Speed of Sound
Fig 39.1: Propagation of a sound wave
(a) Wave Propagating into still Fluid (b) Stationary Wave
This shows that
(a) dv = 0 if dρ is positive.
(b) A compression wave leaves behind a fluid moving in the direction of the wave (Fig. 39.1(a)).
(c) Equation (39.1) also signifies that the fluid velocity on the right is much smaller than the wave speed " a ". Within the framework of infinitesimal strength of the wave (sound wave), this " a " itself is very small.
In the above expression, Aρa is the mass flow rate. The first term on the right hand side represents the rate of outflow of x-momentum and the second term represents the rate of inflow of x momentum.
dp = padV (39.2)
If the wave strength is very small, the pressure change is small.
Combining Eqs (39.1) and (39.2), we get
he larger the strength dp/p of the wave ,the faster the wave speed; i.e., powerful explosion waves move much faster than sound waves.In the limit of infinitesimally small strength, dp → 0 we can write
(a) In the limit of infinitesimally strength of sound wave, there are no velocity gradients on either side of the wave. Therefore, the frictional effects (irreversible) are confined to the interior of the wave.
(b) Moreover, the entire process of sound wave propagation is adiabatic because there is no temperature gradient except inside the wave itself.
(c) So, for sound waves, we can see that the process is reversible adiabatic or isentropic.
So the correct expression for the sound speed is
For a perfect gas, by using of = const and p = pRT, we deduce the speed of sound as
For air at sea-level and at a temperature of 150C, a=340 m/s
Pressure Field Due to a Moving Source
Consider a point source emanating infinitesimal pressure disturbances in a still fluid, in which the speed of sound is "a". If the point disturbance, is stationary then the wave fronts are concentric spheres. As shown in Fig. 39.2(a), wave fronts are present at intervals of Δt .
Now suppose that source moves to the left at speed U < a. Figure 39.2(b) shows four locations of the source, 1 to 4, at equal intervals of time , with point 4 being the current location of the source.
at point 1, the source emanated a wave which has spherically expanded to a radius 3aΔt in an interval of time 3Δt During this time the source has moved to the location 4 at a distance of 3uΔt from point 1. The figure also shows the locations of the wave fronts emitted while the source was at points 2 and 3, respectively.
When the source speed is supersonic U > a (Fig. 39.2(c)), the point source is ahead of the disturbance and an observer in the downstream location is unaware of the approaching source. The disturbance emitted at different
Fig 39.2 Wave fronts emitted from a point source in a still fluid when the source speed is
(a) U = 0 (still Source) (b) U < a (Subsonic) (c) U > a (Supersonic)
points of time are enveloped by an imaginary conical surface known as "Mach Cone". The half angle of the cone α is known as Mach angle and given by
Since the disturbances are confined to the cone, the area within the cone is known as zone of action and the area outside the cone is zone of silence .
An observer does not feel the effects of the moving source till the Mach Cone covers his position.
Basic Equations for One-Dimensional Flow
Here we will study a class of compressible flows that can be treated as one dimensional flow. Such a simplification is meaningful for flow through ducts where the centreline of the ducts does not have a large curvature and the cross-section of the ducts does not vary abruptly.
In one dimension, the flow can be studied by ignoring the variation of velocity and other properties across the normal direction of the flow. However, these distributions are taken care of by assigning an average value over the cross-section (Fig. 39.3).
The area of the duct is taken as A(x) and the flow properties are taken as p(x), ρ(x), V(x) etc. The forms of the basic equations in a one-dimensional compressible flow are;
Fig 39.3 One-Dimensional Approximation
For steady one-dimensional flow, the equation of continuity is
p (x).V x). A (x) = Const
Differentiating(after taking log), we get
Consider a control volume within the duct shown by dotted lines in Fig. 39.3. The first law of thermodynamics for a control volume fixed in space is
where is the kinetic energy per unit mass.
Let us discuss the various terms from above equation:
It is to be noted that dA is directed along the outward normal.
The work done on the control surfaces is
The rate of heat transfer to the control volume is
where Q is the heat added per unit mass (in J/kg).
We know that the density p is given by /VA hence the first term on the right may be expressed in terms of (specific volume=1/ρ).
Equation (39.8) can be rewritten as
This is energy equation, which is valid even in the presence of friction or non-equilibrium conditions between secs 1 and 2.
Bernoulli and Euler Equations
Integrating along a streamline, we get the Bernoulli's equation for a compressible flow as
For adiabatic frictionless flows the Bernoulli's equation is identical to the energy equation. Recall, that this is an isentropic flow, so that the Tds equation is given by
Tds = dh - vdp
For isentropic flow, ds=0
Hence, the Euler equation (39.11) can also be written as
Vdv + dh = 0
This is identical to the adiabatic form of the energy Eq. (39.10).
Momentum Principle for a Control Volume
For a finite control volume between Sections 1 and 2 (Fig. 39.3), the momentum principle is
where F is the x-component of resultant force exerted on the fluid by the walls. Note that the momentum principle, Eq. (39.13), is applicable even when there are frictional dissipative processes within the control volume