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The thermo-physical effects of thermal radiation, heat generation and species concentration on Magneto-hydrodynamic free convective heat and mass transfer flow over a vertical plate with variable thermal conductivity is considered. The governing

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© copyright Faculty of Engineering – Hunedoara, University POLITEHNICA Timisoara
ACTA TECHNICA CORVINIENSIS
– Bulletin of Engineering
Tome X [2017] Fascicule 4 [October – December] ISSN: 2067 – 3809
1.
O.A. OYEM,
2.
A.J. OMOWAYE,
2
O.K. KORIKO
THERMO-PHYSICAL EFFECTS OF THERMAL RADIATION AND HEAT GENERATION ON FREE CONVECTIVE HEAT AND MASS TRANSFER OVER A VERTICAL PLATE
1.
Department of Mathematical Sciences, Federal University Lokoja, Kogi State, NIGERIA
2.
Department of Mathematical Sciences, Federal University of Technology Akure, Ondo State, NIGERIA
Abstract:
The thermo-physical effects of thermal radiation, heat generation and species concentration on Magneto-hydrodynamic free convective heat and mass transfer flow over a vertical plate with variable thermal conductivity is considered. The governing partial differential equations are transformed into coupled nonlinear ordinary differential equations with similarity transformations. The resulting coupled nonlinear differential equation is solved numerically by Runge-Kutta forth order with shooting technique. The effects of the thermo-physical properties on velocity, temperature and concentration profiles are illustrated graphically and numerical values of skin friction, Nusselt number and Sherwood number are presented in tables.
Keywords:
MHD free convection; heat generation; thermal radiation; variable thermal conductivity; vertical plate
Introduction
The problem of fluid flow in an electromagnetic field has been studied for its importance in geophysics, metallurgy aerodynamic extrusion of plastic sheets and other engineering processes such as in petroleum engineering, chemical engineering, composite or ceramic engineering and heat exchanger (Sandeep and Sugunamma, 2013). Radiative convective flows are involved in countless industrial and environmental processes. For example, heating and cooling chambers, fossil fuel combustion energy process evaporation from large open water reservoirs, astrophysical flows, solar power technology and space vehicle re-entry. In the context of space technology and in processes involving high temperatures, the effects of radiation are of vital importance which has opened up to various investigators. Many researchers have investigated the interaction of thermal radiation, heat generation and free convection for different geometries, by considering various flow patterns. Muthucumaraswamy et al. (2006) investigated the effect of thermal radiation on the unsteady free convective flow over a moving vertical plate with mass transfer in the presence of a homogenous first order chemical reaction. Their result showed that velocity increases with decreasing radiation parameter or chemical reaction parameter. The interaction of free convection with thermal radiation of a viscous incompressible unsteady MHD flow past a moving vertical cylinder with heat and mass transfer was analysed by Reddy and Reddy (2009) and it was observed that when radiation parameter increases, the velocity and temperature decreases in the boundary layer. Khaleque and Samad (2010) studied radiation and viscous dissipation effects on a steady two-dimensional MHD free convection flow along a stretching sheet with heat generation. They observed that momentum boundary layer and the thermal boundary layer thickness reduce as a result of increasing radiation. Jana and Ghosh (2011) studied the effects of thermal radiation of a viscous incompressible fluid occupying a semi-infinite region of space bounded by an infinite horizontal moving hot flat plate in the presence of indirect natural convection by way of an induced pressure gradient. From their study, velocity slightly increases with increased radiation parameter as temperature increases with increasing values of
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
r
