Knox plot

Fig. 4 The Knox plot h (reduced plate height) versus (reduced velocity). (Reproduced with permission from Preston Publications.)...

The graphical representation of In h as function of In u (Fig. 2-6) is known in the literature as the Knox plot. The dependence of the curve s position on the retention of the compound is disadvantageous. Minima in this kind of illustration are only obtained for compounds having no retention (k = 0). 

Fig. 2.4. A Knox plot. Reproduced from Bristow (1976) with permission.

These four factors cause band broadening and work additively such that their summation contributes to H, the height equivalent to a theoretical plate. This parameter is dependent on flow rate due to the contribution from longitudinal diffusion and mass transfer. Band broadening can therefore be minimised by use of an optimum flow rate. This relationship is usually visualised in a Knox plot (Fig. 2.4) using the reduced plate height (/j) as the vertical axis and the reduced velocity (u) as the horizontal axis. The reduced velocity is given by the formula ... 

The plate number N can be increased by using a longer column, by decreasing the particle size or by optimising the flow rate (from the Knox plot). However, since resolution depends on it can be impractical to significantly increase the resolution by means of this alone. 

Table 6.2 lists the A, B and C values experimentally obtained by several workers [19-22]. To be meaningful, the Knox plots should be obtained on the same column, with the same solute and different mobile... 

The use of the Knox plots to study the causes of micellar reduced efficiencies leads to the following conclusions. The micellar phase flow anisotropy seems to be much higher than the flow anisotropy obtained with a hydro-organic phase of comparable viscosity (increased A term). This is only partly due to the micellar viscosity. The main reason of such differences in flow patterns is the partial clogging of the stationary phase pores by adsorbed surfactant molecules [19, 22]. A temperature raise decreases the mobile phase viscosity and the amount of adsorbed surfactant [22]. Both effects decrease the flow anisotropy and the A term. It will be exposed thereafter that alcohol additions to a micellar phase dramatically reduce the amount of adsorbed surfactant. 

Clearly, the Knox plot study points out that surfactant adsorption on the stationary phase is responsible for the bulk of efficiency loss observed using micellar phases. A slow solute exchange between the micelle apolar core and the aqueous phase is another possible explanation for MLC efficiency loss. 

MLC efficiency can be enhanced (i) by reducing the mobile phase flow rate to work closer to the optimum of the Knox plot, (ii) by increasing the temperature which decreases the viscosities, increases the rate constants and decreases the amount of adsorbed surfactant, (iii) especially by adding an organic modifier such as an alcohol whose alkyl chain has a length such as the ratio Cnon/Cnsurf close to 1/3. The amoimt of acMed alcohol should be increased if the surfactant concentration is increased to keep constant the alcohol to surfactant ratio. 

The curves represent a plot of log (h ) (reduced plate height) against log (v) (reduced velocity) for two very different columns. The lower the curve, the better the column is packed (the lower the minimum reduced plate height). At low velocities, the (B) term (longitudinal diffusion) dominates, and at high velocities the (C) term (resistance to mass transfer in the stationary phase) dominates, as in the Van Deemter equation. The best column efficiency is achieved when the minimum is about 2 particle diameters and thus, log (h ) is about 0.35. The optimum reduced velocity is in the range of 3 to 5 cm/sec., that is log (v) takes values between 0.3 and 0.5. The Knox... 

FIGURE 6.1 A Poppe plot for the required plate number in conventional HPLC. The parameters are taken from Poppe s original paper (Poppe, 1997). The parameters are maximum pressure AP = 4x 107 Pa, viscosity / = 0.001 Pa/s, flow resistance factor

diffusion coefficient D= lx 1CT9 m2/s, and reduced plate height parameters using Knox s plate height model are A — 1, B— 1.5, C = 0.05. 

Further examination of equations (1) and (2) indicates 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 dlffuslvlty. A plot of the (B) term against diffusivity for benzyl acetate and hexarnethyl benzene is shown in figure 4. 

