Chapter 2, Problem 2.23P

A method for determining the thermal conductivity k and the specific heat $c_p$ of a material is illustrated in the sketch. Initially the two identical samples of diameter $D=60\text{mm}$ and thickness $L=10\text{mm}$ and the thin heater are at a uniform temperature of $T_i=23.00°\text{C,}$ while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux $q_o^n$ on each of the sample interfaces, and the heat flux is maintained constant for a period of time, $\Delta t_o.$ A short time after sudden heating is initiated, the temperature at this interface $T_o$ is related to the heat flux as
$T_o\left(t\right)-T_i=2q_o^n{\left(\frac{t}{\pi \rho c_{p}k}\right)}^{1/2}$
For a particular test run, the electrical heater dissipates 15.0 W fora period of $\Delta t_o=120\text{s,}$ and the temperature at the interface is $t_o\left(30\text{s}\right)=24.57°\text{C}$ after 30s of heating. A long time after the heater is deenergized, $t\gg \Delta t_o,$ the samples reach the uniform temperature of $T_o\left(\infty \right)=33.50°\text{C}\text{.}$ The density of the sample materials, determined by measurement of volume and mass, is $\rho =3965{\text{kg/m}}^{\text{3}}.$

Determine the specific heat and thermal conductivity of the test material. By looking at values of the thermophysical properties in Table A.1 or A.2, identify the test sample material.