Value Methods

Another option for solving boundary value problems is to treat them like initial value problems. Since a second-order equation can be reduced to two first-order equations, two initial conditions are necessary. One condition will be known at a boundary. Simply assume a value for the other dependent variable at that same boundary, integrate to the other side and check if the required boundary condition is satisfied. If not, change the initial value and repeat the integration. The success of this method depends upon the skill with which you program the iterations from one trial to the next. 

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Kay K G 1994 Semiclassical propagation for multidimensional systems by an initial value method J. Chem. Phys. 101 2250... 

Walton A R and Manolopoulos D E 1996 A new semiclassical initial value method for Franck-Condon spectra Mol. Phys. 87 961... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Future costs of replacement filters and energy are calculated according to the current value method. The final result for a 1 m /s filter with average pressure loss of 200 Pa may be as shown in Table 9.3, if the calculation is based on a 10-year period. 

Amino Acid Analysis Visual Photometric b Value Method... 

Range Mean Range Mean Value Method and software... 

Solution of the equations in (2.8-2.18) proceeds with an adaptive nonlinear boundary value method. The solution procedure has been discussed in detail elsewhere (10) and we outline only the essential features here. Our goal is to obtain a discrete solution of the governing equations on the mesh Af... 

Brunger AT. Assessment of phase accuracy by cross validation the free R value. Methods and applications. Acta Cryst 1993 049 24-36. 

It should be noted that for Example 10-4 none of the evaluation methods that did not take into account the time value of money could differentiate between these plants, but the net present value method not only could differentiate but determined which was best. 

One of the problems with the Net Present Value method of evaluation is that an interest rate must be chosen, and in cases where the timing of incomes and outlays differs greatly, different interest rates will result in different conclusions. 

In Example 10-17 the net present value for plant 1 is zero at an interest rate of 20%. So this is the rate of return. The return for plant 2 is 23.5%. This indicates that plant 2 is superior to plant 1. This method gives a single answer that does not require the advance choice of an interest rate. This means, for Example 10-17, that the Net Present Value method would give a different answer than the Rate of Return method if an interest rate of 5% were used to determine the former. 

Both methods assume that the money earned can be reinvested at the nominal interest rate. Suppose the rates of return calculated are after tax returns and the company is generally earning a 5% or 6% return on investment. Is it reasonable to expect that all profits can be reinvested at 23% or even 20% No, it isn t Yet this is what is assumed in the Rate of Return method. Sometimes the rate of return may be as high as 50%, while a reasonable interest rate is less than 15%. Therefore if a reasonable value for the interest rate has been chosen (this is discussed later in this chapter) and the two methods differ, the results indicated by the Net Present Value method should be accepted. 

The Rate of Return method can be modified to give the same answer as the Net Present Value method. This can be illustrated by considering Example 10-7. Table 10-9 gives the NPV for various interest rates for both alternatives of the example. 

This says plant 1 is best if less than an 8% interest rate is acceptable. Otherwise plant 2 is best. The Net Present Value method gives the same answer. At an interest rate of just below 8% the net present values of the two plants are equal. 

The disadvantage of this method is that if three or more alternatives are being compared the process is time-consuming unless a digital computer is used. First any two projects are compared, then the best is compared with one of those remaining, and this process is continued until all have been considered and only the best remains. Each comparison involves trial-and-error calculations. On the other hand, the Net Present Value method requires only one calculation for each project. 

Determine whether it is more economical for a 150,000,000 lb/yr polystyrene plant to buy styrene in 3,000-ton or 1,000-ton shipments. The cost of shipping is 0.230/ton mile in the former case and 0.260/ton mile in the latter case. The distance to be shipped is 1,250 miles. Assume the former requires a 26-day storage capacity and the latter a 17-day storage capacity. (See example in Chapter 3.) The value of money is 10%. Use the tank sizes given in Table 5-2 only. The Net Present Value method should be used. 

Using a Net Present Value method, determine which pump should be used if it is estimated that the plant will last... 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

The pixel values are sorted out in ascending order in the median value method whereby ix> i2> i >. .. > i . Therefore, the median value of the intensity of the foreground is ... 

Precision The reproducibility of the method likewise must match the needs of the problem. The hydroxyl value method used in the Bulging Drum Problem might not have been precise enough to detect any differences between the good and bad EOs. 

When the distribntion equation can be expressed in the form y = ao + ai x, from a plot of y vs. X the intercept on the y axis yields the ao parameter and the slope the di parameter. This treatment is referred to as the limiting value method. 

Attachments or Appendices Miscellaneous back-up information such as discussion of rejected or less-probable scenarios, documents of special interest or value, method and conduct of the investigation and team membership, photographs, diagrams, calculations, lab reports, references, noncontributory factors, terms of reference. 

M.D. Smooke. Solution of Burner-Stabilized Premixed Laminar Flames by Boundary-Value Methods. J. Comp. Phys., 48 72-105,1982. 

M.D. Smooke, J.A. Miller, and RJ. Kee. Determination of Adiabatic Flame Speeds by Boundary Value Methods. Comb. Sci. Techn., 34 79-89,1983. 

The extent of deformation from the chair conformation of cyclohexane has been investigated by the R- value method (67JA1836). The R- value is based on vicinal coupling constants and is defined by the expression ... 

In the EPRI Study, a bottom-up approach was developed to define sites and applications where distributed generation may piovide high value. Methods first were developed with the cooperation of Ihe Los Angeles Dept, of Water Power (LADWP). These were tested and refined with the assistance of South West Corporation (CSW) and Oglethorpe Power Corporation (OPC). In the OPC study, the methodology used applied not only to fuel cells, but also to distributed diesels and batteries. More detail is given in the D. Rasller reference listed. 

The iodine value (IV) is used to determine the level of unsaturation in a fat/oil system. It is expressed as the number of grams of iodine that add to/react with 100 g of sample. The traditional iodine value method using the Wijs reagent requires carbon tetrachloride (CC14). For safety reasons, CC14 is no longer considered to be an acceptable chemical and it is not readily available for purchase, and if offered it is extremely expensive. Therefore the traditional method has been modified to a more human-friendly system which uses cyclohexane. 

If a better estimate of the mixture formation pressure is needed, the engineer may use the KySi value method in Section 4.2.2 to obtain an estimated hydrate pressure of 1.26 MPa. 

Section 4.2 deals with the most useful hydrate equilibria—calculations of temperatures and pressures at which hydrates form from gas and free water. In this section, two historical methods, namely, the gas gravity method (Section 4.2.1) and the Kvs, value method (Section 4.2.2), for calculating the pressure-temperature equilibrium of three phases (liquid water-hydrate-vapor, Lw-H-V)1 are discussed. With the gas gravity method in Section 4.2.1.1, a method is given for limits to expansion, as for flow through a valve. In Section 4.2.2 a distribution coefficient (KVSi) method is provided to determine whether a component prefers residing in the hydrate or the vapor phase. These methods provide initial estimates for the calculation and provide a qualitative understanding of the equilibria. A statistical... 

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