Graphical method for solving

In 1954, Brouwer proposed a graphical method for solving these equations. The method has been adopted because of further development by Kroeger and Vink (1956). The method entails dividing the range of... 

For flow with arbitrary exit age distribution E(t) Eq. (144) must be solved directly. A convenient graphical method for doing this has been devised by Schoenemann (S9) who then discusses the application of this method to some industrial reactors. The direct use of Eq. (144) is also illustrated by Levenspiel (L13), Sherwood (S13) and Petersen s (P5) treatment of catalyst-activity levels in regenerator-reactor systems. 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

Try a graphical method and a regression analysis method for solving... 

GRAPHICAL METHODS FOR TWO-COMPONENT SYSTEMS. For systems containing only two components it is possible to solve many mass-transfer problems graphically. The methods are based on material balances and equilibrium relationships some more complex methods require enthalpy balances as well. These more complex methods will be discussed in Chap. 18. The principles underlying the simple graphical methods are discussed in the following paragraphs. Their detailed applications to specific operations are covered in later chapters. 

In this chapter we will deal with the kinetics and equilibrium calculations of heterogeneous systems. Graphical and computational methods for solving equilibrium problems will be presented. The precipitation of calcium carbonate will be discussed in some detail, and the chemistry of phosphorus will be used as a detailed example of several heterogeneous equilibria that are of relevance to natural waters and treatment processes. 

This specific problem shows again the power of following a method for solving problems. Read and understand, represent things graphically, and so on. 

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

By a brilliant physical intuition B. van der Pol succeeded finally (1920) in establishing his equation (which is given in Section 6.11) but, not having any mathematical theory at his disposal, he determined the nature of Ike solution by the graphical method of isoclines. It became obvious that the problem, which was a real stumbling block for many years, had been finally solved, at least in principle. 

Equations 11.21 and 11.22 and the equilibrium relationship are conveniently solved by the graphical method developed by McCabe and Thiele (1925). The method is discussed fully in Volume 2. A simple procedure for the construction of the diagram is given below and illustrated in Example 11.2. 

The SLP subproblem at (4,3.167) is shown graphically in Figure 8.9. The LP solution is now at the point (4, 3.005), which is very close to the optimal point x. This point (x ) is determined by linearization of the two active constraints, as are all further iterates. Now consider Newton s method for equation-solving applied to the two active constraints, x2 + y2 = 25 and x2 — y2 = 7. Newton s method involves... 

They wrote a FORTRAN program which solved all equations but one, that of charge conservation. The pH at electrical neutrality was determined by a graphical method, in which the total positive and negative charge concentrations were calculated and plotted for a series of assumed pH s and the crossing point found. 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

As was already mentioned, in theoretical atomic spectroscopy, while considering complex electronic configurations, one has to cope with many sums over quantum numbers of the angular momentum type and their projections (3nj- and ym-coefficients). There are collections of algebraic formulas for particular cases of such sums [9, 11, 88]. However, the most general way to solve problems of this kind is the exploitation of one or another versions of graphical methods [9,11]. They are widely utilized not only in atomic spectroscopy, but also in many other domains of physics (nuclei, elementary particles, etc.) [13],... 

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