Page 65 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

40 Chapter Three

3
where pis the water density in kg/m , A is the cross-sectional area
(which we are assuming is cylindrical with constant diameter) of the
2
tank in m , Y is the tank level in m, and the quantity pAY is the
amount of the water in the tank.
Altogether now, one gets

(3-3)

For control purposes, F; would probably be adjusted to maintain
Y on target, so there might be another equation describing the con-
troller's dynamic behavior which we will leave to later. On the other
hand, F., depends on the level of water in the tank, so we need to come
up with a "constitutive" equation relating the rate F., to a potential or
tank liquid height Y. A simple relationship is
y
F=-
o R (3-4)
where R is the resistance associated with the tank's outlet piping. If Y
has units of m and F., has units of kg/ sec, then R must have units of
m·sec/kg. This is a bit of an idealization because from high school phys-
ics the reader probably remembers Torricelli's law, which states that the
jet of liquid emerging from a hole in the side of a tank is given by

(3-5)

Therefore, Eq. (3-4) is a linearization of Eq. (3-5). The concept of
linearization is mentioned in App. Din the section about the Taylor's
series. Since we expect our controllers to keep the process "near" its
nominal values, linearization may well suffice.
Combining Eqs. (3-3) and (3-4) gives

Y dY
F=-+pA- (3-6)
' R dt
which, after some simple rearrangement, is

dY
r-+Y=RF (3-7)
dt I
where r = RpA. This last quantity is the time constant and the reader
should check that the units of the time constant work out to be sec.
For the sake of generality rewrite Eq. (3-7) as

dY
r-+Y=gU (3-8)
dt