# Combining On-Line Characterization Tools with Modern Software Environments for Optimal Operation of Polymerization Processes

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## Abstract

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## 1. Introduction

## 2. Model Centric Framework

**Figure 1.**Schematic of the integrated simulation, estimation, optimization and feedback control of polymerization systems.

#### 2.1. Process Modelling

#### 2.1.1. Reaction Mechanisms and Kinetic Equations

_{j}and D

_{j}are the corresponding growing live polymer radical and dead polymer. Under standard assumptions such as well-mixed reactor, quasi steady state assumptions (for the radicals) and long chain hypothesis, the following set of kinetic and dynamic equations describe the system:

_{m}= C

_{m}V, N

_{i}= C

_{i}V, N

_{s}= C

_{s}V

_{m}, C

_{i}and C

_{s}represent the concentrations of monomer, initiator and solvent in the reactor, respectively. V illustrates the volume of the content of the reactor, F

_{m}and F

_{i}are the volumetric flow rate of monomer and initiator respectively which are fed into the reactor in the semi batch mode. F

_{out}is the constant flow rate out of the reactor for the ACOMP extraction stream. C

_{mf}, C

_{if}, C

_{sif}and C

_{smf}are the concentration of monomer in the monomer feed stream, the initiator in the initiator feed stream and solvent in the initiator and monomer flow stream. P

_{0}is the total concentration of live polymer which is obtained from the quasi steady state assumption. λ

_{0}, λ

_{1}and λ

_{2}are the corresponding moments for the dead polymers, and 𝛼 is the probability of propagation. f is the initiator efficiency and ρ

_{m}, ρ

_{i}, ρ

_{s}and ρ

_{p}are the densities of the monomer, initiator, solvent and polymer which are temperature dependent. k

_{p}, k

_{d}, k

_{fm}, k

_{fs}, k

_{tc}and k

_{td}are the propagation, initiation, chain transfer to monomer, chain transfer to solvent, termination by combination and termination by disproportionation rate. Kinetic rate constants are all temperature dependent functions based on Arrhenius equation [28] and as we will see due to strong nonlinearity of the MMA system, the propagation and termination rate depend on conversion as well.

_{m0}is the initial amount of monomer in the reactor. Number average and weight average molar mass of the polymers are calculated by considering only the moment of dead polymers and neglecting the live polymer concentration which is valid for low and medium conversions when the concentration of live polymer is negligible.

#### 2.1.2. Formalism for Gel, Glass and Cage Effects in MMA Polymerization

_{t}, and should be considered in the formulation of the model. At high conversion when even the motion of monomer is severely restricted the propagation rate k

_{p}, is also decreased. This glassy state in which the solution is highly viscous sets a limiting conversion on the polymerization process. In this work the correlation by [28,32] is used for both gel and glass effect. This can be written as:

_{ftc}and v

_{fpc}are the critical free volumes which are calculated as below:

_{f}is the total free volume which is given by:

_{i}and T

_{gi}are the volume fraction and the glass transition temperature of the polymer, solvent and monomer and α

_{i}is a constant. The values of the parameter for this case are shown in Table 1 [28]. For butyl acetate the corresponding glass transition temperature was considered as an adjustable parameter which has to be determined by parameter estimation.

Parameter | MMA | Poly Methyl Methacrylate | Butyl Acetate |
---|---|---|---|

𝛼 | 0.001 | 0.00048 | 0.001 |

T_{g}(K) | 167 | 387 | # |

_{0}is the initial initiator efficiency and C is a constant with the values of 0.53 and 0.006 for Azobisisobutyronitrile Fan et al. [33].

#### 2.1.3. Molar Mass Distribution

_{n}= αP

_{n−1}and P

_{n}= (1 − α)α

^{n−1}P:

#### 2.1.4. Energy Balances

_{r}and T

_{j}denote the reactor and jacket temperature respectively. It was assumed that both reactor and jacket are perfectly mixed and have a constant temperature. ρ

_{r}and C

_{pr}are the average density and specific heat capacity of the reactor. C

_{pm}, C

_{ps}and C

_{pj}are the specific heat capacity of monomer, solvent and coolant flow which consists of water and ethylene glycol. U is the overall heat transfer coefficient and A is the heat transfer area.

