40
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
Gupta, A. S., Thodos, G., Ind. Eng. Chem. Fundam., 3, 218 (1964). Hallmann, V. W., Paas. G., Wolpert, H. D., Vai. Tech., 6, 169 (1973). Houghen. J. O., Piret, E. L., Chem. Eng. Prog., 47, 295 (1951). Kunii, D., Sujuki, M., Int. J . Heat Mass Transfer, IO, 884 (1967). Leva, M., Ind. Eng. Chem.. 39,857 (1947). Leva, M., Grummer, M., Ind. f n g . Chem., 40, 415 (1948). Leva, M., Weintraub, M., Grummer, M..Clark, E. L., Ind. Eng. Chem.. 40, 747 (1948). Lindauer, G. C., AIChE J., 13, 1181 (1967). Nelson, P. A., Galloway, T. R.. Chem. f n g . Sci., 30, 1 (1975). Plautz, D. A,, Johnstone, H. F., AIChE J., 1, 193 (1955). Ranz, W. E., Chem. Eng. Prog., 48, 247 (1952). Rowe, P. N., Chem. f n g . Sci., 30, 7 (1975). Schumann, T. E. W., Voss, V., J . Fuel, 13, 249 (1934). Singer. E., Wilhelm, R. H., Cbem. Eng. Prog., 46, 343 (1950). Sorenson, J. P., Stewart, W. E., Chem. Eng. Sci., 29, 827 (1974). Verschoor, H., Schuit, G. C. A., Appl. Sci. Res., 42, (Part A2, No. 2), 97 (1950).
Votruba. J., Sinkule, J., Hlavacek, V.. Skrivarek, J., Chem. Eng. Sci., 30, 117 (1975a). Votruba, J.. Mikus, O., Nguen, K., Hhvacek, V., Skrivanek. J., Chem. f n g . Sci., 30, 201 (1975b). Wakao, N., Kato, K., J . Chem. Eng. Jpn., 2, 24 (1969). Wakao, N., Vortmeyer, D., Chem. Eng. Sci., 26, 1753 (1971). Wakao, N., Takano. Y.. Pei, D. C. T., J . Chem. Eng. Jpn., 6, 269 (1973). Wilhelm, R. H., Johnson, W. C., Wynkoop, R., Collin, D. W., Chem. Eng. Prog., 44, 105 (1948). Yagi, S.,Kunii, D., AIChEJ., 3, 373 (1957). Yagi, S.,Kunii, D., Wakao, N., AICh€ J., 6, 543 (1960). Yovanovich, M. M., ASME Paper 73-HT-43, ASME-AIChE Heat Transfer Conference, Atlanta, Ga., 1973.
Received for review February 3, 1977 Accepted July 6, 1978
Heat Transfer in Gas-Solid Packed Bed Systems. 2. The Conduction Mode Arcot R. Balakrishnan and David C. T. Pei' Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G I
The conduction mode of heat transfer in packed beds subject to flowing gases has been studied analytically. The analysis is based on the more realistic assumption of a finite contact spot between the spheres in the bed, the dimensions of which may be obtained from the Hertzian elasticity theory, rather than the usual assumption of a point contact. Moreover, the convective effects of the flowing gas were also incorporated in the analysis as boundary conditions. The effect of parameters such as contact spot dimensions, packing geometry, Biot modulus (convective effects), and radiation on the conduction mode have also been examined.
