Theory of the Method

Let us consider the profile of a film at instant of time t, which is characterized 

An increment, dx, per elementary increment of time, dt, is the sum of an increment, dx, resulting from the flow under the action of the tangential shear stress per unit area of the blow-off current, X, and an inaement, dx2, due to the effect of evaporation. It is obvious that the first increment due to the gas flow is 

With the exception of a very short initial period, the film has a very low slope profile, which allows one to substimte 

Summing Eqnation 3.254 and Equation 3.255 and transforming them, we eventually obtain 

Consider conseqnences of the solution obtained. If there is no special boundary viscosity, i.e., ti(/j) = r = const, then 

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As is evident firom the theory of the method, h must be the height of rise above a surface for which AP is zero, that is, a flat liquid surface. In practice, then, h is measured relative to the surface of the liquid in a wide outer tube or dish, as illustrated in Fig. n-6, and it is important to realize that there may not be an appreciable capillary rise in relatively wide tubes. Thus, for water, the rise is 0.04 mm in a tube 1.6 cm in radius, although it is only 0.0009 mm in one of 2.7-cm radius. 

Thus, the major conclusions of tiie early studies by Volkenshtein and his colleagues applicable to the theory of the method of semiconductor gas sensors are the following a) chemisorption of particles on a semiconductor surface can be accompanied by a charge transfer between adsorption-induced surface levels and volume bands of adsorbent and b) only a certain fraction of absorbed particles is charged, the fraction being dependent on adsorbate and adsorbent. 

The subsection, theory, concentrates on the theory of the method(s) of relevance for this work, that is, the methods aiming at the ignition front rate, the conversion gas rate (by many authors referred to as burning rate or combustion rate), the conversion gas... 

A theory of the method to obtain the combustion rate and ignition rate was not described. Only results were presented. Limitations and assumptions of the method were not discussed. 

Koistinen et al [7] present a number of graphs on the grate fluxes (W/m, s). However, no theory of the method to calculate the grate effect was described. It must be based on the measured mass loss rate. Furthermore, no plots on the combustion rate (kg/m, s) are shown. Magnitudes of the air factor is presented, but no theory of how it is determined. No uncertainty analysis is carried out and no verification method is... 

Nine studies regarding the thermochemical conversion of packed-beds of biomass have been reviewed. The review is summarized in Table 7 below. The focus of the survey has been on the theories of the methods applied to measure ignition front rate, conversion rate (combustion rate, burning rate), conversion gas stoichiometry, and air factors. 

This author has classified the theory of the method with respect to physical model and mathematical model. The physical model can be broken down into system definition, descriptive definition of the sought quantity, measurands, and main assumptions. The mathematical model can be divided into mathematical definition of the sought quantities, and mathematical relationship between sought quantity and measurands. 

In studying a system by a nonlinear impedance method, use is made of the system s nonlinear characteristics. A variant of the nonlinear impedance method called the amplitude demodulation method was first applied in the electrochemistry of semiconductors, in particular to diamond electrodes, in [83] (see the quoted paper for the theory of the method and the experimental set-up). A perturbing current signal of a high frequency oo, modulated in amplitude at a low frequency 2, is applied to electrochemical cell the demodulated low-frequency voltage signal is to be measured at the frequency 2. In accordance with the theory of the method [83], under the condition of formation of a depletion layer in a semiconductor electrode, the in-phase component of the cell response Re h is inversely proportional to d(C 2)/dE. Hence, for the acceptor concentration in the semiconductor we have [compare Eq. (1)] ... 

Reference Books. The two series of books listed below describe important experimental techniques and apparatus for a variety of physical measurements. These books also deal with the phenomenological theory of the methods described. 

We will now have a look at two possible ways of viewing a covalent bond between atoms. They are known as the valence bond method and the molecular orbital method. We will consider the theory of the methods separately and judge the merits of both approaches when we come to examples later. 

The theory of the method is as follows Suppose that the solution contains the ions M" " and A" which are not necessarily univalent but are denoted as such for simplicity. The fraction of the total current carried by the cations is r+, and that carried by the anions is f. ... 

The theory of the method is as follows. The instantaneous rate of a reaction of the nth order involving only one reacting substance is proportional to the nth power of its concentration,... 

In a course like this, the question of how much to write about a technique is debatable. One extreme would be to consider only the spectra, and to ignore how they are produced the other is to delve too deeply into the theory of the method, and leave the student uncertain what to look for in the actual spectrum. I have aimed to be closer to the first extreme than the second, since some excellent detailed books on spectroscopy are available. The theory parts of this book cover approximately the same ground as the video tapes on IR, MS, UV, C and H spectra which were made at the University of Liverpool and which are available from the Royal Society of Chemistry. 

