HETP equation

The first alternative HETP equation to be developed was that of Giddings in 1961 [1] of which the Van Deemter equation appeared to be a special case. Giddings did not develop his equation because the Van Deemter equation did not fit experimental data. 

In 1967, Huber and Hulsman [2] introduced yet another HETP equation having a very similar form to that of Giddings. Their equation included a modified multipath term somewhat similar in form to that of Giddings and a separate term describing the resistance to mass transfer in the mobile phase contained between the particles. The form of their equation was as follows ... 

Effect of Mobile Phase Compressibility On the HETP Equation for a Packed GC Column... 

It is seen that the HETP equation, as derived by Van Deemter, now applies only to a point distance (x) from the inlet of the column. Now, it has already been shown that... 

Thus, the complete HETP equation for a packed GC column that takes into account the compressibility of the carrier gas will be... 

Thus the HETP equation given by equation (13) becomes... 

It is seen that the two curves are quite different and, if the results are fitted to the HETP equation, only the data obtained by using the exit velocity gives correct and realistic values for the individual dispersion processes. This point is emphasized by the graphs shown in Figure 5 where the HETP curve obtained by using average velocity data are deconvoluted into the individual contributions from the different dispersion processes. 

The HETP equation is not simply a mathematical concept of little practical use, but a tool by which the function of the column can be understood, the best operating conditions deduced and, if required, the optimum column to give the minimum analysis time calculated. Assuming that appropriate values of (u) and (Dm) and (Ds)... 

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... 

To identify the pertinent HETP equation that describes dispersion in a packed bed, the following logical procedure will require to be carried out. 

It follows that the contribution from the resistance to mass transfer in the stationary phase can be ignored with respect to that in the mobile phase for all values of (k ),and the HETP equation simplifies to... 

Hence the term "HETP equation" for the equation for the variance per unit length of a column. 

To develop an HETP equation it is necessary to first identify the dispersion processes that occur in a column and then determine the variance that will result from each process per unit length of column. The sum of all these variances will be (H), the Height of the Theoretical Plate or the total variance per unit column length. There are a number of methods used to arrive at an expression for the variance resulting from each dispersion process and these can be obtained from the various references provided. However, as an example, the Random-Walk Model introduced by Giddings (5) will be employed here to illustrate the procedure.The theory of the Random-Walk processes itself can be found in any appropriate textbook on probability (6) and will not be given here but the consequential equation will be used. 

Equation (10) is the basic form of the Van Deemter equation and will be expanded and discussed with other HETP equations in the next chapter. 

The Van Deemter equation remained the established equation for describing the peak dispersion that took place in a packed column until about 1961. However, when experimental data that was measured at high linear mobile phase velocities was fitted to the Van Deemter equation it was found that there was often very poor agreement. In retrospect, this poor agreement between theory and experiment was probably due more to the presence of experimental artifacts, such as those caused by extra column dispersion, large detector sensor and detector electronic time constants etc. than the inadequacies of th Van Deemter equation. Nevertheless, it was this poor agreement between theory and experiment, that provoked a number of workers in the field to develop alternative HETP equations in the hope that a more exact relationship between HETP and linear mobile phase velocity could be obtained that would be compatible with experimental data. 

The identification of the pertinent HETP equation must, therefore, be arrived at from the results of a sequential series of experiments. Firstly, all the equations must be fitted to a series of (H) and (u) data sets and those equations that give positive and real values for the constants of the equations identified. The explicit form of those equations that satisfy the preliminary data, must then be tested against a series of data sets that have been obtained from different chromatographic systems. Such systems might involve columns packed with different size particles or employ mobile phases or solutes having different but known physical properties. 

Constant in Strigle s HETP equation for hydrocarbon stripping, Eq. (9.37)... 

The marked differences in flow profiles have a significant effect on the efficiency of separation in open tubes. This is discussed in many publications including Knox and Grant (41. In the case of an open tube, the flow variation across the tube in pressure drive means that as the solute moves along it is dispersed. This is counteracted by transverse molecular diffusion to give a resulting net dispersion that is described by the Taylor equation 16. The additional effects of axial dispersion leads to the HETP equation... 

The plug-Iike profile in the electro-drive system means that the solute undergoes minimal flow-derived dispersion. This effectively removes the diameter-dependent second term in Eq. (4.3), and therefore the HETP equation becomes ... 

However, once retention of the solute occurs, transcolumn equilibration becomes necessary and the HETP equation contains a diameter-dependent mass transfer term. The HETP equations for a retained solute in both pressure- and electro-drive systems are due to Golay [17] (Eq. (4.5)) and Aris [18] (Eq. (4.6)). 

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