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Mathematical analysis of a seeded, nonnucleating continuous zeolite crystallizer
ELSEVIER
Mathematical analysis of a seeded, nonnucleating continuous zeolite crystallizer Brian H. Pittenger* and Robert W. Thompson Worcester Polytechnic Institute, Worcester, Massachusetts USA Recently, a new operational scheme for a zeolite crystallizer was reported which facilitated the experimental investigation of high silica ZSM-5 crystal growth and could be used for nucleation studies. The unique feature of the operation was that seed crystals were placed in an appropriate solution in a gently stirred tank vessel, and fresh clear reagent solution was pumped into the vessel on a continuous basis, with the same volumetric flow rate of fluid being removed on a continuous basis. The effluent stream contained a representative sample of the crystals in the vessel, and, therefore, the crystal concentration in the vessel itself decreased with time. A mathematical model of the system was presented by the authors. A more comprehensive mathematical model of the system is presented which accounts for the kinetic and operational steps in that crystallizer operation. The moment transforms of the population balance equation with constant system volume and size-independent crystal growth rate formed the basis of the model. It is shown that an approximate quasi-steady-state form of the model is informative, and yet flawed, whereas the complete transient model fits the experimental data reported previously. It also is shown that the change of the average crystal size with time represents the crystal growth rate at the instantaneous solution concentration, even when the data are not linear with respect to time. Keywords:
Crystallizer
modelling;
seeded
crystallizer
INTRODUCTION Over the last 35 years various crystallizer operational schemes have been utilized to study molecular sieve zeolite crystallization kinetics. Among these, in chemical engineering nomenclature, have been unseeded or seeded batch crystallizers (by far the most common), semibatch crystallizers, continuous stirred tank crystallizers, tubular flow crystallizers, clear solution batch crystallizers, and several more exotic schemes that defy traditional labels. A new operational scheme was reported recently’ in which seed crystals were introduced into a stirred tank crystallizer with an effluent stream volumetric flow rate that was equal to the clear reagent feed stream on a continuous basis. The concentration of growing seed crystals in the system diminished with time because of their removal in the effluent stream. In one report, the nucleation of new crystals was essentially eliminated by using a low reagent concentration and maintaining a very low flow rate of material through the system.’ In a * Present address: Jenike and Johanson, Inc., One Technology Park Dr., Westford, MA 01886. Address reprint requests to Dr. Thompson at the Dept. of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA 01609. Received 27 February 1995; revised 6 February 1996; accepted 11 February 1996 Zeolites 17:272-277, 1996 0 Elsevier Science Inc. 1996 655 Avenue of the Americas, New York, NY 10010
companion report,’ nucleation was observed to occur when the flow rate and reagent concentration were at higher levels. Although the crystallizer operation described above would not be appropriate for most commercial manufacturing applications, it does have certain distinct advantages for experimental studies. These advantages were described clearly by the original authors,‘,* who also presented a mathematical analysis of the relationship between reagent feed rate and crystal growth rate. It was the purpose of this brief study to provide a more complete and comprehensive mathematical model for this type of zeolite crystallizer operational mode and to investigate further how information on the crystallization kinetics might be extracted from such crystallizer operation. The basis of the modeling effort was the population balance formalism, which has been discussed previously.sa4 The concept behind population balance models is the conservation of the numbers of particles due to nucleation washout and the conservation of mass during crystal growth by consumption of reagents from solution. It will be demonstrated how simplications of the current development result in the model developed previously and what additional utility is contained in the current form of the model. The crystallizer operation described in the second paragraph is inherently transient; that is, it will never operate in a steady state because the concentration of 0144-2449/96/$15.00 PII SO144-2449(96)00022-X
Seeded
particles will decrease over time as particles wash out in the effluent stream. Based on the results from the experiments, however, it was tempting to use a quasisteady-state form of the model and take the solute concentration to be a constant value. This approximate form of the model gave an analytical solution and might have been justified in view of the very long system time constant and the low solute concentration close to the solubility limit. However, it will be shown that even though the seed crystal growth rate agrees with the slope of average crystal size versus time line, there are problems with this mathematical formulation. Furthermore, this approximate form of the model does not describe situations in which the average crystal size change with time is curved instead of linear. It will be demonstrated that the complete transient model addresses these weaknesses.
NOMENCLATURE Lumped constant Crystal birth, or nucleation, rate Solute concentration, solute concentration in the feed stream, equilibrium solute concentration Crystal linear growth rate Crystal growth rate constant Average crystal size Zeroth . . . third moments of the crystal size distribution Initial number of crystals in the crystallizer Time Dimensionless solute concentration Lumped dimensionless parameters Crystal mass density Dimensionless time Dimensionless moments of the crystal size distribution Solution residence time in the crystallizer (volume/flow rate) Crystal surface area shape factor Crystal volume shape factor
THEORETICAL
DEVELOPMENT
The basis of the mathematical model for the seeded, nonnucleating zeolite crystallizer is the population balance, summarized in several sources.3’4 The population balance equation describing this crystallizer operation is a partial differential equation because of the inherent transient nature of the process. However, in this instance, the preferred starting point for the development is the family of ordinary differential equations derived from transforming the original partial differential equation to a set of moment transform equations.334 These moment transforms are typically easier to solve than the original population balance, and sufficient information is generated by their solution. To describe the seeded, nonnucleating system with a crystal-free feed stream and an effluent stream representative of
zeolite crystallizer:
nonnucleating
B.H.
