Modeling of Membrane-assisted Seeded Batch Crystallization

Article Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Modeling of Membrane-Assisted Seeded Batch Crystallization Po-Chen Su and Jeffrey D. Ward* Department of Chemical Engineering, National Taiwan University, Taipei, 106-07, Taiwan

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S Supporting Information *

ABSTRACT: In this work, the operation of membrane-assisted seeded batch crystallization is modeled for five substances: potassium nitrate, potassium sulfate, pentaerythritol, succinic acid, and potassium alum. The effect of seed loading, seed size, and water removal rate on product crystal size is investigated. Seed loading and seed size were found to have a significant impact on product crystal size for a given batch time and product yield. Trajectories for the optimal water removal rate versus time were also determined. Overall, the results show that membrane-assisted crystallization behaves similarly to other types of crystallization in terms of product crystal size and size distribution while having the potential to reduce cost and equipment size.

1. INTRODUCTION Crystallization from solution is an essential unit operation for separation and purification in the chemical and related industries. An important product quality metric for crystallization processes is the product crystal size distribution. Various methods commonly employed to drive crystallization include cooling, evaporation, reaction, and antisolvent addition. Recently, membrane-assisted crystallization has attracted increased interest because, as an intensified process, it has the potential to reduce equipment size and energy consumption and improve performance, controllability, and design space.1−4 Membranes are used on a very large scale in desalination processes;5,6 however, in these applications typically no solid product is produced, and therefore they are not crystallization operations. The term “membrane-assisted crystallization” can be applied to many different types of processes in which membranes are used and crystallization occurs. Crystallization may happen on the membrane surface,7 in the bore of a hollow fiber membrane,8 or in a separate vessel (crystallizer).9 Membranes can also be used in conjunction with antisolvent.10−13 Membraneassisted crystallization processes can be operated in batch, semibatch,13 or continuous14 mode. Several different types of membranes have been proposed for membrane-assisted crystallization by solvent removal, including reverse osmosis membranes15,16 and porous hydrophobic membranes.17−19 The membrane can remove the solvent from solution and thus provide the supersaturation required to drive crystallization. Membranes can provide a greater surface area per unit volume of equipment and may reduce energy consumption and improve the operability and controllability of the process. These potential advantages have attracted the attention of researchers. However, compared with other methods of generating supersaturation, there have been relatively few reports in the literature about the development of operating recipes for © XXXX American Chemical Society

membrane-assisted crystallization. Therefore, in this work such recipes are developed for five representative substances. The effect of seed size and seed mass is considered, as well as the water removal strategy. In batch crystallization processes, it is common to introduce seeds in order to suppress nucleation and control the product crystal size distribution. The supersaturation profile (the value of the supersaturation versus time during the batch) may also have a significant effect on the product crystal properties.20 In this work, the effect of seed crystal properties and supersaturation trajectory (which is related to the water removal strategy) on the nucleated mass ratio (μ3,n/μ3) is investigated for membraneassisted batch crystallization. A constraint is imposed on the production rate, so minimizing the nucleated mass ratio corresponds to maximizing the growth of seeds and therefore improving the product crystal size distribution. Five substances (potassium nitrate, potassium sulfate, pentaerythritol, succinic acid, and potassium alum) are studied and a membrane-assisted crystallization process model is developed based on mass and population balances.

2. MODELING The membrane-assisted batch crystallization process considered in this work consists of a crystallizer, buffer tank, and membrane module. The buffer tank and membrane module are operated at a higher temperature than the crystallizer in order to ensure that solution contacting the membrane remains undersaturated. The membrane module removes water from the undersaturated solution in the buffer tank to provide the driving force for Received: Revised: Accepted: Published: A

May 30, 2019 August 13, 2019 August 19, 2019 August 19, 2019 DOI: 10.1021/acs.iecr.9b02935 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 1. Schematic diagram for membrane-assisted crystallization with clear liquid circulation

