Optimal temperature profile, example

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. 

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

In this section we shall be concerned with more realistic models of tubular reactors. The isothermal reactor is obviously the simplest type, but it implies that either there are no large heat effects or that they can be completely dominated by temperature control. The reactor with an optimal temperature profile is clearly the most desirable, but this means that the rate of heat exchange can be regulated precisely at each point. Between these two extremes there is a range of designs about which something should be said. We shall not always solve the equations in detail but we shall try to show the important features of the behavior of the reactor by means of examples. 

All other things being equal, as they are in this contrived example, the competitive reaction sequence of Equation 6.6 is superior for the manufacture of B than the consecutive sequence of Equation 6.1. The CSTR remains a doubtful choice, but the isothermal PER is now better than the adiabatic PER. This is because the optimal temperature profile is increasing rather decreasing. See Table 6.6 for comparisons. 

The above case of single reversible exothermic reactions was an example of an output problem. Intuitive logic led to the qualitative conclusion that the optimum temperature profile was the one that maximized the rate at each point. This was also the quantitative solution, and led to the design techniques presented. For yield problems, if the kinetics are not too complex, the proper qualitative trends of the optimal temperature profiles can also often be deduced by reasoning. However, the quantitative aspects must usually be determined by formal mathematical optimization methods. Simple policies, such as choosing the temperature for maximum local pointwise selectivity, rarely lead to the maximum final overall selectivity because of the complex interactions between the various rates. 

There is no constant of integration due to the boundary condition that both AG/T and A(l/7 ) are zero at equilibrium. However, AH will be temperature-dependent most of the time. For example, in producing ammonia from hydrogen and nitrogen, the goal is to maximize the output of ammonia at the exit. An approximately constant AT between the optimal path and the equilibrium temperature provides the optimal temperature profile, which reduces the exergy loss by approximately 60% in the reactor. The equipartition of forces principle for multiple, independent rate-controlled reactions and multiphase and coupled phenomena, such as reactive distillations, may lead to the improved use of energy and reduced costs (Sauar et al., 1997). 

This temperature dependency is exploited in optimal control problems of batch reactor where optimal temperature profile is obtained by either maximizing conversion, yield, profit, or minimizing batch time for the reaction. One of the earliest works on optimal control of batch reactor was presented by Denbigh[25] where he maximized the yield. The review paper by Srinivasan et al.[26] describes various optimization and optimal control problems in batch processing and provides examples of semi-batch and fed-batch reactor optimal control. 

Example 5.5 Optimal Temperature Profile for Penicillin Fermentation... 

Example 5.5 Solution of the Optimal Temperature Profile for Penicillin Fermentation. Apply the orthogonal collocation method to solve the two-point boundary-value problem arising from the application of the maximum principle ofPontryagin to a batch penicillin fermentation. Obtain the solution of this problem, and show the profdes of the state variables, the adjoint variables, and the optimal temperature. The equations that describe the state of the system in a batch penicillin fermentation, developed by Constantinides et al.(6], are ... 

Figure 6.2 displays the temperature profile for a 10-zone case and for a 99-zone case. The 99-zone case is a tour de force for the optimization routine that took a few hours of computing time. It is not a practical example since such a multizone design would be very expensive to build. More practical designs are suggested by Problems 6.11-6.13. 

The control variables can be constrained to fixed values (e.g. fixed initial temperature in a temperature profile) or constrained to be between certain limits. In addition to the six variables dictating the shape of the profile, ttotai can also be optimized if required. For example, this can be important in batch processes to optimize the batch cycle time in a batch process, in addition to the other variables. 

Example 14.1 Consider again the chlorination reaction in Example 7.3. This was examined as a continuous process. Now assume it is carried out in batch or semibatch mode. The same reactor model will be used as in Example 7.3. The liquid feed of butanoic acid is 13.3 kmol. The butanoic acid and chlorine addition rates and the temperature profile need to be optimized simultaneously through the batch, and the batch time optimized. The reaction takes place isobarically at 10 bar. The upper and lower temperature bounds are 50°C and 150°C respectively. Assume the reactor vessel to be perfectly mixed and assume that the batch operation can be modeled as a series of mixed-flow reactors. The objective is to maximize the fractional yield of a-monochlorobutanoic acid with respect to butanoic acid. Specialized software is required to perform the calculations, in this case using simulated annealing3. 

Determination of die process parameters that ensure a permissible temperature profile and optimal solidification path is based on the general principles of the theory of batch reactors formulated in Section 2.7. Let us illustrate this approach with the example of solidification of a urethane-based compound for use as a coating.176... 

The operating procedure presented in this section can be further optimized. The objective function should include not only the productivity of the reactor, but also costs related to investment, utilities and manpower. Examples of decision variables that can be considered are the temperature profile, the amounts of initiator that are added, and the timing of initiator addition. We leave this optimization as an exercise for the reader. 

In this paper, a tube of size 1/4" in diameter was considered with styrene monomer preheated to 135 C. The radial variations in temperature are minimal and good control over the concentration profile was possible. Some typical variations in conversion with radial position are shown in Figure 10. The zone temperatures for this example represent a sub-optimal case. However, it is readily seen that as we approach the optimal solution, the first zone temperature converges to an upper limit, while the second zone temperature goes to absolute zero. Figure 11 shows this trend. We also note that as the optimal temperatures are approached, there is a steady drop in the... 

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