Results for the volumetric heat transfer coefficient are also ... bf3u in the ,B-phase, ..... 3. ; Douglas & Arbogast9; Panfilov25;. Auriault & Royer4; Quintard and Whitaker38,39). ... SI. The domain of validity of these two macroscopic equations has been discussed in ...... are usually based on a log-mean temperature driving.
Advances in Water Resources, Vol. 20, Nos 2-3, pp. 77-94, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I: SO 3 0 9 - 1708 (96) 0 0 0 2 4 - 3 0309-1708/97/$17.00+0.00
ELSEVIER
Two-medium treatment of heat transfer in porous media: numerical results for effective properties • M. Quintard LEPT-ENSAM, Espl. des Arts et Metiers, 33405 Talence Cedex, France
M. Kaviany Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI48109, USA
and S. Whitaker Department of Chemical Engineering, University of California at Davis, Davis, CA 95616, USA (Received 5 October 1995; accepted 9 December 1995) This paper presents two- and three-dimensional numerical solutions for the local closure problems associated with the two-equation model for heat transfer in porous media. Results are presented for the thermal dispersion tensors for both phases, in addition to the coupling dispersion tensors that appear naturally in the two-equation model. Results for the volumetric heat transfer coefficient are also given for Pi:clet numbers ranging from 0·1 to 1000. The longitudinal thermal dispersion coefficient for the one-equation model is compared with experimental data and a reasonably good agreement is obtained. The comparison between theory and experiment for the heat transfer coefficient is also promising. Copyright © 1996 Elsevier Science Ltd.
NOMENCLATURE av
Af3e Af3u Aue b
bI brr bf3f3 bf3u bf3u buu cf3f3 ,.cuf3 Cuu
cf3u cp h hf3 hu kw
interfacial area per unit volume. area of entrances and exits for the ,B-phase. area of the ,B~u interface. area of entrances and exits for the u-phase. vector field that maps V' (T) onto t in the oneequation model. = {bf3f3 in the ,B-phase, buf3 in the u-phase.
Keff
Kf3f3
= { bf3u in the ,B-phase, buu in the u-phase. vector field that maps V' (Tf3)f3 vector field that maps V'(Tu)" vector field that maps V' (Tf3)f3 vector field that maps V'(Tu)U = (nf3u kf3V'bf3f36f3u). = (nuf3 ku V'b uf3 6f3u) = -cf3{3= (nuf3 ku V'buu 6f3u),
Kf3u
onto onto onto onto
ff3.
1jJ.
Kuu
~u. Tu.
Kuf3
0
0
0
77
= (nf3u
kf3 V'bf3u6f3u) = -CuU' heat capacity. film heat transfer coefficient. = (n f3u kf3V'sf36f3u), = - (nuf3 ku V' su6f3u) = hf3. thermal conductivity of the w-phase. one-equation model effective thermal conductivity tensor. two-equation model effective thermal conductivity tensor associated with V' (Tf3)f3 in the ,B-phase equation. two-equation model effective thermal conductivity tensor associated with V'(Tu)" in the ,B-phase equation. two-equation model effective thermal conductivity tensor associated with V' (Tu)U in the u-phase equation two-equation model effective thermal cond ucti vi ty tensor associated with V' ( Tf3) f3 in the u-phase equation. 0
o
0
78
r
ro
M. Quintard et al. characteristic length for the w-phase. characteristic length associated with local volume averaged quantities. = -na,6' outwardly directed unit normal vector pointing from the ,8-phase toward the u-phase. position vector. characteristic length associated with averaging volume. scalar field that maps ((Ta)a - (T,6),6) onto T,6. scalar field that maps ((Ta)a - (T,6),6) onto Ta. time. point temperature in the w-phase. = c,6(T,6),6 + ca(Ta)a, spatial average temperature. = cw(Twt, phase average temperature for the w-phase. intrinsic phase average temperature for the w-phase. Tw - (Tw)W, spatial deviation temperature in the w-phase. two-equation model transport coefficient associated with V (T,6),6 in the ,8-phase equation.
