 Open Access
 Total Downloads : 135
 Authors : J. Abdullahi
 Paper ID : IJERTV3IS110426
 Volume & Issue : Volume 03, Issue 11 (November 2014)
 Published (First Online): 11122014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Heat and Mass Transfer with Radiation and Dissipation over a Fixed Vertical Plate
J. Abdullahi
Government Secondary School, Kamba, P.M.B. 1002,
Kebbi State, Nigeria
Abstract:This paper investigates the effect of heat and mass transfer with radiation and dissipation over a fixed vertical plate. The dimensionless governing equations are solved using perturbation techniques. The effects of velocity, temperature and concentration are studied for different parameters like modified Grashof number, Grashof number, Suction/injection parameter, Schmidt number and Prandtl number, Eckert number.
Key words: heat transfer, mass transfer, vertical plate, radiation, dissipation

INTRODUCTION
In the processes involving high temperature, the radiation heat transfer in combination with conduction, convection and also mass transfer plays very important role in the design of pertinent equipments in the areas such as nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites, and space vehicles. Reference [1] presented radiation and mass transfer effects on transfer free convectional flow of a dissipative fluid past semiinfinite vertical plate with inform heat and mass flux. Viscous mechanical dissipation effects are important in geophysical flows and also in certain industrial and are usually characterized by the Eckert number. Reference [2] reported the influence of viscous heating dissipation effects in natural convective flows, showing that the heat transfer rates are reduced by an increase in the dissipation parameter. Soret and Dufour effects are important for intermediate molecular weight gases in coupled heat and mass transfer in binary system, often encountered in chemical process engineering and also in highspeed aerodynamics. Reference [3] used network simulation methods (NSM) to study the effect of viscous dissipation, Soret and radiation on unsteady MHD free convection flow past a vertical porous plate .Reference [4]
circular pipe geometry. Reference [8] analyzed the radiation effects on an unsteady two dimensional hydromagnetic free convective boundary layer flow of a viscous incompressible fluid past a semiinfinite vertical plate with mass transfer in the presence of heat source or sink. Reference [9] studied effects of variable suction and thermophoresis on steady MHD combined free forced convective heat and mass transfer flow over a semiinfinite permeable inclined plate in the presence of thermal radiation. Reference [10] investigated free convection flow past a vertical plate. Reference [11] reported effects of varying viscosity and thermal Conductivity on steady free Convective flow and heat transfer along an isothermal vertical plate in the presence of heat Sink. Reference [12] studied heat transfer to unsteady MHD flow past an infinite vertical moving plate with variable Suction.

MATHEMATICAL FORMULATION Consider the transient free convective flow of a viscous fluid in a vertical channel with the walls at a constant distance d apart. The xaxis is taken along one of the wall of the channel and yaxis is normal to it. It is also considered that there is radiation only from the fluid. The fluid is a grey, emitting, and absorbing radiation, but non scattering medium and the Roseland approximation is used to describe the radiative heat flux in the x direction is assumed negligible in comparison the channel walls ( y 0 ) while the other wall at y d is maintained at a
constant temperature Td , which causes free convection current in the channel. Under usual Boussinesqs approximation the mathematical model for the above free convection flow in the channel is stated below in equations (1), (2) and (3)
presented radiation and mass transfer effects on an unsteady MHD free convection flow past a heated vertical plate with viscous dissipation. Reference [5] investigated
U
t
v 0
U
d d
y
U
y
(1)
the thermal radiation effects on Magnetohydrodynamic (MHD) flow past a semiinfinite vertical plate in the presence of mass diffusion. Reference [6] investigated network simulation method applied to radiation and
g T T
g C C
viscous dissipation and Soret effects on MHD unsteady free convection over vertical porous plate. Reference [7] examined the dispersion of a chemically nonreacting and chemically reacting solute in a micro polar fluid, for
c
t
v c D
0 y
2c
y12
(2)
T
T
k 2T
1 u 2
Where U is the flow mean velocity.
t v0 y c
y12 c
y
(3)
Assume solution of the form
p p
The initial and boundary conditions
U U
t y
2U
y 2
Gr GcC
(6)
U 0,T T (T T ),
d d
C C
1 2C
C Cd
(C Cd ) at y 0 (4)
(7)
U
t y Sc
y 2
0,T Tw ,C Cw at y a
1 2
u 2
Where u is the axial Velocity, t is the time ,T is the fluid
t y Pr y 2
Ec
y
(8)
temperature. T and T0 are walls temperatures P the
pressure, g the gravitational force, q the radiative heat flux,
With the boundary conditions
C the fluid concentration, C and C0 are walls
U 0, 1, C 1
at y 0
concentrations D is the mass diffusivity, d is a constant,
is the frequency of the oscillation, is the coefficient of
thermal expansion, is the coefficient of concentration expansion, the density of the fluid, v is the kinematics
U 0, 0, C 0 at y 1
(9)
viscosity coefficient, v0 means suction velocity, which is a
U (y ,t ) U 0 (y ) U 1
(y )e i t
nonzero positive constant and the minus sign indicate that the suction is toward the plate.
Dimensionless variable and parameters of the above
C ( y ,t ) C
0 1
(y ,t )
(y ) C (y )e i t
0
( y ) ( y )e i t
(10)
three formulated problems are.

