Transient Conduction in Semi-Infinite Solids

Theoretical Foundation of Transient Conduction in Semi-Infinite Solids

Overview and application scenarios

• Immediately after welding heat input: In arc or laser welding, the heat-affected zone (HAZ) temperature distribution during the early phase of heat propagation in the base metal fits the semi-infinite solid solution perfectly.
• Laser processing surface temperature: During the first few milliseconds after laser illumination, surface temperature rise is governed by semi-infinite solid behavior, critical for controlling processing depth.
• Ground freezing depth: When ground surface temperature drops sharply in winter, estimating how deep freezing penetrates is essential. Water pipe burial depth is designed based on this calculation.
• Rapid cooling in casting/forging: When high-temperature metal is quenched in water or oil, the near-surface cooling rate determines whether martensitic transformation occurs.

Governing equations and similarity variable

Here $\alpha = k/(\rho c_p)$ is the thermal diffusivity [m²/s]. For steel, approximately $1.2 \times 10^{-5}$; for aluminum, approximately $9.7 \times 10^{-5}$ m²/s. Larger values mean temperature changes propagate faster.

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

The crucial step is introducing the similarity variable:

This variable transformation converts the PDE into an ODE. Two independent variables ($x$ and $t$) collapse into one ($\eta$). This is called the Boltzmann transformation.

Derivation of analytical solutions

With boundary conditions $\theta(0) = 0$ and $\theta(\infty) = 1$. This solves in terms of the error function:

Converting back, we get the most important formula for semi-infinite solid temperature distribution:

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

Beyond $\eta = 2$ (i.e., $x = 4\sqrt{\alpha t}$), temperature change is negligible. This defines the penetration depth.

Penetration depth and surface heat flux

For example, steel ($\alpha \approx 1.2 \times 10^{-5}$ m²/s) subjected to sudden welding heat:

• At $t = 1$ s: $\delta \approx 14$ mm
• At $t = 10$ s: $\delta \approx 44$ mm
• At $t = 100$ s: $\delta \approx 139$ mm

If the plate thickness is much larger than $\delta$, the semi-infinite approximation is valid. For a 20 mm steel plate, the approximation holds for roughly $t \lesssim 2$ seconds after welding.

Notice the $1/\sqrt{t}$ dependence: enormous heat flux is needed initially, then it decays with time. As $t \to 0$, $q \to \infty$—the initial temperature gradient at the surface is infinite. Real heat sources have finite power, so constant heat flux conditions are often more realistic than constant temperature in the very early stages.

Three patterns of boundary conditions

In Case 2, surface temperature rises as $T(0,t) = T_i + \frac{2q_0''}{k}\sqrt{\frac{\alpha t}{\pi}}$, proportional to $\sqrt{t}$. Laser processing typically uses Case 2 since laser power is approximately constant.

Validity assessment of semi-infinite solid approximation

• $\mathrm{Fo} < 0.05$: Semi-infinite approximation error is <1%. Safe to use.
• $0.05 < \mathrm{Fo} < 0.2$: Back-surface effects beginning. Accuracy requires care.
• $\mathrm{Fo} > 0.2$: Definitively finite-thickness regime. Use Fourier series solutions.

For a 50 mm steel plate ($\alpha = 1.2 \times 10^{-5}$), $\mathrm{Fo} = 0.05$ corresponds to $t \approx 10.4$ seconds. So welding simulations are valid within ~10 seconds for semi-infinite approximation on this thickness.

Numerical Computation Methods for Transient Conduction in Semi-Infinite Solids

Scenarios where analytical solutions are unavailable

• Temperature-dependent material properties: Steel's thermal conductivity drops ~30% from 0°C to 800°C. Nonlinearity prevents closed-form solutions.
• 2D/3D effects: Finite-width welding beads, localized laser spots need multidimensional analysis.
• Time-varying boundaries: Moving welding torch, pulsed laser irradiation create time-dependent boundary conditions.
• Phase changes: Melting/solidification with latent heat (Stefan problem) requires modified approaches.
• Composite/dissimilar materials: Interfacial thermal resistance in bonded structures.

FEM discretization of semi-infinite solids

1. Create a sufficiently large analysis domain: Extend the domain to 2–3 times the penetration depth $\delta = 4\sqrt{\alpha t_{\max}}$, apply $T = T_i$ (Dirichlet condition) at the far boundary. Simplest and most robust.
2. Use infinite elements: Special elements like ANSYS INFIN111 or Abaqus CINPE4 with built-in far-field decay. Dramatically reduces mesh size at the cost of complexity.

The FEM weak form via Galerkin gives the discrete system:

where $[C]$ is the heat capacity matrix, $[K]$ the thermal conductivity matrix, and $\{F\}$ the heat load vector.

Mesh strategy

In ANSYS, set Bias Factor to 10–20 for automatic exponential grading. Abaqus uses "single bias" on edge seeds.

Time integration scheme

• Initial time step: $\Delta t_1 = \Delta x_{\min}^2 / (6\alpha)$ (one-sixth the thermal diffusion time of the smallest surface element)
• Time step growth: Increase by factor 1.2–1.5 each step (geometric progression)
• Automatic control: Use ANSYS AUTO TIME STEPPING or Abaqus automatic scheme. Most reliable for accuracy.

Implicit methods (backward Euler, Crank-Nicolson) are unconditionally stable, but fine initial steps are still needed for accuracy.

Application of infinite elements

Using infinite elements reduces the analysis domain to ~1.5× penetration depth, cutting elements dramatically. However, element orientation must be correct—wrong orientation can amplify instead of decay, producing nonsensical results. Always verify against analytical solutions.

