Minimum Time Problem

A series of minimum time problems (Chapter 5) were solved at different values of q with increasing holdup for each case. Figures 3.18a and 3.18b show the minimum time solution vs. percent total holdup in the column for different mixtures at different q and Figures 3.19a and 3.19b show the corresponding optimum reflux ratio (required to get the separation in minimum time) vs. percent total holdup of the column. The results are summarized in Table 3.3 which shows, for each given separation, the optimum value of holdup to achieve the best performance out of the given column. The corresponding best minimum batch time and the optimum reflux ratio to achieve that are also presented in the table for each case. 

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 ... 

Application to Batch Distillation Minimum Time Problem... 

The minimum time problem (shown as a negative of a maximum time problem) can be written as ... 

Robinson (1970) considered an industrial 10-component batch distillation operation. The feed condition is shown in Table 5.3. The distillation column was currently producing the desired product using constant reflux ratio scheme. Table 5.4 summarises the results of the application of minimum time problem using simple model with and without column holdup. 

Table 5.2. Summary of Minimum Time Problem Using Pontryagin s Maximum Principle...

The Hamiltonian, the adjoint equations and the optimal reflux ratio correlation will be same as those in Equations P.10-P.13 (Diwekar, 1992). However, note that the final conditions (stopping criteria) for the minimum time and the maximum distillate problems are different. The stopping criterion for the minimum time problem is when (D, xq) is achieved, while the stopping criterion for the maximum distillate problem is when t, xo) is achieved. See Coward (1967) for an example problem. 

Mujtaba (1989) considered the minimum time problem with a separation task (D = 1.16 kmol, x D= 0.906). The task is same as those reported for ideal case in Table 4.8 (Chapter 4). The simulation used 4 reflux ratio levels including an initial total reflux operation. Mujtaba also used 4 reflux ratio levels to compare the simulation results. The lower and upper bounds on the reflux ratio are (0.3 and 1.0). 

Table 4.6 in Chapter 4 presents the simulation results for a quaternary batch distillation. The amount of product and the composition of key component of each cut were used by Mujtaba (1989) to formulate and solve a minimum time problem for each cut. Optimal reflux ratio in each operation step is obtained independently of other step with the final state of each step being the initial state of the next step. 

A two-level optimisation solution technique as presented in Chapter 6 and 7 can be used for a similar optimisation problem. For a given product specifications (in terms of purity of key component in each Task) and considering ReTi as the only outer level optimisation variable, the above MDO problem (OP) can be decomposed into a series of independent minimum time problem (Single-period Dynamic Optimisation (SDO) problem) in the inner level. For each iteration of the outer level optimisation, the inner-level problems are to be solved. As mentioned in the earlier chapters, the method is efficient for simultaneous design and operation optimisation especially with multiple separation duties. 

The operation strategies and the arguments presented above decomposes MDO problem (OP) into a series of three independent SDO problems as (1) Maximum productivity problem for Task 1 and (2) Minimum time problems for Tasks 2 and 3. 

Minimum time problem In this problem batch time is minimized for a specific final concentration of methyl ester. 

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