Characteristic Quantities

The advantage of the simulations compared to the experiments is that the correspondence between the tracer diffusion coefficient and the internal states of the chains can be investigated without additional assumptions. In order to perform a more complete analysis of the data one has to look at the quench-rate and chain-length dependence of the glass transition temperature for a given density [43]. A detailed discussion of these effects is far beyond the scope of this review. Here we just want to discuss a characteristic quantity which one can analyze in this context. 

The time scale of a phenomenon whose characteristic quantity has the units seconds is given by the reciprocal of that quantity. In the present context, therefore, we can state that the preequilibrium assumption is valid if the A B reaction is. very fast on the time scale (1/ 2) of the B — C reaction. 

Kenner, m, connoisseur, expert, professional, Kenn-grdsse,/. characteristic quantity, -linie, /. characteristic line, characteristic, graph, curve. -Unienknick, m. break or knee in a characteristic or graph, -marke,/. identification mark, -merkmal, n. characteristic criterion, -nummer, /. identification number. 

Let the initial distance between the particles AH and B be denoted by r. The mutual potential energy of the two charged particles is —c2/r, as in the simple ionic dissociation depicted in Fig. 8a. If the value of r is sufficiently great, the energy associated with the electrostatic fields will not depend appreciably on r. For the proton transfer there is thus a characteristic quantity similar to D or. 

The values of Oj s together with the respective values for the moduli of the composites Ec were introduced in the adequate models expressing the mechanical properties of the composites in terms of their constituents, and gave excellent coincidence with their experimental values based on the cubic variation of ir with of. The values of the characteristic quantities for the three-term unfolding model, as derived from the above-described procedure, are all included in Table 1. 

Two versions of an alternate model were also introduced, where the mesophase layer was assumed as possessing variable mechanical and physical properties, accomodating a smooth transition of the properties of the inclusions with those of the matrix, by assuring in a very short distance the progressive, from the inclusion-matrix boundaries outwards, change of the characteristic quantities of the one phase, in order to match those of the other phase. 

The extent of the solid-like character, i.e. the strength of the samples, can be directly described by the storage modulus G. As both moduli rised during gelation, it seemed to be more efficient to take the ratio of G to G , describing the dominant elastic character of the viscoelastic sample, as a second characteristic quantity than to use the loss modulus G" itself. 

For a specific comparison of the two different reactor types, channels of 300 pm diameter were considered. The equivalent pellet size for that case is 675 pm. As a characteristic quantity, the conversion at the reactor exits was computed for different flow velocities and a range of Damkohler numbers spanning three orders of magnitude. The results for the two different reactor types obtained in such a way were practically indistinguishable. This suggests that the different reactors considered in this study are equivalent as far as chemical conversion is concerned. 

On shrinking the size of micro-channel reactors by reducing the channel dimensions, a number of characteristic quantities such as pressure drop and the degree of chemical conversion are affected. In order to permit a meaningful comparison of the reactor geometry with a scaled geometry, it is important to keep one or a few... 

In some cases, it may not be desirable to reduce the volume of a reactor, and rather a decrease of pressure drop or channel length may be the goal. In Table 1.5, the dependence of several characteristic quantities on channel diameter is given, where the efficiency and at least one specific quantity is kept fixed in each line. 

Table 1.5 Dependence of the number of micro channels N, their length L, the cross-sectional area of the reactor S and the pressure drop AP on the micro-channel diameter, when the efficiency (i.e. a fixed number of transfer units) and at least one specific characteristic quantity are kept fixed in each line. Three cases with operation time-scales varying as (c/m)°. are considered [114],...

In practice, the process regime will often be less transparent than suggested by Table 1.4. As an example, a process may neither be diffusion nor reaction-rate limited, rather some intermediate regime may prevail. In addition, solid heat transfer, entrance flow or axial dispersion effects, which were neglected in the present study, may be superposed. In the analysis presented here only the leading-order effects were taken into account. As a result, the dependence of the characteristic quantities listed in Table 1.5 on the channel diameter will be more complex. For a detailed study of such more complex scenarios, computational fluid dynamics, to be discussed in Section 2.3, offers powerful tools and methods. However, the present analysis serves the purpose to differentiate the potential inherent in decreasing the characteristic dimensions of process equipment and to identify some cornerstones to be considered when attempting process intensification via size reduction. 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

The measurement of the surface potential asa function of pH for an oxide provides valuable information for the determination of the parameters which describe the surface reactions. Ionizable surface site theories of the formation of surface charge and potential at an oxide surface in contact with a liquid electrolyte involve many more parameters than can be directly experimentally determined. Additional assumptions are required to evaluate these parameters, which explains why there is often no agreement in the literature about their value. A mathematical treatment of the amphoteric surface site model is given which exhibits the characteristic quantities which can be experimentally measured. It is shown that the measurement of both the surface potential i/>o and the surface charge 00 are required to completely determine these characteristic quantities. This approach is applied to Si02 and AI2O3, two surfaces for which both charge and potential measurements are available. 

