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Structural characteristics of packed beds of low aspect ratio
Shorter Communications Acknowledgement-Support from the Texas ERAP. No. 003604-019, is acknowledged. RUBEN
D. COHEN
Department of Mechanical Engineering and Materials Science Rice University Houston, TX 77251, U.S.A.
Chnnical En&wring Science, Printed in Great Britain.
Vol.
47, No.
8, pp.
2105
grant
REFERENCES
Eid. K., Gourdon, C., Casamatta, G. and Muratet, G., 1991, Drop breakage in a pulsed sieve-plate column. Chem. Engng Sci. 46, 1595-1608. Prabhakar, A., Sriniketan, G. and Varma, Y. B. G., 1988, Dispersed phase holdup and drop size distribution in pulsed plate columns. Can. J. them. Engng 66, 232-240.
2105S2109,1992.
CQO-2509p2 s5.00 + 0.00 0 1992 Pergamon Press LtcI
Structural characteristics of packed beds of low aspect ratio (Received 30 April 1991; accepted for publication 12 December INTRODUCTION
zz = z,
Hydrodynamics, and mass and heat transfer in a packed bed depend on the structure of the bed, which is in turn determined by the location of the particles in the bed. The location is prescribed by the confining nature of the wall of the column, particularly in the region close to the wall. For randomly packed beds of aspect ratio (ratio of tube diameter to particle diameter) greater than two, determining the exact locations is not possible, although the first two layers of the particles from the wall are well-ordered, because the mechanism of random packing of even the simplest form of particles, namely spheres, is not yet understood. Investigations should therefore aim to analyze relatively simple systems; beds of equal-sized spheres of aspect ratio between one and two constitute such systems. These beds have been the subject of considerable research work, [e.g. Scott et al. (1974), Hsiang and Haynes (1977), Dixon and Creswell (1979). Dixon et al. (1984). Melanson and Dixon (1985), Ahn et al. (1986). Cui et al. (19901, Johnson and Kapner (1990) and Tsotsas and Schliinder (1990)]. An analysis of the structure of the bed has hardly ever been reported. The purpose of the work is to fill this gap by analyzing the location of the particles and then to derive radial profiles of void fraction of specific lateral surface area in beds of low aspect ratio (1 Q a < 2) packed with equal-sized spheres. LOCATION
OF PARTICLES
Beds of aspect ratio 2 When spheres of equal size are dropped one by one in a cylindrical column whose diameter is twice that of the particles and allowed to settle in stable positions, the first two spheres stand on t.he floor and touch the wall. Their centers will be at the same height and are at a distance of rp from the wall. The plane containing the line joining the centers of the two spheres is perpendicular to the axis of the column. The next two spheres will occupy positions such that they rest on the first two and touch the wall. Successive layers of spheres rest on the preceding ones and touch the wall [see Fig. l(a)]. By using the conditions corresponding to such stable positions (that is, each sphere in a layer touches the wall and rests on the two spheres in the preceding layer), we canshow that the bed consists of spheres lying in pairs with centers at locations given by the relations x2 = -x1
(1)
Y2 =
(2)
-YY,
1991)
x i+i
-xi
=
Y,+, = z*+,
=
Z(
=
-Y, =
(3)
= rp -
y*_,
(4)
= xi-2 zj__2
%k -JL
+
for i = 3,5.7..
(5)
.
(6)
or, alternatively. in polar coordinates rl =
r2
=
. . . =
(7)
p-p
ei+ 1 = e, + 7t
eg+z= ei + for i = 1, 3, 5,
(8)
z2 ...
where the coordinate system is chosen such that the z-axis coincides with the axis of the column, and the origin lies on the base of the column. The line through the centers of any layer of spheres is perpendicular (when projected on the x-y plane) to that through the centers of the adjacent pairs of spheres. Centers of the spheres in any two successive layers form the comers of a regular tetrahedron with each side of length d,, as shown in the upper part of Fig. l(a). Beds of aspect ratio 1 d a 4 2 For beds of aspect ratio 1 d a < lp, the spheres are arranged one over the other, with each one touching the wall [as shown in Fig. l(b) and (c)]. The centers of all the spheres lie at a distance of r, from the wall of the column. Considering any two spheres, we can show that the stable position of the second sphere (corresponding to minimum z) is given by XI = -X,_,
(10)
Y, = -YYt-1
(11)
zi = =<_a +fd,
(12)
fori=2,3,4,... or, alternatively. in polar coordinates t-1
-
r* = . . . = b = r&a
-
1)
(13)
Shorter Communications particle. The values xt and y, can be chosen arbitrarily without losing any generality, since by suitable rotation of the coordinate system all choices can be made equal. Also the x- and y-coordinates of the center of each particle, and hence the structures of beds with aspect ratio (a) 2 repeat for every two layers of spheres, (b) between 1 and 19 repeat for every two spheres, and (c) between lg and 2 repeat for every four spheres.
lh* lfl#
h*
That is, the locations of particles show periodic&y. The characteristics such as the number of spheres per unit height, overall void fraction, void fraction profile and lateral surface area profile of such regions of the bed are therefore the same and are equal to the corresponding characteristics of the overall bed. We shall call such sections of the bed, i.e. sections whose characteristics are the same as those of the complete bed, unit cells. The bed may then be considered to be constituted by a number of such unit cells. As will be shown in the next section, the height of a unit cell need not correspond to the height that shows periodicity in the coordinates of the particles.
hu
Ib)
Ia)
h'
Id)
Fig. 1. Structure of low aspect ratio beds packed with equal(c) (I = 19, (b) a = 1.5, spheres; (a) a = 2, (d) a = 1.95.
