##
277375 Evaluating Trade-Offs in the Operation of a Semi-Batch Polymerization Reactor

**Evaluating trade-offs in the operation
of
a semi-batch polymerization reactor**

F. Assassa^{a,c}, F. Logist^{b},
J. Van Impe^{b}, W. Marquardt^{a}

^{a}* AVT –
Process Systems Engineering, Aachener Verfahrenstechnik, RWTH Aachen
University, Turmstraße 46, D-52056 Aachen, Germany*

^{b}* BioTeC
& OPTEC, Department of Chemical Engineering, Katholieke Universiteit
Leuven,
W. de Croylaan 46, B-3001 Leuven, Belgium*

^{c }*German
Research School for Simulation Sciences GmbH, 52425 Jülich, Germany*

{Filip.Logist,Jan.VanImpe}@cit.kuleuven.be

{Fady.Assassa,Wolfgang.Marquardt}@avt.rwth-aachen.de

**Abstract.** Optimizing the design and operation of polymerization reactors often involves
multiple and conflicting objectives. In the current study, the trade-off
between economic profit and energy consumption is evaluated for the operation
of a semi-batch reactor, which produces a
co-polymer of styrene and butyl-acrylate. A set of trade-off or so-called Pareto optimal
solutions is generated based on a dynamic reactor model. Consequently, we have
to solve a multi-objective dynamic optimization problem. A scalarization method
(normal boundary intersection) is employed to tackle the multi-objective aspect.
This method converts the original multi-objective optimal control problem into
a series of single-objective optimal control problems, which are each solved
with direct single shooting (adaptive control vector parameterization). For this series of optimization problems, it can be
assumed that neighboring values of the scalarization parameter result in
neighboring solutions. We exploit this feature by building an accurate
approximation for the solution of the first single-objective optimization by
grid adaptation and structure detection. The detected structure is repeatedly
updated along the Pareto set. The resulting Pareto set as well as the resulting
profiles for controls and states are discussed.

** **

**Case study.**

**Process.** The case study deals with a
semi-batch reactor, which produces a co-polymer of styrene and butyl-acrylate
in solution with pentyl acetate. Di-cumyl peroxide is chosen as the initiator.
Both monomers and the diluted initiator are fed separately. Their feed flow
rates as well as the coolant inlet temperature are time-varying control
variables.

**Model.** Hartwich [1] developed a model of
the reactor from kinetic models of [2, 3] including reaction rates, polymer
distribution moment equations and mass balances. The energy balance including
the cooling jacket has been added. This gives rise to a model with 199
algebraic and 23 differential states.

**Decision variables. **The feed flow rates of the
monomers and the diluted initiator as well as the coolant inlet temperature are
decision variables of the optimal control problem. The optimization algorithm
tries to minimize the objective function by varying these control variables.

**Constraints.** Feasible and safe batch
operation is ensured by specifying bounds along the batch duration for the
reactor content, the reactor temperature set-point, the reactor temperature and
the accumulation of monomer. To guarantee production, lower bounds are imposed
on the reactor content, the conversion, the amount of polymer and the reactor
temperature at the end of the batch. Moreover, pre-specified product quality
bounds on, e.g., the weight
average molecular weight and the polydispersity, have to be fulfilled at the
end time of the batch. The batch duration time is fixed.

**Objectives. **Two conflicting objectives
are evaluated: (i) the profit, i.e., the return due to the product minus the
cost for monomers and initiator, and (ii) the energy consumption, i.e., the
energy needed to heat up the reactor.

**Optimization procedure.**

The approach for computing the Pareto set of the multi-objective dynamic optimization problem is similar to [4] and combines (i) the normal boundary intersection (NBI) method [5] for scalarization of the objectives with (ii) control grid adaptation and structure detection techniques [6-8] to efficiently solve the resulting dynamic optimization problem. The approach was implemented in the software package DyOS.

