Anomalous Axial Dispersion Trends in Anion Exchange Chromatography

I’m trying to track the parameters of a lumped rate model without pores as a function of resin usage for anion exchange chromatography.
My hypothesis is:
As the number of resins usage increases, the performance will decrease so that the Axial_dispersion value will increase, resulting in broader and lower peaks.
However, in my experiments, the value of axial_dispersion actually decreased linearly from 8e-08 to 3e-08.
Do you have any idea why this happened?

For reference, the experimental data used in the inverse fitting is the data where NaCl is continuously added and then wfi (conductivity 0) is added to lower the concentration. As a result, there is a region where the concentration drops off sharply, which I used mainly for fitting.

It’s hard for me to give a definitive answer without seeing what the cycle data and model fits look like—maybe you can share an example?—but I have some suggestions.

You can perform transition analysis on the conductivity drops to directly assess the impact of repeated resin use on column efficiency (see below BPI article). This approach does not involve using CADET but has been employed successfully for similar problems.

You can fit pulse injection data for acetone/salt pulse or blue dextran to determine D_{ax} and assess how it changes over cycles.

You also may want to try using different linear velocities for these evaluations—remember Van Deemter!

I agree with your hypothesis in the sense that as column efficiency decreases with repeated use, fitted values of D_{ax} should increase. However, that does not necessarily imply that axial dispersion increases over cycles because there are other origins of band broadening. Over continued use, there may be changes in packing, compression, and even channeling through the column.



There has been a study on column fouling by the Morbidelli group, which is probably of high relevance for you.

They found that the mass transfer and/or the binding capacity can decrease depending on how you clean the column.
I believe that the changes in the axial dispersion will very low and of little relevance for the overall performance…


If we use the formulation of D_{ax} obtained from moment analysis of the equilibrium dispersive model, if I recall correctly, we can see that it is dependent on the column efficiency.

D_{ax} = \mathrm{HETP} \left(\frac{u}{2\varepsilon_e}\right), where u is the superficial velocity and \varepsilon_e is the interstitial porosity. Here, \mathrm{HETP} can be determined using pulse injections under non-binding conditions.

Using this expression, calculated values of D_{ax} will increase over resin cycles—assuming that column efficiency decreases. This is because plate height will increase with band broadening as the column degrades. This analysis suggests an opposite trend to that stated:

We can also consider the correlation by Rastegar and Gu for the Péclet number and obtain an expression to directly estimate D_{ax}.

D_{a x}=0.7 D_o+\frac{d_p u}{0.18+\left(u d_p \nu^{-1}\right)^{0.59}}, where D_o is the free solution diffusivity, d_p is resin particle diameter, u is the superficial velocity, and \nu is kinematic viscosity.

Assuming D_o and \nu are constant, d_p would have to change to alter D_{ax} for a certain u. Seems like we are missing something.

@jaymax, I think this calls for further investigation. Here are some ideas for you to pursue.

Putting the experimental data aside for a moment, you can simulate the salt concentration transition using a range of D_{ax} values and linear velocities. You can also simulate salt pulse injections and sample the same parameters. You can even vary \varepsilon_e and d_p, if for some reason these parameters change over cycles.

From this analysis, it should be clear what the expected trends should be—from a theoretical standpoint. There are, of course, factors that we cannot easily consider, such as changes in packing heterogeneity, compression, and flow distribution. If you are feeling ambitious, you can also account for the particle size distribution.


First of all, thanks for the very detailed answer to my question! :smile:

@Lukas_Gerstweiler thanks for introducing me to this paper. It was actually the paper I referenced when I started my CADET project and it was very insightful. If you don’t mind, could tell me why you think the axial dispersion is negligible on the overall performance?

@alters Hey, I’ve been trying to understand this phenomenon for the past few days, but no formula explains it, and the behavior of the column suggests that my hypothesis is wrong. I actually plotted the whole peaks, and I can see with the naked eye that as the number of resins usage increases, the sharpness of the conductivity drop during the transition from NaCl to WFI becomes progressively steeper.
I think the packing method is the key of this phenomenon. Since this column used SOURCE series resin, it was necessary to pack the resin with very high pressure, but due to the performance problem of the packing machine, it was not possible to apply such high pressure. Therefore, I came up with the following hypothesis again.


  1. the column was not pressurized enough at the time of packing and the resin was packed with some space left to be pressed further.
  2. the pressure applied during the process pressed the resin more uniformly, which is why the performance of the column actually increased and the axial dispersion decreased, as the number of resin usage increases

Does the above hypothesis make sense to you?

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I think your hypothesis is reasonable, since if the interstitial space is shrunken (\varepsilon_e decreases) you will reduce the number of flow paths and effectively decrease D_{ax}.

Additionally, depending on how compressible the beads are, you could possibly decrease the particle size to some extent which would also decrease D_{ax}. This seems less likely though.

