Heat Transfer in Gas-Solid Packed Bed Systems. 2. The Conduction ...

40 Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
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Yovanovich, M. M., ASME Paper 73-HT-43, ASME-AIChE Heat Transfer
Conference, Atlanta, Ga., 1973.
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
Received for review February 3, 1977
Accepted July 6, 1978
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 con-
vective 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 at a higher temperature than
B. Gas, at a bulk temperature of tf, flows past the sphere
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
hfp is 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
= -ql
0, 5 0 5 R - Bo, r = a
A- 0, 5 0 I R, r = a (3)
where T = t - tf and 6'" = sin-' (rJa). The general solution
is
where x = cos 0.
0019-7882/79/1118-0040$01.00/0 @ 1978 American Chemical Society
(4)

-.-E c
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.
From the orthogonality properties of Legendre polynomials
it follows that
-2
.lPn2(x) dx = -.I__
2n + 1
Moreover, since xo is very close to unity (as Bo is small),
the following approximation may be made (Balakrishnan,
1976)
-2
L?;(X) dx = ~
2n + 1
and furthermore, by definition
L:P,(x) dx = -~
Equation 5 then reduces to
1 Cn
2n + 1 [Pn-,(X,) - Pn+1(~0)1 =
(8)
2 40 cn 41 (-cn) hfP
A, -*
2n + 1 k, 2n + 1 k, 271 + 1 ks
2
2n + 1
and, therefore, the solution to eq 12 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 =
cu 40 + 41
rrc2
2a .f
Rcdl = --
k, rrc2 n=l
d
Pzn-Axo) - PZn(X0)
(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
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
Therefore, the conduction heat transfer per layer of
spheres in the bed is
The corresponding Nusselt number is defined as
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 pa-
rameters. 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 at 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 (hcd in dimensionless form)
against the contact angle do for two different bed materials.
The following observations may be made. (1) The NUcd
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 conduc-
tivity) 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 at 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 con-
duction, and at 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
d
20
Figure 2. Effect of contact size.
1 IRON OXIDE SPHERES
D : 2 a I 0 635 cm
K : 0.61 JlmsK
5
0 004 008
CONTACT SIZE
There are several models (Agro and Smith, 1953; Singer
and Wilhelm, 1950; Wakao and Kato, 1969) in the liter-
ature on the effective conductivity of packed beds (con-
vective 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 to 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 tem-
perature difference at each pair due to the heat flux of all
the pairs can be obtained from the result of a single pair
0
2 STEEL SPHERES
D = 2a-0635 cm
K : 4327 JlrnsK
,012 016 .02
Bed wall Bed wail Bed wll
I t D + I I
I - i D +
I igr
Simple Cubic Body-Centered tUbic Face -Centered Cubic
Figure 3. Thermal contact patterns of different regular packing
arrangements.
by the method of superposition. For example, the tem-
perature difference at b-b' and c-c' due to the heat flux
at a-a' alone is the same and can be obtained from eq 4
and is equal to
(AT)b-bt = (A7'),-, = 2 E AnPn(1/2) (14)
The temperature difference at each pair due to the heat
flux at all the pairs is
= 2eAn[Pn(1) + 2Pn(1/2)1 (15)
n=l
Hence the resistance of the sphere is
Similarly, for the body-centered cubic arrangement, the
expression for the resistance (Balakrishnan, 1976) is
m
n=l

