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Fuzzy logic and rigid control of a seeded semi-batch, anti-solvent, isothermal crystallizer
Chemical Engineering Science 63 (2008) 991 – 1002 www.elsevier.com/locate/ces
Fuzzy logic and rigid control of a seeded semi-batch, anti-solvent, isothermal crystallizer M. Sheikhzadeh, M. Trifkovic, S. Rohani ∗ Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ont., Canada N6A 5B9 Received 26 September 2006; received in revised form 5 October 2007; accepted 13 October 2007 Available online 11 December 2007
Abstract The objective of this study was to investigate the closed-loop control strategy for a batch crystallizer using two different control methods. The isothermal crystallization of paracetamol (PA) in isopropanol–water mixtures at 20 ◦ C was used as the model system. The annuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR) and the focus beam reflectance measurement (FBRM) were used for in situ measurement of the concentration of paracetamol and chord length counts, respectively. The controlled variables were the supersaturation and the difference in the chord length counts between two sampling intervals and the manipulated variable was the anti-solvent (water) flow rate. The rigid logic control (RLC) and the fuzzy logic control (FLC) were applied. The direct objective of this study was to maintain the controlled variables inside their predetermined ranges. The indirect objectives were minimization of the batch time and maximizing the final crystals size and they were closely related to the meeting of the direct objectives. The experiments conducted with FLC resulted in approximately 30% reduction of time of the experiment and an increase of 80 m in the final volume weighted mean size. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Semi-batch crystallization; Fuzzy logic control; In situ measurement; Supersaturation; Paracetamol
1. Introduction Crystallization from solution is an industrially important unit operation due to its ability to provide high purity separation. The crystal size distribution (CSD) is important in the production of high quality products and affects the efficiency of downstream operations, such as filtration and washing. Semi-batch crystallizers are heavily used in industry, and the majority of these crystallizers are seeded. It is well known that control of the supersaturation has a strong influence on the CSD and crystallization time (Chung et al., 1999). Recent improvement of technology for on-line measurement of concentration and size distribution have allowed for the control and optimization of crystallization processes. Recently, Zhou et al. (2006) and Yu et al. (2006) have proposed the feedback control strategy for controlling the supersaturation by manipulating the anti-solvent addition rate. ∗ Corresponding author.
E-mail address: [email protected] (S. Rohani). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.10.024
Fuzzy set theory has been used successfully in all the technical fields, including controls, modeling, image/signal processing and expert systems. The most successful and active field, however, is fuzzy control. The primary thrust of this control scheme is to utilize the human operator’s knowledge and experience to emulate human controller behavior. The important advantage of fuzzy controllers is that a mathematical model of the system to be controlled is not required, and a satisfactory nonlinear controller can often be developed empirically without complicated mathematics (Ying, 2000). Another advantage of fuzzy logic control (FLC) is an easy implementation and ability to control several controlled variables by less number of manipulated variables. This is extremely beneficial in processes where huge interaction between parameters exists. This study is the first comprehensive investigation of the use of a single input/multi output FLC of a semi-batch seeded crystallizer. The performance of the FLC was compared with a rigid logic controller (RLC), using a look-up table. The system considered was the anti-solvent (water) crystallization of paracetamol in isopropanol and water mixture at 20 ◦ C.
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The main aim of this work was to check the use of fuzzy logic to control the supersaturation and stabilize the number of particles in a specific size range, which were considered as direct objectives. The indirect objectives were improving the batch time and CSD. The FLC and RLC methods were used in the feedback control loop, which kept the controlled variables in their preset ranges. Due to the high effect of these variables on batch time and CSD, the controllers’ performances were reflected on achieving the indirect objectives as well.
defined as A. This fuzzy set is characterized by a membership function (A ) that maps A to a finite closed interval [0,1]. This correlation can be described by the following notation:
2. Theory
AND : A∩B (x) = min(A (x), B (x)), OR : A∪B (x) = max(A (x), B (x)),
2.1. Fuzzy controller theory FLC was initially conceptualized and implemented by Mamdani (Gupta and Qi, 1991) in a successful attempt to control a boiler steam engine. FLC can be viewed as a dynamic expert system involving a fuzzy rule base (FRB). The associated fuzzy inference engine (FIE) depicted in Fig. 1 is the tool for performing fuzzy inferences. Information from the system leads to fuzzy decision-making via the firing of rules in the FRB through the application of fuzzy set calculations. The decisions of the fuzzy expert system are then translated into control actions at the actuator level. A brief background to the fuzzy set theory and fuzzy calculus that forms the framework for the design and formulation of the fuzzy logic controller is presented below. 2.1.1. Fuzzy sets and operations Fuzzy controllers perform fuzzy-set operations. If we assume that U denotes the universe of discourse, the fuzzy set in U is
A : A → [0, 1].
(1)
Unlike the binary AND and OR operators whose operations are uniquely defined, the fuzzy counterparts are non-unique. To a large extent, the Zadeh fuzzy AND and OR operators have been found to be most useful for fuzzy control (Takagi and Sugeno, 1985). Their definitions are as follows: (2)
where A and B are fuzzy sets and A and B their membership functions. 2.1.2. Fuzzification Fuzzy control and modeling always involve a process called fuzzification at every sampling time. Fuzzification is a mathematical procedure for converting an element of U into the membership value of a fuzzy set. The result of fuzzification of x ∈ A is represented by A (x). 2.1.3. Fuzzy rules A fuzzy controller uses fuzzy rules, which are linguistic IF–THEN statements. Fuzzy rules link the input variables of fuzzy controllers to output variables. Based on Mamdani (Takagi and Sugeno, 1985), the IF–THEN fuzzy rule can be described as follows: IF (x1 is Ai ) AND (x2 is Bi ) THEN (u1 is Ci ),
Fig. 1. Block diagram of the experimental setup and the fuzzy logic controller.
(3)
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Fig. 2. Experimental setup.
where x1 and x2 are fuzzy input variables and u1 is fuzzy output variable. Ai , Bi and Ci are fuzzy subsets and AND is a fuzzy logic operator. 2.1.4. Fuzzy inference To define the relationship between fuzzy sets of inputs and outputs, fuzzy inference should be established. In Eq. (3), calculating the response to the IF clause is referred to as fuzzy inference. A number of fuzzy inference methods can be used to accomplish this task but only some of them are very popular in fuzzy control. A popular fuzzy inference method is the Mamdani minimum inference method. Eq. (3), in general, can be written:
2.1.5. Defuzzification Defuzzification is a mathematical process used to convert a fuzzy set or fuzzy sets to a real number. It is a necessary step in the implementation of a fuzzy controller. Perhaps the most popular defuzzification method is the centroid calculation, which returns the center of the area under a curve. Assume that the output variable of a fuzzy controller is u, and that evaluating N fuzzy rules using some fuzzy inference method produces N membership values, 1 , . . . , N , for N single output fuzzy sets. If these fuzzy sets are nonzero only at u = 1 , . . . , N , the generalized defuzzifier produces the following defuzzification result (Sarma, 2001):
IF x1 is S1 AND . . . AND xM is SM THEN u1 is W1 , . . . , zp is Wp , where xi (i = 1, . . . , M) is the vector of fuzzy input variables (process outputs) and uj (j = 1, . . . , P ) is the fuzzy output variables (process inputs). Si is the input fuzzy set and Wj is the output fuzzy set. Then, the Mamdani inference rule will be expressed as follows: RM = min(s , W (u)) for all u,
(5)
where RM denotes the Mamdani minimum inference method and is the final membership yielded by fuzzy logic AND operator in the rule antecedent.
(4) N
u=
k=1 k · k N , k=1 k
(6)
where is a design parameter that can vary in the interval 0 ∞. When = 1, the centroid defuzzifier is obtained. Before using defuzzifier, the resulting membership function of the output (W ) has to be calculated in terms of aggregation operator MAX and implication operator MIN (Carvalho and Durao, 2002).
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Fig. 3. FTIR spectra for seeded semi-batch crystallization: (a) during the entire crystallization experiment, (b) at one sampling time.
2.2. The rigid controller theory
3. Experimental setup
Sometimes, rigid information for process variables, constraints and operation conditions exist and can be used to develop a controller. By dividing the input and output values to logic sets, constant value of outputs can be obtained for each logic set of inputs. IF . . . THEN . . . structure can be applied for each logic sets of inputs. Eq. (7) shows the two possible logic structures used in this study
The experiments were performed in a 1-L glass jacketed vessel (Simular, Automated Chemical Reactors and Calorimeters; HEL Ltd., Vinelean, NJ). A Huber bath circulator was used for keeping the temperature inside the vessel constant throughout the crystallization process. This was performed by a two level cascade control system. A Teflon-coated thermocouple was used for reading the temperature in the flasks. A metering pump (Gala 1602) was used for pumping water, which was used as an anti-solvent in the crystallization process. The solution was stirred by a four-bladed propeller. An electronic balance (Oxford, B41002) was used for recording the amount of the anti-solvent added to the solution. The in situ focus beam reflectance measurement (FBRM) probe (Metler Toledo, Redmond, WA) was used to measure the chord length counts of the crystals during the process. An in situ annuated total reflectanceFourier transform infrared spectroscopy (ATR-FTIR) (Hamilton Sundstrand, CA and DMD-270 diamond ATR immersion probe) was used for on-line measurement of the solute concentration in the solution. The probes also served as baffles in
⎧ IF x1 is E1 AND . . . AND xM is EM1 THEN u1 = K, ⎪ ⎨ IF x1 is E2 AND . . . AND xM is EM2 THEN ⎪ ⎩ u1 = f (x1 , . . . , xM )
(7)
where x1 → xM are the inputs, E1 → EM1 , E2 → EM2 are the input conditions and u1 is the output which can be assigned either a constant value C, or be expressed as a function of inputs f (x1 , . . . , xM ).
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002 Table 1 The rigid logic control look-up decision table SS (gsolute /gsolvent )
N1.50
FR (g/min)
0.002 0.002–0.0025
N/A −10 < N1.50 < 10 < − 10 > 10
3 3 3 2.8
0.0025–0.003
> 10 < − 10 −10 < N1.50 < 10
2.6 2.1 2.8
0.003–0.005
−10 < N1-50 < 10 > 10 < − 10
−562.5 SS + 4.1 −562.5 SS + 3.9 −562.5 SS + 4.3
0.005–0.0055
−10 < N1.50 < 10 < − 10 > 10
0.9 1.1 0.95
0.0055–0.006
−10 < N1.50 < 10 < − 10 > 10
0.2 0.4 0.2
N/A
0.2
> 0.006
995
1245.8 cm−1 . The band associated with O.H stretching vibration was found at 1030 cm−1 . The band due to C–H stretching vibration was present at 1515.8 cm−1 . Table 2 shows various calibration curves obtained from correlating the heights of the peaks to the concentration of paracetamol. Although the first two multiple linear regression equations shown in Table 2 produced practically the same deviation from the experimental value, the second equation had better reproducibility. This result agreed with the calibration curve reported by Hojjati et al. (2007). The supersaturation was defined by: SS = C − C ∗ ,
(8)
where C is the measured concentration based on Table 2 and C ∗ is the solubility concentration, which was a function of the mass fraction of water in the water–isopropanol mixture at each temperature. For example at 20 ◦ C, C ∗ is shown in Fig. 4 and given by Hojjati and Rohani (2006): C ∗ = 2.3058m3w − 4.5757m2w + 2.5514mw − 0.2496,
(9)
where mw is the mass fraction of water and C ∗ is in gsolute /gsolvent(IPA+H2 O) . 3.3. Size measurement
Scheme 1. Paracetamol chemical structure.
the system. The experimental setup is shown in Fig. 2. CSD of the final product was measured by Malvern 2000 SCIRRCCO (Malvern Instruments Ltd., UK). 3.1. Materials Paracetamol (acetaminophen or 4-acetamidophenol) 98% was purchased from Aldrich Chemical Company Inc. (MO, USA). HPLC grade isopropanol was purchased from EMD Chemical Inc. (NJ, USA). Form 1 of paracetamol was used and no transformation to other forms was observed during the experiments. The final product was checked by XRPD to confirm the polymorphic composition. 3.2. Concentration measurement The ATR-FTIR was used to identify the presence of certain functional groups in paracetamol and isopropanol. Distinct changes in the spectra due to paracetamol crystallization can be observed in Fig. 3(a). Fig. 3(b) shows single spectra at one sampling time. The quantification of the ATR-FTIR spectra for paracetamol was based on the analysis of the height of four different peaks at 1030, 1245.8, 1515.8 and 1710 cm−1 . Scheme 1 depicts the chemical structure of paracetamol. The strongest band in the spectra was due to C&O stretching vibration at
The FBRM DL600 was used in the present study for in situ measurement of the chord length distribution (CLD). The chord length, or the distance between two edges in a particle, is obtained as the product of the beam speed and the time of the corresponding backscattered signal. The chord lengths acquired every 5 s were organized in 90 channels over the entire range between 1 and 1000 m. The average of 12 consecutive samples using a moving average (MA) was used as in the control algorithm. Therefore, the sampling interval of the control system was t = 1 min. Chord lengths counts in the 1.20 m range were defined to indicate the nucleation event. The difference between the chord length counts of two consecutive sampling intervals in the range 1.50 m was used as the second controlled variable. N1.50 (t) = N1.50 (t) − N1.50 (t − t).
(10)
3.4. Experimental procedure Before running the experiments, the ATR-FTIR chamber was purged for 12 h using filtered air (CO2 and H2 O free) to stabilize the background. The FTIR chamber was filled with liquid nitrogen as inert gas. The solution of 160 g of IPA and 240 g of water saturated with paracetamol at 20 ◦ C was added in the jacketed vessel. Stirrer speed was maintained at 200 rpm at the beginning of the crystallization process and then gradually increased to 300 rpm in order to keep the solids in suspension. This provided good mixing while avoiding excessive splashing. The clear solution was initially kept above the saturation temperature at 24 ◦ C. The temperature was then lowered to 20 ◦ C to start the experiment. At the onset of experiment, FBRM counts showed a nil value. Pre-weighed seeds were added to the crystallizer. The particle size distribution of the seed is shown in
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Table 2 The concentration calibration curve equations based on ATR-FTIR spectra Calibration curve equation C, gsolute /gsolvent(IPA+H2 O)
X definition
R 2 value
C = 0.1616X(1) + 0.9464X(2) + 0.0060
X(1) = P (1515.8) − P (1710) X(2) = P (1245.8) − P (1710)
0.9965
C = −0.3319X(1) + 2.1588X(2) + 0.0308
X(1) = P (1515.8) − P (1030) X(2) = P (1245.8) − P (1030)
0.9964
C = 0.0437X(1) − 0.0081X(2) − 0.0230
X(1) = P (1515.8)/P (1710) X(2) = P (1245.8)/P (1710)
0.9879
C = −0.0171X(1) + 0.1687X(2) − 0.1229
X(1) = P (1515.8)/P (1030) X(2) = P (1245.8)/P (1030)
0.9943
evaluated using the program written in fuzzy module. In the case of the rigid controller the output was calculated according to the several different scenarios written in another M-file. The outputs of Matlab were sent to the WinIso software, and the required flow rate of the anti-solvent was calculated and applied to the crystallization process.
