Parameter estimation in a dynamic gas-liquid film model
Advancing the chemical engineering fundamentals
Chemical Reaction Engineering: Kinetics & Modelling (T2-2a)
Keywords: non-linear regression, gas-liquid dynamic model, model identifiability
The design and simulation of gas-liquid reactors needs, on one hand, of dynamic mathematical models for describing simultaneously the microscopic (gas-liquid mass transfer at the interface level) and the macroscopic (macroscale mixing and chemical reaction at the reactor level) scales of the process. On the other hand, the knowledge of the different coefficients that characterize the model (gas hold-up, Epsilon, Henry’s constant, H, diffusion coefficients DA and DB, liquid film thickness, delta, kinetic rate constant, k2,…) is fundamental. Although generalized correlations exist in the literature for the above coefficients [1], in many cases is necessary to estimate those coefficients from particular experimental data using non-linear regression.
The general procedure that we propose here is based, as the first step, in the structural identifiability analysis of the mathematical model [2, 3] that let us to know their identifiable parameters (volumetric mass transfer coefficient kLa, aDA/delta, the reciprocal of the diffusion time, DA/delta^2,…), that usually consist of a combination of the coefficients characterizing the model. In some cases is possible to rewrite the mathematical model in function of the identifiable parameters. The following step is the estimation of those parameters from experimental data through non-linear regression [4]. Although is not always possible to obtain the value of the original coefficients from the identified parameters only using gas-liquid mass transfer experimental data, the design or simulation of the process is achieved at this point.
In this work we focus on the parameter estimation step. The mathematical model rewritten applying the structural identifiability analysis to the dynamic gas-liquid reactor model previously reported [2], consists of the following seven parameters:
p1=(1-Epsilon)/Epsilon; p2=H; p3=a·DA/delta; p4=DA/delta^2; p5=k2·CB0; p6=DB/delta^2; p7=a·DB/delta
where the first four parameters are related with the gas substance soluble in the liquid phase, A, the last two are related with the non-volatile solute in the liquid phase, B, and p5 is related with the second order kinetic rate constant.
The state variables of the model that are chose as observable variables (concentration of A in the gas phase or in the liquid phase, concentration of B in the liquid phase, or their combinations) and the relative values of the model parameters, that depend on the conditions fixed for the experiments, have a great influence on the optimized parameters obtained after the non-linear regression procedure. Different metrics have been proposed and tested in this work for comparing the effect of the observable variables and for analyzing the sensibility of each parameter on the mathematical model, and a procedure is established for getting the maximum information from the experimental data (i.e. the kinetic rate constant k2).
The authors gratefully acknowledge the Conselleria d’Empresa, Universitat i Ciència de la Generalitat Valenciana for financial support of this work under project ref. GV06/083.
1.- N. Kantarci, F. Borak, and K.O. Ulgen. Process Biochemistry 40 (2005) 2263-83
2.- J. Navarro-Laboulais, S.C. Cardona, J.I. Torregrosa, A. Abad, and F. López. AIChE J. 52 (2006) 2851-63
3.- J. Navarro-Laboulais, S.C. Cardona, J.I. Torregrosa, A. Abad, and F. López. AIChE J. submitted
4.- P. Englezos and N. Kalogerakis. Applied Parameter Estimation for Chemical Engineers. New York, Marcel Dekker, 2001.
Presented Monday 17, 12:11 to 12:30, in session Chemical Reaction Engineering: Kinetics & Modelling (T2-2a).
