Pump Head Calculator

Detailed Description

Pump head is the mechanical energy that the pump adds to each unit weight of fluid, expressed as feet of the pumped fluid. For steady, incompressible flow, that energy appears as changes in elevation, pressure, and velocity and as energy dissipated between the measurement locations and the pump flanges. The calculator combines those contributions to reconstruct the pump head from field data. This approach follows the Pumping System Assessment Tool (PSAT) head-tool method described in [3].

Final Result

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Pump head is the sum of three signed differential heads and two estimated line losses:

\begin{equation}\label{eq:pump-head-total} H_{pump}=\Delta H_z+\Delta H_p+\Delta H_v+H_{f,s}+H_{f,d} \end{equation}

\(H_{pump}\)	Pump head \([\unit{ \foot}]\)
\(\Delta H_z\)	Discharge-minus-suction elevation head \([\unit{ \foot}]\)
\(\Delta H_p\)	Discharge-minus-suction pressure head \([\unit{ \foot}]\)
\(\Delta H_v\)	Discharge-minus-suction velocity head \([\unit{ \foot}]\)
\(H_{f,s}\)	Estimated suction-line friction head \([\unit{ \foot}]\)
\(H_{f,d}\)	Estimated discharge-line friction head \([\unit{ \foot}]\)

The three differential terms describe the change in fluid energy from the selected suction reference point to the discharge gauge. Each may be positive, zero, or negative. The friction terms are added to account for the energy lost between the reference points and the pump flanges. For nonnegative loss coefficients, both friction terms are nonnegative.

Choose the Measurement Configuration

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Use the configuration that matches the available suction-side measurement:

Configuration	Suction reference point	Suction pressure	Suction velocity used in \(\Delta H_v\)
Suction gauge	Center of the suction pressure gauge	Measured suction gauge pressure \(P_s\)	Velocity in the suction pipe
Suction tank	Fluid surface in a tank, well, or reservoir	Gas overpressure above the fluid \(P_t\); use 0 psig when open to atmosphere	Zero at the tank surface

In both configurations, the suction-pipe velocity is still calculated because it is needed for the suction-line friction estimate. The tank assumption only sets the velocity at the tank fluid surface to zero when calculating the differential velocity head.

Measurements and Sign Convention

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• Measure flow in U.S. gallons per minute and pipe inside diameters in inches. For each gauge, use the pipe inside diameter at its pressure-measurement location [3].
• Enter pressures as gauge pressure in psig. Vacuum on the suction side may therefore be entered as a negative gauge pressure.
• Reference every elevation to the same datum. The datum may be arbitrary, such as a plant floor, because only elevation differences are used [3].
• Measure gauge elevation at the center of the gauge, not the center of the pipe. The elevation must correspond to the supplied pressure reading. If a reading has been corrected to another reference elevation, use that elevation instead.
• Use one steady flow rate for the suction and discharge pipes.
• Enter fluid specific gravity relative to the reference water used by the pressure-head conversion.
• Sum the applicable dimensionless loss coefficients separately for the suction and discharge paths between their reference points and the pump flanges.

The calculation proceeds from the field measurements as follows:

1. Convert flow to cubic feet per second and pipe diameters to feet.
2. Calculate suction- and discharge-pipe velocities and velocity heads.
3. Calculate the elevation, pressure, and velocity differences for the selected configuration.
4. Estimate suction- and discharge-line friction heads from their loss coefficients.
5. Sum all five contributions using \eqref{eq:pump-head-total}.

Elevation Head

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Elevation head is the vertical change from the suction reference point to the discharge gauge:

\begin{equation}\label{eq:pump-head-elevation} \Delta H_z=\begin{cases} z_d-z_s & \text{suction gauge}\\ z_d-z_t & \text{suction tank} \end{cases} \end{equation}

\(\Delta H_z\)	Discharge-minus-suction elevation head \([\unit{ \foot}]\)
\(z_d\)	Elevation of the center of the discharge gauge from the common datum \([\unit{ \foot}]\)
\(z_s\)	Elevation of the center of the suction gauge from the common datum \([\unit{ \foot}]\)
\(z_t\)	Suction-tank fluid-surface elevation \([\unit{ \foot}]\)

A discharge gauge below the suction reference point produces a negative elevation-head contribution. This is physically valid; it adjusts for the lower discharge gauge and partially offsets the pressure-head contribution.

