fluid_mechanics

218 Chapter 5 ■ Finite Control Volume Analysis
observed from a frame of reference attached to the fixed and nondeforming control volume. The
fluid velocity measured relative to a fixed control surface is an absolute velocity, V. The velocity
of the nozzle exit flow as viewed from the nozzle is called the relative velocity, W. The absolute
and relative velocities, V and W, are related by the vector relationship
V W U
(5.43)
where U is the velocity of the moving nozzle as measured relative to the fixed control surface.
The cross product and the dot product involved in the moment-of-momentum flow term of
Eq. 5.42,
1r V2rV nˆ dA
cs
The algebraic sign
of r V is obtained
by the right-hand
rule.
can each result in a positive or negative value. For flow into the control volume, V nˆ is negative.
For flow out, V nˆ is positive. The correct algebraic sign to assign the axis component of r V
can be ascertained by using the right-hand rule. The positive direction along the axis of rotation is
the direction the thumb of the right hand points when it is extended and the remaining fingers are
curled around the rotation axis in the positive direction of rotation as illustrated in Fig. 5.5. The direction
of the axial component of r V is similarly ascertained by noting the direction of the cross
product of the radius from the axis of rotation, rê r , and the tangential component of absolute velocity,
V u ê u . Thus, for the sprinkler of Fig. 5.4, we can state that
c cs
1r V2rV nˆ dA d 1r 2 V u2 21m # 2
(5.44)
where, because of mass conservation, m #
axial
is the total mass flowrate through both nozzles. As was
demonstrated in Example 5.7, the mass flowrate is the same whether the sprinkler rotates or not. The
correct algebraic sign of the axial component of r V can be easily remembered in the following
way: if V u and U are in the same direction, use ; if V u and U are in opposite directions, use .
The torque term 3 g 1r F2 contents of the control volume 4 of the moment-of-momentum equation 1Eq.
5.422 is analyzed next. Confining ourselves to torques acting with respect to the axis of rotation
only, we conclude that the shaft torque is important. The net torque with respect to the axis of rotation
associated with normal forces exerted on the contents of the control volume will be very
small if not zero. The net axial torque due to fluid tangential forces is also negligibly small for the
control volume of Fig. 5.4. Thus, for the sprinkler of Fig. 5.4
a B1r F2 contents of the R T shaft
(5.45)
control volume axial
Note that we have entered T shaft as a positive quantity in Eq. 5.45. This is equivalent to assuming
that T shaft is in the same direction as rotation.
For the sprinkler of Fig. 5.4, the axial component of the moment-of-momentum equation 1Eq.
5.422 is, from Eqs. 5.44 and 5.45
r 2 V u2 m # T shaft
(5.46)
We interpret T shaft being a negative quantity from Eq. 5.46 to mean that the shaft torque actually
opposes the rotation of the sprinkler arms as shown in Fig. 5.4. The shaft torque, T shaft , opposes
rotation in all turbine devices.
+
F I G U R E 5.5
Right-hand rule convention.