assumptions: (a) a spiral case of constant height; (b) an evenly distributed
flow into the turbine; (c) no friction losses.
Referring to Fig. 12.13(a), the discharge in the section of the spiral
case defined by an angle is given by qQ/2(, where Qis the total dis-
charge to the runner.
The velocity at any point within the spiral case can be divided into
radial (Vr) and tangential (Vt) components.
The tangential component, VtK/r, where K 30 gH/Nπ(from the
basic Euler equation for the power absorbed by the machine) and the dis-
charge through the strip dqis given by
dqVth 0 drKh 0 dr/r.
Therefore
q
R
r 0
Kh 0 dr/rQ/2π or lnR/r 0 Q/2πKh 0. (12.16)
Equation (12.16) shows that for a given vortex strength, K, a definite
relationship exists between andR.
The most economical design of a power station substructure and the
narrowest spiral case can be obtained by choosing a rectangular section
adjoining the guide vanes (entrance ring) by steep transition (symmetrical
or asymmetrical), as shown in Fig. 12.13(b).
We can write
hh 0 (r r 0 ) (12.17)
wherecot 1 cot 2.
514 HYDROELECTRIC POWER DEVELOPMENT
Fig. 12.13 Typical cross-sections of a spiral case
