D o , Db
= percent deviations defined by Equations 5
and 4 KI-K, = constants in Equations 4a and 5a
L1. Lr, L ; , LI = constants in Equations 4 and 5 P = atmospheric pressure P = critical pressure P, = reduced pressure, P P , R-K = Redlich-Kwong equation T = absolute temperature, K T , = critical temperature T, = reduced temperature, T T , volume in cc / gmol R = 82.06 atm ccigmol OK &!. = acentric factor
v = w
literature Cited
Ackerman, F. J., M.S. thesis, University of California, Berkeley, Lawrence Radiation Lab., UCRL Report No. 70650, 1966. Chueh, P. L.. Prausnitz, J. M., Ind. Eng. Chem. Fundam. 6. 492 (1967). Chueh, P. L., Prausnitz, J. M., Ind. Eng. Chem. 60, 34 (1968). Couch, E . J., Kobe, K. A,, Chem. Eng. Data 6, 229 (1961). Din, F., “Thermodynamic Functions of Gases,” Vols. 1, 2 , 3, Butterworth’s Scientific Publications, London, 1961.1962.
Estes, J. M., Tully, P. C., AIChE J . 13, 192 (1967). Gray, R . D., Rent, N. H., Zudkevitch, D., “A Modified Redlich-Kwong Equation of State,” AIChE 69th Symposium, New Orleans, 1969. Hall, E.A , , Ibele, W. E , Trans. A S M E 57, 1003 (1955). Joffe, J., Zudkevitch, D., Ind. Eng. Chem. Fundam. 5 , 455 (1966). Redlich. O., Dunlop, A . K., Chem. Eng. Progr. Symp. Ser. 59 44,95 (1963). 44, 141 (1965). Sage, B. H., Lacey, W. N., Monographs on API Research Project 37, 1950, 1955. Skamenca, D. G., M.S. thesis, Newark College of Engineering, Newark, N. J., 1970. Shah, K. K., Thodos, G., Ind. Eng. Chem. 57, 30 (1965). Vogl, W. F., M.S. thesis, University of Virginia, Charlottesville, Va., 1968. Wilson, G. M., Aduan. Cryog. Eng. 11, 392 (1966). Zudkevitch, D., Joffe, J., “Correlation and Prediction of Vapor-Liquid Equilibria with the R-K Equation of State,” 69th Symposium, AIChE, New Orleans, 1969.
RECEIVED for review January 15, 1970 ACCEPTED July 6, 1970
Physical Adsorption in a Moving Bed of Fine Adsorbents Sok Moon Yoon and Daizo Kunii Department of Chemical Engineering, University of Tokyo, Tokyo, Japan Adsorption
and
desorption
of
adsorptive
gases
were
studied
theroretically
and
experimentally in a moving bed of fine activated alumina through which the air was flowing upward. Mathematical models were developed and fundamental equations were solved to show the longitudinal concentration distribution of adsorptive gas in a moving bed. A dimensionless factor, a , determined by the operating conditions of the moving bed of adsorptive particles, was utilized to represent the longitudinal distribution of concentration for any adsorptive gas. The adsorptive gases (ethane, ethylene, propane, and freon-12) with different partition coefficients on activated alumina were introduced into the moving bed through an appropriate distributor, to effect a plane source of the injected gas.
By comparison of the experimental distribution of the
adsorptive gas in the longitudinal direction with the theoretical distributions, gas phase mass transfer coefficients were evaluated.
A d s o r p t i o n of gas on a solid particle can be described as a three-step mechanism: diffusion of the component from the main stream of the gas to the external surface of the adsorbent particle (external diffusion), diffusion into the particle through pores (intraparticle diffusion), and adsorption on the intersurface of the pore. Kasten and Amundson (1952) solved the adsorption problem in a moving bed of uniform and porous spheres, which move downward in rodlike flow countercurrent to the fluid flow. They obtained the analytical solution of the initial adsorbate concentration flowing into the 64
Ind. Eng. Chern. Process Des. Develop., Vol. 10, No. 1, 1971
moving bed as a function of parameters, assuming both resistances of internal and intraparticle diffusion, and applying linear isotherm for equilibrium and kinetic expression for nonequilibrium. Furthermore, in a similar way, Siegmund et al. (1956) analyzed two problems of moving beds: the problem for nonadiabatic case of pebble heaters, and the problem for nonlinear equilibrium relation of moving adsorbers. Moro’oka and Miyauchi (1969) discussed the mean residence time of adsorptive gas in a moving bed for a stimulus response condition. According to the authors (Yoon and Kunii, 1970), pres-
sure drop through a moving bed is originated by the slip velocity between the gas flow and the descending solids. The flowing gas measured a t the outlet of a bed is not the same velocity as the gas flowing through the test section of a countercurrent moving bed. With high descending velocity of fine solids, gas may flow downward against the higher pressure a t the bottom of the bed. Considering this information, theoretical and experimental works were planned to investigate the mechanism of physical adsorption and desorption of gas in a moving bed of' fine adsorbents.