Introduction In a previous paper (Balakrishnan and Pei, 1978), a review of the several analytical and empirical models to predict the conduction mode of heat transfer in packed beds was presented. It was pointed out that many of the models, particularly the earlier ones, are highly empirical in nature and need considerable refinement before they can be used with confidence for design purposes. The recent studies by Chan and Tien (1973) and Yovanovich (1973) have taken a more realistic approach of assuming finite contact spots between the particles constituting the bed. This was used in estimating the conduction heat transfer in super insulation problems. However, they are applicable only for the case where there is no fluid flowing through the bed. On the other hand, it is believed (Bhattacharyya and Pei, 1975) that the flowing fluid does have an effect on the conduction mode. This paper presents an analysis which may he used to estimate the conduction heat transfer in packed beds subject to convective effects. As in the models of Chan and Tien (1973) and Yovanovich (1973), the present model makes use of the concept of a finite contact spot between the individual spheres in the bed, the dimensions of the contact spot being evaluated from the Hertzian theory of elasticity. The conductive effects, which are used as boundary conditions in the analysis, are estimated from an earlier experimental correlation of Pei and co-workers (Bhattacharyya and Pei, 1975; Balakrishnan and Pei, 1974). Analysis Consider a single sphere of radius r = a , between two planes A and B where A is a t a higher temperature than B. Gas, a t a bulk temperature of tf, flows past the sphere 0019-7882/79/1118-0040$01.00/0
and heat is transferred by conduction through the sphere between the two planes and by convection between the sphere and the gas. This configuration is shown in Figure 1.
The flux from plane A to the sphere is qo and a flux of q1 leaves the sphere into plane B. The convective flux from the sphere to the air stream is equal to hfp(t- tf), where
hfpis the convective heat transfer coefficient. There is a finite contact spot between the spheres and each of the other two spheres at planes A and B, respectively, and the contact radius is given by the Hertz relation
For the situation described above, the temperature field in the sphere is described by the Laplace equation
with the following boundary conditions
= -hfpT
0, 5 0 5 R - Bo, r = a 0, 5 0 IR , r = a
(3) = -ql where T = t - tf and 6'" = sin-' ( r J a ) . The general solution is A -
(4) where x = cos 0. @ 1978 American
Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
41
conductivity of the bed, k,, can hence be defined by a heat balance on the cube (the cube being a part of the bed has the same k,)
The conduction heat transfer coefficient, hcd, may be obtained from the definition -.-E
c
Therefore, the conduction heat transfer per layer of spheres in the bed is
Figure 1. Coordinates for the analysis.
A , is determined using all the boundary conditions in eq 3 and utilizing the orthogonality properties of Legendre polynomials; i.e.
The corresponding Nusselt number is defined as
From the orthogonality properties of Legendre polynomials it follows that -2 .lPn2(x)dx = 2n + 1 Moreover, since xo is very close to unity (as Bo is small), the following approximation may be made (Balakrishnan, 1976) -2 dx = 2n 1 and furthermore, by definition -.I__
L?;(X)
L:P,(x)
dx = -~ 2n
~
1
+ 1 [Pn-,(X,)
+
Cn
- P n + 1 ( ~ 0 ) 1=
(8)
Equation 5 then reduces to 2n
2
40
+1
k, 2n
cn
41
+1
(-cn)
k, 271 + 1
A,
hfP -*
ks
2 2n + 1 and, therefore, the solution to eq 1 2 with boundary conditions is
where Bi = hfp-a/k,,which is the Biot modulus. By definition, the thermal resistance of the sphere is given by T(a,O)- T(a,n) R-A1 = 4 0 + 41 rrc2 cu
d
Rcdl =
2a .f Pzn-Axo) - P Z n ( X 0 ) k , r r c 2 n=l (2n - 1) + Bi
--
(10)
The resistance of a cubical volume of side 2a, just enclosing one of the spheres in the bed, is also Rcdl. (This is because the only solid phase in this cube through which conduction can take place is the sphere.) The effective thermal
