Cooling Crystallization, PID and MPC Control

O. A Nguepkap Yamba

Coachs: Ing.dr J. Wildenberg Ing.dr J. Landlust Supervisor: Prof.dr T.Backx Professor: Prof.dr M.Steinbuch

Eindhoven 28Th October 2008

Technical University of Eindhoven Department of system and control Dynamics and control technology group

Summary The aim of this project is about designing and comparing two types of batch control cooling crystallizer: PID control and on-line optimal control (MPC), with a focus on both product quality and process performances. The first part starts with the definition and application of cooling crystallization. All aspects of cooling crystallization are discussed, from process development to the operating mechanism. The second part concerns the modeling framework applied for most crystallizer. Mass balance and population balance equations are treated. The method of moment is used as a better and more appropriate description of the system properties. Kinetic models and enthalpy balance are also mentioned. The following part is the design procedure for the batch crystallization control. Requirements in achieving good product quality and process performance are discussed. First a temperature trajectory is build using the INCA environment in other to perform measurements on the process. Subsequently a PID control is implemented to control a constant supersaturation set point towards the crystallization process. Finally, respect to a predefined objective function; an on-line optimal control is implemented. The last part of this project is about comparing, both methods applied for cooling crystallization control, where general conclusions and recommendations are drawn respect to the advantages and disadvantages of both applied method. Introduction

Cooling crystallization is more generally define as the formation of solid with desirable and consistent properties from a solution, where the solubility line is highly sensitive to decrease of temperature. It is widely used in pharmaceutical, petrochemical, semiconductor industry. Crystallization is often conducted in batches rather than continuously because this allows greater flexibility and shorter process development. In solution crystallization, the physical system consists of one or more solute dissolve in a solvent. For crystallization to occur, the system must be supersaturated. The supersaturation is defined as the amount by which the solute concentration exceeds the saturation concentration, is usually expressed as:

)(* TCCC −=Δ

Where C is the actual solute concentration and )(* TC the solubility line (saturation concentration), which is a function of the temperature solution. Process development The first step in developing a control system for solution crystallization is to determine the saturation concentration and metastable limit. The saturation concentration also called solubility represents the minimum solute concentration for which the crystalgrowth can occur. The mestablelimt, on the other, indicates the concentration above which undesirable spontaneous nucleation occur. For a cooling crystallization to take place, the supersaturation must be increase by decreasing the solute temperature; which affects the rate of nucleation, growth and determine the CSD (crystal size distribution)

Figure 1: Solubility curve and metastable zone

Process mechanisms

Any industrial crystallizers should be designed to operate within the metastable zone, at lower supersaturations and with high residence time. During crystallization operation, several mechanisms can occur depending on the define control objectives. Nucleation Nucleation is the formation of new crystalline particles, and it is classified as primary or secondary according to the mechanism through which it occurs. Primary nucleation New solids are form from clear liquid at high supersaturation. This type of nucleation can also be subdivided into two subtype; heterogeneous, where nucleation start from microscopic particles and homogeneous, where no particles is present and nucleation occur as a result of local fluctuations of the concentration in the liquid Secondary nucleation New crystals are generated from parent crystals (low supersaturation). The main source of parent material in the liquid is attrition. Attrition is the discontinuous separation of very small particles from a parent crystal due to collisions with the impeller in the pump, the vessel wall and other crystals. Growth and Dissolution Growth refers to deposition of solid material on existing crystals. In the absence of agglomeration and breakage, growth together with nucleation determines the final crystal size distribution. The driving force for the crystal growth is again the supersaturation. On the other hand, dissolution takes place when the solution is undersaturated; crystals are easily rounded of at their corners and edges. Agglomeration Particles sizes in industrial crystallizers may also be affected by agglomeration and breakage. Agglomerates form when growing crystals collide with each other, the particles stick together to form a new particle. In general agglomeration is related to higher supersaturation, and even operation in the unstable zone. If they are more particles present, which increase the probability of collision, agglomeration will be more extensive. I general, agglomeration is associated with high supersaturation, this means, to minimize agglomeration, supersaturation should be limited. Modeling of cooling crystallization

A Mathematical model of a process is a system of equations that enable prediction of the process behavior and its response to a set of inputs. The model can the be used for equipment design, to determine safe operating conditions, to estimate state that can not be measure, and to improve the control of the process. This chapter discusses the population (method of moment), mass and energy balances for the batch cooling crystallizer. It also explains how the solubility curve is obtained from a solution Population Balance More generally, the population balance for a batch crystallizer with one characteristic growth dimension is defined as:

BLfG

tf

=∂

∂+

∂∂ )(

Where f is the population density function for the entire crystallizer, G is the growth rate of the crystals (m/s), B is the rate of nucleation (#/kg s) and L is the crystal length. Due to the fact that the equation above is difficult to solve even numerically, it has been converted to a series of ordinary differential equations well-know as method of moments. (Miller and Rawlings, 1994), where the ith moment of the crystal-size distribution is defined as:

∫∞

=0

)( dLLfLiiμ

For the current application, we have considered the zero till third moments and the model equations are:

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+

+

+

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

300

300

200

00

.

