J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
Combined Effect of Soret Effect and Temperature Dependent Heat Sources on Hydromagnetic Convective Heat and Mass Flow Through A Porous Medium in A Porous Cylindrical Annulus D. CHITTI BABU1 and Prof. D. R. V. PRASADA RAO2 1
Department of Mathematics, Government College (A), Rajahmundry, INDIA. 2 Department of Mathematics, S. K. University, Anantapur, A. P. ABSTRACT In this paper, we analyze Soret effect on the natural convective heat and mass transfer in a fluid saturated porous medium confined in a porous vertical cylindrical annulus, under the influence of a radial magnetic field. The flow is subjected to the presence of temperature dependent heat generating source. The zeroth and first order perturbation equations obtained by using asymptotic expansion with respect to suction parameter are solved numerically by finite difference technique. Key words: Convective heat transfer, mass transfer, porous medium .
1. INTRODUCTION The analysis of temperature field as modified by the generation of heat and mass transfer in moving fluids is important in Engineering processes pertaining to flows in which as fluid supports an exothermic chemical or nuclear reaction and in problems connected with dissociating fluids8,9. The volumetric rate of heat generation has been assumed to be constant1,7, 12,13,14 or a function of space variable2,4,5,6. Sparrow and Cess15 have obtained solutions of the steady flow and heat transfer of the stagnation point flow taking into account the temperature dependent volumetric heat generation.
Foraboschi and Federico3 have assumed volumetric rate of heat generation of the type Q = Q0 (T-T0) when T â&#x2030;Ľ T0 Q=0 when T < T0 in their study of steady state temperature profiles for laminar parabolic and piston flow in circular tubes. In this paper, we analyse Soret effect on the natural convective heat and mass transfer in a fluid saturated porous medium confined in a porous vertical cylindrical annulus, under the influence of a radial magnetic field. The flow is subjected to the presence of temperature dependent
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
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heat generating source. The zeroth and first order perturbation equations obtained by using asymptotic expansion with respect to suction parameter are solved numerically by finite difference technique. The velocity, temperature and concentration distributions are discussed for different values of Grashoff number G, the suction parameter λ, the heat source parameter α, buoyancy ratio N, Soret parameter So. The shear stress and the rate of heat transfer are evaluated computationally for different variations in the governing parameters. 2. FORMULATION OF THE PROBLEM AND SOLUTION We consider the fully developed, steady laminar free convective flow of an incompressible, viscous, electrically conducting fluid through a porous medium in a annular region between two vertical coaxial porous circular pipes. We choose the cylindrical polar coordinates system 0 (r, 0, z) with the inner and outer cylinders at r =a and r =b respectively. The fluid is subjected to the influence of as radial magnetic field (Ho / r). Pipes being sufficiently long all the physical quantities are independent of the axial coordinate z. The fluid is chosen to be of small conductivity so that the Magnetic Reynolds number is much smaller than unity and hence the induced magnetic field is negligible compared to the applied radial field. Also the motion being rotationally symmetric the azimuthal velocity V is Zero. The equation of motion governing the MHD flow through the porous medium are
ρ e uwr = − p z + µ ( wrr + wr / r − ( µ / k ) w − ρg − (σµ e2 H 02 a 2 / r 2 ) w
