Page 71 - Fluid Mechanics and Thermodynamics of Turbomachinery

52 Fluid Mechanics, Thermodynamics of Turbomachinery
From eqn. (2.56) the change in static pressure in found as

˛ 2 q 2 / .p p /. (2.58)
p 2 p 1 D .˛ 1 q 1
01 02
From eqn. (2.51), with eqns. (2.47a) and (2.47b), the diffuser efficiency (or diffuser
effectiveness) can now be written:

.p 2 p 1 //q 1
D D C p /C pi D .
1 1/A 2
R
Substituting eqn. (2.58) into the above expression,
2
˛ 1 [1 .˛ 2 /˛ 1 //A ] $
R
D D 2 2 (2.59)
.1 1/A / .1 1/A /
R R
where $ is the total pressure loss coefficient for the whole diffuser, i.e.
$ D .p 01 p //q 1 . (2.60)
02
Equation (2.59) is particularly useful as it enables the separate effects due the
changes in the velocity profile and total pressure losses on the diffuser effectiveness
to be found. The first term in the equation gives the reduction in D caused by
insufficient flow diffusion. The second term gives the reduction in D produced by
viscous effects and represents inefficient flow diffusion. An assessment of the relative
proportion of these effects on the effectiveness requires the accurate measurement
of both the inlet and exit velocity profiles as well as the static pressure rise. Such
complete data is seldom derived by experiments. However, Sovran and Klomp
(1967) made the observation that there is a widely held belief that fluid mechanical
losses are the primary cause of poor performance in diffusers. One of the important
conclusions they drew from their work was that it is the thickening of the inlet
boundary layer which is primarily responsible for the reduction in D . Thus, it is
insufficient flow diffusion rather than inefficient flow diffusion which is often the
cause of poor performance.
Some of the most comprehensive tests made of diffuser performance were those
of Stevens and Williams (1980) who included traverses of the flow at inlet and at
exit as well as careful measurements of the static pressure increase and total pressure
loss in low speed tests on annular diffusers. In the following worked example, to
illustrate the preceding theoretical analysis, data from this source has been used.

EXAMPLE 2.5. An annular diffuser with an area ratio, A R D 2.0 is tested at low
speed and the results obtained give the following data:

at entry, ˛ 1 D 1.059,B 1 D 0.109
at exit, ˛ 2 D 1.543,B 2 D 0.364,C p D 0.577

Determine the diffuser efficiency.
NB B 1 and B 2 are the fractions of the area blocked by the wall boundary layers
at inlet and exit (displacement thicknesses) and are included only to illustrate the
profound effect the diffusion process has on boundary layer thickening.