Worked example

Several worked examples of identifying chemical reactivity hazards are presented in this chapter. The objective of this chapter is to illustrate the use of the Preliminary Screening Method for Chemical Reactivity Hazards (Chapter 3) by way of a few, relatively simple examples that show different decision paths. 

In this method, the pressure gradient across a packed bed of known voidage is measured as a function of flow rate. The diameter we calculate from the Carman-Kozeny equation is the arithmetic mean of the surface distribution (see Worked Example 6.1 in Chapter 6). 

This method gives a volume distribution and measures a diameter known as the laser diameter. Particle size analysis by laser diffraction is very common in industry today. The associated software permits display of a variety of size distributions and means derived from the original measured distribution. 

The whole of the moving stream should be taken for many short time increments. 

Since the eventual sample size used in the analysis may be very small, it is often necessary to split the original sample in order to achieve the desired amount for analysis. These sampling rules must be applied at every step of sampling and sample splitting. 

Detailed description of the many devices and techniques used for sampling in different process situations and sample dividing are outside the scope of this chapter. However, Allen (1990) gives an excellent account, to which the reader is referred. 

A non-spherical FCC catalyst particle falls in a column of nitrogen at ambient conditions and attains a terminal fall velocity ut. The density and viscosity of the nitrogen are pf= 1.2 x 10 3 g cm 3 and p = 1.8 x 10 4 g cm-1 s 1 respectively. Calculate the terminal velocity of the particle given its physical characteristics, dp=70 pm, sphericity (f) = 0.7 

Min particle diameter Max particle diameter Mean particle diameter Mass fraction % Mass fraction in sieve fractionj mean diameter Cumulative mass fraction 

Mean particle diameter at 16% 84% of cumulative mass fraction 

A fine spherical silica-based catalyst with the physical characteristics reported below is being investigated for the development of a new fluidised bed process. Fundamental parameters such as the bed voidage and minimum fluidisation velocity, umf, have to be determined. 

Knowing the voidage, the number of particles can be determined. The total volume occupied by the particles in the bed is equal to  

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The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

Galerkin finite element procedure - a worked example... 

To illustrate the basic concepts described in this section we consider the following worked example. 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

Calculation of pore size distribution (Roberts Method"). Worked example from desorption branch of nitrogen isotherm on... 

Guedens, W. J. Yperman, J. Mullens, J. et al. Statistical Analysis of Errors A Practical Approach for an Undergraduate Ghemistry Lab, Part 1. The Goncept, /. Chem. Educ. 1993, 70, 776-779 Part 2. Some Worked Examples, /. Chem. Educ. 1993, 70, 838-841. 

Specific details, including worked examples, are found in the sections of this chapter covering individual gravimetric methods. 

An emphasis on practical applications. Throughout the text applications from organic chemistry, inorganic chemistry, environmental chemistry, clinical chemistry, and biochemistry are used in worked examples, representative methods, and end-of-chapter problems. 

Having assigned symmetry species to each of the six vibrations of formaldehyde shown in Worked example 4.1 in Chapter 4 (pages 90-91) use the appropriate character table to show which are allowed in (a) the infrared specttum and (b) the Raman specttum. In each case state the direction of the transition moment for the infrared-active vibrations and which component of the polarizability is involved for the Raman-active vibrations. 

The A A2 X Ai, n -n system of formaldehyde (see Section 7.3.1.2) is also electronically forbidden since A2 is not a symmetry species of a translation (see Table A.l 1 in Appendix A). The main non-totally symmetric vibration which is active is Vq, the hj out-of-plane bending vibration (see Worked example 4.1, page 90) in 4q and d transitions. 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

An inventor may estabHsh utiHty by providing several working examples which disclose preparation, appHcation, and even some or all of the benefits of the invention. UtiHty may also be substantiated by merely disclosing several appHcations for the invention. One method of determining the breadth or scope of an invention is to define the invention by only those elements essential to performing the intended task. This definition should then become the broadest cl aim of the patent appHcation. 

Safety, Health, and Loss Prevention in Chemical Processes Student Problems Instmctor s Guide for Undergraduate Engineering Curricula Ha2ard Evaluation Procedures, with Worked Examples, 2nd ed. 

CCPS G-1. 1992. Guidelines for Hazard Evaluation Procedures, Second Edition with Worked Examples. American Institute of Chemical Engineers, Center for Chemical Process Safety. New York. 

The praetieal utilization of linear reetifieation is demonstrated later through a worked example. Fitting statistieal distributions to sample data using the linear reetifieation method ean be found in Ayyub and MeCuen (1997), Edwards and MeKee (1991), Kottegoda and Rosso (1997), Leiteh (1995), Lewis (1996), Metealfe (1997), Misehke (1992), Rao (1992), and Shigley and Misehke (1989). 

The varianee equation ean be solved direetly by using the Calculus of Partial Derivatives, or for more eomplex eases, using the Finite Difference Method. Another valuable method for solving the varianee equation is Monte Carlo Simulation. However, rather than solve the varianee equation direetly, it allows us to simulate the output of the varianee for a given funetion of many random variables. Appendix XI explains in detail eaeh of the methods to solve the varianee equation and provides worked examples. 

The varianee equation provides a valuable tool with whieh to draw sensitivity inferenees to give the eontribution of eaeh variable to the overall variability of the problem. Through its use, probabilistie methods provide a more effeetive way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer ean quiekly foeus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the varianee eontribution of eaeh of the design variables in a stress analysis problem. 

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