Algoritimos

Differential Equations
Dr. M.J. Willis
Department of Chemical and Process Engineering,
University of Newcastle e-mail: [e-mail]
Written: 18th November, 1999
Introduction
In order to understand and effectively design chemical processes as well as their associated control systems they must be understood. Mathematical modelling involves the representation of physical devices such as tanks, reactors, columns, etc as well as the control hardware. Developing mathematical models of chemical processes normally involves the use of differential equations. Once written, these equations must be solved to indicate the performance of a particular system.
The aim of these notes is to revise differential equations and their solution.
Attention is restricted to the investigation of the behaviour of systems that may be described using 1st order ordinary differential equations. At the end of this section of the course you should understand the behaviour of 1st order DE's when subject to common process input changes.
What is a differential equation?
Differential equations (DE’s) are equations in the usual sense however at least one of the terms will include a rate of change (differential) term. The following equations are examples of DE’s,
= 3 dt dy and 6 3 2
2
+ + = dt dy dt d y (1) where ‘y’ could represent a process output such as temperature, flow or level and
‘t’ could represent time. A DE provides information regarding the rate of change of a process variable with respect to time. This can provide important information regarding the current and future operation of a process. For a level system this may allow the prediction of liquid overflow from a vessel and for a temperature system prediction of excessive temperature fluctuations, for instance.
The term ‘order of a DE ‘ refers to the highest differential term that is involved. So the two equations are 1st and 2nd order. To understand how ‘y’ changes with respect to ‘t’, the