Heat Extraction

a) A body of mass M has a temperature-independent specific heat C. If the body is heated reversibly from a temperature T_i to a temperature T_f, what is the change in its entropy?

b) Two such bodies are initially at temperatures of 100 K and 400 K. A reversible engine is used to extract heat with the hotter body as a source and the cooler body as a sink. What is the maximum amount of heat that can be extracted in units of MC?

c) The specific heat of water is C = 4.2 J/g K, and its density is 1 g/cm³. Calculate the maximum useful work that can be extracted, using as a source 10³ m³ of water at 100°C and a lake of temperature 10°C as a sink.

SOLUTION

a) For a mass of fixed volume we have

(1)

So, by the definition of C,

(2)

Since C is independent of T, we may rewrite (2) and integrate:

(3)

The change in entropy is then

(4)

b) The maximum heat may be extracted when the entropy remains constant. Equating the initial and final entropies yields the final temperature of the two bodies:

(5)

The heat extracted, Q, is then equal to the difference in initial and final internal energies of the bodies:

(6)

c) Here we may calculate the maximum useful work by using the Carnot efficiency of a reversible heat engine operating between two reservoirs, one starting at a high temperature (100°C) and the other fixed (the lake) at 10°C. The efficiency may be written for a small heat transfer as

(7)

where the heat dQ transferred from the hot reservoir equals its change in internal energy –MC dT. We may then find the total work by integrating dW as follows:

(8)

We may also use the method of part (b) and the fact that the entropy is conserved. Denote the mass of the hot water M and the lake M_L. Equating the initial and final entropies gives

(9)

Writing the final temperature T as T_C + δ, where δ is small (it’s a big lake), and expanding the logarithm, we obtain

(10)

Substituting (10) back into (9) gives

(11)

As before, the work extracted equals the change in internal energy of the bodies, so

which is the same as above.