Modern Engineering Thermodynamics

Summary 399
equation, and a series of differential relations for the nonmeasurable u, h, s properties cast in terms of the measurable
p, v, T properties allow us to develop a general procedure for constructing thermodynamic tables and charts.
In the concluding section of this chapter, the history and philosophy of modeling steam with the ideal gas
equations is discussed. It is seen that this is a reasonably accurate approximation for saturated vapor at low
pressure and superheated vapor (steam gas) at low pressure or high temperature.
Some of the more important equations introduced in this chapter follow. Do not attempt to use them without
understanding their limitations.
1. Two new thermodynamic properties: the specific Helmholtz function, f = u – Ts, and the specific Gibbs
function, g = h – Ts.
2. The Gibbs phase equilibrium condition:
s fg = h fg
T sat
3. The Maxwell equations:
∂T
= − ∂p
∂v s ∂s
v
∂T
∂p
s
= ∂v
∂s p
∂p
∂T
v
= ∂s
∂v T
∂v
= − ∂s
∂T p ∂p
T
4. The Clapeyron equation:
dp
= h fg
dT
sat
T sat v fg
5. General u, h, and s property relations:
Z T2
u 2 − u 1 = c v dT +
T 1
Z T2
h 2 − h 1 = c p dT +
T 1
s 2 − s 1 =
=
Z T2
T 1
Z T2
T 1
T ∂p
v 1
∂T
Z v2
Z p2
p 1
Z v2
c v
T dT + v1
c p
T dT − Z p2
p 1
− p
v
v − T ∂v
∂T
∂p
∂T
∂v
∂T
dv
v
dp
p
p
dv
dp
6. Specific heat relations:
∂c v
= T ∂2 p
∂v T ∂T 2
v
∂c p
∂p
T
= −T ∂2 v
∂T 2 c p − c v = Tβ 2 v/k
p
7. Using the gas tables to find changes in entropy: s 2 − s 1 = ϕ 2 − ϕ 1 − R ln ðp 2 /p 1 Þ
9. The compressibility factor: Z = pv/ðRTÞ:
10. Relations for use with the Generalized charts:
h
h* − h
2 − h 1 = ðh* 2
− h*Þ 1
−
h* − h
Tc
T c T c M
−
2
s 2 − s 1 = ðs* 2
− s*Þ 1
− ðs* − sÞ 2 − ðs* − sÞ 1 ð1/MÞ
where h 2
* − h 1
* and s 2
* − s 1
* are the equivalent ideal gas changes in specific enthalpy and specific entropy of the
real gas.
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