Pressure Head

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Pressure head follows from the hydrostatic relation \(H=p/\gamma\). Pressure in psi is first multiplied by 144 to express it per square foot. Fluid specific weight is represented as the reference-water specific weight multiplied by specific gravity:

\begin{equation}\label{eq:pump-head-pressure-basis} \gamma=SG\,\gamma_w,\qquad k_p=\frac{\gamma_w}{144},\qquad \Delta H_p=\frac{144\,\Delta P}{\gamma_w SG}=\frac{\Delta P}{k_p SG} \end{equation}

\(\gamma\)	Specific weight of the pumped fluid \([\unit{ \lbf\per\foot\cubed}]\)
\(\gamma_w\)	Reference-water specific weight represented by the calculator, 62.316000144 \([\unit{ \lbf\per\foot\cubed}]\)
\(SG\)	Fluid specific gravity relative to the reference water \([\unit{ \unitless}]\)
\(k_p\)	Reference-water pressure gradient, 0.432750001 \([\unit{ \psi\per\foot}]\)
\(144\)	Square inches per square foot \([\unit{ \inch\squared\per\foot\squared}]\)
\(\Delta P\)	Discharge-minus-suction gauge-pressure difference \([\unit{ \psi}]\)
\(\Delta H_p\)	Discharge-minus-suction pressure head \([\unit{ \foot}]\)

Apply the pressure difference for the selected configuration:

\begin{equation}\label{eq:pump-head-pressure-configurations} \Delta P=\begin{cases} P_d-P_s & \text{suction gauge}\\ P_d-P_t & \text{suction tank} \end{cases} \end{equation}

\(\Delta P\)	Discharge-minus-suction gauge-pressure difference \([\unit{ \psi}]\)
\(P_d\)	Discharge gauge pressure \([\unit{ \psig}]\)
\(P_s\)	Suction gauge pressure \([\unit{ \psig}]\)
\(P_t\)	Suction-tank gas overpressure \([\unit{ \psig}]\)

Specific gravity compares the fluid's density with that of water at the reference condition (68 \(\unit{\degreeFahrenheit}\) / 20 \(\unit{\degreeCelsius}\)), where \(SG=1\). At this condition, water has a density of about 62.3 \([\unit{\lbm\per\foot\cubed}]\) [7]. Under standard gravity, 1 lbm weighs 1 lbf, so the pressure-head equation uses the same number as the specific weight of water, \(\gamma_w\) (62.316 \(\unit{\lbf\per\foot\cubed}\) in the calculator).

The retained pressure gradient is consistent with these approximate reference-water properties and gives 2.310803 feet of water per psi at \(SG=1\). The familiar 2.31 feet per psi is an engineering approximation [8] [5].

Pipe Velocity and Velocity Head

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Convert the entered flow and diameter before applying continuity and the velocity-head relation:

\begin{equation}\label{eq:pump-head-flow-conversion} Q=\frac{q}{k_Q},\qquad D_{ft}=\frac{D}{12} \end{equation}

\(Q\)	Volumetric flow rate used by the velocity calculation \([\unit{ \foot\cubed\per\second}]\)
\(q\)	Entered volumetric flow rate \([\unit{ \gallon\per\minute}]\)
\(k_Q\)	Retained flow conversion, 448.8311693 \([\unit{ \gallon\second\per\minute\foot\cubed}]\)
\(D_{ft}\)	Pipe inside diameter used by the area calculation \([\unit{ \foot}]\)
\(D\)	Entered pipe inside diameter \([\unit{ \inch}]\)
\(12\)	Inches per foot \([\unit{ \inch\per\foot}]\)

\begin{equation}\label{eq:pump-head-velocity} A=\frac{\pi D_{ft}^{2}}{4},\qquad v=\frac{Q}{A},\qquad H_v=\frac{v^2}{2g} \end{equation}

\(A\)	Internal pipe cross-sectional area \([\unit{ \foot\squared}]\)
\(\pi\)	Circle ratio \([\unit{ \unitless}]\)
\(D_{ft}\)	Pipe inside diameter \([\unit{ \foot}]\)
\(v\)	Average pipe velocity \([\unit{ \foot\per\second}]\)
\(Q\)	Volumetric flow rate \([\unit{ \foot\cubed\per\second}]\)
\(H_v\)	Pipe velocity head \([\unit{ \foot}]\)
\(g\)	Gravitational acceleration, 32.174 \([\unit{ \foot\per\second\squared}]\)
\(4\)	Geometric denominator for circular area expressed with diameter \([\unit{ \unitless}]\)
\(2\)	Kinetic-energy denominator in the velocity-head relation \([\unit{ \unitless}]\)

