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## Mathematical Representation of Distributed Flow Processes for Automatic Control

May 12th, 1965

### Thesis submitted for the degree of Doctor of Philosophy in the University of Cambridge, May 1965.

S F Bush

#### Summary

The subject of this thesis is the simplification and computation of mathematical representations, or models, of distributed chemical engineering systems in which there is flow and interaction of one or more quantities such as heat and mass. Counterflow heat-exchangers, regenerators, tubular reactors and packed distillation columns are examples of main types for which simplified representations have been obtained through the application of one of two methods. Among those systems for which one or other of the two methods is suitable, the choice of method is found to depend basically on the number of process transfer units in each stream.

One method, the diffusion representation, is based on a method applied by G I Taylor to the problem of solute dispersion in a capillary tube. In its present application the method replaces the set of partial differential equations, one for each stream, by a single parabolic partial differential equation. Such a transformation entails a certain error, mainly in the early part of the transient response, which is smallest for systems with the largest number of transfer units. If the error is acceptable, the transformation greatly simplifies subsequent numerical analysis.

The second method is an iterative method and is suitable for systems with a small number of exchange units. In contrast to the first method, it isolates the wave-like characteristics of exchange flow systems. In favourable cases it provides simple but accurate representations in either transfer function form or explicitly in the time domain. The iterations are shown to converge for a general class of flow system and form the basis of a numerical procedure for integrating the sets of first order hyperbolic differential equations which describe these systems.

The simplified representations obtained by application of the methods are compared with the standard representations by computing the step responses of both representations. The criteria of applicability which are proposed for the two methods are obtained by numerical experiments. Tables are presented comparing the numberical methods used in computing both simplified and standard models.