Ind. Eng. Chem. Res. 2001, 40, 707-713
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Interactive Effects of Particle Mixing and Segregation on the Performance Characteristics of a Fluidized Bed Crystallizer Lie-Ding Shiau* and Tsan-Sheng Lu Department of Chemical and Materials Engineering, Chang Gung University, Kweishan, Taoyuan, Taiwan, Republic of China
A mathematical model that considers the interactive effects of the segregation and mixing of the crystals is developed in this paper to describe the behavior of crystal growth in a fluidized bed crystallizer operated in a batch mode. It is assumed that the liquid phase moves upward through the bed in plug flow, and the solid phase is represented by a series of equal-sized ideal mixed beds of crystals. The crystals in each bed are completely mixed; however, the crystals in different beds are totally segregated. This concept corresponds to a one-parameter method, and the one parameter of this model is the number of mixed beds of crystals in series (M). The developed model can determine the crystal size, the solution concentration, and the bed voidage in each mixed bed of crystals during the operation. Subsequently, it can predict the weightaverage size (L h ) and the coefficient of variation (CV) of the resulting crystals in a fluidized bed crystallizer. Introduction The basic principle of a fluidized bed crystallizer is that a slightly supersaturated solution is passed upward through a bed of crystals and relieves its supersaturation on the suspended crystals.1 The crystals in the suspended bed are then allowed to grow during the operation. A uniform product is thereby obtained because the crystals are not discharged until they have grown to the required size. Because impeller is not used in a fluidized bed crystallizer, there is no contact of crystals with impeller. In addition, the contacts among crystals can be reduced to a lesser degree by the nearplug-flow pattern of the liquid phase in the fluidized bed. As a consequence, the generation of nuclei can be effectively eliminated. For the production of large, uniform crystals, such a fluidized bed type of crystallizer is often recommended on the grounds that a nucleation rate lower than that in an agitated type of mixed flow crystallizer can result. As the supersaturated solution flows upward through a fluidized bed crystallizer, the liquor contacting the bed relieves its supersaturation on the growing crystals, and subsequently, the supersaturation decreases along the upward direction. As a result, the crystals near the bottom grow faster than those near the top of the crystallizer. Such behavior results in a variation in particle size with height. The crystals segregate in the bed with the small ones at the top and the large ones at the bottom. When the bed is composed of particles of different sizes, the opposing effects of classification and mixing interact with each other to determine the overall behavior of the system. However, for simplicity reasons, design methods for fluidized bed crystallizers are generally based on the assumption of perfect size classification.2 Perfect size classification implies that the crystals segregate in the bed with the small ones at the top and the large ones at the bottom, i.e., there is no mixing * Author to whom correspondence should be addressed. Tel.: 011-886-3-3283016 Ext. 5291. Fax: 011-886-3-3283031. E-mail:
[email protected].
effects among crystals. Recently, Shiau et al.3 developed a model to describe the behavior of crystal growth in a fluidized bed crystallizer operated in a batch mode. It is also assumed that the liquid phase moves upward through the bed in plug flow and that the solid phase in the fluidized bed is perfectly classified. This model can determine the variations of the crystal size, the solution concentration, and the bed voidage along the bed height during the operation. In practical applications, it is impossible to ensure that crystals in the fluidized bed are perfectly classified. Many studies have shown that, when the bed is composed of particles of different sizes, the opposing effects of classification and mixing interact with each other to determine the overall behavior of the system.4-10 Thus, the crystals in a fluidized bed crystallizer are more or less vertically mixed. A one parameter model will be proposed in this paper to account for the deviation of the interactive effects of particle classification and mixing from