With the assumption that all crystals remain in the crystallizer, mass balance equations for the buffer tank can be written:

crystallization. The temperature of the crystallizer and buffer vessel remains constant during the batch, and fluid is heated when it is transferred from the crystallizer to the buffer tank and cooled when it is transferred from the buffer tank to the crystallizer. If a higher temperature is required for membrane operation, then the buffer vessel could be operated at a higher temperature or a second heater could be installed on the feed line to the membrane module. This would increase the consumption of energy but not otherwise affect the results. Two limiting cases are considered: One in which only clear liquid is transferred from the crystallizer to the buffer tank, and one in which a slurry with the same crystal density and size distribution as the crystallizer is transferred to the buffer tank. The crystal population balance is solved for these two cases by the method of moments. μi is the ith moment of crystal size distribution, defined as μi =

∫0

∞

Lif (L , t ) dL

i = 0, 1, 2, ...

dVb = −rw dt

(2)

rw = R w /τ

(4)

dC b ij dV y = jj−C b b + qCc − qC bzzz/Vb dt dt (3) k { where Vb and Cb are the solution volume and the concentration in the buffer tank respectively, Cc is the concentration in the crystallizer, and rw is the water removal rate which may be constant or variable during the batch. If it is constant, then it is the total amount of water removed (Rw, with units of volume) divided by the batch time:

The slurry or solution circulation rate between the crystallizer and the buffer tank is denoted by q. In the crystallizer, the mass balance model can be expressed as

(1)

dVc =0 dt

where f(L, t) is the crystal number size distribution function. 2.1. Crystallizer with Clear Liquid Transfer. A schematic diagram of membrane-assisted crystallization with clear liquid transfer is given in Figure 1. In this model, it is assumed that no solid crystals are transferred to the buffer tank. This is only an approximation, but it could approximately be achieved by using a settling zone where solid crystals are allowed to settle and return to the crystallizer while clear liquid flows out. Under these circumstances, all nucleated crystals and growing seeds will remain in the crystallizer. The process includes two vessels: a half-liter crystallizer and a half-liter buffer vessel. The temperature in the crystallizer is maintained at a constant value throughout the batch and this temperature is chosen so that the feed is just saturated at the beginning of the batch. The temperature in the buffer tank is set to be high enough so that the solution remains undersaturated even after passing through the membrane module. A low-pressure pump is placed between the crystallizer and buffer tank and the pump power can be adjusted to determine the circulation rate between the crystallizer and buffer tank. This circulation rate has an effect on the product crystal size distribution. Another pump is located between the buffer tank and the membrane module and it is assumed that this pump provides sufficient pressure for water to traverse the membrane. It is also assumed that the permeate is nearly pure water.

(5)

dCc dV i y = jjj−Cc c − 3ρc k vμ2 GVc − qCc + qC bzzz/Vc dt d t (6) k { where Vc is the solution volume in the crystallizer. In this simulation work, it is assumed that the total volume in the crystallizer is constant. The initial concentration of solute in both tanks is the same and is equal to the saturation concentration at the crystallizer operating temperature. If no crystals are lost to the buffer tank, the population balance equations are similar to those of an ordinary well-mixed batch crystallizer:

dμ0 dt dμi dt

=B = iGμi − 1 ,

(7)

i = 1, 2, ...

(8)

where B and G are crystal nucleation and growth rate, respectively. 2.2. Crystallizer with Slurry Transfer. The crystallizer with slurry circulation is assumed to be well-mixed and therefore B

DOI: 10.1021/acs.iecr.9b02935 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 2. Schematic diagram for membrane-assisted crystallization with slurry circulation.

