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What is L3/D4 (L cubed over D to the fourth)? It is a
measure of pump rotor stiffness, its 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 is the elasticity
of the material and I is moment of inertia. (See Figure 1.)
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
For 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
often is written as L3D4
Thus, L3D4 becomes a criterion for an indirect assessment or
a comparison of a rotor deflection under load. Mechanical seals cannot tolerate
much deflection and are prone to leakage if their faces are displaced by more
than 0.001?0.002 inches.
The lower the L3D4, the less 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
ANSI pumps have L3D4 ratios that range from 20 to 120, but
new designs have been introduced with this ratio below 10.
You easily can 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 center line and the diameter of the shaft under the
bearing. Then, cube the length, raise the diameter to the 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. By tabulating
these ratios for different pumps, you can make your own plant database of L3D4
of different designs.
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.