&Cd
e,=e,_, +n for i = 2, 3,4,
(14)
..._
For the aspect ratio of 19, the two successive spheres are so located that the horizontal plane through the center of one sphere is tangential to the other sphere [Fig. l(c)]. For aspect ratios beyond this, the plane intersects the other sphere [Fig. l(d)]. For such systems, the location of the centers of the spheres can be shown to be given by xi=
-xi_1
(15)
Yi=
-YY,-1
(16)
zi = It--l +fd,
(17)
for i = 2,4,6, . . . (18)
x1+.%= x1
(19)
Yif4 = Y, for i = 1,2. 3,.
z
i
+zi--2 2
=zi-1
Y3 =
+LJd,
(20)
(21)
x1 ~1 ,/(2b*b,
b2
-
bf)
(24)
Beds of aspect ratio lq < a < 2 For beds with aspect ratio in this region, the height of the unit cell is the vertical distance between the horizontal planes passing through the centers of a given sphere and the third sphere from it, as shown in Fig. l(d): h, = f(S+
9).
(25)
(22)
These equations show that the beds may be considered to be formed in groups of four spheres. UNIT
Beds of aspect ratio 1 Q a < 19 A simiiar analysis on beds with aspect ratio in this region has shown that unit cells with a height equal to the vertical distance between the planes passing through centers of any two successive spheres [see Fig. l(b) and (c)] will have the same characteristics as the overall bed, irrespective of the location of the base of the unit cell. Therefore, h. =f:
b2 - bl - YJYI
y,(b’ - b,) +
Beds of aspect ratio 2 Consider sections of the bed with different bases and different heights. Radial void fraction profiles estimated as per the methods discussed later indicate that sections with a base at any arbitrary position of the bed give the same characteristics as the overall bed as long as the height is an integer multiple of h*/2. Sections with other heights give marked deviations. It is therefore concluded that the height of an appropriate unit cell for beds of aspect ratio two is the vertical distance between the planes passing through the centers of any two successive layers of spheres [see Fig. l(a)], i.e.
..
for i = 3, 5.7. . . . X3 =
HEIGHT OF A UNTT CELL The height of a unit cell is an important characteristic that determines the construction and therefore the structure of the complete bed.
CELLS
From the above discussions we note that, for a given aspect ratio in the range 1 < n < 2, the location of all the particles can be estimated relative to the location (x- and y-coordinates, or r and 0 in polar coordinates) of the first
NUMBER
OF SPHERES PER UNIT HEIGHT IN A UNIT CELL
Another parameter of interest in describing the structure and the other characteristics of the bed is N.. the number of spheres per unit height (Govindarao and Froment, 1986). This is a measure of the solid volume contained in a unit cell. We have for a = 2 N-=2*
(26)
Shorter Communications
2107
Rearranging eq. (29). we have
forlda619 N_
!
=
NW=---.
Nl
2
(27)
f
RADIAL VOID FRACTION PROFKLES Void fraction is a volume-averaged property. We can define the void fraction at any radial distance from the wall in terms of a concentric cylindrical channel (CCC) with an arbitrary thickness which is chosen such that the cylindrical surface through the radial position lies exactly at the center of the CCC. This is a very convenient and useful variation of the division into a number of equal-sized CCCs suggested by Govindarao and Froment (1986). Thus, considering a CCC of a dimensionless thickness Ar* in the unit cell with its central surface at r* from the wall, we have for the void fraction in this CCC
OVERALL VOID PRAClXON The overall void fraction E,, is given by volume
of solids in the unit cell
volume
of the unit cell
= 1 -$N;. Combining
eqs (26)+28)
with eq. (29), we have
for a = 2
E,r = 1 E,, = 1 - $
= 0.5286
(30)
sum of volumes
of the sohere segments
in the CCC
volume of the CCC
forl&a
(34)
2
&o=l-andfor
(33)
Thus, N./a2 attains a constant value of 0.915 as the aspect ratio increases beyond 14.
4
f+s
&o=l-
= 1.5(1 - Eo).
(31)
3a’f
lp
8
Cap with radius R::
3a2(f + 8)’
The variation of e,, with the aspect ratio as computed above is compared in Fig. 2 with experimental data of Scott et al. (1974), Hsiang and Haynes (1977), and McGreavy et al. (1986). The agreement is very good. The overall void fraction starting from a value of 0.33 corresponding to a = 1, increases with u, reaching a peak value for a = 1.67 and then decreases to a value of 0.53 for a = 2. There is a sharp shift at a = 19. For aspect ratios beyond a = 2, z,, asymptotically approaches a constant value of 0.39 (McGreavy et al., 1986). The peak in the curve corresponds to the aspect ratio of 1.67, at which the solid fraction in the unit cell (bed) is minimum. Note that so can be determined experimentally for beds of all aspect ratios. However, N. can be calculated rigorously by the methods discussed in the above section only for beds of aspect ratio up to two. For larger beds, N. has to be approximated by methods such as the one suggested by Govindarao and Froment (1986). For such large aspect ratio beds, eq. (29) shows that N. can be more conveniently estimated from the experimental value of eO.
1 .oo
1
.5 _ 0.75 -I
ti
e .?z0.50 9
-
dota from
Z o-25 b
+ q
d
0.00
,
-
a
.oo
1
Scott et a~.
