A JOINT EXPERIMENTAL DESIGN PROCEDURE FOR PARAMETER ESTIMATION

A joint experimental design procedure which aims at obtaining precise parameter estimates of a given mathematical model is proposed and illustrated by means of computer simulated experiments

This criterion is found both to modify the shape of the joint confidence hyperellipsoid and to reduce its volume significantly

Therefore, it behaves as a combination of two of the most representative techniques in use today, known as the minimum volume or the shape criteria. Numerical results are given which serve to display its salient features and to compare it with the above mentioned techniques.     

用序贯法对反应速率模型中的参数作精确估算

对一个确定的模型,不仅要确定参数的估算值,还必须了解其可靠性。为了缩小参数的不定性,降低参数间的相关性,以扩大模型拟合的适定性和准确度,本文讨论了参数精确估算的序贯设计过程。 文内还详细讨论了精估参数的实验设计准则——最小联合置信容积准则和形状准则,以及序贯设计的实用价值。以氨合成为例,用序贯法精确估算了反应速度模型中的参数。这一模型是在序贯判别过程中选定的。把精估结果用于拟合大型生产过程的实测数据,得到了满意的结果。     

Precise parameter estimation in reaction modelling — sensitivity in choosing experimental conditions

A sequential experimental design developed by Box and coworkers was reviewed and the sensitivity of experimental conditions in increasing the preciseness of parameter estimates was examined. Continuous kinetic data were utilized for the examination. It is found that experimental conditions are in some cases rather sinsitive and slightly deviated experimental conditions from those specified by the criterion may induce some difficulty in reducing the size of confidence region of parameter estimates.     

A SEQUENTIAL PROCEDURE FOR PRECISE ESTIMATION OF PARAMETERS IN KINETIC MODEL

For a determined mathematical model,it is not only necessary to obtain the estimates of parameters,but also their reliabilites.The present work deals with a sequential statistical design strategy for anexperimental program to obtain precise parameter estimetes for a nonlinear kinetic model.It aims at de-creasing the indeterminatness of the parameter estimates,in reducing the correlation between parameterestimates,and in inceasing the adequacy and accuracy of the model.The paper discusses the experimental design criterion for precise parameter estimates-the minimun vol-ume design criterion and the spherical shape design criterion,and the practical value of a sequential statis-tical design strategy.At last,an example was performed to precise estimate the parameters in the kinetic model of ammoniasynthesis reaction which was selected as being adequate in the previous work.Comparison of the results from the industrial unit and the corresponding values calculated with themodel using the estimates of parameters reveals the reasonable reliability of the model.     

Characterization and modeling of micromixing performance in micropore dispersion reactors

Micropore dispersion reactors have been considered as one of the most promising micro-structured devices. For reliable design of this kind of microreactors, the micromixing performance in those microreactors with different detailed geometric structures have been investigated in this work. The pore size, pore shape, pore number and pore distance were varied and the micromixing performance was characterized by the Villermaux/Dushman parallel competing reaction. The results showed that the mixing performance was greatly influenced by the geometric structures. The segregation indexes, X_S, were found in the range of 10⁻² to 10⁻³, indicating that the micropore dispersion reactors have high micromixing efficiency. To deeply understand the micromixing process, CFD simulation was carried out to describe the flow fields in the reactors. Based on the simulation results a mathematical model was developed and a new area parameter, S, was defined by considering both mixing region surface and mass transfer distance. A linear relationship between S and lg(X_S) was obtained at last, which is very helpful for optimizing the structure design of micropore dispersion reactors.     

A combined approach to optimize the drying process of flavonoid-rich leaves (Inga edulis) using experimental design and mathematical modelling

