Bernoulli Equation Calculator
Calculate pressure, velocity, elevation, head loss, or flow rate between two points using Bernoulli’s equation and optional diameter-based flow checks.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the unknown variable and unit setup. The required known-value fields update automatically.
Enter the known values
Use the same pressure reference at both points. Optional diameters are used for flow-rate and continuity checks.
Visual Check
Compare HGL, EGL, pressure head, velocity head, elevation head, and head loss.
Solution
Live result, hydraulic checks, warnings, and solution steps.
Quick checks
- Check—
Show solution steps See conversions, equation setup, substitution, checks, and assumptions
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Uses Bernoulli’s equation with optional head loss for steady incompressible flow along a streamline.
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Bernoulli Equation Calculator
The Bernoulli Equation Calculator above helps solve for pressure, velocity, elevation, head loss, or flow rate between two points in a steady incompressible flow. Enter the known values, choose the unknown, and the calculator applies Bernoulli’s equation with unit conversion, solution steps, head checks, and a visual energy interpretation.
Bernoulli’s principle is the physical idea behind the equation: when a fluid speeds up, static pressure often decreases if elevation and losses do not offset the change. This is why Bernoulli’s equation is commonly used for pressure-drop checks, nozzles, restrictions, venturi-style flow problems, siphons, and pipe-flow energy balances.
Quick Answer
Bernoulli’s equation relates pressure energy, kinetic energy, and elevation energy along a streamline. In head form, it is commonly written as \( \frac{P_1}{\rho g}+\frac{V_1^2}{2g}+z_1=\frac{P_2}{\rho g}+\frac{V_2^2}{2g}+z_2+h_L \). Use it when flow is steady, density is approximately constant, both points are on the same streamline, and any real-flow losses are represented by \(h_L\).
Do not rely on a simplified Bernoulli calculation when…
Do not use the simplified result by itself when the flow is highly compressible, strongly unsteady, dominated by pump or turbine work, affected by large friction losses that have not been estimated, or when the two points are not on the same streamline. For final pipe, pump, nozzle, or hydraulic system design, pair Bernoulli checks with friction loss, minor loss, pump curve, fluid property, and field verification.
Inputs and Outputs Used by the Calculator
The calculator uses the known flow conditions at Point 1 and Point 2 to solve the selected unknown. The required inputs change depending on whether you are solving for pressure, velocity, elevation, head loss, or flow rate.
| Type | Value | What It Means | Common Unit |
|---|---|---|---|
| Input or output | Pressure, \(P_1\) or \(P_2\) | Static fluid pressure at Point 1 or Point 2. | Pa, kPa, psi, ft H₂O |
| Input or output | Velocity, \(V_1\) or \(V_2\) | Average fluid speed at each point. | m/s, ft/s |
| Input or output | Elevation, \(z_1\) or \(z_2\) | Vertical height above or below a chosen datum. | m, ft |
| Input or output | Head loss, \(h_L\) | Energy loss from friction, fittings, turbulence, or other real-flow effects. | m, ft |
| Input | Density, \(\rho\) | Fluid mass per unit volume, used to convert pressure to pressure head. | kg/m³, lbm/ft³ |
| Input | Gravity, \(g\) | Acceleration due to gravity used in pressure, velocity, and elevation head terms. | m/s², ft/s² |
| Input or output | Flow rate, \(Q\) | Volume of fluid passing through a section per unit time, usually calculated with \(Q=AV\). | m³/s, L/s, GPM, CFS |
| Input | Diameter, \(d\) | Internal pipe or flow-section diameter used to calculate area for flow-rate checks. | m, mm, in, ft |
Gauge pressure vs. absolute pressure
Bernoulli’s equation can use gauge pressure or absolute pressure, but both points must use the same reference. A common error is entering gauge pressure at one point and absolute pressure at the other point, which shifts the result by atmospheric pressure.
Bernoulli Equation Formula
The calculator uses Bernoulli’s equation with optional head loss. The pressure form is helpful for solving pressure directly, while the head form is usually easier for hydraulic interpretation.
Pressure Form With Head Loss
This form keeps each term in pressure units. It is useful when solving for \(P_1\) or \(P_2\).
