Approximations techniques

in order to speed up and simplify the calculation, the faults and events in a fault tree are sometimes assumed to be mutually exclusive and independent. Under this assumption, probabilities for the OR gates are 

Problem A motion subsystem consists of a motor and a power source. If steady state unavailability of a motor is 0.01 and steady state unavailability of the power source is 0.001, what is the steady state unavailability of the subsystem  

Problem Three thermocouples are used to sense temperature in a reactor. The three signals are wired into a safety PLC, and a trip will occur if only one of the sensors indicates a trip. The probability of failure in the safe mode (causing a spurious trip) for a one-year mission time is 0.005. What is the probability of a spurious (false) trip  

Solution An expanded version of Equation 5-6 is needed (see Appendix B). 

Problem Three thermocouples are used to sense temperature in a reactor as in Example 5-7. Use an approximation technique to estimate the probability of subsystem failure. 

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At this point we may apply well-known approximation techniques. For each decomposition of W, i.e., H = "Hi-I-H2+-the corresponding Lie-generator decomposes accordingly... 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

Researchers must be particularly cautious when using one estimated property as the input for another estimation technique. This is because possible error can increase significantly when two approximate techniques are combined. Unfortunately, there are some cases in which this is the only available method for computing a property. In this case, researchers are advised to work out the error propagation to determine an estimated error in the final answer. 

The calculation of overall heat transfer coefficient U using the equations jireviously presented can be rather tedious. Fleat transfer specialists have computer programs to calculate this value. There are some quick approximation techniques. Table 2-8 comes from the Gas Processors Suppliers Association s Engineering Data Book and gives an approximate value of U for shell and tube heat exchangers. 

Chapter 6 (page 122) the relative accuracies of various model chemistries is discussed in Chapter 7 (page 146). See Appendix A for a discussion of the approximation techniques used by the various methods. 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

An approximation technique can greatly simplify calculations when the change in composition (x) is less than about 5% of the initial value. To use it, assume that x is negligible when added to or subtracted from a number. Thus, we can replace all expressions like A + x or A - 2x, by A. When x occurs on its own (not added to or subtracted from another number), it is left unchanged. So, an expression such as (0.1 — 2x)2x simplifies to (0.1 )2x, provided that 2x 0.1 (specifically, if x < 0.005). At the end of the calculation, it is important to verify that the calculated value of x is indeed smaller than 5% of the initial values. If it is not, then we must solve the equation without making an approximation. 

For the more general case of arbitrary rate constants, the analysis is more complex. Various approximate techniques that are applicable to the analysis of reactions 5.4.1 and 5.4.2 have been described in the literature, and Frost and Pearson s text (11) treats some of these. One useful general approach to this problem is that of Frost and Schwemer (12-13). It may be regarded as an extension of the time-ratio method discussed in Section 5.3.2. The analysis is predicated on a specific choice of initial reactant concentrations. One uses equivalent amounts of reactants A and B (A0 = 2B0) instead of equi-molal quantities. 

It should be noted here that since the original work done at Mobil was completed, there have been new developments published in the literature. Ishida and Wen (1968) analytically solved a special case for the transition region when the reaction rate does not depend on the local solids (coke) concentration. Wen (1968) has also numerically solved the more general problem for certain kinetic forms, and Amundson and co-workers have done much work on the diffusion and reaction in the boundry layer about a carbon particle (Caram and Amundson, 1977). We will not attempt to review the literature or compare the more accurate numerical solutions with our ad hoc approximation technique. However, we note that our technique was simple, fit the experimental and commercial data extremely well, and provided us with valuable insight and understanding... 

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

The Q-factor approach is based upon the weight-to-size ratios (Q-factors) of the calibration standard and the polymer to be analyzed. The Q-factors are employed to transform the calibration curve for the chemical type of the standards (e.g. polystyrene) into a calibration curve for the chemical type of polymer under study. The inherent assumption In such a calibration approach is that the weight-to-size ratio is not a function of molecular weight but a constant. The assumption is valid for some polymer types (e.g. polyvinylchloride) but not for many polymer types. Hence the Q-factor method is generally referred to as an approximation technique. 

The purpose of this study was to evaluate the linear calibration technique employing a single polydisperse standard and the search algorithm described above for non-aqueous and aqueous SEC. Comparison of this calibration technique to peak position, universal calibration, and Q-factor approximation techniques which make use of a series of narrow MWD polystyrene standards was also carried out. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

In the point force approximation technique (see Section Ic), Burgers (BIO) suggested a polynomial approximation for the distributed line force along the axis of a body of large aspect ratio ... 

Here we consider three theoretical approaches. As for rigid spheres, numerical solutions of the complete Navier-Stokes and transfer equations provide useful quantitative and qualitative information at intermediate Reynolds numbers (typically Re < 300). More limited success has been achieved with approximate techniques based on Galerkin s method. Boundary layer solutions have also been devised for Re > 50. Numerical solutions give the most complete and... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

The remaining four chapters discuss theoretical approaches and considerations which have been suggested to include the effects of many-body complications, to use approximate techniques, to use more realistic continuum hydrodynamic equations than the diffusion equation, and to use more satisfactory statistical mechanical descriptions of liquid structure. This work is still in a comparatively early stage of its development. There is a growing need for more detailed experiments which might probe the effects anticipated by these studies. 

Finally, several attempts have been made to solve the Debye—Smoluchowski equation in the time domain using approximate techniques based on uniformly small perturbations (Montroll [74], Abell and Mozumder... 

Now this equation cannot be solved by anything but approximate techniques. To derive the equation used by Clifford et al. [442], let us allow ail the reactants to be statistically independent of each other. The density nN is the product of the individual one-reactant densities, n, (rh fir,-0, f°)... 

Much interest has developed on approximate techniques of solving quantum mechanical problems because exact solutions of the Schrodinger equation can not be obtained for many-body problems. One of the most convenient of such approximations for the solution of many-body problems is the application of the variational method. For instance, with approximate eigen-functions p , the eigen-values of the Hamiltonian H are En... 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

The most general flame problem has been formulated by Klein in such a way that a successive approximation technique may usually be used to... 

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