The differential velocity head is then:

\begin{equation}\label{eq:pump-head-velocity-differential} \Delta H_v=\begin{cases} H_{v,d}-H_{v,s} & \text{suction gauge}\\ H_{v,d} & \text{suction tank} \end{cases} \end{equation}

\(\Delta H_v\)	Discharge-minus-suction velocity head \([\unit{ \foot}]\)
\(H_{v,d}\)	Discharge-pipe velocity head \([\unit{ \foot}]\)
\(H_{v,s}\)	Suction-pipe velocity head \([\unit{ \foot}]\)

Equal suction and discharge pipe diameters give equal pipe velocities and therefore zero differential velocity head in the suction-gauge configuration.

Estimated Line Losses

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Each aggregate loss coefficient multiplies the velocity head in the pipe used as that side's reference diameter:

\begin{equation}\label{eq:pump-head-friction} H_{f,s}=K_sH_{v,s},\qquad H_{f,d}=K_dH_{v,d} \end{equation}

\(H_{f,s}\)	Estimated suction-line friction head \([\unit{ \foot}]\)
\(H_{f,d}\)	Estimated discharge-line friction head \([\unit{ \foot}]\)
\(K_s\)	Aggregate suction-line loss coefficient \([\unit{ \unitless}]\)
\(K_d\)	Aggregate discharge-line loss coefficient \([\unit{ \unitless}]\)
\(H_{v,s}\)	Suction-pipe velocity head \([\unit{ \foot}]\)
\(H_{v,d}\)	Discharge-pipe velocity head \([\unit{ \foot}]\)

A fitting coefficient tabulated for a different diameter must be normalized to the pipe diameter used by the calculator. For the same flow, velocity varies with \(1/D^2\) and velocity head varies with \(1/D^4\). Therefore:

\begin{equation}\label{eq:pump-head-loss-coefficient-scaling} K_{ref}=K_e\left(\frac{D_{ref}}{D_e}\right)^4 \end{equation}

\(K_{ref}\)	Loss coefficient normalized to the calculator's reference pipe \([\unit{ \unitless}]\)
\(K_e\)	Loss coefficient stated for the fitting or element \([\unit{ \unitless}]\)
\(D_{ref}\)	Inside diameter of the reference pipe entered in the calculator \([\unit{ \inch}]\)
\(D_e\)	Inside diameter associated with the stated element coefficient \([\unit{ \inch}]\)

For example, a 12-inch valve with \(K_e=0.4\) located in a system referenced to a 16-inch pipe contributes \(K_{ref}=0.4(16/12)^4=1.26\). This normalization guidance and example are consistent with the PSAT head-tool documentation [3].

Unit Conversions and Reference Properties

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The flow conversion is a unit relationship. One cubic foot contains 1728 cubic inches, one U.S. liquid gallon contains exactly 231 cubic inches, and one minute contains 60 seconds [4]. Thus:

\begin{equation}\label{eq:pump-head-gpm-per-cfs} k_{Q,exact}=\frac{1728}{231}(60)=448.831168831\ldots \end{equation}

\(k_{Q,exact}\)	Unit-derived gallons per minute in one cubic foot per second \([\unit{ \gallon\second\per\minute\foot\cubed}]\)
\(1728\)	Cubic inches per cubic foot \([\unit{ \inch\cubed\per\foot\cubed}]\)
\(231\)	Cubic inches per U.S. liquid gallon \([\unit{ \inch\cubed\per\gallon}]\)
\(60\)	Seconds per minute \([\unit{ \second\per\minute}]\)

The implementation retains 448.8311693 rather than replacing it with the longer unit-derived value so existing calculation results remain unchanged. See physics::conversions::kGallonsPerMinutePerCubicFootPerSecond.

Head is expressed in feet because it is energy per unit weight of fluid: foot-pounds of energy per pound-force reduce to feet. This also relates pressure to the height of a liquid column [5]. For a centrifugal pump at the same flow rate, speed, and impeller geometry, with comparable viscosity and no cavitation, head is approximately independent of fluid density, while the pressure rise for that head changes with specific gravity.