where C, is the concentration of adsorbate inside of the solid, t b is the void fraction of the bed, pb and p 7 are bulk density and solid density respectively, k , is the mass transfer coefficient in gas phase, and m is the partition coefficient of adsorbate, which is defined by (volume of gas adsorbed or captured, cc)/(volume of solids, cc). At steady-state boundary conditions imposed upon the solutions are
c, = cru,c, = 0 C, = C 8 ~ =- CdL- + F u,, c, = CgLI
Mathematical Model
The flow diagram of solids and gases in the moving bed is shown in Figure 1. In the moving bed system, the spherical porous solids, whose average radius is R , are moving downward a t a constant superficial velocity, u,. Carrier gas, such as air, is flowing through a moving bed countercurrently under isothermal and isobaric conditions a t a constant superficial velocity. u ~ An ~ .adsorptive gas is introduced into the middle of the test section as a plane source a t a constant feed rate. F , negligible in respect to uem;the superficial velocity of the gas mixture and the air are assumed to be same. The adsorptive gas introduced into the test section will flow upward with the air flowing upward, and mass transfer will occur between the gas and solids above the feed plane. The solids which have adsorbed gas move down and the gas will be desorbed under the feed plane. Therefore, a t the steady-state operation, the concentration distribution will appear in the bed, above and under the feed plane. Evaluation of the longitudinal concentration distribution of adsorptive gas through a moving bed is required. I t was desirable to develop a model incorporating all of the transfer mechanisms. T o make the model tractable, the following major assumptions were made: Since surface adsorption is so rapid, the pointwise equilibrium is attained between the adsorbed material and that existing within the porous solid. Adsorbate diffuses through the sphere very rapidly because of the small size of solids. Longitudinal dispersion of both gas and solids is negligible with respect to bulk flow. Concentration distribution in the radial direction in the bed is negligible. Only one kind of adsorptive gas is fed into the bed. Each particle is covered uniformly with gas mixture. Average transfer coefficients can be used throughout the course of the adsorption. With these restrictions the differential equations for mass transfer can be formulated and solved analytically to describe the longitudinal concentration distribution, C,, in a moving bed. In the bulk stream of gas within the bed, the mass balance for the adsorbate is
Within the moving solids, the mass balance for the adsorbate is
c, = 0 , c, = C i H
at x = 0
(3)
at x = Lat x = L+ at x = H
(4) (5) (6)
The results of solutions describing the longitudinal concentration distribution in the moving bed are
forx=O
-
for x = L+
L-
1
H
CRoin Equation 7 can be obtained by the mass balance over the bed
F = u,,Cr0
+ (1 - tg)m u , C , ~
(9)
where CsHcan be evaluated from the relation of
Solids
Gas
-
x=o ....: ... , . .
Sampling taps 2 c m apart
c g = Cgo cs=o
x=L CgL-=CgL.+
2 c r n apart
Solids Figure 1. Flow of gases and solids and boundary conditions
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
65
Ctl
0.2
0
0
0.4
OB
0.6
1
0.2
Experimental Procedure
0.4 XIJ
I1
x
0.6
0.8 --z
0.2
:I
0.4
X
I I1
N
x
The theoretical curves for dimensionless concentration of adsorbate in a moving bed C,* us. dimensionless bed length a t various X are shown in Figure 2. We can predict that the concentration distribution in a moving bed is varied by a , the function of the operating conditions.
