(628g) The Exact Equivalence between Diffusion and LDF Models

When designing and optimizing pressure swing adsorption or temperature swing adsorption processes the equivalence between the linear driving force model and the diffusion model is often invoked to reduce the computational effort by an order of magnitude. This is very effective for relatively slow cycles where Glueckauf’s original approximation [1] provides sufficient accuracy.

Over the past 35 years, from Nakao and Suzuki’s [2] contribution, there has been a significant effort by several groups to extend this approach to fast cycles, with some success but no formal exact equivalence has been accepted by the adsorption community.

The starting point of our analysis is the fact that any adsorption process consists of a cyclic sequence of steps and therefore the basis for establishing the correct equivalence comes from studying the response of an adsorption column to a sinusoidal input in concentration, ie the method to be used is based on the frequency response analysis [3]. We show that the typical approach of simulating both models and finding an LDF correction factor [2] is in fact based on matching the amplitude ratios obtained from the two models, but that the corresponding phase lags are not correct. We revisit the equivalence criterion proposed by Rouse and Brandani [4] and prove that this can be presented in the form of explicit relationships, ie no approximate solution is needed, and that the two parameters obtained have physical meaning. Matching the amplitude ratio and the phase lag of the response at cyclic steady state is strictly valid only for linear systems, but the proposed method reduces to Gluekauf’s equivalence for longer cycles, which we know if effective also for non-linear systems [5].

Single particle kinetics, kinetics of a perfectly mixed adsorber and the kinetics of a full adsorption column are shown to yield the exact match between the two models and the basis for the equivalence of non-sinusoidal inputs is discussed. The equivalence is shown to provide the means to accelerate convergence to cyclic steady state for fast cycles, but if the aim is to model also the approach to cyclic steady state, then no exact equivalence is possible and the actual solution of the diffusion equation with appropriate numerical grids [6] should be used.

References

[1] Gluekauf E. Trans. Faraday Soc.; 1955, 51, 1540-1541.

[2] Nakao S., Suzuki M. J. Chem. Eng. Japan; 1983, 16, 114-199.

[3] Stephanopoulos G. Chemical Process Control. An Introduction to Theory and Practice, Prentice-Hall, Englewood Cliff, NJ 1984

[4] Rouse A.J. and Brandani S. A New LDF Approximation for Cyclic Adsorption Processes, In AIChE Separation Technology Topical Conference Vol. 2, P. Bryan and A. Serbezov Eds., 2001, p 739–743.

[5] Sircar S. and Hufton J.R. Adsorption; 2000, 6, 137–147.

[6] Ahn H. and Brandani S. Adsorption; 2005, 11, 113–122.