60 min and the initial seed size and seed loading ratio are selected separately for proper performance for each system, as shown in Table 2. In every case, the seed crystal size distribution is assumed to be parabolic. 2.3. Crystallization Kinetic Models. In this work, kinetic models for all substances are taken from the literature and are listed in Table 3. Supersaturation (ΔC), which appears in the kinetic models for pentaerythritol, succinic acid, and potassium alum is expressed as a function of concentration (kg/(kg-solvent s)):

Table 1. Parameters in the Crystallization Process Model description

symbol

value

unit

gas constant solution circulation rate volume of crystallizer initial volume of buffer tank total water removal

R q Vc Vb Rw

8.3145 0.06 0.5 0.5 0.25

J/(K mol) L/min L L L

the stream that flows into the buffer tank from the crystallizer has the same magma density and size distribution as the crystallizer. Figure 2 shows a schematic diagram of the process with slurry circulation. In the buffer tank, the overall mass balance equation is the same as for the case of clear liquid transfer (eq 2) but the component mass balance (analogous to eq 3) is different:

ΔC = C − Csat

Solubility data are presented in Table 4. Relative supersaturation (S), which appears in the kinetic models for potassium nitrate and potassium sulfate is defined as

dC b ij dV y = jj−C b b + qρc kvμ3 + qCc − qC bzzz/Vb dt dt (9) k { The crystals from the crystallizer that are transported into the buffer tank and then dissolved are accounted for by the term qρckvμ3. For the population balance with slurry transfer, the moment equations can be expressed as follows: i dV y = B − jjjμ0 c + qμ0 zzz/Vc dt k dt {

(12)

S = ΔC /Csat

(13)

Among the nucleation kinetic expressions considered in this work, that of pentaerythritol is the only one in which the nucleation rate is not proportional to μ3 or MT, which means that the nucleation rate does not depend on the volume or mass of crystals in the crystallizer. If the nucleation rate depends on μ3 or MT it is expected that the nucleation rate will rise more rapidly later in the batch when a greater mass of crystals is present in the crystallizer. Therefore, if that term is absent it is expected that the nucleation rate will rise less rapidly toward the end of the batch. The kinetic expressions for succinic acid26 include a dependence on the impeller speed N which is taken to be 550 rpm in this work. 2.4. Sensitivity Analysis. For the case of clear liquid removal from the crystallizer, sensitivity analysis was performed to study the effect of important design variables on the product quality as measured by the nucleated mass ratio. The variables considered in the sensitivity analysis include the seed loading

dμ0

(10)

i dV y = Gμi − 1 − jjjμi c + qμi zzz/Vc i = 1, 2, ... dt (11) k dt { The parameters used in these equations and the properties of the system are listed in Table 1. The initial values of the moments are those of the seeds. The process duration for all cases are dμi

Table 2. Parameters and Conditions of Each Substance component

density (kg/m3)

temp in the crystallizer (K)

volumetric shape factor

initial concentration (kg/kg-solvent)

seed loading ratio (kg/kg-product)

seed mean size (μm)

potassium nitrate potassium sulfate pentaerythritol succinic acid potassium alum

2110 2660 1400 1560 1760

303.15 303.15 343.15 298.15 308.15

1 1.5 0.5921 1.122 0.471

0.4599 0.1303 0.3096 0.0973 0.1659

0.145 0.15 0.09 0.15 0.14

320 490 250 310 540

C


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Industrial & Engineering Chemistry Research Table 3. Kinetic Models substance

nucleation rate

crystal growth rate

11 1.78

−4 1.32

ref

G(t ) = 1.1612 × 10 S

Chung et al.(1999)23

B(t ) = 2.85 × 1020 e−7517/ T S2.25μ3

G(t ) = 144 e−4859/ T S1.5

Sarkar et al.(2006)24

pentaerythritol

B(t ) = 2.3282 × 108ΔC 3.8a

G(t ) = 6.49 × 10−5 e−40/ T ΔC1.9

Bernardo and Giulietti (2010)25

succinic acid

B(t ) = 5.61 × 107ΔC 4.01MT N2.28

G(t ) = 1.04 × 10−6ΔC1.05N 0.63

Qiu and Rasmuson (1990)26

potassium alum

B(t ) = 1.15 × 1028 e−12028/ T ΔC 2.1MT (1 + C)(5.5T − 621.32)−1

G(t ) = 39.94 e−3849/ T ΔC1.38

Corriou and Rohani (2002)27

potassium nitrate

B(t ) = 4.6401 × 10 S

potassium sulfate

μ3

a

Nucleation rate is for 600 rpm impeller speed.