The process of drying of the leaves of Inga edulis, a flavonoid-rich plant from Amazonia, was optimised in a forced convection dryer by combining experimental design and mathematical modelling. First, the monolayer moisture content was determined by hygroscopic studies at 25 °C and the value of this parameter (5.86 g/100 g dry matter) was fixed as a minimum to be achieved. Next, a 3² full-factorial design was applied, with temperature (40, 55 and 70 °C) and airflow velocity (0.6, 1.2 and 1.8 m/s) as the independent variables, the dependent variable being the drying time required to reduce the moisture content until the level determined in the monolayer was achieved. Nine kinetic models, with one to four parameters, were fitted to the experimental data. The fitting performance of these models was assessed using statistical criteria and, from these results, the Henderson and Pabis model was found to be sufficient to predict the monolayer drying times. Subsequently, a multiple linear regression was carried out by applying a second-order model to describe the relationship between the independent and dependent variables, which established the adequacy of this model (R² > 0.97) and, thus, the applicability of the combined statistical and mathematical approaches. Finally, an experimental domain that could minimise the drying times (time < 40 min) was determined. An experimental validation of the model was accomplished in a point of this domain and good agreement was found between the experimental and predicted values. Moreover, no flavonol degradation was observed, whereas a loss of 19.4% for the total flavanols was found.     

Nonlinear parameter estimation through particle swarm optimization

Parameter estimation procedures are very important in the chemical engineering field for development of mathematical models, since design, optimization and advanced control of chemical processes depend on model parameter values obtained from experimental data. Model nonlinearity makes the estimation of parameter and the statistical analysis of parameter estimates more difficult and more challenging. In this work, it is shown that many of these difficulties can be overcome with the use of heuristic optimization methods, such as the particle swarm optimization (PSO) method. Parameter estimation problems are solved here with PSO and it is shown that the PSO method is efficient for both minimization and construction of the confidence region of parameter estimates. Moreover, it is shown that the elliptical approximation of confidence regions of nonlinear model parameters can be very poor sometimes and that more accurate likelihood confidence regions can be constructed with PSO, allowing for more reliable statistical analysis of the significance of parameter estimates.     

Sequential statistical design strategy in an experimental kinetic study

The value of using a sequential statistical design strategy is illustrated through its use in an experimental kinetic study of the oxidation of o-xylene over a vanadium oxide catalyst. The rate of oxidation of o-xylene was determined in a differential reactor over the following ranges of conditions: 270 to 300°C, (0.5 to 3.5) x 10^-4 g.mole o-xylene/1., (1.0 to 10.0) x 10^-3 g.mole oxygen/1. The statistical design strategy for precise parameter estimation was found to be effective in increasing the precision of the parameter estimates, in reducing the correlation between parameter estimates and in showing the magnitude of this problem, and in providing a reliable prediction of the value of carrying out additional runs.     

Comparison of HPLC and NMR characterization of the stereogenic properties of cyclotriphosphazene derivatives containing two equivalent centres of chirality: Cis (meso) and trans (racemic) isomers

The di-spiro derivatives of the reaction of gem-disubstituted cyclotriphosphazenes, N₃P₃Cl₄X₂ (X = Ph, PhS, PhNH, PhO) with 3-amino-1-propanol are expected to exist as cis and trans geometric isomers and exist as meso and racemic, respectively. The geometric isomers were separated by column chromatography on silica gel and analyzed by elemental analysis, mass spectrometry, and ³¹P and ¹H NMR spectroscopies. The stereogenic properties of all the compounds (trans, 2a–5a) and (cis, 2b–5b) were investigated by ³¹P NMR spectroscopy on the addition of a chiral solvating agent (CSA), (S)-(+)-2,2,2-trifluoro-1-(9′-anthryl)ethanol; one example of a derivative with the trans configuration (compound 5a, X = PhO) has been found where the ³¹P NMR/CSA method does not lead to the expected separation of the signals of enantiomers, even up to a molar ratio of CSA:compound of 50:1. On the other hand, chiral HPLC methods have been developed in order to characterize the trans (2a–5a) and cis (2b–5b) forms of the cyclotriphosphazene derivatives and give good separation of enantiomers for the trans disubstituted compounds (2a–5a). It is found that chiral HPLC is more reliable than the ³¹P NMR/CSA method for characterising the stereogenic properties of the cis and trans isomers of di-spiro 3-amino-1-propanoxy cyclophosphazene derivatives.     

Drying of solids; estimation of the mathematical model parameters

In order to determine physical meaning of a thin‐layer mathematical model parameter, the Page model was modified and tested on numerous experimental data. Applicability of the model was tested on the drying kinetics data, X(t), of 11 different types of the porous materials and dried in a convective, vacuum and/or microwave dryers under the predetermined external process conditions. During the entire drying time the drying kinetics of all the investigated materials and heating methods were successfully correlated with the modified Page model. The evaluated values of a new parameter, t_k, corresponded to the time at which diffusion, as a governing mechanism of moisture movement through the material, started. The results were confirmed by the pore size distribution of some materials.     