1974)
Hsiang and 4 McCreavy et i?rfsg
eq. 29
The normalized volume of a sphere segment Y* lying in the CCC is computed by using the following equations, which take the curved boundaries into account as per the method of Govindarao et al. (1990):
$1977) 6)
1.i5 1.50 l.iS Aspect ratio
,
2.60
Fig. 2. Comparison of predicted values of overall void fraction with reported experimental data.
s P
y--R;
[s arcsin fi
+ ,/Fj]
dy+
Slice with radii Rt and Rf : The volume is computed as the difference between the volumes of the caps with radii R: and Rf . Rod-like segment with radius
V&
=
‘2 x
Rf :
[s arcsin (fi)
+ ,/-I
dy*.
(36)
Annular ring with radii RF and Rz: The volume is computed as the difference between the volumes of the rod-like segments with radii R: and RF. The void fractions are estimated for different values of r* from eq. (34) using several values of A+, ranging from l/2000 to l/8 [or in other words, considering the parameter m of Govindarao and Froment (1986), in the range 1000 to 47. At any given radial distance, there is practically no difference between the different values of the void fraction, the largest difference being only of the order of 0.002. All further estimates are therefore made taking Ar* as l/200, corresponding to m = 100. Figure 3 gives the radial void fraction profiles in beds of different aspect ratios where the experimental profiles reported by Benenati and Brosilow (1962) for a = 5.6 and 14.1 are also included for comparison purposes. The profiles in low aspect ratio beds, 1 < 0 G 2, are significantly d&rent from those in high aspect ratio beds. While the latter profiles show damped oscillations, the void fraction in the bed with a = 2 starts from unity at the wall, falls to a minimum of 0.19 at a distance of about O&M, from the wall, and then r&s to reach unity at the axis of the bed. At the axis, the spheres have only point contacts [see Fig. l(a)] just as at the wall, and hence the void fraction is unity. The profiies for 1 < a < 18 show a continuous fall in the void fraction from a value of unity near the wall to zero
Shorter Communications
6
0.00
0.25
0.00
Distance
0.50 from
0.75
1.00
0
woll/(d,/2)
Distance
Fig. 3. Radial void fraction profiles in beds of aspect ratio 1.5, 1.95, 2, 5.6 and 14.1; data for last two beds are from Benenati and Brosilow (1962).
towards the axis. In these beds, the region close to the axis is filled entirely with solids. For beds with l$ -=zLI -z 2 there is again a continuous decrease in the void fraction from unity at the wall to a fairly low value towards the axis, and then there is a slight increase followed by another decrease.
from
Fig. 4. Radial specific lateral surface area profiles in beds of aspect ratio 1.5, 1.95, 2 and 14.1.
the wall). In lower aspect ratio beds (a -z 2), the profile falls steeply in the region near the wall and then approaches a very low value near the axis. CONCLUSIONS Three distinct types of behavior
RADIAL SPECIFIC LATERAL SURFACE AREA PROFILES In analyzing the flow distribution in packed beds, knowledge of radial variation of the lateral surface area is also of interest in addition to information on radial variation of the void fraction, particularly for the estimation of the hydraulic radius. The specific lateral surface area in a CCC is given by
4. =
sum of the normalized
volumes
Cap with radius R::
Ir’ Sx,+Y
s VP--R:
[arcsin -1
dy* + 2rg(r,* -
k*). (38)
Slice with radii Rt and RP : The lateral surface area is computed as the difference between the lateral surface areas of the caps with radii Rf and Rz. Rod-like
segment with radius RF:
s W**R:
Y. - R:
Annular
[arcsin ,&)I
dy*.
(39)
ring with radii R: and RF: The lateral surface area is computed
as the difference between the lateral surface areas of the rod-like segments with radii RF and R?. Figure 4 shows the variation of the specific lateral surface area, q,.. with distance from the wall in beds of different aspect ratios. Again, the surface area protiles in beds of low aspect ratio are significantly different from those in beds of high aspect ratio; the former do not show oscillations. In beds of aspect ratio two, q+ falls steeply from a high value near the wall (or at the axis) as we move towards the axis (or
for
The structure of the bed shows periodicity, and may therefore be described in terms of unit cells, i.e. sections whose characteristics are the same as those of the overall bed. The
of the sphere segments in the CCC
The lateral surface area of each of the four types of segments we may encounter in a CCC of a low aspect ratio bed is evaluated as follows:
are observed-ne
beds of aspect ratio 2, the second for beds of aspect ratio between 1 and I@, and the third for beds of aspect ratio between 18 and 2.
