## Affinity Laws Applied to Centrifugal Pump Applications

**Pump and System Curves**

The pump curve is solely a function of the physical characteristics of the pump. The system curve is completely dependent on the size of pipe, the length of pipe, the number and location of elbows, and other factors. Where these two curves intersect is the natural operating point (see Figure 4). This is where the pump pressure matches the system losses and everything is balanced.

**Figure 4. Sample pump system curves**

If the system is part of a process that changes often or continuously, then some method of altering the pump characteristics or the system parameters is necessary. Two methods can accomplish the continuously varying flow objective. One method is throttling, which changes the system curve by use of a control or throttling valve. The other method is to vary the speed of the pump, which modifies the pump curve.

**Throttling System**

With the throttling method, obstructing the flow increases the head pressure. A system with two different valve settings is shown in Figure 6.

**Figure 5. Throttling System**

**Figure 6. Sample power requirements for the throttling system**

For comparison, let’s use an example to determine the power requirements for the throttling system, then the variable speed system. A pump (with an 8-inch impeller) operating at a base speed of 3,560 rpm is used. This pump is to operate a system requiring a 250-foot head at 250 gpm (see Figure 6).

From the information shown, the horsepower requirements at the throttling system flow rates are shown in Table 1.

**Table 1. Throttling system power requirements**

**Variable Speed System**

In comparison, the variable speed method takes advantage of the change in pump characteristics that occur when the impeller speed is changed (see Figure 7). The lower pump speed changes the pump curve based on the head generated by the velocity of the fluid being pumped. Remember that the head is equal to V^{2}/2g.

**Figure 7. Sample variable speed system**

**Affinity Laws**

A set of formulas that is used to predict the operation of a centrifugal pump at any operating point based on the original pump characteristics is known as the affinity laws.

Where:

N = Pump speed

Q = Flow (gpm)

P = Pressure (feet)

HP = Horsepower

Using the same pump example as the throttling system, power requirements are calculated for the system for different

speeds (see Table 2).

Table 2. Variable system power requirementsNote: Use 25 HP for HP1, 1,750 for N1 and 250 for Q1 to fill in Table 2. |

Use the affinity laws to calculate the values for the remainder of the operating points. Obviously, varying the speed requires much less power. To determine the actual power required, the efficiency of the drive should be factored in. The energy savings will depend on the amount of time the pump is operated at each reduced speed point.

To calculate the actual savings, the brake horsepower must be converted to watts and then multiplied by the hours of operation. The result is then multiplied by the cost per kilowatt hour to show the cost to operate the pump at each flow point. Subtract the variable speed value from the throttling value to show the difference in energy cost.

Using the figures in Table 2, a flow of 200 gpm when throttled takes 22.5 horsepower. With variable speed only 12.8 horsepower is required. If the flow is required for 2,000 hours per year at 7 cents per kilowatt hour, the cost comparison is:

**Throttling system:**

22.5 HP x 0.746 = 16.785 kW

16.785 x 2,000 = 33,570 kWh

33,570 x 0.07 = $2,350

**Variable speed system:**

12.8 x 0.746 = 9.5488 kW

9.5488 x 2,000 = 19,097 kWh

19,097 x 0.07 = $1,337

**Savings:**

$2,350 –$1,337 = $1,013