Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
189
Numerical Design Techniques for Staged Classified Recycle Crystallizers: Examples of Continuous Alumina and Sucrose Crystallizers Alan D. Randolph* and Chung-Sung Tan Department of Chemical Engineering, University of Arizona, Tucson, Arizona 8572 1
Simultaneous population and mass balance equations were solved together with power law kinetics of nucleation and growth rate in computer Programs RECYC and SHUG to predict the crystal-size distribution from a multistage classified recycle crystallization process. Transformation from nonhomogeneous ordinary differential equations to moment equations made it convenient to model many complex configurations and resulted in a significant saving of computer time. The population distribution of the last stage was assumed to be a distribution in these calculations. This critical assumption not only provides enough data to calculate recycle moments but also avoids the numerical errors inherent in other techniques for the inversion of moments to obtain CSD. The effects on CSD of the form of the classification function were studied for hypothetical self-nucleating alumina and externally seeded sugar processes. For the alumina process, the recycling of a large fraction of classified particles increases the yield and mean particle size and narrows the CSD. For the sugar process, mean size is determined by external seed rate. Again, the distribution is narrowed with increasing recycle of classified crystals. Production rate increases when recycling more undersize particles due to the increase in specific surface area. The phenomenon of size dispersion due to growth rate fluctuations limits the ultimate narrowness of CSD that can be obtained in a staged classified recycle sugar crystallizer.
Introduction There are many examples of crystallization processes which require size-classification and recycle of the classified slurry upstream in a staged process. At one extreme, if sufficient seed is not recycled relief of supersaturation drops, while if too much recycle occurs particle size may decline and yield drop due to solids lost through the solids classification overflow. Typically such systems are of the Class I (variable yield) type and have such large volumes of mother liquor relative to solids that total filtration is impractical. Examples of chemicals that utilize such a recycle crystallizer configuration are, e.g., borax decahydrate or alumina (bauxite process). Another example, not yet commercially implemented, which would require such a staged, classified recycle process is continuous raw sugar crystallization. Such a continuous process would be staged and classified to achieve the narrow CSD specification rather than to increase yield by raising solids content. In fact, solids content in the latter stages of a continuous sugar process would have to be lowered by recycle of mother liquor from the centrifugation station. The essential features of such a staged, classified recycle process are shown in Figure 1. In a typical application of such a crystallizer configuration CSD performance is an important consideration in determining yield (supersaturation relief and solids recovery), determining dewatering capacity, and meeting product specification. Current techniques (Randolph and Larson, 1971, Chapter 8) for simulating crystallizers of complex open-loop configuration, e.g., classified product removal, fines destruction, settled liquor advance, and interstage product takeoff, are inadequate to model the recycle configuration of Figure 1.It is conceivable that numerical simulators of the open-loop type could be used iteratively to model recycle systems, as illustrated by the solution algorithm diagram in Figure 2, but the mechanics would be cumbersome and calculation time could be prohibitive. The purpose of this paper is to outline two techniques for the numerical analysis and design of such staged classified systems with recycle. The first technique comprises a total rigorous numerical solution of the distributed CSD equations while the second technique
is an approximate method utilizing a moment transformation of the ODE describing each mixed stage together with an approximate recovery of the size distribution from the last stage. The moment approximation is particularly efficient for calculating average properties of the distribution, e.g., mean size and variance, while the appr,.rimate recovery of the last stage distribution allows for the mathematical implementation of classification and recycle. Development of System Models Technique I. Complete Numerical Solution of Recycle Equations. Figure 1illustrates the classified recycle problem to be solved. The particular system inodeled was comprised of two mixed tanks followed by a solids classifier with recycle of intermediate-sized crystals. Two points regarding this configuration should be made. First, the assumption of ideal mixed stages is not necessary to the development of the complete numerical solution technique. In fact, an individual stage configuration as complex as those developed in previous open-loop simulations may be used. (MSMPR stages must be assumed for the approximate moment algorithm as discussed in a subsequent section.) Recycle is assumed to be from the external classifier at the end of the process, although there is no limitation on recycle from an internal stage classifier. In general, the classification equipment would be such an expensive component of plant that only one tail-end classification system would be provided (e.g., Bayer bauxite process). Second, the classification system as illustrated by two hydroclones in series is arbitrary. Equivalent classification systems in terms of two or three settlers, a wet screen and a settler, two classifying centrifuges, etc., could be devised. Obviously, particle separation is an important cost factor for any given system but all that is needed for CSD analysis is the size-separation efficiency curves of the particular classifying devices. The modeling equations describing the recycle process of Figure 1 can be summarized as follows. Define
C I ( L )= classifier 1separation curve, percent of feed particles of size L reporting to underflow
0019-7882/78/1117-Ol89$01.00/0 0 1978 American Chemical Society
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Growth rate determined f r o m growth kinetics:
CLASSIFIER 8 SEPARATOR
r - - - -- - - - i
for j’th stage
Figure 1. Schematic of continuous multistage classified recycle crystallizer.
C 2 ( L ) = classifier 2 separation curve, percent of feed particles of size L reporting to underflow [Note: The numbering of the classifiers is reversed from their order in the process. Notation is determined by the size of the classification cutpoints, namely (L50)z> (L50)l.l
These several forms of the mass balance were implemented in different versions of the numerical simulators which were developed and are useful for specific crystallization problems.
