6. Agitation

Scale-up Criteria

• Scale-up relation, between diameter of vessel (or diameter of impeller) and rpm of stirrer: \[\frac{n_2}{n_1} = \left(\frac{D_1}{D_2}\right)^x\]

• Scale-up criteria for equal power per unit volume: \(P \propto n^3D_a^5\)  and  \(V \propto D^3\)  (for tank with \(H=D\)). For geometric similarity \(D_{a1}/D_{a2} = D_1/D_2\); where \(D\) is the diameter of vessel. Therefore, \(V\propto D_a^3\). This gives, \(P/V \propto n^3D_a^2\). For equal \(P/V\), \[\frac{n_2}{n_1} = \left(\frac{D_{a1}}{D_{a2}}\right)^{2/3}\]

• Scale-up criteria for constant tip speed: \(nD_a=\text{constant}\)

• Scale-up criteria based on various requirements, are given below in the table:

  Operation	Scale-up exponent (\(x\))
  Blending time (\(1/n\))	0
  Surface behavior (same Froude number)	1/2
  Mass transfer, droplet size (equivalent power per unit volume, which results in equivalent interfacial area per unit volume)	2/3
  Solids suspension	3/4
  Constant tip velocities (equal liquid motion)	1

  
  (This value of \(x\) is based on empirical and theoretical considerations. Ref: Harnby - Mixing in the Process Industries)

Typical Power Consumption for Stirred Tanks

The data given below provide a rough idea about the magnitude of the power requirement for certain duties.