0.2
0.15
0.1
5. The controller configuration
Experimental zone
Solubility (gsolute/gsolvent)
0.25
0.05
0 0
10
20
30
40
50
60
5.1. RLC configuration
70
80
90
100
Water mass percent (%) Fig. 4. Solubility data for paracetamol in water–isopropanol mixture [8].
Fig. 14. The addition of total of 40 g of the anti-solvent (water) in all experiments was initiated 30 min after the addition of the seed. All required data were scanned every minute during the process. At the end of each experiment the solution was withdrawn, filtered through a 1 m filter, dried, and then its size distribution was measured. 4. Implementation of the control system The architecture of the control system used in this study is depicted in Fig. 1. The WinIso data acquisition system was connected with both, the FBRM software and ATR-FTIR software. The WinIso outputs together with the data received from the ATR-FTIR and the FBRM, were printed to a text file. These data were then imported by MATLAB and the supersaturation (Eq. (6)) and the difference in the chord length counts (Eq. (8)) were calculated in an M-file. The flow rate of the anti-solvent, which was the fuzzy output of the control system, was calculated using two different control methods: a fuzzy logic and a RLC. The FLC algorithm was first created using Fuzzy Logic Toolbox and then imported in the same M-file. Thus, the supersaturation and the difference in the number of chord lengths were first computed according to the codes written in an M-file and then the output was exported and
The set of the open loop experiments were performed before designing the feedback controllers. They consisted of applying linear and nonlinear anti-solvent feed rate profiles to the system and measuring the corresponding SS and N1.50 . The ranges of SS and N1.50 with respect to FR were obtained based on the results gained from these sets of experiments. The specified range for each variable in the feedback controller was assigned as the zone of the ranges obtained from the above experiments. Then, this range was divided in different scenarios based on the dominance of the variable in the crystallization process. Table 1 shows the IF–THEN structures which were applied for different logic sets of inputs. Within each scenario of the SS, FR was assigned a value based on the value of N1.50 (below the minimum, above the maximum, or in the specified range). As seen from Table 1, the output flow rate was assigned a specific value when the supersaturation was near the limits of its range, otherwise the flow rate was calculated according to a proportional control law. The higher flow rate value was assigned when the supersaturation was very low, or the lower flow rate value when the supersaturation was very high. This was done in order to provide an extra force to the system, and consequently keep the supersaturation inside the specified range. Table 3 Maximum and minimum of input and output variables Membership function
Min
Max
Unit
SS N1.50 FR
0 −20 0.25
0.008 20 3
(gsolute /gsolvent(IPA+H2 O) ) number s−1 g min−1
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
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Fig. 5. Membership functions for superstauration, chord length counts difference and the anti-solvent flow rate.
The criteria in Table 1, which were acceptable in terms of the controller performance, resulted after performing several experiments with fewer scenarios or different values for inputs and output. 5.2. The FLC configuration The decision table for RLC (see Table 1) was the basis for designing the FLC. The membership functions (A ) involved were supersaturation (SS), difference in the number of chord lengths (N1.50 ) and the flow rate of anti-solvent
(FR). The range for each membership function is shown in Table 3. It was then necessary to choose the appropriate number of fuzzy sets and their division on the universe of the discourse of inputs and output. Fig. 5 depicts the optimal results for fuzzy inputs and fuzzy output membership functions. The overlapping of fuzzy sets was present for all membership functions, and represented the main advantage of using fuzzy logic algorithm in comparison to the RLC. The possibility of overlapping the fuzzy sets enabled the synergistic effect of neighboring scenarios, which consequently provided better flexibility of the
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Fig. 6. Representation of rules of fuzzy logic controller when both inputs are in the middle of their range.
Table 4 The fuzzy logic controller decision table
N1.50 (t)
UR L M H AR
Supersaturation LL
L
M
H
HH
HH HH HH HH HH
HH H H MU MU
MU MU M ML ML
ML L L LL LL
LL LL LL LL LL
UR: Under range, L: low M: medium, H: high, AR: above range, HH: high high, MU: medium up, ML: medium low, LL: low low.
controller. As seen in Fig. 5, the largest overlapping of fuzzy sets occurred for SS, at both extremes of its specified range. Fig. 6 depicts 19 rules of the fuzzy controller, and indicates the FR that corresponds to the case when SS and N1.50 are in the middle of their specified ranges. The rules were built using IF and THEN, AND and NOT logical operators. Some of the rules and their effects on the flow rate are shown in Table 4. As a consequence of the rules and fuzzy sets properties, the map surface was created. The map surface was the basis for the performance of the controller and is shown in Fig. 7. 6. Results and discussion The performance of the RLC and FLC was compared according to the indirect objectives which were minimization of the
batch time and obtaining better final CSD of the product, while maintaining the product yield of the crystallization process. The batch time was specified as the time necessary to increase the mass water fraction from 60% to 70%. This corresponded to the addition of 400 g of anti-solvent (water) to the crystallizer. It was very important to keep the supersaturation, which was one of the main objectives, in a specified range within the metastable zone. This range was estimated based on the open loop experiments. Fig. 8 shows the supersaturation profiles obtained with the FLC and RLC, respectively. Although both control methods managed to keep the SS in the specified range, the FLC method on average kept the SS closer to the middle of the specified range. This can be also seen in Fig. 9, which represents the concentration profiles throughout the experiments for the FLC and RLC, respectively. Another important objective in crystallization processes is to minimize nucleation rate and maximize the growth rate. In this study, the procedure for achieving this objective was proposed by controlling the supersaturation and considering the effect of number of particles in the control methodology. This objective was achieved in this study by maintaining the SS and the difference of chord length counts of 1.50 m between two sampling times in their predetermined ranges. The range for this controlled variable was determined based on the results obtained from the set of the open loop experiments as well. The main dependence of flow rate on Delta N is to prevent the occurrence of nucleation events even when the SS is in the desired
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
999
Fig. 7. Surface map for the fuzzy logic controller.
x 10-3 Solubility
9
Supersaturation
Upper Limit
Lower Limit
0.16 8
Concentration (gsolute/gsolvent)
ΔC=C-C* (gsolute/gsolvent)
0.15 7 6 5 4 3 2
0.14 0.13 0.12 0.11 0.1 0.09 0.08
1 100
120
140
160
180
200
220
240
260
280
300
100 120 140 160 180 200 220 240 260 280 300
Time (min)
Time (min)
x 10-3
0.15
ΔC=C-C* (gsolute/gsolvent)
8 7 6 5 4 3 2
Concentration (gsolute/gsolvent)
9
0.14 0.13 0.12 0.11 0.1 0.09
1
0.08 0 100 120 140 160 180 200 220 240 260 280 300 Time (min) Fig. 8. Supersaturation results using: (a) fuzzy logic controller, (b) rigid logic controller.
