Operations, Logistics & Engineering
Carberry Number (Ca) Formula: Observable External Film Mass-Transfer Check in Catalysis
Diagnose external boundary-layer film diffusion falsification in packed-bed catalytic reactors using James J. Carberry’s observable number Ca = R_obs / (k_g * a_p * C_b) and Mears’s < 5% criterion n * Ca < 0.05.
Why you cannot measure intrinsic catalyst kinetics until Carberry’s number passes n * Ca < 0.05!
Whereas the Weisz-Prater modulus diagnoses INTERNAL pore diffusion inside a catalyst pellet, external stagnant-film resistance between bulk fluid concentration C_b (mol/m^3) and external pellet surface concentration C_s can falsify laboratory packed-bed kinetics before reactants ever enter a pore! Because intrinsic rate constant k is unknown prior to testing, Damköhler’s number Da = k / (k_g * a_p) cannot be evaluated directly.
Chemical reaction engineering pioneer James J. Carberry (1976, Chemical and Catalytic Reaction Engineering) solved this by defining the purely observable Carberry Number Ca = R_obs / (k_g * a_p * C_b) = (C_b - C_s) / C_b, where R_obs (mol/(m^3_pellet * s)) is the measured volumetric rate, k_g (m/s) is the external film mass-transfer coefficient, and a_p = 6 / d_p (m^-1 for spheres of diameter d_p) is external surface area per pellet volume! Because C_s / C_b = 1 - Ca exactly at steady state, an nth-order surface reaction has external effectiveness eta_ext = (1 - Ca)^n—and Mears’s criterion guarantees < 5% rate falsification whenever |n| * Ca < 0.05!
| Carberry Number Ca & Reaction Order n | Surface-to-Bulk Ratio C_s / C_b = 1 - Ca | External Effectiveness (1 - Ca)^n & Regime |
|---|---|---|
| Ca = 0.015, Order n = 1 (n*Ca = 0.015 < 0.05) | 0.9850 (1.5% Film Drop) | 0.9850 — Intrinsic Kinetic Regime (<5% Bias) |
| Ca = 0.120, Order n = 2 (n*Ca = 0.240 >= 0.05) | 0.8800 (12% Film Drop) | 0.7744 — Film-Disguised Rate (-22.6% Bias!) |
| Ca -> 1.000 (Mass-Transfer Limiting Ceiling) | 0.0000 (C_s -> 0 at Surface) | External Diffusion Controlled (Apparent E_a -> ~5 kJ/mol) |
Observable Carberry Number Ca = R_obs / (k_g * (6 / d_p) * C_b) for Spherical Pellets
Computes Ca = (C_b - C_s) / C_b from observed pellet rate R_obs (B2), film coefficient k_g (B3), sphere diameter d_p (B4), and bulk concentration C_b (B5).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Catalyst Film Parameters", IF(B2 / (B3 * (6 / B4) * B5) >= 1, "Unphysical Rate >= Max Film Flux (Ca >= 1)", ROUND(B2 / (B3 * (6 / B4) * B5), 4)))Mears-Carberry External Mass-Transfer Diagnostic (n * Ca < 0.05 Criterion)
Tests n * Ca < 0.05 and reports external film effectiveness (1 - Ca)^n when external resistance is significant.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Catalyst Film Parameters", IF(B2 / (B3 * (6 / B4) * B5) >= 1, "Unphysical Rate >= Max Film Flux (Ca >= 1)", IF(B6 * (B2 / (B3 * (6 / B4) * B5)) < 0.05, "Negligible External Film Resistance (n*Ca < 0.05)", "External Film Limitation Present (Increase Linear Velocity u!)")))Try the example
Use the original values or change them to test your understanding.