Engineering & Science
Eucken & Modified Eucken Polyatomic Gas Thermal Conductivity Formula
Estimate dilute polyatomic gas thermal conductivity k = (μ / M) * (C_v + 2.25 * R) and Modified Eucken k_mod = (μ / M) * (1.32 * C_v + 1.77 * R) (mW/(m·K)) from gas viscosity and molar heat capacity.
Separating translational kinetic energy from internal rotational/vibrational modes in polyatomic gases!
For monatomic noble gases (Ar, He, Ne where molar heat capacity C_v = 1.5 * R), kinetic theory proves k = 2.5 * (μ / M) * C_v = 3.75 * R * μ / M! Polyatomic molecules like CO_2, CH_4, and steam ALSO carry internal rotational and vibrational energy (C_v - 1.5 * R) that diffuses at molecular self-diffusivity rather than 2.5× momentum diffusivity. Arnold Eucken (1913) weighted translational modes by 2.5 and internal modes by 1.0 to obtain k_Eucken = (μ / M) * (C_v + 2.25 * R), while Hirschfelder, Curtiss, and Bird (1954) accounted for Lewis-number self-diffusion (ρ * D_AA / μ ≈ 1.32) to derive Modified Eucken: k_Mod = (μ / M) * (1.32 * C_v + 1.77 * R)!
Test gaseous CO_2 at 300 K (viscosity μ = 15.0 μPa·s = 1.50 × 10^-5 Pa·s, molar mass M = 44.01 g/mol = 0.04401 kg/mol, C_v = 28.80 J/(mol·K), R = 8.31446 J/(mol·K)): Eucken predicts 16.19 mW/(m·K) and Modified Eucken predicts 17.97 mW/(m·K)—bracketing the experimental NIST value (~16.8 mW/(m·K)) cleanly!
| Gas at 300 K (μ in μPa·s, M in g/mol, Cv in J/mol·K) | Classic Eucken k (mW/m·K) | Modified Eucken k_mod (mW/m·K) | Experimental Benchmark |
|---|---|---|---|
| CO2 (μ = 15.0, M = 44.01, Cv = 28.80) | 16.19 mW/(m·K) | 17.97 mW/(m·K) | 16.8 mW/(m·K) (Bracketed!) |
| Monatomic Argon (μ = 22.7, M = 39.95, Cv = 12.47) | 17.72 mW/(m·K) | 17.72 mW/(m·K) | Identical when Cv = 1.5*R! |
| Invalid μ <= 0 or M <= 0 | "Invalid Gas Properties" | Error Guard | Positivity Check |
Classic Eucken Gas Thermal Conductivity k_Eucken (mW/(m·K))
With μ in μPa·s (10^-6 Pa·s) and M in g/mol (10^-3 kg/mol), (μ_uPa / M_g) * (C_v + 2.25 * R) * 10^-3 W/(m·K) equals (μ_uPa / M_g) * (C_v + 2.25 * 8.31446) directly in mW/(m·K)!
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Gas Properties", ROUND((B2 / B3) * (B4 + 2.25 * 8.31446), 2))Modified Eucken (Hirschfelder) Thermal Conductivity k_Mod (mW/(m·K))
Evaluates (μ_uPa / M_g) * (1.32 * C_v + 1.77 * 8.31446) in mW/(m·K).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Gas Properties", ROUND((B2 / B3) * (1.32 * B4 + 1.77 * 8.31446), 2))Try the example
Use the original values or change them to test your understanding.