Optimisation simple models

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... 

With these kinetic data and a knowledge of the reactor configuration, the development of a computer simulation model of the esterification reaction is iavaluable for optimising esterification reaction operation (25—28). However, all esterification reactions do not necessarily permit straightforward mathematical treatment. In a study of the esterification of 2,3-butanediol and acetic acid usiag sulfuric acid catalyst, it was found that the reaction occurs through two pairs of consecutive reversible reactions of approximately equal speeds. These reactions do not conform to any simple first-, second-, or third-order equation, even ia the early stages (29). 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

The result of this work with the model clearly shows that there are worthwhile savings to be had from brine optimisation particularly as the cost of implementation will be small. The plant already has a system which optimises its production schedule and the brine optimisation results can be added to the associated program in the form of a simple look-up table. 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

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. 

The input data defining column configurations, feed, feed composition, column holdup, etc. are given in Table 11.10. The reaction is modelled by simple rate equations (Table 11.10). The batch time is 12 hrs (ts). The objective of the study was to maximise the conversion (X) of the limiting reactant and to obtain the main product with purity of 0.7 molefraction by optimising the reboil ratio defined as V/L. The following optimisation problem (PI) was considered. Model type III was considered with chemical reaction. 

Dynamic sets of process-model mismatches data is generated for a wide range of the optimisation variables (z). These data are then used to train the neural network. The trained network predicts the process-model mismatches for any set of values of z at discrete-time intervals. During the solution of the dynamic optimisation problem, the model has to be integrated many times, each time using a different set of z. The estimated process-model mismatch profiles at discrete-time intervals are then added to the simple dynamic model during the optimisation process. To achieve this, the discrete process-model mismatches are converted to continuous function of time using linear interpolation technique so that they can easily be added to the model (to make the hybrid model) within the optimisation routine. One of the important features of the framework is that it allows the use of discrete process data in a continuous model to predict discrete and/or continuous mismatch profiles. 

The model I is very simple, and it is not very sensitive to the physical properties of the bed, but the values of the overall mass transfer coefficients optimised are strongly dependent from the equilibrium relation assumed and it only is able to describe the initial part of the extraction. Ke was determined by mass balance assuming a uniform distribution in solid bed. Ke values of 0.5, 0.2, and 0.6 were obtained for 7, 10 and 15 MPa. Assuming a = 3000 nAn", the mass transfer coefficients calculated with model I are of some orders of magnitude lower than those for external mass transfer coefficients. This type of models have being applied with success to the extraction of edible oils from seeds were the solute is in a... 

Burton K W C and Nickless G 1987 Optimisation via simplex part I. Background, definitions and a simple application Chem. Int. Lab. Syst. 1 135-49 Clifford P K 1983 Homogeneous semiconducting gas sensors a comprehensive model Anal. Chem. Symp. Series 17 (Washington, DC ACS) p 135... 

More efficient is a generic tree representation of the separations based on tasks. The sequencing can be formulated as a structural optimisation problem where standard techniques based on Mixed Integer Linear Programming (MILP) apply. The tasks consist of simple distillation columns, as well as of hybrids for complex column arrangements, modelled by appropriate shortcut or semi-rigorous methods. Details can be found in Doherty and Malone (2001). 

---