Example H1.1

MATLAB code for example 1.1 from the book "Regeltechniek voor het HBO"

Model of a heat proces

Date : 31/03/2022
Revision : 1.0

Table of Contents

Model of a heat proces Static behaviour Dynamic behaviour Moving from one static state to the other

Static behaviour

We consider the dynamic properties of the heat process of figure 1.11. A container of oil is heated by an applied power p(t), see figure 1.11a. The temperature of the oil is T(t). The stirrer provides a uniform temperature in the container. Because T(t) is higher than the (constant) outside temperature To, a heat flow q(t) will pass through the wall to the outside. We call the total thermal resistance of the wall of the container R. In the static state - i.e. when equilibrium has been established and T no longer changes - the following applies:

supplied power = dissipated power, i.e. p = q, or:

, so:

This relationship is shown graphically in figure 1.11b and is called the static characteristic.

Dynamic behaviour

In control engineering, we are always interested in the dynamic behavior as well. We want for example, that when there is a change in the set value, the output of the process, the temperature T, also follows this change as quickly and accurately as possible.

The quantities p(t) and T(t) are then considered as time functions. An at any time valid expression for the process behavior can - just as for the static state - be based on the heat balance: supplied heat = removed heat + stored heat.

For the stored heat, the heat capacity C of vessel and oil must be known. This is equal to the number of joules required to make the process (oil plus barrel) rise one degree in temperature. The number of joules required for a temperature increase of dT degrees for dt seconds is CdT joules. The stored heat per unit time is then given by:

such that the following holds:

or:

and also:

This last expression is a differential equation of the first order, where we imagine that from the time t = 0 a power p(t) is supplied. The input of our system is represented by p(t) and the output by T(t). To determine the relationship between p(t) and T(t), the differential equation must be solved.

Moving from one static state to the other

It can be seen that the already known solution for the static behavior also follows from this differential equation. Since T(t) is then constant - say

- at a supplied constant power p(t) - say

- of the differential equation only remains (derivatives are then zero):

So this is consistent with what we had found earlier and with point 1 of figure 1.11b. Similarly, for point 2 in the same figure:

We move from point 1 to point 2 by increasing the constant power

to the constant

. Eventually (static state, also called stationary operation) the temperature

then changes to

. What the dynamic behavior of this process looks like, causing one static state to change to another, is determined by the solution of the differential equation of the mathematical model of the linear process considered here. This is qualitatively shown in figure 1.12, both for the transition from point 1 to 2 and from 2 to 1. Other transitions are also shown. The rate at which the new stationary states are established is found to be determined by the value of RC, the so-called time constant of the process.