Abstract
The article considers non-stationary heat transfer in a plate under conditions of high-temperature aerosol spraying during the formation of thermal protection on this plate. To solve such a problem in a domain with a moving boundary, which increases the computational domain, a substitution is used that reduces the domain with a moving boundary to a domain with fixed boundaries. The finite cosine Fourier transform was applied to the obtained problem, for which the problem for eigenvalues and eigenfunctions with boundary conditions of the third kind was previously solved. A new analytical solution and results on the thermal state of the entire system with a complex temperature pattern at the moving boundary are obtained.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223060173/MediaObjects/12202_2023_7269_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223060173/MediaObjects/12202_2023_7269_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1134%2FS1995080223060173/MediaObjects/12202_2023_7269_Fig3_HTML.png)
REFERENCES
V. Formalev, ‘‘Modeling of heat and mass transfer in heat-shielding composite materials based on the universal law of binder decomposition,’’ High Temp. 58, 386–392 (2020). https://doi.org/10.1134/S0018151X20030050
V. Formalev, ‘‘Thermal shock waves in nonlinear solid media,’’ High Temp. 50, 744–748 (2012). https://doi.org/10.1134/S0018151X12050033
E. Kartashov and V. Kudinov, Analytical Methods of the Theory of Heat Conduction and its Applications (Editorial URSS, Moscow, 2018) [in Russian].
S. Reznik, E. Mikhailovsky, and E. Prosuntsov, ‘‘Heat and mass transfer in the chemical vapor deposition of silicon carbide in a porous carbon–carbon composite material for a heat shield,’’ J. Eng. Phys. Thermophys. 90, 314–324 (2014). https://doi.org/10.1007/s10891-017-1567-0
A. Chernyshov, ‘‘Solution of the Stefan two-phase problem with an internal source and of heat conduction problems by the method of rapid expansions,’’ J. Eng. Phys. Thermophys. 94, 95–112 (2021). https://doi.org/10.1007/s10891-021-02277-x
K. Efimov, E. Loboda, A. Ovchinnikov, A. Yakimov, and S. Gaar, ‘‘Mathematical simulation of the action of oscillations of a blunt-nosed conic body on the conjugate heat and mass transfer in its coating,’’ J. Eng. Phys. Thermophys. 94, 127–136 (2021). https://doi.org/10.1007/s10891-021-02280-2
L. Rabinskiy and El. Kuznetsova, ‘‘Analytical and numerical study of heat and mass transfer in composite materials on the basis of the solution of a Stefan-type problem,’’ Per. Tche Quim. 15 (Spec. Iss. 1), 339–347 (2018).
V. Formalev, B. Garibyan, and A. Orekhov, ‘‘Mathematical modeling of heat transfer in anisotropic half-space based on the generalized parabolic wave heat transfer equation,’’ Lobachevskii J. Math. 43, 1842–1849 (2022). https://doi.org/10.1134/S1995080222100110
V. Formalev and L. Rabinskiy, ‘‘On the second initial-boundary value problem for parabolic equation, containing mixed derivatives,’’ Lobachevskii J. Math. 40, 964–968 (2019). https://doi.org/10.1134/S1995080219070114
V. Formalev, S. Kolesnik, Ek. Kuznetsova, and L. Rabinskiy, ‘‘On the features of heat transfer in anisotropic regions with discontinuous thermal-physical characteristics,’’ Int. J. Pure Appl. Math. 111, 303–318 (2016). https://doi.org/10.12732/ijpam.v111i2.14
V. Formalev, Thermal Conductivity of Anisotropic Bodies. Analytical Methods for Solving Problems (Fizmatlit, Moscow, 2014) [in Russian].
