Basics of Fluid Mechanics, 2014a

216 CHAPTER 7. ENERGY CONSERVATION
Inserting the potential energy due to the centrifugal forces into the energy equation
yields
Energy Equation in Accelerated Coordinate
˙Q − Ẇ = d ∫ [E u + U 2
dt cv 2 + a x x + a y y +(a z + g)z − ω2 r 2 ]
ρdV
2
∫
+
(h + U 2
cv 2 + a x x + a y y +(a z + g) z − z ω2 r 2 )
U rn ρdA
2
∫
+ PU bn dA
cv
(7.104)
7.4.4 Simplified Energy Equation in Accelerated Coordinate
7.4.4.1 Energy Equation in Accelerated Coordinate with Uniform Flow
One of the way to simplify the general equation (7.104) is to assume uniform flow. In
that case the time derivative term vanishes and equation (7.104) can be written as
˙Q − Ẇ = ∫
cv
Energy Equation in steady state
(h + U 2
2 + a x x + a y y +(a z + g) − z ω2 r 2 )
U rn ρdA
2
∫
+ PU bn dA
cv
(7.105)
Further simplification of equation (7.105) by assuming uniform flow for which
(
˙Q − Ẇ = h + U 2
)
2 + a x x + a y y +(a z + g) − z ω2 r 2
U rn ρdA (7.106)
2
∫
+ P U bn dA
Note that the acceleration also have to be averaged. The correction factors have to
introduced into the equation to account for the energy averaged verse to averaged
velocity (mass averaged). These factor make this equation with larger error and thus
less effective tool in the engineering calculation.
7.4.5 Energy Losses in Incompressible Flow
In the previous sections discussion, it was assumed that there are no energy loss. However,
these losses are very important for many real world application. And these losses
cv