Head Form
This form expresses each energy term as length of fluid head. It is the clearest form for pressure head, velocity head, elevation head, HGL, and EGL.
Flow Rate Relation
Flow-rate mode uses cross-sectional area and average velocity. For a circular pipe, use the internal diameter, not the outside diameter.
Extended Bernoulli Equation With Pump or Turbine Head
\(h_p\) is pump head added to the fluid, and \(h_t\) is turbine head removed from the fluid. The calculator above uses the simpler two-point form unless pump or turbine effects are handled separately.
Why is head loss added to the downstream side?
In this convention, \(h_L\) is added to the downstream side because it represents energy that was available at Point 1 but is no longer available as pressure head, velocity head, or elevation head at Point 2. When solving for downstream pressure, the head-loss term is subtracted from \(P_2\), which lowers the predicted downstream pressure.
Pressure drop from velocity increase
When elevation and head loss are small, a velocity increase usually creates a static pressure drop. This is the common Bernoulli pressure-drop case used for nozzles, restrictions, and venturi-style checks.
Common Bernoulli Equation Rearrangements
The calculator handles the algebra automatically, but these rearrangements help you understand what each solve mode is doing.
| Solve For | Formula | Use Case |
|---|---|---|
| \(P_2\) | \(P_2=P_1+\frac{1}{2}\rho(V_1^2-V_2^2)+\rho g(z_1-z_2)-\rho g h_L\) | Find downstream pressure from upstream pressure and energy changes. |
| \(V_2\) | \(V_2=\sqrt{V_1^2+\frac{2(P_1-P_2)}{\rho}+2g(z_1-z_2)-2gh_L}\) | Find downstream velocity from pressure, elevation, and head loss. |
| \(z_2\) | \(z_2=z_1+\frac{P_1-P_2}{\rho g}+\frac{V_1^2-V_2^2}{2g}-h_L\) | Find required downstream elevation for the selected energy balance. |
| \(h_L\) | \(h_L=\frac{P_1}{\rho g}+\frac{V_1^2}{2g}+z_1-\frac{P_2}{\rho g}-\frac{V_2^2}{2g}-z_2\) | Estimate energy loss between two points when all other terms are known. |
| \(Q\) | \(Q=AV=\frac{\pi d^2}{4}V\) | Calculate volumetric flow rate from internal diameter and average velocity. |
Square-root warning for velocity solves
If the expression inside a velocity square root is negative, there is no real velocity solution for the selected inputs. That usually means the pressure, elevation, and loss terms do not provide enough energy for the requested condition.
What the Variables Mean
Every Bernoulli term represents a form of mechanical energy. Pressure head, velocity head, and elevation head can exchange with one another as the fluid moves through a system.
| Symbol | Meaning | How to Enter It |
|---|---|---|
| \(P_1, P_2\) | Static pressure at Point 1 and Point 2. | Use the same pressure reference at both points: gauge with gauge or absolute with absolute. |
| \(\rho\) | Fluid density. | Use a density appropriate for the fluid and temperature. Water is often approximated as \(1000\,kg/m^3\). |
| \(V_1, V_2\) | Average velocity at each point. | Use the average section velocity, not a local jet speed unless that is the intended control section. |
| \(g\) | Acceleration due to gravity. | Use \(9.80665\,m/s^2\) or \(32.174\,ft/s^2\) for standard Earth gravity. |
| \(z_1, z_2\) | Elevation relative to a chosen datum. | Use the same datum for both points. Elevation can be negative if the point is below the datum. |
| \(h_L\) | Head loss between Point 1 and Point 2. | Use zero only for an ideal lossless case. For real pipes, estimate friction and fitting losses separately. |
| \(Q\) | Volumetric flow rate. | Use \(Q=AV\), where \(A\) is the internal flow area and \(V\) is average velocity. |
| \(d\) | Pipe or section diameter. | Use internal diameter when calculating flow area. |
How to Use the Calculator
Start by choosing the unknown variable. Then enter the known values using consistent units and a consistent pressure reference.
Choose the solve mode
Select whether you want to solve for pressure, velocity, elevation, head loss, or flow rate. The visible input fields update based on that choice.
Enter Point 1 and Point 2 values
Enter the known pressure, velocity, elevation, density, gravity, and head-loss values. Use the unit selectors to match your source data.
