CHEMICAL ENGINEERING TRANSACTIONS
VOL. 57, 2017
A publication of
The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet
Guest Editors: Sauro Pierucci, Jiří Jaromír Klemeš, Laura Piazza, Serafim Bakalis
Copyright © 2017, AIDIC Servizi S.r.l.
ISBN 978-88-95608- 48-8; ISSN 2283-9216
On-Line Model-Based Control of Crystal Size Distribution in
Cooling-Antisolvent Crystallization Processes
Navid Ghadipashaa, Jose A. Romagnoli*a, Stefania Troncib, Roberto Barattib
a Dept. of Chemical Engineering, Louisiana State University, South Stadium Road, Baton Rouge, LA 70803
b Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Universit_a degli Studi di Cagliari, Piazza D’Armi, I-09123,
Cagliari, Italy
jose@lsu.edu
This contribution deals with the formulation and implementation of different nonlinear model-based controllers
for controlling the crystal size distribution (CSD) in non-isothermal antisolvent crystallization processes. Three
different control algorithms are developed and tested experimentally. First, knowing the exact transfer function
of the crystal mean size and CSD variance as a function of antisolvent feed rate and temperature, an internal
model based controller is defined to achieve the desired CSD characteristics. Subsequently, by exploiting the
analytical solution of the CSD mean size, the geometric linearizing controller is developed to track the system
to the target. An alternative configuration of the geometric linearizing controller is also tested using an
observer based PI controller, which is more convenient for experimental implementation. Experimental
validation of the strategies is carried out for the ternary system of water-ethanol-sodium chloride.
1. Introduction
Crystallization is a widely used technique for the separation and purification of crystalline solid products. It is a
ubiquitous unit operation in many industries including food, pharmaceutical and fine chemicals. One of the
important structural property of the crystals which can largely influence the textural and physical
characteristics of the particles is the crystal size distribution (CSD). For example, the shape of the CSD can
have considerable impact on the product features such as dissolution rate, bulk densities and packing
properties.
A great deal of effort on controlling the CSD has been reported in the literature (e.g., Nagy and Braatz, 2012)
most of them using the population balance equations (PBE) for dynamic modelling of the crystallization
systems. One of the main issue using PBE is that it may result in complex dynamic models which cannot be
easily used for model-based control designs and real time implementation. As an alternative, it was recently
shown by Grosso et.al (2011) that it is possible to describe a crystallization process by means of a stochastic
approach which allows the obtainment of the CSD evolution with respect to time using the Fokker−Planck
equation. The proposed formulation provides a global model to describe the parameter dependence on
operation conditions (Cogoni et al., 2013) and recently such approach was generalized applying the noise
induced transition theory, which led to the obtainment of deterministic nonlinear differential equation
describing mean and mode of the CSD (Baratti et al., 2016).
In this contribution, different model based feedback control strategies were applied and compared to achieve a
specific particle size distribution in the crystallization of sodium chloride in water using ethanol as antisolvent.
The proposed methodology utilizes the benefits of combining cooling and antisolvent flow rate as manipulated
inputs, and a novel stochastic model to describe the time evolution of particle characteristics enables the
design of model based controllers and off-line determination of optimal trajectories.
DOI: 10.3303/CET1757181
Please cite this article as: Ghadipasha N., Romagnoli J., Tronci S., Baratti R., 2017, On-line model-based control of crystal size distribution in
cooling-antisolvent crystallization processes, Chemical Engineering Transactions, 57, 1081-1086 DOI: 10.3303/CET1757181
1081
2. Crystallization model
In a previous work (Baratti et al., 2017) the noise induced transition theory was applied to describe the
dynamic behaviour of CSD. Starting from the stochastic equation (1) for the crystal size L
𝑑𝐿(𝑡)
𝑑𝑡
= ℎ(𝐿, 𝑡) + 𝑔(𝐿, 𝑡)𝜂(𝑡) (1)
where h(L,t) is the deterministic growth term, g(L,t) is the random component which takes into account growth
fluctuations and the unknown dynamics, (t) is a random term assumed as Gaussian additive white noise, the
probability density distribution of L is given by the Fokker-Planck equation (2)
∂𝜓(𝐿,𝑡)
𝜕𝑡
=
𝜕
𝜕𝐿
{−[ℎ(𝐿) + 𝐷𝑔′(𝐿)𝑔(𝐿)]𝜓(𝐿, 𝑡) + 𝐷
𝜕
𝜕𝐿
𝑔2(𝐿)𝜓(𝐿, 𝑡)} (2)
with the following initial (2a) and boundary conditions (2b,c):
𝜓(L, t0) = 𝜓0(𝐿), 𝑡 = 𝑡0 (2a)
D {𝑔(𝐿)
𝜕
𝜕𝐿
𝑔(𝐿)𝜓(𝐿, 𝑡)} = ℎ(𝐿)𝜓(𝐿, 𝑡) at L=0, t > t0 (2b)
∂𝜓(𝐿,𝑡)
𝜕𝑡
= 0 at 𝐿 → ∞ and t > t0 (2c)
where g’(L) is the derivative with respect to the state L.
