# Surrogate Models for Online Monitoring and Process Troubleshooting of NBR Emulsion Copolymerization

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. ANNs for NBR Emulsion Copolymerization in a Batch Reactor

_{i}, and the predicted output, X

_{i}. The predicted output X

_{i}, is a function of the weights (W

_{jk}for one hidden layer) used in the network, where the subscripts j and k represent the indices of the input and output neurons. In vector notation, W

_{jk}is usually represented by $\underset{\xaf}{W}=({W}_{12},{W}_{13},{W}_{14},\mathrm{...})$:

_{jk}, are obtained by training the network using Levenberg–Marquardt back-propagation algorithm. In this algorithm, the weights are adjusted using the method of steepest descent with respect to the error E, as defined by Equation (1). The builtin function, trainlm.m, in MATLAB is based on this algorithm and is used in the simulations for the NBR system. The desired output, Y, is obtained from the mechanistic model, which gives good predictions when compared with the experimental data, as established in previous work [4,5]. The original mechanistic model is highly nonlinear with 32 multiscale state variables. The data obtained from the mechanistic model are divided into training, validation, and untested datasets (also called unseen datasets). From the available data sets, 70% of the data is used for training, 15% percent of the data is used for validation, and the remaining 15% of the data is used for testing. To avoid an overfitted model for the training data, a program in MATLAB is written for obtaining the optimum size and complexity of the network with the objective that the training error be comparable to the prediction error. Performance characteristics of the designed ANN for the prediction of conversion, cumulative copolymer composition (CPC), weight-based average molecular weight (MW

_{w}) and tri-functional branching frequency (BN

_{3}) are shown in Figure 1a–d. In these figures, the profiles obtained from the mechanistic model (MM) are compared to those obtained using ANN for a targeted batch time of 700 min (a typical value used in commercial production). A very good agreement was achieved between the predictions using ANN with those obtained using the well-established and tested MM. The designed ANN for NBR emulsion copolymerisation in a batch reactor can, thus, be safely used in conjunction with process control and optimisation algorithms for describing the desired properties of the polymer.

## 3. ANN for Inverse Modeling of NBR Emulsion Copolymerisation in a Train of CSTRs

_{w}) at the exit of the eighth reactor can be simulated using the mechanistic model. Out of the 64 available datasets, 52 datasets are then used for training the network and 12 datasets are left in order to check the performance of the ANN with respect to inverse modeling of the recipe ingredients for obtaining the desired polymer properties. The different levels of the reaction ingredients used for MM simulations are shown in Table 1. The low and high levels of the reaction ingredients can be normalized to −1 and +1, respectively which, in turn, are used as outputs from the network. The inputs to the network, as shown in Figure 2, are X, CPC, and MW

_{w}, while the outputs are the recipe ingredients RA, I, E, M, W, and CTA.

^{6}. Hence, the optimum configuration used in this work was a network with three hidden layers with 20 neurons in each layer. Though the number of weights for such a large network is very high, the simulation time for obtaining the trained network in all simulations was only a few seconds. The trained network (saved as .MAT file in MATLAB) in turn is used as an inverse modeling tool to predict the recipe ingredients (of the first reactor) for targeted properties of the stream exiting the eighth reactor. Figure 3 (a through f) shows the recipe ingredients’ predictions obtained using the ANN-based inverse modeling for desired conversion (X) compared to the data obtained from the mechanistic model. The proposed ANN-based inverse modeling could predict the required recipe ingredients very well for desired conversion levels. Similar results and trends were obtained for desired CPC and MW

_{w}(as shown in Figure 4 and Figure 5). The predictions for all reaction ingredients are precise, except for the initiator in some cases. The prediction capability of the ANN can be quantified using the mean sum of squared errors (MSE) for each dataset. The MSE values for each of the predicted datasets are shown in Table 3.

