Transp Porous Med (2009) 80:117–135 DOI 10.1007/s11242-009-9347-8
Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries J. Prathap Kumar · J. C. Umavathi · I. Pop · Basavaraj M. Biradar
Received: 20 September 2008 / Accepted: 22 January 2009 / Published online: 14 February 2009 © Springer Science+Business Media B.V. 2009
Abstract An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature. Keywords
Mixed convection · Perturbation method · Porous medium
List of Symbols Roman Symbols A Constant defined in Eq. (8) b Thermal expansion coefficient ratio, β2 /β1 Br
(1)2
Brinkman number, µ1 U0
/k1 T
J. P. Kumar · J. C. Umavathi · B. M. Biradar Department of Mathematics, Gulbarga University, Gulbarga 585106, Karnataka, India e-mail:
[email protected] J. C. Umavathi (B) Department of Civil Engineering, National Chi Nan University, University Rd, Puli Nantou, 545, Taiwan, ROC e-mail:
[email protected] I. Pop Faculty of Mathematics, University of Cluj, 3400 Cluj, CP 253, Romania
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Cp Da g Gr GR h h1 h2 K k1 k2 m n p Re T (i) U0 u T1 , T2 X, Y
J. P. Kumar et al.
Specific heat at constant pressure Darcy number based on h 1 , K / h 21 Acceleration due to gravity Grashoff number, gβ1 h 31 T /υ12 Dimensionless parameter, Gr/Re Width ratio, h 2 / h 1 Height of the region-I Height of the region-II Permeability of the porous media Thermal conductivity of the fluid in region-I Thermal conductivity of the fluid in region-II Viscosities ratio, µ1 /µ2 Densities ratio,ρ2 /ρ1 Dimensional pressure (1) Reynolds number, U0 h 1 /υ1 Temperature Reference velocity, −(d P/d X )(h i2 /48µi ) Velocity Temperature of the boundaries Space co-ordinates
Greek Symbols α Thermal diffusivity β Coefficient of thermal expansion ε Dimensionless parameter, (Gr/Re)Br κ Thermal conductivities ratio, k1 /k2 µ Viscosity ν Kinematic viscosity θi Nondimensional temperature, (Ti − T0 )/T ρ Density √ σ Permeability parameter, h 1 / K T Difference in temperature, (T2 − T1 )
Subscripts 1 & 2 Reference quantities for region-I and II respectively
1 Introduction Mixed convection flow through a heated channel has been extensively explored because of its occurrence in many practical applications such as the cooling of modern electronic systems, heat exchangers, solar energy collection, as in the conventional flat plate collector, chemical processing equipments, transport of heated or cooled fluids, etc. Comprehensive reviews have been conducted by Incropera (1986); Aung (1987) and Gebhat et al. (1998). Fluid flow problems in closed ducts, partly filled with porous media, have received great attention recently. This fact is motivated by several engineering applications of this specific
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area. These applications include: porous journal bearing, nuclear reactor, geothermal systems, solid matrix heat exchangers, porous flat plate collectors, thermal insulation, storage of nuclear waste materials, grain storage and drying, and many others. Valuable references can be found in the books by Nield and Bejan (2006); Ingham and Pop (2005); Vadasz (2008), and in the recent papers by Barletta et al. (2007, 2008). Recent technological implications have given rise to increased interest in combined free and forced convection flow in vertical channels in which the objective is to secure a quantitative understanding of a configuration having current engineering applications (Al-Hadharami et al. 2002). Parang and Keyhani (1987) studied the fully developed buoyancy-assisted mixed convection in a vertical annulus by using Brinkman-extended Darcy model. Their results indicated that the Brinkman term could be neglected for lower Darcy number. Muralidhar (1989) performed a numerical calculation for buoyancy-assisted mixed convection in a vertical annulus by using the Darcy model. The results show that the Nusselt number increases with the Rayleigh number and/or Peclet number. Hadim and Chen (1994) investigated the Darcy number effects on the buoyancy-assisted mixed convection in the entrance region of a vertical channel with asymmetric heating at fixed values of Reynolds number, Forchheimer number, Prandtl number, and modified Grashof number. The literature shows that several investigators have studied the steady-state characteristics of the hydrodynamics as well as the heat transfer behavior of flows through closed conduits partly filled with porous material. The steady hydrodynamics behavior of the fluid flow in channels partly filled with porous material is first investigated by Beavers and Joseph (1967) who presented an empirically based correlation for the velocity gradient at the clear fluid/porous interface in terms of the velocities in the fluid layer and the porous region. The same problem is solved analytically using the matched asymptotic expansion technique by Vafai and Thiyagaraja (1987) and solved exactly by Vafai and Kim (1990). The transient hydrodynamic behavior of the fluid flow in channels partly filled with porous material is investigated by Al-Nimr and Alkam (1998), where the unsteadiness in the hydrodynamic behavior is due to a step change in the imposed pressure gradient. Malashetty et al. (2001, 2004) studied two fluid flow and heat transfer in an inclined channel containing porous-fluid layer and composite porous medium. Recently Umavathi et al. (2005, 2006, 2008) studied mixed convection in a vertical porous channel. Keeping in view the practical applications of mixed convection flow and fluid flows in channels partly filled with porous material, the goal of this study is to investigate the fully developed mixed convection flow in a vertical channel with one region filled with porous matrix saturated with a viscous fluid and another region with a clear viscous fluid different from the fluid in the first region using Brinkman model.
