- Email: [email protected]
Crystal Size Engineering in Melt Suspension Crystallization
0263±8762/97/$07.00+ 0.00 Institution of Chemical Engineers
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION P. J. DIEPEN, O. S. L. BRUINSMA and G. M. VAN ROSMALEN Delft University of Technology, Laboratory for Process Equipment, Delft, The Netherlands
M
elt suspension crystallization is used for ultrapuri® cation of caprolactam from caprolactam-water mixtures. The in¯ uence of hydrodynamics, cooling pro® le, impurity concentration, and residence time on the crystal bed permeability were modelled, and measured in a cooling disk crystallizer. Permeability gives a good indication of the e ciency of the solid-liquid separation. Reducing backmixing improves permeability. Permeability increases with increasing residence times and decreasing water content. Keywords: caprolactam; melt crystallization; permeability; CSD
INTRODUCTION Melt suspension crystallization (MSC) is a technique, that can be used for large scale separation and ultrapuri® cation of organic compounds. Applications are the puri® cation of monomers, pharmaceutical compounds and other high added-value materials. The purity of the product is often higher than 99.9 wt%. Since almost 80% of binary organic mixtures form eutectic systems1, the crystals are very pure when growth rates of less than 10- 7 m s- 1 are applied. Therefore, the main bottle-neck for introduction of MSC in industry is solid-liquid separation. In MSC, the crystal size distribution (CSD) itself is not a product speci® cation. Only the e ect of the CSD on solid-liquid separation is of importance. Small crystals have a relatively large surface area and hinder the ® ltration and the washing process. Therefore, the number of small crystals has to be minimized. In principle, this can be achieved by using a cascade of crystallizers or crystallizer compartments. In this study, the in¯ uence of backmixing, residence time, cooling pro® le, and impurity concentration on the permeability of a bed of crystals has been established for such a cascade. Caprolactam was used as a model compound, and was crystallized from caprolactam-water mixtures containing 1.0 to 2.5 wt% water. The cascade used was a 100 litre GMF cooling disk crystallizer (CDC). The horizontal crystallizer trough (Figure 1) is divided into eight compartments by seven cooling elements. The slurry can ¯ ow to the next compartment via the openings in the bottom of the cooling elements. In this way, a cocurrent cascade of eight crystallizer compartments is created without the need for internal slurry handling. The temperature pro® le of the cooling liquid will, at given feed temperature and ¯ ow rates of cooling medium and feed, determine the temperature in each compartment. The cooling elements are equipped with Te¯ on wipers, and each compartment is assumed to be well mixed. Earlier experiments2 showed that there was signi® cant
backmixing in the CDC and that nucleation rates were relatively high in all compartments. A reduction of the back¯ ow in the CDC was attained by diminishing the size of the openings underneath the axis (Figure 1) by Te¯ on plates. Thus, the `plug ¯ ow character’ of the crystallizer was enlarged, in order to obtain a higher mean crystal size and a smaller coe cient of variance of the CSD. Furthermore, experiments were done using cocurrent as well as countercurrent cooling action. In this way, the temperature pro® les over the CDC were changed. In the case of cocurrent cooling, the temperature di erence between the cooling liquid and the suspension is relatively high in the ® rst compartment and low in the last compartment. This could improve the product CSD, because the nucleation rate is lowered in each consecutive compartment. MODELLING Hydrodynamics The CDC can be regarded as a cascade of eight well mixed crystallizers with backmixing. The backmixing is represented by a backmixing factor f: f
u
=u
b
(1)
p
The product stream leaves the CDC via an over¯ ow. Therefore, the crystal mass content in the outlet of the CDC is somewhat smaller than that of the last compartment. This e ect is accounted for by a size independent classi® cation factor, h: M T, overflow (2) h= M T, compartment 8 Crystallization Kinetics The nucleation rate in compartment i is, at a constant 171
172
DIEPEN et al.
