Scaling laws processes

The standard mechanisms of collisional energy transfer for both small and large molecules have been treated extensively and a variety of scaling laws have been proposed to simplify the complicated body of data [58, 59, 75]. To conclude, one of the most efficient special mechanisms for energy transfer is the quasi-reactive process involving chemically bound intennediates, as in the example of the reaction ... 

Steinfeld J I, Ruttenberg P, Millot G, Fanjoux G and Lavorel B 1991 Scaling laws for inelastic collision processes in diatomic molecules J. Phys. Chem. 95 9638—47... 

The WLF approach is a general extension of the VTF treatment to characterize relaxation processes in amorphous systems. Any temperature-dependent mechanical relaxation process, R, can be expressed in terms of a universal scaling law ... 

The purpose is to highlight some of the relevant scaling laws to be taken into account when reducing the size of process equipment. 

The chemical manufacturers defend their negative view by referring to the scaling law, which predicts that processing equipment becomes more cost-efficient... 

A plasma process is characterized by many parameters, and their interrelations are very complex. It is of paramount importance to understand, at least to a first approximation, how the plasma parameters have to be adjusted when the geometrical dimensions of the plasma system are enlarged. Especially of use in scaling up systems are scaling laws, as formulated by Goedheeret al. [148, 149] (see also Section 1.3.2.2). 

A simplified theory of these processes has been established (Audouin et al., 1994). It leads to a simple scaling law for the prediction of the thickness of the degraded layer (TDL) ... 

The effects of miniaturization on the performance of an analytical separation system are often discussed in terms of a reduction of a characteristic length parameter (e.g., column diameter dc or particle diameter dp) and the associated consequences for lateral zone dispersion processes and their interplay with longitudinal (axial) zone dispersion. A rigid discussion of the physical-chemical basis is beyond the scope of this chapter. For a general account in terms of scaling laws and proportionality considerations, the reader is referred to the references [12,14]. A comprehensive and detailed description with emphasis on the underlying physical-chemical processes can be found in the book by Giddings [20]. 

The issue of scaling was touched upon briefly in the previous section. Here, the quantitative features of scaling expressed as scaling laws for fractal objects or processes are discussed. Self-similarity has an important effect on the characteristics of fractal objects measured either on a part of the object or on the entire object. Thus, if one measures the value of a characteristic 9 (cu) on the entire object at resolution cu, the corresponding value measured on a piece of the object at finer resolution 9 (rcu) with r < 1 will be proportional to 9 (cu) ... 

When the scaling law (1.3) of the measured characteristic 6 can be derived from the experimental data (w,0), an estimate of the fractal dimension df of the object or process can be obtained as well. In order to apply this method one has first to derive the relationship between the measured characteristic 6 and the function of the dimension g(df), which satisfies... 

The challenge is therefore to find a theoretical expression for these scaling laws. It will in any case depend upon scaling laws for the statistical distribution of fundamental geometrical reservoir properties. It will also depend upon these hidden processes that arise because of the nonlinear nature of movable boundary flows (quite apart from nonlinearities intrinsic to the continuum relations themselves). There have been some remarkable pioneering attempts to predict continuum properties of porous media from fundamental parameters, mainly by chemical engineers (of whom I wish to single out Howard Brenner and co-workers) and physicists, but they have as yet made little impact on the oil industry. 

The asymptotic scaling laws of MCT describe the crossover from the fast relaxation to the onset of the slow relaxation (a-process)—that is, a power-law decay of 4>(f) toward the plateau with an exponent a, and another power-law decay away from the plateau with an exponent b. For the purpose of the present review, we again ignore the -dependence. 

Unfortunately for most reactor systems, a scale-up process cannot be achieved simply from a knowledge of as a function of J. In Chapter 5 we present elementary forms of the kinetic rate law from which the design equations can be evaluated, either by graphical or numerical integration or with the aid of a table of integrals. 

As in the case of turbulent diffusion, the chemical flux is often expressed by Fick s first law, as shown in Eqs. [1-3] and [1-4], but in this case D is called a mechanical dispersion coefficient. Dispersion also occurs at much larger scales than that of soil particles for example, groundwater may detour around regions of relatively less permeable soil that are many cubic meters in volume. At this scale, the process is called macrodispersion. 

These figures may appear to be daunting economic goals for biomass not to be restricted to essentially captive use within the present biomass industries. An opportunity cost of 3.84 /GJ coupled to a 40% efficient process constrains the capital cost to 1.5 k /TJ/annum output capacity. Only the densified biomass option coupled with gasifiers at the point of use can meet this cost criterion allowing that there will be prepared fuel transportation costs. If the liquid fuel opportunity cost of 5.49 is used then the capital cost for the conversion has to be less than 7.9 k /TJ/annum output capacity. Allowing that the usual scaling law of an exponent to the 0.7 power is likely to apply to methanol plants for example then a 4000 tpd plant would be feasible. 

High-resolution proton DQ MAS NMR is used as a new technique that is capable of revealing complex motional processes in entangled polymer melts. Theoretical analysis shows the connection of quantities relating anisotropic polymer dynamics to data obtained from our DQ-MAS NMR experiment. With this technique, dynamic chain ordering as well as scaling laws consistent with the reptation model was previously observed for polybutadiene (PB). 

A schematic diagram of the jet structure is shown in Fig. 10.9. As discussed in the last chapter, processes occurring in the shear layer of a jet strongly affect particle formation by homogeneous nucleation. Lesniewski and Friedlander (1998) hypothesized that there is a range of operation for which particle formation occurs in the shear layer of the jet but is quenched by dilution, depletion, or nucleation suppression as the particles move down the axis. If nucleation is confined to the shear layer, useful scaling laws can be derived for correlating particle concentration data. 

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