Diffusion from a Line Source

The mean concentration downwind of a continuous, crosswind line source at a height h emitting at a rate (gm l s ) is governed by 

The General Structure of Multiple-Source Plume Models Multiple-source plume (and particularly Gaussian plume) models (Calder 1977) are commonly used for predicting concentrations of inert pollutants over urban areas. Although there are many special-puipose computational algorithms currently in use, the basic element that is common to most is the single-point-source release. The spatial concentration distribution from such a source is the underlying component and the multiple-source model is then developed by simple superposition of the individual plumes from each of the sources. 

The long-term average concentration, which we denote by an uppercase C, is given by 

When the emission intensities S and the dispersion functions Q, that is, the meteorological conditions, can be assumed to be independent or uncorrelated, the average value of the product of S and Q is equal to the product of their average values, and we thus have 

American Society of Mechanical Engineers (1973) Recommended Guide for the Prediction of the Dispersion of Airborne Effluents, 2nd ed. ASME, New York. 

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Nee, V. W., Mass Diffusion from a Line Source in Neutral and Stratified... 

When c is assigned units of particles per unit, length. nd corresponds to the total number of particles in the source, and Eq. 4.40 describes the one-dimensional diffusion from a point source as in Fig. 4.5a. Also, when c has units of particles per unit area, nd has units of particles per unit length and Eq. 4.40 describes the one-dimensional diffusion in a plane in two dimensions from a line source initially containing nd particles per unit length as in Fig. 4.55. Finally, when c has units of particles per unit volume, nd has units of particles per unit area, and Eq. 4.40 describes the one-dimensional diffusion from a planar source in three dimensions initially containing nd particles per unit area as in Fig. 4.5c. These results are summarized in Table 5.1. 

Equation 4.40 gives the solution for one-dimensional diffusion from a point source on an infinite line, an infinite thin line source on an infinite plane, and a thin planar source in an infinite three-dimensional body (summarized in Table 5.1). Corresponding solutions for two- and three-dimensional diffusion can easily be obtained by using products of the one-dimensional solution. For example, a solution for three-dimensional diffusion from a point source is obtained in the form... 

For this calculation we use the problem of neutron diffusion from a point source in an infinite medium. Similar calculations can be made for the plane and line source, but the basic relationship between L and the crow-flight distance will prove to be the same in all cases. Consider then the diffusion of neutrons in an infinite medium from a point source of strength go neutrons per unit time. For convenience, we place the source at the brigin of a suitable coordinate system. As previously noted, this situation has spherical symmetry, and the neutron distribution may be described completely by the radial coordinate r alone. Let us compute now the quantity r for this system. We define as the average square... 

MacCready, P. B., Jr. T. B. Smith M. A. Wolf "Vertical diffusion from a low altitude line source-Dallas tower studies." Final Report, Contract No. DA-42-007-CML-504,... 

Stern s experiment was conceptually simple and, as such, it had a beauty all its own. His approach was based on the method of molecular beams. The molecular beam method was originated in 1911 by Louis Dunoyer. In 1921, it was a relatively novel experimental method. Since that time, the molecular beam method has been the basis for an extremely productive line of physical investigation and, as we shall see in future chapters, has yielded both detailed and precise information about atomic properties. In this method, atoms diffuse from a source at one end of a highly evacuated cylindrical chamber, travel a path along the axis of the chamber, and are detected at the other end. Near the source exit, a sue-... 

Figure 2.16 Distorted plume caused by canyon winds and diffusion from an elevated source. Plume boundaries (defined a fixed ratio, e.g. 1/10 of centerline concentration at given value of x) in horizontal plane, showing components unentrained (dash-dot line), detrained (dashed line) and plume in canyon below building height (shading).

The line-source technique is a transient method capable of very fast measurements. A line source of heat is located at the center of the sample being tested as shown in Fig. 4. The whole is at a constant initial temperature. During the course of the measurement, a known amount of heat is produced by the line source, resulting in a heat wave propagating radially into the sample. The rate of heat propagation is related to the thermal diffusivity of the polymer. The temperature rise of the line somce varies linearly with the logarithm of time. Starting with the Fourier equation, it is possible to develop a relationship which can be used directly to calculate the thermal conductivity of the sample from the slope of the linear portion of the curve ... 

