Bernoulli'ss Equation Calculator
Calculate bernoulli's equation calculator calculate fluid pressure P2, velocity v2, or height h2 along a streamline using Bernoulli's Law (P1 + 0.5*rho*v1^2 + rho*g*h1 = P2 + 0.5*rho*v2^2 + rho*g*h2). Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Fluid Conservation of Energy (Bernoulli Streamline)
Enter your physical parameters below to compute verified Bernoulli fluid metrics.
Calculation Results
Calculated using verified physical methodology: Bernoulli Streamline Equation: P_1 + \1 / 2\rho v_1^2 + \rho g h_1 = P_2 + \1 / 2\rho v_2^2 + \rho g h_2
P_2 = P_1 + \1 / 2\rho(v_1^2 - v_2^2) + \rho g(h_1 - h_2)
Quick Summary
The Bernoulli's Equation Calculator evaluates conservation of energy along a fluid streamline ($P_1 + \1 / 2\rho v_1^2 + \rho g h_1 = P_2 + \1 / 2\rho v_2^2 + \rho g h_2$), solving final pressure ($P_2$), dynamic pressure, and total head in kPa, bar, and PSI.
Formula Explanation
Bernoulli Streamline Equation: P_1 + \1 / 2\rho v_1^2 + \rho g h_1 = P_2 + \1 / 2\rho v_2^2 + \rho g h_2
P_2 = P_1 + \1 / 2\rho(v_1^2 - v_2^2) + \rho g(h_1 - h_2)
How It Works
The Bernoulli's Equation Calculator computes static pressure, dynamic kinetic energy pressure ($\1 / 2\rho v^2$), and elevation potential pressure ($\rho g h$) at Point 1. It subtracts Point 2 kinetic and potential components to find final static pressure ($P_2$) in kPa, Pascals, bar, and PSI.
Step-by-Step Worked Example
Practical Problem: Water ($\rho = 1,000.0\text{ kg/m}^3$) flows in a pipe from Point 1 ($P_1 = 200.0\text{ kPa}$, $v_1 = 2.0\text{ m/s}$, $h_1 = 0.0\text{ m}$) to Point 2 ($v_2 = 6.0\text{ m/s}$, $h_2 = 5.0\text{ m}$). Calculate final static pressure $P_2$.
- Step 1: Identify Input Parameters: $P_1 = 200,000\text{ Pa}$, $v_1 = 2.0\text{ m/s}$, $h_1 = 0.0\text{ m}$, $v_2 = 6.0\text{ m/s}$, $h_2 = 5.0\text{ m}$, $\rho = 1,000.0\text{ kg/m}^3$.
- Step 2: Calculate Dynamic Pressure Change ($\Delta P_{dyn}$): $\1 / 2\rho (v_1^2 - v_2^2) = 0.5 \times 1,000 \times (2^2 - 6^2) = 500 \times (4 - 36) = -16,000\text{ Pa} = -16.00\text{ kPa}$.
- Step 3: Calculate Potential Head Pressure Change ($\Delta P_{pot}$): $\rho g (h_1 - h_2) = 1,000 \times 9.80665 \times (0.0 - 5.0) = -49,033.25\text{ Pa} = -49.03\text{ kPa}$.
- Step 4: Solve Final Static Pressure ($P_2$): $P_2 = 200.00 - 16.00 - 49.03 = 134.97\text{ kPa}$.
- Step 5: Convert and Interpret Final Pressure Units: Final Pressure $P_2 = 134.97\text{ kPa} = 134,966.75\text{ Pa} = 1.350\text{ bar} = 19.58\text{ PSI}$. Total Bernoulli Head $H = 20.60\text{ meters}$.
Real-World Calculation Examples
Scenario 1: Municipal Water Pipe Elevation Rise
Parameters: $P_1 = 200\text{ kPa}$, $v_1 = 2\text{ m/s}$, $v_2 = 6\text{ m/s}$, $h_2 = 5\text{ m}$
Result: $P_2 = 134.97\text{ kPa}$ (1.35 bar, 19.58 PSI). Pipe elevation pressure drop.
Scenario 2: Venturi Tube Flowmeter Constriction
Parameters: $P_1 = 300\text{ kPa}$, $v_1 = 1\text{ m/s}$, $v_2 = 10\text{ m/s}$, $h_1 = h_2 = 0\text{ m}$
Result: $P_2 = 250.50\text{ kPa}$ (2.51 bar, 36.33 PSI, $\Delta P = 49.5\text{ kPa}$). Venturi suction drop.
Scenario 3: Aircraft Wing Airfoil Aerodynamic Lift
Parameters: Air density $\rho = 1.225\text{ kg/m}^3$, $v_{upper} = 70\text{ m/s}$, $v_{lower} = 50\text{ m/s}$
Result: $\Delta P = 1.47\text{ kPa}$ ($0.21\text{ PSI}$ net lifting pressure). Aerodynamic wing lift.
Scenario 4: Torricelli Reservoir Drainage Spout
Parameters: $P_1 = P_2 = 1\text{ atm}$, $v_1 \approx 0$, $h_1 - h_2 = 10.0\text{ m}$
Result: Exit velocity $v_2 = \sqrt{2 g h} = 14.01\text{ m/s}$ (50.4 km/h). Water jet exit speed.
Key Benefits of Using This Calculator
Conservation of Energy Streamline Solver
Applies mechanical energy conservation across static, dynamic, and potential energy heads.
Venturi & Aerodynamic Analysis
Ideal for analyzing Venturi meters, carburetors, pitot tubes, and aircraft wing lift.
Multi-Unit Readouts
Outputs final pressure in kPa, Pascals, Bar, and Imperial PSI ($\text{lb/in}^2$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is Bernoulli's Principle?
Bernoulli's Principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or fluid potential energy.
What is the formula for Bernoulli's Equation?
P1 + 0.5*rho*v1² + rho*g*h1 = P2 + 0.5*rho*v2² + rho*g*h2 = Constant along a streamline.
What assumptions are made in Bernoulli's Equation?
Assumes ideal fluid flow: incompressible (constant density rho), non-viscous (zero friction), steady state, and along a continuous streamline.
What is dynamic pressure vs static pressure?
Static pressure P is thermodynamic fluid pressure; dynamic pressure q = 0.5 * rho * v² represents kinetic energy per unit volume.
What is Total Stagnation Pressure?
P_total = P_static + 0.5 * rho * v² (pressure when fluid is brought to rest frictionlessly).
What is Torricelli's Law?
Torricelli's Law v = sqrt(2*g*h) calculates exit jet velocity from an open tank orifice under gravity, derived directly from Bernoulli's equation.
How does a Pitot tube measure airspeed?
A Pitot tube measures the difference between total stagnation pressure and static pressure (delta P = 0.5 * rho * v² => v = sqrt(2*delta P / rho)).
What is Hydraulic Head in fluid engineering?
Hydraulic Head H = P/(rho*g) + v²/(2*g) + z represents total energy height in meters of fluid column.
Who formulated Bernoulli's Equation?
Swiss mathematician Daniel Bernoulli published the law in his 1738 book Hydrodynamica.
How does pipe constriction affect fluid pressure?
A constriction increases velocity v (via Continuity Equation A1*v1 = A2*v2), which causes static pressure P to drop (via Bernoulli's Equation).