Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamics 155

and so

The central temperature is

T --R 9P^2 +TS=372K.

• 6k

1157
Let H be the flow of heat per unit time per unit area normal to the
isothermal surface through a point P of the body. Assunze the experimental
fact
H=-kVT,
where T is the temperature and k is the coefficient of thermal conductivity.
Finally the thermal energy absorbed per unit volume is given by cpT, where
c is the specific heat and p is the density.
(a) Make an analogy between the thermal quantities HI k, T, c, p and

(b) Using the results of (a) find the heat conduction equation.

(c) A pipe of inner radius rl, outer radius r2 and constant thermal

conductivity k is maintained at an inner temperature TI and outer tem-

perature T2. For a length of pipe L find the rate the heat is lost and the

temperature between rl and r2 (steady state).

the corresponding quantities El J, V, p of steady currents.

(SVNY, Bufiulo)

Solution:

(a) By comparison with Ohm's law J = aE = -0 grad V(V is voltage)

and conservation law of charge dp/at = -V. J, we obtain the analogy

cpT u p;H J; grad T grad V; k tl u.

(b) By the above analogy and charge conservation law, we have

cp- = -grad. (-k grad T) = kV2T.

aT

at

Then the heat conduction equation is

aT k

___ -V2T = 0.

at pc