From equation (3.13) we can deduct a rough approximation of the location where maximum ground-level concentration occurs. It is argued that the turbulent diffusion acts more and more on the emitted substances, when the distance from the point source increases therefore the downwind distance dependency of the diffusion coefficients is done afterwards. If we drop this dependency, equation (3.13) leads to xmax=34,4 m for AK=I (curve a) and xmax=87,7 m for AK=V (curve b), what is demonstrated in fig n The interpolated ranges of measured values are lined in. Curve a overestimates the nondimensional concentration maximum, but its location seems to be correct. In the case of curve b the situation is inverted. Curve c is calculated with the data of AK=II. The decay of the nondimensional concentration is predicted well behind the maximum. Curve d is produced with F—12,1, f=0,069, G=0,04 and g=l,088. The ascent of concentration is acceptable, but that is all, because there is no explanation of plausibility how to alter the diffusivity parameters. Therefore it must be our aim to find a suitable correction in connection with the meteorological input data. 

Another widely used solution is for an instantaneous planeAine/point source (such as spill of a toxic pollutant, or diffusion of rare Earth elements from a tiny inclusion of monazite or xenotime into a garnet host) to diffuse away in either one dimension, two dimensions, or three dimensions (Figure l-6b). If the source is initially in a plane, which may be defined as x = 0 (note that x = 0 represents a plane in three-dimensional space), then diffusion is one dimensional. If the source is initially a line, which may be defined as x = 0 and y=0, then diffusion is... 

Mercury diffusion pumps are normally constructed from quartz or heat-resistant glass and are therefore a possible source of hazard should they Sreak, especially whilst they are hot. However, during over 40 years of working with such pumps, the author has neither experienced nor heard of such an accident. The major real disadvantage of mercury pumps is the relatively high vapour pressure of mercury at room temperature (ca. 10 Torr), which makes its necesssary to ensure that the cold traps prevent efficiently the mercury vapour from diffusing forward into the line. 

In actual practice, any tubular light source will have a finite diameter and will not behave as a true line source. Radiation from an extended light source will emanate from points displaced from the lamp s axis, causing the lamp to appear rather like a diffuse light source. In addition, imperfections in the... 

The initial distribution is simulated by a uniform distribution of point, line, or planar sources placed along x > 0 as in Fig. 5.5. The strength, or the amount of diffusant contributed by each source, must be Co dx. The superposition can be achieved by replacing in Table 5.1 with c(x)dV [c(x)dx in one dimension] and integrating the sources from each point. 

A common transient method is the line source technique, and such an apparatus was developed by Lobo and Cohen41 which could be used with melts. Oehmke and Wiegmann42 used the line source technique for measurements as a function of temperature and pressure. A hot wire parallel technique43 yielded conductivity and specific heat from the same transient, and then diffusivity was calculated. Zhang and Fujii44 obtained conductivity, diffusivity and the product of density and specific heat from a short hot wire method. 

Nuclear reactions producing exotic nuclei at the limits of stability are usually very non-specific. For the fast and efficient removal of typically several tens of interfering elements with several hundreds of isotopes from the nuclides selected for study mainly mass separation [Han 79, Rav 79] and rapid chemical procedures [Her 82] are applied. The use of conventional mass separators is limited to elements for which suitable ion sources are available. There exists a number of elements, such as niobium, the noble metals etc., which create problems in mass separation due to restrictions in the diffusion-, evaporation- or ionization process. Such limitations do not exist for chemical methods. Although rapid off-line chemical methods are still valuable for some applications, continuously operated chemical procedures have been advanced recently since they deliver a steady source of activity needed for measurements with low counting efficiencies and for studies of rare decay modes. The present paper presents several examples for such techniques and reports briefly actual applications of these methods for the study of exotic nuclei. 

A complete theory of turbulence is still lacking, so we must restrict our discussions to two general cases of interest to us (a) the diffusion of particles from point or line sources where the turbulence may be said to be isotropic, and (b) the behavior of particles near large land surfaces— as for example, dust storms. We shall begin our discussion with an explanation of the meaning of eddy-diffusion, which is characteristic of the conditions to be more fully discussed later. 

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