Disturbance attenuation

This result shows that the phase velocity of the disturbance is a = afo/ia cos (p) and that the disturbance attenuates as x increases with a characteristic decay length given by / = ayo/(cua sin (p). The consequent dependence of the phase velocity on the frequency [see equations (115) and (116)] represents sound dispersion, and the attenuation illustrates sound absorption. Various specific results follow directly from equations (114)-(116). As CD 0, a (i o attenuation) as cd oo, a ayo... 

Simplified Analysis for Series CSTRs. Although general problems require optimization of a nonlinear dynamic model as discussed above, the analysis can be greatly simplified for some special cases. The case of particular interest for the problems considered later is that of continuous-flow stirred-tank reactors (CSTRs) in series. In this case, it is desired to add reagent so as to keep variations in the net concentration of effluent and reagent, cnet, at the exit of the last tank below a certain level, 8 ei, in the face of step disturbances in the inlet concentration of magnitude A,.,. This objective can be expressed as a required disturbance attenuation, 5,., where... 

Given the minimum delay in the control response relative to the disturbance response at the treatment system exit (r = tj- where y is the exit measurement), the best possible disturbance attenuation, 6, is given by... 

It is possible to determine the optimal volume of n CSTRs in series to achieve the required disturbance attenuation by equating the required and estimated attenuation (see Section V.A.4). 

If one tank in a multitank chain was uncontrolled, the disturbance attenuation was further increased by about 50%. 

For preliminary design purposes, the estimate from Eq. (52) is an adequate predictor of achievable PI performance when all tanks are tightly controlled. If control on one tank of a multiple CSTR system is rendered ineffective—due to uncertainty, high delay compared to the minimum delay, or simply the absence of a controller—the predicted disturbance attenuation should be increased by 50%. An exception to this is the case of variation in the sensitivity of pH to concentration on the final CSTR. In this case, no degradation of performance from that obtained with the minimum buffering (maximum titration curve slope) will occur as the performance required in terms of concentration deviations relaxes along with the controller performance. 

Summary. It is possible to rapidly estimate the economically optimum number and size of CSTRs in series required to achieve a given disturbance attenuation. The design obtained from this analysis will always have equal-sized tanks due to the dependence of the constraints on the product of the tank residence times. The tanks will typically be between 10 and 30 m in size with residence times between 2 minutes and 1 hour. This is consistent with industrial practice and contradicts the recommendation in the literature, based on frequency response arguments, that tank sizes should be split in a ratio of about 1 4 or greater (Shinskey, 1973 Moore, 1978 McMillan, 1984). This provides a simple illustration of the ability of the integrated approach to balance operability and economic considerations in a way that qualitative argument cannot. 

The optimal series CSTR configuration to minimize cost for a given disturbance attenuation requirement and instantly reacting reagent comprises equal-sized tanks for typical treatment systems. 

If local containment is applied to the worst-case disturbance, the new worst-case disturbance can be assumed to be bounded by startup to maximum flow, giving Lcnet —. 01 M This disturbance may be in either the acid or the alkali direction, so 6c e, is reduced to. 00025 N. This corresponds to a required disturbance attenuation of. 025. The residence time has been doubled and the achievable attenuation decreases by a factor of 2" compared to the table above. The rigorous bound suggests that one tank might be just adequate (.017 versus. 025), while the heuristic estimate indicates that two tanks would be required. The two-tank system has an estimated attenuation of. 004 compared to a requirement of. 025 and should be able to deliver the required performance despite the titration curve variability. 

The mixing delay is 10 seconds. The maximum measurement lag is 30 seconds. The reagent dynamics are complex, but the effective lag associated with the initial response is no more than 10 seconds (based on examination of time for 50% conversion of reagent using the model given above). The effective delay, calculated as f 4 (Section II.B.2), is therefore about 41 seconds. The estimated disturbance attenuation with PI control is 8 X 10 (Eq. 52). Transient control performance is not expected to be a problem for this system and a basic control scheme should be adequate. 

For frequencies where this is not satisfied, acceptable disturbance attenuation can not be achieved using feedback control with the considered input as the manipulated variable. The smallest frequency for which (12) is not satisfied is denoted b . 

As an alternative, or complement, to changing bandwidth limitations, as discussed above, one can also change the bandwidth requirements through process design modifications. This applies in particular to bandwidth requirements imposed by the need for disturbance attenuation. 

Vary the gain to achieve the maximum closed-loop disturbance attenuation. How effective is the controller in rejecting disturbance not already rejected by the natural attenuation of the process ... 

Which combination of gain and level control provides the best disturbance attenuation ... 

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