Impurity dispersion patterns

Impurity dispersion models are mathematical models of the distribution of impurities in the atmosphere.

Gaussian models

Gaussian models are based on the hypothesis that the distribution of particles in a jet or cloud is close to normal.

Non-stationary Gaussian model

Equation describing the distribution of the pollutant for the non-stationary case
$C(x,y,z,t)={\frac {Q}{(2\pi )^{3/2}\sigma _{x}\sigma _{y}\sigma _{z}}}\exp[-{\frac {((x-x_{0})-ut)^{2}}{2\sigma _{x}^{2}}}]\exp[-{\frac {(y-y_{0})^{2}}{2\sigma _{y}^{2}}}]\{\exp[-{\frac {(z-H)^{2}}{2\sigma _{z}^{2}}}]+\exp[-{\frac {(z+H)^{2}}{2\sigma _{z}^{2}}}]\}$ ${\ displaystyle C (x, y, z, t) = {\ frac {Q} {(2 \ pi) ^ {3/2} \ sigma _ {x} \ sigma _ {y} \ sigma _ {z} }} \ exp [- {\ frac {((x-x_ {0}) - ut) ^ {2}} {2 \ sigma _ {x} ^ {2}}}] \ exp [- {\ frac { (yyy_ {0}) ^ {2}} {2 \ sigma _ {y} ^ {2}}}] \ {\ exp [- {\ frac {(zH) ^ {2}} {2 \ sigma _ {z} ^ {2}}}] + \ exp [- {\ frac {(z + H) ^ {2}} {2 \ sigma _ {z} ^ {2}}] \}}$ $C(x,y,z,t)={\frac {Q}{(2\pi )^{3/2}\sigma _{x}\sigma _{y}\sigma _{z}}}\exp[-{\frac {((x-x_{0})-ut)^{2}}{2\sigma _{x}^{2}}}]\exp[-{\frac {(y-y_{0})^{2}}{2\sigma _{y}^{2}}}]\{\exp[-{\frac {(z-H)^{2}}{2\sigma _{z}^{2}}}]+\exp[-{\frac {(z+H)^{2}}{2\sigma _{z}^{2}}}]\}$

$C(x,y,z,t)$ ${\ displaystyle C (x, y, z, t)}$ $C(x,y,z,t)$ - Concentration of the pollutant at the coordinates $x, y, z$ ${\ displaystyle x, y, z}$ $x,y,z$ at the moment of time $t$ ${\ displaystyle t}$ $t$ [g / m ³ ]
$Q$ ${\ displaystyle Q}$ $Q$ - power of a continuous point source of pollution, [g / s] (here it’s just the amount of pollution [g])
$u$ ${\ displaystyle u}$ $u$ - wind speed at a height of H meters, [m / s]
$H$ ${\ displaystyle H}$ $H$ - effective height of the source of pollution, [m]
$t$ ${\ displaystyle t}$ $t$ - time of transport, [s]
$\sigma _{x},\sigma _{y}$ ${\ displaystyle \ sigma _ {x}, \ sigma _ {y}}$ $\sigma _{x},\sigma _{y}$ - horizontal dispersions, [m]
$\sigma _{z}$ ${\ displaystyle \ sigma _ {z}}$ $\sigma _{z}$ - vertical dispersion, [m]
$x_{0},y_{0},H$ ${\ displaystyle x_ {0}, y_ {0}, H}$ $x_{0},y_{0},H$ - coordinates of point source of pollution, [m]

Options $\sigma _{x},\sigma _{y},\sigma _{z}$ ${\ displaystyle \ sigma _ {x}, \ sigma _ {y}, \ sigma _ {z}}$ $\sigma _{x},\sigma _{y},\sigma _{z}$ increase with distance $x-x_{0}$ ${\ displaystyle x-x_ {0}}$ $x-x_{0}$ , the rate of increase depends on the intensity of turbulence and thus on the stability of the atmosphere. For practical use of addiction $\sigma _{x},\sigma _{y},\sigma _{z}$ ${\ displaystyle \ sigma _ {x}, \ sigma _ {y}, \ sigma _ {z}}$ $\sigma _{x},\sigma _{y},\sigma _{z}$ from distance are determined on the basis of experimental data.