N'cd
20
Figure 4. Effect of packing geometry.
-
81 1 0 008
1.02
E: 5 5 x 10" N/m2
1 SIMRE CUEK
2 SCW CENTRED CUSK
3 FACE CENTRED CUBIC
(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 ' 4 cd
1
R,d3 = - R
3 cd
(18)
(19)
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).
To 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 iden-
tically with eq 11. This is because the simple cubic pattern
was assumed (implicitly) in the derivation of eq 11.
Furthermore, as before
1 1
hcd = -.-
Rcdi A,'
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979 43
Table I. Parameter Values of Packing Arrangements
(Chan and Tien, 1973)
simple body- face-
cubic, centered centered
parameter i = 1 cubic, i = 2 cubic, i = 3
A si 4a ' 2a2/3 16a2/3
K' 1 116 314
Nt' 2a 22a/3 2a/3
Here, again, when i = 1, eq 22 and 12 are identical owing
to the implicit assumption of simple cubic packing ge-
ometry in deriving eq 12.
Figure 4 shows the effect of these three packing ar-
rangements 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 NUcd
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
3
2
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. At 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
N”cd
5
4 -
3 -
Figure 5. Effect of Biot modulus on Nucd.
NUc d
6
1
0
0635 crn IRON OXIDE SPHERES
01 02 0 3 0 4 0 5 0 6 07
0 635 crn IRON OXIDE SPHERES
0 100 200 300
Figure 6. Effect of Reynolds number on hTu,d.
The increase of NUcd with Bi can be qualitatively ex-
plained 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,2 entering
the contact spot at (a,O) and a quantity Q1 = q1rr,2 leaving
the contact spot at (a,s). Let the rest of the heat
Qo - Q1 = Qf = hfP.(4~a2)*T
be the fraction that leaves the sphere by convection. By
definition
T= 1
area of sphere
T(a,O) d A/S dA
area of sphere
If hfp is 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
hcd =
(Qo + QJ/2
(4~’) (AT)
it is obvious that hcd will increase with decreasing AT
(caused by increasing hf, as indicated above). Therefore
Nucd increases with increasing Bi.
BI
400 500 600 700
h, is the convective heat transfer coefficient, which
obviously must be a function of Reynolds number.
Therefore, it might be informative to plot both NU,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)
kf
Bi = 0.016~--.[Ar,1°~25[Re,10~5
ksa
Using this for 0.635-cm diameter iron oxide spheres, both
Nucd and 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 NUcd on Re, at
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
NU,d varies almost linearly with Bi and therefore NU,d =
mBi + c (where m and c are constants). Furthermore, from
eq 24, Bi = nRe,o.5 (where n is a constant) and hence NUcd
= (man) Re:.5 + c, and this is the type of behavior ex-
hibited by Nucd with Re, in Figure 6. In comparing the

SPHERE 1
2
SPHERE 2
DlFTUSlMLY
REF LECTlVE
CYLINDRICAL WALL
Figure 7. Coordinates for radiant exchange between two spheres.
behavior of Nucd and 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, hfp itself is independent of hcd.
One more point to be noted here is that an average
convective heat transfer coefficient hfp is used in the model.
However, at the contact point the local heat transfer
coefficient will vary owing to rarefied gas effect. But in
view of the fact that the hfp used 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 hfp is a good
approximation of the situation.
4. Effect of Radiation. At 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-
Figure 8. Effect of high temperature on Nurcd.
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979 45
erage surface temperature T,) circumscribed with the
diffusively reflective wall is expressed as (Hottel, 1954)
Q, = A1F12dT14 - T241
(25)
where the overall exchange factor F12 is
2- = 2 (; - 1) + F,, 1
F12
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)
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
(&y h, = 2
kcal/m2 h K
- - 0.264
e
or in SI units
h, = 2 0'227
- - 0.264
e
(&y J/m2 s K (29)
h, and hd can be combined to obtain hrcd (the factor a/4
is included since h, is based on the projected area of the
spheres and hcd on the area of a face of the cube enclosing
the sphere)
hrcd = hcd + hr (i> (30)
hrcd is plotted against T, the average system temperature
in Figure 8. hd is plotted as Nurd, the dimensionless heat
transfer coefficient of the radiation and conduction modes.
0 400 800 - 1200 1600 2000
T (OK1

46 Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 1, 1979
The 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 at 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. The 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/m2 s K
k,, thermal conductivity, J/m s K
KL, numerical factor in eq 22, dimensionless
n, order of Legendre polynomial
Nt, 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
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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.
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Received for review February 3, 1977
Accepted July 6, 1978