DEVELOPMENT OF METHODS AND ALGORITHMS FOR ESTIMATING THE TEMPERATURE DISTRIBUTION IN THE BODY OF A RECTANGULAR PARALLELEPIPED SHAPE UNDER THE INFLUENCE OF HEAT FLOW AND THE PRESENCE OF HEAT EXCHANGE
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.07Keywords:
variational approach, thermal conductivity, heat flow, rectangular parallelepiped, heat exchange, temperatureAbstract
The article describes methods and computational algorithms for estimating the temperature
distribution law in the body of a rectangular parallelepiped shape under the influence of heat flow
and the presence of heat exchange. It is believed that one of the faces is amenable to heat flow, and
the other faces are insulated or are under the influence of the environment. To use the variational
approach, the total energy functional is calculated, taking into account the boundary conditions.
Minimizing the functional and equating it to zero, we obtain a system of linear equations, the
solution of which gives the temperature of a rectangular parallelepiped at the nodal points. Further,
substituting these nodal temperature values into the approximating function, we obtain the law of
temperature distribution in the body in the form of a rectangular parallelepiped. The temperature
distribution law is obtained by dividing a rectangular parallelepiped into one, two and three
elements. To speed up the process of calculating the temperature distribution law, an algorithm has
been developed that allows you to create a program code that increases the calculation efficiency by
an order of magnitude. This is achieved by the fact that the created code contains only a system
of linear equations, unlike the main program, which forms a general functional of full energies,
calculates derivatives of this functional and obtains a system of equations.
References
[1] Segerlind J.L., Applied finite element analysis (New York-London-Sydney-Toronto, Jonh Wiley and Song, 1976): 422.
[2] Zenkeevich O., Morgan K., Konechnyie elementyi i approksimatsiya [Finite elements and approximation] (M.: Mir, 1986): 318.
[3] Kudaykulov A., Tashev A.A., Askarova A., “Computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section”, From the journal Open Engineering (2018): 170–175.
[4] Arshidinova М., Begaliyeva К., Kudaykulov А., Tashev А., “Numerical Modeling оf Nonlinear Thermomechanical Processes in a rod of variable cross section in the presence of heat flow”, International Conference on Information Science and Control Engineering (2018): 351–354.
[5] Beck J.V., “Verification solution for partial heating of rectangular solid”, International Journal of Heat and Mass Transfer (2004): 4243–4255.
[6] Beck J.V., “Cole K. D. Improving convergence of summations in heat conduction”, International Journal of Heat and Mass Transfer (2007): 257–268.
[7] Wang X.Y., “Local fractional functional decomposition method for solving local fractional Poisson equation in steady heat-conduction problem”, Thermal Science (2016): 785–788.
[8] Isachenko V.P., Osipova V.A., Sukomel A.S., Teploperedacha: ucheb. dlya vuzov [Heat transfer: textbook for universities] (M.: Energiya 1975): 488.
[9] Kraynov A.Yu., Osnovyi teploperedachi. Teploperedacha cherez sloy veschestva: ucheb. posobie [Fundamentals of heat transfer. Heat transfer through a layer of matter: textbook] (Tomsk, 2016): 48.
[10] Gao F., Yang X.J., “Local fractional Euler’s method for the steady heat-conduction problem”, Thermal Science (2016): 735–738.
[11] Petrova L.S., “Matematicheskoe modelirovanie protsessov nagreva kusochno-odnorodnyih tel s uchetom relaksatsii teplovogo potok [Mathematical modeling of heating processes of piecewise homogeneous bodies, taking into account the relaxation of the heat flow]”, Internet-zhurnal «Naukovedenie» 9(1) (2017).
[12] Yilmazer A., Kocar C., “Heat conduction in convectively cooled eccentric spherical annuli: A boundary integral moment method”, Thermal Science (2017): 2255–2266.
[13] Ovchinnikov S.V., Teploprovodnost v pryamougolnyih tverdyih telah s vnutrennim vyideleniem tepla: ucheb. dlya vuzov [Thermal conductivity in rectangular solids with internal heat release: textbook] (M.: SGU imeni N.G. Chernyishevskogo,2015): 104.
[14] Tsega E.G., “Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates”, Journal of Applied Mathematics (2022): 8.
[15] Galanin M. P., Proshunin N.N., Rodin A.S., Sorokin D.L., “Reshenie trehmernogo nestatsionarnogo uravneniya
teploprovodnosti metodom konechnyih elementov s uchetom fazovyih perehodov [Solving the three-dimensional nonstationary heat equation by the finite element method, taking into account phase transitions]”, Preprintyi IPM im. M.V.Keldyisha 27(66) (2016). DOI:10.20948/prepr-2016-66.
