Speed of Sound Notes | EduRev

Fluid Mechanics

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Speed of Sound

  • The so-called sound speed is the rate of propogation of a pressure pulse of infinitesimal strength through a still fluid. It is a thermodynamic property of a fluid.
  • A pressure pulse in an incompressible flow behaves like that in a rigid body. A displaced particle displaces all the particles in the medium. In a compressible fluid, on the other hand, displaced mass compresses and increases the density of neighbouring mass which in turn increases density of the adjoining mass and so on. Thus, a disturbance in the form of an elastic wave or a pressure wave travels through the medium. If the amplitude and theerfore the strength of the elastic wave is infinitesimal, it is termed as acoustic wave or sound wave. 
  • Figure 39.1(a) shows an infinitesimal pressure pulse propagating at a speed " a " towards still fluid (= 0) at the left. The fluid properties ahead of the wave are p,T and p while the properties behind the wave are p+dp, T+dT and  p +dp. The fluid velocity dV is directed toward the left following wave but much slower. 
  • In order to make the analysis steady, we superimpose a velocity " a " directed towards right, on the entire system (Fig. 39.1(b)). The wave is now stationary and the fluid appears to have velocity " a " on the left and (a - dV) on the right. The flow in Fig. 39.1 (b) is now steady and one dimensional across the wave. Consider an area A on the wave front. A mass balance gives 

Speed of Sound Notes | EduRev
 

Fig 39.1: Propagation of a sound wave 
(a) Wave Propagating into still Fluid        (b) Stationary Wave

 

Speed of Sound Notes | EduRev           (39.1)

 

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.

  • Applying the momentum balance on the same control volume in Fig. 39.1 (b). It says that the net force in the x direction on the control volume equals the rate of outflow of x momentum minus the rate of inflow of x momentum. In symbolic form, this yields

Speed of Sound Notes | EduRev

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.

  • Simplifying the momentum equation, we get

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

Speed of Sound Notes | EduRev                   (39.3a)           

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

Speed of Sound Notes | EduRev                                   (39.3b)   

Note that

(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

Speed of Sound Notes | EduRev                      (39.4)

For a perfect gas, by using of Speed of Sound Notes | EduRev   = const and p = pRT, we deduce the speed of sound as

Speed of Sound Notes | EduRev            (39.5)


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 Speed of Sound Notes | EduRev , 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

Speed of Sound Notes | EduRev

Speed of Sound Notes | EduRev

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

Speed of Sound Notes | EduRev

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; 

 

  • Continuity Equation
  • Energy Equation
  • Bernoulli and Euler Equations
  • Momentum Principle for a Control Volume

Speed of Sound Notes | EduRev
Fig 39.3 One-Dimensional Approximation

Continuity Equation

For steady one-dimensional flow, the equation of continuity is

p (x).V  x). A  (x) = Const

Differentiating(after taking log), we get 

Speed of Sound Notes | EduRev            (39.6)

Energy Equation

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

Speed of Sound Notes | EduRev    (39.7)

where Speed of Sound Notes | EduRev  is the kinetic energy per unit mass.

Let us discuss the various terms from above equation:

  • The first term on the left hand side signifies the rate of change of energy (internal + kinetic) within the control volume
  • The second term depicts the flux of energy out of control surface.
  • The first term on the right hand side represents the work done on the control surface
  • The second term on the right means the heat transferred through the control surface.

It is to be noted that dA is directed along the outward normal.

  • Assuming steady state, the first term on the left hand side of Eq. (39.7) is zero. Writing  Speed of Sound Notes | EduRev  (where the subscripts are for Sections 1 and 2), the second term on the left of Eq. (39.7) yields

Speed of Sound Notes | EduRev

The work done on the control surfaces is

Speed of Sound Notes | EduRev

The rate of heat transfer to the control volume is

Speed of Sound Notes | EduRev

where Q is the heat added per unit mass (in J/kg).

  • Invoking all the aforesaid relations in Eq. (39.7) and dividing by Speed of Sound Notes | EduRev  , we get

Speed of Sound Notes | EduRev          (39.8) 

We know that the density p is given by Speed of Sound Notes | EduRev /VA hence the first term on the right may be expressed in terms of Speed of Sound Notes | EduRev(specific volume=1/ρ).
Equation (39.8) can be rewritten as

Speed of Sound Notes | EduRev           (39.9) 

  • NOTE:- Speed of Sound Notes | EduRev is the work done (per unit mass) by the surrounding in pushing fluid into the control volume. Following a similar argument, is the  Speed of Sound Notes | EduRev  work done by the fluid inside the control volume on the surroundings in pushing fluid out of the control volume.
     
  • Since h = e + p Speed of Sound Notes | EduRev. (39.9) gets reduced to 

Speed of Sound Notes | EduRev                 (39.10) 

This is energy equation, which is valid even in the presence of friction or non-equilibrium conditions between secs 1 and 2.

  • It is evident that the sum of enthalpy and kinetic energy remains constant in an adiabatic flow. Enthalpy performs a similar role that internal energy performs in a nonflowing system. The difference between the two types of systems is the flow work Speed of Sound Notes | EduRev  required to push the fluid through a section.
     

Bernoulli and Euler Equations

  • For inviscid flows, the steady form of the momentum equation is the Euler equation,
     

Speed of Sound Notes | EduRev                               (39.11)
 

Integrating along a streamline, we get the Bernoulli's equation for a compressible flow as

Speed of Sound Notes | EduRev       (39.12)

  • 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

Therefore,

Speed of Sound Notes | EduRev

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

Speed of Sound Notes | EduRev                   (39.13)

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

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