# Data-Driven Estimation of Significant Kinetic Parameters Applied to the Synthesis of Polyolefins

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## Abstract

**:**

## 1. Introduction

## 2. Process Modelling

## 3. Experimental Equipment and Setup

#### 3.1. Materials

#### 3.2. Process Description

## 4. Data-Driven Estimation of Significant Kinetic Parameters

#### 4.1. Parameter Selection: Global Sensitivity Analysis

- (1)
- Define an objective function, and the dimension ($D$) for a sample of input parameters. For each parameter, define an uncertainty index. In this case, we adopted 4% of change with respect to the mean value.
- (2)
- Build three random matrices ${\mathit{M}}_{\mathbf{1}}$, ${\mathit{M}}_{\mathbf{2}}$ and ${\mathit{M}}_{\mathbf{3}}$—Equations (33a)–(33c), respectively—of dimension $D\times J$ based on the defined uncertainty: ${\mathit{M}}_{\mathbf{1}}$ is the sampling matrix, ${\mathit{M}}_{\mathbf{2}}$ is the resampling matrix, and ${\mathit{M}}_{\mathbf{3}}$ is the backup matrix.$${\mathit{M}}_{\mathbf{1}}=\left[\begin{array}{ccc}{z}_{11}\cdots & {z}_{1j}\cdots & {z}_{1J}\\ \vdots & \vdots & \vdots \\ {z}_{D1}\cdots & {z}_{Dj}\cdots & {z}_{DJ}\end{array}\right]$$$${\mathit{M}}_{\mathbf{2}}=\left[\begin{array}{ccc}{z}_{11}^{\prime}\cdots & {z}_{1j}^{\prime}\cdots & {z}_{1J}^{\prime}\\ \vdots & \vdots & \vdots \\ {z}_{D1}^{\prime}\cdots & {z}_{Dj}^{\prime}\cdots & {z}_{DJ}^{\prime}\end{array}\right]$$$${\mathit{M}}_{\mathbf{3}}=\left[\begin{array}{ccc}{z}_{11}^{\u2033}\cdots & {z}_{1j}^{\u2033}\cdots & {z}_{1J}^{\u2033}\\ \vdots & \vdots & \vdots \\ {z}_{D1}^{\u2033}\cdots & {z}_{Dj}^{\u2033}\cdots & {z}_{DJ}^{\u2033}\end{array}\right]$$
- (3)
- Evaluate the row vectors of matrices ${\mathit{M}}_{\mathbf{1}}$ and ${\mathit{M}}_{\mathbf{2}}$. If the output is unfeasible, meaning that the combination of inputs in a vector caused the simulation to break or other related problems, use the next available feasible row of the matrix ${\mathit{M}}_{\mathbf{3}}$, and update the matrices to ${\mathit{M}}_{\mathbf{1}}\prime $ and ${\mathit{M}}_{\mathbf{2}}\prime $, which denote the improved sampling and resampling matrices, respectively. Then, calculate the total average (${\widehat{g}}_{0}$) of both evaluations as described in Equation (34).$${\mathit{g}}_{\mathit{S}}=g\left({\mathit{M}}_{\mathbf{1}}\prime \right),{\mathit{g}}_{\mathit{R}}=g\left({\mathit{M}}_{\mathbf{2}}\prime \right)$$$${\widehat{g}}_{0}=\frac{1}{2D}{\displaystyle \sum}_{d=1}^{D}\left({\mathit{g}}_{\mathit{S}}+{\mathit{g}}_{\mathit{R}}\right)$$
- (4)