It is possible to demonstrate that the mean of the distribution can be approximated by (see Baratti et al., 2017)
𝑑�̂�
𝑑𝑡
= ℎ(�̂�) (3)
According to the previous studies (Cogoni et al., 2014; Baratti et al., 2017), the following functions are used to
describe the crystallization system
ℎ(𝐿) = 𝑟𝐿 (1 −
𝐿
𝐾
) , 𝑔(𝐿) = 𝐿 (4a,b)
𝑟(𝑞, 𝑇) = 𝛾0𝑟 + 𝛾1𝑟 𝑞 + 𝛾2𝑟 𝑇 (5)
𝐾(𝑞, 𝑇) = 𝛾0𝐾 + 𝛾1𝐾 𝑞
2 + 𝛾2𝐾 𝑇
2
𝐷(𝑞, 𝑇) = 𝛾0𝐷 + 𝛾1𝐷 𝑞 + 𝛾2𝐷 𝑇
2
with T and q indicating temperature and antisolvent flow rate, respectively, and the following values for the
model parameters
𝛾0𝑟 = 1.69312; 𝛾1𝑟 = 0.47074 ; 𝛾2𝑟 = −0.03831
𝛾0𝐾 = 141.7756; 𝛾1𝐾 = −3.11423; 𝛾2𝐾 = 0.02796
𝛾0𝐷 = 0.28094; 𝛾1𝐷 = 0.06687; 𝛾2𝐷 = −0.0002136
3. Controllers
Three different model based control strategies were applied to track the system to the desired CSD. The
antisolvent flow rate was used ad manipulated variable to control the mean size of the distribution, while
temperature followed a reference trajectory obtained off-line with an optimization procedure. An optimal mean
trajectory was also calculated and used as input for the internal model control.
3.1 Internal model control
A linear approximated model is obtained from Eq. (3) and (4a), using the relationships (5) calculating the
deviation from the reference trajectory. The model in Laplace Transform domain is reported in Eq. (6a)
�̅�(𝑠) =
𝐾𝑝
𝜏𝑠+1
�̅�(𝑠) (6a)
where
1082
𝜏 =
𝛾1,𝐾 𝑞
∗+𝛾0,𝐾
𝛾1,𝑟 𝑞
∗+𝛾0,𝑟
, 𝐾𝑝 = 𝜏 (
𝜕ℎ
𝜕𝑟
𝜕𝑟
𝜕𝑞
|
∗
+
𝜕ℎ
𝜕𝐾
𝜕𝐾
𝜕𝑞
|
∗
) (6b)
The asterisk indicates that the parameters are calculated at the reference trajectory. Eq. (6) can be used to
obtain the IMC controller (Garcia et al., 1989), reported in Eq. (7)
𝐺𝑐 =
𝜏𝑠+1
𝐾𝑝
1
𝜆𝑠+1
(7)
where is the tuning parameter and Kp and are adapted along the reference trajectory.
3.2 Linearizing control
Let consider a stable linear tracking error reference trajectory (Bastin and Dochain, 1990)
𝑑
𝑑𝑡
[�̂�∗(𝑡) − �̂�(𝑡)] = 𝜆1(�̂�
∗ − �̂�) + 𝜆2 ∫(�̂�
∗ − �̂�)𝑑𝑡 (8)
where the coefficients λiiare tuning coefficients which are to be chosen so that the differential equation (8)
is stable. The reference trajectory is reported in (9) where q* and T* are the inputs values leading to the
desired CSD in terms of mean and variance and they have been calculated solving the FPE (2) at steady state
conditions (Cogoni et al., 2014)
𝑑
𝑑𝑡
�̂�∗(𝑡) = 𝑟(𝑞∗, T∗)�̂�∗(𝑡) (1 −
�̂�∗(𝑡)
𝐾(𝑞∗,T∗)
) (9)
Substituting (9) and (3) with 4a and 5 in (8), at given temperature, it is possible to obtain the q value which
allows the system to follow the reference trajectory. Because the parameter dependence on q is nonlinear, the
equation (8) was solved numerically.
