Page 66 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 41

where g is the process gain. The reader should check that the units of
the gain are m·sec/kg.
We can verify that g is in fact the process gain by letting time go
to infinity assuming that the process input U jumps from zero to a
constant value Uc at time zero. At time equals infinity, the process has
supposedly settled out to a new steady state where the rate of change
of all the variables is zero, that is,

or

. dY O
1 I.Dl.,_.dt=

and
lim,_. Y(t) = Y ..

which causes Eq. (3-8) to become

which in turn yields

(3-9)

The definition of the process gain in Sec. 3-1 shows that the gin Eq. (3-9)
is indeed the process gain.
In summary, the model was developed by first applying a conser-
vation law which related flows, that is, Fi and F"' to the rate of change
of the potential, that is, Y. Then, a constitutive equation was used to
replace one of the flows with an expression containing a potential.
This left us with a model equation that gave the response of the pro-
cess output Y to the process input U in terms of the process parame-
tersgand f.
Scaling
As an alternative, the general first-order model equation presented
above could be scaled to have unity time constant and unity gain.
One would start with