Parameter estimation is extensively used in chemical engineering to determine transport properties, kinetics as well as thermodynamic equilibrium constants (Jallut, Thomas, Touré, & Diard, 2001). The treatment of experimental data obtained from chromatographic columns is a typical example of such an approach. This technique is used to characterise liquid or gas phase adsorbents (Golshan-Shirazi & Guiochon, 1992; Gritti, Piatkowski, & Guiochon, 2002; Hufton and Danner, 1993a and Hufton and Danner, 1993b; Jolimaitre, Tayakout-Fayolle, Jallut, & Ragil, 2001; Tondeur, Kabir, Luo, & Granger, 1996), soils (Chen & Wagenet, 1995) as well as polymers (Von Meien, Biscaia, & Nobrega, 1997). One can perform specific experiments to extract the values of some of the parameters. For example, the frontal technique allows to determine the equilibrium isotherm by performing a global mass balance of the adsorbate (Gritti & Guiochon, 2003). Axial dispersion coefficient as well as porosities can be estimated from inert tracer experiments (Hejtmanek & Schneider, 1993). The other approach consists in performing the estimation of all the parameters from only one experiment. Such an approach allows to minimize the sources of errors, to save time and to avoid numerous experiments (Altenhöner, Meurer, Strube, & Schmidt-Traub, 1997). However, as far as the number of parameters to estimate is greater, a structural analysis of the model has to be performed.
In this paper, we propose a way to derive models for chromatographic columns in order to avoid a possible overparametrization and we link this question to the structural identifiability of the parameters. This approach is applied to distributed parameter models. We restrict ourselves to the case of a single component experiment: an adsorbate is transported isothermally by a carrier and is supposed to be adsorbed in a solid phase. When it is introduced according to a step function from an equilibrium state, the response of the system is the breakthrough curve. The models that are used to extract parameters from experimental breakthrough curves are formulated in the state space from suitable material balances of the adsorbate, according to the general form (1):
equation(1)
View the MathML sourcex˙=∂x∂t=f(x,u,θ)geq(x*,x,θ)=0y(t)=h(x,u,θ)x(t=0)=x0(θ)
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x is the state vector, u is the manipulated variable (the inlet adsorbate concentration) and y is the measured variable (the outlet adsorbate concentration). The algebraic equation geq (x*, x, θ) = 0 represents the equilibrium conditions that are generally assumed at interfaces. For an estimation procedure to make any sense, and prior to any experimental considerations, one has to insure that the set of parameters θ is structurally globally identifiable ( Park & Himmelblau, 1982; Vajda, Rabitz, Walter, & Lecourtier, 1989; Walter, Lecourtier, Kao, & Happel, 1989; Walter and Pronzato, 1990, Walter and Pronzato, 1995 and Walter and Pronzato, 1997; Walter, Pronzato, Soong, Otarod, & Happel, 1995). This property ensures the existence and the uniqueness of the set of parameters for a given input–output behaviour of the model. The identifiability study of the general linear monodisperse model of chromatographic columns has been performed in Tayakout-Fayolle, Jolimaitre, & Jallut, 2000. The conclusion of this study is that there exists a state vector x for which the number of parameters is minimal and that these parameters are globally identifiable. In this paper, we extend the approach to some nonlinear cases. An illustrative example is given by using experimental breakthrough curves taken from literature ( Chern & Chien, 2002). The relevance of the change of state coordinates that we propose is also confirmed by performing the structural identifiability analysis of a discretized version of one of the models that are discussed in the paper.
The state variables C f1 and View the MathML sourceCf1* seem to be the best one in term of the number of macro-parameters and lead to the important fact that, for an ordinary chromatographic experiment, the initial conditions are totally measured. The change of variable that we propose to represent the state of the solid is defined as the concentration of a fluid phase in equilibrium with the solid phase. If a spatial variation of the concentration in the solid is considered, it has been shown in the linear case ( Tayakout-Fayolle et al., 2000) that the best choice is also the concentration of the fluid phase in equilibrium with the solid phase at every time and point. If the thermodynamic equilibrium is assumed to be reached at the fluid–solid interface, the driving force for the mass transfer between the external fluid and the solid becomes a continuous function from one side to the other side of the interface. It seems that this property is very important and is a guide for the definition of the well-suited state variable for the solid phase whatever the form of the equilibrium relation. We think that this change of state variable is applicable to other situations like adsorption bidisperse models for example.
As noticed in the previous section, the minimal representation gives rise to a minimal number of macro parameters; it is clear that in this case no numerical problem occurs. The over parameterization leads to an infinity of solutions for some parameters which implies the numerical situation described in the previous section.
Another quality of this representation has been emphasized in Section 3: for non-linear equilibrium laws for which the expression cannot be explicitly inverted, the use of this representation permits to decrease the index of the system.
