Cooling Crystallization of Paracetamol in Hollow Fiber Devices

Publication Date (Web): December 10, 2006 ... Such a capability is not existent in industrial cooling crystallizers; it allows the decoupling .... deg...
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Cooling Crystallization of Paracetamol in Hollow Fiber Devices Dimitrios M. Zarkadas† and Kamalesh K. Sirkar* Center For Membrane Technologies, Otto H. York Department of Chemical Engineering, New Jersey Institute of Technology, 161 Warren Street, Newark, New Jersey 07102

We examine here cooling crystallization of aqueous paracetamol solutions in hollow fiber devices. It is shown that a solid hollow fiber crystallizer (SHFC)-static mixer assembly can be operated successfully up to 3040°C below the metastable zone limit. Such a capability is not existent in industrial cooling crystallizers; it allows the decoupling of nucleation and growth, an opportunity offered currently only by impinging jet mixers for antisolvent crystallization. In addition, it leads to the achievement of very high nucleation rates and hence small crystal sizes. The former were 2-4 orders of magnitude higher than values previously obtained in solid hollow fiber crystallizers for potassium nitrate and salicylic acid and reached values encountered only in impinging jet crystallization. A qualitative comparison with existing literature data showed that the SHFCstatic mixer combination confined the crystal size distribution (CSD) to smaller sizes and a narrower range. Finally, a linear relationship between the mean crystal size and the cooling medium temperature was observed. This relationship is indicative of the simplicity available in SHFCs vis-a`-vis CSD control. Introduction Crystallization and precipitation processes are often the most critical step in the separation, purification, and production of active pharmaceutical ingredients (APIs) and fine chemicals. The most important product characteristics such as crystal size distribution (CSD), crystal shape, and habit are solely determined by the operating conditions during the crystallization step (i.e., temperature, rates of supersaturation generation and depletion, mixing, and agglomeration/breakage phenomena). Good control of the above parameters can yield a product of acceptable and reproducible quality. In addition, downstream processing such as filtration and drying can benefit significantly in terms of robustness, operating simplicity, and cost-effectiveness from a well-designed and controlled crystallization process. Cooling crystallization from solution is probably the most frequently employed crystallization technique in the pharmaceutical industry. It is carried out in stirred vessels operated in a batch or continuous mode. The former is by far the predominant practice in the pharmaceutical industry; the latter is mainly employed in the production of inorganic chemicals.1 Despite the fact that industrial crystallizers can be successfully operated, there is often considerable variability in the principal process outcome, the crystal size distribution. Moreover, they cannot in general meet the targets of a narrow CSD and a small mean crystal size, often desirable in the production of pharmaceuticals and specialty chemicals, due to imperfect mixing and the resulting non-uniform supersaturation conditions inside the crystallizer. Imperfect mixing is an inherent characteristic of industrial crystallizers whose performance is often different from that obtained at laboratory scale and frequently characterized by segregation effects.2 Imperfect mixing is also not uncommon even at a small scale. Significant differences in the CSD obtained at different points of a laboratory crystallizer have been reported.3 Mixing and supersaturation control issues have been addressed in two ways over the years. The first is to improve * To whom correspondence should be addressed. Phone: (973)5968447. Fax: (973)642-4854. E-mail: [email protected]. † Present address: Schering-Plough Research Institute, 1011 Morris Ave., U-13-2-2025, Union, NJ 07083.

existing facilities and process practices by applying new insitu monitoring techniques that can lead to better prediction and control of the applied supersaturation and hence better control of the final CSD.4-10 This approach is very popular nowadays, especially in the pharmaceutical industry where the Process Analytical Technology (PAT) initiative is expected to transform the way the process is operated. Despite its success stories, this approach cannot often address the mixing issue: well-mixed crystallizers are intrinsically inclined toward a spectrum of local conditions in time and space and, consequently, a relatively broad CSD. The second approach is to develop new crystallization techniques where supersaturation can be created and depleted on a microscale, resulting therefore in a narrow CSD and a small crystal size. Several techniques have been developed over the years either at an experimental level or as commercial applications. Impinging jet precipitation11 and spherical crystallization12 have been already applied at a production scale, while the experimental techniques include emulsion crystallization,13-14 membrane-based crystallization,15-18 and antisolvent crystallization/precipitation19 with supercritical fluids, which has been tested at a pilot plant level. However, none of these techniques is applicable for cooling crystallization from solutions. We recently proposed a new cooling crystallization technique,20 solid hollow fiber cooling crystallization (SHFCC), which relies on the use of nonporous hollow fibers for the creation of uniform temperature and hence supersaturation conditions in a constrained environment of sub-millimeter scale. The technique was tested successfully for both aqueous (KNO3) and organic systems (salicylic acid in ethanol) with a high solubility. It was shown that the combination of a solid hollow fiber crystallizer with a stirred tank downstream produced crystals of 3-4 times smaller mean size, while nucleation rates were 2-3 orders of magnitude higher compared to results obtained in mixed suspension mixed product removal (MSMPR) crystallizers. Here we are reporting our results for solid hollow fiber cooling crystallization of paracetamol (4-amidophenol, acetaminophen) from its aqueous solutions. The latter is an example of a system with low solubility. It is also a system representative of crystallization in the pharmaceutical industry, which has been

