15 Air Plume

.title[Emissions and Plume Models] <br> .subtitle[BEE 4750/5750] <br> .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] <br> .date[October 24, 2022]

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# Outline

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1. Homework 4 Release
2. Questions?
3. Emissions of Pollutants
4. Gaussian Plume Models of Point Source/Receptors

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# Poll

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URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

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* Unit Commitment
  * Increased Complexity Switching from LP to MIP

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# Air Pollution

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.center[![Clean Air Act Air Pollutants](https://deq.utah.gov/wp-content/uploads/AQversion2.jpg) .cite[Source: [Utah Department of Environmental Quality](https://deq.utah.gov/communication/news/ask-an-environmental-scientist-what-air-pollutants-does-deq-test-for)]]

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# Greenhouse Gases

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.center[![Human-Emitted Greenhouse Gases](https://assets.nrdc.org/sites/default/files/styles/full_content--retina/public/media-uploads/greenhouseeffect_graphic2_2400x1800_0.png?itok=1SUXMPEl) .cite[Source: [NRDC](https://www.nrdc.org/stories/greenhouse-effect-101)]]

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# Clean Air Act Impacts

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.center[![Clean Air Act Changes in GDP and Criteria Pollutants](https://media.rff.org/images/clean_air_PR.width-1480.png) .cite[Source: [Resources for the Future](https://www.rff.org/news/press-releases/clean-air-act-successes-and-challenges-1970/)]]

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# Modeling Air Pollution

<hr>

There are many approaches to modeling concentrations of pollutants/chemicals, including:

1. Point sources/receptors (plume/puff models)
2. Analyzing an "airshed" (box models)

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# Point Sources/Receptors

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In this case, we're concerned about the dispersal from a point source: .center[![Source-Receptor Geometry](figures/air-pollution.png)]

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# Point Sources/Receptors

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At a given instant, the flow from source to receptor follows some path: .center[![Source-Receptor Geometry](figures/air-pollution-instant.png)]

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# Point Sources/Receptors

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Averaging over time gives us a "plume": .center[![Source-Receptor Geometry](figures/air-pollution-plume-timeavg.png)]

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# Point Source/Receptors

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What is the distribution of pollutant above the receptor? .center[![Source-Receptor Geometry](figures/air-pollution-plume-timeavg.png)]

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# "Gaussian Plume"

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We can reason that the concentration follows a "bell curve": .center[![Source-Receptor Geometry](figures/air-pollution-plume-shape.png)]

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# "Gaussian Plume"

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We can reason that the concentration follows a "bell curve": .center[![Source-Receptor Geometry](figures/air-pollution-plume-dist.png)]

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# "Gaussian Plume"

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Not illustrated here, but this same reasoning applies cross-wind. .center[![Source-Receptor Geometry](figures/air-pollution-plume-dist.png)]

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# Gaussian Plume Model

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For an elevated source **without reflection**:

$$
C(x,y,z) = \frac{Q}{2\pi u\sigma_y \sigma_z} \exp\left[-\frac{1}{2}\left(\frac{y^2}{\sigma_y^2} + \frac{(z - H)^2}{\sigma_z^2}\right)\right].
$$

| Variable  | Meaning                                     |
|:---------:|:------------------------------------------- |
|    $C$    | Concentration (g/m$^3$)                     |
|    $Q$    | Emissions rate (g/s)                        |
|    $u$    | Wind speed (m/s)                            |
|    $H$    | Height of source (m)                        |
| $x, y, z$ | Downwind, crosswind, vertical distances (m) |

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# Gaussian Plume Model Derivation

<hr>

We use the **advective-diffusion equation** to describe the mass-balance:

$$
\frac{\partial C}{\partial t} = [D + K] \nabla^2 C - \vec{u} \cdot \vec{\nabla} C $$

where $D$ is the diffusion coefficient, $K$ is the dispersion coefficient due to turbulent mixing.

