 Open Access
 Total Downloads : 724
 Authors : P. K. Tripathy, S. K. Mishra
 Paper ID : IJERTV1IS8218
 Volume & Issue : Volume 01, Issue 08 (October 2012)
 Published (First Online): 29102012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Two Phase Thermal Boundary Layer Flow
P. K. Tripathy
Deptt. Of Mathematics & Science,
U. C. P. Engg. School, Berhampur (Orissa), INDIA
S. K. Mishra
Deptt. Of Mathematics, R. C. M. Science College, Khallokote (Orissa) , INDIA
ABSTRACT:
Momentum integral method has been employed by using third degree profiles for velocity, temperature and particle density to study the thermal boundary layer characteristics over a flat plate. It is observed that, the particle velocity, the particle density and the temperature on the plate approaches a finite value towards the downstream. The solution is valid throughout the plate unlike previous studies available in the literature. It has been observed that, heat flows from the plate towards the fluid as Nusselt number (Nu) is positive. Irrespective of presence of heavier or lighter material particles, the particles settles down on the plate as expected and the buoyancy force stabilizes the boundary layer growth.
Key words: Twophase flow, Boundary layer characteristics, Buoyancy force, Heat transfer
NOMENCLATURE :
, , 
Space coordinates i.e. distance 
, 
Velocity and thermal 

along the perpendicular to 
equilibrium time 

, 
plate length Velocity components for the 
, 
Specific heat of fluid and SPM respectively 

fluid phase in , and 
Fluid phase Reynolds number 

directions respectively 
Prandtl number 

, 
Velocity components for the 
Eckret number 

particle phase in , and 
Nusselt number 
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 
Grassoff number
Froud number
Skin friction coefficient 

Skin friction (Shear stress for clear fluid) 

p 
Pressure of fluid phase 

Volume fraction of Suspended 

particulate matter (SPM) 

D 
Diameter of the particle 

Boundary layer thickness 

Thermal diffusivity 

Thermal conductivity 

Concentration parameter 
Coefficient of volume
expansion 
Reference / Characteristic 

Diffusion parameter 
length 

Friction parameter the fluid and the 
between particle 
U A 
Free stream velocity 2 2 

= 18 2 

INTRODUCTION:
The boundary layer flow of a gas particulate mixture over a flat plate gives the detailed structure of the flow and estimates the surface characteristics like skin friction coefficient, particulate velocity and density on the surface under various assumptions. Several investigators [19] have derived equations governing the Two Phase flow and reduce them to boundary layer type using Prandtl boundary layer approximations. Marbels [2] solutions is valid for downstream region of the plate and the particulate velocity on the surface remains zero. Singleton [6] has studied compressible gas particulate flow over a flat plate for high and low slip flow regions by employing series solution method. Soo [7] has derived momentum integrals for the gas and particulate phases and solved the same by using linear profiles both for gas phase and particle phase and quadratic profile for particulate density. Tabakoff and Hammed [8] have used fourth degree profiles for both gas and particle velocity and particle density. Soo [3] and Tabakoff and Hammed [9] has pointed out that particle velocity decreases linearly along the plate length x and particle density increases continuously along the plate length x. Their study leads to a surface particle velocity zero and particle density to infinity at a distance along the plate length = 1. No effort has been made for studying the temperature distribution inside the boundary layer. Jain & Ghosh [1] have investigated the structure and surface property of the boundary layer of a gas particulate flow over a flat plate by employing momentum integral method. They have pointed out that the third degree profile for velocity and particle density gives results which is valid to far downstream stations on the plate. With the third degree profile of particulate velocity on the surface continuously decreases from its free stream value and particulate density on surface increases rather slowly from its free stream value at the leading edge to an asymptotic value as we approach far downstream on the plate surface.
The present study is an attempt to study the temperature distribution inside the boundary layer over a flat plate which gives a better understanding of the gas particulate boundary layer flow one. In this case, the momentum integral method is adopted to study the flow and temperature distribution by using a third degree profiles.