( 2.3) In the presence of temperature dependent heat sources the equation of energy takes the form ρC p (uTr ) = k1 (Trr + Tr / r ) + µ (2(u r2 + u 2 / r 2 ) + wr2 ) +
+ ( µ / k + σµ e2 H 02 )(u 2 + w 2 ) + Q (Te − T ) (2.4) 1 1 uc r = D1 (C rr + c r ) + k11 (θ rr + θ r ) ( 2.5) r r ρ =ρe (1-β( T –T0) -β *(C-C0))
(2.6)
where (u,w) are the velocity components along 0(r,z) directions respectively, ρ is the density of the fluid, p is the pressure, T is the temperature, µ is the coefficient of viscosity, Cp is the specific heat at constant pressure, k is the porous permeability, σ is the electrically conductivity, µe is the magnetic permeability and ρe , T0 and co are density, temperature and concentration in the equilibrium state (suffices r and z indicate differentiation with respect to the variables), k1 is the coefficient of thermal conductivity and Q is the strength of the heat source/sink. The boundary conditions are w(a)=w(b)=0 T(a)=Ti and T(b)=To C(a) =Ci and C(b)=Co The equation of continuity gives ru = aua= bub ⇒ u b = (a /b) ua
(2.7a) (2.7b) (2.7c) (2.8)
ur + u / r = 0
(2.1)
In the hydrostatic state equation (2.3) gives -ρe g - pe , z = 0 (2.9)
ρe u ur = -pr + µ (u r r + ur / r-u / r2) – (µ / k) u
(2.2)
where ρe and pe are the density and pressure in the static case and hence
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
-ρg-pz = - ( ρ- ρe) g - pd.z
(2.10)
where pd is the dynamic pressure Also substituting (2.9) in (2.2) we find
889
where
M = (σµe2 H o2 a 2 / ρv)1/ 2 (the
Hartmann
G = ( β ga (T1 −T e) / v 2 ) (the 3
number),
∂pd = f (r ) ∂r
Grashoff number), λ = au a / v (the Suction
Using the relation (2.7a) – (2.10) in (2.1) – (2.6) the equations governing free convection heat and mass transfer flow under no pressure gradient are w r r + (1- aua / v ) wr /r+ ((βg / v ) ((T-Te) + (β*g / v ) (C-Ce))
eter), P = ( µC p / k1 ) (the Prandtl number),
− (σµ e2 H o2 a 2 / v)( w / r 2 ) − ( w / k ) = 0
parameter), D2−1 = ( a 2 / k ) (the Darcy param-
Ee = (v / k1a 2 (T1 − Te ))
(the
Eckert
QL2 (the K1
Heat
Source
number), α =
Sc =
Parameter),
(2.11) Making use of (2.8) in (2.4) and (2.5) the equations reduces to Trr + (1 − au a / v)Tr / r + µ ( 2(u 2 / r 2 ) + wr2 ) + ( µ / k + σµ e2 H o2 )(u 2 + w 2 ) + Q (Te − T ) = 0
(2.12) 1 C rr + (1 − au a / v)C r / r + k11 (θ rr + θ r ) = 0 r (2.13) Introducing the non-dimensional variables
(r′ ,w′ , θ’, C′ ) as r′ = r/a, w′ = w (a/ v ),
T − Te C − Ce θ′ = , C′ = Ti − Te Ci − Ce
(2.14)
Making use of non-dimensional variables, the above equations reduces to
wrr + (1 − λ )(1 / r ) wr − ( D2−1 +
number), N =
S0 =
v (the D1
Schmidt
β * ∆C (the buoyancy ratio), β ∆T
k11 β * (the Soret Parameter) βv
The corresponding boundary conditions are w=0, θ =1, c=1 on r=1 w=0, θ =m1, c=m2 on r=s where m1 =
T0 − Te T1 − Te
, m2 =
Co − Ce Ci − Ce
3. ANALYSIS OF THE FLOW Assuming Ee<<1, we take the solution as w=w0+Eew1+………………… θ = θ0 +Ee θ1 + ……………… C=C0 + Ee C1+………………. (3.1)
( M 2 / r 2 ))W = −G (θ + NC ) (2.15) θ rr + (1 − λ )θ r / r − αθ =
Substituting (3.1) in equations (2.15)–(2.17) and separating the like powers of Ee, the equations to the Zeroth order are
− PE c ( wr2 + λ 2 / r 4 )
M2 wo, rr (1 − λ )(1 / r ) wo, r − ( D + 2 ) w0 = r − G (θ 0 + NC o ) (3.2) θ 0,rr + (1 − λP)(1 / r )θ 0,r − αθ 0 = 0 (3.3)
(2.16) ScS o 1 C rr + (1 − λ )C r / r = − (θ rr + θ r ) N r (2.17)
−1 2
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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
C 0, rr + (1 − λSc)(1 / r )C o, r = −
Sc S0 1 (θ 0, rr + θ 0, r ) N r
(3.4)
+ ( M 2 / r 2 ))) wo,i + (1 +
and to the first order are
w1, rr + (1 − λ )(1 / r )W1, r − ( D 2−1 + − G (θ 1 + NC1 )
M2 ) w1 = r2 (3.5)
θ 1, rr + (1 − λP)(1 / r )θ 1, r − αθ 1 = − P ( wo2, r + λ 2 / r 4 )
(3.6)
C1, rr + (1 − λSc) + (1 / r )C1, r 1 = −
Sc S0 1 (θ 1, rr + θ 1, r ) N r
h(1 − λ ) ) w0,i −1 − (2 + h 2 ( D2−1 2ri
(1 −
= −Gh 2 (θ o,i + NC o,i ) h(1 − λP ) (1 − )θ 1,i −1 − 2θ 1,i + 2ri
