Atmospheric diffusion and pollutants

Atmospheric dispersion,^[3] or diffusion, studies how pollutants are mixed in the environment. There are many factors included in this modeling process, such as which level of atmosphere(s) the mixing is taking place, the stability of the environment and what type of contaminant and source is being mixed. The Eulerian and Lagrangian (discussed below) models have both been used to simulate atmospheric diffusion, and are important for a proper understanding of how pollutants react and mix in different environments. Both of these models take into account both vertical and horizontal wind, but additionally integrate Fickian diffusion theory to account for turbulence. While these methods have to use ideal conditions and make numerous assumptions, at this point in time, it is difficult to better calculate the effects of turbulent diffusion on pollutants. Fickian diffusion theory and further advancements in research on atmospheric diffusion can be applied to model the effects that current emission rates of pollutants from various sources have on the atmosphere.^[4]

Turbulent diffusion flames

Using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) processes, there has been on-going research on the effects of turbulent diffusion in flames. Main areas of study include combustion systems in gas burners used for power generation and chemical reactions in jet diffusion flames involving methane (CH₄), hydrogen (H₂) and nitrogen (N₂).^[5] Additionally, double-pulse Rayleigh temperature imaging has been used to correlate extinction and ignition sites with changes in temperature and the mixing of chemicals in flames.^[6]

Eulerian approach

The Eulerian approach to turbulent diffusion focuses on an infinitesimal volume at a specific space and time in a fixed frame of reference, at which physical properties such as mass, momentum, and temperature are measured.^[7] The model is useful because Eulerian statistics are consistently measurable and offer great application to chemical reactions. Similarly to molecular models, it must satisfy the same principles as the continuity equation below (where the advection of an element or species is balanced by its diffusion, generation by reaction, and addition from other sources or points) and the Navier–Stokes equations:

${\displaystyle {\partial c_{i} \over \partial t}+{\partial \over \partial x_{j}}(u_{j}c_{i})=D_{i}{\partial ^{2}c_{i} \over \partial x_{j}\partial x_{j}}+R_{i}(c_{1},...,c_{N},T)+S_{i}(x,t)}$

${\displaystyle {i=1,2,...,N}}$

where ${\displaystyle c_{i}}$ = species concentration of interest, ${\displaystyle u_{j}}$ = velocity t= time, ${\displaystyle x_{j}}$= direction, ${\displaystyle D_{i}}$ = molecular diffusion constant, ${\displaystyle R_{i}}$ = rate of ${\displaystyle c_{i}}$ generated reaction, ${\displaystyle S_{i}}$ = rate of ${\displaystyle c_{i}}$ generated by source. ^[8] Note that ${\displaystyle c_{i}}$ is concentration per unit volume, and is not mixing ratio (${\displaystyle kg/kg}$) in a background fluid.

If we consider an inert species (no reaction) with no sources and assume molecular diffusion to be negligible, only the advection terms on the left hand side of the equation survive. The solution to this model seems trivial at first, however we have ignored the random component of the velocity plus the average velocity in u_j= ū + u_j’ that is typically associated with turbulent behavior. In turn, the concentration solution for the Eulerian model must also have a random component c_j= c+ c_j’. This results in a closure problem of infinite variables and equations and makes it impossible to solve for a definite c_i on the assumptions stated.^[9]

Fortunately there exists a closure approximation in introducing the concept of eddy diffusivity and its statistical approximations for the random concentration and velocity components from turbulent mixing:

${\displaystyle \langle u_{j}'c'\rangle =-K_{jj}{\partial (c) \over \partial x_{j}}}$

where K_jj is the eddy diffusivity.^[8]

Substituting into the first continuity equation and ignoring reactions, sources, and molecular diffusion results in the following differential equation considering only the turbulent diffusion approximation in eddy diffusion:

${\displaystyle {\partial c_{i} \over \partial t}+{\overline {u}}_{j}{\partial (c) \over \partial x_{j}}={\partial \over \partial x_{j}}{\bigg (}K_{jj}{\partial (c) \over \partial x_{j}}{\bigg )}}$

Unlike the molecular diffusion constant D, the eddy diffusivity is a matrix expression that may vary in space, and thus may not be taken outside the outer derivative.

Lagrangian approach

The Lagrangian model to turbulent diffusion uses a moving frame of reference to follow the trajectories and displacements of the species as they move and follows the statistics of each particle individually.^[7] Initially, the particle sits at a location x’ (x₁, x₂, x₃) at time t’. The motion of the particle is described by its probability of existing in a specific volume element at time t, that is described by Ψ(x₁, x₂, x₃, t) dx₁ dx₂ dx₃ = Ψ(x,t)dx which follows the probability density function (pdf) such that:

${\displaystyle {\boldsymbol {\psi }}(\mathbf {x} ,{\mathit {t}})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} ',{\mathit {t}}'){\boldsymbol {\psi }}(\mathbf {x} ',{\mathit {t}}')d\mathbf {x} '}$ Where function Q is the probably density for particle transition.

The concentration of particles at a location x and time t can then be calculated by summing the probabilities of the number of particles observed as follows:

${\displaystyle \langle c(\mathbf {x} ,{\mathit {t}})\rangle =\sum _{i=1}^{m}{\boldsymbol {\psi }}_{i}(\mathbf {x} ,{\mathit {t}})}$

Which is then evaluated by returning to the pdf integral^[8]

${\displaystyle \langle c(\mathbf {x} ,{\mathit {t}})\rangle ==\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} _{0},{\mathit {t}}_{0})\langle c(\mathbf {x} _{0},{\mathit {t}}_{0})\rangle d\mathbf {x} _{0}+\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{t_{0}}^{t}{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} ',{\mathit {t}}'){\mathit {S}}(\mathbf {x} ',{\mathit {t}}')d{\mathit {t}}d\mathbf {x} '}$

Thus, this approach is used to evaluate the position and velocity of particles relative to their neighbors and environment, and approximates the random concentrations and velocities associated with turbulent diffusion in the statistics of their motion.

Solutions

The resulting solution for solving the final equations listed above for both the Eulerian and Lagrangian models for analyzing the statistics of species in turbulent flow, both result in very similar expressions for calculating the average concentration at a location from a continuous source. Both solutions develop a Gaussian Plume and are virtually identical under the assumption that the variances in the x,y,z directions are related to the eddy diffusivity:

${\displaystyle \langle c(x,y,z)\rangle ={\frac {q}{2\pi \sigma _{y}\sigma _{z}}}exp{\bigg [}-{\bigg (}{\frac {y^{2}}{\sigma _{y}^{2}}}+{\frac {z^{2}}{\sigma _{z}^{2}}}{\bigg )}{\bigg ]}}$

where ${\displaystyle \sigma _{y}^{2}={\frac {2K_{yy}x}{\overline {u}}}\sigma _{z}^{2}={\frac {2K_{zz}x}{\overline {u}}}}$

q= species emission rate, u = wind speed, σ_i² = variance in i direction. ^[8]

Under various external conditions such as directional flow speed (wind) and environmental conditions, the variances and diffusivities of turbulent diffusion are measured and used to calculate a good estimate of concentrations at a specific point from a source. This model is very useful in atmospheric sciences, especially when dealing with concentrations of contaminants in air pollution that emanate from sources such as combustion stacks, rivers, or strings of automobiles on a road.^[2]