Automated Continuous Process Control Part 3 ppsx

to = t0.632DO - t (2-5.3)

The units of t and to are the same time units as those used by the control system.

Now that t and to have been evaluated, we proceed to evaluate K. Following the

definition of gain,

K = ( ) - ∞

( ) - = ∞ = 156 150 ∞

55 50

6

5

1 2 C

CO

C

CO

C

% % CO . %

30 PROCESS CHARACTERISTICS

Figure 2-5.1 Process response curve.

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This gain says that at the present operating condition, a change of 1%CO results in

a change of 1.2°C in outlet process temperature. This gain correctly describes the

sensitivity of the outlet process temperature to a change in controller output.

However, this gain is only a partial process gain and not the total process gain. The

total process gain is the one that says how much the process output, c(t) in %TO,

changes per change in process input, m(t) in %CO; the reader may refer to Fig. 2-

1.1b to understand this point. That is, the process output is given by the transmitter

output and it is not the temperature. Therefore, we are interested in how much the

transmitter output changes per change in controller output, or

(2-5.4)

The change in transmitter output is calculated as follows:

or, in general,

(2-5.5)

Therefore, the total process gain for this example is

If the process variable had been recorded in percent of transmitter output, there

would be no need for any extra calculation.

We can now write the transfer function for this process as

This transfer function describes the relation between the transmitter output and the

controller output. If a transfer function describing the relation between the trans￾mitter output and any other process input (other than the controller output) is

desired, the same procedure is then followed to evaluate K, t, and to. In this case

the units of the K will be different than before; that is, they will not be %TO/%CO.

The units will depend on the units of the particular input.

To illustrate the approximation using the two-point method, consider a third￾order process described by the following transfer function:

C s

M s

e

s

t s o ( )

( ) = +

- 0 8

1

.

t

K = = 4 0 8 % . TO %

5%CO

TO

%CO

D D

c = Ê

Ë

ˆ

¯

Ê

Ë

ˆ

¯

PV

span

TO = change in process variable in engineering units

span of transmitter

100 100 % %TO

Dc = ∞

∞

Ê

Ë

ˆ

¯ = 6 100 4 C

150 C % %TO

K

c

m = = D

D

change in transmiter’s output, %TO

change in controller’s output, %CO

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