For the last compartment: S k, 8V 8 + (1 + f )u pmk, 7 - (h + f )u pmk, 8 = 0
(6)
The S k,i are the source terms for the k-th moment of the i-th compartment: (7) k = 0 : S 0, i = Bi k= ¤ - 0 : S k, i = kmk- 1, i Gi
The overall kinetics of the CDC can be described with an overall nucleation rate, Bov, and an overall growth rate, Gov, that are equal to: m (8) Bov = 0,8
Figure 1. The GMF Cooling Disk Crystallizer.
s
revolution rate of the wipers, given by an empirical expression: (3) Bi = k N(x imp)Gji M T, i So far, attrition, agglomeration, and size-dependent crystal growth have not been incorporated in the model. Each compartment is assumed to be a class II type crystallizer, which means that the yield is independent of the throughput and that the exit concentration of the liquid phase approaches the equilibrium concentration3. Mass and Heat Balances The steady-state mass and heat balances were derived for the CDC, taking into account the backmixing factor f and classi® cation factor h. It was assumed that the caprolactam crystals are 100% pure. The mass and heat balances yield the temperatures and crystal mass contents in each compartment. The backmixing factor f and classi® cation factor h are obtained by ® tting the experimentally obtained temperatures and crystal mass contents. The crystal mass contents are proportional to the third moments of the CSD, and are used to solve the moment balances for each compartment. Moment Balances The moment balances are solved in order to obtain the growth and nucleation rates in each compartment. The steady-state moment balances of the CDC are derived. For the ® rst compartment this reads: (4) S k, 1V 1 - (1 + f )u pmk, 1 + f u pmk,2 = 0 For compartment i, in between the ® rst and the last compartment, this is: S k, i V i + (1 + f )u pmk, i- 1
- (1 + 2f )u pmk, i + f u pmk,i+ 1 = 0
Figure 2. The model of the CDC.
(5)
m l (9) Gov = 1,8 = 10 m0,8 s s The number based coe cient of variation of the CSD in compartment i, CV N,i, can be calculated from the moment balances and is equal to: CV N, i =
Ö
---------------------m 0, i m2, i 1 2 m1, i
(10)
The measured mass based mean crystal size in the last compartment is used to solve the moment balances. This yields the growth rates, Gi, and nucleation rates, Bi, in each compartment i. Then, the population balances can be solved. Population Balances The steady-state population balances of the CDC have been derived. It is assumed that the growth rates, Gi, are a function of supersaturation only, and constant for each compartment. For the ® rst compartment the population balance is equal to: ¶n1(l ) (11) - l G1V 1 - (1 + f )u p n1(l) + f u pn2(l) = 0 ¶ For compartment i, in between the ® rst and the last compartment, this is: ¶ni(l) - ¶l GiV i + (1 + f )u pni- 1(l) (12) - (1 + 2f )u pni(l) + f u pni+ 1(l) = 0 For the last compartment: ¶n8(l) - ¶l G8V 8 + (1 + f )u pn7(l) - (h + f )u pn8(l) = 0
(13)
The above equations can be rewritten in matrix form as: ¶n(l) A n(l) (14) ¶l = The model is completed by the boundary conditions of the system: B (15) n(l = 0) = n0 = i Gi The foregoing system of di erential equations can be solved exactly by a Laplace transformation. The Laplace Trans IChemE, Vol 75, Part A, February 1997
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION
transform of equation (14) reads: (16) sn(s) - n0 = An(s) Cramer’s rule is used to solve the foregoing system of algebraic equations in ni(s). After expanding into partial fractions, and inverting the Laplace solution, we ® nally have: ni (l) =
S
8
Ki, j C exp(- k j l)
j =1
(17)
