Engineering calculator reviewed for preliminary design use · Last updated: March 2026
Apply the Bernoulli equation P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ + ΔP_L to calculate pressure, velocity, or elevation between pipe cross-sections. Apply to nozzle design, orifice sizing, and tank drainage estimation.
What this calculator is used for
Bernoulli's theorem is a foundational principle of fluid mechanics: along a streamline, the sum of pressure, kinetic, and potential energy (the total head) is conserved. It is the starting point for relating pressure, velocity, and elevation between pipe sections and underpins the design of nozzles, orifices, and venturi meters as well as velocity-measurement principles. For real fluids the extended form adds a head-loss term.
Typical engineering use cases
Estimating exit velocity/flow from nozzles, orifices, and venturis
Evaluating static, dynamic, and elevation head changes between sections
Understanding and checking pitot-tube velocity measurement
Approximating free discharge from a tank (Torricelli's theorem)
Equation and methodology
In head form each energy term is expressed in units of length (head):
P/ρg + v²/2g + z = constant (for real fluids, add head loss h_L to the downstream side)
In pressure form: P₁ + ½ρv₁² + ρgz₁ = P₂ + ½ρv₂² + ρgz₂ + ΔP_L, where P is static pressure [Pa], ρ the density [kg/m³], v the velocity [m/s], g = 9.81 m/s², and z the elevation above datum [m]. Solving together with the continuity equation Q = A·v yields the unknown velocity or pressure.
Assumptions and limitations
Inviscid (frictionless) ideal fluid is assumed
Incompressible flow only (constant density)
Valid only along a single streamline under steady flow
No external work (pumps/turbines) or heat transfer in the section
Design notes
In real piping, friction and fitting losses are significant, so always include the head-loss term h_L. Where velocity rises the static pressure drops; if it falls below the vapor pressure, cavitation occurs. For systems containing pumps or turbines, add the corresponding energy terms separately rather than stretching the plain Bernoulli equation beyond its valid range.
Worked Example
Given:
Water ρ = 998 kg/m³, horizontal contraction (z₁ = z₂)
Result: P₂ ≈ 292.5 kPa — a 7.5 kPa drop as the flow speeds up.
Interpretation: The static pressure falls where velocity rises — the basis of venturis and the warning behind flow-induced low-pressure zones. This is the ideal (loss-free) value; real friction makes the true P₂ a little lower, so use the extended Bernoulli with a ΔPL term for actual piping.
Common Mistakes & Misuse
Using the frictionless Bernoulli form for real piping — without the head-loss term ΔPL it over-predicts downstream pressure or velocity.
Applying it to compressible gas at high velocity, where density is not constant and the incompressible energy balance fails.
Mixing units between the pressure form (Pa) and the head form (m) within one equation, so terms no longer add up.
Evaluating the two points off the same streamline / across a pump or fitting that adds or removes energy not in the equation.
Frequently Asked Questions
When can I apply Bernoulli's equation?
Bernoulli's equation applies to steady, incompressible flow along a streamline. For real piping, add a head loss term (ΔPL) for friction. It does not apply to compressible flow, unsteady flow, or flow across shock waves.
What is the relationship between Bernoulli and pressure loss?
The extended Bernoulli equation includes a head loss term: P₁ + ½ρv₁² + ρgz₁ = P₂ + ½ρv₂² + ρgz₂ + ΔPL. The ΔPL term represents energy dissipated by friction and is calculated using Darcy-Weisbach or similar methods.
For preliminary estimation and educational use only. Results may depend strongly on assumptions, input data, fluid or material properties, and the range of validity of the underlying equation. Verify critical calculations independently and follow the applicable code, specification, and formal engineering review process before using any result for design, procurement, fabrication, operation, or safety decisions.