Practical Applications of Transient Conduction in Semi-Infinite Solids

Temperature prediction of welding HAZ

For thick plates where semi-infinite applies, the simplified form is:

Example: Heat input $Q = 1.5$ kJ/mm, no preheat ($T_0 = 20$°C), plate thickness 25 mm, mild steel → $t_{8/5} \approx 12$ seconds. If $t_{8/5} < 6$ s, hydrogen cracking risk rises. If $> 30$ s, toughness drops. This directly drives welding procedure design.

Surface temperature in laser processing

For example, SUS304 ($k = 16$ W/(m·K), $\alpha = 4.2 \times 10^{-6}$ m²/s) hit with power density $I_0 = 10^8$ W/m² for 1 ms:

$T(0, 1\text{ms}) = 20 + \frac{2 \times 10^8}{16}\sqrt{\frac{4.2 \times 10^{-6} \times 10^{-3}}{\pi}} \approx 1,470$°C

Melting point is ~1,400°C, so melting starts. By controlling laser pulse duration, processing depth is controlled. However, once melting begins, latent heat effects and phase change require Stefan problem solutions for precision.

Ground freezing depth estimation

$\operatorname{erfc}(\eta_f) = 0.25$ → $\eta_f \approx 0.81$

With soil thermal diffusivity $\alpha \approx 5 \times 10^{-7}$ m²/s and $t = 1$ week ($6 \times 10^5$ s):

$x_f = 2 \times 0.81 \times \sqrt{5 \times 10^{-7} \times 6 \times 10^5} \approx 0.89$ m

Freezing reaches ~90 cm depth. Water pipe codes mandate 1.0–1.2 m burial depth in cold climates—a direct application of this calculation. Real scenarios involve soil moisture freeze-thaw latent heat, which slows penetration beyond the erfc prediction (Stefan problem).

Contact temperature calculation

where $e = \sqrt{k \rho c_p}$ is thermal effusivity [J/(m²·K·s^{1/2})]. This quantifies how strongly material "imposes" its temperature.

Example: 200°C steel ($e \approx 14,000$) contacting 20°C copper ($e \approx 37,000$):

$T_c = (14000 \times 200 + 37000 \times 20)/(14000 + 37000) \approx 69$°C

Interface temperature leans toward copper—the material with higher effusivity "wins." This explains why stainless handrails feel colder than wood at the same room temperature: metal has high effusivity, wood has low. Heat conducts away from your hand faster in metal. Casting/quenching dynamics and heat sink selection depend critically on effusivity.

Transient Conduction in Semi-Infinite Solids: Software & Solver Comparison for Transient Conduction in Semi-Infinite Solids

Implementation methods in commercial tools

1. Compare FEM temperature profiles against erfc solution for Case 1
2. Check both surface temperature history and depth-wise temperature distribution
3. Verify surface heat flux: $q_s'' = k(T_s - T_i)/\sqrt{\pi\alpha t}$ (numerical differentiation accuracy check)
4. Demonstrate mesh convergence across 3+ mesh densities

The NAFEMS T2 benchmark is exactly this problem.

Implementation in open source

import numpy as np
from scipy.special import erfc

alpha = 1.2e-5   # Steel thermal diffusivity [m²/s]
Ti, Ts = 20, 800  # Initial and surface temps [°C]
t = 5.0           # Time [s]
x = np.linspace(0, 0.05, 100)  # Depth [m]

T = Ti + (Ts - Ti) * erfc(x / (2 * np.sqrt(alpha * t)))
delta = 4 * np.sqrt(alpha * t)  # Penetration depth
print(f"Penetration depth: {delta*1000:.1f} mm")

For FEM, use FEniCS + Python: 1D mesh + Dirichlet BC + backward Euler in ~50 lines. Excellent for validating commercial results.

Advanced Research in Transient Conduction in Semi-Infinite Solids

Non-Fourier heat conduction

where $\tau_q$ (thermal relaxation time, ~10⁻¹¹–10⁻¹³ s for metals) replaces the pure diffusion behavior with a "heat wave" propagating at finite speed. This is a hybrid between diffusion and wave equations. Relevant for femtosecond laser machining, semiconductor hotspot dynamics, cryogenic physics.

Application to inverse problems

• Casting interfacial heat transfer coefficient: From mold interior thermocouples, deduce molten metal–mold heat transfer
• Rocket engine heat loads: Measure wall temperature, back-calculate surface heat flux
• Quench analysis: From internal temperature history, infer surface cooling rate

Because analytical erfc solutions exist, the Green's function (kernel) is known, and regularized inversion (Tikhonov regularization) is stable. Recent ML approaches (Physics-Informed Neural Networks / PINNs) combine this known analytical structure with neural networks for robust inverse estimation.

Transient Conduction in Semi-Infinite Solids: Common Issues & Debugging for Transient Conduction in Semi-Infinite Solids

Common failures and countermeasures

Verification benchmarks

1. NAFEMS T2: Standard 1D transient conduction. Constant boundary at surface, compare FEM to erfc at specified depth/time.
2. Case 1 custom: Steel ($k=50$, $\rho=7850$, $c_p=500$), $T_i=20$, $T_s=500$°C. At $t=10$ s and $x=5$ mm, exact answer: $T \approx 434$°C
3. Case 2 constant flux: Same material, $q_0''=10^6$ W/m². Compare surface temperature time history.
4. Surface heat flux validation: Compute $q_s'' = k(T_s-T_i)/\sqrt{\pi\alpha t}$ from theory; compare with FEM output. Verifies numerical differentiation accuracy.

All should agree within 2% for proper meshing/time stepping.