SCALING LAW

A specified overpressure is produced by two quantities of the same explosive at distances that are related by the equation  

An approximate measure of explosive yield may be obtained by determining the amount of energy that is transferred to the blast wave by the explosion products. An upper limit may be found by evaluating 

This actually represents the reversible work performed by the explosive system so that it overestimates the energy transferred to the blast wave. However, if each system is treated in a similar fashion, then 

TNT equivalents can also be obtained for compressed-gas systems in which a chemical reaction does not occur. For the ideal gas case in which a sudden expansion occurs from pressure to pressure P2 at temperature T, we have 

Arrival times, pulse durations, and impulses also follow the cube-root scaling law. Thus, 

The time taken to produce a certain change by dififiisional flow of matter can be expressed as  

Equation (5.125) can be used to consider matter transport by volume diffusion. 

The volume of the matter transported is proportional to R, where R is the radius of the sphere shown in Fig. 5.13 [1]. As a result, V2 is proportional to (XRf, or V2 = For lattice diffusion, the area over which the matter diffuses is proportional to R. Therefore, A2 is proportional to (A/ ), or A2 = A Ai. The flux J is proportional to V/x, which is the gradient of chemical potential. For a curved surface with a radius of curvature r, fi varies as a function of 1/r. Therefore, J varies with V(l/r) or 1/r. Because J2 is proportional to l/(Ar), there is J2 = Therefore, the parameters for lattice diffusion are  

Therefore, according to Eq. (5.127), the time taken to produce geometrically similar changes by a lattice diffusion mechanism increases as the cube of the particle size. The scaling laws for the other mass transport mechanisms can be 

To determine the relative rates of the different mechanisms, it is more useM to express Eq. (5.128) in terms of rate. For a given change, the rate is inversely proportional to the time, so that Eq. (5.128) can be written as  

The relationship between physical quantity under consideration and the object size, I may or may not be influenced by other physical quantities depending on the situations. Let us consider an example of fluid flow circulating through a microchannel at a fixed pressure difference. For a laminar flow, inside a tube of radius a, we have the flow rate Q as 

the order of magnitude of velocity inside the capillary can be written as 

We note that for this situation, the average velocity is inversely proportional to the length of the channel. Similarly, keeping AP, p, and L constant, we observe different relationships between the average velocity and the transverse dimension of the channel. 

It follows from equation (28) that the partition function of the chain is given by 

In going to the continuous notation of equation (27) we have allowed every point along the contour of the chain to undergo excluded volume interaction with every other point along the chain. To justify that this simplification is valid we can introduce a cut-off length along the chain below which the monomers do not have excluded volume interaction, i.e. 

A is thus an additional parameter introduced to characterize the model chain, but we will show that it plays no significant role for L large, i.e, the global properties of the chain are independent of A in 

Thus the partition function is a function of n, v and A/l only (where v = wP Let us now perform the following changes of variables 

Thus the functional nature of the partition function is unchanged by the transformation only the variable are regrouped. It is easy to calculate i from Z and the result is 

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An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

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... 

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

It has become fashionable to prefix the names of disciplines with bio , as in biophysics, bioinfonnatics and so on, giving the impression that in order to deal with biological systems, a different kind of physics, or infonnation science, is needed. But there is no imperative for this necessity. Biological systems are often very complex and compartmentalized, and their scaling laws may be different from those familiar in inanimate systems, but this merely means that different emphases from those useful in dealing with large unifonn systems are required, not that a separate branch of knowledge should necessarily be developed. 

Aiivisatos A P 1997 Scaling law for structural metastability in semiconductor nanocrystais Ber. Bunsenges Phys. Chem. 101 1573... 

The time constants characterizing heat transfer in convection or radiation dominated rotary kilns are readily developed using less general heat-transfer models than that presented herein. These time constants define simple scaling laws which can be used to estimate the effects of fill fraction, kiln diameter, moisture, and rotation rate on the temperatures of the soHds. Criteria can also be estabHshed for estimating the relative importance of radiation and convection. In the following analysis, the kiln wall temperature, and the kiln gas temperature, T, are considered constant. Separate analyses are conducted for dry and wet conditions. 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180. 

The values of m given above conform to Hemng s scaling law (1950) which states that since the driving force for sintering, the transport length, the area over which uansport occurs and the volume of matter to be transported are proportional to a, and respectively, the times for equivalent change in two powder samples of initial particle size ai q and 2,0 are... 

The molecular properties H t), determined by the minor chain model (Table 1), are interrelated and have a convenient common scaling law. The dynamic proper-... 

If the chain ends are segregated on the surface at the start of the welding, then the same general scaling law applies but with different values of r and s for some of the number properties. The average properties remain unaffected [1]. 

Scaling laws If a full-scale test is not possible, reduced-scale experiments are a good alternative. However, certain scaling laws must be observed (see Section 12.4, Scale model experiments ). Correct scaling for isothermal flows is usually possible. However, scaling of buoyant flows in large rooms may be difficult, if not impossible. Then numerical simulation is the better choice. 

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

The relaxation time in Eq. (15) and the scaling law Z — 2v+ for the dynamic critical exponent Z are then understood by the condition that the coil is relaxed when its center of mass has diffused over its own size... 

VII. POLYMER CHAINS IN RANDOM POROUS MEDIA A. Scaling Laws in Equilibrium... 

T. Irisawa, M. Uwaha, Y. Saito. Scaling laws in thermal relaxation of fractal aggregates. Europhys Lett 50 139, 1985. 

Figures 32.12 and 32.13, and Figure 32.14 shows how the effect of changing speed or diameter of a pump impeller may be predicted, using the scaling laws ...

Karrasik et al and has been modified to appear in metric form. The role used is often called the Scaling Laws,... 

Figure 32.17 Scaling laws applied to diameter change (after Karas-sik et aP.)...

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 ... 

Possible Sets or Scaling Laws for Forced Convection Burn-Out and the Implied Scaling Factors for Water and Freon-12... 

Most of the tests made so far have used water and Freon-12 (CC12F2), and the scaling factors implied by the various possible sets of scaling laws may be calculated from the physical properties for these two fluids. The appropriate scaling factors based on water at 1000 psia, for which pL/pv = 20.63, are listed in Table VII. As an example of how the scaling factors are calculated, the group Ahjl in Eq. (39) will have the same value for water and Freon-12 if... 

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