Since the SPV method has been intensively used for diffusion length measurement in undoped a-Si H, we shall discuss the theory of the method in some detail. The approach is to contrast and compare the theory to that already given for conventional semiconductors. The differences arise from two basic facts (1) Undoped a-Si H is a photoconductive semi-insulator rather than an extrinsic semiconductor (2) The thickness of the surface space-charge region (the surface barrier) may be comparable to the diffusion length, whereas in the SPV theory for conventional semiconductors it is assumed that W c Lp. 

In the same way as for a calorimeter for which the scanning mode has brought some interesting advantages, it seemed to the authors that an attanpt should be made with the MDR in using it with a linear increase in temperature [20]. Thus, the theory of the method has been developed, leading to results of concern, proving at least that this method is feasible. 

W.J. Lorenz, C. Mayer, H. Fisher, Galvanostatic current change method for differentiation between primary and secondary surface boundary inhibition of electrode reactions I. Theory of the method, Z. Phys. Chem. 52 (1967) 180-192. 

Polymers are not, however, always randomly oriented and the phenomenon of orientation is discussed in the next two chapters, chapters 10 and 11. One of the ways used to obtain information about the degree of orientation of the polymer is to measure either the refractive indices of the polymer for light polarised in different directions or its birefringences, i.e. the differences in refractive index for light polarised in different directions. In chapter 11 the theory of the method is described in terms of the polarisability of structural units of the polymer. This polarisability is a second-rank tensor like the molecular polarisability referred to in the earlier sections of this chapter and, insofar as the assumption of additivity in section 9.2.3 holds, it is in fact the sum of the polarisability tensors of all the bonds in the unit. Since, however, the whole basis of the method is that the structural units are anisotropic, the tensors must be added correctly, taking account of the relative orientations of the bonds, unlike the treatment used to calculate the refractive index of PVC in example 9.1, where scalar bond refractions are used. 

The theory of the method is rather complicated, because the amplitude of Raman scattering is described by a second-rank tensor, so it will not be discussed here. Just like for fluorescence, P2 cosff)) and (P4(cosd)), or cos 6) and (cos" 6), can, at least in principle, be obtained for the simplest type of uniaxial orientation distribution and these values now refer directly to the molecules of the polymer itself. In practice it is often necessary to make various simplifying assumptions. For biaxially oriented samples several other averages can be obtained. 

Solving Eqs. (7) and (10) simultaneously by the least squares method is the theory of the method for simultaneous adjustment. Assuming the correction value for the ground coordinates is the difference between the most probable value and the approximate value, the least squares equation can, in this case, be written as follows ... 

The theory of the method is as follows. Consider the effect of temperature upon the proportion of the tube occupied by liquid. Let F be the fraction occupied by liquid, V the internal volume of the tube, m the mass of substance present, py and p are the densities of vapour and liquid respectively, and a the cubical coefficient of expansion of glass, and let the subscripts 0 and T refer to room temperature and temperature T respectively. 

The theory of the method was developed to include, in addition to the non-parallelism of the absorption and emission vectors and the effects of... 

In 1955 Walsh established the foundations of modern analytical atomic absorption spectroscopy. In the same year Alkemade and Milatz also published a paper suggesting similar procedures. The work of Walsh, however, was much more detailed, since he examined the theory of the method, the basic principles involved, the instrumentation requirements, and its advantages over flame emission. 

Concise introductions to the theory of the methods discussed and the measurement procedures are provided (in Chapter 2) to make it easier to understand the instructions given for the methods selected. 

In this chapter some of the more important methods of studying coordination compounds have been reviewed. The detailed interpretation of the experimental results is often rather difficult, and, for the phenomena of optical activity and NQR, for example, the theory of the method has only been worked out incompletely. Discussion has therefore been confined to the qualitative level, it being considered important that the student should have a pictorial idea of the phenomena considered. In this way he or she should both have been made aware of which technique is likely to be of use in tackling a particular problem and also of some of the difficulties associated with its application. Inevitably, it has been necessary to be selective and this has been done on the basis either of techniques which the student may well meet in the laboratory or of techniques which are of particular importance. Finally, Chapter 16 will provide examples of the application of some of the techniques described in the present chapter, as well as a few more which have proved to be of particular value in the study of bioinorganic molecules. 

This chapter is divided into two large sections, one on small-angle neutron scattering and the second on neutron reflectometry. Within each section the theory of the method will be discussed and some basic aspects of the experimental features set out. Finally, selected applications of each method in polymer science will be described. 

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