adiation parameter. Uddin and Kumar (2011) investigated the heat and mass transfer characteristics and the flow behaviour on MHD flow near the lower stagnation point of a porous isothermal horizontal circular cylinder. It was observed that with the increase in Schmidt number, skin friction and Nusselt number decreases while, Sherwood number increases. Shit and Haldar (2012) studied the effect of thermal radiation and temperature dependent viscosity on free convective flow and mass transfer of an electrically conducting fluid over an isothermal sheet. The result of their study showed that effects of thermal radiation and variable viscosity can lead to the decrease of heat transfer, whereas, the presence of viscoelasticity of the fluid and magnetic field causes increase of heat transfer. Sandeep and Sugunamma (2013) analysed the effects of inclined magnetic field and radiation on free convective flow of dissipative fluid past a vertical plate through porous medium in the presence of heat source. Their results showed that decrease in inclination angle reduces the magnetic field effect but increase in radiation term does not show much effect on increase in skin friction, while, increase in heat source parameter causes the increase in velocity of fluid flow. Omowaye and Ajayi (2014) studied the effects of some thermo-physical properties on force convective stagnation point on a stretching sheet with convective boundary conditions in the presence of thermal radiation and magnetic field. They observed that increasing the value of magnetic field parameter resulted in the increase in skin friction coefficient, whereas, the Nusselt number decreases with increasing values of the magnetic parameter and radiation number. A number of interesting applications of the principles of heat transfer are concerned with systems in which heat may be generated internally. Nuclear reactors, electrical conductors, exothermic and endothermic chemically reacting systems are some examples (Holman, 2002). Mixed convection MHD flow of nano-fluid over a non-linear stretching sheet with thermal radiation and suction and variable magnetic field is investigated by Keshtekar et al. (2014). Their results showed that thermal radiation, heat generation and the permeability parameter decreases with an increase in Prandtl number, suction and magnetic parameter. Animasaun and Oyem (2014) investigated the effect of variable viscosity, Dufour, Soret and thermal conductivity on free convective heat and mass transfer of non-Darcian flow past flat surface. Their result showed that with increase in Dufour and Soret Parameter, fluid velocity increases and temperature increases with increase in variation of Dufour but decreases with increase in Soret. Olajuwon and Baoku (2014) investigated the combined influence of internal heat generation, suction, thermal radiation and chemical reaction of nth order on an unsteady free convective heat and mass transfer flow of a chemically reactive third grade fluid over an infinite vertical porous plate with oscillating temperature applied to the plate. They noted that suction parameter has the influence of reducing velocity, temperature and species concentration fields, as the fluid velocity increases as the value of the second grade viscoelastic parameter increases.
GOVERNING EQUATION
A steady two-dimensional free convective heat and mass transfer fluid flow over a vertical plate with variable thermal conductivity, heat generation, thermal radiation and species concentration are considered. The flow in the
x
-axis is taken along the vertical plate in the upward direction and the
y
-axis is normal to the plate. Here,
T
w
and
C
w
are temperature and concentration of the fluid at the plate and
T
and
C
are the temperature and concentration outside the boundary layer as shown in figure 1.
Figure 1. Problem schematics
Under the Boussinesq’s approximation, the boundary layer is governed by the mass, momentum, energy and concentration equations as;
∂
u
∂
x+
∂
v
∂
y=0 (1)
u
∂
u
∂
x+v
∂
u
∂
y=
υ∂
u
∂
y
+g
β
(T
T
)
+g
β
∗
(C
C
)
σβ
0
u
ρ
(2)
u
∂
T
∂
x+v
∂
T
∂
y=1
ρ
c
p
∂∂
y
�κ
(T)
∂
T
∂
y
1
ρ
c
p
∂
q
r
∂
y + Q
0
ρ
c
p
(T
T
)(3)
u
∂
C
∂
x+v
∂
C
∂
y=D
∂
C
∂
y
(4)
The radiative heat flux
q
r
is describes by the Rosseland approximation given by
q
r
=
4
σ
∗
3k
∗
∂
T
4
∂
y (5)
where
σ
∗
is the Stefan-Boltzmann constant and
k
∗
is the Rosseland mean absorption coefficient. Following
152 | Fascicule 4
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
Chamkha (1997), the temperature difference within the flow is assumed to be sufficiently small so that
T
4
may be expressed as a linear function of temperature
T
. This is accomplished by expanding
T
4
in a Taylor’s series about the free stream temperature
T
such that
T
4
≈
4T
3
T
3T
4
(6)
In this research, the author shall assume that the fluid thermal conductivity
κ
vary as a linear function of temperature (Prasad et al. 2010) as
κ
(T)=
κ
∗
[1+
δ
∗
(T
T
)] (7)
where
κ
(T)
is the thermal conductivity of the fluid depending on the fluid temperature
T
and
δ
∗
being a constant which may either be positive or negative. In this research,
δ
>0
is thermal conductivity generation and
κ
∗
is the thermal conductivity of the ambient fluid. The corresponding boundary conditions are
u=0 v=0 T=T
w
C=C
w
at y=0u
→
0 T
→
T
C
→
C
as y
→ ∞
(8)
The velocity components along the axes can be expressed as:
u=
∂ψ∂
y and v=
∂ψ∂
x (9)
substituting equation (9) into equation (1), equation (1) is satisfied. Defining similarity variable and the dimensionless variables as
η
=y
U
0
2υ
x ,
θ
(
η
)=T
T
T
w
T
,
ϕ
(
η
)=C
C
C
w
C
,
ψ
=
2x
υ
U
0
f (
η
) (10)
where
ψ
is the stream function,
f
is dimensionless velocity,
θ
is the dimensionless fluid temperature and
ϕ
is the dimensionless concentration. Introducing equation (10) into equations (1)-(9), the coupled nonlinear ordinary differential equation is obtained
f
+ff
+Gr
θ
+Gm
ϕ
Mf
=0 (11)
�
1+
γθ
+43N
θ
+Prf
θ
+
γθ′
+PrQ
θ
=0 (12)
ϕ
+Scf
ϕ
=0 (13)
subject to the boundary conditions
η
=0 ; f=0 f
=0
θ
=1
ϕ
=1
η → ∞
; f
→
0
θ →
0
ϕ →
0 (14)
Equations (11) – (13) with boundary conditions (14) describe the heat and mass transfer over a vertical plate in the presence of magnetic field under variable thermal conductivity, thermal radiation, heat generation and species concentration. It is noteworthy that the local parameters
,
,
and
in equations (11) – (13) are functions of
. Here, prime denotes the differentiation with respect to
and:
Gr=
xg
(
T
−T
)
U
(local thermal Grashof number),
Gm=
xg
∗
(
C
−C
)
U
(local modified thermal Grashof number),
M=
x
U
(local magnetic field parameter),
Pr=
α
(Prandtl number),
Q=
xQ
c
U
(heat generation parameter),
Sc=
D
(Schmidt number),
γ
=
δ
(T
w
T
)
(variable thermal conductivity parameter),
N=
∗
4
∗
T
(thermal radiation parameter of the flow).
NUMERICAL COMPUTATION
The set of equations (11) - (13) under the boundary conditions (14) are solved numerically by Runge-Kutta forth order technique with shooting method. The basic idea is to reduce the higher order nonlinear differential equations (11) - (13) to system of first order linear differential equations and they are further transformed into initial value problem and then apply shooting technique with
Gr
,
Gm
,
M
,
Pr
,
Q
,
Sc
,
γ
and
N
as prescribed parameters. The results are presented in figures 2 - 25. Let
f=x
1
,
f
=x
,
f
=x
3
,
θ
=x
4
,
θ
=x
5
,
ϕ
=x
6
,
ϕ
=x
7
. Then equations (11) – (14) are transformed into a system of first order differential equations as:
x
1
=x
x
=x
3
x
3
=Mx
Grx
4
Gmx
6
x
1
x
3
x
4
=x
5
x
5
=
�
3N3N(1+
γ
x
4
)+4
(
Prx
1
x
5
γ
x
5
PrQx
4
)x
6
=x
7
x
7
=
Scx
1
x
7
(15)
subject to the following initial conditions:
x
1
(0)=0
,
x
(0)=0
,
x
3
(0)=s
1
,
x
4
(0)=1
,
x
5
(0)=s
,
x
6
(0)=1
,
x
7
(0)=s
3
(16)
The computations are carried out by a program which uses a symbolic and computational computer language MATLAB with a step-size of
Δη
=0.001
, chosen to satisfy the convergence criterion of
10
−6
. From the process of computation, the skin friction coefficient
f
′′
(0)
, Nusselt number
θ′
(0)
and mass transfer coefficient in terms of Sherwood number
ϕ′
(0)
respectively are also worked out and their numerical values are presented in a tables 1 – 3.
RESULTS AND DISCUSSION
In order to get a physical insight into the problem, velocity, temperature and concentration profiles have been computed and displayed graphically. Skin-friction coefficient, Nusselt and Sherwood numbers are also computed for various values of the governing parameters: thermal radiation parameter
(N)
, local thermal Grashof number
(Gr)
, local modified thermal Grashof number
(Gm)
, Prandtl number
(Pr)
, local
153 | Fascicule 4
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
mag
netic field parameter
(M)
, Schmidt number
(Sc)
, heat generation parameter
(Q)
and variable thermal conductivity parameter
(
γ
)
. The velocity, temperature and concentration profiles are displayed in figures 2 – 4 respectively. It was observed from figure 2 that as
γ
increases in value, velocity increases steeply to a peak along the plate and decreases gradually away from the plate towards the free stream. In figure 4, concentration profiles decreases with an increase in variable thermal conductivity, while, in figure 3, temperature profiles increases with increasing values of thermal conductivity variation parameter
γ
.