Pittenger
and
R. W. Thompson
the mixed contents, utilized in the original script,’ the first four moment equations are: dm, -=--
m,
dt
T
manu-
(1)
--dm1 %+w dt --T --dm, ?+2Gm, dt --T
(3)
dm, -=-dt
(4)
m, + 3Gm, T
The first four moments are related to the total number of crystals, the total length of crystals, their total su@ce area, and the cumulative maSsof crystals. The differential equation for the third moment, %, is not necessary to complete the analysis of this problem and so is not considered further, although including it in the solution poses no difficulty. To be complete, however, the following solute balance is required: dC dt -=--2-
G-C
c$crG (5)
T
The growth rate is assumed to be independent of crystal size and dependent on the solution driving force for growth to the first power; that is, G= g(C-
C,)
(6) The initial value of each dependent variable is required to solve the differential equations. These are determined from the amount of seed crystals in the crystallizer at the beginning of the experiment, the size distribution of those crystals, and the initial solute concentration in solution. The values of the other parameters also are required to be known before solving the differential equations or must be determined from the solutions. It will be shown later that the parameters can be grouped into only two constants and that their values can be estimated from consideration of the experimental data. The solution of Equation (1) is an exponentially decreasing function, which does not depend on any of the crystallization parameters; that is, the total number of crystals in the system decreases exponentially with time, and the rate of that decrease depends only on the residence time, 7. In circumstances in which the residence time is quite long, days in some of the experiments reported,’ the decrease is quite slow but still predictable. A feature of the set of moment equations describing this crystallizer system can be illustrated by considering the evolution of the average crystal size with time. The average crystal size, &, is determined from the ratio of the first moment (total length) to the zeroth moment (total number), that is, 2x2
(7)
Thus, the time derivative of the average crystal size would be found by taking the time derivative of both
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Seeded
nonnucleating
zeolite
crystallizer:
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and
R. W. Thompson
sides of Equation (7). By introducing the quantities from the right sides of Equations (1) and (2) where appropriate, the evolution of the average size may be demonstrated to be
the crystal growth rate, G, depends on solute concentration (so G can be presumed constant as well) the following expressions for the moments can be derived. m, ( t) = No e-*”
;=
G(t)
(8)
which means that the slope of the experimental curves is the crystal growth rate at the instantaneous solute concentration in the system at that time, even if the experimental data are not linear. In most cases shown by Cundy et al.’ the experimental curves were, in fact, straight lines, since the crystal growth rate during the experiment was held approximately constant by suitable variation in the reagent feed rate. Before proceeding to the results, it is worthwhile to reiterate the advantages of using this modeling framework. Most importantly, by using the population balance and moment transform formalism, a coherent set of equations arises from fundamental principles. From this set of equations, other advantages can be noted as follows. 1.
2.
3.
4.
5.
A nucleation mechanism may be included by adding either B(t) or B(C) to the right side of Equation (1). Any initial seed crystal size distribution may be included in the model as initial conditions rather than requiring that all crystals be the same size, as in other approaches.’ All statistical indicators (average size, standard deviation, cumulative surface area, etc.) of the crystal population are readily predictable using this method. The complete transient operation may be determined, rather than only the quasi-steady-state performance, including cases in which the solute concentration changes quite noticeably. As a corollary of 4, instantaneous changes in the average crystal size of the product crystals may be used to determine crystal growth rates at the solute concentration in the vessel at that time.
RESULTS
AND
DISCUSSION
Equations (l)-(6) may be solved using any convenient numerical solution package on a variety of computers, since an analytical solution of the entire system of equations is unlikely to be found. However, before advancing to the complete numerical solution, analytical or quasi-analytical solutions are illustrated for two simplified versions of the problem and are discussed below.
The Quasi-steady-state
case
An analytical solution can be obtained to the problem posed by noting that in several cases reported by Cundy et al. the solution concentration remained almost constant. By approximating the time derivative of the solute concentration in equation (5) as zero (the quasi-steady-state approximation) and recalling that
274
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(9)
m, (t) = (ml (0) + No Gt] em*”
9(t) S(t)
= {n+,(O) + 2 Gm,(O) + (No G*&/2} = S(O) + G(t)
(10) 8’
(11)
(12)
The terms N,, rni (0)) m, (0) , and f(0) are the initial values of the dependent variables given for each experiment. Note that the expression for the average size in Equation (12) was found by taking the ratio of the first moment to the zeroth moment (Equations (10) and (9)) but also can be developed directly from the integration of Equation (8) presuming that G(t) was a constant value. It also should be pointed out that the slope of the & versus time curve will be the growth rate, as per Equation (8), and for the quasi-steady-state case it is predicted to be purely linear. For the quasi-steady-state case, the solution to Equation (5) can be solved for the growth rate, G, by setting the derivative equal to zero. This gives:
Equation (13) is the point at which one notes the apparent discrepancy between the assumptions of constant solute concentration and constant crystal growth rate, and the corresponding derived growth rate. The value of the second moment, m,(t) , is predicted to increase with time, because of increased particle growth and then decrease exponentially, because crystals are washed out of the system. The value of G will vary inversely with m,?, i.e., G will decrease initially and then increase as the number of particles decreases. Therefore, the crystal growth rate is not predicted to be a constant value, in spite of the constant solute concentration. This trend was explored by Cundy et al.’ in their appendix, in which the effect was demonstrated in a series of graphs based on the output from their mathematical model. Results generated with Equations (9)(13) were the same and are not presented for brevity. Physically, the reason the quasi-steady-state model predicts such behavior is that the crystal growth rate has been assumed to be constant, when in fact the growth rate for each crystal changes with time according to the relationship shown in Equation (13). Thus, the supersaturation introduced in the feed stream is essentially completely discharged by growth of the crystals in the system at any time. As the number of crystals in the system decreases with time because of their continual washout, the cumulative crystal surface area changes with time. Consequently, the crystal growth rate changes with time, as shown by Equation (13), at least insofar as this approximate analysis demonstrates. This effect was the reason the feed rate needed to be adjusted to keep the linear growth rate approximately constant.
