Two Phase Thermal Boundary Layer Flow

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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 stabilizes the boundary layer growth. Key words: Two-phase flow, Boundary layer characteristics, Buoyancy, Heat transfer NOMENCLATURE : x, y, Space co-ordinates i.e. distance τ p, τ T Velocity and thermal along the perpendicular to equilibrium time plate length c p, c s Specific heat of fluid and SPM q u, v Velocity components for the respectively fluid phase in x, y and z Re Fluid phase Reynolds number directions respectively Pr Prandtl number q p u p, v p Velocity components for the Ec Eckret number particle phase in x, y and z Nu Nusselt number directions respectively Gr Grassoff number T, T p Temperature of fluid and Fr Froud number particle phase respectively c f Skin friction coefficient T w, T Temperature at the wall and free-stream respectively τ w Skin friction (Shear stress for clear fluid) υ, υ p Kinematic coefficient of p Pressure of fluid phase viscosity of fluid and particle phase respectively φ Volume fraction of Suspended particulate matter (SPM) ρ, ρ p Density of fluid and particle D Diameter of the particle phase respectively Boundary layer thickness ρ s, ρ m Material density of particle a Thermal diffusivity and mixture respectively κ Thermal conductivity μ, μ p Coefficient of viscosity of fluid α Concentration parameter and particle phase respectively β Coefficient of volume

expansion ε Diffusion parameter F Friction parameter between the fluid and the particle F = 8μ ρ p d 2. INTRODUCTION: L Reference / Characteristic length U Free stream velocity A 2 L 2 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 co-efficient, particulate velocity and density on the surface under various assumptions. Several investigators [-9] have derived equations governing the Two Phase flow and reduce them to boundary layer type using Prandtl boundary layer approximations. Marbel s [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 x =. No effort has been made for studying the temperature distribution inside the boundary layer. Jain & Ghosh [] 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. 2. MATHEMATICAL FORMULATION & SOLUTION : The governing equations of two dimensional gas particulate flow within the boundary layer on a flat plate are 2

u + v y = () ρ pu p + y ρ pv p = (2) ρ u u u + v y = p + u μ y φ φ ρ s τ p u u p ρgβ T T (3) ρ p u p u p + v p u p y = φ y μ s u p y + ρ p τ p u u p + φ ρ s ρ g (4) ρc p u T T + v y = y T κ y + μ u y 2 + φ φ ρ s c s τ T T p T (5) T φρ s c s u p p + v T p p y = φ k p 2 T p y 2 + φρ sc s τ T T T p (6) The boundary conditions are At y = : u =, v =, u p = a 2 (x), v p =, ρ p = a 3 x, T =, T p = a 4 (x) (7) At y = : u = U, u p = U, ρ p = ρ p, T = T, T p = T (8) Clearly > t and > p It may be noted that, the thickness of the thermal boundary layer ( t ), particle velocity boundary layer ( p ), particle thermal boundary layer ( pt ) 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. y = (wall) to y =, we get d dx u u dy = U U μ u + ρu 2 y F ρ p φ ρu u p U dy F ρ p φ ρu u U dy + U 2 gβ T T dy (9) ρ p u p U u p u p dy = φ μ s F ρ y p u u p dy ρ g ρ y = ρ p dy s () u T T dy = a T y y = + μ ρc p u y 2 dy + φ c s τ T ρc p ρ p T p T dy () 3

ρ p u p T p T dy = φ k p c s T p y y= + τ T ρ p T p T dy (2) By introducing the non- dimensional quantities like x = x L, y = y, u = u U, u p = u p U, T = T T T w T, ρ p = ρ p ρ p, T p = T p T T pw T (3) The equations (9) to (2) reduces to d dx L u u dy = μ ρu u y y = + F ρ p φ ρ U ρ p u p dy F ρ p φ ρ U ρ p u dy + L Gr Re 2 T dy (4) ρ p u p u p dy u T = L2 dy = al U ε Re u p y y = FL U T + μ UL y y = ρc p T w T ρ p u u p dy Fr u y 2 dy ρ ρ ρ p dy s (5) + c s ρ p L φ τ T c p ρ U ρp T p T dy (6) ρp T p T dy = τ TU L ρ p u p T p dy L2 ε Pr Re T p y y = (7) Subject to the boundary conditions y = : u =, v =, u p = a 2 (x ), v p =, ρ p = a 3 (x ), T =, T p = a 4 (x ) (8) y = : u = u p = ρ p =, T =, T p = (9) To make the equation consistent, we use the auxiliary condition that the flux of particulate mass across any control volume is zero. i.e. ρ p U = ρ p u p dy (2) 4