Stationary Gaussian model

By integrating over time the concentration of pollution emitted from a continuous source, it is possible to obtain a steady-state concentration distribution for the stationary Gauss model

$C(x,y,z)={\frac {Q}{2\pi u\sigma _{y}\sigma _{z}}}\exp[-{\frac {(y-y_{0})^{2}}{2\sigma _{y}^{2}}}]\{\exp[-{\frac {(z-H)^{2}}{2\sigma _{z}^{2}}}]+\exp[-{\frac {(z+H)^{2}}{2\sigma _{z}^{2}}}]\}$ ${\ displaystyle C (x, y, z) = {\ frac {Q} {2 \ pi u \ sigma _ {y} \ sigma _ {z}}} \ exp [- {\ frac {(yyy_ { 0}) ^ {2}} {2 \ sigma _ {y} ^ {2}}}] \ {\ exp [- {\ frac {(zH) ^ {2}} {2 \ sigma _ {z} ^ {2}}}] + \ exp [- {\ frac {(z + H) ^ {2}} {2 \ sigma _ {z} ^ {2}}] \}}$

In both cases, the wind direction coincides with the direction of the axis. $x$ ${\ displaystyle x}$ In the Gaussian model, it is also assumed that there is a reflection of the pollutant from the surface of the earth. Reflection is characterized by a member in braces. The model was built on the assumption that the atmosphere is homogeneous and stable.

The presented model has several disadvantages:

Does not take into account the surface relief
Does not take into account changes in meteorological parameters in space and time
Does not describe the work of sources of pollution for a limited time.
Used characteristics obtained for terrestrial, not elevated sources.
Does not take into account the vertical structure of the boundary layer

Gaussian models can adequately describe the distribution of the pollutant only in the horizontal direction; they are applicable to very short distances for calculating the vertical profile.

Pasquill-Briggs model

The variance values are given in the form:

\sigma _{y}=p_{1}x(1+q_{1}x)^{-0.5}

{\ displaystyle \ sigma _ {y} = p_ {1} x (1 + q_ {1} x) ^ {- 0.5}}

\sigma _{z}=p_{2}x(1+q_{2}x)^{-1}

{\ displaystyle \ sigma _ {z} = p_ {2} x (1 + q_ {2} x) ^ {- 1}}

$p_{i},q_{i}$ ${\ displaystyle p_ {i}, q_ {i}}$ - set table for each class of atmospheric stability

For distances from 100 m to 10 km in the case of a flat open area ^[1]
$\sigma _{y}={\frac {\alpha _{x}x}{\sqrt {1+10^{-4}x}}}$ ${\ displaystyle \ sigma _ {y} = {\ frac {\ alpha _ {x} x} {\ sqrt {1 + 10 ^ {- 4} x}}}}$
$\sigma _{z}={\frac {\alpha _{z}x}{s_{z}(x)}}$ ${\ displaystyle \ sigma _ {z} = {\ frac {\ alpha _ {z} x} {s_ {z} (x)}}}$

Pasquill Resilience Class Table

Wind speed, m / s	Atmospheric stability classes AF
	Daytime. Solar illumination level			Night time. Cloud cover
	Strong	Average	Weak	> 50%	<50%
<2	A	AB	B	E	F
2-3	AB	B	C	E	F
3-5	B	BC	C	D	E
5-6	C	CD	D	D	D
> 6	C	D	D	D	D

