J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
Non- darcian convective flow of heat and mass transfer through a porous medium in a coaxial duct with the effect of radiation D. CHITTI BABU1 and T. KOTESWARA RAO2 1
2
Department of Mathematics, Government College(A), Rajahmundry, INDIA. Department of Mathematics,VRS &YRN College of Engg. & Tech., Chirala, A.P. ABSTRACT We investigate in this paper non-Darcian free and forced convective heat &mass flow through a porous medium in a co-axial cylindrical duct with radiation effect, where the boundaries are maintained at constant temperatures.The velocity, temperature and concentration enter flow field have been analytically evaluated using Gauss-siedel iteration method and their behaviour is discussed computationally for variations in the governing parameters. Key words: Convective heat transfer, mass transfer, porous medium, radiation.
INTRODUCTION The interaction of radiation with laminar free convection heat transfer from a vertical plate was investigated by Cess2 for an absorbing emitting fluid in the optically thick region, using the singular perturbation technique. Arpaci1 considered a similar problem in both the optically thin and optically thick regions and used the approximate integral technique and first-order profiles to solve the energy equation. Raptis and Perdikis8 studied the effects of thermal radiation and free convective flow past moving plate. Das et al4 analyzed the radiation effects on the flow past an impulsively started infinite isothermal vertical plate. Chamkha et al3 considered the effect of radiation on free convective flow past a semi-infinite vertical plate with mass transfer. Similarity solution to this problem was given by Ostrach7, Siegel9 studied the transient free convective flow past a semi-infinite vertical plate by integral method. The same problem was studied by Gebhart5 by an approximate
method. Kim and Fedorov6 studied transient mixed radiative convection flow of a micro polar fluid past a moving, semi-infinite vertical porous plate. In this paper we investigate nonDarcian free and forced convective heat and mass flow through a porous medium in a coaxial cylindrical duct with radiation effect, where the boundaries are maintained at constant temperatures. The velocity, temperature and concentration enter flow field have been analytically evaluated using Gausssiedel iteration method and their behaviour is discussed computationally for variations in the governing parameters. The shear stress on the inner and outer cylinder and the Nusselt number,Sherwood number on the boundaries have been analytically evaluated and their behaviour is discussed for different variations in the governing parameters. FORMULATION OF THE PROBLEM We consider a fully developed laminar convection flow in co-axial cylindrical duct
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
741
filled with porous medium. The axis of cylindrical duct is taken to be z-axis in the vertical upwards direction and r–axis is the radial direction. The inner and outer cylinders of duct are maintained at constant temperatures Ti and To respectively. The temperature gradient in the flow field is sufficient to cause natural convection in the flow region. A constant axial pressure gradient is also imposed so that this resultant flow is mixed convection flow. The Brinkman–Forchheinner extended Darcy Equation which account for boundary inertial effects in the momentum equation is used to obtain the velocity field. Based on the above assumptions the governing equations the vector form are
be in local thermal equilibrium. Since the flow is unidirectional, the equation of continuity (2.1) reduce to
∂u =0 ∂z
The momentum and energy equation in the scalar form reduces to
∂p µ ∂ 2 u 1 ∂u µ ρδF 2 + + − u− u ∂z δ ∂r 2 r ∂r k k * + βρ 0 g (Τ − Τ0 ) + β ρ 0 g (C − C 0 ) = 0 (2.6) ∂ 2 Τ 1 ∂Τ ∂Τ + = λ 2 + ρcp u ∂z r ∂ r ∂ r ∂q ∂u + Q(T0 − T ) − ( r ) ∂r ∂r 2
∇.q = 0
(2.1)
( )
ρ ∂q ρ + 2 q. ∇ q = − ∇ p − ρ g δ ∂t δ µ ρF − q − q. q + µ ∇ 2 q ( 2 .2 ) k
k
µ
µ
∂ 2 C 1 ∂C ∂C + = D 2 + ∂z r ∂r ∂r ∂ 2T 1 ∂T k11 2 + r ∂ r ∂ r
( )
∂T ρcp + q. ∇ Τ = λ ∇ 2 Τ + ∂t
( 2.7)
( 2.8)
1 ∂ ∂q r ∂u µ + Q(T0 − T ) − ( ) ( 2.3) r ∂r ∂r ∂r
The boundary conditions are u(a) = u(b) = 0 T(a) = Ti, T(b) = To, C(a) = Ci, C(b) = Co (2.9)
ρ − ρ 0 = − βρ 0 (Τ − Τ0 )
The axial temperature gradient
2
(2.4)
∂Τ assumed to ∂z
be a constant by A.
( )
∂C 2 2 ∂t + q. ∇ C = D1 ∇ C + k11∇ T
(2.5)
Here the thermo physical properties of solid and fluid have been assumed to be constant except for the density variation in the body force term (Boussinesq approximation) and the solid particles, fluid are considered to
We define the following non-dimensional variables,
z ∗ r a , r = , u∗ = u , a a υ 2 bβ pa δ ∗ p∗ = ,δ = , θ∗ 2 ρr a z∗ =
=
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Τ − Τi Τo − Τi
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
=
C∗
C − Ci Co − Ci
u (1) = u (s) = 0,
Approximation
for
4 qr = - 4σ ∂T 3β r ∂u r r
Introducing these non-dimensional Variables then the governing equation in the non-dimensional form reduce to
d 2 u 1 du + = P1 + δ D −1u + δ 2 Au 2 2 r dr dr −δ G (θ + NC ) + δ
µ2 r2
u (2.10)
d 2θ 1 + dr 2 r d 2C 1 + dr 2 r
dθ = P1 N1 u + α1θ (2.11) dr dC = Sc1 N1u (2.12) dr gβ (Τ1 − Τ0 ) a 3 Where G = (Grashoff 2
γ
number),
a2 (inverse darcy parameter), k ρ cp γ Aa N= , P= (Prandtl number) , Τ1 − Τ0 λ
D −1 =
N1 =
a2
4σ
λ
(Heat source parameter)
∗
βR λ
(Radiation parameter),
SOLUTION OF THE PROBLEM The governing equations of flow and temperature are coupled non – linear differential equations. Assuming the porosity δ to be small we write (3.1) u = u0 + δ u1 + δ 2 u2 +….…… 2 θ = θ 0 + δθ 1 + δ θ 2 + ........... (3.2)
C = C 0 + δC1 + δ 2 C 2 + ..........
(3.3) Substituting the above in equations (2.10) (2.12) and equating the like powers of δ , we obtain equations to the zeroth order as
d 2 u 0 1 du 0 + = P1 (3.4) r dr d r2 d 2θ 0
1 d θ0 = P1 N 1 u 0 + α 1θ (3.5) r dr
+
d r2 d 2C0
1 d C0 = ScN 1 u 0 r dr
+
d r2
(3 . 6 )
The first order equations are
d 2 u1 1 du1 + = D −1 u 0 − G (θ 0 + NC0 ) (3.7) 2 r dr dr
d 2θ 1 2
+
dr d 2C1 d r
1 d θ1 = P1 N u1 + α 1θ 1 (3.8) r dr +
2
1 d C1 = ScN r d r
1
u 1 (3 .9 )
The second order equations are
d 2u2 dr
2
+
1 du 2 = D −1 u1 − r dr
G (θ 1 + NC1 ) + Au 0
3N1 P 3 N1 α ν , P1 = , Sc = 3N1 + P D 3 N1 + P
d 2θ 2
The corresponding boundary conditions are
d r2
α1 =
θ (s)=0, C(s) =0
θ (1) =1, C(1) = 1,
Taking Rosseland radiation as
α =Q
742
+
2
(3.10)
1 dθ 2 = P1 N u 2 + α 1θ 2 (3.11) r dr
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
743
d 2C2 d r2
+
1 d C2 = ScN 1 u 2 (3.12) r dr
0.15 0.1 0.05 I
The corresponding boundary conditions are u0 (1) = u0 (s) = 0, θ 0 (1) = 1, θ 0 (s) =0, C0(1) =1, C0(S)=0 u1 (1) = u1 (s) = 0, θ 1 (1) = θ 1 (s) = 0, C1 (1) =C2(S) =0 u2 (1) = u2 (s) = 0 , θ 2 (1) = θ 2 (s) = 0, C2(1) = C2(S)=0 The differential equations (3.4) to (3.12) have been discussed numerically by reducing the differential equations into difference equations which are solved using Gauss–Seidel iteration method.
II
0 1
u
1.04
1.08
1.12
1.16
III
1.2
IV
-0.05
V -0.1 -0.15 -0.2
r
Fig. 1 Velocity u with G, s=1.2
I II III 102 5X102 8X102
G
IV -102
V -5X102
0.3 0.2 0.1 u
I II
0 1
1.2
1.4
1.6
1.8
III
2
IV
-0.1
SHEAR STRESS, NUSSELT NUMBER AND SHERWOOD NUMBER
V -0.2 -0.3
The shear stresses on the inner and outer cylinders are evaluated using the formula
du du Z inner = , Z outer = − dr r =1 dr r = s The rate of the heat transfer (Nusselt number) on the inner and outer cylinders are given by
-0.4
r
Fig 2 Velocity u with G, s=2
I II III 102 5X102 8X102
G
IV -102
V -5X102
0.3 0.25 0.2
Nu inner
dθ = dr r =1
Nu outer
dθ = − dr r = s
The rate of mass transfer (Sherwood number) on the inner and outer cylinder are given by
dc dc Shinner = Shouter = − dr r =1 , dr r = s
I II
u 0.15
III IV
0.1 0.05 0 1
1.04
N1
I II 0.5 2.5
1.08
r
1.12
1.16
Fig 3 u with N1,s=1.2
III 4
IV 10
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
1.2
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010) 1
1.2
0.8
1
0.6
I
744
0.8
I II
II
0.4 u 0.2
III
III
C 0.6
IV
IV 0.4
0 1
1.2
1.4
1.6
1.8
2
0.2
-0.2 -0.4
0
r
Fig 4
N1
I II 0.5 2.5
1
1.04
u with N1,s=2
III 4
1.08
Fig. 7
IV 10
r
1.12
I II III N1 0.5 2.5 4
1.2
1.2
1
1
I
θ 0.8
1.16
1.2
C with N1,s=1.2
IV 10
I
0.8
II 0.6
IIi
II III
C 0.6
IV
IV
0.4
0.4
0.2 0.2
0 1
1.04
1.08
1.12
1.16
1.2
0 1
r
θ with N1,s=1.2
Fig 5
N1
1.2
I II III 0.5 2.5 4
1.4
Fig.8
IV 10
N1
r
1.6
1.8
2
C with N1,s= 2
I II III 0.5 2.5 4
IV 10
1.2
1
0.8
I
θ
II 0.6
III IV
0.4
0.2
0 1
1.2
1.4
Fig 6
N1
I II 0.5 2.5
r
1.6
1.8
θ with N1,s= 2
III 4
IV 10
2
DISCUSSION The analysis has been carried out when the outer cylinder is at a higher temperature than that of the inner cylinder. The actual flow is along the gravitational field hence the axial velocity (u) in the vertical direction represents the actual flow u>0 indicates the reversal flow. For computational purpose, we have chosen the non-dimensional temperature(θ) on the inner cylinder to be 1 and on the outer cylinder at continuous to be 0 and concentration (c) on the inner cylinder to
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
745
be 1 and on the outer cylinder at approaches to be 0. The velocity, temperature and concentration are exhibited in figures. We observe in the wide gap case from figs 1 & 2 that no reversal flow appears in the region for
the narrow gap case and enhances larger in wide gap case (figs.5&6). An increase in the radiation parameter N1, depreciates C in the flow region in both narrow & wide gap case (figs.7&8). The Nusselt number which measures the rate of heat transfer at the boundaries has been contributed in tables 1-4. It is observed that the rate of heat transfer is negative at the inner cylinder r=1 for all variations. Nu
>
any value of |G|( < 0 ). The velocity depreciates in heating of the cylindrical wall and enhances in the cooling case with maximum attained at r=1.15 in the narrow gap case and r=1.8 in the wide gap case. The effect of radiation parameter is to enhance u in the narrow gap
enhances with G and heat source parameterα. At the outer cylinder r=s, Nu depreciates with
and depreciates in the wide gap case (figs. 3&4). An increase in the radiation parameter N1, enhances slightly θ in the flow region in
D-1, G and α.
