18th Mediterranean Conference on Control & Automation Congress Palace Hotel, Marrakech, Morocco June 23-25, 2010
Dynamic Optimization of a Polymerization Reactor Radoslav Paulen, Miroslav Fikar, and M. A. Latifi
Abstract— This paper presents an approach to dynamic optimization of an emulsion polymerization reactor. The reactor can be described as a strongly non-linear hybrid system with three dynamical modes. We describe a detailed process model described by 7 differential equations. To solve the problem, we apply control vector parametrization (CVP) method and calculate necessary information for non-linear programming (NLP) solvers using theory of optimal control according to Pontryagin’s maximum principle.
I. INTRODUCTION Following the rising performance of computers there is a significant effort to advance in field of computer science. Numerical integrators can now handle more complex systems and that is why more detailed process models might be established. Hybrid models can be viewed as models which can fight with imprecision of classical continuous or discrete models. They keep character of both continuous and discrete models. Phenomena appearing in real world might be caught better to process model due to this property. By hybrid processes we denote processes which are described by a hybrid model. In our study we treat emulsion copolymerization of styrene and α-methylstyrene taking place in batch reactor. This process model can be described by ordinary differential equations (continuous character) with different right-hand sides (discrete character) according to the stage of the process. Emulsion polymerization refers to a unique process employed for some radical chain polymerizations. It involves the polymerization of monomers in the form of emulsions. This process has several distinct advantages. The physical state of the emulsion (colloidal) system makes it easy to control the process. The product of an emulsion polymerization, referred to as a latex, can in many instances be used directly without further separations [1]. An approach used to find an optimal control of a plant or process is termed dynamic optimization [2], [3]. It encompasses several techniques, which can be divided in two broad frameworks, variational methods and discretization methods. In variational methods [4], a classical calculus of variations together with Pontryagin’s maximum principle [5] is applied. These methods address the dynamic optimization problem in its original infinite dimensional form. A big advantage of this is that we are looking for an exact solution to problem without any transformations. But it is disadvantageous if we want to solve a problem for not quite simple systems. R. Paulen and M.Fikar are with Faculty of Chemical and Food Technology, Slovak University of Technology, Bratislava, Slovakia
{radoslav.paulen,miroslav.fikar}@stuba.sk M.
A.
Latifi
is
with
the
ENSIC,
INPL,
[email protected]
978-1-4244-8092-0/10/$26.00 ©2010 IEEE
Nancy,
France
Then a discretization plays important role, since original infinite dimensional problem is transformed to a non-linear programing (NLP) problem. Sequential discretization, usually achieved via control parametrization [6], [7], [8], is a discretization approach in which the control variable profiles are approximated by a sum of basis functions in terms of a finite set of real parameters. These parameters then become the decision variables in a dynamic embedded NLP. Function evaluations are provided to this NLP via numerical solution of a fully determined initial value problem (IVP), which is given by fixing the control profiles. This method has the advantages of yielding a relatively small NLP and exploiting the robustness and efficiency of modern IVP and sensitivity solvers [9]. Dynamic modeling and optimization of this process was previously studied in [10] and [11]. The first approach in [10] used complete discretization approach. As it is an infeasible type of optimization, it can have problems in application in real-time predictive control. The second study [11] searches for optimal profiles of reactor cooling jacket inlet temperature (a more realistic type of control). However, the process model considered contained only two hybrid stages. Thus, this paper represents improvement over these works both in model complexity and attractiveness of optimization method. In next sections we introduce the emulsion polymerization