## 4. Surrogate Modeling for NBR Emulsion Copolymerization

## 5. Transfer Function Models for the First CSTR in the Reactor Train

_{p}). These are the typical outputs (in principle, measurable) which are, in turn, used in the controlled production of NBR latex. Surrogate modeling is conducted in the Laplace domain by programming interactive simulations between SIMULINK and MATLAB. The performance of the corresponding fitted transfer function model is evaluated in terms of the coefficient of determination, R

^{2}. The input variables chosen to be related to any output variable are restricted to the states that have higher impact than others and that can also be used as practical manipulated variables in control applications. For example, when the reactor is started full of batch recipe (for other types of start-up policies refer to [4]), the conversion obtained at the exit of the reactor is expressed as a function of initiator, and acrylonitrile and butadiene (monomers) flow rates, as shown in Equation (2):

_{1}denotes conversion, X

_{1}represents initiator flowrate, X

_{2}is the acrylonitrile flow rate, and X

_{3}is the butadiene flow rate, all in the Laplace domain. τ

_{1}, τ

_{2}, and K

_{1}are the parameters of the model. Since the input and output variables in the transfer function models represent perturbations from initial steady states, the final model constitutes an initial value problem with all variables (outputs) to be zero at time t = 0. The structure of this model basically consists of a combination of a second order and two first order systems. The parameters τ

_{1}, τ

_{2}, and K are obtained by fitting the model response to the data obtained from the mechanistic model for a given step change in the input variables X

_{1}, X

_{2}, and X

_{3}. Similarly, for other output variables such as CPC (Y

_{2}), MW

_{w}(Y

_{3}), and N

_{p}(Y

_{4}), the corresponding models are given by Equations (3)–(5):

_{2}, Y

_{3}and Y

_{4}represent the output variables CPC, MW

_{w}, and N

_{p}, respectively, whereas X

_{4}, X

_{5}, and X

_{6}represent the ratio of flow rates of the two monomers (AN to Bd), chain transfer agent flow rate, and emulsifier flow rate, respectively. K

_{2}, K

_{3}, K

_{4}, and τ

_{3}through τ

_{8}are parameters of the empirical models. Equations (2)–(5) represent the empirical surrogate models for the dynamics of the first reactor in the reactor train. When the reactor is started full of recipe, the inflow to the first reactor is equivalent to giving a step input to all input variables X

_{1}through X

_{6}and the output responses Y

_{1}through Y

_{4}are used to obtain the corresponding parameters of the empirical models. The final comparison of the responses obtained for X, CPC, MW

_{w}, and N

_{p}using the data from the mechanistic model (MM) and the proposed empirical models (EM) are shown in Figure 6a–d, respectively.

^{2}values (close to unity) reported on the corresponding figures. Since the weight-based average molecular weight (MW

_{w}) in the mechanistic model is obtained from a set of very highly nonlinear model equations that originate using the method of moments, the corresponding empirical model had a lower R

^{2}value compared to the values of the other output variables. In general, the performance of the surrogate models is very good. The proposed empirical models are not only simple and amenable to use for online purposes but also have very few parameters. With the success of using this method for a single CSTR (the first reactor of the CSTR train), the properties at the exit of the eighth reactor can also be empirically modeled and further used in online applications, such as control and grade transitions.

## 6. Optimal CTA Profile for Minimizing off-Spec Product

_{8}is the MW

_{w}obtained at the exit of the eighth reactor, X

_{5}is the flow rate of the CTA to the first reactor in the reactor train, and K, τ

_{1}, and τ

_{2}are model parameters. The empirical model consists of eight second-order transfer functions in series, each of them corresponding to the dynamics of each individual reactor in the reactor train. From the R

^{2}values reported in Figure 7a,b, it is evident that the proposed second order transfer function in series fits very well the corresponding mechanistic model in addition to the benefit of employing three parameters only (K, τ

_{1}, and τ

_{2}).

_{w}. For simultaneous control of conversion, CPC, and MW

_{w}, a multiobjective function based on a weighted sum or ε-constrained methods can be used [19,20]. While grade changes can involve an increase or decrease in several product specifications, such as conversion, CPC, N

_{P}, MW

_{w}, etc., the application to the scenario where a decrease in MW

_{w}by adding extra CTA to the reactor train is discussed here as an example. In general, the inflows to the last few reactors are manipulated rather than manipulating the inflows to the first few reactors, due to the fact that the monomer droplets are absent in the last few reactors. For example, in a reactor train started up full of recipe, the monomer droplets will disappear in the sixth reactor of the train [5]. Especially for grade changes involving MW

_{w}, the corresponding manipulated variable is the flow rate of CTA added to the reactors with monomer droplets present. The optimal flow rate of CTA to be added to the first reactor in the reactor train is obtained using the optimisation function represented by Equation (7):