[16] Goloviznin V.M., Koterov V.N., Krivtsov V.M., “Raschet uravneniya teploprovodnosti na nestrukturirovannyih
krivolineiynyih setkah [Calculation of the heat equation on unstructured curvilinear meshes]”, Zhurnal vyichislitelnoy matematiki i matematicheskoy fiziki (2011): 2075–2083.
[17] Brata S., Maduta C., Pescari S., “Steady-State Three - Dimensional Numerical Simulation of Heat Transfer for Thermal Bridges Assessment”, Politehnica University of Timisoara, Faculty of Civil Engineering, Civil Engineering and Services Department. Journal of applied engineering sciences (2016): 17–22.
[18] Zhalnin R.V., Ladonkina M.E., “Reshenie trehmernyih uravneniy teploprovodnosti s pomoschyu razryivnogo metoda Galerkina na nestrukturirovannyih setkah [Solving 3D Heat Equations Using the Discontinuous Galerkin Method on Unstructured Meshes]”, Vestn. Samarskiy gosudarstvennyiy universitet. Ser. Fiziko-matematicheskie nauki 19(3) (2015): 523–533.
[19] Povstenko Y., Kyrylych T., “Fractional heat conduction in solids connected by thin intermediate layer”, Continuum Mechanics and Thermodynamics (2019): 1719–1731.
[20] Dozhdikov V.I., Poryadin S.V., Dozhdikov K.V., “Optimizatsiya upravleniya protsessom teploprovodnosti v tverdom tele [Optimization of the control of the heat conduction process in a solid]”, Vestn. TGU 4(14) (2009): 704–705.
[21] Ivanov V.V., Karaseva L.V., “Odin iz vozmozhnyih variantov priblizhennogo resheniya zadach nelineynoy teploprovodnosti [One of the possible options for the approximate solution of problems of nonlinear heat conduction]”, Vestn. Inzhenernyiy Dona 2 (2019).
[22] Pavlov V.P., Kudoyarova V.M., “Analiz temperaturnogo polya v tverdom tele metodom splaynov [Analysis of the
temperature field in a solid by the spline method]”, Vestn. Ugatu 22(2) (2018): 10–17.
[23] Kutateladze S.S., Osnovyi teorii teploobmena [Fundamentals of the theory of heat transfer] (M.: Atomizdat, 1979): 416.
[24] Lyikov A.V., Teoriya teploprovodnosti: ucheb. posobie [Theory of thermal conductivity: textbook] (M.: Vyisshaya shkola, 1967): 600.
[25] Haji-Sheikh A., Beck J.V., “Temperature solution in multi-dimensional multi-layer bodies”, International Journal of Heat and Mass Transfer (2002): 1865–1877.
[26] Carslaw H.S., Jaeger J.C., Conduction of heat in solids. Second edition (Oxford clarendon press, 1959).
[27] Rogi´e B., “Practical analytical modeling of 3D multi-layer Printed Wired Board with buried volumetric heating sources”, International Journal of Thermal Sciences (2018): 404–415.
[2] Zenkeevich O., Morgan K., Konechnyie elementyi i approksimatsiya [Finite elements and approximation] (M.: Mir, 1986): 318.
[3] Kudaykulov A., Tashev A.A., Askarova A., “Computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section”, From the journal Open Engineering (2018): 170–175.
[4] Arshidinova М., Begaliyeva К., Kudaykulov А., Tashev А., “Numerical Modeling оf Nonlinear Thermomechanical Processes in a rod of variable cross section in the presence of heat flow”, International Conference on Information Science and Control Engineering (2018): 351–354.
[5] Beck J.V., “Verification solution for partial heating of rectangular solid”, International Journal of Heat and Mass Transfer (2004): 4243–4255.
[6] Beck J.V., “Cole K. D. Improving convergence of summations in heat conduction”, International Journal of Heat and Mass Transfer (2007): 257–268.
[7] Wang X.Y., “Local fractional functional decomposition method for solving local fractional Poisson equation in steady heat-conduction problem”, Thermal Science (2016): 785–788.
[8] Isachenko V.P., Osipova V.A., Sukomel A.S., Teploperedacha: ucheb. dlya vuzov [Heat transfer: textbook for universities] (M.: Energiya 1975): 488.
[9] Kraynov A.Yu., Osnovyi teploperedachi. Teploperedacha cherez sloy veschestva: ucheb. posobie [Fundamentals of heat transfer. Heat transfer through a layer of matter: textbook] (Tomsk, 2016): 48.
[10] Gao F., Yang X.J., “Local fractional Euler’s method for the steady heat-conduction problem”, Thermal Science (2016): 735–738.
[11] Petrova L.S., “Matematicheskoe modelirovanie protsessov nagreva kusochno-odnorodnyih tel s uchetom relaksatsii teplovogo potok [Mathematical modeling of heating processes of piecewise homogeneous bodies, taking into account the relaxation of the heat flow]”, Internet-zhurnal «Naukovedenie» 9(1) (2017).
