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Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length

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Abstract

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank–Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.

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Abbreviations

c(r, z):

Heat capacity of functionally graded material

c 0 :

Constant value

c1, c2:

First ceramic and second ceramic

h i , h o , h L , h T :

Convection coefficients at inner, outer, lower and upper surfaces

k r (r, z), k z (r, z):

Thermal conductivity in the radial and axial directions

[k 1]e, [k 2]e, [k 3]e :

Characteristic matrices of elements

[K 3], [K]:

Global characteristic matrices

[N(r, z)]:

Matrix of linear interpolation functions

n r , n z :

Radial and axial power law exponents

m1, m2:

First metal and second metal

{q}e :

Nodal temperature vector of element

{Q}:

Global nodal temperature

r i , r o :

Inner and outer radii

T c :

Constant value

T ∞ :

Surrounding temperature

\( V_{{c_{1} }} \), \( V_{{c_{2} }} \), \( V_{{m_{1} }} \), \( V_{{m_{2} }} \):

Volume fractions of basic materials

ρ(r, z):

Mass density of functionally graded material

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Authors and Affiliations

  1. Department of Mechanical Engineering, Amirkabir University of Technology, P.O. Box 15875-4413, Hafez Ave., Tehran, Iran

    Masoud Asgari & Mehdi Akhlaghi

Authors
  1. Masoud Asgari
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  2. Mehdi Akhlaghi
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Correspondence to Mehdi Akhlaghi.

Appendix

Appendix

The matrix of linear interpolation functions is

$$ [N(r,z)] = [\begin{array}{*{20}c} {N_{i} (r,z)} & {N_{j} (r,z)} & {N_{k} (r,z)} \\ \end{array} ]^{{}} $$
(32)

where subscripts i, j, k are related to three nodes of each element. And its components are

$$ N_{i} = {\frac{{a_{i} + b_{i} r + c_{i} z}}{2A}},\quad N_{j} = {\frac{{a_{j} + b_{j} r + c_{j} z}}{2A}},\quad N_{k} = {\frac{{a_{k} + b_{k} r + c_{k} z}}{2A}} $$
(33)

where the constants a, b and c are defined in terms of the nodal coordinates as:

$$ \begin{gathered} a_{i} = r_{j} z_{k} - r_{k} z_{j} \hfill \\ a_{j} = r_{i} z_{k} - r_{k} z_{i} \hfill \\ a_{k} = r_{i} z_{j} - r_{j} z_{i} \hfill \\ b_{i} = z_{j} - z_{k} \hfill \\ b_{j} = z_{k} - z_{i} \hfill \\ b_{k} = z_{i} - z_{j} \hfill \\ c_{i} = r_{j} - r_{k} \hfill \\ c_{j} = r_{k} - r_{i} \hfill \\ c_{k} = r_{i} - r_{j} \hfill \\ \end{gathered} $$
(34)

And A is the area of the element given by

$$ A = 1/2(r_{i} z_{j} + r_{j} z_{k} + r_{i} z_{k} - r_{k} z_{j} ) $$
(35)

Vector of nodal temperature (degrees of freedom) is

$$ \{ q\}^{e} = \left\{ \begin{gathered} T_{i} \hfill \\ T_{j} \hfill \\ T_{k} \hfill \\ \end{gathered} \right\} $$
(36)

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Asgari, M., Akhlaghi, M. Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length. Heat Mass Transfer 45, 1383–1392 (2009). https://doi.org/10.1007/s00231-009-0515-8

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