Schmidt Number (Sc): Momentum-to-Mass Diffusivity Ratio
The Schmidt number (symbol Sc) is a dimensionless quantity defined as the ratio of momentum diffusivity to mass diffusivity: Sc = \u03bd / D, where \u03bd is kinematic viscosity (m²·s⁻¹) and D is the binary mass diffusivity (m²·s⁻¹). High Sc values indicate that momentum diffuses much faster than mass, while low Sc values signal the opposite. Sc links fluid properties to convective mass transfer and is central to similarity analysis in chemical engineering and environmental flows.
Definition and Units
Because \u03bd and D share the same units, Sc is dimensionless. Typical gases at ambient conditions have Sc near 0.7–1.0 because molecular diffusion and momentum diffusion proceed at similar rates. Liquids usually exhibit Sc values from hundreds to thousands due to much lower molecular diffusivities. Using SI units for \u03bd (m²·s⁻¹) and D (m²·s⁻¹) keeps Sc calculations consistent across datasets and correlations.
Engineers often compute \u03bd from dynamic viscosity \u03bc (Pa·s) and density \u03c1 (kg·m⁻³) via \u03bd = \u03bc/\u03c1. Diffusivities come from property databases or measurement, sometimes estimated with correlations such as the Wilke–Chang or Fuller methods. Recording temperature, pressure, and composition alongside Sc maintains traceability for audits and design reviews.
Historical Background
The Schmidt number is named after Ernst Schmidt, a German engineer who, along with Ludwig Prandtl and others in the early twentieth century, advanced boundary-layer theory. Their work linked the mathematics of heat, mass, and momentum transfer through analogies that remain staples of transport curricula. As chemical engineering matured, Sc emerged as a key parameter for correlating mass-transfer coefficients across equipment and scales.
Standardisation efforts, including ISO 80000-11 on characteristic numbers, codify the symbol Sc and its usage, ensuring consistency across disciplines from aerospace to wastewater treatment. Modern handbooks and computational fluid dynamics (CFD) software include Sc-dependent correlations as defaults for diffusive transport models.
Conceptual Foundations
Similarity and Boundary Layers
Sc enters directly into film and boundary-layer theories. For laminar flow over a flat plate, the ratio of concentration to velocity boundary-layer thickness scales as Sc¹ᐟ³. High Sc fluids develop thin concentration boundary layers relative to momentum boundary layers, increasing sensitivity to surface roughness and interfacial instabilities. These relationships allow engineers to adapt velocity profiles to predict mass-transfer rates.
Analogy with Heat Transfer
The Chilton–Colburn analogy extends heat-transfer correlations to mass transfer by substituting Sc for the Prandtl number. In turbulent pipe flow, a common relation is Sh = 0.023 Re⁰·⁸ Sc¹ᐟ³, mirroring the Dittus–Boelter equation for Nusselt numbers. Such analogies make Sc indispensable when designing absorbers, strippers, and humidifiers.
Applications and Importance
In gas absorption, Sc guides packing selection and solvent choice by indicating how quickly solutes diffuse relative to momentum. In multiphase reactors, matching Sc across pilot and full-scale units helps preserve interfacial renewal rates. Environmental engineers use Sc when modelling pollutant dispersion in water bodies, ensuring laboratory diffusion measurements translate to field conditions.
CFD practitioners specify Sc to tune turbulence models for scalar transport, while microfluidics designers rely on high Sc values to predict diffusion-limited mixing. Across these domains, documenting Sc alongside Reynolds and Sherwood numbers ensures reproducibility and regulatory compliance.
Working with Schmidt Numbers
When publishing or sharing results, cite the property sources for \u03bc, \u03c1, and D, and state whether Sc varies with temperature or composition across the domain of interest. Use the Reynolds calculator to pair Sc with flow conditions, and map the resulting Sh predictions to mass-transfer coefficients. Cross-linking to the Prandtl number explainer reinforces the analogy between mass and heat transfer.
Finally, retain significant figures and SI units throughout. Reporting Sc with two to three significant digits typically suffices, but note uncertainties when property data are estimated. Doing so keeps calculations transparent and aligns with ISO guidance on characteristic numbers.