I n d . Eng. Chem. Res. 1993,32, 2781-2788
2781
Design Charts for the Fines-Dissolving Crystallizer Bhagya
C. Sutradhart and A l a n D. Randolph’
Department of Chemical Engineering, The University of Arizona, Tucson, Arizona 85721
The principle of particle size increase in continuous crystallization processes is to remove fine particles at a faster rate than large ones. Two technologies based on this principle are the double draw-off (DDO) and the fines-dissolving (FD) crystallizer configurations. Design charts were developed for the F D crystallizer. It was found that the particle size increase passes through a maximum value as the fines classification cut size is increased. Based on this information, a design protocol was suggested for the FD crystallizer. A limited amount of data are required for this method of design. Particle size increase was demonstrated in the laboratory for sodium chloride crystallization in caustic liquor using an FD crystallizer. A fines/feed flow modification as well as actual fines dissolution was used in the FD experiments. Significant size increase was observed in both cases. These experiments indicate that appreciable dissolution cost would be involved in FD operation. Experimental results were in good agreement with values predicted from design charts. Introduction
Fines-Dissolving Crystallizer
The crystal size distribution (CSD) or particle size distribution (PSD) of a solid product is often important in many process operations and determines properties associated with the handling of the solid product. In addition, the CSD can affect the operation of a crystallizer. A summary of these CSD-related properties is given in Randolph and Larson (1988). Some of these properties are improved if the particle size is larger. For example, larger particle size will give better dewatering properties by enhancing filtration and drying rates and will give less mother liquor entrainment. This is particularly important in a process where the mother liquor is the final product. Concentration of alkali and removal of salt in chlor-alkali processes is an example of such a process. There are several fertilizer processes which require large product crystal size,e.g., ammonium sulfate and potassium chloride. A discussion of the importance of particle size increases and the potential techniques for this will be restricted to sodium chloride crystallization in caustic liquor (chlor-alkali process) in this paper. Two important technologies, namely, double drawoff (DDO) and fines dissolving (FD), are available for making large crystal particles. Both of these are based on the principle of size-dependent particle residence time, i.e., removal of the smaller crystals at a faster rate. A stream of classified fines is removed along with a mixed underflow in both of these designs. They differ in the way the fines are handled. In the FD crystallizer, the fines are dissolved and solute returned to the crystallizer, whereas in the DDO crystallizer, the fines are combined with the mixed underflow to form the final product. Either mechanism produces larger particle size. Their relative advantages and disadvantages help select the appropriate configuration. There is an extensive literature on both of these technologiesfor crystal size increase. White and Randolph (1989) developed design charta for the DDO crystallizer and showed that there is an optimum fines classification cut size for DDO operation. The objective of this work was to develop similar easy-to-use design charts for the FD crystallizer as well as to demonstrate the efficacy of fines dissolving in caustic liquor NaCl crystallization.
The widest application of fines dissolving crystallizers is in the fertilizer industry, e.g., potassium chloride and ammonium sulfate. Swenson Process Equipment Co. developed a draft tube baffle (DTB) crystallizer with fines dissolving in the 1950s (Bennett, 1984). Particles smaller than a certain size are removed at a fast rate from the crystallizer, dissolved,and then returned to the crystallizer (this is also called fines destruction with solute recycle). This action allows the same production on a smaller number of particles and thus leads to an increase in mean particle size. The operating growth rate is forced to a higher level in FD crystallizers, often causing crystallizer vessel fouling. Slurry density does not increase in this type of crystallizer. The fines dissolving crystallizer,unlike the DDO crystallizer, is suitable for high natural slurry density systems. However, there is a penalty involved in the form of dissolution cost, unless the process uses an unsaturated feed to dissolve the fines. This configuration is not the alternative if the solute is sparingly soluble. The technique for and the consequences of increasing particle size by fines dissolving was first analyzed by Saeman (1956). Since then, there has been a growing interest in the study and the modeling of the effect of fines dissolving on the CSD (Lei et al., 1971; Randolph, et al., 1973; Juzaszek and Larson, 1977). Larson and Randolph (1969) presented a general population balance for particles in an arbitrary suspension and illustrated the principles in the context of classification and fines destruction systems. Nauman and Szabo (1971) characterized the steadystate performance of continuous crystallizers with nonselective fines destruction in terms of productivity and CSD. Nauman (1971) analyzed the case of selective fines destruction. Fines destruction has also been shown to improve the final CSD in a batch crystallizer (Jones et al., 1984; Zipp, 1986; Zipp and Randolph, 1989). A rigorous analysis of the effect of fines destruction is available in Randolph and Larson (1988). A schematic diagram of fines destruction with solute recycle is shown in Figure 1. Similar to the DDO crystallizer, an overflow stream containing fines (cut size, LF)is withdrawn at a flow rate Q,, = (R - 1)Q, where the mixed underflow (product) rate is Q. R is called the draw-off ratio. Both R and LF are adjustable parameters, and the size increase is influenced by both. The fines are dissolved by adding heat and/or solvent (or a part of the feed if the feed is unsaturated)
t Present address: E.I. du Pont de Nemours & Co., Victoria Development Laboratory, Victoria, TX 77902-2626.
