Basics of Fluid Mechanics, 2014a

380 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
Solution
The momentum equation written for the control volume shown in Figure (11.2) is
R
P F
cs
{ }} { { U (ρUdA)
}} {
(P + dP ) − P = (ρ + dρ)(c − dU) 2 − ρc 2 (11.8)
Neglecting all the relative small terms results in
(
)
dP =(ρ + dρ) c 2 ∼ 0 ∼ 0
−✘✘✘✿
2cdU +✘✘✘✘✘✿
dU 2 − ρc 2 (11.9)
And finally it becomes
dP = c 2 dρ (11.10)
This yields the same equation as (11.5).
End Solution
11.3.2 Speed of Sound in Ideal and Perfect Gases
The speed of sound can be obtained easily for the equation of state for an ideal gas (also
perfect gas as a sub set) because of a simple mathematical expression. The pressure
for an ideal gas can be expressed as a simple function of density, ρ, and a function
“molecular structure” or ratio of specific heats, k namely
and hence
c =
P = constant × ρ k (11.11)
P
√
{ }} {
dP
dρ = k × constant × constant × ρ k
ρk−1 = k ×
ρ
= k × P ρ
(11.12)
Remember that P/ρ is defined for an ideal gas as RT , and equation (11.12) can be
written as
Ideal Gas Speed Sound
c = √ kRT
(11.13)
Example 11.2:
Calculate the speed of sound in water vapor at 20[bar] and 350 ◦ C, (a) utilizes the
steam table (b) assuming ideal gas.