Ind. Eng. Chem. Res. 1989, 28, 276-284
276
Literature Cited
Foundation for support of this work.
Nomenclature C = cumulative weight percent coke on the solid sample
R = activity of solid catalyst as percentreactionof ,,,-hexadecane pulse
V = cumulative reactant volume (liquid basis), p L V , = cumulative reactant volume leaving the matrix component of the composite when cumulative volume contacted the composite (liquid basis)
v has
Greek Letters = weight fraction of component in the composite catalyst qz = Partitioned-Flow model adjustable parameter representing the cumulative fraction of material leaving the matrix and contacting the zeolite, when cumulative volume V has contacted the composite $z = Partitioned-Flow model parameter representing the instantaneous fraction of material leaving the matrix and contacting the zeolite, for incremental volume AV contacting the composite = summation of all incremental volumes t
Subscripts c = composite catalyst m = silica-alumina matrix component z = zeolite component Superscript * = value calculated from model Registry No. AGZ-50, 118681-47-3;hexadecane, 544-76-3; silica, 7631-86-9; alumina, 1344-28-1.
Abbot, J.; Wojciechowski, B. W. The Effect of Temperature on the Product Distribution and Kinetics of Reactions of n-Hexadecane on NY Zeolite. J. Catal. 1988, 109, 274-283. Chowdhary, V. R.; Srinivasan, K. R. Desorptive Diffusion of Benzene in H-ZSM5 Under Catalytic Conditions Using Dynamic Sorption/Desorption Technique. J. Catal. 1986, 202, 316-337. Dean, J. W. Composite Catalyst Activity From Component Coking. Ph.D. Dissertation, West Virginia University, Morgantown, 1987. Dean, J. W.; Dadyburjor, D. B. An Ambient-PressurePulse Microreactor with Continuous Thermoeravimetricand On-Line Chromatographic Analyses for Catalytic Cracking. Ind. Eng. Chem. Res. 1988, 27, 1754. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. Hughes, R. Deactivation of Catalysts; Academic: New York, 1984. Masui, M.; Shimadzu, S.; Sashiwa, T.; Sawa, S.; Mimura, M. In Catalyst Deactivation; Delmon, B., Froment, G. F., Eds.; Elsevier: New York, 1980; p 261. Miale. J. C.: Chen. N. Y.: Weisz. P. B. Catalvsis bv" Crvstalline Alu" " minosilicates. 3. Catul. 1966, 6, 278. Nace, D. M. Catalytic Cracking over Crystalline Aluminosilicates. Ind. Eng. Chem. Prod. Res. Deu. 1969,8, 24-31. Oblad, A. G. Molecular Sieve Cracking Catalysts. Oil Gas J. 1972, 70, 84. Plank, C. J.; Rosinski, E. J.; Hawthorne, W. P. Acidic Crystalline Aluminosilicates. Ind. Eng. Chem. Prod. Res. Deu. 1964,3, 165. Satterfield, C. N. Heterogeneous Catalysis in Practice; McGraw-Hill: New York, 1980. Voorhies, A., Jr. Carbon Formation in Catalytic Cracking. Ind. Eng. Chem. 1945,37, 318. Received for review February 8, 1988 Revised manuscript received September 1, 1988 Accepted November 29, 1988
SEPARATIONS Optimum Fines Size for Classification in Double Draw-Off Crystallizers Edward T. White* and Alan D. Randolph Department of Chemical Engineering, University of Arizona, Tucson, Arizona 85721
The crystal size distribution (CSD) equations were solved for the double draw-off (DPO) crystallizer configuration and compared to the mixed suspension mixed product removal (MSMPR) configuration to give dimensionless design charts for mass and Sauter mean crystal sizes, crystal growth rate, slurry density, and supersaturation. The DDO configuration gives higher yields (lower supersaturation) and larger particle sizes than MSMPR operation by removing a classified overflow stream (particle size < LF) and a mixed underflow stream. These design charts enable prediction of particle size and yield improvement using the DDO configuration, knowing only the mean size produced in an MSMPR crystallizer together with estimates of nucleation and growth are kinetics. The optimum classification size, LF, to achieve these benefits can be selected. A crystallizer configuration that has utility in many applications is the double draw-off (DDO) design, wherein a classified overflow stream (classified at some fines cut size, LF)is removed from the mixed magma concurrently with a mixed suspension underflow stream (Figure 1). These overflow and underflow streams are then combined as the crystal product. This crystallizer configuration has been discussed extensively in the literature (e.g., Randolph
* Permanent address: Department of Chemical Engineering, University of Queensland, Brisbane, Australia. 0888-5885/89/2628-0276$01.50/0
and Larson Chapter 7, (1988); Hulburt and Stephango (1969); Randolph e t al. (1984); and Chang and Brna (1986)). Advantages that may result are (a) increased mean particle size, (b) less vessel fouling (because of lower supersaturation), and (c) higher yields of solute (important for slow-growing crystal systems). The product size distribution by weight is typically of bimodal form, which may be a disadvantage (less than expected filtration rates based on particle size) or an advantage (higher product bulk density). Revently, Randolph e t al. (1984) showed t h a t the mean Q 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 277 MS M PR CRY STALL1Z E R ...., i = 3
CONFIGURATION
0
,
L-
POPULATI~N DENSITY PLOT
DDO CRYSTALLIZER
4
CONFIGURATION
O
0
2
4
8
8
1
2
4
8
I
10
0
1
2
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POPULATION DENSITYPLOT
Figure 1. DDO (double draw-off) configuration and resulting size distribution.
