Continuous Membrane-Assisted Crystallization To Increase the

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Continuous Membrane-Assisted Crystallization to Increase the Attainable Product Quality of Pharmaceuticals and Design Space for Operation Jiayuan Wang, and Richard Lakerveld Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 26 Apr 2017 Downloaded from http://pubs.acs.org on April 29, 2017

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Industrial & Engineering Chemistry Research

Continuous Membrane-Assisted Crystallization to Increase the Attainable Product Quality of Pharmaceuticals and Design Space for Operation Jiayuan Wang, Richard Lakerveld* Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Email: [email protected]

KEYWORDS Crystallization, Membrane-Assisted Crystallization, Attainable Region, Quality by Design, Design Space

ABSTRACT

Continuous manufacturing is an important paradigm shift in pharmaceutical industries and has renewed the interest in continuous crystallization. The combination of crystallization and membranes is a promising hybrid technology for separation and purification of pharmaceuticals. The impact of membranes as an extension to conventional continuous crystallization processes

ACS Paragon Plus Environment

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on attainable product quality and design space is investigated systematically using model-based optimization. The proposed model is based on a full population balance such that all relevant crystallization phenomena can be included and is solved using a first-order discretization scheme with a hybrid grid. A case study involving continuous crystallization of paracetamol using a series of mixed suspension, mixed product removal (MSMPR) crystallizers is presented to illustrate the approach. The results show that the attainable size and design space can be enlarged significantly by extending conventional crystallization with membranes. In particular, larger crystals or shorter residence times can be achieved. Furthermore, to obtain a crystal size within a desired range, a broader range of temperatures can be applied, which increases operational flexibility. 1. Introduction Pharmaceutical industry is challenged by the need for more efficient manufacturing processes due to the increasing competition,1,

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demand for more sustainable processes, a changing

mindset,3 and new regulatory incentives.4-6 Continuous manufacturing, as opposed to batch-wise manufacturing, is a novel paradigm in pharmaceutical manufacturing to improve efficiency.7-9 Compared to conventional batch-wise processing, potential advantages of continuous processing include improved product consistency, smaller inventories, reduced costs,3,

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and new

opportunities for on-demand drug production with modularized, small-scale and mobile equipment.8, 11 Continuous crystallization is expected to play an important role within many of the future continuous pharmaceutical processes, because many of the active pharmaceutical ingredients (APIs) are produced in crystalline form and intermediate compounds can often be separated effectively with a single crystallization step while still meeting stringent requirements for material attributes.12-14 Continuous crystallization is a common and well-studied unit


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operation for bulk and fine chemicals. The shift towards continuous manufacturing in pharmaceutical industry has renewed the interest in continuous crystallization. Much research has been devoted to the improvement of continuous crystallization by developing dedicated control strategies15-17 and design methods18-25 for different types of continuous crystallizers (e.g. plug flow,26, 27 a single or a cascade of mixed-suspension mixed-product removal (MSMPR) crystallizers28, 29 and oscillatory baffled crystallizers30). For pharmaceutical applications, rather than simply operating batch equipment in continuous flow mode, the manufacturing paradigm shift also provides opportunities to introduce new technologies for crystallization that especially work well in continuous flow mode. One of such new technologies is membrane-assisted crystallization,31-33 which is a hybrid separation process combining crystallization and membranes. Membranes are mainly designed for continuous operation with constant permeate flow and have been coupled with crystallization processes to fulfil various functions such as heat exchange,34 assisting nucleation,35 solvent removal,35-37 anti-solvent injection,38 and impurity removal.39 Potential benefits of continuous membrane-assisted crystallization (cMAC) compared to stand-alone continuous crystallization include a higher yield,39, consumption,36,

37

and higher operational flexibility.31,

37

40

reduced energy

In particular, solvent removal via

membrane provides several advantages over traditional evaporation methods in terms of their ability to create a large and better defined interfacial area leading to better local supersaturation control and potential energy savings. The properties of API crystals are strongly related to the product safety, bio-availability,41 as well as the cost of downstream processes.42 Therefore, especially for pharmaceutical applications, the product quality is of utmost importance when investigating the potential benefits of cMAC compared to conventional crystallization processes.


