Page 52 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 33
1 2
1 2
If the difference between the inlet and outlet kinetic energies is small, i.e. c + c ,
2 1 2 2
then
h 2s / (2.21a)
tt D .h 1 h 2 //.h 1
When the exhaust kinetic energy is not usefully employed and entirely wasted,
the relevant adiabatic efficiency is the total-to-static efficiency ts . In this case the
ideal turbine work is that obtained between state points 01 and 2s. Thus

1 2
h 02s C c /
ts D .h 01 h 02 //.h 01
2 2s
h 2s /. .2.22/
D .h 01 h 02 //.h 01
If the difference between inlet and outlet kinetic energies is small, eqn. (2.22)
becomes
1 2
h 2s C c /. (2.22a)
ts D .h 1 h 2 //.h 1
2 1
A situation where the outlet kinetic energy is wasted is a turbine exhausting directly
to the surroundings rather than through a diffuser. For example, auxiliary turbines
used in rockets often do not have exhaust diffusers because the disadvantages of
increased mass and space utilisation are greater than the extra propellant required
as a result of reduced turbine efficiency.

Hydraulic turbines
When the working fluid is a liquid, the turbine hydraulic efficiency h , is defined
as the work supplied by the rotor in unit time divided by the hydrodynamic energy
difference of the fluid per unit time, i.e.

W x W x
h D D . (2.23)
g.H 1 H 2 /
W x max
Efficiency of compressors and pumps

The isentropic efficiency c of a compressor or the hydraulic efficiency of a pump
h is broadly defined as,

useful (hydrodynamic) energy input to fluid in unit time
c .or h / D .
power input to rotor
The power input to the rotor (or impeller) is always less than the power supplied
at the coupling because of external energy losses in the bearings and glands, etc.
Thus, the overall efficiency of the compressor or pump is
useful (hydrodynamic) energy input to fluid in unit time
o D
power input to coupling of shaft

Hence the mechanical efficiency is

m D o / c .or o / h /.