2.6. Numerical Methods and Initial Conditions. The mathematical model developed in this work can be applied to any type of membrane separation system that can produce a permeate stream with high purity, including reverse-osmosis membranes and porous hydrophobic membranes. All the differential equations in this simulation are solved with the ode45 algorithm in Matlab r2018a. Table 2 shows the parameters and conditions in the crystallizer of all substances. Under the consideration of clear liquid circulation, only clear liquid is transported from the crystallizer to the buffer tank and all crystals remain in the crystallizer. Supersaturation changes with time because pure water is removed continuously from the buffer tank by the membrane. Therefore, the crystal growth rate and the nucleation rate vary with time. The crystal size distribution in the crystallizer can be determined by population balance and the product number and volume distributions at the end of the batch can be determined using the method described by Ward et al.29,30 The process begins with a saturated solution in the crystallizer, therefore the crystal growth and nucleation rate are initially zero.

Table 4. Solubility for Each Substance solubility (saturated concentration)a potassium nitrate Csat = 1.721 × 10−4T 2 + 5.88 × 10−3T + 0.1286 potassium sulfate Csat = −7.14 × 10−6T2 + 2.46 × 10−3T + 0.0629 pentaerythritol logCsat = 0.01377T − 5.23443 succinic acid

Csat = 1.7797 × 10−6T3 − 4.7631×10−5T2 + 4.1772 × 10−3T − 5.7983 × 10−3

potassium alum

Csat = 5.85 × 10−5T 2 − 0.031T + 4.1636

a

The unit of solubility is kg/kg-solvent. The unit of temperature is centigrade.

ratio, seed size, water removal rate, and the circulation rate between the buffer tank and the crystallizer. 2.5. Effect of Water Removal Strategy. To study the effect of water removal rate on the crystal size distribution, different removal policies are discussed. Besides a constant water removal rate, three additional profiles are considered: linearly decreasing removal rate, linearly increasing removal rate, and quadratic increasing removal rate. Finally, for succinic acid crystallization, an optimal trajectory for the water removal rate is determined from an optimal supersaturation trajectory determined previously.28

3. RESULTS AND DISCUSSION 3.1. Base-Case Results. Plots of key process variables versus time during the batch under base-case conditions for different

Figure 3. Dynamic results for the potassium nitrate process under base-case conditions (Table 2): (a) temperature in the crystallizer; (b) concentration in the crystallizer; (c) relative supersaturation in the crystallizer; (d) crystal growth rate; (e) crystal nucleation rate; (f) third moment of crystals in the crystallizer. D


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size and seed loading were determined by trial and error for each species so that the nucleated mass ratio (μ3,n/μ3) at the end of the batch was around 0.1. Table 2 shows seed size and seed loading for all substances. Under this condition, growth of seeds is the dominant factor in determining the product crystal size distribution. 3.2. Sensitivity Analysis. Sensitivity analyses were performed to determine the effect of various process properties on the nucleated mass ratio (μ3,n/μ3) for the different chemical systems. 3.2.1. Effect of Seed Loading Ratio for Constant Water Removal Rate. In the first sensitivity analysis, the seed mass is increased while the seed size and water removal rate are fixed. The circulation rate between the crystallizer and buffer tank is adjusted to meet the production rate constraint. Because the total yield (production rate) is fixed in the simulation, the mass of solute that is crystallized during the batch decreases as the seed loading ratio increases. Furthermore, when the number of seeds initially present in the crystallizer is increased, the surface area available for growth also increases. Therefore, the supersaturation and nucleation rate decrease when a larger seed mass is utilized. Figure 5 shows how the nucleated mass ratio changes versus seed loading for the different chemical systems. 3.2.2. Effect of Seed Loading Ratio for Constant Circulation Rate. In this analysis, the seed mass is increased while the seed size and circulation rate are fixed. As in the previous case, nucleated mass ratio decreases when more seeds are added and the circulation rate remains constant. Figure 6 shows how the

