💨 Bernoulli Equation Flow Calculator

Last updated: May 14, 2026

💨 Bernoulli Equation Flow Calculator

Steady, incompressible, inviscid flow along a streamline — solve for any unknown at Point 1 or Point 2.

P₁ + ½ρv₁² + ρgz₁ = P₂ + ½ρv₂² + ρgz₂
kg/m³
Point 1 (Upstream / Reference)
Pa (Pascals)
m/s
m (above datum)
Point 2 (Downstream / Target)
Pa (Pascals)
m/s
m (above datum)
P₂ = — Pa
Step-by-step breakdown
—
Pressure Head Δ (m)
—
Velocity Head Δ (m)
—
Elevation Head Δ (m)

Assumes steady, incompressible, inviscid, irrotational flow along a single streamline. g = 9.81 m/s²

Bernoulli's Equation: The Energy Budget of a Moving Fluid

Every pipe network, aircraft wing, venturi meter, and irrigation channel operates under the same governing principle: energy along a streamline is conserved. Daniel Bernoulli codified this observation in 1738, and the resulting equation — deceptively compact, enormously powerful — became the cornerstone of classical fluid mechanics. Understanding how to apply it correctly, and where its assumptions break down, is what separates a competent engineer from someone who merely memorizes formulas.

What the Equation Actually Says

The full Bernoulli equation for steady, incompressible, inviscid flow along a streamline is written as:

P₁ + ½ρv₁² + ρgz₁ = P₂ + ½ρv₂² + ρgz₂

Each term has units of Pascals (N/m²) and represents an energy per unit volume stored in a different form. The first term, static pressure P, is the thermodynamic work energy of the fluid pushing against its surroundings. The second term, ½ρv², is the kinetic energy density — how much energy the fluid carries by virtue of its motion. The third term, ρgz, is the potential energy density due to elevation above a chosen datum. Bernoulli's insight is that these three can trade value back and forth along a streamline without net loss — as one rises, others must fall proportionally.

This is more than an algebraic trick. It is a direct statement of the work-energy theorem applied to a fluid parcel: the net work done by pressure forces on a moving element equals its change in kinetic plus potential energy. No heat transfer, no friction losses, no shaft work — those require the extended Bernoulli equation or full Navier-Stokes treatment. In its basic form, the equation applies to the ideal fluid that real engineers use as a well-calibrated starting point.

Pressure, Velocity, and Elevation: The Three Unknowns

In practice, you always know five of the six quantities (P₁, v₁, z₁, P₂, v₂, z₂) and need to solve for the sixth. Each mode of the equation tells a different physical story.

Solving for pressure is the most common industrial application. A pipe narrows from 100 mm diameter to 50 mm diameter. Continuity (ṁ = ρAv = constant for incompressible flow) forces v₂ = 4v₁. Bernoulli then tells you exactly how much the static pressure drops across the contraction. This is the operating principle behind every venturi meter, carburetor throat, and differential pressure flow sensor. The pressure transducer reads the drop; Bernoulli converts it to a flow rate.

Solving for velocity is central to anemometry and discharge coefficient analysis. Torricelli's theorem — the velocity of fluid exiting a hole in a tank at depth h is √(2gh) — is simply Bernoulli applied between the free surface (P = P_atm, v ≈ 0) and the orifice exit (P = P_atm, z = 0). Setting those values and solving for v gives v = √(2gh), the famous result that makes tank drainage calculable from geometry alone.

Solving for elevation appears frequently in piping layout problems. If you know inlet pressure and velocity and the downstream conditions, you can determine the maximum height the fluid can reach — critical when sizing pump head requirements or determining whether a siphon will sustain flow without cavitating at its apex.

