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 Authors : S. S. Bishoyi, P. K. Tripathy, S. K. Mishra
 Paper ID : IJERTV1IS10444
 Volume & Issue : Volume 01, Issue 10 (December 2012)
 Published (First Online): 28122012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Two Phase Free Convection Flow and Heat Transfer From A Vertical Flat Plate
S. S. Bishoyi1, P. K. Tripathy2 and S. K. Mishra3
1Department of Mathematics, Gandhi Institute of Industrial & Technology, Golanthra, Berhampur, Dist. Ganjam, Odisha, INDIA
2Department of Mathematics & Science, U.C.P. Engineering School, Berhampur 760010, (Odisha) INDIA
3Department of Mathematics, R.C.M. Science College, Khallikote761030,(Odisha)INDIA
Abstract
Momentum integral method has been employed to study the effect of suspended particulate matter (SPM) on twophase free convection heat transfer from a vertical plate. The presence of SPM increases the velocity of the carrier fluid where as diffusion of particles and concentration of particles in the fluid has little effect on it. The Grashoff number has a effect to increase the carrier fluid velocity but Prandtl number has a effect to decrease the magnitude of particle velocity and temperature. It is observed that, the heat is transferred from plate to fluid.
Key words: Particulate suspensions; boundary layer characteristics; volume fraction; heat transfer.
Nomenclature:
Space coordinates i.e. distance along the perpendicular to plate length
Velocity components for the fluid phase in and directions respectively
Velocity components for the particle phase in and and
directions respectively
Temperature of fluid and particle phase respectively
Temperature at the wall and freestream respectively
Kinematic coefficient of
viscosity of fluid and particle phase respectively
Density of fluid and particle phase respectively
Material density of particle and mixture respectively
Coefficient of viscosity of fluid
and particle phase respectively Velocity and thermal
equilibrium time
Specific heat of fluid and SPM respectively
Gravitational acceleration vector
Fluid phase Reynolds number Prandtl number
Eckret number Nusselt number Grassoff number Froud number
Skin friction coefficient
p Pressure of fluid phase
Volume fraction of Suspended particulate matter (SPM)
d Diameter of the particle Radius of each particle Boundary layer thickness Thermal diffusivity
Thermal conductivity Concentration parameter Loading ratio
Coefficient of volume expansion
Diffusion parameter Particle mass fraction
,the fluid density at STP
1.67x mass of hydrogen at STP
Molecular weight of fluid
Friction parameter between the fluid and the particle
Reference / Characteristic length
U Free stream velocity
A
Subscripts:
Values at free stream
w Values on the plate/wall
1. Introduction
The study of a flow situation where free and forced convention effects are of comparable order, finds application in several industrial and technical processes such as nuclear reactors cooled during emergency situation, solar central receivers exposed to winds, electronics devices cooled by fans and heat exchanges placed in a lowvelocity environment. The simplest physical model convention flow is the two dimensional laminar mixed convection flow along a vertical flat plate and extensive studies have been conducted on this type flow [ 1 4].
The effect of buoyancy forces in flow and heat transfer is usually ignored when a forced convection flow over a cooled or heated surface is studied. But it is not justified as the buoyancy forces influences the flow and temperature functions despite the presence of forced convention flow.
The steady, laminar, free convective two phase boundary layer flow of a viscous incompressible fluid with suspended particulate matter along a semi infinite vertical flat plate is considered.
It has shown Rudinger [5], that the particle contribution towards the pressure can be neglected even at extremes of the density ratio and mass fraction as long as the particle diameter d is larger than a few hundredths of a micron i.e.
Further the particle volume fraction
may become significant, if the density
ratio or the loading ratio or both become sufficienly large. If we consider the gravitational force as a body
force, we should also include buoyancy force as a part of the interaction force between the pseudofluid and gas phases. Buoyancy force on pseudofluid is . If we combine buoyancy force with the corresponding
The boundary layer equations for two dimensional two phase flow after simplifications are given by
body force due to gravitational acceleration, we get for the pseudofluid, the total force due to gravitational
acceleration as .
Soo[6] has clarified the condition under which the gravity effect is negligible. the conditions are the gravity effect on the fluid phase must be small, compared with the effect of friction. In addition to the above conditions one of the following conditions:

The density of the material of the fluid phase and the particle phase are equal

In case of Turbulent flow, the Intensity of the random particle motion due to fluid drag is much greater than
that due to the gravity effect.

