Page 50 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 31
Comparing the above definitions it is easily deduced that the mechanical efficiency
m , which is simply the ratio of shaft power to rotor power, is
m D 0 / t .or 0 / h /.
In the following paragraphs the various definitions of hydraulic and adiabatic effi-
ciency are discussed in more detail.
For an incremental change of state through a turbomachine the steady flow energy
equation, eqn. (2.5), can be written
P
2
1
dQ d P W x DPm[dh C d.c / C gdz].
2
From the second law of thermodynamics

1
P
dQ 6 PmTds DPm dh dp .

Eliminating dQ between these two equations and rearranging

1 1 2
d P W x 6 Pm dp C d.c / C gdz . (2.19)
2
For a turbine expansion, noting P W x D P W t > 0, integrate eqn. (2.19) from the initial
state 1 to the final state 2,
1 1 1
Z
2
P W x 6 Pm dp C .c 2 c / C g.z 1 z 2 / . (2.20)
2 2 1 2
For a reversible adiabatic process, Tds D 0 D dh dp/ . The incremental
maximum work output is then
1
2
D Pm[dh C d.c / C gdz]
2
d P W x max
Hence, the overall maximum work output between initial state 1 and final state 2 is
Z 1 1
2
DPm dh C d.c / C gdz
P W x max
2
2
z 2 /] .2.20a/
DPm[.h 01 h 02s / C g.z 1
where the subscript s in eqn. (2.20a) denotes that the change of state between 1 and
2 is isentropic.
For an incompressible fluid, in the absence of friction, the maximum work output
from the turbine (ignoring frictional losses) is
DPmg[H 1 H 2 ], (2.20b)
P W x max
1 2
where gH D p/ C c C gz.
2
Steam and gas turbines
Figure 2.5a shows a Mollier diagram representing the expansion process through an
adiabatic turbine. Line 1 2 represents the actual expansion and line 1 2s the ideal
or reversible expansion. The fluid velocities at entry to and at exit from a turbine