Infinite column efficiency

For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other nvass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In aU other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. 

The ideal model (Chapter 7) assiunes an infinite column efficiency. This makes the band profiles that it predicts unrealistically sharp, especially at low concentrations. This sharpness is explained by the fact that the ideal model propagates concentration discontinuities or shocks. For a hnear isotherm, the elution profile would be identical to the input profile, clearly an unacceptable conclusion. The effects of a nonideal column are significant in three parts of the band profile. The shock is replaced by a steep boimdary, the shock layer, whose thickness is related to the coefficients of the column HETP (axial dispersion and mass transfer resistance see Chapter 14). The top of the band profile is roimd, instead of being... 

A second difficulty arises from the diffusion term. This, too, renders the equations intractable even to current numerical methods of solution. The problem is avoided by the assumption of infinite column efficiency. This instantly introduces a difficulty in that the model is then far from representing real chromatographic systems. The equations, however, may now be solved numerically. In fact, for a single solute, there is an analytical solution of the equations resulting from these assumptions [3]. When two solutes are of interest, similar equations are written for both and the set of equations is solved numerically. The above assumptions reduce the differential Eq. (A 2.1) to a simpler equation ... 

The required number of actual plates, A/p, is larger than the number of theoretical plates, because it would take an infinite contacting time at each stage to estabhsh equihbrium. The ratio is called the overall column efficiency. This parameter is difficult to predict from theoretical... 

The basis for determining the required number of plates or column efficiency at infinite dilution is to solve the simplified Knox equation in terms of the optimum d /L (Eq. (7.19)). 

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

If it is assumed that the equilibrium between the two phases is instantaneous and, at the same time, that axial dispersion is negligible, the column efficiency is infinite. This set of assumptions defines the ideal model of chromatography, which was first described by Wicke [3] and Wilson [4], then abundantly studied [5-7,32-39] and solved in a number of cases [7,33,36,40,41]. In Section 2.2.2, which deals with the equilibrium-dispersive model, it is shown how small deviations from equilibrium can be handled while retaining the simplicity of Eq. 2.4 and of the ideal model. 

This model of nonlinear chromatography, the simplest model, was formulated and studied first by Wicke [3], Wilson [4], and DeVault [5]. It assumes that the column efficiency is infinite. There is no axial dispersion and the two phases are constantly at equilibrium. 

As observed by DeVault [6], there can be only a single value of the concentration in any given point of the (t, z) space. In the framework of the ideal model, in which the column efficiency is infinite, this propagation phenomenon results in a concentration discontinuity or shock appearing at the band front. If the isotherm is convex downward, which occms less frequently, the derivative d q/dC is positive, then the velocity associated with a concentration decreases with increasing concentrations. Therefore, the converse effect occurs a shock appears on the rear part of the band profile, since the low concentrations now move faster than the high ones but cannot pass them either. For this t5q>e of isotherm, the profile obtained is a diffuse front and a rear discontinuity. 

Because the ideal model is based upon the assiunption that the column efficiency is infinite, we expect serious discrepancies to arise between the band profiles it... 

In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written ... 

The boxmdary condition corresponds to the injection of a rectangular pulse of finite width, tp, and height, C°, C - Because the column efficiency is infinite in the ideal model, we can write the boxmdary condition ... 

The derivation of the separation conditions is based on the ideal or equilibrium model, i.e., on the assumption that axial dispersion and the mass transfer resistances are all negligible and that the column efficiency is practically infinite. In conventional studies of SMB, it is further assumed that the solid phase flow rate through each column and the void fraction of each column are the same. In the Hnear case, the ratio of the internal flow rate and the solid-phase flow rate can be combined with the slope of isotherm (a,) by using a safety margin, jSy [25,27] ... 

Ideal model of chromatography A model of chromatography assuming no axial dispersion and no mass transfer resistance, i.e., that the column efficiency is infinite (fi = 0). This model is accurate for high-efficiency, strongly overloaded columns. It permits an easy study of the influence of the thermodynamics of phase equilibrium (i.e., of the isotherm) on the band profiles and the separation. See Chapters 7 to 9. 

