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Low Rayleigh number flow in a heat generating porous media
INT. O C ~ . }{~%TMASS TI~%NSFER Vol. 13, pp. 281-294, 1986 ~
0735-1933/86 $3.00 + .00 Press Ltd. Printed in the United States
LOW RAYLEIGHNUMBERFLOWIN A HEAT GENERATINGPOROUSMEDIA W.E. Stewart, Jr. and C.L.G. Dona University of Missouri-Kansas City Department of Mechanical and Aerospace Engineering Energy Research Laboratory Truman Campus Independence, Missouri 64050 (Com~cated
by J.P. Hartnett and W.J. M i n d - - z )
ABSTRACT A numerical analysis has been performed for the steady-state temperature and stream function distributions in a short cylinder, having an isothermal side and top, an insulated bottom, for a uniform heat generating porous medium. The analysis uses the stream function formulation of Darcy's equation in cylindrical coordinates and the Boussinesq approximation. A single energy equation was used for the f l u i d and solid, since conduction was the expected mode of heat transfer at low heat generation rates for a lead sphere air porous media system. The solution of the non-dimensionalized mo~ntum and energy equations res~Ited in small Rayleigh numbers (2x10"b to 0.2) indicating the heat transfer is by conduction. Solutions for the stream lines and isotherms were obtained using a transient e x p l i c i t finite-difference approximation using a mean bed thermal conductivity.
Introduction Fluid currents formed in a f l u i d saturated porous media during conduction heat transfer has many important applications, such as in oil and gas production, cereal grain storage, geothermal energy, and porous insulation. This investigation considers the numerical solution of steady-state stream lines of air and the isotherms by conduction heat transfer in an air and solid lead sphere porous medium in an enclosed in a short cylinder. The porous medium was considered to be generating heat uniformly throughout the volume. The situation analyzed is for the
281
282
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
cylinder wall and top end isothermal and at the same temperature. The bottom end of the cylinder is adiabatic. Natural convection heat transfer in enclosures due to uniform heat generation in a porous media saturated with a fluid has been studied for several different geometries. The work of Gasser and Kirimi [1] was done on a horizontal porous layer of infinite extent with heat generation represented the analytical treatment of the problem. Hardee and Nilson [2] carried out experimental work which deals with the case of uniform heat generation in the fluid.
The situations investigated
included both rectangular and cylindrical enclosures having the vertical walls and bottom insulated and the top held at a constant temperature. Hardee and Nilson presented the following correlation for the case of a heat generating fluid,
Nu (T~ Ra)½
(i)
:
A comparison of the studies of [2-5] reveal a similarity between the relationship of the Nusselt number as a function of the Rayleigh number, although the boundary conditions and configurations differ. The study of Saatjian and Caltagirone [6] concerned a numerical study of porous media between two horizontal impermeable planes which was heated from below.
As the media was heated, the porous matrix de-
composed exothermically into gaseous products, simulating heat generation and mass transfer.
The heat generation was not steady and
continuous because the reaction eventually vanished due to depletion of the reactant. The investigation of Somerton, et al., [7], concerned the numerical solution of heat generation in a horizontal porous layer, essentially presenting the numerical solution to some of the experimental work of
Sun [4]. The work of Beukema, et a l . , [8], was concerned with a similar situation to that considered here, modeling the heat transfer in stored grain except the container was a rectangular parallelepiped.
The
parallelepiped, though, had all surfaces isothermal, not really typical of grain storage containers. The results were presented as a function of average internal temperature, where the porous media and surrounding
Vol. 13, No. 3
FIEWINAHEATG~qERATIIk~POR0t~M~3IA
fluid were assumed to be at equilibrium.
283
No velocity results were
given. The study by Rhee, et a l . , [9] concerned the experimental determination of heat transfer rates for a vertical cylinder in which a liquid layer existed above the solid/liquid porous bed, where the heat was generated inductively in the solid (steel spheres). Basically, the onset of convection was determined to be a function of the depth of the liquid layer above the porous bed. Two Rayleigh numbers were defined by Rhee, et a l . , one separately for the porous bed, an "internal" value, as
RI =
kfgBf q'''L3K 2kmZ~fvf
(2)
where L was the height of the bed. The cylinder was a glass beaker representing an insulated side and insulated bottom. The porous bed and liquid layer were cooled at the top surface of the liquid layer•
The
cylindrical geometry, though, is irrelevant in the study• Also an "external" Rayleigh number for the liquid layer was defined in [g] from Katto and Masouka [10] as kfgBfaT L K RE = km~fVf
(3)
which was determined to be approximately equal to 472 at the onset of convection• For the four data points of Rhee, et al., for a fluid/bed depth ratio of one, the critical internal Rayleigh number was approximately 12 for a Nusselt number defined as
(41 Nu = 2km (TB.Tm) where TB is the temperature at bottom of the bed and Tm is the temperature at the bed - liquid interface•
A temperature variation across the
bottom of the porous bed, where the bottom was essentially insulated are to be expected• Especially in the presence of convection TB is not a uniform value and as to how TB was determined was not stated.