Seeded
The quasi-saturated-solution case If one begins with equation (13) and reconsiders the growth rate to be not constant but inversely proportional to the cumulative crystal surface area in the crystallizer, %, then the solutions of Equations (l)-(4) are not so straightforward. If the change in solute concentration from the feed to the effluent stream, ( C, - G), is essentially constant, then G may be introduced into the ordinary differential equations as A/T, where A is the collection of constants in Equation (13). Then, in fact, analytical solutions for only Equations (1) and (4) may be obtained, whereas the differential equations for the cumulative crystal length and the cumulative surface area must be solved numerically. Cundy et al.’ solved a special case of this system of equations by assuming that the crystals were of uniform size, which was appropriate since that was the situation that applied to their experiments. It was decided to solve the complete system of equations rigorously rather than solve this approximate system numerically. The complete solution Before proceeding to the solution of the relevant ordinary differential equations, which should describe the transient nature of the experimental facility,’ it is beneficial to change the differential equations to a dimensionless form. Doing so makes the solutions easier to comprehend and reduces the number of adjustable parameters to two, thereby reducing the number of constant terms that must be varied to understand their effect on the solution. Thus, the following dimensionless variables are defined for convenience.
mo po = q(O) ml
"= p*=-
m,(O) %!(O)
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
with p,(O) = 1.0. (16) with pp (0) = 1.0.
(17) with y(0) = yO. It is clear from examination of Equations (14)-(17) that the only two adjustable parameters for the system are now the lumped parameters, cr and B, which are defined as follows.
a-Rwo-cs~ 50
The set of ordinary differential equations was solved using the software package Mathematics, which permitted direct interaction with the solver package to change the two adjustable parameters to note their effect on the solutions. The software also permitted the user to draw figures at the console for viewing the results almost instantaneously. It should be noted that any of several differential equations solvers currently available commercially could solve this system of equations. Solutions were generated with no difficulties due to stiffness or steep gradients in the transient profiles. Experiment Bl was selected at random to test the solution of the model. According to the manuscript, experiment Bl was conducted with the following parameters: a residence time of 9.5 days, an initial size of the seed crystals placed in the crystallizer vessel of 1.3 pm, and a growth rate of approximately 0.11 pm/day, and uniform, over the 22-day test. The feed and effluent solute concentrations were noted to be approximately 0.33 M and 0.0’7 M, respectively, from which, according to its definition, the initial value of y was estimated to be y(O) = 0.60. The values reported above were sufficient to define the first parameter, (Y, as: OL= 1.33 (dimensionless),
These definitions result in the dimensionless solute concentration, y, being bounded between 0 and 1, the dimensionless moments, the nis having initial values of 1, and the dimensionless time, 0, measuring the number of residence times, 7, which the system has been operating. The differential equations that must be solved can now be written in terms of these dimensionless variables as: ho x-=-bo
with ~~(0) = 1.0.
which left the value of B to be determined. B was computed from its definition, but there was some uncertainty in the initial value of the number of crystals present in the crystallizer solution. The initial number of seed crystals introduced was estimated by using the following relation for the third moment of the seed crystal population, i.e., the relation for the mass per unit volume of uniform sized seed crystals: (18)
%3(O) = dw%mw”
where c$, is the volume shape factor for the silicalite crystals and set equal to l/2. The term m(0) in the definition of B was replaced by the expression obtained
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Seeded
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
3 d (pm)
2
1
0.5 l
Figure 1 of p. Data
Average points
0
l-5
size versus time for Q = 1.33 replotted from Ref. 1.
