Purity constraint

We shall now follow the unrestricted Hartree-Fock (UHF) formalism to obtain a restricted high-spin open-shell functions as proposed in [34], [35]. In order to eliminate spin contamination in the UHF function f i, the following spin purity constraint is imposed on the spatial orbitals ... 

Mujtaba and Macchietto (1997) have considered a maximum conversion problem for BREAD, subject to given product purity constraints. The reflux ratio is selected as the control parameters to be optimised for a fixed batch time so as to maximise the conversion of the limiting reactant. The optimal product amount, condenser and reboiler duties are also calculated. Referring to Figure 4.5 for CBD column the optimisation problem can be stated as ... 

Owing to the practical conditions for the production of fructose syrup no purity constraints exist. The purity of the extract in respect of fructose is ... 

The objective of meeting the product specifications is reflected by the purity constraint over the control horizon Hn which is corrected by a bias term 8Pue1 resulting from the difference between the last simulated and the last measured process output to compensate un-modeled effects ... 

The second purity constraint over the whole prediction horizon acts as a terminal (stability) constraint, forcing the process to converge towards the optimal cyclic steady state. The goal of feedback control in a standard control approach (i.e. to fulfill the extract purity) is introduced as a constraint here. A feasible path SQP algorithm is used for the optimization (Zhou et al., 1997), which generates a feasible point before it starts to minimize the objective function. 

The desired purity for the experiment reported below was set to 55.0% and the controller was started at the 60th period. As in the simulation study, a diagonal matrix R = 0.02 I (3,3) was chosen for regularization. The control horizon was set to Hr = 1 and the prediction horizon was Hv = 60 periods. Figure 9.10 shows the evolution of both the product purity and the controlled variables. In the open-loop mode, where the operating point was calculated based on the initial model, the product purity constraint was violated at periods 48 and 54. After one cycle, the controller drove the purity above 55.0% and kept it there. The controller first reduces the desorbent consumption. This action seems to contradict the intuitive idea that more desorbent injection should enhance separation. However, in the presence of a reaction, this is not true, as shown by this experiment. The controlled variables converge towards a steady state, but they still change from period to period, due to the non-ideality of the plant. 

Figure 18.19 Optimization results for the later eluting component in a tertiary mixture at three different purities on a 90 m FF Sepharose stationary phase 91%, 95%, and 99% purity constraints. Column conditions diameter 1.6 cm length 10.5 cm. Feed conditions ribonuclease A, a chymotrypsinogen A, and the artificial component at 0.5 mM each. All optimal results are presented as a function of column loadings (dimensionless column volume). (a) Optimal production rate times yield (mmol/min/mL). (b) Optimal production rate (mmol/min/mL). (c) Optimal yield. Reproduced with permission from D. Nagrath et ah, Biotechnol. Prog., 20 (2004) 163 (Fig. 8).

When the purity constraints are included in the objective function, the problem to be solved may be stated as follows. The economic objective function... 

In the formulation of a typical problem of this type, consider again the complex column shown in Fig. 9-2, and suppose that the purity specifications or constraints are taken to be (bh/dh)L, (bjd u, (wr/dr)L, and (ws/ds)LI. Then the problem to be solved consists of finding the number of plates ku k2, and k5 and the corresponding values of the operating variables Lx/D, D, and W that minimize the operating costs and capital costs per mole of the most valuable product (D, , or W). When the purity constraints are included in the objective function, the problem to be solved may be stated as follows. The economic objective function... 

Ziomek et al. (2005a) and Ziomek and Antos (2005b) developed a procedure based on a random search routine to optimize the productivity and eluent consumption for purity constraints of PUext.raff >90%. They applied a stage model and... 

The purity constraints (14) were written as linear constraints as follows ... 