1 INTRODUCTION
The macroscopic description of heat transfer in porous media by a single energy equation implies the assumption of local thermal equilibrium between the moving fluid phase and the solid phase. This hypothesis has been investigated by several authors (Truong & Zinsmeister47 . Whitaker 52 . S6zen & Vafai 46. Kavi any l8. Nield & Bej~n22; GObb6 & Quintard 13 ; Quintard &, Whitaker 37 - 39 ). For situations in which local thermal equilibrium is not valid, models have been proposed based on the concept of two macroscopic continua, one for the fluid phase, the ,8-phase in this paper and the other for the solid-phase, i.e. the u-phase. If intrinsic average temperatures for the ,8-phase and the u-phase are denoted by (T,6),6 and (Tat, respectively, the resulting macroscopic model is a two-equation thermal transport model such as (V ortmeyer & Schaefer4o ; Schliinder44; Vortmeyer48 ; Glatzmaier & Ramirez l2 ) ,8-phase
8(T,6),6 ,6,6 c,6(pcp),6 ~ + c,6(pc p),6(v,6) • V (T,6)
= V . (K~. V (T,6),6) - ayh( (T,6),6 - (Tan
(la)
u-phase
two-equation model transport coefficient associated with V(Tat in the ,8-phase equation. two-equation model transport coefficient associated with V(Ta)a in the u-phase equation. two-equation model transport coefficient associated with V (T,6),6 in the u-phase equation. velocity of the ,8-phase. V,6 (v,6) averaged velocity. (v,6),6 intrinsic phase averaged velocity. averaging volume. V volume of unit cell. Vceli volume of the w-phase contained within V. Vw X position vector of the centroid of the averaging volume. position vector relative to the centroid of the y averaging volume.
U,6a
Pw
w-phase indicator function. volume fraction of the w-phase. Dirac distribution associated with the interface A,6a' density in the w-phase.
concept of two macroscopic continua. The volumetric heat transfer coefficient ayh is generally determined experimentally, or by using correlations based on boundary layer theory. Also, no comprehensive approach is available for the determination of the thermal dispersion tensors. Since all these coefficients are coupled in the two energy equations, their experimental determination is not straightforward, and can lead to inaccurate results (Quintard & Whitaker28 ). Correlations for the heat transfer coefficient, based on boundary layer theory, do not provide good approximations because the thermal transport process within the pores and through the solid does not closely resemble a boundary layer process. Because of this, it is important to derive representations for these macroscopic coefficients on the basis of a comprehensive analysis of the relation between the pore-scale structure and the macroscopic description. Non-equilibrium models, based on local volume averaging, have been obtained by Carbonell & Whitaker,7 Zanotti & Carbonell,54-56 Levec & Carbone1l20 ,21 and Quintard & Whitaker?8 Starting from the pore-scale conservation equations, they obtain the following form for the macroscopic equations ,8-phase
8(T,6),6
C,6(pcp),6~
+ c,6(pcp)(v,6)
,6,6 . V(T,6)
6 - u,6,6· V (T,6) - U,6a· V (Tat
This model is intuitive, since it is not derived from the pore-scale transport equations through some scaling-up theory. However, it is the simplest form that reflects the
= V· (K,6,6. V(T,6),6 + K,6a· V(Ta)a) - ayh((T,6),6 - (Ta)a)
(2a)
79
Heat transfer in porous media u-phase o(T(J)(J ()f3 )(J E(J (pCp ) (J---ar- - U(Jf3· \7 Tf3 - U(J(J· \7(T(J = \7. (K(Jf3. \7(Tf3)f3
+ K(J(J. \7(T(Jn (2b)
- avh((T(Jr - (Tf3)f3)
