22
1 Working Principles
through the elementary streamtube. Within the mean line representation, the result
for the average streamsurface is considered to be representative for the entire flow.
We thus note co
m r
M
m 2 r 1 .1 dm
W
= − ∫
.
With
m r
dm1 .1 dr
=
it follows
After multiplication by Ω and division by
m , the moment of momentum equation in
the relative frame results in Eq. (1.25).
Equation (1.26) may be written as
M is the moment term associated to the turning of the relative flow (change of
rw u ), in other words, the lift. The expression demonstrates, for a driven machine,
that the moment required to drive the rotor must balance the moment of two forces,
namely Coriolis force and lift force. In Eq. (1.25) this means that the term u u
2
2
1
2
−
represents the work supplied against the Coriolis force and u w
u w
u
u
2 2
1 1
−
the work
supplied against the lift force. The interpretation with a driving machine is similar,
but for work delivered by the Coriolis and lift forces.
The above analysis implies that two forces intervene in the rotor work. The effect
of lift is the simplest to understand. For the axial pump and the axial turbine, analysed earlier (Figs. 1.2 and 1.3), it is obvious that turning of the flow within the rotor
generates lift on the blades. Physically, this lift is a consequence of a pressure difference between both sides of a blade or a rotor blade channel, generated by turning the
flow. We will analyse the origin of this pressure difference in Chap. 2. Figures 1.2
and 1.3 demonstrate that lift intervenes in the rotor work. In the axial examples in
Figs. 1.2 and 1.3, u 1 = u 2 , but, meanwhile, we understand that the moment of forces
intervenes with work and that a change of radius affects the resulting moment.
The role of the Coriolis force in the rotor work only follows in an abstract way
from the above analysis. At this stage, it is obvious that rotation induces forces in
the fluid, namely the Coriolis force and the centrifugal force. Only the Coriolis
force has a component in the tangential direction. This component intervenes in the
moment of momentum. The concrete way rotation-induced forces intervene is by
causing pressure differences between the two sides of a blade. Associated to the Coriolis force, there is a pressure difference, with a moment of the resulting rotor force
around the rotation axis. This is analysed further for radial machines in Sect. 1.7 and
in Chap. 3, Sect. 3.3.2.
For the time being, confusion might arise by the finding that there is a work
contribution by the centrifugal force in the work and energy equations on a streamline in a rotating frame, namely expressions (1.13) and (1.14). With the above
analysis it is clear that the centrifugal force does not contribute to the energy exchange between the rotor and the fluid (the rotor work), as the centrifugal force has
no moment around the rotation axis. It becomes obvious from expressions (1.13)
and (1.14) that the centrifugal force intervenes in the change of kinetic energy and
2
2
co
2
1
M
m 2 r dr
m ( r
r ).
W
W
= −
= −
−
∫
M
M
M
co
= −
− .
1 Working Principles
through the elementary streamtube. Within the mean line representation, the result
for the average streamsurface is considered to be representative for the entire flow.
We thus note co
m r
M
m 2 r 1 .1 dm
W
= − ∫
.
With
m r
dm1 .1 dr
=
it follows
After multiplication by Ω and division by
m , the moment of momentum equation in
the relative frame results in Eq. (1.25).
Equation (1.26) may be written as
M is the moment term associated to the turning of the relative flow (change of
rw u ), in other words, the lift. The expression demonstrates, for a driven machine,
that the moment required to drive the rotor must balance the moment of two forces,
namely Coriolis force and lift force. In Eq. (1.25) this means that the term u u
2
2
1
2
−
represents the work supplied against the Coriolis force and u w
u w
u
u
2 2
1 1
−
the work
supplied against the lift force. The interpretation with a driving machine is similar,
but for work delivered by the Coriolis and lift forces.
The above analysis implies that two forces intervene in the rotor work. The effect
of lift is the simplest to understand. For the axial pump and the axial turbine, analysed earlier (Figs. 1.2 and 1.3), it is obvious that turning of the flow within the rotor
generates lift on the blades. Physically, this lift is a consequence of a pressure difference between both sides of a blade or a rotor blade channel, generated by turning the
flow. We will analyse the origin of this pressure difference in Chap. 2. Figures 1.2
and 1.3 demonstrate that lift intervenes in the rotor work. In the axial examples in
Figs. 1.2 and 1.3, u 1 = u 2 , but, meanwhile, we understand that the moment of forces
intervenes with work and that a change of radius affects the resulting moment.
The role of the Coriolis force in the rotor work only follows in an abstract way
from the above analysis. At this stage, it is obvious that rotation induces forces in
the fluid, namely the Coriolis force and the centrifugal force. Only the Coriolis
force has a component in the tangential direction. This component intervenes in the
moment of momentum. The concrete way rotation-induced forces intervene is by
causing pressure differences between the two sides of a blade. Associated to the Coriolis force, there is a pressure difference, with a moment of the resulting rotor force
around the rotation axis. This is analysed further for radial machines in Sect. 1.7 and
in Chap. 3, Sect. 3.3.2.
For the time being, confusion might arise by the finding that there is a work
contribution by the centrifugal force in the work and energy equations on a streamline in a rotating frame, namely expressions (1.13) and (1.14). With the above
analysis it is clear that the centrifugal force does not contribute to the energy exchange between the rotor and the fluid (the rotor work), as the centrifugal force has
no moment around the rotation axis. It becomes obvious from expressions (1.13)
and (1.14) that the centrifugal force intervenes in the change of kinetic energy and
2
2
co
2
1
M
m 2 r dr
m ( r
r ).
W
W
= −
= −
−
∫
M
M
M
co
= −
− .