DOI: https://doi.org/10.15587/1729-4061.2017.118930

### Development of the method of approximate solution to the nonstationary problem on heat transfer through a flat wall

Olexander Brunetkin, Maksym Maksymov, Oksana Maksymova, Anton Zosymchuk

#### Abstract

In the present work, we propose a method for approximate analytical solution to the nonstationary problem of heat transfer through a flat wall in the concentrated statement. In the course of the study, three issues were consistently addressed: 1. symmetrical heating of a body, 2. asymmetrical heating, 3. nonstationary heat transfer.

At the first stage, we solved in approximate analytical statement the problem on symmetrical heating of a plate. The solution obtained has an error. The availability in the scientific literature of exact analytical solution, in a distributed (one-dimensional) statement, allowed us to assess the accuracy of the obtained approximate solution. It does not exceed the limits permissible for engineering calculations. A special feature of the developed method is the possibility of its application as an integral part in solving the problems on nonstationary heat transfer.

At the second stage, we solved a problem on asymmetrical heating of a plate. By using numerical study, the character of displacement of the minimum in temperature profile for thickness of the plate was identified. This made it possible, when applying the method developed at the previous stage, to obtain a solution to the problem on asymmetrical heating of the body. A special feature of the solutions is the developed approach to determining position of the temperature minimum for thickness of the plate. Such an approach was employed as another constituent part for solving a problem on nonstationary heat transfer.

At the third stage, numerical study allowed us to identify a characteristic point of varying temperature profile and the trajectory of its motion in the process of nonstationary heat transfer. Based on these data and applying the developed method, we demonstrated the possibility of approximate analytical solution to the problem on nonstationary heat transfer through a flat wall.

In all cases, when the exact solutions were lacking, assessment of error in approximate representation was conducted by comparing with the results of numerical calculations. The error did not exceed 6 %

#### Keywords

non-stationary heat transfer, energy accumulation, flat wall, analytical calculation, approximate solution

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#### References

Brunetkin, А. I., Maksymov, M. V. (2015). Method for determining the composition of combustion gases when burned. Naukovyi visnyk Natsionalnho hirnychoho universytetu, 5, 83–90.

Carslaw, H. S., Jaeger, J. C.; Pomerantsev, A. A. (Ed.) (1964). Conduction of heat in solids. Мoscow: Nauka, 488.

Lykov, A. V. (1967). The theory of heat conduction. Мoscow: Higher School, 600.

Grysa, K., Maciag, A., Adamczyk-Krasa, J. (2014). Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems. Journal of Heat Transfer, 136 (9), 091302. doi: 10.1115/1.4027770

Hoshan, N. A. (2009). The triple integral equations method for solving heat conduction equation. Journal of Engineering Thermophysics, 18 (3), 258–262. doi: 10.1134/s1810232809030084

Zarubin, V. S., Kuvyrkin, G. N., Savelyeva, I. Y. (2018). Two-sided thermal resistance estimates for heat transfer through an anisotropic solid of complex shape. International Journal of Heat and Mass Transfer, 116, 833–839. doi: 10.1016/j.ijheatmasstransfer.2017.09.054

Damle, R. M., Ardhapurkar, P. M., Atrey, M. D. (2016). Numerical investigation of transient behaviour of the recuperative heat exchanger in a MR J–T cryocooler using different heat transfer correlations. Cryogenics, 80, 52–62. doi: 10.1016/j.cryogenics.2016.09.003

Kang, Z., Zhu, P., Gui, D., Wang, L. (2017). A method for predicting thermal waves in dual-phase-lag heat conduction. International Journal of Heat and Mass Transfer, 115, 250–257. doi: 10.1016/j.ijheatmasstransfer.2017.07.036

Karvinen, R. (2012). Use of Analytical Expressions of Convection in Conjugated Heat Transfer Problems. Journal of Heat Transfer, 134 (3), 031007. doi: 10.1115/1.4005129

Shupikov, A. N., Smetankina, N. V., Svet, Y. V. (2007). Nonstationary Heat Conduction in Complex-Shape Laminated Plates. Journal of Heat Transfer, 129 (3), 335. doi: 10.1115/1.2427073

Brunetkin, O., Maksymov, M., Lysiuk, О. (2017). A simplified method for the numerical calculation of nonstationary heat transfer through a flat wall. Eastern-European Journal of Enterprise Technologies, 2 (5 (86)), 4–13. doi: 10.15587/1729-4061.2017.96090

Profos, P.; Davydov, N. I. (Ed.) (1967). Regulirovanie parosilovyh ustanovok. Moscow: Energiya, 368.

Brunetkin, O. I., Gusak, A. V. (2015). Determining the range of variation of convective heat transfer coefficient during the combustion of alternative kinds of gaseous fuels. Odes’kyi Politechnichnyi Universytet. Pratsi, 2 (46), 79–84. doi: 10.15276/opu.2.46.2015.15

Kuznetsov, Y. N. (1989). Heat transfer in the problem of the safety of nuclear reactors. Мoscow: Energoizdat, 296.

