Graphic techniques

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

Experimentally, Rg is measured at a series of different C2 s and 0 s, which makes the extrapolations of Kc2/Rg at constant 0 to C2 = 0 and of Kc2/Rg at constant C2 to 0 = 0 equally feasible. In the next section we shall examine a specific graphical technique which combines these two extrapolations in a single procedure. 

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. 

CHAPTER FOUR Graphical techniques for mass integration... 

A fault tree is a graphic technique used to analyze complex systems. Fault tree analysis attempts to describe how and why an accident or other undesirable event lias occurred. It may also be used to describe how and why an accident or otlier undesirable event could take place. 

For shells in series it is necessary to develop a weighted MTD correction factor, and a graphical technique is presented by Gulley. ... 

Zanker has presented a graphical technique for determining the fouling resistance (factor) for process or water fluid systems based on selected or plant data measurements, as shown in Figures 10-43A, 10-43B, and 10-43C. The design determination procedure presented by ZankeD is quoted here and used by permission from Hydrocarbon Processing... 

The upper envelope of E as a function of R maximized over p and p can now be found. We first maximize E over p for each value of p numerical techniques or Eqs. (4-104) and (4-106) can be used. Then we maximize this result over p either by using graphical techniques to find the upper envelope or by using Eqs. (4-97) and (4-98). This optimum E,R curve is generally called the reliability curve for the given channel. 

This book, and many standard texts, emphasizes graphical techniques for htting data. These methods give valuable qualitative insights that may be missed with too much reliance on least-squares analysis. 

The biplot in Fig. 37.3 has been constructed from the factor scores of the 12 compounds and the factor loadings of the five physicochemical and biological variables [42,43]. (The biplot graphic technique is explained in Section 31.2.) It is... 

A graphic technique may be obtained from the polynomial equations, as represented in Fig. 6. Figure 6a shows the contours for tablet hardness as the levels of the independent variables are changed. Figure 6b shows similar contours for the dissolution response, t50%. If the requirements on the final tablet are that hardness be 8-10 kg and t o% be 20-33 min, the feasible solution space is indicated in Fig. 6c. This has been obtained by superimposing Fig. 6a and b, and several different combinations of X and X2 will suffice. 

The last step, which concerns the graphic techniques, may be requested at any time after the regression analysis has been performed and will probably be appropriate at several different stages of a project. 

Van Winkle, M., Todd, W.G. Minimizing Distillation Costs Via Graphical Techniques, Chemical Engineering, Mar. 6, 1972, p. 105. 

The third type of methods treats time in an exact manner by allowing it to vary in search of a true optimum. Worthy of mention at this stage is that none of the graphical techniques bears this crucial capability in batch chemical plants. Ultimately, mainly mathematical techniques are used in order to treat time exactly. This assertion is further justified later in this textbook when graphical techniques are compared to mathematical techniques using a typical problem. 

A Graphical Technique for Wastewater Minimisation in Batch Processes... 

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