Schmidt number

 ( Dimensionless Numbers Of Fluid Mechanics ) 

In fluid dynamics, the Schmidt number (denoted ) of a fluid is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. It was named after German engineer Ernst Heinrich Wilhelm Schmidt (1892–1975).

The Schmidt number is the ratio of the Shear stress for diffusivity (viscosity divided by density) to the diffusivity for mass transfer . It physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer.

It is defined Eq. 6.71. as:

$\backslash mathrm\{Sc\}\; =\; \backslash frac\{\backslash nu\}\{D\}\; =\; \backslash frac\; \{\backslash mu\}\; \{\backslash rho\; D\}\; =\; \backslash frac\{\; \backslash mbox\{viscous\; diffusion\; rate\}\; \}\{\; \backslash mbox\{molecular\; (mass)\; diffusion\; rate\}\; \}\; =\; \backslash frac\{\backslash mathrm\{Pe\}\}\{\backslash mathrm\{Re\}\}$

where (in SI units):

• $\backslash nu\; =\; \backslash tfrac\; \backslash mu\; \backslash rho$ is the kinematic viscosity (m²/s)
• is the mass diffusivity (m²/s).
• is the dynamic viscosity of the fluid (Pa·s = N·s/m² = kg/m·s)
• is the density of the fluid (kg/m³)
• is the Peclet Number
• is the Reynolds Number.

The heat transfer analog of the Schmidt number is the Prandtl number (). The ratio of thermal diffusivity to mass diffusivity is the Lewis number ().

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Turbulent Schmidt Number The turbulent Schmidt number is commonly used in turbulence research and is defined as:

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

where:

• $\backslash nu\_\backslash mathrm\{t\}$ is the eddy viscosity in units of (m²/s)
• $K$ is the Eddy diffusion (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 6.

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Stirling engines For , the Schmidt number is related to the Power density. 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. Schmidt Analysis (updated 12/05/07)

$\backslash mathrm\{Sc\}\; =\; \backslash frac\{\backslash sum\; \{\backslash left\; |\; \{Q\}\; \backslash right\; |\}\}\{\backslash bar\; p\; V\_\{sw\}\}$

where:

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

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