Chapter 22
Simulation and Optimization in Supercritical Fluid Chromatography 1
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F. Van Puyvelde , Ε. H. Chimowitz , and P. Van Rompay 1
Department of Chemical Engineering, K. U. Leuven De Croylaan 46, Heverlee 3001, Belgium Department of Chemical Engineering, University of Rochester, Rochester, NY 14627 2
Gradient programming is an experimentally proven technique for tuning the separating action in supercritical fluid chromatography (SFC). In this study we describe a mathematical framework for sim ulating and optimizing the process. Results are given, showing the utility of simulations and how opti mization ideas may advance this area of technology. Chromatographic separations are usually performed under constant operating conditions of pressure and temperature. However, it is also clear that this may not be the o p t i m a l manner i n which to operate the process. Changing the op erating conditions as a function of time (referred to as gradient programming) is an experimentally proven technique, commonly applied i n chromatography, to improve the quality of the separation. Temperature programming is very common in gas chromatography, while gradient elution is applied i n liquid chromatography. Supercritical fluid chromatography ( S F C ) however offers a greater range of pos sibilities to control solute retention since i n the near-critical region of the mobile phase, physical properties are extremely sensitive to thermodynamic conditions. Here we provide an analysis for simulating and optimizing S F C under dynamic conditions. T h e numerical solution of the mathematical model proposed w i l l enable us to run sensitivity studies which can subsequently be used to direct the optimization of the process.
M o d e l Development In supercritical fluid chromatography, a very small sample of the mixture is i n jected into the stream of the supercritical solvent; the supercritical fluid is used in excess. This means that the solute species are extremely dilute. In this case the assumption of linear chromatography w i t h the local equilibrium assumption
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is justified. T h e use of a more elaborate and detailed models is seldom warranted for modeling linear systems (1), (2). For these systems, the precise nature of the mass transfer effects (film resistance, intraparticle mass transfer resistance, etc.) has only a very modest effect on the transient response curves of reasonably long chromatographic columns (2 - 5). T h e basic equation used i n this analysis is the mass balance. T h i s balance equation states that the difference between the amount of solute flowing into a shell minus the amount of solute flowing out of the shell must equal the rate of accumulation of the solute i n the differential shell. O r written mathematically: d(uc) dx where c is the concentration,of the solute i n the mobile phase, q the concentration of the solute on the stationary phase, ε the porosity of the packed bed and D the dispersion coefficient. Appropriate boundary conditions are: 2
c(x,0)
=
0
( )
c(0,
