Page 57 - Entrophy Analysis in Thermal Engineering Systems

The common source of entropy increase 49

the air gained heat, whereas the pipe and the second tank lost heat. Inter-
esting to note is that the heat gained was almost equal to the total heat lost.
To correctly determine entropy generation, one would need to account for
the heat transfer processes taking place between the different parts of the
apparatus and the calorimeters.
Consider now a rigid tank, which, as before, contains a compressed ideal
gas. It is then allowed to escape to the atmosphere. An expansion like this
with air as the gaseous medium was also experimented by Joule [4]. The
escaping air passed through a long coil of pipe in order to reduce the tem-
perature of the air to its initial temperature. The entire assembly was placed
in a water calorimeter. Joule’s observation was that heat was given off to the
expanding air yielding a cooling effect in the calorimeter. These experiments
led Joule to determine the mechanical equivalent of heat. The external work
performed by the escaping air to overcome the atmospheric pressure was
found to be the same as the amount of heat absorbed by the air. The ana-
lytical formulation of Joule’s experiment was presented by Thomson [5] and
later Clausius [6]. They both concluded that in the expansion process of air
(treated as an ideal gas), it is necessary to supply heat to maintain a constant
temperature. From these arguments, it should be obvious now that the
entropy increase associated with the isothermal expansion of an ideal gas
is indeed due to the heat transferred to the gas from an external source.
For an infinitesimal amount of work done in the expansion of an ideal
gas, we have δQ¼δW, where δQ denotes the infinitesimal amount of heat
supplied to maintain an isothermal expansion of the gas. Substituting
δQ¼TdS and δW¼pdV gives

(4.12)
TdS ¼ pdV
Integrating Eq. (4.12) from an initial volume to a final volume with the use
of the ideal gas equation, one obtains

ΔS ¼ nR ln V f (4.13)
V i
where ΔS is the increase in the entropy of the expanding gas.
Assume that the total heat Q required in the expansion process is supplied
from a source whose temperature during the infinitesimal heat transfer δQ is
T s . The net decrease in the entropy of the source is

ð
δQ
ΔS s ¼ T s (4.14)