity increases with the increase of Sc, Pr, Re or K. Further the effects of the Grashof number

     1.2

     0.8

     f

     0.4

     0

     0 0.5 1

     for heat transfer Gr or Grashof number for mass transfer Gc are found from the figure 5 and figure 6. Which show that the fluid velocity enhances with the increment of Gr or Gc. It is observed from figure 7 that with the increase in parameter S, the fluid velocity increases.It is observed from figures 8, 9 and

     10 that the fluid temperature increases with the increase in heat source parameter S or Schmidt number Sc, Prandtl number Pr. Figure 11 shows that species concentration increases with the increase in Schmidt number Sc.

     It is observed from table 1 that increment in rate of heat generation/absorption parameter causes increase in skin friction coefficient and Nusselt number at first plate and decrease in the same at second plate. The skin friction coefficient initially increasing and decreasing later with the increment in Schmidt number. Again it is increases with the increase in Prandtl number whereas decreases with the increment of permeability parameter. The Sherwood number increases at first plate and decreases at the second plate with the increase in Schmidt number.

     Figure 2. Velocity distribution versus when Gr = 5.0,Gc = 2.0,Sc = 0.22, Re = 1.0, m = 2.0, n = 2.0, K = 2.0, S = 0.0.

     3

     Re = 2.0

     2

     f Re = 1.5

     1 Re = 1.0

     1.6

     0.8

     f

     Gc = 3.0

     Gc = 2.0

     Gc = 1.0

     0

     0 0.5 1

     Figure 3. Velocity distribution versus when

     Gr = 5.0,Gc = 2.0,Sc = 0.22, Re = 1.0, m = 2.0, n = 2.0, K = 2.0, S = 0.0.

     0

     0 0.5 1

     Figure 6. Velocity distribution versus when

     Gr = 5.0, K = 2.0, Sc = 0.22, Pr =1.0, m = 2.0, n = 2.0, S=0.0.

     1.6

     1.6

     1.2

     K = 1, 2

     S = 1.0

     S = 1.0

     S = -1.0

     0.8

     f

     0.4

     0

     0 0.5 1

     0.8

     f

     0

     0 0.5 1

     Figure 4. Velocity distribution versus when Gr = 5.0,Gc = 2.0, Sc = 0.22, Pr = 1.0, m = 2.0, n = 2.0, S = 0.0.

     Figure 7. Velocity distribution versus when

     Gr = 5.0,Gc = 2.0, Pr = 1.0, Re = 1.0, m = 2.0, n = 2.0, K = 2.0, Sc = 0.22.

     2.5

     1.6

     0.8

     Gr = 5.0

     Gr = 3.0

     2

     1.5

     1

     S = 1.0

     S = 0.0

     S =- 1.0

     f Gr = 1.0

     0

     0 0.5 1

     0.5

     0

     0 0.5 1

     Figure 5. Velocity distribution versus when

     K = 1.0,Gc = 2.0, Sc = 0.22, Pr = 1.0, m = 2.0, n = 2.0, S = 0.0.

     Figure 8. Temperature distribution versus when

     Pr = 1.0, Sc = 0.22, Gr = 5.0, Gc = 2.0,m = 2.0, n = 2.0.

     2.5

     2

     1.5

     1

     0.5

     0

     Sc = 0.22, 1.2

     0 0.5 1

     2.5

     2

     1.5

     1

     0.5

     0

     Pr = 1, 2, 3

     0 0.5 1

     Figure 9. Temperature distribution versus when

     Pr = 1.0, S = 1.0, Gr = 5.0, Gc = 2.0, m = 2.0,

     n = 2.0.

     Figure 10. Temperature distribution versus when

     S = 1.0, Sc = 0.22, Gr = 5.0, Gc = 2.0, m = 2.0,

     n = 2.0.

     2.5

     2

     1.5

     1

     Sc = 1.2

     Sc = 0.6

     0.5

     0

     0 0.5 1

     Figure 11. Species concentration versus when

     Re = 1.0, n = 2.0.

     Table 1. Numerical values of skin friction coefficient (Cf ) , Nusselt number (Nu) and Sherwood number (Sh) at both the plates for various values of physical parameters.

