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