Figure 2: Velocity profiles over a vertical plate for various values of variable thermal conductivity
γ
Figure 3: Temperature profiles over a vertical plate for various values of thermal conductivity variation parameter
γ
Figure 4: Concentration profiles for various values of
γ
Prandtl number
Pr
is found to be the parameter that relates the relative thickness of the hydrodynamic and thermal boundary layer. Thereby making it the link between the velocity and the temperature fields (Holman, 2004). The effects of Prandtl number
Pr
on velocity, temperature and concentration distributions are displayed in figures 5, 6 and 7 respectively. Figures 5 and 6 shows that velocity and concentration profiles increases with the increase in Prandtl number
Pr
. It was also observed in figure 5 that there was an initial rise at
Pr=0.71,1.0
and later, a sharp rise in velocity and concentration boundary layers away from the plate towards the free stream was observed at
Pr=3.0
. Figure 6 shows that temperature profiles decreases as Prandtl number
Pr
increases. It is also interesting to note that increase in
Pr
means a decrease in variable thermal conductivity.
Figure 5: Prandtl number
(Pr)
effects on velocity profiles Figure 6: Prandtl number
(Pr)
effects on temperature profiles Figure 7: Prandtl number
(Pr)
effect on concentration profiles
It was observed from figure 6 that for
Pr=0.71,1.0
, there is a slight rise in the thermal boundary layers near the plate but decreases sharply for
Pr=3.0
towards the plate (Khaleque and Samad, 2010). In figure 7, increase in Prandtl number results in the increase in concentration profiles. It was observed that it increases sharply away from the plate towards the free stream. Figures 8 - 10 shows the variation of velocity, temperature and concentration profiles respectively with local thermal Grashof number
(Gr)
. From figure 8,
154 | Fascicule 4
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
it was observed that velocity profiles increases spontaneously at an initial state near the vertical plate as local thermal Grashof number increased but moving away from the plate, a cross flow in the velocity is induced as velocity gently decreases away from the plate at a slower rate with increasing values of
Gr
. Since
Gr>0
corresponds to externally cooled plate, substantial increase in velocity distribution profiles is observed near the plate, thereby indicating greater cooling results in velocity increase and thermal boundary layer thickness. This situation reveals that the buoyancy force accelerates the velocity field and no flow reversal occurs to prevent separation (Khaleque and Samad, 2010, Jana and Ghosh, 2011).
Figure 8: Grashof number
(Gr)
effect on velocity profiles Figure 9: Grashof number
(Gr)
effect on temperature profiles Figure 10: Grashof number
(Gr)
effect on concentration profiles
From figures 9 and 10, the thermal boundary layer on temperature profiles and concentration boundary layers reduces with increasing values of
Gr
, causing the fluid temperature to reduce at every point other than the plate. When a heated surface is in contact with the fluid, the result of temperature difference causes buoyancy force, which induces the natural convection (Abah et al. 2012). Velocity, temperature and concentration distribution profiles for different values of thermal radiation parameter with
Gr=1.0
,
γ
=0.5
,
Sc=0.2
,
Q=0.01
,
Gm=1.5
,
Pr=0.71
,
M=1.2
are displayed in figures 11, 12 and 13. It was observed from figure 11 that velocity profiles increases initially to a peak near the plate as
η
increases but decreases away towards the free stream as the thermal radiation parameter
N
increases. It also reveals that increasing values of
N
causes less interaction of radiation with the momentum boundary layer.
Figure 11: Velocity profiles against
η
for varying values of
N
Figure 12: Temperature profiles against
η
for various values of thermal radiation parameter
N
Figure 13: Concentration profiles for various values of thermal radiation parameter
N
Figure 12 reveals that temperature profiles increase as radiation parameter
N
increases. This is because thermal radiation in the fluid increases the temperature within the boundary layer which accelerates the convection as well as increases the flow within the boundary layer. It was observed from figure 13 that the concentration profiles decreases in boundary layers away from the plate as thermal radiation parameter
155 | Fascicule 4

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