Page 36 - Fluid Mechanics and Thermodynamics of Turbomachinery
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Introduction: Dimensional Analysis: Similitude 17
the machine and
, the ratio of specific heats C p /C . In the following analysis the
compressible fluids under discussion are either perfect gases, or else, dry vapours
approximating in behaviour to a perfect gas.
Another choice of variables is usually preferred when appreciable density changes
occur across the machine. Instead of volume flow rate Q, the mass flow rate Pm is
used; likewise for the head change H, the isentropic stagnation enthalpy change
h os is employed.
The choice of this last variable is a significant one for, in an ideal and adiabatic
process, h 0s is equal to the work done by unit mass of fluid. This will be discussed
still further in Chapter 2. Since heat transfer from the casings of turbomachines
is, in general, of negligible magnitude compared with the flux of energy through
the machine, temperature on its own may be safely excluded as a fluid variable.
However, temperature is an easily observable characteristic and, for a perfect gas,
can be easily introduced at the last by means of the equation of state, p/ D RT,
C , m being the molecular weight of the gas and R 0 D
where R D R 0 /m D C p
8.314 kJ/(kg mol K) is the Universal gas constant.
The performance parameters h 0s , and P for a turbomachine handling a
compressible flow, are expressed functionally as:
h 0s , ,P D f. , N, D, Pm, 01 ,a 01 ,
/. (1.14a)
Because 0 and a 0 change through a turbomachine, values of these fluid variables
are selected at inlet, denoted by subscript 1. Equation (1.14a) express three separate
functional relationships, each of which consists of eight variables. Again, selecting
01 , N, D as common factors each of these three relationships may be reduced to
five dimensionless groups,
2
h 0s P P m 01 ND ND
, , D f , ,
. (1.14b)
3
2
N D 2 01 N D 5 01 ND 3 a 01
3
Alternatively, the flow coefficient DPm/. 01 ND / can be written as D
2
P m/. 01 a 01 D /.As ND is proportional to blade speed, the group ND/a 01 is regarded
as a blade Mach number.
For a machine handling a perfect gas a different set of functional relationships is
often more useful. These may be found either by selecting the appropriate variables
for a perfect gas and working through again from first principles or, by means
of some rather straightforward transformations, rewriting eqn. (1.14b) to give more
suitable groups. The latter procedure is preferred here as it provides a useful exercise.
As a concrete example consider an adiabatic compressor handling a perfect gas.
T 01 / for the
The isentropic stagnation enthalpy rise can now be written C p .T 02s
perfect gas. This compression process is illustrated in Figure 1.9a where the stag-
nation state point changes at constant entropy between the stagnation pressures
p 01 and p 02 . The equivalent process for a turbine is shown in Figure 1.9b. Using
the adiabatic isentropic relationship p/ D constant, together with p/ D RT, the
expression
.
1//
T 02s p 02
D
T 01 p 01