Shock Concentration discontinuity arising at the front of a chromatographic band when the isotherm is convex upward, at its rear when it is convex downward, if the column efficiency is infinite (ideal model. Chapter 7). The discontinuity is stable and forms because in this case a velocity is associated with each concentration, and this velocity increases with increasing concentration for a convex upward isotherm. Points on the front profile at high concentrations move faster than points at low concentrations, and pile up at the front of the band. However, the area of the band is proportional to the sample size and is finite. So, a discontinuity must form. 

Touching Bands Degree of separation between two bands. Touching bands separation is achieved when, under ideal model conditions (i.e., with an infinitely efficient column), the retention time of the front shock of the second component is equal to the retention time of a zero concentration of the first component. Unless the column efficiency is poor, this corresponds to a 100% recovery yield of each component. 

By deflnition, the column has not quite reached the overload level if the change in the A value is less than 10% with respect to infinitely small amounts of sample. Most phases are not overloaded if the sample mass is less than 10 pg per gram of stationary phase. For the determination of column efficiency by injecting test compounds and calculating the number of theoretical plates, it is best to use sample sizes of 1 pg per gram of stationary phase it can be assumed that the column contains 1 g of phase (in reality from 0.5 to 5.0 g, depending on the column dimensions). 

The internal diameter of the column affects several chromatographic aspects. In the beginnings of modern liquid chromatography there was much discussion of the infinite diameter effect (20-22). Due to slow radial mass transfer, for certain combinations of particle diameter and column diameter, solute injected directly onto the center of a column will traverse the length of the column without ever approaching the column walls. For poorly packed columns this significantly increases column efficiency, by eliminating wall effects. However, for well-packed columns the effect is rather small. The practical utilization of this phenomenon also requires specialized injection apparatus and decreased column sample capacity. For these reasons, this concept is now little discussed. 

Prominent models for estimating peak profiles carry out a differentiation of the equilibrium isotherm with approximations for the mass transfer contribution. The equilibrium-dispersive model, above, assumes that all contributions due to nonequilibrium can be lumped into an apparent axial dispersion term. It further assumes that the apparent dispersion coefficient of the solutes remain constant, independent of the concentration of the sample components. For small particles, these approximations are reasonable for many applications. The ideal model assumes that the column efficiency is infinite. There is no axial diffusion, and the two phases are constantly at equilibrium. The band profiles obtained as solutions are in good agreement with experimental chromatograms for columns with N > 1000 having high loading factors. On the other... 

The specific retention volume, a chromatographic property of a given solute of interest, is related to the column temperature such that plots of log Vg versus l/T should yield a straight line whose slope is related to either the heat of vaporization of the solute at infinite dilution or, if u = 1 (if Raoult s Law holds), the heat of vaporization of the pure solute. Studies of column efficiency via a consideration of the effect of on H (N = H/L) have shown that an optimum can be found that minimizes H or maximizes N. The interested reader can refer to the excellent text by Harris and Hapgood on PTGC (60). Hinshaw has offered some additional insights into PTGC (61). 

For theoretical discussions, the Langmuir isotherm q = aCI l+bC) is convenient. For a review of isotherms used in liquid chromatography, see [23]. With Da = 0 this assumption would be equivalent to assuming that the column efficiency is infinite. However, the finite efficiency of an actual column can be taken into account by including in the axial dispersion coefficient the influences on band profiles due to both axial dispersion and the kinetics of mass transfer ... 

Guiochon et al. [3] also arrived at equations for retention and efficiency, based upon the solution of differential mass balance equations for chromatography using the Ideal Model of chromatography. This makes the major assumption that the column efficiency is infinite, under which conditions it is possible to reach an analytical solution of the equations. Their equation for capacity factor converges with that of Snyder et al. at high values of efficiency and has the virtue of simplicity ... 

This gives a value of 340 mg for the column saturation capacity. As noted earlier, this method is used when column efficiencies are low and the approximation of an infinite efficiency is no longer possible. For most HPLC separations, Eq. (8) can be used with sufficient accuracy. 

---