Process Nonlinearities. The numerical value of K, t, and to depend on the process

operating conditions. Processes where the numerical values of K, t, and to are con-

stant over the entire operating range, known as linear processes, occur very infre-

quently. Most often, processes are nonlinear. In these processes the numerical values

of K, t, and to vary with operating conditions. Nonlinear processes are the norm.

Figure 2-3.7 shows a simple example of a nonlinear process. A horizontal tank

with dished ends is shown with two different heights, h1 and h2. Because the cross

section of the tank at h1 is less than at h2, the level at h1 will respond faster to changes

in inlet, or outlet, ¬‚ow than the level at h2. That is, the dynamics of the process at

h1 are faster than at h2. A detailed analysis of the process shows that the gain

depends on the square root of the pressure drop across the valve. This pressure drop

depends on the liquid head in the tank. Thus the numerical value of the gain will

vary as the liquid head in the tank varies.

The tank process is mainly nonlinear because of the shape of the tank. Most

processes are nonlinear, however, because of their physical“chemical characteris-

tics. To mention a few, consider the relation between the temperature and the rate

of reaction (exponential, the Arrhenius expression); between the temperature and

the vapor pressure (another exponential, the Antoine expression); between ¬‚ow

through a pipe and the heat transfer coef¬cients; and ¬nally, the pH.

The nonlinear characteristics of processes are most important from a process

control point of view. As we have already discussed, the controller is always adapted

to the process. Thus, if the process characteristics change with operating conditions,

the controller tunings should also change, to maintain control performance.

Mathematical Description of Single-Capacitance Processes. Mathematics

provides the technical person with a very convenient communication tool. The equa-

tion that describes how the output variable, O(t), of a single-capacitance process,

with no dead time, responds to a change in input variable, I(t), is given by the dif-

ferential equation

dO(t )

+ O(t ) = KI (t )

t (2-3.2)

dI (t )

We do not usually use differential equations in process control studies, but rather,

transform them into the shorthand form

24 PROCESS CHARACTERISTICS

O( s) K

= (2-3.3)

I ( s) t s + 1

This equation is referred to as a transfer function because it describes how the

process “transfers” the input variable to the output variable. Some readers may

remember that the s term refers to the Laplace operator. For those readers that may

not have seen it before, don™t worry: s stands for “shorthand.” We will only use this

equation to describe single-capacitance processes, not to do any mathematics. Equa-

tion (2-3.3) develops from Eq. (2-3.2), and because this equation is a ¬rst-order dif-

ferential equation, single-capacitance processes are also called ¬rst-order processes.

The transfer function for a pure dead time is given by the transfer function

O(s)

= e - to s (2-3.4)

I (s)

Thus, the transfer function for a ¬rst-order-plus-dead-time (FOPDT) process is

given by

O(s) Ke - to s

= (2-3.5)

I (s) t s + 1

Transfer functions will be used in these notes to facilitate communication and to

describe processes.

2-3.2 Multicapacitance Processes

The following two examples explain the meaning of multicapacitance.

Example 2-3.3. Consider the tanks-in-series process shown in Fig. 2-3.8. This

process is an extension of the single tank shown in Fig. 2-3.1. We are interested in

learning how the outlet temperature from each tank responds to a step change in

inlet temperature to the ¬rst tank, Ti(t); each tank is assumed to be well mixed.

Figure 2-3.8 also shows the response curves. The response curve of T1(t) shows the

¬rst tank behaving as a ¬rst-order process. Thus its transfer function is given by

T1 (s) K1

= (2-3.6)

Ti (s) t1 s + 1

The T2(t) curve shows the steepest slope occurring later in the curve, not at the

beginning of the response. What happens is that once Ti(t) changes, T1(t) has to

change enough before T2(t) starts to change. Thus, at the very beginning, T2(t) is

barely changing. When the process is composed of the ¬rst two tanks, it is not of

¬rst order. Since we know there are two tanks in series in this process, we write its

transfer function as

T2 (s) K2

= (2-3.7)

Ti (s) (t1 s + 1)(t 2 s + 1)

25

SELF-REGULATING PROCESSES

60

Ti (t ) 55

Ti (t )

50

45

0 10 20 30 40 50

60

T1 (t ) 55

T1 (t ) 50

45

0 10 20 30 40 50

60

T2 (t ) 55

T2 (t ) 50

45

60

T3 ( t ) 55