size from a continuous crystallizer initially operating at low solids density ( M T ) can be considerably increased using the DDO configuration. The system studied was calcium sulfate in a simulated flue gas desulfurization (FGD) liquor producing gypsum crystals. With normal MSMPR operation, by use of such weak feed concentrations, a suspension density of 6 kg/m3 resulted, with a product crystal mass mean size of 60 pm. Considerable vessel wall fouling was observed. However, with DDO operation and a flow ratio of R (8,+ Q,)/Q, = 10 (i.e., 90% of the offtake is as a classified overflow), the mean crystal size was increased to 200 pm (a factor of over 3). The suspension density in the crystallizer increased to 60 kg/m3, and the crystallizer vessel was virtually free of wall fouling. A classifier cut size LF = 27 pm was used. Chang and Brna (1986) determined similar CSD improvement using a pilot scale FGD unit. There is an optimum cut size for DDO operation which results in a maximum product size, and this study provides charts which allow this optimum value to be selected. Double Draw-Off Configuration. The process flows and nomenclature of the DDO crystallizer are illustrated in Figure 1. As the overflow to underflow ratio increases, the suspension density and the solute yield increase, while the growth rate decreases. Note that the only practical limitation on this ratio is the magnitude of the slurry density, MT,which builds to intolerable levels if R is increased too much. At the other limit, as Q, 0 (or LF a),the system reduces to the simple MSMPR crystallizer. The equations for DDO operation are derived in detail in the references, particularly in Randolph and Larson (1988). These relations are summarized in the Appendix. A general analysis is done for variable recovery (class I) systems assuming growth and nucleation rates follow power law relations of the general form
-
-
G = kGSa
(1)
B o = kNG iMT'
(2)
Note that size-independent growth rate is assumed. Deviations from this assumption might occur, especially at small sizes. Such deviations at small sizes would not greatly affect calculations of mass size. Assuming G #
0
12
14
18
xF, DIMENSIONLESS CUT SIZE, LF/GoT,
Figure 2. Effect of cut size and draw-off ratio, R , on the mass mean size of the product for high-recovery systems.
G(L)greatly simplifies the calculations. This assumption was shown to be satisfactory in an experimental study of DDO operation for the gypsum system (Randolph et al., 1984). Qualitative changes in the CSD would not be expected to change in any case. The results are expressed as ratios compared to MSMPR operation. This allows generalized presentation. Use of these ratios has the advantage that the results are independent of k G and kN. That is, they do not depend on the absolute values of the growth and nucleation rates, but only on the form of their dependence on operating conditions. To obtain results for high yield (class 11) systems, it is necessary merely to consider results where the recovery approaches 100%. This, in fact, gives a simpler analysis, and the results for high yield systems will be discussed first. The results for MSMPR operation are obtained simply by letting Q, = 0 ( R = 1) in the equations. '
Results for High-Recovery (Class 11) Systems For high-recovery systems, the results can be related to two dimensionless operating quantities, R the draw-off ratio; and xF = Lp/G0r1, the dimensionless cut size. The graphs that follow solve th_e equations in the Append@ for (a) the mass mean size, L4,3;(b) the Sauter mean size, L,$; (c) the crystal growth rate, G; (d) the crystallizer suspension density, MT;and (e) the fraction of the product leaving in the overflow. These will be presented in turn, followed finally by an example showing their use. (a) Mass Mean Size, L4,3.Figure 2 shows the increase in mass mean product size due to DDO operation. Different values of j, the exponent for the power law dependence of nucleation rate on suspension density, are used
278 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 I
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Figure 3. Optimum size for fines classifier versus draw-off ratio, R, for high-recovery systems (i = 2). Optimal values for mass mean size and Sauter mean size are shown.