- Generate a matrix ${\mathit{N}}_{\mathit{q}}$ formed by all columns of matrix ${\mathit{M}}_{\mathbf{2}}\prime $, except the column of the ${z}_{\mathrm{q}}$ parameter, which is pulled from ${\mathit{M}}_{\mathbf{1}}\prime $, as explained in Equation (35a). Subsequently, generate another matrix ${\mathit{N}}_{\mathit{T}\mathit{j}}$ formed with all columns of ${\mathit{M}}_{\mathit{1}}\prime $ and with the column of the ${z}_{j}^{\prime}$ parameter, pulled from ${\mathit{M}}_{\mathbf{2}}\prime $ as denoted in Equation (35b).$${\mathit{N}}_{\mathit{j}}=\left[\begin{array}{ccc}{z}_{11}^{\prime}\cdots & {z}_{1j}\cdots & {z}_{1J}^{\prime}\\ \vdots & \vdots & \vdots \\ {z}_{D1}^{\prime}\cdots & {z}_{Dj}\cdots & {z}_{DJ}^{\prime}\end{array}\right]$$$${\mathit{N}}_{\mathit{T}\mathit{j}}=\left[\begin{array}{ccc}{z}_{11}\cdots & {z}_{1j}^{\prime}\cdots & {z}_{1J}\\ \vdots & \vdots & \vdots \\ {z}_{D1}\cdots & {z}_{Dj}^{\prime}\cdots & {z}_{DJ}\end{array}\right]$$
- (5)
- Evaluate the row vectors of matrices ${\mathit{N}}_{\mathit{j}}$ and ${\mathit{N}}_{\mathit{T}\mathit{j}}$. If an evaluated function is unfeasible, the output is replaced by ${\widehat{g}}_{0}$. The outputs are obtained in column vectors.$${{\mathit{g}}^{\prime}}_{\mathit{j}}=g\left({\mathit{N}}_{\mathit{j}}\right),{\mathit{g}}_{\mathit{R}}^{\prime}{}_{\mathit{j}}=g\left({\mathit{N}}_{Tj}\right)$$
- (6)
- A sample generates the following estimates, which are calculated based on scalar products of the vectors from above.$${\gamma}_{q}{}^{2}=\frac{1}{2D}{\displaystyle \sum}_{d=1}^{D}\left({\mathit{g}}_{\mathit{S}}\xb7{\mathit{g}}_{\mathit{R}}+{{\mathit{g}}^{\prime}}_{\mathit{j}}\xb7{\mathit{g}}_{\mathit{R}}^{\prime}{}_{\mathit{j}}\right)$$$$\widehat{V}=\frac{1}{2D}{\displaystyle \sum}_{d=1}^{D}\left({\mathit{g}}_{\mathit{S}}{}^{2}+{\mathit{g}}_{\mathit{R}}{}^{2}\right)-{\widehat{g}}_{0}{}^{2}$$$${\widehat{V}}_{q}=\frac{1}{2D}{\displaystyle \sum}_{d=1}^{D}\left({\mathit{g}}_{\mathit{S}}\xb7{\mathit{g}}_{\mathit{R}}^{\prime}{}_{\mathit{j}}+{\mathit{g}}_{\mathit{R}}\xb7{{\mathit{g}}^{\prime}}_{\mathit{j}}\right)-{\gamma}_{j}{}^{2}$$$${\widehat{V}}_{-q}=\frac{1}{2D}{\displaystyle \sum}_{d=1}^{D}\left({\mathit{g}}_{\mathit{S}}\xb7{{\mathit{g}}^{\prime}}_{\mathit{j}}+{\mathit{g}}_{\mathit{R}}\xb7{\mathit{g}}_{\mathit{R}}^{\prime}{}_{\mathit{j}}\right)-{\gamma}_{j}{}^{2}$$The selection of sensitive parameters relies on the first and total sensitivity indices.Equation (40) introduces the objective function, defined in this case as:$$G={\displaystyle \sum}_{i=1}^{i}\frac{{y}_{i}-{h}_{i}\left(z\right)}{{\sigma}_{{y}_{i}}^{2}}$$

#### 4.2. Data-Driven Parameter Estimation

#### 4.2.1. Estimation Problem

#### 4.2.2. Retrospective Cost Model Refinement (RCMR) Algorithm

#### 4.3. Framework Implementation

**,**it does not provide knowledge for the parameter updates. The estimated parameter $\widehat{\mathit{z}}$ is updated by the estimator, which seeks the minimization of the error signal $\mathit{e}$.

