(730e) Dynamic Modeling and Control of Multi-Stage Slug-Flow Crystallization

The
design of robust pharmaceutical crystallization processes has been of high
interest in academia and industry for the last decade (Nagy and Braatz, 2012).
Continuous-flow tubular crystallizers have shown potential for high
reproducibility and process efficiency at low capital and production cost
(e.g., Alvarez & Myerson, 2010; Lawton et al., 2009; Vacassy
et al., 2000). A recent advance that combines the advantages of continuous and
batch crystallizers is the air/liquid slug-flow crystallizer (Eder et al.,
2010; 2011; 2012; Jiang et al., 2014; 2015) which has unique advantages
including narrow residence time distribution, no stirrer generating secondary
nucleation, and easy post-crystallization separation. The potential application
of slug-flow crystallization in the final stage of pharmaceutical manufacturing
is the focus of this design study.

Similar
to batch crystallization, the design of a robust slug-flow crystallization
process comes from a good understanding of how the supersaturation
profile is affected by the design variables, such as the method and rate of supersaturation generation, and the corresponding
implementation. For example, supersaturation should
be minimized so as to avoid impurity incorporation and secondary nucleation
(e.g., Jiang et al., 2012; 2014), which is important for slug-flow
crystallization with its possibly short residence time (on the order of
minutes) (Eder et al., 2012; Jiang et al., 2014) and with fast heat transfer
associated with the large surface area-to-volume of tubular crystallizers.

Unlike
past studies that involve mathematical modeling or analysis of similar
continuous crystallizers (e.g., Eder et al., 2010; Kubo et al., 1998), this
study only analyzes an experimental demonstration (Jiang et al., 2014) but also
investigates the effect of and optimizes numerous design variables (e.g.,
number of heat exchangers, length of tubing in each heat bath/exchanger, mass
flow rate through the heat exchangers) while minimizing the total
equipment/material use in the design. This presentation describes the first
dynamic simulation of slug-flow crystallization for a series of counterflow, concentric shell-and-tube heat exchangers.

The
method of moments is combined with the numerical method of lines with a finite
difference approximation to reduce the model described by four partial
differential equations (balances on population, solute, and energy in both the
shell and tube) to a system of ordinary differential equations. The system is to
be designed so that, under normal operating conditions, the temperature drop in
each heat exchanger produces an environment to minimize the maximum supersaturation, to promote molecular purity, while
applying a constraint that ensures a nearly maximum yield.

Operating
so that the exit concentration is near the minimum concentration provides the
desired yield. The optimal crystal size is specified by the upstream nucleation
rate. Excessive length of the tubes should be avoided, as that would introduce
a time delay in the process dynamics that is undesirable for closed-loop
control. The control objective is to minimize the maximum supersaturation
in the system by manipulating the flowrate and inlet temperature of cooling water
in each heat exchanger to suppress the effects of process disturbances.

Optimal
steady-state spatial profiles for the four-stage slug-flow crystallizer are
shown in Figure 1. The supersaturation reaches the
minimized maximum value in less than 5 meters, and returns to that maximum once
in each heat exchanger. The drop in supersaturation
near the outlet also occurs in less than 5 meters.

Figure
1: Supersaturation profile under steady, optimal
conditions in the slug-flow crystallizer.

Investigating
the dynamic response of the measured and product quality variables to
disturbances, manipulations, and uncertainties provides a great deal of information
and insight not available from steady-state models. The solute concentrations
in the slug as it leaves each heat exchanger (?exit concentrations?) are
observed to be highly sensitive to cooling water temperatures and variation in
the growth kinetics. The cooling water flowrates have relatively low
effectiveness as manipulated variables for rejection of some disturbances.
Increasing the cooling water flowrate or decreasing the cooling water
temperature decreases the exit concentrations. The model shows that the effect
of most input disturbances on concentration, yield, and average particle size
are dampened by the process design automatically.

The
response of maximum and average supersaturation is of
vital importance as the driving force for growth as well as nucleation. While
this system is designed to minimize secondary nucleation, supersaturation
levels must be kept under control to avoid the incorporation of impurities into
the crystalline matrix. Cooling water temperature and flowrate are seen to have
a large effect locally. Inlet conditions for the slug stream have a decreasing
effect with each subsequent heat exchanger. Persistent sources of relatively
large deviation are again variation in the growth kinetics.

Outlet
temperatures are also compared and evaluated in terms of value as measurements
for feedback control, due to the relative ease and accuracy at which
temperature can be measured. Not surprisingly, the greatest effect by far on
the outlet slug stream temperature is the temperature of the cooling water.

Manipulating
only the cooling water flowrates is shown to be insufficient to reject process
disturbances that result from the deviation of growth kinetic parameters from
nominal values. Additionally, without a tight control of the inlet temperature
of cooling water, manipulation of the cooling water flowrates will be
insufficient to suppress that resulting deviation as well. Changes in the inlet
temperature and concentration for the slug stream are less problematic; the
system design is inherently robust to disturbances which occur far upstream.
The robustness allows the on-line measurement of particle size, a relatively
expensive and inaccurate endeavor, to be eliminated from the design of a
control algorithm; particle size is not observed to be affected greatly by
upstream disturbances.

In
an effort to increase the output response to changes in manipulated variables,
the cooling water inlet temperature should be a manipulated variable rather
than allowing it to be a disturbance. That is, a lower level regulatory control
system should be implemented so that the cooling water inlet temperature is
controlled, with the setpoint to the cooling water
inlet temperature being a manipulated variable of the higher level control
system. The extra degrees of freedom from controlling not only the cooling
water flowrates, but also the cooling water temperatures, greatly increase the
controllability of the slug-flow crystallization process. From a design
standpoint, sensitivity analysis indicates the tubing length should be
increased beyond nominal design conditions to serve as a buffer to
disturbances.

Several
control strategies motivated by the dynamic analysis are investigated for the
multivariable control of the slug-flow crystallizer (e.g., Figure 2).
Neglecting particle size measurements for the reasons mentioned above, there
are two measurements of interest that can be taken at the outlet of each heat
exchanger: temperature and solute concentration. There are also two manipulated
variables for each heat exchanger: cooling water flowrate and cooling water
temperature. A variety of factors are considered in selecting the best control
strategy in terms of the number of measurements, number of controllers, and
which controllers receive which measurements (and control which manipulated
variables). Of the eight possible measurements, the four temperature
measurements are inexpensive and accurate while four measurements of solute
concentration would be relatively expensive and difficult. The design of this
process does not provide evidence that intermediate solute concentration
measurements would be useful process control elements, and it is unlikely that
the final implemented control systems design would contain those measurements.
With respect to the eight manipulated variables, the inlet temperature of
cooling water could be altered by applying energy to a thermal bath or, perhaps
more likely, combining multiple feeds from multiple baths at Y-mixers. The
cooling water temperature displays a greater effect on process dynamics than
the cooling water flowrate, but incompressible fluid flowrates have the
advantage of being easily and quickly controlled. After comparing the pros and
cons of different control strategies, an optimal control system design will be
described.

Figure
2: Example control strategy for the slug-flow crystallizer, which uses the
complete set of eight measured and eight manipulated variables.

References:

Alvarez,
A.J., Myerson, A.S., 2010. Crystal Growth & Design, 56, 349-369.

Eder, R.J.P., Radl,
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