fluid_mechanics

12.8 Turbines 681
U
Section (1)
W 2
W 1
V 2
U 2
Section (2)
V 1 U 1 V θ1
F I G U R E 12.30 Inlet and exit velocity
triangles for the impulse turbine shown in Fig. 12.29.
as indicated during its passage across the blade, the blade must push on the fluid in the direction
opposite of the blade motion. Hence, the fluid pushes on the blade in the direction of the blade’s
motion—the fluid does work on the blade 1a turbine2.
E XAMPLE 12.8
Non-Pelton Wheel Impulse Turbine (Dental Drill)
GIVEN An air turbine used to drive the high-speed drill used
by your dentist is shown in Figs. 12.29 and E12.8a. Air exiting
from the upstream nozzle holes forces the turbine blades to move
in the direction shown. The turbine rotor speed is 300,000 rpm,
the tangential component of velocity out of the nozzle is twice the
blade speed, and the tangential component of the absolute velocity
out of the rotor is zero.
FIND Estimate the shaft energy per unit mass of air flowing
through the turbine.
SOLUTION
We use the fixed, nondeforming control volume that includes the
turbine rotor and the fluid in the rotor blade passages at an instant
of time 1see Fig. E12.8b2. The only torque acting on this
control volume is the shaft torque. For simplicity we analyze this
problem using an arithmetic mean radius, r m , where
r m 1 2 1r 0 r i 2
A sketch of the velocity triangles at the rotor entrance and exit is
shown in Fig. E12.8c.
Application of Eq. 12.5 1a form of the moment-of-momentum
equation2 gives
w shaft U 1 V u1 U 2 V u2
where w shaft is shaft energy per unit of mass flowing through
the turbine. From the problem statement, V u1 2U and V u2 0,
where
U vr m 1300,000 revmin211 min60 s212p radrev2
10.168 in. 0.133 in.22112 in.ft2
394 fts
(1)
(2)
is the mean-radius blade velocity. Thus, Eq. 112 becomes
w shaft U 1 V u1 2U 2 21394 fts2 2
310,000 ft 2 s 2
1310,000 ft 2 s 2 2132.1741ft # lbm21lb # s 2 22
9640 ft # lblbm
(Ans)
COMMENT For each lbm of air passing through the turbine
there is 9640 ft # lb of energy available at the shaft to drive the
drill. However, because of fluid friction, the actual amount of energy
given up by each slug of air will be greater than the amount
available at the shaft. How much greater depends on the efficiency
of the fluid-mechanical energy transfer between the fluid
and the turbine blades.
Recall that the shaft power, W # shaft, is given by
W # shaft m # w shaft
Hence, to determine the power we need to know the mass flowrate,
m # , which depends on the size and number of the nozzles. Although
the energy per unit mass is large 1i.e., 9640 ft # lblbm2, the
flowrate is small, so the power is not “large.”