B016721 by journal.ijrtem

Invention Journal of Research Technology in Engineering & Management (IJRTEM) ISSN: 2455-3689 www.ijrtem.com Volume 1 Issue 6 ǁ August. 2016 ǁ PP 07-21

Effect of Thermal Radiation and Radiation Absorption On Unsteady Convective Heat And Mass Transfer Flow In A Vertical Channel With Travelling Thermal Waves C Sulochana1, Ramesh H2,Tayappa H3.

(1Dept. of mathematics, Gubarga University, Kalaburgi. India.) (2Govt. P U College Chikkanargunda, Gadag. India.) 3 ( Asst. Professor, SSA Govt. First Grade College Bellary. India.)

Abstract: In this paper, we discuss the combined influence of thermal radiation, chemical reaction and radiation absorption on free connective heat and mass transfer flow of a viscous electrically conducting fluid in a non-uniformly heated vertical channel .The walls are maintained at non-uniform temperature and a uniform concentration is maintained on the walls. The coupled equations governing the flow, heat and mass transfer have been solved by using a perturbation technique with the slope  of the boundary temperature as a perturbation parameter. The expression for the velocity, the temperature, concentration, the rate of heat and mass transfer are derived and are analysed for different variations of the governing parameters G,R,M,, Sc,Q1,, 1, N, N1,k, P and x.

Key words: Heat and Mass transfer, Radiation Absorption, Thermal Radiation and Travelling Thermal waves. I. INTRODUCTION Transport of momentum and thermal energy in fluid – saturated porous media with low porosities with commonly described by Darcy’s model for conservation of momentum and by an energy equation based on the velocity field found from this model Kaviany [13]. In contrast to rocks, soil, sand and other media that do fall in this category, usually have high porosity, Vajravelu [31] examined the steady flow of heat transfer in a porous medium with high porosity. Raptis [23] studies mathematically the case of time varying two-dimensional natural convective heat transfer of an incompressible electrically conducting, viscous fluid through a high porous medium bounded by an infinite vertical porous plate. Jaiswal and Soundalgekhar [11] studied the natural convection in a porous medium with high porosity. A phenomenological theory of combined heat and mass transfer in porous media was previously established by Devries [6, 7] and Philip and Devries [20] commonly known as Devries mechanism model, its practical usefulness is widely recognized in describing the simultaneous heat and mass transfer with in a wide range of porous media. Experimental studies for mixed convection heat and mass transfer in the horizontal channel has been studied by Kamotani et al [9] and Maughan and Incorporal [16]. The heat and mass transfer through porous medium has been carried out by several authors [1, 2, 18, 26, 28, 30] under different conditions. In the above mentioned investigations the boundary walls are maintained at constant temperature. However, there are a few physical situations which warrant the boundary temperature to be maintained non-uniform. It is evident that in forced or free convection flow in a channel (pipe) a secondary flow can be created either by corrugating the boundaries or by maintaining non-uniform wall temperature. Such a secondary flow can be of interest in a few technological processes. For example, in drawing optic glass fibers of extremely low loss and band width the processes of modified chemical vapour deposition (MCVD) [14, 27] has been suggested in recent times. Performs from which these fibers are drawn are made by passing a gaseous mixture into a fused-silica tube which is heated locally, by an oxy-hydrogen flame particulate of So2Geo2 composition are formed from the mixture and collect on the interior of the tube. Subsequently these are fined to form a vitreous deposit as the flame traversed along the tube. The deposition is carried out in the radial direction through the secondary flow created due to non–uniform wall temperature. The application of electromagnetic fields in controlling the heat transfer as in aerodynamic heating leads to the study of magneto hydrodynamics heat transfer. The MHD heat transfer has gained significance owing to advancement of space technology. The MHD heat transfer can be divided into sections. One contains problems in which the heating is an incidental by product of the electromagnetic fields as in the MHD generators and pumps etc. and the second contains of problems in which the primary use of electromagnetic fields is to control the heat transfer [30]. With the fuel crisis