Have you noticed any increases in pressure over the course of your cycles? If we refer to the Kozeny-Carman equation,

\frac{\Delta P}{L} = \frac{-180 \mu}{d_p^2}\frac{\left(1-\varepsilon_e\right)^2}{\varepsilon_e^3}u

we can see that if \varepsilon_e decreases and/or d_p decreases, pressure drop will increase. So if you observe a change in pressure over resin lifetime, it may confirm your hypothesis.

Additonally, if you perform tracer pulse injections (acetone and blue dextran) after each cycle you can monitor how the HETPs are changing. If HETP is increasing, it would be consistent with your findings.



I want to add that I actually have seen a similar behavior on conducting fouling studies on prepacked captoQ columns, in which the breakthrough curve became steeper during the first few experiments before it starts to become worse as expected. It was quite significant and I have observed this on 5 different columns and also cannot explain it. My hypothesis was that the bed might rearranges and becomes more evenly packed, but havent found time to further investigate this. So I agree your hypothesis can be an explanation.

Regarding the axial dispersion, I think it depends a bit what you try to express with it. Based on Van Deemter the peak broadening is based on Eddy Diffusion (caused by different flow paths, depending on the packing quality), mass transfer, and diffusion in the liquid. To my understanding (and please correct me here if I am wrong), D_ax in CADET is time dependent and hence represents peak broadening by diffusion. This should be little affected by column fouling, and is only relevant at very low flow rates. Usually the peak broadening caused by Eddy Diffusion/packing quality and mass transfer is much larger.
As suggested by @alters trying a range of different flow rates would allow to further investigate this.


The equations for describing molecular diffusion and column dispersion are actually the same, even though based on different mechanisms or driving forces. By default, the coefficient does not change over time, but the band broadening effect increases as a peak or front moves from the column inlet to outlet.


@Lukas_Gerstweiler Thanks for sharing your experience! I have now completed fitting the entire usage cycle, and I can see that the trend of axial dispersion is similar to yours, in which the value decreased to some extent and then went up. In terms of axial dispersion, I believe it represents A (Eddy diffusion) and B (Axial diffusion), based on the article written by Cytiva (Understanding fluid dynamics within a chromatography column | Cytiva). Since I used the Lumped rate model with pore instead of the General rate model, film mass transfer isn’t relevant to my study, so I thought I could skip it. As a result, the two remaining terms of the Van Deemter equation (A, B) are affected by the axial dispersion value, meaning the performance of the column is 100% influenced by axial dispersion itself. I might have some logical errors due to my incomplete knowledge about chromatography :joy: Do you think my logic is reasonable?

@alters Thank you for your suggestion. I followed your advice and found a clear trend of increasing pressure drop along the resin usage cycle, as well as in axial dispersion. So I think this follows the Kozeny-Carman equation, which means the hypothesis is correct. (One thing that bothers me is that the bed porosity derived from the inverse fitting method of CADET showed no significant trend of going up or down.) One thing I want to ask you about is the equation you showed me in your previous answer:


I tried to find the source of this equation, but I couldn’t. Could you tell me the source of this equation or its name?

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Nice! That is encouraging to hear that the pressure values over cycles change as dictated by the theory.

I would not recommend inverse fitting the porosity, or any physical parameter for that matter, because it can be directly measured experimentally (e.g., injection of blue dextran or large, excluded proteins). You may obtain nonphysical or unrealistic values when fitting these parameters and possibly introduce issues into the model w.r.t. simulations under extrapolative settings.

Additionally, interstitial porosity \varepsilon_e should be close to 0.36 as dictated by the theory of random close packing. From Wikipedia (Random close pack - Wikipedia): “Experiments and computer simulations have shown that the most compact way to pack hard perfect same-size spheres randomly gives a maximum volume fraction of about 64%”. Anecdotally, I have tested maybe 40 columns in my PhD and never found \varepsilon_e to lie outside the range of 0.35 - 0.45. Generally you can assume 0.4 and it is a good assumption.

You can find this equation in the Schmidt-Traub textbook in equation 6.65 in the 2005 version of the book. If you PM me, I can possibly share the pdf of the book. But, I highly recommend purchasing this textbook because it is extremely useful resource w.r.t. chromatography modeling. This equation is obtained from moment analysis of the equilibrium dispersive model, so its application to the lumped rate and general rate isn’t quite correct; but, modelers still use it due to its convenience and simplicity.

Name of textbook:
“Preparative Chromatography of Fine Chemicals and Pharmaceutical Agents”
Edited by Henner Schmidt-Traub


Just a follow-up on this… Although the theory dictates that \varepsilon_e should not be smaller than 0.36, it assumes that the spheres are uniform in their radius, have sphericity \Psi = 1, and are incompressible. We know that chromatographic resins are present in a particle size distribution—which is often quite broad—may not be perfectly spherical, and may be compressible (e.g., agarose media). Thus, it may be that \varepsilon_e < 0.36. The smallest value I have seen in publications, that I can recall, was \varepsilon_e \approx 0.31.