perfect size classification in a fluidized bed crystallizer. Through this study, one can gain better insight into the complex crystallization behavior in a fluidized bed crystallizer. Model Development A fluidized bed crystallizer isothermally operated in a batch mode is discussed in this section. Seed crystals are initially placed in the crystallizer and allowed to grow as the supersaturated solution is pumped continuously into the bottom of the crystallizer. The supersaturation of the inlet solution is at a maximum at the bottom of the crystallizer and decreases along the upward direction. The solution velocity is controlled so that the crystals are uniformly suspended in the fluidized state. At the end of a run, the crystals are withdrawn as product. A mathematical model that considers the interactive effects of segregation and mixing of the crystals, will be developed to determine the resulting crystal size distribution (CSD) of a fluidized bed based on the following assumptions: (1) The system is maintained at isothermal conditions. (2) The
10.1021/ie0005250 CCC: $20.00 © 2001 American Chemical Society Published on Web 12/19/2000
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liquid phase moves upward through the bed in a plug flow pattern. (3) The solid phase is represented by a series of equal-sized ideal mixed beds of crystals. (4) The seeding crystals are uniform-sized. (5) No nuclei formation occurs (operation within the metastable region). (6) Crystal breakage is neglected. Note that assumption 3 is included here to take into account the interactive effects of segregation and mixing of the crystals in this fluidized bed crystallizer. It is assumed that the solid phase is represented by a series of equal-sized ideal mixed beds of crystals. The crystals in each bed are completely mixed; however, the crystals in different beds are totally segregated. This concept corresponds to a one-parameter method, and the one parameter of this model is the number of mixed beds of crystals in series (M). A limiting case is represented by a crystallizer with a plug flow of solution and a mixed bed of crystals (M ) 1), where all of the crystals are well mixed in the bed.11 For the intermediate cases, it can be assumed that the liquid phase moves upward in plug flow and that the solid phase is represented by M equal-sized ideal mixed beds of crystals in series (M g 2). As M approaches infinity, this model corresponds to perfect size segregation model developed by Shiau et al.3 In this section, first, the limiting case, M ) 1, is discussed. This case provides a reasonable starting point, although complete mixing of solids is, in practice, not compatible with a plug flow of liquid because of the back-mixing of liquid caused by the movement of the solids. To simulate the crystallization process of the fluidized bed crystallizer described above, the fluidized bed is subdivided into N stages, each of which contains solutions of equal supersaturation. Note that each stage has the same volume and contains the same number of crystals at the start of a run. As N approaches infinity, assumption 2 will be satisfied. Because the crystals are completely mixed, it is assumed that the crystals circulate very rapidly in the various stages and are of the same size. Although crystals continue to grow during the operation, the variations of crystal size, bed voidage, and bed height need to be derived. The unsteady-state mass balance of the solute in stage j can be described by
1 d (AHsCj) ) Qj-1Cj-1 - QjCj - NpFpβL2G dt 2 j ) 1, 2, ..., N (1) where AHs denotes the volume of the liquid phase in stage j. The term on the left-hand side of eq 1 is the rate of solute accumulation in stage j. The first and second terms on the right-hand side of eq 1 indicate the inlet and outlet solute mass flow rates in stage j, respectively, and the last term denotes the rate of growth on the surface of the suspended crystals in stage j. The factor of 1/2 on the last term of eq 1 arises because G is the growth rate of the crystal diameter. As the crystals are completely mixed in the fluidized bed, it is assumed that the crystals circulate very rapidly among various stages from j ) 1 to j ) N. Thus, the growth rate G can be defined as
G)
dL dt
N
)
kgL (Cj - C*) /N ∑ j)1 l
n
(2)
Similarly, the unsteady-state mass balance of the
solvent in stage j can be described by
[ ( )]
Cj d AHs 1 dt Fp
(
) Qj-1 1 -
) ( )
Cj-1 Cj - Qj 1 Fp Fp j ) 1, 2, ..., N (3)
