AVERAGE example

FIGURE 8.7 Male and female sample averages. Example 8.1. 

Table B.l Parameter values used in the higher-order averaging example...

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

If the composition of the waste stream is known, then the theoretical oxygen demand can be calculated from the appropriate stoichiometric equations. As a first level of approximation, we can assume that this theoretical oxygen demand would be equal to the COD. Then, experience with domestic sewage indicates that the average ratio of COD to BOD will be on the order 1.5 to 2. The following example will help to clarify these relationships. 

Table 4.5 shows the results for an example. These results differ significantly from those obtained by the method of Riazi for the initial and 10% distilled points. The reported average error for this method is about 3°C, except for the initial point where it reaches 12°C. 

The average error of this simplified method is about 3°C and can reach 5°C. Table 4.22 shows an application of this method calculating the temperature of hydrate formation of a refinery gas at 14 bar. Table 4.23 gives an example applied to natural gas at 80 bar. Note that the presence of H2S increases the hydrate formation temperature. 

A European Directive, 85/210/EEC, limits benzene content to 5% by volume in all gasolines regular, premium, with or without lead. This level is easily achieved, since the average value in 1993 was less than 3%. in France, for example, average benzene concentrations of 1.7% and 2.6% were reported for leaded and unleaded premium fuels, respectively, in 1993. 

Within the same geographicai region, the crude specific gravity varies from one reservoir to another, in Saudi Arabia, for example, the crude from the Ghawar field has an average standard specific gravity on the order of 0.850 (34° API) while the specific gravity of the crude from the nearby Safaniyah field is 0.893 (27° API). 

Each of the input parameters has an uncertainty associated with it. This uncertainty arises from the inaccuracy in the measured data, plus the uncertainty as to what the values are for the parts of the field for which there are no measurements. Take for example a field with five appraisal wells, with the following values of average porosity for a particular sand ... 

In the example, development without appraisal leads to an NPV which is the weighted average of the outcomes m (-40+6+40) / 3 = + 2 million. Development after appraisal allows the decision not to develop in the case of the low STOMP, and the weighted average of then outcomes is m (0+6+66) / 3 = + 24 million. 

In the above example, the discount rate used was the annual compound interest rate offered by the bank. In business investment opportunities the appropriate discount rate is the cost of capital to the company. This may be calculated in different ways, but should always reflect how much it costs the oil company to borrow the money which it uses to invest in its projects. This may be a weighted average of the cost of the share capital and loan capital of a company. 

Let us consider the calculation of sensitivity threshold in the case when the cracks are revealing by PT method. Constant distance H between crack s walls along the whole defect s depth is assumed for the simplicity. The calculation procedure depends on the dispersity of dry developer s powder [1]. Simple formula has to be used in the case when developer s effective radius of pores IC, which depends mainly on average particle s size, is smaller than crack s width H. One can use formula (1) when Re is small enough being less than the value corresponding maximum sensitivity (0,25 - 1 pm). For example. Re = 0,25 pm in the case when fine-dispersed magnesia oxide powder is used as the developer. In this case minimum crack s width H that can be detected at prescribed depth lo is calculated as... 

We are interested in < E (0[,(t)i)E3(62,, where <> means the average over the ensemble of surfaces, the subindexes 1 and 2 refer to two different points of observation and the subindexes A and B belong to two different conditions of illumination, which for example arise from two different wavelengths, two different incident angles, etc.. If A = B and 1 = 2, the above expression gives the angular distribution of the mean scattered intensity, otherwise it turns to be the intensity correlation coefficient y from < E Eb >, assuming that we deal with fully developed speckle. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

For example, if one finds that 100 cars pass a fixed position on a highway in an hour, then the average number of cars per minute is 100/60. The probability that two cars pass in a minute is given by = 0.10. The... 

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