Setton Model

Initially, Setton obtained a formula for ground-based sources of pollution, which was confirmed by the results of observations in Porton (England) under equilibrium conditions for relatively short distances (several hundred meters). The distribution of impurities near a point source in different directions is described by a Gaussian law. Impurity concentration at the point $(x,y,z)$ ${\ displaystyle (x, y, z)}$ from a source located at the origin, proportional to the product ^[2]
$p_{y}={\frac {1}{\sigma _{y}{\sqrt {2\pi }}}}\exp(-{\frac {y^{2}}{2\sigma _{y}^{2}}})$ ${\ displaystyle p_ {y} = {\ frac {1} {\ sigma _ {y} {\ sqrt {2 \ pi}}}} \ exp (- {\ frac {y ^ {2}} {2 \ sigma _ {y} ^ {2}}})}$
on similar functions $p_{z}$ ${\ displaystyle p_ {z}}$ and $p_{x}$ ${\ displaystyle p_ {x}}$

$\sigma _{y}^{2}$ ${\ displaystyle \ sigma _ {y} ^ {2}}$ dispersion distribution of impurities in the direction $y$ ${\ displaystyle y}$

$\sigma _{i}^{2}={\frac {1}{2}}c_{i}^{2}({\overline {u}}t)^{2-n}$ ${\ displaystyle \ sigma _ {i} ^ {2} = {\ frac {1} {2}} c_ {i} ^ {2} ({\ overline {u}} t) ^ {2-n}}$

$c_{i}$ ${\ displaystyle c_ {i}}$ some coefficients
${\overline {u}}$ ${\ displaystyle {\ overline {u}}}$ average wind speed
$t$ ${\ displaystyle t}$ time after the source action (in the case of the instant source), for a continuous source, it is assumed that $t=x/{\overline {u}}$ ${\ displaystyle t = x / {\ overline {u}}}$
$i=1,2,3$ ${\ displaystyle i = 1,2,3}$ corresponds to $x, y, z$ ${\ displaystyle x, y, z}$
parameter $n$ ${\ displaystyle n}$ You can define the vertical profile of the wind speed, thereby indirectly take into account the conditions of stratification

Turbulent Diffusion Model

The general equation of mass transfer is described in general form by the equation of turbulent diffusion
${\frac {\partial C}{\partial t}}+{\frac {\partial }{\partial x}}u\cdot C+{\frac {\partial }{\partial y}}v\cdot C+{\frac {\partial }{\partial z}}\omega \cdot C={\frac {\partial }{\partial x}}D_{x}{\frac {\partial C}{\partial x}}+{\frac {\partial }{\partial y}}D_{y}{\frac {\partial C}{\partial y}}+{\frac {\partial }{\partial z}}D_{z}{\frac {\partial C}{\partial z}}$ {\ displaystyle {\ frac {\ partial C} {\ partial t}} + {\ frac {\ partial} {\ partial x}} u \ cdot C + {\ frac {\ partial} {\ partial y}} v \ cdot C + {\ frac {\ partial} {\ partial z}} \ omega \ cdot C = {\ frac {\ partial} {\ partial x}} D_ {x} {\ frac {\ partial C} {\ partial x }} + {\ frac {\ partial} {\ partial y}} D_ {y} {\ frac {\ partial C} {\ partial y}} + {\ frac {\ partial} {\ partial z}} D_ { z} {\ frac {\ partial C} {\ partial z}}}

Boundary condition
$D_{z}{\frac {\partial C}{\partial z}}+\omega C=\beta C$ ${\ displaystyle D_ {z} {\ frac {\ partial C} {\ partial z}} + \ omega C = \ beta C}$