Nucleation Kinetics Bj0 = kNjGjlMTJJCj’th stage, Class 11system)
(7)
Free Liquor V o l u m e
nj = population density in the j’th stage
V , = volume in the j’th stage Qj = liquor discharge from the j’th stage M T ~= solids concentration in the j’th stage The CSD modeling equations for this system in their conventional form (Randolph and Larson, 1971) are set forth below. Population Balance
dn. G . V . I = Qj-lnj.-l - Qjnj dL
’
with boundary conditions n j ( 0 ) = njo = Bj”/Gj Mass Balance. Several useful forms of the mass balance can be written, depending on system constraints. Thus slurry
som
density maintained constant: MT, = pk,
njL3
dL
(4)
Growth rate constrained by solute resources:
In general the above set of equations will contain more unknowns than equations and certain of the variables must be treated as externals (constrained) to obtain a solution. The choice of constrained and dependent variables gives rise to the various forms of the numerical simulators that were developed. This paper discusses three numerical CSD simulators having the various forms as outlined in Table I, namely Programs MARK 1 (Nuttal et al., 1975), S H U G , and RECYC. The former program simulates open-loop systems while the latter two programs simulate the classified recycle configuration of Figure 1with complete numerical and approximate moment algorithms, respectively. A particularly useful selection of variables for a sugar crystallizer is to fix Gj, Vj, and M T ~and treat Qj and PT as dependent variables, e.g., Program SHUG-2. This choice is dictated by the fact that M T must be limited to a maximum value which will allow circulation in the pan while supersaturation (growth rate) can be controlled independently by evaporation rate and must be limited to a maximum value to avoid formation of false grain (secondary nucleation). The choice of externals vis-a-vis dependent variables is obviously a designers choice and determines the specific form in which the equations are solved. Useful design properties of the CSD are then obtained from solutions of eq 1-8 as follows. production = Q k p k v
(5) for j’th stage
Figure 2. Series solution algorithm used in Program MARK 1.
solids losses = QkpkV
sorn
sorn -
CznkL3 dL
(1 C2)(l - C l ) n k L j dL
(9)
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
191
Table I. Implemented Forms of Recycle Equations Program
Version
Constrained variablesa
Dependent variablesa
Function
G, MT, B o ,V Class I1 open-loop systems MARK 1 A P,Q P, M T , BO, V , s Class I open-loop systems MARK 1 C Ci,Q G, B O , V, P Class I1 evaporative open-loop systems D MT, Q Mark 1 Q, PT Evaporative, Class I or 11, SHUG 2 G, MT, V Q, V Classified recycle systems, seeded SHUG 3 G, MT, PT G, MT, B o ,V Class I1 classified recycle systems RECYC 1 P, Q P,M T , B O , V, s Class I classified recycle systems RECYC 2 Ci,Q G, BO, V , P Class I1 evaporative classified recycle 3 MT, Q RECYC Q, PT Evaporative, Class I or 11, RECYC 4 G, MT, V Q, V Classified recycle systems, seeded RECYC 5 G, MT,PT a Note: Unsubscripted P is production from given stage; PT is total production from unit.
supersaturation = s = s ( G )
0
(11)
start
where the function s(G) is the inverse of any empirical growth rate-supersaturation relationship. Such growth rate relationships are probably best obtained from batch relief-ofsupersaturation experiments using weighed amounts of carefully sized seed crystals.
soL
n Read d a t a , initialize
C2nkL3 dL
cumulative CSD = W , =
moments (mk), =
som
LkC2nxdL
(12)
La
LknjdL
1
I
(for product) 1
A
or
( m k ) 2=
I
(for j'th tank)
(13')
Mean (mass) size
Coefficient of variation
The statistical averages and CV can of course be evaluated in any stage or for the product. The usefulness of the numerical simulation described by eq 1-15 depends strongly on the algorithm used to implement the solution. Figure 2 outlines an algorithm in which an open-loop CSD simulator (e.g., MARK 1) for complex configurations is used iteratively to solve the recycle case. In such iterative applicatons of an open-loop simulator the recycle distribution is treated as an external. The initial guess for the function n~ is updated after each pass. The solution experiences the slow convergence typical of such recycle systems. The MARK 1 program has been used for such recycle calculations in industrial applications but a t great computation expense and limited success. An alternate solution algorithm is to solve the stage equations in parallel, e.g., with a multivariable Runge-Kutta integration routine. Thus for the two-stage SHUG simulator the central equations
are solved simultaneously in a single integration routine. In
Calculate d e s i r e d properties of f i n a l d i s t r i b u t i a i
1
n W Figure 3. Parallel solution algorithm used in Program SHUG. the above, y1 and y2 are the dimensionless distributions representing nl and n2 and SI and S2 are dimensionless third moments representing M T and ~ M T Figure ~ 3 illustrates this parallel solution algorithm which was implemented in program SHUG for study of a continuous sugar crystallizer. In both the MARK 1 and SHUG algorithms convergence of the system equations was by direct substitution of a weighted average of the previous and just-calculated variables. Convergence for the parallel algorithm could quite likely be speeded up using a multivariable directed search algorithm, e.g., Newton-Raphson, but this was not attempted. A typical execution of Program SHUG would require 25 direct substitution iterations requiring 400 s of CPU time on a CDC-6400. Convergence sensitivity was observed; a maximum weighted average of 40% of the just-calculated value could be used for the next iteration. Higher weights produced unbounded numerical oscillations. Technique 11.Moment Approximation. If each stage of the process is of MSMPR configuration and the recycle stream
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is not classified, the process of Figure 1can be readily simulated by a closed set of algebraic moment equations in terms of moments of the distribution in each stage, where
represents the k’th moment for the j’th stage (Broadfoot and White, 1975). These algebraic moment equations must be solved together with the mass balance constraints, eq 4,5, or 6, which introduces nonlinearities but retains the algebraic character of the problem. At worst this formulation reduces to a nonlinear set of algebraic equations which can be solved by standard iterative techniques. Thus, for the staged problem of Figure 1 the moment equations become mo,] = mo,J-l ml,] = mlJ-1
+ rBJo+ ol-l@mo,R
+ GTmo,] + Oj-l@ml,R
(16)
For j = 1,i.e., the first stage, the additional term @mk,Ris included in the right-hand term to account for the slurry recycle, where @ is the fraction recycled. However, when size-classification is introduced, either in the recycle stream or individual stages, the simple constant-coefficient moment transformation is no longer applicable. The numerical problem is that the recycle moments mR (necessary for the first stage calculation) cannot be obtained simply in terms of the last stage distribution due to the nonconstant coefficients introduced by the classification functions. Thus
In the following development it was assumed that each stage had the MSMPR configuration but that the recycle stream was classified. Classified recycle was accounted for while still using the numerically efficient moment transformation by the following technique. (a) The weight-size density distribution leaving the last (k’th) stage was assumed to be a r distribution. Thus
The above integration was implemented with a Simpson’s rule algorithm. Having the recycle moments ( m k ) ~the , entire solution can be carried out efficiently in the moment domain. Note that as all the I?-distribution parameters are readily available from simple algebraic relationships, the only numerical penalty of this technique is the computer time required for integration to get mR. The technique gives orderof-magnitude reductions in computation time compared with complete integration of the ODE’Sfor each stage. For example, a two-stage simulation required 36 s of computing time on a CDC-6400 computer compared with 8-10 min using the SHUG-I11 algorithm. A series convergence algorithm similar to that shown in Figure 2 was used to solve the mass balance constraint for each stage. The technique proved to be quite accurate for a number of cases as shown in the following section. The assumption of MSMPR stages with tail-end classification is valid for most staged processes (e.g., bauxite) because duplicating the classification system for each stage would be prohibitively expensive. The assumption of a r distribution for the last stage appears to be quite accurate; errors due to this assumption are most likely much less than the ability to define the size performance of a classification system. Effects of Size Dispersion on Model. White and Wright (1971) have shown that the phenomenon of size spreading due to random fluctuations in growth rate produces a widening of the CSD in configurations that would otherwise produce a very narrow distribution. Randolph and White (1977) model this size spreading with a size dispersion coefficient. Thus population flux is given by both convection and dispersion terms as
where DG is the growth rate diffusivity or size dispersion coefficient. When eq 18 is introduced into the population balance equation for an MSMPR stage a second-order ODE results. Thus dn
n
d*n
G -dL+ - =7 DG -rjdL (b) The r-distribution parameters are determined simply from the known k’th stage moments so as to match the weight mean and CV between the two distributions. Thus a=
Fortunately, eq 19 is still acceptable for the moment transformation. The MSMPR moment equations modified for size dispersion are shown as follows (Randolph and White, 1977).
