Optimization optimal batch time

In the reaction scheme in series (sixth row in Table 2.1), the required product is often the intermediate I, and its concentration has a maximum at time t, which can be taken as the optimal batch time, When the system follows a first-order kinetics not affected by chemical equilibrium (Fig. 2.5), it can be easily shown that t depends on the values of the rate constants through the following expression ... 

The value of q, the optimal batch time with and without recycle, the percentage time savings due to recycle and the optimal amount and composition of the recycle are presented in Table 8.7 for cases wherever it is applicable. The accumulated and instant distillate composition curves with and without recycle cases are shown in Figures 8.14-8.17. These figures also show the optimal reflux ratio profiles for each case. 

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. 

Once the optimal batch time is obtained, the corresponding optimal reflux ratio and the maximum achievable conversion can be calculated directly using the polynomial equations presented in Figure 9.9 and 9.11, respectively. For frequently changed market prices, the same methodology can be applied efficiently without any difficulties to find the optimal batch time, amount of products, energy costs, etc. to calculate the maximum attainable profit. However, note the same exercise will have to be carried out for each product specification (x D). 

For a given product purity of x D = 0.70, Mujtaba and Macchietto (1997) solved the maximum profit problem for a number of cost parameters using the method described above. The results are presented in Table 9.3. For each case, Table 9.3 also shows the optimal batch time, amount of product, reflux ratio, total reboiler duty and maximum conversion (calculated using the polynomial equations). 

Case 1 of Table 9.3 is the base case. It shows the optimisation results using the cost parameters presented in Table 9.2. The maximum profit and optimal batch time obtained by optimisation shows very good agreement to those shown in Figure 9.8. The maximum profit shown in Figure 9.8 is between 3.99-4.13 ( /hr) with an optimum batch time between 12-14 hr. Each of the optimisation problems (i.e. solution of P2 with Equation 9.6) presented in Table 9.3 requires approximately 3- 4 iterations and about 3- 4 cpu sec using a SPARC-1 Workstation (Mujtaba and Macchietto, 1997). 

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. 

The overall cost optimization criterion (5.4-137) to be minimized is composed of two terms. The first, which is called proportional, is related to the yield of both C and E one needs to maximize the yield of C, Yc, while minimising the ratio of yields, Y c. Yc and Kg are nc.f/nB.o and nE./nBA, respectively, n is number of moles, and y is a factor expressing the relative weight of the two terms (y was assumed to be one). The second term, called the non-proportional or fixed cost of operation, is the reciprocal of the ratio of yield of C per batch time, s, and as such it should be minimized, p is the weighing factor (equal to 174 in the process under consideration) in this term. 

For a fixed molar ratio (ns/riAh equal to 0.05887, the temperature as applied in experiment E4, and a batch time of 347.8 dimensionless units, the feed rate of B (and thus the feed time) was optimized by computation to find tj = 323.19 dimensionless units. A run was carried out at these conditions. The data collected from this experiment were then used for re-estimation of the kinetic parameters. The new kinetic model was used to evaluate the new optimum feed rate for the same total amount of B. The optimum batch time reduced to 275.36 and the feed time to 242.75 units. Table 5.4-19 summarizes the results for three successive optimizations and re-estimations. Evidently, even a very simplified kinetic model can be successfully used in search for an optimum provided that kinetic parameters are updated based on every subsequent run carried out at the optimum conditions evaluated from the preceding set of kinetic parameters. 

Run Feed time Optimal Obtained Batch time tB Yield Yc Ye Selectivity YdYo J Improvement in J [%]... 

Marchal-Brassely et al. (1992) proposed the use of tendency modelling to optimize the batch time, the amount of initial reactants, the feed time of reactants, the temperature profile, and the feed-rate profile. The method proposed is an iterative one. Its principle is as follows ... 

The interval of the batch time is split in two equal intervals. Temperature and feed rate at the boundaries of sub-intervals are subjected to optimization together with the other variables. Temperature and feed rate between the boundaries of sub-intervals are assumed to be straight lines connecting the initial and final values. The optimum values of variables obtained in step two are taken as initial guesses for optimization. The new profiles consist of two ramps joining optimized points. 

