FOURIER_HEAT
Overview
The FOURIER_HEAT function calculates the Fourier number (Fo) for heat transfer, a dimensionless quantity used to characterize transient heat conduction in solids. Named after J.B.J. Fourier, who formulated the modern understanding of heat conduction, the Fourier number represents the ratio of the actual time elapsed to the characteristic time scale for heat diffusion through a material.
This implementation uses the fluids library, a comprehensive Python package for fluid dynamics and heat transfer calculations. For detailed API documentation, see the fluids.core documentation.
The Fourier number is defined as:
\text{Fo} = \frac{\alpha t}{L^2} = \frac{k t}{C_p \rho L^2}where \alpha is the thermal diffusivity (m²/s), t is time (s), L is the characteristic length (m), k is thermal conductivity (W/m/K), C_p is heat capacity (J/kg/K), and \rho is density (kg/m³). The function accepts either thermal diffusivity directly or the individual material properties (density, heat capacity, and thermal conductivity) to compute the result.
The Fourier number physically represents:
\text{Fo} = \frac{\text{Heat conduction rate}}{\text{Rate of thermal energy storage}}
When Fo ≪ 1, insufficient time has passed for heat to diffuse significantly through the material. When Fo ≅ 1, significant temperature change occurs throughout the characteristic length. When Fo ≫ 1, the system approaches thermal equilibrium. The Fourier number is commonly used alongside the Biot number to analyze convective heating or cooling of solids, with applications in materials processing, thermal system design, and heat exchanger analysis.
This example function is provided as-is without any representation of accuracy.
Excel Usage
=FOURIER_HEAT(t, L, rho, Cp, k, alpha)t(float, required): Time (s)L(float, required): Characteristic length (m)rho(float, optional, default: null): Density (kg/m³)Cp(float, optional, default: null): Heat capacity (J/kg/K)k(float, optional, default: null): Thermal conductivity (W/m/K)alpha(float, optional, default: null): Thermal diffusivity (m²/s)
Returns (float): Fourier number for heat (float), or error message string.
Examples
Example 1: Demo case 1
Inputs:
| t | L | rho | Cp | k |
|---|---|---|---|---|
| 1.5 | 2 | 1000 | 4000 | 0.6 |
Excel formula:
=FOURIER_HEAT(1.5, 2, 1000, 4000, 0.6)Expected output:
5.625e-8
Example 2: Demo case 2
Inputs:
| t | L | alpha |
|---|---|---|
| 1.5 | 2 | 1e-7 |
Excel formula:
=FOURIER_HEAT(1.5, 2, 1e-7)Expected output:
3.75e-8
Example 3: Demo case 3
Inputs:
| t | L | rho | Cp | k |
|---|---|---|---|---|
| 2 | 1 | 800 | 3800 | 0.5 |
Excel formula:
=FOURIER_HEAT(2, 1, 800, 3800, 0.5)Expected output:
3.289e-7
Example 4: Demo case 4
Inputs:
| t | L | alpha |
|---|---|---|
| 0.5 | 0.5 | 2e-7 |
Excel formula:
=FOURIER_HEAT(0.5, 0.5, 2e-7)Expected output:
4e-7
Python Code
import micropip
await micropip.install(["fluids"])
from fluids.core import Fourier_heat as fluids_fourier_heat
def fourier_heat(t, L, rho=None, Cp=None, k=None, alpha=None):
"""
Calculate the Fourier number for heat transfer.
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)
rho (float, optional): Density (kg/m³) Default is None.
Cp (float, optional): Heat capacity (J/kg/K) Default is None.
k (float, optional): Thermal conductivity (W/m/K) Default is None.
alpha (float, optional): Thermal diffusivity (m²/s) Default is None.
Returns:
float: Fourier number for heat (float), or error message string.
"""
try:
t_val = float(t)
L_val = float(L)
except Exception:
return "Error: t and L must be numeric values."
# If alpha is provided, use it
if alpha is not None:
try:
alpha_val = float(alpha)
except Exception:
return "Error: alpha must be a numeric value."
try:
result = fluids_fourier_heat(t_val, L_val, alpha=alpha_val)
except Exception as e:
return f"Error: Failed to calculate Fourier number for heat: {str(e)}"
return result
# Otherwise, need rho, Cp, k
if None in (rho, Cp, k):
return "Error: must provide either alpha or all of rho, Cp, and k."
try:
rho_val = float(rho)
Cp_val = float(Cp)
k_val = float(k)
except Exception:
return "Error: rho, Cp, and k must be numeric values."
try:
result = fluids_fourier_heat(t_val, L_val, rho=rho_val, Cp=Cp_val, k=k_val)
except Exception as e:
return f"Error: Failed to calculate Fourier number for heat: {str(e)}"
return result