Interactive Bernoulli Equation Solver
Enter known values and leave exactly one of the six flow variables blank. The calculator will solve for the missing variable using:
P1 + 1/2 ρv12 + ρgh1 = P2 + 1/2 ρv22 + ρgh2
What Is the Bernoulli Equation?
The Bernoulli equation is a core relationship in fluid mechanics. It links pressure energy, kinetic energy, and potential energy along a streamline for an ideal flowing fluid. Engineers use it to estimate flow speed in pipes, pressure drop in nozzles, tank discharge behavior, and many other practical problems.
In its common form:
P + 1/2 ρv2 + ρgh = constant
Where:
- P = pressure (Pa)
- ρ = fluid density (kg/m³)
- v = flow velocity (m/s)
- g = gravitational acceleration (m/s²)
- h = elevation above a reference point (m)
How to Use This Bernoulli Equation Calculator
Step-by-step workflow
- Enter the fluid density and gravity (defaults are water at 1000 kg/m³ and 9.81 m/s²).
- Fill in five of the six state variables: P1, v1, h1, P2, v2, h2.
- Leave one field blank for the unknown.
- Click Calculate to solve the missing value.
If all six variables are entered, the tool compares the left-hand side and right-hand side of the equation and reports the residual difference.
Worked Example
Suppose water flows from point 1 to point 2 with:
- ρ = 1000 kg/m³
- g = 9.81 m/s²
- P1 = 250000 Pa, v1 = 2 m/s, h1 = 8 m
- P2 = 180000 Pa, h2 = 2 m
- Unknown: v2
Enter all known values, leave v2 blank, and calculate. The solver returns the velocity required at point 2 to satisfy Bernoulli balance.
Assumptions Behind Bernoulli’s Equation
This equation is powerful, but it depends on simplifying assumptions. Results are best when:
- The flow is steady (doesn’t change with time).
- The fluid is incompressible (density approximately constant).
- Viscous losses are negligible (or separately accounted for).
- You evaluate points along the same streamline.
- No pumps/turbines are adding or removing shaft work between points.
In real systems, friction and minor losses can be significant. For high-accuracy engineering design, add head-loss terms (Darcy-Weisbach and minor-loss coefficients) to the energy equation.
Common Mistakes to Avoid
- Unit mismatch: Mixing kPa with Pa or cm with m causes large errors.
- Wrong density: Always use the fluid and temperature-appropriate ρ.
- Ignoring elevation: Height differences can dominate pressure terms in some systems.
- Negative velocity square: If the solved v2 is negative, your input set is physically inconsistent under Bernoulli assumptions.
Where This Calculator Is Useful
- Pipe flow quick checks
- Nozzle and venturi estimates
- Tank outlet and siphon analysis
- Hydraulic lab exercises and homework verification
- Sanity-checking simulation or field measurements
Final Notes
This Bernoulli equation calculator is designed for fast, practical analysis. For compressible gases, strong turbulence, cavitation risk, or systems with large losses, use extended energy equations and detailed fluid models. Still, as a first-pass engineering tool, Bernoulli remains one of the most useful equations in applied physics.