Criteria Air Pollutants
- Pollutants which have an ambient air quality standard.
- Most common examples:
- PM2.5, PM10
- O3
- CO
- SO2
- NO2
Impact of the Clean Air Act
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Decoupling of GDP and Air Pollution
Some Approaches For Modeling Air Pollution
Last Class: Box (or “airshed”) model of air pollution
- Looks at overall mass-balance.
- Total inputs/outputs in a particular boundary.
Today: Point sources and receptors (plume/puff models)
Point Sources/Receptors
Concerned about dispersal from a point source:
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Point Source/Receptor Schematic
Point Sources/Receptors
At a given instant, flow from source to receptor follows a path:
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Point Source Flow Traces
Point Sources/Receptors
Averaging those traces over time gives us a “plume”:
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Point Source Average Plume
Point Sources/Receptors
What is the distribution of pollutant above the receptor?
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Point Source Average Plume
“Gaussian Plume”
We can reason that the concentration follows a “bell curve”:
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Gaussian Plume
“Gaussian Plume”
We can reason that the concentration follows a “bell curve”:
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Gaussian Plume Distribution
“Gaussian Plume”
Not illustrated, but this reasoning also applies cross-wind (\(y\)).
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Gaussian Plume Distribution
Gaussian Plume Model
\[\begin{equation}
C(x,y,z) = \frac{Q}{2\pi u \sigma_y \sigma_z} \exp\left[-\frac{1}{2} \left(\frac{y^2}{\sigma_y^2} + \frac{(z - H)^2}{\sigma_z^2}\right)\right]
\end{equation}\]
| \(C\) |
Concentration (g/m\(^3\)) |
| \(Q\) |
Emissions Rate (g/s) |
| \(H\) |
Effective Source Height (m) |
| \(u\) |
Wind Speed (m/s) |
Gaussian Plume Derivation
Use the advective-diffusion equation to describe the mass-balance of a small air parcel :
\[ \frac{\partial C}{\partial t} + \color{blue}\overbrace{[D + K] \nabla^2 C}^\text{diffusion} \color{black} - \color{red}\overbrace{\overrightarrow{u} \cdot \overrightarrow{\nabla} C}^\text{advection}\]
- \(D\) is the diffusion coefficient
- \(K\) is the dispersion coefficient (turbulent mixing)
Diffusive Flux
Concentration gradient + Diffusion ⇒ Flux
Fick’s law: mass transfer by diffusion
\[F_x = D \frac{dC}{dx}\]
Turbulent Flux
Concentration gradient + Turbulent mixing ⇒ Flux
\[F_x = K_{xx}\frac{dC}{dx}\]
The dispersion coefficient \(K_{xx}\) depends on flow/eddy characteristics.
Gaussian Plume Model Derivation
\[\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C - [D + K] \nabla^2 C = 0\]
Assumptions:
\[ \frac{\partial C}{\partial t} = 0\]
Gaussian Plume Model Derivation
\[\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C - [D + K] \nabla^2 C = 0\]
Assumptions:
- Wind only in \(x\)-direction:
\[\vec{u} \cdot \vec{\nabla} C = u_x \frac{\partial C}{\partial x} + \cancel{u_y \frac{\partial C}{\partial y}} + \cancel{u_z \frac{\partial C}{\partial z}}\]
Gaussian Plume Model Derivation
\[\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C - [D + K] \nabla^2 C = 0\]
Assumptions:
- Turbulence \(\gg\) diffusion, e.g. \(K \gg D\), and \(K\) is unimportant along \(x\)-direction:
