in which θm is the mixing time, V is the liquid volume, N is the agitation speed, D is the impeller diameter, NQ is the impeller flow number (provided by the impeller manufacturer), and k is a constant related to the number of volume
turnovers required to achieve homogeneity. Typically, five volume turnovers are assumed to result in homogeneity, but the
actual impeller pumping rate is higher than the impeller discharge rate because of momentum transfer to the surrounding fluid.
Therefore a multiplier, k, of three is often used. More recently, it has been suggested, based on turbulence theory, that
the mixing time should be independent of impeller type9
in which T is the tank diameter (m), D is the impeller diameter (m), P is the impeller power input (W), V is the liquid volume (m3), and ρ is the fluid density (kg/m3). More accurate mixing times can be calculated using more complex models, for example, those based on CFD as described earlier.
Shear rate is another agitation-related parameter that is often evaluated, especially for cell culture applications. Because
mammalian cells lack a cell wall, they are more susceptible to shear damage than microbial cells. Impeller tip speeds can
be correlated to maximum impeller shear rates, and therefore, constant tip speed has been suggested as a scaling criterion
for mammalian cell culture agitation.10 However, it has been shown by multiple investigators that impeller shear rates commonly used for cell culture applications
are orders of magnitude below the shear rates required to cause cell damage.11 Therefore, use of tip speed as a primary agitation scaling parameter is not recommended.
Power input (P/V)or mean specific energy dissipation rate is a parameter that is more commonly used for scaling agitation
across sites or scales. The mean energy dissipation can be calculated from:
in which ε-meanT is the mean energy dissipation rate (W/kg), P is the power input (W), V is the liquid volume (m3) and Np is the impeller power number. Mean impeller energy dissipation rates typically used for mammalian cell cultures range from
0.001 to 0.050 W/kg (1 to 50 W/m3).10 A consequence of scaling agitation at constant energy dissipation is that the length scale of turbulent eddies is expected
to be constant. The Kolmogorov microscale of turbulence is related to energy dissipation through the following equation:
in which λ is the Kolmogorov length scale (cm), υ is the kinematic viscosity (cm2/s) and ε-meanT is expressed as cm2/s3.