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.
Similar content being viewed by others
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
References
Fujii M., Zhang X., Imaishi N., Fujiwara S., Sakamoto T. (1997) . Int. J. Thermophys. 18, 327
Zhang X., Fujiwara S., Qi Z., Fujii M. (1999) . J. Jpn. Soc. Micrograv. Appl. 16, 129
Zhang X., Hendro W., Fujii M., Tomimura T., Imaishi N. (2002) . Int. J. Thermophys. 23: 1077
N. Sakoda, E. Yusibani, P.L. Woodfield, K. Shinzato, M. Kohno, Y. Takata, M. Fujii, in Proceedings 8th Asian Thermophysical Properties. Conference, Fukuoka, Japan (August 2007)
P.L. Woodfield, J. Fukai, M. Fujii, Y. Takata, K. Shinzato, Int. J. Thermophys., doi:10.1007/s10765-008-0468-z
Assael M.J., Karagiannidis L., Malamataris N., Wakeham W.A. (1998) . Int. J. Thermophys. 19: 379
Healy J.J., de Groot J.J., Kestin J. (1976) . Physica 82C: 392
M.J. Assael, C.A. Nieto de Castro, H.M. Roder, W.A. Wakeham, in Experimental Thermodynamics, Vol. III, Measurement of the Transport Properties of Fluids, IUPAC Chemical Data Series, No. 37, ed. by W.A. Wakeham, A. Nagashima, J.V. Sengers (Blackwell Scientific Publications, Great Britain, 1991), p. 163
Kierkus W.T., Mani N., Venart J.E.S. (1973) . Can. J. Phys. 51: 1182
Kannuluik W.G., Martin L.H. (1934) . Proc. Royal Soc. Lond. A, 144: 496
Kannuluik W.G. (1931) . Proc. Royal Soc. Lond. A 131: 320
Carslaw H.S., Jaeger J.C. (2003) Conduction of Heat in Solids, 2nd edn. Oxford University Press, Great Britain
Jaeger J.C. (1940) . J. Proc. Royal Soc. New South Wales 74: 342
Jaeger J.C. (1942) . J. Proc. Royal Soc. New South Wales 75: 130
Abramowitz M., Stegun I.A. (eds) (1972) Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables. Dover, New York
Kakac S., Yener Y. (1985) . Heat Conduction, 2nd edn. Hemisphere, New York, pp. 244–276
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10765-008-0469-y