lateral surface areas of the sphere segments in the CCC
sum of the normalized
woll/(d,/2)
. (37)
i
height of such a unit cell is seen to be half of the height of the bed which shows periodic&y in the location of the particles, and is independent of the position of the base of the unit cell. This also indicates the complete reproducibility of the packing of low aspect ratio beds, which is of particular advantage in many applications. The various expressions derived here may be directly used to obtain a rigorous simulation of flow, and mass and heat transfer in fixed-bed systems of low aspect ratio and should be of considerable value in the design and operation of industrial packed beds, particularly those involving high rates of heat and mass transfer, and in the more realistic interpretation of laboratory and pilot-plant data from such systems. Also, the possibility of this rigour makes it expedient to use the low aspect ratio beds as laboratory multiphase reactors. VENNETI M. H. GOVINDARAO KANDIRAJU V. S. RAMRAO AMMAVAJJALA V. S. RAO
Department of Chemical Engineering lndian Institute of Science Bangalore 560012, India NOTATION aspect ratio defined by eq. (13) [=d,2--(zs-z*)*]
Shorter Communications column diameter particle diameter (=
k* N. 4P r ‘P r* z* R:,
R4
JK=z,
(= J2 + 20 - a*) height of a unit cell, normalized with dp height at which coordinate periodicity occurs, normalized with dp [=(rz’ - R:’ + w*‘)/~w*] number of spheres per unit height in a unit cell specific lateral surface area at r* r-coordinate of the sphere particle radius dimensionless radial coordinate (= r/d,) thickness of a CCC dimensionless thickness of a CCC ( = Arid,) radius of cylinder used in defining cap, slice, rodlike or annular ring segment of a sphere, normllized with d,
(= r;* - y*q
lateral surface area of a sphere segment, normalized with the lateral surface area of the sphere [= E:+ - (y* - w+P] volume of a sphere segment, normalized with the volume of the sphere horizontal distance between the axis of the column and the center of the sphere, normalized with
t V* W*
d, x-coordinate of the center of the ith sphere y-coordinate of the center of the ith sphere dimensionless Y-coordinate ( = y/d,) z-coordinate of the center of the ith sphere
xi
Yl Y’ =i
J., Zoulalian,
Chemica! Engineering Science, Printed in Great Britain.
Mass Transfer 28, 383-394.
A. and Smith, J. M.,
Vol.
47, No.
Engng Sci. 32, 678-681. Johnson, G. W. and Kapner, R. S., 1990, The dependence of axial dispersion on non-uniform flows in beds of uniform packing. Chem. Engng Sci. 45, 3329-3339. McGreavy, C., Foumeny, E. A. and Javed, K. H., 1986, Characterization of transport properties for fixed bed in terms of local bed structure and flow distribution. Chem
Scott, D. S., Lee, W. and Papa, J., 1974, The measurement of transport coefficients in gassolid heterogeneous reactions. Chem. Engng Sci. 29, 2155-2167. Tsotsas, E. and Schliinder, E. U., 1990, Measurements of mass transfer between particles and gas in packed tubes at very low tube to particle diameter ratios. W&me- und
void fraction at r* &coordinate of the center of the ith sphere-
3.
J. 25, 663-676.
Engng Sci. 41, 787-797.
REFERENCES Ahn,
A.I.Ch.E.
Dixon, A. G., Dicostanzo, M. A. and Saucy, B. A., 1984, Fluid-phase radial transport in packed beds of low tubeto-particle diameter ratio. Int. J. Heat Mass Transfer 27, 1701-1713. Govindarao, V. M. H. and Froment, G. F., 1986, Voidage profiles in packed beds of spheres. Chem. Engng Sci. 41, 533-539. Govindarao, V. M. H., Manjunath, S., Rao, A. V. S. and Ramrao, K. V. S., 1990, Voidage profile in packed beds by multi-channel model: effects of curvature of the channels. Chem. Engng Sci. 45, 362-364. Hsiang, T. C. S. and Haynes, H. W., 1977, Axial dispersion in small diameter beds of large, spherical particles. Chem
Melanson, M. M. and Dixon, A. G., 1985, Solid conduction in low d,/d, beds of spheres, pellets and rings. Int. J. iYeut
Greek letters overall void fraction e0 er* 01
2109
dispersion in packed beds with large wall effect. A.I.Ch.E. J. 32, 170-174. Benenati, R. F. and Brosilow, C. B., 1962, Void fraction distribution in beds of spheres. A.I.Ch.E. J. 8, 359-361. Cui, L. C., Schweich, D. and Villermaux, J.. 1990, Consequence of flow nonuniformity on the measurement of effective diffusivity. A.I.Ch.E. J. 36, 86-92. Dixon, A. G. and Cresswell, D. L., 1979, Theoretical prediction of effective heat transfer parameters in packed beds.
8, pp.
ZLO!-2113.
1986,
Axial
Stofibertragung
25, 245-256.
1992. 0
OOC%2509/92 SSM 1992 Persamon h
+ 0.00 Ltd
Relationships between surface diiusivity and pore diffusivity in Fatch adsorption: measurements of the diffusivities for n-hexane and n-decane in 5 A molecular sieves (First received 27 May
1991; accepted
INTRODUCMON Various models have been proposed for diffusion in an adsorbent [e.g. Costa and Rodrigues (1985) and Do and Rice (1987)]. Among them, the two most frequently used models are surface diffusion and pore diffusion. If the adsorption isotherm is of the Henry type, the difhtsivities for these two models are easily converted, one from the other. In general, the diffusivity is considered to be a function of intrapellet concentration (Ruthven, 1984). The diffusivities (average, not concentration-dependent) for the two models are easily determined from a batch adsorption experiment. Our interest is whether these two diffusivities may always be converted reciprocally, or under what conditions of batch adsorption experiment the conversion may be made.
in revised form 6 December
1991)
The system considered comprises spherical adsorbent particles (of weight W,, radius R, density p$ exposed to a mixture of sorbate and inert fluid in a mixing vessel (of volume V). The overall adsorption process is assumed to be controlled by the intraparticle mass transfer, and not by the particle-fluid mass transfer. Also, the surface diffusivity and the pore diffusivity are assumed to be constant, not depending on adsorbed and solute concentrations. respectively. The adsorption isotherm is assumed to be of the Langmuir type:
(1)
D. COHEN
Department of Mechanical Engineering and Materials Science Rice University Houston, TX 77251, U.S.A.