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Figure 1. Solid hollow fiber cooling crystallization: operating principle.

studied extensively either by cooling21-25 or by antisolvent crystallization.26-31 This study had multiple objectives. The first one was to illustrate the performance of SHFCC at high supersaturation levels for which the contribution of homogeneous nucleation to the overall nucleation rate is either significant or dominant. This was not possible during the already reported potassium nitrate and salicylic acid experiments, which were operated close to or below the metastable zone boundary.20 The second goal was to clearly show that SHFCC can be used to effectively decouple nucleation and growth, the former being performed by the SHFC and the latter by a mixing device (e.g., static mixer) downstream. A third objective was to compare to the extent possible the performance of SHFCC with existing literature data. Finally, the choice of system here ensures that the technique has been studied for compounds covering completely the solubility range encountered in practice. Solid Hollow Fiber Cooling Crystallization: Method Description An illustration of the proposed technique is provided in Figure 1. The solution to be crystallized is fed through the lumen side of a SHFC, which essentially is a shell-and-tube heat exchanger. A suitable cooling solution is circulated in counter-current, cocurrent, or cross-flow mode through the shell side of the device. In such an arrangement, the feed solution, subdivided into numerous fluid packets, travels through each hollow fiber with the same velocity and under the same cooling conditions. If cooling proceeds below the feed saturation temperature, crystallization will take place in each fiber under the same conditions leading to a narrow CSD. Solid hollow fiber crystallizers offer various advantages as compared to conventional cooling crystallization equipment. First, their scale-up is straightforward and is not anticipitated to present any problems in contrast to stirred crystallizers, for which an exact translation of mixing conditions from small to large scale is a known issue.32 In the devices described here, supersaturation creation and crystallization occur always in the same environment: the exact identical interior of each hollow fiber. Moreover, our heat transfer studies in hollow fiber devices33 have shown that the temperature profiles obtained along the fiber depend primarily on three factors: the flow rate through each fiber, the polymeric wall material, and the temperature and hydrodynamic conditions of the shell-side fluid. Of the above factors, the first two will be exactly the same upon scale-up and the same will also be true for the shell-side fluid

