Page 64 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 39

The second parameter, called the process time constant r is defined
as the time required for the process output to reach 63% of its final
value in response to a step change in the process input. Careful exam-
ination of Fig. 3-2 will show that the time constant for the water tank
is 10.0 time units.
For the time being, these two parameters, the process gain and
the process time constant, will suffice for our characterization of the
process. Note that the process gain is a static characterization param-
eter. It can tell the analyst where the process will ultimately settle
after a step-change to the input. On the other hand, the time constant
is a dynamic characterization parameter for it tells us how the process
gets from one state to another.

3-2 Mathematical Descriptions of the
First-Order Process
To describe the dynamic behavior of a process we must decide in
which domain we will work. The obvious first choice is the continu-
ous time domain where for our purposes time will vary continuously
from zero to infinity. There are many reasons why one might want to
"transform" the domain of analysis into something else, say, the
Laplace domain, or the frequency domain, or the discrete time
domain. However, first things first! Let us now delve into the con-
tinuous time domain. Beware! There will be a lot of math. It will be
my challenge to minimize it and keep it simple.

3-2-1 The Continuous Time Domain Model
How can we develop an equation that will describe the behavior of a
process? What do we have to work with? Often one has to start with
the fundamental conservation laws for mass, momentum, and energy.
For our tank of water the apparent fundamental law would be the con-
servation of mass; in crude terms, what goes in has to either go out or
accumulate:

Rate of water in= rate of water out+ rate of accumulation of water
in the tank

The rate of water flowing in can be represented by F the inlet flow
1
rate in kg/ sec. The rate of water leaving the tank can be represented by F.,.
The rate of accumulation of water in the tank is the rate of change of the
mass in the tank with respect to time. Does this sound familiar? Yes, it is
the derivative from first-year calculus (also reviewed inApp. A), namely,

(3-2)