Chapter 2, Problem 2.18P

An engineer desires to measure the thermal conductivity of an aerogel material. It is expected that the aerogel will have an extremely small thermal conductivity.

1. Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aerogel's thermal conductivity.
2. The engineer designs a new apparatus for which an electric heater of diameter $D=150\text{mm}$ issandwiched between two thin plates of aluminum. The steady-state temperatures of the 5-mm-thickaluminum plates, $T_1$ and $T_2,$ are measured with thermocouples. Aerogel sheets of thickness $t=5\text{mm}$ are placed outside the aluminum plates, while a coolant with an inlet temperature of $T_{c,i}=25°\text{C,}$ maintains the exterior surfaces of the aerogel at a low temperature. The circular aerogel sheets areformed so that they encase the heater and aluminum sheets. providing insulation to minimize radialheat losses. At steady state, $T_1=T_2=55°\text{C,}$ and the heater draws 125mA at Determine the value of theaerogel thermal conductivity $k_a.$
3. Calculate the temperature difference across the thickness of the 5-mm-thick aluminum plates. Comment on whether it is important to know the axial locations at which the temperatures of the aluminum plates are measured,
4. If liquid water is used as the coolant with a total flow rate of $\stackrel{.}{m}=1\text{kg/min}\left(0.5\text{kg/min}\right)$ for each of the two streams), calculate the outlet temperature of the water, $T_{c,o}.$