The calculation of the motor efficiency for a variable-speed centrifugal pump can be complicated and requires first the determination of the pump speed, efficiency, and brake horsepower (bhp) required by the pump at various points on the water system's flow head curve or in the system head area. Some pumping systems have just one pump while others can have as many as five or six pumps. To simplify this evaluation, we will consider single pump operation only.

Figure 11 describes a water system with a uniform system head curve while Figure 22 describes a system with a system head area. The system head area is caused by load shifting in the water system. This means that various water loads on the system are active at different times. Loads far from the pumping installation may be active at one time while water loads near the pump may be active at other times.

The evaluation provided herein can be used for any point on the system head curve of Figure 1 or on the system head area of Figure 2. The above pump performance requirements will be determined by use of the Affinity Laws for variable torque machines such as centrifugal fans and pumps. The simple system head curve of Figure 1 will be used for this calculation of the motor efficiency as various points on the curve.

## Pump Efficiency and Speed

The basic Affinity Laws for centrifugal pumps are:

For fixed diameter impellers, the pump flow varies directly with the speed:

Equation 1 Q1 = S1
Q2 S2

The pump head varies as the square of the speed:

Equation 2 h1 = S12
h2 S22

Where the symbols are "Q" for flow in gpm, "h" for pump head in ft, and "S" for pump speed in rpm.

By combining equations 1 and 2 into equation 3, we can secure the relationship between head and flow. This equation enables us to determine the speed and efficiency at any flow and head if we know the flow, head, and speed at another point.

Equation 3
Q12 = h1 or Q1 = [Q22/h2 • h1]0.5
Q22 h2

Q1 and h1 determine the equivalent pump operating point on a known pump curve. Equation 3 enables us by trial and error to find the pump speed and efficiency for Q2 and h2. Q2 and h2 determine the pump operating point where we want know the pump speed and efficiency.

Figure 1 describes this procedure. Assume that it is desired to determine the pump speed and efficiency when it is operating at point 2, which is 500 gpm at 50 ft head. Insert these values in equation 3 and continually change values for h1 until the values for point 1 land on the known pump curve. In this case, the values for point 1 are 561 gpm at 63 ft. We thus have determined the pump efficiency, which is 83% from Figure 1. Now we need the pump speed.

From Equation 1, we can determine the pump speed for point 2. If the pump speed for point 1 is 1,750 rpm, the pump speed for point 2 is 500 gpm/561 gpm multiplied by 1,750 rpm or 1,560 rpm. We now have the information necessary to determine the pump bhp. This procedure should be adequate for the normal operating range of variable-speed centrifugal pumps which is from 30% to 100% of full speed.

## Pump bhp

Equation 4 provides the calculation of the pump bhp.

Pump bhp = ___ Flow, gpm x head, ft.
3,960 x pump efficiency (as a decimal)

For our example:

Pump bhp = 561 x 50 = 7.6 bhp
3,960 x 0.83

From Figure 1, we can calculate the bhp at the design condition of 600 gpm at 60 ft where the pump efficiency is 84%. Using equation 4:

Pump bhp at design = _ 600 x 60 = 10.7 bhp
3,960 x 0.84

It would be the designer's decision whether a 10- or 15-hp motor would be required for this application. If a 15-hp motor is selected, its efficiency at full speed and 15 hp load condition would be 91.0%.

## Multiple Point Selection

We have now completed the evaluation of the pump at one point, namely 500 gpm at 50 ft. To complete this evaluation, we should determine the pump performance at other points on the pump curve of Figure 1. Table 1 lists these points and the pump conditions using the above procedures. This will give us the total analysis of this variable-speed pump as it varies from 100 to 600 gpm.

## Electric Motor Efficiencies

Since we have determined the bhp and speed at our design condition, we must now determine the motor efficiency. This is the most difficult part of our calculations. Figure 43 describes an efficiency curve for a 10-hp NEMA Design B induction motor at 1,760 rpm. Unfortunately, there is no published data for such a motor at reduced speeds and loads. At present, there is no published program for determining the efficiency of such a motor at reduced loads and speeds.

Various motor manufacturers have internal programs where they can provide you with an estimate for their motors at reduced loads and speed, but the author was unable to secure such a program in computer form. The data shown in Table 2 for motor efficiency was developed from an old development project. This data includes the effect of a VSD on the electric motor.

## VSD Efficiency

Almost all VSDs for centrifugal pumps are of the variable-frequency type. This is due to their cost, compactness, and efficiency. Most of these drives have efficiencies in the range of 95% to 98%. Older drives disturbed the sine wave and caused lower drive and motor efficiencies. Contemporary drives do not do this and can be credited with much higher efficiencies.

It has been difficult to acquire drive efficiency data that is related to the type and speed of the motor. There is so much inexactness in the computation of pump head and other factors in variable speed pumping that, for most applications of 10 hp and above, a drive efficiency of 97% can be assumed. For precise calculations, the VSD manufacturer should be consulted for exact efficiencies. The type and quality of the motor should be provided to this manufacturer.

## Computation and Use of kW Input

Table 2 provides the kW input for the model pump using the data of Tables 1 and 2 and a VSD percentage of 97%. kW input is derived from Formula 4:
kW input = Pump bhp x 0.746
Motor efficiency x VFD efficiency

The energy input to a pumping system has proved to be the simplest and most efficient method of programming systems where there are several pumps. Figure 42 describes a two-pump system with the addition and subtraction of the standby pump being accomplished through the use of kW input to the pump drives. The dashed curves indicate the calculated energy input while the solid lines are for the actual, measured kW input.

The pump controller in this case is equipped with adaptive control, which adjusts the point of addition to achieve the lowest possible energy consumption. This enables the system to operate at the highest efficiency even though the original kW input calculations were not accurate.

## Conclusions

It is obvious from the above that the computation of kW input for an electric motor at various loads at reduced speeds is a complex subject requiring calculations and estimates. It is the author's experience that it is difficult to get within 5% to 10% accuracy for calculated kW inputs of variable-speed pumps when comparing them afterwards with electrical inputs secured from actual operation.

Considering the thousands of variable-speed motors that are installed each year, it is the writer's opinion that an independent organization such as NEMA or IEEE should develop a program for determining the estimated efficiencies of induction motors at reduced speeds and loads so that we do not need to depend upon a motor manufacturer for motor efficiency for each application of such a motor. If we are to estimate energy consumptions of variable speed pumps, it is imperative that we have readily available, estimated motor efficiencies at reduced loads and speeds. ES

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Works Cited
1. Rishel, James B. P.E., HVAC Pump Handbook, McGraw-Hill, New York City, 1996.
2. Rishel, James B. P.E., Water Pumps and Pumping Systems, McGraw-Hill, New York City, 2002.
3. Jordon, Howard E., Energy-Efficient Electric Motors and Their Applications, Second Edition, Plenum Press, New York City, 1994.