Calculate the Thermal Conductivity of a Bar of Material

Question

The following arrangement is used to determine the thermal conductivity of a bar of material. At one end of the bar, an electrical heater is attached. The horizontal surfaces of the bar are covered in insulation. At the other end of the bar, a heat sink is located. The heat sink maintains the respective surface at 20°C. Three thermocouples are located along the length of the bar, one at the mid-point. At steady-state, the temperatures are as indicated below.

The bar has a square cross-section of side 2 cm. Note that the second thermocouple is located at the mid-point along the length of the bar. Only the metal surface is connected to the electrical heater. All the heat flow enters the bar at the metal surface.

(a) What do you understand by steady-state, given that initially, before the heated is switched on, the entire bar is at a temperature of 20°C?

(b) Using Excel (preferably), sketch the temperature profile along the bar. What can you deduce from this temperature profile? What does this tell you about the insulation?

(c) Write down the integrated form of Fouriers Law for plane-wall conduction. Explain why, for the above situation. it is not appropriate to use the integrated form of Fourier's Law to determine the thermal conductivity of the metal.

(d) Calculate the heat-transfer area (A) for plane-wall conduction through the bar.

(e) Using your graphical plot of the profile. estimate the thermal conductivity (k) of the metal bar. You are advised to plot the temperature data as an Excel or Mathcad file. By comparing your calculated value of k with published data, what do you estimate to be the material of the bar?

(f) Estimate the rate of heat loss through the insulation.
Hint: This is the difference between the value of Q entering the bar at the heater and the value of Q leaving at the heat sink.

(g) How do you define pettect insulation? Sketch the temperature profile along the bar if the present insulation were replaced with a perfect insulation. Also, for perfect insulation, sketch the temperature profile across the bar (in a vertical direction. as shown by the red line).

Summary

The question belongs to Chemical Engineering and the question is about thermal conductivity of a material. The material is being heated. Questions such as steady state, temperature profile, insulation, integrated form of Fouriers Law, etc have been answered clearly in the solution.