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# Diffusive Flux
<hr>

$$F_x = D\frac{dC}{dx}

Concentration gradient + diffusion $\Rightarrow$ Flux]]

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# Turbulent Flux
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$$F\_x = K\_{xx} \frac{dC}{dx}
$$

The dispersion coefficient $K_{xx}$ depends on flow/eddy characteristics.

Concentration gradient + turbulent mixing $\Rightarrow$ Flux]]

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# Gaussian Plume Model Derivation

<hr>

$$
\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C  - [D + K] \nabla^2 C  = 0
$$

Assumptions:

* Steady-state, e.g. $\frac{\partial C}{\partial t} = 0$

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# Gaussian Plume Model Derivation

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$$
\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C  - [D + K] \nabla^2 C  = 0
$$

Assumptions:

* Wind only in the $x$-direction:

$$
\vec{u} \cdot \vec{\nabla} C = u_x \frac{\partial C}{\partial x} + \cancel{u_y \frac{\partial C}{\partial y}} + \cancel{u_z \frac{\partial C}{\partial z}}
$$

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# Gaussian Plume Model Derivation

<hr>

$$
\frac{\partial C}{\partial t} + \vec{u} \cdot \vec{\nabla} C  - [D + K] \nabla^2 C  = 0
$$

Assumptions:

* Turbulence $>>$ diffusion, e.g. $K >> D$, and $K$ is unimportant along $x$-direction:

$$
-[\cancel{D} + K] \nabla^2 C = \cancel{K\_{xx}} \frac{\partial^2 C}{\partial x^2} + K\_{yy} \frac{\partial^2 C}{\partial y^2} + K\_{zz} \frac{\partial^2 C}{\partial z^2}
$$

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# Gaussian Plume Model Derivation

<hr>

With these assumptions, the equation simplifies to:

$$
u \frac{\partial C}{\partial x} = K\_{yy} \frac{\partial^2 C}{\partial y^2} + K\_{zz}\frac{\partial^2 C}{\partial z^2}
$$

$$
Q = \iint u C dy dz
$$

]

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# Gaussian Plume Model Derivation

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Solving this PDE:

$$
C(x,y,z) = \frac{Q}{4\pi x \sqrt{K\_{yy} + K\_{zz}}} \exp\left[-\frac{u}{4x}\left(\frac{y^2}{K\_{yy}} + \frac{(z-H)^2}{K\_{zz}}\right)\right]
$$

To deal with $K$, substitute:

$$
\begin{aligned}
\sigma\_y^2 &= 2 K\_{yy} t = 2 K\_{yy} \frac{x}{u} \\\\
\sigma\_z^2 &= 2 K\_{zz} \frac{x}{u}
\end{aligned}
$$

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# Gaussian Plume Model Derivation

<hr>

Solving this PDE:

$$
\begin{aligned}
C(x,y,z) &= \frac{Q}{4\pi x \sqrt{K\_{yy} + K\_{zz}}} \exp\left[-\frac{u}{4x}\left(\frac{y^2}{K\_{yy}} + \frac{(z-H)^2}{K\_{zz}}\right)\right] \\\\
\Rightarrow C(x,y,z) &= \frac{Q}{2\pi u \sigma\_y \sigma\_z} \exp\left[-\frac{1}{2}\left(\frac{y^2}{\sigma\_y^2} + \frac{(z-H)^2}{\sigma\_z^2}\right) \right],
\end{aligned}
$$

which looks like a Gaussian distribution probability distribution if we restrict to $y$ or $z$.

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# Gaussian Plume Model with Reflection

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Last piece: the ground dampens vertical dispersion.

.right-column[The mass that would be dispersed below the ground is reflected back, increasing the concentration.]

