We wish to examine the time temperature variation of a fluid, Tf , enclosed in a container with a heating element and a thermostat. The walls of the container are pure copper. The fluid is engine oil and its temperature Tf varies with time. The thermostat is set to cut off power from the heating element when Tf reaches 65°C and to resume supplying power when Tf reaches 55°C. Wall properties:

The inside size of the container is (0.5 m × 0.5 m × 0.5 m) The wall thickness is 0.01 m. Thus, the inside surface area As,i = 1.5 m2, outside surface area As,o = 1.5606 m2, engine oil volume V_oil = 0.125 m3, and wall volume V_wall = 0.0153 m3. The power, Q, of the heating element = 20,000 W. The inside convective heat transfer coefficient hi = 560 W/m2-C. The outside convective heat transfer coefficient ho = 110 W/m²-C. Using a lump parameter analysis (assume that the engine oil is well mixed) and the first law of thermodynamics, the governing equations describing the time temperature variation of both materials are as follows: