fluid_mechanics

612 Chapter 11 ■ Compressible Flow
maximum entropy difference occurs at a pressure of 2.62 psia and
a temperature of 432.1 °R. p T s s 1
(psia) ( R) [( ft lb) ( lbm R)]
COMMENT Note that for Fanno flow the entropy must increase
in the direction of flow. Hence, this flow can proceed either
from subsonic conditions upstream to a sonic condition 1Ma 12
downstream or from supersonic conditions upstream to a sonic
condition downstream. The arrows in Fig. 11.11 indicate in which
direction a Fanno flow can proceed.
7 502.3 33.6
6 496.8 39.8
5 488.3 46.3
4 474.0 52.6
3 447.7 57.3
2.62 432.1 57.9
2 394.7 55.4
1.8 378.1 53.0
1.5 347.6 47.0
1.4 335.6 44.2
Fanno flow properties
can be obtained
from the second
T ds equation combined
with the continuity
and energy
equations.
We can learn more about Fanno lines by further analyzing the equations that describe the
physics involved. For example, the second T ds equation 1Eq. 11.182 is
For an ideal gas
and
or
T ds dȟ dp
r
dȟ c p dT
r p
RT
dp
p dr r dT
T
Thus, consolidating Eqs. 11.1, 11.7, 11.18, and 11.77 we obtain
T ds c p dT RT a dr r dT T b
Also, from the continuity equation 1Eq. 11.402, we get for Fanno flow rV constant, or
dr
r dV V
(11.18)
(11.7)
(11.1)
(11.77)
(11.78)
(11.79)
Substituting Eq. 11.79 into Eq. 11.78 yields
T ds c p dT RT a dV V dT T b
or
ds
dT c p
T R a1 dV
V dT 1 T b
By differentiating the energy equation 111.742 obtained earlier, we obtain
dV
dT c p
V
(11.80)
(11.81)