Mathematical modelling of styrene drying by adsorption onto activated alumina

This work is focused on the analysis and modelling of styrene drying, raw material in the manufacture of synthetic rubber, by means of adsorption onto activated alumina. Equilibrium experiments, carried out under isothermal conditions at 10°C, correlated to the equation q (kg/kg)=2.659×10⁻⁴ C (mg/kg). Fixed bed column experiments were performed working with different flow rates and using different bed lengths in order to obtain experimental breakthrough curves. A mathematical model that considers the influence of both film and pore mass transfer resistances described satisfactorily well the experimental results. Finally a value of D_p=6.101×10⁻⁹ m²/s was obtained from correlation of experimental data to simulated curves and using the minimum weighted standard deviation as optimisation criterion.     

Ozonation Kinetics of Acid Red 27 Azo Dye: A Novel Methodology Based on Artificial Neural Networks for the Determination of Dynamic Kinetic Constants in Bubble Column Reactors

A procedure for the determination of initial parameter values for quadratically convergent optimization methods is proposed using artificial neural networks coupled with a non-stationary gas-liquid reaction model. The evaluation of the regression and the mean squared error coefficients of the neural network during its training process allow the parameter sensitivity analysis of the gas-liquid model. This analysis examines how many and which parameters of the model will be available depending on the observable information of the mathematical model. Numerical simulations show the relevance of the initial values and the non-linearity of the objective function. The methodology has been applied to the study of the reaction of the azo-dye Acid Red 27 with ozone in acid media. The rate constant is in the order of (1.6 ± 0.1) 10³ M^?1 s^?1 under the experimental conditions.     

Sequential experimental design procedures for precise parameter estimation in ordinary differential equations

In this paper, it is shown how an experimental program for precise parameter estimation can be designed sequentially for the case that the mathematical model is given in the form of a set of ordinary differential equations. Two strategies are proposed. The first aims at minimizing the volume of the joint confidence region associated with the parameter estimates. The second attempts to alter as much as possible the shape towards a spherical region, by shortening the length of the longest principal axis of the confidence region to the maximum extent. The application of both criteria is illustrated by means of examples, representative for real problems in chemical reaction engineering. The techniques are easily applicable with our present day computing facilities. Qualitative indications are derived concerning the question when the use of an experimental design will result in an appreciable gain in significance for the parameter estimates.     

Methyl methacrylate-chloroprene copolymerization: an evaluation of copolymerization models

The composition and triad fraction data of Ebdon (Polymer 1974, 15, 782) for the copolymerization of methyl methacrylate with chloroprene have been re-evaluated using non-linear least squares curve-fitting procedures and statistical testing of conclusions. Using the compositions and triad fractions separately, best estimates of reactivity ratios together with the joint confidence intervals for these parameters for terminal and penultimate models have been calculated. It has been shown that the terminal model cannot describe adequately the experimental compositions or triad fractions. While the allowance of a penultimate effect for methyl methacrylate chain end radicals provides a significant improvement over the fit of the terminal model to the copolymer compositions, there is only a small probability that both the triad fraction and the composition data can be described adequately by one set of penultimate reactivity ratios.     

Necessary conditions for steady-state multiplicity for (m,n)-th order reactions and analysis of multiplicity patterns for a class of single reactions in perfectly mixed reactors

The first part of the paper presents exact necessary conditions for steady-state multiplicity for a general (m,n)-th order reaction as considered by Lin (1980, Chem. Engng Sci.35, 1537–1543). The effect of changes in the various parameters on the location of the cusp point is investigated.The second part presents an analysis of multiplicity patterns for a class of single reactions described by rate equations of the type = kc^pe^-E/RT/(1 + Kc)^q, as considered by Tsotsis et al. (1982, Chem. Engng Sci.37, 1235–1243), and which may exhibit up to five steady states. The construction of the hysteresis and double limit varieties allows the parameter space to be subdivided into subregions corresponding to all possible multiplicity patterns, 1, 1-3-1, 1-3-1-3-1, 1-3-5-3-1, and within the latter pattern corresponding to the four possible different bifurcation diagrams. Exact necessary and sufficient conditions are presented for all these types of hysteresis behaviour. Some numerical data are given which should be helpful in gaining some insight into this phenomenon.     