$C$ ${\ displaystyle C}$ - concentration of pollutant [g / m ³ ]
$D_{x},D_{y},D_{z}$ ${\ displaystyle D_ {x}, D_ {y}, D_ {z}}$ - turbulent diffusion coefficients [m ² / s]
$u$ ${\ displaystyle u}$ - average wind speed along the axis $x$ ${\ displaystyle x}$ [m / s]
$v$ ${\ displaystyle v}$ - average wind speed along the axis $y$ ${\ displaystyle y}$ [m / s]
$\omega$ ${\ displaystyle \ omega}$ - average sedimentation rate of particles of a pollutant, [m / s]
$\beta$ ${\ displaystyle \ beta}$ - constant [m / s]. With $\beta =0$ ${\ displaystyle \ beta = 0}$ the boundary condition means that the flux on the surface is zero, all pollutant remains in the atmosphere "reflected" from the surface of the earth. With $\beta =\infty$ ${\ displaystyle \ beta = \ infty}$ the contaminant "sticks" to the surface. In the intermediate case $0<\beta <\infty$ ${\ displaystyle 0 <\ beta <\ infty}$ the substance is partially "reflected" partially "sticks", usually only two extreme possibilities are considered - "reflection" or "sticking".

The analytical solution of the turbulent diffusion equation is in particular cases under the assumptions of specific functions of the diffusion coefficients of the coordinates.

An example of solving a 3D turbulent diffusion equation - finite element method

Solution of the turbulent diffusion equation with constant diffusion coefficients and homogeneous boundary conditions.

The solution of the turbulent diffusion equation with constant turbulent diffusion coefficients $D_{x},D_{y},D_{z}$ ${\ displaystyle D_ {x}, D_ {y}, D_ {z}}$ under the action of a constant point source of pollution with regard to homogeneous boundary conditions
$u{\frac {\partial C}{\partial x}}=D({\frac {\partial C}{\partial x}}+{\frac {\partial C}{\partial y}}+{\frac {\partial C}{\partial z}})+Q\delta (r)$ ${\ displaystyle u {\ frac {\ partial C} {\ partial x}} = D ({\ frac {\ partial C} {\ partial x}} + {\ frac {\ partial C} {\ partial y}} + {\ frac {\ partial C} {\ partial z}}) + Q \ delta (r)}$

$Q\delta$ ${\ displaystyle Q \ delta}$ - The effect of a permanent point source of pollution, $\delta$ ${\ displaystyle \ delta}$ - Dirac delta function
$Q$ ${\ displaystyle Q}$ - Power point source of pollution, [g / s]
$r$ ${\ displaystyle r}$ - Distances from the source, [m]
$D=D_{x}=D_{y}=D_{z}$ ${\ displaystyle D = D_ {x} = D_ {y} = D_ {z}}$ - Turbulent diffusion coefficient, [m ² / s]

Equation solution
$C(x,y,z)={\frac {Q}{4\pi Dr}}\exp[-{\frac {u}{2D}}(r-x)]$ ${\ displaystyle C (x, y, z) = {\ frac {Q} {4 \ pi Dr}} \ exp [- {\ frac {u} {2D}} (rx)]}$
According to this model, the dependence of concentration on the distance to the source is hyperbolic in nature, while according to the Gauss model, this dependence has the character of an exponential decay law.

Solution of the turbulent diffusion equation with constant diffusion coefficients under the boundary condition "reflection"

The solution of the turbulent diffusion equation at $u=const$ ${\ displaystyle u = const}$ and presence at the point $x=0,y=0,z=h$ ${\ displaystyle x = 0, y = 0, z = h}$ , stationary point source of pollution and at the boundary condition of “reflection” at the level of $z=0$ ${\ displaystyle z = 0}$ :
$D_{z}{\frac {\partial C}{\partial z}}+\omega C=0,z=0$ ${\ displaystyle D_ {z} {\ frac {\ partial C} {\ partial z}} + \ omega C = 0, z = 0}$
$C(x,y,z)={\frac {Q}{2\pi x{\sqrt {D_{y}D_{z}}}}}\exp[-{\frac {uy^{2}}{4D_{x}\cdot x}}]\{\exp[-{\frac {u(z-h)^{2}}{4D_{z}\cdot x}}]+\exp[-{\frac {u(z+h)^{2}}{4D_{z}\cdot x}}]\}$ ${\ displaystyle C (x, y, z) = {\ frac {Q} {2 \ pi x {\ sqrt {D_ {y} D_ {z}}}}} \ exp [- {\ frac {uy ^ { 2}} {4D_ {x} \ cdot x}}] \ {\ exp [- {\ frac {u (zh) ^ {2}} {4D_ {z} \ cdot x}}] + \ exp [- { \ frac {u (z + h) ^ {2}} {4D_ {z} \ cdot x}}] \}}$