The Head Form and Why Engineers Prefer It

Dividing every term in the Bernoulli equation by ρg yields the head form, where each term carries units of meters:

P/(ρg) + v²/(2g) + z = constant = Total Head (H)

The three components are called pressure head, velocity head, and elevation head. Their sum, the total head H, is constant along a streamline in ideal flow and equals the height to which the fluid would rise in a piezometer tube open to atmosphere. Engineers think in head because it is geometrically intuitive — a pump that delivers 30 m of head lifts water 30 m against gravity, regardless of pipe diameter or fluid velocity. The calculator's head breakdown panel shows how each head component changes between your two points, making it immediately visible whether a pressure drop is being traded for kinetic energy (as in a nozzle) or elevation (as in an uphill pipe).

Assumptions and Their Real-World Consequences

Bernoulli's equation rests on four assumptions that engineers must evaluate before trusting its output:

Steady flow. The velocity field does not change with time. Transient events — water hammer, pump startup, valve slam — violate this and require separate treatment using the unsteady Bernoulli equation or method of characteristics.

Incompressible flow. Density is constant. For liquids, this is almost always valid. For gases, it holds when the Mach number stays below roughly 0.3 (about 100 m/s for air at sea level). Above that, compressibility effects become significant and isentropic flow relations must replace Bernoulli.

Inviscid (frictionless) flow. This is the most routinely violated assumption. Real pipes generate friction losses that increase with velocity, pipe length, and roughness. The Darcy-Weisbach equation quantifies these losses as a head loss term h_L, extending Bernoulli to: H₁ = H₂ + h_L. For short, smooth sections at moderate Reynolds numbers, the ideal Bernoulli result may be accurate within 1-5%. For long pipe runs, the viscous correction is dominant.

Flow along a single streamline. The equation connects two points on the same streamline, not arbitrary points in the flow field. In irrotational flow this restriction relaxes (the constant is the same for all streamlines), but in rotational flow — wakes, separated boundary layers, turbulent cores — the streamline constraint is real and must be respected.

Common Engineering Applications

The Venturi meter calculates volumetric flow rate Q from a measured pressure differential ΔP: Q = Cd · A₂ · √(2ΔP / [ρ(1 − (A₂/A₁)²)]). The discharge coefficient Cd (typically 0.98–0.99 for a well-machined Venturi) accounts for the small real-fluid deviation from ideal Bernoulli behavior.

Pitot tubes measure airspeed on every commercial aircraft. The stagnation point at the tube's inlet converts all kinetic energy to pressure; comparing stagnation pressure to static pressure gives dynamic pressure q = ½ρv², from which v is extracted. At cruise, a Boeing 737 sees roughly 25,000 Pa of dynamic pressure — Bernoulli translated into airspeed indication on the flight deck.

Sprinkler system design relies on Bernoulli to calculate head loss requirements so that the farthest head still receives adequate pressure. Irrigation engineers use it to size distribution pipes. Hydraulic jump analysis in open channels begins with the specific energy equation, which is Bernoulli adapted for free-surface flow.

Using This Calculator Effectively

Select which quantity you want the equation to solve for — the corresponding input field is disabled and displays "SOLVE" to keep track of your unknown. Choose your fluid from the preset list or enter a custom density; note that for air problems, velocities above 100 m/s may push you into compressible territory where results require verification against isentropic relations. All pressures are absolute (gauge pressure users should add atmospheric pressure, typically 101,325 Pa). Elevations can be negative if a point is below your chosen datum. The step-by-step output lets you trace exactly which energy transformation drove the result — useful when verifying hand calculations or explaining a design to a client.

The head breakdown at the bottom of the result panel shows the net transfer between pressure head, velocity head, and elevation head. In a well-designed system with no viscous losses, these three changes sum to zero. Any imbalance between the check values for LHS and RHS in the steps would indicate a data entry error — the calculator flags this with a verification row so discrepancies are visible immediately.

