Dispersed plug flow model continuity equation

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

Liquid-liquid extraction is carried out either (1) in a series of well-mixed vessels or stages (well-mixed tanks or in plate column), or (2) in a continuous process, such as a spray column, packed column, or rotating disk column. If the process model is to be represented with integer variables, as in a staged process, MILNP (Glanz and Stichlmair, 1997) or one of the methods described in Chapters 9 and 10 can be employed. This example focuses on optimization in which the model is composed of two first-order, steady-state differential equations (a plug flow model). A similar treatment can be applied to an axial dispersion model. 

The basic plug-flow model, with or without dispersions, is a continuous" model because the concentrations and temperature are described by differential equations. An alternative representation is by a discrete model - the so-called "cell" model (16, 17), in which it is assumed that the reactor can be broken down into several connected cells. It had long been assumed that the continuous and discrete models are equivalent ways of representing a reactor however, this assumption has recently been questioned in two different contexts (18, 19). 

The continuity equations for mass and energy will be used to derive the hyperbolic partial differential equation model for the simulation of moving bed coal gasifier dynamics. Plug flow (no axial dispersion) and adiabatic (no radial gradients) operation will be assumed. 

Axial dispersion model In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. The mathematical derivation is similar to that for plug flow except that a term is now included for diffusive flow in addition to that for convective flow. This term appears as (d[A]ldz), where is the effective axial dispersion coefficient. The continuity equation in the absence of radial variations takes the form... 

Modelling both the continuous countercurrent and simulated moving bed processes has been considered by a number of authors. The continuous countercurrent separation process has been addressed by Ching and Ruthven (1984), who assumed axially dispersed flow of fluid and counter-current plug flow of solids in a column. The fundamental differential equation describing the steady state operation of such a system is, for each component. 

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