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Two-dimensional temperature distribution in FGM sectors with the power-law variation in radial and circumferential directions

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Abstract

This study aimed at presenting a steady-state analytical solution for the two-dimensional heat conduction in a cylindrical segment made of functionally graded materials. It is acquired by taking advantage of the Fourier transform and separation of variables rather than numerical methods. Sturm–Liouville theory is employed to find the proper and adequate Fourier transformation. Continuous variations along the radial and circumferential directions based on the power-law function are taken into account, and non-homogeneous boundary conditions are applied to the problem. The obtained formulation is verified by the available solutions. Through solving an illustrative example, the temperature distribution is deliberated for a combination of boundary conditions. It is to be emphasized that mathematical robustness and generality of the solution are its primary advantage which is not often seen in the previously published literature.

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Abbreviations

a :

Inner radius

b :

Outer radius

C n, D n :

The constant coefficient order n

c p :

Specific heat capacity

F :

Fourier transformation

\(k_{{\text{R}}} ,k_{\varPhi }\) :

The coefficients of heat conductivity in both R and \(\varPhi\) directions

q″:

Heat flux (W m−2)

X :

Independent function of R

\(R, \varPhi\) :

Cylindrical segment coordinate system

\(\varTheta\) :

Temperature distribution

\(\alpha , \mu\) :

Constant coefficient (W m−2 K−1)

\(\eta , \beta , \nu\) :

Constant coefficient (W m−1 K−1)

\(\varGamma\) :

Independent function of \(\varPhi\)

\(\varrho\) :

Density

\(\varDelta_{{\text{n}}}\) :

Separation constant

\(\lambda\) :

Conductivity ratio

\(\gamma\) :

Material constant coefficient

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Acknowledgements

Dr. Jing gratefully acknowledges the financial support of the National Key Research and Development Program of China (Grant No. 2018YFB1502000), National Natural Science Foundation of China (Grant No. 51961130386), Royal Society-Newton Advanced Fellowship (Grant No. NAF\R1\191163).

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Bojnord, 945 3155111, Bojnord, Iran

    Amin Amiri Delouei & Hasan Sajjadi

  2. Mechanical Engineering Department, Shahrood University of Technology, 361 9995161, Shahrood, Iran

    Amin Emamian & Sajjad Karimnejad

  3. International Research Center for Renewable Energy, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Dengwei Jing

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  1. Amin Amiri Delouei
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  2. Amin Emamian
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Correspondence to Amin Amiri Delouei or Dengwei Jing.

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Amiri Delouei, A., Emamian, A., Karimnejad, S. et al. Two-dimensional temperature distribution in FGM sectors with the power-law variation in radial and circumferential directions. J Therm Anal Calorim 144, 611–621 (2021). https://doi.org/10.1007/s10973-020-09482-5

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