temperature. Hence, the main source of uncertainty remaining is the hydrodynamic conditions on the shell side. The latter are a known source of variability for membrane contactors, especially if parallel flow is used at the shell side.34 However, cross-flow configurations can improve the situation and have already been applied in commercial membrane contactors. Second, the polymeric nature of the hollow fibers makes them ideal to control the temperature of the inside wall, which is the cooling surface exposed to the crystallizing solution. The thermal conductivity of the wall can be tailored to desired levels by choosing different polymeric construction materials and/or varying the wall thickness. The heat transfer theory developed previously33 predicts that the result of such action will be a different inside wall temperature. Essentially, this is the net result of distributing the largest and controlling part of the heat transfer resistance across the wall and not across the fluid as in metal tubing. It has also been shown that the difference between the inside wall and the crystallizing solution temperature can be kept as low as 1-2 °C in hollow fiber devices.35 As a result, supersaturation creation at each fiber cross section is relatively uniform. Therefore, supersaturation control, the objective for the application of numerous in-situ monitoring techniques in stirred crystallizers, is an inherent characteristic of SHFCs. Close temperature control is a characteristic absent in the case of conventional cooling crystallizers, for which, due to the negligible heat transfer resistance of the metal wall, the inside wall temperature is equal to the coolant temperature. In addition, the temperature difference between the coolant and the crystallizing solution has to be maintained high during scale-up of jacketed crystallizers in order to compensate for the surface area/ volume loss accompanying the larger vessel diameter. This leads to severe incrustation problems.36 The same temperature difference is also relatively high, about 5-10 °C, in shell and tube heat exchangers used in forced circulation crystallizers and can lead to uncontrolled nucleation.37 From the above discussion, it is apparent that solid hollow fiber devices offer the ability to decouple up to a certain degree the choice of the coolant temperature from the crystallization kinetics, as expressed by the metastable zone width. The most distinctive characteristic of SHFCs, however, is the small diameter of the hollow fibers, typically in the submillimeter range. It leads to high values of surface area/volume ratio, up to 4000 m2/m3. These values are at least 1 order of magnitude higher than shell-and-tube heat exchangers typically used in forced circulation cooling crystallizers. For conventional cooling coils, typically used in batch or continuous stirred tanks, the same ratio is 314 m2/m3 for a coil of 1/2 in. diameter and if only the pipe volume is considered. However, in a real arrangement, the total volume occupied by the coil depends also on its width; hence, the above value should be divided by a factor of at least 3-5. For jacketed vessels the comparison is even more favorable since for a vessel of 60 cm diameter the surface-to-volume ratio is only 7 m2/m3. The large surface/volume ratio that hollow fiber devices offer can have an impact on the overall device performance at several levels. First, it can improve heat transfer. Our results33 for singlephase heat transfer show that the hollow fiber devices used in this study can transfer 3-10 times more heat on a volumetric basis than conventional shell-and-tube heat exchangers for both aqueous or organic streams. Moreover, it was shown that these levels of efficiency were accompanied by lower pressure drop.The performance of hollow fiber devices in terms of heat transfer volumetric efficiency is also 3-10 times higher as compared to cooling coils and up to 60 times higher as compared

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to jacketed vessels. Second, process performance dependence on fouling of the heat transfer area is appreciably reduced since there is a large amount of surface area available to compensate for any local losses. Finally, the very large surface area available can lead to higher nucleation rates as compared to conventional crystallizers of the same volume and operated under the same supersaturation conditions. As far as crystallization kinetics are concerned, solid hollow fiber crystallizers offer yet another, more fundamental potential advantage over conventional cooling crystallizers stemming from the small diameter of the hollow fibers used. It is the possibility to control the nucleation rate by varying the diameter of the fiber. This can be explained as follows. Schubert and Mersmann38 carried out precipitation experiments with Ba(OH)2 and H2SO4 in the presence of nanoparticles of SiO2, Al2O3, and TiO2. They found that the rate of heterogeneous nucleation was proportional to the volumetric surface area Rfor of foreign particles present in the solution in the range 5 × 103-2.5 × 105 m2/m3. On the other hand, the inside surface of the hollow fiber can be considered as foreign surface available for heterogeneous nucleation. The volumetric surface area for a hollow fiber or in general a tube based on the inside area (the area available for nucleation) can be calculated by

afor,t )

πDiL 2

Di π L 4

)

4 Di

(1)

Equation 1 indicates that values inside the range quoted by Schubert and Mersmann38 can be obtained if hollow fibers with diameter between 100 and 800 µm were used. As a result, it can be expected that higher nucleation rates will be achieved by using fibers of decreasing diameter. Equation 1 also explains why such an issue does not arise in conventional cooling crystallizers; the volumetric surface of conventional tubes is very low, and heterogeneous nucleation can occur only due to suspended solids inside the crystallizing solution. The role of suspended particles will also be important for hollow fiber devices. However, unless a certain concentration of foreign solids in the crystallizing solution is exceeded, the volumetric surface of the fiber will be equally contributing or the controlling factor in heterogeneous nucleation. Experimental Section Chemicals and Materials. Paracetamol (4-acetamidophenol, 98%, Fisher Scientific) solutions were prepared by dissolving the compound in deionized water. A target concentration of about 0.055 g/g water was used in all experiments, which corresponds to a saturation temperature of about 65 °C. Polypropylene solid hollow fibers of 420/575 µm i.d./o.d. (Celgard, Charlotte, NC) were used for the fabrication of two almost identical modules. Module 1 was made of 35 fibers of 21.9 cm length, while module 2 had the same number of fibers and an active length of 20.3 cm. The shell side in all cases was 3/ in. FEP tubing (Cole Parmer, Vernon Hills, IL). The modules 8 were fabricated by connecting the FEP tubing with two polypropylene male run tees (Cole Parmer). The tube sheet was formed by potting the two ends of the male run tees with an epoxy resin (C-4 resin with activator D from Armstrong, Easton, MA). The resin was left to cure for at least 24 h. The outside fiber surfaces were treated with an aqueous potassium dichromate solution for 5-7 min to enhance bonding with the epoxy. The solution was prepared by dissolving 5 g of potassium dichromate in 10 mL of water and 80 mL of 95.7% sulfuric acid.