FAQ

What is the difference between static pressure, dynamic pressure, and total pressure in Bernoulli's equation?
Static pressure (P) is the actual thermodynamic pressure the fluid exerts on its container walls — it is what a pressure gauge measures when moving with the fluid. Dynamic pressure (½ρv²) represents the additional pressure created when the flow is brought to rest (stagnated); it is proportional to kinetic energy per unit volume. Total (stagnation) pressure is their sum: P_total = P + ½ρv². Bernoulli's equation in its simplest form states that total pressure plus the elevation pressure head (ρgz) is constant along a streamline — or equivalently, that total head H = P/(ρg) + v²/(2g) + z is conserved. A pitot tube measures stagnation pressure directly; subtracting the local static pressure gives the dynamic pressure, and hence the velocity.
Can Bernoulli's equation be used for air and gases, or only liquids?
Bernoulli's equation in its standard form assumes incompressible flow, meaning constant density. For liquids like water, oil, or mercury this is essentially always valid. For gases like air, the equation is accurate as long as the flow velocity stays below roughly Mach 0.3 — about 100 m/s at sea-level air conditions. At higher speeds, compressibility causes density to change significantly as the flow accelerates or decelerates, and the isentropic flow equations (which account for density variation) must be used instead. For HVAC duct analysis, low-speed wind tunnel work, and most pneumatic conveying systems well below 100 m/s, standard Bernoulli gives results accurate within engineering tolerances.
Why does pressure decrease when a fluid speeds up through a narrow pipe section?
This is the core physical insight of Bernoulli's equation, and it follows directly from energy conservation. For an incompressible fluid in a horizontal pipe, Bernoulli reduces to P + ½ρv² = constant. Continuity (mass conservation) requires that if the pipe cross-section decreases, the velocity must increase proportionally: A₁v₁ = A₂v₂. Since the kinetic energy term ½ρv² increases with velocity, the static pressure P must decrease by an equal amount to keep the total constant. Energy is not lost; it is converted from pressure form to kinetic form. This is why a garden hose nozzle produces a fast jet at low pressure, why aircraft wings generate lift (the faster-moving air over the curved upper surface is at lower pressure), and why venturi meters can infer flow rate from a measured pressure drop.
What happens when the calculator returns a 'no real solution' error for velocity?
This error occurs when the algebraic rearrangement for velocity produces a negative value under the square root (v² < 0), which has no physical meaning. It indicates the input conditions are energetically inconsistent: you have specified a pressure drop, elevation change, or combination that would require the fluid to gain more kinetic energy than the available pressure and elevation head can supply. Check whether the pressure at the high-velocity point is plausible — if it drops below the fluid's vapor pressure, cavitation occurs and Bernoulli no longer applies. Also verify the sign convention: pressures must be in absolute Pascals, and elevation should be positive upward from your datum. Swapping Point 1 and Point 2 assignments sometimes resolves directional confusion.
How do I account for friction losses in a real pipe when using the Bernoulli calculator?
The standard Bernoulli equation assumes an ideal, frictionless fluid. Real pipes generate viscous friction losses quantified by the Darcy-Weisbach equation: h_L = f · (L/D) · v²/(2g), where f is the Darcy friction factor (from the Moody chart or Colebrook equation), L is pipe length, and D is diameter. To use this tool for a real pipe, compute h_L separately, then subtract it from your known upstream total head before solving. Equivalently, if you know inlet and outlet pressures from measurements, the calculator will correctly reflect the actual energy state at each point — the friction loss is implicitly embedded in the lower-than-ideal downstream pressure you measured. For preliminary design, ideal Bernoulli gives an upper-bound velocity and a lower-bound pressure drop; real conditions will be somewhat more conservative.
Why must I choose a datum for elevation, and does the choice affect my results?
The datum is an arbitrary reference height from which elevations z₁ and z₂ are measured — sea level, the pipe centerline, the pump inlet, or any convenient fixed point. The choice does not affect the calculated result because Bernoulli's equation only uses the difference (z₁ − z₂), not the absolute values. If you shift your datum by 5 m, both z₁ and z₂ increase by 5 m and the difference stays the same. Common engineering practice is to set the datum at the lowest point in the system (making all elevations positive) or at the reference point of interest (such as the pump inlet, giving z = 0 there). The important rule is consistency: once you choose a datum, all elevations in the same calculation must be measured from that same reference.