Page 182 - Entrophy Analysis in Thermal Engineering Systems

Exergy 177

11.6 Maximum efficiency

The maximum thermal efficiency of a heat engine is defined as the
maximum theoretical work per unit thermal energy supplied to the engine.
The form of expression for the engine efficiency depends on whether heat is
supplied from a thermal reservoir, a hot stream, or fuel. The relations that we
derived for maximum work in Sections 11.2–11.4 will now be used to
derive expressions for the maximum efficiency of a heat engine.
For an engine, which receives a given quantity of heat from p thermal
reservoirs while communicating with n+1 reservoirs (Fig. 11.1), dividing
the maximum work obtained from Eq. (11.9) with Ψ de ¼0 by the total
amount of heat imparted to the engine yields

p
n
X X
Ψ Ψ th
th
j k
j¼1 k¼p +1
η ¼ (11.30)
max p
X
j¼1 Q j
In the simplest case with two thermal reservoirs (n¼1 and p¼1), Eq. (11.30)
reduces to the Carnot efficiency.

T 0
Ψ th 1 Q 1
η ¼ 1 ¼ T 1 ¼ 1 T 0 (11.31)
max
Q 1 Q 1 T 1
If the engine, as another example, receives a given quantity of Q 1 +Q 2 heat
from two thermal reservoirs (p¼2) and rejects heat to only one reservoir
(n¼2), we have
2

Ψ th T 0 T 0
X
1 Q 1 +1
j Q 2
η ¼ j¼1 ¼ T 1 T 2 (11.32)
max 2
X Q 1 + Q 2
j¼1 Q j
The expression for the maximum efficiency of an engine that receives heat
from a hot stream obeys