\[-[\cancel{D} + K] \nabla^2 C = \cancel{K_{xx}} \frac{\partial^2 C}{\partial x^2} + K_{yy} \frac{\partial^2 C}{\partial y^2} + K_{zz} \frac{\partial^2 C}{\partial z^2}\]
Gaussian Plume Model Derivation
With these assumptions, the equation simplifies to:
\[u \frac{\partial C}{\partial x} = K_{yy} \frac{\partial^2 C}{\partial y^2} + K_{zz}\frac{\partial^2 C}{\partial z^2}\]
Assume mass flow through vertical plane downwind must equal emissions rate \(Q\):
\[Q = \iint u C dy dz\]
Gaussian Plume Model Derivation
Solving this PDE:
\[C(x,y,z) = \frac{Q}{4\pi x \sqrt{K_{yy} + K_{zz}}} \exp\left[-\frac{u}{4x}\left(\frac{y^2}{K_{yy}} + \frac{(z-H)^2}{K_{zz}}\right)\right]\]
Now substitute
\[\begin{aligned}
\sigma_y^2 &= 2 K_{yy} t = 2 K_{yy} \frac{x}{u} \\
\sigma_z^2 &= 2 K_{zz} \frac{x}{u}
\end{aligned}\]
Gaussian Plume Model Derivation
This results in:
\[
C(x,y,z) = \frac{Q}{2\pi u \sigma_y \sigma_z} \exp\left[-\frac{1}{2}\left(\frac{y^2}{\sigma_y^2} + \frac{(z-H)^2}{\sigma_z^2}\right) \right]\]
which looks like a Gaussian distribution probability distribution if we restrict to \(y\) or \(z\).
Gaussian Plume Model with Reflection
Last piece: the ground dampens vertical dispersion.
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Reflected Mass Off Ground
Gaussian Plume Model with Reflection
We can account for this extra term using a flipped “image” of the source.
Final Model: Elevated Source with Reflection
\[\begin{aligned}
C(x,y,z) = &\frac{Q}{2\pi u \sigma_y \sigma_z} \exp\left(\frac{-y^2}{2\sigma_y^2} \right) \times \\\\
& \quad \left[\exp\left(\frac{-(z-H)^2}{2\sigma_z^2}\right) + \exp\left(\frac{-(z+H)^2}{2\sigma_z^2}\right) \right]
\end{aligned}\]
Final Model Assumptions
Assumptions:
- Steady-State
- Constant wind velocity and direction
- Wind >> dispersion in \(x\)-direction
- No reactions
- Smooth ground (avoids turbulent eddies and other reflections)
Dispersion and Atmospheric Stability
Estimating Dispersion “Spread”
Values of \(\sigma_y\) and \(\sigma_z\) matter substantially for modeling plume spread downwind. What influences them?
Estimating Dispersion “Spread”
Main contribution: atmospheric stability
Estimating Dispersion “Spread”
Contributors to atmospheric stability:
- Temperature gradient
- Wind speed
- Solar radiation
- Cloud cover
- Richardson number (buoyancy / flow shear)
Atmospheric Stability Classes
| A |
Extremely unstable |
Sunny summer day |
| B |
Moderately unstable |
Sunny & warm |
| C |
Slightly unstable |
Partly cloudy day |
| D |
Neutral |
Cloudy day or night |
| E |
Slightly stable |
Partly cloudy night |
| F |
Moderately stable |
Clear night |
Estimating Dispersion “Spread”
\[\begin{aligned}
\sigma_y &= ax^{0.894} \\
\sigma_z &= cx^d + f
\end{aligned}\]
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Dispersion Coefficients
Note: here \(x\) is in km, while in the plume equation \(y\), \(z\) are in m!
Estimating Dispersion “Spread”
Then take estimates for \(\sigma_y\) and \(\sigma_z\) and plug into plume dispersion equation
\[\begin{aligned}
C(x,y,z) = &\frac{Q}{2\pi u {\color{red}\sigma_y \sigma_z}} \exp\left(\frac{-y^2}{2{\color{red}\sigma_y^2}} \right) \times \\\\
& \quad \left[\exp\left(\frac{-(z-H)^2}{2{\color{red}\sigma_z^2}}\right) + \exp\left(\frac{-(z+H)^2}{2{\color{red}\sigma_z^2}}\right) \right]
\end{aligned}\]