+ rBjo + oj-l@mo,R m l j j = ml,j-1 + Grmo,j t rDGn(0) + oJ-l@ml,R mo,j = m0,j-l
m24,k -1 m3,km5,k - m24,k
and
mk,j
mkj-1
+ kGrrnk-l,]
+ k(k - l ) r D ~ m k - 2 ,+j oj-l@mk,R (c) The above r distribution is converted to the associated population r distribution. Thus nk = cr,(L, a’, 6’) where a’ = a - 3 and b‘ = [ ( a - 3)/a]b. The constant c is determined so as to give the same slurry density in the k’th stage. Thus C =
m3’k
+
= ma,k(C~’/b’)~/[(U’3)(U’
(19)
+
2)(U’
(20)
Introduction of the second-order term due to size dispersion requires two boundary conditions as follows.
L = 0; Bo = noG - D ~ r i ( 0 ) L-. m ; n - + O The liquid phase mass balance (eq 6’) must be modified slightly when size dispersion is considered in the model. Thus
-k I)]
(d) The recycle moments are then calculated from the distribution and the classification functions as
rp
Equation 6’’ was obtained from the total liquids plus solids mass balance by substituting the population balance recursion
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Table 11. List of Computer Input Data for Comparison between RECYC, MARK 1, and SHUG Saturation concentration Growth kinetics Nucleation kinetics
Inlet concentration Growth kinetics Nucleation kinetics
Feed flow rate Parameters in r distribution of the seed Growth kinetics Nucleation kinetics Classification function
Feed flow rate Parameters in r distribution of the seed Growth, nucleation kinetics and classification as run 3. Feed flow rate Seed Growth kinetics Nucleation kinetics
Run 1 0.1 g/cm3; 0.1 g/cm3 G = k,(C - C*); h , = 1025 B o = k ~ G ' , ~ h ' kf N~ =~ 10 , ~ n0./Cm3, pm
Run 2 1.5 g/cm3;sat. concn 0.1 g/cm3;0.1 g/cm3 G = k,(C - C*); h , = 50 Bo = ~ N G ~ . ~ M kNT =~ 10, no./cm3~, Fm Run 3 277 gpm, seed 0.1 g/cm3 a' = 4; b' = 400 pm
? ? To mm mm N N NN mm mm
bb
~m
l d : m,? ? " ? m m or- m a w mr- m N m m m
m m mm
ar-
m m ?d: o o m , ~ ? o m 0 e 0 0 0 "N r-mm e* m o o 3 - 3
o0 0o 0o o0 &A
N N
,,A, N N
a m
a 0 N O
mm xx m a m m Tt- X N
0 -
_ c;L$;a;
lnmm
0c N m x
G = k,(s - 1)2;h , = 54.9; 13.7
B o = ~ N Gh~; = 0.1 no./cm3-pm C&) = 0.5; L < 500 pm 0.5-1.0 L in (500,700) 1.0 L > 700 pm C I ( L ) = 1.0 for all L
000
00
0 0 0 0 0 0
bb
b N b N
0 0
N m m O 0 0 0
(OW
m a
0 0
mcl
Run 4 277 gpm, seed 0.1 g/cm3 a' = 10; b' = 400 prn
Run 5 100 gpm; inlet concn 0.35 g/cm3 0.0 g/cm3 G = k,(s - 1)2;k , = 0.38 B o = ~ N Gh~; = 1.0 X 106no./cm3wn
0 0 0
relationship for r n 3 in terms of r n 2 and m l . The diffusivity term in eq 6" represents a small correction due to the randomly fluctuating growth velocity. Size dispersion was added to the approximate moment equations of Program RECYC to test its effect in limiting the ultimate narrowness of CSD that could be obtained in a staged classified recycle sugar process. Numerical Results A computer program titled RECYC was develop which simulated multistage classified recycle processes using the approximate moment technique. Five forms of RECYC were implemented as described in Table I. This program was compared for appropriate case studies with the rigorous SHUG and MARK 1simulators to demonstrate accuracy and then used for predictive simulations of continuous bauxite and sugar processes. A large set of algebraic equations are solved numerically in the RECYC program and iterations are continued until acceptable convergence is achieved. The convergence criteria used in Program RECYC were that I AG I /G and I AmRI / r n R less than 5 X and 2 X respectively, for any two consecutive iterations. In order to test the accuracy of simulator RECYC the program was compared with existing program MARK 1 and the rigorous classified recycle algorithm of program SHUG. Two-stage processes are chosen for comparison since simulator SHUG is only implemented for two-stage systems. No stage limitation (other than total computer running time) exists for program RECYC.
t:
193
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Table IV. Conditions Employed for Five-Stage Alumina Process Variable
0.1)
Quantity
Crystal density, g/cm3 Shape factor Inlet concentration, g/cm3 Temperature, "C Saturation concentration, g/cm3 Volume, gal
t I
Run gpm
75,70,60, 50,40 0.11, 0.087, 0.068, 0.048, 0.042
20 30 40 50
Yp,
%
%
91.99 90.15 88.57 87.16
91.89 90.00 88.40
86.99
P, tondday
r4,3',
34.00 49.95 65.42 80.47
73.81 73.79 73.77 73.76
pm
(CV)Pw,t. R 0.218 0.217 0.215 0.214
1.25 1.25 1.25 1.26
Table VI. Effect of Recycle Fraction on Yield and CSD for Five-Stage Alumina Process (See Figure 4) ( L 1= 30 pm, Lz = 40 pm, L3 = 65 pm, L4 = 70 pm)
I\
. "
.
