Scaling law, equation

In order to calculate the molar mass distribution, the scaling law (Equation (52))... 

In the excluded volume limit / = 1 we find the linear scaling laws. Equations (11,43), (11,48) yield... 

Figure 8 Comparison between the measured distance betweenforce branches (open symbols) and calculated mesh sizes for the polymer network calculated using the scaling law (Equation 5) (closed symbols) (diamonds) PAMPS (squares) PSS (circles) xan-than. Data from ref. 29...

Taylor series, in first approximation, Levelt-Sengers et. al. obtained the following scaling law equation [2] ... 

The scaling law (Equation 170) is valid in the limit p 0. From the definition of p (Equation 159) it follows that p decreases with increasing N at any fixed temperature e. I his means that the region of the realization of scaling (Equation 169) increases with molecular weight. This prediction agrees with Dobashi et al. s results (1980ab). 

Knowing the scaling laws [equations (5.3) and (5.4)] enables us to handle most viscous interface hydrodynamics problems dimensionally. Even though that approach may overlook the details of flows, the scaling laws provide us with a quick picture of the behavior of such flows (in terms of the pertinent parameters and how important quantities vary as functions of these parameters). In this sense, these laws provide a valuable path for problem solving. Whenever questions arise that involve cumbersome calculations, we will not hesitate to resort to this approach, all the while encouraging the reader to consult original papers for the details of the calculations. 

The scaling law, equation (86), suggests that the ratio x = /( h is a universal number. From their... 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Equation (13) gives the basic scaling law for dissipation. But if a more rigorous treatment is applied, we obtain [7] ... 

The specific viscosity )jsp of a dilute solution of spheres is directly related to their hydrodynamic volume VV Nl denotes Avogadro s number. Typically the intrinsic viscosity [tj] follows a scaling law, the so-called Mark-Houwink-Sakurada equation ... 

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

Horio s scaling law derivation was based on the requirement that two similar circulating fluidized beds have equal values of voidage distribution, dimensionless core radius, gas splitting to core and annulus, solid splitting to core and annulus, and cluster voidage. The CAFM equations were then examined to determine how these requirements could be met. 

The phenomenological magnetic equation of state, based on the Brillouin formula, has a form related to a scaling law ... 

Use the scaling law given by Equation 6-21 and Figure 6-23 (or Equation 6-23) to estimate the peak side-on overpressure. 

Because of the way it was derived, the last term in the diffusion equation due to the gradient of <1> has the same scaling laws as the first one. Therefore it does not... 

By converting the governing hydrodynamic equations for a particular system into nondi-mensional ones, Horio et al. (1986) and Glicksman (1988) derived the so-called scaling laws for fluidized beds. These laws should be seen as a guide to design small-scale, cold-flow models, which simulate the hydrodynamic behavior of the commercial units (Knowlton et al., 2005). 

We have already implied that underwater expin parameters depend on the distance from the charge at which the parameters are observed and on charge weight. We will now consider the quantitative dependence of these parameters on both distance from and weight of the expl. These relationships are known as scaling laws or similitude equations... 

In a similar way the contribution for all the different modes to the three transport coefficients can be calculated. Equations (58) and (61) are the classic mode coupling theory expressions that provide general expressions for the shear viscosity and thermal conductivity, respectively. Using these general expressions and the ideas of static scaling laws, Kadanoff and Swift have calculated the transport coefficients near the critical point. 

A comparison of the power law description of oxide thickness (equation 68) and the Deal-Grove model is shown in Figure 28. The data of Deal and Grove taken at 700 °C are plotted on linear scales, and equation 66 is plotted with measured values of B/A and t. Shown for comparison is equation 68 with the appropriate constants obtained by a fractional weighted least-square... 

Equation (11,85) is the nonlinear scaling law, The linear scaling law results at the fixed point / = 1. As is easily derived with the help of Eq. (11.43) it takes the form discussed in Sect. 9.1 V becomes a function of s = CpRg and... 

This calculation illustrates a general feature we may write down scaling laws for normalized quantities in terms of scaled momenta qRg (or qf e, equivalently), scaled concentrations cpRand ip replacing the coupling. Such relations involve only physically observable macroscopic quantities. They must have a uniquely defined -expansion, where ip = 0(e) acts as an expansion parameter. The result is necessarily independent of any conventions of the renormalization scheme. Not even the form of the RG flow equations matters. Furthermore, in establishing such results, we never have to invoke a condition like hr = 1. These are the great virtues of consistent e-expansion. 

Widom9 and others have tied down the relationships between the critical exponents still further. They proposed that the singular portion of the thermodynamic potential was a homogeneous functionv of the reduced temperature and the other variables. This assumption leads to the observance of the power-law behavior for the various thermodynamic properties and produces the scaling laws as equalities rather than inequalities of the type developed above [equation (13.5)]. 

Leiva et al. [65] have reported for poly(itaconates) monolayers the surface behavior at the air - water interface at different surface concentrations. They have found that for these type of polymers, the air - water interface at 298 K, is a bad solvent, very close to the theta solvent. At the semidilute region concentration, the surface pressure variation was expressed in terms of the scaling laws as a power function of the surface concentration. According to equation (3.3), the log it vs log T plot shows a linear variation with slope 2 v/(2 u-1). 

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