108
3 Fans
control volume with dimension dx in flow direction and average dimension dy in the
normal direction. The intervening body forces per mass unit are:
•  centrifugal force by rotor rotation: 
2
1 r
Cf
r
W
=
.
•  Coriolis force by rotor rotation: 
2
2 1 y
Co
w
w
W
W
= − × = −
.
•  centrifugal force by streamline curvature for an observer attached to a fluid particle (streamline radius of curvature is R): Cu
w
R
y
=
2
1 .
Pressure forces onto the infinitesimal control volume in x and y directions are
where b is the axial rotor width.
The x-oriented momentum balance for steady flow within the relative frame, for
an observer attached to the rotor, is
The y-oriented force balance for relative standstill of an observer attached to a fluid
particle is
The changes of r with a change of x or y are given by
and
,
p
p
dxdyb
dydxb
x
y
∂
∂
−
−
∂
∂
2
1 .1
.
r x
w
p
wdyb dx
dxdyb
r
dxdyb
x
x
r
W
r
∂
∂
= −
+
∂
∂
2
2
0
1 .1 2
.
r y
p
w
dydxb
r
w
dxdyb
y
R
W
W
r


∂
= −
+
−
+


∂


1 .1
and
1 .1 .
r x
r y
dr
dx
dr
dy
=
=
Fig. 3.11 Curved streamline and infinitesimal streamtube part (  left); contour for determining the
velocity difference over a blade (  right)