Figure 2. Experimental setup for solid hollow fiber cooling crystallization of paracetamol from aqueous solutions.

Apparatus and Procedure. The experimental setup used is shown in Figure 2. It illustrates the various operating schemes implemented for SHFCC. The feed solution held to a constant temperature by means of a thermostatic bath (Haake A81) is pumped through the lumen side of the SHFC by a diaphragm pump (Cole Parmer). An aqueous cooling solution of 33 vol % ethylene glycol is circulated through the shell side by a Polystat chiller (Cole Parmer). The inlet and exit temperatures of the two streams were recorded with a four-channel temperature recorder (Sper Scientific LTD, Scottsdale, AZ) with an accuracy of ( 0.2 °C. Flow rates were measured by flowmeters (Cole Parmer) and confirmed by measuring the time required to collect a certain volume of liquid. The SHFC was used in combination with a 3/8 in. in-line static mixer (Cole Parmer). Two samples were obtained during each run: one was mixed magnetically for 15-30 s and then filtered (called “mixed”) while the other was filtered immediately as it was coming out of the static mixer (called “unmixed”). This practice was dictated by the experimental observation that the stream exiting the static mixer was only mildly turbid and that upon further stirring the slurry became more turbid, a sign of continuing supresaturation depletion. The CSD differences between the two samples can be used to illustrate the effect of mixing in the SHFCC device. The duration of filtration was 30-45 s for the mixed samples and 1 min or higher for the unmixed samples. Therefore, local supersaturation depletion is probably more intense for the unmixed samples. Crystals obtained on the filters were thoroughly dried and then weighed. The volume of filtrate collected was also measured in order to obtain the magma density of the suspension exiting the experimental setup. Two kinds of filters were used throughout this study: glass fiber filters (Whatman 934-AH) with a cutoff size of 1.5 µm and hydrophilized PVDF membranes (Pall Corp., NY) with a cutoff size of 0.2 µm. Analytical Methods. The feed concentration of paracetamol solutions was determined either gravimetrically or by a UV method. Paracetamol crystals were sized by optical microscopy since they are known to agglomerate heavily.21,24-25 An optical microscope (Swift Instruments International, M4000-D) equipped with a digital camera and a stage micrometer was used to obtain sample pictures. Prior to sizing the particle contour line was corrected manually to facilitate the accuracy of the measurement. Crystal sizing was performed on the corrected images by available free imaging software (Image Tool version 3, University of Texas Health Science Center in San Antonio, TX) automatically. When sizing is performed automatically, the software measures the area of the crystal/particle and assigns it to an ellipse. Therefore, the crystal size computed this way is based on area and not on volume as in the case of laser

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Figure 3. Operating conditions during the paracetamol runs in the SHFCC device.

Figure 4. Paracetamol crystallization from aqueous solutions. % number differential CSD of mixed samples.

