Centrifugal Pumps Design and Application 2nd ed - Val S. Lobanoff, Robert R. Ross (Butterworth-Heinemann, 1992)

442 Centrifugal Pumps: Design and Application
Lateral Critical Speed Analysis
Pump rotordynamics are dependent on a greater number of design variables
than are many other types of rotating equipment. Besides the journal
bearing and shaft characteristics, the dynamic characteristics of the
seals and the impeller-stationary lip interaction can have significant effects
on the critical speed location, rotor unbalance sensitivity, and rotor
stability [9, 19]. In this context, a seal is an element having a liquid film
within a tight clearance. The liquid film has dynamic characteristics similar
to a bearing. There are a variety of seal configurations including
floating ring seals, grooved seals, and others. Several seal geometries
will be discussed.
For modeling purposes, seals can be treated as bearings in the sense
that direct and cross-coupled stiffness and damping properties can be calculated
based on the seal's hydrostatic and hydrodynamic properties [20],
Seal clearances, geometry, pressure drop, fluid properties, inlet swirl,
surface roughness, and shaft speed are all important in these calculations.
The high pressure liquid being pumped also flows (or leaks)
through the small annular spaces (clearances) separating the impellers
under different pressures, such as wear rings and interstage bushings,
and creates a hydrodynamic bearing effect that transforms the pump rotor
from a two-bearing system to a multi-bearing system. The additional
stiffness generated by the pumped liquid as it lubricates these internal
bearings (seals, etc.) is referred to as the "Lomakin effect" [21].
The Lomakin stiffness effect minimizes the shaft deflections when the
pump is running, and in some cases, the Lomakin effect can be of sufficient
magnitude to prevent the critical speed of the rotor from ever being
coincident with the synchronous speed. Since the pressure drop across
seals increases approximately with the square of the pump speed, the seal
stiffness also increases with the square of the speed.
The amount of support derived from the seals as bearings depends
upon (a) the pressure differential, and therefore disappears completely
when the pump is at rest, and (b) the clearance that increases significantly
as the sealing surfaces wear. Consequently, contact between the rotor and
stationary parts may take place each time the pump is started or stopped.
In consideration of these facts, the rotordynamic analyses should include
the effects of worn seals (loose clearances) as well as new seals (tight
clearances).
Analytical techniques [22, 23, 24] have been developed whereby the
seal geometry can be specified and the characteristics calculated for specific
assumptions with regard to inlet swirl, groove design, etc. A series
of grooved seal designs used in commercial pumps has been tested to verify
the adequacy of the techniques.