Chnnical En&wring Science, Printed in Great Britain.
Vol.
47, No.
8, pp.
2105
grant
REFERENCES
Eid. K., Gourdon, C., Casamatta, G. and Muratet, G., 1991, Drop breakage in a pulsed sieve-plate column. Chem. Engng Sci. 46, 1595-1608. Prabhakar, A., Sriniketan, G. and Varma, Y. B. G., 1988, Dispersed phase holdup and drop size distribution in pulsed plate columns. Can. J. them. Engng 66, 232-240.
2105S2109,1992.
CQO-2509p2 s5.00 + 0.00 0 1992 Pergamon Press LtcI
Structural characteristics of packed beds of low aspect ratio (Received 30 April 1991; accepted for publication 12 December INTRODUCTION
zz = z,
Hydrodynamics, and mass and heat transfer in a packed bed depend on the structure of the bed, which is in turn determined by the location of the particles in the bed. The location is prescribed by the confining nature of the wall of the column, particularly in the region close to the wall. For randomly packed beds of aspect ratio (ratio of tube diameter to particle diameter) greater than two, determining the exact locations is not possible, although the first two layers of the particles from the wall are well-ordered, because the mechanism of random packing of even the simplest form of particles, namely spheres, is not yet understood. Investigations should therefore aim to analyze relatively simple systems; beds of equal-sized spheres of aspect ratio between one and two constitute such systems. These beds have been the subject of considerable research work, [e.g. Scott et al. (1974), Hsiang and Haynes (1977), Dixon and Creswell (1979). Dixon et al. (1984). Melanson and Dixon (1985), Ahn et al. (1986). Cui et al. (19901, Johnson and Kapner (1990) and Tsotsas and Schliinder (1990)]. An analysis of the structure of the bed has hardly ever been reported. The purpose of the work is to fill this gap by analyzing the location of the particles and then to derive radial profiles of void fraction of specific lateral surface area in beds of low aspect ratio (1 Q a < 2) packed with equal-sized spheres. LOCATION
OF PARTICLES
Beds of aspect ratio 2 When spheres of equal size are dropped one by one in a cylindrical column whose diameter is twice that of the particles and allowed to settle in stable positions, the first two spheres stand on t.he floor and touch the wall. Their centers will be at the same height and are at a distance of rp from the wall. The plane containing the line joining the centers of the two spheres is perpendicular to the axis of the column. The next two spheres will occupy positions such that they rest on the first two and touch the wall. Successive layers of spheres rest on the preceding ones and touch the wall [see Fig. l(a)]. By using the conditions corresponding to such stable positions (that is, each sphere in a layer touches the wall and rests on the two spheres in the preceding layer), we canshow that the bed consists of spheres lying in pairs with centers at locations given by the relations x2 = -x1
(1)
Y2 =
(2)
-YY,
1991)
x i+i
-xi
=
Y,+, = z*+,
=
Z(
=
-Y, =
(3)
= rp -
y*_,
(4)
= xi-2 zj__2
%k -JL
+
for i = 3,5.7..
(5)
.
(6)
or, alternatively. in polar coordinates rl =
r2
=
. . . =
(7)
p-p
ei+ 1 = e, + 7t
eg+z= ei + for i = 1, 3, 5,
(8)
z2 ...
where the coordinate system is chosen such that the z-axis coincides with the axis of the column, and the origin lies on the base of the column. The line through the centers of any layer of spheres is perpendicular (when projected on the x-y plane) to that through the centers of the adjacent pairs of spheres. Centers of the spheres in any two successive layers form the comers of a regular tetrahedron with each side of length d,, as shown in the upper part of Fig. l(a). Beds of aspect ratio 1 d a 4 2 For beds of aspect ratio 1 d a < lp, the spheres are arranged one over the other, with each one touching the wall [as shown in Fig. l(b) and (c)]. The centers of all the spheres lie at a distance of r, from the wall of the column. Considering any two spheres, we can show that the stable position of the second sphere (corresponding to minimum z) is given by XI = -X,_,
(10)
Y, = -YYt-1
(11)
zi = =<_a +fd,
(12)
fori=2,3,4,... or, alternatively. in polar coordinates t-1
-
r* = . . . = b = r&a
-
1)
(13)
Shorter Communications particle. The values xt and y, can be chosen arbitrarily without losing any generality, since by suitable rotation of the coordinate system all choices can be made equal. Also the x- and y-coordinates of the center of each particle, and hence the structures of beds with aspect ratio (a) 2 repeat for every two layers of spheres, (b) between 1 and 19 repeat for every two spheres, and (c) between lg and 2 repeat for every four spheres.
lh* lfl#
h*
That is, the locations of particles show periodic&y. The characteristics such as the number of spheres per unit height, overall void fraction, void fraction profile and lateral surface area profile of such regions of the bed are therefore the same and are equal to the corresponding characteristics of the overall bed. We shall call such sections of the bed, i.e. sections whose characteristics are the same as those of the complete bed, unit cells. The bed may then be considered to be constituted by a number of such unit cells. As will be shown in the next section, the height of a unit cell need not correspond to the height that shows periodicity in the coordinates of the particles.
hu
Ib)
Ia)
h'
Id)
Fig. 1. Structure of low aspect ratio beds packed with equal(c) (I = 19, (b) a = 1.5, spheres; (a) a = 2, (d) a = 1.95.