In this
investigation we are concerned with convective flow present only in the conductive heat transfer mode• Although the convective flow does not augment the heat transfer, Nu=l, the flow patterns are of concern in certain applications•
284
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
Numerical Model The system modeled is a vertical right circular cylinder of radius R and height H, shown schematically in Figure 1, assumed to be f i l l e d with 2mm lead spheres having a uniform porosity of 0.37 throughout the bed and a permeability of 2.8x10"9m2. The heat generation rates, ~ ' " , are relatively small. As such, the f l u i d velocities are expected to be small and the neighboring f l u i d and solid are expected to be in thermal equilibrium.
Thus, the temperature distribution can be represented by a
single energy equation, as in cylindrical coordinates by ..(pCp~m ~aT + (pCp)f vvT : km v2T + q ' "
(5)
or,
LT+ Vz aZ) + (pCP)ma _ZT az at
(pCp)f (v r aT
(6)
: km (1 a (r aT) + a2T) + q'" ar
ar
az 2
The momentumequations for assumed Darcy model flow are
al~ + ~ V r = 0 ar
(7)
a.__~.P_ Pgz + p Vz = 0
(8)
where K is permeability and gz = - g" An incompressible f l u i d was assumed, except in the buoyancy term, in the momentumequation using the Boussinesq approximation. The continuity equation is satisfied by the stream function, ¢, in cylindrical coordinates as,
~r -- ! ~, r az
Vz =
_ ! ~
r ar
(9)
VO1. 13, NO. 3
FIOWINAHEATG~RATINGPOBOUSMEDIA
285
Non-dimensionalizing the equations with r
=
e
=
T/R, z
=
~IR, ~
=
$I(= R),
(T-T®) / (q'"RH/km), and t = ~ / R z
results in the single non-dimensional energy equation as, 1 @_~ae
I ~
ae _ 1 a (r ae ) + a2e + R
+ ~ ~z a-'f- ~ ar ~z
o
r
ar
Ef
(1o)
a-~
By taking the cross derivatives of equations (8) and (9), the non-dimensional momentumequation becomes, a
1
a
1
g6Kq'k' 'R2H Be vf~ m ar
(11)
The coefficient of a0/ar is the Rayleigh number, Ra = gBK(q'")R2H vf = km
(12)
and equation (12) becomes a 1
a
1
ae
(13)
The boundary conditions assumed were an isothermal cylinder side and top and adiabatic cylinder bottom.
Stream function values at the
cylinder surfaces and axial centerline were assumed zero, which results in a s l i p flow condition at a l l surfaces• The boundary conditions and i n i t i a l condition for equation (10) are e(1,z,t) = 0 e(r,H/R,t) = 0 ae(r,O,t)/az
= 0
ae(O,z,t)/ar
= 0
e(r,z,O) = 0 and the boundary conditions for equation (14) are kb(r,O,t) = 0 ~(r,H/R,t) = 0
286
W.E. Stewart, Jr. and C.L.G. Dona
@(1,z,t)
: 0
@(O,z,t)
: 0
Vol. 13, No. J
Equations (10) and (13) were expressed in transient f i n i t e difference form and ¢ and o solved for e x p l i c i t l y .
For the low heat
generation rates used here, Pe<2 for all heat generatiQn rates and central differencing was used exclusively [13], as upwinding was not necessary. Values of e and ¢ were solved with respect to time by augmenting eq. (10) by a time increment, At.
The e's obtained were then substitut-
ed in eq. (13) to obtain new ¢ values.
This process was continued until
the o and ~ values reached equilibrium.
The time required to reach a
steady-state temperature, Tss, was defined as the time necessary for the o to reach 99.9% to i t s final value, as determined in previous simulations.
The value of 99.9% was used to decrease computer run time.
Discussion and Results The system of two non-linear partial d i f f e r e n t i a l equations, (10) and (13), were solved using a central differencing e x p l i c i t transient f i n i t e - d i f f e r e n c e scheme for ¢ and o.
Since the results yielded
r e l a t i v e l y small velocities, Pe<2, the use of central differencing is appropriate. The solutions obtained were invariant with respect to the grid refinement.
Grids from 11x21 to 51x101 were used for several different
cases. All the results for ¢, e, Nu, and Ra were identical for each test case with the different mesh sizes. The heat generation rates per unit volume of 8.99x103, 3.60x10~, 1.43x10s, and 3.60xlOSW/m3 were used. The simulations were performed for cylinder top and side boundary isothermal temperatures of 250, 287, and 310°K. The thermophysical properties of pure lead and dry a i r were used for the solid spheres and saturating f l u i d , respectively, [17].
