EXERCISES 1.3

1.3.1. Consider a one-dimensional rod, from x=0 to x=L. Assume that the
       heat energy flowing out of the rod at x = L is proportional to
       the temperature difference between the end temperature of the bar
       at x=L and the known external temperature. Derive (1.3.5). In
       this process, explain physically why H is positive.

1.3.2. Two one-dimensional rods of different materials joined at x = x0
       are said to be in perfect thermal contact if the temperature is
       continuous at x = x0, which means u(x0-,t) = u(x0+,t), and no
       heat energy is lost at x = x0, which means the heat energy
       flowing out of one flows into the other. What mathematical
       equation represents the latter condition at x = x0? Under what
       special condition is du/dx continuous at x = x0?

1.3.3. Consider a bath containing a fluid of specific heat c and mass
       density rho that surrounds the end x = L of a one-dimensional
       rod. Suppose that the bath is rapidly stirred in a manner such
       that the bath temperature is approximately uniform throughout,
       equaling the temperature at x = L, which is u(L,t). Assume that
       the bath is thermally insulated except at its perfect thermal
       contact with the rod. where the bath may be heated or cooled by
       the rod. Determine an equation for the temperature in the bath.
       This will be a boundary condition at the end x = L.
       Reference: Exercise 1.3.2.