Calculator · Fluid Mechanics

Bernoulli Equation Calculator

Two stations on one steady incompressible stream, with the option of a pump, a turbine and a stated loss between them. Every term is a head, and the equation carries the signs so you never have to.

Inputs

Fluid density ρ[kg/m³]

Station 1

Pressure p₁ (gauge)[kPa]

Speed V₁[m/s]

Elevation z₁[m]

Station 2

Speed V₂[m/s]

Elevation z₂[m]

Between the stations

Pump head h_p[m]

A pump adds energy to the fluid, so h_p is a nonnegative magnitude.

Turbine head h_t[m]

Head loss h_L[m]

Entered explicitly here. Predicting a loss from pipe geometry belongs to pipe flow, not to this equation.

Gauge pressure is measured relative to the ambient pressure, so 0 kPa means ambient. It may legitimately be negative. No atmospheric pressure is added anywhere in this calculation.

Results

Pressure p₂, from H₁ + h_p − h_t − h_L = H₂

194 kPa

With the solved value substituted back, the two stations balance to within a residual of 0 m.

Total head H₁	20.598 m
Total head H₂	20.598 m
Elevation head z₁ / z₂	0 m / 0 m
Pressure head p/(ρg), 1 / 2	20.394 m / 19.782 m
Velocity head αV²/(2g), 1 / 2	0.20394 m / 0.81577 m
Pump head h_p	0 m
Turbine head h_t	0 m
Head loss h_L	0 m
Pressure p₁	200 kPa
Pressure p₂	194 kPa
Speed V₁ / V₂	2 m/s / 4 m/s
Pressure reference	Gauge

Assumptions

· Steady, incompressible, one-dimensional mean flow along a single path.
· All station pressures use one reference kind; no atmospheric pressure is added.
· Pump, turbine and loss heads are nonnegative magnitudes; the equation supplies the signs.

Explore in the Lab

See these two stations as heads and grade lines in the Bernoulli & Energy Lab

Both stations keep their gauge pressures and the same elements between them.

Technical reference

H = z + p/(ρg) + α V²/(2g)

H₁ + h_p − h_t − h_L = H₂

HGL = z + p/(ρg)

EGL = HGL + α V²/(2g)

H: total mechanical head at a station[m]
z: elevation above the datum, positive upward[m]
p: static pressure in the stated reference kind[Pa]
V: mean speed at the station[m/s]
α: kinetic-energy correction coefficient, 1 here[-]
h_p: head added by a pump[m]
h_t: head extracted by a turbine[m]
h_L: irreversible head loss between the stations[m]
g: gravity, 9.80665 m/s²[m/s²]

The flow is steady, incompressible and along one streamtube from station 1 to station 2. Both pressures must use the same reference kind; nothing is converted silently.

Classical Bernoulli is this equation with h_p = h_t = h_L = 0. Adding a loss does not make it less exact; it makes it an energy balance rather than a statement of conservation.

A speed is recovered from a squared term, so it comes back as a magnitude. If the state would need a negative velocity head, no speed is reported at all.

A negative pump head, turbine head or loss is refused rather than displayed: those three are nonnegative by definition.