3.3 Observer-based controller
Let consider the linear model in deviation variables (10)
𝑑�̃�
𝑑𝑡
= aq̃ + b, a =
∂h
𝜕𝑞
|
0
(10)
where a is a constant and it is obtained evaluating the h derivative at nominal condition, and b considers the
modeling errors due to model linearization. Applying the reference trajectory (11)
𝑑�̃�
∗
𝑑𝑡
= K(𝐿 − 𝐿∗) = −𝐾�̃� (11)
where K is a positive tuning parameter, it is possible to calculate the value of �̃� from Eq. (10) satisfying (11)
−𝐾�̃� = aq̃ + b, q̃ = −𝐾�̃�−b
𝑎
(12)
The parameter b can be reconstructed through the first order observer in (13) as reported in Castellanos-
Sahagún (2005)
�̇̂� = 𝜔(𝑏 − �̂�) = 𝜔(�̇� − 𝑎�̃� − �̂�) with �̂�(0)=0 (13)
where is the observer gain. The coordinate change in (14) is applied to avoid noise sensitiveness due to the
derivative of 𝐿 with respct to time (�̇�).
𝜒 = (�̂� − 𝜔�̃�), �̇� = (�̇̂� − 𝜔�̇�) = −𝜔𝜒 − 𝜔(𝑎�̃� + 𝜔�̃�), 𝜒(0) = 0 (14)
4. Experimental set-up
In order to validate the proposed optimization and control strategies, crystallization of sodium chloride in water
using ethanol as antisolvent is considered as a case study. Sodium chloride (reagent plus 99%) (SIGMA-
ALDRICH, United States), 190 proof ethanol (VWR, United States) and only deionized water are used. The
reactor is made up of a 1 L glass, jacketed cylindrical crystallizer (Ace Glass Incorporated) which is
submerged into a temperature control bath. Temperature is measured using an RTD probe which is wired up
to a slave temperature control system (Thermo Fisher Scientefic) capable of heating and cooling. At the start
up condition, the crystallizer is loaded with 34 g of NaCl in 100 g of water. Ethanol is added to the solution at
different rates using a calibrated peristaltic pump (Masterflex Model 77200-60, Cole-Parmer). Mixing is
1083
provided by a propeller type agitator at the speed of 400 RPM. Along the experiment, particles are circulated
throw a second pump (Masterflex Model 77201-60, Cole-Parmer) into a cell where they are lighted up by an
illumination system. The cell provides a wide and thin layer of solution causing effective reflection from the
crystals which makes it suitable for taking pictures. Images are captured continuously using a USB
microscope camera (BASLER Model MD900) which fits into the side tube on the side of the microscope with
one of the supplied adapters and connects to a computer. Using image-based texture analysis which is
explained in more detail in Zhang et al. (2014), crystal size distribution at each sampling time is determined
and the results are sent to the data acquisition and control computer which can communicate to the slave
temperature and flow rate controllers using Labview (2009).
In a parallel measurement, on-line monitoring of the solute concentration was also implemented. This
additional measurement is not used specifically in the control algorithm but provide practical information
regarding the rate of crystal mass evolution. The same approached applied by Ghadipasha et.al (2015) is
used which is based on simultaneous measurement of conductivity, antisolvent mass fraction and
temperature. This procedure states an inferential measurement of concentration whereby the conductivity,
antisolvent mass fraction, and temperature are the primary measurements which are then translated into the
secondary inferred measurement being the concentration. Conductivity is monitored on-line using an Orion 4
Star probe model 018020 MD which is applicable for solution of high concentration. Figure 1 shows a
schematic representation of the experimental setup with the supervisory computer used to implement the CSD
control.
Figure 1: schematic representation of the experimental setup with the supervisory computer used to
implement the CSD control
5. Results
Initially the proposed mode-based controllers were tested through extensive simulation studies using the full
FPE model as the plant. For realistic purposes uncertainties were introduced by adding an error to the
antisolvent pump feeding the system. The final set-point was set equal to (µ,σ)=(132 µm,59.16 µm) and a
specified trajectory (optimal) for the mean size was defined as the desired objective. An error of -10% on the
initial flow rate was considered to insert some kind of realism to the simulation to fabricate the model-plant
mismatch. Also a comparative analysis was performed between the behaviours of the proposed alternative
controllers. Next, the proposed controllers were tested in the described experimental facilities using the same
target as in simulation studies. For this purpose alternative controller blocks were developed and implemented
within the experimental set-up already in place. Due to space limitation only the experimental results are
shown in this work. The obtained closed-loop trajectories for the main targets and input variables are shown in
Figures 2-4. Clearly all controllers achieve the target mean size with varying degree of performance. They are
able to control the mean size, however the antisolvent flow rate trajectories to achieve the desired set point
are different. Overall compared with the experimental results from Ghadipasha et.al (2015), the model-based
controllers outperform the PI controller being the alternative linearizing control implementation the one
showing the best performance. Smooth approach to target mean size and smooth control action can be
observed in all three cases. Also as expected, the antisolvente flow rate for each of the controllers reach an
asymptotic value very close to the one predicted from the asymptotic map (Ghadipasha et.al 2015). For all
controllers, the standard deviation approaches values around 70-75% of the desired target with the alternative
1084
implementation of the linearizing control showing a smoother trajectory. The different control algorithms
proposed will be also compared considering the CSD at the end of the batch. A number of images are taken
and manually analyzed to produce the corresponding histogram. This representation of the experimental
results requires that a very high number of crystals be considered to adequately describe the population of
crystals, with a prohibitive effort for the experimentalists. According to previous studies (Grosso et al. 2011), a
log-normal distribution is considered as best fit of the experimental data obtained at the end of the batch for
the different runs. This analysis was applied to calculate the final CSD to compare with the target values.
