Page 26 - Fluid Mechanics and Thermodynamics of Turbomachinery

Introduction: Dimensional Analysis: Similitude 7
Because of the large number of independent groups of variables on the right-hand
side of eqns. (1.2), those relationships are virtually worthless unless certain terms
can be discarded. In a family of geometrically similar machines l 1 /D, l 2 /D are
constant and may be eliminated forthwith. The kinematic viscosity, D / is
very small in turbomachines handling water and, although speed, expressed by ND,
is low the Reynolds number is correspondingly high. Experiments confirm that
effects of Reynolds number on the performance are small and may be ignored in a
first approximation. The functional relationships for geometrically similar hydraulic
turbomachines are then,
3
D f 4 [Q/.ND /] .1.3a/
3
D f 5 [Q/.ND /] .1.3b/
3
O P D f 6 [Q/.ND /]. .1.3c/
This is as far as the reasoning of dimensional analysis alone can be taken; the actual
form of the functions f 4 , f 5 and f 6 must be ascertained by experiment.
One relation between , , and O P may be immediately stated. For a pump the
net hydraulic power, P N equals QgH which is the minimum shaft power required
in the absence of all losses. No real process of power conversion is free of losses and
the actual shaft power P must be larger than P N . We define pump efficiency (more
precise definitions of efficiency are stated in Chapter 2) D P N /P D QgH/P.
Therefore

1 Q gH 3 5
P D N D . (1.4)
ND 3 .ND/ 2
Thus f 6 may be derived from f 4 and f 5 since O P D / . For a turbine the net
hydraulic power P N supplied is greater than the actual shaft power delivered by
the machine and the efficiency D P/P N . This can be rewritten as O P D by
reasoning similar to the above considerations.

Performance characteristics
The operating condition of a turbomachine will be dynamically similar at two
different rotational speeds if all fluid velocities at corresponding points within the
machine are in the same direction and proportional to the blade speed. If two
points, one on each of two different head flow characteristics, represent dynamically
similar operation of the machine, then the non-dimensional groups of the variables
involved, ignoring Reynolds number effects, may be expected to have the same
numerical value for both points. On this basis, non-dimensional presentation of
performance data has the important practical advantage of collapsing into virtually
a single curve, results that would otherwise require a multiplicity of curves if plotted
dimensionally.
Evidence in support of the foregoing assertion is provided in Figure 1.3 which
shows experimental results obtained by the author (at the University of Liverpool)
on a simple centrifugal laboratory pump. Within the normal operating range of
3
this pump, 0.03 < Q/.ND /< 0.06, very little systematic scatter is apparent which