Introduction & Context

The transient transfer in a semi-infinite body is a fundamental analytical model in process engineering used to describe diffusion or heat conduction in systems where the penetration depth of the process is small relative to the total thickness of the medium. This model is critical for predicting concentration profiles in soil contamination, gas absorption in liquids, and transient thermal response in thick structural components. By assuming the medium extends to infinity in one direction, we simplify the governing partial differential equations into an analytical solution involving the Gauss Error Function, allowing for rapid estimation of concentration or temperature gradients without the need for complex numerical simulations.

Methodology & Formulas

The calculation relies on the diffusion analogy where the concentration profile is determined by the surface concentration, initial concentration, and the diffusion coefficient over time. The following algebraic steps define the methodology:

First, the time is converted to seconds:

\[ t_{seconds} = t_{hours} \times 3600.0 \]

The penetration depth, which defines the validity of the semi-infinite assumption, is calculated as:

\[ penetration\_depth = 2.0 \times \sqrt{d_{coeff} \times t_{seconds}} \]

The dimensionless argument for the error function is defined as:

\[ \eta = \frac{x}{2.0 \times \sqrt{d_{coeff} \times t_{seconds}}} \]

The concentration at depth x and time t is determined by:

\[ C(x, t) = C_s + (C_0 - C_s) \times \text{erf}(\eta) \]

To evaluate the error function, the following approximation is utilized:

\[ t_{val} = \frac{1.0}{1.0 + p \times | \eta |} \] \[ y = 1.0 - (((((a_5 \times t_{val} + a_4) \times t_{val}) + a_3) \times t_{val} + a_2) \times t_{val} + a_1) \times t_{val} \times e^{-\eta^2} \] \[ \text{erf}(\eta) = \text{sign}(\eta) \times y \]

Condition	Threshold / Criteria	Action
Semi-infinite validity	penetration_depth > soil_thickness	Model invalid: Use finite-body solution
Physical constraint	d_coeff ≤ 0	Error: Diffusion coefficient must be positive
Physical constraint	t_seconds ≤ 0	Error: Time must be greater than zero

The semi-infinite body model is a powerful simplification for transient heat conduction. You can apply this assumption when the following conditions are met:

The thermal penetration depth is significantly smaller than the actual thickness of the material.
The temperature at the far boundary remains unaffected by the surface heat flux during the timeframe of interest.
The process duration is short enough that the thermal wave does not reach the opposite side of the component.

When a constant heat flux is applied to the surface of a semi-infinite solid, the surface temperature increases over time. To determine this, use the following approach:

Identify the thermal conductivity and thermal diffusivity of your material.
Apply the analytical solution derived from the error function, which relates surface temperature to the square root of time.
Ensure your units for time and thermal properties are consistent to avoid scaling errors in the calculation.

While the semi-infinite body model simplifies complex transient problems, process engineers must be aware of these limitations:

It assumes constant thermophysical properties, which may not hold true if the material undergoes significant temperature-dependent changes.
It ignores internal heat generation, which is often present in electrical or chemical processes.
It does not account for complex geometries where edge effects or multi-dimensional heat flow become dominant.

Worked Example

A chemical spill has occurred on the surface of a deep soil bed. An environmental engineer needs to predict the pollutant concentration at a specific depth after 24 hours to assess contamination spread, using the transient diffusion model for a semi-infinite medium.

Known Input Parameters

Constant surface concentration, \(C_s\): 10.000 kg/m³
Initial uniform concentration, \(C_0\): 0.000 kg/m³
Diffusion coefficient, \(D\): \(1.000 \times 10^{-7}\) m²/s
Depth of interest, \(x\): 0.100 m
Time since spill, \(t\): 24.000 hours (equivalent to 86400.000 s)
Total soil bed thickness: 2.000 m

Step-by-Step Calculation

Validate the semi-infinite medium assumption. The penetration depth is approximately \(2\sqrt{Dt}\). From the provided results, the calculated penetration depth is 0.186 m. Since 0.186 m < 2.000 m, the semi-infinite body model is valid for this analysis.
Calculate the dimensionless argument \(\eta\). The formula is \(\eta = \frac{x}{2\sqrt{Dt}}\). Using the provided numerical result: \(\eta = 0.538\).
Evaluate the Gauss Error Function, \(\text{erf}(\eta)\). Using the provided numerical result: \(\text{erf}(0.538) \approx 0.553\).
Solve for the concentration \(C(x,t)\). The governing equation is rearranged: \[ C(x,t) = C_s + (C_0 - C_s) \cdot \text{erf}(\eta) \] Substituting the known values: \[ C(x,t) = 10.000 + (0.000 - 10.000) \times 0.553 = 4.468 \text{ kg/m}^3 \]

Final Answer

The predicted pollutant concentration at a depth of 0.100 m after 24 hours is 4.468 kg/m³.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle