$$H$$
$$K$$
$$C_\mathrm{F}$$
$$z$$
$$\varepsilon_\mathrm{b}$$
$$u_\mathrm{i}$$
$$t_\mathrm{f}$$

#### 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.