Principal differential analysis (PDA) is an alternative parameter estimation technique for differential equation models in which basis functions (e.g., B-splines) are fitted to dynamic data. Derivatives of the resulting empirical expressions are used to avoid solving differential equations when estimating parameters. Benefits and shortcomings of PDA were examined using a simple continuous stirred-tank reactor (CSTR) model. Although PDA required considerably less computational effort than traditional nonlinear regression, parameter estimates from PDA were less precise. Sparse and noisy data resulted in poor spline fits and misleading derivative information, leading to poor parameter estimates. These problems are addressed by a new iterative algorithm (iPDA) in which the spline fits are improved using model-based penalties. Parameter estimates from iPDA were unbiased and more precise than those from standard PDA. Issues that need to be resolved before iPDA can be used for more complex models are discussed.

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A simplified steady‐state model to predict MWDs of ethylene/butene and ethylene/hexene copolymers produced industrially using heterogeneous Z‐N catalysts is developed. Estimability analysis is used to guide model simplification and to determine which parameters can be estimated using the available data. Scaling of response variables and parameters using information about their uncertainties ensures that appropriate results are obtained from the estimability analysis. Parameter estimates are obtained to provide good predictions of the measured MWDs. Although the parameter values obtained are specific to the Z‐N catalyst of our industrial sponsor, the method should be useful for parameter estimation and model simplification in other catalytic polymerization systems.magnified image

Fluidized-bed polyetliylene reactors are prone to unstable behavior and temperature oscillations (Choi and Ray, 1YNSb

A simplified steady‐state model has been developed to predict molecular weight distributions and average compositions of ethylene‐hexene copolymers produced using heterogeneous Ziegler‐Natta catalysts in gas‐phase reactors. The model uses a simplified reaction scheme to limit the number of parameters that must be estimated. The number of parameters is further reduced by assuming that different types of active sites share common rate constants for some reactions. Estimates of kinetic parameters are obtained using deconvolution analysis of industrial copolymer samples produced using a variety of isothermal steady‐state operating conditions. The parameter estimates should prove useful as initial guesses for future parameter estimation in a non‐isothermal model.magnified image

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An empirical correlation, based on conventional forms, has been developed to represent the oxygen mass transfer coefficient as a function of operating conditions and organic fraction in two-phase, aqueous-organic dispersions. Such dispersions are characteristic of two-phase partitioning bioreactors, which have found increasing application for the biodegradation of toxic substrates. In this work, a critical distinction is made between the oxygen mass transfer coefficient, k(L)a, and the oxygen mass transfer rate. With an increasing organic fraction, the mass transfer coefficient decreases, whereas the oxygen transfer rate is predicted to increase to an optimal value. Use of the correlation assumes that the two-phase dispersion behaves as a single homogeneous phase with physical properties equivalent to the weighted volume-averaged values of the phases. The addition of a second, immiscible liquid phase with a high solubility of oxygen to an aqueous medium increases the oxygen solubility of the system. It is the increase in oxygen solubility that provides the potential for oxygen mass transfer rate enhancement. For the case studied in which n-hexadecane is selected as the second liquid phase, additions of up to 33% organic volume lead to significant increases in oxygen mass transfer rate, with an optimal increase of 58.5% predicted using a 27% organic phase volume. For this system, the predicted oxygen mass transfer enhancements due to organic-phase addition are found to be insensitive to the other operating variables, suggesting that organic-phase addition is always a viable option for oxygen mass transfer rate enhancement.

Abstract:The mechanism causing oscillation in continuous ethanol fermentation by Zymomonas mobilis under certain operating conditions has been examined. A new term, ''dynamic specific growth rate,'' which considers inhibitory culture conditions in the recent past affecting subsequent cell behavior, is proposed in this article. Based on this concept, a model was formulated to simulate the oscillatory behavior in continuous fermentation of Zymomonas mobilis. Forced oscillation fermentation experiments, in which exogenous ethanol was added at a controlled rate to generate oscillatory behavior, were performed in order to obtain estimates for the model parameters and to validate the proposed model. In addition, data from a literature example of a sustained oscillation were analyzed by means of the model, and excellent agreement between the model simulation and experimental results was obtained. The lag in the cells' response to a changing environment, i.e., ethanol concentration change rate experienced by the cells, was shown to be the major factor contributing to the oscillatory behavior in continuous fermentation of Zymomonas mobilis under certain operating conditions.