0.6
0.8 1
0
0.2
0.6
0.4
1
0.8
C L
Figure 2. C i v s X at various
CY
obtained from Equation 1. Therefore,
The experimental apparatus is the same unit which was employed for the studies on the pattern of gas flow (Yoon and Kunii, 1970). Four kinds of commercial gases (ethane, propane, ethylene, and freon-12) were used as adsorbates. Adsorptive gas was introduced continuously t o the middle part of the test section, (4.1-cm diam) as a plane source. Details of the injection device are given by Yoon and Kunii (1970). The test section was designed to sample gas through fine tubes a t various places in the longitudinal direction. The longitudinal concentration distributions of a binary system of air and adsorptive gas in a moving bed of fine adsorbent were measured a t room temperature and atmospheric pressure. Gas chromatography was used to analyze the sampled gases. Adsorbent. MS catalyst (A1203 = loo%), (Mizusawa Chemicals, Ltd.) was used as an adsorbent. Physical properties are in Table I. Fine solids with an average diameter of 0.154 mm were used. This may justify the minor effect of the longitudinal dispersion of gas and the mass transfer in the solid phase. Measurement of m. Experimentally the partition coefficients for the present gases and solids were evaluated by measuring the difference of retention time of adsorptive gas and nonadsorptive gas in the chromatographic column. Gas-solid chromatography has recently proved very useful in the evaluation of heats of adsorption (Everly, 1961; Greens and Pust, 1958; Habgood and Hamlan, 1959; Hanson et al., 1964; Moro'oka and Miyauchi, 1969). Assuming that the rate of adsorption is rapid and the pointwise equilibrium between the adsorptive gas and solids is attained instantaneously in the void volume, the partition coefficient m is evaluated by the following relation: m = ( V R- V R ~ ) / Vm,~ = ~ V R / + V m, , (17)
+
VR = V, + m V. VR0= V, + m, V ,
where CY'
=
-
( p - 1)
kg
d, m us
The dimensionless forms of Equations 7 and 8 are
c:, = c,~- C, C,,- - C,
c,*? =
-
G 2 ~
CRL-
-
1 - ealXl
1 - ea' ea2X2
- ,a2
1 - eaz
where
XI= XiL, x,= ( x - L ) / ( H - L ) ,
CY1
=
a'L
CY?
66
where m is the partition coefficient of adsorptive gas, -; m,, the partition coefficient of nonadsorptive gas, which is equivalent to the pore volume of solid, -; V,! the volume of gas in dead space, cc; VR, the retention volume of adsorptive gas, cc; VRa, the retention volume of nonadsorptive gas, cc; V,, the volume of solids, cc. For evaluation of m, two lengths of chromatographic column (4.45-mm diam) were employed: 20 cm and 80 cm, respectively. Helium and nitrogen were used as carrier gases. The superficial velocities of carrier gases through the column were maintained a t about 1.3 cmisec, similar to the range of velocities carried out in the adsorption experiments in the moving bed. In the chromatographic column the effect of pressure could be neglected because the velocity of the carrier gas was too small to liberate the pressure gradient. Therefore, only the effect of temperature was considered as follows:
AVk = (ToT,) AVR
and =
CY'
(H - L)
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
(18) (19)
(20)
where T , is the column temperature and T , = the tem-
Table I. Physical Properties of MS Catalyst Average particle diameter, Bulk density, p 6 Solid density, p g Chemical density, pI Void fraction of solid, es Void fraction of bed, C *
ZD
0.154 mm 0.87 gJcc 1.50 g/cc 3.50 gicc 0.55 0.42
Table II. Heats of Adsorption and Condensation Gases ~
Heats of adsorption, kcalimole Heats of condensation, kcal I mole"
Ethane
Ethylene
Propane
Freon-12
3.20
3.82
4.24
5.06
3.51
3.24
4.49
4.82
Lange (1961)
perature of flow measuring device, room temperature in OK. T o maintain the linear isotherm, the partial pressure of adsorptive gas in a mixture was held within 10%. As reported by Moro'oka and Miyauchi (1969), the same values of m were obtained for the different carrier gas, helium or nitrogen. Figure 3 shows the plots of the partition coefficients of adsorptive gases as a function of the reciprocal column temperature utilizing 20- and 80-cm MS catalyst columns. With each gas, a straight line is obtained under 30" C. Using the data from Figure 3, heats of adsorption were evaluated by Arrhenius equation. I n Table I1 heats of adsorption for the present system are given with the heats of condensation for each gas (Lange, 1961) for comparison.
