Editing Archimedes number

{{Short description|Dimensionless number in fluid dynamics}}
{{distinguish|Archimedes constant}}
In viscous [[fluid dynamics]], the '''Archimedes number''' ('''Ar'''), is a [[dimensionless number]] used to determine the motion of [[fluid]]s due to [[density]] differences, named after the ancient Greek scientist and mathematician [[Archimedes]].

It is the ratio of  gravitational forces to viscous forces<ref>{{Cite book|title=Handbook of Solvents, Volume 2 - Use, Health, and Environment|last=Wypych|first=George|publisher=ChemTec Publishing|year=2014|edition=2nd|pages=657}}</ref> and has the form:<ref name=":1">{{Cite book|title=Mixing in the Process Industries|last1=Harnby|first1=N|last2=Edwards|first2=MF|last3=Nienow|first3=AW|publisher=Elsevier|year=1992|edition=2nd|pages=64}}</ref>

:<math>\begin{align}\mathrm{Ar} & = \frac{g L^3 \frac{\rho - \rho_\ell}{\rho_\ell}}{\nu^2} \\
& = \frac{g L^3 \rho_\ell (\rho - \rho_\ell)}{\mu^2} \\
\end{align}</math>

where:

* <math>g</math> is the local external field (for example [[gravitational acceleration]]), {{math|m/s<sup>2</sup>}},
* <math>L</math> is the [[characteristic length]] of body, {{math|m}}.
* <math>\frac{\rho - \rho_\ell}{\rho_\ell}</math> is the [[submerged specific gravity]],
* <math>\rho_\ell</math> is the [[density]] of the fluid, {{math|kg/m<sup>3</sup>}},
* <math>\rho</math> is the density of the body, {{math|kg/m<sup>3</sup>}},
* <math>\nu = \frac{\mu}{\rho_\ell}</math> is the [[kinematic viscosity]], {{math|m<sup>2</sup>/s}},
* <math>\mu</math> is the [[dynamic viscosity]], {{math|Pa·s}},

== Uses ==
The Archimedes number is generally used in design of tubular [[Chemical reactor|chemical process reactors]]. The following are non-exhaustive examples of using the Archimedes number in reactor design.

=== Packed-bed fluidization design ===
The Archimedes number is applied often in the engineering of [[packed bed]]s, which are very common in the chemical processing industry.<ref name=":0">{{Cite book|title=Chemical Reactor Design, Optimization, and Scaleup|last=Nauman|first=E. Bruce|publisher=John Wiley & Sons|year=2008|edition=2nd|pages=324}}</ref> A packed bed reactor, which is similar to the ideal [[plug flow reactor model]], involves packing a tubular [[Chemical reactor|reactor]] with a [[solid]] [[Catalysis|catalyst]], then passing [[Incompressible flow|incompressible]] or [[Compressible flow|compressible]] [[fluid]]s through the solid bed.<ref name=":0" /> When the solid particles are small, they may be "fluidized", so that they act as if they were a fluid. When fluidizing a packed bed, the pressure of the [[working fluid]] is increased until the [[pressure drop]] between the bottom of the bed (where fluid enters) and the top of the bed (where fluid leaves) is equal to the weight of the packed solids. At this point, the [[velocity]] of the fluid is just not enough to achieve fluidization, and extra pressure is required to overcome the [[friction]] of particles with each other and the wall of the reactor, allowing fluidization to occur. This gives a minimum fluidization velocity, <math>u_{mf}</math>, that may be estimated by:<ref name=":1" /><ref>{{Cite book|title=Multiphase Catalytic Reactors - Theory, Design, Manufacturing, and Applications|last1=Önsan|first1=Zeynep Ilsen|last2=Avci|first2=Ahmet Kerim|publisher=John Wiley & Sons|year=2016|pages=83}}</ref>

:<math>u_{mf}=\frac{\mu}{\rho_ld_v}\left((33.7^2+0.0408\text{Ar})^\frac{1}{2}-33.7\right)</math>

where:
* <math>d_v</math> is the diameter of sphere with the same volume as the solid particle and can often be estimated as <math>d_v\approx 1.13d_p</math>, where <math>d_p</math> is the diameter of the particle.<ref name=":1" />

=== Bubble column design ===
Another use is in the estimation of gas holdup in a [[Bubble column reactor|bubble column]]. In a bubble column, the gas holdup (fraction of a bubble column that is gas at a given time) can be estimated by:<ref>{{Cite journal|last1=Feng|first1=Dan|last2=Ferrasse|first2=Jean-Henry|last3=Soric|first3=Audrey|last4=Boutin|first4=Olivier|date=April 2019|title=Bubble characterization and gas–liquid interfacial area in two phase gas–liquid system in bubble column at low Reynolds number and high temperature and pressure|journal=Chem Eng Res Des|volume=144|pages=95–106|doi=10.1016/j.cherd.2019.02.001 |s2cid=104422302 |doi-access=free|bibcode=2019CERD..144...95F }}</ref>

:<math>\varepsilon_g=b_1\left[\text{Eo}^{b2}\text{Ar}^{b3}\text{Fr}^{b4}\left(\frac{d_r}{D}\right)^{b5}\right]^{b6}</math>

where:

*<math>\varepsilon_g</math> is the gas holdup fraction
*<math>\text{Eo}</math> is the [[Eötvös number|Eötvos number]]
* <math>\text{Fr}</math> is the [[Froude number]]
* <math>d_r</math> is the diameter of holes in the column's [[Bubble column reactor|spargers]] (holed discs that emit bubbles)
* <math>D</math> is the column diameter
* Parameters <math>b1</math> to <math>b6</math> are found empirically

=== Spouted-bed minimum spouting velocity design ===
A [[spouted bed]] is used in drying and coating. It involves spraying a liquid into a bed packed with the solid to be coated. A fluidizing gas fed from the bottom of the bed causes a spout, which causes the solids to circle linearly around the liquid.<ref>{{Cite book|title=Fluidization, Solids Handling, and Processing - Industrial Applications|last=Yang|first=W-C|publisher=William Andrew Publishing/Noyes|year=1998|pages=335}}</ref> Work has been undertaken to model the minimum velocity of gas required for spouting in a spouted bed, including the use of [[artificial neural network]]s. Testing with such models found that Archimedes number is a parameter that has a very large effect on the minimum spouting velocity.<ref>{{Cite journal|last1=Hosseini|first1=SH|last2=Rezaei|first2=MJ|last3=Bag-Mohammadi|first3=M|last4=Altzibar|first4=H|last5=Olazar|first5=M|date=October 2018|title=Smart models to predict the minimum spouting velocity of conical spouted beds with non-porous draft tube|journal=Chem Eng Res Des|volume=138|pages=331–340|doi=10.1016/j.cherd.2018.08.034 |bibcode=2018CERD..138..331H |s2cid=105461210 }}</ref>

==See also==
*[[Stokes flow|Viscous fluid dynamics]]
*[[Convection]]
*[[Convection (heat transfer)]]
*[[Dimensionless quantity]]
*[[Galilei number]]
*[[Grashof number]]
*[[Reynolds number]]
*[[Froude number]]
*[[Eötvös number]]
*[[Sherwood number]]

== References ==
<references />

{{NonDimFluMech}}

{{DEFAULTSORT:Archimedes Number}}
[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Fluid dynamics]]