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J. F. Parcher, T. H. Ho, and H. W. Haynes, Jr.
A Numerical Solution for the Material and Momentum Balance Equations for Finite Concentration Chromatography Jon F. Parcher,” T. H. Ho, and Henry W. Haynes, Jr. Depadment of Chemistryand Department of Chemical Engineering, The University of Mississippi, University, Mississippi 38677 (Received August 20, 1975; Revised Manuscript Received May 3, 1976) Publication costs assisted by the National Science Foundation
The material and momentum balance equations are solved for a chromatographic system with a single solute at finite concentrations, composition dependent gas phase viscosity, and a finite pressure drop across the column. The equations are solved numerically and shown to be reasonably accurate for a set of seven chromatographic systems which had been studied previously by static methods or which could be accurately described by a theoretical model.
Introduction There are several different forms of chromatographic retention theories. Each theory utilizes some form of material or momentum balance equations written for a concentration gradient in the column, ad the theories differ only in the simplifying assumptions used to reduce the mathematical complexity of the system. The complete description of a chromatographic system with finite solute concentrations is too complex for any analytical solution and it is common practice to utilize several simplifying assumptions regarding the partition or adsorption kinetics, the predominant dispersion mechanism(s), and the pressure gradient across the length of the column. It is usually assumed that the kinetics of the adsorption or partition processes are very rapid compared to the velocity of the concentration gradient. This is generally a valid assumption and can be verified by studying the effect of carrier gas flow rate on the measured experimental parameters. In the special case of finite concentration chromatography, it is commonly assumed that the shape of the solute peak or “break through” curve is determined solely by the shape of the equilibrium isotherm and the pressure variations caused by the condensation of the solute in the liquid phase. The validity of this assumption depends upon the isotherm and the concentration of solute in the system. Valentin and Guiochonl have recently presented a convincing argument for the second-order nature of diffusion and mass-transfer contributions to the shape of large concentration boundaries in a chromatographic column. The secondary effects can be minimized but never completely removed. One method used to minimize diffusion and nonequilibrium effects is to operate a t a velocity corresponding to a minimum plate height (HETP) and this is possible only if the column is operated at a significant pressure drop. The primary distinguishing feature of the various extant retention theories is the treatment of the pressure gradient across a column and the composition dependence of the gas phase viscosity. Usually, the viscosity of the gas phase is assumed to be constant, even though it may actually vary drastically with composition. The assumptions regarding the pressure gradient are usually (i) the total pressure is uniform
* Author to whom correspondence should be addressed a t the Department of Chemistry. The Journal of Physical Chemistry, Vol. 80, No. 24, 1976
throughout the column, (ii) the total pressure is a linear function of the distance from the inlet, or (iii) the square of the total pressure is a linear function of the distance from the inlet. In many cases, none of these assumptions concerning the gas phase viscosity and the pressure gradient are realistic. It is the purpose of this paper to show that the momentum and material balance equations can be solved under these conditions by numerical techniques. Today’s widespread usage of high-speed computers means that it is not necessary to resort to physically unrealistic assumptions for the sole purpose of obtaining a solution to a complex system of equations. Material a n d Momentum Balance Equations The situation which we wish to describe mathematically is one in which finite concentrations of a single solute are introduced into the column a t large carrier gas flow rates, which necessitates a finite pressure drop across the length of the column. In addition, the viscosity of the solute and carrier gas may differ appreciably causing the viscosity of the gas phase to be composition dependent. It will be assumed that any heat effects are small so that the column operates isothermally. In general, the concentrations, velocity, and quantities dependent upon concentration are all functions of time, t , and position, z, in the column. Under these conditions the mass balance equations for the solute concentration and total (solute t carrier) concentration are
b (CtV) a2
+ 0,-+ac at
01-
at
=0
where CAand Ct are the concentrations (M) of the solute and the solute plus carrier gas in the mobile phase, respectively; V is the superficial velocity (cm/s); 0, and 01 are the ratios of the areas of the gas and liquid phases to the area of the empty column; and of pure liquid phase. The superficial velocity, V, is defined as the volume flow rate divided by the area of the empty column and is related to the interstitial velocity, u, by the relation V = 4.We have expressed the mass balance equations in terms of 0 and V;however, the equations can be expressed equally well using the interstitial velocity and the volumes of the gas and liquid phases.
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Momentum Balance Equations for Finite Concentration Chromatography The momentum balance equation for flow in packed beds is usually written in the form of Darcy’s law: -8.314 X 1O6K,T dC (3) (1 BCt)217 where K , is the permeability of the column (cm2);T is the absolute temperature (K); 9 is the gas phase viscosity (cP);and B is the second virial coefficient of the gas. If the usual carrier gases and pressures encountered in gas chromatography are assumed, the term BCT