## 5. Results

#### 5.1. Global Sensitivity Analysis

#### 5.2. Homopolymerization

#### 5.3. Copolymerization

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Notation | |

$C$ | catalyst precursor, [mol] |

${C}^{*}$ | active catalyst site, [mol] |

$D$ | 1,9-decadiene, [mol] |

$DC$ | dead catalyst site, [mol] |

$d$ | total amount of 1,9-decadiene inserted into the growing polymer chains |

${E}_{a}{}_{p}$ | activation energy for ethylene propagation reaction rate, [J·mol^{−1}] |

${F}_{M}$ | ethylene feed flow rate, [mol·s^{−1}] |

$K$ | total number of pendant unsaturation’s present in the dead chains |

${k}_{a}$ | catalyst activation constant, [s^{−1}] |

${k}_{b}$ | macromonomer reincorporation rate constant, [L·mol^{−1}·s^{−1}] |

${k}_{dP}$ | living chain deactivation rate constant, [L·mol^{−1}·s^{−1}] |

${k}_{{p}_{11}}$ | propagation rate constant for ethylene, [L·mol^{−1}·s^{−1}] |

${k}_{{p}_{12}}$ | propagation rate constant for 1,9-decadiene, [L·mol^{−1}·s^{−1}] |

${k}_{t}$ | termination rate constant, [s^{−1}] |

${k}_{0}{}_{p}$ | pre-exponential factor for ethylene propagation reaction rate, [L·mol^{−1}·s^{−1}] |

$lcb$ | long chain branching |

${L}_{i}^{=}$ | dead polymer chain that contains a terminal unsaturation and has chain size i, [mol] |

${L}_{i}$ | dead polymer chain with chain size i and without a terminal unsaturation, [mol] |

$M$ | ethylene, [mol] |

$m$ | total amount of ethylene inserted into the growing polymer chains |

$MM$ | average molar mass of the repeating unit, [g·mol^{−1}] |

$M{M}_{D}$ | molar mass of 1,9-decadiene, [g·mol^{−1}] |

$M{M}_{M}$ | molar mass of ethylene, [g·mol^{−1}] |

${M}_{n}$ | number average molecular weight, [g·mol^{−1}] |

${M}_{w}$ | weight average molecular weight, [g·mol^{−1}] |

$PDI$ | polydispersity index |

${P}_{i}^{*}$ | living polymer chain with chain size i, [mol] |

$R$ | ideal gas constant, [J·mol^{−1}·K^{−1}] |

$T$ | reaction temperature, [K or °C] |

${T}_{r}$ | reference temperature, [K or °C] |

$V$ | reaction mixture volume, [L] |

Greek letters | |

${\lambda}_{k}$ | k^{th} moment for the dead chain |

${\mu}_{k}$ | k^{th} moment for the living chain |

${\rho}_{PE}$ | Polyethylene density, [g·L^{−1}] |

$\phi $ | Average frequency of pendant double bonds in the polymer chains |

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**Figure 3.**Global sensitivity indices: (

**a**) first-order sensitivity index; (

**b**) total sensitivity index.

**Figure 4.**Estimation of a single significant parameter. Comparison between theoretical (dashed line), and estimated kinetic parameters (continuous line) at 120 °C (blue), 130 °C (green) 140 °C (orange): (

**a**) Dynamic estimation of ${k}_{7}$; (

**b**) Dynamic estimation of ${k}_{\mathrm{p}}{}_{11}$.

**Figure 5.**Estimation of two significant parameters. Comparison between theoretical (dashed line), and estimated kinetic parameters (continuous line) at 120 °C (blue), 130 °C (green) 140 °C (orange): (

**a**) Dynamic estimation of ${k}_{5}$; (

**b**) Dynamic estimation of ${k}_{dP}$; (

**c**) Dynamic estimation of ${k}_{7}$, (

**d**) Dynamic estimation of ${k}_{\mathrm{p}}{}_{11}$.

**Figure 6.**Comparison of the monomer flow rate (${F}_{\mathrm{M}}$) between the experimental values (circles), fundamental model (dashed line), estimated with a single parameter (continuous red line), and estimated with two parameters (continuous green line) at: (

