Example Problem

Some of the problems often encountered during ultrasonic inspection of plane specimens are also found on cylindrical specimens. For example, problems associated with the directional characteristic of the ultrasonic transducer. Furthermore, the discontinuity influences the shape and propagation direction of a reflected pulse, causing wave mode transformation. In addition, the specimen influences the shape and amplitude of the reflected pulse by sound absorption. 

A marvellous and rigorous treatment of non-relativistic quantum mechanics. Although best suited for readers with a fair degree of mathematical sophistication and a desire to understand the subject in great depth, the book contains all of the important ideas of the subject and many of the subtle details that are often missing from less advanced treatments. Unusual for a book of its type, highly detailed solutions are given for many illustrative example problems. 

It is wise to gain some experienee using the SC rules, so let us eonsider a few illustrative example problems. 

Each example problem includes a detailed solution that helps students in applying the chapter s material to practical problems. 

Over 50 solved example problems are sprinkled throughout the book. 

Combinatorial. Combinatorial methods express the synthesis problem as a traditional optimization problem which can only be solved using powerful techniques that have been known for some time. These may use total network cost direcdy as an objective function but do not exploit the special characteristics of heat-exchange networks in obtaining a solution. Much of the early work in heat-exchange network synthesis was based on exhaustive search or combinatorial development of networks. This work has not proven useful because for only a typical ten-process-stream example problem the alternative sets of feasible matches are cal.55 x 10 without stream spHtting. 

Example 3 Calculation of TG Method The TG method will he demonstrated hy using the same example problem that was used above for the approximate methods. The example column was analyzed previously and found to have C -I- 2N + 9 design variables. The specifications to be used in this example were also hstedat that time and included the total number of stages (N = 10), the feed-plate location (M = 5), the reflux temperature (corresponding to saturated liquid), the distillate rate (D = 48.9), and the top vapor rate (V = 175). As before, the pressure is uniform at 827 kPa (120 psia), but a pressure gradient could be easily handled if desired. 

Top-down calculations for tbe example problem are shown in Table 13-10 and bottom-up calculations in Table 13-11. Top-down and bottom-up calculations have provided values of and respectively. For a bubble-... 

Calculated values of d from tbe first column iteration in tbe example problem are as follows ... 

The equations are presented in their primitive form to keep them more universal. Consistent units must be used, as appropriate, at the time of application. The example problems will include conversion values for the units presented. The symbol g will be used for the universal gravity constant to maintain open form to the units. 

Thus, the length of a settling chamber is inversely proportional to the square of the particle diameter. For example, if it is desirable to separate out particles that are two times smaller than the selected size, then the length of the chamber must be increased by a factor of four. The equation may also be used to determine the smallest particle diameter that can be removed by a chamber of specified dimensions. The following example problem illustrates some of these design principles. 

For a unidirectionally reinforced composite material subject to uniaxial load at angle 0 to the fibers (the example problem in Section 2.9.1 on the maximum stress criterion), the allowable stresses can be found from the allowable strains X, Y , etc., in the following manner. 

Table 1 -1 describes a gas field. The example problems that are worked in many of the sections of this text are for sizing the individual pieces ot equipment needed for this field. 

D. Hendershot, "A Simple Example Problem Illustrating tlie Methodology of Chemical Process Quantitative Risk Assessment," paper presented at AICliE Mid-Atlantic Region "Day in Industry" for Chemical Engineering Faculty, Apr. 15, 1988. 

Figure 2-28. Line sizing sheet for example problem, Example 2-7.

However, the example problem does not require a calculation for vapor density. Instead, the superficial vapor mass velocity G can be substituted into Equation 8-73 because ... 

Wilke-Chang reported the recommended values for i ) as follows water, 2.6 benzene, heptane and ether, 1.0 methanol, 1.9 ethanol, 1.5 una.ssociated solvents, 1.0. I he mixture parameter for the example problem is considered unity. 

Figure 11-51. Example problem 300-ton ammonia refrigeration system. Example 11-7.

This method is explained by solving two example problems. 

As an energy source, nuclear fusion possesses several additional advantages over nuclear fission. In particular, light isotopes suitable for fusion are far more abundant than the heavy isotopes required for fission. You can calculate, for example (Problem 73), that the fusion of only 2 X 10-9 % of the deuterium ( H) in seawater would meet the total annual energy requirements of the world. 

The intramolecular Hiyama reaction is excellently suited for the construction of carbocyclic and heterocyclic ring systems of different size. In most examples, problems of induced diastereoselectivity are involved (Sections 1.3.3.3.9.2.2. and D.2.3.). In the total synthesis of the antitumor germacranolide ( )-costunolide from ( ,.E)-farnesol, the correctly substituted ( , )-cyclodecadiene moiety was constructed by an intramolecular Hiyama reaction22. 

Nonlinear least-squares programs have made time lag methods much less important. They are less accurate, for one thing. For another, the linearity of the appropriate plots, although a necessary consequence of first-order kinetics, does not constitute a proof of first-order kinetics. Certain other kinetic equations also lead to linear plots of either function. For example, Problem 2-11 presents data for a product-catalyzed reaction. The data in this case can be plotted linearly according to the Guggenheim equation, although the reaction does not follow first-order kinetics and the plot of In [A] versus time is decidedly nonlinear. 

The current undergraduate curriculum in chemical engineering, although it provides an excellent conceptual base for graduates who move into the electronics industries, could be improved by the introduction of instractional material and example problems relevant to the challenges outlined in this chapter. This would not require the creation of new courses, but rather the provision of material to emich existing ones. This theme is echoed, more broadly, in Chapter 10. 

---