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Effct of Thermal Radiation And Radiation… deepening all over the world there is great concern to utilize the enormous power beneath the earth’s crust in the geothermal region [28]. Liquid in the geothermal region is an electrically conducting liquid because of high temperature. Hence the study of interaction of the geomagnetic field with the fluid in the geothermal region is of great interest, thus leading to interest in the study of magneto hydrodynamic convection flows through porous medium. Ravindra [24] has investigated the mixed convection flow of an electrically conducting viscous fluid through a porous medium in a vertical channel. Nagaraja [17] has investigated the combined heat and mass transfer effects on the flow of a viscous fluid through a porous medium in a vertical channel. Ravindra Reddy [25] has analyzed the effects of magnetic field on the combined heat and mass transfer in channels using finite element techniques. Since many industrially and environmentally relevant fluids are not pure, it has been suggested that more attention should be paid to convective phenomena which can occur in mixture, but are not present in common fluids such as air or water. Applications involving liquid mixtures include the casting of alloys, ground water pollutant, migration and separation operations. In all these situations multi component liquids can undergo natural convection driven by temperature and species gradients. In the case of binary mixtures, species gradients can be established by the applied solute boundary conditions such as species rejection associated with alloys casting or can be induced by coupled transport mechanisms such as Soret (thermo) diffusion. In the case of Soret diffusion, gradients are established in otherwise uniform concentration mixtures in accordance with the on sager reciprocal relationships. Recently some importance has been attended to the Benard problem in a two component system in which an initially homogeneous mixture is subjected to a temperature gradient. Then thermal diffusion known as Soret effect takes place and as a result of mass fraction distribution is established in the liquid layer [11]. The sense of migration of the molecular species is determined by the sign of soret coefficient. Keeping this in view several authors have investigated the soret effect under varied conditions [3, 8, 9, 10, 15, 19] Prasad [21] has discussed the convective heat and mass transfer of a viscous electrically conducting fluid through a porous medium in a vertical channel taking into account the dissipative effects. Using a perturbation technique, the velocity, the temperature and the concentration, the rate of heat and mass transfer have been analyzed for different variations in the governing parameters. Srinivasa Reddy [29] has analyzed the Soret effect on the convective heat and mass transfer of a viscous fluid through a porous medium in a vertical channel, the walls being maintained at non-uniform temperature. The coupled equations governing the flow, heat and mass transfer have been solved by assuming that the Eckert Ec is much less than 1. Ramakrishna Reddy [22] has analyzed the Soret effect on mixed convective Heat and mass transfer flow of an electrically conducting fluid through a porous medium in a vertical channel. Vijayabhaskar Reddy [32] has studied the heat and mass transfer in a vertical channel with non-uniform heated vertical walls. Chin Yong Cheng [4, 5] have studied the convective heat and mass transfer flow through a porous medium with variable wall temperature and concentration. In this problem, we discuss the combined influence of thermal radiation, chemical reaction and radiation absorption on free connective heat and mass transfer flow of a viscous electrically conducting fluid in a non-uniformly heated vertical channel .The walls are maintained at non-uniform temperature and a uniform concentration is maintained on the walls. The coupled equations governing the flow, heat and mass transfer have been solved by using a perturbation technique with the slope  of the boundary temperature as a perturbation parameter. The expression for the velocity, the temperature, concentration, the rate of heat and mass transfer are derived and are analysed for different variations of the governing parameters G,R,M,, Sc,Q1,, 1, N, N1,k, P and x.

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Effct of Thermal Radiation And Radiation…

II.

Formulation of the problem

We consider the motion of viscous, incompressible fluid through a porous medium in a vertical channel bounded by flat walls. The thermal buoyancy in the flow field is created due to the non-uniform temperature on the walls. y =  L while both the walls are maintained at uniform concentration. The Boussinesq approximation is used so that the density variation will be considered only in the buoyancy force. The viscous and Darcy dissipations are neglected in comparison to the heat conduction in the energy equation. Also the kinematic viscosity , the thermal conducting k are treated as constants. We choose a rectangular Cartesian system 0 (x, y) with x-axis in the vertical direction and y-axis normal to the walls. The walls of the channel are at y =  L. The equations governing the unsteady flow, heat and mass transfer are Equation of continuity

u v  0 x y

(1)

Equation of linear momentum

 e (u

u u p  2u  2u  v )     ( 2  2 )  g  ( e2 H o2 )u x y x x y

 e (u

v v p  2v  2v  v )    ( 2  2 ) x y y x y

(2) (3)

Equation of Energy:

 (q R ) T T  2T  2T  e C p (u  v )   ( 2  2 )  Q(T  Te )  Q1' (C  Ce )  x y y x y

(4)

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Effct of Thermal Radiation And Radiation… Equation of diffusion

C C  2C  2C (u  v )  D1 ( 2  2 )  k1 (C  Ce ) x y x y

(5)

Equation of state

   e   e (T  Te )     e (C  Ce ) where

e

(6)

is the density of the fluid in the equilibrium state, Te, Ce are the temperature and concentration in the equilibrium

state, (u, v) are the velocity components along O(x, y) directions, p is the pressure, T, C are the temperature and Concentration in the flow region,is the density of the fluid,  is the constant coefficient of viscosity, Cp is the specific heat at constant pressure,  is the coefficient of thermal conductivity,  is the electrically conductivity, e is the magnetic

permeability,  is the coefficient of thermal expansion, Q is the strength of the constant internal heat source,  is the volume expansion with mass fraction and D1 is the molecular diffusivity, Q1 is the radiation absorption coefficient and q R is the radiative heat flux. 

In the equilibrium state

pe  e g x where p  pe  p D , p D being the hydrodynamic pressure. 0

(7)

By Rosseland approximation (Brewester [5] the radiative heat fluxis given by

qr   Expanding

where obtain

4  (T  4 ) 3 r y

(8)

T  4 about Te by Taylor expansion and neglecting the higher order terms we get T 4  4TTe3  3Te4

(9)

 is the Stefan-Boltzmann constant and  r is the mean absorption coefficient. Substituting (8) & (9) in (4) we 

16 R k f  2T T   2T  2T  T ' C p  u v    ( 2  2 )  Q(Te  T )  Q1 (C  Ce )  x  x y 3 Te3 y 2  x 

(10)

The flow is maintained by a constant volume flux for which a characteristic velocity is defined as L

1 Q u d y. 2 L L

(11)

The boundary conditions for the velocity and temperature fields are u = 0, v = 0 , T  Te   (x / L) ,C=C1 on y = -L

u  0 , v  0 , T  Te   (x / L), C  C2 Sin(mx  nt ) is the imposed traveling thermal wave

on y = L

(12)

In view of the continuity equation we define the stream function  as u = - y, v =  x (13) Eliminating pressure p from equations (2) & (3) and using the equations governing the flow in terms of  are

[ x ( 2 ) y  y ( 2 ) x ]   4   g ( T  Te ) y    g ( C  Ce ) y  (

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 e2 H o2  2 ) 2 e y

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(14)

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Effct of Thermal Radiation And Radiation…

16 R k f  2T   T  T   2T  2T    ( 2  2 )  Q(Te  T )  Q1' (C  Ce )  C p   x z 3 Te3 y 2  y x x z  

(15)

 C  C (  )  D1 2 C  k1 (C  Ce ) y x x y

(16)

Introducing the non-dimensional variables in (12) & (13) as

x   x / L , y   y / L ,     / ,  

T  Te C  C2 ,C  T C1  c2

(17)

the governing equations in the non-dimensional form (after dropping the dashes) are

( ,  2 ) G  2 )   4  ( )( y  NC y )  M 2 2 ( x, y) R y

 y  RP ( y RP (

   x x C   x x

 4  2 )   2     Q1C  y 3N y 2 c )   2 C  C y

(18)

(19) (20)

where

R

 gTe L3 G 2  Cp   

Sc 

(Reynolds number) (Grashof number) ( Prandtl number), (Schmidt Number)





K1 L2 D1

QL2



(Heat source parameter)

(Chemical reaction parameter)

 e2 H o2 L2 (Hartmann Number) 2 Q  (C  C 2 ) L2 Q1  1 1 (Radiation absorption parameter) D1 (T  Te ) 4  (Radiation parameter) N1   R 3  Te M2 

N2 

3N1 , P1  PN 2 ,  1  N 2 3N1  4

The corresponding boundary conditions are

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 (1)  (1)  1  

 0,  0 at y  1 x y  ( x, y )   (x) , C  1 on y  1

 ( x, y )   (x)  C  0, 0 y y

,C  0

(21)

y  1

y 0

(22)

The value of  on the boundary assumes the constant volumetric flow in consistent with the hypothesis (11). Also the wall temperature varies in the axial direction in accordance with the prescribed arbitrary function t.

III.

Analysis of the flow

The main aim of the analysis is to discuss the perturbations created over a combined free and forced convection flow due to non-uniform boundary temperature imposed on the boundaries. Introduce the transformation such that

x  x, Then

   x x

   O( )   O(1) x x

For small values of <<1, the flow develops slowly with axial gradient of order  and hence we take

  O(1) x

Using the above transformation the equations (4.18-4.20) reduce to

( , 12 ) G  2 )  14  ( )( y  NC y )  M 2 2 ( x, y) R y     RP1 (  )  12   1   Q1 N 2 C y x x y  C  C ScR (  )  12 C  C y x x y

R(

12   2

(23) (24) (25)

2 2  x 2 y 2

We adopt the perturbation scheme and write

 ( x, y)  0 ( x, y)   1 ( x, y)   2 2( x, y)  ...............  ( x, y)  0 ( x, y)   1 ( x, y)   2  2 ( x, y)  ................  ( x, y)  0 ( x, y)  1 ( x, y)   2 2 ( x, y)  ......................

(26)

On substituting (26) in (23) - (25) and separating the like powers of  the equations and respective conditions to the zeroth order are

 0, y y y y y  M 12 0, y y   0, yy  1  0  Q1 C0

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G ( 0, y  NC0, y ) R

(27) (28)

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Effct of Thermal Radiation And Radiation…

C0, yy   C0  0

(29)

with  0(+1)-(-1) = 1,  0, y = 0 ,  0 , x =0

o   ( x ) ,

at y = 1

o   ( x ) ,

(30)

C0  1

y  1

C0  0

y 1

(31)

The first order are



11, yy

G  1, y y y y y  M 12 1, y y  (1, y  N C1, y )  R( 0, y  0, x y y   0, x  0, y y y ) R  1 1  P1 R ( 0, x o, y  0, y ox )  Q1C1

(32) (33)

C1, yy   1 C0  ScR ( 0, xCo, y  0, yCox )

(34)

 1(+1) -  1(-1 ) = 0  1, y = 0 ,  1 , x = 0 at y = 1 1(1) = 0 C1(1) = 0

(35)

with at y =  1

The equations to the second order are

G ( 2 y  N C 2, y )  R( 0, yyt   0, x  1, y y y  R   1, x  0, y y y   oy 1, xyy   1. y 0, xyy )

 2, y y y y y  M 12 2, y y 



21, yy

   2  P1 R ( 0, x1, y  0, y1x  1, y o, x   1, x o, y )  Q1C2

C2, yy   1C2  ScR ( 0,xC1, y  0, yC1x  1, yCo,x  1,xCo, y )

(36) (37) (38)

with  2(+1) -  2(-1 ) = 0  2, y = 0 ,  2 , x = 0 at y = 1 2(1) = 0 C2(1) = 0 at y =  1

IV.

(39) (40)

Solution of the problem

Solving the equations (27) - (38) subject to the relevant boundary conditions we obtain

Ch( 1 y ) Sh( 1 y )  ) Ch( 1 ) Sh( 1 ) Ch(  2 y) Sh(  2 y) Ch(  2 y) 0    a6 ( Sh( 1 y)  Sh( 1 ) )  a5 (Ch( 1 y)  Ch( 1 ) ) Ch(  2 ) Sh(  2 ) Ch(  2 )  o  a21Ch(M 1 y)  a22 Sh(M 1 y)  a20 y  a21  f1 ( y) C0  0.5(

f1 ( y)  a15 Sh(1 y)  a16Ch( 1 y)  a17 Sh( 2 y)  a18Ch( 2 y)

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C1  a45 (Ch(  3 y )  Ch(  3 )  a47 ( Sh(  3 y )  Sh(  3 )

Ch( 1 y ) Ch( 1 y ) )  a46 (Ch(  4 y )  Ch(  4 ) ) Ch( 1 ) Ch( 1 )

Sh( 1 y ) Sh( 1 y ) )  a48 ( Sh(  4 y )  Sh(  4 ) ) Sh( 1 ) Sh( 1 )

 a49 (Ch(  5 y )  Ch(  5 )

Ch( 1 y ) Ch( 1 y ) )  a50 (Ch(  6 y )  Ch(  6 ) ) Ch( 1 ) Ch( 1 )

 a51 ( Sh(  5 y )  Sh(  5 )

Sh( 1 y ) Sh( 1 y ) )  a52 ( Sh(  6 y )  Sh(  6 ) ) Sh( 1 ) Sh( 1 )

 a53 ( Sh(  2 y )  Sh(  2 )

Sh( 1 y ) Sh( 1 y ) )  a54 ( Sh(2 1 y )  Sh(2 1 ) ) Sh( 1 ) Sh( 1 )

 a55 (Ch(2 1 y )  Ch(2 1 )  Ch(  3 )

Ch( 1 y ) )  a57 ( y 2Ch(  3 y )  Ch( 1 )

Ch( 1 y ) Ch( 1 y ) )  a58 ( ySh (  3 y )  Sh(  3 ) )  a59 ( y 2Ch(  4 y )  Ch( 1 ) Ch( 1 )

 Ch(  4 )

Ch( 1 y ) Sh( 1 y ) )  a60 ( ySh (  4 y )  Sh(  4 ) )  a61 ( yCh ( 1 y )  Ch( 1 ) Sh( 1 )

 Ch( 14 )

Sh( 1 y ) Ch( 1 y ) )  a62 ( ySh ( 1 y )  Sh( 1 ) )  a63 ( y 2  1)Ch( 1 y )  Sh( 1 ) Ch( 1 )

 a64 ( y 2  1) Sh( 1 y )

1  b29Ch( 2 y)  b30 Sh( 2 y)  f 2 ( y) f 2 ( y )  b5 Ch( 1 y )  b6 Sh( 1 y )  b7 ySh ( 1 y )  b8 yCh ( 1 y )   b9 ySh (  2 y )  b10 yCh (  2 y )  b11 y 2 Ch( 1 y )  b12 y 2 Sh( 1 y )   b13Ch(2 1 y )  b14 Sh(2 1 y )  b15 Sh(  4 y )  b16 Sh(  3 y )   b17Ch(  3 y )  b18Ch(  4 y )  b19Ch(  5 y )  b20Ch(  6 y )   b21Ch(  7 y )  b22Ch(  8 y )  b23 y 2 Ch(  3 y )  b24 yCh (  3 y )   b25 ySh (  3 y )  b26 y 2 Ch(  4 y )  b27 yCh (  4 y )  b28 ySh (  4 y ) 

 1  k92Ch(M1 y)  k93Ch(M1 y)  f 3 ( y)

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Effct of Thermal Radiation And Radiation…

f 3 ( y )  (k 44  k 77 ) Sh( 1 y )  k 45  k 79 ) Sh(  2 y )  k 46 Sh(  3 y )   k 47 Sh(  4 y )  k 48 Sh(  5 y )  k 49 Sh(  6 y )  k 50 Sh(  7 y )   k 51Sh(  8 y )  k 52 Sh(  9 y )  k 53Sh( 10 y )  k 54 Sh( 11 y )   k 55 Sh( 12 y )  (k 56  k 73 )Ch(  2 y )  k 58Ch(  3 y )   k 59Ch(  4 y )  k 60Ch(  5 y )  k 61Ch(  6 y )  k 62Ch(  7 y )   k 63Ch(  8 y )  k 64Ch(  9 y )  k 65Ch( 10 y )  k 66Ch( 11 y )   k 67Ch( 12 y )  k 68 Sh(21 y )  k 69 Sh(2 2 y )  k 70Ch(21 y )   k 71Ch(2 2 y )  k 72 ySh ( 1 y )  k 73 ySh (  2 y )  k 76 yCh ( 1 y )   k 78 yCh (  2 y )  k 79 yCh ( M 1 y )  k81 yCh ( M 3 y )  k82 yCh ( M 4 y )   k83 ySh ( M 2 y )  k84 ySh ( M 1 y )  k85 ySh ( M 4 y )  k86 yCh (2M 1 y )   k87 ySh (2M 1 y )  k88 y 2 Sh( M 1 y )  k89 y 2 Ch( M 1 y )  k 90 y 2 Sh( M 2 y )   k 91 y 2 Where a1,a2,………………,a64,b1,b2,…….,b28,k1,k2,……..,k93 are constants. Nusselt number and sherwood number : The local rate of heat transfer coefficient (Nusselt number Nu) on the walls has been calculated using the formula

Nu 

1  ( ) y 1  m   w y

where 1

 m  0.5   dy 1

and the corresponding expressions are

( N u ) y  1 

(d 9   d11 ) ( m  Sin ( x  t )) ,

( N u ) y  1  Where

(d 8   d10 ) ( m  1)

 m  d14   d15 The local rate of mass transfer coefficient (Sherwood Number Sh) on the walls has been calculated using the

formula

Sh 

1 C ( ) y 1 C m  C w y

Where 1

C m  0.5  C dy 1

and the corresponding expressions are

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Effct of Thermal Radiation And Radiation…

( Sh ) y  1 

(d 4   d 6 ) (C m )

( N u ) y  1 

(d 5   d 7 ) (C m  1)

Cm  d12   d13 Where d1 .d 2 ,.........................., d14 are constants. Where

Particular Case: In the absence of chemical reaction (k=0) the results are in good agreement with (32) V. Discussion of the results In this analysis we discuss effect of chemical reaction and radiation absorption on the heat and mass transfer flow of viscous, electrically conducting fluid in a non-uniformly heated vertical channel in the presence of heat generating sources. The axial velocity (u) is shown in figs. 1-4 for different values of , Q1, N1 and k. The axial velocity enhances with 4 and reduces with higher 6 (fig. 1). The variation of u with radiation absorption parameter Q 1 shows that higher the radiation absorption larger |u| in the flow region (fig. 2). The variation of u with chemical reaction parameter k shows that higher the chemical reaction parameter larger |u| in the entire flow region (fig.3). The variation of u with radiation parameter N1 shows that the axial temperature reduces with increase in N1. Thus higher the radiative heat flux smaller the axial velocity (fig. 4). The secondary velocity (v) which is due to the non-uniform boundary temperature is shown in figs.5-8 for different parametric values. An increase in 4 enhances |v| in the left half and reduces in the right half while for higher 6, it reduces in the left half and enhances in the right half of the channel (fig. 5). An increase in Q14 results in an enhancement in |v| and for higher Q16, we notice a depreciation in |u| in the left half and enhancement in the right half of the channel (fig. 6). From fig. 7 we find that an increase in the chemical reaction parameter k2.5 enhances |v| in the left half and depreciates in the right half for higher k3.5 we notice depreciation in |v| in the entire flow region (fig. 8). With respect to the radiation parameter N1 we find that the secondary velocity enhances with increase in N 1 (fig.24). The non-dimensional temperature () is shown in figs. 9-12 for different parametric values. With respect to  we find that the actual temperature enhances with increase in 3 and depreciates with higher 6 (fig. 9). Lesser the molecular diffusivity smaller the actual temperature. Also it enhances with increase in the radiation absorption parameter Q 1 (fig. 10). With respect to the chemical reaction parameter k we find that higher k results in a depreciation of the actual temperature (fig. 11). The variation of θ with N1 shows that the actual temperature depreciates with increase in N1.Thus higher the radiative heat flux smaller the actual temperature (fig. 12). The non-dimensional concentration (C) is shows in figs. 13-16 for different parametric values. An increase in 4 enhances the actual concentration and depreciates with higher 6 (fig. 13). An increase in Q1 results in an enhancement in C (fig. 14). From fig. 15 we find a depreciation in the concentration with chemical reaction parameter k. The variation of C with radiation parameter N1 shows that the actual concentration enhances with increase in N1≤1.5 and reduces with higher N1≥5 (fig.16). The rate of heat transfer for (Nusselt number) at y = 1 is shown in tables 1-3 for different values of , Q1 and K. The variation of Nu with heat source parameter  shows that |Nu| depreciates with increase in  at y = +1 and at y = -1, it enhances with 4 and for higher 6, we notice an enhancement for G>0 and depreciation for G<0. (tables 1). An increase in Q1 enhances |Nu| at y = 1. The variation of Nu with radiation parametyer N1 shows that the rate of heat transfer reduces at y=1 and enhances at y=-1(tables 2). The variation of Nu with chemical reaction parameter k shows that |Nu| enhances with increase in k1.5 and depreciates with higher k2.5 at both the walls (tables 3). The rate of mass transfer Sherwood number (Sh) at y = 1 is shown in tables 4-5 for different parametric values. |Sh| experiences an enhancement with increase in Q1 at y = 1. With respect to the radiation parameter N1 we find the rate of