reaction model, give hybrid representation of this model and then we describe dynamic optimization approach used to compute optimal control profiles for this process. Finally we give and discuss obtained results. II. PROCESS MODEL The overall process is described by a kinetic model, a molecular weight distribution model, and heat balance equations in the reactor and in the jacket. A. Kinetic Mechanism The copolymerization of styrene and α-methylstyrene has been previously studied both in the bulk and in solution. Similarities with the homopolymerization behavior of styrene or α-methylstyrene are often emphasized. The kinetic model of an emulsion polymerization usually divided into three stages. The experimental validation was carried out by [10]. The three stages of the model are as follows: 1) In the first stage, free radicals are produced in the aqueous phase by initiator decomposition. They are captured by the micelles swollen with monomer. The polymerization begins in these micelles. This stage, corresponding to particle heterogeneous nucleation
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(the polymerization rate is increasing), stops when all of the micelles have disappeared. 2) Particle growth occurs during the second stage. Monomer diffuses rapidly from monomer droplets toward the particles, which are saturated with monomer as long as monomer droplets exist. This stage ceases when all of the monomer droplets have disappeared (the polymerization rate is constant). 3) The third and final stage is characterized by the decrease of the monomer concentration in particles, also the polymerization rate is decreasing. The following assumptions are used in the modeling of the kinetic process: Styrene and α-methylstyrene are both hydrophobic monomers; thus, only micellar nucleation is taken into account. Because of this hydrophobic character, the reactions of propagation, transfer to monomer, termination in the aqueous phase, and radical desorption are neglected. Termination in particles is considered to be very rapid compared to radical entry into particles; thus, it can be assumed that no more than one radical is present in each polymer particle (zero-one system). This allows us to write Pj• = N • , where variable Pj• represents number of activated polymers of chain length j and N • represents number of active particles. The maximum conversion rate considered is generally around 65%, so that gelation has not yet occurred. Thus, the gel effect is not included in the model to avoid unnecessary complexity. Coagulation is assumed to be negligible. The kinetic mechanism is then written as follows: • Initiator decomposition A → 2R• •
R +m→N •
•
•
•
where a is a constant, fMS is molar fraction of α-methylstyrene at the beginning of a reaction and kp′ is the propagation rate constant for styrene homopolymerization. The quasi-stationary state approximation allows to calculate the initiator radicals concentration Ra R• = k1 m + k2 Np where Np = N + N • is overall number of polymer particles. Then the rate of particles formation may be written as1 Ra NA N˙ p = k1 m k1 m + k2 Np Here NA stands for Avogadro’s number. Like [12], we introduce the factor which is similar to an effectiveness for the particles relative to the micelles in collecting an initiator radical k2 SNA ε= k1 m The rate of particle formation is now expressed by Ra NA N˙ p = k1 m (1) εNp 1+ SNA With the classical hypothesis that the emulsifier molecules are adsorbed in a monomolecular layer on the polymer particles surface, the emulsifier concentration may be written S = So − kv (XMo )2/3 Np1/3
Rn = k1 mR
•
Initiation N + R• → N•
Ri = k2 N R•
N• + R• → N
Rt = k2 N • R•
Termination
Propagation P•j + M → P•j+1
•
kp = kp′ exp(−a.fMS )
Ra = 2f kd A
Particle formation •
B. Kinetic Model Only the global monomer concentration is described. The depropagation is taken into account by including in the propagation rate constant a factor depending on the αmethylstyrene molar fraction in the initial charge
Rp = kp Mp N •
Transfer to monomer P•j + M → M• + P•j
RtrM = ktrM Mp N •
Here A means initiator concentration, R• stands for initiator radical concentration, m represents number of micelles per unit volume, M is global monomer concentration and finally f is initiator efficiency. Variables R and k differing by index of partial reaction of kinetic mechanism denote reaction rates and reaction kinetic coefficients.
