Page 47 - Entrophy Analysis in Thermal Engineering Systems

38 Entropy Analysis in Thermal Engineering Systems

A similar relation can be derived for the adiabatic expansion process.

T 4 v 3
c v ln ¼ R g ln (3.16)
T 3 v 4
Since T 1 ¼T 4 ¼T L and T 2 ¼T 3 ¼T H , it can be deduced by comparing
Eqs. (3.15) and (3.16) that

v 2 v 3
¼ (3.17)
v 1 v 4
The heat transfer ratio Q L /Q H can now be found using Eqs. (3.11), (3.12),
and (3.17). Hence,

Q L
¼ T L (3.18)
Q H
T H
Eq. (3.18) is valid for a Carnot cycle as well as the class of heat engines under-
going two isothermal processes, i.e., ideal Sterling and Ericson cycles.
Given the definition of the thermal efficiency of a heat engine as

net work
η ¼ ¼ W net (3.19)
heat input Q in
th
where W net ¼Q H Q L , Q in ¼Q H , and using Eq. (3.18), the expression
obtained for the Carnot efficiency is

η ¼ 1 T L (3.20)
C T H

3.3.2 Derivation of the Clausius integral
Eq. (3.18) is a remarkable result. For a Carnot cycle communicating with
two thermal reservoirs, Eq. (3.18) can be rearranged to read

+ Q L ¼ 0 (3.21)
Q H
T H
T L
Clausius called the second law the theorem of the equivalence of transformations.
The first kind of transformation introduced by Clausius is the generation of
heat at temperature T from work whose equivalence-value is Q/T. The sec-
ond form of transformation is related to the transference of heat from a body
at temperature T 1 to another at temperature T 2 [10]. In this case, the
equivalence-value is

Q Q

T 2 T 1