Algebraic methods

Binary systems of course can be handled by the computer programs devised for multicomponent mixtures that are mentioned later. Constant molal overflow cases are handled by binary computer programs such as the one used in Example 13.4 for the enriching section which employ repeated alternate application of material balance and equilibrium stage-by-stage. Methods also are available that employ closed form equations that can give desired results quickly for the special case of constant or suitable average relative volatility. 

Minimum Trays. This is found with the Fenske-Underwood equation. 

A mixture of chlorinated phenols can be represented as an equivalent binary with 90% 2,4-dichlorphenol (DCP) and the balance 2,4,6-trichlorphenol with a relative volatility of 3.268. Product purity is required to be 97.5% of the lighter material, and the residue must be below 20% of 2,4-DCP. It is proposed to use a batch distillation with 10 theoretical stages. Vaporization rate will be maintained constant. 

The btms compositions at a particular value of R are found by successive applications of the equations 

Start with = yo The calculations are performed with the given computer program and the results are tabulated. The values of L/L, are found by material balance  

This is less than the constant reflux, / = 0.647, to be found in part b. 

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The algebraic methods of reconstruction give result at incomplete and complete set of initial projection data. But the iterative imhlementation of these methods requires large computing resources. Algebraic method can be used in cases, when the required accuracy is not great. 

This is a comprehensive survey of algebraic methods for internal molecular motions. 

Kellman M E 1995 Algebraic methods in spectroscopy Ann. Rev. Rhys. Chem. 46 395... 

This survey compares algebraic methods with more standard approaches, and the bifurcation approach in this article. Bunker P R 1979 Moleoular Symmetry and Speotrosoopy (New York Academic)... 

Algebraic methods - in these techniques calculation of grid coordinates is based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. 

Algebraic Method for Dilute Gases By assuming that the operating and equilibrium curves are straight hues and that heat effects are negligible. Senders and Brown [Ind. Eng. Chem., 24, 519 (1932)] developed the following equation ... 

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... 

These results are identical to those obtained using the gra]riiical and the algebraic methods. 

Underwood Algebraic Method Adjacent Key Systems Variable a... 

Underwood Algebraic Method Split Key Systems Constant Volatilily [72]... 

Time from start of distillation to fill receiver, or value of relative volatility (Underwood Parameter) to satisfy Underwood Algebraic Method... 

Underwood Algebraic Method, 71 Example 8-23 Minimum Reflux Ratio Using Underwood Equation, 73 Minimum Reflux Colburn Method, 74 Example 8-24 Using the Colburn Equation to Calculate Minimum Reflux Ratio,... 

BalK83 Balasubramanian, K. Operator and algebraic methods for NMR spectroscopy II. NMR projection operations and spin functions. J. Chem. Phys. 78 (1983) 6369-6376. 

The reaction coordinate defined in Section 2.8 provides an algebraic method for calculating equilibrium concentrations. For a single reaction. 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

The formal, algebraic, method. The presence of recycle implies that some of the mass balance equations will have to be solved simultaneously. The equations are set up with the recycle flows as unknowns and solved using standard methods for the solution of simultaneous equations. 

The next question is how to find the partial fractions in Eq. (2-25). One of the techniques is the so-called Heaviside expansion, a fairly straightforward algebraic method. We will illustrate three important cases with respect to the roots of the polynomial in the denominator (1) distinct real roots, (2) complex conjugate roots, and (3) multiple (or repeated) roots. In a given problem, we can have a combination of any of the above. Yes, we need to know how to do them all. 

When the algebraic methods are used, care must be taken that the constraints are obeyed. This usually means following a boundary until the search leaves the vicinity of the constraints. This should be kept in mind while reading about the various procedures. 

Again, equation 66-A16 is cubic in Z4 and can be solved by algebraic methods. For higher powers of the variable we can derive similar expressions. After the sixth power, algebraic methods are no longer available to solve for the Z , but after evaluating the summations, computerized approximation methods can be used. 

There is a need for large-scale differential-algebraic methods for simulating systems at multiple scales (e.g., fluid mechanics and molecular dynamics), a capability that is still at a very early stage. 

Find equations for xlf jc2, and x3 in terms of zl9 z2, z3. Use an algebraic method first check the result using A-1. 

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... 

Three-dimensional coupled roto-vibrators by algebraic methods... 

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