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

(a) Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aero-gel's thermal conductivity.

(b) The engineer designs a new apparatus for which an electric heater of diameter D = 150 mm is sandwiched between two thin plates of aluminum. The steady-state temperatures of the 5-mm-thick aluminum plates, T₁, and T₂, are measured with thermocouples. Aero-gel sheets of thickness t = 5 mm are placed outside the aluminum plates, while a coolant with an inlet temperature of T_c.i = 25°C maintains the exterior surfaces of the aero-gel at a low temperature. The circular aero-gel sheets are formed so that they encase the heater and aluminum sheets, providing insulation to minimize radial heat losses. At steady state, T, = T 2 = 55°C and the heater draws 125mA at 10 V. Determine the value of the aero-gel thermal conductivity ka'

(c) 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.

(d) If liquid water is used as the coolant with a total flow rate of m = 1 kg/min (0.5 kg/min for each of the two streams), calculate the outlet temperature of the water, T_c,o.

Heater leads Tei Coolant in Aerogel sample D -Heater Aluminum plate