Non-Steady-State Heat Conduction Model of a Semi-Infinite Large Body

In the study of transient heat transfer, the semi-infinite solid model is a cornerstone for analyzing thermal processes

In the study of transient heat transfer, the semi-infinite solid model is a cornerstone for analyzing thermal processes where the material's depth is significantly greater than the thermal penetration depth. This model is particularly useful when a surface is suddenly subjected to a thermal disturbance—such as rapid heating, cooling, or a sudden heat flux—and the heat does not reach the far boundary of the material within the timeframe of interest.

This approximation is widely applied in engineering disciplines to model phenomena such as laser surface processing, welding, quenching of metals, and the thermal response of electronic components. By assuming the body is "infinitely" thick in one direction, we simplify the complex boundary value problems into more manageable mathematical forms.

Fundamental Assumptions

To maintain the analytical tractability of the model, several key assumptions are typically made:

  • Geometric Symmetry: The body is treated as a homogeneous, isotropic, semi-infinite slab occupying the region $x \ge 0$. The thickness is assumed to be large enough that $\lim_{x \to \infty} T(x, t) = T_i$, meaning the far end remains unaffected by surface perturbations.
  • Constant Material Properties: The thermal conductivity ($k$), density ($\rho$), and specific heat capacity ($c$) are assumed to be independent of temperature. Consequently, the thermal diffusivity $\alpha = \frac{k}{\rho c}$ is treated as a constant.
  • Initial State: At time $t = 0$, the entire medium is at a uniform initial temperature $T_i$.
  • One-Dimensional Flow: Heat conduction is assumed to occur solely along the $x$-axis (perpendicular to the surface).

2. Mathematical Formulation and Similarity Transformation

2.1 The Governing Equation

For a one-dimensional, non-steady-state conduction process in a homogeneous medium, the temperature distribution $T(x, t)$ is governed by the following partial differential equation (PDE):

$$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}, \quad 0 < x < \infty, \quad t > 0$$

2.2 The Similarity Variable Approach

Solving a PDE directly can be mathematically intensive. However, for the semi-infinite model, we can employ a similarity transformation to convert the PDE into an ordinary differential equation (ODE). We introduce a dimensionless similarity variable, $\eta$:

$$\eta = \frac{x}{2\sqrt{\alpha t}}$$

By defining a dimensionless temperature $\theta(\eta)$ as:

$$\theta(\eta) = \frac{T(x, t) - T_i}{T_s - T_i}$$

The original heat equation simplifies to the following second-order ODE:

$$\frac{d^2\theta}{d\eta^2} + 2\eta\frac{d\theta}{d\eta} = 0$$

The general solution to this equation is expressed in terms of the error function ($\text{erf}$) or the complementary error function ($\text{erfc}$), where $\text{erfc}(\eta) = 1 - \text{erf}(\eta)$.

3. Analytical Solutions for Common Boundary Conditions

Depending on how the surface at $x=0$ is manipulated, different analytical solutions emerge.

3.1 Constant Surface Temperature (Dirichlet Condition)

If the surface temperature is instantaneously raised to a constant value $T_s$ at $t=0$, the temperature distribution is given by:

$$\boxed{T(x, t) = T_i + (T_s - T_i) \operatorname{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right)}$$

This solution describes how the "thermal wave" penetrates the material over time.

3.2 Constant Surface Heat Flux (Neumann Condition)

In many industrial processes, such as constant-power heating, a constant heat flux $q_0$ is applied to the surface. The boundary condition is defined as $-k \frac{\partial T}{\partial x}\big|_{x=0} = q_0$. The resulting temperature profile is:

$$\boxed{T(x, t) = T_i + \frac{2q_0\sqrt{\alpha t}}{k\sqrt{\pi}} \exp\left(-\frac{x^2}{4\alpha t}\right) - \frac{q_0 x}{k} \operatorname{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right)}$$

3.3 Arbitrary Time-Varying Surface Temperature

For complex scenarios where the surface temperature $T_s(t)$ changes according to a specific function, we utilize Duhamel's Principle (or Laplace transforms) to derive a convolution integral:

$$T(x, t) = T_i + \int_{0}^{t} \frac{x}{2\sqrt{\pi\alpha (t-\tau)^3}} \exp\left(-\frac{x^2}{4\alpha (t-\tau)}\right) [T_s(\tau) - T_i] , d\tau$$

This integral allows for the modeling of any arbitrary thermal history at the boundary.

