Influence coefficient method

Practical applications of the influence coefficient method to multiplane, multispeed balancing are presented by Badgley and the author. The separate problem of choosing balancing planes is discussed at some length by Den Hartog, Kellenberger, and Miwa for the (N + 2)-plane method, and by Bishop and Parkinson in the V-plane method. 

The influence coefficient method is simple to apply, and data are now easily obtainable. Consider a rotor with n discs. The method of influence coefficients provides the means for measuring the compliance characteristics of the rotor. 

Tessarzik, J.M., Badgley, R.H., and Anderson, W.J., Flexible Rotor Balancing by the Exact-Point Speed Influence Coefficient Method, Transactions... 

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

Alternatively, fundamental parameter methods (FPM) may be used to simulate analytical calibrations for homogeneous materials. From a theoretical point of view, there is a wide choice of equivalent fundamental algorithms for converting intensities to concentrations in quantitative XRF analysis. The fundamental parameters approach was originally proposed by Criss and Birks [239]. A number of assumptions underlie the application of theoretical methods, namely that the specimens be thick, flat and homogeneous, and that, for calibration purposes, the concentrations of all the elements in the reference material be known (having been determined by alternative methods). The classical formalism proposed by Criss and Birks [239] is equivalent to the fundamental influence coefficient formalisms (see ref. [232]). In contrast to empirical influence coefficient methods, in which the experimental intensities from reference materials are used to compute the values of the coefficients, the fundamental influence coefficient approach calculates... 

The most complex case is the analysis of all, or most, of the elements in a sample about which little or nothing is known. In this case a full qualitative analysis would be required before any attempt is made to quantify the matrix elements. Once the qualitative composition of the sample is known, again, one of three general techniques is typically applied use of type standardization, use of an influence coefficient method, or use of a fundamental parameter technique. 

To determine major and minor elements in complex samples, more elaborate matrix correction algorithms need to be appHed. They can be roughly divided into two categories the influence coefficient methods and the fundamental parameter method. 

Influence coefficient methods AU these models have essentially the same form ... 

The simplest quantitative analysis situation to handle is the determination of a single element in a known matrix. In this instance, a simple calibration curve of analyte concentration versus line intensity is sufficient for quantitative determination. A slightly more difficult case might be the determination of a single element where the matrix is unknown. Three basic methods are commonly employed in this situation use of internal standards, addition of standards, and use of a scattered line from the X-ray source. The most complex case is the analysis of all, or most, of the elements in a sample, about which little or nothing is known. In this case a full qualitative analysis would be required before any attempt is made to quantitate the matrix elements. Once the qualitative composition of the sample is known, one of three general techniques is typically applied type standardization, influence coefficient methods, or fundamental parameter techniques. Both the influence coefficient and fundamental parameter techniques require a computer for their application. 

In principle, an empirical correction procedure can be described as the correction of an analyte element intensity for the influence of interfering elements, using the product of the intensity from the interfering element line and a constant factor, as the correction term [14], This constant factor is today generally referred to as an influence coefficient, since it is assumed to represent the influence of the interfering element on the analyte. Commonly employed influence coefficient methods may use either the intensity or the concentration of the interfering element as the correction term. These methods are referred to as intensity correc-... 

The major advantage of influence coefficient methods is that a wide range of concentration ranges can be covered by using a relatively inexpensive computer for the calculations. A major disadvantage is that a large number of well-analyzed standards may be required for the initial determination of the coefficients. However, where adequate precautions have been taken to ensure correct separation of instrument- and matrix-de-... 

Alternatively, standardless fundamental parameter (FP) techniques are based on built-in mathematical algorithms that describe the physics of the detector response to pure elements. In this case, the typical composition of a sample must be known, while the calibration model may be verified and optimized by one single standard sample. The techniques include the fundamental parameter method,the influence coefficient method, and the empirical coefficient method. 

Having the influence coefficients obtained, the normal surface deformations can be obtained from the multisummation as described in Eq (27). The computation may be implemented using different numerical approaches, including direct summation (DS), MLMI, and DC-FFT based methods, which will be briefly described in this section. 

Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

TABLE 2—Relative errors for DS, FFT-based method and MLMI over different grids (%) (Green, Constant and Bilinear stand, respectively, for the schemes based on Green s function, constant function, and linear interpolation in determining the influence coefficients). ... 

XRF nowadays provides accurate concentration data at major and low trace levels for nearly all the elements in a wide variety of materials. Hardware and software advances enable on-line application of the fundamental approach in either classical or influence coefficient algorithms for the correction of absorption and enhancement effects. Vendors software packages, such as QuantAS (ARL), SSQ (Siemens), X40, IQ+ and SuperQ (Philips), are precalibrated analytical programs, allowing semiquantitative to quantitative analysis for elements in any type of (unknown) material measured on a specific X-ray spectrometer without standards or specific calibrations. The basis is the fundamental parameter method for calculation of correction coefficients for matrix elements (inter-element influences) from fundamental physical values such as absorption and secondary fluorescence. UniQuant (ODS) calibrates instrumental sensitivity factors (k values) for 79 elements with a set of standards of the pure element. In this approach to inter-element effects, it is not necessary to determine a calibration curve for each element in a matrix. Calibration of k values with pure standards may still lead to systematic errors for unknown polymer samples. UniQuant provides semiquantitative XRF analysis [242]. 

The alpha coefficients method based on simpler algorithms for calculating inter-element influence coefficients. 

Unlike thermally developing flow, the superposition method cannot be applied directly to the simultaneously developing flow because of the dependence of the velocity profile on the axial locations. However, certain influence coefficients are introduced to determine the local Nusselt number for simultaneous developing flow in concentric annuli with thermal boundary conditions that are different from the four fundamental conditions the influence coefficients 0 through 0 2, determined by Kakacj and Yiicel [104] are listed in Tables 5.24 and 5.25. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

On the basis of data obtained the possibility of substrates distribution and their D-values prediction using the regressions which consider the hydrophobicity and stmcture of amines was investigated. The hydrophobicity of amines was estimated by the distribution coefficient value in the water-octanole system (Ig P). The molecular structure of aromatic amines was characterized by the first-order molecular connectivity indexes ( x)- H was shown the independent and cooperative influence of the Ig P and parameters of amines on their distribution. Evidently, this fact demonstrates the host-guest phenomenon which is inherent to the organized media. The obtained in the research data were used for optimization of the conditions of micellar-extraction preconcentrating of metal ions with amines into the NS-rich phase with the following determination by atomic-absorption method. 

The most ubiquitous method of transmission spectroscopy, in which the amount of light passing through a sample is determined. Very often the influence of reflection and scattering is neglected and the ratio of incident and transmitted intensity ( / ) is linked to the absorption coefficient (a) and the sample thickness (d) by Lambert-Beer s law (see Eq. (9.11)). 

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