Formula Wilke

The Best Estimate Plus Uncertainties (BEPU) analysis is recommended by IAEA in addition or alternatively to the deterministic approach in the safety analysis of nuclear components and systems (IAEA 2009). The aim of the BEPU analysis is to determine a quantile of an output measure of interest (noted R in the following) with a certain level of confidence (usually the 95% quantile obtained with a 95% confidence, denoted Rg gf and to verify that this quantile is below an acceptable limit (acceptance criteria). In order to obtain this quantile, the uncertainty space of input parameters is sampled at random according to their combined probability distribution and a code calculation is performed for each sampled set of parameters. The number of code calculations is determined by the requirement to estimate a tolerance and confidence interval for the quantity of interest. Wilks formula (Wilks 1941) (or Wald formula (Wald 1943) when several criteria must be respected simultaneously) is used to determine the number of calculations to obtain the uncertainty bands and the associated quantile with a given confidence level. In classical BEPU analysis, there is no separation between the aleatory variables and the epistemic variables the epistemic variables, which are often model uncertainties, are generally modeled by uniform probability distributions within intervals provided by expert opinion and propagated in the same way that the aleatory variables by Monte Carlo simulation. 

The correlation energy of a uniform electron gas has been determined by Monte Carlo methods for a number of different densities. In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula. This has been constructed by Vosko, Wilk and Nusair (VWN) and is in general considered to be a very accurate fit. It interpolates between die unpolarized ( = 0) and spin polarized (C = 1) limits by the following functional. 

Pressure Dependencies Equation 3.95 predicts the binary diffusion coefficient to scale as p l, which is generally true except as the pressure approaches or exceeds the critical pressure. The Takahashi formula [392], which can be used to describe the high-pressure behavior, is discussed below. The Chapman-Enskog theory also predicts that Vji, increases with temperature as T3/2. However, it is often observed experimentally the temperature exponent is somewhat larger, say closer to 1.75 [332], An empirical expression for estimating T>jk is due to Wilke and Lee [433]. The Wilke-Lee formula is [332]... 

In the case of viscosity, the semi-empirical formula due to Wilke [432] and modified by Bird et al. [35] can be used. The Wilke formula for mixture viscosity is given by... 

The molecular diffusion coefficient of species i in a multi-component mixture can be calculated by the Wilke formula 50... 

The spread of the reflections around the reciprocal lattice points due to the finite particle dimensions and to cumulative lattice disorders of the "ideally paracrystalline" type is calculated according to the formula given by Hosemann and Wilke (7). They showed that for a one-dimensional crystal the integral breadth (8) varies with the Miller index (h) of the reflection and can be approximated by the expression... 

In fact, for quick practical calculations, the formula of Wilke and Chang (W8) is often sufficient. It has been verified with many systems, both electrolyte and nonelectrolyte. For water at 10°-60°C, Shrier (S26) has... 

In the case of COa absorption, often used to determine interfacial parameters in gas-liquid reactors, Ratcliff and Holdcroft (R2) have found that the diffusivities in aqueous solutions of chlorides, nitrates, and sulfates of sodium and magnesium vary roughly as (rather than as in the Wilke and Chang formula). Nijsing et al. (N11) and Danckwerts and Alper (D7) found the diffusivity of COa in sulfates of sodium and magnesium, and in NaaCOa, to vary as However, all these results... 

SOLUTION As in the preceding example, none of the simple effective diffusivity formulas are applicable to the situation in the Loschmidt tube. We will proceed with the effective diffusivity formula of Wilke that gives... 

Estimate the rate of evaporation of ethyl propionate(l) into mixtures of air(2) and hydro-gen(3) using the method of Burghardt and Krupiczka. This problem is based on experiments conducted by Fairbanks and Wilke (1950) with a view to assessing the validity of Wilke s effective diffusivity formula (Eq. 6.1.14). 

Repeat Example 8.6.1 (diffusion in a Stefan tube) using Wilke s effective diffusivity formula (Eq. 6.1.14). 

The simplest approximation, employed for very many years until the most recent developments, is known as Local Spin Density Approximation (LSDA) and does not depend on the gradients of the electronic density but only on the electronic density itself. One of the variants of LSDA, commonly employed in the applications to molecular systems in the last years, is the one called SVWN. In this exchange-correlation functional, the exchange is provided by Slater s formula (3) for the uniform electron gas, whereas the correlation is evaluated according to the expression derived by Vosko, Wilk and Nusair (4) from an interpolation of previous Monte-Carlo results for the spin-polarized homogeneous electron gas... 

Example 13-3 To demonstrate the use of the predictive method, Example 13-3, which is taken from Ref. 24, is presented. A statement of this example appears in Table 13-4. Physical properties which are listed in the table were estimated as follows. The viscosity of each pure component i in the vapor phase was predicted by use of the correlation proposed by Bird et al.1,2 20 The pure component viscosities so obtained were used to predict the viscosities of the vapor mixtures by use of the semi-empirical formula proposed by Wilke.40... 

The effective diffusion coefficient is obtained from the molecular diffusion coefficient (A), the catalyst particle porosity e and tortuosity. The approximate formula of Wilke and Chang was... 

Since the non-parametric approach is the subject of part of a previous work (Mullor et al., 2008) we will just comment some particularities about it as initial methodology to improve through the parametric one. When variables in the output are independent the method consists in applying the univariate version to each one. This version, when nothing except continuity is known about g(y), consists in generating, via Monte Carlo simulation, a random sample of size N from which minimum and maximum are taken as lower and upper tolerance limits respectively. Wilks formula... 

The parameter is the viscosity of the mixture, and for gases at moderate pressures (less than 10 atm) it is adequately calculated from the semi-empirical formula of Wilke et al. (Bird et al, 1960)... 

The Stokes-Einstein formula for diffusion coefficients is limited to cases in which the solute is larger than the solvent. Predictions for liquids are not as accurate as for gases. The Wilke and Chang [16] correlation for diffusion in liquids is an empirical correlation and is given by... 

International Union of Pure and Applied Chemistry, Graphic Representations (Chemical Formulae) of Macromolecules (Recommendations 1994) Pure Appl. Chem., 66,2469-2482 (1994). Wilks, E. S. Macromolecular Nomenclature Note No. 17 Whither Nomenclature Polym. Prepr. 40(2), 6-11 (1999) also available at www.chem.umr.edu/ poly/nomenclature.html. 

Transport properties such as viscosity and thermal conductivity of a gas mixture are estimated based on the mixture rules. A simplified mixfure model developed based on fhe kinefic theory model is widely used (Bird et. al., 1960 Mills, 2001 Wilke, 1950). These formulae are given as follows ... 

Viscosity is a measure of fluid resisfance to mofion, and if relates fhe strain rate to applied shear stress. A functional dependence of gas viscosify on temperature at low density is given by Chapman-Enskog based on kinetic theory (Bird et al., 1960) using Lennard-Jones potentials. The theory has been also extended to multicomponent gas mixtures. For most common applications, however, a simplified semiempirical formula of Wilke (1950) is used ... 

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