Optimal feeding policy

A complex reaction is run in a semi-batch reactor with the purpose of improving the selectivity for the desired product P, compared to that of the waste Q, which is costly to treat and dispose. The kinetics are sequential with respect to components A, P and Q but are parallel with respect to B. The relative magnitudes of the orders of the two reactions determine the optimal feeding policy. 

Recently, many batch operations have been transformed into fed-batch (semicontinuous) operations by the gradual introduction of nutrient into the reactor. The rationale is to control the feed optimally to maximize a composite performance index. For the case of penicillin fermentation, for example, for which the specific growth rate and the specific penicillin formation rate are mutually disposed, the optimal feed policy is carried out in two phases. During the first phase, cell biomass is quickly built up to the allowable maximum level. During the second product formation phase, the feed is controlled such that... 

It often becomes necessary in biochemical reactions to continuously add one (or more) substrate(s), a nutrient, or any regulating compound to a batch reactor, from which there is no continuous removal of product. A reactor in which this is accomplished is conventionally termed the semibatch reactor (Chapter 4) but is referred to as a fed-batch reactor in biochemical language. The fed-batch mode of operation is very useful when an optimum concentration of the substrate (or one of the substrates in a multisubstrate system) or of a particular nutrient is desirable. This can be achieved by imposing an optimal feed policy. 

Clearly, the feed rate Q t) can be manipulated to give an optimal feed policy (with respect to time) to maximize performance. Usually, a constant or exponential addition policy is the most practical and the simplest. Yet, in realistic situations, the feed rate must be judiciously varied so that a robust optimization criterion can be met. Thus first a suitable objective function must be evolved. Many such objective functions are possible, for example, maximization of yield and minimization of production cost. 

The embedded model approach represented by problem (17) has been very successful in solving large process problems. Sargent and Sullivan (1979) optimized feed changeover policies for a sequence of distillation columns that included seven control profiles and 50 differential equations. More recently, Mujtaba and Macchietto (1988) used the SPEEDUP implementation of this method for optimal control of plate-to-plate batch distillation columns. 

Srinivasan, B., Ubrich, O., Bonvin, D. and Stoessel, F. (2001) Optimal feed rate policy for systems with two reactions, in DYCOPS, 6th IFAC Symposium on Dynamic Control of Process Systems. International Federation of Automatic Control, 455 460, Cheju Island Corea. 

This example is taken from Mujtaba and Macchietto (1996). The problem is to design a column for 2 binary separation duties. One of the separations is very easy compared to the other one. The fraction of production time for each duty is specified together with the still capacity (B0) and the vapour load (V). Each binary mixture produces only one main distillate product and a bottom residue (states MPf= Dl, Bfl] and MP2=[D2, Bf2]) from feed states EFt= Fl and EF2= F2], respectively, with only one distillation task in each separation duty. Desired purities are specified for the two main-cuts (x Di and xID2). Also obtain the optimal operating policies in terms of reflux ratio for the separations. 

Once the optimal feed rates were obtained, they were applied to the actual process (i.e. simulation by the mechanistic model of the process). Table 2 shows the difference between the amounts of the final product and by-product on neural network model and the actual process. It can be seen from Table 2 that the actual amounts of product and by-product rmder these optimal control policies are quite different from the neural network model predictions. This indicates that the single neural network based optimal control policies are only optimal on the neural network model and are not optimal on the real process. Hence, they are not reliable. This is mainly due to the model plant mismatches, which is rmavoidable in data based modelling. 

The optimal temperature profiles obtained pose the practical problem of how to implement this optimal temperature policy. The optimal temperature profiles obtained suggest that the reactor should be operated with quite a high feed temperature followed by a very efficient cooling to decrease the temperature sharply at the first 15% of the reactor depth. The implementation of this optimal policy is technically difficult and expensive. A number of suboptimal policies were suggested and discussed (Elnashaie et ai, 1987b) to overcome the technical difficulties associated with the implementation of the absolute optimal policy. One of the main technical problems associated with optimal temperature is the very high temperature at the reactor inlet part which the catalyst may not be able to withstand. 

