The various calculations used in the power cost balance sheets in June’s Pump System Improvement column, which were shown in Tables 1 and 2 on page 22 of the June issue of *Pumps & Systems* (see them below), are documented in industry standards, along with methods covered in fluid dynamics texts. This article will demonstrate these calculations and provide additional details that explain how the conclusions were derived.

## Calculating the Pump Elements

The manufacturer’s supplied pump curve is the primary document for pump operation. A curve for the spray pump is shown here as Figure 2.

Looking at the curve we can see that the pump with a 7.375-inch impeller operating at 3,550 revolutions per minute (rpm) develops 213.3 feet of head with an efficiency of 78.1 percent. The nameplate on the attached motor shows an efficiency of 94 percent.

With the pump operating 8,000 hours per year and an electrical power cost of $0.10 per kilowatt (kW), the annual operating costs of the spray pump can be calculated using Equation 1.

Inserting the known values into Equation 1 results in the following calculation:

An important point to remember is all the head developed by the pump is consumed by the process and control elements. We will demonstrate how Equation 1 can be used to determine the annual operating cost for each item found in the power cost balance sheet.

## Calculating the Process Elements

Next, we will look at the process elements using energy to provide the static head across the system; overcome the head loss of the process fluid passing through the pipe, valves and fittings; and provide the differential pressure across the spray nozzle to achieve the proper fluid distribution. All of the system energy is provided by the pump, so if we can determine the head loss across these devices along with the flow rate through them, we can determine the amount of power provided by the pump that is used for each process element item.

We will first look at the energy and power required for the fluid to overcome differences in elevation and pressure across the system.

Figure 3 shows that the stock tank is the inlet boundary, and the outlet boundary is the discharge of the spray nozzles into the atmosphere. The static head is calculated using Equation 2.

Inserting the know values into Equation 2 gives us the following calculation:

The spray pump supplies the energy to each item in the system and the static head is only a part of the pump’s total head required, so the annual operating cost can be determined for the static head by using the h_{S} value calculated in Equation 2 with the head inserted into Equation 1.

Next we will determine the annual operation cost for the system pipelines. The head loss in each pipeline is calculated using the Darcy-Weisbach method found in fluid dynamic texts or Crane Technical Paper 410. The head losses for all pipelines are displayed in Figure 3. The three pipelines are in series, so the head loss for all pipelines in the system can be determined by summing the individual head losses for each pipeline.

The annual operating cost attributed to the pipelines can be determine by using Equation 1 and inserting the pipeline head loss in the equation. This results in an annual operating cost of $10,663 for the pipelines.