Dimensionless Numbers in Unsteady Heat Conduction: Fourier Number and Biot Number

In the study of transient (unsteady) heat conduction, temperature is a dynamic variable that evolves not only across spatial coordinates but also through time. Mathematically, this behavior is governed by complex partial differential equations (PDEs) that can be daunting to solve directly. To navigate this complexity, heat transfer engineers rely on dimensionless numbers. These parameters serve as mathematical bridges, stripping away specific units to reveal the underlying physics.

Among these, the Fourier Number ($Fo$) and the Biot Number ($Bi$) are the most critical. While the Fourier number characterizes the temporal progress of heat diffusion, the Biot number evaluates the competition between different modes of thermal resistance. Together, they dictate which mathematical models are appropriate for describing a given thermal process.
The Fourier number is a dimensionless parameter that describes the rate at which heat diffuses through a material. It essentially provides a measure of "thermal progress"—how far the heat wave has penetrated the body at a given moment.

1. Mathematical Definition

The Fourier number is defined as:
$$Fo = \frac{\alpha t}{L^2}$$

Where:

• $\alpha$ is the thermal diffusivity ($\text{m}^2/\text{s}$), representing the material's ability to conduct thermal energy relative to its ability to store it. It is calculated as $\alpha = \frac{k}{\rho c_p}$, where $k$ is thermal conductivity, $\rho$ is density, and $c_p$ is specific heat capacity.
• $t$ is the elapsed time ($\text{s}$).
• $L$ is the characteristic length ($\text{m}$), a geometric parameter specific to the shape of the object.

2. Physical Interpretation

At its core, $Fo$ represents the ratio of the actual time elapsed to the characteristic time required for heat to diffuse across the object's scale.

• High Thermal Diffusivity ($\alpha$): A material with a high $\alpha$ (like copper) will have a larger $Fo$ for the same amount of time, meaning heat spreads rapidly.
• Small Characteristic Length ($L$): Smaller objects reach thermal equilibrium much faster because the denominator $L^2$ is smaller, leading to a higher $Fo$.
• Temporal Evolution:
  • When $Fo \ll 1$, the heat has only just begun to penetrate the surface; the core of the object remains largely unaffected by the external temperature change.
  • As $Fo$ increases, the thermal energy penetrates deeper, and the temperature distribution within the body begins to evolve toward a more uniform state.

The Biot Number ($Bi$): The Competition of Thermal Resistances

While the Fourier number focuses on the dimension of time, the Biot number focuses on the dimension of spatial resistance. It quantifies the relationship between the resistance to heat transfer at the surface (convection) and the resistance to heat transfer within the body (conduction).

1. Mathematical Definition

The Biot number is defined as:
$$Bi = \frac{h L_c}{k}$$

Where:

• $h$ is the convective heat transfer coefficient ($\text{W}/\text{m}^2\cdot\text{K}$), representing how effectively the surrounding fluid removes heat from the surface.
• $k$ is the thermal conductivity of the solid ($\text{W}/\text{m}\cdot\text{K}$).
• $L_c$ is the characteristic length, typically defined as the ratio of the object's volume ($V$) to its surface area ($A$): $L_c = V/A$.

2. Physical Interpretation

The Biot number can be viewed as the ratio of two thermal resistances:
$$Bi = \frac{\text{Convective Resistance}}{\text{Conductive Resistance}} = \frac{1/hA}{L_c/k}$$

• Low Biot Number ($Bi \ll 1$): This indicates that internal conduction is much faster than surface convection. The "bottleneck" for heat transfer is the surface. Consequently, the temperature inside the object remains nearly uniform throughout the process.
• High Biot Number ($Bi \gg 1$): This indicates that surface convection is very efficient, but internal conduction is slow. The "bottleneck" is inside the material, leading to significant temperature gradients between the surface and the core.

The Lumped Capacitance Method (LCM)

The most significant practical application of the Biot number is determining whether a complex transient problem can be simplified using the Lumped Capacitance Method.

