Scaling equations

This is Widom s scaling assumption. It predicts a scaled equation of state, like the law of corresponding states, that has been verified for fluids and magnets [102]. 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

G. D. J. Phillies. Universal scaling equation for self-diffusion by macromolecules in solution. Macromolecules 79 2367-2376, 1986 Dynamics of polymers... 

Compare two different explosive charge weights of the same material. For an observed overpressure of 40 psi from a specific charge using the scaling equation above, the scaled distance is Z = 5 ft/lb /. WTiat is the distance for an overpressure of 40 psi with a charge of 500 lb ... 

Series Scaleup of Laminar Gas Flows. The scaling equations are similar to those used for turbulent gas systems but the exponents are different. The different exponents come from the use of Equation (3.22) for rather than Equation (3.23). General results, valid for any form of scaleup that uses a single tube, are... 

Phillies, GDJ, Universal Scaling Equation for Self-Diffusion by Macromolecules in Solution, Macromolecules 19, 2367, 1986. 

Phillies, GDJ, Dynamics of Polymers in Concentrated Solutions The Universal Scaling Equation Derived, Macromolecules 20, 558, 1987. 

For conservation of mass it is required that 6 = 2. For the scaled equation (56) to be time-invariant, the separation constant (w) given by... 

Karabash. Successive testing of derivative hypotheses showed that hypothesis HI is true for any set of analytical data of all groups involved in the investigation [1]. A linear correlation was established between the concentrations of elements in sets of snow-melt samples when comparing then in the logarithmic scale, equation (1)... 

R. S. Drago, B. B. Way land. A Double-Scale Equation for Correlating Enthalpies of Lewis Acid-Base Interactions. J. Am. Chem. Soc. 1965, 87, 3571-3577. 

Figure 8.6B shows a wider P-T portion with the location of the critical region for H2O, bound by the 421.85 °C isotherm and the p = 0.20 and 0.42 glcvci isochores. The PVT properties of H2O within the critical region are accurately described by the nonclassical (asymptotic scaling) equation of state of Levelt Sengers et al. (1983). Outside the critical region and up to 1000 °C and 15 kbar, PVT properties of H2O are accurately reproduced by the classical equation of state of Haar et al. (1984). An appropriate description of the two equations of state is beyond the purposes of this textbook, and we refer readers to the excellent revision of Johnson and Norton (1991) for an appropriate treatment. 

In 1965, the following double-scale equation was proposed (39) to correlate enthalpies of adduct formation in the gas phase and poor solvating solvents ... 

Fig. 3.8 McLeod compression vacuum gauge with square-law scale (equation 3.11)...

We have written Eq. (5.4) with variables grouped as they are in order to define two very important quantities. The first quantity in parentheses is called the modulus—or in this case, the tensile modulus, E, since a tensile force is being applied. The tensile modulus is sometimes called Young s modulus, elastic modulus, or modulus of elasticity, since it describes the elastic, or recoverable, response to the applied force, as represented by the springs. The second set of parentheses in Eq. (5.4) represents the tensile strain, which is indicated by the Greek lowercase epsilon, e. The strain is defined as the displacement, r — rp, relative to the initial position, rp, so that it is an indication of relative displacement and not absolute displacement. This allows comparisons to be made between tensile test performed at a variety of length scales. Equation (5.4) thus becomes... 

The concept of generalized homogeneity [see, e.g., H. E. Stanley. Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971), pp. 178ff] refers to a more general class of functions that are envisioned to satisfy scaling equations of the form... 

The correlation length ( ) corresponding to simulations at the critical density and slightly supercritical temperature was calculated from the scaling equation = 0 (Tr -l)-o.63 where Tr is the reduced temperature, and a substance-specific amplitude (47). At Tr = 1.02, and using = 1.5 A for carbon dioxide (47), the... 

The last comment is about the finite-size scaling equation (1) related to SIT. Certainly, expression (2) cannot be reduced to the form of equation (1) and no genuine scaling exists. However, in a restricted region of values of T and B representation of the theoretical curves in the form (1) can be done. This was demonstrated in [7]. This means that the scaling presentation cannot be the decisive argument in favor of a specific model. 

Note that the electric field as a function of ion position remains unchanged. The solution, + x> of the scaled equation... 

However, it is not quite clear whether this relation can be applied for non-Euclidean, ramified structures. Simulation results of carbon black formation under ballistic conditions by Meakin et al. [14] indicate that a scaling equation is fulfilled, approximately, between the number of particles Np in a primary aggregate and the relative cross section area A/Ap ... 

To describe T,x and p,x diagrams of phase equilibrium in the region of a critical transition we used scaling equations ... 

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