which gives after non dimensionalisation d dx ρ p u p Using the profiles u = ( y ) 3 dy = (2) u p = a 2 ( y ) 3 ρ p = a 3 ( y ) 3 (22) T = ( y ) 3 T p = a 4 ( y ) 3 So, the two-phase boundary layer non-dimensional equations after using the third degree profiles are given by, da = 56μ 2 FL αaa dx ρul 3 U 2 4a 3 + 3 + 4 3 da 2 dx = da 4 dx = da dx 8 6a 2+2a 3 2a 2 2 +6a 2 a 3 28a 2 2 a 3 + 2A 2+6a 2 28a 2 2 Gr Re 2 FL U A a 2 4a 3+3 68 ε Re a 2 3 da 56 dx + 3 Pr Re α a4 φ 42 Pr da 3 = 4a 3+3 dx 4a 2 +3 2 A (23) + 4 A Fr 2A 6+24a 2 6a 3 +56a 2 a 3 + 9 Ec 5Re α Aa4 3 da 2 φ 5 Pr 9+6a 2 +6a 3 +4a 2 a da 3 φ αa 2 Pr ρ ρ s 3+a 3 dx +3da 3 dx +7a 2 da 3 dx +7a 3 da 2 dx dx 2 α ε φ Pr 2 Re a 4 9+6a 2 +6a 3 +4a 2 a 3 da 2 dx (26) 3. DISCUSSION OF THE RESULTS: Equations (23) to (26) with boundary conditions (8) and (9) are integrated numerically by Runge- Kutte 4 th order scheme. The solutions are obtained for different Prandtl number (Pr), volume fraction (φ), material density of SPM (ρ s ), diameter or size of the particle (D), diffusion parameter (ε), concentration parameter (α) 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. () & (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 x i.e. towards the downstream of the plate. In fig. () & (4) the profiles for carrier fluid temperature display a simple shape which is found in the thermal da3 dx (24) (25) 5

u & T ----> Particle velocity -------> 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, arising due to stress present in the particle phase and the heat due to conduction through the particle phase in the modeling of two-phase 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 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.7,. and 7. 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. () 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. ). Inclusion of Buoyancy increase the magnitude of the particle velocity and temperature, but the temperature assumes negative value (Fig. 2 & 3). Inclusion of Buoyancy decrease the skin friction and also heat transfer from plate fluid as can be observed from table-..2..8.6.4.2. u T. 5.. Fig. : Variation of u & T with y.6.4.2..8.6.4.2. x =.2 x = 75.2. 5.. Fig.2: Variation of particle velocity with y 6

Particle velocity ----> Tp -------> Particle velocity -------> Particle density -------> Particle density --------> Particle temp. -------->.2..8 x =.2 x = 75.2.. -.. 5. y ------->..6.4.2. y ------->. 5.. Fig.3: Variation of particle density with y -2. -3. -4. -5. Fig.4: Variation of particle temperature with y x =.2 x = 75.2.6.4.2..8.6.4.2. Rhop = 8 Rhop = 243 Rhop = 8. 5.. Fig.5: Variation of particle velocity with y.2..8.6.4.2. Rhop = 8 Rhop = 243 Rhop = 8 y -------->. 2. 4. 6. 8.. Fig.6: Variation of particle density with y 5. 4. 3. D = micron D = 6 micron. -2. y -----> -.. 5.. 2.... 2. 4. 6. 8.. Fig.7 : Variation of particle velocity with y -3. -4. -5. -6. -7. Fig.8 : Variation of particle temperature with y D = micron D = 6 micron 7