Check pressure reference and datum
Use gauge pressure at both points or absolute pressure at both points. Also use one elevation datum for \(z_1\) and \(z_2\).
Review result, heads, and warnings
Use the pressure head, velocity head, HGL, EGL, flow-rate, and continuity checks to see whether the result is physically reasonable.
How to Interpret the Result
A Bernoulli result is an energy-balance result. The number is most useful when you also understand which energy term increased, which term decreased, and whether the assumptions make sense.
| Result Type | What It Means | What to Check Next |
|---|---|---|
| Lower downstream pressure | Static pressure dropped, often because velocity increased, elevation increased, or energy was lost. | Check velocity change, elevation change, and \(h_L\). |
| Higher downstream pressure | Static pressure increased, often because velocity decreased or elevation decreased. | Check if the selected point order matches the flow direction. |
| High velocity | Kinetic energy dominates the energy balance. | Check pipe diameter, flow rate, cavitation risk, and whether compressibility matters. |
| Positive head loss | Energy is dissipated between Point 1 and Point 2. | Compare with friction-loss and minor-loss estimates. |
| Negative head loss | The calculation implies energy addition or inconsistent inputs. | Check pump/turbine effects, pressure reference, units, and point order. |
| No real velocity solution | The required downstream condition needs more energy than the inputs provide. | Reduce head loss, change pressure/elevation inputs, or verify units. |
What to do with the result
Use the solved value as an energy-balance check first. Then compare it with pipe friction calculations, flow-rate continuity, operating pressure limits, vapor pressure, pump curves, and field measurements before treating it as a final design value.
What changes the result most?
Velocity often has the strongest effect because velocity head depends on \(V^2\), not just \(V\). Doubling velocity increases velocity head by a factor of four. Elevation differences also matter strongly in water systems because each meter of elevation is one meter of elevation head.
Quick sanity check
For water, \(10.2\,m\) of pressure head is approximately \(100\,kPa\), and \(2.31\,ft\) of water is approximately \(1\,psi\). If a small velocity or elevation change creates an enormous pressure change, recheck units, density, gravity, and pressure reference.
Input Quality Checklist
Bernoulli calculations are sensitive to unit consistency and physical setup. Check these items before relying on the output.
Same Streamline
Use two points along the same streamline or a well-defined control path. Do not compare unrelated locations in a complex flow field.
Pressure Reference
Use gauge pressure at both points or absolute pressure at both points. Mixing them causes a large offset.
Elevation Datum
Measure \(z_1\) and \(z_2\) from the same reference elevation. The absolute datum is less important than consistency.
Internal Diameter
For \(Q=AV\), use internal flow diameter. Outside pipe diameter can significantly overestimate flow area.
Head Loss
Use \(h_L=0\) only for an ideal lossless estimate. Real pipe systems usually need friction and fitting losses.
Fluid Properties
Use density appropriate for the fluid. Water, oil, air, and seawater have very different densities.
Diameter errors grow quickly
Flow area depends on \(d^2\), so diameter mistakes are amplified. A 10% diameter error creates about a 21% area error because \(1.10^2=1.21\). For flow-rate mode, always use the internal flow diameter.
Step-by-Step Worked Example
A common Bernoulli problem is solving for downstream pressure when velocity increases through a smaller section and elevation stays the same.
Use the pressure rearrangement
Substitute the values
Solve
Check the velocity head change
For water, \(0.612\,m\) of head is about \(6.0\,kPa\), which matches the pressure drop from \(200\,kPa\) to \(194\,kPa\).
Result
Pressure at Point 2: approximately \(194\,kPa\).
Why this answer is reasonable
The velocity increased from \(2\,m/s\) to \(4\,m/s\), so static pressure dropped as pressure energy converted into velocity head. Because elevation and head loss were zero, the pressure change comes only from the kinetic energy difference.
Mini Example: Flow Rate From Diameter and Velocity
If \(V=2\,m/s\) and \(d=100\,mm=0.1\,m\), then:
This flow-rate check is useful when you know pipe diameter and average velocity but do not need a full pressure-energy calculation.
