Page 33 - Entrophy Analysis in Thermal Engineering Systems

24 Entropy Analysis in Thermal Engineering Systems

Clausius, who uses it fundamentally in his mathematical investigations” [23].
Indeed, Rudolf Clausius derived an equation in 1850 similar to that given
by Eq. (2.13) in the following form [24]:

ð
C ¼ Aa + tÞ (2.14)
where C denotes the inverse of Carnot’s function, A designates the equiv-
alent of heat for the unit of work, i.e., inverse of J in Eq. (2.13), and a¼273 is
the inverse of the expansion coefficient of air, i.e., 0.003665 determined
from the experiments of Magnus and Regnault. The relation between
the absolute temperature scale and centigrade, [K]¼[°C]+a, was given
by William Rankine in a paper read on February 4, 1850 before the Royale
Society of Edinburgh [25]. Clausius also arrived at the same relation [26]
with a difference that in the Clausius relation a¼273, whereas in that of
Rankine a¼274.6.
Rankine, in a subsequent paper, presented an analytical formulation to
derive an expression for the efficiency of the Carnot cycle [27].

T 1 T 0
W
¼ (2.15)
Q 1 T 1
where Q 1 is the total heat supplied, T 1 and T 0 denote the temperature at
which heat is supplied to and rejected by the working substance,
respectively.
A remarkable conclusion enunciated by Rankine is that “Carnot’s Law is
not an independent principle in the theory of heat; but is deducible, as a consequence,
from the equations of the mutual conversion of heat and expansive power.” An equa-
tion analogous to Eq. (2.15) was also obtained by Thomson using the com-
bined gas laws; i.e., Eq. (2.12), and Poisson’s equations, which was published
as a note to a paper authored by Joule on air engines [28].
After presenting a modified definition for the absolute temperature scale,
Thomson presented mathematical expressions of the first and second laws of
thermodynamics for a system undergoing a reversible process [21].

ð 0 n 1 + H t Þ ¼ 0 (2.16)
n
W + JH t + H t + … + H t
0 n 1 n
+ H t + … + H t + H t ¼ 0 (2.17)
H t
0 n 1 n
t t t t
0,
where W denotes the work done, H t , H t …, H t n 1, H t n designate the quan-
0 n 1 , t , respectively.
n
tities of heat taken in at temperatures t, t , …, t