Plate height equation

Because of the complex central term representing mobile phase band broadening, Eq. 12.1 has a rather awkward form, not simple to use. The awkwardness reflects the complexity of packed-bed processes, specifically the complexity caused by (i) the many types of velocity states and (ii) the competition (coupling) between diffusion and flow in controlling random displacements. 

We will use two approaches to skirt the complexity of Eq. 12.1. First, we will show how various simplified expressions, each applicable over a limited range, emerge as special cases of Eq. 12.1. Second, we will find that by using reduced variables, important conclusions can be drawn from Eq. 12.1 without regard to its awkward form. 

Several approximate forms of Eq. 12.1 can be used for packed columns. Over a limited low-velocity range where diffusion controls the mobile phase random walk, the terms containing drop out and Eq. 12.1 reduces to 

At higher velocities, flow dominates diffusion in the central (coupling) term on the right of Eq. 12.1, giving the limiting form 

Another alternative to the complete coupled expression of Eq. 12.1 is the somewhat simplified coupling equation found by replacing the summation by a single term. This can be written in several forms, including 

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To minimize the multiple path and mass transfer contributions to plate height (equations 12.23 and 12.26), the packing material should be of as small a diameter as is practical and loaded with a thin film of stationary phase (equation 12.25). Compared with capillary columns, which are discussed in the next section, packed columns can handle larger amounts of sample. Samples of 0.1-10 )J,L are routinely analyzed with a packed column. Column efficiencies are typically several hundred to 2000 plates/m, providing columns with 3000-10,000 theoretical plates. Assuming Wiax/Wiin is approximately 50, a packed column with 10,000 theoretical plates has a peak capacity (equation 12.18) of... 

The general approach for kinetic optiaization of open i tubular columns has been to adopt the familiar Golay equation T (equation 1.34) and to assuae that the aobile phase can be approximated by an incompressible fluid with ideal gas properties, (44-50). Circumstances that are approximate at best but serve adequately to demonstrate some of the fundamental characteristics of open tubular columns operated at low fluid densities. The column plate height equation can be written in the form given in M equation (6.1)... 

In simpler formulations,1811 the fourth and fifth terms are neglected, and the temperature is taken to be constant, so the plate height equation is written as... 

In order to compare data obtained with otherwise similar chromatographic systems in which only the particle size of the column packing and solute diffusivity may vary, Eq. (21) should be written in dimensionless form. Using an approach taken from chemical engineering, Knox (7, 34) has shown that a corresponding reduced plate height equation is given by Eq. (22)... 

In general, optimization calculations are considerably easier when they are carried out at the maximum efficiency or when the reduced plate height equation is used. 

The most detailed model is the general rate model [2]. This model has been studied by several authors [3]. The moments calculated from the general rate model allow the derivation of a most detailed plate height equation for both particulate and monolith columns [4]. 

Many modifications to the original van Deemter plate height equation have appeared in the literature (19-23). Some account for mass transfer in the gas phase (19,20) and other modifications were made for velocity distribution because of retarded... 

Electroosmotic flow, described in Section 4.9, is another complicating factor in electrophoresis. The electroosmotic flow process is often responsible for nonselective ion transport superimposed on the electrophoretic transport. When electroosmotic displacement is significant, it must be kept in mind that the distance X in the preceding plate height equations is the displacement distance due to electrophoresis alone it does not include electroosmotic displacement. Also the voltage V must be calculated as that applied over the path of electrophoretic displacement only, not including the distance of electroosmotic displacement [41]. 

Equation 9.17 can be used to obtain a plate height equation for field-flow fractionation. (The longitudinal diffusion term, Eq. 9.14, is generally negligible in FFF.) We assume that the relative flow dispersion hYIVis approximately unity. Most diffusion processes are limited to paths not much larger than thus we can write d 2(. With these assumptions we get the equation [24]... 

A more detailed treatment of chromatography (see Chapter 11) suggests a plate height equation of the form [2]... 

The last two equations provide a good summary of how the various plate height terms are assembled into an overall plate height equation for chromatography. Whether or not all the constants (y, qy A,) of the equations are known, these expressions show the nature of the dependence of H on the controllable system parameters vy dp, d, Ry Dmy and Ds. The expressions consequently have profound consequences for guiding practical chromatography. These consequences will be discussed in the next chapter. 

The optimization procedure outlined above is independent of the specific plate height equations chosen to represent chromatography. It depends only on the validity of the scaling relationships used to define reduced plate height and reduced velocity, and the emergence of universal curves from the scaling relationships. 

Transform the low-velocity form of the plate height equation... 

Reduced Plate Height Equation A Common Link Between Chromatographic Methods, J. C. Giddings, J. Chromatogr., 13, 301 (1964). 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

It can be seen that as soon as retention takes place there will be a loss in performance. This will be most evident in latger bore tubes. This impo.ses a restriction on tube diameter and makes the use of narrow bore tubes necessary to achieve the maximum performance, which limits, due to practical constraints, the use of pressure drive. In CEC the packed bed introduces further contributions to the plate height equation. 

The band-broadening contributions can be described in the form of a plate-height equation, where one usually assumes, as in chromatography, mutual independence of terms. 

Guiochon, G. Siouffi, A. Band broadening and plate height equation, in. Elow velocity. J. Chromatogr. Sci. 1978, 16, 470-481, 598-609. 

Plate Height Equation in the General Rate Model.313... 

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