0888-5885/93/2632-2781~04.00~0
1993 American Chemical Society
2782 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Overflow, Heat and/or
Solvent
I It-----
I *
~ P r o d u c t l P (01
Loa
-
I
0
LF
~--..---. I
I
I
I
L
0
(b)
Figure 1. (a) Fine-dissolving crystallizer configuration and (b) population density distribution in the FD crystallizer.
and returned to the crystallizer. Fines classification is usually achieved by using an internal fines trap, often a skirted baffle or an external hydroclone. The appropriate equations are shown in Appendix A. A simplified model to predict size increase in FD crystallhers was presented by Randolph and Larson (19881, who expressed the size increase with fines destruction as Ld,PILd,l
\-------_-
= (1/p)1/(3+i)
(1)
where
p = e-(R-l)L~/G, (2) The principal assumption behind eq 1is that the mass of fines destroyed is negligible compared to the production rate. Thus, fines destruction does not appreciably perturb the form of the exponential distribution. For efficient operation, the separation of fines should be of large enough size to materially decay the population density (Randolph, 1970; Mullin, 1972). A generalized equation was presented by Kraljevich (1977) and Kraljevich and Randolph (1978) as
where h = -In
0
(4a)
o ( t ) = i c e - P p 3 dp
The results were graphically expressed for various values of i from which size increase was predicted for potassium chloride crystallization. The FD configuration can be successfully simulated using existing CSD simulators (Mark I, Nuttal, 1971; or Crystal Ball, Sharnez, 19871, but these simulators need trained personnel to use them. Development of easy-to-
Figure 2. Effects of fines cut size and draw-off ratio on mass mean size in high-yield systems for the FD configuration.
use design charts was first attempted by Kraljevich (1977) and Kraljevich and Randolph (1978). Their procedure used an approximation for the dominant size in the crystallizer. A design procedure was advanced in the above work to predict size improvement for a given fines cut size. Selection criteria for fiies cut size was not attempted. In the present work, a set of design charts was prepared (in the spirit of CSD analysis and/or improvement by the operating engineer) without resorting to detailed simulation or process modeling. Criteria for the selection of a fines classification cut size were investigated. Applicability of the FD configuration in low-yield systems was also examined.
Particle Size Increase in High-Yield Systems White and Randolph (1989) showed that the DDO configuration exhibits an optimum fines classification cut size for an increase in mean particle size. The present work shows that the FD design also has an optimum fines cut size. Figures 2 and 3 show the effects of fines cut size and the draw-off ratio on the mass mean and the Sauter mean sizes, respectively, in the FD crystallizer. As in the case with DDO, the size increase is larger for a larger fines draw-off ratio. However, unlike in the DDO configuration, slurry density has no influence on the change in mean size with fines destruction. This is because slurry density is constant in high-yield systems operated in the FD configuration.
Fines Dissolution Rate in High-Yield Systems Figure 4 shows the variation of the amount of fines to be dissolved in the overflow stream with the fines cut size. It can be seen that the amount increases sharply with an increase in cut size. Dissolution of the fines is often costly and may sometimes be difficult. Therefore, a fines cut size lower than the optimum or a lower draw-off ratio may have to be chosen, depending on the dissolution cost and capacity.
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2783 inn
-
10.0
R-IO
5.0
5.0
P
/
0
I
J
ui 0)
c
0
a f
3
2
(3 L.
