Loop Optimisation Problems

The column initialisation is only required for the first inner loop optimisation problem (described in section 6.2). The liquid composition on the plates, condenser holdup tank and in the distillate accumulator (differential variables) at time t = 0 are set equal to the fresh charge composition (xB0) to the reboiler. The DAE model equations are solved at time t = 0 to provide a consistent initialisation of all the remaining variables. The final values of all these variables at the end of the distillation task in each inner loop problem are stored and used for column initialisation for the subsequent inner loop optimisation problems. At the beginning of each task, the distillate accumulator holdup is set/reset to zero. 

---

The constraints in the outer loop optimisation problem are simple bounds on the decision variables and their gradients can be easily calculated. Mujtaba and Macchietto solved the outer loop optimisation problem using the efficient SQP technique due to Chen (1988). Mujtaba and Macchietto (1993) used 1.0E-3 and 1.0E-2 as the tolerances for the inner and outer loop optimisations respectively. These were quite tight considering the fact that all the optimisation variables and constraints were scaled within the range 0-10. 

Having design parameters fixed in the outer problem and with a specific choice of D° (discussed in section 7.2) the inner loop optimisation can be partitioned into M independent sequences (one for each mixture) of NTm dynamic optimisation problems. This will result to a total of ND = 2 NTm problems. In each (one for each task) problem the control vector m for each task is optimised. This can be clearly explained with reference to Figure 7.3 which shows separation of M (=2) mixtures (mixture 1 = ternary and mixture 2 = binary) and number of tasks involved in each separation duty (3 tasks for mixture 1 and 2 tasks for mixture 2). Therefore, there are 5 (= ND) independent inner loop optimal control problems. In each task a parameterisation of the time varying control vector into a number of control intervals (typically 1-4) is used, so that a finite number of parameters is obtained to represent the control functions. Mujtaba and Macchietto (1996) used a piecewise constant approximation to the reflux ratio profile, yielding two optimisation parameters (a control level and interval length) for each control interval. For any task i in operation m the inner loop optimisation problem (problem Pl-i) can be stated as ... 

The optimum number of plates, the optimum values of the decision variables for both outer and inner loop optimisation problems, and optimal amounts and composition of all products are shown in Table 7.2. Typical composition profiles in the product accumulator tank are shown in Figure 7.6. Bold faced mole fractions in Table 2 are the specifications (all satisfied) and underlined mole fractions are decision variables which were optimised. Although the optimum number of plates is almost close to that of the base case, the optimal total operation time is 14% lower than the base case. The profit with the optimal design and operation is 35% higher than that for the base case (calculated using the same cost model). This is obtained... 

Problem P2-3 is formally similar to P2-1 and P2-2. These are all solved using the same solution procedure. Any of the outer loop variables which have known or fixed values may be dropped from the outer loop optimisation. 

Re i, xRi, Re D2) to consider in the outer optimisation problem. At the function evaluation stage the solution (reflux profiles, column profile, duration of task) of each inner loop is stored as A, B, and C, respectively, as shown in Figure 6.3a. 

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 ... 

In this separation, there are 4 distillation tasks (NT-4), producing 3 main product states MP= D1, D2, Bf) and 2 off-cut states OP= Rl, R2 from a feed mixture EF= FO. There are a total of 9 possible outer decision variables. Of these, the key component purities of the main-cuts and of the final bottom product are set to the values given by Nad and Spiegel (1987). Additional specification of the recovery of component 1 in Task 2 results in a total of 5 decision variables to be optimised in the outer level optimisation problem. The detailed dynamic model (Type IV-CMH) of Mujtaba and Macchietto (1993) was used here with non-ideal thermodynamics described by the Soave-Redlich-Kwong (SRK) equation of state. Two time intervals for the reflux ratio in Tasks 1 and 3 and 1 interval for Tasks 2 and 4 are used. This gives a total of 12 (6 reflux levels and 6 switching times) inner loop optimisation variables to be optimised. The input data, problem specifications and cost coefficients are given in Table 7.1. 

In this problem, there are 3 outer loop decision variables, N and the recovery of component 1 from each mixture (Re1 D1B0, Re D2,BO)- Two time intervals for reflux ratio were used for each distillation task giving 4 optimisation variables in each inner loop optimisation making a total of 8 inner loop optimisation variables. A series of problems was solved using different allocation time to each mixture, to show that the optimal design and operation are indeed affected by such allocation. A simple dynamic model (Type III) was used based on constant relative volatilities but incorporating detailed plate-to-plate calculations (Mujtaba and Macchietto, 1993 Mujtaba, 1997). The input data are given in Table 7.3. 

Also with these assumptions and specifications the whole multiperiod optimisation problem shown in Figure 8.12 will now be decomposed into a series of independent dynamic optimisation problems (Figure 8.13). Referring to Figure 8.12 and Figure 8.13 the optimisation problem for Recycle Loop 1 may now be described as follows ... 

The optimisation problem is now the same as P4 (Equation 8.9) for binary mixtures. Also because of the assumptions made earlier, the formulation will automatically account for the quasi-steady state operation. The optimisation problems for all other recycle loops can be described similarly. 

The formulation of the s-constraint technique is performed as one of the objectives is assigned as the objective function while the others are constrained within specified upper limits. The selected process parameters are assigned as the decision variables of the optimisation problem. The optimiser searches over the process variables, within the feasibility and constraints regions and feeds these selected variables to the model in HYSYS. Then, it waits for the process in HYSYS to converge and then recalculate the objectives and evaluate the optimisation results. This search loop between the optimiser in Excel and the model in HYSYS continues until a global optimum point is found which represents a point on the Pareto curve. The above optimisation process is repeated for different bounds of the constrained objectives to develop the entire Pareto curve. 

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. 

For each outer loop function and gradient evaluation 4 and 14 inner loop problems were solved respectively (a total of 124 inner loop problems). For the inner loop problems 12-14 iterations for Tasks 1 and 3 and 5-7 iterations for Tasks 2 and 4 were usually required. For this problem size and detail of dynamic and physical properties models the computation time of slightly over 5 hrs (using SPARC-1 Workstation) is acceptable. It is to note that the optimum number of plates and optimum recovery for Task 1 (Table 7.2) are very close to initial number of plates and recovery (Table 7.1). This is merely a coincidence. However, during function evaluation step the optimisation algorithm hit lower and upper bounds of the variables (shown in Table 7.1) a number of times. Note that the choices of variable bounds were done through physical reasoning as explained in detail in Chapter 6 and Mujtaba and Macchietto (1993). 

The results presented in Table 8.3 are in good agreement. The small differences between the results might be due to the different accuracy set for the optimisation (see Table 8.4). Since the gradients in the two-level formulation were solved by finite difference the inner loop problems PI and P2) were to be solved very tightly (accuracy for the optimiser = 1.0E-4). Whereas, the outer loop problem (P0) of the two-level formulation and the one level problem (P4) were solved using the optimiser accuracy = 1.0E-2. 

In in-line analysis, the chemical analysis is done in situ, directly in the main process stream or reactor, using a chemically sensitive probe. A condition is that the equipment has to be placed in the plant (with consequences for maintenance and safety aspects). In this case, there still is physical contact between probe and sample. Consequently, an in-Une process-monitoring device must often deal with hostile industrial processing conditions elevated p, T, fluctuating conditions, chemically aggressive environments, electrical noise, dust, and vibrational problems. Sampling delays are very short, or non-existent for in-Une devices. The feedback and control loop can be optimised in real-time manner. However, an in-line apparatus may interfere with the... 

---