Bernoulli's Equation Calculator
Understanding Bernoulli's Equation
Bernoulli's equation is a fundamental principle in fluid dynamics that describes the relationship between pressure, velocity, and elevation in a moving fluid. It's a statement of the conservation of energy for an inviscid, incompressible fluid in steady flow. Essentially, it states that for a fluid flowing horizontally, one with higher velocity will have lower pressure than one with a lower velocity.
The Equation
The most common form of Bernoulli's equation is:
P + ½ρv² + ρgh = constant
Where:
- P is the static pressure of the fluid (in Pascals, Pa). This is the pressure exerted by the fluid at rest.
- ρ (rho) is the density of the fluid (in kilograms per cubic meter, kg/m³).
- v is the flow velocity of the fluid (in meters per second, m/s).
- g is the acceleration due to gravity (approximately 9.81 m/s² on Earth).
- h is the height or elevation of the fluid above a reference point (in meters, m).
The equation implies that the sum of these three terms (pressure energy, kinetic energy per unit volume, and potential energy per unit volume) remains constant along a streamline. This means if one term increases, at least one of the other terms must decrease to maintain the constant sum.
Applications
Bernoulli's principle has numerous practical applications across various fields:
- Aerodynamics: It explains how airplane wings generate lift. The curved upper surface of the wing forces air to travel faster than the air flowing under the flatter bottom surface. This higher velocity on top results in lower pressure, creating an upward force (lift).
- Venturi Meters: These devices use Bernoulli's principle to measure the flow rate of a fluid. A constriction in a pipe causes the fluid velocity to increase and pressure to decrease, which can be measured and related to the flow rate.
- Pitot Tubes: Used to measure airspeed in aircraft and wind speed, these tubes measure the difference between static pressure and stagnation pressure (where the fluid velocity is zero) to determine the flow velocity.
- Chimneys: Wind blowing over the top of a chimney creates an area of lower pressure, helping to draw smoke out of the building.
- Carburetors: In older internal combustion engines, carburetors used the Venturi effect to draw fuel into the air stream.
How the Calculator Works
This calculator helps you compute the total pressure (or the constant value in Bernoulli's equation) at a specific point in a fluid flow. You need to input the fluid's density, its velocity at that point, the static pressure at that point, its elevation, and the local acceleration due to gravity. The calculator will then sum up the pressure term (P), the dynamic pressure term (½ρv²), and the hydrostatic pressure term (ρgh) to give you the total pressure head.
Example Calculation
Let's consider water flowing through a pipe. Suppose:
- The density of water (ρ) is 1000 kg/m³.
- The velocity of the water (v) is 5 m/s.
- The static pressure (P) at a certain point is 150,000 Pa.
- The elevation (z) at that point is 2 meters above a reference.
- The acceleration due to gravity (g) is 9.81 m/s².
Using the calculator:
Total Pressure = P + ½ρv² + ρgh
Total Pressure = 150000 Pa + ½ * 1000 kg/m³ * (5 m/s)² + 1000 kg/m³ * 9.81 m/s² * 2 m
Total Pressure = 150000 Pa + ½ * 1000 * 25 Pa + 19620 Pa
Total Pressure = 150000 Pa + 12500 Pa + 19620 Pa
Total Pressure = 182120 Pa
The calculator would output 182120 Pa for these inputs.