0
.-0
c
B I.o
I
3.0
I
I
6.O
9.0
Dimensionless
I
12.0 Fines Cut Size, xo
0.7
I
3
3.0
12.0
9.0
6.0
15.0
Dimensionless Fines Cut Size, xo
Figure 3. Effects of fines cut size and draw-off ratio on Sauter mean size in high-yield systems for the FD configuration. 12.0
5.0
i.2
-
i.j.2 a=l
---
c
0
Mass Mean Size Per-pass Yield
10.0 R = IO
d
7
c
0
a C
.c s! 8.0 0
a 0 .c
a
L.
U
2 n \ c 0)
0 1.0 a
6.0
B C
.-0
-2a 4.0 c
.-n In In W
I
c
i i
2.0
\,0.6 I
2.0
\
\0.7 I
4.0
\
I
6.0
I
I
I
8.0
10.0
12.0
Dimensionless Fines Cut Size, xo 0.0 0.0
3.0
6.0
9.0
12.0
I!
Dimensionless Fines Cut Size, xo
Figure 6. Effects of initial (MSMPR) per-pass yield on mass mean size and actual per-pass yield in low-yield system in the finesdissolving crystallizer.
Figure 4. Relative fines dissolution rate in the FD crystallizer.
Fines Dissolving in Low-Yield Systems
Growth Rate in the F D Crystallizer for High-Yield Systems
The effects of fines cut size and draw-off ratio on the mass mean size of the product crystals for low per-pass yield systems are shown in the Figures 6 and 7. Both kinetic parameters i and j (response factors to supersaturation and slurry density, respectively) influence the outcome of fines destruction, and in these figures i = j = 2. Figure 6 shows the effect of dimensionlesscut size and initial (MSMPR)per-pass yield on the per-pass yield of the fines destruction system and the ratio of mean size with fines destruction without destruction. The effect of draw-off ratio for a system having an initial per-pass yield of 0.8 is shown in Figure 7.
It has already been discussed that the growth rate in the fines dissolving crystallizer is forced to a higher level to achieve a particle size increase. Such increases ingrowth rate are displayed in Figure 5. The growth rate increases with an increase in both the fines cut size, LF,and the draw-off ratio, R. The increase in growth rates (more fouling and/or poor crystal habit) and the increase in the rate of fines dissolution often limit the extent of particle size increase in a fines dissolving crystallizer.
2784 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
Low-Yield Systems. For a low-yield system (e.g., c0
< 50%), the use of fines dissolving is not advisable because ,R=8 Mass Mean Size Per-pass yield
of a drop in production rate as aresult of fines destruction. is moderate (>75%),the FD design may be However, if lo used for size increase with some sacrifice in per-pass yield. Assume the system mentioned above (i = j = 2) has an initial yield of only 80 % . Suppose amaximum 10% decline in per-pass yield could be allowed (any unrelieved supersaturation of course is not “lost” but recycled upstream to the feed). If the fines classification system can operate only for xo I1.0, it is observed from Figure 7 that the permissible recycle ratio is 13.0. By interpolation from Figure 7, it is found that the maximum size increase is for a draw-off ratio of -1.5 (this can be better determined with the computer program we used). The corresponding fines cut size would be -75 pm. The expected mass mean product crystal size would be 119 pm. If the amount of fines to be dissolved is within the range of dissolution capacity, these numbers are acceptable. Otherwise,a lower fines cut size and/or a lower draw-off ratio should be chosen. Note that in many cases a small increase in particle size at the cost of significant fines dissohtion may not be attractive. For low-yield systems, application of fines dissolving design is limited.
-
0.2
I
0.0
I
2.0
\
\
I
\,
I
I
4.0 6.0 8.0 10.0 Dimensionless Fines Cut Size, x o
12.0
Figure 7. Effects of draw-off ratio on mass mean size and per-pass yield in low-yield systems in the fines-dissolving crystallizer.
Unlike crystallizers operated in a DDO configuration, the yield in these systems decreases as a result of fines dissolution. In fact, if the initial yield is low, the results of fines destruction could be devastating as the yield would diminish to a negligible value even for a low classification size and draw-off ratio, even though a significant size increase might be attained.
