Page 200 - Fluid Mechanics and Thermodynamics of Turbomachinery
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Three-dimensional Flows in Axial Turbomachines 181
therefore
2
/.
1/
1 EK
p D constant (6.24)
2
r 2
For this free-vortex flow the pressure, and therefore the density also, must be larger at
the casing than at the hub. The density difference from hub to tip may be appreciable
in a high-velocity, high-swirl angle flow. If the fluid is without swirl at entry to the
blades the density will be uniform. Therefore, from continuity of mass flow there
must be a redistribution of fluid in its passage across the blade row to compensate
for the changes in density. Thus, for this blade row, the continuity equation is,
Z
r t
2 rdr, (6.25)
P m D 1 A 1 c x1 D 2 c x2
r h
where 2 is the density of the swirling flow, obtainable from eqn. (6.24).
Constant specific mass flow
Although there appears to be no evidence that the redistribution of the flow across
blade rows is a source of inefficiency, it has been suggested by Horlock (1966) that
the radial distribution of c for each blade row is chosen so that the product of axial
velocity and density is constant with radius, i.e.
d Pm/dA D c x D c cos ˛ D m c m cos ˛ m D constant (6.26)
where subscript m denotes conditions at r D r m . This constant specific mass flow
design is the logical choice when radial equilibrium theory is applied to compressible
flows as the assumption that c r D 0 is then likely to be realised.
Solutions may be determined by means of a simple numerical procedure and, as
an illustration of one method, a turbine stage is considered here. It is convenient
to assume that the stagnation enthalpy is uniform at nozzle entry, the entropy is
constant throughout the stage and the fluid is a perfect gas. At nozzle exit under
these conditions the equation of radial equilibrium, eqn. (6.20), can be written as
2
dc/c D sin ˛dr/r. (6.27)
From eqn. (6.1), nothing that at constant entropy the acoustic velocity a D
p
.dp/d /,
2 2
1 dp 1 dp d a d c 2
D D D sin ˛,
dr d dr dr r
2
2
∴ d / D M sin ˛dr/r .6.28/
where the flow Mach number
p
M D c/a D c/ .
RT/. (6.28a)
The isentropic relation between temperature and density for a perfect gas is
T/T m D . / m /
1
