Modern Control Engineering

152 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

Since thermal resistance Rmay be written as

heat flow rate qmay be given, in terms of thermal resistance R,as

where is the bath temperature and is the thermometer temperature. Hence, we

can rewrite Equation (4–48) as

or

(4–49)

Equation (4–49) is a mathematical model of the thermometer system.

Referring to Equation (4–49), an electrical analog for the thermometer system can be writ-

ten as

An electrical circuit represented by this last equation is shown in Figure 4–41.

RC

deo

dt

+eo=ei

RC

du

dt

+u=ub

C

du

dt

=

ub-u

R

Q–+ub Q–+u

q=

AQ–+ubB-AQ–+uB

R

=

ub-u

R

R=

d(¢u)

dq

=

¢u

q

R

ei Ceo

Figure 4–41

Electrical analog of

the thermometer

system shown in

Figure 4–40.

Problems

B–4–1.Consider the conical water-tank system shown in

Figure 4–42. The flow through the valve is turbulent and is

related to the head Hby

whereQis the flow rate measured in m^3 sec and His in

meters.

Suppose that the head is 2 m at t=0. What will be the

head at t=60sec?

Q=0.005 1 H

2m

3m

2m

H

r

Figure 4–42Conical water-tank system.

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