VOl. 13, NO. 3
FIEW IN A H E A T ~ ~ S M I ~ D I A
287
The results for all of the heat generation rates used in the simulations yielded small values of the defined Rayleigh numbers and Nusselt numbers of unity.
The very small values of Rayleigh number are
due to the use of air as the fluid, as opposed to a fluid with large values of specific heat and density.
Even though the boundary
conditions here differ from that of the experiments of Rhee, e t . a l . , their critical Rayleigh number of 33 should be of the same order of magnitude for this situation.
The only difference between the defini-
tion of Rayleigh number in Eq. (2) from [9] and that defined in this investigation by Eq. (12), is the factor of 2 in the denominator of Eq. (2).
The definition of the Rayleigh number forces its value to remain
constant with time for each set of boundary conditions and heat generation rates. Steady-state results for the dimensionless isotherms B are shown in Figure 2.
For all of the simulations the shape and values of the
isotherms did not change significantly for different heat generation rates and boundary temperatures. These invariant results for B are to be expected since the heat transfer is primarily that of conduction. The adiabatic boundary condition at r=O, @B(o,z,t)/Br=O, was not adhered to precisely, since the values of ~ and B at r=O along the axis were obtained by extrapolation from the neighboring nodes. The extrapolation resulted in isotherms slightly skewed from normal, especially for B1, B2, and B3. Similar results were obtained for the dimensionless values of stream function ~.
The streamlines shown in Figure 3 did not change
significantly in shape or position for all the boundary conditions, effective conductivities, and heat generation rates used in the analyses. The actual numerical values did change for the different parameters by several orders of magnitude, and are plotted in Figure 4.
The
differences in streamlines, though, represent very small real velocities, on the order of lxlO-4m/s. The stream function values are shown in Figure 4 for T = 287°K.
288
W.E. Stewart, Jr. and C.L.G. Dona
I
I
H
I
1
I i
I
FIG. I Schematic of Cylinder Coordinates and Dimensions
!
0
r
FIG. 2 Approximate Dimensionless Isotherms for All Simulations
Vol. 13, No. 3
Vol. 13, NO. 3
FIEWINAHF_ATG~%]~tTINGPOBOt~MEDIA
r FIG. 3 Approximate Dimensionless Streamlines for All Simulations As T increases, S decreases which results in a decreasing value of
Ra. This, in turn, decreases the absolute value of the right hand side of eq. (10), leading to smaller values of negative ~. This relationship is evident in the results shown in Figure 5, where the variation of ~1 through ~5 for the different boundary temperatures are plotted for q'"=3.6xlOSW/m 3.
289
290
W.E. S ~ ,
--,--
Jr. and C.L.G. Dona
~z
kin=k=/I.59
//
//.'5
km~=k=/15.9 ....
17ol. 13, No. 3
I// I'~,
km ~- k , / 2 5 , 0
I/
,'"
165
I
'
///~4
,,//,'/ ,,,,'.,',//,~% ,:;" l/l/ "~ ,,, ,'
,,:,///
¢/
164
/
III
"~'~
/
.
;//,y///
q, ,
o65
I
I
//2"/%
//.//
//.// /// /
166
///
/
//// // / / o67
/
/
/
/%
/
/
/
/
/ / /
lllill
, 10 ~
,I',
, ,,,,I
,
i0 =
L ......
I i03
'~iw:
PIG. 4 Variation of Dimensionless Stream Function with Effective Thermal Conductivities and Heat Generation Rate for T = 287°K
Vol. 13, NO. 3
FT-OWINAHEATGIKkIERATINGPCBOUSMEDIA
4--
5-
% -~xto ~ 2-
I
I
240
I 260
I
I
I
280
i 500
I
I 520
T,~ (°K)
FIG. 5 Variation of Dimensionless Stream Function with Boundary Temperature for km 2 ks/25.0 and q ' " = 3.6 x 105 W/m3. Conclusions Numerical results have been obtained for a stream function formulation of free convection in a saturated heat generating porous media contained in a short vertical cylinder.
The fluid was air and the
porous solid was lead spheres. The governi.g differential equations were non-dimensionalized and resulted in the definition of a modified Rayleigh number for this investigation.
The stream function formulation
of the governing equations were solved using the explicit transient finite difference technique. The solutions were performed for different approximate values of the effective value of thermal conductivity for the bed of lead spheres and air.
Since Rayleigh numbers and the
resulting Peclet numbers were small only central differencing was used. A constant value of calculated Nusselt number over the range of Rayleigh investigated confirm that the primary mode of heat transfer was by conduction. Also the shapes of the streamlines and isotherms remained essentially invariant.
