Archimedes number

In viscous

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

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

References