$2\mathrm{.}15$ 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

$D=50mm$ and thickness $L=10mm$ and the thin

heater are at a uniform temperature of $T_i=23\mathrm{.}00C,$

while surrounded by an insulating powder. Suddenly

the heater is energized to provide a uniform heat flux

$q_o^{\text{'}\text{'}}$ on each of the sample interfaces, and the heat flux

is maintained constant for a period of time, $t_o.$ A short

time after sudden heating is initiated, the temperature at

this interface $T_o$ is related to the heat flux as

$T_o(t)-T_i=2q_o^{\text{'}\text{'}}(\frac{t}{c_{p}k}{)}^{\frac{1}{2}}$

For a particular test run, the electrical heater dissipates

$20\mathrm{.}0$ W for a period of $t_o=100s,$ and the temperature

at the interface is $T_o(60s)=26\mathrm{.}77C$ after $60$ s of heat$-$

ing. A long time after the heater is deenergized, $t{t}_0,$

the samples reach the uniform temperature of $T_o()=$

$39\mathrm{.}80C.$ The mass of each sample is $m=78$ grams.

Determine the specific heat and thermal conduc$-$

tivity of the test material. By looking at values of the

thermophysical properties in Table A$.1$ or A$\mathrm{.}2,$ identify

the test sample material.