In case of Turbulent flow of electrically charged
particles, the intensity of the particle motion due to
electrostatic force is greater than

Mathematical Formulation
In the present problem, to show the free convections effects, we have considered a very long
(1)
(2)
(3)
(4)
(5)
(6)
(7)
and wide plane perpendicular to the floor, which is either heated or cooled. In either case the mathematical solution is same. The coordinate and coordinate are chosen along and perpendicular to the plate. Further the
In free convection, outside the boundary layer , where as and approaches to zero, equation (3) becomes
temperature of the heated or cooled surface be taken as and the temperature decreases asymptotically to the
value of ambient fluid far away. The fluid viscosity
on the plate increases from zero at the plate surface to a maximum value in the immediate vicinity of the surface and there after decreases asymptotically to a zero value far away. The velocity and temperature of the particle phase differs from fluid velocity on the plate and approaches asymptotically to the fluid velocity and temperature.
Assuming the boundary layer flow laminar, and since the flow is two dimensional in the present consideration and steady the boundary layer
Or, (8)
Putting (8) in (3), we get
(9)
The coefficient of thermal expansion is given by
simplification gives the following governing equations for the present flow problems. Here in free convection
the buoyancy force sustains the fluid motion, and hence
the gravity term must be included in momentum integral equations.
Or,
Or, (10)
Substituting (10) in (9), we get
Also, for smooth velocity and temperature profiles we assume a , .
Integrating (14) to (17) between the limit to
, we get
Or,
(11)
(20)
Hence the governing equations for the present problem are
(21)
(12)
(13)
(14)
(22)
(23)
By introducing nondimensional quantities
(15)
(16)
(17)
We need one more auxiliary condition to make the system consistent. Here we use the condition that flux of the particulate mass acrossany control volume is zero
In equations (20) to (23), we get after dropping bars
(24)
(25)
i.e (18)
Boundary condition for dimensional quantities
(19)
(26)
(27)
(28)
Further the equation (18) reduces to
and the boundary condition are
(29)
Where
(39)
(30)
and
(40)
(31)
Selecting the velocity profiles subjected to the boundary conditions (30) and (31) as
(32)
(41)
(42)
and substituting (32) in (25) to (29), we get
Simplifying equations (33) to (37), we get
(33)
(34)
(35)
(36)
(37)
(38)

Discussion of Results and conclusion
The 1st order differential equation (38) to (42) are solved by using 4th order RungeKutta method for various values of the nondimensional parameters, Reynolds number(Re), Grassoff number(Gr), Volume fraction , Species density of the particle , density of the particle , Prandtl number , Froud number , Eckert number
Fig.1 depicts the pattern of the carrier fluid phase and temperature T. From Table1, it can be observed that the presence of SPM results in increase in magnitude of the carrier fluid velocity. The diffusion and concentration parameter has a little effect on velocity, temperature of the carrier fluid and as well as particle phase.
From Figs.2, 3, 4; it is observed that magnitude of the carrier fluid velocity u increase with the increase of where as particle phase velocity and temperature decreases with the increase of . From Table2, it is observed that the Nusselt number decreases with the increase of . Further heat is transferred from plate to fluid in case of air and from fluid to plate in case of electrolyte solution and water .
Table3 depicts the presence of SPM increase the skin friction and decrease the Nusselt number.
1.20E+00
4.50E+03
.00 5.00 10. 