_{w}(t) is the measured value of the weight-based molecular weight at any time t, and ${F}_{CTA}^{*}$ is the steady state value of the flow rate of CTA. Assuming a control valve with a rangeability of 50:1 is used to manipulate the CTA flow rate, the manipulated flow rate is constrained between $\frac{1}{7}{F}_{CTA}^{*}$ and $7{F}_{CTA}^{*}$. In Equation (7), time t refers to the operation time of the eighth reactor. The mean residence time for each CSTR in the train is 60 min; hence, the total time for a reactor train of eight CSTRs will be 480 min. Since it takes three times the total mean residence time for the reactor train to reach steady state operation, the corresponding operational time used in the simulations was set to 1500 min (approximately). The optimum value for F

_{CTA}is obtained by minimizing the cost function (as shown in Equation (7)) at different time steps simultaneously. This procedure can be extended for other grade change applications with specifications on other variables, such as CPC or X, with manipulated variables being the flow rates of monomers, initiator, and/or emulsifiers.

_{w}at the exit of the eighth reactor for the cases where a regular CTA flow rate based on a “full of recipe” start-up is compared to that of the CTA flow rate obtained from optimisation using Equation (7). In both cases (refer to Figure 8a), the area under the solid curve and the dashed curve with respect to the steady state value of MW

_{w}is an indirect measure of the amount of off-specification product generated during the operation of the reactor train. Figure 8a clearly shows that using the proposed optimisation method, the amount of off-specification product/material can be minimized by several folds compared to the base case where a constant CTA flow rate is used. The corresponding CTA flow rate profile to be added to the first reactor obtained from the above mentioned optimisation procedure is shown in Figure 8b. This CTA flow profile can be practically achieved by using an automatic flow controller installed on the CTA flow line. The proposed online method for the adjustment of the manipulated CTA flow rate can also be applied to the flow rates of monomers and initiator to control CPC and/or X.

## 7. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

NBR | Nitrile Butadiene Rubber |

AN | Acrylonitrile |

Bd | Butadiene |

CSTR | Continuous-Stirred Tank Reactor |

ANN | Artificial Neural Network |

MM | Mechanistic Model |

EM | Empirical Model |

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**Figure 1.**Comparison of (

**a**) conversion; (

**b**) cumulative copolymer composition; (

**c**) weight-average molecular weight; and (

**d**) tri-functional branching frequency profiles obtained using ANN and mechanistic models. For process conditions, refer to Dube et al. [2].

**Figure 2.**ANN structure used for inverse modeling with X, CPC, and MW

_{w}as inputs; RA, I, E, M, W, and CTA as outputs.

**Figure 3.**Comparison of the predictions between the unseen targeted values (o) and the values obtained using ANN (+) for desired conversion levels vs. (

**a**) reducing agent; (

**b**) initiator; (

**c**) emulsifier; (

**d**) monomer; (

**e**) water; and (

**f**) chain transfer agent.

**Figure 4.**Comparison of the predictions between the unseen targeted values (o) and the values obtained using ANN (+) for desired cumulative copolymer composition levels vs. (

**a**) reducing agent; (

**b**) initiator; (

**c**) emulsifier; (

**d**) monomer; (

**e**) water; and (

**f**) chain transfer agent.

**Figure 5.**Comparison of the predictions between the unseen targeted values (o) and the values obtained using ANN (+) for desired weight-based average molecular weight levels vs. (

**a**) reducing agent; (

**b**) initiator; (

**c**) emulsifier; (

**d**) monomer; (

**e**) water; and (

**f**) chain transfer agent.

**Figure 6.**Comparison of the responses obtained from the proposed empirical models (EM) and mechanistic model (MM) for (

**a**) conversion, (

**b**) CPC, (

**c**) MW

_{w}, and (

**d**) N

_{p}.