In these equations, the effective transport coefficients are determined by solving three local boundary value problems for unit cells representative of the porous medium under consideration. They have been solved for fully developed flow and heat transfer through capillary tubes by Zanotti & Carbonell,54-46 and for two-dimensional and three-dimensional periodic unit cells by Quintard & Whitaker28 for the case of pure diffusion. The capillary tube model, although instructive, is not representative of the complex geometries found in real porous media. In particular, this model gives a Peclet-number squared dependence for the longitudinal thermal dispersion coefficient, and it gives a heat transfer coefficient that is independent of the local velocity. Both results do not agree well with experimental values for packed beds and other porous media, thus there is a need for the use of more realistic unit cells. This paper presents numerical results for periodic arrays of cylinders and spheres for the general convective-diffusion problem. These results are used to emphasize the dependence of the effective properties on the Peclet number and thermal conductivity ratio. 2 THEORETICAL BACKGROUND In this section we briefly review the introduction of the two-equation model. More comprehensive presentations can be found in papers by Carbonell & Whitaker,7 Zanotti & Carbonell, 54-56 Kavi any 18 and Quintard & Whitaker. 28 A rendering of the averaging problem under consideration is given in Fig. 1. The thermal transport problem at the pore-scale is given as follows oTf3 (pc p)f3 7it + (pcp)f3vf3. \7Tf3
= \7. (kf3\7Tf3) ,
in ,B-phase
(3a) (3b)
B.C.2.
Df3(J.kf3\7Tf3=Df3(J.k(J\7T(J,
(pcp)(J 0:r(J = \7. (k(J\7T(J)'
atAf3(J
in u-phase
(3c) (3d)
The complete description of the thermal transport requires the introduction of the continuity and momentum equations for the fluid phase, along with the no-slip boundary condition. It is assumed that the physical properties of the fluid and the solid phases are constant, thus the velocity field can be determined independently. Macroscopic equations can be derived from several points of view (see for instance Ene & Polisevski,1O for a
avernging volurre V
Fig. 1. Averaging volume for a two-phase system.
review of results obtained from homogenization theory). In this paper we make use of the method of volume averaging in which macroscopic quantities are spatially regularized fields derived from pore-scale quantities by an averaging procedure. Well-behaved macroscopic quantities for ordered or disordered systems, i.e. spatially smoothed fields devoid of small-scale fluctuations, are obtained by using conveniently chosen weighting functions. This problem has been studied extensively by Quintard & Whitaker 28 ,30-36 in which weighting functions were used to provide results that were valid for both ordered and disordered systems; however, in this development we will make use of the classical definitions of averaged quantities for the sake of simplicity. The intrinsic average temperature for the ,B-phase is given by (Tf3) f3
1 = -V
J
f3 vi!
Tf3 d V
(4)
and the pore-scale temperature deviation in the ,B-phase is defined by f3 Tf3 = (TfJ) + Tf3 (5) One can introduce this decomposition into the porescale equation for the ,B-phase and then form the volume average in order to obtain the macroscopic equation. After extensive use of the averaging theorem, the following energy equation emerges for the ,B-phase Ef3(pC p)f3
= \7.
o(Tf3)f3 f3 ---ar+ (pc p)f3(v(3) • \7 (Tf3) + (pcp)fJ(v f3. \7T(3)
[kf3(Ef3\7(Tf3)f3
+ ~ L+f3 Df3(JTf3 dA )] (6)