Tp ---------> Tp ---------> Particle velocity --------> Tp --------> Particle velocity ----->.. 2. 4. 6. 8.. -. -2. -3. Pr =.7 Pr =. Pr = 7. 2..5..5 Alpha =. Alpha =.2 Alpha =.3-4. -5. Fig. 9: Variation of particle temperature with y. y ------->. 2. 4. 6. 8.. Fig. : Variation of particle velocity with y.. 5.. -. -2. -3. -4. -5. Fig. : Variation of particle temperature with y y -------> 2.5 2..5. Alpha =..5 Alpha =.2 Alpha =.3. without Bouyancy with Bouyancy. 5.. Fig. 2: Comparision of particle velocity with and without Bouyancy 2... y -----> -.. 5.. -2. -3. -4. without Bouyancy with Bouyancy -5. Fig. 3: Comparision of particle temperature with and without Bouyancy 8

Table : Comparison of Skin friction & Nusselt number with and without Buoyancy x C f Without Buoyancy C f With Buoyancy Nu Without Buoyancy Nu With Buoyancy. 9.63E- 9.8E- 7.8E+ 7.53E+ 9.9 2.38E-4 9.68E-6.38E+3 5.63E+ 9.7.73E-4.59E-7 2.E+3.84E+ 29.5.45E-4 2.6E-9 2.5E+3 4.5E-2 39.3.28E-4 4.26E- 2.96E+3 9.84E-4 49..8E-4 6.98E-3 3.39E+3 2.2E-5 58.9.E-4.4E-4 3.8E+3 3.96E-7 68.7.4E-4.88E-6 4.2E+3 7.57E-9 78.5.E-4 3.7E-8 4.62E+3.42E- 88.3 9.67E-5 5.4E-2 5.2E+3 2.6E-2 98. 9.4E-5 8.26E-22 5.43E+3 4.76E-4 Table 2 : Comparison of plate values with and without Buoyancy Plate values without Buoyancy Plate values with Buoyancy x u P ρ p T p u P ρ p T p..2e+ 9.84E-.5E+.2E+ 9.84E-.3E+ 9.9 2.84E+.3E- 4.38E+.48E+ 6.25E- -4.49E+ 9.7 7.3E+ -2.27E+.9E+.5E+ 6.4E- -4.62E+ 29.5 2.57E+ -4.42E+ 2.28E+.5E+ 6.4E- -4.63E+ 39.3 2.5E+ -5.2E+.9E+.5E+ 6.4E- -4.63E+ 49..98E+ -5.22E+.5E+.5E+ 6.4E- -4.63E+ 58.9.97E+ -5.24E+.3E+.5E+ 6.4E- -4.63E+ 68.7.97E+ -5.24E+.3E+.5E+ 6.4E- -4.63E+ 78.5.97E+ -5.24E+.3E+.5E+ 6.4E- -4.63E+ 88.3.97E+ -5.24E+.3E+.5E+ 6.4E- -4.63E+ 98..97E+ -5.24E+.3E+.5E+ 6.4E- -4.63E+ 9

REFERENCES :. Jain A.C. & Ghosh A., Gas particulate laminar boundary layer on a flat plate, Z.F.W., 979, 3, pp.29-37. 2. 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, 963. 3. Mishra S.K. & Tripathy P.K., Mathematical and Numerical modeling of two phase flow and heat transfer using non-uniform grid, Far East journal of Applied Mathematics, 2,54(2),pp.7-26. 4. Mishra S.K., Tripathy P.K., Approximate solution of two phase thermal boundary layer flow, Reflections des ERA, 2,6 (2)pp.3-48. 5. 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, 97. 6. Singleton R. E., The compressible gas solid particles flow over a semi infinite flat plate. Z. angew. Math. Phys., 965,6, pp. 42 449. 7. Soo S.L., Fluid Dynamics of multiphase systems. Blaisdell publishing company, London, 967. 8. Tabakoff W., Hamed A., Analysis of cascade particle gas boundary layer flows with pressure gradient. AIAA th Aerospace Science meeting. San Diego, 972. pp. 72 87. 9. Tabakoff W., Hamed A., The boundary layer of particulate gas flow. Z. Flugwiss., 972, 2, pp. 373 379.

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