Ki,j is the eigenvector of matrix A, and the k js are the eigen values of the homogeneous solution of the partial di erential equations. The constant matrix C can be determined from: (18) C = Ki-, j1 n0, i In this way all size distributions, ni (l), in all compartments of the CDC are known. These distributions can now be used to predict the ® ltration properties, like the crystal bed porosity and the crystal bed permeability. Porosity and Permeability Theporosity model derived byOuchiyama and Tanaka4 is based on a simpli® ed packing model. The porosity of a random packing of particles with a given particle size distribution is a function of this distribution. Under the assumption that the crystals are spherical, the average porosity is equal to:
e
173
Solving the Model The heuristics of solving the heat, mass, moment and population balances have been described by Jansens2. The input parameters of the model are the temperatures, the crystal mass contents and the permeabilities of the product crystal beds in the compartments, and the mass based mean crystal size of the product. The parameters to be determined from these experimental input parameters are the backmixing factor (f ), the classi® cation factor (h), the kinetic parameters k N and j, and the MacDonald’s constant (Km). The mass based mean crystal size was determined by sieve analysis of the crystals, obtained by centrifuging a sample of the slurry. The permeability of the crystal bed is measured by ® ltration at a constant pressure di erence6. RESULTS AND DISCUSSION Crystal Morphology The morphology of the caprolactam crystals a ects the ® lterability of the crystal bed. Figure 3(a) shows a typical example of a caprolactam crystal grown at a water concentration of 3 wt%. This morphology closely resembles the predicted equilibrium morphology of caprolactam grown via dimer growth units7. The produced crystals can be characterized by their length/width (l/w) and length/thickness (l/t) ratios8, Figure 3(b). At 3wt% water, the l/w ratio is approximately equal to 2, and the l/t ratio is approximately 6. This yields
l=¥
2 i = 1-
e
l=¥
6
l=0
l=0
1 ni (l)(l ~ ln, f )3 dl + ni
[(l + l
n,i
Where:
ni (l)l 3 dl
(l ~ ln,i) = 0
e
l=0
for l £ ln,i ,
l=0
ni (l)
)3 - (1 ~ ln, i)3]dl
= l - ln i for l > ln i ni =1 + 134 (7 - 82 0)ln i l=¥
e
(19)
l=¥
(20)
,
,
ni (l)(l + ln, i
e
l=¥
f
)
2
(
1-
ni (l) l3 - (l ~ ln, i
l=0
)
3 ln, i dl 8 l + ln, i
) g dl
(21)
3
The permeability of a crystal bed in each compartment can be calculated from the average porosity of the CSD, with the modi® ed Kozeny-Carman equation proposed by MacDonald et al.5 Pm,i =
3 1 2 i 2 l212 , i Km (1 - 2 i)
(22)
The unknown parameter Km is obtained by ® tting with the measured permeabilities of the compartments. Trans IChemE, Vol 75, Part A, February 1997
Figure 3. (a) Caprolactam crystal grown at a water concentration of 3wt%. (b) Predicted equilibrium morphology of caprolactam.
174
DIEPEN et al. Table 1. Cocurrent cooling versus countercurrent cooling pro® le. CVN,8 2 8 Bov Gov P m,8 cooling pro® le (´ 10- 6#/m3 /s) (´ 108 m/s) (- ) (- ) (´ 1012 m2) cocurrent countercurrent
1.621 1.608
5.198 5.151
0.76 0.32 0.79 0.31
1.88 1.87
Figure 5. Clearly, the mass based distribution does not di er much per compartment, but the number based size distribution does. This explains why Jansens2 found that the mass based mean crystal size and coe cient of variation do not change in the CDC, but that the permeability does. The change in permeability, as can also be seen from Figure 4, is caused by an increase in number based mean crystal size, and a decrease of the number based coe cient of variation. Figure 4. Crystal content and permeability in the CDC.
