Modern Control Engineering

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

Q

Qi

H

CapacitanceC

Figure 4–27

Liquid-level system.

Example Problems and Solutions

A–4–1. In the liquid-level system of Figure 4–27 assume that the outflow rate Qm^3 sec through the out-

flow valve is related to the head Hm by

Assume also that when the inflow rate Qiis 0.015 m^3 sec the head stays constant. For t<0the

system is at steady state AQi=0.015m^3 secB. At t=0the inflow valve is closed and so there is

no inflow for t0. Find the time necessary to empty the tank to half the original head. The

capacitanceCof the tank is 2 m^2.

Solution.When the head is stationary, the inflow rate equals the outflow rate. Thus head Hoat

t=0is obtained from

or

The equation for the system for t>0is

or

Hence

Assume that, at t=t 1 , H=1.125m. Integrating both sides of this last equation, we obtain

It follows that

or

Thus, the head becomes half the original value (2.25 m) in 175.7 sec.

t 1 =175.7

21 H^2

1.125

2.25

= 21 1.125- 21 2.25=-0.005t 1

3

1.125

2.25

dH

1 H

=

3

t 1

0

(-0.005)dt=-0.005t 1

dH

1 H

=-0.005dt

dH

dt

=-

Q

C

=

- 0.01 1 H

2

• CdH=Q dt

Ho=2.25 m

0.015=0.01 1 Ho

Q=K 1 H=0.01 1 H

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