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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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