MATHEMATICAL FORMULATION & SOLUTION :
The governing equations of two dimensional gas particulate flow within the boundary layer on a flat plate are
+ = 0 (1)
+
= 0 (2)
+ = +
(3)
1
+
=
+
+
(4)
+ = + 2 +
(5)
1
+
=
2 +
1 (6)
2
The boundary conditions are
At y = 0 : u = 0, v = 0, = 2(), = 0, = 3 , T = 0, = 4() (7) At y = : u = U, = , = , T = , = (8)
Clearly > and >
It may be noted that, the thickness of the thermal boundary layer ( ), particle velocity boundary layer ( ), particle thermal boundary layer ( ) are the same as that of the velocity boundary layer . Strictly speaking, they are different, in general. This assumption has its justification in that it simplifies
the computational work and the results obtained are very near to the experimental results and to the exact solutions.
Now, on integration equations from (2) to (6) w. r. t. = 0 (wall) to = , we get
1 =
+ 1
1
0
2 0
1
0
1
1 +
1 (9)
1
0
0 2
=
1
(10)
+
0
= 0
0
0
=
+ 2 1 1
(11)
0
= 0
0
1 0
=
+ 1
(12)
0
=0
0
By introducing the non dimensional quantities like
= , = , = , = , =
, =
, =
(13)
0
The equations (9) to (12) reduces to
1 1 =
+ 1 0 1 1
0
=0
1
0
1 0 1 1 + 1 (14)
1 1
1
0
0 2
0
= 2
1
1
1
(15)
= 0
0
0
1
=
+ 1 2
0
=0
0
+ 1 1
0
1
(16)
1
0
1 1
2
=
(17)
0
0
Pr
=0
Subject to the boundary conditions
= 0 : = 0, = 0, = (), = 0, =
(), = 1, =
() (18)
2
3
4
= 1 : = = = 1, = 0, = 0 (19)
To make the equation consistent, we use the auxiliary condition that the flux of particulate mass across any control volume is zero.
0
i.e. 0 =
(20)
which gives after non dimensionalisation
1
= 0 (21)
0
Using the profiles
= 1 (1 )3
= 1 1 (1 )3
2
= 1 1 (1 )3 (22)
3
= (1 )3
= (1 )3
4
So, the twophase boundary layer nondimensional equations after using the third degree profiles are given by,
= 56 2 2 43 + 3 + 14
(23)
3
3 2
1862 +123 122 +162 3 282 3 + 2 12+162 28 2 3
2 2
2
20 2 43 +3 1680 12 + 140 1 1 3+3
2 =
(24)
2 6+242 163 +562 3
3 + 3 + 1 4 3 2 +3 3 +72 3 +73 2
56
1 105
1 4 9+62 +63 +142 3
1 2 4
210
4 =
1 420
1 2
(25)
3 = 43 +3
1
1
2
9+62 +63 +142 3
(26)
42 +3

DISCUSSION OF THE RESULTS:
Equations (23) to (26) with boundary conditions (18) and (19) are integrated numerically by Runge Kutte 4th order scheme. The solutions are obtained for different Prandtl number (Pr), volume fraction (), material density of SPM (), diameter or size of the particle (D), diffusion parameter (), concentration parameter () for uniform plate temperature. The temperature, velocity and particle density profiles are presented in figures for different values of above parameters. It is seen from fig. (1) & (2) that the carrier fluid velocity satisfied the no slip condition but the particle velocity profiles do not satisfy no slip condition at the wall and go on increasing with i.e. towards the downstream of the plate. In fig. (1) &