&Cd
e,=e,_, +n for i = 2, 3,4,
(14)
..._
For the aspect ratio of 19, the two successive spheres are so located that the horizontal plane through the center of one sphere is tangential to the other sphere [Fig. l(c)]. For aspect ratios beyond this, the plane intersects the other sphere [Fig. l(d)]. For such systems, the location of the centers of the spheres can be shown to be given by xi=
-xi_1
(15)
Yi=
-YY,-1
(16)
zi = It--l +fd,
(17)
for i = 2,4,6, . . . (18)
x1+.%= x1
(19)
Yif4 = Y, for i = 1,2. 3,.
z
i
+zi--2 2
=zi-1
Y3 =
+LJd,
(20)
(21)
x1 ~1 ,/(2b*b,
b2
-
bf)
(24)
Beds of aspect ratio lq < a < 2 For beds with aspect ratio in this region, the height of the unit cell is the vertical distance between the horizontal planes passing through the centers of a given sphere and the third sphere from it, as shown in Fig. l(d): h, = f(S+
9).
(25)
(22)
These equations show that the beds may be considered to be formed in groups of four spheres. UNIT
Beds of aspect ratio 1 Q a < 19 A simiiar analysis on beds with aspect ratio in this region has shown that unit cells with a height equal to the vertical distance between the planes passing through centers of any two successive spheres [see Fig. l(b) and (c)] will have the same characteristics as the overall bed, irrespective of the location of the base of the unit cell. Therefore, h. =f:
b2 - bl - YJYI
y,(b’ - b,) +
Beds of aspect ratio 2 Consider sections of the bed with different bases and different heights. Radial void fraction profiles estimated as per the methods discussed later indicate that sections with a base at any arbitrary position of the bed give the same characteristics as the overall bed as long as the height is an integer multiple of h*/2. Sections with other heights give marked deviations. It is therefore concluded that the height of an appropriate unit cell for beds of aspect ratio two is the vertical distance between the planes passing through the centers of any two successive layers of spheres [see Fig. l(a)], i.e.
..
for i = 3, 5.7. . . . X3 =
HEIGHT OF A UNTT CELL The height of a unit cell is an important characteristic that determines the construction and therefore the structure of the complete bed.
CELLS
From the above discussions we note that, for a given aspect ratio in the range 1 < n < 2, the location of all the particles can be estimated relative to the location (x- and y-coordinates, or r and 0 in polar coordinates) of the first
NUMBER
OF SPHERES PER UNIT HEIGHT IN A UNIT CELL
Another parameter of interest in describing the structure and the other characteristics of the bed is N.. the number of spheres per unit height (Govindarao and Froment, 1986). This is a measure of the solid volume contained in a unit cell. We have for a = 2 N-=2*
(26)
Shorter Communications
2107
Rearranging eq. (29). we have
forlda619 N_
!
=
NW=---.
Nl
2
(27)
f
RADIAL VOID FRACTION PROFKLES Void fraction is a volume-averaged property. We can define the void fraction at any radial distance from the wall in terms of a concentric cylindrical channel (CCC) with an arbitrary thickness which is chosen such that the cylindrical surface through the radial position lies exactly at the center of the CCC. This is a very convenient and useful variation of the division into a number of equal-sized CCCs suggested by Govindarao and Froment (1986). Thus, considering a CCC of a dimensionless thickness Ar* in the unit cell with its central surface at r* from the wall, we have for the void fraction in this CCC
OVERALL VOID PRAClXON The overall void fraction E,, is given by volume
of solids in the unit cell
volume
of the unit cell
= 1 -$N;. Combining
eqs (26)+28)
with eq. (29), we have
for a = 2
E,r = 1 E,, = 1 - $
= 0.5286
(30)
sum of volumes
of the sohere segments
in the CCC
volume of the CCC
forl&a
(34)
2
&o=l-andfor
(33)
Thus, N./a2 attains a constant value of 0.915 as the aspect ratio increases beyond 14.
4
f+s
&o=l-
= 1.5(1 - Eo).
(31)
3a’f
lp
8
Cap with radius R::
3a2(f + 8)’
The variation of e,, with the aspect ratio as computed above is compared in Fig. 2 with experimental data of Scott et al. (1974), Hsiang and Haynes (1977), and McGreavy et al. (1986). The agreement is very good. The overall void fraction starting from a value of 0.33 corresponding to a = 1, increases with u, reaching a peak value for a = 1.67 and then decreases to a value of 0.53 for a = 2. There is a sharp shift at a = 19. For aspect ratios beyond a = 2, z,, asymptotically approaches a constant value of 0.39 (McGreavy et al., 1986). The peak in the curve corresponds to the aspect ratio of 1.67, at which the solid fraction in the unit cell (bed) is minimum. Note that so can be determined experimentally for beds of all aspect ratios. However, N. can be calculated rigorously by the methods discussed in the above section only for beds of aspect ratio up to two. For larger beds, N. has to be approximated by methods such as the one suggested by Govindarao and Froment (1986). For such large aspect ratio beds, eq. (29) shows that N. can be more conveniently estimated from the experimental value of eO.
1 .oo
1
.5 _ 0.75 -I
ti
e .?z0.50 9
-
dota from
Z o-25 b
+ q
d
0.00
,
-
a
.oo
1
Scott et a~.