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Key outstanding questions are how the addition of membranes to a continuous crystallization process may enlarge the range of attainable product qualities or whether the addition of membranes can enlarge the range of operating conditions for which a desired range of product qualities can be produced. The range of attainable product qualities of a continuous crystallization process without membranes has been identified systematically in various studies by mapping the so-called attainable regions of tubular crystallizers and cascades of MSMPRs.43-47 Those studies are valuable to design flexible continuous crystallization processes enabling a wider range of attainable product qualities. Membranes have the potential to further increase the range of attainable product qualities, which has not been studied systematically to the best of our knowledge. In addition, for pharmaceutical applications, typically a crystalline product with specific requirements for solid-state properties such as a desired mean crystal size and width of the crystal size distribution (CSD) has to be produced to facilitate easy downstream processing and to guarantee a targeted product functionality and to satisfy regulatory requirements. The challenge for design and operation of a pharmaceutical crystallization process is to operate at conditions such that those specific solid-state properties can be produced in a verifiable and robust way. Traditionally, product quality in pharmaceutical industry is controlled by strictly following batch recipes and off-line validation. A more flexible approach is based on a so-called design space, which aligns well with a quality-by-design (QbD) framework.48,

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A design space is

defined as the multi-dimensional range of input parameters and operating conditions for which an acceptable product quality can be guaranteed.5, 50 Besides conventional approaches such as risk analysis and Design-of-Experiment,51-53 the popularity of using model-based techniques to


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construct design spaces and relief the burden of experimental work has increased in recent years.54-56 The main advantage of a design space is that a range of operating conditions can be implemented without further need for regulatory approval, which increases operational flexibility. Consequently, a large design space is preferred to allow flexible manufacturing. In principle, the integration of membranes with crystallization can provide more operational flexibility and robustness.57,

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However, systematic studies that map a design space of a

pharmaceutical cMAC process have not been published to the best of our knowledge. Therefore, the potential of cMAC to enlarge a design space compared to conventional crystallization processes remains largely unknown. The objective of this paper is to investigate how continuous membrane-assisted crystallization can enlarge the range of attainable product qualities and design spaces compared to conventional continuous crystallization. Model-based optimization is applied to systematically identify those two spaces. The model is based on a full population balance to include various crystallization phenomena that are relevant for cMAC processes such as dissolution of small crystals, which often happens inevitably when integrating membranes with crystallization. The feed to the membranes is typically heated above the saturation temperature of the solute to prevent membrane fouling, which often causes uncontrolled dissolution of fines.31, 36 The remainder of this paper is organized as follows: In Section 2, the process model is presented and the optimization problems for attainable region and design space are formulated for a selected case study. Section 3 presents the results illustrating the influence of membranes on attainable region and design space. The paper ends with conclusions and outlook for future work. 2. Approach


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2.1 Process model An MSMPR crystallizer is the most common type of continuous crystallizer, which offers advantages related to control, simplicity, and easiness when converting existing batch equipment to continuous crystallizers.59 Moreover, various mixing behaviors and process configurations can be modeled by combining multiple instances of the MSMPR model, e.g., by changing the number of stages, adding recycle loops, and combination with unit operations for classification and dissolution.29, 60, 61 Therefore, a model of an MSMPR crystallizer including solvent removal via a membrane is used here as a basic building block to construct various cMAC process configurations (Figure 1).

Figure 1. Elementary building block of a cMAC process model Supersaturation is the driving force for crystallization. In addition to solvent removal over the membrane, cooling can be applied in a cMAC process to generate additional supersaturation. Membranes are sensitive to fouling when brought into contact with a supersaturated solution. Therefore, the operating temperature of the membrane is chosen to be larger compared to the crystallizer temperature to avoid severe fouling. The mother liquor sent to the membrane is


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ideally free from any crystals to avoid dissolution. However, a complete solid-liquid separation can practically not be obtained without any rigorous solid-liquid separation device. Therefore, some degree of classification is assumed with only the small crystals entering the membrane feed (see Figure 1). The development of a CSD as function of operating conditions can be described by a population balance, which for a single MSMPR crystallizer at steady state is given by:62, 63

∂(Gi ni ) F n − Fn = Bi − Di + i−1 i −1 i i , ∂L Vi

(1)

where ni denotes the number density distribution in vessel i with volume Vi . The growth rate Gi is multiplied with ni to give the convective flux ∂ ( G i ni ) / ∂ L along the crystal length axis ( L ).