The conduction heat transfer coefficient obtained by eq 12 can now be used in obtaining the total heat transfer in packed beds subject to gas flow. Therefore, the validity of this analysis can be substantiated by comparing the total heat transfer so obtained with experimental data on total heat transfer of packed beds subject to flowing gas. This will be dealt with in part 3 of this series. Parameter Studies It is apparent that the conduction coefficient, hcd, as derived in the analysis will depend on a variety of parameters. This section examines this dependence of hcd on parameters such as packing arrangements (contact patterns), contact size (bed heights), Biot modulus (convective effects), and thermal radiation a t elevated temperatures. 1. Effect of Contact Size. Equation 11 for Rcdl,has for an independent parameter the contact size xo (= cos 0,) where 0,, called the contact angle, is the ratio of rc/a. It is strongly dependent on the pressure applied on the contact spot. In fact, Chan and Tien (1973) have shown that for a material with a Young's modulus of 5.51 X 1O'O N/m2 and a Poisson's ratio of 0.22, an order of magnitude increase in the applied pressure increases r,/a by about half an order of magnitude. Figure 2 shows plots of Nud (hcdin dimensionless form) against the contact angle do for two different bed materials. The following observations may be made. (1)The N U c d varies linearly with Bo for a particular bed material with a constant Biot modulus. (2) Increasing the Biot modulus, for a particular bed material, increases the slope of the linear plot. (3) Different bed materials produce different slopes. Iron oxide (which has a lower thermal conductivity) has a much lower slope than steel although the Biot modulus used with the iron oxide spheres is much larger than that used with steel (0.2 and 0.4 vs. 0.003 and 0.008). It may also be noted from Figure 2 that a t the limiting case Bo 0, Nud approaches zero. This is to be expected as the resistance to conduction through the sphere will increase with decreasing contact angle. As Bo becomes smaller, the bending of the heat flow lines in the sphere will be greater, thereby increasing the resistance to conduction, and a t the limiting case of Bo 0 the resistance will tend toward infinity. Moreover, the limiting case Bo 0 corresponds to the case where the spheres are in point contact.
-
-
-
42
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
1
2
IRON OXIDE SPHERES
D K
2a
:
0.61 J l m s K
I
STEEL SPHERES
0 635 cm
:
D
= 2 a - 0 6 3 5 cm
K
:
4327 J l r n s K
5
20
d
0
008
004
CONTACT
SIZE
016
,012
.02
0
Figure 2. Effect of contact size.
There are several models (Agro and Smith, 1953; Singer and Wilhelm, 1950; Wakao and Kato, 1969) in the literature on the effective conductivity of packed beds (convective effects are neglected in these studies) which consider the spheres to be in point contact. However, this assumption which overlooks the size and shape of the contact spots between the spheres is compensated in the models through empirical constants. 2. Effect of Packing Geometry. Three basic regular packing patterns-the simple cubic, face-centered cubic, and body-centered cubic-are used as convenient physical models to analyze the conduction heat transfer in a bed of uniform spheres. The three regular packing patterns are shown in Figure 3. It may be noted that in actual beds the porosity or void fraction is generally less than the simple cubic pattern but greater than the other two packing patterns (Chan and Tien, 1973). For a regular packing, each layer of the arrangement is isothermal normal to the direction of heat flow. Strictly speaking, this is true only when the heat flow is in one direction. In other words, this is a good assumption only when axial conduction predominates over the heat transfer in the radial direction (which is the case when the bed walls are adiabatic). Furthermore, each particle has an identical contact pattern with the particles immediately above and below itself as shown in Figure 3. So the resistance to heat flow of any one sphere is the same as the other spheres in the bed. For a given bed only those points in physical contact with a layer above or below are of interest to the analysis. This is because, as stated earlier, each layer of a regular packing is isothermal. These thermal contact points can be grouped into pairs. Each pair is composed of a heat supply region in the upper hemisphere and a heat removal region in the lower hemisphere. These two regions are diametrically opposite t o each other. In the case of the simple cubic there is only one such pair of contact point pairs. In this case Rdl can be determined using eq 10. In the case of the face-centered cubic there are three pairs of contact points, symmetrical about an axis of the sphere that is parallel to the axis of the bed, namely a-a', b-b', and c-c' as shown in Figure 3. The temperature difference and heat flux will be the same for each pair. The temperature difference a t each pair due to the heat flux of all the pairs can be obtained from the result of a single pair
Bed wail
Bed wall
I
Bed wll
II
I
t D +
I
Simple Cubic
igr
Body-Centered tUbic
- i D +
Face -Centered Cubic
Figure 3. Thermal contact patterns of different regular packing arrangements.
