Optimization temperature profile

Example 6.5 Find the optimal temperature profile, T z), that maximizes the concentration of component B in the competitive reaction sequence of Equation (6.1) for a piston flow reactor subject to the constraint that F=3h. 

FIGURE 6.2 Piecewise-constant approximations to an optimal temperature profile for consecutive reactions (a) 10-zone optimization (b) 99-zone optimization. 

Compare the (unconstrained) optimal temperature profiles of 10-zone PFRs for the following cases where (a) the reactions are consecutive as per Equation (6.1) and endothermic (b) the reactions are consecutive and exothermic (c) the reactions are competitive as per Equation (6.6) and endothermic and (d) the reactions are competitive and exothermic. 

Can the calculus of variations be used to find the optimal temperature profile in Example 6.5 ... 

Figure 5.4-70. Optimal temperature profile for parallel reactions of the same order.

Figure 5.4-72. Optimal temperature profile for consecutive reactions of equal order.

Nonisothermal reactors with adiabatic beds. Optimization of the temperature profile described above assumes that heat can be added or removed wherever required and at whatever rate required so that the optimal temperature profile can be achieved. A superstructure can be set up to examine design options involving adiabatic reaction sections. Figure 7.12 shows a superstructure for a reactor with adiabatic sections912 that allows heat to be transferred indirectly or directly through intermediate feed injection. 

Semibatch with optimized constant addition rate of chlorine and optimized temperature profile 92.7... 

Fig. 6. Constant V optimal temperature profiles. 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.

Kokossis and Floudas (1994) extended the MINLP approach so as to handle nonisothermal operation. The nonisothermal superstructure includes alternatives of temperature control for the reactors as well as options for directly or indirectly intercooled or interheated reactors. This approach can be applied to any homogeneous exothermic or endothermic reaction and the solution of the resulting MINLP model provides information about the optimal temperature profile, the type of temperature control, the feeding, recycling, and by-passing strategy, and the optimal type and size of the reactor units. 

The purpose of this optimization problem is to determine the optimal temperature profiles to achieve the desired final product concentration in minimum batch time, thus the performance index is the final time whereas the desired production concentration is defined as a terminal constraints. The formulation of the minimum batch time problem can be shown as... 

Fig. 4. Optimal temperature profile 1 interval (1), 5 intervals (2), 10 intervals (3), 20 intervals (4), 40 intervals (5)...

N. Aziz, M.A. Hussain, I.M. Mujtaba, Performance of different types of controllers in tracking optimal temperature profiles in batch reactors, Comp. Chem. Eng. 24 (2000) 1069-1075. 

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. 

Pz has a practical range of 10 to 50, it becomes easy to vary these parameters in order to study the entire range of allowable optimal temperature profiles. 

Fig. 13 Optimal temperature profile calculated from a first-principle model for maximizing the mean crystal size for unseeded crystallization of paracetamol in water and the simulated change in mean crystal size during crystallization.

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). 

Fig. 1. The resulted optimal temperature profile (continuous fine with plusses) and the linear cooling profile (dashed line).

Fig. 2 Off-line calculated optimal temperature profile (dashed line) and corresponding level (continuous line) based on the master recipe catalvst feedintr nolicv tnominal easet...

Fig. 4 Simulation using the off-line calculated optimal temperature profile with a new catalyst dosing strategy.

This work presents the on-line level control of a batch reactor. The on-line strategy is required to accommodate the reaction rate disturbances which arise due to catalyst dosing uncertainties (catalyst mass and feeding time). It is concluded that the implemented shrinking horizon on-line optimization strategy is able to calculate the optimal temperature profile without causing swelling or sub-optimal operation. Additionally, it is concluded that, for this process, a closed-loop formulation of the model predictive controller is needed where an output feedback controller ensures the level is controlled within the discretization intervals. 

Fig 9.6 The optimal temperature profile for a single exothermic reaction. 

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