form factors that are equal to k A = 5 and kV = 0.5 respectively, compared to form factors of kA = p and k V = p/6 for spherical particles. Hydrodynamics and Crystallization Kinetics At strong backmixing conditions (f ³ 1), the permeability did not change signi® cantly over the compartments in the CDC, at low crystal mass content in the last compartment (£15 V%)2. The in¯ uence of reduced backmixing can be seen from Figure 4. The openings are partly closed, which causes a gradual built-up of the crystal mass and an increase in permeability over the compartments in the CDC, due to the increased plug ¯ ow character of the crystallizer. The experiment was done at a water concentration of 2.5 wt%, and a residence time of 0.9 h. The backmixing factor (f ) was determined to be equal to zero, and the classi® cation factor (h) is equal to 0.89. The kinetic parameters are equal to: k N = 1.2 ´ 1021 # s m- 2 kg- 1, j = 2. Crystal Size Distribution The mass based and number based crystal size distributions in the compartments are shown in
Process Parameters The permeability increases with increasing residence time (Figure 6). Therefore, the costs of a solid-liquid separation device will be lower at higher residence times. However, the volume of the crystallizer will increase. The costs of both devices will determine the economically optimal con® guration and residence time in the crystallizer. Therefore, with the above described model for the porosity and permeability, an optimization can be done for the CDC, but also for other crystallizer types. At cocurrent cooling (Table 1) the overall nucleation rate, Bov, increases and the overall growth rate, Gov, decreases compared to countercurrent cooling. Also, the number based coe cient of variation in compartment 8, CV N,8, decreases, and the porosity increases slightly. Therefore, the permeability of a crystal bed of the product, Pm,8, should be higher in the case of a cocurrent cooling pro® le. This modelled e ect is also con® rmed experimentally, but the increase is low. This means that countercurrent cooling will be preferred, because of energy advantages. Furthermore, it was found that at increasing water content of the feed, the permeability decreases. The permeability decreases with a factor of two from 1 wt% water to 2.5wt%. This is due to the increasing nucleation and decreasing growth rate with increasing impurity concentration.
Figure 5. Mass based and number based crystal size distribution in each compartment.
Trans IChemE, Vol 75, Part A, February 1997
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION l kN MT n P V
175
length, m nucleation constant, #/(kgs) (m/s)- j crystal content, wt% population density, #/m4 permeability, m2 volume, m3
Greek letters porosity 2u ¯ ow, m3 s- 1 k eigen value, m- 1 residence time, h s
REFERENCES
Figure 6. Permeability versus residence time.
CONCLUSIONS Partly closing the openings in the cooling elements reduces the backmixing, which has a positive e ect on the permeability. Cocurrent cooling medium ¯ ow reduces nucleation slightly, but does not increase the permeability signi® cantly. The permeability increases with decreasing water content, and increasing residence time. NOTATION A B f G h Km K
i ´ i matrix nucleation rate, # m- 3 s- 1 backmixing factor growth rate, m s- 1 classi® cation factor MacDonald constant eigen vector
Trans IChemE, Vol 75, Part A, February 1997
1. Matsuoka, M. and Fukushima H., 1986, Bunri Gijutsi ( Separation Process Engineering), 16 4. 2. Jansens, P. J., Bruinsma, O. S. L. and van Rosmalen, G. M., 1995, AIChEJ., 41 (4): 828. 3. Randolph, A. D. and Larson, M. A., 1988, Theory of Particulate Processes, 2nd Ed. (Academic Press Inc., San Diego). 4. Ouchiyama, N. and Tanaka, T., 1984, Ind Eng Chem Fundam, 23: 490. 5. MacDonald, M. J., Chu, C. F., Guillot, P. P. and Ng, K. M., 1991, AIChEJ, 37 (10): 1583. 6. Coulson, J. M. and Richardson, J. F., 1978, Chemical Engineering, (Pergamon Press). 7. Geertman, R. M. and van der Heijden, A. E. D. M. , 1992, J Crystal Growth, 125: 363. 8. Jansens, P. J. et al., 1996, J Crystal Growth, 155: 126.
ACKNOWLEDGEMENT The support of this research project by GMF-Gouda, NWO, and DSM is gratefully acknowledged.
ADDRESS Correspondenceconcerningthis paper should be addressed to Dr P. J. Diepen, Laboratory for Process Equipment, Delft University of Technology, Leeghwaterstraat 44, 2628-CA Delft, The Netherlands.
