Scale geometric similarity

In the normal case of a geometrically similar scale-up, it can be readily shown that the surface area per unit volume varies inversely with the vessel diameter. Thus larger vessels are more difficult to cool, since the heat generated by a reaction in a potential runaway situation is proportional to the vessel volume, whereas the surface area available to dissipate a given heat output is decreased. Vigorous reactions may require the reactor to be detuned by operating with more dilute feedstock in order to reduce the full-scale reaction intensity. This... 

Furthermore, it is dear that the scale-up of an injector inherently lessens its efficiency. This is caused by the fact that the dispersing effect of the liquid propulsion jet is restricted to its circumference which, in the case of geometrically similar scale-up, increases only linearly (u = Jtd) while its cross-sectional area increases quadratically (S = it d2/4). This means that with increasing diameter of the device, an increasingly smaller fraction of the liquid throughput is dispersed The dispersion efficiency of injectors inherently diminishes with increasing scale. 

The propellant charge is also geometrically similarly scaled r, = Sto, — Sr o, where is the radius of the charge, and r is the radius of the vessel. 

So time scales with the scaling factor. Without belaboring the point with more complicated equations, it can be shown that for scaled guns with scaled propellants as well as scaled bullets, the muzzle velocities would be equal. Geometrically similar scaled bullets with equal mass densities have the same drag coefficients. It will be shown later that the entire trajectory is scaled such that velocity loss is equal over scaled distances. 

Figure 9.20 The data show that the inlet jet Reynolds number (Re ) is a good scale up criterion for geometrically similar scaled CIJ mixers CIJ-d2 data are also shown to evidence that a variation of the inlet tube/chamber diameter ratio may influence the final particle size. Both PEGylated copolymer nanospheres and nanocapsules are shown, evidencing different dependence on Re. quenched samples (quench ratio = 1) in the upper graph, non-quenched in the lower graph, measured after solvent (acetone) evaporation. Nanocapsules (containing Miglyol , MR=1.26) , scale down , reference (CIJ-dl) , scale up A, CIJ-d2. Polymer nanospheres , scale down O, reference (ClJ-dl) O, scale up , CIJ-d2.

Having estabhshed the residence time and power input, the scale-up can Be now done using the principle of geometric similarity together with equal power per unit volume discussed earlier. 

Scale-Up of Continuous Mixers Although scaling up on the basis oFconstant power per unit Feed rate [kWh/lcg or (hp-h)/lb] is iisii-allv a good First estimate, several other Factors rnav have to be considered, As the equipment scale is increased, geometric similarity being... 

Scale-Up and Operation Scale-up of pharmaceutical mixer granulators is difficult because geometric similarity is often not preserved. Kristensen recommends constant relative swept volume ratio... 

As Fig. 8.3 shows, grain boundary nuclei will be geometrically similar at all temperatures. The volume of the lens-shaped nucleus will therefore scale as (r f, i.e. 

Figure 1.3.1a, b illustrates an idea proposed by the author (Berty 1974) the basis for geometric similarity should be what the catalyst sees on the inside, instead of the gross features of a reactor that can be observed from the outside. Identity as well as similarity can be achieved on various scales if the investigator bases the criteria on the direct surroundings of the catalyst where flow, partial pressures, and temperature all are important. 

Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups, which are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachines can be employed to (1) compare data from various types of machines—it is a useful technique in the development of blade passages and blade profiles, (2) select various types of units based on maximum efficiency and pressure head required, and (3) predict a prototype s performance from tests conducted on a smaller scale model or at lower speeds. 

Various methods of scale-up have been proposed all based on geometric similarity between the laboratory equipment and the full-scale plant. It is not always possible to have the large and small vessels geometrically similar, although it is perhaps the simplest to attain. If geometric similarity is achievable, dynamic and kinematic similarity cannot often be predicted at the same time. For these reasons, experience and judgment are relied on with aspects to scale-up. 

To scale-up a reactor from Vj to Vj with geometrically similar systems having similar bulk average temperatures (i.e., the physical properties of the fluids are identical), Equation 7-82 becomes... 

Because there are advantages in maintaining geometric similarity between the pilot plant and the full-scale plant reactors, the larger unit often has the same aspect ratio as the small unit. That is, Rj = Rj. Equation 13-27 then becomes... 

The requirement of identical dimensionless boundary conditions is met when the model is geometrically similar to full scale in all details that are important for the volume flow, the energy flow and the contaminant flow see Fig. 12.24. 

Rice, R.W. and Baud, R.E., 1990. The role of micromixing in the scale-up of geometrically similar batch reactors. American Institution of Chemical Engineers Journal, 36, 293-298. 

Figure 5-5Y. A-310 Impeller. Develops 50% more action than ordinary propellers and is geometrically similar for accurate scale-up. By permission, Lightnin, a unit of General Signal. (Formerly Mixing Equipment Co.)...

Geometric similarity is often considered the most important feature to establishing similarity in mixing, basing the scaled-up larger unit on the smaller initial model or test unit. 

The dynamic response used to describe fluid motion in the system is bulk velocity. Kinematic similarity exists with geom.etric similarity in turbulent agitation [32]. To duplicate a velocity in the kinematically similar system, the kno m velocity must be held constant, for example, the velocity of the tip speed of the impeller must be constant. Ultimately, the process result should be duplicated in the scaled-up design. Therefore, the geometric similarity goes a long way in achieving this for some processes, and the achievement of dynamic and/or kinematic similarity is sometimes not that essential. 

Because the most common impeller type is the turbine, most scale-up published studies have been devoted to that unit. Almost all scale-up situations require duplication of process results from the initial scale to the second scaled unit. Therefore, this is the objective of the outline to follow, from Reference [32]. The dynamic response is used as a reference for agitation/mixer behavior for a defined set of process results. For turbulent mixing, kinematic similarity occurs with geometric similarity, meaning fixed ratios exist between corresponding velocities. 

Determine geometric similarity to develop a single scale ratio R, for the relative magnitudes for all linear dimensions [32]. 

To select a turbine, there must also be geometric similarities for the type of turbine, blade width, number of blades, impeller diameter, etc. From the geometric similarity determination of the turbine diameter, the mixer speed can be established to duplicate. The Scale Ratio R, ... 

For geometric similarity of liquid motion (n = 1.0) the linear scale ratio of volume is... 

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. 

A general rale, which is often applied in scale-up, is that of geometric similarity between the small and large vessels. However, as shown in Table 6.3, the relevant parameters that... 

Table 6.3. Various parameters on scale-up using geometric similarities...

Based on the practical history of scale-up, most fermentation processes for alcohol and organic acid production have followed the concepts of geometric similarity and constant power per unit volume. From the above concept, and as a strong basis for translation of process criteria, only physical properties of the process were considered in the scale-up calculation. For power consumption in an agitated vessel, there is a fixed relation between impeller speed, N, and impeller diameter, l)t. The constant power per unit volume, for a mechanical agitated vessel is given by ... 

There has to be a relation between kLa with aeration rate and agitation speed, and scale-up factor has to be determined. To eliminate the effect of viscous forces, the rheology of the media and broth for a large vessel have to be similar to that of a bench-scale vessel. For scale-up based on geometric similarity, the constant values a and b are proposed for the mass-transfer correlation in Table 13.1. 

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