FOURIER_MASS

Overview

The FOURIER_MASS function calculates the Fourier number for mass transfer (Fo or Fo_m), a dimensionless quantity that characterizes the ratio of time to the characteristic time scale for mass diffusion. Named after French mathematician and physicist Jean-Baptiste Joseph Fourier, this number is fundamental in analyzing time-dependent mass transport phenomena.

The mass-transfer Fourier number is derived by nondimensionalizing Fick’s second law of diffusion. It provides a measure of how much diffusive transport has occurred relative to the characteristic diffusion time scale. The dimensionless group is defined as:

\text{Fo}_m = \frac{D \cdot t}{L^2}

where D is the mass diffusivity (m²/s), t is the elapsed time (s), and L is the characteristic length scale (m). This can also be interpreted as:

\text{Fo} = \frac{\text{Diffusive transport rate}}{\text{Storage rate}}

When Fo ≪ 1, insufficient time has elapsed for significant concentration changes to propagate through the characteristic length. When Fo ≈ 1, concentration changes have affected the entire domain. When Fo ≫ 1, the system approaches steady-state conditions.

This implementation uses the fluids library, a comprehensive collection of fluid mechanics and thermodynamics functions. For additional details, see the fluids.core documentation.

The mass Fourier number is commonly used alongside the Biot number when analyzing transient diffusion problems with convective boundary conditions, such as drying processes, catalyst pellet diffusion, and drug delivery systems.

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

Excel Usage

=FOURIER_MASS(t, L, D)

• t (float, required): Time (s)
• L (float, required): Characteristic length (m)
• D (float, required): Mass diffusivity (m²/s)

Returns (float): Fourier number for mass (float), or error message string.

Examples

Example 1: Demo case 1

Inputs:

t	L	D
1.5	2	1e-9

Excel formula:

=FOURIER_MASS(1.5, 2, 1e-9)

Expected output:

3.75e-10

Example 2: Demo case 2

Inputs:

t	L	D
0.5	2	1e-9

Excel formula:

=FOURIER_MASS(0.5, 2, 1e-9)

Expected output:

1.25e-10

Example 3: Demo case 3

Inputs:

t	L	D
1.5	4	1e-9

Excel formula:

=FOURIER_MASS(1.5, 4, 1e-9)

Expected output:

9.375e-11

Example 4: Demo case 4

Inputs:

t	L	D
1.5	2	2e-9

Excel formula:

=FOURIER_MASS(1.5, 2, 2e-9)

Expected output:

7.5e-10

Python Code

import micropip
await micropip.install(["fluids"])
from fluids.core import Fourier_mass as fluids_fourier_mass

def fourier_mass(t, L, D):
    """
    Calculate the Fourier number for mass transfer (Fo).

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

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

    Args:
        t (float): Time (s)
        L (float): Characteristic length (m)
        D (float): Mass diffusivity (m²/s)

    Returns:
        float: Fourier number for mass (float), or error message string.
    """
    try:
        t_val = float(t)
        L_val = float(L)
        D_val = float(D)
    except Exception:
        return "Error: All parameters must be numeric values."
    if L_val == 0:
        return "Error: L must not be zero."
    try:
        result = fluids_fourier_mass(t_val, L_val, D_val)
    except Exception as e:
        return f"Error: {str(e)}"
    return result

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