100 120 140 160 180 200 220 240 260 280 300 Time (min) Fig. 9. Concentration profiles using: (a) fuzzy logic controller, (b) rigid logic controller.
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0.75 Fuzzy Logic Control Rigid Logic Control
100-251 µm
500 400 300 200 100
Water fraction (gwater/gtotal solvent)
Number of particles (Number / s)
1-50 µm 600
0.7
0.65 Time difference
0.6 0 100 120 140 160 180 200 220 240 260 280 300
0
40
60
80
100 120 140 160 180 200
Time (min)
Time (min)
Fig. 10. Output from two FBRM channels throughout an experiment with the fuzzy logic controller.
20
Fig. 12. Comparison of the crystallization batch time required to add 40 g water using the fuzzy and rigid logic controllers.
15 3.5
10 3 Flow Rate (gsolvent/min)
ΔN1-50(t)
5 0 -5 -10
2.5 2 1.5 1 0.5
-15 100 120 140 160 180 200 220 240 260 280 300
0 100 120 140 160 180 200 220 240 260 280 300
Time (min)
Time (min)
15 3.5
10
ΔN1-50(t)
5
0
-5
-10
-15 100 120 140 160 180 200 220 240 260 280 300 Time (min)
Fig. 11. The difference of the chord length counts in two successive sampling intervals (1.50 m) using: (a) fuzzy logic controller, (b) rigid logic controller.
Flow Rate (gsolvent/min)
3 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 220 240 260 280 300 Time (min)
Fig. 13. The anti-solvent flow rate using: (a) fuzzy logic controller, (b) rigid logic controller.
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
range. Thus, if the number of particles starts increasing in the specific range, before the SS reaches the limits of the predetermined range, the FR is changed in order to keep it within the desired limits. Fig. 10 shows the output from two FBRM channels throughout the experiment with the FLC. As seen from the figure, the number of small crystals decreased and the number of larger crystals increased throughout the experiment. The difference in the number of chord length counts obtained by the FLC and RLC methods is shown in Figs. 11(a) and (b), respectively. Both controllers kept the number of particles in the desired range, although the FLC performed slightly better, since the FLC controller did not saturate at its limits. 8 Seed RLC FLC (run1) FLC (run2)
7
Volume %
6 5 4 3 2 1 0 0
200
400
600
800
1000
1200
Size (μm) Fig. 14. Comparison of seed and final size distributions obtained by the rigid logic and fuzzy logic controllers.
Table 5 Batch times for RLC and FLC controlled methods Experiment
Batch time RLC
Batch time FLC
1 2 3 4 5
195 200 204 197 211
135 138 140 150 143
Mean
201.4
141.2
1001
Fig. 12 depicts the time required for the two controllers to reach 70% water mass fraction. The FLC needed significantly shorter time (an hour) to reach the specified final mass water fraction while maintaining the product yield. This corresponded to approximately 30% reduction of time in comparison with the RLC. The seed loading expressed as ratio of seed mass with respect to the final product mass was equal to 0.11 and the yield of the process was approximately 51%. The result for yield would be improved if the operation was carried out in the wider solubility range. The flow rate of the anti-solvent (water) was the only input to the process. As shown in Fig. 13, the FLC resulted in smaller variations in the anti-solvent flow rate (FR) without reaching the saturation point. The higher average flow rate of the antisolvent for the FLC explains the shorter required crystallization time. This can be explained by the fact that FLC is based on the surface map as oppose to the rigid values for RLC, which in turn provides more flexibility of the controller over the prespecified range for FR. The CSD was performed after filtration and drying of the final product using the Malvern MasterSizer. The experiments conducted with the fuzzy logic controller resulted in the larger mean particle size in terms of the CSD vol%. Fig. 14 depicts the CSD for the seeds and the final product obtained with the controllers. The volume weighted mean sizes for seeds, RLC, and two runs with the FLC were 209.2, 256.1, 336.1, and 350.2 m, respectively. As mentioned above, the batch time for the FLC was considerably shorter. Table 5 shows batch times for five repeated experiments using both, FLC and RLC method. The ANOVA table for the comparison of means for batch times obtained by the two controlled methods is shown in Table 6. The large F value, which represents the ratio of variance between the two groups and the variance within the group, indicates that the means for batch times for two different controlled methods are different statistically. This significant decrease in a batch time and increase in crystal size can be attributed to the fact that nucleation events were minimized in the case of the FLC, and consequently the supersaturation was used more effectively for the growth of particles. As the supersaturation was used more for the growth, the higher flow rate was needed to make up for this loss in the driving force of the crystallization process. Hence, the experiments with the FLC resulted in the shorter
Table 6 ANOVA table for comparison of batch times means Source of variation
SS (sum of squares)
d.f.
MS = SS/d.f. mean square
F
F > Prob
Treatment (or between two controlledmethods) Error (or within each controlled method)
9.0601e + 003
1
9.0601e + 003
248.2219
2.6318e−007
292.0000
8
36.5000
9.3521e + 003
9
Total
d.f. is degrees of freedom.
–
–
–
–
–
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M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
temperature, ◦ C fuzzy output universe of discourse fuzzy input
batch time and larger particles. This also explains the increase in the volume weighted mean size of CSD for the FLC cases. The FLC method had slightly wider CSD and larger mean particle size in comparison with the RLC method. However, both FLC and RLC experiments result in different shape of CSD in comparison to one for the seeds, which indicates that some nucleation or agglomeration occurred during both experiments.
T u U x
7. Conclusions
References
This paper describes a real time control system of a semibatch crystallization process for the paracetamol–isopropanol– water system. The two controlled variables were the supersaturation and the difference in the chord length counts between two sampling intervals. The manipulated variable was the flow rate of the anti-solvent (water). A fuzzy logic and a rigid logic controller were used. The fuzzy logic controller showed better performance in terms of a shorter batch time and larger mean particle size of the final product.
Carvalho, M.T., Durao, F., 2002. Control of a floatation column using fuzzy logic inference. Fuzzy Sets and Systems 125, 121–133. Chung, S.H., Ma, D.L., Braatz, R.D., 1999. Optimal seeding in batch crystallization. The Canadian Journal of Chemical Engineering 77. Gupta, M.M., Qi, J., 1991. Theory of T-norms and fuzzy inference methods. Fuzzy sets and systems 40, 431–450. Hojjati, H., Rohani, S., 2006. Measurement and prediction of solubility of paracetamol in water–isopropanol mixture. Part 1: measurement. Organic Process Research and Development 10 (6), 1101–1109. Hojjati, H., Sheikhzadeh, M., Rohani, S., 2007. Control of supersaturation in a semi-batch antisolvent crystallization process using a fuzzy logic controller. Industrial and Engineering Chemistry Research 46 (4), 1232–1240. Sarma, P., 2001. Multivariable gain-scheduled fuzzy logic control of an exothermic reactor. Engineering Applications of Artificial Intelligence 14, 457–471. Takagi, T., Sugeno, M., 1985. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics 15, 116–132. Ying, H., 2000. Fuzzy Control and Modeling. IEEE Press Series, New York. Yu, Z.Q., Chow, P.S., Tan, R.B.H., 2006. Application of attenuated total reflectance-Fourier transform infrared (ATR-FTIR) technique in the monitoring and control of antisolvent crystallization. Industrial & Engineering Chemistry Research 45, 438–444. Zhou, G.X., Fujiwara, M., Woo, X.Y., Rusli, E., Tung, H., Starbuck, C., Davidson, O., Ge, Z., Braatz, R.D., 2006. Direct design of pharmaceutical antisolvent crystallization through concentration control. Crystal Growth and Design 6 (4), 892–898.
