Optimized profile

It might be possible to reduce the inventory significantly by changing reactor conversion and recycle inert concentration without a large cost penalty if the cost optimization profiles are fairly flat. 

The optimal profile for the competitive reaction pair is an increasing function of t (or z). An adiabatic temperature profile is a decreasing function when the reactions are endothermic, so it is obviously worse than the constant temperature, isothermal case. However, reverse the signs on the heats of reactions, and the adiabatic profile is preferred although still suboptimal. 

The resulting optimal profiles for the operating conditions will need to be evaluated in terms of their practicality from the point of view of control and safety. If a complex profile offers only a marginal benefit relative to a fixed value, then simplicity (and possibly safety) will dictate a fixed value to be maintained. But an optimized profile might offer a significant increase in the performance, in which the complex control problem will be worth addressing. 

The optimization is now constrained to be at a fixed (optimized) chlorine addition rate, but the temperature profile optimized. Profile optimization is used for the temperature, as discussed in Chapter 3. The batch cycle time required is 1.42 h. The resulting fractional yield of MBA from BA now reaches 92.7%. 

The final option is to allow both the chlorine addition profile and temperature profile to be varied through the batch. The optimization shows a further improvement of the objective to 99.8%. It requires 1.35 h of batch cycle time and 75.0 kmol of chlorine. The optimized profiles for reaction temperature and feed addition rate of chlorine are shown in Figure 14.5. 

Semibatch with optimized profiles of addition rate of chlorine and temperature 99.8... 

Find out an optimal profile when using the step feeding alternative. After setting CONTROLSET=0 and STEPF = 1 first try to find an optimal profile by manually adjusting FI to F20. Make sure that TMRa(j is not dropping below TMRlim. 

Fig. 7. Optimal profile for U(t). Reprinted with permission from Comp. Chem. Eng., 14, No. 10, 1083-1100, S. Vasantharajan and L. T. Biegler, Simultaneous Optimization of Differential/Algebraic Systems with Error Criterion Adjustment, Copyright 1990, Pergamon Press PLC.

Models are also useful m strategic project planning to compare the costs and benefits of competing therapies, such as the current standard therapy for schizophrenia and a theoretical drug profile of an antipsychotic designed to improve the therapeutic effect for schizophrenia patients. The costs and benefits can be compared to determine the optimal profile (one that would produce the best possible results) as well as the minimal profile (the properties that a compound must demonstrate to show a clinically meaningful improvement ova the standard therapy). 

For the optimal strategy of maintaining operational stability, Lee et al. have calculated the optimal profile of addition of fresh, non-deactivated enzyme into a CSTR under different deactivation kinetics. If a CSTR is charged initially with an amount of enzyme of initial activity N0, at time t under deactivation, the amount remaining is given by Eq. (5.83), where k(t) denotes an arbitrary deactivation function (J. Y. Lee, 1990). 

The implementation of the on-line optimization strategy requires the knowledge of current states and/or parameters in nonlinear process models in order to modify a new optimal profile defined as a set point for a controller. It is known that some measurements i.e. concentration are available at low sampling rate with significant time delay. To overcome this difficulty, state and parameter estimation is incorporated into the proposed on-line optimization algorithm. 

Except for the cases where the optimal temperature profile is of the bang-bang type, analytical solutions for axially varying optimal profiles are almost impossible. Denn et. al. (11) used a variational approach for a wide class of distributed parameter systems where the optimizing decisions may enter into the state equations or boundary conditions. When intermediate control is involved, one can only obtain numerical approximations to the optimal solution. 

If the set of admissible controls is bounded, closed and convex, then it follows (12) that a unique minimizing optimal profile T (z) always exists. Denn et. al. (11) employed the variational approach in determining the necessary and sufficient conditions for an optimal solution. An iterative scheme for approximating the optimal profile was then developed based on the solution of the system equations in the forward direction... 

Here, we consider two alternatives. First, we consider the sequential approach, where we optimize the reactor network with an optimal temperature profile, then integrate the maintenance of this optimal profile with the energy flows in the rest of the flowsheet. In the second case, we solve the above problem with the simultaneous formulation proposed in (PIO). 

---