Process modelling and optimization aid FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory

Process modelling and optimization aid Parametric identification from experimental data FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory

Parametric identification Likelyhood Maximum Method Number of measurement i={1, 2, …, n j } Number of component j={1, 2, …, m} example : concentration, temperature, molecular weight… Operating conditions (i th measurement of j th component) : example for continuous experiments : time of measurement Measured value (i th measurement of j th component) : Corresponding predicted value : Corresponding unknown true value : Parameters unknown true values :  * Measurement error :

Parametric identification Likelyhood Maximum Method Measurement error modelling (replications) : independant gaussian errors with average =0 multiplicative errors : additive errors : Probability density : Likelyhood maximum : Identified parameters values :

Parametric identification Likelyhood Maximum Method Likelyhood function : Estimation of the measurement errors : Parameters estimation :

Parametric identification Likelyhood Maximum Method Parameters estimation by minimization of : Total number of freedom degree : total number of measurements - parameters number - variances number + 1 Unbiased estimation of measurement error variances :

Parametric identification Likelyhood Maximum Method Example 1 : unknown average and variance of n gaussian hazards y i Parametric identification : Likelyhood maximum method gives biased variance :

Parametric identification Likelyhood Maximum Method Example 2 : unknown C (and T), measurement of P and T Parametric identification :

Parametric identification Likelyhood Maximum Method Example 3 : terpolymerization in tubular reactors (69 parameters) styrene/a-methylstyrene/acrylic acid 1 rst step : simultaneously indentification of 23 parameters (3 times) 2 nd step : simultaneously indentification of the 69 parameters

Parametric identification Parameters confidence domain Vectors of n m elements (independant for e) : Confidence domain calculation : projection of e on a tangential plane to the model b is the projection of e and h the distance between experiments and model e 2 = b 2 + h 2 b and h are independant (orthogonal)

Parametric identification Parameters confidence domain Measurement 1 Measurement 2 Experimental point Model

Parametric identification Parameters confidence domain Definitions and properties : Fisher Snedecor test for confidence domain :

Parametric identification Parameters confidence domain Determination of parameters confidence domain : 1 rst to identify the estimated parameters by optimization 2 nd to determine the confidence domain of parameters by optimization of the same function than in identification

Parametric identification Parameters confidence domain Example 1 : speed identification of a bullet 1rst measurement : length of shot 2nd measurement : time to reach this length Y 1 and Y 2 are the coordinates of the projection of the measurements on the tangent plane to the model

Parametric identification Parameters confidence domain Example 1 :

Parametric identification Parameters confidence domain Example 2 : application to a simple enzymatic reaction An enzyme E with a substrate S transitorily gives a specific complex enzyme-substrate C before the researched product P (Michaelis-Menten kinetics)

Parametric identification Parameters confidence domain Example 2 : application to a simple enzymatic reaction

Parametric identification Identification quality Parameter estimation by evolutionary algorithm (or genetic) = set of solutions (defined number) Confidence domain determination by evolutionary algorithm = set of solutions (defined number) with end test by corresponding Fischer Snedecor test Set of parameters vector (confidence domain representation) = possibility to calculate correlations between parameters Detection of high correlations

Parametric identification Identification quality To vizualize the confidence domain = projection of the solutions set on 2 parameters space Non elliptical confidence domain = non linear model Estimated parameters not at the confidence domain center = non linear model Correlations between parameters can be reduced by new experiments

Parametric identification Identification quality Confidence interval Parameter 1 Parameter 2 Reduced Confidence interval Estimated parameter Confidence domain Inclined confidence domain = correlation between the 2 parameters

Parametric identification Identification quality Confidence interval = overall range of the corresponding parameter Reduced confidence interval = range of the parameter when the others take their estimated value If the reduced confidence interval contains the 0 value, the corresponding parameter is not significantly different to 0 When a parameter is not significantly different to 0, a model reduction is possible

Parametric identification Identification quality Comparison between experimental data with corresponding simulations from model and estimated parameters Confidence interval of model prediction from Student test : If an experimental data is not include in the corresponding confidence interval the measurement is maybe deviating

Parametric identification Identification quality Estimated value Measured value Confidence interval of model predictions Experimental data versus prediction Deviating data

Parametric identification Identification quality Example : Modelling of polymer blend Young modulus Correlations between parameters

Parametric identification Identification quality Example : Modelling of polymer blend Young modulus Comparison between experimental and calculated young modulus values