where So denotes initial emulsifier concentration, Mo means initial monomer concentration, X signifies monomer conversion and 1/3 2 36πMM kv = ωP2 (as NA )3 ρ2P where MM denotes monomer molecular weight, ωP means polymer weight fraction in particles, variable as represents surface area occupied by an emulsifier molecule and ρP stands here for polymer particle density. The rate of monomer consumption is expressed according to Np n ¯ (2) M˙ = −Rp = −kp Mp NA with average number of radicals per particle n ¯ =0.5, and monomer concentration in particles Mp which can be expressed as (1 − Xc )ρM Mp = Mpc = X ≤ Xc (3a) [(1 − Xc ) + Xc ρM /ρP ] MM (1 − X)ρM Mp = X > Xc (3b) [(1 − X) + XρM /ρP ] MM 1 through the whole paper a dot over a variable represents its time derivative
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there constant Mpc denotes critical monomer concentration in particles, Xc stands for critical monomer conversion and ρM represents monomer density. C. Molecular Weight Distribution Model Besides it is important to evaluate the polymer properties. They are linked to the polymer structure and can be described in terms of average molecular characteristics (number and weight average molecular weights, polydispersity index, . . . ). The used approach is an extension of the tendency model [13] to emulsion polymerization. According to the kinetic mechanism, the rates of production of the zeroth, first and second order moments of the molecular weight distribution may be written Q˙ 0 =Rt + RtrM Q˙ 1 =L(Rt + RtrM ) Q˙ 2 =2L2 (Rt + RtrM )
(4)
E. Hybrid Representation of the Process Model Majority of this section uses notation introduced by [14]. Dealing with the hybrid systems of ODEs2 means dealing with a system of a form mode Si
x˙ (i) = f (i) (x(i) , u(i) , p, t) Snk (i) (i) ∀t ∈ [t0 , tf ], Si ∈ k=1 Sk (i)
(i)
where x(i) ∈ Rnx , u(i) ∈ Rnu and p ∈ Rnp . Func(i) (i) tion f (i) is such that f (i) : Rnx × Rnu × Rnp × (i) nx R → R . Switching from one stage (mode Sk ) to another (mode Sj ) occurs when a unique transition condi(k) tion Lj (x˙ (k) , x(k) , u(k) , p, t), Sj ∈ P (k) is satisfied. Set P (k) contains all possible successive stages of mode Sk . (k) Transition function Tj (x˙ (k) , x(k) , u(k) , x˙ (j) , x(j) , u(j) , p, t) determines possible discontinuity in variables participating in model. There is a special case of transition function defined (k) at mode initial time t = t0 where
where
(0)
RtrM =ktrM Mp
(k)
(k)
Tk (x˙ (k) , x(k) , u(k) , p, t) = 0 ⇒ x(k) (t0 ) = x0
Ra n ¯ Np Rt = Np + Sε
Rp Rt + RtrM Here variable L denotes kinetic chain length. Once these moments are known, the number-average molecular weight ¯ n ), weight-average molecular weight (M ¯ w ) and polydis(M persity index (Ip ) can be calculated according to ¯ ¯ w = MM Q2 Ip = Mw ¯ n = M M Q1 M (5) M ¯n Q0 Q1 M
mode Si
L=
In order to achieve temperature control, it is necessary to describe the reactor temperature dynamics. The reaction takes place in a glass stirred tank reactor. Heat control is realized thanks to a cooling fluid circulating in a jacket at constant flow rate. The inlet temperature of the cooling fluid is controllable. Energy balances on the jacket and the reactor contents give the following equations for the reactor and jacket temperature dynamics
x˙ (i) = f (i) (x(i) , u, p) (i) (i) ∀t ∈ [t0 , tf ], Si ∈ {S1 , S2 , S3 }
(10)
where state vector is defined as x = (M, Np , Q0 , Q1 , Q2 , T, Tj )T
(11)
