Archimedes number

Not to be confused with Archimedes constant.

In viscous fluid dynamics, the Archimedes number (Ar), is a dimensionless number used to determine the motion of fluids due to density differences, named after the ancient Greek scientist and mathematician Archimedes.

It is the ratio of gravitational forces to viscous forces ^[1] and has the form: ^[2]

${\displaystyle {\begin{aligned}\mathrm {Ar} &={\frac {gL^{3}{\frac {\rho -\rho _{\ell }}{\rho _{\ell }}}}{\nu ^{2}}}\\&={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}\\\end{aligned}}}$

where:

• ${\displaystyle g}$ is the local external field (for example gravitational acceleration), m/s²,
• ${\displaystyle L}$ is the characteristic length of body, m.
• ${\displaystyle {\frac {\rho -\rho _{\ell }}{\rho _{\ell }}}}$ is the submerged specific gravity,
• ${\displaystyle \rho _{\ell }}$ is the density of the fluid, kg/m³,
• ${\displaystyle \rho }$ is the density of the body, kg/m³,
• ${\displaystyle \nu ={\frac {\mu }{\rho _{\ell }}}}$ is the kinematic viscosity, m²/s,
• ${\displaystyle \mu }$ is the dynamic viscosity, Pa·s,

Uses

The Archimedes number is generally used in design of tubular 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 beds, which are very common in the chemical processing industry. ^[3] A packed bed reactor, which is similar to the ideal plug flow reactor model, involves packing a tubular reactor with a solid catalyst, then passing incompressible or compressible fluids through the solid bed. ^[3] 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, ${\displaystyle u_{mf}}$, that may be estimated by: ^{[2] [4]}

${\displaystyle u_{mf}={\frac {\mu }{\rho _{l}d_{v}}}\left((33.7^{2}+0.0408{\text{Ar}})^{\frac {1}{2}}-33.7\right)}$

where:

• ${\displaystyle d_{v}}$ is the diameter of sphere with the same volume as the solid particle and can often be estimated as ${\displaystyle d_{v}\approx 1.13d_{p}}$, where ${\displaystyle d_{p}}$ is the diameter of the particle. ^[2]

Bubble column design

Another use is in the estimation of gas holdup in a 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: ^[5]

${\displaystyle \varepsilon _{g}=b_{1}\left[{\text{Eo}}^{b2}{\text{Ar}}^{b3}{\text{Fr}}^{b4}\left({\frac {d_{r}}{D}}\right)^{b5}\right]^{b6}}$

where:

• ${\displaystyle \varepsilon _{g}}$ is the gas holdup fraction
• ${\displaystyle {\text{Eo}}}$ is the Eötvos number
• ${\displaystyle {\text{Fr}}}$ is the Froude number
• ${\displaystyle d_{r}}$ is the diameter of holes in the column's spargers (holed discs that emit bubbles)
• ${\displaystyle D}$ is the column diameter
• Parameters ${\displaystyle b1}$ to ${\displaystyle b6}$ 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. ^[6] Work has been undertaken to model the minimum velocity of gas required for spouting in a spouted bed, including the use of artificial neural networks. Testing with such models found that Archimedes number is a parameter that has a very large effect on the minimum spouting velocity. ^[7]

See also

• Viscous fluid dynamics
• Convection
• Convection (heat transfer)
• Dimensionless quantity
• Galilei number
• Grashof number
• Reynolds number
• Froude number
• Eötvös number
• Sherwood number

References

1. Wypych, George (2014). Handbook of Solvents, Volume 2 - Use, Health, and Environment (2nd ed.). ChemTec Publishing. p. 657.
2. Harnby, N; Edwards, MF; Nienow, AW (1992). Mixing in the Process Industries (2nd ed.). Elsevier. p. 64.
3. Nauman, E. Bruce (2008). Chemical Reactor Design, Optimization, and Scaleup (2nd ed.). John Wiley & Sons. p. 324.
4. Önsan, Zeynep Ilsen; Avci, Ahmet Kerim (2016). Multiphase Catalytic Reactors - Theory, Design, Manufacturing, and Applications. John Wiley & Sons. p. 83.
5. Feng, Dan; Ferrasse, Jean-Henry; Soric, Audrey; Boutin, Olivier (April 2019). "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". Chem Eng Res Des. 144: 95–106. doi: 10.1016/j.cherd.2019.02.001 . S2CID   104422302.
6. Yang, W-C (1998). Fluidization, Solids Handling, and Processing - Industrial Applications. William Andrew Publishing/Noyes. p. 335.
7. Hosseini, SH; Rezaei, MJ; Bag-Mohammadi, M; Altzibar, H; Olazar, M (October 2018). "Smart models to predict the minimum spouting velocity of conical spouted beds with non-porous draft tube". Chem Eng Res Des. 138: 331–340. doi:10.1016/j.cherd.2018.08.034. S2CID   105461210.