291
292
W.E. Stewart, Jr. and C.L.G. Dona
Nomenclature Cp
-
specific heat
d g
-
diameter of sphere
gz H
-
k
-
thermal conductivity
k m Nu
-
mean bed thermal conductivity
-
-
Pe
-
gravitational constant gravitational acceleration in positive z direction height of cylinder
modified Nusselt number dimensional
pressure
Peclet number, VzaZ, Vrar :f
~f
qlll -
uniform heat generation rate per unit volume
r
dimensional radial coordinate direction -
dimensionless radial coordinate direction
R
-
radius of cylinder
Ra
-
modified Rayleigh number dimensionless time
t
dimensional time T
-
temperature
TO Tco
-
area averaged cylinder bottom temperature
Tss v
-
steady-state temperature
Vr
-
dimensionless radial velocity
Vr
-
dimensional radial velocity
isothermal boundary temperature velocity
dimensionless axial velocity
vz
dimensional axial velocity z
-
dimensionless axial coordinate direction
~
-
dimensional axial coordinate direction
0.
-
13
-
km/(pCp) f c o e f f i c i e n t of thermal expansion, l/T®
Vol. 13, No. 3
Vol. 13, NO. 3
¢
FIEW IN A HEAT G~%~ATING ~
MEDIA
porosity dimensionless stream function dimensional stream function
B K
-
dimensionless temperature, (T-T=)/(q'"RH/k m) permeability, d2c2/180(1-E) 3 dynamic viscosity
vf p
-
o
-
¢
kinematic viscosity of f l u i d , ~®/p= density (pCp)m/(pCp) f
temperature difference, T-T
¢o (To'T=) (pCp)m - mean bed value, (l-¢)(pCp) s + E(pCp)f
Subscripts f
-
fluid
i
-
i n i t i a l value
m
-
mean value
o
-
averagevalue
s
-
solid (spheres)
=
-
boundaryvalue
References 1. 2. 3. 4. 5. 6. 7. 8.
R.D. Gasser and M.S. Karimi, Onset of Convection in a Porous Medium with Internal Heat Generation, J. Heat Transfer, Feb. 1976, 49-54. H.C. Hardee and R.H. Nilson, Natural Convection in Porous Media with Heat Generation, Nuclear Science and Engineering, 63, 119-132 (1977). R.J. Buretta and A.S. Berman, Convective Heat Transfer in a Liquid Saturated Porous Layer, ASME J. Applied Mechanics, 43, 2 (1976). W. Sun, Convection I n s t a b i l i t y in Superposed Porous and Free Layers, Ph.D. Dissertation, University of Minnesota (1973). J.W. Elder, Steady Free Convection in a Porous Media Heated from Below, J. Fluid Mech., 27, 1, 29-48 (1967). E. Saatdjian and J.P. C~tagirone, Natural Convection in a Porous Layer under the Influence of an Exothermic Decomposition Reaction, J. Heat Transfer, 102, 654-658, Nov. 1980. C.W. Somerton, J.M.--l~cDonough, and I. Catton, Natural Convection in a Volumetrically Heated Porous Layer, ASME 22, 43-47 (1982). K.J. Beukema, S. Bruin, and J. Schenk, Thre~Dimensional Natural Convection in a Confined Porous Medium with Internal Heat Generation, Int. J. Heat Mass Transfer, 26, 3, 451-458 (1983).
293
294
9. 10. 11. 12.
13. 14. 15. 16. 17. 18.
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
S.J. Rhee, V.K. Dhir and I. Catton, Natural Convection Heat Transfer in Beds of Inductively Heated Particles, J. Heat Transfer, 100, 78-85, Feb. 1978. Y. Katto, and T. Masouka, Criterion for the Onset of Convective Flow in a Fluid in a Porous Medium, I n t ' l . J. Heat Mass Transfer, 10, 297-309 (1967). V. Arpaci, Conduction Heat Transfer, Addison-Wesley, Reading, MA, 235 (1966). W. E. Stewart, Jr. and C.L.G. Dona, Free Convection at Low Rayleigh Numbers in a Heat Generating Porous Media in a Short Cylinder, ASME 46, 217 (1985) be presented at the Nat'l. Heat Transfer Conf., Denver, Colorado, Aug. 1985. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Hemisphere, NY (1981). M. Combarnous, Natural Convection in Porous Media and Geothermal Systems, Proc. of 1978 I n t ' l . Heat Transfer Conference, Toronto, 45-59. M.A. Combarnous and S.A. Bories, Hydrothermal Convection in Saturated Porous Media, Adv. in Hydro Science, V.T. Chow, Ed., Academic Press, 10, 231-307 (1975). W.E. Stewart, Jr. and D.R. Smith, Transient Heat Transfer in Air Saturated Porous Media in a Short Cylinder, I n t ' l . Comm. Heat Transfer, 1__1, 4, 369-376, July/August 1984. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat Transfer, J. Wiley & Sons, Inc., N.Y., Tab. A.1 and A.4 (1981). J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publ. Co., N.Y., 133 (1972).