for (a), (b), and (c). A strong dependence on j is seen in these results. The ratio of the mass mean size with DDO operation to that without (i.e., MSMPR) is plotted against the dimensionless classifier cut size, xF, with the draw-off ratio R = (Q, + Q,)/Q, as a parameter. For R = 2 and 16, three curves are shown. These are for different values of i (the power law exponent for the dependence of nucleation rate on growth rate). The effect of i is not severe for j = 0 and only somewhat larger for higher j . For the other R values, only the curves for i = 2 are shown. As R increases at a fixed cut size in all cases, a coarser product results. This is as expected. Varying the cut size, xF, however, at a given R has a marked effect. First, for a zero cut size (clear liquor advance), there is usually (for i > j ) an increase in mean size. This occurs because the increased solids results in a lowered supersaturation, which reduces nucleation if the G factor in the B o correlation outweighs the M+ factor. When i = j , however, there is no change in size, and when i < j , a decrease in mean size is observed. This results because the factor M+ outweights the factor G and the nucleation rate then rises. Now, if the cut size is increased, a further, and substantial, increase in the mean size occurs. This results because the overflow exponentially decays the population density up to size LF, so the fines are removed at a faster rate, with the larger particles left for further growth. However, if the cut size is increased too much, excessive material is removed in the overflow, which when remixed with the underflow gives a smaller average size. Thus, there is an optimum cut size which gives product with the maximum mass mean size. On Figure 2 , the loci of these optima are drawn (for i = 21, giving the optimum cut size for any particular value of R. These optimum values are replotted in the upper curves on Figure 3. The corresponding maximum size ratios from Figure 2 are shown as a parameter. The optimum cut size is typically in the region 2-4 times G,T~, Le., of the order of the median mass size of the original MSMPR distribution. This is a substantial value and somewhat higher than might have been expected intuitively. (b) Sauter Mean Size, L 3 , Z . The filtering characteristics of product crystals depend primarily on the square of the Sauter mean size, L3,2. It also depends on the packing void fraction which will change to some extent with changes in the shape of the size distribution. The product size distribution from DDO operation will usually be bimodal on a mass basis (Figure 4). The first
15
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35
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L / G o ~ l , DIMENSIONLESS CRYSTAL SIZE
R, DRAWOFF RATIO = (Q,+Q,)/Q,
Figure 4. Typical mass mean size distribution for DDO operation. Note discontinuity at classifier cut size, L/G,rl = 3.
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peak is associated with the overflow fine particles and the second with the underflow coarse. The nature of this distribution will cause significant differences between the various mean sizes, depending on the magnitude of the contribution from the smaller sizes. The changes in the Sauter mean, L3,2, due to the draw-off ratio, R , and cut size, xF, are shown in Figure 5 , again as (a), (b), (c) for j = 0, 1, 2. Again, values are shown for i = 2, while curves for i = 1 and 3 are only shown for R = 2 and 16, to demonstrate differences. For the Sauter mean, i has a larger effect than for the mass mean. The curves are markedly different in appearance from those for the mean always E4,3 mean (Figure 2). For j = 0, the
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 279 2
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Figure 6. Growth rate changes for DDO operation with high-recovery systems.
Figure - 7. Suspension density changes for DDO operation with high-recovery systems.
