Page 35 - Entrophy Analysis in Thermal Engineering Systems

26 Entropy Analysis in Thermal Engineering Systems

As noted by Clausius, Eq. (2.22) is used for determination of the magnitude
of S. For the case of perfect gases, he derived the following differential equa-
tions for a unit weight of the substance [30].

(2.23)
dU ¼ c v dT
dT dV
(2.24)
+ A R g
dS ¼ c v
T V
where R g denotes the gas constant. Eq. (2.24) may be obtained by combi-
nation of Eqs. (2.21), (2.23) and the differential form of Eq. (2.18), i.e.,
(2.25)
dQ ¼ dU + A dW
where dW¼pdV.
Recall the coefficient A in Clausius’ formulation that denotes the equiv-
alent of heat for a unit of work. If the heat, work, and energy terms in
Eq. (2.25) are described with the same unit (e.g., J), the coefficient A will
then vanish. In a book that Clausius published years after his ninth memoir,
the coefficient A is dropped off from his formulation. For instance,
Eq. (2.24) after integration is expressed as [30]

S ¼ S 0 + c v ln T + R g ln V (2.26)
T 0 V 0
where the subscript “0” designates an initial or reference state.

2.5 Remarks
The first and foremost conclusion that can be drawn from the forgoing
discussions is that the foundation of thermodynamics is due to the labor of
many ingenious men, although the individual contribution may differ from
one to another. It would be unfair to entitle mere one or two individuals the
merit of discovering or founding the main laws of thermodynamics, which
evolved through several decades. The role of ideal gas law in the theoretical
development of thermodynamics, overlooked in many sources, should espe-
cially be highlighted. The way that the second law is taught in introductory
classes does not follow the same logic as it was developed by the founders.
This issue will further be discussed in the next chapter.
An example is the absolute temperature scale that is usually introduced
without a clear justification. If the ratio of the heat received-to-heat rejected
in a Carnot cycle is proportional to the ratio of the corresponding (absolute)
temperatures, it is not merely due to adopting a new temperature scale, as the