((1 + (1 −
(3.7)
The corresponding boundary conditions are wo(1)=0,wo(s)=0, θ0(1)=0,θ0(s)=m1, C0(1)=0,C0(s)=m2 w1(1)=0,w1(s)=0, θ1(1)=0, θ1(s)=0, (3.8) C1(1)=0,C1(s)=0 The differential equations (3.2)-(3.7) have been discussed numerically by reducing the differential equations into difference equations which are solved using GaussSeidel Iteration method. The differential equations involving θ0, C0, w0, θ1, C1 and w1 are reduced to the following difference equations h(1 − λP ) (1 − )θ 0 ,i −1 − (2 + αh 2 )θ 0,i 2ri h(1 − λP ) + ((1 + )θ o ,i +1 = 0 2ri h(1 − λSc) (1 − )C 0,i −1 − 2C 0 ,i + 2ri − Sc S 0 h(1 − λSC ) ((1 + )C o ,i +1 = 2ri N h h [(1 − )θ 0,i −1 − 2θ 0,i + (1 + )θ 0 ,i +1 ] 2ri 2ri
h(1 − λ ) ) wo,i +1 2ri
h(1 − λP ) )θ 1,i +1 = Ph(λ 2 / r 4 + w02, r ). 2ri
h(1 − λSc) )C1,i −1 − 2C1,i + 2ri
((1 +
h(1 − λSc) )C1,i +1 = 2ri
− ScS 0 h h ((1 − )θ 1,i −1 − 2θ 1,i (1 + )θ 1,i +1 ) N 2ri 2ri h(1 − λ ) (1 − ) w1,i −1 − (2 + h 2 ( D2−1 2ri
+ M 2 / r 2 )))w1,i + (1 + − Gh 2 (θ 1,i +1 + NC1,i )
h(1 − λ ) ) w1,i +1 = 2ri (i = 1,2,............21)
where h is the step length taken to be 0.05 together with the following conditions θ0,0 = 1,θ0,17 =m1, θ1,0 = 0, θ0,17 =m1, w0,0 = 0,w0,17 =0, w1,0 = 0, w1,17 =0,C0,0 = 0,C0,17 =0, C1,0 = 0, C1,17 =0 All the above difference equations are solved using Gauss-Seidel iterative method to the fourth decimal accuracy. 4. SHEAR STRESS AND NUSSELT NUMBER The shear stress on the pipe’s given by
τ 1 = µ(
∂w ) r = a ,b ∂r
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
which in the non-dimensional 1 2 2 reduces to τ = τ /( µ / a ) = ( wr ) r =1, s
0 -0.01 1
= ( w0, r + E e w1, r ) r =1, s
1.4
1.6
1.8
2
-0.02
The heat transfer through the pipe to the flow per unit area at the pipe surface is given by q = k1 (
1.2
891
∂T ) r =a ,b ∂r
-0.03
Series1
W -0.04
Series2
-0.05
Series3
-0.06
which in the non dimensional form is
-0.07
Nu = (
-0.08
qa ∂θ ) = ( ) r =1,s k1 (T1 − Te ) ∂r
Fig 2 Variation of w with α
We investigate the MHD convective heat and mass transfer of a viscous electrically conducting fluid through a porous medium confined in a porous vertical circular annulus in the presence of temperature dependent heat source. We suppose that the outer cylinder is at a higher temperature than the inner cylinder.
I 2
α
5. DISCUSSION OF THE RESULTS
II 4
III 6
2.5000
θ
2.0000
Series1
1.5000
Series2 Series3
1.0000
Series4
0.5000
Series5
0.0000 1
0.08 0.06 0.04 0.02 0 W -0.02 1 -0.04 -0.06 -0.08 -0.1
1.2
1.4
1.6
1.8
2
Fig 3 Variation of θ with G Series1
I II III 103 3X103 5X103
G
IV -103
V -3X103
Series2 1.2
1.4
1.6
1.8
2
Series3
2.5
Series4
2
Series5 θ
Series1
1.5
Series2 1
Series3
0.5 0 1
Fig 1 Variation of w with G
I II III G 103 3X103 5X103
IV -103
V -3X103
1.2
1.4
Fig 4
α
1.6
1.8
2
Variation of θ with α
I 2
II 4
III 6
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
892 2.5000 2.0000
Series1 1.5000
Series2
1.0000
Series3
C
Series4 0.5000 0.0000 1
1.2
1.4
1.6
1.8
2
Fig 5 Variation of C with α
I 2
α
II 4
III 6
2.5 2 Series1
1.5 C
Series2
1
Series3
0.5 0 1
1.2
1.4
1.6
1.8
2
Fig. 6 Variation of C with S0
S0
I 0.5
II 1.0
III -0.5
IV -1.0
The actual flow in all the cases is along the gravitational field and hence the axial velocity (w) in the vertical downward direction represent, the actual flow, w>0 indicates the reversal flow. For computational purpose, we have chosen the non-dimensional temperature (θ) on the outer cylinder to be m1=-1, on the inner cylinder θ continues to be 1. We observe
