## Thermal conductivity problems

Data for these problems:
Thermal conductivities (Wm-1K-1):
Copper 385
Air 0.0241
Brick 0.6
Glass 0.8
Iron 80

1. Calculate the rate of flow of heat down the following specimens if the temperature difference between the two ends of the specimens is maintained at 100 oC: (a) a copper rod 10 cm long with a diameter of 0.4 cm;
(b) an iron rod of the same dimensions;
(c) a 'slab' of air 5 cm thick and with an area of 0.5 m2;
(d) a brick 20 cm long with ends 10 cm by 7 cm.
Assume in each case that no heat energy is lost from the sides of the specimen.

2. In a domestic heating system, a room is warmed by a 'radiator' through which water passes at a rate of 0.12kg s-1. The steady-state difference between the inlet and outlet temperatures of the water is 6.0 K.

The radiator is made of iron of thermal conductivity 80 W m-1 K-1 and has an effective surface area of 1.5 m2 with walls 2.0 mm thick.
(a) At what rate is heat supplied to the room?
(b) What is the mean temperature difference between the inner and outer surfaces of the radiator walls?
[Specific heat capacity of water = 4.2 x 103 J kg-1 K-1]

3. The Earth has a crust (thin compared with the radius of the Earth) of thermal conductivity approximately 2 W m-1 K-1. On average the temperature of the crust increases by about 20 oC for each kilometre below the surface.
Estimate the total rate at which heat leaves the Earth's surface by conduction through the crust. Take the radius of the Earth to be 6.4 x 106 m.)

4. An insulated wire whose resistivity is 5 x 10-7 Ωm and diameter 1 mm carries a current of 5 A. The insulation material is 1 mm thick and has a coefficient of thermal conductivity of 0.2 W m-1 K-1 Find the temperature difference between the inner and outer surfaces of the insulating material when steady state has been reached.

5. Explain how the temperature varies along a uniform metal bar which is:
(a) unlagged,
(b) lagged all along its length.
Assume that heat is supplied at a constant rate to one end and that a steady state has been reached.

In case (b) suppose heat flows along the bar, which is of 20 cm2 cross-section, at the rate of 600 J s-1
If the temperature at a certain distance (x) from the heated end is 60 oC, where will the temperature be 40 oC? The coefficient of thermal conductivity of the bar is 400 W m-1 K-1.)

6. Bars of copper and iron of equal length are welded together end to end and lagged.
(a) Determine the tem¬perature of the interface when the free end of the copper is at a temperature of 100 oC and the free end of the iron is at 0 oC and conditions are steady.

(b) Sketch the variation down the two bars of (a) the rate of flow of heat through the bars, and (b) the temperature of the bars. (The thermal conductivity of copper is 385 W m-1 K-1 and that of iron is 80 W m-1 K-1.)

7. (a) A sheet of glass has an area of 2.0 m2 and a thickness of 8.0 x 10-3 m. The glass has a thermal conductivity of 0.80 W m-1 K-1. Calculate the rate of heat transfer through the glass when there is a tem¬perature difference of 20 K between its faces.
(b) A room in a house is heated to a temperature 20 K above that outside. The room has 2 m2 of win¬dows of glass similar to the type used in (a). Suggest why the rate of heat transfer through the glass is much less than the value calculated above.
(c) Explain why two sheets of similar glass insulate much more effectively when separated by a thin layer of air than when they are in contact.

8. A cubical copper tank with sides 0.5 m long and 3 mm thick is full of water initially at 85 oC. The temperature of the surroundings is 18oC.
(a) Calculate the initial rate of loss of heat from the tank
(b) Calculate the initial rate of cooling in oC per second if the specific heat capacity of water in 4200 Jkg-1 oC-1 and the density of water is 1000 kgm-3.
(c) Repeat parts (a) and (b) if the tank is covered with a layer of insulation 1 cm thick and of thermal conductivity 1.0 Wm-1K-1.