Chemical, Process & Catalysis Engineering
Wheeler Catalyst Poisoning Formula: Uniform vs Selective Pore-Mouth Activity Ratio F
Predict heterogeneous catalyst deactivation severity when fraction alpha_p of active sites is poisoned using Ahlborn Wheeler’s (1951) Uniform Poisoning Ratio F_unif and Selective Pore-Mouth ("Shell") Poisoning Ratio F_mouth across Thiele modulus h_0.
Why 20% sulfur poisoning at the pore mouth can destroy > 80% of a diffusion-limited catalyst’s rate!
When feed trace poisons (H2S, As, Pb, or basic nitrogen) deactivate fraction alpha_p (0 <= alpha_p < 1) of internal catalyst sites, naive intuition expects fractional remaining reaction rate F = Rate_poisoned / Rate_clean = 1 - alpha_p! In his landmark catalysis monograph, Ahlborn Wheeler (1951, Advances in Catalysis) proved that pore-diffusion resistance (clean slab Thiele modulus h_0) dramatically alters poisoning sensitivity depending on whether poison deposits UNIFORMLY or SELECTIVELY at the exterior pore mouth!
Under Non-Selective Uniform Poisoning, intrinsic rate constant drops to k * (1 - alpha_p) so Thiele modulus shrinks to h_0 * SQRT(1 - alpha_p): at high h_0, F_uniform -> SQRT(1 - alpha_p) > (1 - alpha_p) (slower reaction lets reactant penetrate deeper into previously unused pore interiors)! Conversely, under Selective Pore-Mouth ("Shell-Progressive") Poisoning, reactant must diffuse through a dead outer shell of fractional thickness alpha_p before reaching clean core (1 - alpha_p), yielding Wheeler’s exact slab ratio (writing TANH(u) = (1 - EXP(-2*u)) / (1 + EXP(-2*u))): F_mouth = (TANH(h_0 * (1 - alpha_p)) / TANH(h_0)) / (1 + alpha_p * h_0 * TANH(h_0 * (1 - alpha_p)))!
| Catalyst Regime (Poisoned Fraction alpha_p = 0.20 = 20% of Sites) | Uniform Poisoning Ratio F_unif | Pore-Mouth Poisoning Ratio F_mouth | Physical Deactivation Contrast |
|---|---|---|---|
| Kinetic Regime (h_0 = 0.20, No Diffusion Limit) | 0.802 (Linear ~1 - alpha_p) | 0.798 (Linear ~1 - alpha_p) | Nearly identical ~80% remaining activity |
| Moderate Pore Diffusion (h_0 = 3.00) | 0.888 | 0.622 | Pore-mouth dead shell adds series resistance |
| Strong Pore Diffusion (h_0 = 15.00) | 0.894 (~SQRT(0.80)) | 0.250 (~1 / (1 + alpha_p * h_0)) | 20% shell poison kills 75% of reactor rate! |
Wheeler Non-Selective Uniform Catalyst Poisoning Activity Ratio F_uniform
Evaluates SQRT(1 - alpha_p) * TANH(h_0 * SQRT(1 - alpha_p)) / TANH(h_0) via exact exponential TANH(u) = (1 - EXP(-2*u)) / (1 + EXP(-2*u)) for poisoned site fraction B2 (0 <= alpha_p < 1) and clean slab Thiele modulus B3 > 0.
=IF(OR(B2 < 0, B2 >= 1, B3 <= 0), "Requires 0 <= Poisoned Fraction alpha_p < 1 and Thiele h_0 > 0", ROUND(SQRT(1 - B2) * ((1 - EXP(-2 * B3 * SQRT(1 - B2))) / (1 + EXP(-2 * B3 * SQRT(1 - B2)))) / ((1 - EXP(-2 * B3)) / (1 + EXP(-2 * B3))), 4))Wheeler Selective Pore-Mouth ("Shell") Poisoning Activity Ratio F_mouth
Evaluates (TANH(h_0 * (1 - alpha_p)) / TANH(h_0)) / (1 + alpha_p * h_0 * TANH(h_0 * (1 - alpha_p))) across B2 and B3.
=IF(OR(B2 < 0, B2 >= 1, B3 <= 0), "Requires 0 <= Poisoned Fraction alpha_p < 1 and Thiele h_0 > 0", ROUND((((1 - EXP(-2 * B3 * (1 - B2))) / (1 + EXP(-2 * B3 * (1 - B2)))) / ((1 - EXP(-2 * B3)) / (1 + EXP(-2 * B3)))) / (1 + B2 * B3 * ((1 - EXP(-2 * B3 * (1 - B2))) / (1 + EXP(-2 * B3 * (1 - B2))))), 4))Try the example
Use the original values or change them to test your understanding.