.
10 15
L,
L2
'
I
!
L3
L4
Run
ye,
yp,
a
%
%
tons/day
pm
(CV)P,,,,
R
9 10 11 12
0.1 0.2 0.3 0.4
87.16 86.78 86.47 86.10
86.99 86.63 86.37 85.78
80.47 80.42 79.99 79.34
73.76 69.63 64.93 57.53
0.214 0.287 0.357 0.469
1.26 0.98 0.73 0.41
P,
L,3P9
I
SIZE, MICRONS
OO
6 7 8 9
ye,
2500,2500,2500,2500,2500
Ili 1 '1
0.01
Q,
2.42 1.0 0.35
IT
II
Table V. Effect of Volumetric Flow Rate on Yield and CSD ( L 1 = 30 pm, Lz = 40 pm, L3 = 65 pm, L4 = 70 pm, a =
11i
10 I S
S I Z E , MICRONS
Figure 4. Classification functions for a five-stage alumina crystallizer.
Open-Loop Comparison of RECYC with MARK 1. Since the MARK 1 simulator does not consider a recycle stream, the classification function C z ( L )was set equal to 1.0 for all particle sizes (no classified recycle; see eq 17). Runs 1 and 2 in Tables I1 and I11 show the input data and results, respectively, for this open loop test of the two simulators. Run 1 represents a high yield (Class 11) system while Run 2 tests a variable yield system. The kinetic parameters are arbitrary, but are typical of inorganic systems, e.g., KCl. Good agreement is found between simulators RECYC and MARK 1as shown in Table 111. Comparison of RECYC and SHUG for Two-Stage Classified Recycle Configuration. Runs 3 through 5 in Table I1 summarize external variables and parameters used for the comparison between RECYC and SHUG for a classified recycle process. The parameter h~ in the nucleation rate model is 0.1 in Runs 3 and 4, indicating that nucleation is suppressed in this sucrose process. For Run 5 , k~ is 106, indicating self-seeding by secondary nucleation in both stages. Table I11 shows the results from both simulators for Runs 3-5. The error terms in Table I11 are calculated based on eq 5 . The good agreement between these three computer programs suggests that Program RECYC is equivalent to the rigorous numerical calculations of Programs MARK 1and SHUG. As the only approximation in Program RECYC was in the assumption of a best-fitting r distribution for the k'th stage (best-fitting in the sense of matching the CV and mean on a weight basis) this approximation seems well justified. The use of algebraic (moment) equations rather than differential (as
in MARK 1 and SHUG) gave a t least a 10-fold reduction in computational time for all comparison runs studied. In fact, the five-stage alumina process to be considered in a subsequent section would not be feasible without using the moment transformation. Predictive Simulation of Continuous Alumina and Sucrose Crystallizers. Program RECYC, having been demonstrated to be equivalent and more efficient than Programs MARK 1 or SHUG, was used to predictively model the CSD, yield, and/or production performance of multistage classified recycle alumina and sugar crystallizers. The alumina system (Bauxite process) was chosen because it represents an important commercial application of this particular crystallizer configuration. Further, published nucleation and growth rate data exist which make yield calculations meaningful. The raw sucrose crystallization system was chosen because of the importance and difficulty of producing a narrow CSD in a continuous process. The staged recycle process of this paper offers the possibility of solving this problem with a processfeasible configuration. Analysis of a Five-Stage Alumina Crystallizer. Table IV lists the conditions assumed for alumina crystallization. The growth rate kinetics proposed by Misra and White (1969) are used, as follows. where G is in pm/h and s is in g/L. Most literature sources discuss nucleation in caustic aluminate solution qualitatively and fail to offer a reasonable nucleation mechanism. Though Misra and White (1969) proposed a power law model for nucleation rate kinetics, they observed that the parameters in their model depend on temperature and seed quantity. Hence, their nucleation rate model was not used in the present work. However, they show that a t temperatures of 45 to 60 "C and under a wide range of supersaturation and seed quantity the nuclei density is in the range of 106 to 5 X lo6 no./cm3-pm. Because of a lack of accurate information on power-law nucleation kinetics it was assumed that the nuclei density approached 106 no./cm3-pm in all stages. This constant value of 106 was used in all subsequent alumina crystallizer calculations. The solids recycle ratio in a five-stage unit is defined by
For this cooling process, the discharge flow rate from each
Ind. Eng. Chern. Process Des. Dev., Vol. 17, No. 2 , 1978
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Table VII. Effect of Small Particle Recycle on Yield and CSD for Five-Stage Alumina Process (See Figure 5 ) ( L 3 = 65 Fm, L4 = 70 Fm, a = 0.1)
L1, Pm
Lz, Pm
YB,
yp,
%
%
P, tonslday
t 4 . 8 ,
Run
Pm
(cv)pw.t.
R
13 14 9 15
20 20 30 30
30 40 40 55
81.46 87.32 87.16 86.30
87.25 87.31 86.99 86.25
80.70 80.76 80.47 79.78
74.47 74.19 73.76 65.55
0.i72 0.191 0.214 0.388
1.43 1.34 1.26 0.69
Table VIII. Effect of Large Particle Recycle on Yield and CSD for Five-Stage Alumina Process (L1 = 30 bm, Lz = 40 bm, a = 0.1)
L, fim
L, wm
YP,
%
YB, %
P, tonslday
L P ,
Run
pm
(CV)PW.t.
