🧽 Porous Media Effective Conductivity

Calculate effective thermal conductivity of porous media using Parallel, Series, Hashin-Shtrikman, EMT/Bruggeman, and Geometric Mean models.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

Heat Transfer

📊 Solver Telemetry ● ACTIVE

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📅 Released Jun 2026

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Parameters Input

Solid Thermal Conductivity ($k_s$) [W/m·K]:

Fluid Thermal Conductivity ($k_f$) [W/m·K]:

Porosity ($\epsilon$):

Hot Temperature ($T_{hot}$) [°C]:

Cold Temperature ($T_{cold}$) [°C]:

Bed Thickness ($L$) [m]:

Bed Cross-Section Area ($A$) [m²]:

Effective Medium Bounds: $$k_{par} = (1-\epsilon)k_s + \epsilon k_f \quad \text{(Voigt)}$$ $$k_{ser} = \left[\frac{1-\epsilon}{k_s} + \frac{\epsilon}{k_f}\right]^{-1} \quad \text{(Reuss)}$$

Results & Analysis

Results and visualizations will be displayed here upon completion of the computation.

Calculation Methodology

Mathematical Model & Theory

Determining the effective thermal conductivity of porous media (e.g. soil, metal foams, packed beds) requires combining the conductivities of the solid and fluid phases based on their geometric distributions. The principal models implemented are:

Parallel (Voigt): Assumes phases are oriented parallel to heat flow. Represents the upper bounds limit: $$k_{par} = (1 - \epsilon)k_s + \epsilon k_f$$
Series (Reuss): Assumes phases are arranged in layers perpendicular to heat flow. Represents the lower bounds limit: $$k_{ser} = \left[\frac{1 - \epsilon}{k_s} + \frac{\epsilon}{k_f}\right]^{-1}$$
Geometric Mean: Empirical model matching random distributions: $$k_{geo} = k_s^{(1 - \epsilon)} k_f^\epsilon$$
Hashin-Shtrikman Bounds: Tightest possible bounds for isotropic composite materials: $$k_{HS}^+ = k_s + \frac{\epsilon}{\frac{1}{k_f - k_s} + \frac{1 - \epsilon}{3k_s}}$$
EMT (Bruggeman): Self-consistent model for spherical inclusions: $$(1-\epsilon)\frac{k_s - k_{EMT}}{k_s + 2k_{EMT}} + \epsilon\frac{k_f - k_{EMT}}{k_f + 2k_{EMT}} = 0$$

Academic References:

Kaviany, M. (1995). Principles of Heat Transfer in Porous Media. Springer.
Nield, D. A., & Bejan, A. (2017). Convection in Porous Media. Springer.

Worked Engineering Example

Problem Statement:
A glass-fiber bed saturated with air has the following parameters: solid thermal conductivity $k_s = 1.4$ W/m·K, fluid conductivity $k_f = 0.026$ W/m·K, and bed porosity $\epsilon = 0.4$. Calculate the effective thermal conductivity using the Parallel, Series, and Geometric Mean models.

Step-by-step Solution:
1. Calculate the Parallel effective conductivity $k_{par}$:
$$k_{par} = (1 - 0.4) \times 1.4 + 0.4 \times 0.026 = 0.84 + 0.0104 = 0.8504 \text{ W/m·K}$$ 2. Calculate the Series effective conductivity $k_{ser}$:
$$1/k_{ser} = \frac{1 - 0.4}{1.4} + \frac{0.4}{0.026} = 0.4286 + 15.3846 = 15.8132$$ $$k_{ser} = 1 / 15.8132 = 0.0632 \text{ W/m·K}$$ 3. Calculate the Geometric Mean effective conductivity $k_{geo}$:
$$k_{geo} = 1.4^{(1-0.4)} \times 0.026^{0.4} = 1.4^{0.6} \times 0.026^{0.4}$$ $$k_{geo} = 1.2203 \times 0.2325 = 0.2837 \text{ W/m·K}$$
Final Result Summary:
• Parallel (Max Bound) = 0.850 W/m·K
• Series (Min Bound) = 0.063 W/m·K
• Geometric Mean = 0.284 W/m·K