Influid dynamics, theSchmidt number (denotedSc) of afluid is adimensionless number defined as theratio ofmomentum diffusivity (kinematic viscosity) andmass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusionconvection processes. It was named after German engineerErnst Heinrich Wilhelm Schmidt (1892–1975).

The Schmidt number is the ratio of theshear component for diffusivity (viscosity divided bydensity) to the diffusivity for mass transferD. It physically relates the relative thickness of the hydrodynamic layer and mass-transferboundary layer.^[1]

It is defined^[2] as:

${\displaystyle \mathrm{S}\mathrm{c}=\frac{\nu }{D}=\frac{\mu }{\rho D}=\frac{\text{viscous diffusion rate}}{\text{molecular (mass) diffusion rate}}=\frac{\mathrm{P}\mathrm{e}}{\mathrm{R}\mathrm{e}}}$

where (inSI units):

• ${\displaystyle \nu =\frac{\mu }{\rho }}$ is thekinematic viscosity (m²/s)
• D is themass diffusivity (m²/s).
• μ is thedynamic viscosity of the fluid (Pa·s = N·s/m² = kg/m·s)
• ρ is thedensity of the fluid (kg/m³)
• Pe is thePeclet Number
• Re is theReynolds Number.

Theheat transfer analog of the Schmidt number is thePrandtl number (Pr). The ratio ofthermal diffusivity tomass diffusivity is theLewis number (Le).

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

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

where:

• ${\displaystyle {\nu }_{\mathrm{t}}}$ is theeddy viscosity in units of (m²/s)
• ${\displaystyle K}$ is theeddy 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 theturbulent 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 6.^{[4][5][6][7][8]}

ForStirling engines, the Schmidt number is related to thespecific power.Gustav Schmidt of the German Polytechnic Institute of Prague published an analysis in 1871 for the now-famousclosed-form solution for an idealized isothermal Stirling engine model.^[9][10]

${\displaystyle \mathrm{S}\mathrm{c}=\frac{\sum \left|Q\right|}{\overline{p}{V}_{sw}}}$

where:

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

• Dimensionless numbers of fluid mechanics
• Dimensionless numbers of thermodynamics
• Fluid dynamics

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