Use of Design Charts for the FD Configuration As in the case of the DDO crystallizer (White and Randolph, 1989),the results for the FD configuration also are expressed relative to the MSMPR configuration. If the kinetic parameters (i, j , and a) and the initial (MSMPR) yield are known, these charts can be used to predict the size improvement for a given set of conditions. An optimum design strategy can be formulated following the procedure mentioned below. High-Yield Systems. In this case, the suspension density and the external production rate remain unaltered. Draw-off ratio and fines cut size can be obtained from a balance of pump cost and availability, dissolution cost, and product-handling cost. Consider a high-yield system with i = j = 2. Product mass mean size obtained from an MSMPR experiment is 100 pm (i.e,, Go7 = 25 pm). Existing facilities allow a dissolution rate corresponding to @ = 0.8. The corresponding draw-off ratio giving a maximum in mass mean size is approximately equal to 7 (Figures 2 and 4). Using an interpolation in Figure 2, it is found that the optimum fines cut size (dimensionless) is 3.8. This means that the actual fines cut size should be LF = Gorxo= 95 pm. The expected product mass mean size is 291 pm. The system would operate with an apparent growth rate -3.6 times the MSMPR value. Thus, fouling problems might be encountered. It can be seen in Figure 2 that, close to the maximum, the change in mass mean size is insensitive to fines cut size. Without a large sacrifice in particle size improvement, a fines cut size (dimensionless), xo = 3.0, may be chosen for this case. The corresponding fines classification size is 75 pm. The mass mean size would be 285 pm, and the system would operate at a growth rate -3.1 times the MSMPR value.
Chlor-Alkali Process The chlor-alkali i n d r t r y is one of the most important heavy chemical industries in the United States. The manufacture of chlorine and caustic soda is done almost entirely by an electrolytic method in which pretreated brine is electrolyzed in specially designed cells (e.g., diaphragm cell, mercury cell) to produce sodium hydroxide, chlorine, and hydrogen. The overall chemical reaction may be written as 2NaC1+ 2H20 = 2NaOH + H2t
+ Cl,t
(5)
The decomposition efficiency of the cells is usually low (approximately 50%, Shreve and Brink, 1977), and the catholyte contains approximately 9% NaOH and 16% NaC1. This weak sodium hydroxide solution containing unreacted NaCl is evaporated to about 50% NaOH in a multiple-effect evaporator system (e.g., in DOW’S operation at Freeport, Texas). The caustic concentration increases as the liquor passes countercurrently to the steam flow. The final product from the triple-effect evaporator typically contains about 50% (by weight of solution) sodium hydroxide. The solubility of sodium chloride decreases as the alkali concentration increases. Consequently, most of the unreacted salt crystallizes out during evaporation. The salt so recovered is filtered off and usually reused. The crystal size distribution (CSD) of NaCl in the product is of importance because caustic losses and production rate depend on filter-cake entrainment and filtration (or centrifuge) rate, respectively. It is observed that as the caustic concentration increases, the average salt crystal size decreases (Scrutton, 1985).Of particular concern therefore is blinding of the screens due to small size particles, especially for the product from the first-effect evaporator (highest alkali concentration). A larger particle size would result in higher production rate and less product loss.
Particle Size Increase Studies The fines dissolving crystallizer is suitable for particle size increase in salt crystallization from aqueous solutions, because NaCl can be dissolved easily in water. Also, salt crystallization is carried out with high suspension density.
Ind. Eng. Chem. Res., Vol. 32,No. 11, 1993 2785 Overflow
9
P@Tt n #iT=== Coolant Out Coolant In
Crystallizer
Figure 8. Modified fines classificatior. ,evice for boiling magma.
Crystallization of NaCl in caustic liquor is performed with a high slurry density too. However, dissolution of fines is difficult in concentrated alkali. The objective of the present experiments was to explore the particle size increase as well as to estimate the amount of water needed to dissolve the fines for different process conditions. Particle size increases (with respect to MSMPR experiments) for various draw-off ratios (R)and fines were estimated from the previously mencut sizes (LF) tioned design charts using the nucleation kinetics (i = 1.53,j = 1.24) obtained by Sutradhar (1992). It was observed that for an optimum fines cut size, about a 40% increase in mass mean particle size (23% increase in Sauter mean) should be obtained for a draw-off ratio as low as 2. For the range of growth rates obtained in the MSMPR experiments, the optimum fines cut size (for R = 2) was found to be in the range 150-300 pm. To obtain this increase in mean particle size, the rate of fines to be dissolved was predicted to be -23% of the production rate of salt. A higher draw-off ratio will produce particles of even larger mean size with a corresponding increase in the amount of fines to be dissolved.
