Weiss approximation

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Curie-Weiss approximation for the Ising model. Both these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, p = 1/2, 6 = 3 and y = 1. 

Transverse relaxation under incomplete motional narrowing conditions in multispin systems can be treated with the aid of the Anderson/Weiss approximation [2]. Precession phase shifts caused by spin interactions with many coupling partners is assumed to be subject to the central limit theorem. That is, the distribution of the local fields i-e., the distribution of the corresponding angular frequency offsets and hence the distri-... 

The synthesis of the Ni(n) complex of the 13-membered (anionic) macrocycle (78) is also achieved using an in situ procedure (Cummings Sievers, 1970) in which triethylenetetramine, acetic acid, acetylacetone, and nickel acetate are heated in water at the reflux. Addition of iodide ion and adjustment of the pH of the solution to approximately 10, leads to crystallization of the Ni(n) complex of the required cyclized product (78) as its iodide salt. The reaction type has been extended to include Cu(ii) as the template metal (Martin, Wei Cummings, 1972) and has also been... 

Figure 2. Solvent-averaged potential for charged hard-sphere ions in a dipolar hard-sphere solvent. MC approximation by Patey and Valleau (16) and LHNC approximation by Levesque, Weis, and Patey (11). Also shown are the primitive model functions for solvent dielectric constants 9.6 and 6.

Wei, J. and Realff, M.J. (2004) Sample average approximation methods for stochastic MINLPs. Computers el Chemical Engineering, 28, 336. 

The electron diffraction study was complemented by an all-electron theoretical calculation of Lu, Wei, and Zunger (LWZ) (1992), using the local density approximation for the exchange and correlation terms in the Hamiltonian. They find agreement within x0.6% between the calculated and dynamic structure factor values for the lowest three reflections, (100), (110), and (111). But for (200), with sin 0/A = 0.3464 A-1, the discrepancy is as large as 1.7%. The discrepancy is attributed to insufficiently accurate knowledge of the temperature factors in this diatomic crystal, which affect the derivation of the X-ray structure factor from the electron diffraction measurement, as well as the calculation of the dynamic theoretical structure factors needed for the comparison with experiment. For the monoatomic Si crystal for which the B values are well known, the agreement is... 

We summarize results of relaxation analysis and describe the algorithm of approximation of steady state and relaxation in Section 4.3. After that, several examples of networks are analyzed. In Section 5 we illustrate the analysis of dominant systems on a simple example, the reversible triangle of reactions A2<- A3 A. This simplest example became very popular for the lumping analysis case study after the well-known work of Wei and Prater (1962). The most important mathematical proofs are presented in the appendices. 

The propellant sample size shall be approximately 50 ml. Determine the weight taken by wei ghing the sampler to the nearest O.lg before and after the liquid is removed Calculate the results as follows ... 

Caseinomacropeptide (CMP), also called glycomacropeptide (GMP), is a hydrophilic gjycopeptide with molecular wei t (MW) of approximately 9 kDa (I), released from... 

This expression describes the fastest and most important mode of transport in groundwater. In fact, an important task of the hydrologist is to develop models to predict the effective velocity u (or the specific flow rate q). Like the Darcy-Weis-bach equation for rivers (Eq. 24-4), for this purpose there is an important equation for groundwater flow, Darcy s Law. In its original version, formulated by Darcy in 1856, the equation describes the one-dimensional flow through a vertical filter column. The characteristic properties of the column (i.e., of the aquifer) are described by the so-called hydraulic conductivity, Kq (units m s"1). Based on Darcy s Law, Dupuit derived an approximate equation for quasi-horizontal flow ... 

Matrix [K] follows from the linear approximation suggested by Wei and Prater [75] ... 

TABLES 1 and 2 show the calculated and measured results of splitting energies in WZ and ZB structures, respectively. Suzuki et al derived the values of A and Ar for WZ and ZN GaN and AIN from a full-potential linearised augmented plane wave (FLAPW) and band calculation [3,4], Another result with LAPW calculation was given by Wei and Zunger [5], Kim et al [6] determined them by the linear muffm-tin orbital (LMTO) method within the atomic sphere approximation (ASA). Majewski... 

The Records Conservation Section has five years of satisfactory experience treating a variety of unique works with Wei To solutions. A survey of conservators and scientists in other institutions verified the chemistry was sound and the results were aesthetically acceptable. The problems to be resolved were mechanical, involving equipment choice rather than the chemistry of the treatment (11). The hazards inherent to the system could be isolated and controlled at the treatment site. The solvent, approximately 90% dichlorodifluoromethane and 10% methanol by volume, has a maximum allowable concentration of 982 ppm in air (12), a level many times more than expected in workroom air. Incorporation of solvent recovery equipment not only reduces unit treatment cost but avoids a potential detrimental effect on the ozone layer in the upper atmosphere by dichlorodifluoromethane. 

Cross-flow monoliths have been explored by Degnan and Wei (11-12) as cocurrent and countercurrent reactor-heat exchangers. Four cross-flow monoliths in series were employed the individual blocks were analyzed by a one-dimensional approximation. They found good agreement between theory and experiment. 

Application of Additives to Silk. Deacidifying Agent. This material is ethoxymagnesium ethyl carbonate dissolved in trichlorotrifluoroethane (Wei To Associates). It has been extensively used as an alkaline buffering agent to protect paper and cellulosic textiles from aging (7,8,21). Samples were dipped one at a time in the solution for 30 s and then dried flat on a sheet of poly(methyl methacrylate). The treated samples had an add-on of approximately 3% and were relatively stiff. 

It is in the area of approximate lumping that the distinction between the meaning of lumping, as used in this article, and of overall kinetics, possibly becomes a fuzzy one. For instance, the analysis of many parallel irreversible first-order reactions, to be briefly discussed in the next section, has been called lumping in the literature, though it is not in any sense an exact (Wei-Kuo) lumping, nor an approximate one One chooses to consider only one lump, as given later by Eq. (97) but the qualitative kinetic behavior of the lump in no way resembles that of the... 

As explained in [106] condition (30) may be satisfied either by imposing D[X/,j C Tk, suggesting the use of discontinuous T, or by giving a sufficient number of interior nodes in each iif of 7 in case of continuous Th- This last fact weis first observed numerically in [98], where a sixteen node Qi finite element approximation of r is used. 

Approximate reaction networks have become customary for modeling reactions in which the species are too numerous for a full accounting or chemical analysis. Lumped components or continuous distributions commonly take the place of single components in process models for refinery streams (Wei and Kuo 1969 Weekman 1969 Krambeck 1984 Astarita 1989 Chou and Ho 1989 Froment and Bischoff 1990). Polymerization processes are described in terms of moments of the distributions of molecular weight or other properties (Zeman and Amundson 1965 Ray 1972, 1983 Ray and Laurence 1977). Lumped components, or even hypothetical ones, are also prevalent in models of catalyst deactivation (Szepe and Levenspiel 1968 Butt 1984 Pacheco and Petersen 1984 Schipper et al. 1984 Froment and Bischoff 1990). 

The relationship between molecular wei t and intrinsic viscosity for each copolymer composition was not established. An approximate value for was (htained from Eq. (1) derived for polyisbbutylene (J9) ... 

We employed Ihgashimura s method to calculate an A value for the isobutylene- -pinene system. Dam are shown in Fig. 9. The necessary DP values have been estimated firom available Afy data by dividing by the average monomer molecular wei t [0-pinene content] x 136 + [isobutylene content] x 56 g ole. The errors in My are expectedly large, since the equation we used to obtain Hfy is strictly valid only for polyisobut>dene (19). However, the error introduced by this approximation ritould... 

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