Bi and D i are the birth and death rate, respectively, which may include terms for nucleation, agglomeration, breakage and dissolution. Fi −1 and Fi denote the inlet and the outlet volumetric flow rates of vessel i , respectively. The following assumptions and modifications are made to simplify and adjust Equation (1) to describe a cMAC process: (i) Growth rate Gi is independent of crystal length; (ii) Nucleation introduces new crystals at zero length only; (iii) Agglomeration and breakage are negligible; (iv) The fines dissolution in the membrane feed is complete and subject to the following step classification function:64, 65 1 0

φ ( L) = 

L ≤ Lcrit

(2)

L > Lcrit .

Consequently, Equation (1) can be simplified to

Gi

∂ni Fi −1ni −1 − Fn i i − Fi , M ,inφ ( L)ni = , ∂L Vi

(3)


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where Fi , M ,in is the volumetric flow rate of the feed stream to the membrane module (see Figure 1). The nucleation term B0,i appears in the boundary conditions as follows:

ni ( L = 0) =

B0.i , Gi

(4)

A discretization scheme is used to solve the population balance described by Equation (3). Discretization strategies for population balance models have been studied extensively.66-69 Highorder discretization strategies are usually applied if more accurate results are required. However, such higher-order schemes may suffer from numerical instabilities.69,

70

To prevent such

instabilities and to obtain sufficient simplicity for optimization, a first-order discretization scheme along the length axis with a total of n eq points is used as follows:

ViGi (0 − N j ,i ) − ( Fi + Fi ,M ,inφ ( L j ) ) N j ,i + Fi−1 N j ,i −1 + B0,iVi = 0, j = 1, ∆L ViGi ( N j −1,i − N j ,i ) − ( Fi + Fi ,M ,inφ ( L j ) ) N j ,i + Fi −1N j ,i −1 = 0, ∀j = 2,3,..., neq . ∆L

(5)

Equation (5) can also be seen as the crystal number balance over each slot. More details on derivation can be found in the supporting information or literature.69 N

j ,i

is the crystal number

in the j th slot of the CSD in the i th vessel, which is defined by

N j ,i = ∫

Lj

L j −1

ni ( L)dL.

(6)

Furthermore, let ai = − bi =

Vi Gi − ( Fi + Fi , M ,inφ ( L j ) ) , ∆L

(7)

ViGi , ∆L

(8)

so that Equation (5) can be written in the following matrix form for the i th vessel:


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 ai b  i      

  N1,i   B0,iVi   N1,i −1 Fi −1    N 2,i   0   N 2,i −1 Fi −1          N 3,i   0   N 3,i −1 Fi −1   = 0.  M  +  + M M            N N F  neq −1,i  0  neq −1,i −1 i −1       ai   N neq ,i   0   N neq ,i −1Fi −1     

ai bi

ai O O bi

ai bi

(9)

th

The k moments of the CSD in the i th vessel can be obtained via numerical integration by assuming all crystals are in the middle of each slot: k

L +L  mi ,k = ∑ j−1 j  N j,i . 2  j 

(10)

The process model is completed by the solute and total mass balance over each vessel as follows: Fi −1ε i −1ρ sol ,i −1ci −1 + Fi −1 (1 − ε i −1 ) ρ s + Fi , M ,out ρ sol ,i , M ,out ci , M ,out − Fiε i ρ sol ,i ci − Fi (1 − ε i ) ρ s − Fi , M ,inε i , M ,in ρ sol ,i ci − Fi , M ,in (1 − ε i , M ,in ) ρ s = 0, Fi −1ε i −1ρ sol ,i −1 + Fi −1 (1 − ε i −1 ) ρ s + Fi , M ,out ρ sol ,i , M ,out − Fiε i ρ sol ,i − Fi (1 − ε i ) ρ s − Fi , M ,inε i , M ,in ρ sol ,i − Fi , M ,in (1 − ε i , M ,in ) ρ s = 0,

