Schmidt number

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

The Schmidt number is the ratio of the shear component for diffusivity viscosity/density to the diffusivity for mass transfer D. It physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer.[1]

It is defined[2] as:

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

where:

• ${\displaystyle \nu }$ is the kinematic viscosity or (${\displaystyle {\mu }}$/${\displaystyle {\rho }\,}$) in units of (m²/s)
• ${\displaystyle D}$ is the mass diffusivity (m²/s).
• ${\displaystyle {\mu }}$ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/m·s)
• ${\displaystyle \rho }$ is the density of the fluid (kg/m³).

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

Turbulent 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] An assessment of the existing literature on the subject still indicates significant uncertainty concerning the correct specification of this variable.[8] Stemming from the experimental and numerical evidence of its local variability, a new formulation for the turbulent Schmidt number, consisting in computing it locally, was proposed.[9] Through the latter, directly depending on the strain-rate and the vorticity invariants, a stronger relation between the concentration and the turbulence fields was finally ensured.[9]

Stirling engines

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.[10][11]

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

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.

References

1. tec-science (2020-05-10). "Schmidt number". tec-science. Retrieved 2020-06-25.
2. Incropera, Frank P.; DeWitt, David P. (1990), Fundamentals of Heat and Mass Transfer (3rd ed.), John Wiley & Sons, p. 345, ISBN 978-0-471-51729-0 Eq. 6.71.
3. Brethouwer, G. (2005). "The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. Linear theory and direct numerical simulation". J. Fluid Mech. 542: 305–342. Bibcode:2005JFM...542..305B. doi:10.1017/s0022112005006427.
4. Colli, A. N.; Bisang, J. M. (January 2018). "A CFD Study with Analytical and Experimental Validation of Laminar and Turbulent Mass-Transfer in Electrochemical Reactors". Journal of the Electrochemical Society. 165 (2): E81–E88. doi:10.1149/2.0971802jes.
5. Colli, A. N.; Bisang, J. M. (July 2019). "Time-dependent mass-transfer behaviour under laminar and turbulent flow conditions in rotating electrodes: A CFD study with analytical and experimental validation". International Journal of Heat and Mass Transfer. 137: 835–846. doi:10.1016/j.ijheatmasstransfer.2019.03.152.
6. Colli, A. N.; Bisang, J. M. (January 2020). "Coupling k Convection-Diffusion and Laplace Equations in an Open-Source CFD Model for Tertiary Current Distribution Calculations". Journal of the Electrochemical Society. 167: 013513. doi:10.1149/2.0132001JES.
7. Contigiani, C. C.; Colli, A. N.; González Pérez, O.; Bisang, J. M. (April 2020). "The Effect of a Conical Inner Electrode on the Mass-transfer Behavior in a Cylindrical Electrochemical Reactor under Single-Phase and Two-Phase (Gas-Liquid) Swirling Flow". Journal of the Electrochemical Society. 167 (8): 083501. doi:10.1149/1945-7111/ab8477.
8. Longo, R.; Fürst, M.; Bellemans, A.; Ferrarotti, M.; Deurdi, M.; Parente, A. (May 2019). "CFD dispersion study based on a variable Schmidt formulation for flows around different configurations of ground-mounted buildings" (PDF). Building and Environment. 154: 336–347. doi:10.1016/j.buildenv.2019.02.041.
9. Longo, R.; Bellemans, A.; Deurdi, M.; Parente, A. (2020). "A multi-fidelity framework for the estimation of the turbulent Schmidt number in the simulation of atmospheric dispersion". Building and Environment. 185: 107066. doi:10.1016/j.buildenv.2020.107066.
10. Schmidt Analysis (updated 12/05/07) Archived 2008-05-18 at the Wayback Machine
11. http://mac6.ma.psu.edu/stirling/simulations/isothermal/schmidt.html

Notes