Sherwood Number Calculator

Sh = k_cL/D_AB — dimensionless convective mass transfer. The mass-transfer analogue of the Nusselt number; relates convective mass transfer coefficient to molecular diffusivity over a characteristic length.

Mass Transfer · Dimensionless Number

Inputs

Solve for:

Characteristic length L (m)

Mass transfer coefficient kc (m/s)

Typical: gas-phase 10⁻³–10⁻¹ m/s · liquid-phase 10⁻⁶–10⁻⁴ m/s

Mass diffusivity D (m²/s)

Theory

Definition:

Sh = k_c × L / D_AB [dimensionless]

k_c = Sh × D_AB / L [m/s]

D_AB = k_c × L / Sh [m²/s]

Sh = 1 means convection provides no enhancement over pure molecular diffusion. Higher Sh means the convective mass transfer coefficient is that many times larger than the pure-diffusion reference D_AB/L.

Heat–mass transfer analogy (Nu → Sh, Pr → Sc):

Nu = h × L / k_f ↔ Sh = k_c × L / D_AB

h = Nu × k_f / L ↔ k_c = Sh × D_AB / L

Pr = ν / α ↔ Sc = ν / D_AB

Replace Pr with Sc and h with k_c in any forced-convection Nu correlation to get the corresponding Sh correlation.

Common Sh correlations:

Geometry / regime	Correlation	Validity
Laminar pipe, const-c	Sh = 3.66	Fully developed, Re < 2300
Laminar pipe, const-flux	Sh = 4.36	Fully developed, Re < 2300
Turbulent pipe	Sh = 0.023 Re⁰·⁸ Sc⅓	Re > 10 000, 0.6 < Sc < 3000
Flat plate, laminar avg	Sh = 0.664 Re^½ Sc^⅓	Re < 5×10⁵, Sc > 0.6
Sphere (Ranz-Marshall)	Sh = 2 + 0.6 Re^½ Sc^⅓	2 < Re < 200, 0.6 < Sc < 250
Pure diffusion (stagnant)	Sh = 2	Re → 0 (sphere), or stagnant film

Sh vs Nu vs Bi:

Sh = k_c L / D_AB convective mass transfer — uses fluid diffusivity

Nu = h L / k_f convective heat transfer — uses fluid conductivity

Bi = h L / k_s solid resistance — uses solid conductivity