Last month’s column described the process of creating an energy cost balance sheet for a piping system (see Figure 1). The manufacturer’s curve was used to calculate the cost of operation for the centrifugal pump. This column explains how to perform the detailed cost calculation for other items in the system.
Figure 1 shows the operating data from the installed plant instrumentation. This operating data and the manufacturer’s equipment data sheets will be used to calculate the differential pressure and flow rate through each item. With that information, the cost of electrical power and the annual rate of operation, the annual operating cost and energy use can be calculated for each item in the system’s energy cost balance sheet.
To calculate the operating cost for an item in the system, its head loss and corresponding flow rate must be determined. Insufficient installed instrumentation means these values need to be calculated with data from the equipment manufacturers.
Determining Pump Flow Rate
Last month’s column calculated a pump head of 235 feet (ft) and a flow rate of 4,000 gallons per minute (gpm). From the pump curve, it was determined that the pump had an efficiency of 83 percent at its flow rate. The pump’s operating costs were calculated after looking at the pump’s motor efficiency, annual hours of operation and cost of electrical power.
So how was the flow rate through the pump of 4,000 gpm established? The flow to the destination tank is 2,500 gpm, but because there is no flow element in the bypass circuit, the pump’s total flow rate cannot be calculated. Without this value, the pump efficiency and power consumed also cannot be calculated.
The pump curve is the key to this process because it provides manufacturer-supplied test data on how the pump operates for its flow range. Knowing the pump’s total head, the pump curve can be entered on the head axis and move across until the known pump head value intersects the pump curve. Dropping straight down on the pump curve gives the flow rate, and the intersection point provides the pump efficiency.
In Figure 1, the process pump’s discharge pressure is 102.8 pounds per square inch (psig). Centrifugal pump curves use head instead of pressure, so the discharge pressure must be converted to feet of fluid using Equation 1.
H = Head in feet of fluid
P = Pressure in lb/inch2
ρ = Fluid density in lb/ft3
144 is a conversion factor for ft2 to in2
A pressure gauge is not installed on the pump suction, so a temporary gauge was mounted on a suction vent. The temporary suction pressure gauge reads 1.7 psig, resulting in 3.95 ft of fluid. Subtracting the discharge and suction heads (239 ft minus 3.95 ft), the pump’s total head is 235 ft. Reading from the pump curve (see Figure 2), the resulting flow rate through the pump is 4,000 gpm, and the pump is 83 percent efficient.
Next, the cost to operate the system’s process elements will be evaluated using their flow rates and head loss.
Static Head Cost
The static head accounts for the difference in fluid energy between the destination and supply tanks. Per the Bernoulli equation, fluid energy is composed of three types of head: elevation, pressure and velocity. Because the fluid is at rest in the supply and destination tanks, the velocity head has no value (see Equation 2).
ZB = Elevation of tank bottom in ft
LFS = Level of fluid surface above tank bottom in ft
PFS = Pressure on fluid surface in psi
ρ = Fluid density in lb/ft3
1 = Supply tank
2 = Destination tank
The static head is the energy difference between the supply and destination tanks. It remains the same regardless of the flow rate between the tanks. The static head can be used to calculate the annual operating cost due to the head component within the process circuit.
The 2,500 gpm flow rate through the process circuit is used in the annual operating cost formula (see Equation 3). The 83 percent pump efficiency comes from the pump curve with a 4,000 gpm flow rate. The pump efficiency applies to the 4,000 gpm flow rate because the pump supplies both the process and bypass circuits. Inserting the static head pump efficiency, flow rate, hours of operating and cost of power results in an annual energy cost of $22,800 for the static head.
Q = Flow rate
H = Head
ρ = Fluid density
ηρ = Pump efficiency
ηm = Motor efficiency
Process Element Cost
Process equipment—characterized by a head loss that is a function of flow rate—is the next item in the energy cost balance sheet. In the system shown in Figure 1, the process element has no installed pressure gauges. The manufacturer’s equipment data sheet will be used to calculate head loss.
The manufacturer’s supplied data sheet could have a curve showing the pressure drop and head loss for a range of flow rates, a Cv value or a single value of pressure drop versus a given flow rate. For this example, the manufacturer provided a 14.4 psi pressure drop with a flow rate of 3,000 gpm.