the profiles for carrier fluid temperature display a simple shape which is found in the thermal
boundary layers of pure fluid flow, but the particle temperature on the plate becomes negative towards the downstream of the plate. Fig. (3) displays the profile for the particle densities, which shows that the density of the particle on the plate go on decreasing towards the downstream. Table 2 shows that the particle density and particle velocity on the plate assumes a finite value towards the downstream station of the plate. Physically it indicates that the consideration of finite volume fraction force, arising due to stress present in the particle phase and the heat due to conduction through the particle phase in the modeling of twophase flow may not stabilize the boundary layer growth.
From fig.(5) & (6), we conclude that irrespective of presence of heavier or lighter material particles, the particles settles down on the plate as expected and the buoyancy force stabilizes the boundary layer growth. Fig. (7) & (8) shows the presence of coarser particles decrease the magnitude of velocity and increase the magnitude of temperature of the particle phase in comparison with the presence of finer particles inside the boundary layer.
The values of Prandtl number (Pr) are taken as 0.71, 1.0 and 7.0 which physically corresponds to air, electrolyte solution and water respectively. The magnitude of the particle temperature of water is very low as compare to air and electrolyte solution. Fig. (11) shows the particle temperature increases as the number of particles per unit volume of the mixture increases, where as the magnitude of the particle velocity increases (Fig. 10).
Inclusion of Buoyancy force increase the magnitude of the particle velocity and temperature, but the temperature assumes negative value (Fig. 12 & 13). Inclusion of Buoyancy force decrease the skin friction and also heat transfer from plate fluid as can be observed from table1.
u
1.20
1.60
Particle velocity ——>
1.40
x = 0.2
1.00
u & T —>
0.80
0.60
0.40
0.20
T x = 75.2
1.20
1.00
0.80
0.60
0.40
0.20
0.00
y ——>
0.00 5.00 10.00
0.00
y ——>
0.00 5.00 10.00
Fig. 1 : Variation of u & T with y
Fig.2: Variation of particle velocity
with y
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
1.00
2.00
3.00 x = 0.2
x = 75.2
4.00
5.00/p>
Fig.4: Variation of particle temperature with y
5.00 y ——> 10.00
0.00
1.00
0.00
Fig.3: Variation of particle density
with y
y ——>
0.00 5.00 10.00
0.80
0.60
0.40
0.20
0.00
x = 0.2
x = 75.2
1.20
1.00
Particle density ——>
Particle temp. ——>
1.20
Particle velocity ——>
Particle density ——>
1.00
0.80
Rhop = 800
Rhop = 2403
Rhop = 8010
y ——>
0.00 5.00 10.00
0.60
0.40
0.20
0.00
Fig.5: Variation of particle velocity with
y
>
Rhop = 800
Rhop = 2403
Rhop = 8010
y ——
0.00 2.00 4.00 6.00 8.00 10.00
Fig.6: Variation of particle density with y
5.00
Particle velocity —>
4.00
0.00
y —>
3.00
2.00
D = 100 micron
D = 60 micron
1.000.00 5.00 10.00
Tp ——>
2.00
3.00
1.00
0.00
y ——>
0.00 2.00 4.00 6.00 8.00 10.00
4.00
5.00
6.00
7.00
D = 100 micron
D = 60 micron
Fig.7 : Variation of particle velocity
with y
Fig.8 : Variation of particle
temperature with y
0.00
y ——>
2.50
2.00
without Bouyancy force with Bouyancy force
1.50
1.00
0.50
0.00 y ——>
0.00 5.00 10.00
Fig. 12: Comparision of particle velocity with and without Bouyancy force
Fig. 11: Variation of particle
temperature with y
5.00
Alpha = 0.1
Alpha = 0.2
Alpha = 0.3
1.00
2.00
3.00
4.00
10.00
5.00
0.00
Fig. 10: Variation of particle velocity
with y
y ——>
0.00 2.00 4.00 6.00 8.00 10.00
1.00
0.50
0.00
Alpha = 0.3
Alpha = 0.2
1.50
Alpha = 0.1
2.00
10.00
Pr = 0.71
20.00 Pr = 1.0
Pr = 7.0
30.00
40.00
50.00
Fig. 9: Variation of particle temperature with y
y ——>
0.00 2.00 4.00 6.00 8.00 10.00
0.00
Tp ———>
Tp ——>
Particle velocity ——>
Particle velocity —>
without Bouyancy force
with Bouyancy force
y —>
.00 5.00 10.
2.00
1.00
Tp ———>
0.00
1.000 00
2.00
3.00
4.00
5.00 Fig. 13: Comparision of particle temperature with and without Bouyancy
force
Table 1 : Comparison of Skin friction & Nusselt number with and without Buoyancy force
Without Buoyancy
force
With Buoyancy
force
Without Buoyancy
force
With Buoyancy
force
0.10
9.63E01
9.80E01
7.81E+01
7.53E+01
9.90
2.38E04
9.68E06
1.38E+03
5.63E+01
19.70
1.73E04
1.59E07
2.00E+03
1.84E+00
29.50
1.45E04
2.60E09
2.51E+03
4.51E02
39.30
1.28E04
4.26E11
2.96E+03
9.84E04
49.10
1.18E04
6.98E13
3.39E+03
2.02E05
58.90
1.10E04
1.14E14
3.81E+03
3.96E07
68.70
1.04E04
1.88E16
4.21E+03
7.57E09
78.50
1.00E04
3.07E18
4.62E+03
1.42E10
88.30
9.67E05
5.04E20
5.02E+03
2.61E12
98.10
9.41E05
8.26E22
5.43E+03
4.76E14
Table 2 : Comparison of plate values with and without Buoyancy force
Plate values without Buoyancy force
Plate values with Buoyancy force
0.10
1.02E+00
9.84E01
1.05E+00
1.02E+00
9.84E01
1.03E+00
9.90
2.84E+00
1.03E01
4.38E+00
1.48E+00
6.25E01
4.49E+00
19.70
7.30E+00
2.27E+00
1.19E+01
1.50E+00
6.14E01
4.62E+00
29.50
2.57E+00
4.42E+00
2.28E+00
1.50E+00
6.14E01
4.63E+00
39.30
2.05E+00
5.12E+00
1.19E+00
1.50E+00
6.14E01
4.63E+00
49.10
1.98E+00
5.22E+00
1.05E+00
1.50E+00
6.14E01
4.63E+00
58.90
1.97E+00
5.24E+00
1.03E+00
1.50E+00
6.14E01
4.63E+00
68.70
1.97E+00
5.24E+00
1.03E+00
1.50E+00
6.14E01
4.63E+00
78.50
1.97E+00
5.24E+00
1.03E+00
1.50E+00
6.14E01
4.63E+00
88.30
1.97E+00
5.24E+00
1.03E+00
1.50E+00
6.14E01
4.63E+00
98.10
1.97E+00
5.24E+00
1.03E+00
1.50E+00
6.14E01
4.63E+00
REFERENCES :

