Creation mathematical model of a film solar collector using evolutionary search algorithm

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Prydniprovska State Academy of Civil Engineering and Architecture

Creation mathematical model of a film solar collector using evolutionary search algorithm

Irodov V.F.,

doctor of engineering, professor

Abstract

collector dimensionless mathematical model

The construction of a mathematical model of a film solar collector based on an evolutionary search algorithm is considered. The aim of the work is to build a mathematical model based on a limited set of experimental data from a range of permissible parameters. The dimensionless complexes characterizing the work of the collector are used. The total array of experimental data is divided into two arrays - a training sequence and a test sequence. As a criterion for the adequacy of the model, the criterion of the minimum deviation of the simulation values from the experimental data was used. The model was built on the points of the training sequence and tested on the test. As an example, a numerical solution to the problem of optimizing the work of the collector is shown under restrictions on permissible parameters. To solve the problem of constructing a model and the optimization problem, an evolutionary search algorithm was used.

Keywords: solar collector, heating systems, hot water systems, experimental studies, evolutionary algorithms, dimensionless appearance, math search.

Main part

To create complex hot water supply or heating systems based on film solar collectors, it is necessary to have a mathematical model of the collector. The resulting mathematical model should be as close as possible to the physical model in a variety of parameters, which is necessary for further research and construction of a mathematical model of a common hot water supply and heating system. Film solar collectors are attracted by their simplicity of construction and the corresponding low cost. One of the possible designs of film solar collector is available in the technical solution [1].

Fig. 1 Design of solar collector

1. liquid inlet;

2. liquid outlet;

3. heat exchange surface;

4. slit liquid dispenser;

5. protrusions on the surface;

6. fluid storage;

7. thermal insulation material;

8. translucent surface; air gap;

9. three-way valve;

10. fluid pump;

11. exit from the collector;

The article [2] describes the experimental technique for this collector and gives experimental results. In this paper, the task was to construct a mathematical model of film solar collector based on experiments [2].

Table 1. The results of the experiment with a solar film collector

де: Тн.вод. (0С) - water temperature at the inlet to the solar collector;

Тк.вод. (0С) - water temperature at the outlet of the solar collector;

Тпов. (0С) - air temperature in the shade;

Н (cm) - the difference in the height of the water between the storage tank and the inlet;

h(cm) - the distance between the translucent and the sorption surfaces;

Pсон. (mW/cm2) - intensity of solar radiation;

Qвод. - the amount of energy that came into the collector;

G (gram/seс) - water consumption;

Qкін. (W) - the amount of solar energy supplied to an area equal to the collector area

As a result of an experiment with film solar collector, three dimensionless complexes of 20 elements were created: p₁ - temperature complex, p₂ - geometric complex, p₃ - complex efficiency of the device [2].

Table 2. Experimental results of film solar collector in the form of dimensionless complexes

Formulation of the problem

Based on experimental data, it is necessary to obtain a mathematical model of a film solar collector.

We will search for a mathematical model of the solar collector in the form ??3=?? (??1, ?? 2), where is the function ?? (??1, ?? 2) characterizes the efficiency of the solar collector. It is necessary to find the type of function ?? (??1, ?? 2), at which the deviation is minimized

??=?|??3????? (??1??,??2??)|????=1>??????(1)

where: p₁i, p₂i, p₃i - dimensionless complexes from experimental data.

Solving the problem

Using the experimental results in a dimensionless form Table 2, two-dimensional Fig.2

Fig. 2 Two-dimensional diagram of dimensionless complexes pt, p2, p3

Fig. 3 Three-dimensional diagram of dimensionless complexes ??1, ?? 2, ?? 3

Subsequently, the type of function was selected that reflects the dependence of ?? 3on ?? 1and ?? 2.

?? (??1,??2)= ?? 1+ ?? 2• ?? 1+ ?? 3•(1? ?? 2)+ ?? 4• ?? 12+ ?? 5•(1? ?? 2) 2 (2

The numerical coefficients a₁, a₂, a₃, a₄, a₅ were determined from the minimization condition (1).

To ensure the adequacy of the modeling object, the entire array of experimental data of (Table 1) is divided into an array of the training sequence - points 1,3,5,7,8,9,11,13,15,16,17,19 from table 1 and an array of the check sequence of the points 2,4,6,10,12,14,18,20 from table 1. The coefficients a₁, a₂, a₃, a₄, a₅ were determined from the minimization condition

And then, the constructed model with the found coefficients a₁, a₂, a₃, a₄, a₅ was checked on a test sequence by calculating

where Ј_u - relative error of the model on the test sequence.