TABLE-1 Nusselt number (Nu) at the inner cylinder r=1,s=1.2 D-1 103 3x103 5x103 7x103 G
I -1.06297 -1.07380 -1.09153 -1.11613 102
II -1.06297 -1.07373 -1.09101 -1.11478 5x102
III -1.06297 -1.07368 -1.09054 -1.11357 8x102
IV -1.06296 -1.07363 -1.09014 -1.11247 103
V -1.06295 -1.07359 -1.08975 -1.11149 5x103
VI -1.11357 -0.86706 -0.88147 -0.90113 5x102
VII -0.38173 -0.68954 -0.70185 -0.71866 5x102
VIII -0.53304 -0.53970 -0.55024 -0.56458 5x102
α
2
2
2
2
2
4
6
8
TABLE-2 Nusselt number (Nu) at the outer cylinder r=s, s=1.2 D-1 103 3x103 5x103 7x103 G
I -1.0040 -1.9883 -0.96988 -0.94359 102
II -1.0039 -0.98889 -0.97043 -0.94503 5x102
III -1.00039 -0.98894 -0.97092 -0.94633 8x102
IV -1.00038 -0.98899 -0.97136 -0.94750 103
V -1.00039 -0.98903 -0.97177 -0.94855 5x103
VI -0.94633 -1.12664 -1.10918 -1.08535 5x102
VII -1.28162 -1.27089 -1.25398 -1.23089 5x102
VIII -1.4304 -1.41967 -1.40329 -1.38097 5x102
α
2
2
2
2
2
4
6
8
VII -0.55518 -0.56918 -0.59078 5x102 6
VIII -0.41543 -0.42737 -0.44583 5x102 8
TABLE-3 Nusslet number (Nu) at the inner cylinder r=1 s=2 D-1 103 3x103 5x103 G α
I -0.91825 -0.93688 -0.96721 102 2
II -0.91814 -0.93715 -0.96700 5x102 2
III -0.91810 -0.93739 -0.96641 8x102 2
IV -0.91743 -0.93763 -0.96693 103 2
V -0.91712 -0.93782 -0.96686 5x103 2
VI -1.00698 -0.73877 -0.76405 5x102 4
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
D. Chitti Babu et al., J. Comp. & Math. Sci. Vol. 1(6), 740-746 (2010)
746
TABLE-4 Nusselt number (Nu) at the outer cylinder r=s, s=2 D-1 103 3x103 5x103 G Îą
I -1.12319 -1.10318 -1.07077 102 2
II -1.12317 -1.10287 -1.07087 5x102 2
III -1.12316 -1.10259 -1.07096 8x102 2
IV -1.12314 -1.10233 -1.07104 103 2
REFERENCES 1. Arpaci V.S.: Effect of thermal radation on the laminar free convection from a eated vertical plate, Int. J. Heat Mass Transer V.11, 871-881,(1968). 2. Cess R.D: The interaction of Thermal radiation with free convection heat transfer, Int. J. Heat Mass Transfer, vol.9,1269-1277 (1966). 3. Chamka A.J, Takhar H.S and Soundagekar V.M: Radiation effects on free convection flow past a semi-infinite vertical plate with mass transfer, Chem Engg. J, Vol.84,335-342,(2001). 4. Das V.N.,Deka R and Soundalgekar V.M. :Radiation effects on flow past An impulsively started vertical plate- An exact solution, J. Theo. Appl. fluid Mech. V1(2),111-115 (1996).
V -1.12313 -1.10211 -1.07112 5x103 2
VI -1.02827 -1.24228 -1.21166 5x102 4
VII -1.40742 -1.38820 -1.35852 5x102 6
VIII -1.55647 -1.53790 -1.50920 5x102 8
5. Gebhart,B.: Transient natural convection from Vertical elements, J. Heat Transfer, Vol.83c,61-70(1961). 6. Kim Y.J,Andrei G. and Fedorov: Transient mixed radiative convection flow of a micro polar fluid past a moving, Semi-infinite vertical porous plate, Int. J. Heat Mass Transfer, Vol.46,1751-1758(2003). 7. Ostrach. S: An analysis of laminar free convection flow and heat transfer along a flat plate parallel to the direction of the generating body force, NACA ReportTRIIL, 63-79,(1953). 8. Raptis A. and Perdikis C.: Radiation and free convection flow past moving plate, Apple. Mech. Engg. Vol. 4, 817-821 (1999). 9. Seigel, R: Transient free convection From a vertical plate, J. Heat Transfer Vol.1, 347-359, (1958).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)