Modern Control Engineering

Section 4–5 / Thermal Systems 139

or

which may be rewritten as

Note that the time constant of the system is equal to RCorM/Gseconds. The transfer

function relating uandhiis given by

where and

In practice, the temperature of the inflowing liquid may fluctuate and may act as a

load disturbance. (If a constant outflow temperature is desired, an automatic controller

may be installed to adjust the heat inflow rate to compensate for the fluctuations in the

temperature of the inflowing liquid.) If the temperature of the inflowing liquid is sud-

denly changed from to while the heat input rate Hand the liquid flow rate

Gare kept constant, then the heat outflow rate will be changed from to and

the temperature of the outflowing liquid will be changed from to The heat-

balance equation for this case is

or

which may be rewritten

The transfer function relating uanduiis given by

where and

If the present thermal system is subjected to changes in both the temperature of the

inflowing liquid and the heat input rate, while the liquid flow rate is kept constant, the

changeuin the temperature of the outflowing liquid can be given by the following

equation:

A block diagram corresponding to this case is shown in Figure 4–26(b). Notice that the

system involves two inputs.

RC

du

dt

+u=ui+Rhi

Q (s)=lCu(t)D Qi(s)=lCui(t)D.

Q (s)

Q (^) i(s)

=

1

RCs+ 1

RC

du

dt

+u=ui

C

du

dt

=Gcui-ho

Cdu=AGcui-hoBdt

Q

–

Q o+u.

–

o

H

–

H +ho ,

–

Q

–

Q i+ui

–

i

Q (s)=lCu(t)D Hi(s)=lChi(t)D.

Q (s)

Hi(s)

=

R

RCs+ 1

RC

du

dt

+u=Rhi

C

du

dt

=hi-ho