стр. 11 |

150

0 10 20 30 40 50

Time, sec

59

56

c( t )

53

50

0 10 20 30 40 50

Time, sec

Figure 2-3.9 Exothermic chemical reactor.

28 PROCESS CHARACTERISTICS

temperature, TJ(t). For this example we have assumed the jacket to be well mixed,

and thus the temperature responds as a п¬Ѓrst-order process. The second variable that

responds is the temperature of the metal wall, TM(t). The amount that TM(t) changes

depends on the volume of the metal, density of the metal, heat capacity of the metal,

and so on. Also, how fast TM(t) changes depend on the thickness of the wall, thermal

conductivity of the metal, and so on. That is, the process characteristics depend on

the physical parameters (material of construction and sizes) of the process. This is

what we discussed in Section 2-3.1. The temperature in the reactor, TR(t), responds

next. Finally, the output signal from the sensor/transmitter, c(t), starts to react. How

fast this signal changes depends on whether the thermocouple (sensor) is a bare

thermocouple or if it is inside a thermowell.

The important thing to learn from this example is that every time a capacitance

is encountered, it slows the dynamics (longer t and to) of the process.

To summarize, multicapacitance, or higher-order processes, are most often

encountered. The reason for this is that processes are usually formed by single

capacitances in series.

2-4 TRANSMITTERS AND OTHER ACCESSORIES

Let us look at the characteristics of transmitters and transducers. Consider an elec-

tronic (4 to 20 mA) pressure transmitter with a calibration of 0 to 50 psig process

pressure. To calculate the gain of this transmitter, we follow the deп¬Ѓnition of gain:

DO (20 - 4) mA TO mA TO

KT = = = 0.32

(50 - 0) psi

DI psi

or

DO (100 - 0)%TO %TO

KT = = =2

(50 - 0) psi

DI psi

depending on whether the output from the transmitter is considered in mA or in

percent.

The dynamics (t and to) of sensor/transmitters are often, although not always, fast

compared to the process unit. The method we learn to characterize the process will

be such that it considers the dynamics of the valve, process unit, and sensor/trans-

mitter all together, as one. Thus, there is no need to discuss in detail the dynamics

of sensors/transmitters. There are, however, some units, such as chromatographs, that

may add signiп¬Ѓcant dead time to the process. As mentioned earlier, dead time has

a signiп¬Ѓcantly adverse effect on the controllability of processes.

Consider a current-to-pneumatic (I/P) transducer. The gain of this transducer is

DO (15 - 3) psi psi

KT = = = 0.75

DI (20 - 4) mA mA

29

OBTAINING PROCESS CHARACTERISTICS FROM PROCESS DATA

or

DO (100 - 0)%output %output

KT = = = 1.0

(100 - 0)%input

DI %input

depending on the units desired.

2-5 OBTAINING PROCESS CHARACTERISTICS FROM PROCESS DATA

In this section we learn how to obtain the process characteristics (K, t, and to) from

process data for self-regulating processes. We have already learned that most

processes are self-regulating and of higher order, with a general transfer func-

tion as

O(s) K

= (2-5.1)

I (s) (t1 s + 1)(t 2 s + 1) L (t n s + 1)

As mentioned earlier, however, higher-order processes can be approximated by

a second-order-plus-dead-time (SOPDT) transfer function, Eq. (2-3.10). What

happens in practice, though, is that there is no easy, reliable, and consistent method

to approximate a higher-order process by this type of transfer function. What it is

usually done, therefore, is to approximate a higher-order system by a п¬Ѓrst-order-

plus-dead-time (FOPDT) transfer function, Eq. (2-3.11). Thus we approximate

higher-order processes by a low-order-plus-dead-time model. The method presented

next is the most objective of all those available, the one that gives the best approx-

imation, and the easiest one to use. (The day this method was developed, Murphy

was sleeping!)

To use a concrete example, consider the heat exchanger shown in Fig. 2-1.1a.

Assume that the temperature transmitter has a calibration of 100 to 250В°C. To

obtain the necessary process data, the following steps are used:

1. Set the controller to manual mode. Effectively, the controller is removed.

2. Make a step change in the controller output.

3. Record the process variable.

For example, suppose that these steps are followed in the heat exchanger example

and the results are those shown in Fig. 2-5.1. The response curve indicates that this

exchanger is a higher-order process.

To obtain the dynamic terms t and to, we make use of the two-point method (or

п¬Ѓt 3 in Ref. 1). The method consists in obtaining two data points from the response

curve (process reaction curve). These two points are the time it takes the process

to reach 63.2% of the total change in output, or t0.632DO, and the time it takes the

process to reach 28.3% of the total change in output, or t0.283DO; these two points are

shown in Fig. 2-5.1. Time zero is the time when the step change in controller output

occurs. With these two data points, t and to are obtained from the following

equations:

t = 1.5(t0.632DO - t0.283DO ) (2-5.2)

30 PROCESS CHARACTERISTICS

Figure 2-5.1 Process response curve.

стр. 11 |