Jain A.C. & Ghosh A., Gas particulate laminar boundary layer on a flat plate, Z.F.W., 1979, 3, pp.2937.

Marble F.E., Dynamics of a gas combining small solid particles. In : R. P. Hagarty , A.L. Jaumotte, O. Lutz, S.S. Penner (Eds.), Combustion and propulsion. Fifth AGARD colloquium. Pergamon press, Orfird/ London / New York / Paris, 1963.

Mishra S.K. & Tripathy P.K., Mathematical and Numerical modeling of two phase flow and heat transfer using nonuniform grid, Far East journal of Applied Mathematics, 2011,54(2),pp.107126.

Mishra S.K., Tripathy P.K., Approximate solution of two phase thermal boundary layer flow, Reflections des RA, 2011,6 (2)pp.113148.

Pai S. I., A review of fundamental equation of the mixture of gas with solid particles. Institute for fluid Dynamics and Applied mathematics, University of Maryland, Techno Note BN 668, 1970.

Singleton R. E., The compressible gas solid particles flow over a semi infinite flat plate. Z. angew . Math. Phys. , 1965,16 , pp. 421 449.

Soo S.L., Fluid Dynamics of multiphase systems. Blaisdell publishing company, London, 1967.

Tabakoff W., Hamed A., Analysis of cascade particle gas boundary layer flows with pressure gradient. AIAA 10th Aerospace Science meeting. San Diego, 1972. pp. 72 87.

Tabakoff W., Hamed A., The boundary layer of particulate gas flow. Z. Flugwiss., 1972, 20, pp. 373 379.
