## Multiperiod Operation Optimisation

6.1. Introduction

In batch distillation, as the overhead composition varies during operation, a number of main-cuts and off-cuts are made at the end of various distillation tasks or periods (see Chapter 3). Purities of the main-cuts are usually determined by the market or downstream process requirements but the amounts recovered must be selected based on the economic trade off between longer distillation times (hence productivity), reflux ratio levels (hence energy costs), product values, etc. Increasing the recovery of a particular species in a particular cut may have strong effects on the recovery of other species in subsequent cuts or, in fact, on the ability to achieve at all the required purity specifications in subsequent cuts. The profitable operation of such processes therefore requires consideration of the whole (multiperiod) operation.

Recall from Chapter 1 that, a single mixture (binary or multicomponent) can be separated into several products (single separation duty) and multiple mixtures (binary or multicomponent) can be processed, each producing a number of products (multiple separation duties) using only one CBD column thus leading to multiperiod operation in both cases.

For single separation duty, Diwekar et al. (1989) considered the multiperiod optimisation problem and for each individual mixture selected the column size (number of plates) and the optimal amounts of each fraction by maximising a profit function, with a predefined conventional reflux policy. For multicomponent mixtures, both single and multiple product options were considered. The authors used a simple model with the assumptions of equimolal overflow, constant relative volatility and negligible column holdup, then applied an extended shortcut method commonly used for continuous distillation and based on the assumption that the batch distillation column can be considered as a continuous column with changing feed (see Type II model in Chapter 4). In other words, the bottom product of one time step forms the feed of the next time step. The pseudo-continuous distillation model thus obtained was then solved using a modified Fenske-Underwood-Gilliland method (see Type II model in Chapter 4) with no plate-to-plate calculations. The short-cut method, also used by Diwekar et. al. (1989) and Diwekar and Madhavan (1991) requires the specification of the mole fraction of all components in each product cut in addition to that of the key component recovered in that cut. In practice it is very difficult to achieve a specification of this type with a multicomponent mixture and considerable differences in the results may be noticed compared with the case of variable non-key product composition (Diwekar and Madhavan, 1991). In fact, for a particular mixture and operating policy (reflux ratio, column pressure, etc.) the residue/distillate composition will follow well defined distillation maps (Bernot et al., 1990) and it is in general not possible to independently specify more than one distillate mole fraction. Furthermore, the shortcut method is limited to columns with large number of plates and with no holdup.

For single separation duty, Farhat et al. (1990) considered the operation of an existing column for a fixed batch time and aimed at maximising (or minimising) the amount of main-cuts (or off-cuts) while using predefined reflux policies such as constant, linear (with positive slope) and exponential reflux ratio profile. They also considered a simple model with negligible liquid holdup, constant molar overflow and simple thermodynamics, but included detailed plate to plate calculations (similar to Type III model).

For single separation duty, Al-Tuwaim and Luyben (1991) proposed a shortcut method to design and operate multicomponent batch distillation columns. Their method, however, required a great number of simulations, which must be computationally very expensive, before they could arrive at an optimum design and find an optimum reflux ratio. Further details are in Chapter 7.

For single separation duty, Bernot et al. (1991) presented a method to estimate batch sizes, operating times, utility loads, costs, etc. for multicomponent batch distillation. The approach is similar to that of Diwekar et al. (1989) in the sense that a simple short cut technique is used to avoid integration of a full column model. Their simple column model assumes negligible holdup and equimolal overflow. The authors design and, for a predefined reflux or reboil ratio, minimise the total annual cost to produce a number of product fractions of specified purity from a multicomponent mixture.

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 manipulating carefully selected decision variables. These are chosen in such a manner that the need for specifying the mole fractions of all the components in the products, as required by previous methods is avoided. For values of the decision variables fixed by the outer loop, the multiperiod operation is decomposed into a sequence of independent optimal control problems, one for each distillation task. In the inner loop, a minimum time problem is then solved for each task to generate the optimal reflux ratio values, reflux switching times, and duration of the task. The procedure permits the use of very general distillation models (as those presented in Chapter 4) described by Differential and Algebraic Equations (DAEs), including rigorous thermodynamics if desired. The model equations are integrated by using an efficient Gear's type method; the inner loop optimal control problems are solved using NLP based optimisation technique described in Chapter 5.

Mujtaba and Macchietto (1996) extended the work or Mujtaba and Macchietto (1993) to include multiple separation duties. Logsdon et al. (1990) also considered multiple separation duties using short-cut model. Some of these works are presented in Chapter 7.

In sections 6.2 to 6.4, the formulation and solution method proposed by Mujtaba and Macchietto (1993) will be presented in detail. Several example problems (involving binary and multicomponent mixtures) from Mujtaba and Macchietto are also presented to demonstrate the idea. Operational alternatives involving separations of binary and multicomponent mixtures are presented in detail in Chapter 3.

In section 6.5, the multiperiod optimisation problem formulation considered by Farhat et al. (1990) is presented with typical example problems.

6.2. Optimisation Problem Formulation- Mujtaba and Macchietto

The dynamic optimisation problem formulation is illustrated for representative multiperiod operations. The STNs in Figures 6.1 and 6.2 for binary and ternary mixtures undergoing single separation duty describe the multiperiod operations (see Chapter 3). For other networks, mixtures with larger number of components and other constraints the problem formulation requires only simple modifications of that presented in this section.

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