Bernoulli's equation says that along a streamline in a steady, incompressible, frictionless flow, the sum of static pressure, dynamic pressure and elevation pressure is the same at every point. Write it out and you get P + ½ρv² + ρgh = constant. This Bernoulli equation calculator takes the two points you describe, applies that constant, and solves for whichever term you left unknown — pressure, velocity or height at the second point.
Arb Digital builds free calculators that tell you what they assume, because the assumptions are where fluid problems go wrong. Almost every misapplication of Bernoulli's equation comes from using it in a situation that violates one of its four conditions, and no amount of correct arithmetic saves an answer in that case. This page states those conditions in full, explains what each one rules out, and describes what to use instead when the flow does not meet them.
The Assumptions This Result Depends On
Every number this tool produces rests on four assumptions. Read them before you use the answer for anything real.
Steady flow. Conditions at any fixed point do not change with time. That rules out starting transients, water hammer, pulsating pumps and anything oscillating. If the flow rate is still changing while you measure, the equation does not apply in this form.
Incompressible fluid. Density is the same at both points. That is a good assumption for liquids under almost all conditions, and for gases only while the flow stays well below about Mach 0.3, where density changes remain within a few percent. Beyond that, the compressible form of the equation is required, which is why NASA's Glenn Research Center page on Bernoulli's equation is explicit that the incompressible form is invalid at supersonic speeds.
Inviscid flow. No viscosity, therefore no friction loss along the pipe or wall. Real flow always loses energy to friction, so real downstream pressure is always lower than this calculator predicts. The shortfall is the head loss, and over any appreciable pipe run it dominates the answer.
Along a streamline. Both points must lie on the same streamline, or the flow must be irrotational so that the constant is the same everywhere. Comparing a point inside a jet with a point in a stagnant region beside it is not a valid application, and it is the mistake behind most incorrect explanations of aerodynamic lift.
What This Bernoulli Equation Calculator Does
Enter the fluid density and the state at point 1 — pressure, velocity and elevation — plus the two known quantities at point 2. Select which quantity at point 2 you want, and the tool rearranges the equation to isolate it. Because the equation is quadratic in velocity, solving for v₂ can produce a negative value under the square root, which means no physical solution exists for the numbers you gave. The tool says so rather than returning NaN, and explains that the pressure drop you specified is not large enough to sustain that velocity.
The grid reports the total head in metres, which is the whole left-hand side of the equation divided by ρg and is the most convenient single number for comparing points in a piping system. It also gives the dynamic pressure at point 2, the pressure change between the two points, and the velocity at point 2 — so whichever quantity you solved for, the supporting figures are all present.
How to Use It
- Choose the unknown first. The selected field stops being read as an input, so leftover values in it cannot affect the result.
- Use absolute pressures, or gauge pressures at both points. Mixing the two adds one atmosphere to one side of the equation and quietly wrecks the answer.
- Set the density for your actual fluid and temperature. Water at 4 °C is 1000 kg/m³ and at 80 °C is about 972. If you need a value for another substance, the density calculator has a substance library.
- Measure both elevations from one datum. Any datum works — the pipe centreline, the pump floor, sea level — because only the difference enters the equation.
- Treat the result as the frictionless bound. Real pressure at point 2 will be lower, and real velocity will be lower, by whatever the friction loss amounts to.
The Formula: How It Is Calculated
The equation in full is P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, with g taken as 9.80665 m/s². Solving for pressure at the second point gives P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂).
With the defaults — water at 998 kg/m³, 200,000 Pa and 2 m/s at point 1, accelerating to 6 m/s at the same height — the calculation is 200,000 + 0.5 × 998 × (4 − 36) + 0 = 200,000 − 15,968 = 184,032 Pa. Speeding the water up by 4 m/s costs almost 16 kPa of static pressure. That trade is the whole content of the equation: pressure energy converts into kinetic energy and back, and the total stays fixed.
Solving for velocity requires rearranging into v₂ = √(v₁² + 2(P₁ − P₂)/ρ + 2g(h₁ − h₂)). The square root is where a physically impossible input reveals itself: if the bracket is negative, you have asked for a velocity that would require more energy than the pressure and elevation difference can supply. Solving for elevation is linear and always has a solution: h₂ = h₁ + (P₁ − P₂)/(ρg) + (v₁² − v₂²)/(2g). The OpenStax University Physics treatment of Bernoulli's equation derives all three forms from conservation of energy.
Why the Real Answer Is Always Lower
Bernoulli's equation has no loss term, so it describes the best case a flow could achieve. Every real installation falls short. Friction against the pipe wall converts mechanical energy into heat continuously along the run; fittings, bends, valves and sudden changes of area each add a local loss; and turbulence dissipates energy wherever the flow separates.
Engineers handle this by adding a head loss term hL to the downstream side, computed from the Darcy-Weisbach relation and a friction factor that depends on Reynolds number and pipe roughness. That calculation is outside the scope of this page, but the implication is not: use the result here as an upper bound on downstream pressure and a check on whether a design is even theoretically possible. If Bernoulli says the pressure at the far end is marginal, the real pressure will be worse.