increases, but the optimum cut size giving the best value is considerably lower than that for the L , , mean (also shown on Figure 5 ) . For j = 1and 2, L3,2values lower than that for MSMPR operation (Le., ratios less than 1)result at the higher cut sizes, even though the mean was larger. This is a direct result of the bimodal nature of the size distrubution. The optimum cut size giving the maximum increase in Sauter mean size has also been plotted against R as the lower curves of Figure 3. This optimum cut size is now only about 1-2 times Go71,typically half of that giving the optimum L4,0mean. ( c ) Growth Rate. Figure 6 shows the effect of R and cut size on the growth rate for three values of j (0, 1, 2). Superimposed are two lines corresponding to the optimum mass size and Sauter mean size conditions. For low values of cut size (clear liquor advance), there is considerable reduction in the operating growth rate in the crystallizer, compared to MSMPR values. For optimum cut sizes, the growth rate reduction is not as large. For xF = 0 and 4, two extra curves are shown (for i = 1 and 3). All other curves are for i = 2. The parameter i does not have a controlling effect. (a) Suspension Density. Figure 7 shows the increase in suspension density for j = 0, 1,2. For low cut sizes (xF values up to the optimum), the suspension density increases nearly proportionally to R. For high cut sizes, there are considerable deviations from this proportionality. This increase in MT limits the extent to which DDO operation can be applied to a crystallizer. If MT becomes high (typically several hundred kilograms per cubic meter),
agitation and suspension will be lost. (e) Fraction of P r o d u c t Mass i n Overflow. The fraction of the product mass carried out by the overflow is shown in Figure 8. At a low cut size, the fines crystal mass approaches zero. As the cut size is increased (for a given R), the fraction increases, until the cut size becomes so large that the underflow is identical with the overflow, and the fraction mass carried is just the ratio of the flow rates. For a given cut size, as R is increased, the fraction mass carried out first rises (because the flow rate increases) but then decreases as the crystals-become of larger size. A t optimum conditions for L4,3,about 20% of the mass is carried in the overflow stream, which is a somewhat larger value than might be expected prior to this analysis. At optimum conditions for .f;3,2, only 1-2% of the mass is carried in the overflow stream. Also shown on Figure 8, for xF = 4, are the corresponding curves for two other values of i, which illustrate the effect of this parameter. (f) Use of Results. The above graphs can be used to predict the effect of DDO operation. First MSMPR conditions (Go, MT,) have to be established, either experimentally or by calculation from measured kinetics. The form of the kinetic relationship (i.e., i, j ) should be known. Once values of R and LF are chosen, the changes in mean size and suspension density can be read directly from Figures 2, 5, and 7. Since suspension density is approximately proportional to R, this ratio is the first quantity to select. R should be chosen so that suspension densities do not exceed values which are difficult to handle. Having chosen R, a near
280 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 1.0
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Figure 9. Effect of draw-off ratio on recovery using a zero cut size (clear liquor advance) for low-recovery systems.
optimum value of LF could be chosen using Figure 3. From Figures 2 and 5, it can be seen that, around the optimum, the mean size is insensitive to changes in LF, so a value a little below the optimum could be selected. As an example, consider a high yield crystallization system where i = 2 and j = 1 and where the MSMPR crystallizer gave a mass mean size of 100 pm (i.e., Go7 = 25 pm) with a suspension density of 50 kg/m3. Supposing that the suspension density for this system could be increased to 400 kg/m3 without suspension problems; then R would be chosen as about 8. From Figure 3, the optimum cut size is XF = 2.9 for a product having maximum mass mean size, or 1.25 for one with maximum Sauter mean. From Figure 2b it can be seen that the maximum mean size is not greatly affected if the cut size is reduced to XF = 1.6. for this value, Figure 7 shows that the crystal content will be 376 kg/m3, while the mass mean size is 300 pm (Figure a), the Sauter mean size is increased by 2.25 times (Figure 5), and the growth rate decreases to 39% of the MSMPR values (Figure 6). With the lowered growth rate (and thus a lowered supersaturation) and a higher suspension density, scaling problems in the crystallizer should also decrease. It should be noted that the analysis on which these graphs are based assumes an ideal classifier with a sharp cut size. This is not achieved in practice, so there may be some slight variation of these results in practical implementation.