Notation A, B, C, S, W fuzzy subsets C actual solution concentration, gsolute /gsolvent C∗ saturated solution, concentration or solubility, gsolute /gsolvent FR flow rate, gwater / min K constant mw water mass fraction (solute free) Mw total water mass, g N1.50 the difference in chord length counts R real number SS supersaturation, gsolute /gsolvent
Greek letter
membership function
Fuzzy logic and rigid control of a seeded semi-batch, anti-solvent, isothermal crystallizer M. Sheikhzadeh, M. Trifkovic, S. Rohani ∗ Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ont., Canada N6A 5B9 Received 26 September 2006; received in revised form 5 October 2007; accepted 13 October 2007 Available online 11 December 2007
Abstract The objective of this study was to investigate the closed-loop control strategy for a batch crystallizer using two different control methods. The isothermal crystallization of paracetamol (PA) in isopropanol–water mixtures at 20 ◦ C was used as the model system. The annuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR) and the focus beam reflectance measurement (FBRM) were used for in situ measurement of the concentration of paracetamol and chord length counts, respectively. The controlled variables were the supersaturation and the difference in the chord length counts between two sampling intervals and the manipulated variable was the anti-solvent (water) flow rate. The rigid logic control (RLC) and the fuzzy logic control (FLC) were applied. The direct objective of this study was to maintain the controlled variables inside their predetermined ranges. The indirect objectives were minimization of the batch time and maximizing the final crystals size and they were closely related to the meeting of the direct objectives. The experiments conducted with FLC resulted in approximately 30% reduction of time of the experiment and an increase of 80 m in the final volume weighted mean size. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Semi-batch crystallization; Fuzzy logic control; In situ measurement; Supersaturation; Paracetamol
1. Introduction Crystallization from solution is an industrially important unit operation due to its ability to provide high purity separation. The crystal size distribution (CSD) is important in the production of high quality products and affects the efficiency of downstream operations, such as filtration and washing. Semi-batch crystallizers are heavily used in industry, and the majority of these crystallizers are seeded. It is well known that control of the supersaturation has a strong influence on the CSD and crystallization time (Chung et al., 1999). Recent improvement of technology for on-line measurement of concentration and size distribution have allowed for the control and optimization of crystallization processes. Recently, Zhou et al. (2006) and Yu et al. (2006) have proposed the feedback control strategy for controlling the supersaturation by manipulating the anti-solvent addition rate. ∗ Corresponding author.
E-mail address: [email protected] (S. Rohani). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.10.024
Fuzzy set theory has been used successfully in all the technical fields, including controls, modeling, image/signal processing and expert systems. The most successful and active field, however, is fuzzy control. The primary thrust of this control scheme is to utilize the human operator’s knowledge and experience to emulate human controller behavior. The important advantage of fuzzy controllers is that a mathematical model of the system to be controlled is not required, and a satisfactory nonlinear controller can often be developed empirically without complicated mathematics (Ying, 2000). Another advantage of fuzzy logic control (FLC) is an easy implementation and ability to control several controlled variables by less number of manipulated variables. This is extremely beneficial in processes where huge interaction between parameters exists. This study is the first comprehensive investigation of the use of a single input/multi output FLC of a semi-batch seeded crystallizer. The performance of the FLC was compared with a rigid logic controller (RLC), using a look-up table. The system considered was the anti-solvent (water) crystallization of paracetamol in isopropanol and water mixture at 20 ◦ C.
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The main aim of this work was to check the use of fuzzy logic to control the supersaturation and stabilize the number of particles in a specific size range, which were considered as direct objectives. The indirect objectives were improving the batch time and CSD. The FLC and RLC methods were used in the feedback control loop, which kept the controlled variables in their preset ranges. Due to the high effect of these variables on batch time and CSD, the controllers’ performances were reflected on achieving the indirect objectives as well.
defined as A. This fuzzy set is characterized by a membership function (A ) that maps A to a finite closed interval [0,1]. This correlation can be described by the following notation:
2. Theory
AND : A∩B (x) = min(A (x), B (x)), OR : A∪B (x) = max(A (x), B (x)),
2.1. Fuzzy controller theory FLC was initially conceptualized and implemented by Mamdani (Gupta and Qi, 1991) in a successful attempt to control a boiler steam engine. FLC can be viewed as a dynamic expert system involving a fuzzy rule base (FRB). The associated fuzzy inference engine (FIE) depicted in Fig. 1 is the tool for performing fuzzy inferences. Information from the system leads to fuzzy decision-making via the firing of rules in the FRB through the application of fuzzy set calculations. The decisions of the fuzzy expert system are then translated into control actions at the actuator level. A brief background to the fuzzy set theory and fuzzy calculus that forms the framework for the design and formulation of the fuzzy logic controller is presented below. 2.1.1. Fuzzy sets and operations Fuzzy controllers perform fuzzy-set operations. If we assume that U denotes the universe of discourse, the fuzzy set in U is
A : A → [0, 1].
(1)
Unlike the binary AND and OR operators whose operations are uniquely defined, the fuzzy counterparts are non-unique. To a large extent, the Zadeh fuzzy AND and OR operators have been found to be most useful for fuzzy control (Takagi and Sugeno, 1985). Their definitions are as follows: (2)
where A and B are fuzzy sets and A and B their membership functions. 2.1.2. Fuzzification Fuzzy control and modeling always involve a process called fuzzification at every sampling time. Fuzzification is a mathematical procedure for converting an element of U into the membership value of a fuzzy set. The result of fuzzification of x ∈ A is represented by A (x). 2.1.3. Fuzzy rules A fuzzy controller uses fuzzy rules, which are linguistic IF–THEN statements. Fuzzy rules link the input variables of fuzzy controllers to output variables. Based on Mamdani (Takagi and Sugeno, 1985), the IF–THEN fuzzy rule can be described as follows: IF (x1 is Ai ) AND (x2 is Bi ) THEN (u1 is Ci ),
Fig. 1. Block diagram of the experimental setup and the fuzzy logic controller.
