Similarities geometric

There are several methods to achieve appropriate scale-up of mixing. The first method involves geometric similarity. This technique employs proportional scale-up of geometric parameters of the vessel. The scaled-up parameters may include such geometric ratios as T)/r ratio, where D is diameter of the impeller and T is diameter of the tank, and Z/T ratio, where Z is the height of the liquid in the vessel. Similar ratios are compared for both the small-scale equipment DiTi) and the larger size equipment (D2T2). For example, 

After R has been determined, other required parameters such as the rotational speed of the larger equipment can then be calculated by power law relationships. In the above example, the required rotational speed, N, can be calculated as 

Rotational speeds may be expressed either in terms of rpm or in terms of hertz. The power law exponent, n, has a definite physical significance. The value of n and the corresponding significance are determined either empirically or through theoretical means. Table 1 lists the most common values assigned to n. 

Scale-up can be completed by using predicted values of Nj to determine the horsepower requirements of the large-size system. In most designs, D/T will be in the following range  

This exponent is rarely used due to excessively large equipment requirement to hold speed constant 

Two bodies are geometrically similar when for every point in one body, there exists a corresponding point in the other. Thus, if X, Y, and Z are the coordinates of a given point, then the relation that defines geometric similarity between two bodies denoted by 1 and 2 is 

We will discuss two of these types of physical similarities in the following sections. 

Geometric similarity is the most obvious requirement in a model system designed to correspond to a given prototype system. It is the similarity of shape. Two systems are geometrically similar when the ratio of any length in one system to the corresponding length in another system is everywhere the same. 

Perfect geometric similarity is not always easy to achieve and difficulties can arise due to 

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Two different types of dynamic test have been devised to exploit this possibility. The first and more easily interpretable, used by Gibilaro et al [62] and by Dogu and Smith [63], employs a cell geometrically similar to the Wicke-Kallenbach apparatus, with a flow of carrier gas past each face of the porous septum. A sharp pulse of tracer is injected into the carrier stream on one side, and the response of the gas stream composition on the other side is then monitored as a function of time. Interpretation is based on the first two moments of the measured response curve, and Gibilaro et al refer explicitly to a model of the medium with a blmodal pore... 

The specific speed compares the adiabatic head and flow rate in geometrically similar machines at various speeds. 

The specific diameter compares head and flow rates in geometrically similar machines at various diameters... 

Geometric similarity. Two vessels are geometrically similar if the ratio of all corresponding dimensions is the same... 

Kinematic similarity. Two vessels are kinematically similar if they are first geometrically similar and have the same ratio of velocities in corresponding positions of the vessel... 

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. 

Based on this design-specific throughput and the required production column rates, the diameter of the commercial column can be calculated. The stage geometry is next set by maintaining geometric similarity to the pilot column. 

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. 

Another distinction in impellers is the way the liquid traverses and leaves the impeller blades. This is called the Specific Speed, Ns. It is another index used by pump designers to describe the geometry of the impeller and to classify impellers according to their clesign type and application. By definition, the Specific Speed, Ns is the revolutions per minute (rpm) at which a geometrically similar impeller would run if it were of such a size as to discharge one gallon per minute at one foot of head. 

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. 

Equality of all groups in Equation 7-13 assures similarity between systems of different sizes. The types of similarity are geometrie, kinematie, and dynamie. The last tliree terms of Equation 7-13 represent the eonditions for geometric similarity, whieh require that all eor-responding dimensions in systems of different sizes have the same ratio to eaeh other. Eor geometrie similarity, Equation 7-13 beeomes... 

Rushton et al. [10] performed extensive measurements of the power requirements for geometrically similar systems and found that for baffled tanks, the Froude number plays no part in determining the power requirements, as vortices do not form in such systems. For unbaffled systems, the Froude number plays a part above N g of about... 

In geometrically similar systems, the characteristic velocity becomes... 

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. 

The principle of dynamical similarity expresses the fact that tivo pumps geometrically similar to each other will... 

For change in speed with a geometrically similar family of fixed impeller design, diameter and efficiency, the following conditions and characteristics change simultaneously [25] ... 

For geometrically similar impellers operating at the same specific speed, the affinity laws are [25,11] ... 

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. 

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