by manipulating Equation puted in this way was:
2
and four
values
(18). The value of B com-
B = 3.27 (dimensionless). B also was estimated by noting that the slope of the solute concentration versus time in the experiment was almost zero at early times. Then, from Equation (17)) the value of B could be estimated by setting the derivative equal to zero and solving for B from the right side of Equation (17) :
P=
1 - 1’(O) r(o)*km
(1%
Since the initial values of y(0) and u2 (0) are known, the value of B for the respective experiment could be estimated. The value of B for experiment Bll was estimated by Equation (19) to be: B = 0.642 (dimensionless). Figure 1 shows the predictions of the average crystal size from the numerical solution of Equations (14)(17) with the initial conditions specified and the values of the parameters noted. The data points for the average crystal size versus time, also shown in the figure, were extracted from the original manuscript’ and replotted for comparison. It will be noted that the fit of the theory to the data is reasonably good for the simulation with B = 0.642. Also for comparison, the predictions are shown for the average crystal size versus time using the same value of (Y but three different values of B. It is observed that the simulations using these other values of B do not fit the data as well; however, it should be pointed out that rather large changes in the value of /3 appear to result in small changes in the simulation results, i.e., the model calculations do not appear to be very sensitive to changes in B, at least with the value of cxused. Some of the deviation between theory and data may be because changes were made in the feed and effluent flow rates during the experiment to ensure that the average crystal size increased linearly with time, i.e., +rwas not strictly constant.’ These manipulations in the feed flow rate make the prior development, restated
276
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in equations (9)-( 11) , not rigorously correct, since 7 was assumed constant to derive them. l3 is a lumped parameter that is different from (Y primarily by the terms m(O)&,‘, whereas (Ycontains the solute driving force for crystal growth. As expected, if the number of seed crystals introduced into the vessel initially was larger the increase of the average crystal size would be expected to be slower since the increased number of seed crystals would compete for limited reagents introduced in the feed stream. Similarly, with larger initial seed crystal size, more solute must be incorporated into each crystal to increase the average crystal size. Therefore, as noted in Figure I, and as expected, the increase in average seed crystal size with time increases more slowly with larger values of l3. Figure 2 shows the corresponding values of the first three moments and the solute concentration with B = 0.642 and demonstrates that the solute concentration was indeed almost constant for quite some time, whereas the cumulative crystal number, length, and surface area changed with time. The total number of crystals in the vessel, po, decreased exponentially with time, as expected, because of their washout in the effluent stream. The cumulative length of crystals in the system, pt, also decreased with time, at a rate that was sufficient to result in an approximately linear increase in the average size, i.e., ul/pO. The cumulative crystal surface area, p2, was noted to increase initially because of crystal growth, then decrease because of the washout of seed crystals from the system. This would not be expected to be a general trend but would depend on the relative magnitudes of the crystal growth rate and the fluid flow rate through the system. Figure 3 shows the change of the average size of crystals in the vessel with run time for three values of CY,and B = 0.642. The middle curve is the same as the middle curve in Figure 1, and the data points are the same as in Figure 1, i.e., replotted from reference (1). The unique feature of the lumped parameter, ~1,is the solute driving force that increases the value of (Y as the crystal growth rate increases. The results shown indicate that the average size of crystals in the vessel should increase more rapidly with increasingly larger values of CY,which i.41.2: 1: 0.8. 0.6. 0.4. 0.2.
-
0
0.5
1.5
2
lo
Figure 2 Transient profiles for the dimensionless number particles (lo,,), cumulative length (p,), cumulative area (p2), solute concentration (y) for (Y = 1.33 and p = 0.642.
of and
Seeded
3 di (Pm)
2
(1
0.5 ld5
Figure 3 Average size versus ues of (Y. Data points replotted
time from
2
for f3 = 0.642 Ref. 1.
and
three
val-
is consistent with the crystals having faster growth rates. The fact that the results appear to be more sensitive to changes in c1 than to changes in l3 reflects the sensitivity of crystal growth to increased solute driving force compared with changes in the initial size of seed crystals or their initial number.
CONCLUSIONS The population balance modeling framework has been used to simulate the performance of a seeded continuous zeolite crystallizer, which was maintained at constant volume by using equal feed and effluent stream flow rates. The moment transformation of the population balance equation resulted in a family of four moment equations which, when coupled with a solute mass balance, formed a closed set of differential equations which could be solved quite easily with Mathematics.
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
The solutions were shown to simulate data from the literature collected with a continuous zeolite crystallizer operated in the manner described.’ Changes in the parameter values of the system were easily incorporated in the simulation, and predicted changes in performance were evaluated in a straightforward way. The utility of this approach is that any such crystallizer operation may be evaluated, even if the change of average crystal size with time is not linear or if the solute concentration changes appreciably during the experiment. In fact, it has been shown that the slope of the experimental curve of average crystal size versus time is the crystal growth rate at the solute concentration in the vessel at that time, regardless of the curvature. Furthermore, as shown, all statistical indicators of the crystal population may be predicted and compared with the experimental data. Lastly, it has been shown that it is not essential to seed such an experimental reactor with uniform sized crystals to conduct such studies, since the more complete model discussed here can be used to gain the same information with a polydisperse seed crystal population.
ACKNOWLEDGMENTS We express our gratitude to Dr. helpful and constructive comments
Colin S. Cundy for on this manuscript.
