Thermophysical Properties Research Laboratory, Inc.

Step-Heating Apparatus

The flash technique has proven to be a fast and accurate method for determining the thermal diffusivity (and hence conductivity) of a wide range of homogeneous materials from cryogenic temperatures into the molten region. Extensions of its use for some types of insulators has been limited due to the larger temperature rise which occurs on the front face of highly insulating materials, partial in-depth absorption of the laser energy by porous or translucent samples and problems associated with rear face temperature transient measurements on such materials. There are also difficulties involved in measuring the thermal diffusivity of large-grained heterogeneous materials where the grain size is of the order of the usual sample thicknesses used. Substituting step heating for the laser pulse tends to overcome problems associated with both large-grain heterogeneous materials and many insulating materials.

The step heating apparatus with the furnace and sample assembly is shown schematically in Figure 8.. The method involves subjecting one face of a specimen into a uniform heat flux and recording the temperature responses at various locations.

A 600 Watt, quartz-iodide tungsten element bulb mounted within an aluminum parabolic reflector is the heat flux source. The reflector is cooled using both a water cooling system and force convection by an air stream. The source has been experimentally verified to be reasonably constant over 30 minutes and requires approximately 2 s to reach maximum output. The heat flux can be controlled between zero and maximum output using a Variac. The flux intensity was measured 30 cm from the heat source and found to vary less than 2% across a 5-cm diameter.

Typical rise times curves are displayed in Figure 9. Diffusivity values are determined using the temperature response data, specimen dimensions and the method of parameter estimation.

The inverse problem of solving for the thermal diffusivity in the heat equation from temperature measurements has been addressed by Beck, who has discussed the problem in detail in "Parameter Esimation in Engineering and Science", (J.V. Beck and K.J. Arnold, published 1977 by John Wiley and Sons) and developed one-dimensional numerical analysis, which was used in obtaining the present results.

In the method, the sum of squares function S

is minimized with respect to the thermal diffusivity, thus the diffusivity value is obtained which produces the best possible agreement between the experimentally measured temperatures Y(ij) and the temperatures T(ij) produced by a finite difference solution of the heat equation subjected to the measured specimen front face temperatures and interior initial conditions. In the equation, i represents time and j refers to the number of thermocouples not on the boundary. In the analysis, a Crank-Nicolson finite difference solution of the heat equation is implemented.

In addition to accounting for interior temperature measurements and allowing front face temperatures to be a function of time, the parameter estimation technique also allows sequential calculation of the sensitivity of the experiment. Sensitivity analysis produces criteria for best locations for interior thermocouples and experiment times that produce theoretical optimum estimates of the diffusivity .

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