External Mass Transfer in Adsorption Systems

Key Insight

External mass transfer resistance, the hindrance to transport in the fluid surrounding an adsorbent particle, is quantified by a film mass transfer coefficient, k_f. This coefficient defines the mass transfer flux at the particle surface, J = k_f * (C - C_s), where C is the bulk concentration and C_s is the surface concentration. The equivalent stagnant film thickness, delta, is inversely related to k_f by delta = D naught / k_f, where D naught is the diffusivity in free solution.

Engineering correlations, expressed in terms of dimensionless Sherwood (Sh), Reynolds (Re), and Schmidt (Sc) numbers, predict k_f values for packed adsorption beds under laminar flow conditions, which are prevalent in protein chromatography. Recommended equations, such as Sh = 1.09 * Re^0.33 * Sc^0.33, or Sh = 1.85 * (epsilon / (1 - epsilon))^0.33 * Re^0.33 * Sc^0.33, are used despite limited protein-specific experimental data, relying on extrapolation enabled by dimensionless parameters. Correlations are also available for agitated contactors, involving agitation power input per unit mass of liquid.

For process design, estimates based on these recommended equations are often sufficiently accurate, assuming a one-third power dependence of Sherwood on Reynolds for low Reynolds values and a Schmidt number dependence of 0.33. For example, a 150 kilodalton protein on 100 micrometer particles in a chromatographic column at 300 cm/h yields k_f values between 2.6x10^-4 and 7.0x10^-4 cm/s, corresponding to film thicknesses of 2.1 to 5.7 micrometers, which are small compared to the particle diameter. In practice, external mass transfer is rarely the controlling factor in preparative and process protein chromatography.