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

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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.

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Abbreviations

a :

Thermal diffusivity of sample

a w :

Thermal diffusivity of wire

b n :

Constant specified after Eq. 11

i :

Square root of  − 1

I 0 :

0 th-Order modified Bessel function of the first kind

I 1 :

1st-Order modified Bessel function of the first kind

J 0 :

0 th-Order Bessel function of the first kind

J 1 :

1st-Order Bessel function of the first kind

K 0 :

0 th-Order modified Bessel function of the second kind

K 1 :

1st-Order modified Bessel function of the second kind

L :

Half the length of the wire

m n :

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

r 0 :

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

T w :

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

α n :

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

ϕ n :

Root of Eq. 17

θ n :

Unsteady part of the solution for the nth eigenvalue

Θ n :

Steady part of the solution for the nth eigenvalue

ζ n :

Function given by Eq. 16

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Woodfield, P.L., Fukai, J., Fujii, M. et al. A Two-Dimensional Analytical Solution for the Transient Short-Hot-Wire Method. Int J Thermophys 29, 1278–1298 (2008). https://doi.org/10.1007/s10765-008-0469-y

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