Handbook of Turbomachinery Second Edition Revised - Ventech!

change of angular momentum:
Fy ¼ dðmVyÞ
dt
Performing the derivative, assuming constant Vy and mass flow rate _m,
results in
G ¼ _mðVy1r1 Vy2r2Þ ð5Þ
The energy transfer per time (power) is obtained by multiplying both sides
of Eq. (5) by the rotational velocity o:
P ¼ Go ¼ o _mðVy1r1 Vy2r2Þ ð6Þ
The power P can also be calculated from the h–s diagram for the actual
process as
P ¼ _mðhin hdisÞ ¼ _mDhactual ð7Þ
Combining Eqs. (6) and (7) and defining the wheel speed U as
U ¼ or ð8Þ
results in the Euler equation for energy transfer in a turbomachine:
Dhactual ¼ U1Vy1 U2Vy2 ð9Þ
The Euler equation, as derived here, assumes adiabatic flow through the
turbine, since enthalpy change is allowed only across the rotor. The Euler
equation relates the thermodynamic energy transfer to the change in
velocities at the inlet and exit of the rotor. This leads us to examine these
velocities more closely, since they determine the work extracted from the
turbine.
Velocity Diagrams
Euler’s equation shows that energy transfer in a turbine is directly related to
the velocities in the turbine. It is convenient to graphically display these
velocities at the rotor inlet and exit in diagrams called velocity or vector
diagrams. These diagrams are drawn in a single plane. For an axial-flow
turbine, they are drawn in the x–y plane at a specific value of r. At the inlet
of a radial-inflow turbine, where the flow is generally in the r–y plane, the
diagram is drawn in that plane at a specific value of x. The exit diagram for
a radial-inflow turbine is drawn in the x–y plane at a specific value of r.
Figure 8(a) shows the velocity diagram at the inlet to an axial-flow
rotor. The stator and rotor blade shapes are included to show the relation
Copyright © 2003 Marcel Dekker, Inc.
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