R
16 17 9
85 70 65 55
95 80 70 65
87.13 87.23 87.16 86.82
87.12 87.28 86.99 86.82
80.60 80.73 80.47 80.31
94.49 80.79 73.76 65.71
0.176 0.190 0.214 0.280
2.05 1.60 1.26 0.87
18
I
io
20
40 5s SIZE, MICRONS
30
I
1
6s m
Figure 5. Classification function C z ( L ) of small particle recycle in five-stage alumina crystallizer. stage is assumed to be the same. Thus eq 22 reduces to
Figure 6. Cumulative weight distributions for five-stage alumina crystallizer with Ls = 65pm, L4 = 70bm, and a = 0.1.
In eq 23 the variable R is the ratio of recycle solids to the total per-pass relief of supersaturation through the process. The production yield is defined by
or
where C* is saturated concentration, YB is calculated based on the relief of supersaturation and Yp on the production rate. YB should be greater than Yp for the alumina system, since some fines will not be captured by the classifier and will report to the mother liquor overflow rather than the product filter. The classification function for this system is shown in Figure 4. (a) Effect of Volumetric Flow Rate. Referring to Figure 4, the classification sizes L1 to L4 were fixed a t 30,40,65, and 70 Mm, and a was set to 0.1. For the parameter choices listed above (and in Table IV) the effect of changing volumetric flow rate on the product yield and CSD is found in Table V. The coefficient of variation and mean size of the product weight distribution shows almost no change for different flow rates; hence, despite different flow rates a narrow size distribution with similar mean size can be obtained. Total product rate
IO
40 55 6570 eo 85 95 SIZE, MICRONS
Figure 7. Classification function C z ( L ) of large particle recycle for five-stage alumina crystallizer. increases with increasing flow rate, but yield drops. In subsequent calculations, volumetric flow rate is fixed at 50 gpm. These further calculations utilized different classification parameters and attempted to find conditions to improve product yield while keeping a narrow size distribution. (b) Effect of Classifier Efficiency. The effect of classifier cut fraction a on the product yield and size distribution is significant, as illustrated in Table VI. For these calculations the classification sizes L1 to L4 weretaken as 30, 40, 65, and
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2
- 0.35
n 3
n - 0.30: 0
- 0 . 2 5s9 D
z! z
- 0.202
- 0.17 I
0.40.5
1.0
I I .s
1
-
10.15
2.0
RECYCLE RATIO, R S I Z E , MICRONS
Figure 8. Cumulative weight distributions for five-stage alumina crystallizer with L1 = 30wm, Lz = 40wm, and a = 0.1.
05
I O
I 5
20
RECYCLE RATIO, R
Figure 9. Calculated relation between recycle ratio and mean particle size for five-stage alumina crystallizer. 70 pm, respectively. Moving the horizontal line represented by a in Figure 4 downward (increasing recycle) gives a small but definite increase in yield while producing a narrower distribution. (c) Effect of Small Particle Recycle. Table VI1 shows the effect of small particle recycle on yield and CSD. Recycling most of the undersized particles (shown in Figure 5 in the direction of the arrow) increases the product yield and results in a narrow size distribution. Product yield increases over 1% under such operation, a significant improvement in the crystallization process. The calculated recycle ratio also increases from 0.69 to 1.43. Figure 6 shows the cumulative product weight distribution for these four runs. Note the increase in CV and decrease in size and yield when the classification size L2 is raised to a value near the product size. Thus, yield in the calculated five-stage alumina crystallizer is increased and the product CSD narrowed by the use of increased solids recycle, whether this increase is achieved by lowering the classification size or increasing the classifier efficiency (lowering a ) . Such process improvements of course result in higher capital and operating costs. (d) Effect of Large Particle Recycle. Extending the absolute size range of recycle particle sizes (as shown in Figure 7) results in a higher recycle ratio and a narrow size distribution with a large product mean particle size. However, the product yield is not improved significantly by recycling such large particles. The results of these calculations are shown in Table VIII. The mean particle size in Run 16 increases to 94.5 pm with a CV of 0.18 (a very narrow size distribution). Thus recycling particles (undersize or oversize) plays an important
Figure 10. Calculated relation between recycle ratio, product yield, and coefficient of variation for five-stage alumina crystallizer. role in determining the crystal-size distribution. The cumulative distributions for these runs are shown in Figure 8. (e) Optimal Operating Conditions. Figures 9 and 10 show the relations between recycle ratio, mean particle size, coefficient of variation, and product yield YB.It was found that if the recycle ratio is larger than 1.2, product yield is higher than 87%, CV is below 0.22, and the mean particle size is larger than 70 pm. Thus a higher product yield and a narrow size distribution with a large mean particle size can be obtained with the correct operating conditions. In order to make the recycle ratio as large as possible, the following conditions are suggested: (i) low values of the cut fraction a , i.e., high classifier efficiency; (ii) extend the size range of recycle particles, Le., shift L1 and Lz to the left and L3 and L4 to the right as shown in Figures 5 and 7. Additional runs in Table IX show that these two conditions satisfy the criteria for producing a desirable CSD and yield in a crystallization process. From Table IX it is observed that Yp is somewhat higher than YB. This is contrary to the relation YB > Yp stated previously and is probably due to slight numerical error in the integration of eq 9 to obtain production rate. Since most of the fines ( L I 15 pm) cannot be separated from the main stream, a certain amount of slurry is lost from the system. This mass loss can be expressed as follows. mLoss =
Q(Cc - C 5 ) - P + QiMT,
(26)
Alternately
If the r distribution assumption matches the true population distribution, mLoss should be equal to m’Loss.These two quantities are 0.045 and 0.021 tonlday, respectively, for Run ton/day, respectively, for 14, and 7.3 X 10-3 and 8.6 X Run 16. The big difference between these two calculated mass losses occurs due to the deviation of the assumed r distribution from the real distribution for small particle size. The r distribution predicts zero population as L 0, contrary to the true distribution for a self-nucleating system. Thus, the model fails in the prediction of product overflow losses. However, in the specific alumina example this mass loss is negligible compared with total product rate, and the error in overflow losses does not significantly influence total calculated yield. In general, if classifier overflow particles are small enough to cause a serious error in the r distribution data fit, overflow losses will be negligible. Analysis of a Three-Stage Classified Recycle Sucrose Crystallizer. Two major goals for a continuous raw sugar
-
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
197
Table IX. Selected Operating Conditions on Yield and CSD for Five-Stage Alumina Process
L1, Run
pm
Lz, pm
19 20 21
25 25 25
35 35 30
L3, ym
L4, pm
80 85
90 95 95
85
YB,
YP,
a
%
96
p, tonslday
0.1 0.1 0.1
87.22 87.15 87.15
81.25 87.19 87.20
80.71 80.64 80.70
Table X. Input Parameters for Three-Stage Sucrose Process Parameter
L4,3P, pm
(CV)PW,t.