(3)
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Fig. 2. Experimental setup.
where x1 and x2 are fuzzy input variables and u1 is fuzzy output variable. Ai , Bi and Ci are fuzzy subsets and AND is a fuzzy logic operator. 2.1.4. Fuzzy inference To define the relationship between fuzzy sets of inputs and outputs, fuzzy inference should be established. In Eq. (3), calculating the response to the IF clause is referred to as fuzzy inference. A number of fuzzy inference methods can be used to accomplish this task but only some of them are very popular in fuzzy control. A popular fuzzy inference method is the Mamdani minimum inference method. Eq. (3), in general, can be written:
2.1.5. Defuzzification Defuzzification is a mathematical process used to convert a fuzzy set or fuzzy sets to a real number. It is a necessary step in the implementation of a fuzzy controller. Perhaps the most popular defuzzification method is the centroid calculation, which returns the center of the area under a curve. Assume that the output variable of a fuzzy controller is u, and that evaluating N fuzzy rules using some fuzzy inference method produces N membership values, 1 , . . . , N , for N single output fuzzy sets. If these fuzzy sets are nonzero only at u = 1 , . . . , N , the generalized defuzzifier produces the following defuzzification result (Sarma, 2001):
IF x1 is S1 AND . . . AND xM is SM THEN u1 is W1 , . . . , zp is Wp , where xi (i = 1, . . . , M) is the vector of fuzzy input variables (process outputs) and uj (j = 1, . . . , P ) is the fuzzy output variables (process inputs). Si is the input fuzzy set and Wj is the output fuzzy set. Then, the Mamdani inference rule will be expressed as follows: RM = min(s , W (u)) for all u,
(5)
where RM denotes the Mamdani minimum inference method and is the final membership yielded by fuzzy logic AND operator in the rule antecedent.
(4) N
u=
k=1 k · k N , k=1 k
(6)
where is a design parameter that can vary in the interval 0 ∞. When = 1, the centroid defuzzifier is obtained. Before using defuzzifier, the resulting membership function of the output (W ) has to be calculated in terms of aggregation operator MAX and implication operator MIN (Carvalho and Durao, 2002).
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Fig. 3. FTIR spectra for seeded semi-batch crystallization: (a) during the entire crystallization experiment, (b) at one sampling time.
2.2. The rigid controller theory
3. Experimental setup
Sometimes, rigid information for process variables, constraints and operation conditions exist and can be used to develop a controller. By dividing the input and output values to logic sets, constant value of outputs can be obtained for each logic set of inputs. IF . . . THEN . . . structure can be applied for each logic sets of inputs. Eq. (7) shows the two possible logic structures used in this study
The experiments were performed in a 1-L glass jacketed vessel (Simular, Automated Chemical Reactors and Calorimeters; HEL Ltd., Vinelean, NJ). A Huber bath circulator was used for keeping the temperature inside the vessel constant throughout the crystallization process. This was performed by a two level cascade control system. A Teflon-coated thermocouple was used for reading the temperature in the flasks. A metering pump (Gala 1602) was used for pumping water, which was used as an anti-solvent in the crystallization process. The solution was stirred by a four-bladed propeller. An electronic balance (Oxford, B41002) was used for recording the amount of the anti-solvent added to the solution. The in situ focus beam reflectance measurement (FBRM) probe (Metler Toledo, Redmond, WA) was used to measure the chord length counts of the crystals during the process. An in situ annuated total reflectanceFourier transform infrared spectroscopy (ATR-FTIR) (Hamilton Sundstrand, CA and DMD-270 diamond ATR immersion probe) was used for on-line measurement of the solute concentration in the solution. The probes also served as baffles in
⎧ IF x1 is E1 AND . . . AND xM is EM1 THEN u1 = K, ⎪ ⎨ IF x1 is E2 AND . . . AND xM is EM2 THEN ⎪ ⎩ u1 = f (x1 , . . . , xM )
(7)
where x1 → xM are the inputs, E1 → EM1 , E2 → EM2 are the input conditions and u1 is the output which can be assigned either a constant value C, or be expressed as a function of inputs f (x1 , . . . , xM ).
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002 Table 1 The rigid logic control look-up decision table SS (gsolute /gsolvent )
N1.50
FR (g/min)
0.002 0.002–0.0025
N/A −10 < N1.50 < 10 < − 10 > 10
3 3 3 2.8
0.0025–0.003
> 10 < − 10 −10 < N1.50 < 10
2.6 2.1 2.8
0.003–0.005
−10 < N1-50 < 10 > 10 < − 10
−562.5 SS + 4.1 −562.5 SS + 3.9 −562.5 SS + 4.3
0.005–0.0055
−10 < N1.50 < 10 < − 10 > 10
0.9 1.1 0.95
0.0055–0.006
−10 < N1.50 < 10 < − 10 > 10
0.2 0.4 0.2
N/A
0.2
> 0.006
995
1245.8 cm−1 . The band associated with O.H stretching vibration was found at 1030 cm−1 . The band due to C–H stretching vibration was present at 1515.8 cm−1 . Table 2 shows various calibration curves obtained from correlating the heights of the peaks to the concentration of paracetamol. Although the first two multiple linear regression equations shown in Table 2 produced practically the same deviation from the experimental value, the second equation had better reproducibility. This result agreed with the calibration curve reported by Hojjati et al. (2007). The supersaturation was defined by: SS = C − C ∗ ,
(8)
where C is the measured concentration based on Table 2 and C ∗ is the solubility concentration, which was a function of the mass fraction of water in the water–isopropanol mixture at each temperature. For example at 20 ◦ C, C ∗ is shown in Fig. 4 and given by Hojjati and Rohani (2006): C ∗ = 2.3058m3w − 4.5757m2w + 2.5514mw − 0.2496,
(9)
where mw is the mass fraction of water and C ∗ is in gsolute /gsolvent(IPA+H2 O) . 3.3. Size measurement
Scheme 1. Paracetamol chemical structure.
the system. The experimental setup is shown in Fig. 2. CSD of the final product was measured by Malvern 2000 SCIRRCCO (Malvern Instruments Ltd., UK). 3.1. Materials Paracetamol (acetaminophen or 4-acetamidophenol) 98% was purchased from Aldrich Chemical Company Inc. (MO, USA). HPLC grade isopropanol was purchased from EMD Chemical Inc. (NJ, USA). Form 1 of paracetamol was used and no transformation to other forms was observed during the experiments. The final product was checked by XRPD to confirm the polymorphic composition. 3.2. Concentration measurement The ATR-FTIR was used to identify the presence of certain functional groups in paracetamol and isopropanol. Distinct changes in the spectra due to paracetamol crystallization can be observed in Fig. 3(a). Fig. 3(b) shows single spectra at one sampling time. The quantification of the ATR-FTIR spectra for paracetamol was based on the analysis of the height of four different peaks at 1030, 1245.8, 1515.8 and 1710 cm−1 . Scheme 1 depicts the chemical structure of paracetamol. The strongest band in the spectra was due to C&O stretching vibration at
The FBRM DL600 was used in the present study for in situ measurement of the chord length distribution (CLD). The chord length, or the distance between two edges in a particle, is obtained as the product of the beam speed and the time of the corresponding backscattered signal. The chord lengths acquired every 5 s were organized in 90 channels over the entire range between 1 and 1000 m. The average of 12 consecutive samples using a moving average (MA) was used as in the control algorithm. Therefore, the sampling interval of the control system was t = 1 min. Chord lengths counts in the 1.20 m range were defined to indicate the nucleation event. The difference between the chord length counts of two consecutive sampling intervals in the range 1.50 m was used as the second controlled variable. N1.50 (t) = N1.50 (t) − N1.50 (t − t).