Fines Classification/Removal in Boiling Magma In laboratory experiments, fines classification is conveniently accomplished by using a fines trap (an inverted cone). The flow of liquid in the trap is quiescent; thus, the large particles are allowed to settle under the action of gravity. However, if the magma is under boiling conditions, such gravity settling of particles is prevented by the agitation provided by boiling. Also, a vapor-liquid mixture is normally withdrawn through the fines trap, which leads to fluctuations in the rate of slurry removal. These problems were solved by cooling the fines trap so that boiling was suppressed locally inside the trap. The arrangement is shown in Figure 8. Ice-cold water was circulated through a U-tube fitted inside the trap (see Figure 8). This allowed a quiescent flow inside the trap, and the desired fines classification was obtained. The rate of flow of the fines slurry could also be maintained constant.
Fines Dissolution: The Inclined Settler
A cost-effective way of accomplishing fines dissolution would be to concentrate the fines and handle a smaller volume of slurry for dissolution while the supernatent liquid is returned directly to the crystallizer. This would involve a particle classification device for segregating the fines. Common commercial equipment used for particle classification includes hydroclones, centrifuges, cone classifiers, sieves, or screens, etc. Particle classification may also be accomplished by exploiting differences in particle settling velocities under the action of gravity. Classifiers using gravity settling are simple in design. They exert very little shear on the particles. However, the sedimentation process is often slow, especially when the particles are small. Settling rate can be enhanced considerably in vessels having inclined walls (Acrivos and Herbolzhiemer, 1979). Lamella settlers are built using this principle. Lamella settlers are composed of one or more narrow channels that are inclined from the vertical. As such, the particles settle onto the upward facing side of each plate and then slide down to the bottom of the settler where they are collected. The review of Davis and Acrivos (1985)described the operation of these inclined settlers along with mathematical models and experimental verifications. In particular, the coarse fraction contains particles of all sizes, enriched in the larger particles. The fine fraction is essentially free of the larger particles. For our experiments, a simple cylindrical inclined settler (12in. long, 4-in. diameter, and inclined at a 30° angle from the vertical) was used. The classification was good, and the liquor from the top of the settler contained practically no particles. Experimental Section Efficacy of the FD configuration for particle size increase was demonstrated in the following way. In the FD configuration, the fines are dissolved and the resulting solution is fed back to the crystallizer. It was found that the composition of this solution was comparable to (or could be adjusted to) the composition of the feed solution. Thus, a modification of the feed/ fines withdrawal was used. Fines were removed, and an equivalent amount of feed solution was added to the crystallizer. The arrangement is shown in Figure 9. One advantage of this rearrangement was that the flow rates of fines withdrawal could be measured continuously. Also, the amount of solid in the overflow stream (to be dissolved) could be measured. Further experiments were performed to have an estimate of the amount of water to be added for fines dissolution (hence estimate additional heat load). An inclined settler was used in order to concentrate the fines and thereby facilitate dissolution. The experimental setup is shown in Figure 10. For both these arrangements, the crystallizer used was that described in the context of MSMPR experiments. It was equipped with the fines classification/removal device described earlier. Steady-state samples from the underflow stream were used for particle size and slurry density measurements. As in the case of MSMPR experiments, six to eight residence times were allowed before collecting the sample for analysis. Treatment of the sample before CSD analysis and slurry density measurement was done as described in the context of MSMPR experiments. A summary of process conditions for all the fines dissolving experiments is given in Table I.
2786 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Fines Collector
ne--
i
p-IJ
I
FeedTank I I
Table 11. Summary of Results from the Fines Dissolving Experiments res. time, draw-off size increase, expt min ratio Lm,rm L ~ . ~ , r m % 317 222 43 SI01 75 5.5 310 250 24 SI05 65 2.3 365 250 46 SI07 65 2.33 400 282 42 SI09 58 1.85 405 250 62 SI13 65 2.76 SI14 65 2.77 3000 250 20 This experiment had fluctuating fines flow rate due to a pump problem.
--
Pump
r Heater Product Sample
---- Vacuum
v
Condensate
Liquid Flow Line
PI, P2, P 3 Pcriatallic Tube Pumps, TCI' Temperature IndicalorlController, TC2: Temperalure Indicator.
