Knox equation

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 appears that the equation introduced by Van Deemter is still the simplest and the most reliable for use in general column design. Nevertheless, all the equations helped to further understand the processes that occur in the column. In particular, in addition to describing dispersion, the Kennedy and Knox equation can also be employed to assess the efficiency of the packing procedure used in the preparation of a chromatography column. 

Stating the Van Deemter and Knox equations in the explicit form, the Van Deemter equation is... 

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 5 Knox equation, with reduced plate height, h reduced velocity (m dpID ), V, coefficient B, describing axial diffusion (typical value 2) coefficient A, describing bed homogeneity (typical value 1-2) and coefficient C, describing mass transfer (typical value 0.05). 

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). 

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. 

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. 

Equation (1.27), which has been expanded to different types of liquid chromatography (Knox equation), shows that there is an optimum flow rate for each separation and that this does indeed correspond to the minimum on the curve represented by equation (1.27). The loss in efficiency that occurs when the velocity is increased represents what occurs when trying to rush the chromatographic separation by increasing the flow rate of the mobile phase. However, intuition can hardly predict the loss in efficiency that occurs when the flow is too slow. To explain this phenomenon, the origins of the terms A, B and C have to be considered. Each of these parameters has a domain of influence that can be seen in Fig. 1.9. Essentially, this curve does not depend on the nature of the solute. 

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) ... 

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