REFERENCES 1 2 3 4
Cundy, C.S., Henty, M.S. and Plaisted, R.J. Zeolites 1995, 15, 342 Cundy, C.S., Henty, MS. and Plaisted, R.J. Zeolites, 1995, 15, 353 Randolph, A.D. and Larson, M.A. Theory of Particulate Processes, 2nd ed., Academic Press, London, 1988 Thompson R.W. Modelling of Structure and Reactivity in Zeolites Chap. 10, (Ed. C.R.A. Catlow) Academic Press, New York, 1992 p. 231
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277
Mathematical analysis of a seeded, nonnucleating continuous zeolite crystallizer Brian H. Pittenger* and Robert W. Thompson Worcester Polytechnic Institute, Worcester, Massachusetts USA Recently, a new operational scheme for a zeolite crystallizer was reported which facilitated the experimental investigation of high silica ZSM-5 crystal growth and could be used for nucleation studies. The unique feature of the operation was that seed crystals were placed in an appropriate solution in a gently stirred tank vessel, and fresh clear reagent solution was pumped into the vessel on a continuous basis, with the same volumetric flow rate of fluid being removed on a continuous basis. The effluent stream contained a representative sample of the crystals in the vessel, and, therefore, the crystal concentration in the vessel itself decreased with time. A mathematical model of the system was presented by the authors. A more comprehensive mathematical model of the system is presented which accounts for the kinetic and operational steps in that crystallizer operation. The moment transforms of the population balance equation with constant system volume and size-independent crystal growth rate formed the basis of the model. It is shown that an approximate quasi-steady-state form of the model is informative, and yet flawed, whereas the complete transient model fits the experimental data reported previously. It also is shown that the change of the average crystal size with time represents the crystal growth rate at the instantaneous solution concentration, even when the data are not linear with respect to time. Keywords:
Crystallizer
modelling;
seeded
crystallizer
INTRODUCTION Over the last 35 years various crystallizer operational schemes have been utilized to study molecular sieve zeolite crystallization kinetics. Among these, in chemical engineering nomenclature, have been unseeded or seeded batch crystallizers (by far the most common), semibatch crystallizers, continuous stirred tank crystallizers, tubular flow crystallizers, clear solution batch crystallizers, and several more exotic schemes that defy traditional labels. A new operational scheme was reported recently’ in which seed crystals were introduced into a stirred tank crystallizer with an effluent stream volumetric flow rate that was equal to the clear reagent feed stream on a continuous basis. The concentration of growing seed crystals in the system diminished with time because of their removal in the effluent stream. In one report, the nucleation of new crystals was essentially eliminated by using a low reagent concentration and maintaining a very low flow rate of material through the system.’ In a * Present address: Jenike and Johanson, Inc., One Technology Park Dr., Westford, MA 01886. Address reprint requests to Dr. Thompson at the Dept. of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA 01609. Received 27 February 1995; revised 6 February 1996; accepted 11 February 1996 Zeolites 17:272-277, 1996 0 Elsevier Science Inc. 1996 655 Avenue of the Americas, New York, NY 10010
companion report,’ nucleation was observed to occur when the flow rate and reagent concentration were at higher levels. Although the crystallizer operation described above would not be appropriate for most commercial manufacturing applications, it does have certain distinct advantages for experimental studies. These advantages were described clearly by the original authors,‘,* who also presented a mathematical analysis of the relationship between reagent feed rate and crystal growth rate. It was the purpose of this brief study to provide a more complete and comprehensive mathematical model for this type of zeolite crystallizer operational mode and to investigate further how information on the crystallization kinetics might be extracted from such crystallizer operation. The basis of the modeling effort was the population balance formalism, which has been discussed previously.sa4 The concept behind population balance models is the conservation of the numbers of particles due to nucleation washout and the conservation of mass during crystal growth by consumption of reagents from solution. It will be demonstrated how simplications of the current development result in the model developed previously and what additional utility is contained in the current form of the model. The crystallizer operation described in the second paragraph is inherently transient; that is, it will never operate in a steady state because the concentration of 0144-2449/96/$15.00 PII SO144-2449(96)00022-X
Seeded
particles will decrease over time as particles wash out in the effluent stream. Based on the results from the experiments, however, it was tempting to use a quasisteady-state form of the model and take the solute concentration to be a constant value. This approximate form of the model gave an analytical solution and might have been justified in view of the very long system time constant and the low solute concentration close to the solubility limit. However, it will be shown that even though the seed crystal growth rate agrees with the slope of average crystal size versus time line, there are problems with this mathematical formulation. Furthermore, this approximate form of the model does not describe situations in which the average crystal size change with time is curved instead of linear. It will be demonstrated that the complete transient model addresses these weaknesses.
NOMENCLATURE Lumped constant Crystal birth, or nucleation, rate Solute concentration, solute concentration in the feed stream, equilibrium solute concentration Crystal linear growth rate Crystal growth rate constant Average crystal size Zeroth . . . third moments of the crystal size distribution Initial number of crystals in the crystallizer Time Dimensionless solute concentration Lumped dimensionless parameters Crystal mass density Dimensionless time Dimensionless moments of the crystal size distribution Solution residence time in the crystallizer (volume/flow rate) Crystal surface area shape factor Crystal volume shape factor
THEORETICAL
DEVELOPMENT
The basis of the mathematical model for the seeded, nonnucleating zeolite crystallizer is the population balance, summarized in several sources.3’4 The population balance equation describing this crystallizer operation is a partial differential equation because of the inherent transient nature of the process. However, in this instance, the preferred starting point for the development is the family of ordinary differential equations derived from transforming the original partial differential equation to a set of moment transform equations.334 These moment transforms are typically easier to solve than the original population balance, and sufficient information is generated by their solution. To describe the seeded, nonnucleating system with a crystal-free feed stream and an effluent stream representative of
zeolite crystallizer:
nonnucleating
B.H.
