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National Advisory Committee for Aeronautics, Technical Notes - Two Dimensional Compressible Flow in Conical Mixed Flow Compressors

A general method of analysis is developed for the two-
dimensional, steady, compressible flow through mixed-flow compres-
sors in which the center line of the passage generates a right cir-
cular cone about the axis of the compressor. 'lhe two-dimensional,
radial-discharge compressor is a special case in which the’cone
angle is equal to 180°. Tue variables taken into account are:
(l) impeller tip Mach number, (2) compressor flow rate, (5) blade
shape (curvature), (4) passage height, (5) number of blades, and
(6) compressor cone angle. Relaxation methods are used to solve
the resulting nonlinear differential equation for the stream
function. @ecial attention is paid to logarithmic-spiral blades,
of which the straight blade (lying on a conic radius) is a partic-
ular case.

As a result of the analysis, it is concluded that the solution
obtained for a given cone angle also applies to certain other cone
angles (that is , other mixed-flow compressors) with a fewer or a
greater number of passages but’with the same included passage angle,
and so forth. It is also concluded that mixed-flow compressors
with the same number of flow passages as radial-discharge com-
pressors and therefore with smaller included passage angles have
lower peak blade loadings and lower mximlm relative velocities
than the corresponding radial-discharge compressors.

The general analysis also applies to imiard-flow turbines.
In fact, the solution obtained for a centrifugal comressor with
smooth (shockless) entry is also the solution (with the flow direc-‘
tion and rotation reversed) for an inward-flow turbine with the
same design characteristics (that is, the same rotor) and with
shockless entry.

A numerical example is presented consisting of a radial-
discharge compressor (cone angle equal to 180°) with constant flow
area and 20 straight, radial blades operating at a tip Mach number
of 1.5. The results of this example are given by plots of the
streamlines, lines of constant pressure ratio, and lines of constant
Mach number. For the conditions of this example, a wheel-type eddy
forms on the driving face of the blade; the velocities and pres-
sures at the impeller tip are reasombl;r uniform; any nonuniformity
in the flow leaving the impeller tip adjusts itself rapidly; the
maximum local Maich number is 0.64 and occurs along the trailing
face of the blade at 68 percent of the tip radius; and the computed
value of the slip factor is 0.90.