It was 1958. I was fresh out of college and working in New York City for a major manufacturer of industrial machinery. One of my early assignments was to calculate the shaft deflections of a number of pumps being bid for hydrocarbon processing to a major contractor who required calculations demonstrating that the wear rings would not rub when operated at low capacities. I calculated the radial thrust on the impellers probably using data from Stepanoff [1]. His book offered the following equation for calculating the radial thrust:

Where:

D2  = impeller diameter [outside diameter—OD], inches

B2  = impeller overall width including shrouds [at the impeller OD], inches

K   = a constant that varies with capacity, determined experimentally

Notice that the product D2 x B2 is the projected area of the discharge of the impeller, and H/2.31 is the total differential pressure produced by the pump. The product of pressure x area, therefore, calculates a force. The K factor is intended to adjust that force to the actual radial thrust. The absence of specific gravity in the equation indicates that its use was intended only for cool water. Although the Stepanoff data was precise and detailed, it reported thrust characteristics for only one size pump. In 1959, Agostinelli, et al, reported radial thrust test results for 16 different pumps [2], while continuing the procedure of considering the effective pressure area being the impeller OD x width (D2 x B2).

 Figure 1. Radial thrust factor at shutoff for single-volute (constant-velocity) pumps (From Reference 3. Courtesy of the Hydraulic Institute, www.pumps.org, Parsippany, N.J.)

 Figure 2. The current Hydraulic Institute radial thrust factor graph for single-volute (constant-velocity) pumps (Courtesy of the Hydraulic Institute, www.pumps.org, Parsippany, N.J.)

## Hydraulic Institute Curve

In 1969, the Hydraulic Institute (HI) published a curve showing K values at shut-off for single-volute pumps as a function of specific speed [3] (see Figure 1). Although the HI values agreed with Agostinelli [2] in the higher specific speed range (around 3,000), they were almost twice the Agostinelli values in the lower range (around 600). The current HI K value graph, as seen in Figure 2, shows shut-off K values lower than the 1969 graph, and which compare favorably with Agostinelli [2]. The procedure continued to consider the effective area to be D2 x B2.

## Actual Radial Thrust of Vertical,  In-line Pumps

I learned that, using Equation 1—an accepted, published equation—and K values, produces significantly inaccurate radial thrust values for some pumps. I was asked by a pump manufacturer to determine the actual radial shaft deflection, at the mechanical seal, for a line of vertical, in-line centrifugal pumps, similar to that in Figure 3. The pump shaft was rigidly coupled to the motor shaft so that, when the pump was equipped with a mechanical seal, the motor bearings absorbed both axial and radial thrusts from the pump.

 Figure 3. A vertical in-line pump with semi-open impeller and rigid coupling

The pumps were equipped with semi-open impellers. The impeller faces were machined at an angle of 20 degrees, resulting in vanes that got wider (an increasing B2 dimension) as the diameter was reduced. Such design is common for impellers in pumps provided to the chemical industry, although uncommon for enclosed impellers and some semi-open impellers. Because the casings were volutes, maximum radial thrust occurred at shut-off (zero gallons per minute).