# Klinkenberg model

Modeling adsorption kinetics using Klinkenberg model.

#### Notes

Klinkenberg provides an useful approximation to the analytical solution of the Convection-Dispersion model proposed by Anzelius for the case of a single solute, an initially clean bed, frontal loading and negligible axial dispersion. According to the Klinkenberg approximation the solute concentration respect to axial distance and time is given by:

\[ \frac{C}{C_\mathrm{F}} \approx \frac{1}{2} [ 1 + \text{erf}( \sqrt{\tau} - \sqrt{\xi} + \frac{1}{8 \sqrt{\tau}} + \frac{1}{8 \sqrt{\xi}} ) ] \]

\[ \tau = K (t - \frac{z}{u_i}) \]

\[ \xi = \frac{K H z}{u_i} (\frac{1 - \varepsilon_b}{\varepsilon_b}) \]

where \(C\) is the concentration in the fluid phase, \(H\) is the linear adsorption constant (also called Henry constant), \(K\) is the global mass transfer coefficient, \(u_\mathrm{i}\) is the interstitial velocity, \(\varepsilon_\mathrm{b}\) is the bulk porosity, \(C_\mathrm{F}\) is the feed concentration, \(z\) is the axial distance, and \(t\) is time. Resistances due to external transport, pore diffusivity, and kinetics are included in \(K\).

This approximation is accurate to \( < 0.6 \% \) error for \(\xi > 2.0 \).

#### References

A. Klinkenberg, Ind. Eng. Chem., 46, 2285–2289 (1954).

J.D. Seader, E.J. Henley, D.K. Roper. Separation Process Principles. John Wiley & Sons, Inc., 3rd edition, 2011.