#### 2.2. Parameter Estimation

_{w}(t)] which are the outputs of the parameter estimation model, u(t) = [T(t), F

_{m}(t), F

_{i}(t)] which are the time-varying inputs and 𝜃 the set of model parameters to be estimated which in this case are [A

_{d}, A

_{p}, A

_{td}, f

_{0}, T

_{gs}]. A

_{d}, A

_{p}and A

_{td}are the pre exponential factors of the decomposition rate, propagation rate and termination rate respectively. The selection of these parameters are justified as the most sensitivity in conversion and weight average molar mass data is with respect to the termination and propagation rate of a polymeric chain. Proper estimation of the initiator efficiency factor is also important since it controls the effective radical concentration. Since the transition temperature for butyl acetate is not available in the literature this parameter should also be estimated. In this work the parameter estimation scheme is based on maximum likelihood criterion. The gEST function in gPROMS is used as the software to estimate the set of parameters using the data gathered from the different experimental runs. Each experiment is characterized by a set of conditions under which it is performed, which are:

- The overall duration.
- The initial conditions which are the initial loading of initiator, solvent and monomer.
- The variation of the control variables. For the batch experiment temperature is the only variable, while in semi batch both temperature and flow rate of monomer and/or initiator have to be considered.
- The values of the time invariant parameters.

_{ijk}with zero means and standard deviations, σ

_{ijk}this maximum likelihood goal can be captured through the following objective function:

_{i}and NM

_{ij}are respectively the total number of experiments performed, the number of variables measured in the ith experiment and the number of measurements of the jth variable in the ith experiment. ${\mathsf{\sigma}}_{\text{ijk}}^{2}$ is the variance of the kth measurement of variable j in experiment i while ${\tilde{z}}_{\text{ijk}}$ is the kth measured value of variable j in experiment i and z

_{ijk}is the kth model-predicted value of variable j in experiment i.

_{w}as part of the optimization.

#### 2.3. Dynamic Optimization

- Duration of each control interval and the values during the interval are selected by the optimizer
- Starting from the initial condition the dynamic system is solved in order to calculate the time-variation of the states of the system
- Based on the solution, the values of the objective function and its sensitivity to the control variables and also the constraints are determined.
- The optimizer revises the choices at the first step and the procedure is repeated until the convergence to the optimum condition is achieved.

_{0}is the initial condition of the system including the initial loading in the reactor and t

_{f}stands for the time horizon while u(t) indicates the control variables which are the temperature, monomer and initiator flow rates subjected to their lower and upper bounds. v

_{t}represents the time variant parameters being the volume of the contents of the reactor. The formulation of the objective function consists of four terms. X

_{f}, M

_{w,f}and f

_{i,f}are the values of the monomer conversion, molar mass and weight fraction of polymer within a chain length at the final time t

_{f}respectively and X

_{t}, M

_{w,t}and f

_{i,t}are their corresponding desired values. w

_{1}–w

_{4}are the weighting factors, determining the significance of each term in the objective function. A schematic representation of the optimization problem for the polymerization problem is given in Figure 2.

## 3. Experimental System

#### 3.1. Experimental Apparatus—ACOMP System

**Figure 3.**Automatic continuous on-line monitoring of polymerization setup for the monitoring of methyl methacrylate solution polymerization used in this work.

#### 3.2. Experimental Procedure

## 4. Results and Discussion

#### 4.1. Validation Using Literature Data

**Figure 4.**Comparison between model simulations and literature results. (

**a**) Panel-Conversion profile. (

**b**) Panel-Molar mass distribution at the end of the batch.

#### 4.2. Experimental Validation for Batch and Semi-Batch Free Radical Polymerization of MMA Using Butyl Acetate as Solvent and AIBN Initiator

**Table 2.**Original and estimated value of the kinetic rate parameters for the free radical polymerization of MMA (first iteration).

Parameter | Description | Original Value | Estimated Value | Confidence Interval | 95% t-value | Standard Deviation | ||
---|---|---|---|---|---|---|---|---|

90% | 95% | 99% | ||||||

A_{d} | Decomposition (1/min) | 1.58 × 10^{15} | 1.37 × 10^{15} | 1.25 × 10^{14} | 1.49 × 10^{14} | 1.96 × 10^{14} | 9.19 | 7.60 × 10^{13} |

A_{p} | Propagation (m^{3}/mol∙min) | 4.2 × 10^{5} | 9 × 10^{5} | 5.23 × 10^{4} | 6.23 × 10^{4} | 8.20 × 10^{4} | 14.43 | 3.17 × 10^{4} |

