Optimal control problems batch reactor

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. 

An optimal control problem for the batch reactor is to find the temperature versus time function, the application of which maximizes the product concentration at the final time tf. That function is the optimal control T t) among all possible control functions, such as those shown in Figure 1.2. 

In this section, an approach to solving the optimal control problem is introduced for reactor-separator processes. The approach involves the simultaneous determination of the batch times and size factors for both of the process units. Furthermore, the interplay between the two units involves trade-offs between them that are adjusted in the optimization. It should be noted that simpler models, than in normal practice, are used here to demonstrate the concept and, in the first example, provide an analytical solution that is obtained with relative ease. 

Figure 3.6 shows dynamics of cellular growth and substrate for a particular species in the batch reactor (above equations) and a fed-batch reactor. As can be seen, these reactions provide better yield in fed-batch reactors. As stated earlier, in a fed-batch reactor, feed is added continuously. Determining optimal feeding profile is a commonly addressed optimal control problem for fed-batch reactors [26]. 

The optimal control problem encountered in biodiesel production in a batch reactor can be classified into three categories depending on the objective function as given below. The decision variable or the control variable is the temperature profile for the reactor. 

P. Benavides and U. Diwekar. Studying various optimal control problems in biodiesel production in a batch reactor under uncertainty. Fuel, 103 585, 2013. 

Batch processes present challenging control problems due to the time-varying nature of operation. Chylla and Haase [4] present a detailed example of a batch reactor problem in the polymer products industry. This reactor has an overall heat transfer coefficient that decreases from batch to batch due to fouling of the heat transfer surface inside the reactor. Bonvin [5] discusses a number of important topics in batch processing, including safety, product quality, and scale-up. He notes that the frequent repetition of batch runs enables the results from previous runs to be used to optimize the operation of subsequent ones. 

The optimal temperature policy in a batch reactor, for a first order irreversible reaction was formulated by Szepe and Levenspiel (1968). The optimal situation was found to be either operating at the maximum allowable temperature, or with a rising temperature policy, Chou el al. (1967) have discussed the problem of simple optimal control policies of isothermal tubular reactors with catalyst decay. They found that the optimal policy is to maintain a constant conversion assuming that the decay is dependent on temperature. Ogunye and Ray (1968) found that, for both reversible and irreversible reactions, the simple optimal policies for the maximization of a total yield of a reactor over a period of catalyst decay were not always optimal. The optimal policy can be mixed containing both constrained and unconstrained parts as well as being purely constrained. 

An application of the above algorithm to the batch reactor problem of Example 3.4 results in the optimal control temperature (T), states (x and y), and costates (Ai and A2) versus time, as shown in Figure 3.1 for the parameters listed in Table 3.3. The corresponding maximum value of the objective functional I is 6.67 kmol/m. ... 

Figure 3.2 Three other optimal control functions for the batch reactor problem...

The degrees of freedom in the problem (Equations 14.1 through 14.11) are represented by u(t) and p. The first vector, u t), depends on t and represents control or optimization variables. In chemical engineering examples, they may correspond to profiles for process stream flow rates, utility fluid flow rates, etc. Vector p corresponds to parameters, degrees of freedom that do not depend on t, which can represent the reactor volume in a batch reactor design problem or any parameter in kinetic expressions within a parameter estimation problem. 

Numerical optimization plays an important role in batch processing. Whether to find maximum yield in the reactor, or maximum distillation in batch distillation, or optimal schedule for batch processing, optimization and optimal control methods are extensively used. In general, the problems in batch and bio processing are large scale problems where analytical solntions are difficult. Hence nnmerical optimization methods are necessary. 

Spreadsheet Applications. The types of appHcations handled with spreadsheets are a microcosm of the types of problems and situations handled with fuU-blown appHcation programs that are mn on microcomputers, minis, and mainframes and include engineering computations, process simulation, equipment design and rating, process optimization, reactor kinetics—design, cost estimation, feedback control, data analysis, and unsteady-state simulation (eg, batch distillation optimization). 

The method proposed for improving the batch operation can be divided into two phases on-line modification of the reactor temperature trajectory and on-line tracking of the desired temperature trajectory. The first phase involves determining an optimal temperature set point profile by solving the on-line dynamic optimization problem and will be described in this section. The other phase involves designing a nonlinear model-based controller to track the obtained temperature set point and will be presented in the next section. 

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