I
ANDRE W. POLLOCK, MILLARD F. BROWN, and CARL W. DEMPSEY Sun Oil Co., Philadelphia 3,‘ Pa.
Machine Solution of a Boundary Value Problem for a Continuous Arosorb Process Adsorption equations based on a Langmuir isotherm were derived for a simplified model of the adsorption zone of the column and solved by numerical methods on an IBM 650 electronic computer
Arosorb process for separating aromatics from saturates by selective adsorption on silica gel (7, 3, 4 ) essentially consists of two steps-an adsorption step where the aromatic components in a feed stream are preferentially adsorbed on the gel, and a desorption step where the aromatics are desorbed by a material boiling in a different range from the feed and which is subsequently recovered by distillation. The first step is very similar to an established analytical method for separating organic compounds by silica gel adsorption. The second step represents, for a variety of operations, the most economical method of recovering the charge aromatics from the gel and regenerating the gel in order to repeat the first step. In the original process, feed and desorbent were alternately run through a packed gel bed, The schematic diagram of this cyclic operation is based on a commercial installation in Texas, which produces a series of aromatic solvents from a heavy catalytic reformate feed. Although the fixed-bed process has proved itself on a plant scale, considerable effort has been directed toward a countercurrent moving-bed system ( 2 , 4, 7, 8, 70),which might offer savings in investment and operation. This is rather analogous to the development of moving-bed and fluidized-bed catalytic cracking processes which eventually supplanted the original Houdry fixed-bed cases. T H E
Advantages of Continuous Moving Bed, Less piping, valves, and instruments required for a fully continuous than for a cyclic operation. Less surge capacity necessity to maintain good control in the desorbent recovery section. T o achieve the same product purities, less gel and desorbent required for the moving-bed process.
Design of Moving-Bed Arosorb Process
A possible design for a moving-bed Arosorb process is shown in Figure 1. I n the top section of the column, a rising stream of feed comes in contact with a falling bed of silica gel which adsorbs the aromatic components in the feed. A “reflux” stream of feed aromatics is injected below the feed point to remove any saturates that may still be adsorbed
in the gel before the gel enters the lower desorbent section. I n this lower section, the falling gel meets rising desorbent and the pure feed aromatics are recovered. The regenerated gel is then carried back to the top of the column, by a rising stream of desorbent, to repeat the operation. The column is obviously the heart of the process as well as the major piece of process equipment. Its diameter is determined by the maximum rising
D E S O R E E N T ~ TANK E
I
A+ DISTILLATION TOWERS
>{
PRODUCT AROMATIC PRODUCT
~
.
DI C
Fixed-bed cyclic Arosorb process, based on a commercial installation VOL. 50, NO. 5
M A Y 1958
725
DISENGAGE?--
+_ _ _ ,
/
I 1 I
I
liquid velocities which will still permit a desired downward flow rate of gel particles. Considerable work in this phase of the investigation has been reported by Mertes and Rhodes (6). The height of the column depends on the residence time necessary for a desired separation. It, therefore, depends mainly on the kinetics of the adsorption process. In contrast to solutions based on a linear adsorption isotherm, this article shows how adsorption equations based on a Langmuir isotherm were derived for a simplified model of the adsorption zone of the column, and how these equations were solved by numerical methods using an IBM 650 electronic computer.