1974)
Hsiang and 4 McCreavy et i?rfsg
eq. 29
The normalized volume of a sphere segment Y* lying in the CCC is computed by using the following equations, which take the curved boundaries into account as per the method of Govindarao et al. (1990):
$1977) 6)
1.i5 1.50 l.iS Aspect ratio
,
2.60
Fig. 2. Comparison of predicted values of overall void fraction with reported experimental data.
s P
y--R;
[s arcsin fi
+ ,/Fj]
dy+
Slice with radii Rt and Rf : The volume is computed as the difference between the volumes of the caps with radii R: and Rf . Rod-like segment with radius
V&
=
‘2 x
Rf :
[s arcsin (fi)
+ ,/-I
dy*.
(36)
Annular ring with radii RF and Rz: The volume is computed as the difference between the volumes of the rod-like segments with radii R: and RF. The void fractions are estimated for different values of r* from eq. (34) using several values of A+, ranging from l/2000 to l/8 [or in other words, considering the parameter m of Govindarao and Froment (1986), in the range 1000 to 47. At any given radial distance, there is practically no difference between the different values of the void fraction, the largest difference being only of the order of 0.002. All further estimates are therefore made taking Ar* as l/200, corresponding to m = 100. Figure 3 gives the radial void fraction profiles in beds of different aspect ratios where the experimental profiles reported by Benenati and Brosilow (1962) for a = 5.6 and 14.1 are also included for comparison purposes. The profiles in low aspect ratio beds, 1 < 0 G 2, are significantly d&rent from those in high aspect ratio beds. While the latter profiles show damped oscillations, the void fraction in the bed with a = 2 starts from unity at the wall, falls to a minimum of 0.19 at a distance of about O&M, from the wall, and then r&s to reach unity at the axis of the bed. At the axis, the spheres have only point contacts [see Fig. l(a)] just as at the wall, and hence the void fraction is unity. The profiies for 1 < a < 18 show a continuous fall in the void fraction from a value of unity near the wall to zero
Shorter Communications
6
0.00
0.25
0.00
Distance
0.50 from
0.75
1.00
0
woll/(d,/2)
Distance
Fig. 3. Radial void fraction profiles in beds of aspect ratio 1.5, 1.95, 2, 5.6 and 14.1; data for last two beds are from Benenati and Brosilow (1962).
towards the axis. In these beds, the region close to the axis is filled entirely with solids. For beds with l$ -=zLI -z 2 there is again a continuous decrease in the void fraction from unity at the wall to a fairly low value towards the axis, and then there is a slight increase followed by another decrease.
from
Fig. 4. Radial specific lateral surface area profiles in beds of aspect ratio 1.5, 1.95, 2 and 14.1.
the wall). In lower aspect ratio beds (a -z 2), the profile falls steeply in the region near the wall and then approaches a very low value near the axis. CONCLUSIONS Three distinct types of behavior
RADIAL SPECIFIC LATERAL SURFACE AREA PROFILES In analyzing the flow distribution in packed beds, knowledge of radial variation of the lateral surface area is also of interest in addition to information on radial variation of the void fraction, particularly for the estimation of the hydraulic radius. The specific lateral surface area in a CCC is given by
4. =
sum of the normalized
volumes
Cap with radius R::
Ir’ Sx,+Y
s VP--R:
[arcsin -1
dy* + 2rg(r,* -
k*). (38)
Slice with radii Rt and RP : The lateral surface area is computed as the difference between the lateral surface areas of the caps with radii Rf and Rz. Rod-like
segment with radius RF:
s W**R:
Y. - R:
Annular
[arcsin ,&)I
dy*.
(39)
ring with radii R: and RF: The lateral surface area is computed
as the difference between the lateral surface areas of the rod-like segments with radii RF and R?. Figure 4 shows the variation of the specific lateral surface area, q,.. with distance from the wall in beds of different aspect ratios. Again, the surface area protiles in beds of low aspect ratio are significantly different from those in beds of high aspect ratio; the former do not show oscillations. In beds of aspect ratio two, q+ falls steeply from a high value near the wall (or at the axis) as we move towards the axis (or
for
The structure of the bed shows periodicity, and may therefore be described in terms of unit cells, i.e. sections whose characteristics are the same as those of the overall bed. The
of the sphere segments in the CCC
The lateral surface area of each of the four types of segments we may encounter in a CCC of a low aspect ratio bed is evaluated as follows:
are observed-ne
beds of aspect ratio 2, the second for beds of aspect ratio between 1 and I@, and the third for beds of aspect ratio between 18 and 2.