80
M. Quintard et al.
and the procedure can be repeated to generate an analogous equation for the a-phase. In order to obtain a closed form for eqn (6) and the transport equation for the a-phase, one needs to develop a boundary value problem for the spatial deviation temperatures, T{3 and Ta. To derive a governing differential equation for Tf3 we divide eqn (6) by cf3 and subtract the result from eqn (3a). The resulting pore-scale equation for the ,a-phase can be written as 8Tf3
f3 (pcp)f3 8t + (pcp)f3vf3 • '\7 (Tf3)
= '\7. (kf3'\7Tf3)
-
+ (pcp)f3vf3 • '\7Tf3
c~l'\7. (kf3 ~ L~ Df3aTf3dA)
-c~lkf3~J Df3a·'\7Tf3dA V v~
(7)
A similar equation can be developed for the a-phase and when decompositions of the form given by eqn (5) are used with eqns (3b) and (3c) the pore-scale boundary conditions become B.C. 1 Tf3 = Ta - ((Tf3)f3 - (Ta)a) B.C. 2 Df3a' kf3 '\7Tf3 =
at Af3a
(8a)
+ Df3a' kf3 '\7 (Tf3) f3
Df3a' ka '\7Ta
+ Df3a' k a'\7 (Ta)a
at Af3a (8b)
Several macroscopic source terms involving averaged temperatures can be found in these equations. A closed form of the averaged equations, such as eqn (6), can be obtained if the deviations are represented in terms of these macroscopic source terms. This is achieved in different ways, depending on several length-scale and time-scale constraints. These requirements are discussed in detail in several papers, and here we present only a summary of the results.
2.1 Local thermal equilibrium
contains an effective thermal dispersion tensor, Keff , which is given by Keff
=
«
ro
«
L
In addition, the time scales associated with the porescale equations must be much smaller than the time scales associated with the averaged equations. This allows for the introduction of a quasi-steady representation of the temperature deviations in terms of the gradient of the average temperature, i.e.,
(10) Here we have used the condition of local thermal equilibrium represented by (T) = (Tf3)f3 = (Tat. The mapping vector b" is referred to as a closure variable and it is only a function of position within the averaging volume. The macroscopic equation for the condition of local thermal equilibrium represents the sum of eqn (6) and the analogous equation for the a-phase and it
ka) L/lo" Df3a bf3 dA
(11)
2.2 Local DOD-equilibrium In this case the time and length scales are such that a unique macroscopic or effective medium cannot represent the macroscopic behavior of the two phases. This problem is similar to the double porosity, double permeability problems which have been investigated by several authors for purely diffusive cases (Barenblatt et al. 5 ; Arbogast et a1. 3 ; Douglas & Arbogast 9; Panfilov25 ; Auriault & Royer4; Quintard and Whitaker 38 ,39). Different strategies have been proposed for the solution of this problem and the two-equation model discussed in this paper represents a quasi-steady estimation of the pore-scale temperature field obtained by representing the spatial deviation temperatures in terms of the macroscopic source terms (Carbonell & Whitaker 7 ; Zanotti & Carbone1l 54 - 56 ; Kaviany18; Quintard & Whitaker28 ) according to Tf3 = bf3f3' '\7 (Tf3)f3 + bf3a' '\7 (Tat - sf3( (Tf3)f3 - (Tan
+ ... ,
Ta = baf3 •'\7 (Tf3)f3 + bo-o-. '\7 (Tat + sa((Ta)a - (Tf3)f3)
+ ... , (12)
Here the mapping vectors and the scalars are obtained from the solution of steady, pore-scale closure problems which will be discussed in the next section. Introducing this representation in eqn (6) gives the required macroscopic equation for the ,a-phase, which is expressed as cf3(pcp)f3
(9)
+ caka)I + ~ (kf3 -
- (pc p)f3(vf3bf3)
This requires that pore scales and macroscopic scales are conveniently separated, i.e. Cf3 , Ca
