Optimal batch operation time

Optimal batch operation time for the sequence of operations in a given reactor. 

Optimal temperature (or other variable) variations during the course of the reaction, aiming at minimizing the reactor size. 

Only the main features will be discussed here — more extensive details are given in Aris [1961]. To simplify the mathematical details only constant volume reactors are considered, but for most practical situations this is not a serious 

The discussion follows that of Aris [1961, 1965, 1969). The price per kilomole of chemical species Aj is Wj, so that the net increase in value of the reacting mixture 

The cost of operation is usually based on four steps  

Preparation and reactor charging time Of, with cost per unit time, of Wp. 

Since our interest is in the reactor operation, all the other times will be taken to be constant, and the main question is to determine the optimal reaction time, with its corresponding conversion. The net profit is 

The actual optimum reaction time, 6r, must still, of course, be found from Eq. 8.1-4 evaluated at x r = x (6r) found from Eq. 8.3.a-9  

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In general, an objective function in the optimization problem can be chosen, depending on the nature of the problem. Here, two practical optimization problems related to batch operation maximization of product concentration in a fixed batch time and minimization of batch operation time given amount of desired product, are considered to determine an optimal reactor temperature profile. The first problem formulation is applied to a situation where we need to increase the amount of desired product while batch operation time is fixed. This is due to the limitation of complete production line in a sequential processing. However, in some circumstances, we need to reduce the duration of batch run to allow the operation of more runs per day. This requirement leads to the minimum time optimization problem. These problems can be described in details as follows. 

Several simulations have been carried out under process parameter uncertainties e.g. in pre-exponential rate constant (ko) and activation energy (Ea). In all case studies we considered 10 time intervals when reactor temperature and switching time are optimized while minimizing the final batch operation time. Results, reported in the value of minimum batch time to obtain the desired product C and the amount of the desired product C at the end of batch operation, from on-line dynamic optimization strategy are also compared with those from the off-line strategy. 

Optimization of Cycle Times. In batch filters, one of the important decisions is how much time is allocated to the different operations such as filtration, displacement dewatering, cake washing, and cake discharge, which may involve opening of the pressure vessel. Ah. of this has to happen within a cycle time /. which itself is not fixed, though some of the times involved may be defined, such as the cake discharge time. 

The most limiting factor for enzymatic PAC production is the inactivation of PDC by the toxic substrate benzaldehyde. The rate of PDC deactivation follows a first order dependency on benzaldehyde concentration and reaction time [8]. Various strategies have been developed to minimize PDC exposure to benzaldehyde including fed-batch operation, immobilization of PDC for continuous operation and more recently an enzymatic aqueous/octanol two-phase process [5,9,10] in which benzaldehyde is continuously fed from the octanol to the enzyme in the aqueous phase. The present study aims at optimal feeding of benzaldehyde in an aqueous batch system. 

Operating conditions. Optimization variables such as batch cycle time and total amount of reactants have fixed values for a given batch reactor system. However, variables such as temperature, pressure, feed addition rates and product takeoff rates are dynamic variables that change through the batch cycle time. The values of these variables form a profile for each variable across the batch cycle time. 

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. 

Remark 3 Batch and semibatch reactors can also be studied by considering their space equivalent PFRs. The space equivalent of a single batch reactor can be regarded as a PFR with no side streams. The optimal holding time for the batch operation can be determined by the optimal PFR length and the assumed linear velocity of the fluid in the PFR. 

As a batch reactor is utilized for the production of a wide variety of high value products, an optimization of batch operating conditions, e.g. temperature, operating time, etc. is... 

The selectivity issues discussed in the previous section can easily be avoided by an excess of acid. An optimized process should ran in semi-batch operation (continuous feeding of the alcohol to keep its concentration low). In this way, the effective water removal would have the greatest impact on the space-time yield. 

Mayur and Jackson (1971) simulated the effect of holdup in a three-plate column for a binary mixture, having about 13% of the initial charge distributed as plate holdup and no condenser holdup. They found that for both constant reflux and optimal reflux operation, the batch time was about 15-20% higher for the holdup case compared to the negligible holdup case. Rose (1985) drew similar conclusion about column holdup but mentioned that the adverse effects of column holdup depends entirely on the system, on the performance required (amount of product, purity), and on the amount of holdup. Logsdon (1990) found that column holdup had a small but positive effect on their column operation. 

The minimum time problem is also known as the time optimal control problem. Coward (1967), Hansen and Jorgensen (1986), Robinson (1970), Mayur et al. (1970), Mayur and Jackson (1971), Mujtaba (1989) and Mujtaba and Macchietto (1992, 1993, 1996, 1998) all minimised the batch time to yield a given amount and composition of distillate using conventional batch distillation columns. The time optimal operation is often desirable when the amount of product and its purity are specified a priori and a reduction in batch time can produce either savings in the operating costs of the column itself or permit improved scheduling of other batch operations elsewhere in a process. Mathematically the problem can be written as ... 

Using the problem formulation and solution given in Logsdon et al. (1990) the optimal design, operation and profit are shown in Table 7.6. The optimal reflux ratio profiles are shown in Figure 7.8. Period 1 refers to Task 1 and Period 2 refers to Task 2. The results in Table 7.6 and Figure 7.8 clearly show the benefit of simultaneous design and operation optimisation for multiple separation duties. The benefit has been obtained due to reduction in batch time. 

Using the above profit function, the solution of problem P2 will automatically determine the optimum batch time (tf), conversion (C), reflux ratio (r) and the amount of product (Di). However, as the cost parameters (CDh CB0, etc.) can change from time to time, it will require a new solution of the dynamic optimisation problem P2 (as outlined in Mujtaba and Macchietto, 1993, 1996), to give the optimal amount of product, optimal batch time and optimal reflux ratio. And this is computationally expensive. To overcome this problem Mujtaba and Macchietto (1997) calculated the profit of the operation using the results of the maximum conversion problem (PI) which were obtained independent of the cost parameters. 

Also Mujtaba (1997) considered the separation of binary mixtures into one distillate product of specified purity. The objectives were to find out whether it was possible to replace conventional dynamic operation of batch columns by steady state operation using continuous columns for a comparable recovery, energy consumption, operation time, productivity, etc. and to obtain optimal operating policy in terms of reflux ratio. The following strategy was considered to compare the performances of the two types of operations ... 

Because of the need to avoid mutations and maintain the superior qualities of the genetically developed strain, batch or fed-batch operations are used in most applications. Continuous culture operations, however, provide a time-invariant environment that facilitates greatly the study of a biological process in research laboratories. Moreover, some industrial operations employ continuous reactors, such as the single-cell protein facility of ICI in Billingham, England (total reactor volume of about 2,300 m3), all waste treatment processes, and others. It should be noted that it is relatively common to follow a batch process with a period of fed-batch or continuous operation. Also, in most cases batch cultivation is the optimal start-up procedure for continuous or fed-batch cultivation (Yamane et al, 1977). 

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. 

Exercise 10,2,4, Draw a careful figure for the extent of an exothermic reaction as a function of isothermal batch time for various temperatures. Then show how to find the optimal batch time and temperature in the kind of batch operations considered in this section. 

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