EULER

Overview

The EULER function calculates the Euler number (Eu), a dimensionless quantity used in fluid dynamics to express the relationship between pressure drop and the kinetic energy of a flowing fluid. Named after the Swiss mathematician Leonhard Euler, who made foundational contributions to fluid mechanics, this number characterizes energy losses in fluid flow where a perfect frictionless flow corresponds to an Euler number of 0.

The Euler number represents the ratio of pressure forces to inertial forces in a fluid system. It is particularly useful in pressure drop calculations for flow through pipes, valves, heat exchangers, and other restrictions. The inverse of the Euler number is sometimes referred to as the Ruark number.

This implementation uses the fluids library, an open-source Python package for fluid dynamics calculations. For complete documentation, see the fluids.core module reference.

The Euler number is calculated as:

Eu = \frac{\Delta P}{\rho V^2}

where:

• \Delta P is the pressure drop experienced by the fluid (Pa)
• \rho is the fluid density (kg/m³)
• V is the characteristic velocity of the fluid (m/s)

Note that an alternative definition divides the pressure drop by the dynamic head (\frac{1}{2}\rho V^2), which yields a value twice as large. The formula used here follows the convention from Perry’s Chemical Engineers’ Handbook and Cengel & Cimbala’s Fluid Mechanics textbook.

The Euler number is closely related to the Cavitation number and can be interpreted through the Darcy–Weisbach equation framework for pipe flow analysis.

This example function is provided as-is without any representation of accuracy.

Excel Usage

=EULER(dP, rho, V)

• dP (float, required): Pressure drop (Pa)
• rho (float, required): Fluid density (kg/m³)
• V (float, required): Characteristic velocity (m/s)

Returns (float): Euler number (float), or error message string.

Examples

Example 1: Demo case 1

Inputs:

dP	rho	V
100000	1000	4

Excel formula:

=EULER(100000, 1000, 4)

Expected output:

6.25

Example 2: Demo case 2

Inputs:

dP	rho	V
500	1.2	10

Excel formula:

=EULER(500, 1.2, 10)

Expected output:

4.167

Example 3: Demo case 3

Inputs:

dP	rho	V
2000	850	2

Excel formula:

=EULER(2000, 850, 2)

Expected output:

0.5882

Example 4: Demo case 4

Inputs:

dP	rho	V
200000	2000	5

Excel formula:

=EULER(200000, 2000, 5)

Expected output:

4

Python Code

import micropip
await micropip.install(["fluids"])
from fluids.core import Euler as fluids_euler

def euler(dP, rho, V):
    """
    Calculate the Euler number (Eu) for a fluid flow.

    See: https://fluids.readthedocs.io/fluids.core.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        dP (float): Pressure drop (Pa)
        rho (float): Fluid density (kg/m³)
        V (float): Characteristic velocity (m/s)

    Returns:
        float: Euler number (float), or error message string.
    """
    try:
        dP_val = float(dP)
        rho_val = float(rho)
        V_val = float(V)
    except Exception:
        return "Error: All parameters must be numeric values."
    if rho_val == 0 or V_val == 0:
        return "Error: rho and V must be nonzero."
    try:
        result = fluids_euler(dP_val, rho_val, V_val)
    except Exception as e:
        return f"Error: {str(e)}"
    return result

Online Calculator