Page 28 - Entrophy Analysis in Thermal Engineering Systems

Birth and evolution of thermodynamics 19

but in almost all introductory thermodynamics textbooks published untill
the present day. In Chapter 3, we will treat this subject in detail and discuss
the shortcomings of the rational reasoning introduced by Carnot. Neverthe-
less, what is remarkable with respect to the Carnot’s contribution
unappreciated, perhaps, in all textbooks is the analytical formulation he pres-
ented using the laws of Boyle-Mariotte and Dalton-Gay Lussac; see
pp. 74–76 in Ref. [12] with the English translation given in Appendix B
of Ref. [13]. Carnot expressed the combined laws, i.e., Eqs. (2.1) and
(2.2), in a single equation as
t + 267
(2.4)
V ¼ c
p
where V is the volume, p the pressure, t the temperature (in centigrade), c a
constant, and the number 267 is the inverse of the expansion coefficient
obtained from the experiments of Gay-Lussac.
Eq. (2.4) is an early version of the ideal gas equation which was used by
Carnot to establish a relation for the isothermal expansion work of air from a

unit of volume to any given volume like V at constant temperature t.
ð (2.5)
r ¼ c 267 + tÞ ln V
where r, in Carnot’s notations, refers to the quantity of work produced in the
process.
Carnot then derived another relation for the quantity of heat, designated
by e, required to maintain the temperature constant during the expansion
process.

c
e ¼ ln V (2.6)
0
F
0
where F is a function that depends on temperature only.
A careful examination of Carnot’s analysis reveals that he had a correct
understanding of the principle of the equivalence of heat and work. It is from
this analytical formulation that he was led to state some of his propositions;
for instance “When a gas varies in volume without change of temperature,
the quantities of heat absorbed or liberated by this gas are in arithmetical pro-
gression, if the increments or the decrements of volume are found to be in
geometrical progression” (Ref. [13], p. 81), or “The motive power of heat is
independent of the agents employed to realize it; its quantity is fixed solely
by the temperatures of the bodies between which is effected, finally, the
transfer of the caloric” (Ref. [13], p. 68). It is interesting to note that Carnot