by the method of superposition. For example, the temperature difference a t b-b' and c-c' due to the heat flux a t a-a' alone is the same and can be obtained from eq 4 and is equal to m
(AT)b-bt
= (A7'),-, = 2 E AnPn(1/2) n=l
(14)
The temperature difference a t each pair due to the heat flux a t all the pairs is = 2eAn[Pn(1) n=l
+ 2Pn(1/2)1
(15)
Hence the resistance of the sphere is
Similarly, for the body-centered cubic arrangement, the expression for the resistance (Balakrishnan, 1976) is
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
43
81 1 0 008
1.02 20
-
N'cd
E: 5 5 x 10"
N/m2
CUEK
1
SIMRE
2
SCW CENTRED
CUSK
3
FACE CENTRED
CUBIC
3 2
Figure 4. Effect of packing geometry.
(The superscript of RcdL,refers to the packing pattern; i = 1 for simple cubic, i = 2 for body-centered cubic, and i = 3 for face-centered cubic.) Equations 16 and 17 may be compared with equation 10, which is the expression for the simple cubic arrangement. Numerical evaluation of eq 16 and 17 suggests that the resistances of face-centered and body-centered arrangements can be directly determined from the resistance of the simple cubic arrangement for the same contact spot dimensions by 1 Rcd2= - R (18) 4 cd
'
1
R,d3 = - R (19) 3 cd and Red' determined by eq 10. From the above it appears that the resistance in the simple cubic arrangement is three times the resistance of the face-centered and four times the body-centered cubic arrangements. This is true only if the dimensions of the contact spot in each of the three arrangements are identical. However, for a given sphere the contact size, xo, varies from arrangement to arrangement. If the pressure P is defined as half the weight of the bed divided by the cross-sectional area of the bed, xo for each packing pattern may be defined (Chan and Tien, 1973) by
where K' is a numerical factor (Table I) which expresses the vertical force P-A: in the direction normal to the contact spot and A: is the cross-sectional area of the bed per sphere (Table I). T o obtain the effective conductivity of the bed consider an elemental volume in the bed of cross-sectional area A,' and height N,'. N,' is the height per layer of spheres and its values for i = 1, 2, and 3 are also listed in Table I. Hence, as in eq 11 1 Nti ke=--
Red'
AsL
It may be noted that for i = 1, eq 21 corresponds identically with eq 11. This is because the simple cubic pattern was assumed (implicitly) in the derivation of eq 11. Furthermore, as before
-.-
hcd = 1 1 Rcdi A,'
Table I. Parameter Values of Packing Arrangements (Chan and Tien, 1973)
parameter
A si
K' Nt'
simple cubic, i=1
bodycentered cubic, i = 2
facecentered cubic, i = 3
4a ' 1 2a
2a2/3 116 22a/3
16a2/3 314 2a/3
Here, again, when i = 1, eq 22 and 12 are identical owing to the implicit assumption of simple cubic packing geometry in deriving eq 12. Figure 4 shows the effect of these three packing arrangements on hcd in nondimensional terms (Nusselt number) for various values of pressure P on the contact spots. In other words, for a particular bed material whose properties are listed in Figure 4,P is varied in eq 20 and three different values of Bo (= rc/a) are obtained-one for each packing arrangement for each value of P. From this Rc; is evaluated and hence hcd values are obtained using (22). The significance of increasing P is that it is equivalent to using a bed with a larger heightldiameter ratio. From the figure it can be seen that the face- and body-centered configurations yield larger Nud values than the simple cubic case. Actual beds with randomly packed spheres have a porosity between that of the simple cubic and face-centered cubic arrangements (Chan and Tien, 1973). Therefore, it can be expected that the actual N U c d will be in between that of the simple and face-centered cubic patterns. 3. Effect of Biot Modulus. The purpose of this section is to study the effect of the convective mode on the conduction. For simplicity, the simple cubic arrangement alone is considered. Equation 12 for computing hcd may be rewritten as
z
n=l
(2n - 1) + Bi
It is clearly seen that Bi (E h , , a / k , ) is an independent variable that affects hcd. To visually see the effect of Bi on hcd, Figure 5 was made, showing a plot of hcd vs. Bi. Property values used were those of 0.635-cm iron oxide spheres. As can be seen from the figure, Nucd increases steadily with Bi. A t Bi = 0, the Nucd value corresponds to the conductance of the bed when convective effects are absent.