(11)

(12)

and over the membrane module with the assumption of total solute rejection by the membrane:

Fi ,M ,inεi ,M ,in ρsol ,i ci + Fi ,M ,in (1 − εi ,M ,in ) ρs − Fi,M ,out ρsol ,i ,M ,out ci ,M ,out = 0,

(13)

Fi,M ,inεi ,M ,in ρsol ,i + Fi,M ,in (1 − εi ,M ,in ) ρs − Fi ,M ,out ρsol ,i ,M ,out − Ri ρsolvent = 0,

(14)

where ρsol , ρsolvent and ρs denote the density of solution, solvent and solid phase, respectively. R is the volumetric flowrate of the removed solvent. c is the weight fraction of the solute in the solution. ε represents the liquid volume fraction of the suspension, which can be related to the third moment of CSD to provide the connection between the population balance and material balances as follows:

ε i = 1 − kv mi ,3 ,

(15)


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where kv is a shape factor and mi,3 is the third moment of the CSD defined by Equation (10). A potential disadvantage of solvent removal via membranes is that the concentration of impurities may rise, which is often a concern for pharmaceutical applications. Therefore, the effect of impurities has to be taken into account when identifying the attainable product size and design space for operation. Equations (11) to (14) still hold when impurities are present if the crystal purity, expressed as the mass fraction of the solute in the solid phase, is assumed equal to one. The purity of crystals is often very high because of the nature of a crystallization process, which justifies this assumption. Moreover, constraints on process operation (i.e., supersaturation) can be imposed to prevent excessive uptake of impurities in the crystal lattice. Therefore, it is expected that the assumption of a pure crystalline phase in the derivation of Equations (11) to (14) will not have a significant impact on the results. The mass balance on the impurity over each vessel can be given by: 1 / SC ⋅ xi −1,impurity Fi −1ε i −1 ρ sol ,i −1ci −1 + xi −1,impurity Fi −1 (1 − ε i −1 ) ρ s − 1 / SC ⋅ xi ,impurity Fiε i ρ sol ,i ci − xi ,impurity Fi (1 − ε i ) ρ s = 0,

(16)

where xi ,impurity is the mass fraction of the impurity in the solid phase in the i th vessel. A segregation coefficient (SC), which is defined as the ratio of the impurity fraction in the crystals to that in solution on a solvent-free basis, is introduced to describe the incorporation of impurities into the crystal lattice.71 2.2 Optimization framework and case study The boundaries of the attainable product quality can be found by maximizing and minimizing the product attributes of interest within the window of critical process parameters.44 In case of identifying the design space, a reverse mapping with respect to identifying an attainable region for product quality can be applied. Therefore, rather than mapping the design space via


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exhaustive discretization of the critical process parameters,72 a similar optimization formulation as used for identifying the attainable product qualities is applied to map the design space systematically by maximizing and minimizing the critical process parameters with a specified range for critical product qualities as constraints. Either single-objective or multi-objective optimization problems can be formulated depending on the number of critical process parameters of concern. A case study is introduced in this section to illustrate the proposed approach and to investigate the influence of membranes on the attainable region for product quality and design space for operation of a cMAC process. Paracetamol crystallized from ethanol is selected as the model system with physical properties and crystallization kinetics as given in Table 1. A detailed study on the attainable region for product quality for a cascade of MSMPRs without membranes has been reported in literature for this system,44 which serves as a reference to analyze the impact of membranes and as a source for validation of the implemented discrete population balance model in this work. The feed to the cMAC process is chosen as a saturated paracetamol-ethanol solution at 340 K and the impurity mass fraction on a solvent-free basis is 0.04 g/g . Table 1. Physical properties and crystallization kinetics for paracetamol in ethanol Parameters

