21
1.5 Basic Laws for Rotating Duct Parts
(change of angular momentum multiplied by rotational speed). Further, we note
that the rotor work equation has been derived here based on a one-dimensional
flow representation (mean line representation) but that formulation for a real threedimensional flow follows from the same principles.
1.5.3 Moment of Momentum in the Relative Frame: Forces
Intervening in the Rotor Work
From a velocity triangle (Fig. 1.8) follows geometrically:
v u w
= + , or v u w
u
u
= + .
The velocity triangle in Fig. 1.8 is drawn deliberately with negative w u . It follows
that
So:
(1.25)
The Eq. (1.25) also follows from a moment of momentum balance, applied in the
relative frame. Fictitious forces to be introduced are
Co
2
w
W
= −
×
and
2
Cf
r
W
=
.
The relative velocity
w has an axial component, a radial component and a tangential one. The axial component does not intervene in the Coriolis force. A tangential unit vector in the sense of the rotation may be noted as
so that
r r
u
r
r u
u r
Co
2
( w 1 w 1 1 )
2 w 1 2 w 1 .
W
W
W
W
= −
×
+
×
= −
+
According to Fig. 1.7, r
m m r
w
w 1 .1
=
, with
1 m a unit vector in the meridional direction with positive axial and radial components. The centrifugal force does not
contribute to the force moment. Neither does the radial component of the Coriolis
force. The moment of momentum balance in the relative frame results in
(1.26)
with
co
m m r
M
2 w 1 .1 dA dm r
W
r
= −
∫
,
where dA is the cross-section area of an elementary annular streamtube around the
average circumferential streamsurface in Fig. 1.7. The elementary length in the meridional plane is indicated by dm. So the term ρ w m dA represents the mass flow rate
uv u uw
u
2
u
= +
.
2
2
2
1
2 2u
1 1u
W u u u w
u w .
D = − +
−
u
r
1 1 1 ,
W
= ×
2 2u
1 1u
co
m( r w
r w ) M M ,
−
= +
Fig. 1.8 Velocity triangle
1.5 Basic Laws for Rotating Duct Parts
(change of angular momentum multiplied by rotational speed). Further, we note
that the rotor work equation has been derived here based on a one-dimensional
flow representation (mean line representation) but that formulation for a real threedimensional flow follows from the same principles.
1.5.3 Moment of Momentum in the Relative Frame: Forces
Intervening in the Rotor Work
From a velocity triangle (Fig. 1.8) follows geometrically:
v u w
= + , or v u w
u
u
= + .
The velocity triangle in Fig. 1.8 is drawn deliberately with negative w u . It follows
that
So:
(1.25)
The Eq. (1.25) also follows from a moment of momentum balance, applied in the
relative frame. Fictitious forces to be introduced are
Co
2
w
W
= −
×
and
2
Cf
r
W
=
.
The relative velocity
w has an axial component, a radial component and a tangential one. The axial component does not intervene in the Coriolis force. A tangential unit vector in the sense of the rotation may be noted as
so that
r r
u
r
r u
u r
Co
2
( w 1 w 1 1 )
2 w 1 2 w 1 .
W
W
W
W
= −
×
+
×
= −
+
According to Fig. 1.7, r
m m r
w
w 1 .1
=
, with
1 m a unit vector in the meridional direction with positive axial and radial components. The centrifugal force does not
contribute to the force moment. Neither does the radial component of the Coriolis
force. The moment of momentum balance in the relative frame results in
(1.26)
with
co
m m r
M
2 w 1 .1 dA dm r
W
r
= −
∫
,
where dA is the cross-section area of an elementary annular streamtube around the
average circumferential streamsurface in Fig. 1.7. The elementary length in the meridional plane is indicated by dm. So the term ρ w m dA represents the mass flow rate
uv u uw
u
2
u
= +
.
2
2
2
1
2 2u
1 1u
W u u u w
u w .
D = − +
−
u
r
1 1 1 ,
W
= ×
2 2u
1 1u
co
m( r w
r w ) M M ,
−
= +
Fig. 1.8 Velocity triangle