Optimisation Problem Formulation and Solution

Mujtaba and Macchietto (1996) formulated a two-level optimisation problem with Dd and D° optimised in an outer loop and U optimised in the inner loop, using an extension of the method of Mujtaba and Macchietto (1993). The outer loop problem can be written as  

Equality constraints h(D°, D°) = 0 may include, for example, a ratio between the amounts of two products, etc. Inequality constraints g(u, D°) 0 for the overall operation include Equations 7.14-7.18 (the first two of which are easily eliminated when m and H are specified) and possibly bounds on total batch time for individual mixtures, energy utilisation, etc. Any variables of D° and D° which are fixed are simply dropped from the decision variable list. Here, Strategy II was adopted for the multiple duty specification, requiring B0 to be fixed a priori. Similar considerations hold for V, the vapour boilup rate. The batch time is inversely proportional to V for a specified amount of distillate. Also alternatively, for a given batch time, the amount of product is directly proportional to V. This can be further explained through Equations 7.24-7.26)  

Batch time, t (hr) for a given amount of distillate ( S D, kmol)  

An increase of V will increase LD (Equation 7.25) and vice versa for a given reflux ratio profile r. An increase in LD will decrease batch time t and vice versa 

As the operating and capital costs grow with V by an economic factor less than 1, a column with large V will always be profitable (Diwekar et al., 1989) and the problem becomes unbounded (also proven in Logsdon et al., 1990). Hence V is also fixed a priori. This leaves just N (the number of internal ideal separation stages or plates) as the only design variable to be optimised (UP = N ). Out of all possible operation decision variables, it is common to specify the mole fraction of key components in main-cuts and sometimes some recoveries or amounts for off-cuts. Assuming NSP such specifications are made, there are (NV + 1 - NSP) outer optimisation problem decision variables. 

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In this chapter first, the optimisation method of Al-Tuwaim and Luyben (1991) for single separation duty is presented. Then the optimisation problem formulation and solution considered by Mujtaba and Macchietto (1996) is explained. Finally, the optimisation problem formulations considered by Logsdon et al. (1990) and Bonny et al. (1996) are presented. 

Mujtaba and Macchietto (1996) presented a general design and operation optimisation problem formulation and solution for single and multiple separation duties. 

Here, a dynamic optimisation (also known as optimal control) problem formulation and solution proposed by Morison (1984) based on Sargent and Sullivan (1979) is presented. The process model can be described by a system of DAEs (model types III, IV and V presented in Chapter 4) as ... 

Design and Operation Optimisation for Single and Multiple Separation Duties Problem Formulation and Solution... 

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. 

In this section optimal operation problem of BREAD processes is presented as a proper dynamic optimisation problem incorporating a detailed dynamic model (Type V- CMH). The problem formulation and solution exploit the methods developed for non-reactive batch distillation by Mujtaba and Macchietto (1991, 1993,1998). These methods are also discussed in Chapters 5 and 6. 

The performance criteria of a batch distillation column can be measured in terms of maximum profit, maximum product or minimum time (Mujtaba, 1999). In distillation, whether batch, continuous or extractive, purity of the main products must be specified as it is driven by the customer demand and product prices. The amount of product and the operation time can be dictated by economics (maximum profit) or one of them can be fixed and the other is obtained (minimum time with fixed amount of product or maximum distillate with fixed operation time). The calculation of each of these will require formulation and solution of optimisation problems. A brief description of these optimisation problems is presented below. Further details will be provided in Chapter 5. 

Thus the multiperiod optimisation problem is formulated as a sequence of two independent dynamic optimisation problems (PI and P2), with the total time minimised by a proper choice of the off cut variables in an outer problem (PO) and the quasi-steady state conditions appearing as a constraint in P2. The formulation is very similar to those presented by Mujtaba and Macchietto (1993) discussed in Chapter 5. For each iteration of PO, a complete solution of PI and P2 is required. Thus, even for an intermediate sub-optimal off cut recycle, a feasible quasi-steady state solution is calculated. The gradients of the objective function with respect to each decision variable (Rl or xRl) in problem PO were evaluated by a finite difference scheme (described in previous chapters) which again requires a complete solution of problem PI and P2 for each gradient evaluation (Mujtaba, 1989). 

