Page 39 - Fluid Mechanics and Thermodynamics of Turbomachinery

20 Fluid Mechanics, Thermodynamics of Turbomachinery
p p
machine is plotted as a function of Pm. T 01 //p 01 for fixed values of N/. T 01 /,
this being a customary method of presentation. Notice that for both machines
subscript 1 is used to denote conditions as inlet. One of the most striking features
of these performance characteristics is the rather weak dependence of the turbine
p
performance upon N/ T 01 contrasting with the strong dependence shown by the
compressor on this parameter.
p
For the compressor, efficient operation at constant N/ T 01 lies to the right of
the line marked “surge”. A discussion of the phenomenon of surge is included in
Chapter 5; in brief, for multistage compressors it commences approximately at the
p
point (for constant N/ T 01 ) where the pressure ratio flattens out to its maximum
value. The surge line denotes the limit of stable operation of a compressor, unstable
operation being characterised by a severe oscillation of the mass flow rate through
the machine. The choked regions of both the compressor and turbine characteristics
may be recognised by the vertical portions of the constant speed lines. No further
p
increase in Pm. T 01 //p 01 is possible since the Mach number across some section
of the machine has reached unity and the flow is said to be choked.

The inherent unsteadiness of the flow within
turbomachines

A fact often ignored by turbomachinery designers, or even unknown to students,
is that turbomachines can only work the way they do because of unsteady flow
effects taking place within them. The fluid dynamic phenomena that are associated
with the unsteady flow in turbomachines has been examined by Greitzer (1986) in
a discourse which was intended to be an introduction to the subject but actually
extended far beyond the technical level of this book! Basically Greitzer, and others
before him, in considering the fluid mechanical process taking place on a fluid
particle in an isentropic flow, deduced that stagnation enthalpy of the particle can
change only if the flow is unsteady. Dean (1959) appears to have been the first
to record that without an unsteady flow inside a turbomachine, no work transfer
can take place. Paradoxically, both at the inlet to and outlet from the machine the
conditions are such that the flow can be considered as steady.
A physical situation considered by Greitzer is the axial compressor rotor as
depicted in Figure 1.12a. The pressure field associated with the blades is such that
the pressure increases from the suction surface (S) to the pressure surface (P). This
pressure field moves with the blades and, to an observer situated at the point * (in the
absolute frame of reference), a pressure that varies with time would be recorded,
as shown in Figure 1.12b. Thus, fluid particles passing through the rotor would
experience a positive pressure increase with time (i.e. ∂p/∂t > 0). From this fact it
can then be shown that the stagnation enthalpy of the fluid particle also increases
because of the unsteadiness of the flow, i.e.

Dh 0 1 ∂p
D ,
Dt ∂t

where D/Dt is the rate of change following the fluid particle.