Unit

Value/Expression

Crystal density ρs 73

kg/m3

1332

Shape factor kv

-

0.866

Solvent density ρsolvent 74

kg/m3

781

Solution density ρ sol 74

kg/m3

1912.4 2.448-c

Impurity SC1

-

0.1


11


kgsolute kg solvent

Solubility75

 T  c* = 0.1025    273 

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5.325

1.602

2.448c*   −4878   2.448c G = 138.7exp  −  *   T   2.448 − c 2.448 − c 

Crystal growth73

m /s

Primary nucleation76

m-3s-1

B0, prim = 2.662 ×108 ( c − c* )

Secondary nucleation77

m-3s-1

B0,sec = 2.656 × 10 7 ( c − c* )

1

2.276

2.232

m2

A lumped segregation coefficient is assumed for all impurities.71

2.2.1 Attainable region for crystal size Crystal size is one of the most important product quality characteristics for many pharmaceutical applications. Therefore, the attainable region for crystal size is investigated here for the cMAC process. The single-objective optimization problem for the attainable size of the cMAC process is formulated as:

mN ,4 mN ,3

max/ min τ i ,Ti , Ri

N

∑τ

subject to

i

( P1)

= τ total

( P1.1)

i =1

N

∑R = R

( P 1.2)

Ti ≤ Ti −1

( P1.3)

i

total

i =1

Ti ,M ≤ 340 ( P 1.4) Ti ≥ 273 ( P1.5) FN (1 − ε N ) ρ s ≥ 0.76 ( P1.6) Ffeed ρ sol , feed c feed mi ,3 ≤ 0.25 1 − xM ,impurity ≥ 0.98

( P 1.7) ( P1.8)

∀i = 1,2,K, M


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The volume-based mean size (i.e. m4 / m3 ) is selected as the representative size characteristic of which the boundaries of the attainable region can be identified by solving (P1). The total residence time ( τ total ) and the total amount of solvent removed over the membranes ( Rtotal ) are fixed. A monotonically decreasing temperature profile along the cascade of crystallizers is enforced for the cooling crystallization process to avoid dissolution in any crystallizer. Moreover, a maximum operating temperature of the membranes and minimum temperature in the crystallizers are specified. Yield is an important economic criterion for a continuous pharmaceutical crystallization process, which is specified by the solute recovery fraction in P1.6. The process yield is typically limited from a practical point of view by a high suspension density (P1.7) or by the requirement of the product purity (P1.8). The decision variables of the optimization problem (P1) are the residence time, the temperature and the solvent removal flow rate of each crystallizer. The remaining process parameters including the number of crystallizers in the cascade, the total residence time, the total flowrate of the removed solvent and the cutoff size ( Lcrit ) for the dissolution are fixed.

2.2.2 Design space A design space usually has a high dimension, which poses challenges for systematic analysis and visualization due to the curse of dimensionality. Therefore, only the two most important process variables are selected here. However, in principle, the optimization-based method presented in this work can be applied to higher-dimensional design spaces to account for all critical process parameters. The choice of critical process parameters to establish the design space usually requires systematically analysis via risk assessment tools.78-81 For illustrative purposes, a cMAC cascade consisting of two MSMPR crystallizers is selected. In addition, the


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residence time for each vessel is fixed at two hours and a membrane module is added to the second vessel only. Consequently, the temperature in the two crystallizers are the critical process parameters to construct the design space. The following optimization problems, which are similar to (P1), are formulated to identify the design space of both crystallizer temperatures:

T1（T2）

max/ min T2 （T1）

subject to

900 ≤

mN ,4 ≤ 1000 mN ,3

( P 2) ( P 2.1)

Ti ≤ Ti −1

( P 2.2)

Ti ≥ 273

( P 2.3)

Ti ,M ≤ 340

( P 2.4)

FN (1 − ε N ) ρ s ≥ 0.6 ( P 2.5) Ffeed ρ sol , feed c feed mi ,3 ≤ 0.25

( P 2.6)

1 − x2,impurity ≥ 0.98

( P 2.7)