diffraction. At least 500-600 crystals were measured for each run to ensure the statistical significance of the acquired CSD. Results The operating conditions during the runs performed are shown in Figure 3, which is a concentration-temperature diagram. The metastable zone was based on data from the literature.21 The latter were available up to 45 °C; the rest of the curve was estimated based on the assumption that as the solution concentration increases, the metastable zone width decreases.32 Figure 3 shows that in the case of a system with a relatively low solubility it is possible to operate the SHFC deep inside the labile zone, as low as 40 °C below the solubility curve. Consequently, high nucleation rates, typical of primary nucleation, can be achieved. Such high values for the degree of undercooling can be encountered only for unmixed systems. It seems therefore that the SHFC and static mixer assembly do not offer significant mixing. An alternative explanation is that the metastable zone is depressed considerably due to the extremely high cooling rates (up to 20-30 K/s) developed in the SHFCC experimental setup, although an exact and unambiguous definition of the cooling rate in a continuous shell and tube crystallizer does not seem to exist. The crystal size distributions obtained for four runs performed at different relative supersaturations are shown in Figures 4 and 5, respectively, for the magnetically mixed and the unmixed samples, respectively. The crystal size presented was obtained by optical microscopy and corresponds to the circle equivalent diameter (CED), namely, the diameter of a circle having the same area as the measured crystal. Optical microscopy was chosen over laser diffraction measurements due to the fact that paracetamol crystals are known to agglomerate heavily.21 This was also confirmed by the optical microscope images taken and by measuring via laser diffraction the CSD of the mixed sample from run 1. The mode of the distribution as measured by laser diffraction occurred at a size of about 400 µm. As shown in Figure 4, no crystal was measured larger than 150 µm by optical microscopy. Figure 4 shows that for all runs except run 4 the number distributions obtained were very similar. The CSD is practically confined between 3 and 150 µm. These numbers compare well with the results of Nagy et al.22 and Fujiwara et al.,21 who reported sizes as large as 400-600 µm for crystallization from

Figure 5. Paracetamol crystallization from aqueous solutions. % number differential CSD of unmixed samples for the SHFC-in-line static mixer in series operation.

aqueous solutions. However, no quantitative comparison can be made with the results of this study due to the fact that special algorithms have to be employed to obtain the true CSD from the chord length distributions obtained by FBRM measurements.23 Figure 5 shows similar trends; reproducibility between runs is good, and the CSD is confined below 200 µm. However, for the unmixed samples, the tail of the distribution at sizes higher than 100 µm is significant. It also reminds, albeit to a lesser extent, the tail of the curves obtained previously for potassium nitrate.20 The tail in the CSDs in Figure 5 and its absence in Figure 4 reveal different degrees of mixing between the two samples obtained during each run. The same conclusion can be drawn from Figure 6, which shows a plot of the experimentally obtained magma densities together with the thermodynamic limits calculated from

MT )

/ m˘ c Cf,in - Cf,out ) m˘ solv V˙ f V˙ f

(2)

Figure 6 shows trends similar to the ones reported previously,20

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Figure 6. Paracetamol crystallization from aqueous solutions: comparison of experimental and calculated magma density values.

namely, magma densities smaller than anticipated. Therefore, the SHFCC apparatus can still be considered a class I system, in which not all supersaturation created is relieved. For the unmixed samples this is straightforward; for the mixed samples this probably can be explained in terms of the small time of mixing (15-30 s). In addition, the solids losses on the filter were significant as compared to the small solid sample size collected (∼1 g). It is also apparent from Figure 6 that the magma densities of the unmixed samples are closer to their thermodynamic limit, as predicted by eq 2, as compared to the reported values for potassium nitrate runs in an once through operation mode.20 Consequently, the presence of the in-line static mixer serves the purpose of supersaturation depletion. However, it does not provide complete supersaturation depletion. A longer and narrower static mixer would perform better and would be able to induce the necessary degree of mixing. The reason this was not realized was that the SHFCs used in this study were small compared to the in-line static mixer, which had an inside diameter of around 9 mm. Ideally, the static mixer would be chosen in such a way that its cross section is at least 4-5 times smaller than the cross section of the fibers used in the SHFC. Translating this requirement to the experimental setup of this study, the diameter of the static mixer to be used would have to be smaller than 1 mm. Consequently, the CSDs of Figure 4 and the results of Figure 5 cannot be considered optimal. Rather, they are indicative of the potential of SHFCC if operated at a different scale. Figure 7 illustrates the % number cumulative undersize CSDs obtained for the mixed samples. The curves for the unmixed samples are similar with the exception that the endpoint of the distribution is shifted to higher sizes. Figure 7 shows that, for all runs, 90% of the crystals produced are smaller than 50 µm; furthermore, the CSD for all purposes can be considered confined below 100 µm. The d90 size for the unmixed samples is higher, about 70 µm. Another observation that can be made from Figure 7 is that the crystal sizes obtained for run 1 are markedly smaller than the rest of the runs. This can be explained on the basis of the much higher supersaturation used during this run. Hence, higher nucleation rates were obtained, with the contribution of homogeneous nucleation being more significant. The above argument becomes clear by inspecting Figure 8. The latter shows a plot of the nucleation rate with respect to the inverse of the square of the logarithm of the supersaturation ratio, the functional dependence valid for primary nucleation rate as shown in the Appendix. The nucleation rates shown in

Figure 7. Paracetamol crystallization from aqueous solutions. % number cumulative undersize CSD of mixed samples.

