fluid_mechanics

652 Chapter 12 ■ Turbomachines
The axial component of this equation applied to the one-dimensional simplification of flow
through a turbomachine rotor with section 112 as the inlet and section 122 as the outlet results in
The Euler turbomachine
equation is the
axial component of
the momentof-momentum
equation.
T shaft m # 11r 1 V u1 2 m # 21r 2 V u2 2
(12.2)
where T shaft is the shaft torque applied to the contents of the control volume. The “” is associated
with mass flowrate into the control volume and the “ ” is used with the outflow. The sign of the V u
component depends on the direction of V u and the blade motion, U. If V u and U are in the same direction,
then V u is positive. The sign of the torque exerted by the shaft on the rotor, T shaft , is positive
if T shaft is in the same direction as rotation, and negative otherwise.
As seen from Eq. 12.2, the shaft torque is directly proportional to the mass flowrate, m # rQ.
1It takes considerably more torque and power to pump water than to pump air with the same volume
flowrate.2 The torque also depends on the tangential component of the absolute velocity, V u .
Equation 12.2 is often called the Euler turbomachine equation.
Also recall that the shaft power, W # shaft, is related to the shaft torque and angular velocity by
W # shaft T shaft v
(12.3)
By combining Eqs. 12.2 and 12.3 and using the fact that U vr, we obtain
W # #
shaft m 1 1U 1 V u1 2 m # 21U 2 V u2 2
(12.4)
Again, the value of is positive when and U are in the same direction and negative otherwise.
Also, W # V u V u
shaft is positive when the shaft torque and are in the same direction and negative otherwise.
Thus, W # v
shaft is positive when power is supplied to the contents of the control volume 1pumps2
and negative otherwise 1turbines2. This outcome is consistent with the sign convention involving
the work term in the energy equation considered in Chapter 5 1see Eq. 5.672.
Finally, in terms of work per unit mass, w shaft W # shaftm # , we obtain
w shaft U 1 V u1 U 2 V u2
(12.5)
where we have used the fact that by conservation of mass, m # 1 m # 2. Equations 12.3, 12.4, and 12.5
are the basic governing equations for pumps or turbines whether the machines are radial-, mixed-,
or axial-flow devices and for compressible and incompressible flows. Note that neither the axial
nor the radial component of velocity enter into the specific work 1work per unit mass2 equation.
[In the above merry-go-round example the amount of work your friend does is independent of how
fast you jump “up” 1axially2 or “out” 1radially2 as you exit. The only thing that counts is your u
component of velocity.]
Another useful but more laborious form of Eq. 12.5 can be obtained by writing the righthand
side in a slightly different form based on the velocity triangles at the entrance or exit as shown
generically in Fig. 12.5. The velocity component V x is the generic through-flow component of velocity
and it can be axial, radial, or in-between depending on the rotor configuration. From the
large right triangle we note that
or
V 2 V 2 u V 2 x
V 2 x V 2 V 2 u
(12.6)
x
θ
V
W
V x
U
V θ
F I G U R E 12.5 Velocity triangle: V
absolute velocity, W relative velocity, U blade velocity.