Modern Engineering Thermodynamics

4.6 Mechanical Work Modes of Energy Transport 111
or
p 1 V n 1 = p 2V n 2
where the exponent n is a constant. Such processes are called polytropic processes. 6 The moving system boundary
work of any substance undergoing a polytropic process is
For n = 1, this integral becomes
ð 1 W 2 Þ polytropic =
moving boundary
Z 2
1
pdV =
Z 2
1
constant
V n dV
and for n ≠ 1, it becomes
ð 1
W 2 Þ polytropic ðn=1Þ = p 1 V 1 ln V 2
= p 2 V ln V 2
(4.28)
moving boundary
V 2 1 V 1
ð 1
W 2 Þ polytropic ðn≠1Þ, = p 2V 2 − p 1 V 1
moving boundary
1 − n
(4.29)
If the material undergoing a polytropic process is an ideal gas, then it must simultaneously satisfy both of the
following equations:
1. The ideal gas equation of state, pV = mRT:
2. The polytropic process equation, pV n = constant:
Combining these two equations by eliminating the pressure p gives
or, for a fixed mass system,
or
mRTV n−1 = constant
T 1 V1 n−1 = T 2 V n−1
2
T 2
= V ! 1−n
2
= v 1−n
2
(4.30)
T 1 V 1 v 1
Similarly, eliminating V in these two equations (for a fixed mass system) gives the polytropic process equations
for an ideal gas:
Polytropic process equations for an ideal gas
T 2
= p ðn−1Þ/n
2
= v 1−n
2
(4.31)
T 1 p 1 v 1
Finally, if we have an ideal gas undergoing a polytropic process with n ≠ 1, then its moving system boundary work
is given by Eq. (4.29), with p 2 V 2 − p 1 V 1 = mRðT 2 − T 1 Þ as the polytropic work equation for an ideal gas (n ≠ 1):
Polytropic work equation for an ideal gas ðn ≠ 1Þ
ð 1 W 2 Þ polytropic ðn≠1Þ
ideal gas
moving boundary
= mR
1 − n ðT 2 − T 1 Þ
(4.32)
6 The term polytropic comes from the Greek roots poly meaning “many” and trope meaning “turns” or “paths.”