(cf3kf3
8(Tf3)f3 f3 f3 --ar+ cf3(pcp)f3(vf3) . '\7(Tf3) - uf3f3' '\7(Tf3)f3 - uf3a' '\7(Ta)a
=
'\7 • (Kf3f3 • '\7 (Tf3)f3
+ Kf3a • '\7 (Tan
- ayh((Tf3)f3 - (Ta)a)
(13)
In this equation, the effective properties such as K f3f3 , Kf3l1' uf3f3' uf3a and the volumetric heat exchange (or transfer) coefficients ayh, are obtained explicitly from the mapping vectors and scalars and representations for these coefficients are given in the next section. An equation analogous to eqn (13) describes the intrinsic average temperature for the a-phase, and this equation is given by ca(pcp)a 8(~;)a - uaf3' '\7(Tf3)f3 -
Uo-o-'
'\7(Tat
= '\7. (K af3 • '\7(Tf3)f3 - Kaa' '\7(Tan
- ayh((Tat - (Tf3)f3)
(14)
Heat transfer in porous media
The domain of validity of these two macroscopic equations has been discussed in several papers (Quintard & Whitaker 28 ,37-39; Grangeot et aIY). When timedependent mechanisms become more important, the assumption that the closure problem is quasi-steady fails and the validity of eqns (13) and (14) breaks down. Additional source terms, such as time derivatives of the averaged temperatures, must be included in the representations given by eqns (12). This time dependency of the local problems that appear in the averaging theory (or in the homogenization process) has been discussed before (Quintard & Whitaker27 ; Douglas & Arbogast9 ; Arbogast et a1. 3 ; Auriault & Royer4 ). This two-equation model is fully compatible with the one-equation model. Indeed, if we formally write (T(3)(3 = (T"),, = (T) and add eqns (13) and (14), we obtain a one-equation model with the effective thermal dispersion tensor of eqn (11) given by
(15)
B.C. 2.
SI
b(3" 2
0= k" Y' bM b(3,,(r + £;)
bM
-
c.;'cM
=
,
(I7c)
at A(3"
=
,
(I7d)
in V"
b(3,,(r), (I7e)
b",,(r + £;) = b",,(r),
i = 1,2,3
(17f)
= (0(3" 0k(3Y'b(3,,6(3,,); C,," = (0,,(30 k" Y'b",,6(3,,) = -c(3,,' C(3"
(I7g)
Problem III (pcp )(3v(3 0Y's(3
= k(3Y' 2s(3 - c(i'h(3, in V(3
(ISa)
B.c. 1.
0(3" 0k(3 Y's(3 = 0(3" 0k" Y's", at A(3"
B.C. 2.
s(3
=
1 + S,,' at A(3"
(ISc)
0=k"Y'2 s"+c.;'h,,, in V" s(3(r + Pi) = s(3(r),
(ISd)
s,,(r + £;) = s,,(r),
i = 1,2,3
(ISe)
3 PORE-SCALE CLOSURE PROBLEMS The macroscopic coefficients in eqns (13) and (14) are determined by three pore-scale closure problems to be solved for representative, periodic unit cells. We refer to the literature cited in the previous sections for a complete description of the theory and we list below the three closure problems
Problem I (pcp )(3v(3 + (pcp )(3v(3 oY'b(3(3 = k(3Y' 2b(3(3 - c(i'c(3(3, in V(3 (I6a) B.c. 1.
0(3" 0k(3 Y'b(3(3
+ o(3"k(3 =
0(3" 0k" Y'b,,(3' at A(3=
\'..... '\\ ........~ .. ............... \\
-..-..-..-..-..-..-..-..-..-..-..-..
..............................................•....
--0.01
.0.05
co.
...
\
"'\.\,
'ci. ;:..
..._.. _.._.. 0.1
\.\
'-'"
.~
co.
.------ 10.
~ U
················100.
-0.1
\ ....
a.
'-"
--.
~
'ci. co.
-0.15
kol kr>=
;:s
'\
..\
'-"
--0.01 ·0.2
0.01
_.. _.. _.. _.. 0.1
[!l\0
------- 10. -0.25 ···········....··100. 0.001 "-_ _ _---'-_ _ _ _..L-_ _ _- - '_ _ _ _-" 0.1
10
100
-0.3 "-_ _ _.....L_ _ _ _...L_ _ _ _'---_ _ _--'
1000
0.1
(b)
(iJ
100,-------------------,
k.lk,=
10
r-==~ ________ --«-~-
- - t~· · · · · · ·
10
100
1000
Cell Peelet Number
Cell Peelet Number
100. ----------------------
Fig. 9. Convective coefficient i,u(3(3 (array of in-line cylinders, E(3 = 0,38).
since they can be discarded for the diffusive regime and at high P6clet numbers. Similar conclusions can be drawn from the results obtained for staggered cylinders. 5.3 Volumetric heat transfer coefficient
The volumetric heat transfer coefficient is obtained when 0.15.------------------0.1 0.1
0.05
0.01 co.
