Dimensionless groups mixing

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

This is probably the most important dimensionless group used to represent the actual flow during mixing in a vessel. Flow Number, Nq (or pumping number) ... 

Thus, the ratios of the various forces occurring in mixing vessels can be expressed as the above dimensionless groups which, in turn, serve as similarity parameters for scale-up of mixing equipment. It can be shown that the existence of geometric and dynamic similarities also ensures kinematic similarity. 

On the assumption that the power required for mixing in a stirred tank is a function of the variables given in equation 7.12, obtain the dimensionless groups which are important in calculating power requirements for geometrically similar arrangements. 

Verify that the unusual mixed units used here actually give the dimensionless groups as stated and that the dimensionless balances are consistent. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

In the design of liquid mixing systems the following dimensionless groups are of importance. 

A power curve is a plot of the power function 4> or the power number Po against the Reynolds number for mixing ReM on log-log coordinates. Each geometrical configuration has its own power curve and since the plot involves dimensionless groups it is independent of tank size. Thus a power curve used to correlate power data in a 1 m3 tank system is also valid for a 1000 m3 tank system provided that both tank systems have the same geometrical configuration. 

The principle of similarity [Holland (1964), Johnstone and Thring (1957)] together with the use of dimensionless groups is the essential basis of scale-up. The types of similarity relevant to liquid mixing systems together with their definitions are listed as follows. 

Single screw extruder. Let us take the case of a single screw extruder section that works well when dispersing a liquid additive within a polymer matrix. The single screw extruder was already discussed in the previous section. However, the effect of surface tension, which is important in dispersive mixing, was not included in that analysis. Hence, if we also add surface tension as a relevant physical quantity, it would add one more column on the dimensional matrix. To find the additional dimensionless group associated with surface tension, as, and size of the dispersed phase, R, two new columns to the matrix in eqn. (4.32) must be added resulting in ... 

It has been shown [2] that, with reference to the mixing effectiveness, the reactor behavior can be described in terms of a dimensionless group x, defined as the ratio of the length of the batch cycle % to the residence time Vc/Fc of the fluid in each compartment,... 

The rate constant K3 which appears in the dimensionless group A5 is also unknown. It corresponds to the combustion of the unstable polymeric residue which is assumed to be very fast, i.e., mass transfer controlled. There are two ways to account mathematically for the destruction of the polymeric residue by gaseous oxygen when it becomes unstable. The first is to use equation (14) with a larger but finite rate constant K3 (or A5) together with the parameter o defined above. If this approach is taken there exists a minimum integration step of order I/A5 that can be used in order to account for the finite mixing time in the reactor and also to account for the assumption that the combustion of the polymer is mass transfer controlled. 

Henzler (1982) was able to correlate the same experimental results by using the dimensionless group Y = (kLaL/v)(ul/g)113 as ordinate and X = PJ(Vpgv) as abscissa. The dimensionless group Y represents the sorption number for bubble columns, while X represents the ratio of the mechanical power of the stirrer to the hydraulic power of the gas throughput. The sorption number Y also allows comparison between mixing tanks and bubble columns (Henzler... 

An extensive compilation was published by Rushton et al. (R13), giving the results of a program of measurements carried out by the Mixing Equipment Company. The data cover a wide range of impeller types, impeller and tank sizes, liquid properties, and operating conditions. They also presented their results in the form of a function of dimensionless groups ... 

It is evident that this method of correlation is not adequate to represent, by a single curve, the data from more than one kind of system the relation between the dimensionless groups is different for each system tested. In terms of numerical values, the power required at the same value of Reynolds number for two different agitators may vary by a factor of 10 or more. Thus these correlation curves should be employed with caution, and preferably applied to mixing systems physically similar to those studied. 

The Froude number described above is frequently used for the description of radial and axial flotvs in liquid media when the pressure difference along a mixing device is important. When cavitation problems are present, the dimensionless group (Pj — p,) /pw - called the Euler number - is commonly used. Here p is the liquid vapour saturation pressure and p is a reference pressure. This number is named after the Swiss mathematician Leonhard Euler (1707-1783) who performed the pioneering work showing the relationship between pressure and flow (basic static fluid equations and ideal fluid flow equations, which are recognized as Euler equations). 

The energy consumption in agitation depends on the basic principles of fluid mechanics however, the flow patterns in a mixing vessel are much too complex for their rigorous application. Therefore, empirical relationships based on dimensionless groups are used. Here, because most fluid foods are non-Newtonian in nature, the... 

A dimensionless group called the power number is commonly used to predict the power required to rotate a mixing impeller. The power number is defined as F/(pAPD ), where P is power, p is fluid density, N is rotational speed, and D is impeller diameter. To be dimensionless, the units of the variables must Be coherent, such as SI metric otherwise appropriate conversions factors must be used. The conversion factor for common engineering units gives the following expression for power number ... 

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