Figure 5 (left) provides a comparative analysis of the experimental CSD obtained at the end of the batch for
each controller. There is quite a good agreement in all cases between the final CSD and the desired target
with the linearizing controller having the best performance. The salt concentration data are also shown in
Figure 5 (right) for the three control configurations. Results show similar trends for all cases which can be
explained by the fact that all controllers attempt to produce a step change in the antisolvent flow rate toward
the asymptotic value as indicated in the operating map.
Figure 2: Results of the IMC controller
Figure 3: Results of the linearizing controller
Figure 4: Results of the observer-based controller
1085
Figure 5: End of the batch distribution and salt concentration for all the controller experiments
6. Conclusions
Different control strategies which take into account the nonlinear and time varying nature of the system were
applied to the non isothermal crystallization of sodium chloride in water with antisolvent. The proposed
algorithms were based on a novel model, which allowed the obtainment of a deterministic nonlinear differential
equation to describe the CSD mean time evolution as function of the model parameters. The proposed
strategies were applied to an experimental plant, furnished with a sensor to measure mean and variance of
the distribution. The model-based controllers outperformed the traditional PI controller reported in our previous
study, evidencing the importance of adapting the controller parameter along the system trajectories and the
efficiency of the proposed model.
Reference
Baratti R., Tronci S., Romagnoli J.A., 2017, A generalized stochastic modelling approach for crystal size
distribution in antisolvent crystallization operations, AIChE J, 63, 551–559.
Bastin, G., Dochain D., Eds. 1990, On-line estimation and adaptive control of bioreactors: Elsevier,
Amsterdam, the Netherlands
Castellanos-Sahagún E., Alvarez-Ramírez J., Alvarez J. , 2005, Two-point temperature control structure and
algorithm design for binary distillation columns Ind. Eng. Chem. Res., 44(1), 142-152.
Cogoni G., Tronci T., Mistretta G., Baratti R., Romagnoli J.A., 2013, Document Stochastic approach for the
prediction of PSD in nonisothermal antisolvent crystallization processes, AIChE J, 59(8), 2843-2851.
Cogoni, G., Tronci S., Baratti R., J.A. Romagnoli, 2014, Controllability of semibatch nonisothermal antisolvent
crystallization processes, Ind. Eng. Chem. Res., 53(17), 7056-7065.
Garcia C.E., Prett D.M., Morari M., 1989, Model predictive control: theory and practice—a survey, Automatica
25(3), 335-348.
Ghadipasha N., Romagnoli J.A., Tronci S., Baratti R.,2015, On‐line control of crystal properties in
nonisothermal antisolvent crystallization, AIChe J., 61(7), 2188-2201.
Grosso M., Cogoni G., Baratti R., Romagnoli J.A, 2011, Stochastic approach for the prediction of PSD in
crystallization processes: Formulation and comparative assessment of different stochastic models,
Industrial and Engineering Chemistry Research, 50(4), 2133-2143.
Nagy, Z.K., Braatz R.D., 2012, Advances and new directions in crystallization control, Annual review of
chemical and biomolecular engineering, 3, 55-75.
Zhang B., Willis R., Romagnoli J.A, Fois C., Tronci S., Baratti R, 2014, Image-Based Multiresolution-ANN
Approach for Online Particle Size Characterization, Ind. Eng. Chem. Res., 53, 7008-7018.
0 100 200 300 400 500 600
0
0.002
0.004
0.006
0.008
0.01
0.012
L [m]
P
r
o
b
a
b
il
it
y
d
e
n
s
it
y
f
u
n
c
t
io
n
IMC
Linearizing Controller
Observer-based controller
Target
1086