Parallel models for arborescent polyisobutylene synthesized in batch reactor

Zhao, Yutian R.; McAuley, Kimberley B.; Puskás, Judit E.

doi:10.1002/aic.14655

Cited by 8 publications

(37 citation statements)

References 56 publications

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“…Different from our previous study, not only [IB] and

{\overset{M}{true}}_{normaln}

data are used for parameter estimation, but

{\overset{M}{true}}_{normalw}

data are also used. Note B kin “data” were not used for parameter estimation, because B kin data were calculated by using

{\overset{M}{true}}_{normaln}

data in Equations and and as a result, there is no extra information in the B kin data that would help the parameter estimation.…”

Section: Experimental Datamentioning

confidence: 99%

“…The weighting factors in the denominator of Equation are pooled variances calculated from three replicate experiments with [IM] 0 = 0.00114 mol L −1 ( s IB 2 = 0.01485 mol 2 L −2 ,

s_{{\overset{M}{true}}_{normaln}}^{2}

= 313.255 kg 2 mol −2 , and

s_{{\overset{M}{true}}_{normalw}}^{2}

= 437.59 kg 2 mol −2 ). Parameter estimation was performed 21 times, starting at the initial guesses from our two previous parameter estimation studies, the converged parameter values from our most recent study, and 18 other randomly chosen sets of parameter values that were between the upper and lower bounds specified in our previous parameter estimation studies . Many of the parameter estimation attempts resulted in convergence to local optima.…”

Section: Experimental Datamentioning

confidence: 99%

“…The average branching level can be calculated according to Paulo and Puskas

B_{kin} = \frac{{\overset{M}{true}}_{normaln}}{{\overset{M}{true}}_{n,theo}} - 1

{\overset{M}{true}}_{n,theo} = M_{IM} + \frac{{[IB]}_{0} - [IB]}{{[IM]}_{0} - [IM]} M_{IB}

where

{\overset{M}{true}}_{normaln}

is the number weight average molecular weight and

{\overset{M}{true}}_{n,theo}

is the theoretical molecular weight when IM behaves only as an initiator, i.e., the vinyl groups on IM molecules are not permitted to react. Note that in calculation, it is assumed that the IM is consumed immediately after the reaction starts (i.e., [IM] = 0 at any time during the batch), since the initial value of IM is small in the experiments, but this assumption is not used in our models …”

Section: Introductionmentioning

confidence: 99%

“…Preliminary estimation of the remaining four apparent rate constants was performed using experimental data, revealing that the assumption of ≤15 IM units per molecule was not valid for typical batch experiments. To address this difficulty, we developed a second PREDICI model that can predict

{\overset{M}{true}}_{normaln}

, but not

{\overset{M}{true}}_{normalw}

nor the MWD . This model was used to estimate all six apparent propagation rate constants for the first time, giving a better match to the experimental data than our previous PREDICI model.…”

Section: Introductionmentioning

confidence: 99%

“…Typical simulation times for a 2 h batch reactor experiment were less than 0.2 s. As expected, the computational times for solving the ODEs was fast compared to our previous Monte Carlo simulations, because the Monte Carlo simulations needed to simulate a large number of individual polymer molecules to obtain accurate results. The model was also implemented in PREDICI, which is convenient for parameter estimation, especially since our previous parameter estimation studies were performed using PREDICI . Material balances on the vinyl groups and end groups were handled automatically by PREDICI by specifying reactions 1 to 6 in Table .…”

Section: Introductionmentioning

confidence: 99%

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Applying Multidimensional Method of Moments for Modeling and Estimating Parameters for Arborescent Polyisobutylene Production in Batch Reactor

Zhao

Arriola

Puskás

et al. 2016

Macro Theory & Simulations

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A mathematical model is developed for the arborescent polyisobutylene system in a batch reactor, using multidimensional method of moments, to predict the concentrations of monomer and inimer as well as number and weight average molecular weight. This model is significantly efficient in computation, making parameter estimation practical. Simulation results agree with results obtained by Monte Carlo simulations. Parameter estimation results show that using the weight average molecular weight data provide better overall fit than leaving them out in the previous model.

