M A C H I N E C O M P U T A T I O N IN, PETROLEUM RESEARCH I
ii = n,,l
Comparison of Batch and Moving Bed
To find the required height of a moving-bed column for a given aromatic removal, one must first find the internal diffusivity, D, the particle radius, R, and the equilibrium isotherm for the system. The isotherm may be determined by breakthrough curves for the charge at various aromatic concentrations passing through a fixed bed or by finding the limiting aromatic concentrations of the liquid in contact with varying amounts of gel. The diffusivity may be determined by batch experiments, plotting liquid aromatic content against time and fitting the curve to the u,, us. T plot. However, T also includes the square of the particle radius. As the real system involves a distribution of particle sizes and the particles are not actually spherical, this may pose some difficulties. However, Mertes and Hirschler report that when spherical particles were assumed and sieve data used for particle radius distribution, the internal diffusivity, calculated by weighting the contribution of the differentsized particles to the aromatics removal found by batch studies, remained constant with time. On the other hand, when an average particle radius was used, the internal diffusivity appeared to drop off with time as the smaller particles became aromatic-saturated and the effect of the large particles became more pronounced. A quicker, more direct method has been partially explored. If the up,,% for the batch and the moving bed are transformed to fractional approaches to the limiting concentration at infinite time, a cross plot can be made to find the ratio of moving-bed time to batch time required to obtain the same fractional approach. This has been done for the linear isotherm case (C = 0) and is shown in Figure 8 for fractional approaches above 50%. Below 50%, when the T values are below 0.05, the numerical solutions lose considerable accuracy, particularly for the higher values of p.
Example. Assume a kerosine containing 20% aromatics, which must be reduced to 4% for a stable jet fuel in the adsorption section of a moving-bed column. Assume that its adsorption isotherm is known to be linear and is 3.0. The material balance equation in the adsorption zone is V(c -
GO)
= S(E
- ED)
=
The average aromatic concentration of the particle at zero time, EO, equals zero and the average particle concentration at infinite time is
=
Kzc
m
Therefore
n
V ( n - no)?,
and dividing by n,
= SKzn,,
1
1
no
-,
1 - -1 Ul.-1
ii
= -P = 1 3
60
R r
_ - o at infinite time ur- 1
S
and the fractional approach will be F=
-
0.20 0.04 0.20 - l / U , - l
s(
= o.80
t
By performing a batch experiment on this charge stock and using the same solid-liquid ratio as the solid flow to liquid flow rates for the moving bed, one can readily measure the batch time necessary to reach an 0.80 fractional approach, F. Referring to Figure 8, for p = 3.0 and F = 0.80, the time ratio would be 3.0. Thus the moving-bed time would be three times the measured batch time and the column would be designed so the gel would be in contact with the liquid for that residence time. This method avoids the necessity of calculating a diffusivity based on an estimated particle-size distribution. I t has not been extended to the nonlinear case as yet. However, it is known that for a given 0, the greater the value of C, the Iower the time ratio for a given fractional approach. Summary
By using a simplified model of the adsorption zone of a moving-bed column, rate equations based on a Langmuir isotherm have been solved on an IBM 650 computer. Comparison of these solutions with the analogous solutions for batch adsorption suggests a method of finding directly the height of a movingbed adsorption zone from the results of a batch experiment.
u
+
Xr u,
V B
= value of u at particle surface =
rate of liquid flow parameter
= dimensionless
(5) \
T
=
3 KZ = dimensionless time parameter = Dt/ R2 I
Literature Cited (1) Davis,
W. H., Harper, J. I., Weatherly, E. R., Am. Petroleum Inst. meeting, San Francisco, Calif., May 1952. (2) Harper, J. I., U. S. Patent 2,644,018
(1953). (3) Hirper,‘J. I., Olsen, J. L., Shuman, F. R., Chem. Eng. Progr. 48, 276 (1952). ’(4) Hirschler, A. E., Mertes, T. S., Chan 8. “Chemistrv of Petroleum Hyd>ocarbons,” vol. I, ed. by B. T. Brooks, others, Reinhold, New York, 1954. (5) Kasten, P. R., Amundson, N. R., IND.ENG.CHEM. 44,1704 (1952). (6) Mertes, T. S., Rhodes, H. B., Chem. Eng. Progr. 51,429 (1955). (7) Olsen, J. L., U. S. Patent 2,564,712 (19511. Zbid., 2,585,490 (1952). Paterson, S., Proc. Phys. SOC.59, 5 5 (1947). Rommel, R. H., U. S. Patent 2,646,451 (1953).
Siegmund, C. W., Munro, W. D., Amundson, N. R., IND. ENC. CHEM.48,43 (1956).
RECEIVED for review October 30, 1957 ACCEPTED February 13, 1958
Acknowledgment
The authors %ish to thank P. Frank Hagerty, Sun Oil Co., for his assistance and suggestions. Nomenclature a
positive root of eq.iation, m ctnh m = 1 m 2 / P = point aromatic concentration in particle , = point aromatic concentration at particle surface at zero time = average aromatic concentration in particle = average aromatic concentration in particle at zermtime = radius of particle = fractional distance from center ofparticle = a/R = rate of particle flow = ith positive root of equation, s c o t s = 1 - sz/p = time particle has spent in adsorption system = dimensionless parameter = n/no =
= distance of point in spherical
particle from center = measure of nonlinearity of adsorption equilibrium, = X i K7, no = aromatic concentration of liquid c in contact with particle D = particle internal diffusivity K1, Kz = parameters in Langmuir-isotherm equation, c =
G‘
n
Kz
- Kl n
Division of Petroleum Chemistry, Symposium on Application of Machine Computation to Petroleum Research. 132nd Meetinq, ACS, New York, N. Y . , September 1957.
Correction Bonding of Teflon In the article on “Bonding of Teflon” by E. R. Nelson, T. J. Kilduff, and A. A. Benderly [I/EC 50, 329 (1958)], two minor errors should be corrected. On page 330, Table I, the a in front of the footnote should be deleted. In column 1, in the first paragraph below the tables, line ll, “are aper” should read “area per.” VOL. 50, NO. 5
0
MAY 1958
729-