(10)
3.4. Experimental procedure Before running the experiments, the ATR-FTIR chamber was purged for 12 h using filtered air (CO2 and H2 O free) to stabilize the background. The FTIR chamber was filled with liquid nitrogen as inert gas. The solution of 160 g of IPA and 240 g of water saturated with paracetamol at 20 ◦ C was added in the jacketed vessel. Stirrer speed was maintained at 200 rpm at the beginning of the crystallization process and then gradually increased to 300 rpm in order to keep the solids in suspension. This provided good mixing while avoiding excessive splashing. The clear solution was initially kept above the saturation temperature at 24 ◦ C. The temperature was then lowered to 20 ◦ C to start the experiment. At the onset of experiment, FBRM counts showed a nil value. Pre-weighed seeds were added to the crystallizer. The particle size distribution of the seed is shown in
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Table 2 The concentration calibration curve equations based on ATR-FTIR spectra Calibration curve equation C, gsolute /gsolvent(IPA+H2 O)
X definition
R 2 value
C = 0.1616X(1) + 0.9464X(2) + 0.0060
X(1) = P (1515.8) − P (1710) X(2) = P (1245.8) − P (1710)
0.9965
C = −0.3319X(1) + 2.1588X(2) + 0.0308
X(1) = P (1515.8) − P (1030) X(2) = P (1245.8) − P (1030)
0.9964
C = 0.0437X(1) − 0.0081X(2) − 0.0230
X(1) = P (1515.8)/P (1710) X(2) = P (1245.8)/P (1710)
0.9879
C = −0.0171X(1) + 0.1687X(2) − 0.1229
X(1) = P (1515.8)/P (1030) X(2) = P (1245.8)/P (1030)
0.9943
evaluated using the program written in fuzzy module. In the case of the rigid controller the output was calculated according to the several different scenarios written in another M-file. The outputs of Matlab were sent to the WinIso software, and the required flow rate of the anti-solvent was calculated and applied to the crystallization process.
0.2
0.15
0.1
5. The controller configuration
Experimental zone
Solubility (gsolute/gsolvent)
0.25
0.05
0 0
10
20
30
40
50
60
5.1. RLC configuration
70
80
90
100
Water mass percent (%) Fig. 4. Solubility data for paracetamol in water–isopropanol mixture [8].
Fig. 14. The addition of total of 40 g of the anti-solvent (water) in all experiments was initiated 30 min after the addition of the seed. All required data were scanned every minute during the process. At the end of each experiment the solution was withdrawn, filtered through a 1 m filter, dried, and then its size distribution was measured. 4. Implementation of the control system The architecture of the control system used in this study is depicted in Fig. 1. The WinIso data acquisition system was connected with both, the FBRM software and ATR-FTIR software. The WinIso outputs together with the data received from the ATR-FTIR and the FBRM, were printed to a text file. These data were then imported by MATLAB and the supersaturation (Eq. (6)) and the difference in the chord length counts (Eq. (8)) were calculated in an M-file. The flow rate of the anti-solvent, which was the fuzzy output of the control system, was calculated using two different control methods: a fuzzy logic and a RLC. The FLC algorithm was first created using Fuzzy Logic Toolbox and then imported in the same M-file. Thus, the supersaturation and the difference in the number of chord lengths were first computed according to the codes written in an M-file and then the output was exported and
The set of the open loop experiments were performed before designing the feedback controllers. They consisted of applying linear and nonlinear anti-solvent feed rate profiles to the system and measuring the corresponding SS and N1.50 . The ranges of SS and N1.50 with respect to FR were obtained based on the results gained from these sets of experiments. The specified range for each variable in the feedback controller was assigned as the zone of the ranges obtained from the above experiments. Then, this range was divided in different scenarios based on the dominance of the variable in the crystallization process. Table 1 shows the IF–THEN structures which were applied for different logic sets of inputs. Within each scenario of the SS, FR was assigned a value based on the value of N1.50 (below the minimum, above the maximum, or in the specified range). As seen from Table 1, the output flow rate was assigned a specific value when the supersaturation was near the limits of its range, otherwise the flow rate was calculated according to a proportional control law. The higher flow rate value was assigned when the supersaturation was very low, or the lower flow rate value when the supersaturation was very high. This was done in order to provide an extra force to the system, and consequently keep the supersaturation inside the specified range. Table 3 Maximum and minimum of input and output variables Membership function
Min
Max
Unit
SS N1.50 FR
0 −20 0.25
0.008 20 3
(gsolute /gsolvent(IPA+H2 O) ) number s−1 g min−1
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997
Fig. 5. Membership functions for superstauration, chord length counts difference and the anti-solvent flow rate.
The criteria in Table 1, which were acceptable in terms of the controller performance, resulted after performing several experiments with fewer scenarios or different values for inputs and output. 5.2. The FLC configuration The decision table for RLC (see Table 1) was the basis for designing the FLC. The membership functions (A ) involved were supersaturation (SS), difference in the number of chord lengths (N1.50 ) and the flow rate of anti-solvent
(FR). The range for each membership function is shown in Table 3. It was then necessary to choose the appropriate number of fuzzy sets and their division on the universe of the discourse of inputs and output. Fig. 5 depicts the optimal results for fuzzy inputs and fuzzy output membership functions. The overlapping of fuzzy sets was present for all membership functions, and represented the main advantage of using fuzzy logic algorithm in comparison to the RLC. The possibility of overlapping the fuzzy sets enabled the synergistic effect of neighboring scenarios, which consequently provided better flexibility of the
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Fig. 6. Representation of rules of fuzzy logic controller when both inputs are in the middle of their range.
Table 4 The fuzzy logic controller decision table
N1.50 (t)
UR L M H AR
Supersaturation LL
L
M
H
HH
HH HH HH HH HH
HH H H MU MU
MU MU M ML ML
ML L L LL LL
LL LL LL LL LL
UR: Under range, L: low M: medium, H: high, AR: above range, HH: high high, MU: medium up, ML: medium low, LL: low low.
controller. As seen in Fig. 5, the largest overlapping of fuzzy sets occurred for SS, at both extremes of its specified range. Fig. 6 depicts 19 rules of the fuzzy controller, and indicates the FR that corresponds to the case when SS and N1.50 are in the middle of their specified ranges. The rules were built using IF and THEN, AND and NOT logical operators. Some of the rules and their effects on the flow rate are shown in Table 4. As a consequence of the rules and fuzzy sets properties, the map surface was created. The map surface was the basis for the performance of the controller and is shown in Fig. 7. 6. Results and discussion The performance of the RLC and FLC was compared according to the indirect objectives which were minimization of the
batch time and obtaining better final CSD of the product, while maintaining the product yield of the crystallization process. The batch time was specified as the time necessary to increase the mass water fraction from 60% to 70%. This corresponded to the addition of 400 g of anti-solvent (water) to the crystallizer. It was very important to keep the supersaturation, which was one of the main objectives, in a specified range within the metastable zone. This range was estimated based on the open loop experiments. Fig. 8 shows the supersaturation profiles obtained with the FLC and RLC, respectively. Although both control methods managed to keep the SS in the specified range, the FLC method on average kept the SS closer to the middle of the specified range. This can be also seen in Fig. 9, which represents the concentration profiles throughout the experiments for the FLC and RLC, respectively. Another important objective in crystallization processes is to minimize nucleation rate and maximize the growth rate. In this study, the procedure for achieving this objective was proposed by controlling the supersaturation and considering the effect of number of particles in the control methodology. This objective was achieved in this study by maintaining the SS and the difference of chord length counts of 1.50 m between two sampling times in their predetermined ranges. The range for this controlled variable was determined based on the results obtained from the set of the open loop experiments as well. The main dependence of flow rate on Delta N is to prevent the occurrence of nucleation events even when the SS is in the desired
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
999
Fig. 7. Surface map for the fuzzy logic controller.
x 10-3 Solubility
9
Supersaturation
Upper Limit
Lower Limit
0.16 8
Concentration (gsolute/gsolvent)
ΔC=C-C* (gsolute/gsolvent)
0.15 7 6 5 4 3 2
0.14 0.13 0.12 0.11 0.1 0.09 0.08
1 100
120
140
160
180
200
220
240
260
280
300
100 120 140 160 180 200 220 240 260 280 300
Time (min)
Time (min)
x 10-3
0.15
ΔC=C-C* (gsolute/gsolvent)
8 7 6 5 4 3 2
Concentration (gsolute/gsolvent)
9
0.14 0.13 0.12 0.11 0.1 0.09
1
0.08 0 100 120 140 160 180 200 220 240 260 280 300 Time (min) Fig. 8. Supersaturation results using: (a) fuzzy logic controller, (b) rigid logic controller.