R
90.0 94.4 94.5
0.173 0.173 0.173
1.93 2.08 2.06
100
Quantity
Crystal density, g/cm3 Shape factor Seed slurry density, gIcm3 Volumes, gal Growth rates, pmlmin Solids concentration, g/cm3
-
I-
1.59 1.0 0.1 2000,4000,6000 4.3, 3.1, 1.9 0.93,0.97, 1.11
E80 K W
s
so
W
?
0
$40
0
W t h Classified Recycle Without Classified Recycle
3 A
a
Table XI. Effect of Number of Stages on Production and CSD for Staged Sucrose Process (L1 = 400 pm, Lp = 800 pm, a = 0.5)
3
u
PO
-
Run
Totalstages
P, L4,3p, tonslday pm (CV)p,,t,
22 1 635.7 23 2 818.3 24 3 172.4 25 4 785.1 a Recycle ratio ( M T R I M T 3 ) .
771 828 795 793
0.275 0.263 0.222 0.213
0
Ra 0.195 0.150 0.135 0.153
400
600
800
1000
1200
1400
SIZE, M I C R O N S
Figure 12. Comparison of cumulative weight distribution for threestage sucrose crystallizer with and without classified crystal recycle.
Table XII. Effect of Seed Rate on Production and CSD for Three-Stage Sucrose Process ( L 1= 400 pm, L2 = 800 pm,a = 0.3) lot
I I
0 0
I
I
I
I I
I I
I
I
Ll L2 SIZE, M I C R O N S
Run
gpm
Qi,
P, tonslday
L,3p, pm
(CV)p,.t.
R
26 27 28 29
280 320 350 396
680.5 717.0 743.0 782.6
845.0 820.6 805.5 784.5
0.220 0.214 0.211 0.207
0.119 0.135 0.147 0.165
B
Table XIII. Effect of Classifier Fraction on Production and CSD for Three-Stage Sucrose Process [See Figure 131 ( L , = 400 um. L , = 800 um)
00
B
SIZE, MICRONS
Figure 11. Classification functions for a three-stage sucrose crystallizer.
crystallizer are to obtain a narrow size distribution (CV 5 0.23) and a mean particle size around 800 Fm (Randolph and White, 1976). Calculations were performed with Program RECYC to design a high grade system having three pans in series of 2000, 4000, and 6000 gal working volume, respectively. The seed to the first pan was assumed to have a mean size of 400 wm with a CV of 35%, both on weight basis. The evaporation rate was assumed to be adjusted so as to give supersaturations of 1.14, 1.11, and 1.08 in the three stages, corresponding to growth rates of 4.3, 3.1, and 1.9 pm/min (White and Wright, 1971). Since supersaturation is controlled in the metastable
a
P, tonslday
L,3p,
Run
pm
(CV)pw,t,
R
30 29 31 32
0.2 0.3 0.4 0.5
786.3 782.6 779.7 775.5
780.7 784.5 787.3 791.8
0.198 0.207 0.215 0.223
0.196 0.165 0.135 0.108
range, nucleation does not occur in any of the stages. Thus seed particles provide the only grain source in the system. Table X lists the input parameters used in the simulation of this three-stage sucrose process. (a) Effect of Number of Stages. An increase in the number of stages primarily affects the production rate and crystal-size distribution. Table XI lists the calculated results assuming different numbers of stages in series having the same total volume. For this comparison total working volume and seed rate are equal. 151, Lz and a in Figure 11 are 400 ym, 800 pm, and 0.5, respectively. Larson and Wolff (1971) examined the effect on CSD of different allocations of the total working volume between the different stages for two- and three-stage systems without classification. In the present work, no analysis was made of the optimal allocation of total volume in order to
198
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Table XIV. Effect of Classifier Sharpness on Production and CSD for Three-Stage Sucrose Process [See Figure 141 (Lz = 900 pm,a = 0.5)
L1,
P,
Run
pm
tons/day
33 34 35 36
300 400 500 600
770.5 772.4 773.2 772.1
L.2,
Table XV. Effect of Large Particle Recycle on Production and CSD for Three-Stage Sucrose Process (L1 = 500 pm, a = 0.5)[See Figure 151
(CV)pw.t.
R
Run
L 21 pm
P, tons/day
L,3P9
pm
pm
(CV)P,,,
R
797.0 794.9 793.6 794.9
0.228 0.220 0.214 0.207
0.126 0.155 0.196 0.249
37 38 39 35
700 800 850 900
780.6 778.0 775.8 773.2
786.8 789.0 791.1 793.6
0.218 0.213 0.213 0.214
0.095 0.146 0.171 0.196
795
-
-022
v1
z
D
x
K 0
u
j790
n -021 E
-
-
m i z
N
0 W
-0207
u 785
400 SIZE, M I C R O N S
5
-
?? D
2
800
-014
z
q
Figure 13. Classification function Cz(L) for three-stage sucrose crystallizer (Runs 29-32).
780
1, 772
,
,
,
775
780
785
5
p
,+I8 790
PRODUCT RATE, l o n / d a y
Figure 16. Relation between product rate, mean particle size, and coefficient of variation for three-stage sucrose crystallizer (81= 396 gpm).
1
,
I
I
I
300 500 SIZE, MICRONS
900
Figure 14. Classification function C*(L) for three-stage sucrose crystallizer (Runs 33-36).
