The performance of a stirred tank to periodic disturbances can be evaluated by consideration of the dead time and time constant properties of the tank.
For example, if the total system dead time is Tdt, it can be defined as:
Td1 — tank dead time, inlet to outlet Td2 — remaining loop dead time (sampling system and control valve motor)
The time constant (t1) for an agitated vessel with dead time (Td1) can be expressed as:
Assuming that the stirred tank has the minimum 3.0-minute time constant previously mentioned and that the total dead time is divided 80% to (Td1) and 20% to Td2, Equation 7.40(11) can be restated:
Expressing t1 in terms of dead time by combining Equations 7.40.10 and 7.40.12:
The dynamic gain of a stirred tank to periodic disturbances is given by Equation 7.40(14):
Gd— dynamic gain of the stirred tank 5 percent change in output percent change in input t0 — period of oscillation of the disturbance t1 — first-order time constant of the tank; approximately equal to (tank volume/flow through the tank system dead time)
To visualize the effect of dynamic gain, consider a flowing stream whose pH falls from 7 to 4 and returns to 7 in one minute. If the stream flowed through a tank with one minute retention time (volume/flow), the spike in pH would pass through virtually unchanged, and the effluent pH would closely track the influent pH. If, however, the stream flowed through a tank with 60 minutes retention time, practically no upset would be observed in the effluent pH because of the capacity effect of the large volume.
The period of oscillation, t0, of a typical composition process under closed-loop control with an optimally tuned (controller settings adjusted to match the process it con trols) three-mode controller can be approximated as a function of the system dead time.
Substituting for t1 from Equation 7.40(13) and t0 from Equation 7.40(15) into Equation 7.40(14):
In this example the stirred tank has reduced the overall process gain by a factor of 30 (1/0.033). Two tanks used in series reduce the process gain (slow the process down) by the product of their individual gains. Assuming a second tank identical to the first, two tanks in series would reduce the process gain by a factor of 302, or 900. With the stirred tank, therefore, it is possible to reduce the process gain to a controllable level. An added benefit of an increased tank capacity is to smooth out high-frequency errors in reagent delivery caused by measurement noise.
This example is readily related to Figure 7.40.12, in which the output of the reaction vessel is the input disturbance in the attenuation vessel. If the frequency or period (t0) of the input disturbance can be kept short (on the order of seconds)by virtue of a tight control loop around the reaction vessel, then the dynamic gain number of the attenuation vessel will be very low (0.033 for the example), thereby increasing its attenuation capability. This results in a stable effluent pH that averages the input disturbance.
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