Engineering & Science
Higbie Penetration & Danckwerts Surface-Renewal Mass Transfer (kL)
Calculate liquid-side gas-absorption mass-transfer coefficients k_L (mm/s) using Ralph Higbie’s unsteady penetration model 2*SQRT(D_AB / (π * t_e)) and Peter Danckwerts’ random surface-renewal theory SQRT(D_AB * s).
Why k_L scales with SQRT(D_AB) instead of D_AB^1.0 in real packed towers!
Whitman’s 1923 stagnant two-film theory predicted k_L = D_AB / δ, implying mass-transfer coefficients should scale linearly with diffusivity D_AB^1.0—yet packed-tower and bubble-column experiments consistently measure k_L ~ D_AB^0.5! Ralph Higbie (1935) solved that puzzle by modeling liquid packets sliding over packing pieces for a brief unsteady contact exposure t_e before remixing, deriving k_L,Higbie = 2 * SQRT(D_AB / (π * t_e))!
Peter V. Danckwerts (1951) replaced uniform contact time t_e with an exponential distribution of turbulent eddy surface-renewal rate s (s^-1), arriving at the even simpler law k_L,Danckwerts = SQRT(D_AB * s)! For CO_2 absorption into water at 25 °C (D_AB = 1.92 × 10^-9 m^2/s) over packed rings with contact time t_e = 0.08 s (mean renewal rate s = 1 / t_e = 12.5 s^-1): Higbie predicts k_L = 0.1748 mm/s and Danckwerts predicts k_L = 0.1549 mm/s (differing only by 2 / SQRT(π) = 1.1284)!
| Turbulent Liquid Contact Regime (DAB = 1.92E-9 m^2/s) | Higbie kL (mm/s) | Danckwerts kL at s = 1/te | Diffusivity Scaling |
|---|---|---|---|
| Standard Packing Exposure (te = 0.08 s, s = 12.5 s^-1) | 0.1748 mm/s | 0.1549 mm/s | kL ∝ D_AB^0.5 |
| High-Shear Structured Packing (te = 0.02 s, s = 50 s^-1) | 0.3496 mm/s (2.0x!) | 0.3098 mm/s | 4x Faster Renewal Doubles kL |
| Zero / Negative Exposure Guard (te <= 0) | "Invalid Mass Transfer Inputs" | "Invalid Mass Transfer Inputs" | Requires DAB > 0, te > 0 |
Higbie Unsteady Penetration Liquid Mass-Transfer Coefficient k_L (mm/s)
Evaluates 1000 * 2 * SQRT(D_AB / (PI() * t_e)) in mm/s from diffusivity B2 (m^2/s) and bubble/packing exposure time B3 (s).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Mass Transfer Inputs", ROUND(1000 * 2 * SQRT(B2 / (PI() * B3)), 4))Danckwerts Random Surface-Renewal Coefficient k_L (mm/s)
Evaluates 1000 * SQRT(D_AB * s) in mm/s from diffusivity B2 (m^2/s) and eddy renewal frequency B4 (s^-1).
=IF(OR(B2 <= 0, B4 <= 0), "Invalid Mass Transfer Inputs", ROUND(1000 * SQRT(B2 * B4), 4))Try the example
Use the original values or change them to test your understanding.