0735-1933/86 $3.00 + .00 Press Ltd. Printed in the United States
LOW RAYLEIGHNUMBERFLOWIN A HEAT GENERATINGPOROUSMEDIA W.E. Stewart, Jr. and C.L.G. Dona University of Missouri-Kansas City Department of Mechanical and Aerospace Engineering Energy Research Laboratory Truman Campus Independence, Missouri 64050 (Com~cated
by J.P. Hartnett and W.J. M i n d - - z )
ABSTRACT A numerical analysis has been performed for the steady-state temperature and stream function distributions in a short cylinder, having an isothermal side and top, an insulated bottom, for a uniform heat generating porous medium. The analysis uses the stream function formulation of Darcy's equation in cylindrical coordinates and the Boussinesq approximation. A single energy equation was used for the f l u i d and solid, since conduction was the expected mode of heat transfer at low heat generation rates for a lead sphere air porous media system. The solution of the non-dimensionalized mo~ntum and energy equations res~Ited in small Rayleigh numbers (2x10"b to 0.2) indicating the heat transfer is by conduction. Solutions for the stream lines and isotherms were obtained using a transient e x p l i c i t finite-difference approximation using a mean bed thermal conductivity.
Introduction Fluid currents formed in a f l u i d saturated porous media during conduction heat transfer has many important applications, such as in oil and gas production, cereal grain storage, geothermal energy, and porous insulation. This investigation considers the numerical solution of steady-state stream lines of air and the isotherms by conduction heat transfer in an air and solid lead sphere porous medium in an enclosed in a short cylinder. The porous medium was considered to be generating heat uniformly throughout the volume. The situation analyzed is for the
281
282
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
cylinder wall and top end isothermal and at the same temperature. The bottom end of the cylinder is adiabatic. Natural convection heat transfer in enclosures due to uniform heat generation in a porous media saturated with a fluid has been studied for several different geometries. The work of Gasser and Kirimi [1] was done on a horizontal porous layer of infinite extent with heat generation represented the analytical treatment of the problem. Hardee and Nilson [2] carried out experimental work which deals with the case of uniform heat generation in the fluid.
The situations investigated
included both rectangular and cylindrical enclosures having the vertical walls and bottom insulated and the top held at a constant temperature. Hardee and Nilson presented the following correlation for the case of a heat generating fluid,
Nu (T~ Ra)½
(i)
:
A comparison of the studies of [2-5] reveal a similarity between the relationship of the Nusselt number as a function of the Rayleigh number, although the boundary conditions and configurations differ. The study of Saatjian and Caltagirone [6] concerned a numerical study of porous media between two horizontal impermeable planes which was heated from below.
As the media was heated, the porous matrix de-
composed exothermically into gaseous products, simulating heat generation and mass transfer.
The heat generation was not steady and
continuous because the reaction eventually vanished due to depletion of the reactant. The investigation of Somerton, et al., [7], concerned the numerical solution of heat generation in a horizontal porous layer, essentially presenting the numerical solution to some of the experimental work of
Sun [4]. The work of Beukema, et a l . , [8], was concerned with a similar situation to that considered here, modeling the heat transfer in stored grain except the container was a rectangular parallelepiped.
The
parallelepiped, though, had all surfaces isothermal, not really typical of grain storage containers. The results were presented as a function of average internal temperature, where the porous media and surrounding
Vol. 13, No. 3
FIEWINAHEATG~qERATIIk~POR0t~M~3IA
fluid were assumed to be at equilibrium.
283
No velocity results were
given. The study by Rhee, et a l . , [9] concerned the experimental determination of heat transfer rates for a vertical cylinder in which a liquid layer existed above the solid/liquid porous bed, where the heat was generated inductively in the solid (steel spheres). Basically, the onset of convection was determined to be a function of the depth of the liquid layer above the porous bed. Two Rayleigh numbers were defined by Rhee, et a l . , one separately for the porous bed, an "internal" value, as
RI =
kfgBf q'''L3K 2kmZ~fvf
(2)
where L was the height of the bed. The cylinder was a glass beaker representing an insulated side and insulated bottom. The porous bed and liquid layer were cooled at the top surface of the liquid layer•
The
cylindrical geometry, though, is irrelevant in the study• Also an "external" Rayleigh number for the liquid layer was defined in [g] from Katto and Masouka [10] as kfgBfaT L K RE = km~fVf
(3)
which was determined to be approximately equal to 472 at the onset of convection• For the four data points of Rhee, et al., for a fluid/bed depth ratio of one, the critical internal Rayleigh number was approximately 12 for a Nusselt number defined as
(41 Nu = 2km (TB.Tm) where TB is the temperature at bottom of the bed and Tm is the temperature at the bed - liquid interface•
A temperature variation across the
bottom of the porous bed, where the bottom was essentially insulated are to be expected• Especially in the presence of convection TB is not a uniform value and as to how TB was determined was not stated.