     K	S	Pr	Re	Sc	(Sh)0	(Sh)1	(Cf )0	(Cf )1	(Nu)0	( Nu)1
     1	1	1	1	.22	1.114030	0.89403	4.19308	-6.206771	2.38399	-0.134329
     1	1	1	1	.62	1.341830	0.721830	4.768734	-5.564032	2.404193	-0.147145
     1	1	1	1	.92	1.529558	0.609558	4.725550	-5.485690	2.149317	-0.156291
     1	1	1	1	1.2	1.717215	0.517215	3.647557	-5.740068	2.433275	-0.164404
     1	1	5	1	.22	1.114030	0.894030	4.841201	-6.451462	6.198525	-0.319794
     1	1	10	1	.22	1.114030	0.894030	5.115754	-6.470849	11.32797	-0.190340
     1	1	15	1	.22	1.114030	0.894030	5.205654	-6.458614	16.38902	-0.129296
     1	1	1	1.5	.62	1.536057	0.606057	9.209730	-7.155386	2.831617	-0.277488
     1	1	1	2	.62	1.744966	0.504965	15.09481	-8.325213	3.290634	-0.310145
     1	0	5	1	.62	1.341830	0.721830	5.122525	-5.649926	5.033918	0.033918
     1	-1	5	1	.62	1.341830	0.721830	4.822035	-5.493314	3.840644	0.391983
     1.5	1	5	1	.22	1.114030	0.894030	4.277225	-7.177588	6.198525	-0.319794
     2	1	5	1	.22	1.114030	0.894030	3.714125	-7.769261	6.198525	-0.319794

  5. Appendix

     f1 () A1 exp(m1) A2 exp(m2) ,

     B5 m B1 B2 B3 exp( Re Sc) ,

     f 2 () Re Gr{C1

     C2

     exp{( Pr Re)}

     ,

     C1 KA4 ,

     C3 C4 C5 C6 exp{( Re Sc)}

     C2 A3

     {(Pr Re) 2 Pr Re 2 1 K},

     f3 ( ) Re Gc{C7 C8 exp{( Re Sc ))} , C3 KB1 ,

     A D

• A ,

C K 2 B

Re ,

1 1 2 4 1

A2 {D1 D2 exp(m2 {1 exp(m1 m2 )},

C5 KB2 ,

A3 (B4

B5 {1 exp( Pr Re)} ,

C6 B3

{(Re Sc) 2 Re 2 Sc 1 K} ,

A4 B4 A3 , C7 KA2 ,

A5 (1 {1 exp(Re Sc)},

C8 A1

{(Re Sc) 2 Sc Re2 1 K},

A6 1 A5 , C9 C1 C2 C4 C5 C6 ,

B1 SA6

Pr Re ,

C10

C1

C3

C4

C5 ,

B (SA ) (Pr Re) 2 ,

2 6 D1 Re GrC9 Re Gc(C7 C8 )} ,

B3 (SA5 )

{(Re Sc) 2 (Re2 Pr Sc) ,

D2 Re Gr{C10

C2

exp( Pr Re)

B4 1 B2 B3 ,

C6 exp( Re Sc)} Re Gc{C7 C8 exp( Re Sc)}

1. References

   1. Y. Jaluria, Natural Convection, Heat and mass transfer, Pergamon Press, New York, 1980.

   2. G. Singh, P. R. Sharma and A. J. Chamkha, Effect of volumetric heat generation/absorption on mixed convection stagnation point flow on an iso-thermal vertical plate in porous media, Int. Journal of Industrial Mathematics, Vol. 2, 2010, pp 59-71.

   3. K. Vajravelu, J. Nayfeh, Hydromagnetic convection at a cone and a wedge, International Communication in Heat and Mass Transfer, Vol. 19, 1992,pp 701-710.

   4. K. Vajravelu, A. Hadjinicolaou, Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation, International Communication in Heat and Mass Transfer, Vol. 20, 1993, pp 417-430.

   5. J.C. Crepeau, R. Clarksean, Similarity solution of natural convection with internal heat generation, ASME Journal of Heat Transfer, Vol. 119, 1997, pp 183-185.

   6. A.J. Chamkha, A.R.A. Khaled, Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption, Heat and Mass Transfer, Vol. 37, 2001, pp 117-123.

   7. M. A. Samad and M. Mohebujjaman, MHD heat and mass transfer free convection flow along a vertical stretching sheet in presence of magnetic field with heat generation, Research Journal of Applied Sciences, Engineering and Technology, Vol. 1, 2009, pp 98-106.

   8. Basant K. Jha and Abiodun O. Ajibade,Free convective flow of a heat generating/absorbing fluid between vertical porous plates with periodic heat input, Int. Communication in heat and mass transfer, Vol. 36, 2009, pp 624-631.

   9. I.A.Abdallah, Analytic solution of heat and mass transfer over a permeable stretching plate affected by chemical reaction, internal heating, Dufour-Soret effect and Hall effect, Thermal Science, Vol. 13 (2), 2009, pp 183-197.

   10. N.Ahmed, D. Kalitha, and D. Berman, MHD free convective poiseuille flow and mass transfer through a porous medium bounded by two infinite vertical porous plates, Int. Journal of Applied Engineering Research, Vol. 5, 2010, pp 25-35.

   11. O. D. Makinde., Heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in highly porous medium with radiation and internal heat generation. Meccanica, DOI 10, 1007/s1 1012-011-9502-5, 2011.