Diffusion from a point source

A solution of (18.130) has been obtained by Huang (1979) in the case when the mean windspeed and vertical eddy diffusivity can be represented by the power-law expressions 

Equation (18.133) can be used to obtain some special cases of interest. If it is assumed that p = n 0, then (18.133) reduces to 

The case of a point source at or near the ground can also be examined. We can take the limit of (18.133) as h — 0 using the asymptotic form of Iv(x) as x — 0 

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Diffusion from a Point Source into a Mdving Fluid. 218... 

The problem of diffusion from a point source has been studied under more general conditions by Klinkenberg, Krajenbrink, and Lauwerier (K12). These authors discuss the solution of the equation... 

The diffusion from a point source in a tube has also been studied by... 

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... 

The probability distribution of a random walk shows that the mean-square displacement after NT jumps is (R2) = NT r2) = rY(r2) (Eq. 7.47). Comparison of the probability distribution (Eq. 7.45) to the point-source solution for one-dimensional diffusion from a point source (Table 5.1) indicates that... 

Hunt, J.C.R., Puttock, J.S., and Snyder, W.H. (1979) Turbulent diffusion from a point source in stratified and neutral flows around a three-dimensional hill. Part... 

Concentration profiles for diffusion from a point source. When a concentrated bolus of solute is deposited within a small region of an infinitely long cylinder, as shown in Figure 3.4a, the molecules slowly disperse along the axis of the cylinder. The curves shown here are realization of Equation 3-34 for a solute with = 10-R = 0.1 cm, N = 1Vav and r = 6, 24, and 72 h. 

Figure 7c shows 2D fluorescence intensity profiles at various times after the start of pulsatile release. The profile was governed by diffusion from a point source and reached steady state after 10 min. The measured fluorescence intensity profiles after 10 and 30 min fit well with the diffusion profile predicted by simulation. The concentration gradient produced by repetitive ejection of the solution was calculated by summing the concentration profiles from each pulse. The concentration profile after n injections was expressed as... 

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... 

EXAMPLE 18.3 Diffusion from a point source. Put a drop of dye in water and observe how the dye spreads. To make the math simple, consider the spreading in one dimension. Begin by putting the dye at x = 0. To determine the dye profile c(x, t), you can solve Equation (18.9) by various methods. The solutions of this equation for many different geometries and boundary conditions are given in two classic texts on diffusion [1], and on heat conduction [2]. 

Fig. 5. Diffusion of pollutants from a point source. PoUutant concentrations have separate Gaussian distributions in both the horizontal (j) and vertical directions. The spread is parameterized by the standard deviations ( O ) which are related to the diffusivity (fQ.

The presumption of a Gaussian distribution for the mean concentration from a point source, although demonstrated only in the case of stationary, homogeneous turbulence, has been made widely and, in fact, is the basis for many of the atmospheric diffusion formulas in common use. Based on the developments of Section IV, we present in this section the Gaussian point source diffusion formulas that have been used for practical calculations. 

Two particular aspects of the transport of degradable contaminants were considered in laboratory experiments that used soil originating from the field experiments described in the previous sections. Studies on diffnsion of degradable insecticides were performed in diffusion cells, while the spatial redistribntion of pesticides from a point source was measured in specially designed pans (60 cm high, 40 cm diameter). Periodic sampling and contaminant analysis enabled visnaUzation of the contaminant transport pathway. 

Gerstl Z, Yaron B (1983) Behavior of bromacU and napropamide in soils. 11. Distribution after application from a point source. Amer J Soil Sci 47 478 83 Gerstl Z, B Yaron, Nye PH (1979a) Diffusion of a biodegradable pesticide as affected by microbial decomposition. Soil Sci Soc Am J 43 843-848... 

A rough estimate of the diffusion penetration distance from a point source is the location where the concentration has fallen off by as 1/e of the concentration at x = 0. This occurs when... 

The diffusion equations thus far developed assume that the particles are colloidal and not affected by any motion of the fluid itself which is regarded as stationary. If we limit our discussion to particles in the size range from 0.5 to 5 p, which remain in suspension for rather long periods of time, and if these particles are emitted from a point source and not subject to disturbance by the surrounding fluid, it is obvious that the concentration of particles at any point must be proportional to the diffusion constant and inversely proportional to the square of the distance from the source. Let C be the concentration per unit time, passing a point at any distance R from the source, then... 

Diffusion Pattern from a Continuous Point Source—The distribution of particles from a point source in a moving fluid can be determined provided we assume that the concentration gradients in the direction of fluid motion are small compared to those at right angles to it. If C, is defined as the concentration of particles over a unit area of a plane horizontal surface downstream and to one side of the mean path of the diffusing stream from a point source, then the equation of diffusion at any point x downstream and at a distance y from the mean path is... 

Diffusion, convection, and dispersion all contribute to the spread of a front. Let us see how much each mechanism contributes to the spread. First, let us see when the diffusion transport is important as compared to the convective transport. We use v2Dot to calculate the spreading distance from a point source 68% of the injected source is within this distance. Table 2.2 shows the results for different time periods compared with the traveled distances during the same time periods by a convective flow of 1 m/day. A typical flow rate in petroleum reservoirs is 1 m/day (interstitial velocity). A typical value of diffusion coefficient of 4 X 10 mVs in a porous medium is used. In the first 5 seconds, the diffusive transport is more important than the convective transport. Soon after, the convective flow becomes the dominant mechanism. 

An aerosol issuing from a point source is dispersed in a steady turbulent plume in the atmosphere. Derive an expression for the variation of the extinction coeflicieni, h (Chapter 5), with position in the plume assuming that (a) the only mechanism affecting the light-scattering portion of the size distribution is turbulent diffusion and (b) the only mechanisms are turbulent diffusion and growth. 

We have seen that under certain idealized conditions the mean concentration of a species emitted from a point source has a Gaussian distribution. This fact, although strictly true only in the case of stationary, homogeneous turbulence, serves as the basis for a large class of atmospheric diffusion formulas in common use. The collection of Gaussian-based formulas is sufficiently important in practical application that we devote a portion of this chapter to them. The focus of these formulas is the expression for the mean concentration of a species emitted from a continuous, elevated point source, the so-called Gaussian plume equation. 

Figure 3.10 Sample bands at the column entrance. In the case on the left, the band resulting from a point source in the center of the column is shown. In the middle, the sample was distributed in a ring. The example on the right shows the band profile resulting from the combination of a circular injection and a diffuser.

Let us next consider the instantaneous release from a point source at the center of a living cell 100 pm in radius. Although the geometry here is a finite one, the initial stages of the diffusion process can be viewed as taking place into an infinite medium. If we assume D = 10" ° m"/s, typical of a protein, then the time it takes for the maximum concentration to reach r = 10 pm equals one-half the value given by Equation 4.5d ... 

Some researchers (e.g., Abramovich,Baturin,Rajaratnam,- and Nielsen and Moller ) consider x to be the distance from a point located at some distance Xq upstream from the diffuser face. Equations for the jet boundaries and velocity profile used in the centerline velocity derivation assume that the jet is supplied from the point source. Addition of the distance Xq to the distance from the outlet corrects for the influence of the outlet size on the jet geometry. For practical reasons some researchers neglect Xq. 

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