LI
1
I
500
1
I
l
l
700 BOO 9 0 0
SIZE, M I C R O N S
Figure 15. Classification function C&L) for three-stage sucrose crystallizer L1 = 500 pm (Runs 35-39). produce the narrowest size distribution. For the cases studied, the stage volumes were arbitrarily varied linearly between the first (smallest) and last (largest) stages. From Table XI it is seen that as the number of stages increases, the size distribution narrows. (b)Effect of Classified Recycle. For a three-stage crystallizer without classified crystal recycle the mean particle size was 808 pm with a CV of 26%. Using a classifier with L1, Lz and a 500 pm, 700 pm,and 0.2, respectively, the CV dropped to 19% and the mean size decreased about 10 pm. Under such operation, 18% of net production is recycled. Figure 12 shows the size distribution for these two runs. It should be noted that all sucrose calculations assume a narrow seed distribution (CV = 0.35); a wider seed would produce a wider product andlor a larger recycle ratio.
(c) Effect of Seed Rate. A seed density of 0.1 g/cm3 was assumed in the previous calculations. The effect of seed rate on the production rate and CSD is shown in Table XII. Mean particle size decreases with increasing seed rate since mean retention time must decrease as seed rate increases (in order to maintain slurry density at a constant level). These results show a tendency toward a narrower size distribution as seed rate increases. Product rate increases as the quantity of the seed increases, but with smaller mean size. In the remaining calculations, seed rate was set a t 396 gpm. (d) Effect of Recycle Fraction. The effects on production and CSD of the recycle underflow fraction, a , for a fixed classification size range of crystals are shown in Table XIII. An increase in particle recycle increases product rate, narrows the size distribution, and decreases the mean particle size. Figure 13 shows the classifier underflow parameters that were simulated. More production (holding MT and G constant) was obtained with increased recycle because crystal specific surface area increased. (e) Effect of Classifier Sharpness. A sharper classification cut is obtained by moving L1 close to Lp. This shift in L1 to the right (Figure 14) with L 2 and a fixed does not change production rate and mean particle size significantly, but a narrower product size distribution is obtained. More recycle is obtained with a sharper classifier cut. Table XIV shows the result of these four runs. (f) Effect of Large Particle Recycle. The recycling of more large particles (Figure 15) increases mean particle size but decreases the total product rate. The decrease in production (at constant MT and G ) is due to the decrease in specific crystal surface area. Table XV shows results for these runs. (g) Optimal Operating Conditions. The relations between product rate, mean particle size, and coefficient of vziation are shown in Figure 16. The relation between P and (L)4,sPis nearly linear, with increasing total production rate corresponding to decreasing mean particle size. The relationship between P and CV appears random; however, each CV is below 23% which is sufficiently narrow for raw sugar. CV does
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Table XVI. Selected Operating Conditions for ThreeStage Sucrose Process
L1, Run 40 41 42
43
pm
L2, pm ’ a
400 800 400 800 500 700 500 700
0.4 0.2 0.4 0.2
P,
E4,3p,
tons/day
pm
(CV)p,,t,
R
779.7 786.3
783.7 780.7 781.7 771.0
0.215 0.198 0.204 0.187
0.135 0.196 0.121 0.182
185.7
796.6
Table XVII. Effect of Size Dispersion on Production and CSD for Three-Stage Sucrose Process Run
L1, pm
L2, pm
a
P, tons/day
L J P 3
pm
400 700 0.3 784.1 783.1 400a 700 0.3 853.3 844.1 500 700 0.4 781.7 785.7 45 500n 700 0.4 853.6 843.9 794.9 400 900 0.5 772.4 46 400n 900 0.5 836.1 858.1 Size dispersion included in model. 44
(CV)p,,t,
R
0.216 0.261 0.209 0.255 0.222 0.264
0.100
0.095 0.121 0.112 0.155 0.133
199
parameter in the model. Broadfoot and White (1975) provided a chart which presents the effects of growth dispersion on CV in a staged crystallizer without classified recycle. In the present study, the ratio of the dispersion parameter to mean ~ less size is less than 0.125 for all stages and the term G T I L is than 0.14 in each stage. From this information it is roughly estimated, using the chart provided by Broadfoot and White (1975), that the increase of CV would be in the range of 3 to 8%. This estimation is in agreement with the results calculated by RECYC. This calculation indicates that the adverse effects of size dispersion (widening the distribution) are of equal or greater magnitude than the favorable effects (narrowing the distribution) of classified recycle. As the distribution narrows, population gradients steepen, and the relative effects of size dispersion increases, This suggests that there may be an absolute limit to the narrowness of a distribution if size dispersion occurs. Acknowledgments The author (A. D. Randolph) is deeply indebted to Dr. E. T. White of the University of Queensland, Australia, for many discussions, ideas, and much encouragement in the modeling of classified crystallizers. Support was given by the University of Queensland and the Australian Research Grants Council for development of Program SHUG. Computer time for this study was provided by the University of Arizona Computer Center. Nomenclature
r distribution, dimensionless b = parameter in r distribution, pm B o = nucleation rate, nuclei/cm3-min C = solute concentration, g/cm3 C, = inlet solute concentration, g/cm3 C* = saturated concentration, g/cm3 CV = coefficient of variation C I ( L ) = classification function C2(L) = classification function DG = growth diffusivity G = growth rate, pm/min h , = growth rate parameter k~ = nucleation rate parameter h~ = surface shape factor hv = volume shape factor L = mean particle size, pm L = particle size, pm = the j’th moment in the i’th unit T - solid concentration, g/cm3 n = nuclei density, [number]/[(unit length)(unit volume)] P = product rate, tons/day Q, = feed rate,gpm Q, = volumetric flow rate of the j’th unit, gpm R = recycle ratio s = supersaturation, g/cm3 S = dimensionless third moment T = temperature V = clear liquor volume, gal VT = total crystallizer volume, gal w = weight density distribution W , = cumulative weight distribution for product x = dimensionless size y = dimensionless population density Y B = product yield based on the relief of supersaturation YP = product yield based on product rate /3 = fraction recycled p = crystal density, g/cm3 T = retention time, min a = parameter in
-
0
400
600
BOO
1000
1200
I400
SIZE, MICRONS
Figure 17. Effect of size dispersion on crystal-size distribution for three-stagesucrose crystallizer. tend to lower values as total product rate increases. In order to obtain a high production rate with a narrow crystal size distribution, the following conditions are suggested: (i) Lower the value of the cut fraction a. (ii) Sharpen the classifier size-cut range, Le., shift L1 to a larger size around 500 pm. Both items (i) and (ii) result in increased crystal recycle. (iii) Minimize the retention time of oversize crystals by maintaining L2 < 800 pm. Some runs made with these conditions are shown in Table XVI. In comparing Run 43 with 40, it is seen that the production rate increases almost 16 tons/day, CV drops from 22% to 19%, and the mean size decreases to 770 pm. The amount of crystal recycle is still acceptable at 18%. This would be an attractive and marketable raw sugar CSD. Effect of Size Dispersion. Table XVII and Figure 17 show that a wider size distribution will be obtained if size dispersion, due to fluctuation in growth rate, is considered in the model. Size dispersion was set equal to 50G (Randolph and White, 1977) for all dispersion calculations. Table XVII shows that size dispersion increases CV about 5% and mean size is increased about 60 pm above those values expected with the diffusive flux not introduced in the model. White and Wright (1971) also indicated that a wider size distribution with a larger mean size would be obtained by including the dispersion
221
Literature Cited Broadfoot, R., White, E. T., Proc Queens/. SOC. Sugar Cane Techno/.. 42 (1975). Larson. M. A., Wolff, P. R , Chem. Eng. Prog., Symp. Ser., 67, No. 110, 97 (1971).
