PARAMETER-ESTIMATION IN NONLINEAR CHROMATOGRAPHY

The method of application sensitivity functions to parameters estimation in nonlinear chromatography, using model (Eqs. 1 to 5), has been presented. Concentration of binding sites A(m) and products K(r)A(m), k1A(m) were the parameters estimated. They have been determined by minimization of least-squares function (9). Using Taylor expansion of C(obl) with respect to p and truncate it after first degree differential term, equation (9) may be rewritten as (Eq. 10). The computation of a new value of the vector p(l+1) at every iteration l requires the knowledge of matrix elements H (I 1) for each experimental time t(j), j = 1, .. , m. To calculate H matrix, equations (12 and 13) were differentiated with respect to p. In this way, the se of differential equations for matrix elements H was obtained. The set of equations (1) and (2) and (12) and (13) with boundary conditions (3), (4), (5) and (17) and (14),(15), (16) and (18) was solved using splines collocation method [12]. To minimize equation (9), Marquardt procedures, modified by Fletcher [13], was applied. To provide verification of the method proposed, some experiments using apparatus presented in Fig, 1 were conducted. Measurements were done using 0.5 m height column with 0.008 m diameter. The column was packed with MERC porous silica particles (0.04-0.063 mum. Density of pellets was rho = 1960 kg/m3 and column porosity epsilon = 0.68. Benzene (compound) and n-hexane (solvent) constitute a testing mixture, Initial values of optimizing coefficients should be proposed first in nonlinear estimating method. The appropriate determination of initial parameters set not only leads to short time of calculation, but it also allows us to overcome problems due to starting point far from optimum. The initial values of A(m) and K(r)A(m) were calculated according to ECP method [1]. The initial value of k1A(m) product was arbitrary chosen as 10(-3) after some testing calculations. In Figure 2 experimental and theoretical peaks are compared. The theoretical peak was calculated using optimal values of the following parameters: K(r)A(m) = 9.66-10(-4), k1A(m) = 1.29 . 10(-3) and A(m) = 2.32 . 10(-3). Initial values of estimated parameters were: K(r)A(m) = 1.05 . 10(-3), k1A(m) = 1.10(-3) and A(m) = 1.7 10(-3). Comparison between theoretical peak, calculated using initial values of optimizing parameters, and experimental peak is illustrated in Fig. 3. One can see (Fig. 2) that the method of sensitivity function enables an excellent prediction of experimental concentration distribution at the column outlet. The number of iterations in Marquardt procedure were in this case equal to 5 and the set of equations (1) and (2) has to be solved 7 times. The parameters estimated at 95% confidence limits were expressed by Eq. (20), and parameter correlation matrix - by Eq. (21). Under another experimental conditions, the number of iterations for optimal parameter chosen and the agreement between experimental and theoretical peaks were similar.