Gr=25 

Gr=30 Gr=50 
0.00E+00
0
1.00E+00
4.00E+03
5.00E+03
3.50E+03
8.00E01
3.00E+03
u
1.00E+04
T
2.00E+03
1.50E+03
6.00E01
2.50E+03
Particle velocity ——>
1.50E+04
4.00E01
2.00E+04
5.00E+02
0.00E+00
2.00E01
1.00E+03
2.50E+04
0 2 4 6 8 10
y —>
0.00E+00
3.00E+04
00
y —>
u —>
Figure 1: Variation of u & T with y with SPM
9.00E+03
8.00E+03
7.00E+03
Gr=25
Gr=30
Gr=50
6.00E+03
5.00E+03
4.00E+03
3.00E+03
2.00E+03
1.00E+03
y —>
0.00E+00
0.00
5.00
10.00
Figure 2: Variation of u with y for different Gr
Figure 3: Variation of particle velocity with y for different Gr
0.00E+00
0.00
5.00
10.00
1.00E+04
2.00E+04
6.00E+04
y —>
Gr=25
Gr=30
Gr=50
4.00E+04
5.00E+04
3.00E+04
Particle Temp.
—>
Figure 4: Variation of particle temperature with y for different Gr.
Table 1: Magnitude of carrier fluid velocity with and without SPM
y 
With out SPM 
With SPM 
0.00E+00 
0.00E+00 
0.00E+00 
2.00E01 
2.22E+01 
6.27E+02 
6.00E01 
6.00E+01 
1.69E+03 
1.00E+00 
8.95E+01 
2.53E+03 
1.40E+00 
1.11E+02 
3.14E+03 
1.80E+00 
1.26E+02 
3.57E+03 
2.20E+00 
1.35E+02 
3.81E+03 
2.60E+00 
1.38E+02 
3.91E+03 
3.00E+00 
1.37E+02 
3.86E+03 
3.40E+00 
1.31E+02 
3.71E+03 
3.80E+00 
1.22E+02 
3.45E+03 
4.20E+00 
1.11E+02 
3.13E+03 
4.60E+00 
9.71E+01 
2.74E+03 
5.00E+00 
8.22E+01 
2.32E+03 
5.40E+00 
6.67E+01 
1.88E+03 
5.80E+00 
5.13E+01 
1.45E+03 
6.20E+00 
3.67E+01 
1.04E+03 
6.60E+00 
2.36E+01 
6.67E+02 
7.00E+00 
1.28E+01 
3.61E+02 
7.40E+00 
4.87E+00 
1.37E+02 
7.80E+00 
5.71E01 
1.61E+01 
8.00E+00 
1.63E10 
4.59E09 
Table 2: Variation of Skin friction & Nusselt number
Variation of 
Skin friction 
with SPM 
Variation Nusselt number 
with SPM 

