fluid_mechanics

11.5 Nonisentropic Flow of an Ideal Gas 615
For Fanno flow, the
Mach number is a
function of the distance
to the critical
state.
5.0
The application of the energy equation 1Eq. 5.692 to Fanno flow gave Eq. 11.74. If Eq. 11.74 is
differentiated and divided by temperature, the result is
Substituting Eqs. 11.14, 11.36, and 11.46 into Eq. 11.91 yields
which can be combined with Eq. 11.90 to form
We can merge Eqs. 11.77, 11.79, and 11.90 to get
Consolidating Eqs. 11.94 and 11.89 leads to
Finally, incorporating Eq. 11.93 into Eq. 11.95 yields
(11.91)
(11.92)
(11.93)
(11.94)
(11.95)
(11.96)
Equation 11.96 can be integrated from one section to another in a Fanno flow duct. We elect to
use the critical 1*2 state as a reference and to integrate Eq. 11.96 from an upstream state to the critical
state. Thus
Ma*1
Ma
(11.97)
where / is length measured from an arbitrary but fixed upstream reference location to a section in
the Fanno flow. For an approximate solution, we can assume that the friction factor is constant at
an average value over the integration length, /* /. We also consider a constant value of k. Thus,
we obtain from Eq. 11.97
1 11 Ma 2 2
k 1
k Ma 2 2k
dT
T d1V 2 2
2c p T 0
dT
T k 1 Ma d1V 2 2
2 0
2 V 2
d1V 2 2 d1Ma 2 2Ma 2
V 2 1 31k 1224Ma 2
dp
p 1 d1V 2 2
d1Ma2 2
2 V 2 Ma 2
1
2 11 kMa2 2 d1V 2 2
d1Ma2 2
fk V 2 Ma 2 2
11 Ma 2 2 d1Ma 2 2
51 31k 1224Ma 2 6kMa f dx
4 D
Ma2
dx
D 0
11 Ma 2 2 d1Ma 2 2
51 31k 1224 Ma 2 6kMa /*
4
31k 1224Ma 2 f 1/* /2
ln e
2
f
1 31k 1224Ma D
/
f dx
D
(11.98)
________ f(* – )
D
0.0
0.1
1.0
Ma
10
For a given gas, values of f 1/* /2D can be tabulated as a function of Mach number for
Fanno flow. For example, values of f 1/* /2D for air 1k 1.42 Fanno flow are graphed as a
function of Mach number in Fig. D.2 in Appendix D and in the figure in the margin. Note that the
critical state does not have to exist in the actual Fanno flow being considered, since for any two
sections in a given Fanno flow
f 1/* / 2 2
D
f 1/* / 12
D
f D 1/ 1 / 2 2
(11.99)
The sketch in Fig. 11.20 illustrates the physical meaning of Eq. 11.99.
For a given Fanno flow 1constant specific heat ratio, duct diameter, and friction factor2 the
length of duct required to change the Mach number from Ma 1 to Ma 2 can be determined from Eqs.
11.98 and 11.99 or a graph such as Fig. D.2. To get the values of other fluid properties in the Fanno
flow field we need to develop more equations.