Article
4: What Is L3/D4 and Why Is It Good If It Is low?
L3/D4
("L-cubed over D-to-the-fourth") is a measure of pump rotor
stiffness, it's ability to resist radial load and to minimize deflection.
It
comes from the basic cantilevered beam deflection formula, which you can find
in any book on mechanics: y = F x L3 / (3 x E X I), where F is
radial load, L is cantilevered length, E modulus of the elasticity of the
material, and I is moment of inertia.
Figure
4-1 Pump shaft
deflects under load, as a structural beam
Load
F could be a cantilevered weight of the overhung load, a centrifugal force
created by the end load unbalance, a hydraulic radial thrust of a centrifugal
pump, or a combination of forces. These forces can be static and not changing
direction (such as weight), or dynamic (such as rotating unbalance).
For
a circular shafts I = 3.14 x D4 / 64, and thus a deflection at a
given force is proportional to:
y ~ L3/D4, or abbreviated it is often written
as L3D4
Thus
L3D4 becomes a criterion for an indirect assessment, or a comparison, of a
rotor deflection under load. Mechanical seals can not tolerate much deflection,
and are prone to leakage if their faces are displaced by more then 0.001"
- 0.002".
Figure
4-2: Seals will leak
as excessive loads causes shaft deflection and seal faces misalignment
The
lower L3D4, the less is shaft deflection, which is better for the seals. If
L3D4 becomes too large, a pump shaft can snap, especially if operated close to
shut-off, where hydraulic radial loads are excessive (see other related
articles in the Pump Magazine on this subject).
ANSI
pumps have L3D4 ratios range from 20 to 120, but new designs have been
introduced with this ratio below 10.0 (see related articles, such as
"Barrier" design, that combines mag-drive and gas seal technologies).
You
can easily determine the L3D4 ratio of your pump by measuring the length of the
shaft from the center of the bearing closest to the impeller and impeller
centerline, and the diameter of the shaft under the bearing. Then, cube the
length, raise the diameter in to fourth power, and obtain the ratio. A shaft
diameter changes from the bearing towards the impeller, but its value under the
bearing is taken nominally. Tabulating these ratios for different pumps, you
can make you own plant database of L3D4 of different designs.
Figure
4-3: L and D
parameters for a pump rotor
As
a word of caution, the ultimate manifestation of pump reliability is its
operating history, which could at times conflict with what a L3D4 number would
indicate. Nevertheless, it is a good guide, and helpful as one of several
design factors that may have an effect on the pump reliability.
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