In this
investigation we are concerned with convective flow present only in the conductive heat transfer mode• Although the convective flow does not augment the heat transfer, Nu=l, the flow patterns are of concern in certain applications•
284
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
Numerical Model The system modeled is a vertical right circular cylinder of radius R and height H, shown schematically in Figure 1, assumed to be f i l l e d with 2mm lead spheres having a uniform porosity of 0.37 throughout the bed and a permeability of 2.8x10"9m2. The heat generation rates, ~ ' " , are relatively small. As such, the f l u i d velocities are expected to be small and the neighboring f l u i d and solid are expected to be in thermal equilibrium.
Thus, the temperature distribution can be represented by a
single energy equation, as in cylindrical coordinates by ..(pCp~m ~aT + (pCp)f vvT : km v2T + q ' "
(5)
or,
LT+ Vz aZ) + (pCP)ma _ZT az at
(pCp)f (v r aT
(6)
: km (1 a (r aT) + a2T) + q'" ar
ar
az 2
The momentumequations for assumed Darcy model flow are
al~ + ~ V r = 0 ar
(7)
a.__~.P_ Pgz + p Vz = 0
(8)
where K is permeability and gz = - g" An incompressible f l u i d was assumed, except in the buoyancy term, in the momentumequation using the Boussinesq approximation. The continuity equation is satisfied by the stream function, ¢, in cylindrical coordinates as,
~r -- ! ~, r az
Vz =
_ ! ~
r ar
(9)
VO1. 13, NO. 3
FIOWINAHEATG~RATINGPOBOUSMEDIA
285
Non-dimensionalizing the equations with r
=
e
=
T/R, z
=
~IR, ~
=
$I(= R),
(T-T®) / (q'"RH/km), and t = ~ / R z
results in the single non-dimensional energy equation as, 1 @_~ae
I ~
ae _ 1 a (r ae ) + a2e + R
+ ~ ~z a-'f- ~ ar ~z
o
r
ar
Ef
(1o)
a-~
By taking the cross derivatives of equations (8) and (9), the non-dimensional momentumequation becomes, a
1
a
1
g6Kq'k' 'R2H Be vf~ m ar
(11)
The coefficient of a0/ar is the Rayleigh number, Ra = gBK(q'")R2H vf = km
(12)
and equation (12) becomes a 1
a
1
ae
(13)
The boundary conditions assumed were an isothermal cylinder side and top and adiabatic cylinder bottom.
Stream function values at the
cylinder surfaces and axial centerline were assumed zero, which results in a s l i p flow condition at a l l surfaces• The boundary conditions and i n i t i a l condition for equation (10) are e(1,z,t) = 0 e(r,H/R,t) = 0 ae(r,O,t)/az
= 0
ae(O,z,t)/ar
= 0
e(r,z,O) = 0 and the boundary conditions for equation (14) are kb(r,O,t) = 0 ~(r,H/R,t) = 0
286
W.E. Stewart, Jr. and C.L.G. Dona
@(1,z,t)
: 0
@(O,z,t)
: 0
Vol. 13, No. J
Equations (10) and (13) were expressed in transient f i n i t e difference form and ¢ and o solved for e x p l i c i t l y .
For the low heat
generation rates used here, Pe<2 for all heat generatiQn rates and central differencing was used exclusively [13], as upwinding was not necessary. Values of e and ¢ were solved with respect to time by augmenting eq. (10) by a time increment, At.
The e's obtained were then substitut-
ed in eq. (13) to obtain new ¢ values.
This process was continued until
the o and ~ values reached equilibrium.
The time required to reach a
steady-state temperature, Tss, was defined as the time necessary for the o to reach 99.9% to i t s final value, as determined in previous simulations.
The value of 99.9% was used to decrease computer run time.
Discussion and Results The system of two non-linear partial d i f f e r e n t i a l equations, (10) and (13), were solved using a central differencing e x p l i c i t transient f i n i t e - d i f f e r e n c e scheme for ¢ and o.
Since the results yielded
r e l a t i v e l y small velocities, Pe<2, the use of central differencing is appropriate. The solutions obtained were invariant with respect to the grid refinement.
Grids from 11x21 to 51x101 were used for several different
cases. All the results for ¢, e, Nu, and Ra were identical for each test case with the different mesh sizes. The heat generation rates per unit volume of 8.99x103, 3.60x10~, 1.43x10s, and 3.60xlOSW/m3 were used. The simulations were performed for cylinder top and side boundary isothermal temperatures of 250, 287, and 310°K. The thermophysical properties of pure lead and dry a i r were used for the solid spheres and saturating f l u i d , respectively, [17].