2 Mathematical Formulation The geometry under consideration illustrated in Fig. 1 consists of two infinite parallel plates maintained at different or equal constant temperatures T1 (left wall) and T2 (right wall), where T1 ≥ T2 , extending in the X and Z directions. The region −h 1 /2 ≤ Y ≤ 0 is occupied by a fluid-saturated porous medium of density ρ1 , viscosity µ1 , thermal conductivity k1 , thermal expansion coefficient β1 , and the region 0 ≤ Y ≤ h 2 /2 is occupied by another viscous fluid of density ρ2 , viscosity µ2 , thermal conductivity k2 , and thermal expansion coefficient β2 . The fluids are assumed to have constant properties except the density in the buoyancy term in momentum equation. A fluid rises in the channel driven by buoyancy forces. The transpose properties of both the fluids are assumed to be constant. We consider the fluids to
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Fig. 1 Physical model and the coordinate system
X
Region-I Region-II
Y Y =−
h1 2
Y=
h2 2
be incompressible and the flow is steady, laminar, and fully developed. It is assumed that the only non-zero component of the velocity is the X −component Ui (i = 1, 2). Thus, as a consequence of the mass balance equation, one obtains ∂Ui =0 ∂X
(1)
so that Ui depends only on Y . The stream wise and the transverse momentum balance equations using Brinkman model yields (Arpaci and Larsen 1984). Region-I gβ1 (T1 − T0 ) −
d 2 U1 1 ∂P v1 − U1 = 0 + v1 ρ1 ∂ X dY 2 K
(2)
and the Y –momentum balance equation can be expressed as ∂P =0 ∂Y
(3)
Region-II d 2 U2 1 ∂P =0 (4) + v2 ρ2 ∂ X dY 2 ∂P =0 (5) ∂Y where K is the permeability of the porous medium and P = p + ρ0 gx (for P1 = P2 = P) is the difference between the pressure and the hydrostatic pressure. The physical meaning of other quantities is mentioned in the List of Symbols. On account of Eqs. 3 and 5, P depends only on X so that Eqs. 2 and 4 can be rewritten as Region-I gβ2 (T2 − T0 ) −
gβ1 (T1 − T0 ) −
d 2 U1 1 dP v1 − U1 = 0 + v1 2 ρ1 dX dY K
(6)
d 2 U2 1 dP =0 + v2 ρ2 dX dY 2
(7)
Region-II gβ2 (T2 − T0 ) −
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Let us assume that the walls of the channel are isothermal. In particular, the temperature of the boundary at Y = −h 1 /2 is T1 , while the temperature at Y = h 2 /2 is T2 , with T2 ≥ T1 . These boundary conditions are compatible with Eqs. 6 and 7, if and only if, d P/dX is independent of X . Therefore, there exists a constant A such that dP =A (8) dX On account of Eq. 8 and by evaluating the derivatives of Eqs. 6 and 7 with respect to X , one obtains dT1 dT2 = 0, =0 (9) dX dX so that the temperature also depends only on Y . By taking into account the effect of viscous dissipation, the energy balance equation can be written as Region-I d2 T1 v1 dU1 2 v1 α1 U2 = 0 + + (10) dY 2 C p dY CpK 1 Region-II α2
v2 d2 T2 + dY 2 Cp
dU2 dY
2 =0
(11)
Equations 6 and 10, 7 and 11 allow one to obtain differential equation for Ui namely, Region-I d 4 U1 1 d 2 U1 ρ1 gβ1 dU1 2 ρ1 gβ1 2 = + + (12) U dY 4 K dY 2 k1 dY k1 K 1 Region-II d 4 U2 ρ2 gβ2 = dY 4 k2
dU2 dY
2 (13)
The boundary conditions on Ui are both no-slip conditions and those induced by the boundary conditions on T and by Eqs. 6 and 7 are U1 = 0 at Y = − U1 (0) d 2 U1 dY 2 d 2 U2 dY 2 dU1 µ1 dY 1 d 2 U1 − U1 dY 2 K
h1 h2 , U2 = 0 at Y = 2 2
= U2 (0) A gβ1 T RT h1 = + at Y = − µ1 2υ1 2 A gβ2 T RT h2 = − at Y = µ2 2υ2 2 dU2 = µ2 at Y = 0 dY ρ1 β1 µ2 ρ1 β1 d2 U2 A 1 − at Y = 0 = + µ1 ρ2 β2 dY 2 µ1 ρ2 β2
d 3 U1 µ2 ρ1 k2 β1 d3 U2 1 dU1 = − at Y = 0 3 dY K dY µ1 ρ2 k1 β2 dY 3
(14)
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Further, Eqs. 12 and 13 subject to the boundary conditions (14) can be written in the dimensionless form by employing the dimensionless quantities u1 = Gr =
U1 (1) U0
, u2 =
gβ1 T h 31
υ12 T2 − T1 RT = , T
U2 (2) U0
, θ1 =
T1 − T0 T2 − T0 , θ2 = , T T
(1)
,
y1 =
Y1 , h1
y2 =
Y2 h2
(1)
U0 h 1 µ1 U0 Gr , GR = , Br = υ1 k1 T Re 1 K Da = 2 , σ = Da h1 Re =
(15) (i)
where Da is the Darcy number based on h 1 and the reference velocity U0 and the reference temperature T0 are given by, (1)
U0
=−
Ah 21 Ah 22 T1 + T2 (2) , U0 = − , T0 = 48µ1 48µ2 2
(16)
Moreover, the temperature difference T is given by T = T2 − T1 if T1 < T2 . As a consequence, the dimensionless parameter RT can only take the values 0 or 1. That is RT is 1 for asymmetric heating, i.e., T1 < T2 , while RT is 0 for symmetric heating, i.e., T1 = T2 , respectively. Equation (8) implies that A can be either positive or negative. If A > 0, then (i) U0 , Re and G R are negative, i.e., the flow is downward. On the contrary, if A < 0, the flow (i) is upward, so that U0 , Re, and G R are positive. Using variables (15) and (16), Eqs. 12 and 13 become Region-I 2 d4 u 1 2 d u1 − σ = G R Br dy 4 dy 2
du 1 dy
2 + G R Br σ 2 u 21
(17)
Region-II d4 u 2 = mnκbh 4 Br G R dy 4
du 2 dy
2 (18)
where κ = k1 /k2 . The boundary and interface conditions (14) become
d2 u 1 dy 2
1 1 u 1 = 0 at y = − , u 2 = 0 at y = 4 4 u 1 (0) = mh 2 u 2 (0) 1 RT G R d2 u 1 at y = − = −48 + dy 2 2 4 nb RT G R du 2 1 = −48 − at y = 2 dy 2 4 du 1 du 2 =h at y = 0 dy dy 1 d2 u 2 − σ 2u1 = + 48(1 − nb) at y = 0 nb dy 2
d3 u 1 du 1 1 d3 u 2 − σ2 at y = 0 = 3 dy dy nbhκ dy 3
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(19)
Fully Developed Mixed Convection Flow
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3 Solution 3.1 Special Cases 3.1.1 Case-I We shall solve first Eqs. 17 and 18 subject to the boundary conditions (19) for the case when the viscous dissipation is absent, that is, when the parameter, Br = 0. Thus, in this case the dimensionless velocity components, u 1 and u 2 , can be expressed as Region-I u 1 = C1 + C2 y + C3 cos h (σ y) + C4 sin h (σ y)
(20)
u 2 = C5 + C6 y + C7 y 2 + C8 y 3
(21)
Region-II
where Ci , i = 1, . . ., 8 are constants of integration. Further, using the variables (15) in Eqs. 6 and 7, we obtain the energy balance equations as Region-I 1 d2 u 1 2 θ1 = − − σ u1 48 + (22) GR dy 2 Region-II 1 θ2 = − nbG R
d2 u 2 48 + dy 2
(23)
With the help of the velocities given by (20) and (21), the analytical expressions for θ1 and θ2 can be expressed as, when Br = 0 Region-I θ1 = −
1 48 − σ 2 C1 − σ 2 C2 y GR
(24)
θ2 = −
1 (48 + 2C7 + 6C8 y) G Rnb
(25)
Region-II
3.1.2 Case-II Solutions of Eqs. 17 and 18 can also be obtained when buoyancy forces are negligible and viscous dissipation is dominating, i.e., G R = 0 and Br = 0, so that a purely forced convection occurs. For this condition solutions of Eqs. 17 and 18 for u 1 and u 2 , using boundary and interface conditions given by (19), are given by Region-I u 1 = B1 + B2 y + B3 cos h (σ y) + B4 sin h (σ y)
(26)
u 2 = B5 + B6 y + B7 y 2 + B8 y 3
(27)
Region-II
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where Bi , i = 1, . . . , 8 are constants of integration. Using again (15) in Eqs. 6 and 7, we obtain the energy balance equations when Br = 0 as Region-I d 2 θ1 = −Br dy 2
du 1 dy
2 + σ 2 u 21
(28)
Region-II d 2 θ2 = −Br κmh 4 dy 2
du 2 dy
2 (29)
The boundary and interface conditions for temperature are 1 RT RT 1 θ1 − =− , θ2 = , θ1 (0) = θ2 (0) 4 2 4 2 dθ1 1 dθ2 = at y = 0 dy hκ dy
(30)
Using (26) and (27), and solving Eqs. 28 and 29 with the boundary conditions (30) we obtain Region-I θ1 = − Br P1 cos h(2σ y) + P2 sin h(2σ y) + P3 cos h(σ y) + P4 sin h(σ y) + P5 y 2 + E y+F (31) Region-II θ2 = −Br κmh 4 P7 y 4 + P8 y 3 + P9 y 2 + Gy + H
(32)
where E, F, G, H and Pi , i = 1, . . . , 9 are constants of integration. 3.2 Perturbation solution We define the dimensionless parameter ε=
Gr Br = G R Br Re
(33)
and assume that its values are small (|ε| 1.
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GR =500 R egion I-P orou s
R egion II-V iscou s
ε = 0 .1
R q t= 1 m =1 n=1 κ=1 b=1 h=1
1 .0 σ = 2 ,4 ,6 ,8
θ 0 .8
0 .6
-0 .2
-0 .1
0 .0
0 .1
0 .2
y Fig. 10 Temperature profiles for different values of σ for isoflux–isothermal wall condition
0
R egion I-P orou s
R egion -V iscou s
-2
u
-4
8 GR =500
-6
-8
-1 0
6
ε = 0 .1
R tq = 1 m =1 n=1 κ=1 b=1 h=1
-0 .2
4 σ=2
-0 .1
0 .0
0 .1
0 .2
y Fig. 11 Velocity profiles for different values of σ for isothermal–isoflux wall condition
5 Conclusions The problem of steady, laminar mixed convection flow in an infinite vertical channel filled with porous and viscous fluids in the presence of viscous dissipation is discussed. Three different combinations of thermal left–right wall conditions were presented. Various analytical solutions on the flow and temperature fields for different special cases with isothermal– isothermal, isoflux–isothermal, and isothermal–isoflux wall heating conditions were obtained. Graphical results were displayed for different parameters governing the flow and heat transfer. Considering equal values for viscosities, widths, and conductivities for fluids in the both regions we obtain the results reported by Umavathi et al. (2006) and Barletta (1998, 1999) for one fluid model. It is also found that the permeability parameter decreases the velocity field in the both regions, viscosity ratio increases the velocity for the porous
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R e g io n I-P o ro u s
R e g io n II-V isc o u s
GR =500 ε = 0 .1
R tq = 1 m =1 n=1 κ=1 h=1 b=1
-0 .6
θ -0 .8
σ= 2 ,4 ,6 ,8 -1 .0
-0 .2
-0 .1
0 .0
0 .1
0 .2
y Fig. 12 Temperature profiles for different values of σ for isothermal–isoflux wall condition
20 0
Nu+,Nu_
-20 -40 -60
Nu
-80 GR=500
-100
ε=0.1
R T=1 m=1 n=1 κ =1 h=1 b=1
-120 -140 -160 -180
0.5
1.0
1.5
2.0
2.5
3.0
σ Fig. 13 Nusselt values for different values of σ
region, but decreases the velocity in the viscous fluid region. Width ratio and conductivity ratio suppresses the flow in both the regions. Acknowledgements The authors would like to express their very sincere thanks to the reviewer for his valuable comments and suggestions. They also wish to thank Professor Donald A. Nield for some of his generous suggestions.
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