Optimization reflux

Thus the optimal reflux ratio for an appropriately integrated distillation column will be problem-specific and is likely to be quite different from that for a stand-alone column. 

Illustration of optimal reflux for different fuel costs. 

Fractionating towers optimal reflux ratio, heat exchange, and so forth... 

EXAMPLE 12.4 DETERMINATION OF THE OPTIMAL REFLUX RATIO FOR A STAGED-DISTILLATION COLUMN... 

In this example we illustrate the application of a one-dimensional search technique from Chapter 5 to a problem posed by Martin and coworkers (1981) of obtaining the optimal reflux ratio in a distillation column. 

Logsdon, J. S. and L. T. Biegler. Accurate Determination of Optimal Reflux Policies for the Maximum Distillate Problem in Batch Distillation. Ind Eng Chem Res 32 (4) 692-700 (1993). 

While the reduced SQP algorithm is often suitable for parameter optimization problems, it can become inefficient for optimal control problems with many degrees of freedom (the control variables). Logsdon et al. (1990) noted this property in determining optimal reflux policies for batch distillation columns. Here, the reduced SQP method was quite successful in dealing with DAOP problems with state and control profile constraints. However, the degrees of freedom (for control variables) increase linearly with the number of elements. Consequently, if many elements are required, the effectiveness of the reduced SQP algorithm is reduced. This is due to three effects ... 

A third type of operation is a trade off between the above two types of operation. Here an optimal reflux policy is chosen so that some objective function is satisfied minimum time, maximum product, maximum profit, etc.), subject to any constraints (product amount and purity) at the end of the process (Coward, 1967 Hansen and Jorgensen, 1986 Diwekar et al., 1987 Mujtaba, 1989 Farhat et al., 1990 Mujtaba and Macchietto, 1992a, 1992b Logsdon and Biegler, 1993 Mujtaba and Macchietto, 1993, 1996, 1997 Mujtaba, 1997, 1999 Sorensen and Skogestad, 1996). This type of operation is known as optimal operation. 

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. 

Kerkhof and Vissers showed that for difficult separations an optimal reflux control policy yields up to 5% more distillate, corresponding to 20-40% higher profit, than either constant distillate composition or constant reflux ratio policies. 

Robinson (1969) considered the following example problem. A binary feed mixture with an initial amount of charge, B0 = 100 kmol and composition xB0 = <0.50, 0.50> molefraction, having constant relative volatility of 2.0 was to be processed in a batch distillation column with 8 theoretical stages. The aim was to produce 40 kmol of distillate product (D) with composition (xd) of 0.5 molefraction for component 1 in minimum time (tF) using optimal reflux ratio (/ ). 

Table 5.2 summarises the results for two cases (i) constant vapour boil-up rate, (ii) variable vapour boilup rate. The initial and final time optimal reflux ratio values are shown in Table 5.2 for both cases. The optimal reflux ratios between these two points follow according to Equation P.13 for each case. See details in the original reference (Robinson, 1969). 

Case Boilup Rate Optimal Reflux Ratio at Minimum... 

Table 5.5. Implementation of Optimal Reflux Profiles in the Real Plant...

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. 

Note using a holdup model, Logsdon and Biegler also reported an optimal reflux ratio profile for the same example but it was significantly different than that obtained by using no holdup model. Refer to the original reference for further details. 

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. 

It is clear from Table 5.9 that the results obtained are in very good agreement with the objectives set for each individual optimisation problem. Table 5.9 also deary shows the advantages of optimal reflux policies over the conventional constant reflux operation. Table 5.9 shows that the time optimal control policy (variable reflux) saves about 63% of the operation time compared to that required in the simulation (Table 4.6). Even the time optimal constant reflux policy saves about 33% of the operation time compared to the original simulation... 

For single separation duty, Mujtaba and Macchietto (1993) proposed a method, based on extensions of the techniques of Mujtaba (1989) and Mujtaba and Macchietto (1988, 1989, 1991, 1992), to determine the optimal multiperiod operation policies for binary and general multicomponent batch distillation of a given feed mixture, with several main-cuts and off-cuts. A two level dynamic optimisation formulation was presented so as to maximise a general profit function for the multiperiod operation, subject to general constraints. The solution of this problem determines the optimal amount of each main and off cut, the optimal duration of each distillation task and the optimal reflux ratio profiles during each production period. The outer level optimisation maximises the profit function by... 

The above-mentioned strategy requires the solution of just 8 inner loop problems for calculating the gradients with respect to the 4 decision variables. Note, additional efficiency can be achieved by using the corresponding optimal reflux ratio profiles from the previous pass as the initial estimate of the optimisation variables for each inner loop problem. This will significantly reduce the number of iterations required for each inner loop problem, and in particular for gradient evaluation. 

Figure 6.4. Optimal Reflux Ratio Profiles for Binary Distillation. [Mujtaba and Macchietto, 1993]e...

Figure 6.12. Optimal Reflux Ratio Profiles in Different Tasks of Figure 6.10...

Case Optimal reflux ratio profile Optimal amount (per batch) (kmol) D1/D2... 

Two binary mixtures are being processed in a batch distillation column with 15 plates and vapour boilup rate of 250 moles/hr following the operation sequence given in Figure 7.7. The amount of distillate, batch time and profit of the operation are shown in Table 7.6 (base case). The optimal reflux ratio profiles are shown in Figure 7.8. It is desired to simultaneously optimise the design (number of plates) and operation (reflux ratio and batch time) for this multiple separation duties. The column operates with the same boil up rate as the base case and the sales values of different products are given in Table 7.6. 

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. 

Figure 7.8. Optimal Reflux Ratio Policies for the Base Case (N=15) and Optimal Design Case (N=20). [Logsdon et al., 1990]1...

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