Figure 4. Final product crystal size distribution for the potassium nitrate process under base-case conditions: (a) number size distribution; (b) volume size distribution.

chemical species are shown in Figure 3 and Figures S1−S4. The third moment of the crystals is strictly increasing with time corresponding to an increase in solid mass in the crystallizer during the batch. Figure 4 and Figures S5−S8 show the predicted product size distribution of the five substances considered in this work. Seed

Figure 5. Sensitivity analysis showing the effect of seed loading on nucleated mass ratio for constant water removal rate for different chemical systems. Other variables are the same as in Table 2: (a) potassium nitrate; (b) potassium sulfate; (c) pentaerythritol; (d) succinic acid; (e) potassium alum. E


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Figure 6. Sensitivity analysis showing the effect of seed loading on nucleated mass ratio for constant circulation rate for different chemical systems. Other variables are the same as in Table 2: (a) potassium nitrate; (b) potassium sulfate; (c) pentaerythritol; (d) succinic acid; (e) potassium alum.

nucleated ratio changes as the seed loading ratio is changed. The trends are similar to the previous case. When more seeds are added in the crystallizer, a lower supersaturation is required to achieve the desired yield, so the water removal will decrease with increasing seed loading. 3.2.3. Effect of Seed Mean Size on Nucleated Mass Ratio for Constant Circulation Rate. The influence of seed size on the nucleated mass ratio is also studied. Conditions in the crystallizer are the same as those described previously. The range of seed size was chosen from 200 to 650 μm for potassium nitrate and pentaerythritol, from 250 to 700 μm for succinic acid, and from 300 to 750 μm for potassium sulfate and potassium alum. Other parameters including the total yield of crystal mass are fixed to the same values as those that were used previously. In Figure 7, a similar trend is observed for all substances: the smaller is the seed crystal mean size, the lower is the product nucleated mass ratio. Since the seed mass is fixed, a larger seed size corresponds to a smaller number of seeds and less surface area for the crystals to grow, causing a higher supersaturation and nucleation rate. Therefore, smaller seeds cause a reduction in the nucleated mass. Among the chemical species considered in this work, succinic acid is the most sensitive to the seed size: the nucleated ratio changes from nearly 0 to around 0.52 when the seed size varies within a range of 450 μm. 3.2.4. Influence of Water Removal Strategy. In previous sections, it is assumed that water is removed across the membrane at a constant rate during each batch. To study the

effect of changing the water removal rate during the batch, different water removal policies are considered. Aside from a constant water removal rate, three additional profiles are considered: linearly decreasing removal rate, linearly increasing removal rate, and quadratic increasing removal rate. In all these cases, product yield and total amount of water removed are fixed. The different water removal rate trajectories result in different supersaturation trajectories and different product crystal size distributions. Figure 8 and Figures S9−S12 show the results for the five substances considered in this work. The results show that, in most of the cases, the constant water removal rate provides the smallest nucleated mass ratio. However, in general, a lower supersaturation at the beginning and a greater supersaturation at the end of the batch reduce the nucleated mass. This is because nuclei formed near the end of the batch have less time to grow and therefore the nucleated mass is lower. 3.2.5. Determination of Optimal Water Removal Trajectory. Nearly optimal supersaturation trajectories were determined previously for the chemical systems considered in this work for a batch cooling crystallizer.28 The trajectory is shown in Figure 9 for succinic acid. This trajectory was determined by sequential iterative optimization using control vector parametrization with a linear spline and supersaturation as the (timevarying) control input. Unlike other cases considered in this work, for the optimal trajectory the initial supersaturation is not zero. The parameters used in this case are listed in Table 5. The membrane crystallization model developed in this work was F


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Figure 7. Sensitivity analysis showing the effect of initial seed mean size on nucleated mass ratio for different chemical systems. Other variables are the same as in Table 2: (a) potassium nitrate; (b) potassium sulfate; (c) pentaerythritol; (d) succinic acid; (e) potassium alum.

