Engineering & Science
Weisz-Prater Criterion (CWP): Observable Internal Pore Diffusion Check
Diagnose internal catalyst pore diffusion limitations directly from measured laboratory reaction rate (-r_A')_obs without knowing intrinsic rate constant k via C_WP = (-r_A')_obs * ρ_c * R^2 / (D_e * C_As).
Why Thiele modulus has a chicken-and-egg problem—and how Weisz-Prater solves it!
To compute classic Thiele modulus Φ = R * SQRT(k / D_e), you need the true intrinsic rate constant k—which you CANNOT measure directly if your lab catalyst pellets are already choked by internal pore diffusion! Paul B. Weisz and Charles D. Prater (Advances in Catalysis, 1954) eliminated k altogether by multiplying η * Φ^2 into an observable modulus built strictly from measured quantities: C_WP = (-r_A')_obs * ρ_c * R^2 / (D_e * C_As)!
When C_WP << 1 (engineering threshold C_WP < 0.3 for first-order reactions), concentration gradients inside the pores are negligible and η ≈ 1; when C_WP > 1, the pellet interior is reactant-starved! Test a R = 0.25 cm (5 mm diameter) hydrogenation pellet with measured rate (-r_A')_obs = 2.5 × 10^-5 mol/(g_cat·s), pellet density ρ_c = 1.40 g/cm^3, effective diffusivity D_e = 0.008 cm^2/s, and surface concentration C_As = 1.2 × 10^-4 mol/cm^3: C_WP = 2.279 (> 1, severe pore diffusion!), whereas crushing the catalyst 5x finer to R = 0.05 cm slashes C_WP 25-fold to 0.091 (kinetic regime confirmed)!
| Catalyst Pellet Radius R | Weisz-Prater Modulus CWP | Diagnostic Threshold Rule | Catalyst Utilization Regime |
|---|---|---|---|
| Commercial Extrudate (R = 0.25 cm) | 2.279 (> 1.0) | CWP >= 1.0 | Severe Internal Pore Diffusion! |
| Transition Sieve Cut (R = 0.12 cm) | 0.525 (0.3–1.0) | 0.3 <= CWP < 1.0 | Moderate Gradient Onset |
| Crushed Lab Powder (R = 0.05 cm) | 0.091 (< 0.3) | CWP < 0.3 | Negligible Resistance (True Kinetics) |
Weisz-Prater Observable Modulus C_WP = r_obs * ρ_c * R^2 / (D_e * C_As)
Evaluates dimensionless C_WP from measured weight-based rate B2 (mol/(g·s)), pellet density B3 (g/cm^3), pellet radius B4 (cm), effective diffusivity B5 (cm^2/s), and surface reactant concentration B6 (mol/cm^3).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Catalyst Inputs", ROUND((B2 * B3 * (B4 ^ 2)) / (B5 * B6), 3))Automated Pore-Diffusion Regime Classification
Applies Fogler’s engineering cutoffs (< 0.3 Kinetic Regime, 0.3–1.0 Moderate Resistance, >= 1.0 Severe Pore Diffusion).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Catalyst Inputs", IF((B2 * B3 * (B4 ^ 2)) / (B5 * B6) < 0.3, "Negligible Diffusion (True Kinetics)", IF((B2 * B3 * (B4 ^ 2)) / (B5 * B6) < 1, "Moderate Pore Resistance", "Severe Internal Pore Diffusion")))Try the example
Use the original values or change them to test your understanding.