**a**) 120 °C, (

**b**) 130 °C, (

**c**) 140 °C.

**Figure 7.**Comparison of the final average properties at different temperatures for experimental values (circles), fundamental model (squares–dashed line), estimated with a single parameter (triangles–continuous line), and estimated with two parameters (diamonds–continuous line): (

**a**) weight-average molecular weight (${M}_{\mathrm{w}}$); (

**b**) number-average molecular weight (${M}_{\mathrm{n}}$).

**Figure 8.**Estimation of a single significant parameter. Comparison between theoretical (dashed line), and estimated kinetic parameters (continuous line) at different initial diene concentrations for the copolymerization experiment A (purple), and copolymerization experiment B (gold); (

**a**) Dynamic estimation of ${k}_{7}$; (

**b**) Dynamic estimation of ${k}_{\mathrm{p}}{}_{11}$.

**Figure 9.**Estimation of two significant parameters. Comparison between theoretical (dashed line), and estimated kinetic parameters (continuous line) at different initial diene concentrations for the copolymerization experiment A (purple), and copolymerization experiment B (gold): (

**a**) Dynamic estimation of ${k}_{7}$; (

**b**) Dynamic estimation of ${k}_{\mathrm{p}}{}_{11}$; (

**c**) Dynamic estimation of ${k}_{5}$; (

**d**) Dynamic estimation of ${k}_{\mathrm{d}}{}_{P}$.

**Figure 10.**Comparison of the monomer flow rate (${F}_{\mathrm{M}}$) during copolymerization between the experimental values (circles), fundamental model (dashed line), estimated with a single parameter (continuous red line), and estimated with two parameters (continuous green line) at different initial diene concentrations: (

**a**) copolymerization experiment A, (

**b**) copolymerization experiment B.

**Figure 11.**Comparison of average polymer properties at different initial diene concentrations between experimental values (circles), fundamental model (dashed line), estimated with a single parameter (red continuous line), and estimated with two parameters (green continuous line). (

**a**) Weight-average molecular weight (${M}_{\mathrm{w}}$) for copolymerization A; (

**b**) number-average molecular weight (${M}_{\mathrm{n}}$) for copolymerization A; (

**c**) weight-average molecular weight (${M}_{\mathrm{w}}$) for copolymerization B; (

**d**) number-average molecular weight (${M}_{\mathrm{n}}$) for copolymerization B.

Rate Constant | Arrhenius Equation |
---|---|

Catalyst activation | ${k}_{a}=exp\left[{k}_{1}+{k}_{2}\left(\frac{T-{T}_{r}}{T}\right)\right]$ |

Propagation | ${k}_{{p}_{11}}={k}_{0}{}_{p}exp\left(-\frac{{E}_{a}{}_{p}}{RT}\right),{k}_{0}{}_{p}={10}^{{k}_{7}}$ |

Monomer transfer & β-hydride elimination | ${k}_{t}=exp\left[{k}_{3}+{k}_{4}\left(\frac{T-{T}_{r}}{T}\right)\right]$ |

Living chain deactivation | ${k}_{dP}=exp\left[{k}_{5}+{k}_{6}\left(\frac{T-{T}_{r}}{T}\right)\right]$ |

T, [C°] | [C_{2}H_{4}], [mol L^{−1}] | $\left[\mathit{M}\right]$, [mol] |
---|---|---|

120 | 0.49472 | 0.07420 |

130 | 0.43732 | 0.06560 |

140 | 0.38141 | 0.05721 |

**Table 3.**Parameters of the copolymerization of ethylene with 1,9-decadiene using dimethylsilyl (N-tert-butylamido) (tetramethylcyclopentadienyl) titanium dichloride (CGC)/MAO.