where Cj/Fp denotes the volume fraction of the solute in the liquid phase and 1 - Cj/Fp denotes the volume fraction of the solvent in the liquid phase. The additivevolume property is assumed to be applicable in the solublization process of the system being studied. Combining eqs 1 and 3 to eliminate Qj yields
dCj 1 dL 1 Q (C - Cj) - Np(Fp - Cj)βL2 ) dt AHs j-1 j-1 2 dt j )1, 2,...., N (4)
[
]
As the crystals grow during bed operation, the bed voidage will vary with the operating time. Information on the variation of bed voidage with particle size is important, because it determines the size of the crystallizer, which can be a significant factor affecting the total cost of the process. In the bed expansion experiments,6,10,12 the relationship between crystal size and bed voidage was found to be in agreement with the Richardson-Zaki equation.l3 Therefore, the RichardsonZaki equation is adopted to describe the expansion characteristic of the fluidized bed
us ) uiz
(5)
ui ) ut10-L/d
(6)
where us is the superficial velocity of the solution and ut is the terminal free-fall velocity of particles of size L placed in a column of diameter d. The superficial velocity normally lies between the terminal velocity and the minimum fluidization velocity. For instance, ut lies between 0.01716 and 0.0858 m/s, and Ret lies between 2.91 and 72.7 for potassium alum crystals of size ∼200-1000 µm suspended in aqueous saturated solution. The expansion index z is a function of Ret and is given by
L z ) 4.4 + 18 Re-0.1 for 1 < Ret < 200 t d
(
)
Ret )
utFL µ
(7) (8)
The term ut in eqs 6 and 8 is the terminal free-fall velocity of crystals of size L which can be evaluated as14,15
[
]
2 2 4 (Fp - F) g ut ) 225 Fµ
1/3
L for 0.4 < Ret < 500 (9)
By differentiating eqs 5-9 with respect to t, one obtains
Ind. Eng. Chem. Res., Vol. 40, No. 2, 2001 709
d 1 dui dz )+ ln dt z ui dt dt
(
)
(10a)
1 dui 1 dut ln 10 dL ) ui dt ut dt d dt
(10b)
Table 1. Conditions Employed in the Example
Re-0.1 dRet dzj dL t L ) 18 - 0.44 + 1.8 Re-1.1 t dt d dt d dt
(
)
(
(10c)
)
(10d)
dL dt
(10e)
dRet F dut dL ) L + ut dt µ dt dt
[
]
2 2 dut 4 (Fp - F) g ) dt 225 Fµ
1/3
L0, µm Nt W0, kg C0, kg/m3 Q0, m3/s d, m n in eq 217 l in eq 217 kg in eq 2,17 m3n+1 s-1 kg-n us, m/s H0, m 0 operating temperature, °C N
Table 2. Basic Physicochemical Properties of Potassium Alum at 23.5 °C
Substituting eqs 10b-10e into eq 10a yields
[
{
FutRe-1.1 t L d ln 10 1 ) - + ln 0.88 + 3.6 dt z d L µ d -0.1 Ret dL 18 (11) d dt
(
]}
Hs )
NpRL
3
(12)
A(1 - )
It should be noted that AHs(1 - ) is the volume of the solid phase in each stage, which equals NpRL3, the total volume of crystals in each stage. By definition, we have Ht ) NHs. Differentiating Hs in eq 12 with respect to t gives
dHs d dL 1 AHs ) + 3NpRL2 dt dt dt A(1 - )
(
)
R β C*, kg/m3 FP, kg/m3 F, kg/m3 µ, kg m-1 s-1
)
The above equation determines the bed voidage as a function of the operating time. As the bed voidage changes, the bed height of each stage can be calculated by
(13)
Equations 2, 4, 11, and 13 constitute a set of N + 3 differential equations with initial value problems that can be solved simultaneously by numerical methods with time increments. Note that the solution flowrate Qj-1 in eq 4, which will vary with the stage number and the operating time, needs to be evaluated by eq 3. At t ) 0, the values of (L, Hs, , Cj) are set to be (L0, H0/N, 0, C0,), where H0/N is the initial bed height of each stage at the start of a run. The initial values of 0 and H0 can be calculated by using eqs 5-9 and 12 when the seed size and the liquid superficial velocity are given. The above derivation can be generalized to any intermediate cases (M g 2). For example, in the case of M ) 2, it implies that the solid phase is divided into two bedssa lower bed and an upper bed. Each bed is of equal bed volume and contains an equal number of crystals at the start of a run. The crystals in each bed are completely mixed, and the crystals in different beds are totally segregated. Because the liquid phase moves upward in plug flow, the derivation of the flow behavior of each bed can then be treated in a manner similar to that described above. Thus, two sets of eqs 1-13 can be written to describe the flow behavior of the upper bed and the lower bed. It should be noted that the solution outlet from the lower bed is the solution inlet of the upper bed. The values of (L, Hs, , Cj) in each bed can then be determined as a function of the operating time.
