On The Dynamic Behavior Of Fixed-Bed Adsorption Columns
Authors: Lapidus Leon 
Published: December 2007

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The present paper reviews the current status of the a priori design and analysis of fixed-bed contacting devices. These include both catalytic reactors and adsorption columns in which coupled physical and chemical phenomena are involved. Of prime consideration is the quantitative specification representing heat, mass and momentum transport or the construction of computationally feasible mathematical models in conjunction with complex adsorption or chemical reaction. This is equivalent to saying that the optimal, stable operation and control of full scale units can be predicted to a considerable degree without the need for extensive pilot plants. Alternately, these models may be used to assist in the design of a pilot unit, to guide the experimental work or to interpret the data from the unit.

CLASSICAL FORMULATION
We shall consider here the classical formulation of the fixed-bed reactor model. In later discussion the application of the necessary techniques to adsorption columns will be deliniated. On a broad conceptual basis the model should recognize and account for such items as axial and radial gradients in the bed proper, bulk flow terms, internal particle gradients, chemical reaction, dynamic accumulation of heat and mass plus the appropriate boundary conditions. These latter include such obvious considerations as zero flux of mass but a finite flux of heat at the bed wall. By applying the equations of change to the flowing stream external to the particles, to the interior of the particles and then coupling the results with appropriate surface rate mechanisms the necessary model equations may be obtained. While an attempt will not be made here to write out these equations it may be stated that even with gross assumptions on what terms to delete because they are unimportant the solution is extremely difficult to obtain even with the largest available digital computers. This follows from the fact that the system equations may involve 4 independent variables, that the P.D.E. are nonlinear and may be boundary-value in form. Thus, as so frequently happens, it is necessary to look for another means of model representation. Such an alternate which does not compromise any of the physical and chemical fundamentals and thus does not lead to a loss in rigor, is fortunately available.

FINITE-STAGE MODEL FORMULATION
Since a detailed derivation of the finite-stage model is given elsewhere we only concern ourselves here with the form of the resultant equations and the implications of same. For ease in presentation we first consider the steady-state heat and mass distribution for a single reacting component with chemical reaction occurring within the porous packing structure. Dispersion of heat and mass in the axial and radial directions is assumed to be characterized by a two-dimensional array of continuous stirred tank reactors (CSTR). The physical arrangement of CSTR is a series of axial rows with each row in an off-set from the row above and below it. Any one CSTR is fed from 2 CSTR in the row above it and in turn feeds into 2 others in the row below (the nomenclature here is consistent with feed to the top of a fixed-bed and subsequent flow down). This arrangement agrees in a broad sense with considering each CSTR as equivalent to the mixing space between the particles. The mean residence time for any one stage (assuming turbulent conditions) is given by the ratio of the particle diameter to the axial linear superficial flow velocity,.