Figure 8. Effect of water removal rate strategy on relative supersaturation profile and product volume crystal size distribution for potassium nitrate: (a) water removal rate; (b) relative supersaturation; (c) final volume size distribution.

used to determine a water removal rate trajectory that approximately achieves this optimal supersaturation trajectory during the batch. This water removal trajectory was simulated, and the results are shown in Figure 10.

The optimal trajectory calls for a small decrease in the supersaturation during the beginning part of the batch. This could be achieved by adding solvent to the buffer tank or crystallizer at the beginning of the batch. However, in this work this possibility was G


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not considered and instead the water removal rate was set to be zero during the early part of the batch, as shown in Figure 11 (the blue solid line). 3.3. Modeling of Crystallizer with Slurry Circulation. When the stream between the crystallizer and buffer tank is considered to be a slurry with the same magma density as the crystallizer, the behavior of the process is significantly different. Many of the seed crystals are removed and dissolved before they have a chance to grow. Figure 12 and Figures S13−S16 show the values of key process variables versus time during the batch when the slurry is circulated between the crystallizer and buffer tank. The third moment of the crystal size distribution in the crystallizer decreases at the beginning of the process as seed crystals are removed to the buffer tank and dissolved. Later, after the supersaturation has risen to an undesirably high value, a burst of nucleation occurs followed by growth of the nucleated crystals. Since a large amount of the initial seeds are lost, the nucleated mass ratio is high in all cases compared to that without slurry circulation. Figure 13 and Figures S17−S20 show the number and size distribution for each system in this case. The nucleated mass ratio of potassium nitrate and succinic acid are over 0.55 and 0.67, respectively (Figure 13 and Figure S19), meaning most of the seeds were lost through the circulation in this extreme assumption. In this situation, nucleation dominates for this species. The model predicts that the nucleated ratio of

Figure 9. Optimal supersaturation trajectory of succinic acid.

Table 5. Parameters in the Optimal Trajectory Model description

symbol

value

unit

seed loading ratio seed mean size process duration volume of crystallizer impeller rotation speed solution circulation rate initial volume of buffer tank

ms L0 t Vc N q Vb

2.3 × 10−4 3.025 × 102 90 2.5 400 1.2 2.5

kg/kg-solvent μm min L rpm L/min L

Figure 10. Dynamic results for the succinic acid process with the optimal water removal trajectory: (a) the third moment of crystals in the crystallizer; (b) crystal growth rate; (c) crystal nucleation rate; (d) solute concentration in the buffer tank; (e) solute concentration in the crystallizer; (f) solution volume in the buffer tank. H


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twice as high as the case of clear liquid circulation (Figure S20). This again suggests that slurry circulation with seeds is not practical. 3.4. Discussion. In general, results are similar for different chemical systems, which is consistent with previous findings for cooling crystallization. Figure 3 and Figures S1−S4 show that if the water removal rate is constant and the initial supersaturation is zero, the supersaturation increases at the beginning of the batch, leading to a peak in the supersaturation and the growth and nucleation rates. Eventually, crystal growth (and to a lesser extent nucleation) increase the surface area available for growth and the supersaturation decreases. The total crystal mass increases slowly at first and then more rapidly toward the end of the batch. Figure 4 and Figures S5−S8 show that for each case the product crystal size distribution has two peaks. The peak at smaller crystal size corresponds to nucleated crystals, and the peak at larger crystal size corresponds to seed-grown crystals. In general the nucleated crystals dominate the number crystal size distribution (CSD) but the seed-grown crystals dominate the volume size distribution. This indicates that although there

