3: Thermal conductivity

3: Thermal Conductivity

Thermal conductivity ($k$) is a fundamental thermophysical property that quantifies a material's inherent ability to conduct heat via conduction. It represents the rate of heat transfer ($Q$) per unit area ($A$) per unit temperature gradient ($dT/dx$) under steady-state conditions. This definition arises directly from Fourier's Law of Heat Conduction:

$q_x'' = -k \frac{dT}{dx}$

where $q_x''$ is the heat flux (heat transfer rate per unit area, W/m²) in the x-direction, and $dT/dx$ is the temperature gradient (K/m). The negative sign indicates heat flows from regions of higher temperature to lower temperature. Essentially, $k$ tells us how readily heat energy diffuses through a material due to molecular interactions (vibrations in solids, collisions in fluids).

Factors Influencing Thermal Conductivity:

• Material Type: Values vary drastically:
  • Metals (High $k$): Excellent conductors due to free electrons (e.g., Copper ~400 W/m·K, Aluminum ~240 W/m·K).
  • Non-Metallic Solids (Moderate/Low $k$): Heat transfer relies on lattice vibrations (phonons). Crystalline solids (e.g., Diamond ~2000 W/m·K) conduct better than amorphous solids (e.g., Glass ~1 W/m·K).
  • Liquids (Low $k$): Molecular collisions dominate (e.g., Water ~0.6 W/m·K, Engine Oil ~0.15 W/m·K).
  • Gases (Very Low $k$): Sparse molecules limit conduction (e.g., Air ~0.026 W/m·K). Insulators like foam leverage trapped air pockets.
• Temperature: For pure metals, $k$ generally decreases with increasing temperature as lattice vibrations scatter electrons more effectively. For gases and most non-metals, $k$ increases with temperature due to higher molecular energy/activity.
• Phase & Composition: Impurities, alloying elements, moisture content, and phase changes (solid vs. liquid) significantly alter $k$. Porous materials have effective conductivity dependent on porosity and pore fluid.

Units: The SI unit is Watts per meter-Kelvin (W/m·K). Imperial units (Btu·in/hr·ft²·°F) are sometimes used; conversion is essential (1 W/m·K ≈ 6.933 Btu·in/hr·ft²·°F).

Distinction from Thermal Diffusivity: While thermal conductivity ($k$) measures how well a material conducts heat, thermal diffusivity ($\alpha = k / (\rho c_p)$)) measures how fast heat propagates through it by incorporating density ($\rho$) and specific heat ($c_p$). High $k$ enables more heat flow, but high $\alpha$ means temperature changes occur more rapidly.