The Sherwood number (Sh) (also called the mass transfer
Nusselt number) is a
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
used in mass-transfer operation. It represents the ratio of the
convective
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the ...
mass transfer to the rate of
diffusive mass transport,
and is named in honor of
Thomas Kilgore Sherwood
Thomas Kilgore Sherwood (July 25, 1903 – January 14, 1976) was a noted American chemical engineer and a founding member of the National Academy of Engineering.
Biography
Sherwood was born in Columbus, Ohio, and spent much of his youth in ...
.
It is defined as follows
:
where
* ''L'' is a characteristic length (m)
* ''D'' is
mass diffusivity
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...
(m
2 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:
:
For example, for a single sphere it can be expressed as:
:
where
is the Sherwood number due only to natural convection and not forced convection.
A more specific correlation is the Froessling equation:
:
This form is applicable to molecular diffusion from a single spherical particle. It is particularly valuable in situations where the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
and
Schmidt number
Schmidt number (Sc) 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 convec ...
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
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
and
Prandtl number
The Prandtl number (Pr) or Prandtl group is a dimensionless number, named after the German physicist Ludwig Prandtl, defined as the ratio of momentum diffusivity to thermal diffusivity. The Prandtl number is given as:
: \mathrm = \frac = \fr ...
. 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
Schmidt number (Sc) 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 convec ...
, 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.
See also
*
Churchill–Bernstein equation In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. The need for the equation arises from the inability to solve the Navier–St ...
References
{{Dimensionless numbers in fluid mechanics
Dimensionless numbers
Dimensionless numbers of fluid mechanics