Numerical examples

FIGURE 6.7 Example calculation of a sum of 4 addends characterized by p-boxes. 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

Summary statistical measures resulting from hypothetical calculations 

Notice that, while the Monte Carlo simulation produces point estimates, the bounding analyses yield intervals for the various measures. The intervals represent sure bounds on the respective statistics. They reveal just how unsure the answers given by the Monte Carlo simulation actually were. If we look in the last column with no assumption, for instance, we see that the variance might actually be over 6 times larger than the Monte Carlo simulation estimates. 

At first the orbits were in the x-y plane, and then out of it but close to it, as the theorem states. However, as the parameter was varied, the periodic orbits continued to exist (as one might expect). Moreover, as the parameter was continued, the periodic orbits moved near the x-z plane and collapsed into it. The figure also illustrates this point. 

A dynamic simulation of this column using HYSYS was used to explore the dynamics of the process for the two cases where different tray temperatures are controlled. Either tray 6 or tray 14 temperature is controlled by manipulating reboiler heat input. Reflux flowTate is held constant. Disturbances are step changes at time equals 5 minutes in feed flowrate (25 percent increase) or feed composition. The feed composition disturbance is a drop in the HHK component in the feed (normal octane changed from 10 mol % to 0 mol % while normal pentane changed from 45 mol % to 55 mol %). 

As shown in Fig. 6.15, the 25 percent increase in feed flowrate, with the reflux flowrate constant, results in a slight increase in the impurity in the bottoms product when tray 6 is controlled. But when tray 14 is controlled, there is a very large change in bottoms composition. 

These dynamic simulations confirm that the control tray selection criteria discussed earlier yield reasonably effective control systems. 

The use of a single tray temperature is only viable if there is a section of the column where sufficient changes in temperature occur from tray to tray. In difficult (low relative volatility) separations, there is very little change in temperature from tray to tray. The effects of pressure and feed composition variations can swamp the effect of key component compositions on this fiat temperature profile. If the temperature change between the top and the bottom of the column is less than about 20 to 30 F. a single tray temperature may prove to be ineffective for control. 

Let us consider the following nonlinear model (Bard, 1970). Data for the model are given in Table 4.3. 

This model is assumed to be able to fit the data given in Table 4.3. Using our standard notation [y=f(x,k)] we have, 

The elements of the (/x )-dimensional sensitivity coefficient matrix G are obtained by evaluating the partial derivatives  

Equations 4.21 and 4.22 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

For the model described earlier, with N = 2, daily demand for a part that follows a normal distribution with a mean (fp) of 50 and a standard deviation (cr) of 25. Assume that the parts stations face a replenishment cost (K) of 125, a holding cost (/ ) of 0.2/day/part, a backorder cost (b) of 5/day/part. Also, suppose each station faces a supply lead time (7) 

Thus the echelon faces a demand with a mean of 100 units and a standard deviation of 141.42 units. Notice that the increased standard deviation of demand at the echelon reflects both order uncertainty as well as the lumpiness caused by the batched ordering from locations. 

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

Turning now to the 2-component relativistic Hamiltonians we observe that the eXact 2-Component (X2C) Hamiltonians reproduces the DC orbital energies quite well, although there is a difference of about 10 Ei, for the lsi/2 level. This discrepancy stems entirely from picture change errors in the two-electron operator since the one-electron X2C Hamiltonian by construction reproduces exactly the positive-energy spectrum of the parent 4-component Hamiltonian. In the present calculations the one-electron X2C Hamiltonian 

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Useful, engineering-oriented monograph with a variety of numerical examples. 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

The use of these equations is perhaps best illustrated by means of a numerical example. In a measurement of the surface tension of benzene, the following data are obtained ... 

Referring to the numerical example following Eq. 11-18, what would be the surface tension of a liquid of density 1.423 g/crc (2-bromotoluene), the rest of the data being the same ... 

Students and instructors). Each chapter presents first the basic surface chemistry of the topic, with optional material in small print. Derivations are generally given in full and this core material is reinforced by means of problems at the end of the chapter. A solutions manual is available to instructors. It is assumed that students have completed the usual undergraduate year course in physical chemistry. As a text for an advanced course, the basic material is referenced to fundamental, historical sources, and to contemporary ones where new advances have been incorporated. There are numerous examples and data drawn from both the older and from current literature. 

This is an inverse lengtli k is known as tire Debye screening lengtli (or double layer tliickness). As demonstrated below, it gives tire lengtli scale on which tire ion distribution near a surface decays to tire bulk value. Table C2.6.4 gives a few numerical examples. 

In table C2.6.5, a few numerical examples for are shown. Smaller colloids are found to aggregate much faster and stabilizing them is therefore more difficult. The validity of equation (C2.6.15) has been confinned experimentally (e.g. [58]). 

Olender and Elber, 1996] Olender, R., and Elber, R. Calculation of classical trajectories with a very large time step Formalism and numerical examples. J. Chem. Phys. 105 (1996) 9299-9315... 