Solution of the stationary turbulent diffusion equation with a power dependence of the vertical turbulent diffusion coefficient

Mathematical formulation of the problem
$u{\frac {\partial C}{\partial x}}=D_{y}{\frac {\partial ^{2}C}{\partial y^{2}}}+{\frac {\partial }{\partial z}}D_{z}(z){\frac {\partial C}{\partial z}}$ ${\ displaystyle u {\ frac {\ partial C} {\ partial x}} = D_ {y} {\ frac {\ partial ^ {2} C} {\ partial y ^ {2}}} + {\ frac { \ partial} {\ partial z}} D_ {z} (z) {\ frac {\ partial C} {\ partial z}}}$

The boundary condition is either "reflection" or absorption.

The equation is written in neglect of diffusion along the wind direction (axis $x$ ${\ displaystyle x}$ )
$D_{y}=const$ ${\ displaystyle D_ {y} = const}$ coefficient of horizontal turbulent diffusion, [m ² / s]
$D_{z}(z)=D_{1}({\frac {z}{z_{1}}})^{1-1/p}$ ${\ displaystyle D_ {z} (z) = D_ {1} ({\ frac {z} {z_ {1}}}) ^ {1-1 / p}}$ vertical turbulent diffusion coefficient m ² / s]
$p$ ${\ displaystyle p}$ air thermal stability parameter $p=\infty$ ${\ displaystyle p = \ infty}$ - indifferent stratification; $p>0$ ${\ displaystyle p> 0}$ - stable stratification; $p<0$ ${\ displaystyle p <0}$ - convection

OND Technique - 86

In Russia and some other countries of the former USSR, the OND-86 method is used to calculate local atmospheric pollution by emissions from industrial enterprises, which reduces to a sequence of analytical expressions obtained as a result of approximation of the difference solution of the turbulent diffusion equation. Method OND-86 allows you to calculate the maximum possible distribution of emission concentrations in a moderately unstable atmosphere and averaged over a 20-30 minute interval, but does not take into account such factors as the stability class of the atmosphere and the roughness of the underlying surface. The technique is applicable to calculate impurity concentrations at a distance from source no more than 100 km.

Notes

↑ 18) Berlyand M. Ye. Modern Problems of Atmospheric Diffusion and Atmospheric Pollution. L .: Gidrometeoizdat, 1975. 448 p.
↑ Berlyand M.E. "Modern problems of atmospheric diffusion and air pollution", 1975

The US Environmental Protection Agency website presents numerous alternative impurity dispersion models, mainly based on Gaussian dispersion models.
Alternative impurity dispersion models
The special module Flotran of the ANSYS software package allows solving various problems of impurity propagation based on the solutions of the Navier – Stokes equations , the continuity equation, the heat transfer equation, and the mass transfer equation.

Links

Materials of the IAEA Meeting, 1987, Chapter 3 p. 26
Sun W.-Y. and C.-Z. Chang. Diffusion model for a convective layer. Part 2: Plume released from a continuous point source. J. Climate Appl. Meteorol. 1986, vol. 25, No 10, pp. 1454-1463
Pasquill F. Atmospheric dispersion parameters in gaussian plume modeling: [part II. Possible Requirements for Workers Values]. / F. Pasquill // EPA-600 / 4-76-030b, US Environmental Protection Agency, Research Triangle Park, North Carolina 27711. - 1976.

[berland_1975-1] 18) Berlyand M. Ye. Modern Problems of Atmospheric Diffusion and Atmospheric Pollution. L .: Gidrometeoizdat, 1975. 448 p.

[autogenerated1-2] Berlyand M.E. "Modern problems of atmospheric diffusion and air pollution", 1975