Plate Model of a Chromatographic Column in the Case of a Nonlinear Sorption Isotherm K. I. Sakodynsky, L. V. Streltzov, V.
Yu. Zelvensky, S. A. Volkov, and I. N. Rozhenko
Karpov lnstitute of Physical Chemistry, Moscow, U.S.S. R.
The progress of industrial-scale chromatography depends in many respects on a successful investigation of the peculiar features of the chromatographic process related to an increase in efficiency and dimensions of the column. The consideration of these peculiarities substantially complicates a mathematical description of the chromatographic process. The high concentrations of the substances injected into the column results in nonlinear effects. Calculation of the shape of the chromatographic peak in the case of a nonlinear sorption isotherm requires the solution of a system of nonlinear differential equations. The integration of such a system is achieved by numerical methods with the use of a computer. T o achieve a mathematical model of the chromatographic process, two approaches may be adopted. One of them consists of considering the chromatographic separation in a column as a process with distributed parameters and describing it in terms of differential equations with partial derivatives (1-3). Such a description of the process seems to be quite natural but it is a very complicated matter to compare the model with the experiment because of the difficulties encountered in the determination of several parameters ( e . g , coefficients of longitudinal diffusion and mass exchange, and characteristics of the sorptive bed). It would seem that this is the reason that there are no comparisons of calculated data with experimental data in the papers cited above. Moreover numerical integration of equations rn partial derivatives is a difficult problem related to the problems of stability and concurrence of the solution. The other approach which was used and is part of the present work is based on the theoretical plate concept. A chromatographic column is conceived in the form of consecutively united equilibrium steps, upon each of which, by definition, thermodynamic equilibrium exists between the gas and the liquid phase. It is presumed that a t any fixed moment of time the concentration of the sorbate a t all the points of the phase within one step is the same; that is, the assumption is made that complete mixing of the gas phase on the theoretical plate occurs. In spite of the somewhat formal nature of this approach. it has definite advantages that are particularly important in practical work. The principal parameter determining band broadening in this case is the height equivalent to a theoretical plate, which can be readily determined experimentally. All of the other parameters of the equations are also easily determined. The equations themselves are simple differentials, the solution of which by numerical methods is easier than that of partial derivatives.
( 1 ) I . E. F u n k a n d G . Houghton, Nature. 188, 4748 (1960). ( 2 ) F . T. Dunchhorst and G . Houghton. Ind. Eng. Chem. Fundarn., 5 , 93 (1966). (3) G . Houghton. J . Phys Chem., 7 , 84 (1963).
Material balance equations for a system of plates with a sample size of A were derived in ( 4 ) . The equation for the first step is as foliows:
where C, and Cl are concentrations of sorbate in the gas and in the liquid phase related by an equilibrium dependence ( 3 ) , V, and VLare volumes of the gas and the liquid phase of one plate, S is the volume of the gas that has passed through the column, and A is the volume of sample with a concentration C,. For all the other steps, the following equation is valid: VgdC,,,
+
VldCL,,+ C,,,dS
=
C,,-,dS(i
=
2 , 3,..., n) (2)
Using the equation of a sorption isotherm approximated in the form of a parabola, =
K O + K,Cg,i + K2Cg,;
(3)
and introducing relative concentrations,
the system of Equations 1 and 2 can be reduced to
The system of n common differential equations with the initial conditions y1 = y2 =
...
=
yn = 0
(6)
gives a full mathematical description of the plate model of the chromatographic process in a column. The solution of the system 5 was obtained on a “Minsk22” computer. The numerical integration method of Runge-Kutt-Merson was used with automatic selection of the step. The time of computation depends to a substantial degree on the number of plates in the column and on curvature of the sorption isotherm. As the curvature increases, the steepness of the front or the rear of the chromatographic wave also increases, i. e . , the derived function of concentration distribution along the column increases.
(4) I . I . Van Deemter. F. I . Zutderweg. and A . Klinkenberg. Chem. Eng. Sci., 5, 2 7 1 (1956).
ANALYTICAL CHEMISTRY, VOL. 45, NO. 8, JULY 1973
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Q
b
0.3
0.2
0
I
A
10.000
600
400
S. crn'
Figure 1. Effect of curvature of the sorption isotherm ( a ) and plate number in the column ( b ) on the shape of the elution
curve
:or S = 34 crn'
0.5
10,500
0
100
50
Theoretical Plates
Figure 2.