VOl. 13, NO. 3
FIEW IN A H E A T ~ ~ S M I ~ D I A
287
The results for all of the heat generation rates used in the simulations yielded small values of the defined Rayleigh numbers and Nusselt numbers of unity.
The very small values of Rayleigh number are
due to the use of air as the fluid, as opposed to a fluid with large values of specific heat and density.
Even though the boundary
conditions here differ from that of the experiments of Rhee, e t . a l . , their critical Rayleigh number of 33 should be of the same order of magnitude for this situation.
The only difference between the defini-
tion of Rayleigh number in Eq. (2) from [9] and that defined in this investigation by Eq. (12), is the factor of 2 in the denominator of Eq. (2).
The definition of the Rayleigh number forces its value to remain
constant with time for each set of boundary conditions and heat generation rates. Steady-state results for the dimensionless isotherms B are shown in Figure 2.
For all of the simulations the shape and values of the
isotherms did not change significantly for different heat generation rates and boundary temperatures. These invariant results for B are to be expected since the heat transfer is primarily that of conduction. The adiabatic boundary condition at r=O, @B(o,z,t)/Br=O, was not adhered to precisely, since the values of ~ and B at r=O along the axis were obtained by extrapolation from the neighboring nodes. The extrapolation resulted in isotherms slightly skewed from normal, especially for B1, B2, and B3. Similar results were obtained for the dimensionless values of stream function ~.
The streamlines shown in Figure 3 did not change
significantly in shape or position for all the boundary conditions, effective conductivities, and heat generation rates used in the analyses. The actual numerical values did change for the different parameters by several orders of magnitude, and are plotted in Figure 4.
The
differences in streamlines, though, represent very small real velocities, on the order of lxlO-4m/s. The stream function values are shown in Figure 4 for T = 287°K.
288
W.E. Stewart, Jr. and C.L.G. Dona
I
I
H
I
1
I i
I
FIG. I Schematic of Cylinder Coordinates and Dimensions
!
0
r
FIG. 2 Approximate Dimensionless Isotherms for All Simulations
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FIEWINAHF_ATG~%]~tTINGPOBOt~MEDIA
r FIG. 3 Approximate Dimensionless Streamlines for All Simulations As T increases, S decreases which results in a decreasing value of
Ra. This, in turn, decreases the absolute value of the right hand side of eq. (10), leading to smaller values of negative ~. This relationship is evident in the results shown in Figure 5, where the variation of ~1 through ~5 for the different boundary temperatures are plotted for q'"=3.6xlOSW/m 3.
289
290
W.E. S ~ ,
--,--
Jr. and C.L.G. Dona
~z
kin=k=/I.59
//
//.'5
km~=k=/15.9 ....
17ol. 13, No. 3
I// I'~,
km ~- k , / 2 5 , 0
I/
,'"
165
I
'
///~4
,,//,'/ ,,,,'.,',//,~% ,:;" l/l/ "~ ,,, ,'
,,:,///
¢/
164
/
III
"~'~
/
.
;//,y///
q, ,
o65
I
I
//2"/%
//.//
//.// /// /
166
///
/
//// // / / o67
/
/
/
/%
/
/
/
/
/ / /
lllill
, 10 ~
,I',
, ,,,,I
,
i0 =
L ......
I i03
'~iw:
PIG. 4 Variation of Dimensionless Stream Function with Effective Thermal Conductivities and Heat Generation Rate for T = 287°K
Vol. 13, NO. 3
FT-OWINAHEATGIKkIERATINGPCBOUSMEDIA
4--
5-
% -~xto ~ 2-
I
I
240
I 260
I
I
I
280
i 500
I
I 520
T,~ (°K)
FIG. 5 Variation of Dimensionless Stream Function with Boundary Temperature for km 2 ks/25.0 and q ' " = 3.6 x 105 W/m3. Conclusions Numerical results have been obtained for a stream function formulation of free convection in a saturated heat generating porous media contained in a short vertical cylinder.
The fluid was air and the
porous solid was lead spheres. The governi.g differential equations were non-dimensionalized and resulted in the definition of a modified Rayleigh number for this investigation.
The stream function formulation
of the governing equations were solved using the explicit transient finite difference technique. The solutions were performed for different approximate values of the effective value of thermal conductivity for the bed of lead spheres and air.
Since Rayleigh numbers and the
resulting Peclet numbers were small only central differencing was used. A constant value of calculated Nusselt number over the range of Rayleigh investigated confirm that the primary mode of heat transfer was by conduction. Also the shapes of the streamlines and isotherms remained essentially invariant.
