Pi-space

X-Ray analysis of crystals of 2-methyl-5-phenyltriazolo[4,5-, triazole 38, obtained by diffusion from methane/ethanol (PI space group), indicates there are two molecules per asymmetric unit. The first is essentially planar, with a mean deviation (non-H-atoms) of 0.03A and a maximum absolute torsion angle, C(2)-C(l)-N(5 )-N(4 ), of 3.4°. The second molecule has a more pronounced twist with C(2A)-C(1A)-N(5 A)-N(4 A) being 9.6° <1993ZK133>. 

We realize that in scale-up the comprehensive knowledge of the functional dependency/(ff ) = 0 (like that in Fig. 3) is not necessary. All we need to know is which pi space describes the process. 

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 process characteristics of a crossbeam stirrer was established in this pi space by evaluation of corresponding measurements in two differently sized mixing vessels (D = 0.3 and 0.6m) using different liquid mixtures (gAp/p = 0.01-0.29 and = 1-5300). It reads (8) ... 

For the sake of simplicity, will be replaced by n and qo by q in the following. For each type of foamer we obtain the following pi space ... 

To prove this pi space, measurements in differently sized model equipment are necessary to produce reliable process characteristics. For a particular foamer (Mersolat H of Bayer AG, Germany) the results are given in Figure 4. They fully confirm the pi space, Equation (22). 

Is one model scale sufficient or should tests be carried out in models of different sizes One model scale is sufficient if the relevant numerical values of the dimensionless numbers necessary to describe the problem (the so-called process point in the pi space describing the operational condition of the technical plant) can be adjusted by choosing the appropriate process parameters or physical properties of the model material system. If this is not possible, the process characteristics must be determined in models of different sizes, or the process point must be extrapolated from experiments in technical plants of different sizes. 

As discharge velocity at the nozzle outlet increases, the following states appear in succession dripping, laminar jet breakup, wave disintegration, and atomization. These states of fiow are described in a pi space Re, Fr, Wep, whereby Wep = pv dp/a represents the Weber number formed by the droplet diameter, dp. To eliminate the fiow velocity, v, these numbers are combined to give... 

To examine the similarity in particle strength of these materials, a standard representation of this physical property must be calculated. Figure 21 shows it in the pi space... 

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 ... 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

This example also shows that the pi-set compiled on the basis of the relevance list does no more than define the maximum pi-space, which may well shrink at the insight gained by measurements. 

Later on, Nikuradse [9] examined these correlations for artificially roughened pipes (by sticking sand particles onto the inner wall surface) and represented them in a pi-space extended by the geometrical number dp/d. He and later researchers, for example [10], were primarily interested in the transition range of the Re number, where the wall roughness is of the same order of magnitude as the wall boundary layer. 

This statement is supported by the results shown in Fig. 1. The researchers carried out their measurements in smooth pipes with diameters d = 0.36 - 12.63 cm, thereby changing the scale in the range of 1 35. Furthermore, the physical properties of the fluid tested (water or air) varied widely. Nevertheless, the relationship i (Re) did not display this change Every numerical value of Re still corresponded to a specific numerical value of The pi-space is scale-independent, it is scale invariant The pi-relationship presented is therefore valid not only for the examined laboratory devices but also for any other geometrically similar arrangement ... 

With these facts the distinction between the pi-space and the original x-space is particularly clear. In a x-space/(x ) = 0, which is constituted of dimensional quantities in the representation... 

Two realizations of the same physical interrelation are considered similar (complete similarity), when m - 1 dimensionless numbers of the m-dimensional pi-space have the same numerical value (pi = idem), because the m-th pi-number will then automatically also have the same numerical value. 

According to our knowledge of the pertinent pi-space (Fig. 1), Re = idem implies Eu = idem. The numerical value of the Euler number EuM, measured in the model-scale at the given ReM value, therefore corresponds to that of the full-scale plant. This then allows us to determine the numerical value of Ap, in the industrial plant from the numerical value of EuM in the model and the given operational parameters ... 

The following example has been chosen because it impressively demonstrates the scale-invariance of the pi-space. Besides this, in the matrix transformation we will encounter a reduction of the rank r of the matrix. This will enable us to understand why, in the definition of the pi-theorem (section 2.7), it was pointed out that the rank of the matrix does not always equals the number of base dimensions contained in the dimensions of the respective physical quantities. 

In this pi-space, the process equation has been evaluated for the cross-beam stirrer, whereby the scale was altered by p = 1 2 and the process parameters were widely varied [15]. The process equation found reads... 

In this pi-space, measurements were evaluated which were performed in a bench-scale flotation cell (Fig. 3 a) of D = 0.6 m. The flotation cell input consisted of biologically purified waste water, containing = 3 g TS/1 activated sludge (TS - total solids), which was processed in the 30 m high bubble columns, the so-called Tower Biology of BAYER AG/Leverkusen, Germany. 

To be able to adjust the process point of the pertinent pi-space in the model experiment, one has to chose an appropriate model material system. 

Each unit operation in process engineering obeys specific laws which demand a separate pi-space. It cannot be expected that different processes can be depicted by the same pi-space. 

Reliable scaling-up of the desired operating conditions from the model to the full-scale plant. This is based on the scale invariance of the pi-space. According to the model theory, two processes may be considered to be similar if they take place under geometrically similar conditions and all dimensionless numbers which describe the process have the same numerical value. 

The variability of physical properties widens both the dimensional x- and the dimensionless pi-space. The process is not determined by the original material quantity x, but by its dimensionless reproduction. (Pawlowski [27] has clearly demonstrated this situation by the mathematical formulation of the steady-state heat transfer in an concentric cylinder viscometer exhibiting Couette flow). It is therefore important to carry out the dimensional-analytical reproduction of the material function uniformly in order to discover possibly existing, but under circumstances concealed, similarity in the behavior of different substances. This can be achieved only by the standard representation of the material function [5, 27]. 

In general, standard representation depends upon the choice of the reference point. The question is posed Do mathematical functions exist whose standard representations do not depend on the choice of the reference point and therefore could be named reference-invariant functions In case of an affirmative answer on the one hand the reference point p0 - here T0 - could be omitted (constriction of the pi-space by one pi-number) and on the other hand the dimensionless representation of the material function would stretch over the entire recorded range. 

If the scale-up is performed in the pi-space with constant physical properties, the requirement ident concerns all pi-numbers involved, whereby the dimensional quantities contained in them can be deliberately varied. The dimensional-analytical validity range includes all physically convenient numerical values of these dimensional quantities. 

The original relevance list now contains two additional quantities, yo and T0. Furthermore, p0 has to replace p. By this it follows that the 3-parametric pi-space... 

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