Analytical solutions and moment analysis of general rate model for linear liquid chromatography

• The linear general rate model (GRM) is analyzed for different boundary conditions.
• The first four moments of the model are analytically and numerically calculated.
• Relationships are derived to match moments of GRM and the lumped kinetic model (LKM).
• These relations can be used to estimate parameters of LKM from GRM parameters.
• The finite volume scheme is applied to validate the analytical results.

The general rate model (GRM) is considered to be a comprehensive and reliable mathematical model for describing the separation and mass transfer processes of solutes in chromatographic columns. However, the numerical solution of model equations is complicated and time consuming. This paper presents analytical solutions of the GRM for linear adsorption isotherms and different sets of boundary conditions at the column inlet and outlet. The analytical solutions are obtained by means of the Laplace transformation. Numerical Laplace inversion is used to transform back the solution in the time domain because analytical inversion cannot be obtained. The first four temporal moments are derived analytically using the Laplace domain solutions. The moments of GRM are utilized to analyze the retention times, band broadenings, front asymmetries and kurtosis of the elution profiles. Relationships are derived among the kinetic parameters to match the first four moments of GRM and the simpler lumped kinetic model (LKM). For validation, the analytical solutions are compared with numerical solutions of a second order finite volume scheme. Good agreements in the results verify the correctness of analytical solutions and the accuracy of the numerical scheme.