**Normal boundary intersection (NBI). **Das and Dennis [5] have
proposed NBI, a geometrically intuitive approach, to mitigate the drawbacks of
the weighted sum approach for scalarization of multiple objectives. NBI first builds a plane in the
objective space which contains all convex combinations of the individual
minima, i.e., the convex hull of individual minima (CHIM), and then constructs
(quasi-)normal lines to this plane. The rationale is that the intersection
between the (quasi-)normal from any point on the CHIM, and the boundary of the
feasible objective space closest to the utopia point (i.e., the point which
contains the minima of the individual objectives) is expected to be Pareto
optimal. Hereto, the multi-objective optimization problem is reformulated as to
maximize the distance from a point on the CHIM along the quasi-normal through
this point, without violating the original constraints.

**Grid adaptation and structure
detection. **The scalarized multi-objective dynamic optimization problem is converted
into a nonlinear programming problem (NLP) by adaptive control vector
parameterization developed by Schlegel and Marquardt [6]. Here, the control
profiles are successively refined employing a wavelet-based analysis of the
optimal solution obtained in the preceding optimization step. Furthermore, a control structure
detection algorithm [7-8] was used to improve computational performance of the
optimal solution.

**Results. **

Results have been obtained for some points along the Pareto
set. Figure 1
shows an exemplarily Pareto set for (i) the negative value of the scaled profit
vs. (ii) the scaled energy consumption with the scalarization vector as **w
= **[1-*A*, *A*]^{T} in which *A* varies from 0 to 1
in steps of 0.1. In general, the process is profitable
but requires energy supplied from outside. More profit can be made at the
expense of consuming more energy. Hence, an engineer can use this information
to select a suitable operation according to the current economic situation.
Grid adaptation
techniques, structure detection and multi-stage reformulations allow a lean
control parameterization, which is updated going along the Pareto set.

Figure 1: Sampled Pareto set of -profit vs. energy consumption with 11 points. For increasing A-values in w = [1-A,A]T optimizations proceed along the set from left to right. |

** **

** **

**Acknowledgements.**
Work supported in part by
Projects OPTEC (Center-of-Excellence Optimization in Engineering) PFV/10/002
and SCORES4CHEM KP/09/005 of the KULeuven. FL acknowledges the FWO travel
grant. JVI holds the chair Safety Engineering sponsored by the Belgian
chemistry and life sciences federation essenscia.

This work has been partially funded by the German Research School for Simulation Sciences.

** **

** **

**References**

1. A. Hartwich (2012), PhD Thesis, RWTH Aachen University, AVT Prozesstechnik

2. A.E. Hamielec, J.F. MacGregor and A. Penlidis (1987). Multicomponent free-radical polymerization in batch, semibatch and continous reactors. Macromolecular symposia, 10:521–570.

3. J. Gao and A. Penlidis (2002). Mathematical modeling and computer simulator/database for emulsion polymerizations. Progress in Polymer Science, 27:403–535.

4. F. Logist, F. Assassa, J. Van
Impe and W. Marquardt (2012), Exploiting grid adaptation and structure
detection in multi-objective dynamic optimization problems, Accepted for oral
presentation in the 22^{nd} European Symposium on Computer-Aided
Process Engineering (ESCAPE 22), Londen (UK), June 17-20 2012.

5. I. Das and J. Dennis (1998). Normal-Boundary Intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM Journal on Optimization 8:631–657.

6. M. Schlegel, K. Stockmann, T. Binder, and W. Marquardt (2005). Dynamic optimization using adaptive control vector parameterization. Computers & Chemical Engineering, 29: 1731–1751.

7. M. Schlegel and W. Marquardt (2006). Detection and exploitation of the control switching structure in the solution of dynamic optimization problems. Journal of Process Control, 16:275–290.

8. M. Schlegel and W. Marquardt (2006). Adaptive switching structure detection for the solution of dynamic optimization problems. Industrial and Engineering Chemistry Research, 45:8083–8094.

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