0263±8762/97/$07.00+ 0.00 Institution of Chemical Engineers
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION P. J. DIEPEN, O. S. L. BRUINSMA and G. M. VAN ROSMALEN Delft University of Technology, Laboratory for Process Equipment, Delft, The Netherlands
M
elt suspension crystallization is used for ultrapuri® cation of caprolactam from caprolactam-water mixtures. The in¯ uence of hydrodynamics, cooling pro® le, impurity concentration, and residence time on the crystal bed permeability were modelled, and measured in a cooling disk crystallizer. Permeability gives a good indication of the e ciency of the solid-liquid separation. Reducing backmixing improves permeability. Permeability increases with increasing residence times and decreasing water content. Keywords: caprolactam; melt crystallization; permeability; CSD
INTRODUCTION Melt suspension crystallization (MSC) is a technique, that can be used for large scale separation and ultrapuri® cation of organic compounds. Applications are the puri® cation of monomers, pharmaceutical compounds and other high added-value materials. The purity of the product is often higher than 99.9 wt%. Since almost 80% of binary organic mixtures form eutectic systems1, the crystals are very pure when growth rates of less than 10- 7 m s- 1 are applied. Therefore, the main bottle-neck for introduction of MSC in industry is solid-liquid separation. In MSC, the crystal size distribution (CSD) itself is not a product speci® cation. Only the e ect of the CSD on solid-liquid separation is of importance. Small crystals have a relatively large surface area and hinder the ® ltration and the washing process. Therefore, the number of small crystals has to be minimized. In principle, this can be achieved by using a cascade of crystallizers or crystallizer compartments. In this study, the in¯ uence of backmixing, residence time, cooling pro® le, and impurity concentration on the permeability of a bed of crystals has been established for such a cascade. Caprolactam was used as a model compound, and was crystallized from caprolactam-water mixtures containing 1.0 to 2.5 wt% water. The cascade used was a 100 litre GMF cooling disk crystallizer (CDC). The horizontal crystallizer trough (Figure 1) is divided into eight compartments by seven cooling elements. The slurry can ¯ ow to the next compartment via the openings in the bottom of the cooling elements. In this way, a cocurrent cascade of eight crystallizer compartments is created without the need for internal slurry handling. The temperature pro® le of the cooling liquid will, at given feed temperature and ¯ ow rates of cooling medium and feed, determine the temperature in each compartment. The cooling elements are equipped with Te¯ on wipers, and each compartment is assumed to be well mixed. Earlier experiments2 showed that there was signi® cant
backmixing in the CDC and that nucleation rates were relatively high in all compartments. A reduction of the back¯ ow in the CDC was attained by diminishing the size of the openings underneath the axis (Figure 1) by Te¯ on plates. Thus, the `plug ¯ ow character’ of the crystallizer was enlarged, in order to obtain a higher mean crystal size and a smaller coe cient of variance of the CSD. Furthermore, experiments were done using cocurrent as well as countercurrent cooling action. In this way, the temperature pro® les over the CDC were changed. In the case of cocurrent cooling, the temperature di erence between the cooling liquid and the suspension is relatively high in the ® rst compartment and low in the last compartment. This could improve the product CSD, because the nucleation rate is lowered in each consecutive compartment. MODELLING Hydrodynamics The CDC can be regarded as a cascade of eight well mixed crystallizers with backmixing. The backmixing is represented by a backmixing factor f: f
u
=u
b
(1)
p
The product stream leaves the CDC via an over¯ ow. Therefore, the crystal mass content in the outlet of the CDC is somewhat smaller than that of the last compartment. This e ect is accounted for by a size independent classi® cation factor, h: M T, overflow (2) h= M T, compartment 8 Crystallization Kinetics The nucleation rate in compartment i is, at a constant 171
172
DIEPEN et al.
