Page 124 - Mechanical Engineers' Handbook (Volume 4)

7 Analysis of Engineering System Components 113

corresponding to the saturated vapor and liquid states found at temperature T (and pressure
sat
P ). Built into the last equation is the identity
sat
h h T (s s )
g ƒ g ƒ
which is equivalent to the statement that the Gibbs free energy is the same for the saturated
states and their mixtures found at the same temperature, g g ƒ .
g
The properties of a two-phase mixture depend on the proportion in which saturated
vapor, m , and saturated liquid, m , enter the mixture. The composition of the mixture is
ƒ
g
described by the property called quality,
m g
x
m m g
ƒ
The quality varies between 0 at state ƒ and 1 at state g. Other properties of the mixture can
be calculated in terms of the properties of the saturated states found at the same temperature,
u u xu ƒg s s xs ƒg
ƒ
ƒ
h h xh ƒg v v xv ƒg
ƒ
ƒ
with the notation ( ) () () . Similar relations can be used to calculate the properties
ƒg g ƒ
of two-phase states other than liquid and vapor, namely, solid and vapor or solid and liquid.
For example, the enthalpy of a solid and liquid mixture is given by h h xh , where
s sƒ
subscript s stands for the saturated solid state found at the same temperature as for the two-
phase state, and where h is the latent heat of melting or solidification.
sƒ
In general, the states situated immediately outside the two-phase dome sketched in Figs.
3 and 4 do not follow very well the limiting models discussed earlier in this section (ideal
gas, incompressible substance). Because the properties of closely neighboring states are usu-
ally not available in tabular form, the following approximate calculation proves useful. For
a moderately compressed liquid state, which is indicated by the subscript ( ) , that is, for a
*
state situated close to the left of the dome in Fig. 4, the properties may be calculated as
slight deviations from those of the saturated liquid state found at the same temperature as
the compressed liquid state of interest,
h
(h ) (v )[P (P )]
ƒ T*
ƒ T*
ƒ T*
*
*
s
(s )
ƒ T*
For a slightly superheated vapor state, that is, a state situated close to the right of the dome
in Fig. 4, the properties may be estimated in terms of those of the saturated vapor state
found at the same temperature:
h
(h )
gT

s
(s ) (P )
P v
g T
gg

gT
T ln P
g T
In these expressions, subscript ( ) indicates the properties of the slightly superheated vapor

state.
7 ANALYSIS OF ENGINEERING SYSTEM COMPONENTS
This section contains a summary of the equations obtained by applying the first and second
1
laws of thermodynamics to the components encountered in most engineering systems, such