Euler Number Equations Formulas Calculator

Fluid Mechanics Dynamics Dimensionless Value

Problem:

Solve for Euler number.

Enter Calculator Inputs:

pressure change (Δp)

density (ρ)

characteristic flow velocity (v)

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	Euler number
	pressure change
	density
	characteristic velocity

Background

The Euler number (Eu) is a dimensionless quantity used in fluid mechanics to describe the relationship between pressure forces and inertial forces within a fluid flow. It is cited after the mathematician Leonhard Euler and critically evaluates how pressure gradients affect flow in systems like pipes, valves, nozzles, and open channels. Engineers and scientists use the Euler number when analyzing systems where pressure drops significantly affect performance or behavior. It's beneficial for comparing different flow situations regardless of size, scale, or units since dimensionless numbers enable direct system comparisons.

Equation

The Euler number is calculated using the following formula:

Eu = Δp / (ρ × v²)

Where:

Eu = Euler number (dimensionless)
Δp = Pressure change or pressure difference (Pascals, Pa)
ρ = Fluid density (kg/m³)
v = Characteristic flow velocity (m/s)

This ratio shows how much pressure force is acting on the fluid relative to the inertial force created by the fluid's motion.

How to Solve

To solve for the Euler number, follow these steps:

Identify or measure the pressure drop, Δp, across the interest section.
Determine the density of the fluid, ρ.
Measure or estimate the characteristic velocity of the fluid, v.
Plug the values into the formula: Eu = Δp / (ρ × v²)

This calculation helps determine if the flow is controlled primarily by pressure forces (high Eu) or inertia (low Eu).

Example

Let's say water flows through a restriction in a pipe, causing a pressure drop of 15,000 Pascals. The fluid is water (ρ = 1000 kg/m³), and the velocity is 5 m/s.

Using the formula:

Eu = 15000 / (1000 × 5²)

Eu = 15000 / 25000

Eu = 0.6

In this case, the Euler number is 0.6. This suggests that inertial forces play a significant role, though pressure forces are still relevant.

Five Fields or Degrees It's Used In

Mechanical Engineering - Analyzing fluid systems like pumps, turbines, and piping networks.
Aerospace Engineering - Studying airflow over aircraft surfaces, especially for drag and lift behavior.
Civil Engineering - Designing hydraulic structures, spillways, and drainage systems.
Chemical Engineering - Modeling flow behavior in reactors and process equipment.
Environmental Science - Understanding river flow, groundwater systems, and pollution transport.

Five Real-Life Applications

Pump and Valve Design - Helps determine optimal pressure drop and sizing.
Aircraft Wing Testing - Used in wind tunnel analysis for pressure distribution.
HVAC Systems - Evaluating airflow resistance in ducts and filters.
Water Supply Systems - Calculating head loss across fittings and bends.
Medical Devices - Designing respiratory equipment and IV systems with accurate pressure-flow balance.

Five Common Mistakes

Forgetting to square the velocity - Always apply v², not just v.
Unit inconsistency - Pressure must be in Pascals, density in kg/m³, velocity in m/s.
Using the wrong velocity - The "characteristic" velocity should match the specific flow area or region being analyzed.
Confusing with Reynolds number - Euler number deals with pressure vs. inertia, not viscous effects.
Ignoring compressibility - The equation assumes incompressible flow; compressible flows need corrections.

Five Frequently Asked Questions

Q1. What does the Euler number represent?
A1. The Euler number measures the fluid's ratio of pressure forces to inertial forces. It's a key indicator of how much influence pressure changes have on the flow.
Q2. Is it used in both gases and liquids?
A2. Yes, it can be applied to both, though additional considerations like Mach number might be needed for compressible gases.
Q3. How is it different from the Reynolds number?
A3. While the Reynolds number compares inertial and viscous forces, the Euler number compares pressure and inertial forces.
Q4. When is a high Euler number important?
A4. High Euler numbers indicate pressure-dominated flows, essential in low-velocity or high-resistance systems like filters or membranes.
Q5. Can the Euler number help predict cavitation?
A5. Yes, indirectly. A very low local pressure (associated with a certain Euler number) may drop below vapor pressure, which is a condition for cavitation.

References - Books:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.