Diffusion equation point source

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

The method used to develop the emission inventory does have some elements of error, but the other two alternatives are expensive and subject to their own errors. The first alternative would be to monitor continually every major source in the area. The second method would be to monitor continually the pollutants in the ambient air at many points and apply appropriate diffusion equations to calculate the emissions. In practice, the most informative system would be a combination of all three, knowledgeably applied. 

It is known that the vertical distribution of diffusing particles from an elevated point source is a function of the standard deviation of the vertical wind direction at the release point. The standard deviations of the vertical and horizontal wind directions are related to the standard deviations of particle concentrations in the vertical and horizontal directions within the plume itself. This is equivalent to saying that fluctuations in stack top conditions control the distribution of pollutant in the plume. Furthermore, it is known that the plume pollutant distributions follow a familiar Gaussian diffusion equation. 

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. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

Figure 8.13 Dispersion of an instantaneous point source [equation (8.5.1)]. A quantity of the diffusing species equivalent to the surface of whatever curve on the diagram is deposited initially at x = 0. The curves are Gauss functions.

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. 

In summary, we conclude that the application of the atmospheric diffusion equation to point sources will introduce an error of the order of 10% into predictions at points reasonably well removed (on the order of 1 km) from the source. 

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... 

Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29], Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form... 

Determine the fundamental solution, c = c(x, X2,t), of the diffusion equation for the point source in this coordinate system. 

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... 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

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... 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

A more advanced theory of the adsorption process at growing drop surfaces was made by MacLeod Radke (1994). In contrast to the theory discussed above they do not assume a point source at the beginning of the process but a finite drop size. On the basis of an arbitrary dependence R(t) a theory of diffusion- as well as kinetically-controlled adsorption was then derived. In addition to the diffusion equation (4.43) the following boundary condition is proposed ... 

This equation for the mean concentration from a continuous point source occupies a key position in atmospheric diffusion theory and we will have occasion to refer to it again and again. 

Before we proceed to solve (18.56), a few comments about the boundary conditions are useful. When the x diffusion term is dropped in the atmospheric diffusion equation, the equation becomes first-order in x, and the natural point for the single boundary condition on x is at x = 0. Since the source is also at x = 0 we have an option of whether to place the source on the RHS of the equation, as in (18.56), or in the x = 0 boundary condition. If we follow the latter course, then the x = 0 boundary condition is obtained by equating material fluxes across the plane at x = 0. The result is... 

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. 

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. 

We begin in this section to obtain solutions for atmospheric diffusion problems. Let us consider, as we did in the previous section, an instantaneous point source of strength 5 at the origin in an infinite fluid with a velocity u in the x direction. We desire to solve the atmospheric diffusion equation, (17.11), in this situation. Let us assume, for lack of anything bet-... 

By considering the solution of the transient diffusion equation for a point source, it may be shown that the definitions of Eqs. 32 and 33 are equivalent [37]. 

Key words Diffusion, Equation, Chemotaxis, Dictyostdium Point source. Pipette, Zigmond... 

Elbicki etal. 984) reviewed the optimum configurations for each of the above electrodes (thin-layer, tubular, and wall-jet) based on a mathematical treatment of the diffusive and convective phenomena in force. Boundary conditions on such physical restraints as electrode area, cell dimensions, and inlet configuration were established. Some confusion in the past has resulted from misinterpreting these equations (Weber, 1983) they are derived for cells in which the boundary layer may freely grow unencumbered. In certain cells (e.g., low-volume wall-jet or long-channel electrodes), walls, nozzles, etc. may impede the growth of the diffusion layer and bias the output current expected. Under these conditions, the wall-jet electrode behaves virtually as a thin-layer cell (if the nozzle spacing is small and the nozzle acts as a point source). Both detectors were concluded to yield output currents of... 

Abstract. The Boltzmann s equation is solved in the case of monoenergetic neutrons created by a plane or point source in an infinite medium which has spherically symmetric scattering. The customary solution of the diffusion equation appears to be multiplied by a constant factor which is smaller than 1. In addition to this term the total neutron density contains another term which is important in the neighborhood of the source. It goes with 1/r in the neighborhood of a point source. 

---