Environmental Engineering Reference
In-Depth Information
In Equation (3.25), the work term is the sum of two contributions. The first is the
“
volume work
”
by the fluid, which can be expressed as
W
vol
=
φ
m
,
out
:
ð
p
out
υ
out
Þ−
φ
m
,
in
:
ð
p
in
υ
in
Þ
ð
Eq
:
3
:
26
Þ
where
υ
is the mass-specific volume (unit volume per unit mass, or 1/
ρ
) and
p
the
pressure.
The second contribution consists of all other types of work
W
:
cv
, such as rotating
shaft work, boundary displacement work, and magnetic and electric (field) work. Thus,
W
:
=
W
:
cv
+
φ
m
,
out
p
out
υ
out
ð
Þ−
φ
m
,
in
p
in
υ
in
ð
Þ
ð
Eq
:
3
:
27
Þ
) has been defined as the enthalpy, h. The energy rate
balance is now simplified by introduction of this quantity:
Now, u +
p
υ
(= u +
p
/
ρ
φ
m
,
in
h
in
+
v
in
−
φ
m
,
out
h
out
+
v
out
dE
cv
dt
W
c
:
+
=
Q
:
cv
−
ð
Eq
:
3
:
28
Þ
2
+gz
in
+gz
out
2
For the enthalpy of a mixture, the following relation holds:
Y
i
h
i
and h
i
=h
re
i
+
ð
T
h=
X
N
c
p
,
i
ðÞ
τ
:
:
d
ð
Eq
3
29
Þ
i
=1
T
ref
The reference condition is usually defined as 1 bar and 298.15 K, and c
p
for each
species is often given as a polynomial function of temperature (see, e.g., Smith et al.
(2005)); the website of the National Institute of Standards and Technology (NIST)
provides useful data for a multitude of (inorganic) species (tinyurl.com/9s23f ); where
experimental data are not available, methods of estimation are employed, as described
by Poling et al. (2001). The heat capacity at constant pressure for a mixture can be
calculated as a mass fraction average:
c
p
=
X
N
:
:
Y
i
c
p
,
i
ð
Eq
3
30
Þ
i
=1
Regarding the density of the fluid mixture, for gases often as equation of state
(EOS), the ideal gas law can be used, though this must be verified. Smith et al.
(2005) give a good overview of the selection of the appropriate EOS. In cases for
which the ideal gas law holds,
!
−1
R
u
T
with the averagemolecular weight MW =
X
N
=
p
MW
Y
i
MW
i
ρ
ð
Eq
:
3
:
31
Þ
i
=1
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