.

3

.

2

.

1

0

.

3

3

2

BrGk

BrG

BrG

BrGB

C vc μρ

μ

μ

μ

μ

μ

μ

μ

Where C id the solution concentration (kg/m3), r0 is the crystal size at nucleation, ist the density of the crystal, Mass Balance

Two mass balances are considered, one for the solute and one for the solvent. Both must take into account the flow in and out of the system and the mass transfer to the solid phase via formation and growth of crystals. Mass in and out via mass production rate due to consecutive transport Liquid phase reaction

∑∑∑===

+Φ−−=NR

PpipLioutiVm

NO

lkioutLm

NI

kkinLm

iL rMtttdt

tdm

1,,,,,

1,,,

1,,,

, .)()()()(

υϕϕ

Kinetic model The kinetic model describes the formation of new crystals, crystals growth and crystals birth (nucleation). For size-independent growth, growth is typically assumed to be proportional to the supersaturation raised to some power.

gg CkG Δ=

For must practical industrial crystallization system, the supersaturation must be kept low enough to avoid primary nucleation, this means new particle are formed almost exclusively by secondary nucleation. The expression used for this nucleation is then:

bb CkB Δ= 3μ

In the expressions above gk , bk , b, g are empirical determined values. Enthalpy balance Finally, the dynamics of the temperature are given by the energy balance. It is shown that the energy balance can be transformed into enthalpy balance by assuming that kinetic energy, potential energy and shaft work can be neglected. We obtain then the relation: Rate of the total enthalpy in and out by convection

)()()()()(,,

1,,

1,, tQttt

dttdH

outVH

NO

lkoutH

NI

kkinH +Φ−Φ−Φ= ∑∑

==

Where Q is the net rate of heat addition. The boundary conditions and initial conditions of the entire above stated model equations are as follow:

0)0(0)0(

)0(0)0(

0)0(,

====

=

BG

CC

m

ini

i

iL

μ

Solubility line The solubility curve defers from on crystallizer to another. It is the equilibrium curve of the system, where the crystals growth and nucleation are zero when the solution is perfectly mixed. It is usually determine empirically, at deferent temperatures. It this way, the solubility can be expressed more over as a quadratic function of temperature.

CbTaTTC ++= 2* )(

Cooling Batch crystallization control Introduction The most concern in must industrial crystallization is to produce crystals that meet specifications in size, shape, composition and internal structure, known as Crystal Size Distribution. However, this can only be achieved by controlling the supersaturation level, which is the major control variable for crystallization processes. The supersaturation level affects the relative rates of nucleation and growth and thus determines the CSD. Since, it is dependent of the solute concentration and the solubility (function of temperature); it can be manipulated in the cooling case, by acting on the flow rate of the cooling medium to the crystallizer medium. Several researchers have proven that the most efficient way to operate a batch crystallizer is to maintain a constant supersaturation. Many approaches have been proposed for designing and operating batch cooling crystallization systems. In this chapter, two of these approaches will be presented; the PID feedback control, base on understanding the basic concept of crystallization to operate the system within the metastable zone and the solubility line and the model-based design approach (MPC), base on developing a detailed model of the system, which is used with optimization problems to determine the optimal temperature over time. Finally, a detailed comparison will be made between the mentioned approaches to choose the one that suite the most for the batch cooling crystallization. NB: For both approaches, a detailed mathematical model have been developed, which represent the real process. Feedback control (PID) A critical question in the design and operation of batch crystallization processes is how the supersaturation should vary in time during the batch. The setpoint in terms of supersaturation (absolute) chosen as constant in this case is defined as:

)(* TCCC −=Δ

Where C is the actual solute concentration and )(* TC t is the saturation concentration. In most application the saturation concentration is determined empirically, defines as a function of the temperature. Thus replacing the measured solute concentration measC for C and an algebraic function for the solubility, solving this equation for T gives the temperature setpoint. At the other hand, replacing the temperature measure for T in the solubility function and solving the equation for C also give the concentration set point. Solubility line

