Constant a policy

To plot DSR trajectories, an initial feed Cj, sidestream concentration C , and a policy are required. Since C = Cf and the four constant a policies have been provided, we can plot four DSR trajectories in Ca-Cb... 

Figure 4.29 also shows the PER trajectory and CSTR locus from the feed. Observe that the DSR trajectory corresponding to a = 0 coincides with the PFR trajectory. DSR residence times have been chosen large enough so as to approximate equilibrium. For values of >0, the DSR trajectory appears to terminate at points on the CSTR locus. The particular termination point is specific to the value of a in the DSR. This is expected since for a constant a policy, the DSR equation at equilibrium is identical in form to the CSTR equation. 

End points C and D therefore represent equilibrium points in the fed-batch for a constant a policy, and are thus also solutions to Equation 7.22. For all values of a in the range 0feed concentration C° set at point O. The locus of x s, given by the curve ACDE, therefore represents the set of equilibrium concentrations achieved in the system. 

In order to achieve the same concentrations of a CSTR in a batch setting, a fed-batch reactor operated with a constant a policy is required. Furthermore, the fed-batch must be initiated and operated, ideally, at the CSTR steady-state concentration. 

A CSTR concentration is obtained in a DSR if the DSR is operated at the equilibrium point with a constant a policy. The CSTR solution appears as the DSR equilibrium concentration with the feed point to the CSTR given by the sidestream composition CP, and the CSTR residence time given by the reciprocal value of a. 

To minimize damage due to power outage, the Celanese Corporation in their plant at Newark, N.J., instituted a policy of always generating half its own power. Merck Company installed additional auxiliary steam power to insure constant refrigeration for its biochemicals at its West Point, Pa., plant. At Allied Chemical s phenol plant in Frankfort, Pa., electric devices on air compressors and pumps were replaced by steam-operated controls, and diesel generators were installed to maintain cooling water circulation. 19... 

Comparability depends on the policy of measuring half-lives the mode of keeping a constant degree of conversion at variable flow rate or enzyme concentration (constant conversion policy) gives better results than the mode of constant flow rate accompanied by varying degree of conversion (constant flow rate policy). 

Right Profiles of decrease in F(t)/F(0) for intraparticle diffusion-influenced zero-order reaction with spherical immobilized enzyme particles packed in the reactor operated under a constant conversion policy (x = 0.99). Enzyme activity decays as E(t)/E(0) = exp ( kd t). 

Two cases are studied. For Case a, single time interval in each step is used, yielding an optimal constant reflux policy. For Case b, five time intervals are used in each step. Table 5.9 summarises the results using single constant (optimised) and variable constant (optimised) reflux ratio profiles. The instant and accumulated distillate profiles together with the reflux ratio profiles are shown in Figures 5.6 and 5.7. 

Note Case a represents optimal constant reflux policy... 

Farhat et al. considered both optimal constant and optimal linear reflux ratio for this problem (Figure 6.15). Final time was fixed and 4 time intervals were considered. The length of each time interval was also optimised. Table 6.12 presents the summary of the optimisation results using both options of reflux ratio profiles. A significant gain of 10.7% in specified products can be observed between the optimal linear reflux policy and the optimal constant reflux policy. 

The constant A is used to render the time dimensionless. In case the kinetic constants /3, are fixed and we consider only one reference composition, there are only two parameters p and k and these are separated in the sense that p is associated with the integral F whilst K is associated with the temperature policy. 

Ogunye and Ray (1971a,b) have formulated the optimal control problem for tubular reactors with catalyst decay via a weak maximum principle for this distributed system. Detailed numerical examples have been calculated for both adiabatic and isothermal reactors. For irreversible reactions, constant conversion policies are found to not always be optimal. A practical technique for on-line optimal control for fixed bed catalytic reactors, has been suggested by Brisk and Barton (1977). Lovland (1977) derived a simple maximum principle for the optimal flow control of plug flow processes. 

With a constant-reflux policy, the instantaneous distillate purity is above the specification at the beginning of the run and below specification at the end. By an overall material balance, the average mole fraction of the light-key component in the accumulated distillate at time t is given by... 

The maintenance of a policy for cutting costs at all levels in order to adapt to a constantly changing technical-economic world, and survive. 

In commercial operation of fixed-bed reactors in which the catalyst is deactivating, it is often necessary to maintain constant conversion by increasing the temperature of the reactor to compensate for catalyst decay. One can think of this as a constant activity policy in which the bed may be considered isothermal at any given time with the overall temperature level increasing with time of operation. 

These two assumptions imply that the vapor rate and the liquid hold-up both decrease along the cascade for a fixed feed flow rate. This produces a policy of decreasing vapor rate along the cascade similar to a decreasing vapor rate policy in simple (batch) distillation, which keeps the instantaneous value of Da approximately constant [47]. 

Rate constants have already been substituted into the rate expressions for convenience (they are kj = 1 s kj = 1 s, and kj = 10 L/(mol.s)). Determine an expression for the critical a policy if the DSR side-stream mixing composition is given by C° = [l,0,0]Tmol/L. 

For Cd = 0.3 mol/L, the recommended operating policy is structure 2. This structure is different compared to structure 1 in that three reaction periods are required. The batch cycle begins with a fed-batch period of constant a in accordance with the equilibrium CSTR concentration, given by point D. The cycle is then brought into a period of varying a, where the sidestream addition is observed to increase sharply to a maximum value of approximately 1.356 s and then completed with a standard batch period lasting approximately 9.24 s the total batch cycle time for this structure is thus roughly 15 s. 

As an alternative approach, a few researchers have used competitive analysis to study DDM policies, where an on-line algorithm A is compared to an optimal off-line algorithm [94]. Given an instance /, let za I) and z I) denote the objective function value obtained by algorithm A and by an optimum off-line algorithm, respectively. We call an algorithm A C -competitive, if there exists a constant a, such that... 

A new method for selecting controlled variables (c) as linear combination of measurements (y) is proposed based on the idea of self-optimizing control. The objective is to find controlled variables, such that a constant setpoint policy leads to near optimal operation in the presence of low frequency disturbances d). We propose to combine as many measurements as there are unconstrained degrees of freedom (inputs, u) and major disturbances such that opi d) = 0. To illustrate the ideas a gas-lift allocation example is included. The example show that the method proposed here give controlled variables with good self-optimizing properties. 

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