200 n
Deformation of the distribution curve with its progress through the column
Hence, in order to retain the stability of solution of system 5 , it is necessary to diminish considerably the integration step, which leads to an increase in the total time of computation. The purpose of modeling was to determine the curve of concentration distribution a t the column outlet. To check the program, computer experiments were made to determine the influence exercised upon the shape of the elution curve by the form of the sorption isotherm and by column efficiency. Figure l a shows peaks obtained on the computer for three values of the parameter K2 characterizing the curvature and form of the sorption isotherm with constant values of the other parameters: K1 = 120, n = 100, Vg = 0.16 cm3, Vi = 0.032 cm3, C, = 0.02 mmole/cm3, A = 50 1558
150
ANALYTICAL CHEMISTRY, VOL. 45, NO. 8, JULY 1973
cm3. The values of K1 = 120 and K2 = -1500 cm3/mmole correspond t o the value of adsorption of n-heptane on Chromosorb with 20% (by weight) of dinonyl phthalate ( 5 ) . The values of K 2 = 0, K Z = 1500 cm3/mmole and of the other parameters are arbitrary. As would be expected for a convex sorption isotherm (K2 < 0), the front of the peak tends to a vertical one, while the rear edge is more diffuse. For a concave isotherm ( K z > 0) the situation is reversed. Figure l b illustrates the influence of column efficiency on the shape of a chromatographic peak. A 4-m, 30-mm ( 5 ) M . S. Vigdergauz and R. N. Maryahin. in "Gazovaya Khromatografiya." Niitekhim, Moscow. 197,l. 15, p . 14.
i.d. column was modeled. The sorbent was a wide-pore silica gel with a surface area of 16 mZ/gram. The size of the n-heptane sample was 4 ml. The length of the column was kept constant while the number of theoretical plates was decreased from 200 to 50, the volume of each plate increasing variable values of the parameters n, V,, V, (for gas-solid chromatography VL is replaced by V, volume of the solid phase). The remaining parameters were constant: K1 = 10, Kz = -200 ml/mmole, C, = 0.022 mmole/ml, A = 700 ml. The decrease in plate number in the column substantially enhanced peak broadening. The retention volume of a peak increases with a decrease in its height, i e., the velocity of progress of high concentrations through the column is greater than that of low ones. This is accounted for by the convex shape (Kz < 0) of the adsorption isotherm of n-heptane on silica gel. The program was prepared in such a way that in addition to the shape of the elution peak, the distribution of concentrations along the column could be determined a t any time desired. This permitted us to follow the gradual deformation of the distribution curve as the wave progressed along the column. In Figure 2 such a situation is shown for a 700-ml volume of a vapor sample. The sorbate has a convex sorption isotherm of moderate curvature. The first curve corresponds to a volume of gas equal to that of the vapor sample which enters the column a t a constant concentration, C,(y = 1). In the column, the sample occupies a volume equal to about 15 theoretical plates. The deviations from a rectangular pulse (the plug injection method) are shown by the fact that upon entering the column the front becomes diffuse. An interesting situation is observed for a linear adsorption isotherm ( K z = 0) when it is generally assumed that the elution peak must be symmetrical. After injection on the column, the front of the band is more diffuse than the rear, because of the effect of the injection mode. Since the concentration gradient determining the propelling force of diffusion is higher on the steeper rear edge, the latter is more intensively broadened, bringing about a decrease in the general asymmetry of the peak. The asymmetry coefficient, defined as the ratio of the width of the frontal half-wave to that of the rear half-wave of the peak a t half its height, for volumes of gas having passed through the column of 80 cm3 (1st curve), 234 cm3 (2nd curve), and 349 cm3 (3rd curve), is equal to 1.35, 1.12, and 1.05, respectively. Thus, the peak is symmetrical when approaching the end of the column. However, since a t the time the peak leaves the column, the rear half-wave stays in the column for a longer period and is, consequently, more in-
n
= 200
10
X
-X
mm
Comparison of the computer peak perimental one (-)
Figure 3.
( - - - ) with
the ex-
tensively broadened than the frontal one, the detector records a somewhat asymmetrical chromatographic peak a t Kz = 0, with a diffuse rear edge. In conclusion, a comparison of the computer curve to experimental data is given. The chromatogram (Figure 3) was obtained on a 4-m, 30-mm i.d. column packed with wide-pore silica gel with a surface area of 16 mZ/gram. The coincidence of t,he fronts is good; the divergence of the experimental curve in the rear may be explained by a certain deviation of the n-heptane absorption isotherm on silica gel from the quadratic form ( 3 )used in the program. It is contemplated that the chromatographic column model proposed here will be used in future computations of the purity of fractions in the separation of multicomponent mixtures. Received for review November 13, 1972. Accepted February 1,1973.
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