200 2.72 × 107 0.18 140 2.43 × 10-5 0.06 1.6 0 9.09 × 10-5 0.00858 0.229 0.842 23.5 40
0.47 3.46 127 1760 1060 1.25 × 10-3
Thus, the resulting CSD in such fluidized bed crystallizer can be calculated. Results and Discussion An example is presented below to demonstrate the application of the developed model. The potassium alum system is chosen herein because its crystals are close to perfect octahedra and its fluidization behavior is similar to that of spheres.16 Basic physicochemical properties of potassium alum are given in Table 1. For seed crystals of size 200 µm settling freely in a saturated solution of potassium alum, Ret ) 2.91 and ut ) 0.01716 m/s from eq 9. The superficial velocity us is set at onefifth of the seed settling velocity in an infinite medium, i.e., us ) 0.00343 m/s. Thus, the conditions employed in the example are listed in Table 2. The initial values of the total bed height and the bed voidage in Table 2 are calculated with the Richardson-Zaki relationship.l3 The initial voidage is assumed to be constant along the bed height because mono-sized seeds are initially suspended in the fluidized crystallizer. The minimum fluidization velocity is given by14-15
[
]
gF(Fp - F)L3 umfFL ) 33.72 + 0.0408 µ µ2
1/2
- 33.7 (14a)
for the whole range of Reynolds number, or for small particles
umf )
g(Fp - F)L2 for Rem < 20 1650µ
(14b)
and for large particles 2
umf )
g(Fp - F)L for Rem > 1000 24.5F
(14c)
umfFL µ
(15)
where
Rem )
In this case, umf lies between 0.000133 and 0.00213 m/s,
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Figure 2. Variation of solution concentration with bed height for M ) 1.
Figure 1. L/L0 and Ht/H versus time for M ) 1.
and Rem lies between 0.023 and 1.45 for potassium alum crystals of size ∼200-800 µm suspended in aqueous saturated solution. As crystals grow larger during bed operation, us must be chosen to fall between umf of the largest crystal attainable and ut of the seed crystals to ensure that all of the crystals can be fluidized and will not be carried away during the operation. The fourthorder Runge-Kutta method is employed to solve the governing equations. The simulation results and discussion of the present example are given below. As crystals grow at different rates in different mixed beds during operation, the crystals in the whole fluidized bed exhibit a CSD. The weight-average size and the coefficient of variation of the resulting crystals are defined as M
L h)
CV )
x
wiLi ∑ i)1
(16)
M
wi(Li - L h )2 ∑ i)1 L h
Figure 3. Variation of bed voidage with bed height for M ) 1.
(17)
where wi is the weight fraction of crystals of size Li in mixed bed i (i ) 1, 2, ..., M) and can be expressed by
wi )
(Nt/M)RL3i Fp Wt
)
(Nt/M)RL3i Fp M
(18)
(Nt/M)RL3i Fp ∑ i)1
Figures1-3 are displayed for the case of M ) 1. In Figure 1, the total bed height (Ht), which determines the size of the crystallizer, increases rapidly with the operating time and reaches about 2.90 times the original bed height after 1 h of operation. Figure 1 also shows that the crystal size (L) increases by about 1.75 times after 1 h of operation. CV equals 0 in this case because the solid phase is completely mixed in the fluidized bed for M ) 1 and the resulting crystals are uniform-sized (note that the seeding crystals are initially uniformsized). Figure 2 shows that C/C0 is a monotonic decreasing function of H/Ht for several operating times. As the solution relieves its supersaturation on the growing crystals, the supersaturation decreases along the up-
ward flow direction of the bed. The supersaturation of the inlet solution drops more rapidly as the crystals grow larger in size, and subsequently, the overall surface areas of crystals available for crystal growth become larger during bed operation. The horizontal dotted line in Figure 2 represents the saturated condition (i.e., C*/C0). Figure 3 depicts that the voidage of the bed drops monotonically with the operating time, which implies that the voidage of the bed decreases as the crystals grow larger in size. The value of the initial voidage 0 () 0.842) is exhibited by the dotted line. Figures 4-8 display results for the case of M ) 2. As shown in Figure 4, the crystals in the lower bed (Ll/2) grow faster then those in the upper bed (Lu/2) as the supersaturation decreases along the upward flow direction of the bed. Because the crystals in the lower bed and in the upper bed have different sizes, L h and CV of the resulting CSDs are plotted against the operating time in Figures 4 and 5, respectively. Figure 6 illustrates the variation of Hl/2/H0, Hu/2/H0, and Ht/H0 with operating time. This figure indicates that the bed height of the lower bed (Hl/2) expands faster than that of the upper bed (Hu/2) during the entire operation. Note that Ht ) Hl/2 + Hu/2. Figure 7 shows that C/C0 is a monotonic decreasing function of H/Ht for several operating times. The deflection point in each line indicates the transition
Ind. Eng. Chem. Res., Vol. 40, No. 2, 2001 711
Figure 4. Ll/2/L0, Lu/2/L0 and L h /L0 versus time for M ) 2.
