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

Abstract

Unlike the conventional transient hot-wire method for measuring thermal conductivity, the transient short-hot-wire method uses only one short thermalconductivity 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 twodimensional 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.

An Accelerated Two-Dimensional Unsteady Heat Conduction Calculation Procedure for Thermal-Conductivity Measurement by the Transient Short-Hot-Wire Method

“…This is a further motivation for the use of this procedure in preference to the analytical solution in Ref. [5].…”

“…For this case, the calculation time taken in this method is less than one-tenth of that taken in the 2D finite-volume method. The analytical solution [5] to the same problem (but neglecting the radial temperature gradient in the wire) is also listed in Table 1. The steadystate numerical results are all within 1 mK of the analytical solution.…”

“…It is worth mentioning that the analytical solution in Ref. [5] is most suitable for fluids with high thermal diffusivity and for large values of t. In the case of Fig. 5 for t < 1 ms, 2000 terms per eigenvalue were required for the analytical solution to converge.…”

Techniques for Accurate Resistance Measurement in the Transient Short-Hot-Wire Method Applied to High Thermal-Diffusivity Gas

“…2 are given in [2] and [13]. A somewhat complicated analytical solution [14] is also available for this problem. Since the analytical solution is more computationally expensive, routine calculations were done using the numerical procedure in [13] and verification of the accuracy of the numerical calculation was done with the analytical solution [14].…”

The heated infinite cylinder with sheath and two thermal surface resistance layers

“…A full analytical solution for a cylinder in a finite external medium, with mixed adiabatic/isothermal boundary conditions, was given by [24]. Further deviations from the considered structure may stem from over-idealization of the external medium (evaporation of moisture, convection and/or Knudsen effects may change the differential equation), the boundary conditions (radiation), or the cylinder (composite cylinders with different surface resistances in between).…”