Energy distribution functions numerical methods

The site energy distribution function f Q) can be calculated by using the experimentally observed overall isotherm (p,T) and a theoretical local isotherm function d(p,T,Q). Here a Langmuir type model equation 9(p,T,Q) with corrections for multilayer adsorption and lateral interactions between the adsorbed molecules is chosen [54—56]. Then the integral equation can be solved by an analytical iterative method based on numerical integration [57]. More details about this procedure are found in [22,53]. 

Several numerical algorithms have been developed in order to solve the Fredholm integral equation and many of them have been applied for determining the adsorption energy distribution function from the experimental adsorption data. The following list includes the most popular and useful numerical methods ... 

Roles and Guiochon (19) developed a numerical method for determining the adsorption energy distribution function from adsorption isotherm data using gas-solid chromatography. They studied a variety of surface heterogeneous solids, such as aluminum oxide ceramic powders, which they coated on the... 

Numerous exact and approximate methods have been proposed to solve the integral equation (10) with Langmuir local adsorption isotherm. Then, an application of a given analytical equation to represent the overall adsorption isotherm determines automatically the form of energy distribution function, the parameters of which are obtained by fitting the isotherm to experimental data [6]. 

The HILDA method developed by House and Jaycock [100] may be considered a modified, numerical version of the iterative procedure proposed by Adamson and Ling [126]. An excellent short presentations of the method can be found in the review by House [127] or in the monograph by Rudzinski and Everett [6]. This procedure can be outlined as follows The form of local isotherm is assumed and the distribution function is evaluated by using the iterative routine for each iterative step appropriate adjustments in distribution are made to bring the calculated and experimental isotherms into the best possible coincidence the condensation-approximation is used to determine the first approximation of the distribution. The Adamson-Ling method was widely applied to evaluate the energy distribution function from the measmed adsorption isotherm [97,122,128-135]. 

A large number of studies have dealt with the applications of numerical methods for the determination of energy distribution functions for selective adsorption systems. These studies have focused on the following topics ... 

Having generated the numerical final state wave function using the method presented above, we determine energy distributions for each partial wave using... 

The moment method can be formulated without reference to the Fourier transform of the flux. It can be regarded simply as a technique of constructing flux distribution functions from their (numerically calculated) moments. When information about the singularities of the flux transform is not utilized, the method becomes less well-founded theoretically, but it gains in flexibility and can be applied even when the singularities of the transform are not well understood. This situation arises in the case of the plane oblique source, as well as in connection with the penetration of fast neutrons whose attenuation coefficient may be a rapidly varying function of the energy with many maxima and minima. 

In 1970s and 1980s several numerical methods were proposed in order to find the distribution energy functions of adsorption on the basis of tabulated data of experimental adsorption isotherm. From a mathematical point of view the integral adsorption equation is the Fredholm integral equation of the first kind. The particular nature of this equation poses severe difficulties to its solution and strict limits to the range of numerical methods that can be used in such a task. 

Mixed QM/MM Car-Parrinello simulation techniques are among the most powerful computational methods for exact numerical calculations of macroscopic and microscopic properties at finite temperatures of a large variety of condensed matter systems of current theoretical and experimental interest. Physical quantities that may be computed exactly and compared directly to experimental results, where available, include the kinetic, potential, and total energy, the radial distribution function, neutron scattering cross-sections, and so on. 

Diffusion theory still remains the main tool for fast reactor numerical analysis. Two reasons are mainly responsible for this (1) in the central part of the reactor, the reactor core, the application of the diffusion approximation is satisfactory for most purposes, (2) the diffusion equation is relatively simple, its properties and the appropriate methods to solve it have long been well known. Although the diffusion equation was applied previously in the other field of theoretical physics, the numerical methods used to solve efficiently the neutron diffusion problems encountered in reactor physics had to be tailored to the properties of the parameters appearing in the equation and to the neutron distribution as a function of space, energy and time. 

Apart from the original method mentioned above, Morrison and eo-workers [143,144] formulated a new iterative teehnique ealled CAEDMON (Computed Adsorption Energy Distribution in the Monolayer) for the evaluation of the energy distribution from adsorption data without any a priori assumption about the shape of this function. In this case, the local adsorption is calculated numerically from the two-dimensional virial equation. The problem is to find a discrete distribution function that gives the best agreement between the experimental data and calculated isotherms. In this order, the optimization procedure devised for the solution of non-negative constrained least-squares problems is used [145]. The CAEDMON algorithm was applied to evaluate x(fi) for several adsorption systems [137,140,146,147]. Wesson et al. [147] used this procedure to estimate the specific surface area of adsorbents. 

Stanley and Guichon [151] have recently proposed the expectation-maximization (EM) method for numerical estimation of adsorption energy distributions. This method does not require prior knowledge of the distribution function or any analytical equation for the total isotherm. Moreover, it requires no smoothing of the adsorption isotherm data and coverages with high stability toward the maximum-likelihood estimate. 

The Lattice-Boltzmann method is a numerical scheme for fluid simulations which originated from molecular dynamics models such as the lattice gas automata. In contrast to the prediction of macroscopic properties such as mass, momentum and energy by solving conservation equations, e.g. the Navier-Stokes equations, the LBM describes the fluid behaviour on a so-called mesoscopic scale [7, 19]. The basic parameter in the Boltzmann statistics is the distribution function f = f(x,, 0, which represents the number of fictitious fluid elements having the velocity at the location x and the time t. The temporal and spatial development of the distribution function is described by the Boltzmann equation in consideration of collisions between fluid elements. 

The solution of (2.8) can only be found numerically in the whole time interval from t = 0, when /(0,0) = 1/2 and 5 = 0, to t 00, when certain limits for / and S have to be observed. When the value of the relative exposure energy A in (2.2) is not small and the initial isotropic angular distribution function /o = 1/2 Is highly distorted, we have to use numerical methods to find the solutions of the diffusion equation (2.1). One of the interesting possibilities here is to consider the molecular interaction potential in the form... 

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