Engineering & Science
Wilke Gas Mixture Viscosity Formula: Kinetic-Theory Binary φ_ij Parameter
Calculate Charles R. Wilke’s (1950) kinetic-theory gas interaction factors φ_12 = [1 + (μ_1/μ_2)^0.5*(M_2/M_1)^0.25]^2 / SQRT(8*(1 + M_1/M_2)) and mixture viscosity μ_m.
Why a 50/50 Hydrogen–Nitrogen mixture has nearly the viscosity of pure Nitrogen (17.0 vs 17.8 μPa·s)!
One of the most counterintuitive phenomena in chemical engineering transport phenomena is adding 50 mol% ultra-light Hydrogen (M_1 = 2.016 g/mol, μ_1 = 8.9 μPa·s) to Nitrogen (M_2 = 28.013 g/mol, μ_2 = 17.8 μPa·s): naive linear averaging predicts 13.35 μPa·s, yet experimental viscosity is 17.02 μPa·s—almost unchanged from pure N2! Charles R. Wilke (1950, Journal of Chemical Physics) derived the exact Chapman-Enskog hard-sphere explanation without empirical fitting constants: φ_ij = [1 + (μ_i/μ_j)^(1/2) * (M_j/M_i)^(1/4)]^2 / SQRT(8 * (1 + M_i/M_j))!
Because M_2 / M_1 = 13.895, fast light H2 molecules are scattered strongly by heavy N2 targets (φ_12 = 1.9103) while heavy N2 momentum transport barely notices light H2 collisions (φ_21 = 0.2750)—yielding μ_m = 17.019 μPa·s at 50/50 H2/N2, and still 14.500 μPa·s even at 80% H2 / 20% N2!
| H2 (1) / N2 (2) Gas Composition | Wilke φ_12 & φ_21 Factors | Wilke Mixture Viscosity μ_m | vs Naive Linear Average |
|---|---|---|---|
| 50% H2 + 50% N2 (Forming/Cracking Gas) | φ_12 = 1.9103 / φ_21 = 0.2750 | 17.019 μPa·s | +27.5% above linear 13.35! |
| 80% H2 + 20% N2 (Hydrotreater Recycle) | φ_12 = 1.9103 / φ_21 = 0.2750 | 14.500 μPa·s | +35.8% above linear 10.68! |
| Invalid y_1 + y_2 <> 1 or M, μ <= 0 | Error Guard | "Invalid Wilke Gas Inputs" | Mass-Balance Guard |
Wilke Binary Gas Mixture Viscosity μ_m (μPa·s)
Combines pure viscosities B4, B5 (μPa·s) and mole fractions B2, B3 with first-principles Wilke φ_12 (B8) and φ_21 (B9).
=IF(OR(B2 < 0, B3 < 0, ABS(B2 + B3 - 1) > 0.0001, MIN(B4:B7) <= 0), "Invalid Wilke Gas Inputs", ROUND((B2 * B4) / (B2 + B3 * B8) + (B3 * B5) / (B3 + B2 * B9), 3))Wilke Asymmetric Collision Interaction Factor φ_12
Outputs [1 + SQRT(μ_1/μ_2)*(M_2/M_1)^0.25]^2 / SQRT(8*(1 + M_1/M_2)) computed in helper cell B8.
=IF(MIN(B4:B7) <= 0, "Invalid Wilke Gas Inputs", ROUND(B8, 4))Try the example
Use the original values or change them to test your understanding.