The Knox Equation

The most striking change is in the A-term that accounts for multipath dispersion (Equation [3.33] in the van Deemter theory) and decreases rapidly with decreasing i.d. This effect was interpreted (Karlsson 1988) as reflecting a decrease in the extent of the flow rate variation over the column cross-section as the i.d. decreases, together with shorter time for solute molecules to diffuse radially through the flow velocity profile (parabolic in the case of a nonpacked column) together these phenomena will decrease the effect of flow anisotropy that is reflected in 

In 1972-1973 Knox et al. [3, 4, 5] examined, in considerable detail, a number of different packing materials with particular reference to the effect of particle size on the reduced plate height of a column. The reduced plate height (h) and reduced velocity (v) were introduced by Giddings [6,7] in 1965 in an attempt to form a rational basis 

It is seen that the reduced plate height is dimensionless. 

The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities then, like the reduced plate height, it also is dimensionless. Employing the reduced parameters, the equation of Knox takes the following form 

The equation of Knox et al. was not derived theoretically from a basic dispersion model, but the constants of the equation were determined by a curve fitting procedure 

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In fact the reduced plate height merely measures the normal plate height in units of particle diameters. It is also seen that the reduced plate height is dimensionless. 

Employing the reduced parameters the equation of Knox takes the following form, 

It should be noted that the constants of the equation were arrived at by a curve fitting procedure and not derived theoretically from a basic dispersion model as a consequence the Knox equation has limited use in column design. It Is, however, extremely valuable in accessing the quality of the packing. This can be seen from the diagram shown in figure 2. 

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Equation 16-183 is qualitatively the same as the van Deemter equation [van Deemter and Zuiderweg, Chem. Eng. Sci., 5, 271 (1956)] and is equivalent to other empiric reduced HETP expressions such as the Knox equation [Knox,y. Chromatogi Set., 15, 352 (1977)]. 

It is interesting to note that the function from the Knox equation... 

Now, equations (1) and (2) indicate that both the Knox equation and the Van Deemter equation predict a linear relationship between the value of the (B) term (the longitudinal diffusion term) and solute diffusivity. 

Similar treatment of the Knox equation does not predict that values of H(min) should be independent of the solute diffusivity neither does it predict that (uopt) should vary linearly with solute diffusivity. Consequently, the relationships shown in Figures 5... 

In summary, it can be said that all the dispersion equations that have been developed will give a good fit to experimental data, but only the Van Deemter equation, the Giddings equation and the Knox equation give positive and real values for the constants in the respective equations. 

The Van Deem ter equation appears to be a special case of the Giddings equation. The form of the Van Deemter equation and, in particular, the individual functions contained in it are well substantiated by experiment. The Knox equation is obtained... 

A more vigorous treatment similar to the van Deemter equation but developed specifically for HPLC is the Knox equation, which uses a number of reduced parameters where h is the reduced plate height (h/dp) and vis reduced velocity (V dJD. ... 

Equation 6 Calculation of optimum ratio of particle size and column length, with selectivity factor, a capacity factor of second component of critical pair under analytical chromatography conditions, fe 02 diffusion coefficient, (cm /s) (typical value for MW 1000 10 cm /s) viscosity, p (cP) specific permeability (1.2 X 10 for spherical particles), feo third term of the Knox equation, C and maximum safe operating pressure, Ap, (bar). 

The curves represent a plot of Log.(/V),(Reduced Plate height)against Log.(v), (Reduced Velocity). The lower the Log.(/7) curve versus the Log.(v) curve the better the column is packed. At low velocities the (B) term dominates and at high velocities the (C) term dominates as in the Van Deemter equation. The best column efficiency is achieved when the minimum is about 2 particle diameters and thus, Log (.ft) Is about 0.35. The minimum value for (H) as predicted by the Van Deemter equation has also been shown to be about two particle diameters. The optimum reduced velocity is in the range of 3 to 5 that is Log.(v ) takes values between 0.3 and 0.5. The Knox equation is a simple and effective method of examining the quality of a given column but, as stated before, is not nearly so useful In column design due to the empirical nature of the constants. 

It is seen that the predicted linear relationship is indeed realized. However, it can be shown that the values for the (B) term from the Knox equation curve fit also give a linear relationship with solute diffusivity so the linear curves shown in figure 4 do not exclusively support the van Deemter equation. 

As in gas chromatography, there is a relationship between the reduced velocity of the mobile phase and the reduced plate height. It is the Knox equation (4) ... 

I2.9( ) Use the form of the Knox equation given by Eq. 12.9, along with values from Table 12.1, to show that... 

Starting with the Knox equation for the reduced plate height... 

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