Schmidt number
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| Name | Schmidt number |
| Quantity | transport coefficient ratio |
Schmidt number The Schmidt number is a dimensionless quantity used in fluid dynamics and transport phenomena to characterize the relative effectiveness of momentum diffusion and mass diffusion in a fluid. It appears in analyses of boundary layers, convection, and mixing in contexts ranging from chemical reactors to atmospheric and oceanic flows. Prominent applications link to engineering practice in Royal Society of Chemistry, process design at DuPont, environmental studies at the National Oceanic and Atmospheric Administration, and standards referenced by organizations such as International Organization for Standardization.
Definition
The Schmidt number is defined as the ratio of kinematic viscosity to mass diffusivity, comparing the rate at which momentum is transported by viscous stresses to the rate at which a scalar species is transported by molecular diffusion. It is invoked in boundary-layer similarity solutions developed in the tradition of work by Ludwig Prandtl and practical correlations used by engineers at General Electric and researchers at Massachusetts Institute of Technology. In industrial and environmental contexts studied by institutions like United States Geological Survey and California Institute of Technology, Sc informs predictions of plume dispersion, sediment transport, and contaminant mixing.
Mathematical Formulation
Mathematically, the Schmidt number is expressed using kinematic viscosity and species diffusivity parameters frequently tabulated in handbooks used at American Society of Mechanical Engineers meetings and in textbooks from Cambridge University Press. The standard formulation involves the kinematic viscosity ν and mass diffusivity D, quantities measured and modeled in laboratories at Max Planck Society institutes and university facilities such as University of Cambridge and Imperial College London.
Physical Significance and Applications
Sc encapsulates the competition between viscous momentum transport and molecular mass transport in flows encountered in apparatus designed by companies like BASF and in atmospheric research by National Aeronautics and Space Administration. Low Schmidt numbers indicate mass diffuses more rapidly than momentum, relevant to gas dynamics studied at CERN and combustion systems researched at Sandia National Laboratories. High Schmidt numbers arise in liquid-phase mass transfer problems common in water treatment plants overseen by agencies like United States Environmental Protection Agency and in oceanographic mixing analyzed by teams at Scripps Institution of Oceanography.
Typical Values and Examples
Typical Schmidt numbers vary widely; gases often have Sc near unity as reported in compilations used at American Chemical Society conferences, while liquids such as aqueous electrolytes can exhibit Sc in the hundreds to thousands, a regime investigated in experiments at Lawrence Berkeley National Laboratory and industrial research at Shell plc. For vapor and atmospheric trace gases monitored by European Space Agency, Sc values influence dispersion modeling; for polymers processed at Bayer, high Sc regimes affect mass transfer in melts.
Measurement and Estimation Methods
Measurement techniques for kinematic viscosity and mass diffusivity used to compute Sc are standardized in test protocols from institutions such as American National Standards Institute and enacted in labs at Brookhaven National Laboratory. Methods include falling-sphere viscometry referenced in studies at ETH Zurich and tracer diffusion experiments akin to work at Johns Hopkins University. Empirical and semi-empirical correlations used by engineers at Siemens and researchers at Princeton University combine measured properties with temperature and pressure dependencies tabulated in sources like publications from Wiley-Blackwell.
Relation to Other Dimensionless Numbers
The Schmidt number relates closely to other dimensionless groups used throughout engineering and geophysics, often appearing alongside the Reynolds number and the Sherwood number in similarity analyses and scaling laws derived in the lineage of Henri Bénard and Osborne Reynolds. It is complementary to the Prandtl number and inversely analogous to parameters used in heat and mass transfer models discussed at forums like International Union of Theoretical and Applied Mechanics conferences. Combined correlations employing Sc and Re are ubiquitous in design guides published by American Institute of Chemical Engineers.
Historical Background and Named After Carl Schmidt
The designation honors the 19th-century chemist and physiologist Carl Schmidt, whose experimental and theoretical work on diffusion and solution chemistry influenced later transport theory developments that intersected with the contributions of contemporaries such as Wilhelm Ostwald and Jacobus Henricus van 't Hoff. Subsequent formalization of the Schmidt number in fluid mechanics drew on foundational concepts propagated through institutions like University of Göttingen and later synthesized in texts from Oxford University Press.
Category:Dimensionless numbers Category:Fluid dynamics