Thursday, March 4, 2010

QbDer, Know Thy Model!

By Dr. Scott Rudge

Resolution in chromatography is critical, from analytical applications to large scale process chromatography. While baseline resolution is the gold standard in analytical chromatography, it is seldom achieved in process chromatography, where “samples” are concentrated and “sample volumes” represent a large fraction of the column bed volume, if not multiples of the bed volume. How do you know if your resolution is changing in process chromatography if you can’t detect changes from examining the chromatogram?

Many use the Height Equivalent to a Theoretical Plate technique to test the column’s resolution. In this technique, a small pulse of a non-offensive but easily detected material is injected onto the column, and the characteristics of the resulting peak are measured. The following equation is used:

Where H is the “height” in distance, tr and tw,1/2 are the retention time and time of the width of the peak at ½ height, L is the length of the column and N is the “number” of theoretical plates in the column. The lower the H, the smaller the dispersion, the greater the resolution.

It is well established that the flowrate and temperature affect the plate height. In fact, when the plate height is plotted against flow rate, we generate what is typically called the “van Deemter plot”, after Dutch scientists (van Deemter et al., Chem. Eng. Sci.,5, 271 (1956)) who established a common relationship in all chromatography (gas and liquid) according to the following equation:

Where A, B and C are constants and v is the linear velocity (flow rate divided by the column cross sectional area) of fluid in the column. It was later proposed by Knox (Knox, J., and H. Scott, J. Chromatog., 282, 297 (1983)) that van Deemter plots could be reduced to a common line for all chromatography if the plate height was normalized to resin particle size, and linear velocity was normalize to the resin particle size divided by the chemical’s diffusivity. While this did not turn out to be generally true, it is very close to true. Chemical engineers will recognize the ratio of linear velocity to diffusivity over particle size as the Peclet number, “Pe”, a standard dimensionless number used in many mass transfer analyses.

Since diffusivity is sensitive to temperature, it is logical that Pe is also sensitive to temperature, decreasing temperature decreases the diffusivity, and increases Pe. Thus, Pe is inversely proportional to temperature. We measured plate height as a function of linear flowrate and temperature in our lab on a Q-Sepharose FF column, using a sodium chloride pulse, and found the expected result, shown in the graph below.

We can easily use this graph as a measurement of van Deemter’s parameters A and C, and find the dependence of the diffusivity of sodium chloride on temperature. Based on these two points, we find the Peclet number proportional to 0.085/T where T is in degrees Kelvin. We also find the dependence of plate height on linear velocity, and we can predict that resolution will deteriorate in the column as flow rate increases.

We can also use Design of Experiments to find the same information. Analyzing the same data set with ANOVA yields significant factors for both flow rate and temperature, as shown in the following table:

Term Coef SE Coef T P

But since the statistics don’t know that the temperature dependence is inverse, or that the Peclet number is a function of both temperature and flowrate, the model yields no additional understanding of the process.

It is possible to use analysis of variance to fit a model other than linear, as the van Deemter model clearly is. But one must know that the phenomenon being measured behaves non-linearly in order to use the appropriate statistics. Using Design of Experiments blindly, without knowing the relationship of the factors and responses, leads to empirical correlations only relevant to the system studied, and should only be extended with caution.

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