fluid_mechanics

216 Chapter 5 ■ Finite Control Volume Analysis
x
z
T
r
T = r × F
y
F
For a system, the
rate of change of
moment-of-momentum
equals the net
torque.
Equation 5.35 is valid for every particle of a system. For a system 1collection of fluid particles2,
we need to use the sum of both sides of Eq. 5.35 to obtain
D
(5.36)
sys Dt 31r V2r dV4 a 1r F2 sys
where
We note that
(5.37)
(5.38)
since the sequential order of differentiation and integration can be reversed without consequence. 1Recall
that the material derivative, D1 2Dt, denotes the time derivative following a given system; see
Section 4.2.1.2 Thus, from Eqs. 5.36 and 5.38 we get
D
(5.39)
Dt 1r V2r d V a 1r F2 sys
sys
or
The sketch in the margin illustrates what torque, T r F, is. For a control volume that is instantaneously
coincident with the system, the torques acting on the system and on the control volume
contents will be identical:
a 1r F2 sys a 1r F2 cv
(5.40)
Further, for the system and the contents of the coincident control volume that is fixed and nondeforming,
the Reynolds transport theorem 1Eq. 4.192 leads to
or
a r dF particle a 1r F2 sys
D
D
Dt 1r V2r dV sys
31r V2r dV4
sys Dt
the time rate of change of the sum of external torques
moment-of-momentum of the system acting on the system
D
Dt sys
1r V2r dV 0 0t cv
1r V2r dV cs
1r V2rV nˆ dA
time rate of change time rate of change net rate of flow
of the moment-of- of the moment-of- of the moment-ofmomentum
of the momentum of the momentum through
system contents of the the control surface
control volume
(5.41)
For a control volume that is fixed 1and therefore inertial2 and nondeforming, we combine Eqs. 5.39,
5.40, and 5.41 to obtain the moment-of-momentum equation:
0
0t cv
1r V2r dV cs
1r V2rV nˆ dA a 1r F2 contents of the
control volume
(5.42)
An important category of fluid mechanical problems that is readily solved with the help of
the moment-of-momentum equation 1Eq. 5.422 involves machines that rotate or tend to rotate around
a single axis. Examples of these machines include rotary lawn sprinklers, ceiling fans, lawn mower
blades, wind turbines, turbochargers, and gas turbine engines. As a class, these devices are often
called turbomachines.
5.2.4 Application of the Moment-of-Momentum Equation 3
We simplify our use of Eq. 5.42 in several ways:
1. We assume that flows considered are one-dimensional 1uniform distributions of average velocity
at any section2.
3 This section may be omitted, along with Sections 5.2.3 and 5.3.5, without loss of continuity in the text material. However, these sections
are recommended for those interested in Chapter 12.