Sherwood Number (Sh): Dimensionless Mass-Transfer Coefficient
The Sherwood number (symbol Sh) expresses the ratio of convective to diffusive mass transfer: Sh = k\u2097 L / D, where k\u2097 is the mass-transfer coefficient (m·s⁻¹), L a characteristic length (m), and D the diffusivity (m²·s⁻¹). Analogous to the Nusselt number in heat transfer, Sh consolidates fluid properties and flow conditions into a dimensionless measure that simplifies scale-up and correlation development.
Definition and Calculation
Sh values depend on geometry and flow regime. For internal flows, L often equals hydraulic diameter; for external flows, it might be particle diameter or characteristic length of a plate. Using SI units for k\u2097 (m·s⁻¹), L (m), and D (m²·s⁻¹) keeps Sh unitless and compatible with ISO guidance on characteristic numbers. Report the chosen L to avoid ambiguity.
Typical correlations take the form Sh = a Re^b Sc^c. In laminar flow over flat plates, theory yields Sh = 0.664 Re¹ᐟ² Sc¹ᐟ³ for local values. Turbulent pipe flow often uses Sh = 0.023 Re⁰·⁸ Sc¹ᐟ³. Always confirm the applicable Reynolds and Schmidt ranges before applying a correlation.
Historical Background
The number honours Thomas Kilgore Sherwood, whose 1930s work on gas absorption established empirical correlations linking Re, Sc, and mass-transfer coefficients. Sherwood's studies, along with those of Chilton and Colburn, cemented the analogy between heat and mass transfer that underpins modern process design. Inclusion of Sh in standards such as ISO 80000-11 formalised its notation and encouraged consistent reporting.
Over time, Sh correlations expanded from simple geometries to packed columns, fluidised beds, and membrane systems. Computational fluid dynamics now supplements experiments, but published Sh correlations remain essential for quick design estimates and equipment rating.
Conceptual Foundations
Film and Boundary-Layer Theory
Sh embodies the balance between convective transport in the bulk fluid and diffusive transport across the concentration boundary layer. Higher Sh values correspond to thinner films and stronger convection. In many analogies, Sh/ReSc = f(Re, Sc) parallels Nu/RePr relations, allowing designers to transfer intuition from heat to mass transfer problems.
Scale-Up and Similarity
Maintaining similar Sh, Re, and Sc between pilot and full-scale equipment helps preserve interfacial fluxes. Deviations can indicate maldistribution, channeling, or regime shifts. Reporting Sh alongside geometric parameters ensures reproducibility across scales and vendors.
Applications and Importance
Chemical absorbers, strippers, and scrubbers rely on Sh to size packing or trays so gas–liquid contact meets removal targets. In drying and coating operations, Sh helps predict solvent evaporation rates and informs airflow design. Bioreactor aeration uses Sh correlations to estimate oxygen transfer coefficients (k\u2097a), vital for cell growth and metabolism.
Environmental engineers deploy Sh when modelling volatilisation from water surfaces or fluxes through sediments. Membrane technologists use Sh to quantify concentration polarisation and design spacer geometries that promote mixing without excessive pressure drop.
Working with Sherwood Numbers
When using Sh correlations, document Reynolds and Schmidt numbers, surface roughness, and whether the reported coefficient is local or average. Validate assumptions with pilot tests or CFD when operating near correlation limits. Use the Reynolds calculator and mass balance tool to keep property data and resulting k\u2097 values consistent across design documents.
Finally, cite the source of any correlation used and note its validity range. Consistent SI notation and transparent assumptions help teams compare alternatives, comply with ISO terminology, and produce auditable design packages.