Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

Centrifugal Pumps, Fans and Compressors 225
FIG. 7.10. Logarithmic spiral vane. Vane angle b¢ is constant for all radii.
Wislicenus (1947) and Stanitz (1952) mentioned earlier. The volume of mathematical
work required to describe these theories is too extensive to justify inclusion here and
only a brief outline of the results is presented.
Busemann’s theory applies to the special case of two-dimensional vanes curved as
logarithmic spirals as shown in Figure 7.10. Considering the geometry of the vane
element shown it should be an easy task for the student to prove that
that the ratio of vane length to equivalent blade pitch is
and that the equivalent pitch is
(7.17a)
(7.17b)
The equi-angular or logarithmic spiral is the simplest form of radial vane system and
has been frequently used for pump impellers in the past. The Busemann slip factor can
be written as
(7.16)
where both A and B are functions of r2/r1, b¢2 and Z. For typical pump and compressor
impellers the dependence of A and B on r 2/r1 is negligible when the equivalent l/s
exceeds unity. From eqn. (7.17b) the requirement for l/s 1, is that the radius ratio
must be sufficiently large, i.e.
(7.17c)