system, so that changes to operating conditions will not significantly affect the mass of crystal produced. However, the DDO configuration is as often utilized to increase per-pass yield as to increase particle size. In many cases, increased recovery is the main benefit that occurs from DDO operation. There are six independent variables involved in considering the analysis of DDO for low-recovery systems. These include the four involved for high-recovery systems, i.e., i, and j , the nucleation kinetics exponents, R, and x F , the DDO operation variables. For low-recovery systems, to these must be added the initial (MSMPR) recovery and a, the growth kinetics exponent (eq 1). It is impossible to cover the effect of all variables exhaustively. Instead, examples of the trends due to several of the variables will be shown, organized as follows: (a) the effect of R on recovery for a classifier cut size of zero; (b) the effect of changing the classifier cut size, x F , on this recovery; (c) the effect of the initial recovery on the mass mean size; (d) the corresponding effect on the Sauter mean size; (e) the effect on suspension density. These will be followed by an example illustrating the use of the plots. (a) Increase in Recovery. The maximum increase in recovery by DDO operation occurs with a zero classifier cut size (i.e., with clear liquor advance). Figure 9 shows for xF = 0 and for j = 0, 1, 2 the increase in recovery as R is increased. A range of lo (initial recovery) values is shown. The recovery increases most rapidly for high values of j . This is due to enhanced nucleation rates resulting from the increased solids content. For each initial recovery, three curves are shown for two
Results for Low-Recovery (Class I) Systems The preceding results assumed a high-recovery (class 11)
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 281 I.o
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values of i and two values of a. The effect of i is not large, though higher recoveries are obtained for smaller i. The effect of a (the growth rate exponent of eq 1)is larger. For a given value for R, higher recoveries result with a smaller value of a. This results because the growth rate falls less rapidly with a drop in supersaturation when a is small. (b) Effect of Classifier Cut Size on Recovery. As the classifier cut size is increased, the recovery will fall. However, the mean crystal size may increase. Figure 10 illustrates the effect on recovery of increasing the cut size for two initial recoveries (20% and 70%) and j = 0, 1, 2. Larger values of j give a lower decrease in recovery. (c) Effect on Mass Mean Size. The effect of cut size on the change in mass mean size is shown in Figure 11. The curves for 100% initial recovery are those for high
Figure 13. Effect of initial recovery on Sauter mean size for DDO operation of low-recovery systems.
recovery from Figure 2. The results do not change greatly for a wide range of initial recoveries. For a given R, for j < 1, the mass mean size is increased slightly for lower initial recoveries and decreases for j > 1. The optimum cut size is not greatly affected by the initial recovery, so as a reasonable approximation, Figure 3 could be used as well for low-recovery cases. Considering Figure 10, for operation a t the optimum cut size for the maximum mass mean, the recovery increase would be halved (approximately) compared with clear liquor advance. The selection of classifier cut size for low-recovery systems involves a trade-off between increased recovery and large-sized product. Figure 12 is similar to Figure 11but illustrates the effect of the exponent on the mean size increase, which also is small. (a) Sauter Mean Size. The effect of initial recovery on the change in the Sauter mean size is also small, as shown by Figure 13. The curves for 100% recovery are those from Figure 5. The critical value of the cut size for the maximum Sauter mean size again is not greatly dependent on the value of the initial recovery. Thus, Figures 2 , 3 , and 5 may be used with reasonable accuracy for both high- and low-recovery systems. (e) Suspension Density. The effect on the suspension density change for low-recovery systems in very substantial. The suspension density will increase with R as it did in the high-recovery system (Figure 7). But also, the suspension density will increase because of the increased recovery. This second contribution can outweigh the first. Figure 14 shows the suspension density as a function of R; the initial recovery, lo;and cut size, xF. Curves are shown for xF = 0,2, and 4 for j = 0, 1, and 2. The curves
282 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
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(Figure 9b), a mass mean size of 177 pm, and a Sauter mean size increase of 5170. The growth rate is decreased to 58% of the MSMPR value. With the reduced growth rate and increased solids content, scaling problems would again be reduced. Figure 15 shows the trends as various cut sizes are used. The value of R necessary to give a suspension density of 400 kg/m3 increases rapidly as X F is increased. The recovery falls, but is still well above the MSMPR value of 20%. The variation of the mass mean and Sauter mean sizes and growth rate is shown. The highest recovery occurs when xF = 0, the largest Sauter mean when X F = 1.5, while the mass mean keeps increasing. The increase in the mass mean is due to the effect of the considerable increases required of R to keep the suspension density at 400 kg/m3.
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Figure 14. Effect of DDO operation on suspension density for low-recovery systems.