Pr=0.71 
Pr=1.0 
Pr=7.0 
Pr=0.71 
Pr=1.0 
Pr=7.0 

1.01E01 
5.17E06 
8.40E06 
2.07E06 
5.80E04 
9.42E04 
2.32E04 
5.05E+00 
1.24E05 
9.18E05 
7.36E08 
6.94E02 
5.15E01 
4.13E04 
1.00E+01 
1.32E05 
1.71E04 
4.56E08 
1.47E01 
1.90E+00 
5.06E04 
1.49E+01 
1.33E05 
2.24E04 
4.24E08 
2.20E01 
3.72E+00 
7.05E04 
1.99E+01 
1.33E05 
2.64E04 
4.12E08 
2.93E01 
5.83E+00 
9.11E04 
2.48E+01 
1.33E05 
2.96E04 
4.05E08 
3.66E01 
8.16E+00 
1.12E03 
2.98E+01 
1.33E05 
3.23E04 
4.01E08 
4.39E01 
1.07E+01 
1.33E03 
3.47E+01 
1.33E05 
3.47E04 
3.97E08 
5.12E01 
1.34E+01 
1.53E03 
3.97E+01 
1.33E05 
3.68E04 
3.95E08 
5.85E01 
1.62E+01 
1.74E03 
4.46E+01 
1.33E05 
3.87E04 
3.93E08 
6.58E01 
1.92E+01 
1.95E03 
4.96E+01 
1.33E05 
4.04E04 
3.91E08 
7.31E01 
2.23E+01 
2.16E03 
5.45E+01 
1.33E05 
4.18E04 
3.90E08 
8.04E01 
2.54E+01 
2.36E03 
5.95E+01 
1.33E05 
4.31E04 
3.89E08 
8.77E01 
2.85E+01 
2.57E03 
6.44E+01 
1.33E05 
4.41E04 
3.88E08 
9.50E01 
3.16E+01 
2.78E03 
6.94E+01 
1.33E05 
4.49E04 
3.87E08 
1.02E+00 
3.46E+01 
2.98E03 
7.43E+01 
1.33E05 
4.56E04 
3.86E08 
1.10E+00 
3.77E+01 
3.19E03 
7.93E+01 
1.33E05 
4.61E04 
3.85E08 
1.17E+00 
4.06E+01 
3.39E03 
8.42E+01 
1.33E05 
4.65E04 
3.85E08 
1.24E+00 
4.35E+01 
3.60E03 
8.92E+01 
1.33E05 
4.68E04 
3.84E08 
1.31E+00 
4.64E+01 
3.81E03 
9.41E+01 
1.33E05 
4.70E04 
3.83E08 
1.39E+00 
4.92E+01 
4.01E03 
9.91E+01 
1.33E05 
4.72E04 
3.83E08 
1.46E+00 
5.20E+01 
4.22E03 
Table 3: Variation of Skin friction & Nusselt number with and without SPM
Skin Friction 
Nusselt number 

Without SPM 
With SPM 
Without SPM 
With SPM 

1.01E01 
0.00E+00 
5.17E06 
0.00E+00 
5.80E04 
5.05E+00 
1.18E05 
1.24E05 
2.77E01 
6.94E02 
1.00E+01 
1.01E05 
1.32E05 
4.74E01 
1.47E01 
1.49E+01 
9.13E06 
1.33E05 
6.43E01 
2.20E01 
1.99E+01 
8.50E06 
1.33E05 
7.99E01 
2.93E01 
2.48E+01 
8.04E06 
1.33E05 
9.44E01 
3.66E01 
2.98E+01 
7.68E06 
1.33E05 
1.08E+00 
4.39E01 
3.47E+01 
7.38E06 
1.33E05 
1.21E+00 
5.12E01 
3.97E+01 
7.14E06 
1.33E05 
1.34E+00 
5.85E01 
4.46E+01 
6.93E06 
1.33E05 
1.46E+00 
6.58E01 
4.96E+01 
6.75E06 
1.33E05 
1.58E+00 
7.31E01 
5.45E+01 
5.66E06 
1.33E05 
2.66E+00 
8.04E01 
5.95E+01 
5.11E06 
1.33E05 
3.60E+00 
8.77E01 
6.44E+01 
4.72E06 
1.33E05 
4.55E+00 
9.50E01 
6.94E+01 
4.47E06 
1.33E05 
5.35E+00 
1.02E+00 
7.43E+01 
4.27E06 
1.33E05 
6.12E+00 
1.10E+00 
7.93E+01 
4.11E06 
1.33E05 
6.85E+00 
1.17E+00 
8.42E+01 
3.98E06 
1.33E05 
7.57E+00 
1.24E+00 
8.92E+01 
3.86E06 
1.33E05 
8.25E+00 
1.31E+00 
9.41E+01 
3.85E06 
1.33E05 
8.32E+00 
1.39E+00 
9.91E+01 
3.84E06 1.33E05 
8.39E+00 1.46E+00 
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