**Figure 7.**Comparison of the model validity for MW

_{w}at the exit of the eighth reactor using mechanistic model (MM) and empirical model (EM) for reactor train start-ups (

**a**) full of recipe and (

**b**) full of water.

**Figure 8.**(

**a**) Comparison of MW

_{w}profiles in the eighth reactor using regular CTA flow rate to that of using optimal flow rate of CTA; and (

**b**) the dynamic CTA flow rate obtained from optimisation.

Ingredient | Low Level (L/min) | High Level (L/min) |
---|---|---|

Sodium Formaldehyde Sulfoxylate (RA) | 0.165 | 0.22 |

p-methane hydroperoxide (I) | 0.046 | 0.062 |

Dresinate/Tamol (E) | 0.89/1.67 | 1.183/2.228 |

Acrylonitrile/Butadiene (M) | 48.6/160.3 | 64.8/213.7 |

Water (W) | 121.36 | 161.81 |

tert-dodecyl Mercaptan (CTA) | 0.33 | 0.44 |

Number of Neurons | Number of Hidden Layers | |||||
---|---|---|---|---|---|---|

Monomer | CTA | |||||

1 | 2 | 3 | 1 | 2 | 3 | |

5 | 0.1532 | 0.1593 | 0.0033 | 0.1370 | 0.0420 | 0.0405 |

10 | 0.0653 | 0.0289 | 0.0836 | 0.0677 | 0.0509 | 0.2600 |

15 | 0.0148 | 0.0659 | 0.0405 | 0.0143 | 0.2070 | 0.1355 |

20 | 0.0245 | 0.2412 | 0.0124 | 0.137 | 0.1175 | 0.0613 |

# | X | CPC | MW_{w} | Mean Squared Error (MSE) | |||||
---|---|---|---|---|---|---|---|---|---|

RA | I | E | M | W | CTA | ||||

1 | 0.5611 | 0.2573 | 1.17 × 10^{5} | 0.1247 | 2.7766 | 0.0136 | 0.0105 | 0.1115 | 0.0255 |

2 | 0.7668 | 0.2811 | 1.51 × 10^{5} | 0.0096 | 4.0446 | 0.0074 | 0.0042 | 0.0240 | 0.0001 |

3 | 0.4743 | 0.2735 | 8.54 × 10^{4} | 0.0518 | 5.6491 | 0.0384 | 0.0014 | 0.0016 | 0.0068 |

4 | 0.5515 | 0.2285 | 7.56 × 10^{4} | 0.2717 | 0.2051 | 0.7459 | 0.0300 | 0.0110 | 0.1170 |

5 | 0.7669 | 0.2811 | 1.95 × 10^{5} | 4.1140 | 1.4764 | 0.0101 | 0.0015 | 0.0002 | 0.0396 |

6 | 0.6454 | 0.2711 | 1.19 × 10^{5} | 5.2109 | 4.4500 | 2.6381 | 0.0287 | 0.0034 | 0.1231 |

7 | 0.5689 | 0.2349 | 1.05 × 10^{5} | 0.1591 | 8.3361 | 0.0016 | 0.0076 | 0.3575 | 0.1394 |

8 | 0.4246 | 0.2676 | 5.45 × 10^{4} | 0.0674 | 2.9982 | 0.0549 | 0.0037 | 0.0053 | 0.0211 |

9 | 0.6895 | 0.2784 | 1.80 × 10^{5} | 0.0325 | 2.6604 | 0.0100 | 0.0094 | 0.0000 | 0.0626 |

10 | 0.7074 | 0.2713 | 1.24 × 10^{5} | 0.0470 | 4.2123 | 0.0217 | 0.0001 | 0.0024 | 0.0424 |

11 | 0.4247 | 0.2676 | 7.22 × 10^{4} | 0.0743 | 2.9586 | 0.0487 | 0.0206 | 0.0027 | 0.1336 |

12 | 0.6703 | 0.2547 | 1.16 × 10^{5} | 0.0014 | 2.6040 | 0.0243 | 0.0321 | 0.0004 | 0.0246 |

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**MDPI and ACS Style**

Madhuranthakam, C.M.R.; Penlidis, A.
Surrogate Models for Online Monitoring and Process Troubleshooting of NBR Emulsion Copolymerization. *Processes* **2016**, *4*, 6.
https://doi.org/10.3390/pr4010006

**AMA Style**

Madhuranthakam CMR, Penlidis A.
Surrogate Models for Online Monitoring and Process Troubleshooting of NBR Emulsion Copolymerization. *Processes*. 2016; 4(1):6.
https://doi.org/10.3390/pr4010006

**Chicago/Turabian Style**

Madhuranthakam, Chandra Mouli R., and Alexander Penlidis.
2016. "Surrogate Models for Online Monitoring and Process Troubleshooting of NBR Emulsion Copolymerization" *Processes* 4, no. 1: 6.
https://doi.org/10.3390/pr4010006