The boundary conditions for n at a = R may be derived from the column material balance. Assuming steadystate conditions, the equation for a particle falling through the column is
Derivation of Equations
7
c = aromatic concentration of liquid in contact with sphere ?i = average aromatic concentration in sphere = ,f(t) S = rate of flow of solids V = rate of flow of,liquid
Substituting Equation 1 into Equation
DES(
Figure 1. Proposed continuous countercurrent Arosorb process The column is the heart of the process and the major piece of equipment
eo
50
c :0.00
40
7 10-
The gel particles are assumed to be uniform, porous spheres entering free of adsorbate and falling in rodlike flow through a rising stream of liquid. The liquid moves at a constant velocity, with a straight profile at right angles to its direction of flow. As the gel falls through the liquid, it preferentially adsorbs aromatics. so that the concentration of aromatics in the liquid decreases as it rises to the top of the column. The postulated mechanism of adsorption involves two assumptions : Film resistance at the surface of the gel particle is negligible compared to resistance in the gel pores. This is supported by considerable experimental evidence (4). In the internal diffusion mechanism, no distinction is made between the adsorbate on the internal surface and the adsorbate in the free pore volume. Such an assumption simplifies the diffusion equation for a nonlinear equilibrium isotherm, as it considers one concentration variable in the particle instead of two-concentration in free pore volume and concentration on adsorbing surface-which may not be linearly related. I t also permits using the experimental equilibrium data of total aromatics taken up by the gel, n, for a given liquid concentration, c, directly, without the necessity of subtracting the gel pore volume. Pore volume determinations are often unsatisfactory because they are strongly affected by the type of material being adsorbed. Based on these assumptions and using spherical coordinates, the equation for diffusion into a single sphere is
/3 z 1 . 5
P ~0.6
Boundary conditions. t = 0,O Ia < R; n=O where
Figure 2. Surface adsorbate concentration of a falling particle in a countercurrent adsorption system increases with time Linear equilibrium in a moving-bed system
726
INDUSTRIAL A N D ENGINEERING CHEMISTRY
D
=
n = a = t = R =
J
Integrating
Substituting Equation 2 and rearranging,
Boundary conditions t = 0; c = co where c g = aromatic concentration of liquid in contact with the gel that has just entered the column. The following parameters are introduced r =
r = a/R
X r
u = n/no
where no = aromatic concentration on the sphere's surface which is in equilibrium with cg. rhe aromatic concentration of the surrounding liquid at zero time. Then Equations 1 and 4 are transformed to
Boundary conditions. r = 0, 0 u=o,~>_O,r=O; u=O
5 r < 1;
Boundary conditions T = 0; u,=~ = 1. All that remains is to sex c as a function of u at the surface of the sphere. As there is assumed no film resistance, this means that c is related to u,,~ by the equilibrium isotherm. For this study a Langmuirtype isotherm is assumed; of the form c =
internal diffusivity point aromatic concentration in sphere = J ( a , t ) distance from center of sphere time radius of sphere
Dt/R2
1
1
K Z - Kl n
--
up=m __-__ Kz - K1 u r F l n o
(5)
When Kl = 0 the isotherm reduces to the linear form
MACHINE COMPUTATION I N PETROLEUM RESEARCH Analytical solutions for the linear isotherm type of adsorption have been reported by Kasten and Amundson (5). There are three equations for various ranges o f p = 3( ;)KZ P 3
where si is the ith positive root of the equation scot s = 1
- s2/p
and m is the positive root of m ctnh m = 1
+
m2/P
Solution of Equations for Nonlinear Isotherm
When K1 # 0, an analytical solution is not possible and numerical methods are necessary ( 7 7). Returning to Equation 5 and differentiating with respect to 7 : de' -
& - Kz
Kino
du,-
- K~u,=lno
1
~
dT
Substituting in Equation 4'
300 UT 1 - Ar 75
U+,l-SAr
+
+ 300
UT.1 - 2Ar
UTj1-4Ar
- 12
-
-
and thus u, 1 will obviously decrease with time. S/V is the ratio of gel to liquid amount in the vessel. As Figure 4 describes the linear isotherm case, these curves have been compared with analytical solutions (9). Here, too, the agreement is within 10% for 7 values above 0.05. Figure 6 shows how the adsorbate is distributed through the gel particle with increasing time in a moving bed. Figure 7 shows the analogous picture for the batch case.