In our case switching between stages passes in order known in advance. Thus, we can define transition conditions as (1)
D. Reactor Temperature Dynamics Model
L2 (x(1) , t) = 0 ⇒ Np(1) (t) = Npst
(12)
(2) L3 (x(2) , t)
(13)
=0⇒X
(2)
(t) = Xc
Npst
where constant represents number of particles which depends on temperature and initial load in reactor. Variable X(t) means monomer conversion which depends on actual monomer concentration M (t) and initial load in reactor. Note that due to known sequence of stages we can write (1)
t0 = t0
(1)
(2)
tf = t0
(2)
(3)
tf = t0
(3)
tf = tf
(14)
Transition functions can be defined as (6)
(0)
(1)
(1)
T1 (x(1) , p, t) = 0 ⇒ x(1) (t0 ) = x0 (1)
(2)
(1)
T2 (x(1) , x(2) , t) = 0 ⇒ x(2) (t0 ) = x(1) (tf )
(7)
where V and Vj are reactor contents volume and reactor cooling jacket volume respectively, ∆H means polymerization reaction enthalpy, term mr Cp represents reactor total heat capacity, constant U means heat-transfer coefficient, A is cooling surface, variables Fj , ρj and Cp,j signify cooling fluid flow rate, cooling fluid density and cooling fluid heat capacity respectively. All the kinetic coefficients are calculated according to Arrhenius law ki = ki,0 exp(−Ej /RT )
(9)
Hence that model represented by (1), (2), (4), (6), and (7) is strongly non-linear and hybrid as well (see [11]). Its hybrid character can be expressed as follows
Np n ¯ NA
V ∆H UA T˙ = − Rp + (Tj − T ) mr Cp mCp UA Fj T˙j = (Tj,in − Tj ) − (Tj − T ) Vj ρj Vj Cp,j
(8)
(2) T3 (x(2) , x(3) , t)
(3)
=0⇒x
(3) (t0 )
(2)
=x
(15)
(2) (tf )
Rest of this section forms conditions for particular stages of the process. As it was mentioned in Section II-A, the number of particles stays constant in the second and third stage. Thus, (2) (3) N˙ p = N˙ p = 0. In the third stage, calculation of variable Mp changes (see (3)). This changes also right hand sides of (2), (4), and (6) where Mp is present. 2 Our studied system consists of explicit differential-algebraic equations (DAE) which can be simply reformulated as ODEs.
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III. PROCESS OPTIMIZATION
B. Optimization Method
A. Problem Definition Dynamic optimization problems can in general be defined for continuous process models as min G(x(tf )) +
u(t),p
Z
tf
F (x(t), u(t), p, t)dt
t0
s.t. x˙ = f (x(t), u(t), p, t) x(t0 ) = x0
∀t ∈ [t0 , tf ]
h(t, x(t), u(t), p) = 0 g(t, x(t), u(t), p) ≤ 0
(16)
u(t) ∈ [uL (t), uU (t)]
From application of Pontryagin’s maximum principle [5] one can obtain necessary conditions of optimality for dynamic optimization problem. These determine definition of adjoint variables and their corresponding boundary conditions (specified at final time). Definition of gradients with respect to controls and parameters can be derived also using this principle. Further details can be found in [15]. Necessary conditions of optimality for hybrid processes were studied in [16] where possible discontinuity in process variables was taken in account. This kind of discontinuity is not present in our case. Detailed analysis of the optimized system showed that discontinuity in adjoint variables is also not present. Therefore, adjoint variable definition is expressed as
p ∈ [pL , pU ] where function G(·) is real-valued function and function F (·) is such that F : Rnx × Rnu × Rnp × R → R. Functions h(·) and g(·) are vector functions of appropriate dimensions that define process constraints. Our objective is to compute such a control u(t) of the system that will drive system to desired terminal state (some state values are prescribed at final time) at minimum time possible. Based on this and Section II-E we can form our optimization problem as min