1. The Criterion for Simplification

If the Biot number is sufficiently small—typically $Bi < 0.1$—we can assume that the internal thermal resistance is negligible. In this regime, we treat the entire object as a single "lump" with a uniform temperature $T(t)$ that changes over time. This allows us to bypass the complex PDEs and use a much simpler first-order ordinary differential equation (ODE).

2. The Mathematical Model

Under the LCM assumption, the energy balance is simplified to:
$$\rho V c_p \frac{dT}{dt} = -h A (T - T_\infty)$$

Integrating this equation yields the analytical solution for the temperature decay:
$$\frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left( -\frac{h A}{\rho V c_p} t \right)$$

By substituting the definitions of $Bi$ and $Fo$, we can see the elegant connection between these two dimensionless numbers:
$$\frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left( -Bi \cdot Fo \cdot \frac{L^2}{L_c^2} \right)$$
(Note: For standard geometries like spheres or plates, this often simplifies to a direct function of $Bi \cdot Fo$.)

Engineering Case Study: Cooling of a Copper Sphere

To illustrate these concepts in a real-world scenario, consider the following problem:

Scenario:
A solid copper sphere with a radius $R = 0.05,\text{m}$ is initially at $T_i = 100^\circ\text{C}$. It is placed in an environment with an ambient temperature $T_\infty = 20^\circ\text{C}$. The air provides a convective heat transfer coefficient of $h = 50,\text{W}/\text{m}^2\cdot\text{K}$.
Properties of Copper: $k = 385,\text{W/m}\cdot\text{K}$, $\rho = 8960,\text{kg/m}^3$, $c_p = 385,\text{J/kg}\cdot\text{K}$.

Goal: Determine if the Lumped Capacitance Method is valid and calculate the sphere's temperature after $100,\text{s}$.

Step 1: Calculate Characteristic Length ($L_c$)
For a sphere:
$$L_c = \frac{V}{A} = \frac{\frac{4}{3}\pi R^3}{4\pi R^2} = \frac{R}{3} = \frac{0.05}{3} \approx 0.0167,\text{m}$$

Step 2: Evaluate the Biot Number ($Bi$)
$$Bi = \frac{h L_c}{k} = \frac{50 \times 0.0167}{385} \approx 0.00217$$
Since $Bi = 0.00217 < 0.1$, the Lumped Capacitance Method is highly accurate for this case. We can assume the temperature is uniform throughout the sphere.

Step 3: Calculate Thermal Diffusivity ($\alpha$) and Fourier Number ($Fo$)
$$\alpha = \frac{k}{\rho c_p} = \frac{385}{8960 \times 385} \approx 1.116 \times 10^{-4},\text{m}^2/\text{s}$$
At $t = 100,\text{s}$:
$$Fo = \frac{\alpha t}{R^2} = \frac{1.116 \times 10^{-4} \times 100}{(0.05)^2} \approx 4.464$$

Step 4: Solve for Temperature $T(100)$
Using the LCM formula:
$$\frac{T(100) - 20}{100 - 20} = \exp\left( -\frac{h}{\rho c_p L_c} \times 100 \right)$$
$$\text{Exponent} = -\frac{50}{8960 \times 385 \times 0.0167} \times 100 \approx -0.0095$$
$$\frac{T(100) - 20}{80} = e^{-0.0095} \approx 0.9905$$
$$T(100) = 20 + (80 \times 0.9905) \approx 99.24^\circ\text{C}$$

Conclusion: Due to copper's high thermal conductivity, the sphere cools very uniformly. After 100 seconds, the temperature has only dropped slightly to approximately $99.24^\circ\text{C}$.

Summary

In the analysis of unsteady heat conduction, these two dimensionless numbers serve distinct but complementary roles:

• The Biot Number ($Bi$) acts as a model selector. It tells the engineer whether they can simplify the problem into a single-variable ODE (Lumped Capacitance) or if they must tackle the full complexity of spatial temperature gradients using PDEs or numerical methods.
• The Fourier Number ($Fo$) acts as a progress tracker. It scales time into a dimensionless format that describes how far the thermal disturbance has penetrated the material.

Mastering these parameters is essential for transitioning from theoretical heat transfer equations to practical, efficient engineering design.