J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
Heat and Mass Transfer Along an Isothermal Vertical Porous Plate in the Presence of Heat Sink BHASKER CHANDRA and MANOJ KUMAR Department of Mathematics, Statistics and Computer Science College of Basic Sciences & Humanities G. B. Pant University of Agriculture & Technology Pantnagar -263145, Uttarakhand India email:bhaskerchandra6@gmail;com mnj_kumar2004@yahoo.com ABSTRACT The objective of present investigation is to investigate the steady, laminar two-dimensional free convective flow with heat and mass transfer along an isothermal vertical porous plate in a viscous incompressible fluid. The study has been done with variable viscosity and thermal conductivity in the presence of heat sink. The governing fluid flow differential equations are transformed into coupled non-linear ordinary differential equations using similarity transformation and then solved numerically using Runga-Kutta method with shooting technique. The velocity, temperature and concentration profiles for various physical parameters have been studied. Skin-friction coefficient and Nusselt number have also been computed and analyzed. It has been observed that velocity increases with an increase in porosity parameter and decreases with increasing Prandle number. Keywords: Free convection, variable viscosity, variable thermal conductivity, heat sink, heat and mass transfer, porosity, heat sink, shooting technique
INTRODUCTION Steady, laminar, natural convection flow in the presence of heat sink has been the subject of great number of investigations due to its importance in many branches of engineering such as combustion modeling, nuclear energy, heat exchangers, petroleum
reservoir etc.Liquid metals having low Prandtl number are generally used as coolants and have applications in manufacturing processes such as the cooling of metallic plate, nuclear reactor etc. Liquid metals have very large thermal conductivity and have ability to transport heat even if small temperature difference exists between
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
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Bhasker Chandra, et al., J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
the surface and fluid. Ostarch (1952) presented the similarity solution of natural convection along vertical isothermal plate. Kay (1966) reported that thermal conductivity of liquids with low Prandtl number varies linearly with temperature in range of 0° F to 400° F. Carey and Mollendorf (1978) studied the effect of temperature dependent viscosity on free convective fluid flow. Chaim (1998) studied heat transfer in fluid flow of low Prandtl number with variable thermal conductivity. Chamkha and Khaled (2001) studied similarity solution of natural convection an inclined plate with internal heat absorption in presence of magnetic field. Mahanti and gaur (2009) presented the effects of varying viscosity and thermal conductivity on steady free convection flow along an isothermal plate in the presence of heat sink. Present study deals with investigation of the effects of varying fluid properties on free convection flow of a viscous incompressible fluid with heat and mass transfer along an isothermal porous plate in the presence of heat sink. Nomenclature: Cp C f g Gr k' K Nu Pr Q S
Skin-friction Coefficient Specific heat at constant pressure Concentration of fluid near plate Dimensionless velocity parameter Acceleration due to gravity Grashof Number Porosity of plate Porosity parameter Nusselt number Prandtl number Heat Sink Heat Sink parameter
Sc Schmidt Number T Temperature of the fluid Tw Temperature of the plate T∞ Ambient temperature of fluid u, v Velocity component along x and y directions β Coefficient of thermal expansion Ă˜ Dimensionless concentration Ρ Similarity variable Îľ Thermal conductivity parameter Îł Viscosity parameter θ Dimensionless temperature Variable viscosity
Âľ Coefficient of viscosiy Îş Thermal conductivity ν Kinematic viscosity Ď Fluid viscosity Formulation Consider steady, laminar, free convection flow of a viscous incompressible fluid along a vertical porous plate in the presence of heat sink Q. The x-axis is taken along the plate and y-axis is perpendicular to the plate. is the temperature of plate and
∞, ∞ are the ambient temperature and concentration of the fluid. The governing fluid flow equations of continuity, momentum, energy and diffusion are as:
∂v ∂u + =0 ∂x ∂y
Ď (u
(1)
∂v ∂u ∂ ∂ +v )=
+ ∂y ∂y ∂x ∂y
∞ C (u
‍כ‏ ΄
(2)
∂ ∂T ∂T ∂T +v ) = (k ) + Q (3) ∂x ∂Y ∂y ∂y
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
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Bhasker Chandra, et al., J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
∂ C ∂C ∂C u +v =D ∂x ∂y ∂y
where
2
K = ఎ ′
(4)
The boundary conditions are u = 0, v = 0, T = , C= C y = 0 u = 0, T ∞, C C∞ as y→ ∞ (5) The fluid viscosity and thermal conductivity are assumed to be temperature dependent and vary linearly with temperature. = Âľ 1 θ 1â „2
6
= κ 1 θ
(7)
Introduce the following non dimensional transformations. ŕ°
Ψ(x,y) = 4 ν f(Ρ) Ρ=
ŕ° ŕ°°
ŕ°°
, Ă˜=
, Gr =
β ŕ°Ż ŕąŕ°ˇ ∞ νఎ
,
∞ ŕłˆ ∞
ೢడ ಎ ఎ
ŕ°
ŕ°Ž $ (
)
Were Ф (x,y) is stream function and Ρ is similarity variable. Substituting these transformations into equations (2), (3) and (4) using equations (6) and (7) the equations (2), (3) and (4) are transformed into nonlinear ordinary differential equations and given as: & ΄΄΄
ఠ!"θ # ఎ ′
3&& ΄΄ % ) = 0
, Pr =
ŕł ,.