∀i = 1,2 2.3 Numerical implementation All optimization problems are solved using the CONOPT solver82 as implemented in the General Algebraic Modeling System (GAMS)83 environment. For each optimization, first, a feasible solution is found via simulation. Subsequently, that solution is used to initialize the optimization solver (i.e., using the so-called “feasible-path” strategy). Furthermore, it is expected that any solution obtained may only be locally optimal due to the high nonlinearity of the model. Therefore, multiple random initial guesses are provided for each optimization case in an attempt to identify the global optimum. The crystal size domain considered in the discrete population balance model ranges from 0 µm to 3000 µm. The choice of the upper limit for the crystal size domain is case-specific, which should ensure that nearly all crystals are included. A typical crystal size distribution for paracetamol is exponentially decreasing in the large size range (see Figure S1 of supporting


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information). By inspecting such representative size distribution, it can be seen that an upper limit of 3000 µm should be sufficiently large to include all the crystals. Furthermore, the same value for an upper limit has also been used by others for similar crystal size distributions.19 Furthermore, different discretization schemes for the population balance have been tested to achieve desired solution accuracy (see Supporting Material for details). The results show that a logarithmic grid for the small size ranges and a uniform grid for the larger size ranges with a total of 350 discretization points gives the best performance, which is applied in this work. In the discrete population balance model, the numerical values of the variables typically differ by several orders of magnitude, which may prohibit successful convergence for optimization. Therefore, the model is scaled to avoid ill-conditioned optimization problems (see Supporting Material for details).

3. Results and Discussion 3.1. Influence of membranes on the attainable crystal size It is expected that membranes can enhance the attainable product mean size since solvent removal offers an additional mechanism for supersaturation generation in addition to cooling. Therefore, the attainable mean crystal size of cMAC should be at least equivalent to conventional continuous crystallization. Figure 2 shows the attainable volume-based mean crystal size for different continuous crystallization configurations with and without membranes. The cases without membranes are closely comparable to literature cases44 despite our different solution methodology of the population balance, which shows consistency for optimization of the full population balance when compared to using the method of moments. The attainable region can be enlarged with every additional crystallizer in the cascade, which, however, shows


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decreasing gains. Furthermore, the results demonstrate that membranes can significantly enlarge the attainable crystal size (red solid line in Figure 2), which illustrates the high potential of cMAC for more flexible operation. For all tested operating conditions, a higher mean crystal size can be achieved for a given residence time and a smaller residence time can be implemented for a given mean crystal size when using cMAC compared to conventional continuous crystallization. When comparing with the same cascade without membranes (blue dashed line in Figure 2), it is clear that the lower boundary of the attainable region remains nearly the same despite the existence of membranes. The optimal operating policies provide additional insights into the observed enlargement of the attainable region upon the addition of membranes. In order to achieve the minimum crystal mean size, all five crystallizers are operated at the lowest temperature (see Figure 3a) whether membranes are present or not, which results in a decreasing supersaturation profile along the cascade. This early growth operating policy84 provides the highest supersaturation at the beginning of the cascade resulting in a high nucleation rate. Due to the large number of nuclei formed in the beginning of the cascade, the solute will be depleted rapidly for further growth. Consequently, a large amount of small crystals is produced resulting in the minimum volumebased mean size. As shown in Figure 2, increasing the total process residence time or adding additional crystallizers has not much impact on the minimal attainable size, because crystals are formed mainly at the beginning of the process with a short residence time. Therefore, the lower boundary of the attainable region is mostly affected by crystallization kinetics and process constraints (e.g., supersaturation constraint44). The optimal operating policy includes solvent removal in the first vessel only, where nucleation is dominant due to the high supersaturation. Therefore, more solute will be consumed to form crystals at the beginning of the cMAC process