Parameter | Value | Units | Original Rate Constants ^{1} |
---|---|---|---|

${k}_{1}$ | −2.92 | - | ${k}_{a}=\alpha \left({k}_{1},{k}_{2}\right)$ |

${k}_{2}$ | 25.00 | - | |

${k}_{3}$ | 2.58 ± 0.08 | - | ${k}_{t}=\alpha \left({k}_{3},{k}_{4}\right)$ |

${k}_{4}$ | 17.2 | - | |

${k}_{5}$ | 10.91 ± 0.95 | - | ${k}_{dP}=\alpha \left({k}_{5},{k}_{6}\right)$ |

${k}_{6}$ | 30.12 ± 2.10 | - | |

${k}_{7}$ | 7.56 ± 0.07 | - | ${k}_{{p}_{11}}=\alpha \left({k}_{7}\right)$ |

${k}_{{p}_{12}}$ | 2039.8 ± 54.7 | L mol^{−1}s^{−1} | - |

${k}_{b}$ | 908.7 ± 69.0 | L mol^{−1}s^{−1} | - |

${E}_{a}{}_{p}$ | 20520.0 | J mol^{−1} | - |

$M{M}_{M}$ | 28.05 | g mol^{−1} | - |

$M{M}_{D}$ | 138.254 | g mol^{−1} | - |

${\rho}_{PE}$ | 940 | g L^{−1} | - |

$R$ | 8.31451 | J mol^{−1} K^{−1} | - |

^{1}${k}_{\iota}=\alpha \left({k}_{\beta},{k}_{\gamma}\right)$ represents ${k}_{\iota}$ as a function of ${k}_{\beta}$ and ${k}_{\gamma}$.

Variable | Homopolymerization | Copolymerization | Units | |
---|---|---|---|---|

A | B | |||

$C\left(0\right)$ | $0.767\times {10}^{-6}\times \mathrm{V}\left(0\right)$ | $0.271\times {10}^{-6}\times \mathrm{V}\left(0\right)$ | $0.271\times {10}^{-6}\times \mathrm{V}\left(0\right)$ | $\mathrm{mol}$ |

${C}^{*}\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

$DC\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

$D\left(0\right)$ | 0 | $0.3\xf7M{M}_{D}$ | $0.4\xf7M{M}_{D}$ | $\mathrm{mol}$ |

$\left(M\right)\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

$\left(d\right)\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

$\left(lcb\right)\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

${\mu}_{0}\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

${\mu}_{1}\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

${\mu}_{2}\left(0\right)$ | 0 | 0 | 0 | $\mathrm{mol}$ |

${\lambda}_{0}\left(0\right)$ | ${10}^{-14}$ | ${10}^{-14}$ | ${10}^{-14}$ | $\mathrm{mol}$ |

${\lambda}_{1}\left(0\right)$ | ${10}^{-14}$ | ${10}^{-14}$ | ${10}^{-14}$ | $\mathrm{mol}$ |

${\lambda}_{2}\left(0\right)$ | ${10}^{-14}$ | ${10}^{-14}$ | ${10}^{-14}$ | $\mathrm{mol}$ |

$V\left(0\right)$ | 0.15 | 0.15 | 0.15 | $\mathrm{L}$ |

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## Share and Cite

**MDPI and ACS Style**

Salas, S.D.; Brandão, A.L.T.; Soares, J.B.P.; Romagnoli, J.A.
Data-Driven Estimation of Significant Kinetic Parameters Applied to the Synthesis of Polyolefins. *Processes* **2019**, *7*, 309.
https://doi.org/10.3390/pr7050309

**AMA Style**

Salas SD, Brandão ALT, Soares JBP, Romagnoli JA.
Data-Driven Estimation of Significant Kinetic Parameters Applied to the Synthesis of Polyolefins. *Processes*. 2019; 7(5):309.
https://doi.org/10.3390/pr7050309

**Chicago/Turabian Style**

Salas, Santiago D., Amanda L. T. Brandão, João B. P. Soares, and José A. Romagnoli.
2019. "Data-Driven Estimation of Significant Kinetic Parameters Applied to the Synthesis of Polyolefins" *Processes* 7, no. 5: 309.
https://doi.org/10.3390/pr7050309