The values of heat of adsorption indicate t h a t physical adsorption occurs in the present gas-solid system: During physical adsorption the heat liberated per mole of gas is generally in the region of 2 to 6 kcal, while the heats of chemisorption are rarely less than 20 kcal/mole (Thomas and Thomas, 1967). Results and Discussion
Figures 4-7 are plots of the concentration of adsorbates a t various places on the longitudinal line of the moving bed operated a t steady-state. All plots in these figures were obtained by gas-chromatographic analysis of the gases, sampled through fine tubes. T o compare them with the theoretical diagram of Figure 2, the experimental data were replotted in dimensionless forms as shown in Figure 8. From Equations 7 and 8 the concentration distributions are determined by the factor CY', which is defined by Equation 13. For every experimental condition, the dimensionless factor, CY,which is the product of CY' by the length of the bed L or H - L , could be determined from Figure 8. Using the determined numerical values of CY, the theoretical lines were obtained. They are presented in Figures 4-7. I n Figures 4-7 the concentrations a t x = L , C,L- and C+, were determined from the experimental data. Figure 4 shows that C g ~ = - 0.240 cc/cc from the extrapolation of the data ( 0 ) with the dotted line and consequently (for CY = -20):
C,,+ = Cp,- - Flu,, = 0.240 - 0.101 = 0.139
Similarly, C g ~ -= 0.156 cc/cc from the extrapolation of the data ( 0 ) with the dashed line, therefore (for CY = 20) : CgL-
I
L = 80 cm
4- 4 45 mrn Flow rates about 1 3 crnlsec C a r r i e r gas helium
20
E
=
CgL-
+ Fiu,,
= 0.156
+ 0.147 = 0.303 C C / C C
The experimental data in Figures 4-7 compared with Figure 8 suggest that the assumptions and the mathematical models developed are reasonable. Furthermore, they show that, whatever the velocities of gases and solids are, the concentration profiles of adsorptive gas in the moving bed are changed according to the value of 0, defined by Equation 14. The mass transfer coefficients in the gas phase, h,, are then evaluated in the present adsorption system. From Equation 13
L.20cm
30
CC/CC
lo
8
The partition coefficients of adsorptive gases, m, can be obtained from Figure 3. At the present experimental condition they are ethane, 2.5 a t 16" C; ethylene, 3.5 a t 19" C; propane, 7.0 a t 19" C; and freon 12, 11.1 a t 21" C. The mass transfer coefficients in solid phase, h,, are obtained from the following definition
6
4
3
k, = De,// R
(22) The effective diffusivity of the adsorptive gas in the solid, De,/,is estimated by the following equation (Wakao and Smith, 1962)
2
3.1
3.2 3.3 3.4
3.5
~
3.6
3.7 3.8 3.9
1 / T X 103("K) Figure 3. rn of adsorptive g a s e s vs 1/T
De,, = tt/ ( ~ I D A+R ~ / D K )
(23) where t s is the void fraction of the solid and DAB and D K are the molecular and Knudsen diffusivity respectively. Knudsen diffusion predominates over this system, because the average pore radius of MS catalyst is 40 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
67
112
I
0
I ... - -.!'9
x.0 . ---
1
1
1
Propane
Ethane
rn=7.0
100
t
I
t
80
c ,/f 40
- - r = d2 0= - 3
I
i
dI
a
0
0.1
0.2
crnlrec
0.3
0.4
a crn/rec
cg, CCICC
0
0.1
0.3
0.2
0.4
0.5
c g , cclcc
Figure 4. Concentration distributions in longitudinal direction (ethane, m = 2.5 a t 15°C)
Figure 6. Concentration distributions in longitudinal direction (propane, rn = 7.0 at 19" C)
112
I
I
I
I
Freon 12
rn = 11.1
1oa Figure 5 . Concentratior distributions in longitudinal direction (ethylene, m = 3.5 a t 19.C)
Figure 7. Concentration distributions in longitudinal direction (freon12,m = 1 1 . 1 a t 21°C)
0
80
t .I/
'D 0,
Theoretical lines -0c
m
-_
20
0
I
9 F/ugm(-)0.126 P
j 1
a
I I
I
I
1
cg,
68
20
a cmlsec
t/
,'
0.1
FhgmI-1 0.0040 0.091 (-1 0.970 2.340
cmlsec
I-----02 0.3
0
0.082 0.149 0.671 1.772
am
I-)
=-0.1 20
oc=
I
0.4
cclcc
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
I
05
ok
/
/
//
0.;
oi2
0:3 cg, c c l c c
0;
0;s
~
~~
Table Ill. Calculated Diffusivities and Mass Transfer Coefficients in Solid Phase" B
ME
Ethylene Ethane
28.05 30.07 44.09 120.93
Propane
Freon-12 " A .air;
T,
K
D ~ R ;
Crn'jSec
1.51 x 1.39 x 1.25 x 9.82 x
292 289 292 294
lo-' 10 10-1 10 '
DK, Crn'/Sec
D,,
1.25 x 1.20 x 1.00 x 6.05 x
3.49 x 3.36 x 2.80 x 1.73 x
10 lo-? lo-' 10
k ~ Cm/Sec ,
Crn-/Sec
0.453 0.436 0.364 0.224
10 10 10 ' 10
28.97
A, and because this is smaller than the other of the mean free path of the molecules of the adsorptive gases (10 ' cm). The predicted Knudsen diffusivity is
D h = 9700 r,,,, ( T ,M B ) '
(24)
The molecular diffusivity was calculated by the theoretical equation based on the modern kinetic theory and the Lennard-Jones expression for intermolecular forces (Bird et a!.,1960). This is I ?