A_{td} | Termination [m^{3}/mol∙min] | 1.06 × 10^{8} | 4.56 × 10^{8} | 5.97 × 10^{7} | 7.12 × 10^{7} | 9.37 × 10^{7} | 6.40 | 3.63 × 10^{7} |

f_{0} | Initial Initiator Efficiency | 0.58 | 0.57 | 0.048 | 0.057 | 0.076 | 9.84 | 0.029 |

T_{s} | Solvent Transition Temperature (K) | 181 | 142.61 | 0.539 | 0.6431 | 0.84 | 221.7 | 0.327 |

Estimated Parameters | A_{d} | A_{p} | A_{t} | f_{0} | T_{s} |
---|---|---|---|---|---|

A_{d} | 1 | - | - | - | - |

A_{p} | 0.117 | 1 | - | - | - |

A_{t} | 0.111 | 0.988 | 1 | - | - |

f_{0} | –0.988 | 0.038 | 0.043 | 1 | - |

T_{s} | 0.104 | 0.179 | 0.233 | –0.070 | 1 |

_{p}and A

_{td}is 0.94 indicating a strong correlation between them and making it difficult to find a unique estimate for these parameters. Unique parameter estimate means that the parameters have an acceptably low correlation to any of the other parameters and a low confidence interval. Thus, in spite of the large covariance mentioned above, a consistent estimation is possible because of the true value of the estimated parameters are located within a very small confidence bands reducing their uncertainty. However, the confidence ellipsoids are large including in most cases negative numbers. Therefore, a second iteration was performed by eliminating two of the correlated parameters A

_{d}and A

_{td}and fixing their values to the estimated ones in the first iteration. The optimal values of the estimated parameters as well as the uncertainty of the parameter represented as 95% confidence interval (CI) are shown in Table 4.

**Table 4.**Original and estimated value of the kinetic rate parameters (second iteration) for the free radical polymerization of MMA.

Parameter | Description | Original Value | Estimated Value | Confidence Interval | 95% t-value | Standard Deviation | ||
---|---|---|---|---|---|---|---|---|

90% | 95% | 99% | ||||||

A_{p} | Propagation Rate (m^{3}/mol∙min) | 3 × 10^{5} | 8.5 × 10^{5} | 2547 | 3035 | 3993 | 280.1 | 1546 |

f_{0} | Initial Initiator Efficiency | 0.58 | 0.56 | 0.001166 | 0.0013 | 0.00182 | 403.2 | 0.00073 |

T_{s} | Solvent Transition Temperature (K) | 142 | 149.94 | 0.3906 | 0.465 | 0.6123 | 322.2 | 0.237 |

**Figure 8.**Comparison between experimental data and simulation with original and estimated parameters: (

**a**) Isothermal experiment; (

**b**) Non-isothermal experiment.

**Figure 10.**Simulation results of the optimal trajectories considering time in the objective function: (

**a**) Input (manipulated) variables; (

**b**) Controlled variables (targets).

Variable | Value | Unit |
---|---|---|

N_{m} | 0.5 | mol |

N_{s} | 0.5 | mol |

N_{i} | 0.01 | mol |

F_{max} | 5 | mL/min |

F_{min} | 0 | mL/min |

T_{max} | 70 | °C |

T_{min} | 50 | °C |

V_{max} | 500 | mL |

V_{min} | 100 | mL |

**Figure 11.**Validation of optimal runs: (

**a**) Input (manipulated) variables; (

**b**) Controlled variables (targets).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ghadipasha, N.; Geraili, A.; Romagnoli, J.A.; Castor, C.A., Jr.; Drenski, M.F.; Reed, W.F. Combining On-Line Characterization Tools with Modern Software Environments for Optimal Operation of Polymerization Processes. *Processes* **2016**, *4*, 5.
https://doi.org/10.3390/pr4010005

**AMA Style**

Ghadipasha N, Geraili A, Romagnoli JA, Castor CA Jr., Drenski MF, Reed WF. Combining On-Line Characterization Tools with Modern Software Environments for Optimal Operation of Polymerization Processes. *Processes*. 2016; 4(1):5.
https://doi.org/10.3390/pr4010005

**Chicago/Turabian Style**

Ghadipasha, Navid, Aryan Geraili, Jose A. Romagnoli, Carlos A. Castor Jr., Michael F. Drenski, and Wayne F. Reed. 2016. "Combining On-Line Characterization Tools with Modern Software Environments for Optimal Operation of Polymerization Processes" *Processes* 4, no. 1: 5.
https://doi.org/10.3390/pr4010005