Engineering & Science
Ranz-Marshall Evaporating Droplet Nusselt & Sherwood Correlation
Calculate spray-drying and fuel-droplet heat and mass transfer via Ranz & Marshall’s 1952 correlation Nu = 2 + 0.6 * Re_d^0.5 * Pr^(1/3) and Sh = 2 + 0.6 * Re_d^0.5 * Sc^(1/3).
Why stagnant spherical droplets have Nu = Sh = 2—and how slip velocity accelerates spray drying!
In pharmaceutical spray dryers, diesel injectors, and atmospheric raindrop evaporation, pure radial conduction/diffusion into a stagnant sphere (Re_d = 0) yields the exact analytical Langmuir limit Nu_0 = Sh_0 = 2.00! William E. Ranz and W. Robert Marshall Jr. (Chemical Engineering Progress, 1952) measured evaporating water and benzene droplets in hot air streams (0 <= Re_d <= 200) to establish the canonical boundary-layer enhancement: Nu = 2 + 0.6 * SQRT(Re_d) * Pr^(1/3) and Sh = 2 + 0.6 * SQRT(Re_d) * Sc^(1/3)!
Watch an atomized slurry droplet moving at droplet Reynolds number Re_d = 144 through hot air (Pr = 0.71, water-vapor Schmidt number Sc = 0.61): convective boundary-layer thinning boosts Nu more than fourfold from 2.00 to 8.42 and Sh from 2.00 to 8.11—slashing droplet drying residence time inside the spray tower!
| Droplet Flow Regime (Air Pr = 0.71, Vapor Sc = 0.61) | Droplet Nusselt Nu | Droplet Sherwood Sh | Spray Dryer Physics |
|---|---|---|---|
| Stagnant Mist Limit (Re_d = 0) | 2.00 | 2.00 | Pure Radial Conduction / Diffusion |
| Nozzle Deceleration Zone (Re_d = 144) | 8.42 (4.21× stagnant) | 8.11 (4.05× stagnant) | Laminar Boundary-Layer Thinning |
| Negative Re_d or Pr <= 0 | Error Guard | "Invalid Droplet Parameters" | Domain Check |
Ranz-Marshall Evaporating Droplet Nusselt Number (Nu)
Evaluates 2 + 0.6 * SQRT(Re_d) * Pr^(1/3) from droplet Reynolds number B2 (>= 0) and gas Prandtl number B3 (> 0).
=IF(OR(B2 < 0, B3 <= 0), "Invalid Droplet Parameters", ROUND(2 + 0.6 * SQRT(B2) * (B3 ^ (1 / 3)), 2))Ranz-Marshall Mass-Transfer Sherwood Number (Sh)
Evaluates the Chilton-Colburn heat/mass analogy partner 2 + 0.6 * SQRT(Re_d) * Sc^(1/3) using gas Schmidt number B4.
=IF(OR(B2 < 0, B4 <= 0), "Invalid Droplet Parameters", ROUND(2 + 0.6 * SQRT(B2) * (B4 ^ (1 / 3)), 2))Try the example
Use the original values or change them to test your understanding.