METHODOLGY
Equation (6) to (8) are coupled, nonlinear partial differential equations and these cannot be solved in closed
Gc
g * C C a2
d d
u ,
form. However, these equations can be reduced to a set of ordinary differential equations,
which can be solved analytically
y T T
Where is the frequency of the oscillation. Substituting
y ,
d ,
(10) into (6) to (9) we obtain
a T Td
U0 U0 Gr0 GcC0
(11)
C C C d
C C ,
, V 0a
U iwU Gr
GcC
d
t u
1 1 1 1 (12)
t 2 , U ,
(5)
C ScC 0
a u
Gr g T Td ,
u
C
0 0
C ScC iScC 0
(13)
Sc
D *
, Pr p ,
k
1 1 1
(14)
U 2
Ec Cp ,
0 0 0
Pr Ec PrU 2
(15)
1 Pr1 iwPr1 2Ec PrU1U1 (16)
Equation (11) to (16) are solved with help of (17), the solutions for fluid temperature , concentration and velocity are given as follows
Ec Grb1 Grb2 Grb3 Grb4
2 2
(2 Pr) ( PrSc ) Pr Pr
U (y ) U (y ) 0, (y ) (y ) 0

Grb5 Grb6

Grb7
0 1 0 1
(2Sc ) (Sc )
Sc
C 0 (y ) C1 (y ) 0
at y 0
(17)
Grb Gr
Gr
a q
U (y ) U
(y ) 0,
(y ) 1, (y ) 0

8 Grb
17
b 16 1 q
2
0 1 0 1
3 10
( Pr)
2 7
Pr
C 0 (y ) 1 ,C1 ( y ) 0 at y 1
(21)
(y ,t )
1 e Pr y e Pr
Shear wood number by differentiating (19) taking y=0
1e Pr
b e 2 Pr y b e (Sc Pr) y b ye Pr y
c eSc
1 2 3
(22)
b e ( Pr ) y b e 2Scy b e Scy
y /
y0
1 eSc
4 5 6
(18)
Ec b e (Sc ) y b y b b e 2 y
Pr
7 8 9 10
e Ec
b e y a16 a e Pr y
y y0 1 e Pr
11 Pr 17 a
Scb (Sc )b b 2b b
16
6 7 9 10 11
Pr
C( y,t) 1
eScy eSc
(19)
Pr a
(23)
1 eSc 17
a e Pr y
a e Scy
Where b1,b2 ,b3 ,b4 ,b5 ,b6 ,b7 ,b8 ,b9 ,b10 ,b11, a16 , a17 are
U ( y ,t )
1
Pr( Pr)
2
Sc (Sc )
constants, their expression are not presented here for sake
of brevity.
a3 y a3 a4 a e y Ec
2 5
4. RESULTS AND DISCUSSION
Except otherwise indicated, we used the following
Grb e 2 Pr y
Grb e ( Pr Sc ) y
1 2
2 Pr(2 Pr) ( PrSc )( PrSc )
3 3 4

Grb ye Pr y Grb e ( Pr ) y Grb e ( Pr ) y
Pr( Pr) 2 Pr2 ( Pr) ( Pr ) Pr
5 6 7
Grb e 2Scy Grb e Scy Grb e (Sc ) y
2Sc (2Sc ) Sc (Sc ) (Sc )Sc
parameters values for the computation Pr = 0.71, Sc =0.01,
= 20, Ec = 0.009, Gr = 2.6, Gc = 1
The Concentration Profiles have been studied and presented in figure 4.2.1 to 4.2.2. The Concentration Profiles for different values of Schmidt number (Sc = 0.1, 0.2, 0.3, 0.4.) are shown in figure 4.2.1. It is observed that the Concentration decreases with the increases of Schmidt
number. The Concentration Profiles for different values of
Grb y 2
Grb y Grb Grb e 2 y
Grb y
Eta number ( = 0.5 , 1.0, 1.5, 2.0.) are shown in figure
8 8 8 10 11
2 2
Gr e Pr y
3
Gr
2 2
a

It is observed that the Concentration decreases with the increases of Eta number. The Temperature Profiles
17
Pr( Pr)
b7
16 y
Pr
have been studied and presented in figure 4.2.3 to 4.2.9.
The Temperature Profiles for different values of Schmidt
Gr a q
number (Sc = 0.1, 0.2, 0.3, 0.4.) are shown in figure 4.2.3.
2 b7 16
1 q2
It is observed that the Temperature decreases with the
Pr
(20)
increases of Schmidt number. The Temperature Profiles for different values of Schmidt number (Sc = 1.0, 1.5, 2.0, 2.5.) are shown in figure 4.2.4. It is observed that the Temperature decreases with the increases of Schmidt number. The Temperature Profiles for different values of Schmidt number (Sc = 0.1, 0.2, 0.3, 0.4.) are shown in
Skin friction number by differentiating (20) taking y=0
u a1 a2 a3 a4 a
figure 4.2.5. It is observed that the Temperature decreases with the increases of Schmidt number. The Temperature Profiles for different values of Suction/injection parameter
y y0
( Pr) (Sc ) 2 5
( = 4.0, 6.0, 8.0, 10.0.) are shown in figure 4.2.6. It is
observed that the Temperature decreases with the increases of Eta number. The Temperature Profiles for different
values of Eckert number (Ec = 0.01, 0.03, 0.05, 0.07.) are shown in figure 4.2.7. It is observed that the Temperature decreases with the increases of Eckert number. The Temperature Profiles for different values of Prandtl number (Pr = 0.64, 0.68, 0.71, 0.85.) are shown in figure 4.2.8. It is observed that the Temperature increases with the increasing of Prandtl number. The Temperature Profiles for different values of Prandtl number (Pr = 0.64, 0.68, 0.71, 0.85.) are shown in figure 4.2.9. It is observed that the Temperature increases with the increasing of Prandtl number. The Velocity Profiles for different values of Schmidt number (Sc = 0.11, 0.12, 0.13, 0.14.) are shown in figure 4.2.10. It is observed that the Velocity increases with the increasing of Schmidt number. The Velocity Profiles for different values of Prandtl number (Pr = 0.60, 0.64, 0.68, 0.71.) are shown in figure 4.2.11. It is observed that the Velocity decreases with the increasing of Prandtl number. The
Velocity Profiles for different values of Eta number ( =
0.11, 0.12, 0.13, 0.14.) are shown in figure 4.2.12. It is observed that the Velocity decreases with the increasing of Eta number. The Velocity Profiles for different values of Grashof number (Gr = 3.0, 3.2, 3.4, 3.6.) are shown in figure 4.2.13. It is observed that the Velocity decreases with the increasing of Grashof number. The Velocity Profiles for different values of modified Grashof number (Gc = 2.0, 2.2, 2.4, 2.6.) are shown in figure 4.2.14. It is observed that the Velocity increases with the increasing of modified Grashof number.
Figure 4.2.1 Concentration profiles for different values of Schmidt number
Figure 4.2.2 Concentration profiles for different values of
Figure 4.2.3 Temperature profiles for different values of Schmidt number
Figure 4.2.4 Temperature profiles for different values of Schmidt number
Figure 4.2.5 Temperature profiles for different values of Schmidt number
Figure 4.2.6 Temperature profiles for different values of number
Figure 4.2.7 Temperature profiles for different values of Eckert number
Figure 4.2.8 Temperature profiles for different values of Prandtl number
Figure 4.2.9 Temperature profiles for different values of Prandtl number
Figure 4.2.11 Velocity profiles for different values of Prandtl number
Figure 4.2.12 Velocity profiles for different values of Eta number
Figure 4.2.13 Velocity Profiles for different values of Grashof number
Figure 4.2.14 Velocity profiles for different values of Modified Grashof
number
Figure 4.2.10 Velocity profiles for different values of Schmidt number
5. CONCLUSION
This paper studied heat and mass transfer with radiation and dissipation over a fixed vertical plate. The dimensionless governing equations are solved using perturbation techniques. The effect of different parameters such as modified Grashof number, Grashof number, Eckert, Schmidt number, Prandtl number and Eta are studied. The conclusions of the study are as follows

The velocity decreases with increases of ,Gr
and Pr

The velocity increases with the increases of Sc and Gc

The Concentration decreases with the increases of Sc and

The temperature decreases with increases of Sc,
and Ec.

The temperature increases with increases of Pr


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