Fourier number (Fo) — Dimensionless time for unsteady heat conduction

Simply put, the Fourier number is a dimensionless time representing "how far unsteady heat conduction has progressed". The defining equation is:

Here, $\alpha$ is the thermal diffusivity [m²/s], $t$ is the elapsed time [s], and $L$ is the characteristic length (e.g., half the plate thickness, cylinder radius) [m].

Good question. Thermal diffusivity is thermal conductivity divided by density and specific heat:

While thermal conductivity $k$ indicates "how easily heat is conducted," thermal diffusivity $\alpha$ represents "how quickly temperature changes propagate". Copper has a large $k$, so it conducts heat easily, but its $\rho c_p$ is also quite large. On the other hand, aluminum has a larger $\alpha$ than copper, so temperature changes actually propagate faster in aluminum.

Exactly. To use an analogy, right after entering a bath, only the surface is warm (Fo is small), but after soaking for a while, the core becomes warm (Fo is large). The Fourier number is an indicator of "how uniformly the temperature has equalized throughout the object".

Let's look at the one-dimensional unsteady heat conduction equation (no internal heat generation, isotropic, constant properties):

Now, introduce dimensionless variables. Define temperature as $\theta = (T - T_\infty)/(T_i - T_\infty)$, position as $\xi = x/L$, and time as $\tau = \alpha t / L^2$. Then the governing equation becomes:

As you can see, the dimensionless time $\tau$ is precisely the Fourier number $\mathrm{Fo}$. After nondimensionalization, all parameters related to material properties and dimensions disappear, meaning for the same Fo and the same boundary conditions, the dimensionless temperature distribution $\theta(\xi, \mathrm{Fo})$ will be the same regardless of material or size. This is the essence of the similarity principle.

If the boundary conditions are matched with the same Bi (Biot number), yes. That's exactly the idea used to create scaled experimental models and why universal charts like the Heisler chart are valid.

The analytical solution for unsteady heat conduction is generally expressed as an infinite series. For example, for an infinite plate with initial temperature $T_i$ suddenly exposed to an environment at $T_\infty$, the dimensionless temperature at the center is:

Here, $\zeta_n$ are eigenvalues determined from the Biot number. The exponential function $\exp(-\zeta_n^2 \, \mathrm{Fo})$ decays more rapidly for larger $n$. When Fo exceeds 0.2, all terms with $n \geq 2$ become less than 2% and can be ignored. So, keeping only the first term gives sufficient accuracy:

Exactly. The Heisler chart is a graphical representation of the one-term approximation results, so it cannot be used in the region Fo < 0.2. If the initial short-time region is needed, you must sum many terms of the series or use the semi-infinite body solution.

For example, for steel ($\alpha \approx 1.2 \times 10^{-5}$ m²/s) with a plate thickness of 20mm ($L = 10$ mm = 0.01 m):

For a 20mm steel plate, the one-term approximation becomes usable after about 1.7 seconds. On the other hand, for a 200mm concrete wall ($\alpha \approx 5 \times 10^{-7}$ m²/s, $L = 0.1$ m), $t \approx 4000$ seconds = over 1 hour. It varies by orders of magnitude depending on material and dimensions.

Exactly. When deciding the analysis method for an unsteady heat conduction problem, the Biot number Bi and Fourier number Fo are considered as a pair. The Biot number is:

where $h$ is the surface heat transfer coefficient, $L$ is the characteristic length, and $k$ is the solid's thermal conductivity. Bi represents the ratio of "internal thermal resistance of the solid" to "convective thermal resistance at the surface".

Yes. If Bi < 0.1, the temperature gradient inside the object is negligible, so the entire object can be treated as having a uniform temperature—this is the Lumped Capacitance Method. In this case, the temperature change follows a simple exponential decay:

The decision flow in practice can be summarized as follows:

It does. In numerical analysis, the element Fourier number (mesh Fourier number) becomes important. Instead of the characteristic length $L$, we use the element size $\Delta x$, and instead of time $t$, we use the time step $\Delta t$:

When using an explicit method (forward Euler method, forward difference method), the solution diverges unless $\mathrm{Fo}_e \leq 0.5$ (1D). This is the famous CFL condition for heat conduction. For 2D, it becomes $\mathrm{Fo}_e \leq 0.25$, and for 3D, $\mathrm{Fo}_e \leq 1/6$.

Yes, for explicit methods, $\Delta t \leq \mathrm{Fo}_{e,\max} \cdot \Delta x^2 / \alpha$, so if the mesh size $\Delta x$ is halved, the allowable time step becomes one-fourth. The number of calculation steps doubles, so the total computational cost balloons to 8 times. This is the biggest weakness of explicit methods.

A Fourier number-based approach is convenient. First, estimate the overall time scale of the analysis:

• Characteristic time $t_c = L^2 / \alpha$ — an estimate of the time for temperature changes to propagate throughout the object.
• Analysis end time — until it approaches steady state (Fo $\approx$ 1~2) or until the time of interest.

Next, guidelines for the time step:

In terms of stability, yes, but accuracy suffers. If the time step is too large, you miss capturing transient temperature changes. Especially when looking at the initial response to thermal shock, "automatic time step control" is standard practice: use a small $\Delta t$ for the first few steps, then increase it once the temperature change becomes gradual.

For general unsteady heat conduction, implicit methods are overwhelmingly advantageous. The reason is simple: heat conduction is a diffusion-type equation, so the time scale is long. Explicit methods force you to use a very small $\Delta t$ for stability, but physically, temperature changes are slow, so using a larger $\Delta t$ with an implicit method is more efficient.

However, there are exceptions:

• Ultra-short-time local heating like laser heating or plasma irradiation → Explicit methods are