Pittenger
and
R. W. Thompson
the mixed contents, utilized in the original script,’ the first four moment equations are: dm, -=--
m,
dt
T
manu-
(1)
--dm1 %+w dt --T --dm, ?+2Gm, dt --T
(3)
dm, -=-dt
(4)
m, + 3Gm, T
The first four moments are related to the total number of crystals, the total length of crystals, their total su@ce area, and the cumulative maSsof crystals. The differential equation for the third moment, %, is not necessary to complete the analysis of this problem and so is not considered further, although including it in the solution poses no difficulty. To be complete, however, the following solute balance is required: dC dt -=--2-
G-C
c$crG (5)
T
The growth rate is assumed to be independent of crystal size and dependent on the solution driving force for growth to the first power; that is, G= g(C-
C,)
(6) The initial value of each dependent variable is required to solve the differential equations. These are determined from the amount of seed crystals in the crystallizer at the beginning of the experiment, the size distribution of those crystals, and the initial solute concentration in solution. The values of the other parameters also are required to be known before solving the differential equations or must be determined from the solutions. It will be shown later that the parameters can be grouped into only two constants and that their values can be estimated from consideration of the experimental data. The solution of Equation (1) is an exponentially decreasing function, which does not depend on any of the crystallization parameters; that is, the total number of crystals in the system decreases exponentially with time, and the rate of that decrease depends only on the residence time, 7. In circumstances in which the residence time is quite long, days in some of the experiments reported,’ the decrease is quite slow but still predictable. A feature of the set of moment equations describing this crystallizer system can be illustrated by considering the evolution of the average crystal size with time. The average crystal size, &, is determined from the ratio of the first moment (total length) to the zeroth moment (total number), that is, 2x2
(7)
Thus, the time derivative of the average crystal size would be found by taking the time derivative of both
Zeolites
17:272-277.
1996
273
Seeded
nonnucleating
zeolite
crystallizer:
B./i.
Pittenger
and
R. W. Thompson
sides of Equation (7). By introducing the quantities from the right sides of Equations (1) and (2) where appropriate, the evolution of the average size may be demonstrated to be
the crystal growth rate, G, depends on solute concentration (so G can be presumed constant as well) the following expressions for the moments can be derived. m, ( t) = No e-*”
;=
G(t)
(8)
which means that the slope of the experimental curves is the crystal growth rate at the instantaneous solute concentration in the system at that time, even if the experimental data are not linear. In most cases shown by Cundy et al.’ the experimental curves were, in fact, straight lines, since the crystal growth rate during the experiment was held approximately constant by suitable variation in the reagent feed rate. Before proceeding to the results, it is worthwhile to reiterate the advantages of using this modeling framework. Most importantly, by using the population balance and moment transform formalism, a coherent set of equations arises from fundamental principles. From this set of equations, other advantages can be noted as follows. 1.
2.
3.
4.
5.
A nucleation mechanism may be included by adding either B(t) or B(C) to the right side of Equation (1). Any initial seed crystal size distribution may be included in the model as initial conditions rather than requiring that all crystals be the same size, as in other approaches.’ All statistical indicators (average size, standard deviation, cumulative surface area, etc.) of the crystal population are readily predictable using this method. The complete transient operation may be determined, rather than only the quasi-steady-state performance, including cases in which the solute concentration changes quite noticeably. As a corollary of 4, instantaneous changes in the average crystal size of the product crystals may be used to determine crystal growth rates at the solute concentration in the vessel at that time.
RESULTS
AND
DISCUSSION
Equations (l)-(6) may be solved using any convenient numerical solution package on a variety of computers, since an analytical solution of the entire system of equations is unlikely to be found. However, before advancing to the complete numerical solution, analytical or quasi-analytical solutions are illustrated for two simplified versions of the problem and are discussed below.
The Quasi-steady-state
case
An analytical solution can be obtained to the problem posed by noting that in several cases reported by Cundy et al. the solution concentration remained almost constant. By approximating the time derivative of the solute concentration in equation (5) as zero (the quasi-steady-state approximation) and recalling that
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(9)
m, (t) = (ml (0) + No Gt] em*”
9(t) S(t)
= {n+,(O) + 2 Gm,(O) + (No G*&/2} = S(O) + G(t)
(10) 8’
(11)
(12)
The terms N,, rni (0)) m, (0) , and f(0) are the initial values of the dependent variables given for each experiment. Note that the expression for the average size in Equation (12) was found by taking the ratio of the first moment to the zeroth moment (Equations (10) and (9)) but also can be developed directly from the integration of Equation (8) presuming that G(t) was a constant value. It also should be pointed out that the slope of the & versus time curve will be the growth rate, as per Equation (8), and for the quasi-steady-state case it is predicted to be purely linear. For the quasi-steady-state case, the solution to Equation (5) can be solved for the growth rate, G, by setting the derivative equal to zero. This gives:
Equation (13) is the point at which one notes the apparent discrepancy between the assumptions of constant solute concentration and constant crystal growth rate, and the corresponding derived growth rate. The value of the second moment, m,(t) , is predicted to increase with time, because of increased particle growth and then decrease exponentially, because crystals are washed out of the system. The value of G will vary inversely with m,?, i.e., G will decrease initially and then increase as the number of particles decreases. Therefore, the crystal growth rate is not predicted to be a constant value, in spite of the constant solute concentration. This trend was explored by Cundy et al.’ in their appendix, in which the effect was demonstrated in a series of graphs based on the output from their mathematical model. Results generated with Equations (9)(13) were the same and are not presented for brevity. Physically, the reason the quasi-steady-state model predicts such behavior is that the crystal growth rate has been assumed to be constant, when in fact the growth rate for each crystal changes with time according to the relationship shown in Equation (13). Thus, the supersaturation introduced in the feed stream is essentially completely discharged by growth of the crystals in the system at any time. As the number of crystals in the system decreases with time because of their continual washout, the cumulative crystal surface area changes with time. Consequently, the crystal growth rate changes with time, as shown by Equation (13), at least insofar as this approximate analysis demonstrates. This effect was the reason the feed rate needed to be adjusted to keep the linear growth rate approximately constant.
