Fitting mass transfer coefficients

The rate of adsorption is typically modelled by the linear driving force (LDF) model:

$$\frac{\partial q_i}{\partial t} = k_i(q_i^* - q_i)$$

Fitting a single mass transfer coefficient per component (e.g. $k_i = constant$) is typically not good enough for most applications. In reality, $k_i$ depends on pressure, temperature, and, in systems with >2 components, mole fraction.

Therefore, $k_i$ should instead account for molecular diffusion and Knudsen diffusion. Film diffusion can be neglected so long as the gas velocity is sufficiently high (which is true in any practical application). The mass transfer coefficient is then calculated by:

$$k_i = \frac{60 D_{i,eff}}{d_p^2}$$

Where the effective diffusion coefficient accounts for molecular diffusion and Knudsen diffusion:

$$\frac{1}{D_{i,eff}} = \frac{1}{D_{i,m,eff}} + \frac{1}{D_{i,k,eff}}$$

Molecular diffusion resistance occurs when molecules collide with each other. Think of a busy corridor with lots of people moving in opposite directions, bumping into each other. In a single-component system, diffusion has no real meaning as there is no reference frame by which to measure the molecular movement. In a binary system, the diffusion coefficient depends on the molecular weight of the two molecules, temperature and pressure. Wolfram has a nice calculator to visualise the binary diffusion coefficient of different species.

In the binary system, molecular diffusivity is essentially independent of composition. For multicomponent systems, the molecular diffusion coefficient is more complex, but a reliable estimate can be made from:

$$D_{m,i} = \frac{1 - y_i}{\sum_{j=1}^{K} \frac{y_j}{D_{i,j}}}$$

Knudsen diffusion resistance arises from molecular collisions with the pore walls. Therefore, this mode dominates when the pore diameter is small and comparable to the size of the diffusing molecule. The Knudsen diffusion coefficient depends on the molecular weight of the molecule and the temperature:

$$D_{k,i} = 97r_{pore} \sqrt{\frac{T}{Mr_i}}$$

Therefore, at high pressure, when a large number of molecules occupy the space in the pores, molecular diffusion becomes limiting. At low pressures, Knudsen diffusion becomes the limiting factor.

The effective molecular and Knudsen diffusion coefficients are obtained by:

$$D_{k,i,eff} = D_{k,i,eff} \frac{\epsilon_p}{\tau}$$ $$D_{m,i,eff} = D_{m,i,eff} \frac{\epsilon_p}{\tau}$$

Importantly, $\epsilon_p$ is the macropore volume fraction, not just the total particle pore volume fraction, which also includes micropores. That is because the mechanism for micropore diffusion is different and should not be accounted for in this equation. However, in reality, the fraction $\frac{\epsilon_p}{\tau}$ is a parameter which is fitted to experimental data, and isn't necessarily physically meaningful.

Using this more fundamental calculation of the mass transfer coefficient leads to a mass transfer coefficient that considers the effect of temperature, pressure and mole fraction. This is important in PSAs where the pressure varies by an order of magnitude. Assuming a constant mass transfer coefficient will likely not yield a model that accurately represents the data. One may obtain a constant mass transfer coefficient that predicts performance at one scale, but it would lead to erroneous large-scale design.

Instead of having one mass transfer parameter per component, there are two parameters to fit: $\frac{\epsilon_p}{\tau}$ and $r_{pore}$. The easiest way to get enough data for fitting these values accurately is to vary the pressure in multicomponent breakthrough experiments. Knudsen diffusion is not affected by varying the pressure, whereas molecular diffusion is inversely proportional to pressure. By varying the pressure, Knudsen diffusion does not change, yet the molecular diffusion does. I.e. you are varying one of the parameters at a time, allowing a better fit to be obtained.

There is actually a lot more to be said about fitting adsorption model parameters. I'll be covering those topics in later articles, so subscribe to the newsletter at the bottom of this page so that you don't miss the next one.