Geometric Scaling Relationships

The simplified scaling relationships, Eq. (53), offer some flexibility in the model design since fewer parameters must be matched than with the full set of scaling relationships. When the fluidizing gas, the pressure and temperature of the scale model are chosen, the gas density and viscosity for the scale model are set. The model must still be geometrically similar to the commercial bed. There is still one free parameter. Generally this will be the linear scale of the model. For the simplified scaling relationships, the gas-to-solid density ratio must be maintained constant... 

Fig. 10.10 Rejection mask used in Visual Empirical Region of Influence (VERI). The shape of the mask is mathematically derived from the visual cognitive factors (a) geometrical scaling relationships used in the derivation of the shape (b) final mask which defines the exclusion region, that is, objects that do not belong to the group (Osbourn and Martinez, 1995)...

The simplified scaling relationships were used by Glicksman et al. (1993b) to compare two geometrically... 

If the scale ratio is not the same along each axis, the relationship among the two bodies is of a distorted geometric similarity and the axial relationships are given by... 

Thus, in some cases, such as scaling-up of geometrically similar stirred tanks, the estimation of power requirement can be simplified using the above relationship. 

The third factors to control and keep constant are the gas pressure and superficial gas velocity. This probably will involve gas recirculation with either a small compressor, or through a hollow shaft or some other pumping device. As seen before, the bubble diameter, the mass transfer area, the gas hold-up, and the terminal bubble-rise velocity, all depend on the superficial velocity of the gas and the power input per unit volume. When these are kept constant, the various mass transfer resistances in the pilot plant and in the large unit will be the same, hence the global rate will be conserved. The last factor is the input power to the agitator. As required for mass transfer, the scale-up must be made on the basis of constant power input per unit volume. If turbulent conditions and geometrical similarity prevail, this rule imposes the following relationship ... 

Two screw profiles that are described by the same geometric parameters are known as geometrically similar. They can be converted into one another by scaling. The pressure and power numbers are dependent only on the throughput number if the screws are geometrically similar. The relationships are linear in the case of fluids of constant viscosity ( Newtonian fluids ). The complete pressure and energy behavior of a class of geometrically similar screws can thus be traced back to a simple linear relationship. 

This statement is supported by the results shown in Fig. 1. The researchers carried out their measurements in smooth pipes with diameters d = 0.36 - 12.63 cm, thereby changing the scale in the range of 1 35. Furthermore, the physical properties of the fluid tested (water or air) varied widely. Nevertheless, the relationship i (Re) did not display this change Every numerical value of Re still corresponded to a specific numerical value of The pi-space is scale-independent, it is scale invariant The pi-relationship presented is therefore valid not only for the examined laboratory devices but also for any other geometrically similar arrangement ... 

Apart from the estimation of df from experimental data for mass, density, and purely geometric characteristics, the calculation of df for a plethora of studies dealing with various characteristics like frequency, electrical conductivity, and intensity of light is also based on the exact relationship that is applicable in each case between df and the scaling exponent a, (1.9). 

Since impeller diameter is established by geometric similarity, the agitator speed for the large-scale equipment must be determined to satisfy process requirements. Large-scale speed NL can be computed by the following relationship ... 

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