On Relaxation Phenomena in Field-Flow Fractionation

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Abstract
The two-dimensional unsteady convective diffusion equation satisfied by the local concentration of the colloid introduced in a field-flow fractionation (FFF) column is solved by the method of finite differences. The alternating direction implicit (ADI) method proposed by Peaceman and Rachford is used. The axial convection term is approximated by a backward difference approximation to obtain a stable and convergent scheme. Numerical results are obtained for various values of the transverse Peclet number for the case of steady laminar flow and a slug input. The numerical results from the ADI method are validated by comparison with numerical solutions obtained using an explicit scheme as well as by internal consistency checks. The results of this work show that the transverse concentration profiles depend in a complex fashion on axial position along the cloud during relaxation. In the presence of a field, asymptoticity in the transverse profiles is approached first in the rear of the colloid cloud, and progresses gradually through the axial extent of the cloud. Ultimately, at a sufficiently large value of time, almost all of the colloid relaxes to asymptotic exponential distributions in the transverse coordinate as predicted from theory. The local concentration of colloid in the system is observed to reach a global maximum value at intermediate values of time during relaxation. The area average concentration distribution is observed to exhibit strong asymmetry when plotted against the axial coordinate at intermediate times both in the presence and in the absence of a field. This asymmetry is in accord with pure convection theory. In contrast, truncated two-term dispersion equations only predict symmetric distributions for symmetric initial conditions. Thus there may be a need to retain higher order terms in the application of generalized dispersion theory in order to predict the observed results.