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On the quasi-stationary problem of heat conduction for a homogeneous half-space with composite coating

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Abstract

The quasi-stationary problem of heat conduction for a homogeneous half-space and composite layer being a coating is considered in the paper. Heat fluxes distributed in a strip and moving on the boundary plane of the considered body with a constant velocity are taken into consideration. The composite layer being the coating is assumed to be composed of four repeated periodical beams with rectangular cross sections. The homogenized model with microlocal parameters is used for the description of the composite layer. Some special cases of an influence of composite components and their locations on the temperature field are analysed.

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Abbreviations

(x, y):

The Cartesian dimensionless moving coordinates related to dimension c

2c :

The width of the heating strip (m)

t :

The time (s)

\(a_1^*, a^{*}, b_1^*, b^{*}\) :

The dimensions of the unit cell of coating (m)

\(T_0 (x, y)\) :

Temperature in the homogeneous substrate at the point (x, y) (K)

\(T_1 (x, y)\) :

The macrotemperature in the composite coating (macrotemperature is an approximation of the temperature) at the point (x, y) (K)

\(\mathbf{h}_i \) :

Heat flux vector in the ith kind of component of the composite coating (\(\hbox {W m}^{-2}\))

\(\mathbf{q}\) :

Heat flux vector (\(\hbox {W m}^{-2}\))

\(K_0\) :

Coefficient of thermal conductivity of the substrate (\(\hbox {W m}^{-1}\,\hbox {K}^{-1}\))

\(K_i\) :

The thermal conductivity coefficients of the coating components (\(\hbox {W m}^{-1}\,\hbox {K}^{-1}\)), \(i=1,2,3,4\)

\(c_0\) :

The specific heat of the substrate (\(\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1}\))

\(c_i\) :

The specific heats of the components of a unit cell (\(\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1}\)), \(i=1,2,3,4\)

\(K_x , K_y\) :

The effective thermal moduli in the homogenized model with microlocal parameters (\(\hbox {W m}^{-1}\,\hbox {K}^{-1}\))

V :

The constant velocity of heating sources (\(\hbox {m}\,\hbox {s}^{-1}\))

\({\hbox {Pe}}_0\) :

The Peclet number for the substrate

\({\hbox {Pe}}_i\) :

The Peclet number of the ith component of the coating, \(i=1,2,3,4\)

\(\rho _0\) :

The mass density of the substrate (\(\hbox {kg}\,\hbox {m}^{-3}\))

\(\rho _i\) :

The mass densities of the components of the unit cell (\(\hbox {kg}\,\hbox {m}^{-3}\)), \(i=1,2,3,4\)

\(\eta _1 =a_1 /a\) :

The saturation coefficient of fundamental unit in the direction of x (–)

\(\eta _2 =b_1 /b\) :

The saturation coefficient of fundamental unit in the direction of y (–)

\(\varDelta _i\) :

The regions occupied the the ith kind of materials, \(i=1,2,3,4\)

i :

The number of the ith component of the coating, \(i=1,2,3,4\).

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Acknowledgements

This investigation was supported under the programme of the Ministry of Science and Higher Education under the name “Regional Initiative of Excellence” in 2019–2022 (project number 011/RID/2018/19; financing amount: 12 million PLN).

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Authors and Affiliations

  1. Department of Mechanics and Applied Computer Science, Faculty of Mechanical Engineering, Białystok University of Technology, 45C Wiejska Str., 15-351, Białystok, Poland

    Roman Kulchytsky-Zhyhailo & Dariusz M. Perkowski

  2. Institute of Hydrogeology and Engineering Geology, Faculty of Geology, University of Warsaw, Al. Żwirki i Wigury 93, 02-089, Warsaw, Poland

    Stanisław J. Matysiak

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  1. Roman Kulchytsky-Zhyhailo
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Correspondence to Stanisław J. Matysiak.

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Kulchytsky-Zhyhailo, R., Matysiak, S.J. & Perkowski, D.M. On the quasi-stationary problem of heat conduction for a homogeneous half-space with composite coating. Acta Mech 231, 1241–1251 (2020). https://doi.org/10.1007/s00707-019-02591-9

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