Figure 7. Variation of solution concentration with bed height for M ) 2.
Figure 5. Coefficient of variation versus time for M ) 2.
Figure 8. Variation of bed voidage with bed height for M ) 2.
Figure 6. Hl/2/H0, Hu/2/H0 and Ht/H0 versus time for M ) 2.
points shift gradually toward the top of the fluidized bed with the increasing operating time as the bed height of the lower bed (Hl/2) expands faster than that of the upper bed (Hu/2) during the operation. Figure 8 shows that the voidages of the lower and upper beds both drop monotonically with operating time. As illustrated in Figures 9 and 10, although the value of M has little influence on the L h value of the resulting CSD, it has great effect on the CV of the resulting crystals. Based on the simulation, the resulting crystals are monodisperse for M ) 1, made up of two different sizes for M ) 2, made up of three different sizes for M ) 3, and so on. As the value of M increases, the resulting crystals become more polydispersed leading to the increase in CV. As shown in Figure 10, CV equals 0 for M ) l, increases from M ) 1 to M ) 4, and then remains nearly constant for M g 4. Figure 9 also shows that Ht decreases quite significantly from M ) 1 to M ) 4 and then remains nearly unchanged for M g 4.
of the upward liquid flow from the lower bed to the upper bed. For any operating time, C/C0 drops faster along the bed height in the lower bed than in the upper bed because the crystals in the lower bed are larger in size and subsequently consume supersaturation faster than those in the upper bed. In addition, the deflection
Conclusions A model is presented in this work to investigate the interactive effects of the segregation and mixing of crystals in a fluidized bed crystallizer operated in a batch mode. The case of M ) 1 implies that the solid
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Acknowledgment The financial support of the National Science Council of the Republic of China under Grant NSC 89-2214-E182-003 is greatly appreciated. Notation
Figure 9. Effect of M on L h /L0 and Ht/H0 after 1 h of operation.
Figure 10. Effect of M on coefficient of variation after 1 h of operation.
phase is completely mixed in the fluidized bed without any segregation effect. As the value of M increases, the segregation effect becomes more important in the flow behavior of the crystals while the mixing effect becomes less important. The case of M f ∞ implies that the solid phase is completely segregated in the fluidized bed without any mixing effect. The simulation results reveal that, although the value of M has little influence on the L h value of the resulting CSD, it has great effect on the CV of the resulting crystals. The resulting crystals are monodisperse for M ) 1. As the value of M increases, the resulting crystals become more polydisperse, leading to the increase in CV. For the potassium alum system presented in the example, the interactive effects of particle mixing and segregation in a fluidized bed crystallizer can be classified into four categories, i.e., M ) 1, M ) 2, M ) 3, and M ) 4. The case of M g 5 can be approximated by the case of M ) 4. This finding should not be extended to other material because the flow behaviors of a fluidized bed crystallizer is greatly influenced by the flow conditions and the physicochemical properties of the crystallizing material. Nevertheless, this model presents an efficient method for studying the interactive effects of particle mixing and segregation in a fluidized bed crystallizer.
