Example a simple illustration

We take the level Aneso 1 to be the level discussed in Section 3.1.1 (we shall now use X eM instead of xeM to denote the state variables) and the level 

The slow time evolution on Mi (i.e., the time evolution on Mi restricted to Msiow) is governed by 

If we compare (127) with (122), we see that the reduction process brought us the following relation between the material parameters A introduced on the level Cmeso i and A2 introduced on the level Cmeho 2  

Everything that we have done so far in this example is completely standard. The next step in which we identify the slow fundamental thermodynamic relation in the state space M2 (i.e., we illustrate the point (IV) (see (120))) is new. Having found the slow manifold Msiow in an analysis of the time evolution in Mi, we now find it from a thermodynamic potential. We look for the thermodynamic potential ip(q,p, e, //. q, e, v) so that the manifold Msiow arises as a solution to 

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A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a... 

From a human reliability perspective, a number of interesting points arise from this example. A simple calculation shows that the frequency of a major release (3.2 x lO"" per year) is dominated by human errors. The major contribution to this frequency is the frequency of a spill during truck unloading (3 X10" per year). An examination of the fault tree for this event shows that this frequency is dominated by event B15 Insufficient volume in tank to imload truck, and B16 Failure of, or ignoring LIA-1. Of these events, B15 could be due to a prior human error, and B16 would be a combination of instrument failure and human error. (Note however, that we are not necessarily assigning the causes of the errors solely to the operator. The role of management influences on error will be discussed later.) Apart from the dominant sequence discussed above, human-caused failures are likely to occur throughout the fault tree. It is usually the case that human error dominates a risk assessment, if it is properly considered in the analysis. This is illustrated in Bellamy et al. (1986) with an example from the analysis of an offshore lifeboat system. 

Heuristic Explanation As we can see from Fig. 20-30, the DEP response of real (as opposed to perfect insulator) particles with frequency can be rather complicated. We use a simple illustration to account for such a response. The force is proportional to the difference between the dielectric permittivities of the particle and the surrounding medium. Since a part of the polarization in real systems is thermally activated, there is a delayed response which shows as a phase lag between D, the dielectric displacement, and E, the electric-field intensity. To take this into account we may replace the simple (absolute) dielectric constant e by the complex (absolute) dielectric constant = e - ie" = e - ic/w, where co is the angular frequency of the applied field. For treating spherical objects, for example, the replacement... 

Most spectroscopic properties are related to second derivatives of the total energy. As a simple illustrative example, vibrational modes, which arise from the harmonic oscillations of atoms around their equilibrium positions, are characterized by the quadratic variation of the total energy as a function of the atomic displacements SRy... 

Total inverse produces cumbersome sets of equations, especially when errors are taken into account. As usual, not considering errors amounts to taking uncorrelated variables with unit variances, but offers attractive illustrative properties. Examples of application abound in literature, but do not usually give enough detail for the student to use them as practical illustrative references. For these reasons, a simple illustration with no errors will be presented, and readers interested in a complete treatment should refer to Tarantola (1987). 

Table 17 lists a number of commercially available pigments, along with their chemical structures, in order to illustrate the different structural types of Naphthol AS pigments. Fastness to solvents and migration resistance improve from top to bottom, i.e., with increasing number of CONH groups in the molecule. The first example, a simple (3-naphthol pigment, is the skeleton from which all other species are derived. 

It is usual that the first reduction wave of commonly studied POMs he monoelec-tronic or, at the utmost, hielectronic [89]. As a consequence, energetically favorable catalytic processes that require larger numbers of electrons can only be accomplished at fairly negative potentials where the necessary number of charges is accumulated and delivered by the POM framework hence the search for strategic parameters that could favor apparently multiple electron uptake on the first wave of POMs. The present case deserves emphasis as a simple illustrative example that proved to be very beneficial in the electrocatalytic reduction of nitrite. 

The above example is only a simple illustration of a weight balance. Similarly, the reaction between elements and compounds may be symbolically expressed to portray the principle of conseivation of matter. For example, if hydrogen is completely burned to water, the reaction between it and oxygen can be represented as follows ... 