Otos could be computed. All the Hixson and Smith data plot well in this fashion, and the straightness of the lines indicate the utility of the time-of-a-transfer-unit concept. Hixson, Drew, and Knox (H3) showed that a characteristic agitation number may be defined as the product of 0tOE and a velocity term for the agitated system. If then the mass transfer coefficient varies as the first power of the chosen velocity term, the agitation number would be constant for a given ratio of interfacial surface to total number of moles of extract phase. In liquid extraction, speed of agitation influences both terms of the quantity Ke[Pg.307]

Figure 5.4 shows plots of the reduced rate equation (on a log-log basis) taken from Knox and Saleem.4 GC is compared with LC for two stationary phases and several nonretained analytes. While all the points do not fall on one smooth curve, the plot clearly shows the similarities between GC and LC plots when reduced parameters are used. 

The methyl hydroperoxide concentration increased to a maximum at the time of maximum rate of reaction and it was concluded that this compound was responsible for chain-branching. This was subsequently confirmed [35] using the theoretical treatment developed by Knox [36]. Values of the rate coefficient of the branching reaction at several temperatures were obtained from the intercepts of the plots of the acceleration constant (0) plotted against acetone concentration. The variation of rate coefficient with temperature was expressed by the equation [35]... 

The van Deemter equation is a useful approximation however, the experimental H u plots often show some downward curvature on the right-hand branch, unpredicted by Eq. (1.10). Giddings explained this behaviour by coupling the flow and the diffusion effects which demonstrates that it is not strictly correct to consider the simple additivity of their contributions to band broadening and he suggested more sophisticated equations to account for this phenomenon [3. For practical purpose, a simple empirical equation, which accounts for the experimental behaviour and is only slightly different from the van Deemter expression was introduced by Kennedy and Knox [4. ... 

Figure 6.13 Comparison of the plots of the reduced axial plate height vs. the reduced flow velocity obtained (a) by PFGNMR and (b) using a conventional chromatographic method. Column packed with 50 m particles of porous C18 silica. The lines shown are the best fits of the experimental data (symbols) to the correlations suggested by Giddings (x = 1), Huber (x = 0.5), and Horvath and Lin (x = 0.33). In either case, the best fit of the data to the Knox equation coincides with that to the Horvath and Lin correlation. Reproduced with permission from U. Tallarek, E. Bayer, G. Guiochon, J. Am. Chem. Soc., 120 (1998) 1494 (Fig. 6). 1998 American Chemical Society.

A t5q ical plot of Eq. 6.105c is shown in Figure 6.11 and compared to the classical Van Deemter and Knox equations. As in liquid chromatography dy is of the order of a tenth of dp and Dm is not much larger than D , the coefficient of the v term in Eq. 6.105c is between 0.01 (fc(, = 0) and 0.1 (at large values of fcg). Thus, the coefficient selected for the curve in Figure 6.11 corresponds to a rather poor column. 

Although there are separate optima for the column length and the particle size, the production rate varies slowly with either L or dp when the ratio dp/L is kept constant. The plot of the production rate versus L and dp has a ridge which is nearly parallel to the direction dp/L = constant (see Figure 18.12). This confirms the theoretical results of Knox and Pyper [17] and of Golshan-Shirazi and Guiochon [20,21]. 

Generally, if we plot the HETP against a linear velocity, we obtain a curved relationship with a minimum and a nearly linear increase of the HETP with linear velocity (Fig. 2.4) at high linear velocity. This relationship can be described by several equations the van Deemter equation, the Giddings equation, and the Knox equation. We will explore all of them in the following, starting with the van Deemter equation. 

Giddings coupling theory does not result in a practical approach to describe the properties of a packed bed. Therefore Kenney and Knox (13) introduced an empirical equation that contains a term useful for capturing the observed curvature of a HETP-velocity plot and so overcomes one of the shortcomings of the van Deemter equation. The term introduced by Knox contains the third... 

Plotting k versus [H], a parabolic curve is obtained when, according to Eq. (6.30), (1)/A3 [H] <11/ 2. Such dependences have been observed by Knox and coworkers [5,12]. 

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