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Effct of Thermal Radiation And Radiationâ&#x20AC;Ś mass transfer periences at both the walls with increase in N1 (tables 4). With respect to k, we find an enhancement in |Sh| with increase in k at y = ď&#x201A;ą1 (tables 5).

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Effct of Thermal Radiation And Radiationâ&#x20AC;¦

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Effct of Thermal Radiation And Radiation…

Nusselt number (Nu) at y = +1 (Table 1) G 100 300 -100 -300 α

I -1.6362 -1.6550 -1.6174 -1.5986 2.5

II -1.5718 -1.6366 -1.5179 -1.3357 4.5

III -1.6257 -1.9507 -1.4228 -0.6884 6.5

IV -0.2850 -0.6769 -1.0753 -2.2547 -2.5

V -0.9993 -0.6721 -1.7310 -2.6902 -4.5

VI -1.0199 -0.5842 -1.8602 -2.9123 -6.5

Nusselt number (Nu) at y = +1 (Table 2) G 100

I -1.6362

II -1.5052

III -1.4955

IV -1.4815

V -1.4654

VI -0.8592

VII -0.7833

VIII -0.7914

300

-1.6550

-1.1731

-1.0459

-0.7675

-0.1291

0.4080

0.3518

0.7742

-100

-1.6174

-1.6317

-1.6362

-1.6429

-1.6508

-1.1581

-1.1043

-1.1411

-300

-1.5986

-2.1060

-2.2997

-2.9268

-13.4456

-1.1324

-1.0750

-1.1013

1.5

3.5

100

1.5

0.5

1.5

2.5

3.5

Nusselt number (Nu) at y = +1 (Table 3) G 100 300 -100 -300 Q1

G 100 300 -100 -300 Q1

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I -1.6362 -1.6550 -1.6174 -1.5986 0.5

II -2.0284 -1.3087 -3.2798 -6.3028 1.5

III -3.3203 -3.2420 -5.6409 2.7378 2.5

Nusselt number (Nu) at y = -1 (Table 4) I II III -1.7085 -0.4428 -1.5439 -1.7390 -0.5734 -1.5949 -1.6776 -0.3476 -1.7091 -1.6464 -0.5050 0.5268 0.5 1.5 2.5

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IV -4.6579 -4.9022 -4.0005 -2.3190 3.5

IV -1.8747 -2.5118 14.5546 0.6143 3.5

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Effct of Thermal Radiation And Radiation… Sherwood number (Sh) at y = +1 (Table 5) Sc 0.24 0.6 1.3 2.01 γ

I -2.5044 -2.5190 -3.3985 -4.9870 0.5

II -2.6016 -4.1394 -10.7174 -9.4170 1.5

VI.

    

III -3.1457 -6.8194 -8.8910 -1.3648 2.5

IV -3.8601 -10.5469 -2.4593 -0.5496 3.5

V -2.5003 -2.4976 -3.2565 -4.3683 0.5

VI -2.4946 -2.4678 -3.0700 -3.6875 0.5

VII -2.5060 -2.5273 -3.4555 -5.2659 0.5

CONCLUSION

In this analysis we discuss effect of chemical reaction and radiation absorption on the heat and mass transfer flow of viscous, electrically conducting fluid in a non-uniformly heated vertical channel in the presence of heat generating sources. The following conclusions are made. Axial velocity enhances in the flow region when the radiation absorption is higher. Higher the radiative heat flux smaller the actual temperature. Concentration reduces with increase in the chemical reaction parameter. The rate of heat transfer (Nusselt number) with respect to radiation parameter reduces at y=1 enhances at y=-1. The rate of mass transfer (Sherwood number) with respect to radiation parameter depreciates at both walls.

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