Sherwood Number (Sh): Convective Mass Transfer Scaling

Definition and Key Relationships

The Sherwood number Sh is a dimensionless group expressing the ratio of convective mass-transfer resistance to diffusive resistance. For a characteristic length L, mass-transfer coefficient k_m, and molecular diffusivity D, it is defined as:

The larger the Sherwood number, the more dominant convection is compared with diffusion in transporting species. Engineers often rewrite Sh in terms of the Reynolds number (Re) and the Schmidt number Sc = ν / D, where ν denotes kinematic viscosity. Many empirical correlations take the general form Sh = C · Re^m · Scⁿ, mirroring the structure of the Nusselt number in heat transfer. Because Sc and the Prandtl number share similar definitions, engineers routinely leverage heat–mass analogies to estimate Sherwood values when direct data are scarce.

Historical Background

Thomas K. Sherwood’s work in the 1930s and 1940s on gas absorption and distillation columns established modern mass-transfer theory. He collaborated with J.A. Danckwerts and Robert Pigford to systematize empirical data, culminating in the Sherwood–Pigford–Wilke compilations still cited today. Their pioneering correlations linked dimensionless groups such as Re, Sc, and Sh, giving chemical engineers reliable tools for designing absorption towers, humidifiers, and evaporators.

Subsequent researchers expanded Sherwood’s framework to packed beds, spray towers, rotating disk contactors, and biological systems. The American Institute of Chemical Engineers codified many Sherwood correlations in its Design Handbooks during the 1950s and 1960s. Today, the Sherwood number appears in International Union of Pure and Applied Chemistry (IUPAC) recommendations and ISO 80000-11 discussions of characteristic numbers, underscoring its foundational status in transport phenomena.

Theoretical Foundations

Boundary Layers and Film Theory

Film theory models mass transfer as diffusion through a stagnant layer of thickness δ adjacent to the interface. Within this layer, molecular diffusion obeys Fick’s law, while outside it, bulk convection dominates. The Sherwood number becomes Sh = L / δ under the film approximation, revealing that high Sh corresponds to thin boundary layers generated by vigorous mixing or turbulence. Empirical correlations effectively estimate δ by relating Sh to Re and Sc.

Penetration and Surface Renewal Theories

Danckwerts’ surface renewal theory treats the interface as continually refreshed fluid elements that contact the surface for a finite exposure time. This approach explains unsteady mass-transfer coefficients in turbulent flows and complements film theory. Sherwood numbers derived from surface renewal models depend on the frequency of renewal events, linking Sh to turbulence intensity and, in ventilation contexts, air-change rates measured with tools like the air changes per hour calculator.

Heat–Mass Transfer Analogies

Because the governing differential equations for heat and mass transfer are analogous, engineers translate well-established Nusselt correlations into Sherwood correlations by substituting thermal diffusivity with mass diffusivity. The Chilton–Colburn j-factor analogy, for instance, defines j_D = Sh / (Re · Sc^1/3) and sets it equal to the friction factor-based heat-transfer j-factor. This approach simplifies design calculations, particularly when combined with resources such as the heat exchanger NTU calculator to cross-check heat and mass transfer performance simultaneously.

Measurement and Correlation Development

Determining Sherwood numbers experimentally involves measuring mass-transfer coefficients through techniques such as controlled evaporation, absorption, or dissolution. Researchers monitor concentration changes over time using gas chromatography, spectroscopy, or electrochemical probes. Data reduction requires accurate property values—diffusivity, viscosity, density—often sourced from correlations or measured directly. Dimensionless groups are then regressed to produce constants C, m, and n for the chosen geometry.

For example, the classical Ranz–Marshall correlation for spherical particles in low-turbulence flow states Sh = 2 + 0.6 Re^1/2 Sc^1/3. Packed tower correlations add complexity by incorporating packing factors and wetting efficiency. Engineers validate correlations against pilot-scale data before scaling up to industrial equipment. Sensitivity analyses quantify how uncertainties in diffusivity or velocity propagate into Sherwood number predictions—a critical step when designing pharmaceutical fermenters or emissions scrubbers where safety margins are tight.

Digital design environments increasingly couple computational fluid dynamics with species transport models to compute spatially varying Sherwood numbers. These simulations resolve local boundary-layer behaviour, capturing phenomena such as channeling in packed beds or oxygen gradients in tissue-engineering scaffolds. Nevertheless, empirical correlations remain indispensable for quick estimates and sanity checks during early project phases.

Applications Across Industries

Chemical absorption and stripping. Sherwood numbers govern the design of gas absorption towers that remove pollutants such as sulfur dioxide or volatile organic compounds. Engineers select packings and operating velocities that deliver the target Sh, ensuring adequate mass-transfer rates without excessive pressure drop.

Bioprocessing and cell culture. Oxygen transfer to microbial or mammalian cultures depends on bubble size, agitation, and medium properties—all encapsulated in Sherwood correlations. Maintaining dissolved oxygen above critical thresholds requires balancing mixing energy with shear-sensitive organisms, a task simplified by pairing Sherwood estimates with Reynolds number calculations.

Environmental engineering. Indoor air quality models compute Sherwood numbers to estimate deposition of aerosols onto surfaces or biofilms within HVAC systems. Higher Sh indicates faster removal of contaminants from the air, complementing building ventilation assessments that rely on air-change metrics.

Energy systems. Electrochemical devices, including fuel cells and redox-flow batteries, use Sherwood analyses to size flow channels and diffusion layers. Optimising Sh helps balance reactant distribution with pumping losses, supporting the broader move toward energy-efficient decarbonisation technologies.

Why the Sherwood Number Matters

The Sherwood number translates complex mass-transfer physics into actionable design rules. By reporting Sh alongside Re and Sc, engineers communicate how operating conditions, material properties, and geometry interact. This transparency speeds up troubleshooting—whether diagnosing underperforming scrubbers, scaling lab fermenters, or validating building ventilation retrofits.

As sustainability and public health priorities elevate the importance of mass transfer—from carbon capture to infection control—the Sherwood number remains essential. It provides a common language for multidisciplinary teams, enabling chemical engineers, microbiologists, and HVAC specialists to collaborate effectively using a shared framework rooted in similarity analysis.

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