APPLICATION OF CUBIC-SPLINES COLLOCATION IN THE SIMULATION OF CHROMATOGRAPHY COLUMN WORK

The cubic splines collocation method for solving homogeneous model of chromatography column (Eqs. 1 and 2) or (Eq. 5) was presented and compared to several finite difference methods. During calculation by means of finite difference methods, a spatial derivatives in the set of equations (1) and (2) were replaced by difference equations (Eq. 6) and (Eq. 7a) or (Eq. 7b) or (Eq. 7c). The set of ordinal derivative equations with boundary conditions (Eqs. 8, 9, and 10) was solved according to Adams method in Gear version [20]. Relative error of calculation was set to be 10(-3). Ideal chromatography column model was solved on the basis of instructions given in [11] for difference scheme II. The partial differential equations (1) and (2) or (5) can be approximated using splines collocations with equations (16) and (17). Set of equations (16) and (17) together with boundary conditions (8), (9), (18) and (21) were solved according to Adams method [20] assuming relative error equal to 10(-3). Values of collocation coefficients y(i) in every integration time step were calculated from equations (19), (22) and matrix (24). Comparison of both numerical methods was based on the analysis of numerical diffusion value generated by these methods, which for linear adsorption isotherm can be calculated from equation (25), when dispersion coefficient D(e) is equal to zero. Figure 2. presents results of simulations for homogeneous model (Eqs. 1 and 2) for De = 0, H(k) = 0.15 m, K(r)A(m) = 1, C0 = 1, epsilon = 0.5 and for impulse duration time equal to: t(imp) = 2 s for w = 0.008 m/s, and t(imp) = 16 s for w = 0.001 m/s. In finite difference method, convection term was approximated with equation (7a). Table shows the comparison of splines collocation method and finite difference method (7b) on the basis of compound mass balances at column outlet and inlet. Using approximation of convection part of model (Eqs. 1 and 2) with equation (7c), the results obtained were worse than those presented in Table. Finally, it can be stated that using finite difference method one needs from three to over ten times more nodal points compared with splines collocation.