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compared to the conventional process. Although the solvent removal has little influence on the minimal attainable size, it does result in an increase in yield without violating constraints on purity. In contrast to the early-growth policy for achieving a small mean crystal size, an increasing supersaturation trajectory is optimal for achieving the maximum crystal mean size (see Figure 3b). In this late-growth scenario, less nuclei are formed due to the low driving force for nucleation in the early stage such that the solute will be consumed slowly and the initial nuclei can grow to a larger size. Clear differences in optimal operating policy are observed between cMAC and conventional continuous crystallization and, consequently, in the attainable mean crystal size. Since solvent removal offers an additional mechanism for supersaturation generation other than cooling, the system can be operated at a higher temperature (see Figure 3b) while still fulfilling the constraint on yield. Moreover, the vessels with a high temperature have more residence time in the optimal policy, because the growth rate of paracetamol is exponentially dependent on temperature. The solvent is removed only in the first vessel, so that more solute will be consumed in the growth dominant environment at higher temperature. The secondary nucleation can be further suppressed in the cMAC process at the same crystal yield, because the second moment of the crystal size distribution becomes smaller when the crystal mean size increases. In summary, the results show that the addition of membranes to a cascade of continuous crystallizers can significantly enhance flexibility to produce crystals with a larger mean size, whereas, any increase in flexibility to produce small crystals is negligible. Finally, a parameter sensitivity study is carried out to investigate the impact of fines dissolution on the attainable size region. As shown in Figure 4, the maximum attainable size shifts upwards as a function of the cutoff size ( Lcrit ) in the classification function and the total


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solvent removal ratio ( Rtotal / F feed ). An increase in both parameters will lead to a larger attainable crystal size, because more small crystals can be dissolved. Note that even a small Lcrit can already give a substantial rise to the attainable maximum size, which stresses the importance to utilize or at least consider the dissolution effect in cMAC processes.

Figure 2. Region of attainable mean crystal size for different cascades of MSMPR crystallizers. For the MSMPR cascade with membranes (red solid line), the total solvent removal ratio Rtotal / F feed is set to 0.2 and the dissolution effect is neglected by fixing Lcrit as 0 µm .


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Figure 3. Operating policies at the points marked in Figure 2. (a) Residence time, temperature and solvent removal policy to achieve the minimum mean crystal sizes marked by the red squares and blue diamonds in Figure 2. (b) Residence time, temperature and solvent removal policy to achieve the maximum mean sizes marked by the red dots and blue triangles.


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Figure 4. The influence of the total solvent removal ratio ( Rtotal / F feed ) and the cutoff size ( Lcrit ) on the attainable particle size in a cMAC process with 5 stages. The total residence time is 5.5 hours. 3.2. Influence of membranes on the design space of continuous crystallization The design space of a two-stage MSMPR cascade is identified by solving optimization problem (P2). The filled green regions in Figure 5 represent the design space of the conventional cascade without membranes (a) and the cMAC process (b). It is clear that membranes can significantly enlarge the design space. The constraints on yield (P2.5) and product mean size (P2.1) are active at the boundaries of the design space. Therefore, to understand the impact of the addition of membranes on the design space, the contour plots of the yield (blue dashed curves) and the mean size (red solid lines) are compared in Figure 5 for the process with and without membranes. It can be seen that in general the yield is mainly dependent on the temperature in the second crystallizer. This can be understood by the large residence time in the second vessel,


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which allows the system to approach the solid-liquid equilibrium. In contrast, the crystal size shows a more complicated temperature dependency, because of the interplay between various kinetic phenomena. For a low temperature in the first crystallizer ( T1 ≤ 315 K ), most of the crystals are formed in the first vessel, but there is insufficient driving force for crystal growth and nucleation in the second vessel due to the fast depletion of the solute in the first vessel. Therefore, the crystal volume-based mean size is mainly determined by temperature in the first crystallizer. For a higher temperature in the first crystallizer ( T1 > 315 K ), the crystallization behavior of the second vessel influences the crystal size most. When comparing the conventional cascade of crystallizers (Figure 5a) with the cMAC process Figure 5b, it can be seen that the contour lines representing the yield shift upward along the T2 axis for the cascade with membranes, because solvent can be removed to reach a higher yield. Therefore, for the same yield, a higher temperature in the second crystallizer is permissible. Furthermore, the addition of membranes also changes the size contour lines significantly compared to the conventional cascade. Due to the dissolution of fines in the second crystallizer, the crystal average size will increase, which explains why the contour lines representing the mean product size move downward along T2 axis. All of the aforementioned changes in the contour plot may result in a significant expansion of the design space for the selected cMAC process with the biggest contribution from the constraint on product mean size. However, the design space may not always be enlarged in cMAC process, e.g. the design space for crystals with size

Continuous Membrane-Assisted Crystallization To Increase the

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