The gap between the two is also a useful diagnostic. If you measure a pressure drop far larger than the ideal calculation predicts and the geometry has not changed, something is obstructing the line. A blocked strainer, a partly closed valve or a collapsed hose liner all show up as an unexplained excess loss.
Venturi, Pitot and the Two Classic Applications
A venturi meter is Bernoulli's equation used deliberately. Narrowing a pipe forces the fluid to speed up, and the equation says the static pressure must fall by an amount that depends only on the two velocities and the density. Measure the pressure difference between the wide and narrow sections and you can infer the flow rate without putting anything in the stream. Combine that with continuity — area times velocity is constant for an incompressible fluid — and you have the working principle of a whole class of flow meters. Our flow rate calculator handles the continuity side of that pairing.
A pitot tube is the same physics used the other way round. Bring the flow to rest at a stagnation point and its dynamic pressure converts entirely into static pressure. The difference between that stagnation pressure and the undisturbed static pressure is ½ρv², so the airspeed follows directly. This is how aircraft measure speed, and the failure modes are instructive: block the pitot port with ice and the indicated airspeed becomes a function of altitude rather than speed, because the trapped pressure no longer tracks the flow.
Both applications share a requirement that is easy to overlook. The two measurement points must be on the same streamline, in a region where the flow is smooth and attached. Put a static port in a separated wake and the pressure it reads has no defined relationship to anything.
Static, Dynamic and Total Pressure
The three terms of the equation have names worth keeping straight. Static pressure is what a gauge flush with the pipe wall reads — the pressure the fluid exerts regardless of its motion. Dynamic pressure, ½ρv², is the additional pressure that appears if the flow is brought to rest. Their sum is total or stagnation pressure, and the elevation term ρgh accounts for height.
Confusing static with total pressure is the most frequent error in practical work. A gauge on a flowing line reads static pressure. A gauge on a dead-ended tapping facing upstream reads something closer to total. The difference is small at low speeds and large at high ones: water at 2 m/s has a dynamic pressure of only 2 kPa, but at 20 m/s it is 200 kPa, which is more than two atmospheres. For unit work on any of these, the pressure converter moves between Pa, bar, psi and metres of water.
When the fluid is not moving at all, every velocity term vanishes and Bernoulli's equation collapses to the hydrostatic relation, P₂ = P₁ + ρg(h₁ − h₂). That case is common enough to deserve its own tool, and the hydrostatic pressure calculator handles it directly.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing gauge and absolute pressure — use one convention for both points. Mixing them injects an error of about 101 kPa, which is larger than many real pressure differences.
- Applying it across a pump or turbine — those machines add or remove energy, so the constant is not the same on both sides. Bernoulli's equation only holds where no work is done on the fluid.
- Ignoring friction over a long run — the equation gives the frictionless bound. Over tens of metres of pipe, head loss can exceed the entire pressure difference the equation predicts.
- Comparing points on different streamlines — valid only when the flow is irrotational. A point in a jet and a point in the still air beside it are not comparable in general.
- Using it for compressible gas flow — above roughly Mach 0.3 the density changes materially between the two points, and the incompressible form no longer applies.
Related Free Tools From Arb Digital
Fluid problems rarely stop at one equation. Pair this with the flow rate calculator for the continuity side, the hydrostatic pressure calculator when the fluid is static, and the buoyancy calculator for submerged bodies. Fluid properties come from the density calculator, and unit work from the pressure converter, the flow rate converter or the viscosity converter. The complete free online tools hub lists everything else.
Frequently Asked Questions
Four: the flow is steady, the fluid is incompressible so density is the same at both points, the flow is inviscid so no energy is lost to friction, and both points lie on the same streamline. Violating any one of them invalidates the result regardless of the arithmetic.
Because the equation contains no loss term. Real flow loses energy to wall friction, fittings and turbulence, so downstream pressure is always below the ideal figure. The difference is the head loss for that section of pipe.
Yes, while the air speed stays below roughly Mach 0.3, around 100 m/s at sea level. Above that, density changes between the two points become significant and the compressible form of the equation is required instead.
Because the velocity form contains a square root. If the pressure and elevation differences you entered cannot supply the kinetic energy required, the expression under the root is negative and no physical solution exists for those numbers.
Either, as long as both points use the same one. The equation only involves the pressure difference, so a consistent offset cancels. Mixing conventions adds about 101 kPa of error to one side.
No. A pump adds energy to the fluid, so the Bernoulli constant differs on either side of it. The extended energy equation with a pump head term is the correct tool for that case.
Total head is the whole left-hand side of the equation divided by density times gravity, which expresses the fluid's energy as an equivalent height of fluid in metres. It lets you compare pressure, velocity and elevation contributions in one common unit.
This tool is provided for educational and study use. It computes the ideal frictionless case and is not a substitute for a full piping or hydraulic design carried out by a qualified engineer.