[12] Yilmazer A., Kocar C., “Heat conduction in convectively cooled eccentric spherical annuli: A boundary integral moment method”, Thermal Science (2017): 2255–2266.
[13] Ovchinnikov S.V., Teploprovodnost v pryamougolnyih tverdyih telah s vnutrennim vyideleniem tepla: ucheb. dlya vuzov [Thermal conductivity in rectangular solids with internal heat release: textbook] (M.: SGU imeni N.G. Chernyishevskogo,2015): 104.
[14] Tsega E.G., “Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates”, Journal of Applied Mathematics (2022): 8.
[15] Galanin M. P., Proshunin N.N., Rodin A.S., Sorokin D.L., “Reshenie trehmernogo nestatsionarnogo uravneniya
teploprovodnosti metodom konechnyih elementov s uchetom fazovyih perehodov [Solving the three-dimensional nonstationary heat equation by the finite element method, taking into account phase transitions]”, Preprintyi IPM im. M.V.Keldyisha 27(66) (2016). DOI:10.20948/prepr-2016-66.
[16] Goloviznin V.M., Koterov V.N., Krivtsov V.M., “Raschet uravneniya teploprovodnosti na nestrukturirovannyih
krivolineiynyih setkah [Calculation of the heat equation on unstructured curvilinear meshes]”, Zhurnal vyichislitelnoy matematiki i matematicheskoy fiziki (2011): 2075–2083.
[17] Brata S., Maduta C., Pescari S., “Steady-State Three - Dimensional Numerical Simulation of Heat Transfer for Thermal Bridges Assessment”, Politehnica University of Timisoara, Faculty of Civil Engineering, Civil Engineering and Services Department. Journal of applied engineering sciences (2016): 17–22.
[18] Zhalnin R.V., Ladonkina M.E., “Reshenie trehmernyih uravneniy teploprovodnosti s pomoschyu razryivnogo metoda Galerkina na nestrukturirovannyih setkah [Solving 3D Heat Equations Using the Discontinuous Galerkin Method on Unstructured Meshes]”, Vestn. Samarskiy gosudarstvennyiy universitet. Ser. Fiziko-matematicheskie nauki 19(3) (2015): 523–533.
[19] Povstenko Y., Kyrylych T., “Fractional heat conduction in solids connected by thin intermediate layer”, Continuum Mechanics and Thermodynamics (2019): 1719–1731.
[20] Dozhdikov V.I., Poryadin S.V., Dozhdikov K.V., “Optimizatsiya upravleniya protsessom teploprovodnosti v tverdom tele [Optimization of the control of the heat conduction process in a solid]”, Vestn. TGU 4(14) (2009): 704–705.
[21] Ivanov V.V., Karaseva L.V., “Odin iz vozmozhnyih variantov priblizhennogo resheniya zadach nelineynoy teploprovodnosti [One of the possible options for the approximate solution of problems of nonlinear heat conduction]”, Vestn. Inzhenernyiy Dona 2 (2019).
[22] Pavlov V.P., Kudoyarova V.M., “Analiz temperaturnogo polya v tverdom tele metodom splaynov [Analysis of the
temperature field in a solid by the spline method]”, Vestn. Ugatu 22(2) (2018): 10–17.
[23] Kutateladze S.S., Osnovyi teorii teploobmena [Fundamentals of the theory of heat transfer] (M.: Atomizdat, 1979): 416.
[24] Lyikov A.V., Teoriya teploprovodnosti: ucheb. posobie [Theory of thermal conductivity: textbook] (M.: Vyisshaya shkola, 1967): 600.
[25] Haji-Sheikh A., Beck J.V., “Temperature solution in multi-dimensional multi-layer bodies”, International Journal of Heat and Mass Transfer (2002): 1865–1877.
[26] Carslaw H.S., Jaeger J.C., Conduction of heat in solids. Second edition (Oxford clarendon press, 1959).
[27] Rogi´e B., “Practical analytical modeling of 3D multi-layer Printed Wired Board with buried volumetric heating sources”, International Journal of Thermal Sciences (2018): 404–415.
Downloads
How to Cite
Tashev, A., Kazykhan, R., & Aitbayeva, B. (2023). DEVELOPMENT OF METHODS AND ALGORITHMS FOR ESTIMATING THE TEMPERATURE DISTRIBUTION IN THE BODY OF A RECTANGULAR PARALLELEPIPED SHAPE UNDER THE INFLUENCE OF HEAT FLOW AND THE PRESENCE OF HEAT EXCHANGE. Journal of Mathematics, Mechanics and Computer Science, 117(1). https://doi.org/10.26577/JMMCS.2023.v117.i1.07
Issue
Section
Applied Mathematics
