To understand the benefits that the method of moments can offer, we base this part of our discussion on Figure 4, experimental
data for a representative globular protein (molecular weight of about 150 kDa). The number of plates is calculated both by
the method of moments (Equation 15), and by the widely used short-cut technique (at 50% of peak height from Table 1). The
method of moments is seen to approach a straight-line asymptote on this figure, indicating that dispersion increases a bit
faster than linearly with percolation velocity. This is in agreement with well-established theory as well as with common observation.
To characterize the extent of dispersion, redefine plate number as per Equations (15) and (16).11
The slope of a straight line from the origin to any point on this correlation (defined in Equation 17) can provide useful
information for column scale-up. If we substitute Equation (17) and (18) into Equation (19), we are left with Equation (20).
The truest measure of column effectiveness is the number of plates, and Schneider found this to be proportional to solvent
residence time, and inversely proportional to the slope g. It suggests, correctly, that solvent residence time is a safe scale-up
criterion if velocity is no higher in the larger column, and the penalty for higher velocity can be predicted from the van
Deemter plot of HETP as a function of u. Maintaining both solvent residence time and percolation velocity is almost always
a reliable scale-up criterion — even for affinity columns with their notoriously slow kinetics. Moreover the effect of increasing
velocity at constant solvent residence time is modest under most commonly observed conditions (high Pé).
On the other hand, the short-cut method falsely suggests that higher velocities yield a very large dividend on scale-up. They
do not, and high velocities require expensive pumps and massive column walls. This is especially true for compressible particles.
CHARACTERIZING COLUMN FLOW DISTRIBUTION
We consider the problem of characterizing column packing. Poor flow distribution, resulting from either the column flow distributors
or packing heterogeneity, can cause substantial deviations from the optimal performance of a chromatographic column. In a
typical chromatographic process, flow in the distributors and the packed bed operates at very low Reynolds numbers, or in
a laminar mode called creeping flow. Creeping flow through porous media is described by the pseudo-continuum Blake-Kozeny equation, presented here as Equation (21).10
Reversing the pressure drop, which corresponds to a reversal of the direction of flow, corresponds only to a change in the
sign but not the magnitude of the velocity. For this reason, creeping flow is said to be reversible. It should be noted that
this equation and the derived reversibility is valid only for a distance scale that is large in comparison to d
. Characterizing heterogeneity can therefore be made with surprising ease via order-of-magnitude analysis because it usually
occurs on a size scale, which is large compare to packing particle diameters.
The intrinsic broadening of a solute pulse, as predicted from linear chromatographic models, operates on a distance-scale
comparable to a particle diameter, and it is irreversible. Thus, we may conclude that plate height is the sum of the dispersive
contributions. All these dissipative processes occur independently of the flow direction.