291
292
W.E. Stewart, Jr. and C.L.G. Dona
Nomenclature Cp
-
specific heat
d g
-
diameter of sphere
gz H
-
k
-
thermal conductivity
k m Nu
-
mean bed thermal conductivity
-
-
Pe
-
gravitational constant gravitational acceleration in positive z direction height of cylinder
modified Nusselt number dimensional
pressure
Peclet number, VzaZ, Vrar :f
~f
qlll -
uniform heat generation rate per unit volume
r
dimensional radial coordinate direction -
dimensionless radial coordinate direction
R
-
radius of cylinder
Ra
-
modified Rayleigh number dimensionless time
t
dimensional time T
-
temperature
TO Tco
-
area averaged cylinder bottom temperature
Tss v
-
steady-state temperature
Vr
-
dimensionless radial velocity
Vr
-
dimensional radial velocity
isothermal boundary temperature velocity
dimensionless axial velocity
vz
dimensional axial velocity z
-
dimensionless axial coordinate direction
~
-
dimensional axial coordinate direction
0.
-
13
-
km/(pCp) f c o e f f i c i e n t of thermal expansion, l/T®
Vol. 13, No. 3
Vol. 13, NO. 3
¢
FIEW IN A HEAT G~%~ATING ~
MEDIA
porosity dimensionless stream function dimensional stream function
B K
-
dimensionless temperature, (T-T=)/(q'"RH/k m) permeability, d2c2/180(1-E) 3 dynamic viscosity
vf p
-
o
-
¢
kinematic viscosity of f l u i d , ~®/p= density (pCp)m/(pCp) f
temperature difference, T-T
¢o (To'T=) (pCp)m - mean bed value, (l-¢)(pCp) s + E(pCp)f
Subscripts f
-
fluid
i
-
i n i t i a l value
m
-
mean value
o
-
averagevalue
s
-
solid (spheres)
=
-
boundaryvalue
References 1. 2. 3. 4. 5. 6. 7. 8.
R.D. Gasser and M.S. Karimi, Onset of Convection in a Porous Medium with Internal Heat Generation, J. Heat Transfer, Feb. 1976, 49-54. H.C. Hardee and R.H. Nilson, Natural Convection in Porous Media with Heat Generation, Nuclear Science and Engineering, 63, 119-132 (1977). R.J. Buretta and A.S. Berman, Convective Heat Transfer in a Liquid Saturated Porous Layer, ASME J. Applied Mechanics, 43, 2 (1976). W. Sun, Convection I n s t a b i l i t y in Superposed Porous and Free Layers, Ph.D. Dissertation, University of Minnesota (1973). J.W. Elder, Steady Free Convection in a Porous Media Heated from Below, J. Fluid Mech., 27, 1, 29-48 (1967). E. Saatdjian and J.P. C~tagirone, Natural Convection in a Porous Layer under the Influence of an Exothermic Decomposition Reaction, J. Heat Transfer, 102, 654-658, Nov. 1980. C.W. Somerton, J.M.--l~cDonough, and I. Catton, Natural Convection in a Volumetrically Heated Porous Layer, ASME 22, 43-47 (1982). K.J. Beukema, S. Bruin, and J. Schenk, Thre~Dimensional Natural Convection in a Confined Porous Medium with Internal Heat Generation, Int. J. Heat Mass Transfer, 26, 3, 451-458 (1983).
293
294
9. 10. 11. 12.
13. 14. 15. 16. 17. 18.
W.E. Stewart, Jr. and C.L.G. Dona
Vol. 13, No. 3
S.J. Rhee, V.K. Dhir and I. Catton, Natural Convection Heat Transfer in Beds of Inductively Heated Particles, J. Heat Transfer, 100, 78-85, Feb. 1978. Y. Katto, and T. Masouka, Criterion for the Onset of Convective Flow in a Fluid in a Porous Medium, I n t ' l . J. Heat Mass Transfer, 10, 297-309 (1967). V. Arpaci, Conduction Heat Transfer, Addison-Wesley, Reading, MA, 235 (1966). W. E. Stewart, Jr. and C.L.G. Dona, Free Convection at Low Rayleigh Numbers in a Heat Generating Porous Media in a Short Cylinder, ASME 46, 217 (1985) be presented at the Nat'l. Heat Transfer Conf., Denver, Colorado, Aug. 1985. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Hemisphere, NY (1981). M. Combarnous, Natural Convection in Porous Media and Geothermal Systems, Proc. of 1978 I n t ' l . Heat Transfer Conference, Toronto, 45-59. M.A. Combarnous and S.A. Bories, Hydrothermal Convection in Saturated Porous Media, Adv. in Hydro Science, V.T. Chow, Ed., Academic Press, 10, 231-307 (1975). W.E. Stewart, Jr. and D.R. Smith, Transient Heat Transfer in Air Saturated Porous Media in a Short Cylinder, I n t ' l . Comm. Heat Transfer, 1__1, 4, 369-376, July/August 1984. F.P. Incropera and D.P. DeWitt, Fundamentals of Heat Transfer, J. Wiley & Sons, Inc., N.Y., Tab. A.1 and A.4 (1981). J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publ. Co., N.Y., 133 (1972).