Figure 8. Experimentally obtained nucleation rates during paracetamol solid hollow fiber cooling crystallization from aqueous solutions.

Figure 8 are approximate; they were estimated as follows. From the number CSD obtained by optical microscopy, the volume of the crystals measured was calculated based on the circle equivalent diameter (CED) for each crystal, Subsequently, the mass of the crystals measured was calculated by multiplying by the density of paracetamol. Finally, assuming that the CSD measured is representative of the true CSD, the total number of crystals obtained was estimated by multiplying the number of crystals measured by the ratio of masses of the whole sample collected and the sample measured. The point that corresponds to run 1 is an obvious outlier compared to the rest of the runs; it corresponds to an order of magnitude higher nucleation rate, a fact indicating that the nucleation mechanism is possibly different for this run. This was confirmed by the calculations presented in the Appendix. These calculations show that, for run 1, the contribution of homogeneous nucleation starts to become appreciable. They also show that the calculated values are close to the experimental ones. Therefore, a combination of homogeneous and heterogeneous nucleation is more possible for high supersaturations. As a final remark, it is noted that the nucleation rates quoted in Figure 8 are very high; to our knowledge, similar values are obtained only during antisolvent or precipitation crystallization.

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Figure 9. Paracetamol solid hollow fiber cooling crystallization from aqueous solutions. Dependence of mean crystal size on coolant temperature.

Finally, a plot of the mean crystal size versus the coolant temperature used is presented in Figure 9. The obtained mean crystal size is proportional to the coolant temperature; the lower the value of the latter, the smaller the size of the crystals produced due to the higher nucleation rates involved. Consequently, controlling the coolant temperature is a simple means to control the mean crystal size, a trend also reported in the literature.39 However, this dependence will be observed primarily in the case of systems with low solubility, for which operation in the labile zone is possible, as shown in this work. For soluble systems such as potassium nitrate, such a trend was not observed. Conclusions The paracetamol experiments presented in this paper illuminated a different operating regime for SHFCC, characterized by high supersaturations and primary nucleation. This regime is important for pharmaceutical molecules or fine chemicals production. It was shown that the mixing conditions inside the SHFC can be used to carry out operation at very high supersaturation levels, deep into the labile zone resulting essentially in a decoupling of the nucleation step from crystal growth. Such a capability does not exist in conventional cooling crystallizers. In stirred vessels, continuous, batch or semi-batch, nucleation and growth occur in the same device. High supersaturations cannot be used due to severe incrustation of the cooling surfaces and a corresponding loss in performance. The high supersaturations applied during the paracetamol runs offer two advantages. First, the nucleation rates achieved are very high, considerably higher than the ones obtained in stirred cooling crystallizers. They are also 2-4 orders of magnitude higher than the results presented previously for potassium nitrate and salicylic acid.20 Consequently, the mean crystal size obtained is decreased substantially. A qualitative comparison with existing literature data obtained in stirred crystallizers showed that smaller crystals are produced by our technique. Second, a high yield can be obtained while operating continuously and for a shorter period of time. High yields accompanied by a controlled crystal size distribution are achievable only in batch crystallizers operated with an optimized cooling program and over a significant period of time. The paracetamol experiments also showed that the mean crystal size can be effectively controlled by manipulating the temperature of the coolant circulated through the shell side. This