for (, = 100% are shown in Figure 4. Over the range considered, for nearly all cases, the increase in the suspension density ratio exceeds R, because of the increased recovery. The increase is larger for higher values of j , for low initial recoveries, and for lower values of x F . As an example, for an initial recovery of lo%, with R = 2 and xF = 0, from Figure 14b, the suspension density would increase almost 10-fold, 2-fold due to the direct effect of R and almost 5-fold from the recovery increase to near 50% (Figure 9b). (f) Use of Plots. The use of these plots can be illustrated by an example. Consider an MSMPR crystallizer giving an initial recovery of 20% for a slow-growing system with i = 2, j = 1, and a = 1. Let the mass mean size be pm (Go7 = 25 pm) with a suspension density of 50 kg/m3. Suppose that a suspension density near 400 kg/m3 is desired. If maximum recovery is required, clear liquor advance (xF = 0) could be used. From Figure 14b, a draw-off ratio, R , of about 2-6 would give a suspension density of 408 kg/m3, a recovery of 63% (Figure 9b), and an increase in the mass and Sauter mean sizes of 21% (either interpolating from Figures 11and 12, approximately from Figures 2 and 5, or from the equations in The Appendix). The growth rate is reduced to 47% of the MSMPR value. However, if a major increase in product size is the goal, a suitable value of XF will have to be chosen. As the value of R depends on the XF value selected, an iterative approach is necessary. Consider xF = 1.6 (from Figures 2, 3, and 5). From Figure lob, a value of R = 3.2 would give a suspension density of 413 kg/m3, a recovery of 54%
Concluding Remarks The DDO crystallizer configuration is perhaps not as extensively used as it could be, although the design is known both in industry and the literature. The first uses of this configuration concentrated on increasing the perpass yield from a system exhibiting slow crystal growth and having a low concentration of solute in the feed liquor. However, it has become clear that the DDO design may be equally useful for increasing particle size and filtration rate, while reducing vessel fouling (e.g., Randolph et al. (1984) and Chang and Brna (1986)). Significantly, particle size can be increased without the energy cost associated with fines dissolving. The design equations for this crystallizer are well-known, so the purpose of this paper is to present useful design charts for a priori analysis of such systems, as well as to suggest a protocol for DDO crystallizer design. In essence, this design protocol consists of running simple bench-scale MSMPR tests (e.g., 7 varying from 0.5 to 2 h), which will easily reveal whether the system is high or low yield (class I1 or I). The necessary parameters for use of these design charts ( G O ~ i,l ,j , a, and can be estimated from several semilog population density plots of these MSMPR data. Analysis of a suitable DDO configuration can then proceed using the design charts and arithmetic calculations without necessarily involving computer simulation or solution of complex ODE'S. The analysis of particle size improvement is shown to be virtually independent of per-pass yield (class I or class I1 behavior); thus, this analysis would be useful in determining size changes obtained from both high- and low-yield DDO operation. The DDO configuration is not useful for every crystallizer application. It cannot be used for systems having a
c0)
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 283 high natural slurry density (i.e., high AC drop across the crystallizer) because the inevitable buildup of slurry density would soon result in an inoperable level of solids. The DDO configuration has typically been used with coolingtype crystallizers (surface or flash-cooled) but could be adapted to salting out or evaporative types.
Nomenclature a = exponent linking growth rate to supersaturation, eq 1 B o = nucleation rate, n0./(m3 min) C2 = constant, eq 4 G = growth rate, pm/min G o = MSMPR value i = exponent linking nucleation rate to growth rate, eq 2 j = exponent linking nucleation rate to suspension density, eq 2 k = moment number, eq 5 k G = growth rate constant, eq 1 k N = nucleation rate constant, eq 2 k, = volume shape factor, eq 9 L = crystal size, wm LF = classifier cut size G3.2 = Sauter mean size, km 43,2,= MSMPR value = mass mean size, pm L4,3,= MSMPR value m = variable number, eq 3 MT = suspension density, kg/m3 MTo = MSMPR value n = population density, n0./(m3 pm) no = nucleic population density: 1, below LF, 2, above LF o = subscript relating to steady-state MSMPR distribution Q = slurry flow rate, m3/s Q, = overflow Q, = underflow R = draw-off ratio, R = (Q, + Q,)/Q, s = supersaturation, kg/m3 si = available inlet supersaturation V = crystallizer working volume, m3
