Thermal Conductivity

Thermal conductivity κ describes how readily a material conducts heat. In insulators it is dominated by phonons; in metals both phonons and electrons contribute.

Kinetic Theory (Phonon Contribution)

From kinetic theory, the lattice thermal conductivity is:

$\kappa = \frac{1}{3} C_V v_s \lambda_{\text{mfp}}$

where:

• C_V — heat capacity per unit volume (J·m⁻³·K⁻¹)
• v_s — sound velocity (m/s)
• λ_mfp — phonon mean free path (m)

Wiedemann–Franz Law (Electronic Contribution)

For metals, the electronic thermal conductivity is linked to electrical conductivity σ by:

$\frac{\kappa_e}{\sigma T} = L_0$

where L₀ is the Lorenz number:

$L_0 = \frac{\pi^2 k_B^2}{3 e^2} \approx 2.44 \times 10^{-8} ; \text{W·Ω·K}^{-2}$

This gives κ_e = L₀ σ T.

Thermal Resistance

For a rod of length L, cross-sectional area A, and conductivity κ:

$R_{\text{th}} = \frac{L}{\kappa A} \quad [\text{K/W}]$

Implement

def kinetic_thermal_conductivity_W_mK(C_V_J_m3K, v_s_m_s, lambda_mfp_m):
    ...

def lorenz_number():
    # Returns L_0 in W·Ω·K⁻²
    ...

def wiedemann_franz_kappa(sigma_S_m, T_K):
    # Returns electron thermal conductivity in W/(m·K)
    ...

def thermal_resistance_K_W(kappa_W_mK, L_m, A_m2):
    # Returns thermal resistance in K/W
    ...