Schmidt number

The turbulent Schmidt number is commonly used in turbulence research and is defined as:[3]

${\displaystyle \mathrm {Sc} _{\mathrm {t} }={\frac {\nu _{\mathrm {t} }}{K}}}$

where:

• ${\displaystyle \nu _{\mathrm {t} }}$ is the eddy viscosity in units of (m²/s)
• ${\displaystyle K}$ is the eddy diffusivity (m²/s).

The turbulent Schmidt number describes the ratio between the rates of turbulent transport of momentum and the turbulent transport of mass (or any passive scalar). It is related to the turbulent Prandtl number which is concerned with turbulent heat transfer rather than turbulent mass transfer. It is useful for solving the mass transfer problem of turbulent boundary layer flows. The simplest model for Sct is the Reynolds analogy, which yields a turbulent Schmidt number of 1. From experimental data and CFD simulations, Sct ranges from 0.2 to 0.5.[4][5][6][7]

For Stirling engines, the Schmidt number is related to the specific power. Gustav Schmidt of the German Polytechnic Institute of Prague published an analysis in 1871 for the now-famous closed-form solution for an idealized isothermal Stirling engine model.[8][9]

${\displaystyle \mathrm {Sc} ={\frac {\sum {\left|{Q}\right|}}{{\bar {p}}V_{sw}}}}$

where,

• ${\displaystyle \mathrm {Sc} }$ is the Schmidt number
• ${\displaystyle Q}$ is the heat transferred into the working fluid
• ${\displaystyle {\bar {p}}}$ is the mean pressure of the working fluid
• ${\displaystyle V_{sw}}$ is the volume swept by the piston.