Appendix—Problem Sets

Extensive tables of standard heats of formation are available, but they are not all at the same reference temperature. The most convenient are the compilations known as the JANAF [1] and NBS Tables [2], both of which use 298 K as the reference temperature. Table 1.1 lists some values of the heat of formation taken from the JANAF Thermochemical Tables. Actual JANAF tables are reproduced in Appendix A. These tables, which represent only a small selection from the JANAF volume, were chosen as those commonly used in combustion and to aid in solving the problem sets throughout this book. Note that, although the developments throughout this book take the reference state as 298 K, the JANAF tables also list A// for all temperatures. 

The thermochemical data for the chemical compounds that follow in this appendix are extracted directly from the JANAF tables [ JANAF thermochemical tables, 3rd Ed., Chase, M. W., Jr., Davies, C. A., Davies, J. R., Jr., Fulrip, D. J., McDonald, R. A., and Syverud, A. N.,./. Phys. Chem. Ref. Data 14, Suppl. 1 (1985)]. The compounds chosen from the numerous ones given are those believed to be most frequently used and those required to solve some of the problem sets given in Chapter 1. Since SI units have been used in the JANAF tables, these units were chosen as the standard throughout. Conversion to cgs units is readily accomplished by use of the conversion factors in this appendix (Table Al). Table A2 contains the thermochemical data. 

Problems are grouped into three categories. Answers to odd-numbered paired problems are provided in Appendix G they enable students to check the answer to the first problem in a pair before tackling the second problem. The Additional Problems, which are unpaired, illustrate further applications of the principles developed in the chapter. The Cumulative Problems integrate material from the chapter with topics presented earlier in the book. We integrate more challenging problems throughout the problems sets and identify them with asterisks. 

Solutions to about one-quarter of the problems in the problem sets are in the Appendix. 

This appendix is organized as follows. First we present a simple iterative algorithm. Then we show that our initial multivectorial problem, set by Eqs. (15a) and (15b), can be translated into a simpler one, which may be solved by a straightforward generalization of the previous iterative algorithm. Finally, we review the technical aspects of our algorithm in detail. 

Problem Sets Each chapter includes roughly 100 problems and exercises, spanning a wide range of difficulty. Most of these exercises are identified with specific sections to provide the practice that students need to master material fi-om that section. Each chapter also includes a number of Additional Problems, which are not tied to any particular section and which may incorporate ideas fi-om multiple sections. Focus on Problem Solving exercises follow, as described earlier. The problems for most chapters conclude with Cumulative Problems, which ask students to synthesize information from the current chapter with what they ve learned fi-om previous chapters to form answers. Answers for all odd-numbered problems appear at the end of the book in Appendix K... 

As presented, the Roothaan SCF proeess is earried out in a fully ab initio manner in that all one- and two-eleetron integrals are eomputed in terms of the speeified basis set no experimental data or other input is employed. As deseribed in Appendix F, it is possible to introduee approximations to the eoulomb and exehange integrals entering into the Foek matrix elements that permit many of the requisite Fj, y elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level eoulomb interaetion integrals that ean be eomputed in an ab initio manner. This approaeh forms the basis of so-ealled semi-empirieal methods. Appendix F provides the reader with a brief introduetion to sueh approaehes to the eleetronie strueture problem and deals in some detail with the well known Hiiekel and CNDO- level approximations. 

The book provides a comprehensive set of examples and case studies that cover a wide variety of process plant situations. Some of these are intended to illustrate the range of situations where human error has occurred in the CPI (see Appendix 1). Other examples illustrate specific techniques (for example. Chapter 4 and Chapter 5). Chapter 7 contains a number of extended case studies intended to illustrate tedmiques in detail and to show how a range of different techniques may be brought to bear on a specific problem. 

Classified problems that start the set and are grouped by type under a particular heading that indicates the topic from the chapter that they address. The classified problems occur in matched pairs, so the second member illustrates the same principle as the first. This allows you more than one opportunity to test yourself. The second problem (even-numbered) is numbered in color and answered in Appendix 6. If your instructor assigns the odd problems without answers for homework, wait until the problem solution is discussed and solve the even problem to satisfy yourself that you understand how to solve the problem of that type. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The problem of controlling the outcome of photodissociation processes has been considered by many authors [63, 79-87]. The basic theory is derived in detail in Appendix B. Our set objective in this application is to maximize the flux of dissociation products in a chosen exit channel or final quantum state. The theory differs from that set out in Appendix A in that the final state is a continuum or dissociative state and that there is a continuous range of possible energies (i.e., quantum states) available to the system. The equations derived for this case are... 

More specifically, the basic notions of a Turing Machine, of computable functions and of undecidable properties are needed for Chapter VI (Decision Problems) the definitions of recursive, primitive recursive and partial recursive functions are helpful for Section F of Chapter IV and two of the proofs in Chapter VI. The basic facts regarding regular sets, context-free languages and pushdown store automata are helpful in Chapter VIII (Monadic Recursion Schemes) and in the proof of Theorem 3.14. For Chapter V (Correctness and Program Verification) it is useful to know the basic notation and ideas of the first order predicate calculus a highly abbreviated version of this material appears as Appendix A. 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

This set of equations can be solved by a variety of approaches. Historically they were solved analytically by a separation-of-variables method, which is tedious, time-consuming, and, for most, an error-prone task. The results presented here were computed using a finite-volume discretization of the momentum equation on a 10 by 10 mesh, which was solved iteratively in a spreadsheet. The programming time was a couple of hours, and the solution is found in about a minute on a typical personal computer. The results are accurate to within one percent of the exact series solutions. The details of the spreadsheet programming for this problem are included in an appendix. 

In the earlier chapters of this book, most problems can be solved with relatively little programming effort using a spreadsheet setting (Appendix D). However, we have not found a similar implementation that can handle the differential-algebraic, method-of-lines al-... 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

E. feU (a) Using the ion-pair equilibrium constant in Appendix J, with activity coefficients = 1, find the concentrations of species in 0.025 M MgS()4. Hydrolysis of the cation and anion near neutral pH is negligible. Only consider ion-pair formation. You can solve this problem exactly with a quadratic equation. Alternatively, if you use SOLVER, set Precision to le-6 (not le-16) in the SOLVER Options. If Precision is much smaller. SOLVER does not find a satisfactory solution. The success of SOLVER in this problem depends on how close your initial guess is to the correct answer. 

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