44
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979 5 0635
crn
IRON OXIDE
0 1
0 2
SPHERES
-
4
3 -
N”cd
1
0
0 3
0 4
0 5
0 6
07
BI
Figure 5. Effect of Biot modulus on Nucd. 6
0 635 crn
IRON OXIDE SPHERES
NUc d
0
200
100
300
400
500
600
700
Figure 6. Effect of Reynolds number on hTu,d.
The increase of N U c d with Bi can be qualitatively explained as follows. Consider the sphere in Figure 1 (along with the coordinates of the analysis described therein). The sphere has a total amount of heat Qo = q0ar,2entering the contact spot at (a,O) and a quantity Q1 = q1rr,2 leaving the contact spot a t ( a , s ) . Let the rest of the heat Qo - Q1 = Qf = hfP.(4~a2)*T
h, is the convective heat transfer coefficient, which obviously must be a function of Reynolds number. Therefore, it might be informative to plot both N U , d and Bi as a function of Reynolds number. For the case of a packed bed subject to gas flow alone, the Biot modulus might be obtained as a function of gas Reynolds number using a literature correlation by (Balakrishnan and Pei, 1974)
be the fraction that leaves the sphere by convection. By definition
Bi = 0.016~--.[Ar,1°~25[Re,10~5
T=
1
area of sphere
T(a,O)d A / S
area of sphere
dA
If hfpis increased, for the same specified heat fluxes, Twill decrease. It is reasonable then to expect AT = [T(a,O)T(a,a)]also to decrease since, qualitatively, if the average temperature of the sphere decreases, the temperature of the “hot” contact spot T(a,O)can be expected to decrease more than the “cold” contact spot T(a,r). Now since (Qo + Q J / 2 hcd
=
( 4 ~ ’(AT) ) it is obvious that hcdwill increase with decreasing AT (caused by increasing hf, as indicated above). Therefore Nucdincreases with increasing Bi.
kf
ksa
Using this for 0.635-cm diameter iron oxide spheres, both Nucdand Bi have been plotted against Re, in Figure 6. It can be seen from the figure that the conductance increases with Reynolds number, rapidly at first and then levelling off. This shows a greater dependence of N U c d on Re, a t lower values of Re . In fact, Nud ultimately seems to reach an asymptotic varue, when further increases in Re, have no appreciable effect on NUc,+ This is to be expected, for the following reason. From Figure 5 it is apparent that N U , d varies almost linearly with Bi and therefore N U , d = mBi c (where m and c are constants). Furthermore, from eq 24, Bi = nRe,o.5 (where n is a constant) and hence N U c d = (man) Re:.5 + c , and this is the type of behavior exhibited by Nucdwith Re, in Figure 6. In comparing the
+
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
erage surface temperature T,) circumscribed with the diffusively reflective wall is expressed as (Hottel, 1954)
SPHERE 1
Q, = A1F12dT14 - T241
DlFTUSlMLY
(25)
where the overall exchange factor F12is =2 - 1) F,, 1
REFLECTlVE
(;
2-
CYLINDRICAL WALL
2 SPHERE
45
+
F12
2
where e is the emissivity of the surface of the spheres. In terms of the radiation heat transfer coefficient h,, eq 25 is approximately rewritten as Qr = A1 hr (TI - 7'2) (27)
Figure 7. Coordinates for radiant exchange between two spheres.