lateral surface areas of the sphere segments in the CCC
sum of the normalized
woll/(d,/2)
. (37)
i
height of such a unit cell is seen to be half of the height of the bed which shows periodic&y in the location of the particles, and is independent of the position of the base of the unit cell. This also indicates the complete reproducibility of the packing of low aspect ratio beds, which is of particular advantage in many applications. The various expressions derived here may be directly used to obtain a rigorous simulation of flow, and mass and heat transfer in fixed-bed systems of low aspect ratio and should be of considerable value in the design and operation of industrial packed beds, particularly those involving high rates of heat and mass transfer, and in the more realistic interpretation of laboratory and pilot-plant data from such systems. Also, the possibility of this rigour makes it expedient to use the low aspect ratio beds as laboratory multiphase reactors. VENNETI M. H. GOVINDARAO KANDIRAJU V. S. RAMRAO AMMAVAJJALA V. S. RAO
Department of Chemical Engineering lndian Institute of Science Bangalore 560012, India NOTATION aspect ratio defined by eq. (13) [=d,2--(zs-z*)*]
Shorter Communications column diameter particle diameter (=
k* N. 4P r ‘P r* z* R:,
R4
JK=z,
(= J2 + 20 - a*) height of a unit cell, normalized with dp height at which coordinate periodicity occurs, normalized with dp [=(rz’ - R:’ + w*‘)/~w*] number of spheres per unit height in a unit cell specific lateral surface area at r* r-coordinate of the sphere particle radius dimensionless radial coordinate (= r/d,) thickness of a CCC dimensionless thickness of a CCC ( = Arid,) radius of cylinder used in defining cap, slice, rodlike or annular ring segment of a sphere, normllized with d,
(= r;* - y*q
lateral surface area of a sphere segment, normalized with the lateral surface area of the sphere [= E:+ - (y* - w+P] volume of a sphere segment, normalized with the volume of the sphere horizontal distance between the axis of the column and the center of the sphere, normalized with
t V* W*
d, x-coordinate of the center of the ith sphere y-coordinate of the center of the ith sphere dimensionless Y-coordinate ( = y/d,) z-coordinate of the center of the ith sphere
xi
Yl Y’ =i
J., Zoulalian,
Chemica! Engineering Science, Printed in Great Britain.
Mass Transfer 28, 383-394.
A. and Smith, J. M.,
Vol.
47, No.
Engng Sci. 32, 678-681. Johnson, G. W. and Kapner, R. S., 1990, The dependence of axial dispersion on non-uniform flows in beds of uniform packing. Chem. Engng Sci. 45, 3329-3339. McGreavy, C., Foumeny, E. A. and Javed, K. H., 1986, Characterization of transport properties for fixed bed in terms of local bed structure and flow distribution. Chem
Scott, D. S., Lee, W. and Papa, J., 1974, The measurement of transport coefficients in gassolid heterogeneous reactions. Chem. Engng Sci. 29, 2155-2167. Tsotsas, E. and Schliinder, E. U., 1990, Measurements of mass transfer between particles and gas in packed tubes at very low tube to particle diameter ratios. W&me- und
void fraction at r* &coordinate of the center of the ith sphere-
3.
J. 25, 663-676.
Engng Sci. 41, 787-797.
REFERENCES Ahn,
A.I.Ch.E.
Dixon, A. G., Dicostanzo, M. A. and Saucy, B. A., 1984, Fluid-phase radial transport in packed beds of low tubeto-particle diameter ratio. Int. J. Heat Mass Transfer 27, 1701-1713. Govindarao, V. M. H. and Froment, G. F., 1986, Voidage profiles in packed beds of spheres. Chem. Engng Sci. 41, 533-539. Govindarao, V. M. H., Manjunath, S., Rao, A. V. S. and Ramrao, K. V. S., 1990, Voidage profile in packed beds by multi-channel model: effects of curvature of the channels. Chem. Engng Sci. 45, 362-364. Hsiang, T. C. S. and Haynes, H. W., 1977, Axial dispersion in small diameter beds of large, spherical particles. Chem
Melanson, M. M. and Dixon, A. G., 1985, Solid conduction in low d,/d, beds of spheres, pellets and rings. Int. J. iYeut
Greek letters overall void fraction e0 er* 01
2109
dispersion in packed beds with large wall effect. A.I.Ch.E. J. 32, 170-174. Benenati, R. F. and Brosilow, C. B., 1962, Void fraction distribution in beds of spheres. A.I.Ch.E. J. 8, 359-361. Cui, L. C., Schweich, D. and Villermaux, J.. 1990, Consequence of flow nonuniformity on the measurement of effective diffusivity. A.I.Ch.E. J. 36, 86-92. Dixon, A. G. and Cresswell, D. L., 1979, Theoretical prediction of effective heat transfer parameters in packed beds.
8, pp.
ZLO!-2113.
1986,
Axial
Stofibertragung
25, 245-256.
1992. 0
OOC%2509/92 SSM 1992 Persamon h
+ 0.00 Ltd
Relationships between surface diiusivity and pore diffusivity in Fatch adsorption: measurements of the diffusivities for n-hexane and n-decane in 5 A molecular sieves (First received 27 May
1991; accepted
INTRODUCMON Various models have been proposed for diffusion in an adsorbent [e.g. Costa and Rodrigues (1985) and Do and Rice (1987)]. Among them, the two most frequently used models are surface diffusion and pore diffusion. If the adsorption isotherm is of the Henry type, the difhtsivities for these two models are easily converted, one from the other. In general, the diffusivity is considered to be a function of intrapellet concentration (Ruthven, 1984). The diffusivities (average, not concentration-dependent) for the two models are easily determined from a batch adsorption experiment. Our interest is whether these two diffusivities may always be converted reciprocally, or under what conditions of batch adsorption experiment the conversion may be made.
in revised form 6 December
1991)
The system considered comprises spherical adsorbent particles (of weight W,, radius R, density p$ exposed to a mixture of sorbate and inert fluid in a mixing vessel (of volume V). The overall adsorption process is assumed to be controlled by the intraparticle mass transfer, and not by the particle-fluid mass transfer. Also, the surface diffusivity and the pore diffusivity are assumed to be constant, not depending on adsorbed and solute concentrations. respectively. The adsorption isotherm is assumed to be of the Langmuir type:
(1)