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“…Different from our previous study, not only [IB] and

{\overset{M}{true}}_{normaln}

data are used for parameter estimation, but

{\overset{M}{true}}_{normalw}

data are also used. Note B kin “data” were not used for parameter estimation, because B kin data were calculated by using

{\overset{M}{true}}_{normaln}

data in Equations and and as a result, there is no extra information in the B kin data that would help the parameter estimation.…”

Section: Experimental Datamentioning

confidence: 99%

“…The weighting factors in the denominator of Equation are pooled variances calculated from three replicate experiments with [IM] 0 = 0.00114 mol L −1 ( s IB 2 = 0.01485 mol 2 L −2 ,

s_{{\overset{M}{true}}_{normaln}}^{2}

= 313.255 kg 2 mol −2 , and

s_{{\overset{M}{true}}_{normalw}}^{2}

Section: Experimental Datamentioning

confidence: 99%

“…The average branching level can be calculated according to Paulo and Puskas

B_{kin} = \frac{{\overset{M}{true}}_{normaln}}{{\overset{M}{true}}_{n,theo}} - 1

{\overset{M}{true}}_{n,theo} = M_{IM} + \frac{{[IB]}_{0} - [IB]}{{[IM]}_{0} - [IM]} M_{IB}

where

{\overset{M}{true}}_{normaln}

is the number weight average molecular weight and

{\overset{M}{true}}_{n,theo}

Section: Introductionmentioning

confidence: 99%

{\overset{M}{true}}_{normaln}

, but not

{\overset{M}{true}}_{normalw}

nor the MWD . This model was used to estimate all six apparent propagation rate constants for the first time, giving a better match to the experimental data than our previous PREDICI model.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Applying Multidimensional Method of Moments for Modeling and Estimating Parameters for Arborescent Polyisobutylene Production in Batch Reactor

Zhao

Arriola

Puskás

et al. 2016

Macro Theory & Simulations

Self Cite

View full text Add to dashboard Cite

show abstract

Parameter estimation and estimability analysis in pharmaceutical models with uncertain inputs

et al. 2023

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A methodology is proposed to aid parameter estimation in fundamental models of pharmaceutical processes. This methodology addresses situations with insufficient data to reliably estimate all parameters, when the estimation is complicated by uncertain independent variables. The proposed method uses an augmented sensitivity matrix to rank the combined set of parameters and uncertain inputs from most estimable to least estimable. An updated mean‐squared‐error criterion is then used to determine the appropriate parameters and inputs that should be estimated, based on the ranked list. A model for one step in a batch pharmaceutical production process with an uncertain initial reactant concentration is used to illustrate the method, revealing that the initial reactant concentration in each batch should be estimated along with three out of six model parameters. Non‐estimable parameters are fixed at their initial values to prevent overfitting. The method will aid error‐in‐variables parameter estimation in many situations involving limited data.

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Advanced Monte Carlo Modeling Using Weight‐Based Selection of Arborescent Polyisobutylene Molecules in a Batch Reactor

Zhao

McAuley

Iedema

et al. 2016

Macro Theory & Simulations

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and copolymerization with one or more different vinyl monomers, which is called self-condensing vinyl copolymerization (SCVCP). [5][6][7][8][9][10][11][12][13] Puskas and Grasmüller have developed a "one-pot" living carbocationic copolymerization of IM and isobutylene (IB) to produce arborescent polyisobutylene ( arb PIB), [ 14,15 ] which greatly reduced the labor required compared with previous multi-step processes. [16][17][18] This one-pot living copolymerization system is illustrated in Figure 2 , where there are two types of vinyl groups, i.e., V I and V M , and three types of chloride end groups, i.e., C I , C M , and C S . V I is the vinyl group on the IM and V M is the vinyl group on the IB monomer. C I is an unreacted chloride group on the IM; C M is a chloride group formed after a reaction with V M , and C S is a chloride group formed after a reaction with V I . Each type of chloride group can be uncapped to form an active carbocation that can react with either type of vinyl group. As a result, there are six different propagation rate constants, which are named k pII , k pIM , k pMI , k pMM , k pSI and k pSM . The fi rst subscript letter after k p is the type of chloride group that is uncapped to produce the reactive carbocation and the second subscript is the type of vinyl group that is consumed. The corresponding propagation reactions are summarized in Table 1 , along with estimated values of the corresponding apparent rate constants. The apparent rate constants in An advanced Monte Carlo (MC) method is developed, using weight-based selection of polymer chains, to predict the molecular weight distribution (MWD) and branching level for arborescent polyisobutylene ( arb PIB) at the end of a batch reaction. This new weight-based MC method uses differential equations and random numbers to determine the detailed structure of arb PIB molecules. Results agree with those from an advanced number-based MC method. The proposed weight-based algorithm requires approximately twice the computation time of the numberbased method, but produces more accurate results in the high-molecular-weight portion of the MWD when the same number of polymer chains is assembled.

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Parallel models for arborescent polyisobutylene synthesized in batch reactor

Cited by 8 publications

References 56 publications

Applying Multidimensional Method of Moments for Modeling and Estimating Parameters for Arborescent Polyisobutylene Production in Batch Reactor

Applying Multidimensional Method of Moments for Modeling and Estimating Parameters for Arborescent Polyisobutylene Production in Batch Reactor

Parameter estimation and estimability analysis in pharmaceutical models with uncertain inputs

Advanced Monte Carlo Modeling Using Weight‐Based Selection of Arborescent Polyisobutylene Molecules in a Batch Reactor

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