feature is in sharp contrast with the existing perception in the crystallization field that operation in the labile zone leads to uncontrolled crystallization and consequently CSD. However, the latter stems from experience with stirred crystallizers; the crystallization in our experimental setup takes place under very different conditions. Except crystal size control, coolant temperature control can be potentially used for crystal morphology control and preferential unseeded polymorph crystallization. The above conclusions are based on short-term operation of the SHFCC setup. Future investigations should clearly show the long-term characteristics of cooling crystallization in SHFCs. Scaling up is another issue that should be addressed during future studies. However, it is our opinion that operation at a larger scale would be less troublesome and more indicative of the advantages SHFCC offers. This is justified at several levels. First, scale-up can be performed easily by maintaining constant the ratio of the feed volumetric flow rate and the fiber cross sectional area. This would introduce minor performance variations in the shell side. Second, the presence of more fibers will make operation more stable; flow disturbances and the associated non-uniform local cooling can be compensated by the large amount of fibers used. Third, properly sized in-line static mixers can be used to deplete the supersaturation created from the SHFC. As already stated, the diameter of the in-line static mixer used in the experiments presented here was far from optimal. This did not allow the achievement of a high degree of mixing downstream the SHFC and the exploitation of all supersaturation generated. For a larger operational scale, such a problem would be easily overcome and the full potential of SHFCC could be realized. Acknowledgment The authors gratefully acknowledge the financial support provided by Pfizer Inc., Global Research & Development, Groton, CT, and the Center for Membrane Technologies at New Jersey Institute of Technology. We also acknowledge the suggestions and encouragement of Dr. David am Ende and Steven J. Brenek from Pfizer Inc. Polypropylene solid hollow fibers were donated by Celgard, Charlotte, NC. Appendix: Calculation of Primary Nucleation Rates The relationships presented below are all extensions or refinements of the classical nucleation theory. As such, they are appropriate only for order of magnitude calculations. However, they can indicate the dominant nucleation mechanism40 or serve as means to provide general relationships for the prediction of the metastable zone width.41,42 Primary nucleation rates can be obtained from the following equation:

Bprimary ) 0.965φhet

( )

DAB ∆C d 5 Cc m

x

7/3

(

[ ]

)

Cc 3 Cc C* exp -1.19 fhet fhet ln (A.1) C* (V ln S)2 ln

where in the absence of data the diffusion coefficient can be obtained from the Stokes-Einstein equation:

DAB )

kBT 3πµdm

and the molecular diameter is approximated by32

(A.2)

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dm )

x

1 CcNA

(A.3)

The heterogeneity factor φhet was found to be approximately equal to 10-11, while the heterogeneity nucleation factor fhet is close to 0.1 for most inorganic systems.32 These two quantities are unity in the case of homogeneous nucleation. The following parameters were used for paracetamol:23 Cc dm φhet fhet DAB (found from eq A.2) n

8.573 7.18 × 10-10 10-11 0.1 7.70 × 10-10 1

Heterogeneous and homogeneous nucleation rates were obtained by eq A.1. In the latter case both the heterogeneity factor and the heterogeneous nucleation factor were set equal to unity. Results are as follows:

run 1 run 2 run 3 run 4

Bhom (m-3s-1)

Bhet (m-3s-1)

2.04 × 4.05 × 10-17 1.36 × 10-40 1.75 × 10-62

2.65 × 1019 4.09 × 1016 1.22 × 1014 6.09 × 1011

108

Nomenclature B ) nucleation rate, m-3 s-1 C ) actual concentration, kg/kg, kmol m-3 ∆C ) concentration difference or supersaturation, kg/kg, kmol m-3 C* ) saturation concentration, kg/kg, kmol m-3 Cc ) crystal molar density, kmol m-3 D ) fiber diameter, m d90 ) crystal size corresponding to 90th percentile of cumulative undersize curve DAB ) molecular diffusion coefficient, m2 s-1 dm ) molecular diameter, m fhet ) heterogeneity nucleation factor, dimensionless K ) constant in eq A.4, dimensionless kB ) Boltzmann’s constant, 1.381 × 10-23 J K-1 L ) length, m m˘ ) mass flow rate, kg s-1 MT ) magma density, kg/m3 NA ) Avogadro’s number, 6.023 × 1023 mol-1 S ) supersaturation ratio, C/C*, dimensionless T ) temperature, °C V˙ ) volumetric flow rate, m3 s-1 Sub-/Superscripts and Greek Letters * ) saturation condition Rfor ) volumetric surface area of foreign particles responsible for heterogeneous nucleation, m2/m3 µ ) viscosity, kg m-1 s-1 ν ) number of ions in a molecule, dimensionless σ ) relative supersaturation, dimensionless φhet ) heterogeneity factor, dimensionless c ) crystal f ) feed fil ) filtrate het ) heterogeneous hom ) homogeneous i ) inside

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ReceiVed for reView April 6, 2006 ReVised manuscript receiVed September 28, 2006 Accepted October 2, 2006 IE060433W