Page 283 - Centrifugal Pumps Design and Application

252 Centrifugal Pumps: Design and Application

the predicted performance by analysis. A modern computer program
may be used to perform the calculations, print out the results, and also
plot the performance curve.
The prediction procedure generally consists of accounting for the vari-
ous components that compose the total head characteristics. These are
friction losses, absorbed head, shock loss, and outlet loss. Power losses
due to internal leakage, disc friction, and mechanical losses are also cal-
culated or estimated as appropriate. The calculations are made using the
required turbine speed, flow capacities, viscosity, specific gravity of the
fluid, and various combinations of mechanical data required for certain
multi-stage turbines. There are many publications that cover basic theory
and design of pumps that show how to calculate the head and loss compo-
nents for pumps. These also apply to hydraulic turbines.

Friction losses. Certain components of the total dynamic head are attrib-
uted to friction losses. These are due to flow through the cases, volute
nozzles, diffusers, guide vanes, and runners (impellers) as appropriate,
These losses may be simply calculated as the resistance to the incompres-
sible flow of fluid in a pipe, using appropriate friction factors, length to
diameter (or hydraulic radius) ratios, and the velocity head.

Absorbed head. The absorbed head is derived from the well-known
Euler's equations and velocity triangles, which have general validity for
all conditions of flow through turbomachines. Refer to Figure 14-4 for
illustration. In practice, the true velocities of flow and direction are
never known. The idealized velocity triangles of the Euler head equation
assume perfect guidance of the flow by the vanes. It is known that there
is a deviation of the fluid from the vane direction, which is the phenome-
non called "slip." This is a consequence of the nonuniform velocity dis-
tribution across the runner channels, boundary-layer accumulation, and
any separation.
Actual prediction of "slip" cannot be predetermined in a practical
manner. However, it has been found that "slip" factors used for pump
design applied to the turbine outlet vectors produce good results. The ab-
solute velocity at the runner inlet (Cj) is the average velocity of the liquid
at the nozzles with a free vortex correction applied to account for the dis-
tance from the nozzles to the runner. The nozzles are the highest velocity
throat areas of the volute cases, diffusers, or guide vanes as appropriate
for the turbine construction.

Shock loss. The shock loss component of the total dynamic head is cal-
2
culated as the velocity head (V s /2g) due to the mismatch of the absolute