A. Orekhov, L. Rabinskiy, G. Fedotenkov, and T. Hein, ‘‘Heating of a half-space by a moving thermal laser pulse source,’’ Lobachevskii J. Math. 42, 1912–1919 (2021). https://doi.org/10.1134/S1995080221080229
A. Orekhov, L. Rabinskiy, and G. Fedotenkov, ‘‘Analytical model of heating an isotropic half-space by a moving laser source with a gaussian distribution,’’ Symmetry 14, 650 (2022). https://doi.org/10.3390/sym14040650
G. Fedotenkov, L. Rabinskiy, and S. Lurie, ‘‘Conductive heat transfer in materials under intense heat flows,’’ Symmetry 14, 1950 (2022). https://doi.org/10.3390/sym14091950
V. Dobryanskiy, G. Fedotenkov, A. Orekhov, and L. Rabinskiy, ‘‘Estimation of finite heat distribution rate in the process of intensive heating of solids,’’ Lobachevskii J. Math. 43, 1832–1841 (2022). https://doi.org/10.1134/S1995080222100079
P. Nikitin, O. Tushavina, and A. Shkuratenko, ‘‘Calculation of heat transfer on the catalytically active surface of high-speed aircraft,’’ INCAS Bull. 11 (Spec. Iss.), 191–201 (2019). https://doi.org/10.13111/2066-8201.2019.11.S.19
L. Rabinskiy, O. Tushavina, and E. Starovoitov, ‘‘Study of thermal effects of electromagnetic radiation on the environment from space rocket activity,’’ INCAS Bull. 12 (Spec. Iss.), 141–148 (2020). https://doi.org/10.13111/2066-8201.2020.12.S.13
B. Antufiev, Y. Sun, O. Egorova, and N. Bugaev, ‘‘Mathematical modeling of the local temperature effect on the deformation of the heat-shielding elements of the aircraft,’’ Adv. Aircr. Spacecr. Sci. 1, 59–68 (2022). https://doi.org/10.12989/aas.2022.9.1.059
N. Bulychev, ‘‘Preparation of stable suspensions of ZnO nanoparticles with ultrasonically assisted low-temperature plasma,’’ Nanosci. Technol.: Int. J. 12 (3), 91–97 (2021). https://doi.org/10.1615/NanoSciTechnolIntJ.2021038033
N. Bulychev, ‘‘Obtaining of gaseous hydrogen and silver nanoparticles by decomposition of hydrocarbons in ultrasonically stimulated low-temperature plasma,’’ Int. J. Hydrogen Energy 47, 21323–21328 (2022). https://doi.org/10.1016/j.ijhydene.2022.04.288
N. Bulychev, ‘‘Synthesis of gaseous hydrogen and nanoparticles of silicon and silica by pyrolysis of tetraethoxysilane in an electric discharge under the ultrasonic action,’’ Int. J. Hydrogen Energy 47, 35581–35587 (2022). https://doi.org/10.1016/j.ijhydene.2022.08.163
N. Bulychev, ‘‘Study of interaction of surface-active polymers with ZnO nanoparticles synthesized in ultrasonically assisted plasma discharge,’’ Nanosci. Technol.: Int. J. 13, 55–65 (2022). https://doi.org/10.1615/NanoSciTechnolIntJ.2021038100
N. Bulychev, ‘‘Obtaining nanosized materials in plasma discharge and ultrasonic cavitation,’’ High Temp. 60 (Suppl. 1), S98–S126 (2022). https://doi.org/10.1134/S0018151X21040076
O. Butusova, ‘‘Surface modification of titanium dioxide microparticles under ultrasonic treatment,’’ Int. J. Pharm. Res. 12, 2292–2296 (2020).
O. Butusova, ‘‘Stabilization of carbon microparticles by high-molecular surfactants,’’ Int. J. Pharm. Res. 12 (Suppl. 2), 1147–1151 (2020).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Formalev, V.F., Garibyan, B.A. & Kolesnik, S.A. Mathematical Modeling of Heat Transfer in a Plate During Plasma Spraying of Thermal Protection on It. Lobachevskii J Math 44, 2292–2298 (2023). https://doi.org/10.1134/S1995080223060173
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223060173