At the start of optimization only the temperature profile, the batch time, and the feed time of G were optimized, while the other variables were kept constant. At the end all the variables specified above were relaxed and optimized. The optimization sequence is shown in Table 5.4-20. The changes in criterion J are shown in Fig. 5.4-38. 

The batch plant shown in Fig. 7.4-6 is to be optimized. The required production capacity is 11070 m per year. The cost coefficients (see Eqn. 7.3-4) are given in Table 7.4-7. The fixed processing times in the batch units, /i. r, are given in Table 7.4-8 together with initial values of processing times in the semi-continuous units, 04-( and those found by optimization. The total batch times, volumes, and costs are also given in this table. 

In continuous processes, parameter profiles might be required to be optimized through space. In batch processes, parameter profiles might need to be optimized through time. How can this be achieved ... 

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. 

The most straightforward way to operate such a process is to maintain a constant chlorine addition rate and a constant temperature. However, both the constant value of the chlorine addition rate and the fixed temperature should be optimized. The temperature of the reaction system is allowed to vary within the set temperature range, but kept constant throughout a batch cycle. The batch time is divided into twenty time... 

In cases where metals or metal ions can contaminate the products, reaction vessels fabricated from inert polymeric materials restrict that possibility. A significant example involved the reaction of maltol with aqueous methylamine to give l,2-dimethyl-3-hydroxypyrid-4-one. The product is a metal chelator employed for the oral treatment of iron overload. Consequently, it is an excellent metal scavenger but must be produced under stringent conditions that preclude metal complexation. Literature conditions involved heating maltol in aqueous methylamine at reflux for 6 h, the product was obtained in 50% yield, but required decolourisation with charcoal135. With the CMR, the optimal reaction time was 1.3 min, and the effluent was immediately diluted with acetone and the near colourless product crystallised from this solvent in 65% yield (Scheme 9.18). A microwave-based batch-wise preparation of 3-hydroxy-2-methylpyrid-4-one from maltol and aqueous ammonia was also developed. 

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. 

If the technique is to be used for the detection of antibodies in serum for example (which will be polyclonal by nature) a polyclonal antibody may be preferable in order to compete more effectively with the antibody in the serum. Batch to batch variation in polyclonal antibodies, does, however, make the establishment of the ELISA more difficult and each the assay must be optimized every time a new batch is prepared. Antibodies may be aliquoted and frozen at -20°C or lower in small, usable quantities. 

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. 

In this type of problem, the objective is to compute the optimal temperature policy maximizing the amount of a desired product concentration for a given fixed batch time subject to bounds on the reactor temperature. The problem can be written mathematically as... 

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... 

Finally, with a change in both k (—50% ko) and k (+20% Ea) in plant model, the results using the on-line optimization strategy show that the GMC controller is able to accommodate this change very well as can be seen in Fig. 8(a). Fig. 8(b) presents the performance of the EKF for estimation of k and k. Since the EKF estimates these parameters close to the true values, the mismatch is eliminated. That leads to high product C obtained at the final batch time (C = 10.2137) compared to the value of C = 8.5827 obtained from the off-line optimization strategy. 

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. 

We assume the desired product to be B. If the reaction is allowed to run for too long, the amount of undesirable C produced may be too high and the yield of B may be lower than if the batch were stopped earlier. If the activation energy of the second reaction is larger than the first, increasing reactor temperature lowers the yield of B but reduces the batch time. Therefore both the reactor temperature and the batch time must be optimized. 

The optimum operation of this fed-batch reactor involves finding the trajectory of feed versus time that maximizes (or minimizes) some economic performance criterion. Of course, the batch time is also an operating optimization variable. 

For simplicity, we restrict ourselves to having a feed flowrate that starts at some value F0 and ramps down with a constant slope S. This practical approach to trajectory optimization is recommended by Smith and Choong6 for batch processes. We want to find the values of F0 and S that achieve the desired conversion and selectivity. There will be many pairs of values that will satisfy the two criteria. Each will have a different batch time and a different amount of C produced. 

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