To exemplify both aspects of the formalism and for illustration purposes, we divide the present manuscript into two major parts. We start with calculations of trajectories using approximate solution of atomically detailed equations (approach B). We then proceed to derive the equations for the conditional probability from which a rate constant can be extracted. We end with a simple numerical example of trajectory optimization. More complex problems are (and will be) discussed elsewhere [7]. 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

Most methods for their preparation convert one class of carboxylic acid derivative to another and the order of carbonyl group stabilization given m Figure 20 1 bears directly on the means by which these transformations may be achieved A reaction that converts one carboxylic acid derivative to another that lies below it m the figure is pracfical a reacfion fhaf converts if fo one fhaf lies above if is nol This is anofher way of saying fhaf one carboxylic acid derivative can be converted to another if the reaction leads to a more stabilized carbonyl group Numerous examples of reacfions of fhis fype will be pre senfed m fhe secfions fhaf follow... 

We 11 see numerous examples of both reaction types m the following sections Keep m mind that m vivo reactions (reactions m living systems) are enzyme catalyzed and occur at far greater rates than those for the same transformations carried out m vitro ( m glass ) m the absence of enzymes In spite of the rapidity with which enzyme catalyzed reactions take place the nature of these transformations is essentially the same as the fundamental processes of organic chemistry described throughout this text... 

Fatty acids occur naturally m forms other than as glyceryl triesters and we 11 see numerous examples as we go through the chapter One recently discovered fatty acid derivative is anandamide... 

This text is similar to that of McCammon and Harvey (see below), but also provides a background for force field-based calculations and a more sophisticated discussion. Includes numerous examples of computing the structure, dynamics, and thermodynamics of proteins. The authors provide an interesting chapter on the complementary nature of molecular mechanics calculations and specific experimental techniques. 

You will come across numerous examples of qualitative and quantitative methods in this text, most of which are routine examples of chemical analysis. It is important to remember, however, that nonroutine problems prompted analytical chemists to develop these methods. Whenever possible, we will try to place these methods in their appropriate historical context. In addition, examples of current research problems in analytical chemistry are scattered throughout the text. 

Several texts provide numerous examples of analytical procedures... 

The applications of Beer s law for the quantitative analysis of samples in environmental chemistry, clinical chemistry, industrial chemistry and forensic chemistry are numerous. Examples from each of these fields follow. 

Numerous examples of standard methods have been presented and discussed in the preceding six chapters. What we have yet to consider, however, is what constitutes a standard method. In this chapter we consider how a standard method is developed, including optimizing the experimental procedure, verifying that the method produces acceptable precision and accuracy in the hands of a single analyst, and validating the method for general use. 

The system considered in this example was chosen for convenience, not as either a best or worst case example. Eyring and co-workers have published numerous examples in which the theory (slightly modified) fits experimental data quite well. 

We conclude this section with a numerical example which serves to review and compare some of the important relationships we have considered. 

This result is known as the Carothers equation. It is apparent that this expression reduces to Eq. (5.4) for the case of f = 2. Furthermore, when f exceeds 2, as in the AA/BB/Af mixture under consideration, then n is increased over the value obtained at the same p for 7= 2. A numerical example will help clarify these relationships ... 

This important equation shows that the stationary-state free-radical concentration increases with and varies directly with and inversely with. The concentration of free radicals determines the rate at which polymer forms and the eventual molecular weight of the polymer, since each radical is a growth site. We shall examine these aspects of Eq. (6.23) in the next section. We conclude this section with a numerical example which concerns the stationary-state radical concentration for a typical system. 

In these unit conversions on H, we have used the facts that 1 atm = 760 Torr and the ratio of densities PHg/ soin - /Psoin t onverts from Torr to millimeters of solution. These numerical examples show that experiments in which Apj, ATf, or ATj, are measured are perfectly feasible for solutes of molecular weight 100, but call for unattainable sensitivity for polymeric solutes of M = 10 . By contrast, osmometry produces so much larger an effect that this method is awkward (at least for 1% concentration) for a low molecular weight solute, but is entirely feasible with the polymer. 

The difficulties with approximations used in theoretical derivations and with the experimental techniques have been pointed out (8), and a series of tutorials have been pubHshed in which the deviations from paraboHc behavior are explained and the correct interpretations of cake filtration data, with many numerical examples, are laid down (8,9). 

There are numerous examples of acryhc NADs, but they are of limited commercial importance. Some typical examples of acryhc NADs are hsted in Table 11, and References 70 and 71 offer exceUent overviews. 

From Other Heteroeyeles by Rearrangement. Although there are numerous examples of pyrazole syntheses from other monocychc heterocycles by chemical, thermal, or photochemical means, such examples ate only of limited practical value on account of high cost. Reaction of diaziridinone (68) with the sodium salt of malondinitrile yields the di-/-butylaminopyrazolinone (69) (eq. 17) (45). 

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