Material function dimensionless representation

Similar behavior of a certain physical property common to different material systems can only be visualized by dimensionless representation of the material function of that property (here the viscosity l). It is furthermore desirable to formulate this function as uniformly as possible. This can be achieved by the standard representation (6,11) of the material function in which a standardized transformation of the material function /i(7) is defined in such a way that the expression produced meets the requirement... 

The type of dimensionless representation of the material function affects the (extended) pi set within which the process relationship is formulated (for more information, see Ref. 11). When the standard representation is... 

Similar behavior of a certain physical property common to different material systems can only be visualized by dimensionless representation of the material function of that property (here the density p). It is furthermore desirable to formulate... 

The type of dimensionless representation of the material function affects the (extended) pi set within which the process relationship is formulated (for more information see Ref. 5). When the standard representation is used, the relevance list must include the reference density po instead of p and incorporate two additional parameters po. Tq. This leads to two additional dimensionless numbers in the process characteristics. With regard to the heat transfer characteristics of a mixing vessel or a smooth straight pipe, Eq. (27), it now follows that... 

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 the dimensional-analytical formulation of processes whose course depends on variable material properties, it is the dimensionless representation of the material function which counts. 

In addition, it should be pointed out that for a dimensionless representation of the dependence i(T) two methods can be used. Only the first possibility has been discussed so far. It consisted of the choice of two additional quantities (transformation parameters) with the dimensions of i and T, these already being contained in the material function. In the case treated above, this has been accomplished by the introduction of i0 and y0 or E/R respectively. Pawlawski [5, 27] terms this representation a genuine one. Due to the fact that its definition is already contained in the material function, it has a higher significance in the dimensionally analytical treatment of a process. 

In contrast to this approach, the parameters with the dimensions of p and T, these being necessary for the dimensionless representation of the process, can also be formed by the parameters influencing the process in question. In this case, one can speak of the process-related representation of the material function. 

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 variable physical properties, the requirement idem also concerns the form of the dimensionless formulated material function. This aspect can make the choice of the model material system considerably more difficult. This requirement is fulfilled, a priori, only if the interesting range Au in the standard representation w = (u) lies close to the standardization point, see the explanation concerning Fig. 9 b in the text. 

In order to compare results of studies that are expressed in different quantities, dimensionless representations are always preferred. Examples of dimensionless quantities are the relative concentration c/c0 already mentioned above and the parameter appearing in the error function z = x / 2 (D t)1/2 in Fig. 7-6. Systems described with help from the same model but differing from one another with respect to material constants, e.g. D values, can have the same z and c/co values at different times. As a result, whole series of curves can be represented by a single, easy to read curve. 

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