/'0..
co.
;:..
0
._.._.._.._.._.. _.._.. _.._.................
'-'" O.OOl"----~----~----~------'
0.1
10 CeO Peelet Number
100
1000
Fig. 8. (a) Longitudinal thermal dispersion coefficient KO"O" (array of in-line cylinders, E(3 = 0,38). (b) Longitudinal thermal dispersion coefficient K(11l" (staggered array of cylinders, E(3 = 0,38).
be summarized as follows • The ratios reach an absolute maximum value for small or intermediate values of the P6clet number. • The asymptotic values are zero for P6clet numbers approaching infinity. • The absolute maximum value is about 0·8 cfl. The results suggest that these corrections might be important for a limited range of the P6clet numbers
: !
I' !
~
u a.
'-"
.
," ;,!
-0.05
.. ,,
--.
I'
~
'";:s1.
j
I'
/ !
.0.1
,/ !
'-"
/
,,'
.0.15
-0.2
!
i
I
--------------_...
........ ... '
"
! i.:
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i/
1
... _ .. _ .. _ ..
::~1
i
i
, .------ 10 I ················100 ~
. o . 2 5 , L . - - - - - ' - - - - - ' - - - - - - L - -_ _---l 0.1
10 Cell Peelet Number
100
1000
Fig. 10. Convective coefficient i,u(3(1 (array of in-line cylinders, E(3 = 0,38).
88
M. Quintard et a1. 0.3,...--------------------,
(a)
1000 r---.---,.------------------,
k/k~= --0.01
0.25
--0.01
._ .. _ .. _.. 0.1
'" -"'-=;~,:;,,:=,,:::~:=~::::::::::=:-
"--_ _ _-'--_ _ _-'--_ _ _-'-_ _----' 0.1
10
100
0.1
1000
10
100
1000
Cell Peclet Number
Cell Peclet Number
Fig. 11. Convective coefficient i,u,,;3 (array of in-line cylinders, c;3 = 0,38).
solving Problem III. It is given by eqn (30) after the sofield has been calculated. The volumetric heat transfer coefficient is represented as a function of the Peelet number in Figs 12(a) and (b). Their dependence on the Peelet number can be roughly estimated by assuming a behavior like ",Pen at large Peelet number. The values of the exponent n at Pe '" 1000 are reported in Table 1. From Figs 12(a) and (b) it is elear that in that range of Peelet number, a Pen dependence is not a very accurate representation. However, it is a convenient form for comparison with heuristic correlations most often used in the literature. For k,,/kf3 < 1, the heat transfer coefficient does not noticeably depend on the Peelet number. This is due to the dominance of the heat transfer resistance in the solid phase. For k,,/kf3 > 1, the heat transfer coefficient is larger and there is some influence on the Peelet number, especially for values of the Peelet number greater than 10. However, at these Peelet numbers the dependence on the Peelet number is much smaller than that which one finds in the elassical correlations for packed beds. While some authors have suggested correlations with weak dependence of the heat transfer coefficient upon the Reynolds number (Sozen & Vafai 46 ), the proposed correlations are generally extracted from boundary layer Table 1. Dependence of the volumetric heat transfer coefficient vs the Peciet number (Pe ~ 1000) 0.01
n, in-line cyl. n, stagg. cyl.
0.1
10
100
0.24 0.24
0.36 0.38
--0.Q1
._ .. _.._.. 0.1 ------ 10.
~
10
0.1
........ ·...... 100.
L -_ _ _
0.1
~
___
~
____
10
~
100
___
~
1000
CeO Peclet Number
Fig. 12. (a) Heat transfer coefficient (array of in-line cylinders, c;3 = 0,38). (b) Heat transfer coefficient (staggered array of cylinders, c;3 = 0·38).
theory (Wakao & Kaguei 50 ) and this implies a relatively strong dependence on the Reynolds number. Similar trends were found by Sahraoui & Kaviany 43 for periodic arrangements (in-line and staggered) of square cylinders. Their results are shown in Fig. 13 and they are based on the use of isothermal solid surfaces (corresponding to k,,/kf3 ~ (0). In the boundary-layer flow and heat transfer characteristics of isolated solid elements, the front stagnation region shows a variation of the local Nusselt number with the square root of the Reynolds
Heat transfer in porous media
(a) In-Line
6=0.9 Pr=0.67
o
5
10
15
20
Nut' (b) Staggered
6=0.9 Pr=0.67
89
However, the coefficient represents the value associated with an isolated sphere and one should not expect to find that type of behavior for a porous medium. In addition, correlations of the type indicated by eqn (34) are usually based on a log-mean temperature driving force that makes use of a local temperature difference of the form (Tf3)f3 - T i . Here T; is an interfacial temperature that could be representative of (Ta)a only when k a/kf3» 1. A second important aspect of our results is the dependence upon the velocity as indicated by the P6clet number. The P6clet number dependence indicated in Table 1 is close to the value of 1/3 which is very near to the value determined experimentally for low Reynolds number flows through bundles of tubes (Whitaker51 Fig. 7.8.5). The proposed theory emphasizes the fact that flow in real porous media is similar to the flow that is established in periodic structures. This is an important aspect that requires more thorough analysis, especially with respect to interpretation of laboratory experiments. So far, there have been no experiments that were designed to test the complete two-equation model theory. In the next section, we present some attempts to interpret the experimental data that are available in the literature.
6 INTERPRETATION OF EXPERIMENTS
5
10
15
20
Fig. 13. Distribution of the local Nusselt number around a square cylinder, for several Reynolds numbers, (a) in-line arrangements, and (b) staggered arrangements.
number (Kaviany 17). As shown in Figs 13(a) and (b), this is more clearly observed in the staggered arrangement of cylinders, where the front stagnation region has a large local Nusselt number which increases noticeably with the Reynolds number. However, these results are different from the empirical correlations typically used in the two-medium model. For example, we cite the correlation proposed by Wakao & Kaguei50 for spherical particles, i.e. (NUd)Asf
= 2
+ l'IRe~'6 Pr 1/ 3
(34)
where (NUd )Asf = avhd k' 13
Red = Pf3(v(3)d / /Lf3
and
Pr = /Lid pf3kf3
with d being the sphere diameter. We first note that our results do not indicate the asymptotic value of 2 for small P6clet numbers.
Our major objective would be to compare experimental data with results obtained with the complete theoretical development. This would require a system allowing the experimental determination of all effective coefficients in eqns (2). In addition, the use of the closure problems to calculate the effective coefficients would require some knowledge of the pore-scale structure. Unfortunately, there are no such data available in the literature. However, results are available for the oneequation model of effective thermal dispersion and some results are available in terms of the solid to fluid heat transfer coefficient. In this section, we present some examples of a comparison between closure problem calculations and experimental data. Our goal is to demonstrate that such comparisons are possible using computational capabilities that are currently available and that these comparisons provide valuable information for the interpretation of experimental data. Several experiments have been performed on thermal dispersion in porous media and reported in the literature. A first attempt to interpret these data were made by Levec & CarboneIl 20,21. In particular, longitudinal thermal dispersion arrangements were compared to the correlations extracted from the theoretical results (Zanotti & CarboneIl 54 - 56 ; Levec & CarboneIl2o ,21) based on a capillary tube model. As was discussed above, the longitudinal dispersion coefficient for a capillary tube model has a P6clet number exponent that is significantly
90
M. Quintard et al. 1000 . - - - - - - - - - - - - - - - - - 7 1
!