Chapter 7 is the climax of the book Here the educated student is asked to apply all that he/she has learned thus far to deal with many common practical industrial units. In Chapter 7 we start with a simple illustrative example in Section 7.1 and introduce five important industrial processes, namely fluid catalytic cracking in FCC units in Section 7.2, the UNIPOL process in Section 7.3, industrial steam reformers and methanators in Section 7.4, the production of styrene in Section 7.5, and the production of bioethanol in Section 7.6. 

Fairbanks, Everitt and Jaeger s modified tape recorder [Fairbanks et al., 1954] is the simplest and oldest example of a pitch-independent time-domain time or pitch scaling system. Although Pitch independent methods are no longer used in practice, they offer a simple illustration of the basic principles underlying most of the more recent algorithms. 

Certain structures of groups of chemical interest which may be difficult to see, can be easily envisaged by using caterpillar trees as models. We use as an example the composition [9] of two permutation groups St and Sj (involving the permutation of i and j objects respectively) which is denoted by S,[SJ (and is read S around Sj). This group is known as the wreath product or the Gruppenkranz . As a simple illustration we show how a caterpillar tree can be used to model the elements of S2[S3]. Then we have two sets, viz.,... 

The Discreet-Element Method for an Assembly of Two-dimensional Disks Example 4.2 serves as a simple illustration of the DEM cycling through a force-displacement constitutive response, F, = kAuj and the law of motion, which relates the F, with Xt and, thus, particle motions. In the general case of an assembly with a very large number of disks the calculation cycle is as follows the F, = kAn, is applied at each contact point of any disk and the vectorial sum of the contact forces is calculated to yield the net force acting on the disk. For such an assembly there are both normal and tangential... 

As a simple illustration of interval methods, consider the example given by Ferson (1996) pertaining to multiplication of two inputs. Input A has an interval of [0.2, 0.4], and Input B has an interval of [0.3, 0.5]. The interval for the model output is [0.06, 0.2]. The output interval is the narrowest possible interval that accounts for all possible forms of dependence between A and B. 

Symmetry considerations are fundamental in any description of molecular vibrations, as will be seen later in detail (Chapter 5). First, however, the molecular symmetries will be discussed, ignoring entirely the motion of the molecules. Various molecular symmetries will be illustrated by examples. A simple model will also be discussed to gain some insight into the origins of the various shapes and symmetries in the world of molecules. Our considerations will be restricted, however, to relatively simple, thus rather symmetrical systems. The importance and consequences of intramolecular motion involving relatively large amplitudes, will be commented upon in the final section of this chapter. 

A simple illustrative example may be taken to indicate the way in which the cage effect can be formulated ([uantitatively. Let us assume that we are investigating the photolysis of I2 in the presence of a scavenger S which can react with I atoms to form the relatively inert radical -SI which does not react with I2 but which may react with I atoms to form stable ST2. The kinetic scheme can be represented by... 

Figure 6.1 shows the structure of the flagellar motor in a simple illustration that reminds us of artificial machines. This machine-like motor is constructed through the self-assembly of proteins. The superior fimctionality and complexity of biological super molecules is quite apparent from this example. The energy for the rotation of the motor is provided by a proton flow from the outside to the inside of the bacteria. When an electrical potential difference is applied between the outside and the inside of the bacteria by immobilizing the bacterial cell on micropipette, the rotation speed can be controlled by al-... 

The example given above provides a simple illustration of the use of moments of inertia and vibration frequencies to calculate equilibrium constants. The method can, of course, be extended to reactions involving more complex substances. For polyatomic, nonlinear molecules the rotational contributions to the partition functions would be given by equation (16.34), and there would be an appropriate term of the form of (16.30) for each vibrational mode. ... 

A method of determination of the envelopes of vibronic molecular spectra has been obtained by a generalization of a similar approach known in the atomic spectroscopy. Its application to diatomic molecules is very easy and has been illustrated by several examples. A simple way of taking into account the Q-dependence of the molecular transition moments has been described. The implementation of the method to more complex cases, involving multidimensional potential hypersurfaces, is straightforward though requires some numerical effort connected with the evaluation of multi-dimensional integrals. 

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