A warm water bath containing 10.0L of water is connected to an ice-water bath with a piece of metal of length L = 2.14 m and cross sectional area A =...

A warm water bath containing 10.0L of water is connected to an ice-water bath with a piece of metal of length L = 2.14 m and cross sectional area A = 1933 cm2. The metal has a thermal conductivity of km = 56.4 Wm-1K-1, a specific heat of cm = 306.4 Jkg-1K-1 and a density of ρm = 3866.6 kgm-3.The warm water bath is initially at a temperature of Th = 69.9 °C.

 What is the initial rate of heat flow along the rod? You may assume it has been in this steady state for quite some time.

b) Now lets consider what will happen if the power source heating the warm water bath is switched off. In this case the temperature of the warm water bath will gradually decrease as heat is transferred to the cool water bath. We can describe the heat lost by the warm water bath and the metal rod (the average temperature is just the averages of the temperatures on either side) in time tf as:

In this equation the final temperature and power are functions of tf, the other variables are not dependent on tf. We can then differentiate with respect to time to get the expression (replacing tfwith t here):

which be rearranged to give:

Use this expression, along with what you know about P to calculate the temperature of the warm water bath after one hour has passed.