Direct solution

The main value of this procedure is that it replaces the direct solution of the n flux relations (11.17) for N.. N, by the solution of... 

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

In order to calculate the distribution function must be obtained in terms of local gas properties, electric and magnetic fields, etc, by direct solution of the Boltzmann equation. One such Boltzmann equation exists for each species in the gas, resulting in the need to solve many Boltzmann equations with as many unknowns. This is not possible in practice. Instead, a number of expressions are derived, using different simplifying assumptions and with varying degrees of vaUdity. A more complete discussion can be found in Reference 34. 

Schreiner etal. (2001) modelled the precipitation process of CaC03 in the SFTR via direct solution of the coupled mass and population balances and CFD in order to predict flow regimes, induction times and powder quality. The fluid dynamic conditions in the mixer-segmenter were predicted using CFX 4.3 (Flarwell, UK). 

Note that the terra e/D is the relative roughness from Figure 2-11. The solution of the above equation is trial and error. Colebrook [6] also proposed a direct solution equation that is reported [7] to have... 

The equation proposed by Churchill [8] is also a direct solution with good accuracy [7]. 

Since the viscosity correction factor is dependent upon the actual orifice area, direct solution is not possible and a trial orifice size must be found before Ihe Ku can be determined accurately. 

The direct-solution method of Akers and Wade [1] is among several which attempt to reduce the amount of trial-and-error solutions. This has been accomplished and has proven quite versatile in application. The adaptation outlined modifies the symbols and rearranges some terms for convenient use by the designer [3]. Dew point and bubble point compositions and the plate temperatures can be determined directly. Constant molal overflow is assumed, and relative volatility is held constant over sections of the column. 

Martinez-Ortiz, J. A., and D. B. Manley, Direct Solution of the Isothermal Gibbs-Duhem Equation for Multicomponent Systems, Ind. Eng. Chem. Process Des. Dev., 17, 3, (1978) p. 346. 

A valuable feature of the Nin/Crn-mediated Nozaki-Takai-Hiyama-Kishi coupling of vinyl iodides and aldehydes is that the stereochemistry of the vinyl iodide partner is reflected in the allylic alcohol coupling product, at least when disubstituted or trans tri-substituted vinyl iodides are employed.68 It is, therefore, imperative that the trans vinyl iodide stereochemistry in 159 be rigorously defined. Of the various ways in which this objective could be achieved, a regioselective syn addition of the Zr-H bond of Schwartz s reagent (Cp2ZrHCl) to the alkyne function in 165, followed by exposure of the resulting vinylzirconium species to iodine, seemed to constitute a distinctly direct solution to this important problem. Alkyne 165 could conceivably be derived in short order from compound 166, the projected product of an asymmetric crotylboration of achiral aldehyde 168. 

Activating agents, such as trifluoroacetic anhydride 1,1 -carbonyldiimidazolc carbodiimides sulfonyl, tosyl, and picryl chlorides and a range of phosphorus derivatives can promote direct solution reactions between dicarboxylic acids and diols or diphenols in mild conditions. The activating agents are consumed during the reaction and, therefore, do not act as catalysts. These so-called direct polycondensation or activation polycondensation reactions proceed via the in situ transformation of one of the reactants, generally the carboxylic acid, into a more... 

The result is rather disappointing the deconvolved image is even worse than the observed one It is easy to understand what happens if we consider the Fourier transform of the direct solution ... 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

In many problems for which no direct solution can be obtained, there is a wave equation which differs but slightly from one that can be solved analytically. As an example, consider die hydrogen atom, a problem that was resolved in Section 6.6. Suppose now that an electric field is applied to the atom. The energy levels of the atom are affected by the field, an example of the Stark effect. If die field (due to the potential difference between two electrodes, for example) is gradually reduced, the system approaches that of the unperturbed hydrogen atom. 

This hybrid approach can significantly extend the domain of applicability of the AIMS method. The use of interpolation significantly reduces the computational effort associated with the dynamics over most of the timescale of interest, while regions where the PESs are difficult to interpolate are treated by direct solution of the electronic Schrodinger equation during the dynamics. The applicability and accuracy of the method was tested using a triatomic model collisional quenching of Li(p) by H2 [125], which is discussed in Section III.A below. 

The simplified procedure with a direct solution can also be used. The excess head loss resulting from the pipe length is given by Equation 4-30. The friction factor/has already been determined ... 

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