The Optimum Velocity

The expression for the optimum mobile phase velocity is given by equation (28) in chapter 12 and is as follows. 

It is seen that the optimum velocity is inversely proportional to the optimum particle diameter and it would be possible to insert the expression for the optimum particle diameter into equation (2) to provide an explicit expression for the optimum velocity. The result would, however, be algebraically cumbersome and it is easier to consider the effects separately. The optimum velocity is inversely 

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It is seen that as the optimum particle diameter is inversely proportional to (cx-1), 

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The Optimum Velocity Profile. The optimum velocity profile (41), that is the velocity profile that yields the maximum value for the flow pattern efficiency, is one in which the mass velocitypv is constant over the radius of the centrifuge except for a discontinuity at the wall of the centrifuge (r = rP). This optimum velocity profile is shown in Figure 14a. For this case the following values for the separation parameters of the centrifuge are obtained... 

Fig. 14. Hypothetical velocity profile models for a countercurrent-flow gas centrifuge, (a) The optimum velocity profile ia a countercurrent gas centrifuge.

It is seen from equation (26) that the optimum velocity is determined by the magnitude of the diffusion coefficient and is inversely related to the particle diameter. Unfortunately, in LC (where the mobile phase is a liquid as opposed to a gas), the diffusivity is four to five orders of magnitude less than in GC. Thus, to achieve comparable performance, the particle diameter must also be reduced (c./., 3-5 p)... 

A number of HETP equations were developed other than that of Van Deemter. Giddings developed an alternative form that eliminated the condition predicted by the Van Deemter equation that there was a finite dispersion at zero velocity. However, the Giddings equation reduced to the Van Deemter equation at velocities approaching the optimum velocity. Due to extra-column dispersion, the magnitude of which was originally unknown, experimental data were found not to fit the Van Deemter... 

Curves relating the optimum velocity to the solute diffusivity are shown in Figure 6. It is seen that the straight lines predicted by the Van Deemter equation are realized for both solutes. 

The basically correct equation appears to be that of Giddings but, over the range of mobile phase velocities normally employed i.e., velocities in the neighborhood of the optimum velocity), the Van Deem ter equation is the simplest and most appropriate to use. 

It should be noted that the velocity employed in equation (7) will be the exit velocity (uq) and not the mean velocity (umean)- Differentiating equation (3) and equating to zero, to obtain an expression for the optimum velocity (uopt). 

Employing the new equations for (C), from equation (8) the optimum velocity (uopt) is given by... 

The efficiency obtained from an open tubular column can be increased by reducing the column radius, which, in turn will allow the column length to be decreased and, thus, a shorter analysis time can be realized. However, the smaller diameter column will require more pressure to achieve the optimum velocity and thus the reduction of column diameter can only be continued until the maximum available inlet pressure is needed to achieve the optimum mobile phase velocity. 

Equation (13) is the first important equation for open tubular column design. It is seen that the optimum radius, with which the column will operate at the optimum velocity for the given inlet pressure, increases rapidly as an inverse function of the separation ratio (cc-1) and inversely as the square root of the inlet pressure. Again it must be remembered that, when calculating (ropt)5 the dimensions of the applied pressure (P) must be appropriate for the dimensions in which the viscosity (r)) is measured. 

Table 1, values for the optimum velocity can be calculated for different values of the separation ratio of the critical pair. The results are shown in Figure 3. 

Although the optimum column radius increases linearly with the separation ratio of the critical pair, this simple relationship is moderated by the ratio of the square of the optimum radius to the optimum velocity, both of which are functions of (a). 

Figure 12. The Linear Curve Relating the Optimum Velocity to the Separation Ratio of the Critical Pair...

The optimum velocity changes linearly from about 1 cm/sec. to 800 cm/sec. and the minimum plate height from 200 pm to about 10 pm over the same separation ratio... 

H) and thus decrease the column efficiency. LC columns are rarely operated below the optimum velocity and thus this situation is the least likely scenario. 

It is interesting to ascertain the effect of temperature at the optimum velocity where the value of (H) is a minimum and the column efficiency a maximum. 

Differentiating equation (2) with respect to (u) and equating to zero to find the optimum velocity,... 

It is seen that when operating at the optimum velocity that provides the minimum value of (H) and thus, the maximum efficiency, solute diffusivity has no effect on solute dispersion and consequently, the column efficiency is independent of temperature. 

However, from equation (3) it is seen that the optimum velocity is directly related to the solute diffusivity and thus, to the temperature. It follows that increasing the temperature increases the optimum velocity and thus provides the same efficiency but a shorter analysis time. 

A column 3 cm long, 4.6 mm in diameter packed with particles 3 Jim in diameter will give about 6,000 theoretical plates at the optimum velocity. This efficiency is typical for a commercially available column. 

Thus, the column should completely resolve about 14 equally spaced peaks. It is seen from figure 1 that a peak capacity of 14 is not realized although most of the components are separated. This means that the column may not have been packed particularly well and/or the flow rate used was significantly above the optimum velocity that would provide the maximum efficiency. The mobile phase that was used was tetrahydrofuran which was sufficiently polar to deactivate the silica gel with a layer (or perhaps bilayer) of adsorbed solvent molecules yet was sufficiently dispersive to provide adequate sample... 

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