Figure 11. Optimal and modified buffer tank holdup trajectory for the succinic acid process.

potassium sulfate and pentaerythritol are 0.30 and 0.44, respectively (Figures S17 and S18). The smallest nucleated mass ratio of all cases is 0.27 of potassium alum, which is also

Figure 12. Dynamic results for the potassium nitrate process with slurry removal: (a) temperature in the crystallizer; (b) concentration in crystallizer; (c) relative supersaturation in the crystallizer; (d) crystal growth rate; (e) crystal nucleation rate; (f) third moment of crystals in the crystallizer. I


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decrease in the nucleated mass. The same behavior is observed in the case for which the product yield and circulation rate are restricted. Conversely, increasing the seed mean size causes an increase in the nucleated mass. All of these trends can be explained by the fact that a larger seed surface area (caused by an increase in seed mass or a decrease in seed size) results in a lower supersaturation and therefore a lower nucleation rate. The effect of the water removal strategy was also studied, and in this case different systems result in somewhat different outcomes. For the succinic acid system, the decreasing water removal strategy gives a much more preferable product size distribution with a smaller nucleated mass ratio. In contrast, in the rest of the systems, the constant water removal strategy provided the best result. The optimal water removal rate for the succinic acid process was also determined and simulated. These results show that adjusting the water removal strategy can have a significant effect on the product crystal size distribution. Finally, an alternative assumption that slurry rather than clear liquid is withdrawn from the crystallizer is considered. In this case, seeds are removed from the crystallizer at the beginning of the batch, resulting in a large nucleated mass and poor overall performance.

Figure 13. Product crystal size distribution for the potassium nitrate process with slurry removal: (a) number size distribution; (b) volume size distribution.

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are many more nucleated crystals, because they are smaller they contribute relatively little to the product crystal mass or volume. Figures 5−7 show how the product crystal size distribution can be improved by changing the seed size or seed loading (initial seed mass). As the seed size is decreased or the seed loading is increased, the nucleated mass decreases, indicating that the peak corresponding to the nucleated crystals has shrunk and the peak corresponding to seed-grown crystals has enlarged, which is desirable. This is consistent with previous studies on cooling crystallization and with intuition: Decreasing the seed size for a given seed mass or increasing the seed mass for a given seed size both increase the seed surface area. A larger seed surface area means that the same amount of mass deposition can be accomplished with a smaller driving force (supersaturation). Lowering the supersaturation during the batch also decreases the nucleation rate, which decreases the nucleated mass. To permit comparison of results for different chemical systems and different water removal rate profiles, it is assumed in this work that the initial supersaturation is zero; that is, the solution is just saturated at the beginning of the batch. In fact, most of the results could be improved slightly by allowing for a nonzero initial supersaturation, but such an allowance would make comparing results for different systems more difficult, therefore it is not considered in this work. It is also assumed in this work that the concentration in the crystallizer remains within the metastable zone at all times during the batch so that primary nucleation can be neglected. Concentration polarization near the membrane surface is also neglected in this work. It is assumed that the buffer tank is sufficiently undersaturated that solution near the membrane does not become supersaturated even though concentration polarization may occur. Concentration polarization may also increase the driving force required to achieve the desired mass flux across the membrane; this should be considered when these calculations are performed.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b02935. Additional figures: dynamic results; final product crystal sizes; effect of water removal rate; product crystal size distributions for the studied processes (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jeff[email protected]. ORCID

Jeffrey D. Ward: 0000-0003-0727-7689 Notes

The authors declare no competing financial interest.

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REFERENCES

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4. CONCLUSIONS Membrane-assisted batch crystallization processes for producing five different chemical products are designed. Sensitivity tests were performed to investigate the effect of changing various properties of the operation recipe. When the product yield and water removal rate are fixed, increasing the seed mass results in a J


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Modeling of Membrane-assisted Seeded Batch Crystallization

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