Diffusion line source

Radiation model involving multi-lamp reactors is provided by Yokota and Suzuki (22). Based on a diffused line source emission model, the light absorption rate in any geometrical photoreactor with multiple lamps was assessed, and the work reveals the existence of optimum light arrangement. 

For two-dimensional (line-source) diffusion, the mean square displacement is 4Dt. For three-dimensional (point-source) diffusion, the mean square distance is 6Dt. 

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... 

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. 

Figure 4.5 One-dimensional diffusion into an infinite domain, (a) Point source diffusing into a line, (b) Line source diffusing into a plane, (c) Planar source diffusing into a volume.

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... 

Analyses of the climbing rates of many other dislocation configurations are of interest, and Hirth and Lothe point out that these problems can often be solved by using the method of superposition (Section 4.2.3) [2]. In such cases the dislocation line source or sink is replaced by a linear array of point sources for which the diffusion solutions are known, and the final solution is then found by integrating over the array. This method can be used to find the same solution of the loopannealing problem as obtained above. 

The solution of the diffusion equation for the quasi-steady state in cylindrical coordinates shows that each dislocation line source will have a vacancy concentration diffusion field around it of the form... 

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. 

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,... 

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. 

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

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

In this problem we wish to examine two aspects of atmospheric diffusion theory (1) the slender plume approximation and (2) surface deposition. To do so, consider an infinitely long, continuously emitting, ground-level crosswind line source of strength qi. We will assume that the mean concentration is described by the atmospheric diffusion equation,... 

The accuracy of this technique is to within 0.3%. Both tungsten and platinum wires are used with diameters of below 7 i im, more typically 4 ixm (for low-pressure gas measurements). Small diameters reduce errors introduced by assuming the wire is a line source (infinitely thin). The effect becomes significant for gases and increases with decreasing pressure. The thermal diffusivity of gases is inversely proportional to the pressure and thus thermal waves may extend to the cell wall. For these cases, a steady state hot-wire technique is used for which the design equation is ... 

Nix, G.H., Lowery, G.W., Vachan, R.I., and Tanger, G.E., Direct determinations of thermal diffusivity and conductivity with a refined line-source technique, in Progress in Aeronautics and Astronautics Therrrwphysics of Spacecraft and Planetary Bodies, Vol. 20, Heller, G. (ed.). Academic Press, New York, 1967, pp. 865-878. 

DIRECT DETERMINATION OF THERMAL DIFFUSIVITY AND CONDUCTIVITY WITH A REFINED LINE-SOURCE TECHNIQUE. FROM THERMOPHYSICS OF SPACECRAFT AND PLANETARY BODIES. PROGRESS IN ASTRONAUTICS AND AERONAUTICS-VOL. 20. 

Stark broadening occurs if there are a considerable number of ions and electrons in the excitation source. Excited atoms therefore are subject to strong electrical fields from nearby ions and electrons and may collide with them. Spectral radiation emitted under these conditions may be sufficiently affected to cause the line to broaden. Since the local electrical fields are continually changing and are not of uniform intensity, a broad, diffuse line results. This effect was observed experimentally by early spectroscopists and led to their identification of certain spectral line series as being diffuse and fundamental. ... 

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 ... 

In order to demonstrate the application of this general method to neutron diffusion in infinite media, the plane-source and line-source distribution problems (Secs. 5.2a and 5.2c) will be treated using (5.97). 

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... 

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