Graphical problems

Convergent paths, where separately elaborated systems are condensed, are preferable to linear paths (compare A (convergent) and e (linear) in Figure 4). Optimal convergence is, of course, not only a graphical problem but also one of yield (57) ... 

I dismissed the original idea entirely, and started to approach the graphic problem in a radically different way. That was when things started to happen. All kinds of dif-... 

They are not good at solving problems where the solution is needed for very close steps (i.e., graphical problems), that is with h much smaller than the ones required by the algorithm s precision. 

This new possibility is very useful even when we need some values of y for several points that are very close to each other (i.e., for graphical problems). Without this possibility (e.g., with the Runge-Kutta algorithms), an integration with a very small step is needed, but the step is much too small if we look at the algorithm s accuracy and stability. 

Professor Q Notice that the equilibrium equations required to solve this problem are actually quite straightforward. As we have seen so many times before in this course, the key to success is careful construction of our all-important graphical problem-solving tool and that tool is... [pause] Everybody "... 

Spencer, H. C. Engineering Graphics Problems. New York Macmillan, 1981. 

The data from Table 7.4 are presented graphically in Fig. 7.11. The optimal is at 10°C, confirming the initial value used for this problem in Chap. 6. 

Structurally benzene is the simplest stable compound having aromatic character, but a satisfactory graphical representation of its formula proved to be a perplexing problem for chemists. Kekule is usually credited with description of two resonating structures which. 

To obviate the tedious graphical iategration, a simplified design procedure was developed on the basis of Colburn s analytical solution, equation 56. Substitution of the ratio presents no problem because this ratio stays fairly constant ia the tower at the low coaceatratioas for which Figure 12 is... 

An important use of the triangular equiHbrium diagram is the graphical solution of material balance problems, such as the calculation of the relative amounts of equiHbrium phases obtained from a given overall mixture composition. As an example, consider a mixture where the overall composition is represented by point M on Figure 2a. If the A-rich phase is denoted by point R (raffinate) and the B-rich phase is denoted by point E (extract), it can be shown that points R, M, and E are coUinear, and also... 

Alternative approaches are to be found in the hterature. Derivations of the above equations are given in numerous texts (2,10—12), which also describe graphical or analytical solutions to the problem. Many of these have direct analogues in other separation processes such as distillation (qv) and hquid—hquid extraction, and use plots such as the McCabe-Thiele diagram or Ponchon-Savarit diagram. 

Image Files. Only the physical problem of moving or storing files has been discussed herein. The format in which the data is actually stored or transmitted is another part of the process. There are a number of file formats, many of which are proprietary, developed to suit the needs of a particular computer platform or software appHcation. There has been some effort to standardize to a few file types, but many others are in use. A thorough discussion of graphics file formats is available (20). 

Whereas Hquid separation method selection is clearly biased toward simple distillation, no such dominant method exists for gas separation. Several methods can often compete favorably. Moreover, the appropriateness of a given method depends to a large extent on specific process requirements, such as the quantity and extent of the desired separation. The situation contrasts markedly with Hquid mixtures in which the appHcabiHty of the predominant distiHation-based separation methods is relatively insensitive to scale or purity requirements. The lack of convenient problem representation techniques is another complication. Many of the gas—vapor separation methods ate kinetically controUed and do not lend themselves to graphical-phase equiHbrium representations. In addition, many of these methods require the use of some type of mass separation agent and performance varies widely depending on the particular MSA chosen. 

It is likely that there will always be a distinction between the way CAD/CAM is used in mechanical design and the way it is used in the chemical process industry. Most of the computations requited in mechanical design involve systems of linear or lineatizable equations, usually describing forces and positions. The calculations requited to model molecular motion or to describe the sequence of unit operations in a process flow sheet are often highly nonlinear and involve systems of mixed forms of equations. Since the natures of the computational problems are quite different, it is most likely that graphic techniques will continue to be used more to display results than to create them. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Various numerical and graphical methods are used for unsteady-state conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berhn, 1924). These methods are very useful because any form of initial temperature distribution may be used. 

To integrate Eq. (11-3), and AT must be known as functions of Q. For some problems, varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate and AT at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient from Eq. (11-2) and define a corresponding mean value of AT,n, such that... 

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