, & , Sc =
' (
The transformed boundary conditions given by (4) are, now f(0) = 0, & ′(0) = 0, θ ( 0 ) = 1, Ă˜(0) = 1 ƒ΄ ∞ = 0, θ ( ∞ , 0, Ă˜(∞) = 0 (11)
Equations (8), (9) and (10) with boundary conditions (11) are solved by using RungeKutta method with shooting technique assuming systematic guess of f″(0) and θ΄(0).The physical quantities of interest are the rate of heat transfer in terms of Nusselt number and the Skin friction coefficient and are given by ఠ= 201 122 34 ర f " 0
(12) డఠ(13) Nu = -(1+ξ) 34 ర θ' 0
RESULTS AND DISCUSSION
Following Crepeau and Clarksean (1997), the heat sink is taken as given below Q = S"#
ŕ°
ŕ°Ž
'& ΄΄ θ΄ θ 2& ′ $
(8)
(1+ θ θ΄΄ θ′$ 3Prθ′ & +θ , 0
(9)
Ă˜Î„Î„ + 3 Sc Ă˜Î„ f = 0
(10)
Runge-Kutta fourth order method with shooting technique has been used to find solution. It is worth mentioning that small values of Prandle number (<<1) which physically correspond to liquid metals have high thermal conductivity but low viscosity. The value of Prandle number (Pr~1) corresponds to di-atomic gases including air. Figure 1 shows the velocity profile for varying heat sink parameter. It is noticed that on increasing heat sink parameter the magnitude of velocity increases. The results in the absence of heat sink have also been shown in the graph and match with Carey and Mollendorf (1978). Figure 2 depicts the effect of porosity parameter on velocity profile and it has been seen that velocity increases with its increasing values for (K=1, 4, 10). Figure 3 represents the effect of Prandle number on velocity of fluid and observed that the velocity decreases
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
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Bhasker Chandra, et al., J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
with increasing value of Prandle number (Pr=`0.023, 1). The authors have also seen that there is no major effect on velocity profile for lower value of Prandle number. It is also observed that flow becomes turbulent for higher values of Pr. Figure 4 depicts the effect of heat sink parameter on temperature profile. It is seen that the temperature increases with increase in heat sink parameter (S=-0.2,-0.1,0) The effect of Prandle number on temperature is illustrated in figure 5. It indicates that on increasing Prandle number (Pr=`0.023, 1), temperature decreases. The computation for effect of porosity parameter on temperature profile was also done and found that there is no significant change in temperature profile with porosity parameter. The effect of Prandle number on concentration is illustrated in figure 6. It is seen that the concentration decreases as Prandle number increases (Pr=`0.023, 1). The variation in concentration profile with respect to Schmidt number is represented
graphically in figure 7. It is noticed that the concentration decreases as Schmidt number increases (Sc=0, 1, 3). Figure 8 shows the effect of heat sink parameter on concentration profile and it has been seen that concentration decreases with increasing values of heat sink parameter (S=-0.2,0.1,0).The effect of porosity parameter on concentration is illustrated in figure 9. It is seen that the concentration decreases as porosity parameter increases (K=1, 4, 10). The values of f''(0) and θ'(0) for different values of physical parameters have been tabulated through table1. It is seen that with increasing value of porosity parameter, heat transfer coefficient and skin friction coefficient both increase. The values of Skin friction coefficient decrease while the heat transfer coefficient increases with increasing values of Prandtl number. While the value of Skin friction coefficient increases and heat transfer coefficient decreases with increasing value of heat sink parameter.