The figures for seven azulenes are summarized in Table 24. Again, the basicity of the unsaturated system increases with the substitution of methyl groups. The differences are such that Mq (the Hq- value for P = 1) represents a characteristic quantity which is useful for the identification of azulene derivatives, and for testing their purity. 

If we combine the escape-rate formula (92) with the result (84) that the escape rate is proportional to the transport coefficient, we obtain the following large-deviation relationships between the transport coefficients and the characteristic quantities of chaos [37, 39] ... 

Thus, we are able to express this constant in terms of system characteristic quantities, making it measurable, at least, in principle. Alternatively, if we know the value of Dq from another source, we can calculate the value of/. 

Nondimensionalizing the energy equation based on the above characteristic quantities (using for nondimensional variables), one obtains ... 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

A characteristic quantity describing the viscous flow state is the dimensionless Reynolds number Re. 

The characteristic quantities to be found in equation 2.17 are only for the combination of the Roots pump and the backing pump, namely maximum compression kg of the Roots pump and gradation k, between Roots pump and backing pump. 

The characteristic quantities of a cryopump are as follows (In no particular order) ... 

Crossover value The crossover value is a characteristic quantity of an already cold refrigerator cryopump. It is of significance when the pump is connected to a vacuum chamber via an HV / UHV valve. The crossover value is that quantity of gas with respect to T =293 K which the vacuum chamber may maximally contain so that the temperature of the cryopanels does not increase above 20 K due to the gas burst when opening the valve. The crossover value is usually slated as a pV value in in mbar I. 

The vendor-estimated cost of remediation using WSWP was 150 to 250 per ton of soil treated in 1995. Soil characteristics, quantity of waste, and target contaminant concentration were cited as having the most significant effect on price (D10361X, p. 25). 

Table 23.6 Characteristic Quantities of the Sediment-Water Exchange and SMSL Model (Fig. 23.4)...

Table 6. Particle size, particle size distribution, and characteristic quantities...

Finally, it should be mentioned that an interesting interpretation of the third eq. (5.4) is obtained by introducing the model of the worm-like chain, as developed by Kratky and Porod (153, 154). As is well-known the characteristic quantity of this model is the persistence length, i.e. the projection of the end-to-end distance of an infinitely long coiled worm... 

A dimensionless variable is the ratio of that variable to a characteristic quantity of the same dimensions (e.g., r = tiff). A dimensionless parameter is the ratio of combinations of characteristic quantities having the same dimensions [Da = k/(l/6)]. We shall go into more detail on the question of rendering equations dimensionless later (See pp. 28-33), but two principles are of sufficient importance that they are worth reiterating.3... 

A. Characteristic quantities used in making variables dimensionless must be capable of being held constant ... 

From our earliest example, we saw that it was advantageous to use dimensionless variables and that the characteristic quantities should be capable of being held constant. In addition, if a parametric study on the effect of varying some input quantity is to be performed, that quantity should appear in only the distinguished parameter. This is no restriction, for the others are proportional to powers of the distinguished parameter, and the proportionality constants are themselves dimensionless numbers. For example, if the viscosity is to be varied, the Reynolds and the Schmidt numbers are both functions of v, but ReSc is not so, if Sc is chosen as the dimensionless viscosity, Re = Cl Sc, where C = ReSc is independent of v. 

Here, the average value of the inlet concentration has been retained as a characteristic quantity and A is the ratio of the amplitude of the forcing to this mean. (The frequency w = 2ttIt will also be used.)... 

Physically this means that the temperature decrease due to incomplete combustion cannot exceed the characteristic quantity RT /E without full extinction or disruption of combustion also taking place. Substituting (16) into (14a) we obtain an equation for the critical reaction time at which extinction occurs ... 

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