Determining the solubility line is a crucial aspect in designing a feedback control for crystallization processes. It is usually determined empirically, which time consuming and high cost for most companies. For this project, a method has been developed to estimate the solubility line by applying a specific cooling profile (temperature). This profile must insure that the process is sufficiently exited for good estimation. The measured parameters are: the temperature in the crystallizer, the solute concentration, and finally the second and third moment of the crystal size distribution. The estimation method is as follow:

• Calculate Crystal growth rate from second and third moment of CSD • Define a Growth-Grid and determine at what time instant the growth rate

intersects the Growth Grid. • Calculate the Temperature and solute concentration at these time instant to related

all three parameters • Extrapolate the G-C-T surface towards growth rate zero; know as equilibrium or

the point of solubility line. The results are shown below.

0 50 100 150 200 250 30018

20

22

24

26

28

30

32

Time[s]

Tem

pera

ture

pro

file

Cooling Profile

Figure 2: Temperature profile

Figure 3: Temperature, concentration, moments (2, 3) measured

Figure 4: Growth rate

Figure 5: Solute concentration

Figure 6: Solubility estimate and calculated

The results shows that the estimated solubility line is quadratic function, which is more or less equal to the one, obtain from data at least up to the temperature of 22 degree. This means in deed that the proposed above method can be used determining the solubility line for feedback control of batch crystallization.

cbTaTTC ++= 2* )( Objectives In general terms, feedback control drives a process in a manner that fulfills a predefined objective. The objective in this project was to maximize crystal growth by manipulating the valve position to maintain a constant supersaturation during the crystallization. The measured outputs for this strategy are the solution temperature, concentration and supersaturation.

Implementation The design implementation is done within INCA online-environment and the architecture is represented as follow:

Figure 7: INCA design architecture

The measurement scheme implemented to fulfill the above control objective is a cascade structure comprised of an integration loop and a PID loop. At each time instant, the supersaturation measurement and the current system temperature is compared to the user-specified supersaturation set point. The primary loop minimizes the error between the two by manipulating the cooling rate set point. This new rate is translated into new temperature set point by the INCA calculator. The secondary loop ensures that the new temperature setpoint specified by the primary loop is achieved by manipulating the valve position.

Figure 8: PID cascade control

PID background The PID controller compares at each sample time the measured process value with the reference setpoint. The difference or error, e, is then processed to calculated a new process input, u. This input will try to adjust the measured process value back to the desired setpoint. The PID is constituted of three terms, proportional, integral and derivative. Proportional term The proportional term P gives a system control input proportional with the error. Using P gives a stationary error in all cases except when the system control input is zero and the process value equals the desired value. Integral term The integral term (I) gives an addition from the sum of the previous errors to the system control input the summing of the error will continue until the process value equals the desired value and this result in no stationary error when the reference is stable. The integral I, is always use in combination with the proportional term. Derivative term The derivative term (D) gives an addition from the rate of change in the error to the system control input. It is typically used with the P, PI as a PD or a PID controller. The time domain representation of a PID controller is: Replacing the integral by the cumulative sum and the derivative by the difference between the current and the previous error approximations gives the discrete form: Where, Ti is the integration time and Td is the derivative time.

⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∫

t

di

p dttdeTde

TteKtu

0

)()(1)()( σσ

( )

TTK

K

TTK

K

pvspne

neneKkeKneKnu

dpd

i

pi

d

n

kip

=

=

−=

−−++= ∑=

)(

)1()()()()(0

To avoid saturation of the actuator (valve; fully closed or open). An anti-windup scheme has been added to the integral term. The anti-windup also reduces both overshoot and the control effort in the feedback system.

Figure 9: Anti-windup scheme

Results and discussions Simulations have been realized for three different supersaturation values; 1, 0.5, and 0.25 in order to investigate, how good the PID performs for each value. The temperature profile is between 32 and 22 degree Celsius.

