Lecture No. 28 - 30. CTS optimization methods

Lecture purpose: studying of optimization methods of CTS

Plan of a lecture:

1. The tasks of optimization of CTS this for the decision.

2. Optimality criterions of calculation of CTS.

3. Analytical and numerical methods of finding of an optimum.

 

Optimization, for example, according to/12/, are called the purposeful activities consisting in receipt of the best results under the corresponding conditions.

Historically, the problem of optimization arose with technical progress and emergence of the competition, i.e. producers of identical goods began to look for the production conditions allowing to issue the same goods with the minimum expenses. Searches of optimal solutions led to creation of special mathematical methods and in the 18th century the mathematical foundation of optimization (calculus of variations, numerical methods and B'day) were laid. However, to the second half of the 20th century optimization methods in many fields of science and technology were applied very seldom as practical use of mathematical optimization methods required huge computing work, which without COMPUTER was extremely difficult to be realized, and in some cases – it is impossible. Especially great difficulties arose in case of the solution of tasks of a process optimization in chemical technology because of a large number of parameters and their difficult interrelation among themselves.

In case of problem definition of optimization it is necessary:

1. Availability of the purpose of optimization. At the same time the formulation of each task of optimization shall require extreme value of only one size, i.e. at the same time two and more criteria of optimization since usually the extremum (a minimum or a maximum) of one criterion doesn't correspond to an extremum of another shan't be attributed to system.

Typical example of the wrong problem definition of optimization: "To receive the maximum performance in case of the minimum cost value". The mistake is that the task of search of an optimum of 2 sizes contradicting each other in essence is set.

The correct problem definition shall be following:

a) to receive the maximum performance in case of the set cost value;

b) to receive the minimum cost value with the set productivity;

In the first case criterion of optimization – performance, and in the second – cost value.

2. Availability of resources of optimization as which understand a possibility of the choice of values of some parameters of the optimized object. An object shall possess certain degrees of freedom - corrective actions.

3. A possibility of quantitative assessment of the optimized size as only in this case it is possible to compare effects of the choice of these or those corrective actions.

4. Usually optimized size is connected with profitability of work of the considered object (the device, the shop, the plant), therefore, the optimized option of work of an object shall be estimated by some quantitative measure – an optimality criterion.

In conclusion, it should be noted that it is accepted to differentiate tasks of static optimization for the processes proceeding in the set modes and tasks of dynamic optimization. In the first case issues of creation and implementation of optimum model of process, and in the second – tasks of creation and implementation of system of optimum control of process are resolved in case of the unsteady modes of operation.

Optimality criterion.

Optimality criterion is called quantitative assessment of the optimized quality of an object. Otherwise, the optimality criterion is the main sign on which judge that, the technological system how well functions, this process works, etc., and also, the optimization task is how well solved.

The optimality criterion is one of system exits, and, the following requirements are imposed to it:

• The optimality criterion shall be expressed quantitatively;

• The optimality criterion shall be the only thing;

• The size of an optimality criterion shall change monotonously (without gaps and jumps);

• The optimality criterion shall reflect the most essential parties of process;

• It is desirable that the optimality criterion made a clear physical sense and easily was calculated.

Based on the chosen optimality criterion the criterion function representing dependence of an optimality criterion on the parameters influencing its value is constituted. The type of an optimality criterion or criterion function is determined by a specific objective of optimization. Thus, the task of optimization comes down to finding of an extremum of criterion function.

The most general statement of an optimum task is expression of an optimality criterion in the form of an economic evaluation (performance, product cost, profit, profitability). However in private tasks of optimization when an object is a part of engineering procedure, it isn't always possible or it isn't always reasonable to allocate a direct economic indicator which completely would characterize overall performance of the considered object. Technical characteristics, indirectly estimating profitability of operation of the aggregate (contact time, a product yield, extent of transformation, temperature) can serve in such cases as an optimality criterion. For example, the optimum temperature profile, cycle duration - "reaction - regeneration" and etc. is established. However, anyway the optimality criterion has the economic nature.

Differentiate simple and difficult criteria of optimization. The optimality criterion is called simple if it is required to determine an extremum of criterion function without task of conditions for any other sizes. Such criteria are usually used in case of the solution of private tasks of optimization (for example, determination of the maximum concentration of a target product, optimum time of stay of reactionary mix in the device, etc.).

operation modes.

The optimality criterion is called difficult if it is necessary to establish an extremum of criterion function under some conditions which are imposed on some other sizes and restrictions. Thus, the procedure of the solution of a task of optimization surely includes, in addition to the choice of managing parameters, also establishment of restrictions for these parameters. Restrictions can be imposed both on technological, and on economic purposes. Differentiate the following main restrictions:

• By quantity and quality of raw materials and products (composition of raw materials, product quality, performance, etc.);

• Under the terms of technology (the device sizes, stay time, temperature of ignition and restructuring of the catalyst, etc.);

• On economic purposes;

• On labor and environmental protection;

Thus, it is necessary for the solution of a task of optimization:

1. to constitute mathematical model of an object of optimization;

2. to choose an optimality criterion and to constitute criterion function;

3. to set possible restrictions which shall be imposed on variables;

4. to choose an optimization method which will allow to find extreme values of required sizes.