DEAN

Overview

The DEAN function calculates the Dean number (De), a dimensionless quantity used in fluid mechanics to characterize flow behavior in curved pipes, channels, and helical coils. Named after British scientist W. R. Dean, who first provided a theoretical solution for fluid motion through curved pipes in 1927–1928, the Dean number quantifies the ratio of centrifugal and inertial forces to viscous forces in curved geometries.

When fluid flows through a curved pipe, the curvature induces a secondary flow pattern superimposed on the primary axial flow. This occurs because fluid in the center of the pipe experiences centrifugal forces that push it toward the outer wall, while fluid near the walls returns along the inner curve. This creates a pair of counter-rotating vortices known as Dean vortices. The Dean number predicts the intensity of these secondary flow patterns.

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

The Dean number is calculated as:

De = \sqrt{\frac{D_i}{D}} \cdot Re

where D_i is the inner diameter of the pipe, D is the diameter of curvature (or spiral diameter), and Re is the Reynolds number. The Dean number characterizes flow regimes in curved pipes: values below 40–60 indicate unidirectional flow, 64–75 shows emergence of secondary flow, 75–200 produces stable Dean vortices, and values above 400 indicate fully turbulent conditions.

For additional theoretical background, see the Dean number article on Wikipedia.

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

Excel Usage

=DEAN(reynolds, inner_diameter, curvature_diameter)

• reynolds (float, required): Reynolds number (dimensionless)
• inner_diameter (float, required): Inner diameter of the pipe (m)
• curvature_diameter (float, required): Diameter of curvature or spiral (m)

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

Examples

Example 1: Demo case 1

Inputs:

reynolds	inner_diameter	curvature_diameter
10000	0.1	0.4

Excel formula:

=DEAN(10000, 0.1, 0.4)

Expected output:

5000

Example 2: Demo case 2

Inputs:

reynolds	inner_diameter	curvature_diameter
20000	0.2	0.5

Excel formula:

=DEAN(20000, 0.2, 0.5)

Expected output:

12650

Example 3: Demo case 3

Inputs:

reynolds	inner_diameter	curvature_diameter
5000	0.05	0.2

Excel formula:

=DEAN(5000, 0.05, 0.2)

Expected output:

2500

Example 4: Demo case 4

Inputs:

reynolds	inner_diameter	curvature_diameter
15000	0.3	0.3

Excel formula:

=DEAN(15000, 0.3, 0.3)

Expected output:

15000

Python Code

import micropip
await micropip.install(["fluids"])
from fluids.core import Dean as fluids_dean

def dean(reynolds, inner_diameter, curvature_diameter):
    """
    Calculate the Dean number (De) for flow in a curved pipe or channel.

    See: https://fluids.readthedocs.io/en/latest/fluids.core.html#fluids.core.Dean

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

    Args:
        reynolds (float): Reynolds number (dimensionless)
        inner_diameter (float): Inner diameter of the pipe (m)
        curvature_diameter (float): Diameter of curvature or spiral (m)

    Returns:
        float: Dean number (float), or error message string.
    """
    try:
        Re = float(reynolds)
        Di = float(inner_diameter)
        D = float(curvature_diameter)
    except (TypeError, ValueError):
        return "Error: All parameters must be numeric values."

    if D == 0:
        return "Error: curvature_diameter cannot be zero."

    try:
        result = fluids_dean(Re, Di, D)
    except Exception as e:
        return f"Error: {str(e)}"

    return float(result)

Online Calculator