Bernoulli Equation Calculator

Solve for pressure, velocity, elevation head, pump head, or head loss between two points using the extended Bernoulli equation.

Point 1
Point 2
Pumps & Losses

Fluid Mechanics Guide

Bernoulli Equation Calculator: Turn Pressures, Velocities, and Head into Clear Decisions

This guide walks through how to use the Bernoulli Equation Calculator, what the terms mean, and how to avoid the most common mistakes when evaluating pressure, velocity, and elevation changes in real fluid systems.

7–10 min read Updated 2025

Quick Start

The Bernoulli Equation Calculator is built around the classic energy form of Bernoulli between two points on the same streamline:

\[ \frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L \]

Here \(p\) is pressure, \(\gamma\) is specific weight, \(V\) is velocity, \(z\) is elevation head, \(h_p\) is pump head added, and \(h_L\) is head loss.

  1. 1 Choose what you want the Bernoulli Equation Calculator to solve for: downstream pressure, velocity, elevation head, required pump head, or total head loss.
  2. 2 Select your fluid and unit system. For most water problems the density is close to \(1000\ \text{kg/m}^3\), but the calculator also supports custom density and specific weight.
  3. 3 Enter the known conditions at point 1 and point 2: pressure, velocity (or flow rate + diameter), and elevations \(z_1\) and \(z_2\) using a consistent datum.
  4. 4 Add any pump head (positive when a pump adds energy) and estimate the head loss \(h_L\) or loss coefficient \(K\) if available.
  5. 5 Run the calculation, then review: the computed unknown, the individual head terms, and the total head balance.
  6. 6 Use the quick stats to sanity-check the Reynolds number (if available), total head change, and whether the sign of the result matches your intuition.
  7. 7 Adjust inputs (for example, pipe diameter or pump head) and re-run to explore “what-if” scenarios before locking in a design.

Tip: Keep all pressures either gauge or absolute on both points. The calculator can work with either, but mixing them in one problem will completely break the energy balance.

Warning: Bernoulli is derived for steady, incompressible, inviscid flow along a streamline. Real systems are never perfect, so you must include appropriate loss terms and confirm that density changes are negligible.

Choosing Your Method

The Bernoulli Equation Calculator primarily works in terms of head (energy per unit weight). Different users prefer slightly different formulations, so the calculator supports several common ways of framing the same physics.

Method A — Classic Bernoulli Between Two Points

Use when you know conditions at both points and want to solve for a single unknown (typically pressure or velocity).

  • Maps directly to the textbook equation.
  • Great for pipes with known velocities at both ends.
  • Easy to visualize using an energy grade line (EGL).
  • Requires you to estimate head losses \(h_L\) from charts or a separate friction calculator.
  • Less intuitive when multiple pumps or turbines lie between the two points.
\(\dfrac{p_1}{\gamma} + \dfrac{V_1^2}{2g} + z_1 = \dfrac{p_2}{\gamma} + \dfrac{V_2^2}{2g} + z_2 + h_L\)

Method B — Extended Bernoulli with Pumps and Losses

Use when pumps, turbines, or significant minor losses exist between locations.

  • Tracks added pump head \(h_p\) and lost head \(h_L\) explicitly.
  • Better for real piping systems with valves, fittings, and equipment.
  • Pairs naturally with Darcy–Weisbach or Hazen–Williams calculators for \(h_L\).
  • Requires more input data (loss coefficients, pump curves, etc.).
  • A bit more bookkeeping for elevations and sign conventions.
\(\dfrac{p_1}{\gamma} + \dfrac{V_1^2}{2g} + z_1 + h_p = \dfrac{p_2}{\gamma} + \dfrac{V_2^2}{2g} + z_2 + h_L\)

Method C — Head Form with Hydraulic & Energy Grade Lines

Use when you want to see how pressure head, velocity head, and elevation interact visually.

  • Great for teaching and design reviews.
  • Helps you identify where head is being lost or added.
  • Makes it easier to spot impossible or inconsistent inputs.
  • A small extra step to interpret results compared with the raw numeric answer.
  • Less common in quick hand calculations, more common in reports.
\(H = \dfrac{p}{\gamma} + \dfrac{V^2}{2g} + z\)   (total head at a point)

What Moves the Number the Most

Bernoulli’s equation groups several physical effects into three main head terms. Knowing which levers matter most makes the Bernoulli Equation Calculator far more useful.

Velocity head \(\dfrac{V^2}{2g}\)

Velocity appears squared, so small changes in speed create large changes in head. Doubling velocity increases velocity head by a factor of four.

Elevation difference \((z_1 – z_2)\)

A 10 m drop adds roughly 10 m of head to the downstream point. Elevation dominates in gravity-fed systems like reservoirs and siphons.

Pressure head \(\dfrac{p}{\gamma}\)

In pressurized pipes, most of the energy often lives in pressure head. Converting pressure units correctly is critical to avoid order-of-magnitude errors.

