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

450 Centrifugal Pumps: Design and Application
rotor and the housing, the average asymptotic tangential velocity is Rw/2,
where R is the seal radius and w is the rotor running speed. The crosscoupled
stiffness coefficient (K) acts in opposition to the direct damping
coefficient (C) to destabilize rotors. Hence, steps that can be taken to reduce
the net fluid rotation within a seal will improve rotor stability [30]
by reducing K.
Childs has defined the dynamic seal coefficients for plain short seals
directly from Hirs' lubrication equations [27] and has included the influence
of fluid inertia terms and inlet swirl. His assumptions are less restrictive
than previous derivations. The derived coefficients are in reasonable
agreement with prior results of Black and Jenssen.
ChiJds [28] has extended the analysis to include finite-length seals.
This analysis includes variable inlet swirl conditions (different from Ro>/
2) and considers variations in the axial and circumferential Reynolds
numbers due to changes in clearances.
A combined analytical-computational method has also been developed
by Childs [28] to calculate the transient pressure field and dynamic coefficients
for interstage and neck ring seals of multistage centrifugal
pumps. The solution procedure applies to constant-clearance or convergent-tapered
geometries that may have different surface-roughness treatments
of the stator or rotor seal elements. The method has been applied to
the calculation of "damper" seals as described by von Pragenau [31] and
several roughened stator designs, such as knurled-indentation, diamondgrid
post pattern, and round-hole pattern, have been tested. These procedures
can be used to calculate serrated or grooved seals of various
geometries,
Critical Spued Map—Considering Seals. The seal configurations at the
balance piston, neck ring, and interstage bushings of the eight-stage
pump are shown in Figure 18-16. The critical speed map (Figure 18-15)
includes the support stiffnesses of the neck ring seals and interstage bushings
combined at each impeller. The seal stiffness and damping coefficients
are listed in Table 18-2 for nominal clearances. Note that a negative
principal stiffness (K) value is predicted for the balance piston.
For this analysis the pump rotor was analyzed as if it had 11 bearings
consisting of two tilted-pad bearings, the balance piston, and eight seals
located at the impellers. For the purposes of developing the critical speed
map, the seal stiffness values were held constant at their maximum levels
(minimum clearances) that represent new seals.
The lateral mode shape of the first critical speed including these seal
effects is shown in Figure 18-17. A bearing stiffness value of 500,000 lb/
in. was again used for the mode shape calculations. Comparing Figure
18-17 with Figure 18-12, it is seen that the seals increase the frequency of