General algebraic method

Simple material-balance problems involving only a few streams and with a few unknowns can usually be solved by simple direct methods. The relationship between the unknown quantities and the information given can usually be clearly seen. For more complex problems, and for problems with several processing steps, a more formal algebraic approach can be used. The procedure is involved, and often tedious if the calculations have to be done manually, but should result in a solution to even the most intractable problems, providing sufficient information is known. 

Algebraic symbols are assigned to all the unknown flows and compositions. Balance equations are then written around each sub-system for the independent components (chemical species or elements). 

Material-balance problems are particular examples of the general design problem discussed in Chapter 1. The unknowns are compositions or flows, and the relating equations arise from the conservation law and the stoichiometry of the reactions. For any problem to have a unique solution it must be possible to write the same number of independent equations as there are unknowns. 

Consider the general material balance problem where there are Ns streams each containing Nc independent components. Then the number of variables, N, is given by  

If Ne independent balance equations can be written, then the number of variables, Nd, that must be specified for a unique solution, is given by  

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Algebraic Method for Concentrated Gases When the feed gas is concentrated, the absorption factor, which is defined in general as A = where K = y°/x, can vary throughout the tower owing... 

The foregoing algebraic method can be generalized using optimization techniques. A particularly useful approach is the transshipment formulation (Papoulias and... 

The only drawback in using this method is that any numerical errors introduced in the estimation of the time derivatives of the state variables have a direct effect on the estimated parameter values. Furthermore, by this approach we can not readily calculate confidence intervals for the unknown parameters. This method is the standard procedure used by the General Algebraic Modeling System (GAMS) for the estimation of parameters in ODE models when all state variables are observed. 

The double degeneracy of the 0(2) case corresponds to the fact that the algebraic method describes in this case two Morse potentials related to each other by a reflection around x = 0. This is a peculiar feature of one-dimensional problems, and it does not appear in the general case of three dimensions. If one uses the 0(2) basis for calculations, this peculiarity can be simply dealt with by considering only the positive branch of M. 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

As in scattering theory in general, one can treat the role of V in either a time independent or a time dependent point of view. The latter is simpler if the perturbation V is either explicitly time dependent or can be approximated as such, say by replacing the approach motion during the collision by a classical path. Algebraic methods have been particularly useful in that context,2 where an important aspect is the description of a realistic level structure for H0. Figure 8.3 is a very recent application to electron-molecule scattering. 

If one assumes that S/8h is independent of x, one arrives again at Eq. (58). If a particular form is chosen for the function F, as one proceeds in the method of polynomials, the calculation of the constants A, B, and E becomes possible. While in this particular problem one can follow a parallelism between the algebraic method and the method of polynomials, the same parallelism can no longer be identified in the other examples examined. It is worth emphasizing that the use of the boundary layer thickness concept in the algebraic method does not imply the existence of a similarity solution. In general, the algebraic method interpolates between the two similarity solutions which are valid in the two asymptotic cases. 

In Chapter 2 we explored some of the methods used for finding the roots of algebraic equations in the form y =f x). In all of the examples given we were seeking to determine the value of an unknown (typically the value of the independent variable, x) that resulted in a particular value for j, the dependent variable. In general, the methods discussed can be used to solve algebraic equations where the dependent variable takes a value other than zero, because the equation can always be rearranged into a form in which y = 0. For example, if we seek the solution to the equation ... 

Algebraic Method for Concentrated Gases When the feed gas is concentrated, the absorption factor, which is defined in general as A = Lm/KGm and where K = t/°/x, can vary throughout the tower due to changes in temperature and composition. An approximate solution to this problem can be obtained by substituting the effective adsorption factors A, and A derived by Edmister [Ind. Eng. Chem. 35, 837 (1943)] into the equation... 

This text also provides a general introduction to vector and matrix -algebraic methods that are highly effective for thermodynamics as well as applications in quantum mechanics and other advanced topics in physical chemistry. 

Finding canonical forms for each of the A s, is an easier task than for any general form A. A technique has been developed at Princeton [189-191] for finding such canonical forms using an algebraic method in nonlinear perturbation theory. For the application of this technique the leading operator Aq must be chosen to be in canonical form. The basic approach is then to find a non-linear transformation C such that the resultant operator. 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

It should be emphasized here that while the discussion of contemporary techniques for decoding spectra for information on the internal molecular motions will largely concentrate on spectroscopic Hamiltonians and bifurcation analysis, there are distinct, but related, contemporary developments that show great promise for the future. For example approaches using advanced algebraic techniques [35, 36] for alternative ways to build the spectroscopic Hamiltonian, and hierarchical analysis using techniques related to general classification methods [37]. 

In excited-state spectroscopies, including fluorescence spectroscopy, spectroscopic intensity is usually linear in functions of each of three or more independent variables, so that a three-way array of data can be fit with a trilinear model. The presence of three or more linear relationships makes algebraic methods for resolving the spectra and other properties of individual components substantially more powerful than in the case of two linear relationships. The use of a general trilinear model is sometimes known as three-way factor analysis, three-mode factor analysis, or threemode principal component analysis. For a review of the mathematics and application to spectroscopy, see our survey article. ... 

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