However, there may be cases when the above constraints are not active. Take, for instance, a case when there are changes to the feed composition, such that the condensation in the HP column is higher than the energy requirements of the LP column. In terms of Figure 2 this means that the minimum vapour flowrate curve for the prefractionator will lie above the curve for the main column, for all recoveries of the middle component. In order to balance the columns it may then be optimal to overpurify the least valuable product in the main column, thus one purity constraint is no longer... 

The optimal results indicate both the thermal separation and chemical reaction effects. The more product alcohol in the entire column, the less reflux ratio we need to satisfy the purity restrictions. The slow increase of the reflux ratio during the first three hours is allowed, since a large amount of product alcohol results from the drastic increase of the feed flow of the educt alcohol. However, when the feed flow has reached its maximum value, the reflux ratio needs to increase drastically in order to ensure the distillate purity constraint. The decrease of the reflux ratio can be explained with the time delay between the feed supply of educt alcohol and the resulting effect of formation of product alcohol caused by the chemical reaction. 

A stochastic dynamic optimization approach has been successfully implemented for a reactive semibatch distillation process. The aim is batch time minimization subject to product purity restrictions. A method for computing the probabilities and their gradients is developed to solve the dynamic stochastic optimization problem. The results obtained by the implementation with a higher probability level show that the consideration of uncertainties with chance constraints leads to a trade-off between the objective value and robustness. A comparison of the stochastic results with the deterministic results is made with respect to the objective values and the reliability of satisfying the purity constraints. We thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support under the contract WO 565/12-1. 

Optimal control of a batch distillation column consists in the determination of the suitable reflux policy with respect to a particular objective function (e.g. profit) and set of constraints. In the purpose of the present work, the optimisation problem is defined with an operating time objective function and purity constraints set on the recovery ratio (90%) and on the propylene glycol final purity (80% molar). Different basis fimctions have been adopted for the control vector parameterisation of the problem piecewise constant and linear, hyperbolic tangent function. Optimal reflux profiles are determined with the final conditions of the previous optimal reactions as initial conditions. The optimal profiles of the resultant distillations are presented on figure 2. 

In this part of the study, optimisation of the production is carried out according to a global approach. The reaction step and the successive distillation step are considered simultaneously in the evaluation of the optimal operating conditions. In order to compare these results with the classical approach ones an operating time criterion has been chosen. Thus, the optimisation problem lies in the minimisation of the operating time required for the propylene glycol synthesis. According to the previous optimisations, two kinds of production have been studied a production with yield and purity constraints and a production with an additional by-products constraint. In order to compare the different approaches, the same constraints have been adopted. 

The top product purity constraint is always active, that is, it is alsways optimal to have Xd = 0.995, so the distillate composition xp> should be selected as a controlled variable (C]). 

Since neither nor A can be infinity in practical calculations, one has to settle on some large finite values. The recommended values are = 100 hartrees for the spin-purity constraint and = 1000 hartrees for the orthogonality constraint. They provide target accuracy close to 10 In concluding this section, it is also worth noting that in... 

Polymerization directly into the finished product, in the form of films or blends, on the surface of molded components or in their morphology. Here there are limitations on practical flexibility and purity, constraints on the shape or design of the finished products and their reproducibility and the chemical process. This approach is not possible for converters (for lack of chemical infra.struc-ture) and is unsuitable even for basic research because of poor reproducibility. 

The surplus diagram captures the purity constraints in the network. If the cumulative net COj is negative at any purity level (i.e. surplus curve crosses y-axis), then the network is not receiving the required amount of CO at the adequate purity. In that case, at least one of the constraints imposed by the demands cannot be satisfied by the sources, rendering the network unfeasible. To make the network feasible, a purified CO stream is required. Then, the options are (i) to import a high-purity stream or (ii) introduce a purification unit to increase the purity of a low-purity stream. Therefore, the second necessary condition for network feasibility is that the balance of CO in the overall system (i.e. the cumulative net CO flow rate) must always be positive. This means that the entire CO surplus curve must lie at or above zero flow rate for the network to be feasible. 

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