Transient Heat Conduction in a Semi-Infinite Solid

Of course, there are no real objects that are "infinitely thick." However, in the initial stage when heat is suddenly applied to a thick wall or the ground, before the heat reaches the back surface, the back surface is "effectively non-existent," so it can be treated as a semi-infinite body.

Here are some common examples in practice:

• Immediately after welding heat input: The initial stage when heat from arc welding or laser welding spreads into the base material. The temperature distribution in the HAZ (Heat-Affected Zone) can be estimated precisely using the semi-infinite body solution.
• Surface temperature in laser processing: The surface temperature rise from the instant of laser irradiation up to a few milliseconds. Directly relates to controlling the processing depth.
• Freezing depth in the ground: Estimating how far freezing progresses when the ground surface temperature drops rapidly in winter. Used as a design basis for the burial depth of water pipes.
• Rapid cooling in casting/forging: The cooling rate near the surface when high-temperature metal is quenched in water or oil. Used to determine whether martensitic transformation occurs.

The one-dimensional unsteady heat conduction equation for a semi-infinite solid ($x \geq 0$) is as follows:

Here, $\alpha = k/(\rho c_p)$ is the thermal diffusivity [m²/s]. For steel, it's about $1.2 \times 10^{-5}$, and for aluminum, about $9.7 \times 10^{-5}$ m²/s. The larger this value, the faster temperature changes propagate.

The point lies in the boundary and initial conditions. In the most basic case:

• Initial condition: $T(x, 0) = T_i$ (uniform temperature everywhere)
• Surface boundary condition: $T(0, t) = T_s$ (surface temperature changes to a constant for $t > 0$)
• Far-field condition: $T(\infty, t) = T_i$ (sufficiently far away remains at the initial temperature)

And the decisive factor is the introduction of the similarity variable:

Applying this variable transformation reduces the partial differential equation to an ordinary differential equation. The two independent variables $x$ and $t$ are combined into a single variable $\eta$. This is called the Boltzmann transformation.

It means "the temperature change at a point twice as far away ($x$) will be the same if the time ($t$) is quadrupled." If $\eta$ is the same, the shape of the temperature profile is the same—this is called self-similarity. Imagine the temperature profile "stretching" deeper proportionally to $\sqrt{t}$ as time passes.

By non-dimensionalizing with $\theta(\eta) = (T - T_s)/(T_i - T_s)$, the governing equation becomes:

The boundary conditions are $\theta(0) = 0$, $\theta(\infty) = 1$. This is solved by the error function erf:

Returning to the original temperature yields the most important formula for the temperature distribution in a semi-infinite solid:

Here, $\operatorname{erfc}(\eta) = 1 - \operatorname{erf}(\eta)$ is the complementary error function.

These are some useful values to remember:

In other words, beyond $\eta = 2$ ($x = 4\sqrt{\alpha t}$), there is almost no temperature change. This forms the basis for the penetration depth.

Exactly. The thermal penetration depth is a guideline for the depth where the temperature change has almost reached:

For example, when heat is instantaneously input into steel ($\alpha \approx 1.2 \times 10^{-5}$ m²/s) during welding:

• After $t = 1$ second: $\delta \approx 4\sqrt{1.2 \times 10^{-5}} \approx 14$ mm
• After $t = 10$ seconds: $\delta \approx 44$ mm
• After $t = 100$ seconds: $\delta \approx 139$ mm

If the plate thickness is sufficiently larger than this $\delta$, it's OK to treat it as a semi-infinite body. For a 20 mm thick steel plate, the semi-infinite body approximation is valid for about $t \lesssim 2$ seconds after welding.

The heat flux at the surface ($x = 0$) is obtained by differentiating the erfc solution with respect to $x$:

It is proportional to $1/\sqrt{t}$, meaning an enormous heat flux is required initially, which decays over time. $q \to \infty$ as $t \to 0$ corresponds to the temperature gradient becoming infinite at the surface initially. In reality, any heat source has finite output, so for the very initial period, a constant heat flux condition is more realistic than a constant temperature condition.

There are three typical boundary condition patterns for semi-infinite solids:

The surface temperature for Case 2 is $T(0,t) = T_i + \frac{2q_0''}{k}\sqrt{\frac{\alpha t}{\pi}}$, which increases proportionally to $\sqrt{t}$. In laser processing, since the output is constant, Case 2 is often closer to reality.

It is judged by the Fourier number $\mathrm{Fo} = \alpha t / L^2$ (where $L$ is the plate thickness). Roughly speaking:

• $\mathrm{Fo} < 0.05$: Error of the semi-infinite body approximation is less than 1%. Safe to use.
• $0.05 < \mathrm{Fo} < 0.2$: Influence of the back surface begins to appear. Be cautious about accuracy.
• $\mathrm{Fo} > 0.2$: No longer a semi-infinite body. Should use the finite thickness solution (Fourier series solution).

For example, for a 50 mm thick steel plate ($\alpha = 1.2 \times 10^{-5}$), the time corresponding to $\mathrm{Fo} = 0.05$ is $t = 0.05 \times 0.05^2 / (1.2 \times 10^{-5}) \approx 10.4$ seconds. That means it's fine to calculate as a semi-infinite body within about 10 seconds after welding.

Good question. Analytical solutions are only usable for cases of "homogeneous, isotropic, temperature-independent properties, simple boundary conditions." In practice, numerical analysis becomes necessary in cases like these:

• Temperature-dependent properties: Steel's thermal conductivity decreases from about 60 W/(m·K) at 0°C to about 30 W/(m·K) at 800°C. It becomes nonlinear, so there's no analytical solution.
• 2D/3D effects: If the heating area is finite in width, like a weld bead, heat diffusion in the lateral direction must be considered.
• Time-varying boundary conditions: Moving welding torches, pulsed laser irradiation, etc.
• Phase change: Cases involving molten pool formation or solidification (Stefan problem).
• Composite materials / dissimilar material joining: Cases where thermal resistance exists at the interface.

Exactly. There are two basic approaches:

1. Take a sufficiently large analysis domain: Secure a domain 2-3 times larger than the penetration depth $\delta = 4\sqrt{\alpha t_{\max}}$, and set $T = T_i$ (Dirichlet condition) at the far boundary. The simplest and most reliable.
2. Use infinite elements: Place special elements like INFIN111 in Ansys or CINPE4 in Abaqus, which have built-in decay at the far boundary, at the boundary. This can significantly reduce the analysis domain.

In the FEM formulation for heat conduction, the Galerkin method yields the following discrete system:

Here, $[C]$ is the heat capacity matrix, $[K]$ is the thermal conductivity matrix, and $\{F\}$ is the heat load vector. It looks similar to $[M]\{\ddot{u}\} + [K]\{u\} = \{F\}$ in structural analysis, right? The difference is whether it's a first-order or second-order time derivative.

Absolutely not a uniform mesh. The erfc temperature profile has a steep gradient near the surface and becomes flat further in. Therefore, a non-uniform mesh (biased mesh) with element sizes increasing exponentially from the surface is essential.

In Ansys, setting the Bias Factor to around 10-20 will automatically create a nice non-uniform mesh. In Abaqus, use single bias for edge seed.

Sharp observation. Near $t = 0$, the surface heat flux diverges as $1/\sqrt{t}$, so the time step also needs to be extremely small initially. Specifically: