Path-lengths

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 echo height of side drilled holes was measured at a constant beam path length as shown in Fig. 1. 

The echo directivity for surface SH Wave probes and SH Wave angle probes was measured. The experiment was carried out by measuring the echo height from side drilled holes of different depths at a constant beam path length. The calculation of echo height was based on a point sound source on the test surface in different phases. The experiment and the calculation were compared. The effects of the frequency, height of... 

The iimnodified temi absorbance usually means this quantity, though some authors use the Napierian absorbance B = -hiT. The absorbance is so iisefiil because it nomially increases linearly with path length, /, tlirough the sample and with the concentration, c, of the absorbing species within the sample. The relationship is usually called Beer s law ... 

The quantity e is called the absorption coefficient or extinction coefficient, more completely the molar decadic absorption coefficient it is a characteristic of the substance and the wavelength and to a lesser extent the solvent and temperature. It is coimnon to take path length in centimetres and concentration in moles per... 

At still shorter time scales other techniques can be used to detenuiue excited-state lifetimes, but perhaps not as precisely. Streak cameras can be used to measure faster changes in light intensity. Probably the most iisellil teclmiques are pump-probe methods where one intense laser pulse is used to excite a sample and a weaker pulse, delayed by a known amount of time, is used to probe changes in absorption or other properties caused by the excitation. At short time scales the delay is readily adjusted by varying the path length travelled by the beams, letting the speed of light set the delay. 

B2.2.2.2 COLLISION RATES, COLLISION FREQUENCY AND PATH LENGTH... 

Table B2.5.2. Examples for pulsed lasers with different pulse durations and corresponding path lengths. For...

S is the path length between the points a and b. The Euler equation to this variation problem yields the condition for the reaction path, equation (B3.5.14). A similar method has been proposed by Stacho and Ban [92]. 

The ortho- and meto-isomers are bulkier than the para-iaomer and diffuse less readily in the zeolite pores. The transport restriction favours their conversion into the /lara-isomer, which is fonned in excess of the equilibrium concentration. Because the selectivity is transport influenced, it is dependent on the path length for transport, which is the length of the zeolite crystallites. 

The path length is set by the experimental configuration while a is known for each transition (such as OO O J—> OO l, J 1 or OO l J—> 00 2, J 1). Thus, a measurement of zi///provides the partial pressure P of molecules produced in probed states such as OO O Jor 00 1 J. (Strictly, optical probing measures the difference in the partial pressures between the upper and lower states of the probed transition however, in practice, the lower state population is always much larger than the upper state population so that the probe senses only the lower state population in the experiment.)... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

It ls not surprising chat such a relation should hold at the Limit of Knudsen diffusion, since Che Knudsen diffusion coefficients are themselves inversely proportional to the square roots of molecular weights, but the pore diameters in Graham s stucco plugs were certainly many times larger chan the gaseous mean free path lengths at the experimental conditions. 

Knudseci s very careful experiments on a long uniform capillary show that N L/ Pj -p ) passes through a marked minimum when plotted as a function of (P +P2)/2, at a value of the mean pressure such that the capillary diameter and the mean free path length are comparable. At higher values of the mean pressure, N L/(pj " 2 rises linearly, as in the case of a porous medium. 

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