&,,
xF Xi
= LF/Gorl
= LF/Grl =
X F / ~
Yk = expression, eq 6
Greek Symbols a = ratio growth rates, DDO/MSMPR, eq 16 & = kth moment r1 = mean residence time = V/(Q, + Q,), min p = crystal density, kg/m3 { = solute fraction recovery lo= MSMPR value
Ci = ~XP[-Q&F/GVI Equation 4 shows that the population distribution of the
DDO crystallizer is piecewise exponential and continuous with respect to particle size, as illustrated in the size distributions shown in Figure 1. For this work we will define a mean residence time r1 = V/(Q, + Q,). This differs from the mean residence time used by Randolph and Larson (1988), r = V/Q, = Rrl. r1 is more convenient, as it does not change when R is altered (for a fixed total throughputrate). Thus, eq 4 may be rewritten as
nl = ( B o / G ) exp(-L/Grl) n2 = ( B o / G ) exp(-L/RGrl) exp(-xl(l - l / R ) ) (4') where x1 = LF/Grl. (b) Product Size Distribution Moments. The moments of the product size distribution are the sum of two truncated y functions and can be expressed generally as & =
k!B "G krlk+lYk
(5)
where k
Yk = 1 + exp(-x1)Cxli(Rk-' - 1)/i! i=O
(6)
Thus, the mass mean and Sauter mean sizes of the product are e4,3
=
= 4GriY4/Y3
(7)
&,2
= & / &= 3G71Y3/Y2
(8)
P4/!43
(c) Mass Balance. The product suspension density, MT, is related to the solute supersaturation drop, si - s; the recovery, z ; and the crystal size distribution by (9) where the fractional recovery of the crystal { = 1 - s/si. (d) Crystallization Kinetics. To obtain a solution to the above equations, relationships between G, B o , and s are required. We will use the common kinetic forms MT
= pkvp3 = si - s = si{
G = kGSa
(1)
B o = kNGiMTJ (2) Combining these kinetic relations with the material and population balances gives
Appendix: Derivation of DDO Model A crystal size distribution model for the DDO crystallizer can be developed from population and mass balances. The derivation is given in greater detail by Randolph and Larson, Chapter 7 (1988). Consider the DDO crystallizer shown in Figure 1. (a) Population Balance. Let subscript 1 refer to the particle distribution below the cut size LF and 2 to that above. Then the population balance can be written as dnm VG - = Q,(L)n, (3)
dL
where for L < LF, m = 1 and Q1 = Q, + Q,, and for L > Lf, m = 2 and Q2 = Q,. Integration of eq 3 shows that the population density decays exponentially at a rate proportional to the removal rate, Q,. Thus,
6pkvk N k G3+irl4y4jy31-js(3+i)~ (10) where Y3 and Y4 are also functions of s. This can be solved for s, and thus the growth rates, recovery, and mean sizes can be evaluated. To obtain a solution, values are required for the kinetic rate constants, kGand kN. These can, however, be eliminated if the results are expressed as ratios to the corresponding MSMPR results. (e) Ratio to MSMPR Values. For MSMPR behavior, R = 1 and thus Yk = 1. Dividing the DDO operation values by the corresponding MSMPR values ( R = 1, all else the same) gives the following relations: (si - s)l-j
[{I - (1 - {,)a1/a)/{,]l-j = (y3+iyjy 4 31-j
(11)
L4,3/L4,3, = ay4/y3
(12)
L3,2/L3,2,
= a y3/
yZ
MT/MT, = (a3+'Y4)1/('-j)
nl = no exp[-(Q, + Q,)L/GVl n2 = C2n0exp[-Q,L/GV]
For continuity of the two distributions a t LF,
(4)
{ = 1 - (1 - "/1.),[
(13) (14) (15)
Ind. Eng. Chem. Res. 1989,28, 284-288
284 a = G / G o = (s/s0)‘
(16)
where the subscript o refers to the corresponding MSMPR operation values and the nonsubscripted variables to DDO operation. Given MSMPR operation values (Go, lo)and kinetic exponents (i, j , a), eq 11 can be solved iteratively for a for selected values of R and XF. In the Yk terms, x1 may be replaced by xF/a. Once a is evaluated, all other ratios can be calculated directly. The results can be expressed in terms of the independent variables i, j , a, R, and xF. For high-yield (class 11) systems, lo 1 and a is not involved, so only the variables i, j , R, and X F are needed. If desired, eq 11 can be programmed simply for iterative solution in a computer. (f) Optimum Values. The values of XF giving maximum values of the mass mean size and Sauter mean size can be obtained by differentiating eq 12 and 13 (and eq 11). Note that aYk/dx, = Yk-1 - Yk (17)
-
c0,
The resulting equations may be solved iteratively. (g) Clear Liquor Advance. For clear liquor advance, xF = 0 (and x1 = 0). Hence, Yk = Rk. With eq 11, this gives
(l/lo)’-’ = [ (1 - () / (1 - l,,)](3+i)”R3+’ which can be solved for the recovery, l.
