Power per volume

More directly related to turbulence and motion at the interface. Includes scale-up for rate of dissolving of solids or mass transfer between liquid phases. Using geometric similarity and equal power per volume results in same n value. 

The factor of 4 increase in (mj) allows Pg/Vi)2 to decrease to 0.3(Eg/F/),. This suggests that the installed horsepower for the full-scale plant would be about a third of that calculated for a conservative scaleup with constant power per volume. This wiU have a major impact on cost and is too large to ignore. The engineer can do any of the following ... 

The fact that the kinetics at the same power per volume are also independent of the reactor size, i. e. of the number of circuits made by the particles, points to... 

Correlation versus power per volume are common for dispersions... 

Subsequently, if power per volume is held constant in two different size systems, the agitator speed must change in relation to the impeller diameter ... 

Note that Pmo/Df) represents the power per volume because the liquid volume is proportional to Df for the geometrically similar vessels. For the constant Pmo/Df,... 

For homogeneous chemical reactions, the power per volume can be used as a scale-up criterion. As a rule of thumb, the intensity of agitation can be classified based on the power input per 1,000 gallon as shown in Table 9.4... 

For the scale-up of the gas-liquid contactor, the volumetric mass-transfer coefficient kLa can be used as a scale-up criterion. In general, the volumetric mass-transfer coefficient is approximately correlated to the power per volume. Therefore, constant power per volume can mean a constant kLa. 

Related Calculations. The majority of gas-dispersion applications are sized on the basis of power per volume. In aerobic fermentation, levels of 5 to 12 hp per 1000 gal (1 to 2.4 kW/m3) are typical, while for aerobic waste treatment, levels of 1 to 3 hp per 1000 gal (0.2 to 0.6 kW/m3) are more common, primarily because of the concentrations and oxygen requirements of the microorganisms. For more on fermentation, see Section 17. 

For equal tip speed and impellers with geometric similarity, the rotational speed remains the same in spite of the added liquid level. Thus, the power and torque remain the same for the same impellers, speed, and fluid properties. However, the volume change affects power per volume 1000(2.66 hp)/7000gal = 0.38 hp/1000 gal (75.0 W/m3). Similarly, the torque per volume is reduced to 1000(3193 in-lb)/7000 gal = 456 in-lb/1000 gal (13.6 N-m/m3). Both changes represent a substantial reduction in agitation intensity from the pilot-scale results. 

These results are consistent with a scaleup from the pilot-scale operation. Although power per volume is reduced and impeller to tank diameter ratio is increased, torque per volume is about half and tip speed is the same. With any realistic scaleup, some factors unavoidably must change, while the important ones are held constant. In a different situation and process, a different variable, such as power per volume, might be held constant. Other variables, such as blend time, could be calculated, at each step in the scaleup. 

It is difficult to predict from fundamental considerations that constant power per unit volume should be a generally significant scale-up criterion. In fact, as Rushton (R8) shows, use of equal power per volume for scaling can result in serious error in many cases. Thus, the successful application of this concept to certain operations must be regarded as a somewhat fortuitous result of the specific interactions present in those particular cases. 

With geometric similarity, equal tip speed means that velocity gradients are reduced and blend times become longer. However, power per volume is also reduced, and viscous heating problems are likely to be more controllable. With any geometric scale-up, the surface-to-volume ratio is reduced, which means that any internal heating, whether by viscous dissipation or chemical reaction, becomes more difficult to remove through the surface of the vessel. 

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