Figure 10: Temperature set point(black), temperature measure(red) and temperature error(blue)

1=Δ setC

Figure 11: Supersaturation: set-point (red) and measured (green)

Figure 12: supersaturation error (red), solubility error (black) and solute concentration (blue)

5.0=Δ setC

Figure 13: Supersaturation: set-point (red) and measured (green)

Figure 14: supersaturation error (red), solubility error (black) and solute concentration (blue)

25.0=Δ setC

Figure 15: Supersaturation: set-point (red) and measured (green)

Figure 16: supersaturation error (black), solubility error (red) and solute concentration (blue)

The results show that the set point profile calculated starts at very lower temperature. Soon after this initial drop, the temperature increase very rapidly, then decreases very slowly at the beginning of cooling and drops down quickly towards the end of the batch. It is also observe that the PID controller can successfully bring the crystallizer temperature to its set point. The temperature profile creates a high supersaturation at the beginning, and then it keeps the supersaturation nearly constant value within the metastable zone during the rest of the batch. Comparing the different tracking supersaturation set points, one can realize that, the more we decrease the supersaturation set point, the more efficiently the PID performed, by continually adjusting the heat exchanger temperature. This can be explained by the fact that, for very high value of the supersaturation, the crystals grow very fast and at a certain moment, it becomes very difficult for the controller to keep up with the growing rate. Towards the end of the three control experiments, the controller has difficulties to maintain the supersaturation at its set-point value. The deviation from the set-point coincides with the onset of the secondary nucleation or due to the integration effect in the crystals volume fraction. At this point, the speed of the controller can not keep-up with the dynamics of this mechanism. However to overcome this undesirable mechanism, one can think of instead of implementing a linear time variant integrator for the PID. This integrator will in fact increase its value at each time the supersaturation error seems to become very high.

Model Predictive Control (MPC) Model predictive control is an optimization strategy that uses a process linear model to predict at each sample instant the best future process output. The key feature of MPC is the moving horizon. At each simulation step, the plant states are sampled and a cost minimizing control strategy is computed. Only the first step of the control strategy is implemented, then the plant states are sampled again and the calculations are repeated starting from the now current state, yielding a control and a new control state path. The prediction horizon keeps being shifted forward, reason why the MPC is also known as receding horizon.

Figure 17: MPC horizons

The prediction horizon should be large enough to allow the controller to predict the future behavior of the process output that is close as possible to the desired behavior. However too large value of the prediction horizon results in very slow response of the process. This means a trade off should be made between the simulation time and the prediction horizon. MPC is usually applied in MIMO industrial process, which has two types of inputs: Disturbances Variables (DV’s) and Manipulated Variables (MV’s) that influence the outputs known as Control variables (CV’s). The basics steps are:

• Prediction of the future process output behavior • Determination of the best future manipulations to drive the process to optimum

conditions • Feedfoward compensation for disturbances • Respecting operating constraints and for determination of optimum conditions

The structure that implemented the MPC strategy is shown as follow:

Figure 18: Structure of MPC

Referring to the batch crystallization, we have defined the temperature of the jacket as the MV, the supersaturation (absolute) and the crystal volume fraction as the CV’s. There are no disturbance variables and the same nonlinear model is used for the process and for the prediction. Again the design is implemented with INCA tools and the architecture is as follow:

Figure 19: Implemented architecture

Linear Model For the cooling crystallization case, we make use of Linear Time Variant model, this means, at every sample instant, a new model is calculated and the prediction is made upon the current model. The mathematical approach to calculate these models are as follow: We consider the general form of nonlinear model and state space linear model

⎩⎨⎧

==+

))(),(()())(),(()1(

kukxgkykukxfkx

Now consider: )()( * kxxkx Δ−= , )()( * kuuku Δ−= , where *u and *x represent the state and input

corresponding to a set of stationary operating conditions and satisfy: 0),( ** =uxf If we expand ))(),(( kukxf by the Taylor series, we have:

termsorderhighkuufkx

xfuxfkukxf ..)()(),())(),(( **

** +Δ∂∂

+Δ∂∂

+=

termsorderhighkuugkx

xguxgkukxg ..)()(),())(),(( **

** +Δ∂∂

+Δ∂∂

+=

The high order terms are usually small compare to the linear terms, thus can be neglected. By doing this we approximate the nonlinear equation by linear one and this leads to the approximating linear state-space model

⎩⎨⎧

Δ+Δ=Δ+Δ=+Δ)()()(

)()()1(kuDkxCky

kuBkxAkx

Where *xfA∂∂

= , *ufB∂∂

= *xgC∂∂

= and *ugC∂∂

= .