Seeded
The quasi-saturated-solution case If one begins with equation (13) and reconsiders the growth rate to be not constant but inversely proportional to the cumulative crystal surface area in the crystallizer, %, then the solutions of Equations (l)-(4) are not so straightforward. If the change in solute concentration from the feed to the effluent stream, ( C, - G), is essentially constant, then G may be introduced into the ordinary differential equations as A/T, where A is the collection of constants in Equation (13). Then, in fact, analytical solutions for only Equations (1) and (4) may be obtained, whereas the differential equations for the cumulative crystal length and the cumulative surface area must be solved numerically. Cundy et al.’ solved a special case of this system of equations by assuming that the crystals were of uniform size, which was appropriate since that was the situation that applied to their experiments. It was decided to solve the complete system of equations rigorously rather than solve this approximate system numerically. The complete solution Before proceeding to the solution of the relevant ordinary differential equations, which should describe the transient nature of the experimental facility,’ it is beneficial to change the differential equations to a dimensionless form. Doing so makes the solutions easier to comprehend and reduces the number of adjustable parameters to two, thereby reducing the number of constant terms that must be varied to understand their effect on the solution. Thus, the following dimensionless variables are defined for convenience.
mo po = q(O) ml
"= p*=-
m,(O) %!(O)
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
with p,(O) = 1.0. (16) with pp (0) = 1.0.
(17) with y(0) = yO. It is clear from examination of Equations (14)-(17) that the only two adjustable parameters for the system are now the lumped parameters, cr and B, which are defined as follows.
a-Rwo-cs~ 50
The set of ordinary differential equations was solved using the software package Mathematics, which permitted direct interaction with the solver package to change the two adjustable parameters to note their effect on the solutions. The software also permitted the user to draw figures at the console for viewing the results almost instantaneously. It should be noted that any of several differential equations solvers currently available commercially could solve this system of equations. Solutions were generated with no difficulties due to stiffness or steep gradients in the transient profiles. Experiment Bl was selected at random to test the solution of the model. According to the manuscript, experiment Bl was conducted with the following parameters: a residence time of 9.5 days, an initial size of the seed crystals placed in the crystallizer vessel of 1.3 pm, and a growth rate of approximately 0.11 pm/day, and uniform, over the 22-day test. The feed and effluent solute concentrations were noted to be approximately 0.33 M and 0.0’7 M, respectively, from which, according to its definition, the initial value of y was estimated to be y(O) = 0.60. The values reported above were sufficient to define the first parameter, (Y, as: OL= 1.33 (dimensionless),
These definitions result in the dimensionless solute concentration, y, being bounded between 0 and 1, the dimensionless moments, the nis having initial values of 1, and the dimensionless time, 0, measuring the number of residence times, 7, which the system has been operating. The differential equations that must be solved can now be written in terms of these dimensionless variables as: ho x-=-bo
with ~~(0) = 1.0.
which left the value of B to be determined. B was computed from its definition, but there was some uncertainty in the initial value of the number of crystals present in the crystallizer solution. The initial number of seed crystals introduced was estimated by using the following relation for the third moment of the seed crystal population, i.e., the relation for the mass per unit volume of uniform sized seed crystals: (18)
%3(O) = dw%mw”
where c$, is the volume shape factor for the silicalite crystals and set equal to l/2. The term m(0) in the definition of B was replaced by the expression obtained
Zeolites
17:272-277,
1996
275
Seeded
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
3 d (pm)
2
1
0.5 l
Figure 1 of p. Data
Average points
0
l-5
size versus time for Q = 1.33 replotted from Ref. 1.
