A Numerical Example

A numerical example is given next to demonstrate the utility of the averaging method described above. 

The equation of motion (4.4) describes the system shown in Fig. 4.1. In this section, results of numerical simulations with both discontinuous and smoothed (continuous) coefficient of friction functions are presented and compared with the averaging results. The numerical values of the parameters are given in Table 4.1. 

For the two cases of r = 20 and r = 200, the averaging results very accurately estimate the amplitude of vibrations when compared with the respective numerical simulation results of the model with smoothed friction function. 

The difference between the results obtained from the system with smoothed coefficient of friction, and the original system (discontinuous friction) is significant for r = 20. This, of course, is the same as the difference shown in Fig. 4.5. For r = 200, on the other hand, the differences among numerical simulation of the original equations, numerical simulation of the smoothed equations, and the averaging results are much smaller. 

Consider the example described earlier. Suppose the manufacturer were to lower the wholesale price to 1.95 but receive a payment for 

Demand Prob Cumulative Revenue Cap Commit Exec Cost Credit Profit X Probability 

Revenue Cost Capacity Credit Profit X Probability 

The associated manufacturer expected profit (sum of the last column) is 11.82, which is larger than the values under the no-coordination system. The corresponding retailer profits are shown in Table 5.6. 

For those wanting to try OC, here is a guide for checking the work. This follows the example given by Whiting and Carr [571]. 

Assume a Cottrell simulation and the use of only five points, giving just three internal points. From Table A.3, this places the internal points at the positions (0.1127, 0.5000, 0.8873), here presenting fewer digits than in the table. Using equation sets (9.82)-(9.84) and the definitions (9.85), we then have 

Matrix V is needed to generate G. and W is now stripped of its outer frame to produce (9.95), 

The above set of odes is now solved, choosing some algorithm. Nothing has been specified about the homogeneous chemical reaction function F(C), but it will add terms to the matrix W when specified. After the time derivative is discretised in some way, the equation can be rearranged into the same form as described in Chap. 8 and solved using the same methods or, as mentioned above, solved using a professional ode or DAE solver. 

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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 ... 

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. 

Figure 6 Thermodynamic cycle for multi-substate free energy calculation. System A has n substates system B has m. The free energy difference between A and B is related to the substate free energy differences through Eq. (41). A numerical example is shown in the graph (from Ref. 39), where A and B are two isomers of a surface loop of staphylococcal nuclease, related by cis-trans isomerization of proline 117. The cis trans free energy calculation took into account 20 substates for each isomer only the six or seven most stable are included in the plot.

Suppose the components are redundant, then their probabilities, if independent, are combined by multiplication (equation 2.7-34). this is generalized by analogy with the preceding (equation 2.7-35). As a numerical example, if the distributions z=(x... 

NUREG/CR-2303 describes the BFR model in some detail and provides a numerical example. 

The effect is best illustrated by a numerical example (Table 31.4). Let us take the case of hard and alkaline deep well water such as that found to the north of London, whose main characteristics are shown in the first column of Table 31.4. The second column shows its quality after de-alkalization has removed nine-tenths of the temporary hardness and converted it into CO2 gas. This is removed from the water by stripping it with air in a packed degassing column, and the product then softened in the third stage to yield the product shown in the third column. 

As a numerical example, suppose F = 0.5 and /c = 0.9. Then Equation (4.8) gives rriout/min = This result may seem strange at first. The density... 

Figure 2.7. Using residuals to judge linearity. Horizontal lines the accepted variation of a single point, e.g., 2 resi thick dashed line perceived trend note that in the middle and near the ends there is a tendency for the residuals to be near or beyond the accepted limits, that is, the model does not fit the data (arrows). For a numerical example, see Section 4.13. The right panel shows the situation when the model was correctly chosen.

Error bars defined by the confidence limits CL(y,) will shrink or expand, most likely in an asymmetric manner. Since we here presuppose near absence of error from the abscissa values, this point applies only to y-transformations. A numerical example is 17 1 ( 5.9%, symmetric CL), upon logarithmic transformation becomes 1.23045 -0.02633. .. 1.23045 + 0.02482. 

Ok) function is sought by repeatedly determining the direction of steepest descent (maximum change in for any change in the coefficients a,), and taking a step to establish a new vertex. A numerical example is found in Table 1.26. An example of how the simplex method is used in optimization work is given in Ref. 143. 

The core parts of program MSD are given in Table 5.8 Table 5.9 displays the results of a numerical example, see file QUOTE RESULT.xls, and compares this to the results obtained using the approximations listed earlier. 

Program CORREL is given in Table 5.10 Tables 5.11-13 display the results of a numerical example. 

Worz et al. give a numerical example to illustrate the much better heat transfer in micro reactors [110-112]. Their treatment referred to the increase in surface area per unit volume, i.e. the specific surface area, which was accompanied by miniaturization. The specific surface area drops by a factor of 30 on changing from a 11 laboratory reactor to a 30 m stirred vessel (Table 1.7). In contrast, this quantity increases by a factor of 3000 if a 30 pm micro channel is used instead. The change in specific surface area is 100 times higher compared with the first example, which refers to a typical change of scale from laboratory to production. 

A numerical example for the estimation of unknown parameters in PDE models is provided in Chapter 18 where we discuss automatic history matching of reservoir simulation models. 

Besides this unique above-described process, there a numerous examples of inter- and intramolecular domino Michael/aldol processes in which the sequence is initiated by the addition of a metalorganic compound to an enone moiety. The Kamimura group [30] synthesized several five- to seven-membered thio- and hy-... 

Therefore this chapter will continue the multiple linear regression (MLR) discussion introduced in the previous chapter, by solving a numerical example for MLR. Recalling... 

A numerical example is given as follows Given A, find its determinant ... 

Whichever way we choose to describe the design, it (and the others of this type) has some attractive features. We will illustrate these features with a numerical example. For our example, we will imagine an experiment where the scientist is interested in determining the influence of temperature and of catalyst on the yield of a chemical reaction. The questions to be answered are does the concentration of catalyst make a difference, and does the type of catalyst make a difference The experiment is to consist of trying each of the four available catalysts and three solvents, and determining the yield. The experiment can be described by Table 10-3. 

The constrained least-square method is developed in Section 5.3 and a numerical example treated in detail. Efficient specific algorithms taking errors into account have been developed by Provost and Allegre (1979). Literature abounds in alternative methods. Wright and Doherty (1970) use linear programming methods that are fast and offer an easy implementation of linear constraints but the structure of the data is not easily perceived and error assessment inefficiently handled. Principal component analysis (Section 4.4) is more efficient when the end-members are unknown. 

Further details and a more quantitative discussion of cyclic voltammetry can be found in more specialized books [332-334], Here, before proceeding to a numerical example, we summarize the reversibility criteria in cyclic voltammtery andpoint out some factors that may lead to unreliable values ofiE R/R-) [335]. 

A numerical example of the carbon dioxide supercritical cycle has been made by Feher (Feher, E.G., The super-critical thermodynamic power cycle. Energy Conversion, vol. 8, pp. 85-90, 1968). The reasons for the neglect of the supercritical cycle until now are not known. 

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