Figure 9. Experimental setup for the fines-dissolvingexperiments with a feed/fines flow modification. x-----------1
250
300
350
400
450
Calculated Mass Median Size, p m
500
Figure 11. Comparison of experimental particle size increases with predicted values for the fines-dissolving experimenta.
----
Pmducl Sample
Liquid Flow Vocuum Line
I: Crytlolllaer, Z*FineaDissolver. 32 Lamello Seltler, 4 ' F n d Pre-heoler, 53 ProduclColleclor, 61Vopor/Liquid Srparotor, 7: Non-return Valve, B. Vacuum Gouqe, 9: Chrontrol. IO: Condenser, 111 Condenrole Ga11rclor, 12. I33 Temperolurr Controller/lndkalorr. 14, Temprrolure Indicator, IS. 163 Hal Plalei. 17,18,19,20,21~ PerislallicTube Pumps. 22.231 Clo$edSystemVacuum Adopters. 241 Fines Trap.
Figure 10. Experimentalsetup for the fines-dissolvingexperiments using water to dissolve fines. Table I. Summary of Process Conditions for the Fines Dissolving Experiments vacuum, boil-up expt temp,OC in.Hg RPM ratioa L~,pm SI01 75 26.0 275 0.402 280 SI05 79 25.0 395 0.286 238 SI07 78 25.0 430 0.195 175 193 360 0.162 SI09 83 22.8 187 325 0.216 SI13 81 23.0 SI14 78 23.2 310 0.182 185 'Boil-up ratio = (rate of evaporation)/(rate of product slurry withdrawal).
Results and Discussion Table I1 provides a summary of all the size increase experiments involving this fines-dissolving configuration. Of these experiments, SI01 was performed using the configuration shown in Figure 10, and the rest of the
experiments were performed with the modified arrangement shown in Figure 9. The rates of fines to be dissolved in experiments SI05, SI07, SIO9, and SI13 were found to be, respectively, 56%, 65 % ,66 7% and 89 % of the corresponding production rate. These are higher than theoretically predicted values. Additional release of salt from the mother liquor due to cooling might have contributed to these high values. However, theoretically calculated values also were high (e.g., 50% for experiment SI05). A significant size increase was observed in experiment SIO1, in which fines dissolution was accomplished by using an inclined settler. In spite of the use of the settler to concentrate the fines before dissolution, an appreciable amount of water was required (approximately 18%v/v of the feed flow rate). This seemed unacceptable. Lower proportions of water were tried, but complete dissolution of fines could not be achieved due to flashing of water under high vacuum. Efficient dissolution (lower proportion of water) can be achieved under low vacuum, in which case the temperature can be increased to aid better dissolution. However, this is impracticable in a small unit like ours, although in large-scale operation this would be feasible. The problem of flashing may also be reduced significantly if the unsaturated liquor from the electrolytic cell (feed to the third-effect evaporator, 9% NaOH, 16% NaC1) is used to dissolve the fines. A plot of the particle sizes obtained from fiies-dissolving experiments against the sizes predicted from the design charta (high-yield system, i = 1.53, j = 1.24) is shown in Figure 11. There was good agreement between experimental and predicted values in three experiments (SI07, SIO9, and SI13). Deviation in experiment SI14 was mainly due to disturbances during the experiment. In experi-
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2787 ments SI01 and SI05, less than expected particle sizes were obtained.
Conclusion Easy-to-use design charts for the FD crystallizer were developed in the spirit of CSD analysislimprovement by the operating engineer without resorting to detailed simulation or process modeling. Such charta might also be helpful in discussions with crystallizer vendors. The present work shows that the FD configuration also has an optimum fines cut size. Based on this criterion, a design protocol has been suggested for the FD crystallizer. A limited amount of information obtainable from simple MSMPRexperimenta is required for the use of these design charts. The analysis underlying the graphs and charts assumes a sharp fines classification cut size which may not be achieved in many cases. Thus, the results may differ slightly from those obtained from charts. The simple nucleation kinetics model BO = kNGiMTi is not a given for every crystallization process, and the results presented could only be expected to hold for systems where the nucleation kinetics can reasonably be forced into the power-law form shown above. Larger-scale crystallization tests in collaboration with a crystallization vendor may be appropriate. Efficacy of the FD configuration for particle size increase in NaCl crystallization from caustic liquor has been experimentally demonstrated. It was shown that an appreciable increase in particle size is possible by using the FD crystallizer, but the size increase involves a considerable amount of water for fines dissolving. Therefore, there is a risk of product (alkali) quality degeneration (note: this is the final stage of evaporationlcrystallization). There was considerable crystallizer vessel fouling in all the FD experiments. Particle size increases in the finesdissolving experiments were compared with the values predicted from the design charts, and agreement between them was good in three experiments.
