Maximum Distillate Problem

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). 

Kerkhof and Vissers (1978) combined the minimum time and the maximum distillate problems into an economic profit function P to be maximised. 

Application to Batch Distillation Maximum Distillate Problem... 

Referring to Simple Model (with zero column holdup) presented in section 5.5.1, a maximum distillate problem can be formulated as ... 

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. 

Logsdon and Biegler (1993) considered a binary separation of cyclohexane-toluene mixture in a conventional batch distillation column. Maximum distillate problem was considered to maximise the amount of distillate with cyclohexane purity of 0.998 molefraction. The input data for the problem is given in Table 5.7. 

Note that for a fixed operation time, t in Equation 9.1, the profit will increase with the increase in the distillate amount and a maximum profit optimisation problem will translate into a maximum distillate optimisation problem (Mujtaba and Macchietto, 1993 Diwekar, 1992). However, for any reaction scheme (some presented in Table 9.1) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for reversible or irreversible reaction scheme and is already explained in the introduction section. Note for batch reactive distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For non-reactive distillation system, of course, the maximum distillate problem has to be solved. 

Mujtaba (1997) used the maximum distillate problem to compare the performances of the two types of distillation columns (CBD and continuous). With the amount of initial charge and the feed flow rate fixed in a continuous column, the operation time (pass time) also becomes fixed. The performance measure using maximum distillate problem allows fixing of the operation time. Other types of optimisation problems such as minimum time or maximum profit problems (presented in the previous chapters) are not suitable for the purpose of comparing the performances of... 

Maximum Distillate Problem where the amount of distillate of a specified concentration for a specified time is maximized [47, 48, 49, 50, 51]. This problem can be mathematically represented as follows ... 

Example 5.6 The maximum distillate problem for separation of a binary mixture involves solution of the following problem ... 

Formulate the maximum distillate problem using the calculus of variations. Solution Since this problem contains equality constraints, we need to use the Euler-Lagrangian formulation. First, all three equality constraints (Equations 5.47 to 5.49) are augmented to the objective function to form a new objective function... 

The following maximum distillate problem illustrates the principle. 

Example 5.7 Formulate the maximum distillate problem presented in Example 5.6 using the maximum principle and compare the formulation with that obtained using variational calculus (solution for Example 5.6). 

Example 5.8 Solve the maximum distillate problem presented in Example... 

Solution Once again the Lagrange multiplier formulation of the maximum distillate problem can be written as... 

Jeffrey S. Logsdon and Lorenzo T. Biegler. Accurate determination of optimal reflux policies for the maximum distillate problem in batch distillation. Ind. Eng. Chem. Res., 32 692-700, 1993. 

---