For the last compartment: S k, 8V 8 + (1 + f )u pmk, 7 - (h + f )u pmk, 8 = 0
(6)
The S k,i are the source terms for the k-th moment of the i-th compartment: (7) k = 0 : S 0, i = Bi k= ¤ - 0 : S k, i = kmk- 1, i Gi
The overall kinetics of the CDC can be described with an overall nucleation rate, Bov, and an overall growth rate, Gov, that are equal to: m (8) Bov = 0,8
Figure 1. The GMF Cooling Disk Crystallizer.
s
revolution rate of the wipers, given by an empirical expression: (3) Bi = k N(x imp)Gji M T, i So far, attrition, agglomeration, and size-dependent crystal growth have not been incorporated in the model. Each compartment is assumed to be a class II type crystallizer, which means that the yield is independent of the throughput and that the exit concentration of the liquid phase approaches the equilibrium concentration3. Mass and Heat Balances The steady-state mass and heat balances were derived for the CDC, taking into account the backmixing factor f and classi® cation factor h. It was assumed that the caprolactam crystals are 100% pure. The mass and heat balances yield the temperatures and crystal mass contents in each compartment. The backmixing factor f and classi® cation factor h are obtained by ® tting the experimentally obtained temperatures and crystal mass contents. The crystal mass contents are proportional to the third moments of the CSD, and are used to solve the moment balances for each compartment. Moment Balances The moment balances are solved in order to obtain the growth and nucleation rates in each compartment. The steady-state moment balances of the CDC are derived. For the ® rst compartment this reads: (4) S k, 1V 1 - (1 + f )u pmk, 1 + f u pmk,2 = 0 For compartment i, in between the ® rst and the last compartment, this is: S k, i V i + (1 + f )u pmk, i- 1
- (1 + 2f )u pmk, i + f u pmk,i+ 1 = 0
Figure 2. The model of the CDC.
(5)
m l (9) Gov = 1,8 = 10 m0,8 s s The number based coe cient of variation of the CSD in compartment i, CV N,i, can be calculated from the moment balances and is equal to: CV N, i =
Ö
---------------------m 0, i m2, i 1 2 m1, i
(10)
The measured mass based mean crystal size in the last compartment is used to solve the moment balances. This yields the growth rates, Gi, and nucleation rates, Bi, in each compartment i. Then, the population balances can be solved. Population Balances The steady-state population balances of the CDC have been derived. It is assumed that the growth rates, Gi, are a function of supersaturation only, and constant for each compartment. For the ® rst compartment the population balance is equal to: ¶n1(l ) (11) - l G1V 1 - (1 + f )u p n1(l) + f u pn2(l) = 0 ¶ For compartment i, in between the ® rst and the last compartment, this is: ¶ni(l) - ¶l GiV i + (1 + f )u pni- 1(l) (12) - (1 + 2f )u pni(l) + f u pni+ 1(l) = 0 For the last compartment: ¶n8(l) - ¶l G8V 8 + (1 + f )u pn7(l) - (h + f )u pn8(l) = 0
(13)
The above equations can be rewritten in matrix form as: ¶n(l) A n(l) (14) ¶l = The model is completed by the boundary conditions of the system: B (15) n(l = 0) = n0 = i Gi The foregoing system of di erential equations can be solved exactly by a Laplace transformation. The Laplace Trans IChemE, Vol 75, Part A, February 1997
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION
transform of equation (14) reads: (16) sn(s) - n0 = An(s) Cramer’s rule is used to solve the foregoing system of algebraic equations in ni(s). After expanding into partial fractions, and inverting the Laplace solution, we ® nally have: ni (l) =
S
8
Ki, j C exp(- k j l)
j =1
(17)
