Hydraulic Structures: Fourth Edition

Equation (12.16) now becomes

Q/2πK

r 1

r 0

hdr/r

R

r 1

H 0 dr/r (12.18)

which, on integration, after replacing hby equation (12.17) gives

Q/2πK(h 0 
r 0 )ln(r 1 /r 0 )(H 0 h 0 )H 0 ln(R/r 1 ). (12.19)

Knowingr 1 from

r 1 (H 0 h 0 )/
r 0 (12.20)

the value of Rdefining the shape of the spiral casing can be determined.
The height H 0 at any angle may be assumed to be linearly increasing from
h 0 at the nose towards the entrance.
The shape of the cross-section is determined at various values of by
assuming the existence of uniform velocity for the entire spiral case
(Mosonyi, 1987) to be equal to the entrance velocity, V 0 0.2 (2gH)1/2. Thus,
knowingqiQi/2π, the area of cross-section at an angle iis given by

Aiqi/V 0 0.18Qi/H1/2. (12.21)

This approximation results in larger cross-sections towards the nose, desir-
able in order to minimize the friction losses (ignored in the theoretical design
development) which are more pronounced in the proximity of the nose.
It is desirable to provide streamlined dividing piers in entrance
flumes of large widths to ensure as even a flow distribution as possible.
The final design of the scroll case should preferably be checked by model
tests, especially in the cases of unconventional arrangements of large
turbine units. An example of the inner shaping of a concrete spiral casing
is shown in Fig. 12.14.

HYDRAULIC TURBINES AND THEIR SELECTION 515

Fig. 12.14 Inner shape of a concrete scroll case with divide piers