The Optimum Particle Diameter

The column length is also defined by the D Arcy equation that describes the flow of a fluid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D Arcy equation is given as follows, 

Substituting for (u0pt) and (Hmin) from equations (9) and (10) respectively, 

It is seen that there will be a unique value for (dp), the optimum particle diameter, that will allow the minimum HETP to be realized when operating at an inlet pressure (P). Rearranging and solving for dp(opt) 

Equation (18) allows the optimum particle diameter to be calculated that will allow the separation to be achieved in the minimum time by utilizing the maximum available inlet pressure and operating at the optimum mobile phase velocity. It is one of the most important equations in column design. 

The characteristics of many of the equations discussed in this chapter will be tested against realistic chromatographic conditions and the typical conditions chosen are given in table 1. 

It should be pointed out that equations (14) and (15) do not give an expression for the minimum column lengths, as the optimum particle diameter has yet to be identified. 

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Substituting for (uopt) and (Hmin.) from equations (8) and (11) respectively. 

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P). Note the expression for (C) is also a function of the particle diameter (dp) and includes known thermodynamic and physical properties of the chromatographic system. Consequently, with the aid of a computer, the optimum particle diameter (dp(opt)) can be calculated as that value that will meet the equality defined in... 

It follows that knowing the optimum particle diameter, the optimum column length can also be identified. It must be emphasized that this optimizing procedure... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

Consider first the equation for the optimum particle diameter. Reiterating equation (18),... 

In a packed column the HETP depends on the particle diameter and is not related to the column radius. As a result, an expression for the optimum particle diameter is independently derived, and then the column radius determined from the extracolumn dispersion. This is not true for the open tubular column, as the HETP is determined by the column radius. It follows that a converse procedure must be employed. Firstly the optimum column radius is determined and then the maximum extra-column dispersion that the column can tolerate calculated. Thus, with open tubular columns, the chromatographic system, in particular the detector dispersion and the maximum sample volume, is dictated by the column design which, in turn, is governed by the nature of the separation. 

The expression that gives the optimum particle diameter is given by equation (27), chapter 12 and is reiterated here. The optimization of the particle diameter will be considered first, as each of the other operating parameters will, directly or indirectly, be determined by the magnitude of the optimum particle diameter. 

The function f(k ) is shown plotted against the thermodynamic capacity ratio in Figure 1. It is seen that for peaks having capacity ratios greater than about 2, the magnitude of (k ) has only a small effect on the optimum particle diameter because the efficiency required to effect the separation tends to a constant value for strongly retained peaks. From equation (1) it is seen that the optimum particle diameter varies as the square root of the solute diffusivity and the solvent viscosity. As, in... 

Figure 1. The Effect of Solute Capacity Ratio on the Magnitude of the Optimum Particle Diameter...

The optimum particle diameter will decrease as the reciprocal of the available pressure and, thus, pressure will have a very significant effect on the magnitude of (dp(opt)). As already stated, the higher the pressure, the smaller the particle diameter,... 

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... 

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

SFE usually requires pre-extraction manipulation in the form of cryogenic grinding, except in cases where analytes are sorbed only on the surface or outer particle periphery. The optimum particle diameter is about 10-50 p,m. Diatomaceous earth is used extensively in SFE sample preparation procedures. This solid support helps to disperse the sample evenly, allowing the SCF to solvate the analytes of interest efficiently and without interference from moisture. 

It is seen from figure (1) that the optimum particle diameter ranges from about 2 micron for very simple separations (a=1.l2) carried out at an inlet pressure of 6000 p.s.i. to about 40 micron for difficult separations (ot= 1.01 ) carried out at an inlet pressure of only 2000 p s.i Furthermore, the curves shown in figure (1) appear to be in conflict with popular opinion, in that, the more difficult separations are best achieved with particles of relatively large diameter, whereas, simple separations require particles of small diameter for optimum performance. This apparent paradox will be discussed more fully later in the chapter. Equation (18) also discloses some interesting properties of the optimized column. 

Consequently as shown in figure (1) the optimum particle diameter will rapidly increase as (a) becomes smaller i e as the separation becomes more difficult. 

Thus, the larger the available inlet pressure the smaller the optimum particle diameter can be. Nevertheless, as a result of the square root function, the sensitivity of the optimum particle diameter to the magnitude of the inlet pressure is much less than it is to the separation ratio of the critical pair. This is confirmed by the curves shown in figure (1) where it is seen that, providing the inlet pressure is above 2000 p.s.i., the effect of pressure on the optimum particle diameter is not nearly as significant as might be expected. 

In the operation of preparative columns, it is necessary to obtain the maximum mass throughput per unit time and, at the same time, achieve the required resolution. Consequently, the column will be operated at the optimum velocity as in the case of analytical columns. Furthermore, the D Arcy equation will still hold and the equation for the optimum particle diameter can be established in exactly the same way as the optimum particle diameter of the analytical column. The equation is fundamentally the same as that given for the optimum particle diameter for a packed analytical column, i.e. (18) In chapter 12, except that (a) and (k ) have different meanings. 

Employing Equation (2) and the data given in table l, the optimum particle diameter was calculated for the preparative separation of a solute where the separation ratio of the critical pair ranged from l.01 to 1.50 at inlet pressures of 1,10,100,1,000 and 10,000 p.s.i. The results obtained are shown in figure I... 

Figure 6 shows the effect of the three limits imposed on the column design, the need to maintain an aspect ratio greater than unity, restricts the column length to a minimum of 5 cm and confines the analysis time to between one minute and one hour, without the restriction, the column with the minimum analysis time would be operated at the maximum pressure available. However, as a result of the limitation on column length, the optimum pressure must be reduced to allow the optimum particle diameter to be ircreased. This, in turn, allows the column length to be increased to... 

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