FIGURE 7.4 Effect of catalyst pellet diameter on the optimal feed temperature policy of the reactor. 

The effect of different design, operating and physico-chemical parameters on the optimal feed temperature policies of the heterogeneous reactor are presented in this section. Table 7.3 gives base values of the parameters. 

Different catalyst particle sizes are considered. The diameters of these particles are 1.5, 2.0 and 2.3 cm. Figure 7.4 shows the effect of dp on the inlet optimal feed temperature policy for the reactor. It is evident from this figure that for dp s 2.0 the optimal dimensionless feed temperature policy >y(T) (m(t)) is an increasing temperature profile with r until the upper constraint y is attained. 

FIGURE 7,5 Effect of thermicity factor on optimal feed temperature policy. 

The optimal feed temperature policy obtained from the pseudo-homogeneous model has been applied to the heterogeneous model the objective function obtained was equal to 0.284134. The objective... 

FIGURE 7.7 Comparison between the optimal feed temperature policies obtained from the pseudo-homogeneous model and the hetero-... 

The practical implementation of the above policies is not necessarily as straightforward as solving the above equations. As can be deduced from Equations 6.70-6.76, Pjjjj is a function of the propagation rate coefficients, the monomer concentrations, and most importantly, the total radical concentration. Hence, to precalculate the optimal monomer feed rates, the radical concentration must be specified in advance and kept constant via an initiator feed policy and/or a heat production policy. This is especially important considering that a constant radical concentration is not a typical polymer production reality. This raises the notion that one could increase the reactor temperature or the initiator concentration over time to manipulate the radical concentration rather than manipulate the monomer feed flowrates, that is, keep P j constant for simpler pump operation. Furthermore, these semibatch policies provide the open-loop or off-line optimal feed rates required to produce a constant composition product. The online or closed-loop implementation of these policies necessitates a consideration of online sensors for monomer... 

There are two basic monomer feed policies (and several modiflcations of the basic ones) that may be used in semibatch polymerization to minimize compositional drift (or optimize other properties). See Hamielec et al. [22] and Fujisawa and Penlidis [43] for more details. 

Lu, Y.P., Dixon, A.G., Moser, W.R. and Ma, Y.H., 1997c. Analysis and Optimization of Cross-Flow Reactors with Staged Feed Policies - Isothermal Operation with Parallel Series, Irreversible Reaction Systems. Chemical Engineering Science, 52(8) 1349-1363. 

Is carried out in an ideal PFR in which optimal temperature policy is maintained with the maximum value of the reactor temperature restricted to 900 K. Calculate the space time required to achieve 70% conversion of A. The concentration of A in the feed solution is 0.5 kmol/m. ... 

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The... 

For single separation duty, Mujtaba and Macchietto (1993) proposed a method, based on extensions of the techniques of Mujtaba (1989) and Mujtaba and Macchietto (1988, 1989, 1991, 1992), to determine the optimal multiperiod operation policies for binary and general multicomponent batch distillation of a given feed mixture, with several main-cuts and off-cuts. A two level dynamic optimisation formulation was presented so as to maximise a general profit function for the multiperiod operation, subject to general constraints. The solution of this problem determines the optimal amount of each main and off cut, the optimal duration of each distillation task and the optimal reflux ratio profiles during each production period. The outer level optimisation maximises the profit function by... 

This is the feed state of the subsequent R — 1) stages which, according to the principle of optimality, must use an optimal R — l)-stage policy with respect to this state. This will result in a value/R i(pR) of the objective functipn, and when qR is chosen correctly this will give/r(Pr+i), the maximum of the objective function. Thus... 

To extract the optimal P-stage policy with respect to the feed state pR+i, we enter section P of this table at the state pR+i and find immediately from the last column the maximum value of the objective function. In the third column is given the optimal policy for stage P, and in the fourth, the resulting state of the stream when this policy is used. Since by the principle of optimality the remaining stages use an optimal (P — 1)-stage pohcy with respect to pR, we may enter section (P — 1) of the table at this state pR and read off the optimal... 

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