^{nd}order relationship between flow rate and head in pump performance.

Mathematically, the affinity rules are very easy calculations to perform. The rules describe how the pump performance will change with a change in speed, but they do not show how the pump will perform in any particular system. This is important when considering how to apply the rules to real world systems to determine the pump speed needed to meet an operating point for variable speed pump applications. Engineered Software's PIPE-FLO Professional and Flow of Fluids programs can be used to determine the pump speed in variable speed operations.

**Systems with Various Amounts of Dynamic and Static Head**Let's consider three systems with varying amounts of static and dynamic head to see real world variable pump applications, all shown in Figure 1 below. The systems consist of a supply tank, variable speed pump, fully open control valve, and product tank, all connected by various lengths of pipe to vary the amount of dynamic head. All equipment elevations are at 0 feet with the exception of the Product Tank, which changes to vary the amount of static head in the system.

Figure 1: Systems with varying amounts of dynamic and static head. At 1000 gpm the pump produces 98.56 feet of TDH at 1735 rpm.

The total pipe length in System 1 is 1570 feet and the Product Tank is at the same elevation and has the same liquid level as the Supply Tank, making it a purely resistive system with no static head. The dynamic head that can be determined using the Darcy method for calculating head loss, as shown in Equation 2 below.

Although it appears that the head loss has a 2^{nd} order relationship to the flow rate, the Darcy friction factor (f) is also a function of the flow rate, which makes the head loss not quite a 2^{nd} order relationship.

System 2 consists of identical components as the first system, but about 25% of the total head at the design flow rate of 1000 gpm is static head, accomplished by reducing the total pipe length to 1168 feet and raising the elevation of the Product Tank to 25 feet.

System 3 has a large amount of static head, about 75% of the total head at the design flow rate, accomplished by reducing the total pipe length to 365 feet and raising the elevation of the Product Tank to 75 feet.

**How a System's Static Head Effects the Pump Speed**Let's determine the pump speed needed to reduce the flow rate in the system down to 500 gpm with the variable speed pump. According to the pump affinity rules, reducing the flow rate by 50% will be accomplished by reducing the speed by 50%, or down to 867.5 rpm.

At this speed, the pump affinity rules can be used to determine the amount of head the pump would produce.

But does the pump produce the amount of head required by the system at 500 gpm at this speed? One way to find out is to calculate the head needed by the system by replacing the pump with a flow leaving the system at the pump suction and a flow entering the system at the pump discharge, then calculating the pressures at the inlet and outlet of the pump. The differential pressure can be converter to feet of fluid, which represents the total head the system requires at 500 gpm. This is done for the three systems in PIPE-FLO as shown below in Figure 2.

Figure 2: Systems shown with flow rates leaving and entering at the pump location to calculate the pressures at the pump suction and discharge, and therefore the system head requirements.

The differential pressure across the pump in System 1 is 18.59 psig – 7.226 psig = 11.36 psid. Convert this to feet of head using the following equation:

The pump speed calculated using the affinity rules does not result in a pump head high enough to meet what the system demands, so the speed must be higher than that calculated with the affinity rule. Figure 3 below shows the three systems calculated in PIPE-FLO and the required pump speed needed to meet the head requirement of each system.

Figure 3: Systems shown with pump speeds calculated in PIPE-FLO to match the head requirements of each system.

In System 1, the pump speed must be 895 rpm as compared to 867.5 rpm calculated with the affinity rule. This difference in pump speed is due to the fact that the system resistance, or head loss, is not quite a 2^{nd} order relationship to flow rate.

The problem is exacerbated by the presence of static head in the system. Notice the higher total head requirements for the second and third systems in Figure 2 and the calculated pump speeds needed to meet the system requirements in Figure 3. When the system flow rate is reduced to 500 gpm by the variable speed pump, the static head is not affected by the reduced flow, so the pump must still overcome the same amount of static head plus the reduced amount of dynamic head.

System 2 requires a total head of 44.6 feet and a pump speed of 1140 rpm to obtain 500 gpm, whereas System 3 requires 81.3 feet of total head and a pump speed of 1530 rpm. Using just the pump affinity rule in these cases would grossly underestimate the pump speed needed to reduce the flow to 500 gpm.

PIPE-FLO uses the pump affinity rules to obtain an initial guess at the required pump speed, but then adjusts that speed using a complex algorithm and an iterative process until the system total head requirements are met by the pump's total developed head.

**Effect of Static Head on Energy Savings**The calculation for pump speed also explains why the use of variable speed pumps in a system with a large amount of static head does not save as much money compared to a system with just dynamic head. Since the power consumption is reduced by the cube of the change in pump speed according to the affinity rules, the completely dynamic head system will consume much less power at the reduced flow rate as compared to the high static head system because of the greater change in pump speed. Pump speed is reduced from 1735 rpm to 895 rpm (840 rpm change) in System 1, from 1735 rpm to 1140 rpm in System 2 (595 rpm change), and from 1735 rpm to 1530 rpm in System 3 (205 rpm change).

Another impact on the energy consumption is where the pump operates on its pump curve at the reduced flow rate. Figure 4 below shows the pump curve (the same pump is used in all three systems) and the operating point for each system at the reduced flow rate. The pump in System 1 is operating at 88.3% efficiency, 83.1% efficiency in System 2, and 72.4% efficiency in System 3.

Figure 4: Pump curves showing operating point and pump efficiency for each system (System 1 on the left, System 2 in the middle, and System 3 on the right).

Let's assume each system is operated for half the year at 1000 gpm and half the year at 500 gpm with energy cost at $0.10 / kWhr and motor efficiency of 93%. Table 1 below summarizes the operating costs for each case as compared to operating the systems with a fixed speed pump at 1750 rpm and using the throttle valve to control the flow rate.

**Table 1: Operating Costs and Energy Savings for Systems with Various Amounts of Static Head**

**Conclusion**The presence of static head in a system reduces the energy savings when comparing variable speed pump operation to operating the system with a fixed speed pump and a throttle valve to control the flow rate. The pump affinity rules describe how the pump performance will change with a change in pump speed, but the actual operating speed of the pump will depend on the head requirements of the system. The more static head the system has, the higher the pump speed will need to be to overcome this head since static head does not change with the flow rate through the system.

In addition, the pump will operate farther back on its pump curve, resulting in lower pump efficiency at reduced flow rates. The net effect of higher operating speeds and lower pump efficiency in systems with static head is a reduction in the energy savings when using variable speed pumps.