Gaussian diffusion equation

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. 

The Gaussian diffusion equation is known as the Pasquill and Gifford model, and is used to develop methods for estimating the required diffusion coefficients. The basic equation, already presented in a slightly different form, is restated below ... 

The reason for calling equation 8.3-1 a "Gaussian diffusion model" is because it has the form of the normal/Gaussian distribution (equation 2.5-2). Concentration averages for long time intervals may be calculated by averaging the concentrations at grid elements over which the plume passes. 

For the case of classical Gaussian diffusion 0=0 and, believing r(t)=2 and t=4 relative units, the equality within the framework of the relationship (1) will be obtained. Such equality assumes p= 1, i.e., each contact of reagents molecules results to reaction product formation. Let s assume, that the value p decreases up to 0,05, i.e., only one from 20 contacts of reagents molecules forms a new chemical species. This means the increase t in 20 times and then at r(t)=2 and =80 relative units from the relationship (1) will be obtained 0=4,33. Since 0 is connected with dimension of walk trajectory of reagents molecules dw by the simple equation... 

The object of this section is to derive the Gaussian equations of the previous section as solutions to the atmospheric diffusion equation. Such a relationship has already been demonstrated in Section IV,B for the case of no boundaries. We extend that consideration now to boundaries. We recall that constant eddy diffusivities were assumed in Section IV,B. 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally had to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy difiusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender-plume approximation), and if the wind direction coincides with the x axis, then = 0. Thus, it is necessary to specify only the lateral (Kyy) and vertical coefficients. It is generally assumed that horizontal homogeneity exists so that u, Kyy, and Ka are independent of y. Hence, Eq. (2.19) becomes... 

In many treatments of free diffusion the propagator is immediately written as a Gaussian function with the argument that it fulfils the diffusion equation. Equation (27) shows the relation with the normal mode solution of the diffusion equation. For diffusion in a bounded region the propagator is no... 

The above treatment shows that the Gaussian curve is a valid solution to the basic diffusion equation. More importantly, it shows in detail how the Gaussian evolves in time and space. In general terms, the last two equations show that the Gaussian pulse becomes broader and lower (more dilute) with the passage of time. 

Inasmuch as we have seen that the basic diffusion equations lead to a Gaussian profile, we may in most cases describe Gaussian spreading as an apparent diffusion process, with an effective diffusion coefficient related to [Pg.93]

To summarize, the observation of a Gaussian profile usually implies that transport is governed mathematically by the diffusion equations and mechanistically by one or more multistep random processes. Below we examine some of the random mechanisms operative in separations. 

This is a plausible way to prove that Eq. (162) is the diffusion equation that applies to the gaussian condition. It is important to point out that a more satisfactory derivation of this exact result can be obtained by using the Zwanzig projection approach of Section III [67,68]. Thus, Eq. (162) as well as Eq. (133) must be considered as generalized diffusion equations compatible with a Liouville origin. 

In the foregoing relationships, by replacing d with the particle coordinate and w with its velocity, we obtain the standard equations of Brownian motion. In the long time limit. Ait) 2Dt with D = foiviO)vit) dt and Eq. (150) becomes the well-known Fick-Einstein diffusion equation. Obviously, the Gaussian process and its long time limit are inherent in this equation. 

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

From the differentiation theorem of Laplace transform, J f /(t) = uP u) —P t = 0), we infer that the left-hand side in (x,t) space corresponds to 0P(x, t)/dt, with initial condition P(x. 0) = 8(x). Similarly in the Gaussian limit a = 2, the right-hand side is Dd2P(x, f)/0x2, so that we recover the standard diffusion equation. For general a, the right-hand side defines a fractional differential operator in the Riesz-Weyl sense (see below) and we find the fractional diffusion equation [52-56]... 

For Gaussian processes with propagator P(x, t) = 1 / V4-nDt exp(—x2/ [4Dt ), one obtains by direct integration of the diffusion equation with appropriate boundary condition the first passage time density [70]... 

Unlike the solution of the diffusion equation, which is positive everywhere, the solution of Eq. 6.134a is equal to 0 for > (m t). Figure 6.16 shows some concentration profiles calculated as solution of Eq. 6.134a. When time increases, the asymmetry of the concentration profiles and the sharpness of the cutoffs at the endpoints, z = ut, increase. Eor sufficiently long times, however, their solution approaches the shifted Gaussian profile predicted by the solution of Eq. 6.22. 

Where F is the variance of analyte molecides about their mean in the analyte broadening zone which have a concentration profile in the Gaussian distribution shape, and the Lz is the distance the zone has moved (please note that Lz does not necessarily refer to column length here). Obviously, this is a more meaningful and useful concept, which views the HETP as the length of column necessary to achieve equihbrium between the Hquid and mobile phase. In addition, equation 27 can be related to the random diffusion process (actually, the movement of analyte molecules between the two phases is hke the molecule motion in a random diffusion process) defined by the Einstein diffusion equation ... 

Hermans and Van Beek626 have recently used the new model of polymer molecules suggested by Rouse.46 At high frequencies the whole molecule cannot follow the field so it is divided into a number of submolecules small enough to follow the field and yet sufficiently large to have a Gaussian distribution. Dielectric relaxation for the case of dipoles parallel to the chain has been calculated by Founder sum transforms. The distribution of relaxation modes appears though the multiplicity of the mathematical solution for the diffusion equation. 

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