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# Gaussian Plume Model with Reflection

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# Final Model: Elevated Source with Reflection

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$$
\begin{aligned}
C(x,y,z) = &\frac{Q}{2\pi u \sigma\_y \sigma\_z} \exp\left(\frac{-y^2}{2\sigma\_y^2} \right) \times \\\\
& \quad \left[\exp\left(\frac{-(z-H)^2}{2\sigma\_z^2}\right) + \exp\left(\frac{-(z+H)^2}{2\sigma\_z^2}\right) \right]
\end{aligned}
$$

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# Final Model: Elevated Source with Reflection

<hr>

Assumptions:

1. Steady-State
2. Constant wind velocity and direction
3. Wind >> dispersion in $x$-direction
4. No reactions
5. Smooth ground (avoids turbulent eddies and other reflections)

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# Gaussian Plume Example

<hr>

SO$_2$ is emitted at a rate of $100 \text{g/s}$ from the top of a $100 \text{m}$-high chimney. The plume initially rises $10 \text{m}$ before being convected horizontally by a wind speed of $15 \text{m/s}$. What is the centerline ground-level concentration $3 \text{km}$ downwind if at this distance $\sigma_y = 80 \text{m}$ and $\sigma_z=30 \text{m}$ under current meteorological conditions?

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# Gaussian Plume Example

<hr>

| Variable  | Value |
|:---------:|:----- |
|    $Q$    | ?     |
|    $u$    | ?     |
|    $H$    | ?     |
| $x, y, z$ | ?     |

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# Gaussian Plume Example

<hr>

| Variable  | Value          |
|:---------:|:-------------- |
|    $Q$    | 40 (g/s)       |
|    $u$    | 2 (m/s)        |
|    $H$    | 110 (m)        |
| $x, y, z$ | 3000, 0, 0 (m) |

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# Gaussian Plume Example

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$$
\begin{aligned}
C(3000,0,0) = &\frac{100 \text{g/s}}{2\pi (15 \text{m/s})(80 \text{m})(30 \text{m})} \exp(0) \times  \\\\
& \quad \left[\exp\left(\frac{-(-110 \text{m})^2}{2(30 \text{m})^2}\right) + \exp\left(\frac{-(110 \text{m})^2}{2(30 \text{m})^2}\right) \right]
\end{aligned}
$$

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# Gaussian Plume Example

<hr>

$$
C(x,y,z) = .27 \text{mg/m}^3,
$$

which is around 2% of the [NIOSH short-term exposure standard for SO$_2$](https://www.cdc.gov/niosh/npg/npgd0575.html).

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# Estimating Dispersion "Spread"

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Values of $\sigma_y$ and $\sigma_z$ matter substantially for modeling plume spread downwind. What influences them?

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# Estimating Dispersion "Spread"

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**Main contribution**: atmospheric stability

Pasquill (1961): Six stability classes, A -> F.

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# Estimating Dispersion "Spread"

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Contributors to atmospheric stability:

* Temperature gradient
  * Wind speed
  * Solar radiation
  * Cloud cover
  * Richardson number (buoyancy / flow shear)

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# Estimating Dispersion "Spread"

<hr>

Can estimate deviations based on stability class,

$$
\begin{aligned}
\sigma_y &= ax^b \\\\
\sigma_z &= cx^d + f
\end{aligned}
$$

where the coefficients are linked to the stability class.

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# Multiple Point Sources

<hr>

| Source | Emissions (kg/day) | Effective Height (m) | Removal Cost ($/kg) |
|:------:| ------------------:| --------------------:| -------------------:|
|   1    |             34,560 |                   50 |                0.20 |
|   2    |            172,800 |                  200 |                0.45 |
|   3    |            103,680 |                   30 |                0.60 |

and five receptors with $u = 1.5$ m/s and air quality standard 150 $ \mu$g/m$^3$. ]

/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_BQWGUc/build/multiple-sources.svg
.right-column[.center[![Schematic for Multiple Sources](figures/multiple-sources.svg)]]

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# Key Takeaways

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* Regulatory controls (or lack thereof) on air pollutants (Clean Air Act, etc) and greenhouse gas emissions (patchwork)
  * Concentrations of contaminants can be modeled in several ways, depending on the goal and assumptions.
  * Point source/receptor: Gaussian plume is common approach.

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# Next Class

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* Multiple source example
  * Box models of airsheds