Impeller geometric similarity

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. 

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 requires all corresponding dimensions of a new system to have the same ratio with a test model which has proven acceptable. These dimensions should include vessel diameter and liquid level, baffle width and number in vessel, impeller diameter, number of blades and width ratio. For example, a tank four times the diameter of the original model also requires a turbine ten times the diameter of the original turbine. 

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. 

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 two geometrically similar, same family, impeller wheels with the same specific speed and operated at the same rpm. 

That is the input power required for one set of impellers. Collection factors for non-geometrical similarity are required to include the effect of known factors in precise calculations. 

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 ... 

Gal-Or and Resnick (G8) measured average residence time in a system that was geometrically similar to those used by Cooper et al (C9) and Yoshida et al (Y4) with air-distilled water and air-sodium sulfite solutions of the same concentration as used by these investigators. The ratio of impeller to tank diameter was 0.4 in one series (as in the work of Cooper and Yoshida) and 0.3 in a second series. Gal-Or and Resnick reported their results as an average residence time in seconds per foot of gas-free liquid, Bh. The average residence time was calculated from the equation... 

It is common practice to use geometric similarity in the scaleup of stirred tanks (but not tubular reactors). This means that the production-scale reactor will have the same shape as the pilot-scale reactor. All linear dimensions such as reactor diameter, impeller diameter, and liquid height will change by the same factor, Surface areas will scale as Now, what happens to tmix upon scaleup ... 

The correlation of Norwood and Metzner shows to be a complex function of the Reynolds number, the Froude number, the ratio of tank-to-impeller diameter, and the ratio of tank diameter to liquid level. However, to a reasonable first approximation for geometrically similar vessels operating at high Reynolds numbers. 

A centrifugal pump with an 8 in. diameter impeller operating at a rotational speed of 1150 rpm requires 1.5 hp to deliver water at a rate of 100 gpm and a pressure of 15 psi. Another pump for water, which is geometrically similar but has an impeller diameter of 13 in., operates at a speed of 1750 rpm. Estimate the pump pressure, flow capacity, and power requirements of this second pump. (Under these conditions, the performance of both pumps is independent of the fluid viscosity.)... 

Two different size pumps are said to be geometrically similar when the ratios of corresponding dimensions in one pump are equal to those of the other pump [Holland and Chapman (1966)]. Geometrically similar pumps are said to be homologous. A set of equations known as the affinity laws govern the performance of homologous centrifugal pumps at various impeller speeds. 

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,... 

Due to the geometric similarity conventions that hold all length ratios constant, tank dimensions are a fixed multiple of impeller diameter. Therefore, the tank volume is proportional to impeller diameter cubed ... 

When a process is dominated by a mixing operation, another gambit for the effective scale-up of geometrically similar systems involves the interrelationships that have been established for impeller-based systems. Tatterson (58) describes a number of elementary scale-up procedures for agitated tank systems that depend upon operational similarity. Thus, when scaling up from levels 1 to 2,... 

Two geometrically similar stirred tanks with flat-blade turbine impellers of the following dimensions are to be operated at 30 °C as pilot-scale and production-scale aerobic fermentors. 

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