Basics of Fluid Mechanics, 2014a

212 CHAPTER 7. ENERGY CONSERVATION
7.3.2 Energy Equation in Frictionless Flow and Steady State
In cases where the flow can be estimated without friction or where a quick solution is
needed the friction and other losses are illuminated from the calculations. This imaginary
fluid reduces the amount of work in the calculations and Ideal Flow Chapter is dedicated
in this book. The second low is the core of “no losses” and can be employed when
calculations of this sort information is needed. Equation (2.21) which can be written as
Using the multiplication rule change equation (7.76)
dq rev = Tds= dE u + Pdv (7.76)
dq rev = dE u + d (P v) − vdP = dE u + d
( P
ρ
)
− vdP (7.77)
integrating equation (7.77) yields
∫ ∫
dq rev =
∫
dE u +
( ) ∫ P
d −
ρ
vdP (7.78)
( ) ∫ P dP
q rev = E u + −
ρ ρ
(7.79)
Integration over the entire system results in
∫
Q rev =
V
h
{(
}} ( )){
P
E u +
ρ
∫ (∫ dP
ρdV −
V ρ
Taking time derivative of the equation (7.80) becomes
h
)
ρdV (7.80)
˙Q rev = D ∫ {(
}} ( )){
P
E u + ρdV − D ∫ (∫ ) dP
ρdV (7.81)
Dt V ρ Dt V ρ
Using the Reynolds Transport Theorem to transport equation to control volume results
in
˙Q rev = d ∫ ∫
hρdV + hU rn ρdA+ D ∫ (∫ ) dP
ρdV (7.82)
dt V
A Dt V ρ
As before equation (7.81) can be simplified for uniform flow as
[
(∫ ∫ )]
dP ˙Q rev = ṁ (h out − h in ) −
dP ρ ∣ −
out
ρ ∣ (7.83)
in
or
(∫ dP
˙q rev =(h out − h in ) −
ρ
∫ dP
∣ −
out
ρ
)
∣
in
(7.84)