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Temperature distributions in a periodically stratified layer with slant lamination

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Abstract

The paper deals with the two-dimensional stationary temperature distribution in a composite layer. The nonhomogenous body is assumed to be composed of periodically repeated two-layered laminae. The layering is inclined with an arbitrary angle to the boundary planes. The lower and upper boundary planes are assumed to be kept at given temperatures. The considered problem is solved within the framework of the homogenized model with microlocal parameters, where the continuity thermal conditions on interfaces are satisfied.

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Abbreviations

(x, y):

Cartesian coordinates connected with the layering (m)

\((\tilde{x},\tilde{y})\) :

Cartesian coordinates connected with the boundary (m)

T(x, y):

Temperature at point (x, y) (K)

h(x):

Shape function

\(h^{\prime } (x)\) :

Derivative of shape function

q(x, y):

Microlocal parameter

q (i) :

Heat flux vector in a layer of the ith kind, i = 1, 2 (W)

q (i) x , q (i) y :

Components of heat flux vector (W)

\(h^{*}\) :

Dimensionless thickness of layer

\((\tilde{x}^{*} ,\,\tilde{y}^{*} )\) :

Dimensionless variables \((\tilde{x},\,\tilde{y})\) determined in Eq. (4.5)

K 1, K 2 :

Coefficients of thermal conductivity of the subsequent component of the body (Wm−1 K−1)

a :

The half of length of heated range (m)

\(\tilde{K},K^{ * }\) :

Effective thermal modulus on the homogenized model with microlocal parameters (Wm−1 K−1)

α:

Angle of inclination of layering to axis \(\tilde{x}\) (rad)

δ 1, δ 2 :

Thickness of the layers being the constituents of composite (m)

δ = δ 1 + δ 2 :

Thickness of fundamental unit (m)

η = δ 1/δ :

Saturation coefficient of fundamental unit by the first kind of material (–)

θ :

Macro-temperature in homogenized model with microlocal parameters (K)

γ :

Parameter of Fourier transform

ω :

Constant determined in Eq. (3.7)

β :

Constant determined in Eq. (3.11)

ϑ 0 :

Intensity of boundary temperature for \(\left| {\tilde{x}} \right| \le a\)

\(\bar{\vartheta }\) :

Fourier transform of the boundary temperature

\(\hat{\gamma }\) :

Variable determined in Eq. (4.7)

i = 1, 2:

Kind of sublayer, i = 1 the first kind or i = 2 the second kind of the subsequent layers

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Authors and Affiliations

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

    Stanisław J. Matysiak

  2. Faculty of Mechanical Engineering, Białystok University of Technology, ul. Wiejska 45 C, 15-351, Białystok, Poland

    Dariusz M. Perkowski

Authors
  1. Stanisław J. Matysiak
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  2. Dariusz M. Perkowski
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Correspondence to Stanisław J. Matysiak.

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Matysiak, S.J., Perkowski, D.M. Temperature distributions in a periodically stratified layer with slant lamination. Heat Mass Transfer 50, 75–83 (2014). https://doi.org/10.1007/s00231-013-1225-9

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  • DOI: https://doi.org/10.1007/s00231-013-1225-9

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