Journal of Chromatography A, 1217 (2010) 8127–8135 Contents lists available at ScienceDirect Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma Martin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatography Krisztián Horváth a,b , Jacob N. Fairchild a , Krzysztof Kaczmarski c , Georges Guiochon a,∗ a University of Tennessee, Department of Chemistry, Knoxville, TN, 37996-1600, USA b University of Pannonia, Department of Analytical Chemistry, P.O. Box 158, Veszprém, H-8200, Hungary c Rzeszów University of Technology, Department of Chemical and Process Engineering, 35-959, Rzeszów, Poland article info Article history: Received 16 February 2010 Received in revised form 4 September 2010 Accepted 6 October 2010 Available online 14 October 2010 Keywords: Mass balance equation Equilibrium-dispersive model Martin-Synge plate model Inverse method abstract An alternative method, called the Martin-Synge algorithm, is introduced to calculate numerical solutions of the equilibrium-dispersive (ED) model. The developed algorithm is based on the earlier work of Friday and Levan [1] and on the continuous plate model of Martin and Synge [2]. The column is divided evenly into a series of virtual vessels in which a simplified mass balance equation is solved accurately by the Runge-Kutta-Fehlberg method and the elution profile is given by the numerical solution for the last vessel. The dispersion of the compound during the elution process is controlled by adjusting the number of virtual vessels into which the column is divided. Solving the ED model under linear conditions with this method gives exactly the same profile as the analytical solution of the Martin-Synge plate model. The Martin-Synge method gives better results than the Rouchon method (1) when the isotherms involved are sigmoidal or anti-Langmuir; and, more importantly, (2) in the case of multi-component problems. Finally, the Martin-Synge method proves to be more robust and faster than the OCFE method that, until now, was considered to be one of the most robust and accurate algorithms. The developed algorithm was used for the calculation of the coefficients of the isotherm of butyl benzoate by the inverse method, using a simplex optimization algorithm. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Several differential models describe the migration of sample zones along chromatographic columns [3]. These models have dif- ferent degree of complexity. The most complex general rate model [4–7], considers all the processes taking place in the mobile phase, in the pores, and on the surface of the stationary phase. This model provides the most detailed information on the chromatographic processes. On the other hand, the simplest ideal model [8–10] does not consider any kinetic process causing band broadening. It just informs on the effects of the thermodynamic process on the evo- lution of band profiles. The equilibrium-dispersive (ED) model is a good compromise for the optimization of chromatographic pro- cesses. In the ED model [3], it is assumed that: (1) the mass transfer kinetics is fast and the mobile and the stationary phases are con- stantly in equilibrium, (2) band dispersion takes place in the column through axial dispersion and nonequilibrium effects (mass trans- fer resistances and finite adsorption-desorption kinetics) and their ∗ Corresponding author. Tel.: +1 8659740733; fax: +1 8659742667. E-mail address: guiochon@ion.chem.utk.edu (G. Guiochon). contributions can be lumped together in an apparent axial disper- sion coefficient, D a . The ED model has no closed-form analytical solution in most cases. Although it has some approximate solu- tions, their validity is limited (see Chapter 10.2 of Ref. [3]). In most cases of practical interest, numerical solutions are needed. A variety of methods are available to derive numerical solutions of Eq. (3) (see later), including finite-difference and finite-element methods. The principle of the finite difference methods consists of replacing the continuous plane (z, t) by the grid obtained by dividing the space and time into a number of small, equal seg- ments and replacing the differential terms by the corresponding finite difference terms. Many combinations of these various finite differences can be used for each term of the mass balance equa- tion and a partial differential equation can be approximated by many different finite-difference schemes. It is essential that the numerical errors made during the calculations be controlled and there are two different approaches for the calculation of numeri- cal solutions of the mass balance equation. First, it can be directly solved by setting the integration increments to minimize the error made [11,12]. Second, the space and time increments can be set on such a way that the numerical error simulates the band dis- persion in the column, as is done in Rouchon [13,14] and Craig [15,16] algorithms. The finite difference methods give fast solu- tions that are easy to use for the solution of the mass balance 0021-9673/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2010.10.035