u(t),p
Z
(3)
tf
(3)
(1)
dt = tf − t0
(1) t0
s.t. x˙ (i) = f (i) (x(i) , u(i) , p) (i)
(i)
∀t ∈ [t0 , tf ], i ∈ {1, 2, 3} (1)
(1)
x(1) (t0 ) = x0 (3)
h(x
(3) (3) (tf ), tf ) L U
(17) =0
u(t) ∈ [u (t), u (t)] p ∈ [pL , pU ] where initial state vector is defined as (1)
x0 = (M0 , 0, 0, 0, 0, T0, Tj,0 )T
(18)
where T0 and Tj,0 are variables which represent initial reactor and cooling jacket temperature, respectively. These are parameters p to be optimized. Terminal state of the reaction is characterized both by quantity and quality terminal conditions. Quantity terminal condition means that we desire final conversion of monomer to be =X
(3)
(3) (tf )
=1−
M (3) (tf ) M0
i ∈ {1, 2, 3}
(20) (21)
(3)
t=tf
where
H (i) (x, λ, u, p, t) = F (·) + (λ(i) )T f (i) (·)
(22)
is Hamiltonian function and λ is vector of adjoint variables. We employ control vector parametrization (CVP) method to solve dynamic optimization problem. In first step, control trajectory u(t) is discretized to final number of intervals with constant control uj , where j ∈ {1, . . . , Nu } indicates control interval number. Hence, an infinite-dimensional optimization problem in continuous control variables is transformed into a finite-dimensional non-linear programming problem that can be solved by any gradient-based method (e.g., a sequential quadratic programming (SQP) method). The performance index evaluation is carried out by solving an initial value problem (IVP) of the original ODE system, and gradients of the performance index as well as the constraints with respect to the parameters (p and approximations of u(t)) can be evaluated by solving the adjoint equations. 1) Computing Gradients: Accurate gradients calculation is one of the most challenging step in solving the NLP problem which was derived from original problem (17) using piecewise-constant control uj applied on specified instant of time ∆tj = tj − tj−1 . Gradients of objective function J can be computed such that ∂J (3) ∂tf
(3)
(3) Xf
∂H (i) λ˙ (i) = − ∂(x(i) )T ∂G (3) (3) λ (tf ) = ∂(x(3) )T
(19)
∂J (i) ∂tj
Quality terminal condition is represented by final value of number-average molecular weight, weight-average molecular weight or polydispersity index of polymer and can be computed by (5). The optimized control variable u(t) is reactor cooling jacket inlet temperature Tj,in .
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(3)
=H (3) (tf ) +
∂G (3)
∂tf
(i) + =H (i) (t− j ) − H (tj ) ∀j ∈ {1, . . . , Nu − 1} (23) (1)
∂G ∂J (1) (1) ∂x = − Jp (t0 ) + (λ(1) )T (t0 ) 0T ∂pT ∂pT ∂p ∂J (i) (i) =Ju (tj−1 ) − Ju (tj ) ∀j ∈ {1, . . . , Nu } ∂uj
where i ∈ {1, 2, 3} and
345
(i)
∂H J˙u = ∂u (i) ∂H J˙p = ∂pT
T Tj,in
340
(24)
335 T,Tj,in [K]
(25)
Any constraint function can be expressed in form of optimization criterion [11], so that constraint gradients can be found in same manner like presented in (23). 2) Optimization Algorithm: We employ the procedure which can be described by following steps: 1) Make initial guess for values of ui , ∆ti and p. 2) Integrate system forward in time. 3) Evaluate objective and constraint functions. 4) Integrate system of adjoint equations backward in time. 5) Calculate gradients. 6) Use NLP solver to get new values of variables ui , ∆ti and p. 7) If the optimality conditions are satisfied then quit. Else, go to Step 2.
