Basics of Fluid Mechanics, 2014a

434 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
It should be noted that P should be replaced by P 0 in the calculations. The speed
of sound at the entrance is
c = √ kRT = √ 1.4 × 287 × 300 ∼ [ m
]
= 347.2
sec
and the density is
ρ = P RT
[ ]
1, 000, 000 kg
=
287 × 300 =11.61 m 3
The velocity at the entrance should be
U = M ∗ c =0.08528 × 347.2 ∼ [ m
]
= 29.6
sec
The diameter should be
√ √
4ṁ
D =
πUρ = 4 × 0.2
π × 29.6 × 11.61 ∼ = 0.027
Nevertheless, for the sake of the exercise the other parameters will be calculated. This
situation is reversed question. The flow rate is given with the diameter of the pipe. It
should be noted that the flow isn’t choked.
End Solution
Example 11.17:
A gas flows of from a station (a) with pressure of 20[bar] through a pipe with 0.4[m]
diameter and 4000 [m] length to a different station (b). The pressure at the exit (station
(b)) is 2[bar]. The gas and the sounding temperature can be assumed to be 300 K.
Assume that the flow is isothermal, k=1.4, and the average friction f=0.01. Calculate
the Mach number at the entrance to pipe and the flow rate.
Solution
First, the information whether the flow is choked needs to be found. Therefore, at first
it will be assumed that the whole length is the maximum length.
with 4 fL
D
4 fL
D
∣ =
max
4 × 0.01 × 4000
0.4
∣ = 400 the following can be written
max
= 400
M
4fL
D
T 0
T 0
∗T
ρ
ρ ∗T
P
P ∗T P 0
P 0
∗T
0.0419 400.72021 0.87531 20.19235 20.19235 12.66915