Engineering & Science
Wen-Yu Minimum Fluidization Velocity (u_mf) & Archimedes Number Formula
Calculate the dimensionless particle Archimedes number Ar = (d_p^3 * rho_f * (rho_p - rho_f) * g) / mu^2, Wen-Yu Reynolds number Re_mf = SQRT(33.7^2 + 0.0408 * Ar) - 33.7, and superficial minimum fluidization velocity u_mf.
Find the exact upward gas speed where a packed catalyst bed lifts into a boiling fluid!
Blow air gently upward through a bed of sand or FCC refinery catalyst pellets and it behaves as a static packed bed; gradually increase superficial gas velocity u0 until upward Ergun drag pressure drop Delta_P / L EXACTLY balances buoyant bed weight (1 - eps_mf) * (rho_p - rho_f) * g—suddenly the particles unlock and bubble like a boiling liquid at the Minimum Fluidization Velocity u_mf!
What if you haven't measured incipient voidage eps_mf or particle sphericity yet? C. Y. Wen and Y. H. Yu (1966) discovered that across hundreds of powders, the shape-voidage combinations (1 - eps_mf)/(Psi^2 * eps_mf^3) ~ 11 and 1/(Psi * eps_mf^3) ~ 14 are nearly universal constants! Substituting those into Ergun's quadratic equation yields the famous Wen-Yu correlation in terms of the dimensionless Archimedes number Ar = (d_p^3 * rho_f * (rho_p - rho_f) * 9.80665) / (mu ^ 2): Re_mf = SQRT((33.7 ^ 2) + (0.0408 * Ar)) - 33.7, and u_mf = (Re_mf * mu) / (rho_f * d_p)! Fluidizing 0.50 mm silica sand (rho_p = 2,600 kg/m^3) in ambient air (rho_f = 1.20 kg/m^3, mu = 1.8e-5 Pa*s) gives Ar = 11,798.9, Re_mf = 6.513, and u_mf = 0.195 m/s!
| Silica Sand Particle Size in Air | Archimedes Number Ar | Wen-Yu Re_mf | Minimum Fluidization u_mf (m/s) |
|---|---|---|---|
| d_p = 0.50 mm (0.0005 m) | 11,798.9 | 6.513 | 0.195 m/s |
| d_p = 1.00 mm (2x coarser) | 94,390.8 (8x Ar!) | 36.918 | 0.554 m/s |
| Invalid Particle Density rho_p <= rho_f | Particles Float! | Guarded Input Error | "Invalid Bed/Fluid Properties" |
Dimensionless Particle Archimedes Number Ar
Evaluates (d_p^3 * rho_f * (rho_p - rho_f) * 9.80665) / (mu ^ 2) when solid density B4 exceeds fluid density B3.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= B3, B5 <= 0), "Invalid Bed/Fluid Properties", ROUND(((B2 ^ 3) * B3 * (B4 - B3) * 9.80665) / (B5 ^ 2), 1))Wen-Yu Superficial Minimum Fluidization Velocity u_mf (m/s)
Solves Re_mf = SQRT(33.7^2 + 0.0408 * Ar) - 33.7 and multiplies by mu / (rho_f * d_p).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= B3, B5 <= 0), "Invalid Bed/Fluid Properties", ROUND(((SQRT((33.7 ^ 2) + (0.0408 * (((B2 ^ 3) * B3 * (B4 - B3) * 9.80665) / (B5 ^ 2)))) - 33.7) * B5) / (B3 * B2), 3))Try the example
Use the original values or change them to test your understanding.