Page 420 - Civil Engineering Formulas

346 CHAPTER TWELVE

A convenient way of determining the highest practicable speed is by the
relation of the specific speed to the head. For Francis turbines this is
n s 900 2 H USCS (12.169)

1,900 2 H SI
For propeller-type turbines, n s 1,000 2 H USCS (12.170)
2,100 2 H SI

Affinity Laws. The relationships between head, discharge, speed, horsepower,
and diameter are shown in the following equations, where Q rate of discharge,
H head, N speed, hp horsepower, D diameter, and subscripts denote
two geometrically similar units with the same specific speed:*

Q 1 Q 2
(12.171)
3 3
N 1 D 1 N 2 D 2
2 2
Q 1 Q 2
(12.172)
4 4
H 1 D 1 H 2 D 2
2 2 2 2
N 1 D 1 N 2 D 2
(12.173)
H 1 H 2
hp 1 hp 2
(12.174)
3 5 3 5
N 1 D 1 N 2 D 2
Most designs used are tested as exact homologous models, and performance is
stepped up from the model by the normal affinity laws given above.
Water Hammer in Penstocks. † If a gate movement is considered as a series
of instantaneous movements with a very small interval between each
movement, the pressure variation in the penstock following the gate
movement will be the effect of a series of pressure waves, each caused by one
of the instantaneous small gate movements. For a steel penstock, the velocity
of the pressure wave a 4,660 2 1 d 100t (USCS) (1,420 1 d 100t)
(SI), where d is the penstock diameter, in (m); and t is the penstock wall
thickness, in (m). The pressure change at any point along the penstock at any
time after the start of the gate movement may be calculated by summing up
the effect of the individual pressure waves.
Approximate formulas (De Sparre) for the increase in pressure h, ft (m), fol-
lowing gate closure, are Eqs. (12.175) through (12.177). They are quite accu-
rate for pressure rises not exceeding 50 percent of the initial pressure, which
includes most practical cases.
h aV g for K 1, N 1 (12.175)

*Karrasik—Pump Handbook, McGraw–Hill.
†
Marks, Mechanical Engineer’s Handbook, McGraw–Hill.