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Analysis of Transient Heat Conduction in a Hollow Cylinder Using Duhamel Theorem

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Abstract

The objective of this paper is to derive the mathematical model of two-dimensional heat conduction at the inner and outer surfaces of a hollow cylinder which are subjected to a time-dependent periodic boundary condition. The substance is assumed to be homogenous and isotropic with time-independent thermal properties. Duhamel’s theorem is used to solve the problem for the periodic boundary condition which is decomposed by Fourier series. In this paper, the effects of the temperature oscillation frequency on the boundaries, the variation of the hollow cylinder thickness, the length of the cylinder, the thermophysical properties at ambient conditions, and the cylinder involved in some dimensionless numbers are studied. The obtained temperature distribution has two main characteristics: the dimensionless amplitude (\(A\)) and the dimensionless phase difference (\(\varphi \)). These results are shown with respect to Biot and Fourier and some other important dimensionless numbers.

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Abbreviations

\(c\) :

Specific heat capacity (\(\text{ J }\,{\cdot }\,\text{ kg }^{-1}\,{\cdot }\,\text{ K }^{-1}\))

\(h\) :

Convective heat transfer coefficient (\(\text{ W }\,{\cdot }\,\text{ m }^{-2}\,{\cdot }\,\text{ K }^{-1}\))

\(k\) :

Thermal conductivity (\(\text{ W }\,{\cdot }\,\text{ m }^{-1}\,{\cdot }\,\text{ K }^{-1}\))

\(r\) :

Radius (m)

\(l\) :

Length (m)

\(T, t\) :

Time (s)

\(\overline{t}\) :

Dimensionless time

\(\overline{r}\) :

Dimensionless radius

\(\overline{z}\) :

Dimensionless length

\(m\) :

Dimensionless thickness

\(x\) :

Ratio of outer radius to the length of hollow cylinder

\(A, A_{jns}\) :

Dimensionless amplitude of temperature

\(Bi\) :

Biot number

\(Fo\) :

Fourier number

\(M\) :

Defined in Eq. 37

\(Q\) :

Stored heat (J)

\(R\) :

Resistance

\(Z\) :

Impedance

\(X_{c}\) :

Reactance

\(\alpha \) :

Thermal diffusivity (\(\text{ m }^{2}{\cdot } \text{ s }^{-1}\))

\(\theta \) :

Temperature field

\(v,\omega , \mu \) :

Eigenvalues

\(\varphi , \varphi _{jns} \) :

Phase difference

\({\varPhi },\eta \) :

Eigen functions

\(\alpha _\mathrm{{n}}^{(\mathrm{{i}})}, \alpha _\mathrm{{n}}^{( \mathrm{{o}})}\) :

Defined by Eq. 18

\(\beta _\mathrm{{n}}^{(\mathrm{{o}} )}, \beta _\mathrm{{n}}^{(\mathrm{{i}})}\) :

Defined by Eq. 18

\(C_{n}^{(i)}, C_{n}^{(o)}\) :

Defined by Eq. 20

\(\varDelta \) :

Defined by Eq. 19

\(\gamma _{jni}, \gamma _{jno}\) :

Defined by Eq. 35

\(\tau \) :

Time (s)

i:

Inner

o:

Outer

\(0\) :

Steady-state

\(1\) :

Transient-state

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Authors and Affiliations

  1. Department of Mechanical Engineering, Malek Ashtar University of Technology (MUT), Tehran, Iran

    H. Fazeli

  2. Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    M. A. Abdous & N. Moallemi

  3. Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    H. Karabi

  4. Department of Mechanical Engineering, Tehran University of Technology, Tehran, Iran

    M. Esmaeili

Authors
  1. H. Fazeli
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  2. M. A. Abdous
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  3. H. Karabi
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  4. N. Moallemi
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  5. M. Esmaeili
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Correspondence to M. A. Abdous.

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Fazeli, H., Abdous, M.A., Karabi, H. et al. Analysis of Transient Heat Conduction in a Hollow Cylinder Using Duhamel Theorem. Int J Thermophys 34, 350–365 (2013). https://doi.org/10.1007/s10765-013-1418-y

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  • DOI: https://doi.org/10.1007/s10765-013-1418-y

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