330 325 320 315 0
0.2
0.4
0.6
0.8
1
time/tf
(a) Nu = 3 345 T Tj,in
340
T,Tj,in [K]
335 330 325
C. Results and Discussion 320
Optimization algorithm proposed in Section III-B was implemented using MATLAB and its NLP solver fmincon which solves constrained NLP problems. Integration was performed using ode45 integrator where 4th order RungeKutta numerical integration method is implemented. Optimization problem was solved for different control discretization scenarios concerning piece-wise constant control. Obtained results are summarized in following table where
315 0
0.2
0.4
0.6
0.8
1
time/tf
(b) Nu = 5 345 T Tj,in
340 335 T,Tj,in [K]
TABLE I C OMPARISON OF DIFFERENT CONTROL PROFILES FOR DESIRED ¯ n,f = 2 × 106 TERMINAL STATE : Xf = 0.7, M
330 325
Nu 1 2 3 4 5 6 7 8 9 10
tf [s] 5340 5234 4950 4859 4673 4609 4550 4539 4502 4445
Xf 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
¯ n,f × 10−6 M 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
¯ w,f × 10−6 M 4.3 4.4 4.8 5.7 5.8 5.8 6.0 6.3 6.3 6.4
320
Ip 2.13 2.20 2.38 2.87 2.87 2.87 3.01 3.13 3.13 3.18
315 0
0.2
0.4
0.6
0.8
1
time/tf
(c) Nu = 7 345 T Tj,in
340
variable Nu represents the number of control segments considered. It can be seen that by the rising number of constant control segments not only the optimization criterion is lowering its value (as expected) but also an overall quality of polymer produced rises (this can be seen in last two columns of Table I). Maximum value of ten constant control segments was chosen as a compromise between quality of approximation of optimal control and computational effort needed to solve the NLP problem. Using of multistart method (i.e. starting algorithm from multiple different starting points) showed that optimization
T,Tj,in [K]
335 330 325 320 315 0
0.2
0.4
0.6
0.8
1
time/tf
(d) Nu = 10 Fig. 1. Trajectories of control variable and reactor temperature for different numbers of control segments (Nu ).
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problem appears to have multiple local minima. However, the cost function reaches similar values in all of them and suggests that the optimum is quite flat. Hence, the sensitivity of the desired terminal quality constraint is small and the optimization problem has good convergence properties. To obtain a global minimum, methods of global optimization have to be applied. It is well known that these methods need to solve the original nonconvex problem repeatedly [17]. Graphically, the results can be represented by Fig. 1, where the control trajectories as well as reactor temperature trajectories are depicted for selected control discretization scenarios. IV. CONCLUSIONS In this paper we studied dynamic optimization of a polymerization process with emulsion copolymerization of styrene and α-methylstyrene. We presented process model and its hybrid representation. Following this representation we derived necessary conditions of optimality of this process. According to these we were able to compute gradients of objective and constraints functions. These gradients were passed to NLP solver which solved NLP obtained by control discretization of original dynamic optimization problem. As the computations suggested there might exist more than one solution to our problem because when the algorithm was launched from different initial points (a multistart method) a different solution was reported at the end of an algorithm run. This suggests that our optimization problem possesses multiple (local) minima. In our future work we will study global optimization methods for further improvement of the obtained solution and we will implement it on our laboratory scale polymerization reactor.
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V. ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants 1/0071/09, 1/0537/10 and the Slovak Research and Development Agency under the project APVV-0029-07. R EFERENCES [1] G. Odian, Principles of Polymerization, 4th ed. John Wiley & Sons, Inc., 2004. [2] D. Chm´urny, R. Prokop, and M. Bakoˇsov´a, Automatic Control of Technological Processes. Bratislava: Alfa, 1988, (in Slovak). [3] J. Mikleˇs and M. Fikar, Process Modelling, Identification, and Control. Berlin: Springer Verlag, 2007. [4] R. Luus and D. Cormack, “Multiplicity of solutions resulting from the use of variational methods in optimal control problems,” Canadian Journal of Chemical Engineering, vol. 50, no. 2, pp. 309–311, 1972. [5] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes. New York: John Wiley & Sons, Inc., 1962. [6] R. G. Brusch and R. H. Schappelle, “Solution of highly constrained optimal control problems using nonlinear programming.” AIAA Journal, vol. 11, no. 2, pp. 135–136, 1973. [7] C. J. Goh and K. L. Teo, “Control parameterization : A unified approach to optimal control problems with general constraints,” Automatica, vol. 24, no. 1, pp. 3–18, 1988. [8] T. Hirmajer and M. Fikar, “Optimal control of a two-stage reactor system,” Chemical Papers, vol. 60, no. 5, pp. 381–387, 2006.
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