A Polymath Approach for the Prediction of Optimized Transesterification Process Variables of Polanga Biodiesel

An attempt has been made to employ an artificial neural network (ANN) combined with a genetic algorithm (GA) in MATLAB 7.0 for predicting the optimized reaction variables for maximum biodiesel production of polanga oil by the transesterification process. The developed ANN is a multilayer feed-forward back-propagation network (5-10-1) with five input, ten hidden and one output layers. The input variables are the molar ratio of ethanol to oil (X ₁ in % v/v), the catalyst concentration (X ₂ in % w/v), the reaction temperature (X ₃ in °C), the reaction time (X ₄ in min), the agitation speed (X ₅ in rpm) and the output parameter is biodiesel yield (% by weight) of polanga oil. The experimental data used in the developed ANN were obtained from response surface methodology (RSM) based on a central composite design. The trained ANN was tested using different training functions from the MATLAB to predict the best correlation coefficients of training, testing and validation. The data generated by trained ANN is used by GA with regards to the best response (for predicting biodiesel yield greater than predicted by RSM) for different combinations of variables (X ₁, X ₂, X ₃, X ₄, and X ₅) to attain optimization. The average biodiesel yield (by performing experiments under optimized conditions) of 92 % by weight was produced against the proposed value of 91.08 % by weight.     

Sequential experimental design for model discrimination: Taking into account the posterior covariance matrix of differences between model predictions

Techniques for experimental design of experiments for model discrimination constitute important tools for scientists and engineers, as analyzed phenomena can very often be described fairly well by different mathematical models. As interpretation and use of available experimental data depend on the model structure, techniques for design of experiments for selection of the best model are of fundamental importance. Besides, experiments must often be designed for estimation of model parameters and reduction of variances of model predictions (or parameter estimates). These two classes of experimental design techniques generally lead to different experimental designs, although model discrimination and reduction of variances of parameter estimates are closely related to each other. In this work the posterior covariance matrix of difference between model predictions is taken into account during the design for model discrimination for the first time. The obtained results show that the model discrimination power becomes much higher when the posterior covariance matrix of difference between model predictions are considered during the experimental design, increasing the capability of model discrimination and simultaneously leading to improved parameter estimates.     

Modeling and simulation of the formation and utilization of microbial products in aerobic granular sludge

A mathematical model is established to simulate the formation of extracellular polymeric substances (EPS), soluble microbial products (SMP), and internal storage products (X_STO) in aerobic granular sludge. The sensitivity of these microbial products concentrations toward the key model parameters is analyzed. Independent experiments are conducted to find required parameter values and to test its predictive ability. The model is evaluated by using one‐cycle operating experimental results of a lab‐scale aerobic granule‐based sequencing batch reactor (SBR) and batch experimental results. Results show that the model is able to describe the microbial product dynamics in aerobic granules and provide further insights into a granule‐based SBR. The effect of the initial substrate and biomass concentrations on the formation of microbial products in aerobic granular sludge can therefore be analyzed by model simulation. A higher substrate concentration results in a greater concentration of EPS, SMP, and X_STO. An accumulation of biomass in the bioreactor leads to an increased production rate of EPS, SMP, and X_STO. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

On the criteria of solids layer inversion in a liquid-solid fluidized bed containing a binary mixture of particles

A simple criterion is proposed for the prediction of solids layer inversion velocity or critical velocity in a liquid-solid fluidized bed containing a binary mixture of particles with solids layer inversion properties. The criterion is established based on the behaviour of the segregation velocity of the particle which locates at the bottom of the bed at low liquid velocities. The criterion is verified satisfactorily with the present experimental results, which were obtained over a wide range of volume ratio of the particles in the binary mixture, and the experimental results reported by Moritoni et al. (1982, Chem. Engng Sci.37, 1751–1757) and Matsuura and Akehata (paper presented at 50th Annual Meeting of Soc. of Chem. Engrs of Japan, 1985). The criterion is also satisfactorily extended to account for the onset velocity of the particle intermixing state before the point of layer inversion.