behavior of Nucdand Bi with Re, in Figure 6, it must be borne in mind that the ordinates for the two parameters are different. One other aspect to be mentioned here is that while the conductive and convective modes are interactive, hf, is independent of hcd. Consider the convective flux Qf = hb(4sa2).(tsdae - tfl& for hcd changes, tsurfae will change, and hence change Qf, but hfp will remain unchanged. Hence, while the conductive and convective fluxes are interactive, hfpitself is independent of hcd. One more point to be noted here is that an average convective heat transfer coefficient hfpis used in the model. However, a t the contact point the local heat transfer coefficient will vary owing to rarefied gas effect. But in view of the fact that the hfpused was obtained from experimental studies, it represents an average value which incorporates this variation. Moreover, this rarefied gas effect is limited to a very small region near the contact spot. Therefore the assumption of constant hfpis a good approximation of the situation. 4. Effect of Radiation. A t high temperatures, heat transfer between the particles can also take place by radiation. The radiant heat transfer between the particles can be analyzed as a parallel path to the conduction mode. It is assumed that the radiation takes place between the bottom hemisphere of one sphere to the upper hemisphere of a sphere immediately above it. It is further assumed that the spheres are circumscribed with a diffusively reflective cylindrical wall. This geometry is shown in Figure 7. The radiant heat exchange Q, between hemisphere 1 (average surface temperature T,) and hemisphere 2 (av-
0
400
800
If ( T , - T,) is small, (T14- T24)/(T1- T,) can be approximated by 4T, where T is the average temperature for radiation, T = ( T , + T2)/2. The average angle factor FI2 in (26) has been shown by Wakao and Kat0 (1969) to be 0.576. Therefore, by substituting the Stefan-Boltzmann constant u = 4.88 X (kcal/m2 h K4) and (26), the expression for h, becomes h, = 2 - - 0.264
kcal/m2 h K
e
or in SI units
(&y
h, = 2 0'227 J/m2 s K (29) - - 0.264 e h, and h d can be combined to obtain hrcd(the factor a / 4 is included since h, is based on the projected area of the spheres and hcdon the area of a face of the cube enclosing the sphere) hrcd
= hcd + hr
(i>
(30)
hrcdis plotted against T , the average system temperature in Figure 8. h d is plotted as Nurd, the dimensionless heat transfer coefficient of the radiation and conduction modes.
-
T (OK1
F i g u r e 8. Effect of high temperature on Nurcd.
(&y
1200
1600
2000
46
Ind. Eng. Chem.
