Representation of the Material Function

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]. 

A standard representation of any material function s(p) with the reference point p0 is therefore given as 

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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... 

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 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. 

The standard representation of the material function p(T) is analogous to p(T). The transformation parameters are... 

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. 

Example 14 Reference-invariant representation of the material function D(T, F)... 

In order to verify these facts, measurements were performed in three geometrically similar copper cylinders with D = 30.0 37.8 and 47.2 mm and five different liquids (glycerol, superheated steam cylinder oil, silicone oil Baysilon M 1.000, Desmophen 1.100 and HD oil SAE 90). The standard representation of their material functions i(T) almost corresponds well (see Example 11). 

One of the drawbacks of the two-microphone transfer function method is that the absorption coefficient determined may not be a true representation of the material s characteristic. In the case of a porous material, such as silica aerogels, the reflected wave from the rigid wall could contribute to a rise in the absorbed energy by the material. To account for this uncertainty, the four-microphone impedance tube setup is usually used to determine the transmission loss (TL) and absorption coefficient (Feng 2013). In the absence of additional microphones downstream of the specimen, a sound meter could be used instead to measure the TL of the specimen under test. However, the sound meter picks up discrete transmitted signals at periodic interval, which could result in a mismatch with the generated signals from the source. 

Magnetic interactions in extended systems can also be studied without creating the more or less approximate representation of the material with an embedded cluster. The approach based on the translational symmetry in the crystal naturally leads to the well-known band stmctures of the Bloch functions, periodic one-electron functions. 

The results of the micromechanics studies of composite materials with unidirectional fibers will be presented as plots of an individual mechanical property versus the fiber-volume fraction. A schematic representation of several possible functional relationships between a property and the fiber-volume fraction is shown in Figure 3-4. In addition, both upper and lower bounds on those functional relationships will be obtained. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

A schematic representation of the combustion wave structure of a typical energetic material is shown in Fig. 3.9 and the heat transfer process as a function of the burning distance and temperature is shown in Fig. 3.10. In zone I (solid-phase zone or condensed-phase zone), no chemical reactions occur and the temperature increases from the initial temperature (Tq) to the decomposition temperature (T ). In zone II (condensed-phase reaction zone), in which there is a phase change from solid to liquid and/or to gas and reactive gaseous species are formed in endothermic or exothermic reactions, the temperature increases from T to the burning surface temperature (Tf In zone III (gas-phase reaction zone), in which exothermic gas-phase reactions occur, the temperature increases rapidly from Tj to the flame temperature (Tg). 

Figure 11.17—Absorption of light by a homogeneous material and representation of% transmittance as a function of the material s thickness.

Crystallizers are made more flexible by the introduction of selective removal devices that alter the residence time distributions of materials flowing from the crystallizer. Three removal functions—clear-liquor advance, classified-fines removal, and classified-product removal— and their idealized removal devices will be used here to illustrate how design and operating variables can be manipulated to alter crystal size distributions. Idealized representations of the three classification devices are illustrated in Fig. 17. 

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