Ki,j is the eigenvector of matrix A, and the k js are the eigen values of the homogeneous solution of the partial di erential equations. The constant matrix C can be determined from: (18) C = Ki-, j1 n0, i In this way all size distributions, ni (l), in all compartments of the CDC are known. These distributions can now be used to predict the ® ltration properties, like the crystal bed porosity and the crystal bed permeability. Porosity and Permeability Theporosity model derived byOuchiyama and Tanaka4 is based on a simpli® ed packing model. The porosity of a random packing of particles with a given particle size distribution is a function of this distribution. Under the assumption that the crystals are spherical, the average porosity is equal to:
e
173
Solving the Model The heuristics of solving the heat, mass, moment and population balances have been described by Jansens2. The input parameters of the model are the temperatures, the crystal mass contents and the permeabilities of the product crystal beds in the compartments, and the mass based mean crystal size of the product. The parameters to be determined from these experimental input parameters are the backmixing factor (f ), the classi® cation factor (h), the kinetic parameters k N and j, and the MacDonald’s constant (Km). The mass based mean crystal size was determined by sieve analysis of the crystals, obtained by centrifuging a sample of the slurry. The permeability of the crystal bed is measured by ® ltration at a constant pressure di erence6. RESULTS AND DISCUSSION Crystal Morphology The morphology of the caprolactam crystals a ects the ® lterability of the crystal bed. Figure 3(a) shows a typical example of a caprolactam crystal grown at a water concentration of 3 wt%. This morphology closely resembles the predicted equilibrium morphology of caprolactam grown via dimer growth units7. The produced crystals can be characterized by their length/width (l/w) and length/thickness (l/t) ratios8, Figure 3(b). At 3wt% water, the l/w ratio is approximately equal to 2, and the l/t ratio is approximately 6. This yields
l=¥
2 i = 1-
e
l=¥
6
l=0
l=0
1 ni (l)(l ~ ln, f )3 dl + ni
[(l + l
n,i
Where:
ni (l)l 3 dl
(l ~ ln,i) = 0
e
l=0
for l £ ln,i ,
l=0
ni (l)
)3 - (1 ~ ln, i)3]dl
= l - ln i for l > ln i ni =1 + 134 (7 - 82 0)ln i l=¥
e
(19)
l=¥
(20)
,
,
ni (l)(l + ln, i
e
l=¥
f
)
2
(
1-
ni (l) l3 - (l ~ ln, i
l=0
)
3 ln, i dl 8 l + ln, i
) g dl
(21)
3
The permeability of a crystal bed in each compartment can be calculated from the average porosity of the CSD, with the modi® ed Kozeny-Carman equation proposed by MacDonald et al.5 Pm,i =
3 1 2 i 2 l212 , i Km (1 - 2 i)
(22)
The unknown parameter Km is obtained by ® tting with the measured permeabilities of the compartments. Trans IChemE, Vol 75, Part A, February 1997
Figure 3. (a) Caprolactam crystal grown at a water concentration of 3wt%. (b) Predicted equilibrium morphology of caprolactam.
174
DIEPEN et al. Table 1. Cocurrent cooling versus countercurrent cooling pro® le. CVN,8 2 8 Bov Gov P m,8 cooling pro® le (´ 10- 6#/m3 /s) (´ 108 m/s) (- ) (- ) (´ 1012 m2) cocurrent countercurrent
1.621 1.608
5.198 5.151
0.76 0.32 0.79 0.31
1.88 1.87
Figure 5. Clearly, the mass based distribution does not di er much per compartment, but the number based size distribution does. This explains why Jansens2 found that the mass based mean crystal size and coe cient of variation do not change in the CDC, but that the permeability does. The change in permeability, as can also be seen from Figure 4, is caused by an increase in number based mean crystal size, and a decrease of the number based coe cient of variation. Figure 4. Crystal content and permeability in the CDC.
