CENTRIFUGAL PUMPS - Modern Design and Practices

Shaft flexibility factor (SFF)

Have you heard of the expression ‘L cube over D fourth’? 

In equation form, it is written as L^3 / D^4 and is called shaft flexibility factor (SFF). Others refer to it plainly as ‘L over D’, or L/D for short.

Shaft flexibility factor (SFF) is a dimensionless index that was popularized by a major American oil company in the 1970s. The company (that was later acquired by a large British conglomerate) analyzed hundreds of bad actor centrifugal pumps in their refinery - pumps with high vibration and noise levels, short MTBF, poor reliability, and costly maintenance. 

They found that bad actor pumps have things in common - they have small shaft diameter, long  span between the centerlines of impeller and radial bearing in overhang pumps, or between the centerlines of their radial bearings in between-bearing pumps. Bad actor pumps have frequent bearing and mechanical seal failures and, in many instances, have shaft breakages.

A large oil refinery complex can easily have hundreds, if not thousands, of centrifugal pumps. Analyzing the shaft deflection of that many pumps was a tedious and time-consuming process. So in lieu of making detailed shaft deflection analysis on each and every pump, they simplified the process by coming up with an index called shaft flexibility factor(SFF).

The shaft flexibility factor (SFF) is represented by the equation:

SFF = (L^3) / (D^4)

where:

L - span between impeller and radial bearing centerlines in overhang pumps, 

or span between centerlines of radial bearings in between-bearing pumps

D - shaft diameter under the shaft sleeve at the stuffing box

^  - exponential symbol

SSF was derived from the simple formula for beam deflection:

Y = [W x L^3] / [C x E x I]

where:

Y - deflection

W- load 

L - beam (or shaft) span

C -  coefficient

E - modulus of elasticity

I  - moment of inertia = [(Pi x D^4) / 64]

In a narrow group of similar pumps under evaluation for a specific service, it is assumed that W, C, and E are the same, or are very close in values. For the purposes of comparing and evaluating the pumps, W, C, and E, are dropped from the formula, resulting in the expression (L^3) / (D^4), henceforth called the shaft flexibility factor (SFF).

In comparing the SFF of one pump with another of similar size and design, the pump with lower SFF is considered to be of more robust shaft design; it has less deflection and longer MTBF than a comparable pump with higher SFF. This comparison is used in evaluating bids for the supply of new pump, or upgrade of existing pump. The lowest bid does not necessarily translate into the best offer if the pump has higher SFF indicating it would eventually have higher operating and maintenance costs. In essence, it was an early form of life cycle cost analysis (LCC) before the use of conventional LCC became widely used in the pump industry. SFF evaluations should be made only among pumps of similar design and are separately done for overhang and between-bearings pumps.

Decades ago, the design of pumps with smaller shaft diameter was influenced by economics – it  means smaller and cheaper seals, bearings, sleeves, etc. It also means less frictional and internal leakage loss resulting in higher pump efficiency. In some high suction pressure applications, it was even necessary to reduce the shaft diameter and seal size to reduce the hydraulic axial thrust in overhang pumps. And in some low NPSHA situations, smaller shaft diameter helps reduce the pump NPSHR by reducing the shaft blockage through the impeller eye.

On the other hand, the longer span design was influenced by the need to design longer stuffing box to accommodate packing rings, as alternative to mechanical seals. Packed pumps typically require several packing rings for effective sealing, thus requiring longer stuffing box. The need for pump covers to have cooling jacket also resulted in longer stuffing box and, by extension, in longer shaft span. Advances in design have now allowed mechanical seals to operate at higher temperature without cooling, allowing the removal of cooling jacket to shorten the stuffing box. 

To some extent, the move towards standardization has also contributed to longer span when a common standard shaft, cover, or bearing bracket, is used among pump of the same design but different sizes. The smaller shaft diameter and longer span do not imply an inferior design, or design flaw. Many pumps with high SFF run as good as their counterparts with low SFF, if selected and used properly. 

The situation becomes problematic when a high SFF pump operates off peak (or away from its BEP), and/or operates at high suction specific speed (NSS) conditions. Operating a pump at off peak capacity increases its radial load that, in turn, increases its shaft deflection. On the other hand, cavitation occurring in high NSS pump increases its vibration that, in turn, also increases shaft deflection. Thus, improving the reliability and MTBF of a pump oftentimes requires not only a reduction of its SFF, but also a hydraulic re-rate to ensure that its hydraulics is optimized for the operating conditions. 

A pump can be modified to reduce its SFF by increasing its shaft diameter, decreasing its shaft span, or both. In some instances, this requires a different or new bearing housing. Making these changes is expensive. A bigger shaft diameter needs bigger mechanical seals and bearings which add significant cost to the equipment. Therefore, the cost-benefit has to be analyzed carefully to find an SSF that will yield an optimum return (ROI) on the added investment. Thus, the company that popularized the use of SFF came up with recommended SSF values for certain pump types and sizes. A reduction in SFF should not be overdone because at some point, a further reduction can start to harm the pump performance.

Example of using SSF assessment in pump evaluation:

A company received three quotes for a single stage overhang pump. The pumps have nearly the same efficiency and are similar in construction, except for the following data:

Option 1 - has SFF of 80 and costs $60,000

Option 2 - has SFF of 90 and costs $55,000

Option 3 - has SFF of 95 and costs $50,000

Based on a typical SFF cost assessment, which Option should be selected?

The answers to these questions, and how the cost assessment is applied, are discussed in the full version of this article.

Test your knowledge:

· There are situations when modifying a pump to reduce its SFF can result in unintended harm to the pump hydraulic and mechanical performance. 

What are these instances?