The estimated DK,D 4 ~De,,, , and h, are presented in Table 111. Comparing h, and h,, the solid phase transfer was so large that it could be neglected and the first assumption was verified. Figure 9 shows the experimental mass transfer results in moving bed system with the group of Shp(k, d , y D A B ) as ordinate and of Re,,(& u,,/u) as abscissa. The reported results in fixed and fluidized beds by previous investigators
X -
041
*
are also included in Figure 9. In a moving bed data y was assumed to be unity and u,, was used to calculate the Sherwood and the Reynolds numbers. For fixed beds, Ranz (1952) gave
Sh, = 2.0
+ 1.8Sc'
'
Re;'
(26)
In Equation 26, the Sherwood number asymptotically approaches a lower limit of 2 as the Reynolds number becomes very small. Resnick and White (1949) studied the sublimation of naphthalene of fine particles (do = 0.21 1.7 mm) and reported the mass transfer coefficients on both fixed and fluidized beds for the low Reynolds numbers. For fluidized beds, Kettenring et al. (1950) studied the vaporization of water from silica gel and alumina (d, = 0.36 ..- 1.0 mm) and Richardson and Szekely (1961) studied the adsorption of CC14and water on activated carbon, charcoal and silica gel (d, = 0.088 2.58 mm) in unsteady-state. I n direct opposition to the prediction by Ranz (1952), the measured Sherwood numbers of investigators do not level off the theoretical value of 2 but keep falling continuously as the Reynolds number is lowered. To explain the contradictory phenomena occurring in the beds of fine particles for the low region of the Reynolds number, Kunii and Suzuki (1967) developed the channeling model and gave
-
-
\ \\
d ,= - 3
~~
Present work (Moving bed) AdSorDtlOn on Activated Alumina I 0 154mm)
N
X
(ap
10-~
~
0 0 0 Ethane
0 A
*
0
10-l
1
A Ethylene 0 Propane
Freon12
10
1 o2
Rep
Figure 8. Concentration distributions in dimensionless form
Figure 9. Experimental gas-solid mass transfer results for a single sphere, fixed beds, fluidized beds, and moving beds for the low Reynolds region
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
69
Sh
-
$5
- 6 (1 - t b ) (
Pep
for Pep
< 10
(27)
where @, is the sphericity, ( is the ratio of average channeling length to the particle diameter, and Pep is the Peclet number, (Zpugm/ DAB). Assuming the proper values of the channeling length factor, (, the Sherwood numbers reported in the previous literatures can be interpreted well with Equation 27 in the range of the low Peclet number. Referring to Equation 27, ( for the present adsorption data in a moving bed is about 100. This suggests that the actual gas flow through a moving bed of fine adsorbents was channeled considerably, because of the descending solids. Conclusions
A dimensionless factor, a , influencing the longitudinal concentration distribution of adsorptive gas in a moving bed was obtained theoretically and the experimental data were compared with the kinetic model. Gas phase mass transfer coefficients were obtained in the region of the low Reynolds number by comparison of the experimental data with the theoretical analysis. Sherwood numbers were comparable to those reported in fixed and fluidized beds by previous investigators.
us = superficial velocity of descending solid,
cm/sec V, = volume of gas in dead space of chromatographic column, cc V R ,V R =~ retention volume of adsorptive gas and nonadsorptive gas respectively, cc IvR = VR- VRo AVh = corrected AVRby temperature V , = volume of solids in chromatographic column, cc x = distance measured in direction of solids flow from the top of the test section, cm XI, X? = dimensionless distance defined by x / L and ( x - L )/ ( H - L ) respectively y = logarithmic mean fraction of inert or nondiffusing component in the Sherwood number GREEKLETTERS a = dimensionless factor defined by a’L or
a’(H - L ) a’
Nomenclature
c, cL‘o,c,,., c,, c,1,
= interparticle concentration of adsorptive =
C,? =
c;, c,”?= c, = C,H = D.AB = Dei, = DK = d p , 3, =
F =
H = k,, k , =
L = m, mo =
M A ,M B
=
P = Pep =
R = rporp= Re,, =
sc
=
Sh, = T = T,, T, = UL, u,,
70
=
gas, cc/cc or moleicc C, a t various places of x = 0, L , and H respectively, ccicc C, above and under the feed plane of adsorptive gas respectively, cc/cc dimensionless C, defined by Equations 15 and 16 respectively concentration of adsorptive gas inside the solid, ccicc or mole/cc C, a t x = H , cc/cc molecular diffusivity, cm’/sec effective diffusivity, cm’isec Knudsen diffusivity, cm2/sec particle diameter and average particle diameter respectively, cm feed rate of adsorptive gas into the moving bed, cmisec or mole/cm2/sec distance from the gas exit to the bottom of the test section, cm mass transfer coefficient in gas phase and in solid phase respectively, cm/sec distance from the gas exit to the feeding plane, cm partition coefficient of adsorptive gas and nonadsorptive gas respectively, cc/cc molecular weight of air and adsorptive gas respectively total pressure, a t m Peclet number, d,u,/ D A B radius of particle, cm pore radius of solid, 4.0 x lo-’ cm for MS catalyst Reynolds number, dpu,/u Schmidt number, U I D O Sherwood number, (k,d,y) / D ~ B absolute temperature, K temperature of chromatographic column and gas respectively, K superficial velocity of gas through fixed bed and through moving bed respectively, cmisec
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
pb,
= factor defined by Equation 13, l / c m
8 = dimensionless factor defined by Equation 14 a, e, = void fraction of bed and solid respectively v = kinematic viscosity, cm2/sec = ratio of average channeling length to particle diameter p s , p t = bulk density, solid density, and chemical density respectively, gicc U A B = collision diameter in Equation 25, cm Q = collision integral in Equation 25
literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 511, Wiley, New York, N.Y., 1960. Everly, P. E., Jr., J . Phys. Chem. 65, 68 (1961). Greens, S. A., Pust, H., J . Phys. Chem. 62, 55 (1958). Habgood, H. W., Hamlan, J. F., Can. J . Chem. 37, 843 (1959). Hanson, R. S., Murphy, J. A., McGee, T. C., Trans. Faraday SOC.60, 597 (1964). Kasten, P. R., Amundson, N. R., Ind. Eng. Chem. 44, 1704 (1952). Kettenring, K. N., Manderfield, E. L., Smith, J. M., Chem. Eng. Progr. 46, 139 (1950). Kunii, D., Suzuki, M., Int. J . Heat Mass Transfer 10, 845 (1967). Lange, N. A., “Handbook of Chemistry,” 3rd ed., p. 1553, McGraw-Hill, New York, N. Y., 1961. Moro’oka, S., Miyauchi, T., Kagaku Kogaku 33, 569 (1969). Ranz, W. E., Chem. Eng. Progr. 48, 247 (1952). Resnick, W., White, R . R., Chem. Eng. Progr. 45, 377 (1949). Richardson, J. F., Szekely, J., Trans. Inst. Chem. Eng. 39, 212 (1961). Siegmund, C. W., Munro, W. D., Amundson, N. R., Ind. Eng. Chem. 48, 43 (1956). Thomas, J. M., Thomas, W. J., “Introduction to the Principles of Heterogeneous Catalysis,” p. 15, Academic Press, New York, 1967. Wakao, N., Smith, J. M., Chem. Eng. Sei. 7, 825 (1962). Yoon, Sok Moon, Kunii, Daizo, IND.ENG.CHEM.PROCESS DES. DEVELOP. 9,559 (1970).
RECEIVED for review January 16, 1970 ACCEPTED June 8, 1970
