Relevance list

First, all physical parameters necessary to describe the problem are listed. This so-called relevance list of the problem consists of the quantity in question and of all the parameters that influence it. In each case, only one target quantity must be considered it is the only dependent variable. On the other hand, all the influencing parameters must be primarily independent of each other. 

After these considerations we are able to precisely make the relevance list ... 

DETERMINATION OF A PI SET BY MATRIX CALCULATION Establishment of a Relevance List of a Problem... 

In the above relevance list, only the density and the viscosity of the liquid were introduced. The material properties of the gas are of no importance as compared to the physical properties of the liquid. It was also ascertained by measurement that the interfacial tension cr does not affect the stirrer power. Furthermore, measurements revealed that the coalescence behavior of the material system is not affected if aqueous glycerol or cane syrup mixtures are used to increase viscosity in model experiments (7). 

In transforming the relevance list Eq. (9) of the above seven physical quantities into a dimensional matrix, the following should be kept in mind to minimize the calculations required ... 

The interdependence of seven-dimensional quantities of the relevance list, Equation (9), reduces to a set of only 7 — 3 = 4 dimensionless numbers,... 

If a natural or universal physical constant has an impact on the process, it has to be incorporated into the relevant list, whether it will be altered or not. In this context, the greatest mistakes are made with regard to the gravitational constant g. Lord Rayleigh (3) complained bitterly saying ... 

Many engineering problems involve several parameters, that impede the elaboration of the pi space. Fortunately, in some cases, a closer look at a problem (or previous experience) facilitates reduction of the number of physical quantities in the relevance list. This is the case when some relevant variables affect the process by way of a so-called intermediate quantity. Assuming that this intermediate variable can be measured experimentally, it should be included in the problem relevance list, if this facilitates the removal of more than one variable from the list. 

The impact that the introduction of intermediate quantities can have on the relevance list will be demonstrated in the following elegant example. 

In material systems with differences in density and viscosity, the relevance list, Equation (13), enlarges by the physical properties of the second mixing component, by the volume ratio of both phases 4> = Vi Vi, and, due to the density differences, inevitably by the gravity difference gAp to nine parameters ... 

By introducing these intermediate quantities v and p, the nine-parametric relevance list, Equation (15) reduces by three parameters to a six-parametric one... 

On the moon the water drainage would be by far less effective. In contrast to the dimensional analysis presented in Reference (9) we are well advised to add g to the relevance list ... 

Enough is known about the physics of the basic phenomenon to compile a first, tentative relevance list. 

Tq (Fig.7), then the relevance list is extended by only two reference parameters, 7o and jSq. This, in turn, leads to only two additional dimensionless numbers. For the above problem, it now follows that... 

In Example 5.1, the working out of the power characteristics of a stirrer in a Newtonian fluid is presented in detail. It is shown how a relevance list containing five parameters stirrer power P, stirrer diameter d, density p, and viscosity p of the liquid and the stirrer speed n... 

In a viscoelastic liquid, the relevance list must be extended by two rheological parameters known in advance shear rate the first differ-... 

The relevance list of this task consists of the target quantity (mixing power P) and the following parameters stirrer diameter d, density p, kinematic viscosity v of the liquid, and stirrer speed ... 

Mixing time 6 is the time necessary to completely homogenize an admixture with the liquid contents of the vessel. It can easily be determined visually by a decolorization reaction (neutralization, redox reaction in the presence of a color indicator). The relevance list of this task consists of the target quantity (mixing time 6) and of the same parameters as in the case of mixing power— on condition that (contrary to Example 3) both liquids have similar physical properties ... 

This relevance list yields in the two parametric mixing time characteristics... 

Relevance list In case of a pure convective mixing and Ap, A Jp 0, the particle size dp is of no influence. 

The third dimensionless number, (j)dp o, resulting from the five parametric relevance list... 

As material parameters, the densities and the viscosities of both phases as well as the interfacial tension a must be listed. We incorporate the material parameters of the dispersion phase and in the relevance list and note separately the material numbers p/pd and rj/rid- Additional material parameters are the (dimensionless) volume ratio of both phases (f) and the mass portion ci of the emulsifler (surfactant) (e.g., given in ppm). 

Therefore, the volume-related energy input E/ V and the throughput-related power input P/q (=P) represent homologous quantities of the same dimension. For the sake of simplicity, Ap will be introduced in the relevance list. 

Now, this 6-parametric relevance list of the dimensional parameters (the dimensionless parameters p/pd, q/f]d, 4>, c are excluded) reads... 

Assuming a quasiuniform power distribution in the throughput or in the volume, a characteristic length of the dispersion space becomes irrelevant. In the relevance list, Equation (66), the parameter d must be cancelled. The target number = 32/ must be dropped and the dimensionless numbers La and Oh must be built by J32 instead of d. At given and constant material conditions pjpd, 9, Ci = const.), the process characteristics will be represented in the following pi space ... 

The mass-related energy input E/pVmust be taken into account as a process parameter. The relevance list reads ... 

In cases involving the ballistic movement of bodies, the formation of vortices in stirring, the bow wave of a ship, the movement of a pendulum, and oher processes affected by the Earth s gravity, the relevance list comprises gp and p. 

The fluid velocity v in pipes—or the superficial gas velocity vq in mixing vessels or in bubble columns—presents a well-known intermediate quantity. Its introduction into the relevance list removes two others (throughput q and diameter D), because v qlD and va qcID, respectively. 

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