100 120 140 160 180 200 220 240 260 280 300 Time (min) Fig. 9. Concentration profiles using: (a) fuzzy logic controller, (b) rigid logic controller.
1000
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002 700
0.75 Fuzzy Logic Control Rigid Logic Control
100-251 µm
500 400 300 200 100
Water fraction (gwater/gtotal solvent)
Number of particles (Number / s)
1-50 µm 600
0.7
0.65 Time difference
0.6 0 100 120 140 160 180 200 220 240 260 280 300
0
40
60
80
100 120 140 160 180 200
Time (min)
Time (min)
Fig. 10. Output from two FBRM channels throughout an experiment with the fuzzy logic controller.
20
Fig. 12. Comparison of the crystallization batch time required to add 40 g water using the fuzzy and rigid logic controllers.
15 3.5
10 3 Flow Rate (gsolvent/min)
ΔN1-50(t)
5 0 -5 -10
2.5 2 1.5 1 0.5
-15 100 120 140 160 180 200 220 240 260 280 300
0 100 120 140 160 180 200 220 240 260 280 300
Time (min)
Time (min)
15 3.5
10
ΔN1-50(t)
5
0
-5
-10
-15 100 120 140 160 180 200 220 240 260 280 300 Time (min)
Fig. 11. The difference of the chord length counts in two successive sampling intervals (1.50 m) using: (a) fuzzy logic controller, (b) rigid logic controller.
Flow Rate (gsolvent/min)
3 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 220 240 260 280 300 Time (min)
Fig. 13. The anti-solvent flow rate using: (a) fuzzy logic controller, (b) rigid logic controller.
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
range. Thus, if the number of particles starts increasing in the specific range, before the SS reaches the limits of the predetermined range, the FR is changed in order to keep it within the desired limits. Fig. 10 shows the output from two FBRM channels throughout the experiment with the FLC. As seen from the figure, the number of small crystals decreased and the number of larger crystals increased throughout the experiment. The difference in the number of chord length counts obtained by the FLC and RLC methods is shown in Figs. 11(a) and (b), respectively. Both controllers kept the number of particles in the desired range, although the FLC performed slightly better, since the FLC controller did not saturate at its limits. 8 Seed RLC FLC (run1) FLC (run2)
7
Volume %
6 5 4 3 2 1 0 0
200
400
600
800
1000
1200
Size (μm) Fig. 14. Comparison of seed and final size distributions obtained by the rigid logic and fuzzy logic controllers.
Table 5 Batch times for RLC and FLC controlled methods Experiment
Batch time RLC
Batch time FLC
1 2 3 4 5
195 200 204 197 211
135 138 140 150 143
Mean
201.4
141.2
1001
Fig. 12 depicts the time required for the two controllers to reach 70% water mass fraction. The FLC needed significantly shorter time (an hour) to reach the specified final mass water fraction while maintaining the product yield. This corresponded to approximately 30% reduction of time in comparison with the RLC. The seed loading expressed as ratio of seed mass with respect to the final product mass was equal to 0.11 and the yield of the process was approximately 51%. The result for yield would be improved if the operation was carried out in the wider solubility range. The flow rate of the anti-solvent (water) was the only input to the process. As shown in Fig. 13, the FLC resulted in smaller variations in the anti-solvent flow rate (FR) without reaching the saturation point. The higher average flow rate of the antisolvent for the FLC explains the shorter required crystallization time. This can be explained by the fact that FLC is based on the surface map as oppose to the rigid values for RLC, which in turn provides more flexibility of the controller over the prespecified range for FR. The CSD was performed after filtration and drying of the final product using the Malvern MasterSizer. The experiments conducted with the fuzzy logic controller resulted in the larger mean particle size in terms of the CSD vol%. Fig. 14 depicts the CSD for the seeds and the final product obtained with the controllers. The volume weighted mean sizes for seeds, RLC, and two runs with the FLC were 209.2, 256.1, 336.1, and 350.2 m, respectively. As mentioned above, the batch time for the FLC was considerably shorter. Table 5 shows batch times for five repeated experiments using both, FLC and RLC method. The ANOVA table for the comparison of means for batch times obtained by the two controlled methods is shown in Table 6. The large F value, which represents the ratio of variance between the two groups and the variance within the group, indicates that the means for batch times for two different controlled methods are different statistically. This significant decrease in a batch time and increase in crystal size can be attributed to the fact that nucleation events were minimized in the case of the FLC, and consequently the supersaturation was used more effectively for the growth of particles. As the supersaturation was used more for the growth, the higher flow rate was needed to make up for this loss in the driving force of the crystallization process. Hence, the experiments with the FLC resulted in the shorter
Table 6 ANOVA table for comparison of batch times means Source of variation
SS (sum of squares)
d.f.
MS = SS/d.f. mean square
F
F > Prob
Treatment (or between two controlledmethods) Error (or within each controlled method)
9.0601e + 003
1
9.0601e + 003
248.2219
2.6318e−007
292.0000
8
36.5000
9.3521e + 003
9
Total
d.f. is degrees of freedom.
–
–
–
–
–
1002
M. Sheikhzadeh et al. / Chemical Engineering Science 63 (2008) 991 – 1002
temperature, ◦ C fuzzy output universe of discourse fuzzy input
batch time and larger particles. This also explains the increase in the volume weighted mean size of CSD for the FLC cases. The FLC method had slightly wider CSD and larger mean particle size in comparison with the RLC method. However, both FLC and RLC experiments result in different shape of CSD in comparison to one for the seeds, which indicates that some nucleation or agglomeration occurred during both experiments.
T u U x
7. Conclusions
References
This paper describes a real time control system of a semibatch crystallization process for the paracetamol–isopropanol– water system. The two controlled variables were the supersaturation and the difference in the chord length counts between two sampling intervals. The manipulated variable was the flow rate of the anti-solvent (water). A fuzzy logic and a rigid logic controller were used. The fuzzy logic controller showed better performance in terms of a shorter batch time and larger mean particle size of the final product.
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Notation A, B, C, S, W fuzzy subsets C actual solution concentration, gsolute /gsolvent C∗ saturated solution, concentration or solubility, gsolute /gsolvent FR flow rate, gwater / min K constant mw water mass fraction (solute free) Mw total water mass, g N1.50 the difference in chord length counts R real number SS supersaturation, gsolute /gsolvent
Greek letter
membership function