Marcenyuk, E. V., Zeleniy, Yu. A., Reznik, S. B. et. al. (2015). Identifikatsiya granichnyh usloviy teploobmena turbiny po rezul'tatam ispytaniy [Identification of turbine heat transfer boundary conditions by using test results]. Vestnik NTU «KhPI», 41 (1150), 72–76.

Brunetkin, O., Maksymov, M., Maksymova, O., Zosymchuk, A. (2017). Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example. Eastern-European Journal of Enterprise Technologies, 5 (4 (89)), 4–11. doi: 10.15587/1729-4061.2017.109569

#### GOST Style Citations

Brunetkin, А. I. Method for determining the composition of combustion gases when burned [Text] / А. I. Brunetkin, M. V. Maksymov // Naukovyi visnyk Natsionalnho hirnychoho universytetu. – 2015. – Issue 5. – P. 83–90.

Carslaw, H. S. Conduction of heat in solids [Text] / H. S. Carslaw, J. C. Jaeger; A. A. Pomerantsev (Ed.). – Мoscow: Nauka, 1964. – 488 p.

Lykov, A. V. The theory of heat conduction [Text] / A. V. Lykov. – Мoscow: Higher School, 1967. – 600 p.

Grysa, K. Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems [Text] / K. Grysa, A. Maciag, J. Adamczyk-Krasa // Journal of Heat Transfer. – 2014. – Vol. 136, Issue 9. – P. 091302. doi: 10.1115/1.4027770

Hoshan, N. A. The triple integral equations method for solving heat conduction equation [Text] / N. A. Hoshan // Journal of Engineering Thermophysics. – 2009. – Vol. 18, Issue 3. – P. 258–262. doi: 10.1134/s1810232809030084

Zarubin, V. S. Two-sided thermal resistance estimates for heat transfer through an anisotropic solid of complex shape [Text] / V. S. Zarubin, G. N. Kuvyrkin, I. Y. Savelyeva // International Journal of Heat and Mass Transfer. – 2018. – Vol. 116. – P. 833–839. doi: 10.1016/j.ijheatmasstransfer.2017.09.054

Damle, R. M. Numerical investigation of transient behaviour of the recuperative heat exchanger in a MR J–T cryocooler using different heat transfer correlations [Text] / R. M. Damle, P. M. Ardhapurkar, M. D. Atrey // Cryogenics. – 2016. – Vol. 80. – P. 52–62. doi: 10.1016/j.cryogenics.2016.09.003

Kang, Z. A method for predicting thermal waves in dual-phase-lag heat conduction [Text] / Z. Kang, P. Zhu, D. Gui, L. Wang // International Journal of Heat and Mass Transfer. – 2017. – Vol. 115. – P. 250–257. doi: 10.1016/j.ijheatmasstransfer.2017.07.036

Karvinen, R. Use of Analytical Expressions of Convection in Conjugated Heat Transfer Problems [Text] / R. Karvinen // Journal of Heat Transfer. – 2012. – Vol. 134, Issue 3. – P. 031007. doi: 10.1115/1.4005129

Shupikov, A. N. Nonstationary Heat Conduction in Complex-Shape Laminated Plates [Text] / A. N. Shupikov, N. V. Smetankina, Y. V. Svet // Journal of Heat Transfer. – 2007. – Vol. 129, Issue 3. – P. 335. doi: 10.1115/1.2427073

Brunetkin, O. A simplified method for the numerical calculation of nonstationary heat transfer through a flat wall [Text] / O. Brunetkin, M. Maksymov, О. Lysiuk // Eastern-European Journal of Enterprise Technologies. – 2017. – Vol. 2, Issue 5 (96). – P. 4–13. doi: 10.15587/1729-4061.2017.96090

Profos, P. Regulirovanie parosilovyh ustanovok [Text] / P. Profos; N. I. Davydov (Ed.). – Moscow: Energiya, 1967. – 368 p.

Brunetkin, O. I. Determining the range of variation of convective heat transfer coefficient during the combustion of alternative kinds of gaseous fuels [Text] / O. I. Brunetkin, A. V. Gusak // Odes’kyi Politechnichnyi Universytet. Pratsi. – 2015. – Issue 2 (46). – P. 79–84. doi: 10.15276/opu.2.46.2015.15

Kuznetsov, Y. N. Heat transfer in the problem of the safety of nuclear reactors [Text] / Y. N. Kuznetsov. – Мoscow: Energoizdat, 1989. – 296 p.

Marcenyuk, E. V. Identifikatsiya granichnyh usloviy teploobmena turbiny po rezul'tatam ispytaniy [Identification of turbine heat transfer boundary conditions by using test results] [Text] / E. V. Martsenyuk, Yu. A. Zeleniy, S. B. Reznik et. al. // Vestnik NTU «KhPI». – 2015. – Issue 41 (1150). – P. 72–76.

Brunetkin, O. Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example [Text] / O. Brunetkin, M. Maksymov, O. Maksymova, A. Zosymchuk // Eastern-European Journal of Enterprise Technologies. – 2017. – Vol. 5, Issue 4. – P. 4–11. doi: 10.15587/1729-4061.2017.109569

Copyright (c) 2017 Olexander Brunetkin, Maksym Maksymov, Oksana Maksymova, Anton Zosymchuk