Page 24 - Fluid Mechanics and Thermodynamics of Turbomachinery

Introduction: Dimensional Analysis: Similitude 5
into a smaller number of dimensionless groups. When the number of indepen-
dent variables is not too great, dimensional analysis enables experimental relations
between variables to be found with the greatest economy of effort. Dimensional
analysis applied to turbomachines has two further important uses: (a) prediction
of a prototype’s performance from tests conducted on a scale model (similitude);
(b) determination of the most suitable type of machine, on the basis of maximum
efficiency, for a specified range of head, speed and flow rate. Several methods of
constructing non-dimensional groups have been described by Douglas et al. (1995)
and by Shames (1992) among other authors. The subject of dimensional analysis was
made simple and much more interesting by Edward Taylor (1974) in his comprehen-
sive account of the subject. It is assumed here that the basic techniques of forming
non-dimensional groups have already been acquired by the student.
Adopting the simple approach of elementary thermodynamics, an imaginary enve-
lope (called a control surface) of fixed shape, position and orientation is drawn
around the turbomachine (Figure 1.2). Across this boundary, fluid flows steadily,
entering at station 1 and leaving at station 2. As well as the flow of fluid there
is a flow of work across the control surface, transmitted by the shaft either to, or
from, the machine. For the present all details of the flow within the machine can
be ignored and only externally observed features such as shaft speed, flow rate,
torque and change in fluid properties across the machine need be considered. To be
specific, let the turbomachine be a pump (although the analysis could apply to other
classes of turbomachine) driven by an electric motor. The speed of rotation N, can
be adjusted by altering the current to the motor; the volume flow rate Q, can be
independently adjusted by means of a throttle valve. For fixed values of the set Q
and N, all other variables such as torque , head H, are thereby established. The
choice of Q and N as control variables is clearly arbitrary and any other pair of
independent variables such as and H could equally well have been chosen. The
important point to recognise is, that there are for this pump, two control variables.
If the fluid flowing is changed for another of different density , and viscosity
, the performance of the machine will be affected. Note, also, that for a turbo-
machine handling compressible fluids, other fluid properties are important and are
discussed later.
So far we have considered only one particular turbomachine, namely a pump of
a given size. To extend the range of this discussion, the effect of the geometric

FIG. 1.2. Turbomachine considered as a control volume.