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International Journal of Thermophysics

, Volume 29, Issue 4, pp 1278–1298 | Cite as

A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method

  • P. L. Woodfield
  • J. Fukai
  • M. Fujii
  • Y. Takata
  • K. Shinzato
Article

Abstract

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermal-conductivity cell. Until now, this method has depended on numerical solutions of the two-dimensional unsteady heat conduction equation to account for end effects. In order to provide an alternative and to confirm the validity of the numerical solutions, a two-dimensional analytical solution for unsteady-state heat conduction is derived using Laplace and finite Fourier transforms. An isothermal boundary condition is assumed for the end of the cell, where the hot wire connects to the supporting leads. The radial temperature gradient in the wire is neglected. A high-resolution finite-volume numerical solution is found to be in excellent agreement with the present analytical solution.

Keywords

Analytical solution Thermal-conductivity measurement Transient short hot wire 

Nomenclature

a

Thermal diffusivity of sample

aw

Thermal diffusivity of wire

bn

Constant specified after Eq. 11

i

Square root of  − 1

I0

0 th-Order modified Bessel function of the first kind

I1

1st-Order modified Bessel function of the first kind

J0

0 th-Order Bessel function of the first kind

J1

1st-Order Bessel function of the first kind

K0

0 th-Order modified Bessel function of the second kind

K1

1st-Order modified Bessel function of the second kind

L

Half the length of the wire

mn

nth Eigenvalue (Eq. 10)

q

Heat supplied per unit time per unit length of wire

Q

Heat supplied to wire per unit time per unit volume

r

Radial coordinate

r0

Radius of wire

R

Radius of hot-wire cell

s

Laplace transform parameter

t

Time

T

Temperature rise in the sample from the initial condition

Tw

Temperature rise in the wire from the initial condition

Y0

0 th-Order Bessel function of the second kind

Y1

1st-Order Bessel function of the second kind

z

Axial coordinate

Greek

αn

Root of Eq. 13

β

Constant specified after Eq. 11

Δ′

Function given by Eq. 14

λ

Thermal conductivity

λw

Thermal conductivity of wire

Λ′

Function given by Eq. 18

ϕn

Root of Eq. 17

θn

Unsteady part of the solution for the nth eigenvalue

Θn

Steady part of the solution for the nth eigenvalue

ζn

Function given by Eq. 16

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • P. L. Woodfield
    • 1
  • J. Fukai
    • 2
  • M. Fujii
    • 1
  • Y. Takata
    • 3
  • K. Shinzato
    • 1
  1. 1.Research Center for Hydrogen Industrial Use and StorageNational Institute of Advanced Industrial Science and TechnologyNishi-kuJapan
  2. 2.Department of Chemical Engineering, Graduate School of EngineeringKyushu UniversityNishi-kuJapan
  3. 3.Department of Mechanical Engineering ScienceKyushu UniversityNishi-kuJapan

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