Head loss \(h_L\)

Represents friction and minor losses. Long, rough, or small-diameter pipes, plus valves and fittings, can easily consume much of the available head.

Pump head \(h_p\)

A pump adds energy to the fluid. If you undersize \(h_p\), the calculator will show insufficient downstream pressure or velocity.

Fluid density \(\rho\) and specific weight \(\gamma\)

Heavier fluids convert a given pressure into less head. Always pair the correct density with your pressure units.

Datum selection

You are free to choose \(z = 0\) anywhere, but it must be consistent. Changing the datum across points breaks the elevation head term.

Steady vs. unsteady flow

Bernoulli is a steady-flow model. Large transients (e.g., water hammer) require more advanced methods beyond a simple Bernoulli calculation.

Worked Examples

The following examples mirror typical use cases for the Bernoulli Equation Calculator. Values are rounded to keep the arithmetic readable; your calculator will carry full precision.

Example 1 — Downstream Pressure in a Horizontal Pipe

  • Fluid: Water, \(\gamma \approx 9.81\ \text{kN/m}^3\)
  • Pipe: Constant diameter, horizontal → \(z_1 = z_2\)
  • Upstream pressure: \(p_1 = 300\ \text{kPa (gauge)}\)
  • Upstream velocity: \(V_1 = 2.0\ \text{m/s}\)
  • Downstream velocity: \(V_2 = 3.0\ \text{m/s}\)
  • Head loss: \(h_L = 4.0\ \text{m}\)
  • Pump head: None, \(h_p = 0\)
1
Write Bernoulli between points:
\[ \frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L \] With \(z_1 = z_2\), the elevation terms cancel.
2
Compute head terms from known data:
\[ \frac{p_1}{\gamma} = \frac{300{,}000}{9{,}810} \approx 30.6\ \text{m} \]
\[ \frac{V_1^2}{2g} = \frac{2.0^2}{2 \times 9.81} \approx 0.20\ \text{m} \]
\[ \frac{V_2^2}{2g} = \frac{3.0^2}{2 \times 9.81} \approx 0.46\ \text{m} \]
3
Solve for downstream pressure head:
\[ \frac{p_2}{\gamma} = \frac{p_1}{\gamma} + \frac{V_1^2}{2g} – \frac{V_2^2}{2g} – h_L \] \[ \frac{p_2}{\gamma} \approx 30.6 + 0.20 – 0.46 – 4.0 \approx 26.3\ \text{m} \]
4
Convert back to pressure:
\[ p_2 = \gamma \frac{p_2}{\gamma} \approx 9{,}810 \times 26.3 \approx 258{,}000\ \text{Pa} \approx 258\ \text{kPa (gauge)} \]
The Bernoulli Equation Calculator would return a downstream pressure near 258 kPa for these inputs.

Example 2 — Required Pump Head Between Two Tanks

  • Fluid: Water, steady and incompressible
  • Tank 1 free surface elevation: \(z_1 = 0\ \text{m}\)
  • Tank 2 free surface elevation: \(z_2 = 12\ \text{m}\)
  • Both surfaces open to atmosphere: \(p_1 = p_2 = 0\ \text{gauge}\)
  • Velocities at surfaces: \(V_1 \approx V_2 \approx 0\ \text{m/s}\)
  • Total head loss in piping: \(h_L = 5\ \text{m}\)
  • Find: Pump head \(h_p\) required to just sustain the flow.
1
Simplify Bernoulli for negligible surface velocity:
\[ \frac{p_1}{\gamma} + z_1 + h_p = \frac{p_2}{\gamma} + z_2 + h_L \] With both surfaces open, \(p_1 = p_2 = 0\) (gauge).
2
Rearrange for pump head:
\[ h_p = (z_2 – z_1) + h_L \] \[ h_p = (12 – 0) + 5 = 17\ \text{m} \]
3
Interpretation:
The pump must provide at least 17 m of head to overcome elevation gain and friction losses. In practice, you would add margin and check the pump curve.
4
Using the calculator:
Set both pressures to 0 (gauge), velocities to 0, enter \(z_1 = 0\ \text{m}\), \(z_2 = 12\ \text{m}\), \(h_L = 5\ \text{m}\), choose “Solve for pump head”, and you should see a result of about 17 m.

Common Layouts & Variations

Bernoulli’s equation appears in many standard layouts. The Bernoulli Equation Calculator can handle each of these if you map your physical system into two representative points and appropriate loss and pump terms.