A ) cross-sectional area of the bed, m2 C0 ) concentration of the influent solution, kg/m3 Cj ) concentration of the solution in stagej, kg/m3 C* ) saturated concentration of the solution, kg/m3 CV ) coefficient of variation defined by eq 18 d ) diameter of the bed, m g ) 9.8 m/s2, gravitational acceleration G ) crystal growth rate, m/s H ) vertical position from the bottom of the bed, m H0 ) total bed height at the start of a run, m Hs ) bed height of each stage given by eq 12, m Ht ) total bed height, m Hl/2 ) bed height of the lower bed for M ) 2, m Hu/2 ) bed height of the upper bed for M ) 2, m kg ) constant in eq 2, m3n+1 s-1 kg-n l ) constant in eq 2 L ) crystal size, m L0 ) seed size, m Ll/2 ) crystal size in the lower bed for M ) 2, m Lu/2 ) crystal size in the upper bed for M ) 2, m L h ) weight-average size of the crystals defined by eq 17, m M ) number of mixed beds of crystals in series n ) constant in eq 2 N ) number of stages in the bed Np ) number of crystals in each stage () Nt/N) Nt ) total number of seeds Qj ) outlet flow rate of the solution from stage j, m3/s Rem ) Reynolds number given by eq 15 Ret ) Reynolds number given by eq 8 t ) operating time, s ui ) velocity given by eq 6, m/s umf ) minimum fluidization velocity, m/s us ) superficial velocity, m/s ut ) terminal velocity, m/s wi ) weight fraction of crystals of size Li in mixed bed i (i ) 1, 2, ..., M) defined by eq 19 W0 ) total mass of seeds, kg Wt ) total mass of crystals, kg z ) the expansion index given by eq 7 Greek Letters R ) volume shape factor of the crystal β ) surface shape factor of the crystal 0 ) initial bed voidage at the start of a run ) bed voidage l/2 ) bed voidage of the lower bed for M ) 2 u/2 ) bed voidage of the upper bed for M ) 2 µ ) viscosity of the solution, kg m-1 s-1 F ) density of the solution, kg/m3 Fp ) density of the crystal, kg/m3
Literature Cited (1) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineers’ Handbook, 6th edition; McGraw-Hill: New York, 1984. (2) Mullin, J. W.; Nyvlt, J. Design of classifying crystallizer. Trans. Inst. Chem. Eng. 1970, 48, T7. (3) Shiau, L. D.; Cheng, S. H.; Liu, Y C. Modelling of a fluidizedbed crystallizer operated in a batch mode. Chem. Eng. Sci. 1999, 54, 865.
Ind. Eng. Chem. Res., Vol. 40, No. 2, 2001 713 (4) Kennedy, S. C.; Bretton, R. H. Axial dispersion of spheres fluidized with liquids. AIChE J. 1966, 12, 24. (5) Carlos, C. R.; Richardson, J. F. Solids movement in liquid fluidized beds. II. Measurement of axial mixing coefficients. Chem. Eng. Sci. 1968, 23, 825. (6) Al Dibouni, M. R.; Garside, J. Particle mixing and classification in liquid fluidized beds. Trans. Inst. Chem. Eng. 1979, 57, 94. (7) Van der Meer, A. P.; Blanchard, C. M. R. J. P.; Wesselingh, J. A. Mixing of particles in liquid fluidized beds. Chem. Eng. Res. Des. 1984, 62, 214. (8) Dutta, B. K.; Bhattacharya, S.; Chaudhury, S. K.; Barman, B. Mixing and segregation in a liquid fluidized bed of particles with different size and density. Can. J. Chem. Eng. 1988, 66, 676. (9) Tavare, N. S.; Matsuoka, M.; Garside, J. Modelling a continuous column crystallizer: dispersion and growth characteristics of a cooling section. J. Crystal Growth 1990, 99, 1151. (10) Frances, C.; Biscans, B.; Laguerie, C. Modelling of a continuous fluidized-bed crystallizer. Chem. Eng. Sci. 1994, 49, 3269. (11) Nyvlt, J. Design of Crystallizer; CRC Press: Boca Raton, FL, 1992.
(12) Chianese, A.; Frances, C.; Di Berardino, F.; Bruno, L. On the behavior of a liquid fluidized bed of monosized sodium perborate crystals. Chem. Eng. J. 1992, 50, 87. (13) Richardson, J. F.; Zaki, W. N. Sedimentation and fluidization, Part I. Trans. Inst. Chem. Eng. 1954, 32, 35. (14) Wen, C. Y.; Yu, Y. H. A generalized method for predicting the minimum fluidization velocity. AIChE J. 1966, 12, 610. (15) Kunii, D.; Levenspiel, O. Fluidization Engineering; Wiley Press: New York, 1969. (16) Mullin, J. W.; Garside, J. Velocity-voidage relationships in the design of suspended bed crystallizers. Br. Chem. Eng. 1970, 15, 773. (17) Mullin, J. W.; Garside, J. Crystallization of aluminum potassium sulphate: a study in the assessment of crystallizer design data. I. Single-crystal grow rates. II. Growth in a fluidized bed. Trans. Inst. Chem. Eng. 1967, 45, 285.
Received for review May 30, 2000 Accepted October 4, 2000 IE0005250