that for the heating of boundaries the axial velocity is negative and it is positive in the case of cooling of boundaries. The magnitude of w enhances with increase in |G| (<0>) with the point of maximum w occurring in the mid region of the flow (fig.1). We notice from fig.2 that |w| depreciates with increase in the strength of temperature dependent heat source. The temperature(θ) enhances in magnitude for an increase in G (fig.3).The variation of θ with α show that a significant decay with an enchase in the strength of temperature dependent heating source (fig.4). From fig.5 the concentration suffers a significant decrease with an increase in the strength of temperature dependent heat source. The concentration experiences an enhancement with an increase in So(>0) and reduces with So(<0)(fig.6). The shear stress( τ ) on the inner and outer cylinders are evaluated for different values of G, D-1, M. It is observed that the stresses on both the boundaries are negative in the heating case and positive in the cooling case. An increase in M or D-1 depreciates the magnitude of the stress on both the cylinders. Thus lower the permeability of the porous medium smaller the magnitude of the stress. (Table 1).The local rate of heat transfer Nu on both the cylinders have been evaluated for different variations in G, D-1, M, α andλ. It is observed that the rate of heat transfer is negative. From Table.2, we notice that an increase in M enhances the magnitude of Nu and as the permeability of the porous medium reduces Nu experiences depreciation. Also an increase in α enhances the magnitude of Nu (Table.3).On the outer cylinder, the rate of heat transfer enhances with an increase in M. Also Nu enhances
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
with D −1 ≤5X103 and depreciates with D −1 ≥7X 103(Table.4). An increase in α, Nu enhances and the magnitude of Nu
893
experiences an enhancement with an increase in suction parameter λ (Table.5).
TABLE .1 Shear stress ( τ ) at the inner cylinder r=1,P=0.71, α=2
G 2X103 5X103 104 -2X103 -5X103 -104 M D-1
I -2.11242 -5.29856 -10.6023 2.11242 5.29856 10.6023 2 2X103
II -1.71212 -4.29505 -8.59695 1.71212 4.29505 8.59695 5 2X103
III -1.06123 -2.66162 -5.32813 1.06123 2.66162 5.32813 10 2X103
IV -2.06291 -5.17347 -10.3413 2.06291 5.17347 10.3413 2 5X103
V -2.03166 -5.09364 -10.17521 2.03166 5.09364 10.17521 2 7X103
TABLE .2 Nusselt number on the inner cylinder r=1,P=0.71, α=2
G 2X103 5X103 104 M D-1
I -0.34353 -0.33798 -0.31812 2 2X103
II -0.34389 -0.34024 -0.32717 5 2X103
III -0.34433 -0.34292 -0.33794 10 2X103
IV -0.34209 -0.32891 -0.28176 2 5X103
V -0.34121 -0.32338 -0.25955 2 7X103
TABLE.3 Nusselt Number on the inner cylinder r=1,P=0.71, M=2
G 2X103 5X103 104 α λ
I -0.34353 -0.33798 -0.31812 2 0.2
II -1.21424 -1.20977 -1.19378 5 0.2
III -1.87747 -1.87372 -1.86027 10 0.2
IV -0.38831 -0.38311 -0.3646 2 0.5
V -0.41538 -0.41043 -0.39273 2 0.7
TABLE. 4 Nusselt number on the outer cylinder r = 2, P=0.71, α=2
G 2x103 5x103 104 M D-1
I 1.71384 1.70933 1.69317 2 2x103
II 1.71409 1.71089 1.69945 5 2x103
III 1.71443 1.71294 1.70766 10 2x103
IV 1.71267 1.70193 1.66351 2 5x103
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (766-)
V 1.71194 1.69739 1.64534 2 7x103
D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol. 1(7), 887-894 (2010)
894
TABLE. 5 Nusselt number on the outer cylinder r = 2, P = 0.71, M = 2
G 2x103 5x103 104 伪 位
I 1.71384 1.70933 1.69317 2 0.2
II 2.74612 2.74248 2.72948 5 0.2
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III 3.59132 3.58826 3.57733 10 0.2
IV 1.97813 1.97362 1.95751 2 0.5
V 2.15341 2.14891 2.13282 2 0.7
7. Inman, R. M., Int. J. Heat Mass Transfer, V.5, 1053 (1962). 8. Lees, L., Jet Propulsion, V. 20, 259 (1959). 9. Lighthill, M. J., Aero Res. Council, July, 1956 (1958). 10. Low, G. M., J. Aero Sci., V.22, 329 (1955). 11. Modejaki, J., Int. J. Heat Mass Transfer, V.6,49 (1963). 12. Ostrach, S., NACA, TN, 2863 (1952). 13. Ostrach,S., NACA, TN, 3141 (1954). 14. Poppendick, F., Chem. Engg. Symp Sec.,V.50, No.11, 93 (1954). 15. Sparrow, E.M and Cess, R. D., Trans, ASME SERE, March (1962).
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