Fluid Mechanics and Thermodynamics of Turbomachinery, 5e

Therefore
Hence,
The diffuser system
Centrifugal Pumps, Fans and Compressors 237
Centrifugal compressors and pumps are, in general, fitted with either a vaneless or
a vaned diffuser to transform the kinetic energy at impeller outlet into static pressure.
Vaneless diffusers
The simplest concept of diffusion in a radial flow machine is one where the swirl
velocity is reduced by an increase in radius (conservation of angular momentum) and
the radial velocity component is controlled by the radial flow area. From continuity,
since m . = rAcr = 2prbrc r, where b is the width of passage, then
(7.30)
Assuming the flow is frictionless in the diffuser, the angular momentum is constant and
cq = cq2r2/r. Now the tangential velocity component cq is usually very much larger than
the radial velocity component cr; therefore, the ratio of inlet to outlet diffuser velocities
c2/c 3 is approximately r 3/r 2. Clearly, to obtain useful reductions in velocity, vaneless
diffusers must be large. This may not be a disadvantage in industrial applications
where weight and size may be of secondary importance compared with the cost of a
vaned diffuser. A factor in favour of vaneless diffusers is the wide operating range
obtainable, vaned diffusers being more sensitive to flow variation because of incidence
effects.
For a parallel-walled radial diffuser in incompressible flow, the continuity of mass
flow equation requires that rcr is constant. Assuming that rc q remains constant, then the
absolute flow angle a2 = tan -1 (c q/c r) is also constant as the fluid is diffused outwards.
Under these conditions the flow path is a logarithmic spiral. The relationship between
the change in the circumferential angle Dq and the radius ratio of the flow in the diffuser
can be found from consideration of an element of the flow geometry shown in
Figure 7.15. For an increment in radius dr we have, r dq = drtan a2 which, upon integration,
gives
(7.31)
Values of Dq are shown in Figure 7.16 plotted against r 3/r 2 for several values of a 2. It
can be readily seen that when a2 > 70°, rather long flow paths are implied, friction
losses will be significant and the diffuser efficiency will be low.