Acknowledgment The experimental part of this work was supported by the Dow Chemical Co., Midland, Michigan. This paper was presented at the AIChE 1993Spring National Meeting, Houston.
Nomenclature a = order of growth kinetics (dimensionless) BO = nucleation rate (L-3.t-l) G = growth rate (Ld-1) Go = growth rate, MSMPR value (L4-I) h(L) = size-dependent removal function (dimensionless) i = growth sensitivity parameter (dimensionless) j = slurry density sensitivity parameter (dimensionless) L = particle size (pm) Ld,l = dominant particle size for residence time T I (pm) Ld,2 = dominant particle size for residence time 7 2 (pm) LF = fines cut size in the crystallizer (pm) L3,2 = Sauter mean particle size (pm) L ~ , Z= ,Sauter ~ mean particle size, MSMPR value (pm) L4,3 = mass mean particle size (pm) L4,3,0= mass mean particle size, MSMPR value (pm) Lm = median size of mass distribution (pm) ~ 5 5 0= , ~median size of mass distribution, MSMPR value (pm) mp = pth moment of CSD, defined in eq 2.6 n = population density nl = population density in the particle size O-LF~
n2 = population density in the particle size range L F ~ - Q = volumetric flow rate (L3.t-1) Qo = overflow rate (L3.t-1) R = draw-off ratio (Q, + Q ) / Q (dimensionless) t = time z = dimensionless fines cut size, Lp/(G,) z, = dimensionless fines cut size, LF/(G,T) Yp = dimensionless part of CSD moment Ypp = dimensionless part of the moment of fines CSD a = ratio of growth rates, G/Go(dimensionless) p = dimensionless quantity, defined in eq 2 5 = fractional per-pass yield (dimensionless) 5, = fractional per-passyield, MSMPR value (dimensionless) X = dimensionless quantity, defined in eq 4 p = density of particle (m.L-3) T = residence time 0 = rate of dissolution of fines relative to production rate (dimensionless) w = dimensionless quantity, defined in eq 5
Appendix. Develoment of Design Equations for the FD Crystallizer The steady-state population balance for the FD configuration can be expressed as G-dn
dL + h(L$ = 0
where the size-dependent removal function, h(L),is given by
h ( L )= R, for L
< L,
(A24
The initial and the continuity conditions are
(A31
n(0)= Bo/G
Solution of eq A1 subject to conditions A2-A4 gives the population density function. Unlike the DDO crystallizer, the population densities in the crystallizer and in the product are the same and are a piecewise continuous line on a semilog plot (Figure 1). Thus
n = n, + n2
645)
where
n, = (Bo/G)e-RL/G7,for L
LF
(A61
Following the method of White and Randolph (1989), the CSD is expressed as a series of its moments:
mp = P!BOGP(~IR)P+~Y~
(AS)
where P
Yp = 1+ e-RRf~(Rx)i(RP+'-i - l)/i!
(A9
i=O
The dimensionless fines cut size, x, is given by
x = LFIGT (A101 Ratios of mass mean and Sauter mean sizes to corresponding MSMPR values are expressed, respectively, as
2788 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 - 7
The quantity a, the dimensionless growth rate, is given by a = GIG, (A131 The rate of dissolution of fines relative to the product rate, 9,can be calculated as '3,F
@ = ( R - 1)-
y3
where D
The fines cut size x is a dependent variable and has therefore been expressed as x = XJff
(A161
x , = L,/G,r
(A171
where The dimensionless fines cut size xo (based on the MSMPR growth rate) is an independent variable. The ratio of growth rates, a, may be obtained from an iterative solution of the dimensionless form of the mass balance equation: [{I - (1 - f,)~~"")/f,,]'-' = ~ Y ~ + ~ Y ~(A18) / R ~
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