To determine the pressure drop through the process element at 2,500 gpm, the Cv must be calculated using the manufacturer’s supplied data points of 3,000 gpm and 14.4 psi (see Equation 4).
Cv = Flow coefficient
Q = Volumetric flow rate in gpm
dP = Differential pressure in psi
SG = Specific gravity
The differential pressure across the process element is 10 psi, after calculating the Cv of the process elements, the specific gravity of the process fluid and the flow rate using Equation 5.
The 10 psi differential pressure converts to 23.3 ft head loss across the process equipment. The head loss results in a $5,592 annual energy cost for the process element as calculated in Equation 6.
The head loss in a pipeline can be calculated using the Darcy equation (see Equation 7).
However, Equation 8 makes calculating the head loss with the flow rate in gallons per minute and pipe inside diameter in inches easier.
hL = pipeline head loss in ft
f = Darcy friction factor (unitless)
L = pipe length in ft
D = pipe inside diameter in ft
v = fluid velocity in ft/s
g = gravitational constant in 32.2 ft/s2
Q = volumetric flow rate in gal/min
d = inside pipe diameter in inches
In this example, we’ll calculate the head loss for a 50-foot suction pipeline made of 16-inch schedule 40 steel. Two gate valves and a sharp edge transition lie along the pipeline from the tank. The pipeline is passing 4,000 gpm of a fluid with a fluid density of 62 lb/ft3 and viscosity of 0.68 centipoise. The steps required to arrive at the pipeline’s head loss are listed below along with the intermediate results.
- Reynolds number = 1.23 x 106 unitless
- Relative roughness = 0.00012 in
- Darcy friction factor = 0.0135 unitless
- Pipe head loss = 0.44 ft
- Valve and fitting loss = 0.57 ft
- Pipeline head loss = 1.01 ft
(The detailed head loss calculation for this example can be found at
The head losses for the circuit’s remaining pipelines are added to obtain the total pipeline head loss in Table 1.
The operating cost for the pipelines is $5,376, after plugging in the sum of the pipeline’s head loss and the flow element, as shown in Equation 9.
Control Valve Cost
Control valves regulate flow rate to achieve the desired set point—in this case, the level in the destination tank. The tank’s outflow (not shown on the drawing) feeds the plant’s demands. When the flow rates into and out of the tank are equal, the level in the destination tank is constant.
To determine the operating cost, the flow rate through and the differential pressure across the level control valve are needed. Most systems do not measure the valve’s differential pressure, which can be calculated knowing the valve position, flow rate and Cv values as a function of valve position. But that requires involved calculations. Instead, the interaction between the pump, process and control elements can be used to calculate the head loss across the control valve.
As shown in past articles, all the energy provided by the pump is used by the process and control elements (see Equation 10).
hPump = ∑ hProcess + hControl Equation 10a
Substituting the calculated value for pump head and the sum of the head loss of the circuit’s process elements, we can determine the head loss across the control valve.
235 = (95 + 23.3 + 22.4) + hControl Equation 10b
The annual operating cost of the level control valve is $22,633, after solving for the known flow rate with 94.3 ft of control valve head loss in Equation 11.
Bypass Circuit Cost
The annual operation costs of the bypass and process circuits follow the same approach. The bypass circuit provides minimum flow through the pump back to prevent pump damage, and its process elements consist of the static head and the pipelines.
The static head in the bypass circuit is the difference in elevations between the end of the pipeline discharging into the supply tank (30 ft) and the liquid level in the supply tank (5 ft). The head losses associated with the pipelines consist of the common pump suction and discharge header, along with the recirculation pipelines returning to the supply tank (3 ft).
The control element in the bypass circuit consists of a back pressure valve (BPV) designed to maintain a 100-psi pressure on the pump discharge regardless of the system flow rate. The set point was selected so the flow rate through the pump is always greater than the manufacturer’s minimum flow rate regardless of the flow to the destination tank.
Like the control valve example, the energy in the bypass circuit must balance (see Equation 12). This results in a head loss of 207 ft for the back pressure valve in the bypass circuit.
235 = (25 + 3) + hControl Equation 12
Next month’s column will explain how to cross-validate the calculated results of the energy cost balance sheet and determine if the system is operating properly.