Basics of Fluid Mechanics, 2014a

392 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
T 0 = constant and P 0 = constant and both of them are known (the condition at
the reservoir). For the point where the static pressure is known, the Mach number
can be calculated by utilizing the pressure ratio. With the known Mach number, the
temperature, and velocity can be calculated. Finally, the cross section can be calculated
with all these information.
In the point where the static pressure known
¯P = P = 3[MPa]
P 0 5[MPa] =0.6
From Table (11.2) or from Figure (11.6) or utilizing the enclosed program, Potto-GDC,
or simply using the equations shows that
M
T
T 0
ρ
ρ 0
A
A ⋆
P
P 0
A×P
A ∗ ×P 0
F
F ∗
0.88639 0.86420 0.69428 1.0115 0.60000 0.60693 0.53105
With these values the static temperature and the density can be calculated.
The velocity at that point is
U = M
T =0.86420338 × (273 + 21) = 254.076K
ρ
{}}{
0
ρ = ρ P 0
=0.69428839 ×
ρ 0 RT 0
[ ] kg
=41.1416
m 3
5 × 10 6 [Pa]
[
287.0
J
kgK
]
× 294[K]
c
{ }} { √
kRT =0.88638317 ×
√
1.4 × 287 × 294 = 304[m/sec]
The tube area can be obtained from the mass conservation as
A = ṁ
ρU =8.26 × 10−5 [m 3 ]
For a circular tube the diameter is about 1[cm].
End Solution
Example 11.4:
The Mach number at point A on tube is measured to be M =2 4 and the static pressure
is 2[Bar] 5 . Downstream at point B the pressure was measured to be 1.5[Bar]. Calculate
the Mach number at point B under the isentropic flow assumption. Also, estimate the
temperature at point B. Assume that the specific heat ratio k =1.4 and assume a
perfect gas model.
5 This pressure is about two atmospheres with temperature of 250[K]