4. Engineering Case Study: Aluminum Alloy Heating

To demonstrate the practical application of this model, consider an aluminum alloy component undergoing rapid surface heating.

Material Properties:

  • Thermal Conductivity ($k$): $237 , \text{W/(m}\cdot\text{K)}$
  • Density ($\rho$): $2700 , \text{kg/m}^3$
  • Specific Heat ($c$): $900 , \text{J/(kg}\cdot\text{K)}$
  • Thermal Diffusivity ($\alpha$): $\frac{237}{2700 \times 900} \approx 9.73 \times 10^{-5} , \text{m}^2/\text{s}$

Scenario:
The surface is instantaneously heated from $T_i = 20^\circ\text{C}$ to $T_s = 200^\circ\text{C}$. We wish to determine the temperature at a depth of $x = 5 , \text{mm}$ after $t = 10 , \text{s}$.

Calculation Steps:

  1. Calculate the similarity variable ($\eta$):
    $$\eta = \frac{0.005}{2\sqrt{9.73 \times 10^{-5} \times 10}} \approx \frac{0.005}{0.06238} \approx 0.254$$

  2. Evaluate the complementary error function:
    Using standard tables or computational tools, $\operatorname{erfc}(0.254) \approx 0.718$.

  3. Compute the final temperature:
    $$T(5,\text{mm}, 10,\text{s}) = 20 + (200 - 20) \times 0.718 \approx 152.6^\circ\text{C}$$

Conclusion: After 10 seconds, the temperature at 5 mm depth has risen significantly to approximately $153^\circ\text{C}$, indicating substantial thermal penetration.

5. Numerical Implementation and Computational Notes

While analytical solutions are elegant, real-world engineering often involves non-linearities (e.g., temperature-dependent $k$ or $\alpha$) or complex geometries that require numerical methods.

Numerical Strategies

  • Finite Difference Methods (FDM): The most common approach for transient problems.
    • Explicit Method (FTCS): Easy to implement but subject to strict stability constraints: $\Delta t \le \frac{\Delta x^2}{2\alpha}$.
    • Implicit Method (Crank-Nicolson): Unconditionally stable and more accurate for larger time steps.
  • Finite Element Method (FEM): Preferred for complex, non-rectangular geometries.

Computational Implementation (Python Example)

Modern programming libraries make it trivial to solve these equations. Below is a Python snippet to compute the Dirichlet solution:

import numpy as np
from scipy.special import erfc

# Material properties for Aluminum
k = 237.0
rho = 2700.0
c = 900.0
alpha = k / (rho * c)

def get_temperature(x, t, Ts=200.0, Ti=20.0):
    """Calculates temperature using the erfc analytical solution."""
    eta = x / (2.0 * np.sqrt(alpha * t))
    return Ti + (Ts - Ti) * erfc(eta)

# Parameters for the case study
depth = 5e-3   # 5 mm
time = 10.0    # 10 s

temp_result = get_temperature(depth, time)
print(f"Temperature at {depth*1e3:.1f} mm after {time}s: {temp_result:.2f} °C")

6. Summary

The non-steady-state heat conduction model for a semi-infinite solid provides a powerful mathematical framework for transient thermal analysis. By leveraging the similarity variable, we can transform complex partial differential equations into practical analytical solutions involving the error function.

Whether dealing with a sudden temperature jump (Dirichlet), a constant heat flux (Neumann), or a time-varying thermal load (Duhamel's Principle), these models allow engineers to predict internal temperature profiles with high precision. For more complex, non-linear, or geometrically irregular problems, these analytical solutions serve as the essential "ground truth" for validating numerical simulations.