• Experiments
~=u~:OO~/'
100
-. ....
••......i'
10
•• 1_ ........" .................. _."!i
Fig. 15. Representation of the 3D Grang.eot et al. 14 ,15l!nit cell (layers of spheres in hexagonal packmg, cross-sectIOn) .
•.•it!i
Input data in our particular case are
air-glass system
I\:
1 L -______
1.
~
________
10.
~
________
100.
1000.
Particle Peclet Number
Fig. 14. Longitudinal thermal dispersion coefficient for airglass systems.
larger than that observed for the staggered arrangements (Sahraoui & Kaviany 42). Some improvements are expected from numerical calculations for more realistic systems, and we present here an example of such a comparison. We chose a liquid-solid system with a moderate thermal conductivity ratio to avoid contactpoint problems (Batchelor & O'Brien6 ; Shonnard & Whitaker45 ). Experiments on air-glass porous systems were carried out by Yagi et al. 53 and Gunn & De Souza. 16 The measured values of the longitudinal thermal dispersion coefficient are plotted in Fig. 14 together with the theoretical predictions of Zanotti & Carbonell. 54-56 Their results correspond to a correlation having the following properties: (1) Diffusive regime asymptotes are extrapolated from
the actual data, (2) Dispersive regime asymptotes are extrapolated from the straight tube results (Zanotti & Carbonell 54-56) and therefore have a P6clet number squared dependence. We solved numerically the three closure problems presented in the previous sections for a simple cubic lattice of spheres with ka/k(3 = 23 and for different P6clet numbers. From these computations it is possible to obtain the longitudinal thermal dispersion coefficient, since the effective thermal dispersion tensor in the oneequation model is given by Keff = K(3(3 + K(3a + K a(3 + Kaa. The normalized numerical results are reported in Fig. 14. For pure conduction, estimates of the effective thermal conductivity can be obtained from Maxwell's theory, i.e. Keff _
k;-
31\: - 2c(3(1\: - 1) 3+c(3(1\:-1) )
= 23;
c(3 = 0·40
~
and Maxwell's result is reported in Fig. 14. For this particular lattice, our results are somewhat lower than the experimental values in the diffusive regime, however, the SC lattice has a higher porosity than the experimental system and when k(3 < ka this produces lower values of the effective thermal conductivity. However, Maxwell's estimate is also lower and thus suggests that the experimental values could be overestimated. Our results in the dispersive regime are relatively good, especially if one notes that there is considerable scatter in the experimental data and there are no adjustable parameters in the theoretical values. Numerical results have been obtained for cases consistent with the experiments of Grangeot et al. 14 ,15 in which the unit cell corresponds to layers of spheres arranged in hexagonal packing. These experiments are particularly interesting since the porous medium corresponds to a truly periodic system, thus there is no ambiguity in the definition of the representative unit cell. In addition, temperature differences between the solid and the fluid phases are available and determination of the fluid-solid heat transfer coefficient were conducted. These results will be discussed later in this section. In the experiments, estimates of the longitudinal thermal dispersion coefficient were made with the one-equation model and they are represented in Fig. 16. Numerical solutions of the closure problems were obtained for a three-dimensional unit cell corresponding to the experiments (Fig. 15). The coefficients K(3(3, K(3a, K a(3 and Kaa are added in order to provide a theoretical estimate of the longitudinal thermal dispersion coefficient. A comparison of these theoretical predictions and the experimental results is presented in Fig. 16. The results are for two different thermal conductivity ratios, namely water-brass system water-nylon system
ka/k(3 = 0-46
The theoretical predictions shown in Fig. 16 are in relatively good agreement with the experimental results for the water-nylon system. Differences for the higher conductivity ratio in the case of the water-brass system
Heat transfer in porous media 1~
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91
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