UT.I-SAT]
The finite-difference Equations 10 and 11 were then programmed for the IBM 650 computer, using the Bell Laboratory interpretive system. The testing of the program and the computation of u us. 7 for 24 parameter sets of p and C took about 5 machine hours. This time could have been reduced by programming in basic machine language. us. Figures 2 and 3 are plots of u, 7 for C values of 0, 0.05, and 0.10. The p parameters are 0.6, 1.5, 3.0, and 6.0. Because in Figure 2, C = 0, these curves may be compared with those for the analytical solution of the linear isotherm case. The u,,~ values by the numerical method agree within 10% with those obtained by the analytical method ( 5 ) for values above 0.05. At very small T values, the u gradient is so steep near the particle surface that numerical approximation admits considerable error. However, as 7 increascs, the numerical solution converges quickly to thd analyticaI solution.
Physical Significance
Figures 2 and 3 show how the surface adsorbate concentration of a falling particle in a countercurrent adsorption system increases with increasing time. The concentration is measured as the ratio of surface concentration at a given value of 7 to the surface concentration a t zero time (at the top of the column). The ratio U,,~ depends on a time parameter 7, an equilibrium parameter, 0, and a nonlinearity parameter, C.
- - _ c =0.10
-c = 0.05 or letting
When Equations 1' and 9 are transformed into finite-difference equations, the following equations are obtained UT + A T , ,
3uIl
=
0
For numerical computation, values of for Ar and l / a for Ar/(Ar)2 were selected. I n testing the program, Ar
'/e
/I
0.1
0.2
0.3 0.4
05
li
0.6
0.7
0.8
09
1.0
Figure 3. Surface adsorbate concentration of a falling particle in aIcountercurrent adsorption system increases with time Nonlinear equilibrium in a moving-bed system
VOL. 50, NO. 5
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727
c * o 00 8'0 6 p'1.5 0.093
p.30
r3.O
0.2
p : 6.0
C
L.I
I
I
1/6
36
4/6
O O
2/6
CENTER
0.2 0
~0.05
I / 5/6
I SURFACE
Y
~
0.1
0.2 0.3
0.4 0.5 0.6
0.7 0.8
0.9
1.0
Figure 7. Relation of u/r to r for batch system
T
/3
= 3.0,
C
= 0.05, in a
Figure 4. Surface concentration decreases with time to a limit determined b y p and C ,
Linear equilibrium in a batch system
I0.C
80
70
80
0.2 0
I 0.1
I
1
I
I
0.4
0.2 0.3
I
I
I
I
0.5 08 0.7 0.8 0.9
i
I
i
f
/+Q=
30
25 3.511
I
m
1.0
T Figure and C
5. Surface concentration decreases with time to a limit determined by /3 I
Nonlinear equilibrium in a batch system
5
c
0.5
4-=!
$
0.7
0.8
F
0.9
1.0
3
I
1 116
Center
728
0.6
Figure 8. Ratio of particle residence times of moving-bed to batch systems for a given fractional approach to equilibrium
0.926
2
Figure 6. system
I
I 36
I
26
I
416
I 516
Surface
I-
Relation o f u / r = ./no to r for /3 = 3.0,
INDUSTRIAL AND ENGINEERING CHEMISTRY
I I
C = 0.05,
in a moving-bed
The ratio of adsorbate concentration in entering liquid to concentration in the effluent from the top of the column can be directly measured as u, only if the adsorption equilibrium is linear. Otherwise, u,, must be calculated by using the Langmuir equation. In Figures 4 and 5 for a batch system, u, I has the same significance, but here the surface concentration decreases with time to a limit determined by the values of p and C.
M A C H I N E C O M P U T A T I O N IN, PETROLEUM RESEARCH I
Comparison Moving Bed
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 GO)
= S(E
=
Kzc
m
Therefore
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. O n 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 u,,,’s 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.
V(c -
ii = n,,l
of Batch and
- ED)
=
The average aromatic concentration of the particle at zero time, EO, equals zero and the average particle concentration at infinite time is
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
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
(1) Davis,
(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 m2/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. O n page 330, Table I, the a in front of the footnote should be deleted. I n column 1, in the first paragraph below the tables, line ll, “are aper” should read “area per.” VOL. 50, NO. 5
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