Process Des. Dev., Vol. 18, No. 1, 1979
T h e figure shows, as expected, that up to about 400 K there is very little contribution by the radiation mode, but beyond that there is a significant contribution by radiation and this keeps increasing rapidly. This clearly indicates the importance of including the radiative effects in beds where temperatures above 400 K or so are expected. In fact a t a temperature of about 950 K the contribution by radiation is almost equal to that by conduction. It is therefore necessary that at temperatures higher than about 500 K a more detailed and sophisticated analysis of the radiation phenomena be used. Effects such as radiation between the spheres and the bed wall, spheres and the flowing medium, and the bed wall and the flowing medium would have to be considered. Such an analysis however is beyond the scope of the present investigation. Conclusions The following conclusions may be drawn from the present investigation. 1. The convective coefficient has a significant effect on the conduction mode. k, (or hcd)is found to increase with the Biot modulus, Bi. At Bi 0, k, obtained is the value for a bed without fluid flow and corresponds with the earlier results of Chan and Tien (1973). 2. T h e effective thermal conductivity, he, depends on the contact spot size. The smaller the contact size, the less is the effective conductance. 3. Depending on the packing geometry, the effective conductivity is different. For the simple cubic packing pattern, k, is less than the body-centered and face-centered cubic packing arrangements. 4. Radiation between the particles is a heat transfer mode parallel to the axial conduction. At about 900 K the magnitude of the two modes is about equal, but beyond this the radiation contribution increases sharply. Nomenclature a, radius of spheres in packed bed, m A,! constant in general solution to Laplace's equation, K A,", cross-sectional area of bed per sphere, m2 Al, area available for radiant exchange, m2 Bi,Biot modulus, (hf,a/k,), dimensionless c,, constant in solution to Laplace's equation [=P,-,(xO) P,,+l(!o)],dimensionless e, emissivity, dimensionless E , Young's modulus, N/m2 F, force acting on contact spot, N Flz, average angle factor for radiation, dimensionless F12,overall exchange factor, dimensionless h, heat transfer coefficient, J / m 2 s K k , ,thermal conductivity, J / m s K KL,numerical factor in eq 22, dimensionless n, order of Legendre polynomial
-
N t , bed height per layer of spheres, m Nucd,conduction Nusselt number = hd.2a/kf, dimensionless P, pressure acting on a cross section of the bed, N/m2 P,, Legendre polynomial of order n, dimensionless q, heat flux entering or leaving the sphere, J / s m2 Qf, heat leaving the sphere by convection, J / s Q, total heat transferred, J / s r, distance coordinate in conduction analysis, m rc, radius of contact spot, m Red*, resistance to heat transfer by conduction, s K / J Re,, particle Reynolds number, 2a.u,-pf/pf, dimensionless; Re, = Re,/(l - e) t, temperature, K T, defined as ( t - tf), K; T : averaged over entire sphere AT, temperature driving force for conduction, K uf, fluid velocity, m/s X, argument of Legendre polynomial [ = cos 81, dimensionless z , distance coordinate along axis of bed, m Greek Letters e, bed porosity, dimensionless 8, angle coordinate in conduction analysis, rad pf, fluid viscosity, N s/m2 u, Poisson's ratio, dimensionless u, Stefan-Boltzmann constant, J/m2 s K Subscripts cd, pertaining to conduction e, effective f, of fluid fp, fluid-particle or convective 0 , pertaining to contact spot r, radiative rcd, radiative-conductive 0, entering sphere 1, leaving sphere Superscripts i, packing geometry; i = 1 for simple cubic, i = 2 for bodycentered cubic and i = 3 for face-centered cubic Literature Cited Agro, W. B., Smith, J. M., Chem. Eng. Prog., 49, 443 (1953). Babkrishnan, A. R., Ph.D. Thesis, University of Waterloo, Waterloo. Ont., Canada, 1976. Balakrishnan. A. R., Pei, D. C. T., Ind. Eng. Chem., Process Des. Dev., 13, 441 (1974). Bahkrishnan, A. R., Pei, D. C. T., Ind. Eng. Chem., Process Des. Dev., preceding article in this issue, 1978. Bhattacharyya, D., Pei, D. C. T., Chem. Eng. Sci., 30, 293 (1975). Chan, C. K., Tien, C. L., J . Heat Transfer, 95, 302 (1973). Hottei, H. C., "Heat Transmission", 3rd ed,Chapter 4, McAdams, Ed., McGraw-Hill, New York, N.Y., 1954. Singer, E., Wiihelm, R. H., Chem. Eng. Prog., 46, 343 (1950). Wakao, N., Kato, K., J . Chem. Eng. Jpn., 2, 24 (1969). Yovanovich, M. M., ASME Paper 73-HT-43, ASME-AIChE Heat Transfer Conference, Atlanta, Ga., 1973.
Received f o r review February 3, 1977 Accepted July 6 , 1978
