Better understanding boosts mixer scale-up

Experience and trials still play a crucial role for rotor/stator devices

By by Chris Ryan and Niraj Thapar, Silverson Machines Ltd.

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Scale-up calculations

 

Figure 3. This unit features fine emulsor screens with circular holes but a variety of other hole types are available for other applications.
Where scale-up is concerned, different manufacturers cite certain criteria as the basis for calculations but, as we’ve noted, the complexity of devices such as rotor/stator mixers makes it more realistic to combine a number of these factors. With the benefit of practical experience, the right combination can give an accurate scale-up recommendation. Traditionally geometric similarity and tank turnover/throughput formed the basis for many calculations. In light of recent research, additional parameters generally considered to provide reliable results include energy dissipation (that is, power per unit volume or power draw), rotor tip speed and shear frequency (residence time, also linked to open area or mixer geometry).

Energy dissipation. In a situation where vessel and mixer geometry are similar across a range of sizes, energy dissipation is considered to be constant and, therefore, a useful tool in scale-up calculations. Power has to be monitored while mixing to determine the effects of the fluid’s physical properties on power draw. The most reliable way of doing this is to incorporate a torque transducer onto the mixer’s shaft to measure the total energy, ε, supplied to the mixture. This is estimated by the formula:

ε = P fluid/V (1)

where V is the volume of the mix (m3) and PFluid is the total power expended to the fluid (W), calculated from torque data as P = 2πNT, with N being the rotational speed (rev/s) and T (N.m) the torque value.

Rotor tip speed. Vtip (m/s) can be used to assess the kinetic energy imparted to the process fluid from the tip of the rotor blade as well as in calculating the shear stress and rates imparted to the fluid:

Vtip = πND (2)

where N is the rotational speed (rev/s) and D is the rotor diameter (m).

Tip speed plays a major role in determining the shear rate in the gap between the rotor and stator, γgap (s-1). This is where shear rates are maximized, therefore quantifying the activity in this region is of considerable importance in predicting the performance of equivalent machines across a range of sizes in scale-up. We can estimate the value for the shear rate in the shear gap via:

γgap = Vtip/δ (3)

where δ is the shear gap (m).

The γgap will become more significant at higher viscosities because the velocity gradient will increase. At lower viscosities the flow patterns in the gap won’t exhibit simple shear flow and will become more turbulent — in which case turbulent energy dissipation may become more significant.

Although scale-up using values of shear rate and tip speed work well in some applications involving agitators and stirrers, they aren’t necessarily always applicable to rotor/stators. The reason is that the flow from the workhead is assumed to travel at the same speed as the rotor tip. It then undergoes a sudden slowdown as it impacts the stator wall; the losses in kinetic energy during this process (hydraulic shear) contribute to functions such as dispersion and emulsification. Therefore, the tip speed in combination with the rotor/stator geometry (and physical properties of the fluids) will have a more significant bearing on scale-up issues than tip speed alone. These factors combined form another parameter, called the shear frequency (sf), that is commonly employed in these calculations.

 

Figure 4. Titanium dioxide slurries from square hole high shear screen (red line)

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