form factors that are equal to k A = 5 and kV = 0.5 respectively, compared to form factors of kA = p and k V = p/6 for spherical particles. Hydrodynamics and Crystallization Kinetics At strong backmixing conditions (f ³ 1), the permeability did not change signi® cantly over the compartments in the CDC, at low crystal mass content in the last compartment (£15 V%)2. The in¯ uence of reduced backmixing can be seen from Figure 4. The openings are partly closed, which causes a gradual built-up of the crystal mass and an increase in permeability over the compartments in the CDC, due to the increased plug ¯ ow character of the crystallizer. The experiment was done at a water concentration of 2.5 wt%, and a residence time of 0.9 h. The backmixing factor (f ) was determined to be equal to zero, and the classi® cation factor (h) is equal to 0.89. The kinetic parameters are equal to: k N = 1.2 ´ 1021 # s m- 2 kg- 1, j = 2. Crystal Size Distribution The mass based and number based crystal size distributions in the compartments are shown in
Process Parameters The permeability increases with increasing residence time (Figure 6). Therefore, the costs of a solid-liquid separation device will be lower at higher residence times. However, the volume of the crystallizer will increase. The costs of both devices will determine the economically optimal con® guration and residence time in the crystallizer. Therefore, with the above described model for the porosity and permeability, an optimization can be done for the CDC, but also for other crystallizer types. At cocurrent cooling (Table 1) the overall nucleation rate, Bov, increases and the overall growth rate, Gov, decreases compared to countercurrent cooling. Also, the number based coe cient of variation in compartment 8, CV N,8, decreases, and the porosity increases slightly. Therefore, the permeability of a crystal bed of the product, Pm,8, should be higher in the case of a cocurrent cooling pro® le. This modelled e ect is also con® rmed experimentally, but the increase is low. This means that countercurrent cooling will be preferred, because of energy advantages. Furthermore, it was found that at increasing water content of the feed, the permeability decreases. The permeability decreases with a factor of two from 1 wt% water to 2.5wt%. This is due to the increasing nucleation and decreasing growth rate with increasing impurity concentration.
Figure 5. Mass based and number based crystal size distribution in each compartment.
Trans IChemE, Vol 75, Part A, February 1997
CRYSTAL SIZE ENGINEERING IN MELT SUSPENSION CRYSTALLIZATION l kN MT n P V
175
length, m nucleation constant, #/(kgs) (m/s)- j crystal content, wt% population density, #/m4 permeability, m2 volume, m3
Greek letters porosity 2u ¯ ow, m3 s- 1 k eigen value, m- 1 residence time, h s
REFERENCES
Figure 6. Permeability versus residence time.
CONCLUSIONS Partly closing the openings in the cooling elements reduces the backmixing, which has a positive e ect on the permeability. Cocurrent cooling medium ¯ ow reduces nucleation slightly, but does not increase the permeability signi® cantly. The permeability increases with decreasing water content, and increasing residence time. NOTATION A B f G h Km K
i ´ i matrix nucleation rate, # m- 3 s- 1 backmixing factor growth rate, m s- 1 classi® cation factor MacDonald constant eigen vector
Trans IChemE, Vol 75, Part A, February 1997
1. Matsuoka, M. and Fukushima H., 1986, Bunri Gijutsi ( Separation Process Engineering), 16 4. 2. Jansens, P. J., Bruinsma, O. S. L. and van Rosmalen, G. M., 1995, AIChEJ., 41 (4): 828. 3. Randolph, A. D. and Larson, M. A., 1988, Theory of Particulate Processes, 2nd Ed. (Academic Press Inc., San Diego). 4. Ouchiyama, N. and Tanaka, T., 1984, Ind Eng Chem Fundam, 23: 490. 5. MacDonald, M. J., Chu, C. F., Guillot, P. P. and Ng, K. M., 1991, AIChEJ, 37 (10): 1583. 6. Coulson, J. M. and Richardson, J. F., 1978, Chemical Engineering, (Pergamon Press). 7. Geertman, R. M. and van der Heijden, A. E. D. M. , 1992, J Crystal Growth, 125: 363. 8. Jansens, P. J. et al., 1996, J Crystal Growth, 155: 126.
ACKNOWLEDGEMENT The support of this research project by GMF-Gouda, NWO, and DSM is gratefully acknowledged.
ADDRESS Correspondenceconcerningthis paper should be addressed to Dr P. J. Diepen, Laboratory for Process Equipment, Delft University of Technology, Leeghwaterstraat 44, 2628-CA Delft, The Netherlands.