ScenarioHow to Model with BernoulliTypical Simplifications & Cautions
Pressurized pipe between two taps Use points at tap centers, include measured pressures and velocities, add \(h_L\) for pipe friction and fittings. Assume steady, fully developed flow. For long pipes, use a friction-factor calculator to estimate \(h_L\).
Reservoir to pipe outlet Take point 1 at the large reservoir surface (\(V_1 \approx 0\)), point 2 at the outlet or vena contracta. Include entrance and exit losses. Use \(\dfrac{p_2}{\gamma} = 0\) for discharge to atmosphere.
Siphon over a crest Use point 1 at upstream free surface, point 2 downstream, with an intermediate check at crest for minimum pressure. Ensure crest pressure does not drop below vapor pressure to avoid cavitation or air release in the siphon.
System with a pump Add \(h_p\) on the left side of Bernoulli (between suction and discharge points), include suction and discharge losses in \(h_L\). Check net positive suction head (NPSH) separately; Bernoulli alone does not protect against cavitation at the pump eye.
System with a turbine Treat the turbine as removing head: use a negative \(h_p\) or add \(h_t\) as a loss term on the right side. Confirm that the extracted head does not exceed the available head difference between inlet and outlet.
  • Verify that both points use the same datum for elevation \(z\).
  • Check that velocity units match the chosen gravitational acceleration \(g\).
  • Confirm that the fluid is effectively incompressible at the given pressures.
  • Estimate whether ignoring minor losses would change results more than a few percent.
  • Compare Bernoulli results with a separate friction or pump-curve calculation.
  • Use the calculator’s “what-if” runs to see sensitivity to pipe size and roughness.

Specs, Logistics & Sanity Checks

Although you are not “buying” a Bernoulli equation, there are inputs and assumptions you must assemble before trusting any result from the Bernoulli Equation Calculator.

Data You Need Up Front

  • Fluid type, temperature, and an appropriate density or specific weight \(\gamma\).
  • Approximate flow rate or velocity in each pipe segment of interest.
  • Pressures at measurement points (gauge or absolute, but consistent).
  • Elevations for points 1 and 2 relative to a convenient datum.
  • Estimated head losses from friction and fittings, or a plan to compute them.

Instrumentation & Field Notes

  • Use calibrated pressure gauges and note their reference level.
  • Record pipe diameters and materials to estimate roughness.
  • Note valve positions and any partially closed fittings that add loss.
  • When possible, take Reynolds number from a companion calculator to justify laminar vs. turbulent assumptions.

Sanity Checks Before You Trust the Answer

  • Does higher downstream elevation always reduce downstream pressure or velocity? It should.
  • Does adding pump head raise energy levels as expected?
  • Do larger head losses reduce available pressure or flow in the right direction?
  • Compare results to back-of-the-envelope estimates, not just to extra decimal places.

Small input errors can easily produce physically impossible results (negative absolute pressures, unrealistically high velocities). Use the calculator iteratively: adjust one input at a time and watch how the individual head terms change.

Frequently Asked Questions

When can I safely ignore head losses in Bernoulli’s equation?
You can sometimes neglect head losses \(h_L\) when the flow path is short, pipe diameters are large, velocities are modest, and you only need a rough estimate. In most real engineering designs, especially with long piping or many fittings, losses are not negligible and should be included. As a quick check, run the Bernoulli Equation Calculator with and without a realistic \(h_L\) estimate to see how much the result shifts.
Should I use gauge or absolute pressure in the Bernoulli Equation Calculator?
You may use either gauge or absolute pressure, as long as you use the same reference at both points in the equation. Many water problems use gauge pressure because free surfaces are at 0 kPa gauge. Absolute pressure becomes important when you are close to vapor pressure and need to check for cavitation risk.
Can I use the Bernoulli equation for compressible gases?
The standard Bernoulli equation assumes incompressible flow with constant density. For low-speed gas flow (Mach number well below 0.3), treating the gas as incompressible can be acceptable. For higher speeds or large pressure changes, you must use compressible-flow relations instead of a simple Bernoulli balance.
What is the difference between total head and static pressure?
Static pressure is just the \(p/\gamma\) term in Bernoulli. Total head \[ H = \frac{p}{\gamma} + \frac{V^2}{2g} + z \] includes pressure head, velocity head, and elevation head. Along a streamline with no pumps or losses, total head remains constant even though pressure, velocity, and elevation can trade off.
Can I mix units in the Bernoulli Equation Calculator?
Inside the calculator, all units are converted to a consistent internal system (usually SI) before computation, but your inputs must still be compatible: for example, if you choose SI, pressure should be in Pa or kPa, velocity in m/s, elevation in meters, and specific weight in N/m³. Mixing imperial and metric at the same time is a fast way to get nonsense results.
How does the calculator handle pump head and turbine head?
Pump head \(h_p\) is treated as a positive term that adds energy between points, while turbine head is treated as energy extracted. In the extended Bernoulli form, the calculator adds \(h_p\) to the left side and includes turbine or other extraction as part of the effective head loss on the right side. Always check the sign convention in the interface to ensure that you are adding or removing head in the intended direction.
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