Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering

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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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering            (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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering    (39.7)

where Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering  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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering  (where the subscripts are for Sections 1 and 2), the second term on the left of Eq. (39.7) yields

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering

The work done on the control surfaces is

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering

The rate of heat transfer to the control volume is

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering

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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering  , we get

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering          (39.8) 

We know that the density p is given by Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering /VA hence the first term on the right may be expressed in terms of Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering(specific volume=1/ρ). 
Equation (39.8) can be rewritten as

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering           (39.9) 

  • NOTE:- Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering 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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering  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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering. (39.9) gets reduced to 

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering                 (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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering  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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering                               (39.11)
 

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

Speed of Sound - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering       (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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering

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 - 2 Notes | Study Additional Documents & Tests for Mechanical Engineering - Mechanical Engineering                   (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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