Engineering & Science
Sherwood Number (Sh), Schmidt Number (Sc) & Frössling Sphere Mass Transfer
Calculate the dimensionless Schmidt number Sc = nu / D_AB, Frössling-Ranz-Marshall sphere Sherwood number Sh = 2 + 0.6 * (Re^0.5) * (Sc^(1/3)), and convective mass-transfer coefficient k_c = (Sh * D_AB) / d_p.
Design spray dryers, evaporating raindrops, and dissolving pharmaceutical pellets!
In convective mass transfer, the Sherwood number Sh = (k_c * d_p) / D_AB plays the exact same role as the Nusselt number in heat transfer—comparing convective mass transfer k_c (m/s) against molecular diffusion D_AB / d_p—while the Schmidt number Sc = nu / D_AB (kinematic viscosity divided by diffusivity) mirrors the Prandtl number!
Nils Frössling (1938) and Ranz & Marshall (1952) proved that around a spherical droplet, gas bubble, or dissolving drug bead of diameter d_p, Sherwood number starts at the exact stagnant-fluid conduction/diffusion floor Sh = 2.00 (at Re = 0!) and grows with flow Reynolds number Re and Schmidt number Sc as: Sh = 2 + 0.6 * (Re ^ 0.5) * (Sc ^ (1/3))! In aqueous dissolution where liquids have high Sc = 1,000 (Sc^(1/3) = 10), even moderate agitation at Re = 400 (Re^0.5 = 20) boosts Sh from 2.00 to 2 + 0.6 * 20 * 10 = 122.00—accelerating tablet dissolution 61-fold over stagnant water!
| Flow & Fluid Regime (dp = 2.0 mm, DAB = 10^-9 m^2/s) | Reynolds Re & Schmidt Sc | Sherwood Number Sh | Mass-Transfer Coeff kc (mm/s) |
|---|---|---|---|
| Stagnant Liquid (Zero Flow) | Re = 0, Sc = 1,000 | Sh = 2.00 (Pure radial diffusion) | 0.0010 mm/s |
| Agitated Dissolution Vessel | Re = 400, Sc = 1,000 | Sh = 122.00 (61x faster!) | 0.0610 mm/s |
| Evaporating Water Drop in Air | Re = 100, Sc = 0.60 | Sh = 7.06 | Gas-phase boundary layer |
Frössling-Ranz-Marshall Sphere Sherwood Number Sh = 2 + 0.6 * Re^0.5 * Sc^(1/3)
Validates Re >= 0 and Sc > 0 (recovering the exact Sh = 2 stagnant limit when Re = 0).
=IF(OR(B2 < 0, B3 <= 0), "Invalid Re/Sc Inputs", ROUND(2 + (0.6 * SQRT(B2) * (B3 ^ (1 / 3))), 2))Convective Mass-Transfer Coefficient k_c = (Sh * D_AB) / d_p in mm/s
Computes k_c from Sh, diffusivity B4 (m^2/s), and sphere diameter B5 (mm), converting m/s to mm/s.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Re/Sc Inputs", ROUND(((2 + (0.6 * SQRT(B2) * (B3 ^ (1 / 3)))) * B4 * 1000000) / B5, 4))Try the example
Use the original values or change them to test your understanding.