DT (33 - 25)∞F outlet temperature ∞F outlet temperature

K= = = 0.8

(35 - 25)∞F inlet temperature

DTi ∞F inlet temperature

18 PROCESS CHARACTERISTICS

Figure 2-3.4 Response of pressure in tank to a change in (a) inlet ¬‚ow and (b) valve

position.

Therefore, the gain tells us how much the outlet temperature changes per unit

change in inlet temperature. Speci¬cally, it tells us that for a 1°F increase in inlet

temperature, there is a 0.8°F increase in outlet temperature. Thus, this gain tells us

how sensitive the outlet temperature is to a change in inlet temperature.

For the gas tank, from Fig. 2-3.4a, the gain is

19

SELF-REGULATING PROCESSES

Figure 2-3.4 Continued.

Dp (62 - 60) psi psi

K= = = 0.2

Dfi (60 - 50)cfm cfm

This gain tells us how much the pressure in the tank changes per unit change in inlet

¬‚ow. Speci¬cally, it tells us that for a 1-cfm increase in inlet ¬‚ow there is a 0.2-psi

increase in pressure in the tank. As in the earlier example, the gain tells us the sen-

sitivity of the output variable to a change in input variable.

Also for the gas tank, from Fig. 2-3.4b, another gain is

20 PROCESS CHARACTERISTICS

(44 - 50) psi

Dp psi

K= = = -1.0

Dvp (46 - 40)% vp %vp

This gain tells us that for an increase of 1% in valve position the pressure in the

tank decreases by 1.0 psi.

These examples indicate that the process gain (K) describes the sensitivity of the

output variable to a change in input variable. The output could be the controlled

variable and the input, the manipulated variable. Thus, in this case, the gain then

describes how sensitive the controlled variable is to a change in the manipulated

variable.

Anytime the process gain is speci¬ed, three things must be given:

1. Sign. A positive sign indicates that if the process input increases, the process

output also increases; that is, both variables move in the same direction. A

negative sign indicates the opposite; that is, the process input and process

output move in the opposite direction. Figure 2-3.4b shows an example of this

negative gain.

2. Numerical value.

3. Units. In every process these are different types of gains. Consider the gas tank

example. Figure 2-3.4a provides the gain relating the pressure in the tank to

the inlet ¬‚ow and consequently, the unit is psi/cfm. Figure 2-3.4b provides the

gain relating the pressure in the tank to the valve position, and consequently,

the unit is psi/%vp. If the sign and numerical value of the gain are given, the

only thing that would specify what two variables are related by a particular

gain are the units. In every process there are many different variables and thus

different gains.

It is important to realize that the gain relates only steady-state values, that is, how

much a change in the input variable affects the output variable. Therefore, the gain

is a steady-state characteristic of the process. The gain does not tell us anything about

the dynamics of the process, that is, how fast changes occur.

To describe the dynamics of the process, the following two terms are needed: the

time constant and the dead time.

Process Time Constant (t). The process time constant (or simply, time constant)

for a single-capacitance processes is de¬ned [1], from theory, as

t = Amount of time counted from the moment the variable starts to respond

that it takes the process variable to reach 63.2% of its total change

Figure 2-3.5, a duplicate of Fig. 2-3.4b, indicates the time constant. It is seen from

this ¬gure, and therefore from its de¬nition, that the time constant is related to the

speed of response of the process. The faster a process responds to an input, the

shorter the time constant; the slower the process responds, the longer the time con-

stant. The process reaches 99.3% of the total change in 5t from the moment it starts

to respond, or in 99.8% in 6t. The unit of time constant is minutes or seconds. The

unit used should be consistent with the time unit used by the controller or control

21

SELF-REGULATING PROCESSES

Figure 2-3.5 Response of pressure in tank to a change in valve position, time constant.

system. As discussed in Chapter 3, most controllers use minutes as time units, while

a few others use seconds.

To summarize, the time constant tells us how fast a process responds once it starts

to respond to an input. Thus, the time constant is a term related to the dynamics of

the process.

Process Dead Time (to). Figure 2-3.6 shows the meaning of process dead time (or

simply, dead time). The ¬gure shows that

to = ¬nite amount of time between the change in input variable

and when the output variable starts to respond

22 PROCESS CHARACTERISTICS

Figure 2-3.6 Meaning of dead time.

The ¬gure also shows the time constant to aid in understanding the difference

between them. Both t and to are related to the dynamics of the process.

As we will learn shortly, most processes have some amount of dead time. Dead

time has signi¬cant adverse effects on the controllability of control systems. This is

shown in detail in Chapter 5.

The numerical values of K, t, and to depend on the physical parameters of the

process. That is, the numerical values of K, t, and to depend on the size, calibration,

23

SELF-REGULATING PROCESSES

h2 , ft

h1 , ft

f i , gpm f, gpm

Figure 2-3.7 Horizontal tank with dished ends.

and other physical parameters of the equipment and process. If any of these changes,

the process will change and this will be re¬‚ected in a change in K, t, and to; the terms

will change singly or in any combination.