The conservation of momentum equation becomes:
ρβ 2 V22 A 2  ρβ1 V12 A 1 = Fx
(2.25)
for steady flow with constant density. With Equation 2.16 the steady flow conservation of linear
momentum equation takes on the familiar form
ρQ (β 2 V2  β1 V1 ) = Fx
(2.26)
2.2.3 Conservation of Energy
The First Law of Thermodynamics can be written:
dE
QW =
(2.27)
dt
where:
=
Rate at which heat is added to a fluid system
Q
=
Rate at which a fluid system does work on its surroundings
W
E
=
Energy of the system
Then dE/dt is the rate of change of energy in the system.
The statement of conservation of energy for a control volume is then:
Flux of energy
Flux of energy
Time rate of change
= QW
out of the

into the control
+
of energy in the
control volume
volume
control volume
The choice of a control volume is arbitrary. To illustrate the procedure, the control volume is
reduced to the size of a streamtube connecting dA1 and dA2 as shown in Figure 2.3. The
streamtube is bounded by streamlines through which no mass or momentum enters.
Figure 2.3. The streamtube as a control volume.
2.8