Page 43 - Entrophy Analysis in Thermal Engineering Systems

34 Entropy Analysis in Thermal Engineering Systems

In the next step, three Carnot cycles operating between (T H , T L ), (T H , T i ),
and (T i , T L ), where T L <T i <T H , are used to show that the function f can be
represented as

ð
gT L Þ
ð
fT H , T L Þ ¼ (3.5)
ð
gT H Þ
It follows that a combination of Eqs. (3.4) and (3.5) yields
gT L Þ
ð
Q L
¼ (3.6)
gT H Þ
ð
Q H
where g(T) is an unknown function of the temperature. In the next step,
Eq. (3.6) is used to introduce the temperature scale, which was first proposed
by William Thomson (Lord Kelvin) [4]; see the discussion of Section 2.3.5.
Finally, g(T)¼T is selected as the simplest functional form for g whereby
concluding Q L /Q H ¼T L /T H and deriving the efficiency of the Carnot
cycle, i.e. η C ¼1 T L /T H . Some texts even skip the intermediate steps relat-
ing to Eqs. (3.5) and (3.6) and introduce the absolute temperature scale right
after Eq. (3.4), e.g., see Ref. [2] Chapter 5 and Ref. [5] Chapter 5.
This method of derivation of the Carnot efficiency suffers from two
issues. First, Eq. (3.4) that plays a major role in the derivation of the Carnot
efficiency is valid subject to the validity of Carnot’s second corollary. How-
ever, as discussed in Section 3.2.2, the proof of the corollaries is not concrete
because certain objections can be raised. Second, the adoption of g(T)¼T
appears to be rather a convention. Most authors state that Eq. (3.6) is the
basis for determination of the absolute temperature scale.
In his 1848 paper [4], Thomson urged the need for a principle that would
serve as a foundation for an absolute temperature scale. He noted that the
temperature 273°C on the air-thermometer would correspond to the vol-
ume of air being reduced to nothing, a point of the scale which could not be
reached at any finite temperature. Notice that the number 273 is the inverse
of the expansion coefficient of air, i.e., 0.00366.
The law that provided a foundation to relate the two temperature scales
(Kelvin and Centigrade) is the combined laws of Boyle-Mariotte and
Dalton-Gay Lussac, which may be expressed as

pV ¼ p o V o 1+ Etð Þ (3.7)