As a result of the evolutionary search for the coefficients, the function is obtained:

?? (??1;??2)=0,51?0.295•??1+0.0027•(1? ?? 2)+3.5• ?? 12?0.23•(1? ?? 2) 2 (5

For many points of the experimental data, the average error of the model is ?? = 0.1267716, and for the points of the test sequence, the average error is ?? = 0.111678. These results indicate sufficient adequacy of the found model for actual data. The following problem was solved as an example of using the constructed model. It was required to find f(p₁`, p₂) max by the evolutionary search method under the conditions:

1???1?0.3;

0.0017???2?0.0043

Figure 5 shows the evolutionary search for a solution to the indicated optimization problem of a film solar collector.

Fig. 5 Graph of the search for points of maximum efficiency of the solar collector, by the method of evolutionary search

At the 20th step of iteration, a complete match was obtained across all branches of the evolutionary search, with p₁ = 0.3; p₂ = 0.0017, which corresponds to max p₁ and min p₂ from the allowable range. The value of the function /(p₁; p₂) at these points is 0.62191, i.e. solar collector efficiency 62%.

A mathematical model of the solar collector of the film type is built. An algorithm and program code for evolutionary search is developed, with the help of which the coefficients and accuracy of the mathematical model are calculated. The results were tested on training and test arrays. The mathematical function has been optimized, as a result of which the points found with the maximum efficiency of the solar collector.

References

1. Chinn D.A., Irodov V.F., Chemoivan A.A. Experimentalnyi doslidgenya sonyachnogo kolektora plivkovogo tipu [Experimental research of a film type solar collector]. // Vcheni zapiski Tavriyskogo nacionalnogo universitetu im. Vernadskogo - Kiev, 2019. №5. - P. 194-197.

2. Stratan F.I., Irodov V.F. Evolyutsionnye algoritmy poiska optimalnykh resheniy [Evolutionary algorithms for finding optimal solutions]. Kishinev, 1984. - P.16-30.

3. Irodov V.F. O postroenii i shodimosti evolyutsionih algoritmov samoorganizacii sluchaynogo poiska [About the construction and convergence of evolutionary algorithms of selforganization and random search]. // Avtomatika. - Kiev, 1987. - №4. - P.34-43.

4. Irodov V. Self-organization methods for analysis of non-linear systems with binary choice relations // Journal Systems Analysis Modelling Simulation. Gordon and Breach Science Publishers, Inc. Newark, NJ, USA. Vol. 18-19, 1995. - 203 - 206 pp.

5. Irodov V.F., Chirm D.A., Dudkin K.V., Chornoivan A.A. Pat. Sonyachniy kolektor z teploobminom u plivci ridini: pat.133072 Ukrayina (UA): MPK F24S 10/00 [Solar collector with heat transfer in a liquid film: pat. 133072 Ukraine (UA): IPC F24S 10/00]. 2018.

6. Emmerich M., Deutz A. Multicriteria Optimization and Decision Making [Virtual Resource] LIACS Master Course. -2006. - 84 p. - Access Mode: URL: http://natcomp. liacs.nl/MOB/material/mco4.pdf. - T itle from Screen. - Date of Access: 28 September 2015.

7. Ivanov S.Y., Ray A.K. Multiobjective optimization of industrial petroleum processing units using Genetic algorithms. XV International Scientific Conference «Chemistry and Chemical Engineering in XXI century» dedicated to Professor L.P. Kulyov / University of Western Ontario, Department of Chemical and Biochemical Engineering - Canada, 2014. - 7-14 p.

8. Zitzler E. Thiele L. An evolutionary algorithm for multiobjective optimization the strength Pareto approach. - Zurich: TIK - Report, 1998. - 43p.

9. Bukatova I.L. Evoltsionnoe modelirovanie i ego prilogeniya [Evolutionary modeling and its applications] - M.: Science, 1979. - 232 p.

10. Ivahnenko A.G., Zaychenko U.P., Dimitrov V.D. Prinyatie resheniy na osnove samoorganizatsii [Self-decision making] - M.: Sov. radio, 1976. - 280 p.

11. Ivahnenko A.G. Systemi evristicheskoy samoorganizacii v tehnicheskoy kibernetike [Heuristic self-organization systems in technical cybernetics] - K.: Technics, 1971. - 392 p.

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