Page 53 - Fluid Mechanics and Thermodynamics of Turbomachinery

34 Fluid Mechanics, Thermodynamics of Turbomachinery
In eqn. (2.19), for a compressor or pump process, replace d P W x with d P W c and
rearrange the inequality to give the incremental work input

1 1 2
P
dW c = Pm dp C d.c / C gdz . (2.24)
2
The student should carefully check the fact that the rhs of this inequality is positive,
working from eqn. (2.19)
For a complete adiabatic compression process going from state 1 to state 2, the
overall work input rate is
Z
2 dp 1
2
P W c = Pm C .c 2 c / C g.z 2 z 1 / . (2.25)
2 2 1
1
For the corresponding reversible adiabatic compression process, noting that Tds D
0 D dh dp/ , the minimum work input rate is
Z 2s 1
P W c min DPm 2 z 1 /]. (2.26)
dh C dc C gdz DPm[.h 02s h 01 / C g.z 2
2
1
From the steady flow energy equation, for an adiabatic process in a compressor
P W c DPm.h 02 h 01 /. (2.27)
Figure 2.5b shows a Mollier diagram on which the actual compression process
is represented by the state change 1 2 and the corresponding ideal process by
1 2s. For an adiabatic compressor the only meaningful efficiency is the total-to-total
efficiency which is
minimum adiabatic work input per unit time
c D
actual adiabatic work input to rotor per unit time
h 02s h 01
D . .2.28/
h 02 h 01
1 2
1 2
If the difference between inlet and outlet kinetic energies is small, c + c and
2 1 2 2
h 2s h 1
c D . (2.28a)
h 2 h 1
For incompressible flow, eqn. (2.25) gives
1
p 1 // C .c 2 2 H 1 ].
4W p D P W p / Pm = [.p 2 2 c / C g.z 2 z 1 /] = g[H 2
1
2
For the ideal case with no fluid friction
H 1 ]. (2.29)
W p min D g[H 2
For a pump the hydraulic efficiency is defined as
g[H 2 H 1 ]
W p min
h D D . (2.30)
W p W p