The Sherwood number (Sh) (also called the mass transfer
Nusselt number
In thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductiv ...
) 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 convec ...
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 enco ...
(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 Reynolds may refer to:
Places Australia
*Hundred of Reynolds, a cadastral unit in South Australia
*Hundred of Reynolds (Northern Territory), a cadastral unit in the Northern Territory of Australia
United States
* Reynolds, Mendocino County, Calif ...
and
Schmidt
Schmidt may refer to:
* Schmidt (surname), including list of people with the surname
* Schmidt (singer) (born 1990), German pop and jazz singer
* Schmidt (lunar crater), a small lunar impact crater
* Schmidt (Martian crater), a List of craters on ...
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 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 convect ...
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 thermal fluid dynamics, the Nusselt number (, after Wilhelm Nusselt) is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection (fluid motion) and diffusion (conduction). The conductiv ...
in terms of the
Reynolds number 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 = \frac ...
. 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 convect ...
, 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