The pressure factor is a reference property rather than a pure unit conversion. The retained 0.432750001 psi per foot implies \(\gamma_w=144(0.432750001)=62.316000144\) \([\unit{\lbf\per\foot\cubed}]\), consistent with approximate water properties near 68 degrees Fahrenheit [7]. Its reciprocal is 2.310803 feet per psi, commonly approximated as 2.31. The calculator retains the exact stored factor so existing results remain unchanged; the numerical relationship does not establish the historical origin of its final digits. See physics::conversions::kPsiPerFootOfWater.

Worked Suction-Gauge Example

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Consider water-like fluid with \(SG=1.0\) flowing at 2000 gpm. The suction measurements are \(D_s=17.9\) inches, \(P_s=5\) psig, \(z_s=5\) feet, and \(K_s=1\). The discharge measurements are \(D_d=15\) inches, \(P_d=50\) psig, \(z_d=1\) foot, and \(K_d=1\).

Contribution	Calculation result (ft)	Interpretation
\(\Delta H_z\)	-4.000000	The discharge gauge is 4 ft below the suction gauge.
\(\Delta H_p\)	103.986135	The pressure rise is the dominant contribution.
\(\Delta H_v\)	0.103859	The smaller discharge pipe has the greater velocity head.
\(H_{f,s}\)	0.101040	Estimated loss from the suction gauge to the pump.
\(H_{f,d}\)	0.204899	Estimated loss from the pump to the discharge gauge.
\(\boldsymbol{H_{pump}}\)	100.395932	Sum of all five contributions.

The negative elevation term does not indicate an error; it offsets part of the pressure-head rise.

WebAssembly Example

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The package exports typed input and result objects from its root entrypoint. The locateFile path depends on where the consuming application deploys client.wasm.

import createModule, {
  type MeasurToolsSuite,
  type PumpHeadSuctionGaugeInput,
} from 'measur-tools-suite';
 
const suite: MeasurToolsSuite = await createModule({
  locateFile: (filename: string) => `/path/to/wasm/${filename}`,
});
 
const input: PumpHeadSuctionGaugeInput = {
  specificGravity: 1.0,
  flowRate: 2000,
  suctionPipeDiameter: 17.9,
  suctionGaugePressure: 5,
  suctionGaugeElevation: 5,
  suctionLineLossCoefficients: 1,
  dischargePipeDiameter: 15,
  dischargeGaugePressure: 50,
  dischargeGaugeElevation: 1,
  dischargeLineLossCoefficients: 1,
};
 
const result = suite.calculatePumpHeadFromSuctionGauge(input);
console.log(result.pumpHead); // approximately 100.396 ft

For a tank-surface reference, provide a PumpHeadSuctionTankInput and call calculatePumpHeadFromSuctionTank.

Assumptions and Limits

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• Flow is steady and incompressible, and the same volumetric flow passes through both pipes.
• Pipe diameters are inside diameters. Specific gravity is constant over the calculation.
• Each gauge measures pressure on pipe of the entered inside diameter [3]. In the tank configuration, the suction diameter is the reference for suction-line losses, while the suction elevation and pressure refer to the tank fluid surface.
• Pressure measurements are gauge pressures and elevations share a common datum. Gauge elevations refer to the centers of the gauges unless the supplied readings have been corrected to another reference elevation; then use the corresponding corrected reference elevations.
• The suction tank or reservoir is large enough that fluid velocity at its surface is negligible.
• Loss coefficients are supplied by the user, are normalized to the entered reference-pipe diameters, and represent only the paths between the reference points and pump flanges.
• The calculation does not enforce physical input ranges. Specific gravity and pipe diameters must be greater than zero, and users must determine whether the measurements and coefficients are suitable for the system being assessed.
• This calculation determines pump head; it does not calculate net positive suction head, pump efficiency, cavitation margin, transient behavior, or a complete system curve.

Related Implementation

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• C++ namespace: pump_head
• Suction-gauge calculation: pump_head::calculateFromSuctionGauge
• Suction-tank calculation: pump_head::calculateFromSuctionTank
• Shared constants: physics::conversions and physics::us

Files
file	pump_head.h
	Declares calculations of the pump head developed from measured operating conditions.

Namespaces
namespace	pump_head
	Calculates pump head from measured system operating conditions.