Page 29 - Fluid Mechanics and Thermodynamics of Turbomachinery

10 Fluid Mechanics, Thermodynamics of Turbomachinery
conditions fluctuate, sophisticated systems of control may incorporate an electronic
computer.
The lines (a) and (c) in Figure 1.5 show the efficiency curves at other blade
settings. Each of these curves represents, in a sense, a different constant geometry
machine. For such a variable geometry pump the desired operating line intersects
the points of maximum efficiency of each of these curves.
Introducing the additional variable ˇ into eqn. (1.3) to represent the setting of the
vanes, we can write

D f 1 . , ˇ/; D f 2 . , ˇ/. (1.5)
Alternatively, with ˇ D f 3 . , / D f 4 . , /, ˇ can be eliminated to give a new
functional dependence

Q gH
D f 5 . , / D f 5 3 , 2 2 (1.6)
ND N D
Thus, efficiency in a variable geometry pump is a function of both flow coefficient
and energy transfer coefficient.

Specific speed

The pump or hydraulic turbine designer is often faced with the basic problem
of deciding what type of turbomachine will be the best choice for a given duty.
Usually the designer will be provided with some preliminary design data such as
the head H, the volume flow rate Q and the rotational speed N when a pump design
is under consideration. When a turbine preliminary design is being considered the
parameters normally specified are the shaft power P, the head at turbine entry H
and the rotational speed N. A non-dimensional parameter called the specific speed,
N s , referred to and conceptualised as the shape number, is often used to facilitate
the choice of the most appropriate machine. This new parameter is derived from the
non-dimensional groups defined in eqn. (1.3) in such a way that the characteristic
diameter D of the turbomachine is eliminated. The value of N s gives the designer
a guide to the type of machine that will provide the normal requirement of high
efficiency at the design condition.
For any one hydraulic turbomachine with fixed geometry there is a unique rela-
tionship between efficiency and flow coefficient if Reynolds number effects are
negligible and cavitation absent. As is suggested by any one of the curves in
Figure 1.5, the efficiency rises to a maximum value as the flow coefficient is
increased and then gradually falls with further increase in . This optimum effi-
ciency D max , is used to identify a unique value D 1 and corresponding unique
values of D 1 and O P D O P 1 . Thus,
Q
D 1 D constant, .1.7a/
ND 3
gH
D 1 D constant, .1.7b/
2
N D 2
P
D O P 1 D constant. .1.7c/
3
N D 5