(18)
Literature Cited Chang, J. C. S.; Brna, T. G. Gypsum Crystallization for Limestone FGD. Chem. Eng. Prog. 1986 (Nov), 51. Hulburt, H. H.; Stephango, D. G . Design Models for Continuous Crystallizers with Double Draw-Off. CEP Symp. Ser. 1969, 65(95), 50. Randolph, A. D.; Larson, M. A. Theory of Particulate Processes, 2nd ed.; Academic Press: New York, 1988. Randolph, A. D.; Vaden, D. E.; Stewart, D. Improved Crystal Size Distribution of Gypsum from Flue Gas Desulfurization Liquors. Chem. Eng. Symp. Ser. 1984, 80(240), 110. Receiued f o r review April 28, 1988 Revised manuscript received November 18, 1988 Accepted December 13, 1988
Chemical Model for Copper Extraction from Acidic Sulfate Solutions by Hydroxy Oximes Jerzy Piotrowicz,? Mariusz B. Bogacki,’ Stanidaw Wasylkiewicz,’ and Jan Szymanowski*,’ Wroclaw Technical University, Wybrzeie Wyspi0Askiego 27, 50-370, Wroclaw, Poland, and Poznafi Technical University, P1. Sklodowskiej-Curie 2, 60-965 Poznari, Poland
Chemical models for copper extraction from acidic sulfate solutions by hydroxy oximes are presented. Extraction constants and hydroxy oxime dimerization constants are calculated. They demonstrate that hydroxy oxime dimerization in the organic phase can be neglected for hydroxy oxime concentrations u p to 20%. Copper extraction from acidic sulfate solutions by commercial hydroxy oxime extractants was discussed in several independent papers. Forrest and Hughes (1975a,b), Hughes et al. (1975), and Robinson and Paynter (1971) discussed empiric and semiempiric models and proposed to use them to correlate equilibrium copper concentration in the organic phase with its equilibrium concentration in the aqueous phase. Szymanowski and Jeszka (1985) used a similar approach to compare simple multistage and countercurrent multistage copper extraction by “pure” isolated fractions of 2-hydroxy-5-nonylbenzaldehyde oxime and 2-hydroxy-5-nonylbenzophenone oxime. Szymanowski and Atamadczuk (1982) used the modified Couchy distribution to model the extraction of copper by pure individual hydroxy oximes from very dilute acidic sulfate solutions. Piotrowicz and Wasylkiewicz (1986) proposed recently the chemical approach, in which activity coefficients of inorganic species present in the aqueous phase were calculated according to the modification of the Pitzer method (Wasylkiewicz, 1988). The aim of this work is to model the extraction of copper from acidic sulfate solutions by “pure” isolation fractions of 2-hydroxy-5-nonylbenzaldehyde oxime and 2-hydroxy5-nonylbenzophenone oxime and to discuss hydroxy oxime association in the organic phase during extraction.
* Author to whom correspondence should be addressed. Wrodaw Technical University.
* PoznaA Technical liniversity.
Extractants and Extraction Conditions Extractant syntheses were described previously (Szymanowski and Jeszka, 1985). The ratio of E to Z isomers in 2-hydroxy-5-nonylbenzophenone oxime (11)was 5.9; in 2-hydroxy-5-nonylbenzaldehyde oxime (I),Z isomer was not found. Extraction data were obtained at 18-20 “C for different hydroxy oxime concentrations and different initial concentrations of sulfuric acid in an aqueous phase. The oximes’ concentrations were approximately equal to 5%, lo%, 15%, and 20%, while the amount of sulfuric acid added to the aqueous phase vriried from 0 to 50 g dm-3 for 2-hydroxy-5-nonylbenzophenoneoxime and from 0 to 200 g dm-3 for 2-hydroxy-5-nonylbenzaldehyde oxime. The equilibrium copper concentrations in the aqueous phase varied from 0 to 30 g dm-3. Results In a previous paper (Szymanowski and Jeszka, 1985), different polynomials were used to match experimental data and to correlate the equilibrium copper concentration in the organic phase with the equilibrium copper concentration in the aqueous phase and the initial concentration of sulfuric acid. Simultaneously, appropriate polynomials correlating the equilibrium sulfuric acid concentration with its initial concentration in the aqueous layer and with the equilibrium copper concentration in the organic phase were derived. These polynomials were statistically significant a t a probability level of 1.0000, and their correlation
0888-5885/89/2628-0284$01.50/00 1989 American Chemical Society
