Решение задачи конвективно-радиационного нагрева (охлаждения) тел простой геометрической формы методом конечных разностей
Journal Title: Математичне моделювання - Year 2017, Vol 1, Issue 1
Abstract
SOLUTION OF THE PROBLEM FOR CONVECTIVE RADIATION HEATING (COOLING) OF BODIES WITH SIMPLE GEOMETRIC FORM BY THE METHOD OF FINITE DIFFERENCES Gorbunov A.D., Ukleina S.V. Abstract The heating or cooling of bodies under the influence of convection and radiation is, at the same time, the most common and often treatment encountered in the practice of of materials heating. The development of a simple and sufficiently accurate technique for calculating temperature fields and thermal stresses in bodies of different geometric shapes is an actual problem in the context of continuous intensification of heat exchange processes in aggregate setups. The aim of the paper is to obtain the improved schemes of finite-difference approximation of the boundary conditions for numerical integration. A finite-difference method for temperature fields and thermal stresses calculation for heating (cooling) simple bodies in the form of a plate, cylinder, or ball is simulated simultaneously by convection and radiation. Using the expansion of the function of the temperature field in the Taylor series, the Lopital rule for uncovering uncertainties, the method of Newton's tangents, the solution of the fourth-degree equation, the idea of the fictitious layer and the elementary balance method of Vanichev, we obtained improved formulas for temperatures calculation at the center and the surface of the body .According to this algorithm, a program was developed using Fortran-77 language.The testing of was carried out by comparison of the results with exact analytical solutions for the case of convective heating of bodies of simple form at Bi = const and Sk = 0. Test calculations showed that the maximal errors occur both at the initial stage and in the determination of surface temperatures. The application of the improved approximation formulas resulted in errors reduction from 5% to 2%. It was found that the circuit choice depends essentially on the value of the heat transfer coefficients - the Stark and Bio numbers. A formula for calculation of the maximum-large heat transfer coefficients was obtained, when the "direct" approximation scheme for the boundary condition on the surface could be efficiently applied. By calculations, the equality of two variants was established: Sk = ∞, Bi = 0 and Bi = ∞, Sk = 0. This fact confirmed the earlier conclusion that in the case of intense nonlinear thermal loading, the determination of the temperature field can be carried out by the linear theory of convective heating (cooling) bodies for Bi = ∞. This technique can be used for thermal calculations, heat engineering equipment, in power metallurgical and other industries. References [1] Vidin Yu.V., Ivanov V.V. Raschet temperaturnykh poley v tverdykh telakh, progrevayemykh konvektsiyey i radiatsiyey odnovremenno [Calculation of temperature fields in solids heated by convection and radion at the same time] Krasnoyarsk: KPU, 1965, 144 p. (In Russ.) [2] A.V. Kavaderov Samoilovich Yu.A. Utochneniye zakonomernostey neogranichennoy plastiny [Refinement of the regularities of an unbounded plate] IFJ, 1960. - t. III, - no. 2. - pp. 57-60. (In Russ.) [3] Gorbunov A.D. K analiticheskomu raschetu termicheskikh napryazheniy pri konvektivnom nagreve tel prostoy formy [To the analytical calculation of thermal stresses in the convective heating of bodies of simple form] Matematica moduluvannya, Dniprodzerginsk: DDТU, 2012, no 1(26), pp. 39-45. (In Russ.) [4] Gol'dfarb E. M. Teplotekhnika metallurgicheskikh protsessov [Heat metallurgical processes]- M.: Metallurgiya, 1967.- 439 p. (In Russian) [5] Samarsky A.A. Vvedeniye v teoriyu raznostnykh skhem [Introduction to the theory of difference schemes] - Moscow: Nauka, 1971. - 552 p. (In Russ.) [6] Ogurtsov AP, Mamaev L. M, Karimov I.K. Matematicheskiye metody i modeli v raschetakh na EVM [Mathematical methods and models in computer calculations] - К .: ISMO, 1997. – 192 p. (In Russ.) [7] Gorbunov A.D. Analiticheskiy raschet protsessov radiatsionnogo nagreva (okhlazhdeniya ) tel na nachal'noy stadii [Analytical calculation of objects radiant heating (cooling) processes at the initial stage]. Matematichne modelyuvannya, 2012, no. 2(27), pp. 90–94. (In Russian) [8] Gorbunov A.D, Ukleina S.V. Analiticheskoye issledovaniye nagreva tverdykh tel radiatsiyey [Analytical study of the heating of solids by radiation] Message 3. Matematichne modelyuvannya, 2015, no 2 (33), pp. 65 - 68. (In Russian)
Authors and Affiliations
А. Д. Горбунов, С. В. Уклеина
Keywords
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