Page 29 - Entrophy Analysis in Thermal Engineering Systems

20 Entropy Analysis in Thermal Engineering Systems

had carefully shown the limitation and the range of the validity of his theory.
He wrote:
The theories that we deduce here would not perhaps be exact if applied outside of
certain limits either of density or temperature. They should be regarded as true only
within the limits in which the laws of Mariotte and of Gay-Lussac and Dalton are
themselves proven (Ref. [13], p. 80).

2.3.2 Clapeyron’s work

Carnot’s work was remained unnoticed for a decade after its publication. It
was then Emile Clapeyron who gave a recognition to the Carnot’s contri-
bution and presented graphically the operation of the Carnot cycle on a
pressure-volume diagram [14]. The analysis of Clapeyron also employs
the ideal gas law, which he expressed as
ð
pV ¼ Rt + 267Þ (2.7)
p 0 V 0
where R ¼ and the subscript “0” denotes a reference state.
t 0 + 267
By applying differential calculus, Clapeyron formulated the mechanical
work and the heat required to produce this effect in a Carnot engine oper-
ating with a perfect gas. He showed that the ratio of the work produced to
the heat supplied obeys where C, a function of temperature, is the inverse
dt
C
0
of function F in Carnot’s notations; see Eq. (2.6). We will later show that C
is indeed the absolute temperature, T. Clapeyron also showed that the dif-
ference between the specific heat at constant pressure and that at constant
volume of (perfect) gases is R C; see pp. 170–171 in Ref. [14].
267 + t

2.3.3 Poisson’s equations

In 1833, Sim eon Denis Poisson presented equations relating the pressure,
volume, and temperature of a perfect gas undergoing an adiabatic process
at two different states [15].
ρ
γ
p
¼ (2.8)
0 ρ 0
p
γ 1
t + 266:67 ρ
¼ (2.9)
t + 266:67 ρ 0
0
where ρ denotes the density and γ is the ratio of the specific heats.