200
Ind.
Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
h a , c., White, E. T.. "Kinetics of Crystallization of Aluminum Trihydroxide from Seeded Caustic Aluminate Solutions", AlChE Sixty-second Annual Meeting, Washington, D.C., Nov 16-20, 1969. Nuttal, H. E., Randobh. A. D.. Simpson, K. 0.. "User Manual for MARK 1-A, C and D for CDS Simulation", University of Arizona, Tucson, Ariz., 1975. Randolph, A. D.. Larson, M. A., "Theory of Particulate Processes", Academic Press, New York. N.Y., 1971. Randolph, A. D., White, E. T., Roc. Queens/. SOC.Sugar Cane Techno/., 43 (1976).
Randolph, A. D., White, E. T., Chem. Eng. Sci., 32, 1067 (1977). White, E. T., Wright, P. G., Chem. Eng. Prog., Symp. Ser., 67, No. 110, 81 (1971).
Receiued for reuzew April 14, 1977 Accepted September 27, 1977
Presented at the AIChE Meeting, Atlanta, Ga., March 1978.
Deactivation Disguised Kinetics S. Krishnaswamy and J. R. Kittrell' Depariment of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
For rapidly deactivating systems, the apparent kinetics of the primary reaction can be disguised by the presence of deactivation effects. It has been shown that the assumed kinetic model for the primary reaction will lead to potentially inaccurate deactivation models which nevertheless adequately fit experimental data. Conversely, the order of the primary reaction can be disguised by the kinetics of the deactivation reaction. For catalytic cracking of gas oil in a transfer line reactor and hydrogen peroxide decomposition in a fixed bed reactor, second-order conversion, concentration-independentdeactivation and first-order conversion, concentration-dependentdeactivation model forms are indistinguishable for conversion of feed. For the case of a moving bed reactor, the model forms are similar and lead to generally equivalent degrees of fit of experimental data.
For the design and optimization of industrial reactors employing a solid catalyst, knowledge of the kinetics of the primary reaction as well as the deactivation reactions is required. A summary of many laboratory investigations of deactivation kinetics has been presented by Butt (1972). Levenspiel and co-workers (1971,1972)have observed that nth order decay models describe many cases of catalyst fouling and have suggested experimental strategies to evaluate deactivation kinetics. A variety of more complex deactivation models have been investigated theoretically, as typified by those of Froment and Bischoff (1961) and by Smith and co-workers (1966, 1967). However, application of concentration-dependent deactivation models to experimental data has been limited. For many reacting systems, such as the catalytic cracking of gas oil, deactivation occurs a t a sufficiently rapid rate that examination of the kinetics of the primary reaction must be interpreted simultaneously with the deactivation kinetics. For example, Weekman and Nace (1970) presented the gas oil cracking rate constant as a function of the catalyst residence time, as shown in Figure 1. Extrapolation of these data to provide information of the kinetics of the primary reaction without the encumbrance of deactivation effects is frequently fruitless. In fact, Weekman and Nace (1970) simultaneously analyzed the kinetics of the primary reaction and of the deactivation reaction. By contrast, Blanding (1953) attempted an analysis of the primary reaction kinetics ignoring deactivation. Furthermore, the use of a transfer line reactor by Paraskos et al. (1976) can result in the reaction time coordinate being equivalent to the deactivation time coordinate, thereby obviously requiring a simultaneous interpretation Of the kinetics of the primary reaction and the deactivation reaction. The purpose of the present paper is to illustrate that the simultaneous presence of deactivation effects can disguise the apparent kinetics of the primary reaction. Specifically, equivalent degrees of fit of experimental data and, in cases, 0019-7882/78/1117-0200$01.00/0
identical equation forms can be achieved by a variety of assumptions regarding the kinetics of the primary and deactivation reactions. In catalytic cracking of gas oils, for example, Voge (1958) and Nace (1969) have reported first-order kinetics for pure compound studies. The somewhat analogous hydrocracking process, which has sufficiently slow deactivation that the primary reaction kinetics can be studied independently of deactivation, exhibits first-order kinetics for both pure compounds and complex gas oil mixtures (Stangeland and Kittrell, 1972). However, the rapidly deactivating system of catalytic cracking of complex gas oil feeds is generally interpreted with second-order kinetics (Weekman and Nace, 1970; Paraskos et al., 1976). This behavior has been attributed to a wide spectrum of cracking rate constants in the gas oil constituents. However, for many cases, the present paper illustrates that this phenomenon could also be explained by deactivation disguised kinetics in rapidly deactivating systems. Catalytic Cracking: Transfer Line Reactor Based on the model of Weekman and Nace (1970), Paraskos et al. (1976) present a general reaction path for gas oil cracking, consisting of a second-order conversion of gas oil to gasoline and a subsequent first-order reaction of gasoline to lighter hydrocarbons and coke, as shown in eq 1 and 2. gas oil
-
-
gasoline
+ (light hydrocarbons and coke)
(1)
gasoline (light hydrocarbons and coke) (2) . If the gas oil conversion is represented by a second-order reaction, and the deactivation rate is represented by a firstorder, concentration-independent model, the balances become (3) (4)
0 1978 American Chemical Society