Mujtaba (1999) considered the conventional configuration of BED processes for the separation of binary close boiling and azeotropic mixtures. Dynamic optimisation technique was used for quantitative assessment of the effectiveness of BED processes. Two distinct solvent feeding modes were considered and their implications on the optimisation problem formulation, solution and on the performance of BED processes were discussed. A general Multiperiod Dynamic Optimisation (MDO) problem formulation was presented to obtain optimal separation of all the components in the feed mixture and the recovery of solvent while maximising the overall profitability of the operation. 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

Referring to Figure 8.2 and given a batch charge (BO, xB0)> a desired amount of distillate DI of specified purity x D1 and final bottom product B2 of specified purity x b2 Mujtaba (1989) determined the amount and composition of the off-cut (Rl, x R1) and the reflux rate policy r(t) which minimised the overall distillation time. In this formulation instead of optimising Rl, xR1) the mixed charge to the reboiler (Bc, xBC) was optimised and at the end of the solution the optimal (Rl, x RI) was evaluated from the overall balance around the mixer in Figure 8.2. The dynamic optimisation problem is formulated as ... 

The algorithmic treatment depends on the architecture of the flowsheeting system. In Equation-Oriented mode, the approach consists of solving all the equations describing the problem simultaneously. In Sequential-Modular approach the mathematical solution must take into account the convergence of units and tear streams, as well as of all design specifications. Supplementary equations must be added, so that the general formulation of the optimisation problem (3.10) becomes ... 

In this paper an industrial semibatch polymerisation process is considered. In order to guarantee the product quality particularly controlled reaction conditions are necessary. The general aim of this work is to ascertain optimal state and control profiles and to develop a model-based control scheme. As a first step, this paper introduces the dynamic model, which is validated with experimental data, and describes the optimisation approach. An aim of the work is to assess the possibilities of the commercial flowsheet simulator CHEMCAD in the optimisation of the performance of semibatch polymerisation processes. Finally the formulation of the mathematical optimisation problem, solution strategies and their implementation in CHEMCAD are discussed. 

A problem is given, and a way to evaluate a proposed solution to it exists in the form of a fitness function. Then a population of individuals made by random guesses to the problem solution is initialised. These individuals are candidate solutions, also known as the particles, hence the name particle swarm. Each particle has a very simple memory of its personal best solution so far, called p The global best solution for each iteration is also found and labelled g est - It is the best value, obtained so far by any particle in the population. Each particle makes this information available to their neighbours, analogous to social interaction. Once set in motion each particle is moved a certain distance from its current location, influenced by a random amount by the p beg/ and g test values to improve fimess. During this iterative process, the particles gradually settle down to an optimum solution. In a minimisation optimisation problem, problems are formulated so that best simply means the position with the smallest objective value. 

The steady state disturbance rejection problem is formulated within an optimisation framework. Given a set of structural and equipment design variables at their optimal values as calculated from the solution of Eq. (1) (e.g. number of stages in a distillation column, total volume of a reactor), a set of set points for the controlled variables, ysp, a set of optimal steady state values for the manipulated variables, u, a set of model parameters and disturbances, e, and symmetric weighting matrices for the deviations from target values for the controlled and manipulated variables, Wy and W , respectively, the following optimisation problem is constructed ... 

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... 

It is quite obvious that a two level optimisation formulation can be very expensive in terms of computation time. This is due to the fact that for any particular choice of R1 and xRi a complete solution (sub-optimal) of the two distillation tasks are required. The same is true for each gradient evaluation with respect to the decision variables (B7and xRj). Mujtaba (1989) proposed a faster one level dynamic optimisation formulation for the recycle problem which eliminates the requirement to calculate any sub-optimal or intermediate solution. In this formulation the total distillation time is minimised directly satisfying the separation requirements for the first distillation task as interior point constraints and for the second distillation task as final time constraints. It was found that the proposed formulation was much more robust and at least 5 times faster than the classical two level formulation. 

The problem is formulated as a multie-objective optimisation model and is solved using Goal Programming technique. The optimal solution includes one oil boiler (B3), one... 

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