At the other hands, by considering realistic perturbations on the inputs and the states denote by px and pu , the matrix of the linear system can be approximate by:

)(2))(),()(())(),()((

* kpxkukpxkxfkukpxkxf

xfA −−+

=∂∂

=

Due to the physical properties of some parameters (can not be lower than zero), we have to clip the states and inputs considering their upper and lower bounds denote by lbub, as:

⎩⎨⎧

+=+=

⎩⎨⎧

+=+=

),max(),min(

,),max(),min(

puxuupuxuu

pxxxxpxxxx

lbl

ubh

lbl

ubh

Finally, the linear approximation becomes:

)()())(),(())(),((

* kxkxkukxfkukxf

xfA

lh

lh

−−

=∂∂

= )()(

))(,)(())(),((* kxkx

kukxgkukxgxgC

lh

lh

−−

=∂∂

=

)()())(),(())(),((

* kukukukxfkukxf

ufB

lh

lh

−−

=∂∂

=)()(

))(),(())(),((* kuku

kukxgkukxgugD

lh

lh

−−

=∂∂

=

Objective function In Crystallization, the final CSD is the principal factor determining the quality of the products. Various properties of the CSD can be used as objective functions, such as:

• Minimize the amount of nucleus-grown crystals • Maximize the average of the total crystals • Maximize the volume fraction of the crystals • Minimize the variation size of the total crystals.

All these word objectives can be translated in mathematical objectives. For cooling crystallization case, two main objectives have been defined which are; maximize the average size by maintaining a supersaturation at its set point, and maximize the total amount of crystals by driving the crystals volume fraction towards its set point. For this reason, the MPC dynamic optimization has been implemented to drive CV’s and the MV from their current value to the solution of the steady state. Thus, the optimal control problem of the batch crystallization process can be stated as follow:

{ } { }⎭⎬⎫

⎩⎨⎧

Δ+−+Δ−Δ= ∑ ∑= =

p mH

j

H

jjjspsp STjVFRVFRjCCJ

1 1

222 )()(min

Where pH is the prediction horizon, mH is the control horizon and jS is the relative weight matrix. Subject to the constraints:

maxmin )( TkTT ≤≤

maxmin )( CkCC Δ≤Δ≤Δ

maxmin )( VFRkVFRVFR ≤≤ Where minT , minT , minCΔ , maxCΔ minVFR , maxVFR are respectively the minimum and maximum temperature, supersaturation and volume fraction values. Results and discussions Simulation has been done for a supersaturation set point of 1 (ideal), a volume fraction crystal set-point of 0.06 (ideal) and a cooling temperature range from 32 till 22. To achieve the optimization criterion, we make use of the dynamic zone (UseDynZone) of the supersaturation to force the controller to operate within the supersaturation constraints [0 1]. In this way, the supersaturation was maintained at its set-point during the crystallization and the crystal volume was driven to the desired value. The results are presented below:

Figure 20: Optimal temperature set point and measured temperature

Figure 21: Supersaturation set point and measured

Figure 22: Volume fraction crystal set-point and measured

Figure 23: Temperature set-point, supersaturation set-point and measured valve position

The results show an optimal temperature trajectory that decreases very slowly at the beginning of the control process and rapidly and the end of the batch. The supersaturation increase very fast at the beginning of the batch towards its set point, then it is maintain at this value almost during the whole control process. However, we also observe that before the end of the batch, the supersaturation starts decrease unexpectedly, due to the limitation in the system dynamics. At this time, the valve is fully open, which means the controller has no more effect on the process. This is shown in figure 22. At the other hands, the crystals volume fraction increase conversely exponential in the direction of its set-point at stays there until the process stops. The predefined control objectives have been more or less achieved, this means the MPC is a good candidate for cooling batch crystallization.

Conclusion and recommendation In this project two methods have been develop in controlling batch cooling crystallization, the PID approach and the MPC approach and both appear to be successful. The PID revels that, it was very difficult to maintain the supersaturation at its set- point during the whole batch for very high value, while good control could be achieved for small value, which results in an increase of the simulation time and small crystals growth rate. Meanwhile, very satisfactory results were obtained with the MPC for very high supersaturation. The question that bears in mind at this point is that, shall we promote the MPC instead of the PID control for crystallization? Performances of every crystallization process always depend of the objectives to be achieved, in terms of product quality, production time and production cost. Looking at the product quality, we have seen so far that better results were obtained with MPC that PID approach. At the other hands, MPC needs very accurate linear model that describe the nonlinear process. In this project we used the same process model as the prediction model and no disturbances were consider, which might explain the obtained amazing results. It also appears that MPC needs much more time for setting up the complete environment and for simulation comparing to PID control. Finally, we can conclude that both methods appear to be adequate in controlling crystallization process. In term of product quality, one might choose the MPC approach and in the terms of production time and cost, the other might choose the PID approach. .