Page 62 - Compression Machinery for Oil and Gas

Centrifugal Compressors Chapter 3 53

If these flow channels are in a rotating system (e.g., in an impeller), mechan-
ical energy is added to or removed from the system. Nevertheless, if the veloc-
ities are considered in a rotating system of coordinates, above principles are
applicable as well.
Another important concept is the conservation of momentum. The change in
momentum M of gas flowing from a point 1 to a point 2 is its mass times its
velocity (mc), and is also the sum of all forces F acting. The change in momen-
tum is thus:

!
dM ! ! !
¼ _ mc 2 c 1 ¼F
dt
To change the momentum of this gas, either by changing the velocity or the
direction of the gas (or both), a force is necessary. Fig. 3.24 outlines this concept
for the case of a bent, conical pipe. The gas flows in through the area A 1 with w 1 ,
p 1 , and out through the flow area A 2 with w 2 , p 2 . The differences in the force is
due to the pressure (p 1 A 1 and p 2 A 2 , respectively), and the fact that a certain mass
flow of gas is forced to change its direction generates a reaction force F R . Split
into x and y coordinates, and considering that

_ m ¼ ρ A 1 w 1 ¼ ρ A 2 w 2
2
1
we get (due to the choice of coordinates, w 1y ¼0)
ð
ð
x : ρA 1 w 1 w 2x w 1 Þ ¼ p 1 A 1 p 2 A 2 Þ + F Rx
x

ð
y : ρA 1 w 1 w 2y ¼ p 2 A 2 Þ + F Ry
y
W
2
p 2
y 2
x
A 2
1

U
W p 1
1
A
1
2
1
(p + QW )
1 1
F x A
F y 1
F
(p + QW )A
2
2 2 2
FIG. 3.24 Conservation of momentum [3].