X u
v
g
Plate
Physical
Y
model
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
Bhasker Chandra,, et al., J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
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Table 1: Variation of f''(0) and - θ'(0) for different values of Pr, S and K. Physical parameters Pr S
K
Values
f"(0)
-θ'(0)
0.023 1.000
0.702989976 0.612696400
-0.437197095 -0.617000000
-0.200 -0.100 0.000 1.000 4.000 10.000
0.702989970 0.737310000 0.780194000 0.702989976 0.831850001 0.863500000
-0.437197095 -0.332500000 -0.202970950 -0.437197095 -0.437527000 -0.437927000
Fig.3 Velocity profile with Prandle number Pr (for K=1,ε=0.1 ,S=-0.2,γ=-0.4,Sc=1) 0.4,Sc=1)
Fig.1 Velocity profile with heat sink parameter S (for K=1,ε=0.1,Pr=0.023,γ=0.4,Sc=1) ε=0.1,Pr=0.023,γ=0.4,Sc=1) Fig.4 Temperature profile with heat sink parameter S (for K=1,ε=0.1,Pr=0.023,γ=0.4,Sc=1) ε γ=0.4,Sc=1)
Fig.2 Velocity profile with porosity parameter K (for ε=0.1,Pr=0.023,γ=0.4,Sc=1,S= =0.4,Sc=1,S=-0.2)
Fig 5. Temperature profile with Prandle number Pr (for K=1,ε=0.1 ,S=-0.2,γ=-0.4,Sc=1) 0.4,Sc=1)
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Bhasker Chandra,, et al., J. Comp. & Math. Sci. Vol.2 (2), 392-398 (2011)
Fig.6 Concentration profile with Prandle number Pr (for K=1,ε=0.1 ,S=-0.2,γ= γ=-0.4,Sc=1)
Fig.9 Concentration profile with porosity parameter K (for ε=0.1,Pr=0.023,γ=-0.4,Sc=1,S= 0.4,Sc=1,S=-0.2 )
REFERENCES
Fig.7 Concentration profile with Schmidt number Sc (for K=1,ε=0.1,Pr=0.023,γ= =-0.4,S=0.2 )
Fig.8 Concentration profile with heat sink parameter S (for K=1,ε=0.1,Pr=0.023,γ ε=0.1,Pr=0.023,γ=0.4,Sc=1)
1. Arunachalam, M. and Rajappa, N. R. “Forced convection on in liquid metals with variable thermal conductivity and capacity”, Acta Mechanica, Vol. Vol 31, pp. 25-31 (1978). 2. Carey, V.P. and Mollendorf, J.C. “Natural convection in liquid with temperature dependent viscosity”, Proc. 6th International Heat Transfer Conference, Toronto Vol.2, .2, pp. 211-217 211 (1978). 3. Chamkha, A. J. and Khaled, A. R. A. “Similarity solutions for hydro magnetic simultaneous heat and mass transfer by natural convection from om an inclined plate with internal heat generation or absorption”, Heat and Mass Transfer, Vol. 37, pp. 117-123 (2001). 4. Chaim, T. C. “Heat transfer in a fluid with variable thermal conductivity over stretching sheet,” Acta Mechanica, Vol. Vol 129, pp. 63-72 (1998). 5. Crepeau, J. C. and R. Clarksean. Similarity solution of natural convection
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with internal heat generation.ASME J. of Heat Transfer 119, 183-185 (1997). 6. Kay, W. M. Convective Heat and Mass Transfer, McGraw-hill Book Co., New York (1966). 7. Mahanti, N. C. and Gaur, Pramod. “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”, Journal of Applied Fluid Mechanics, Vol. 2, pp. 2348 (2009).
398
8. Ostarch, S. “ An analysis of laminar free convective flow and heat transfer about a flat plate parallel to direction of the generating body force”, NACA Technical Report 1111 (1952). 9. Seddeek, M. A. and Salem, A. M. “Laminar mixed convection adjacent to vertical continuously stretching sheet with variable viscosity and variable thermal diffusivity”. Heat and Mass Transfer, Vol. 41, pp. 1048-1055 (2005). 10. Schlichting, H. Boundary Layer Theory, McGraw-Hill Book Co., New York (1968).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)