Page 44 - Entrophy Analysis in Thermal Engineering Systems

Teaching entropy 35

where p o denotes the atmospheric pressure, V o the volume at p o and freezing
temperature, E the expansion coefficient, t the temperature measured in
degree centigrade, p and V represent the pressure and the volume of the
gas at temperature t.
1
Substituting T o ¼ into Eq. (3.7) gives
E
pV ¼ p o V o T (3.8)
T o
where T o ¼273 corresponds to the coefficient of thermal expansion of air
and T¼273+t.
Eq. (3.8) is the simplest state equation that is referred to as the ideal
gas equation, which is used in the analytical arguments of Carnot [1],
Clapeyron [6], Thomson [7], Rankine [8], and Clausius [9]. The numerical
constant T o employed in the analyses of Carnot and Clapeyron was 267,
whereas it was 274.6 (inverse of the expansion coefficient of gases at the
temperature of melting ice) in Rankine’s work. Rankine also noted the
absolute temperature of melting ice being 494.3 degrees in Fahrenheit scale.

3.2.4 Clausius inequality
The last step in the traditional method of teaching entropy is the presentation
of the Clausius inequality given in Eq. (3.9), which then leads to the intro-
duction of entropy as defined in Eq. (1.9). However, the Clausius inequality
is usually introduced without sufficient background and without a clear con-
nection to the previous steps. One would then need to figure out how to
connect the dots, i.e., Carnot cycle, absolute temperature, reversibility,
Clausius inequality, and entropy.

þ
δQ
0 (3.9)
T
In summary, the traditional method of introducing entropy as a thermody-
namic property requires one to go through a lengthy and twisted process as
schematically depicted in Fig. 3.3. The three intermediate steps in Fig. 3.3
are not fully justifiable as discussed in the previous sections. Furthermore, an
introduction of the Clausius inequality as an analytical expression of the sec-
ond law without a background is inappropriate. Among recent authors,
Bejan [3] provides an analysis to derive the Clausius inequality. In the next
section, a simple and straightforward approach is presented for introducing
entropy in thermodynamics classes.