Bernoulli's Equation Calculator
bernoulli's equation calculator fluid pressure, velocity, and elevation head from conservation of energy. Accurate engineering formulas and unit conversions for engineers, students & technicians.
Calculate Bernoulli's Equation Calculator
Enter your engineering parameters below to compute verified physical and mathematical metrics.
Calculation Results
Calculated using verified physical methodology: Bernoulli's Principle: P_2 = P_1 + \1 / 2\rho(v_1^2 - v_2^2) + \rho g(z_1 - z_2)
Quick Summary
The Bernoulli's Equation Calculator applies the conservation of energy principle for frictionless, incompressible streamline flow ($P_1 + \1 / 2\rho v_1^2 + \rho g z_1 = P_2 + \1 / 2\rho v_2^2 + \rho g z_2$), computing downstream static pressure ($P_2$) across SI and Imperial units.
Formula Explanation
Bernoulli's Principle: P_2 = P_1 + \1 / 2\rho(v_1^2 - v_2^2) + \rho g(z_1 - z_2)
Total Head: H = \P / \rho g + \v^2 / 2g + z = \text{constant}
How It Works
The Bernoulli's Equation Calculator balances static pressure energy ($P$), dynamic kinetic energy density ($\1 / 2\rho v^2$), and gravitational potential energy density ($\rho g z$) between two points on a fluid streamline. It computes the resulting downstream static pressure ($P_2$) as fluid velocity and pipe elevation change.
Step-by-Step Worked Example
Practical Problem: Water ($\rho = 1,000\text{ kg/m}^3$) enters a Venturi nozzle at $P_1 = 200\text{ kPa}$, $v_1 = 2.0\text{ m/s}$, $z_1 = 0\text{ m}$. It accelerates to $v_2 = 5.0\text{ m/s}$ while rising to an elevation $z_2 = 2.0\text{ m}$. Calculate downstream pressure $P_2$.
- Step 1: Identify Input Variables: $P_1 = 200,000\text{ Pa}$, $v_1 = 2.0\text{ m/s}$, $z_1 = 0\text{ m}$, $v_2 = 5.0\text{ m/s}$, $z_2 = 2.0\text{ m}$, $\rho = 1,000\text{ kg/m}^3$.
- Step 2: Calculate Dynamic Pressure Change: $\Delta P_{dyn} = \1 / 2 \rho (v_1^2 - v_2^2) = 0.5 \times 1,000 \times (4 - 25) = -10,500\text{ Pa}$ ($-10.50\text{ kPa}$).
- Step 3: Calculate Hydrostatic Elevation Head Pressure Change: $\Delta P_{elev} = \rho g (z_1 - z_2) = 1,000 \times 9.80665 \times (0 - 2) = -19,613.30\text{ Pa}$ ($-19.61\text{ kPa}$).
- Step 4: Execute Bernoulli Equation Addition: $P_2 = 200,000 - 10,500 - 19,613.30 = 169,886.70\text{ Pa}$.
- Step 5: Convert and Interpret Metric Outputs: Downstream Pressure $P_2 = 169.89\text{ kPa}$. Pressure drop: $30.11\text{ kPa}$. Imperial Pressure: $169.89 \times 0.145038 = 24.64\text{ PSI}$. Downstream Bar Pressure: $1.699\text{ Bar}$.
Real-World Calculation Examples
Scenario 1: Horizontal Venturi Flow Meter
Parameters: $P_1 = 300\text{ kPa}$, $v_1 = 1.0\text{ m/s}$, $z_1 = 0$, $v_2 = 4.0\text{ m/s}$, $z_2 = 0$, $\rho = 1,000\text{ kg/m}^3$
Result: $P_2 = 292.50\text{ kPa}$ (42.42 PSI). Flow nozzle pressure drop.
Scenario 2: Vertical Water Riser Pipe
Parameters: $P_1 = 400\text{ kPa}$, $v_1 = 2.0\text{ m/s}$, $z_1 = 0$, $v_2 = 2.0\text{ m/s}$, $z_2 = 10\text{ m}$, $\rho = 1,000\text{ kg/m}^3$
Result: $P_2 = 301.93\text{ kPa}$ (43.79 PSI). Hydrostatic elevation head pressure loss.
Scenario 3: Aircraft Wing Aerodynamic Lift
Parameters: $P_1 = 101.3\text{ kPa}$, $v_1 = 50\text{ m/s}$ (lower), $z_1 = 0$, $v_2 = 60\text{ m/s}$ (upper), $z_2 = 0$, $\rho = 1.225\text{ kg/m}^3$
Result: $P_2 = 100.63\text{ kPa}$. Pressure difference $\Delta P = 0.67\text{ kPa}$ producing upward lift.
Scenario 4: Fire Hose Spray Nozzle
Parameters: $P_1 = 500\text{ kPa}$, $v_1 = 3.0\text{ m/s}$, $z_1 = 0$, $v_2 = 25.0\text{ m/s}$, $z_2 = 0$, $\rho = 1,000\text{ kg/m}^3$
Result: $P_2 = 192.00\text{ kPa}$ (27.85 PSI). High-velocity nozzle kinetic energy conversion.
Key Benefits of Using This Calculator
Complete Energy Head Balance
Evaluates static pressure, dynamic velocity head, and elevation head simultaneously.
Multi-Pressure Unit Conversion
Outputs static pressure in kPa, Pascals, Bar, and Imperial PSI.
Venturi & Nozzle Flow Analysis
Ideal tool for analyzing Venturi meters, spray nozzles, and piping contraction pressure drops.
100% Free Client-Side Engine
Fast interactive calculations running locally in your browser with zero data transmission.
Frequently Asked Questions (FAQ)
What is Bernoulli's Principle?
Bernoulli's Principle states that an increase in fluid velocity occurs simultaneously with a decrease in static pressure or fluid potential energy.
What assumptions are required for Bernoulli's equation?
Bernoulli's equation assumes steady, incompressible, frictionless (inviscid) flow along a single fluid streamline.
What is static pressure vs dynamic pressure?
Static pressure (P) is fluid thermodynamic pressure; dynamic pressure (0.5 * rho * v²) is kinetic energy per unit volume due to fluid motion.
What is total stagnation pressure?
Stagnation pressure P_total = P_static + 0.5 * rho * v² (the pressure when fluid velocity is brought to rest frictionlessly).
How does a Venturi meter measure flow rate?
A Venturi tube narrows pipe area to increase fluid velocity, measuring the resulting static pressure drop to deduce flow rate.
Why does pressure drop when pipe diameter narrows?
Continuity forces fluid velocity to increase in narrow sections; to conserve total energy, static pressure must drop.
How does elevation change affect fluid pressure?
Climbing upward by elevation z decreases static pressure by rho * g * z (hydrostatic head loss).
What is hydraulic head H?
Total hydraulic head H = P / (rho*g) + v² / (2g) + z, expressing total fluid energy in equivalent meters of liquid column.
Can Bernoulli's equation be applied to real pipe systems with friction?
Real piping requires the extended energy equation: P1/rho*g + v1²/2g + z1 = P2/rho*g + v2²/2g + z2 + h_loss (adding friction head loss h_loss).
How convert kPa to PSI?
Multiply kPa by 0.145038 to obtain PSI (e.g. 100 kPa = 14.50 PSI).