Page 107 - Advanced thermodynamics for engineers

92 CHAPTER 4 AVAILABILITY AND EXERGY

4.10 THE VARIATION OF FLOW EXERGY FOR A PERFECT GAS
This derivation is based on Haywood (1980).
The definition of exergy is, from Eqn (4.51)

B 1 ¼ A 1 A 0
while that for the exergy of a flowing gas is, by comparison with Eqn (4.13b) for the availability of a
flowing gas,
B f1 ¼ A f1 A f0 ¼ðH 1 T 0 S 1 Þ ðH 0 T 0 S 0 Þ (4.69)
Equation (4.69) can be expanded to give the specific flow exergy as

b f1 ¼ a f1 a f0 ¼ðh 1 T 0 s 1 Þ ðh 0 T 0 s 0 Þ (4.70)
Now, for a perfect gas,
h ¼ c p T (4.71)
and the change of entropy
T p
s 1 s 0 ¼ c p ln R ln (4.72)
T 0 p 0

Hence, the flow exergy for a perfect gas can be written as

T p T p
b f1 ¼ðh 1 h 0 Þ T 0 c p ln R ln ¼ c p ðT T 0 Þ T 0 c p ln R ln (4.73)
T 0 p 0 T 0 p 0
The flow exergy can be nondimensionalised by dividing by the enthalpy at the dead-state tem-
perature, T 0 ,togive

b f1 T T k 1 p
¼ 1 ln ln (4.74)
c p T 0 T 0 T 0 k p 0
Equation (4.74) has been evaluated for a range of temperature ratios and pressure ratios, and the
variation of exergy is shown in Fig. 4.13.
Also shown in Fig. 4.13, as a straight line, is the variation of enthalpy for the parameters shown. It
can be seen that the exergy is sometimes bigger than the enthalpy, and vice versa. This is because the
enthalpy is purely a measure of the thermodynamic energy of the gas relative to the datum tem-
perature. When the dimensionless temperature, T/T 0 < 1.0 the enthalpy is negative, and when T/
T 0 > 1.0 the enthalpy is positive. This simply means that the thermodynamic energy can be greater
than or less than the datum value. While the enthalpy varies monotonically with the nondimensional
temperature, the exergy does not. The reason for this is that the exergy term, say at p/p 0 ¼ 1.0, is
given by

b f1 T T
¼ 1 ln (4.75)
c p T 0 T 0 T 0