HGL and EGL in Bernoulli’s Equation
The calculator’s visual check separates total head into pressure head, velocity head, and elevation head. This is often more useful than looking at pressure alone because it shows how mechanical energy is distributed between the two points.
| Line or Term | Formula | Meaning | What a Change Means |
|---|---|---|---|
| Pressure head | \(P/(\rho g)\) | Static pressure expressed as an equivalent fluid-column height. | A lower pressure head usually means static pressure has decreased. |
| Velocity head | \(V^2/(2g)\) | Kinetic energy per unit weight of fluid. | A larger velocity head means the fluid is moving faster. |
| Elevation head | \(z\) | Height above the selected datum. | A higher point needs more elevation energy. |
| Hydraulic Grade Line, HGL | \(z+P/(\rho g)\) | Elevation head plus pressure head. | A falling HGL often indicates pressure energy is decreasing along the flow path. |
| Energy Grade Line, EGL | \(z+P/(\rho g)+V^2/(2g)\) | Total mechanical head. | A drop in EGL represents head loss or energy removed from the fluid. |
| Gap between EGL and HGL | \(V^2/(2g)\) | The velocity head. | A larger gap means higher velocity at that section. |
How to think about the energy diagram
If velocity increases through a restriction, the gap between the EGL and HGL increases because velocity head increased. If no pump adds energy, the EGL should generally decrease in the direction of flow when head loss is present.
HGL can be below the pipe centerline
A hydraulic grade line below the pipe centerline indicates negative gauge pressure at that location. That may be acceptable in some siphon-style problems, but it should trigger an absolute-pressure and cavitation check for real liquid systems.
Reference Values and Quick Checks
Use reference values to catch unit mistakes and unrealistic outputs. These values are approximate and should be adjusted for actual fluid properties and conditions.
| Quantity | Approximate Value | How It Helps |
|---|---|---|
| Water density | \(1000\,kg/m^3\) | Common estimate for ordinary water calculations. |
| Standard gravity | \(9.80665\,m/s^2\) | Default gravity for SI calculations. |
| Water pressure head | \(1\,m \approx 9.81\,kPa\) | Quick check for converting pressure to head. |
| U.S. water pressure head | \(1\,psi \approx 2.31\,ft\) of water | Useful field check for water systems. |
| Water-column pressure | \(1\,ft\,H_2O \approx 2989\,Pa\), \(1\,in\,H_2O \approx 249\,Pa\) | Useful when converting between pressure and head units. |
| Velocity head | \(V^2/(2g)\) | At \(2\,m/s\), velocity head is about \(0.20\,m\). |
| Common water pipe velocity check | Often roughly \(1\) to \(3\,m/s\) for many building and distribution checks | Higher velocities can increase head loss, noise, erosion, and transient concerns. |
| Continuity check | \(Q_1 \approx Q_2\) | For a steady incompressible stream, flow rate should be consistent unless flow is added or removed. |
Cavitation and vapor pressure check
If using absolute pressure, compare the solved pressure to the fluid vapor pressure at operating temperature. For water near room temperature, vapor pressure is only a few kPa absolute, but the exact value changes strongly with temperature. If pressure approaches vapor pressure, review cavitation risk before using the result.
Design Ranges and Practical Engineering Checks
Bernoulli’s equation can produce a mathematically valid number that is still incomplete for design. Use the result as one part of a broader hydraulic check.
Low-Speed Liquid Flow
Bernoulli checks are usually most straightforward for steady liquid flow where density is nearly constant and velocity is not extreme.
Pipe Systems
For real pipes, head loss often controls the answer. Pair Bernoulli with friction loss and minor loss calculations.
Nozzles and Restrictions
A restriction can increase velocity and reduce static pressure. Check cavitation risk when liquid pressure gets low.
Pressure Check
If solved absolute pressure is near vapor pressure, the system may be vulnerable to cavitation.
Velocity Check
If velocity is unexpectedly high, check diameter and flow units first. Small diameter errors can create large velocity and head changes.
Head Loss Check
If head loss is a large fraction of total head, friction modeling may dominate the problem more than the ideal Bernoulli terms.
Practical design note
If the system includes a pump, turbine, control valve, long pipe run, or many fittings, the basic equation may need added pump head, turbine head, friction loss, and minor loss terms. A simple two-point Bernoulli calculation should not replace full hydraulic modeling for final design.
Units and Conversion Notes
Unit mistakes are one of the fastest ways to get a wrong Bernoulli answer. Pressure, density, gravity, elevation, and velocity must work together in one consistent unit system after conversion.
| Quantity | Common Units | Conversion Reminder |
|---|---|---|
| Pressure | Pa, kPa, MPa, psi, ft H₂O | \(1\,kPa=1000\,Pa\), \(1\,psi\approx 6894.76\,Pa\) |
| Water-column pressure | ft H₂O, in H₂O | \(1\,ft\,H_2O\approx2989\,Pa\), \(1\,in\,H_2O\approx249\,Pa\) |
| Velocity | m/s, ft/s, mph, km/h | \(1\,ft/s=0.3048\,m/s\) |
| Elevation and head | m, ft, in | \(1\,ft=0.3048\,m\) |
| Density | kg/m³, lbm/ft³, slug/ft³ | \(1\,lbm/ft^3\approx16.0185\,kg/m^3\) |
| Flow rate | m³/s, L/s, GPM, CFS | \(1\,L/s=0.001\,m^3/s\) |
Most common unit trap
Pressure in psi and elevation in feet can be used together only after the units are converted consistently. Do not manually mix SI pressure, U.S. elevation, and density without converting through a consistent base unit system.
U.S. density unit warning
Do not treat \(lbm/ft^3\) and \(slug/ft^3\) as the same density unit. They are different. The calculator converts density internally, but manual U.S. customary Bernoulli calculations require careful handling of gravitational units.
Bernoulli Equation vs. Continuity Equation vs. Head Loss
Bernoulli, continuity, and head-loss equations answer related but different questions. The best hydraulic checks often use all three.
| Method | Main Idea | Best Use | Main Limitation |
|---|---|---|---|
| Bernoulli equation | Conserves mechanical energy along a streamline. | Pressure, velocity, elevation, and head-loss checks. | Needs appropriate assumptions and loss terms. |
| Continuity equation | Conserves mass flow rate. | Relating area, velocity, and flow rate. | Does not calculate pressure by itself. |
| Darcy-Weisbach | Estimates pipe friction head loss. | Long pipe runs and friction-dominated systems. | Requires friction factor and pipe roughness information. |
| Hazen-Williams | Empirical water pipe head-loss equation. | Water distribution estimates in common design ranges. | Not general for all fluids or flow conditions. |
Bernoulli vs. continuity in one sentence
Continuity explains how area and velocity relate through \(A_1V_1=A_2V_2\), while Bernoulli explains how pressure, velocity, elevation, and losses trade energy along the flow path.
Common Mistakes That Cause Wrong Results
Most wrong Bernoulli answers come from setup mistakes, not from the equation itself. The do/don’t list below covers the most common issues.
Common Mistakes
- Mixing gauge pressure and absolute pressure.
- Using different elevation datums for \(z_1\) and \(z_2\).
- Entering outside pipe diameter instead of internal diameter.
- Assuming \(h_L=0\) in a long real pipe system.
- Applying the equation across a pump or turbine without adding the proper energy term.
- Using the incompressible form for high-speed gas flow.
Better Practice
- Use one pressure reference consistently at both points.
- Measure both elevations from the same datum.
- Use average velocity and internal flow area.
- Estimate friction and fitting losses when they are important.
- Check \(Q_1\) and \(Q_2\) with continuity when diameters and velocities are known.
- Use compressible-flow methods when gas density changes significantly.
When head loss dominates
In long pipes, head loss may be much larger than the velocity-head change. Ignoring \(h_L\) can make the Bernoulli result look much more favorable than the real system.
Troubleshooting Unexpected Results
If the calculator result looks wrong, first check units, sign conventions, and whether the physical assumptions match the problem.
| Problem | Likely Cause | Fix |
|---|---|---|
| No real velocity solution | The energy balance does not provide enough pressure/elevation energy for the requested velocity and losses. | Check pressure units, reduce head loss, verify elevations, or reverse point order if appropriate. |
| Negative absolute pressure | Pressure reference or input values are inconsistent, or the requested condition is physically impossible. | Use consistent absolute pressure inputs and check vapor pressure/cavitation risk. |
| Negative head loss | The system may include energy addition, reversed points, or incorrect measurements. | Check for pump head, turbine terms, pressure-reference errors, and direction of flow. |
| Flow rates do not match | Diameter and velocity inputs do not satisfy continuity. | Verify internal diameter, average velocity, and whether flow is added or removed between points. |
| Pressure change seems too large | Possible unit mix-up, density error, or velocity entered in the wrong unit. | Check psi vs kPa, ft vs m, ft/s vs m/s, and density units. |
Suspicious result warning
A mathematically valid answer can still be misleading if head loss was ignored, the selected points are not comparable, or the pressure falls below vapor pressure. For liquids, very low absolute pressure should trigger a cavitation review.
Assumptions, Sources, and Limitations
This calculator is intended for educational use, preliminary engineering checks, and quick fluid mechanics estimates. It uses the standard Bernoulli mechanical energy relationship with optional head loss.
Flow Assumption
The flow is treated as steady and incompressible, with density assumed constant between the two points.
Streamline Assumption
The two points should lie on the same streamline or on a path where the simplified energy balance is appropriate.
Loss Assumption
Real-flow losses are represented only by the entered \(h_L\). The calculator does not automatically derive friction or minor losses unless those values are separately provided.
Equipment Limitation
Pumps, turbines, valves, and equipment energy terms are not included unless manually represented in the setup or treated with a more complete energy equation.
Calculation basis
The calculation is based on the standard Bernoulli mechanical energy equation used in fluid mechanics, including pressure head, velocity head, elevation head, and optional head loss. For a concise engineering reference on Bernoulli’s equation and fluid-flow energy terms, see the Engineering Library reference based on DOE fluid-flow material.
Final design caution
For final pipe, pump, nozzle, hydraulic, or pressure-system design, verify the result with appropriate loss calculations, equipment data, fluid properties, operating conditions, applicable standards, and professional engineering judgment.
Glossary of Terms
These definitions help connect the calculator output to the physical meaning of Bernoulli’s equation.
Pressure Head
The height of fluid column equivalent to static pressure, calculated as \(P/(\rho g)\).
Velocity Head
The kinetic energy per unit weight of fluid, calculated as \(V^2/(2g)\).
Elevation Head
The height of a point above a selected datum, represented by \(z\).
Head Loss
Energy lost between two points due to friction, fittings, turbulence, and other real-flow effects.
Hydraulic Grade Line
The line representing pressure head plus elevation head, \(HGL=z+P/(\rho g)\).
Energy Grade Line
The line representing total mechanical head, equal to HGL plus velocity head.
Continuity
The conservation of flow rate for steady incompressible flow, often written as \(A_1V_1=A_2V_2\).
Cavitation
A damaging condition that can occur when liquid pressure falls low enough for vapor bubbles to form and collapse.
Frequently Asked Questions
What does the Bernoulli Equation Calculator calculate?
It calculates pressure, velocity, elevation, head loss, or flow rate between two points in a steady incompressible flow, depending on the selected solve mode and known inputs.
What formula does the Bernoulli Equation Calculator use?
It uses Bernoulli’s equation with optional head loss: \(P_1+\frac{1}{2}\rho V_1^2+\rho g z_1=P_2+\frac{1}{2}\rho V_2^2+\rho g z_2+\rho g h_L\). For flow rate modes, it uses \(Q=AV\).
Can Bernoulli’s equation calculate flow rate?
Yes. Bernoulli’s equation can support flow calculations when combined with continuity. The calculator’s flow-rate mode uses \(Q=AV\), where \(A\) is area and \(V\) is average velocity.
Should I use gauge pressure or absolute pressure?
Either can be used if both points use the same reference. Do not mix gauge pressure at one point with absolute pressure at another point.
Why is my Bernoulli result negative or impossible?
A negative pressure, negative head loss, or impossible velocity usually means the input energy balance is inconsistent, the units are wrong, head loss is too large, pressure reference is mixed, or the points should be reversed.
Can this calculator be used for final pipe design?
Use it for education and preliminary checks. Final pipe design should also consider friction loss, minor losses, pump or turbine terms, flow regime, fluid properties, applicable standards, field conditions, and engineering judgment.