Sherwood number

{{Short description|Dimensionless number in fluid mechanics}}

{{Refimprove|date=August 2017}}

The Sherwood number (Sh) (also called the mass transfer Nusselt number) is a dimensionless number used in mass-transfer operation. It represents the ratio of the total mass transfer rate (convection + diffusion) to the rate of diffusive mass transport,{{cite book

|last=Heldman

|first=D.R.

|date=2003

|title=Encyclopedia of Agricultural, Food, and Biological Engineering

|publisher=Marcel Dekker Inc.

|isbn=0-8247-0938-1

}} and is named in honor of Thomas Kilgore Sherwood.

It is defined as follows

:$\backslash mathrm\{Sh\}\; =\; \backslash frac\{h\}\{D/L\}\; =\; \backslash frac\{\backslash mbox\{Total\; mass\; transfer\; rate\}\}\{\backslash mbox\{Diffusion\; rate\}\}$

where

• L is a characteristic length (m)
• D is mass diffusivity (m² s⁻¹)
• h is the convective mass transfer film coefficient (m s⁻¹)

Using dimensional analysis, it can also be further defined as a function of the Reynolds and Schmidt numbers:

:$\backslash mathrm\{Sh\}\; =\; f(\backslash mathrm\{Re\},\; \backslash mathrm\{Sc\})$

For example, for a single sphere it can be expressed as {{Citation needed|date=April 2023}}:

:$\backslash mathrm\{Sh\}\; =\; \backslash mathrm\{Sh\}\_0\; +\; C\backslash ,\; \backslash mathrm\{Re\}^\{m\}\backslash ,\; \backslash mathrm\{Sc\}^\{\backslash frac\{1\}\{3\}\}$

where $\backslash mathrm\{Sh\}\_0$ is the Sherwood number due only to natural convection and not forced convection.

A more specific correlation is the Froessling equation:Froessling, N. Uber die Verdunstung Fallender Tropfen (The Evaporation of Falling Drops). Gerlands Beitrage zur Geophysik, 52:107-216, 1938

:$\backslash mathrm\{Sh\}\; =\; 2\; +\; 0.552\backslash ,\; \backslash mathrm\{Re\}^\{\backslash frac\{1\}\{2\}\}\backslash ,\; \backslash mathrm\{Sc\}^\{\backslash frac\{1\}\{3\}\}$

This form is applicable to molecular diffusion from a single spherical particle. It is particularly valuable in situations where the Reynolds number and Schmidt number are readily available. Since Re and Sc are both dimensionless numbers, the Sherwood number is also dimensionless.

These correlations are the mass transfer analogies to heat transfer correlations of the Nusselt number in terms of the Reynolds number and Prandtl number. For a correlation for a given geometry (e.g. spheres, plates, cylinders, etc.), a heat transfer correlation (often more readily available from literature and experimental work, and easier to determine) for the Nusselt number (Nu) in terms of the Reynolds number (Re) and the Prandtl number (Pr) can be used as a mass transfer correlation by replacing the Prandtl number with the analogous dimensionless number for mass transfer, the Schmidt number, and replacing the Nusselt number with the analogous dimensionless number for mass transfer, the Sherwood number.

As an example, a heat transfer correlation for spheres is given by the Ranz-Marshall Correlation:Ranz, W. E. and Marshall, W. R. Evaporation from Drops. Chemical Engineering Progress, 48:141-146, 173-180, 1952.

:

This correlation can be made into a mass transfer correlation using the above procedure, which yields:

:

This is a very concrete way of demonstrating the analogies between different forms of transport phenomena.