by manipulating Equation puted in this way was:
2
and four
values
(18). The value of B com-
B = 3.27 (dimensionless). B also was estimated by noting that the slope of the solute concentration versus time in the experiment was almost zero at early times. Then, from Equation (17)) the value of B could be estimated by setting the derivative equal to zero and solving for B from the right side of Equation (17) :
P=
1 - 1’(O) r(o)*km
(1%
Since the initial values of y(0) and u2 (0) are known, the value of B for the respective experiment could be estimated. The value of B for experiment Bll was estimated by Equation (19) to be: B = 0.642 (dimensionless). Figure 1 shows the predictions of the average crystal size from the numerical solution of Equations (14)(17) with the initial conditions specified and the values of the parameters noted. The data points for the average crystal size versus time, also shown in the figure, were extracted from the original manuscript’ and replotted for comparison. It will be noted that the fit of the theory to the data is reasonably good for the simulation with B = 0.642. Also for comparison, the predictions are shown for the average crystal size versus time using the same value of (Y but three different values of B. It is observed that the simulations using these other values of B do not fit the data as well; however, it should be pointed out that rather large changes in the value of /3 appear to result in small changes in the simulation results, i.e., the model calculations do not appear to be very sensitive to changes in B, at least with the value of cxused. Some of the deviation between theory and data may be because changes were made in the feed and effluent flow rates during the experiment to ensure that the average crystal size increased linearly with time, i.e., +rwas not strictly constant.’ These manipulations in the feed flow rate make the prior development, restated
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in equations (9)-( 11) , not rigorously correct, since 7 was assumed constant to derive them. l3 is a lumped parameter that is different from (Y primarily by the terms m(O)&,‘, whereas (Ycontains the solute driving force for crystal growth. As expected, if the number of seed crystals introduced into the vessel initially was larger the increase of the average crystal size would be expected to be slower since the increased number of seed crystals would compete for limited reagents introduced in the feed stream. Similarly, with larger initial seed crystal size, more solute must be incorporated into each crystal to increase the average crystal size. Therefore, as noted in Figure I, and as expected, the increase in average seed crystal size with time increases more slowly with larger values of l3. Figure 2 shows the corresponding values of the first three moments and the solute concentration with B = 0.642 and demonstrates that the solute concentration was indeed almost constant for quite some time, whereas the cumulative crystal number, length, and surface area changed with time. The total number of crystals in the vessel, po, decreased exponentially with time, as expected, because of their washout in the effluent stream. The cumulative length of crystals in the system, pt, also decreased with time, at a rate that was sufficient to result in an approximately linear increase in the average size, i.e., ul/pO. The cumulative crystal surface area, p2, was noted to increase initially because of crystal growth, then decrease because of the washout of seed crystals from the system. This would not be expected to be a general trend but would depend on the relative magnitudes of the crystal growth rate and the fluid flow rate through the system. Figure 3 shows the change of the average size of crystals in the vessel with run time for three values of CY,and B = 0.642. The middle curve is the same as the middle curve in Figure 1, and the data points are the same as in Figure 1, i.e., replotted from reference (1). The unique feature of the lumped parameter, ~1,is the solute driving force that increases the value of (Y as the crystal growth rate increases. The results shown indicate that the average size of crystals in the vessel should increase more rapidly with increasingly larger values of CY,which i.41.2: 1: 0.8. 0.6. 0.4. 0.2.
-
0
0.5
1.5
2
lo
Figure 2 Transient profiles for the dimensionless number particles (lo,,), cumulative length (p,), cumulative area (p2), solute concentration (y) for (Y = 1.33 and p = 0.642.
of and
Seeded
3 di (Pm)
2
(1
0.5 ld5
Figure 3 Average size versus ues of (Y. Data points replotted
time from
2
for f3 = 0.642 Ref. 1.
and
three
val-
is consistent with the crystals having faster growth rates. The fact that the results appear to be more sensitive to changes in c1 than to changes in l3 reflects the sensitivity of crystal growth to increased solute driving force compared with changes in the initial size of seed crystals or their initial number.
CONCLUSIONS The population balance modeling framework has been used to simulate the performance of a seeded continuous zeolite crystallizer, which was maintained at constant volume by using equal feed and effluent stream flow rates. The moment transformation of the population balance equation resulted in a family of four moment equations which, when coupled with a solute mass balance, formed a closed set of differential equations which could be solved quite easily with Mathematics.
nonnucleating
zeolite
crystallizer:
B.H.
Pittenger
and
R. W. Thompson
The solutions were shown to simulate data from the literature collected with a continuous zeolite crystallizer operated in the manner described.’ Changes in the parameter values of the system were easily incorporated in the simulation, and predicted changes in performance were evaluated in a straightforward way. The utility of this approach is that any such crystallizer operation may be evaluated, even if the change of average crystal size with time is not linear or if the solute concentration changes appreciably during the experiment. In fact, it has been shown that the slope of the experimental curve of average crystal size versus time is the crystal growth rate at the solute concentration in the vessel at that time, regardless of the curvature. Furthermore, as shown, all statistical indicators of the crystal population may be predicted and compared with the experimental data. Lastly, it has been shown that it is not essential to seed such an experimental reactor with uniform sized crystals to conduct such studies, since the more complete model discussed here can be used to gain the same information with a polydisperse seed crystal population.
ACKNOWLEDGMENTS We express our gratitude to Dr. helpful and constructive comments
Colin S. Cundy for on this manuscript.
REFERENCES 1 2 3 4
Cundy, C.S., Henty, M.S. and Plaisted, R.J. Zeolites 1995, 15, 342 Cundy, C.S., Henty, MS. and Plaisted, R.J. Zeolites, 1995, 15, 353 Randolph, A.D. and Larson, M.A. Theory of Particulate Processes, 2nd ed., Academic Press, London, 1988 Thompson R.W. Modelling of Structure and Reactivity in Zeolites Chap. 10, (Ed. C.R.A. Catlow) Academic Press, New York, 1992 p. 231
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277