Static gain

This quantity is also called the static gain or dc gain by electrical engineers. When we talk about the model of a process, we also use the term process gain quite often, in distinction to a system gain. 

Tp is known as the time constant of the process and Kp is called the steady-state gain or static gain or simply the gain of the process. Their physical meaning will become clear in the next three sections. 

Figure 10.5 Effect of (a) time constant and (b) static gain, in the response of first-order lag systems.

Let us return to the heater system discussed in Example 10.2. The time constant and the static gain for the heater were found to be... 

The question then arises as to how one handles first-order systems with variable time constants and static gains in order to find the dynamic response of such systems. There are two possible solutions ... 

Find the dynamic response of a first-order lag system with time constant tp = 0.5 and static gain Kp = 1 to (a) a unit impulse input change, (b) a unit pulse input change of duration S, and (c) a sinusoidal input change, sin 0.51. Determine the behavior of the output after long time (as / - oo) for each of the input changes above. 

The values of the time constant tp and static gain Kp are not well known. Develop an experiment whereby changing m(t) in a certain way and recording the values of y(t) with time you can compute the values of the unknown parameters rp and Kp. 

III.33 Consider the three storage tanks in Figure PHI. 1. For each of these systems, (a) develop the transfer functions between the liquid levels and the inlet streams, (b) determine the time constants and process static gains, and (c) determine which of the three systems have constant and which variable time constants and process gains. Assume that the flow rates of all free effluent streams are linear functions of the corresponding liquid levels [Flowrates in figure are steady state values]... 

Show that the concentration cA of reactant A in an isothermal continuous stirred tank reactor exhibits first-order dynamics to changes in the inlet composition, cA/. The reaction is irreversible, A - B, and has first-order kinetics (i.e., r = kcA). Furthermore (a) identify the time constant and static gain for the system, (b) derive the transfer function between cA and cA (c) draw the corresponding block diagram, and (d) sketch the qualitative response of cA to a unit pulse change in cAj. The reactor has a volume V, and the inlet and outlet flow rates are equal to F. 

The parameters K and Kd are known as closed-loop static gains. 

The proportional control leads to a lower static gain for the closed-loop response compared to the gain of the uncontrolled process [see eqs. (14.20b) and (14.20c)]. Is a lower gain more or less favorable for the controlled process Recall the definition of the static gain from Section 10.4. 

Larger uncertainty in the parameters of a model (static gain, time constant, dead time) requires larger or smaller gain and phase margins for tuning the controller s parameters ... 

IV.S5 Table PIV.3 shows the amplitude ratio and phase lag values of an unknown system at various frequencies, (a) Determine the order of the unknown system, (b) Examine it for dead time, (c) Compute the values of the system parameters (e.g., static gain, time constant or Kp, t, (, etc.) and the value of dead time if the system possesses dead time. 

To simplify the presentation, let us first assume that Gm = G/= 1. Then eqs. (21.9) and (21.10) will be the basis of the controller design. Each of the two process transfer functions, Gp(s) and Gd(s), has two elements (1) the static element, which corresponds to the static gain, and (2) the purely dynamic element, which is a function of s. Thus... 

In Figure 21.9a and b we see two different ratio control configurations. Which one would you prefer and why (Hint Examine the static gain of the control loop in Figure 21.9a and consult Ref. 3 for details.)... 

Cohen-Coon settings (see Section 16.5) From the process reaction curve we can estimate the process static gain K, the dominant time constant r, and the process dead time td Then, from eqs. (16.9) through (16.11c), we can compute the parameters Kc, r/, and rD of a P, PI, or PID control algorithm. The effect of the sampling period T has been accounted for by the nature of the experiment itself, because the reaction curve has been determined using sampled-values of the process output. 

The term (1 + 0.6z" ) in the denominator implies the presence of a pole at -0.6, which will cause controller ringing. Dahlin has suggested that the ringing pole can be eliminated by multiplying D(z) with (1 + 0.6z ), thus canceling the ringing term from the denominator. But in order to keep the static gain K the same, we should divide D(z) by the term lim(l + 0.6z )... 

Here xp and Kp are the process time constant and static gain, respectively, and Kc and T/ are the gain and the integral time constant for the PI controller. 

Show that a tank with variable cross-sectional area along its height also has variable time constant and static gain. 

Since process parameters such as dead times, static gains, and time constants are almost never known exactly, a gain margin larger than 1 (e.g., 1.7) is a safety factor for stable operation. 

Because of different recycle structures, the base-case and the alternatives A, B and C will have slightly different nominal operating points. Table 17.14 shows the scaled static gains for the base-case. The reboiler duty (Q2) has the highest influence on the all... 

Table 17.14 Scaled static gain matrix of the base-case...

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