Page 30 - Fluid Mechanics and Thermodynamics of Turbomachinery

Introduction: Dimensional Analysis: Similitude 11
It is a simple matter to combine any pair of these expressions in such a way as to
eliminate the diameter. For a pump the customary way of eliminating D is to divide
1/2 3/4
1 by 1 . Thus
1/2 1/2
1 NQ
N s D 3/4 D 3/4 , (1.8)
.gH/
1
where N s is called the specific speed. The term specific speed is justified to the
extent that N s is directly proportional to N. In the case of a turbine the power
specific speed N sp , is more useful and is defined by,
O
P 1/2 N.P/ / 1/2
1
N sp D 5/4 D (1.9)
.gH/ 5/4
1
Both eqns. (1.8) and (1.9) are dimensionless. It is always safer and less confusing
to calculate specific speed in one or other of these forms rather than dropping the
factors g and which would make the equations dimensional and any values of
specific speed obtained using them would then depend upon the choice of the units
employed. The dimensionless form of N s (and N sp ) is the only one used in this
book. Another point arises from the fact that the rotational speed, N, is expressed
in the units of revolutions per unit of time so that although N s is dimensionless,
numerical values of specific speed need to be thought of as revs. Alternative versions
of eqns. (1.8) and (1.9) in radians are also in common use and are written
Q 1/2
 s D , .1.8a/
.gH/ 3/4
p
 P/
 sp D . .1.9a/
.gH/ 5/4

There is a simple connection between N s and N sp (and between  s and  sP ). By
dividing eqn. (1.9) by eqn. (1.8) we obtain

N.P/ / 1/2 .gH/ 3/4 P 1/2
N sp
D D .
N s .gH/ 5/4 NQ 1/2 gQH
From the definition of hydraulic efficiency, for a pump we obtain:

N sp  sp 1
D D p , (1.9b)
N s  s
and, for a turbine we obtain:
N sp  sp p
D D . (1.9c)
N s  s
Remembering that specific speed, as defined above, is at the point of maximum
efficiency of a turbomachine, it becomes a parameter of great importance in selecting
the type of machine required for a given duty. The maximum efficiency condition
replaces the condition of geometric similarity, so that any alteration in specific