Managing Multiple Air Pollutant Sources

Lecture 17

October 30, 2024

Review and Questions

Gaussian Plume for Air Pollution Dispersion

Used for point sources.
Typically for continuous emissions from an elevated source.

Gaussian Plume Model

Variable	Meaning
$C$	Concentration (g/m$^3$)
$Q$	Emissions Rate (g/s)
$H$	Effective Source Height (m)
$u$	Wind Speed (m/s)
$y, z$	Crosswind, Vertical Distance (m)

Gaussian Plume Model

Gaussian Plume Model With Ground Reflection

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

Questions

Text: VSRIKRISH to 22333

URL: https://pollev.com/vsrikrish

See Results

Multiple Point Sources

Managing Multiple Plumes

We could use a Gaussian plume to simulate the effect of a single source, but often we:

Have multiple sources that we need to manage;
Care about compliance with regulatory standards at a particular important receptor.

Multiple Point Source Example

Take three sources of SO₂ (air quality standard 13 $\text{mg/m}^3$):

Source	Emissions (kg/day)	Effective Height (m)	Removal Cost ($/kg)
1	86.4	50	0.20
2	216	200	0.45
3	155.52	30	0.60

and five receptors at ground level with $u = 1.5$ m/s.

Aligning Units

Source	Emissions (g/s)	Effective Height (m)	Removal Cost ($/g)
1	10,000	50	0.0002
2	25,000	200	0.00045
3	18,000	30	0.00060

Multiple Point Source Example

Our goal:

Minimize cost of removing SO₂ from the plume sources to ensure all receptors are not exposed beyond the 13 $\text{mg/m}^3$ standard.

Code

sources = [(0, 7), (2, 5), (3, 5)]
receptors = [(1, 1.5), (3, 7), (5, 3), (7.5, 6), (10, 5)]

p = scatter(sources, label="Source", markersize=6, color=:red, xlabel=L"$x$ (km)", ylabel=L"$y$ (km)", legend=:bottomright, ylims=(0, 8), xlims=(-0.5, 10.5))
scatter!(receptors, label="Receptor", markersize=6, color=:black)
for i in 1:length(sources)
    annotate!(sources[i][1], sources[i][2] + 0.3, text(string(i), :red, 14, :center))
    annotate!(sources[i][1], sources[i][2] - 0.3, text(string(sources[i]), :red, 14, :center))    
end
for i in 1:length(receptors)
    annotate!(receptors[i][1], receptors[i][2] + 0.3, text(string(i), :black, 14, :center))
    annotate!(receptors[i][1], receptors[i][2] - 0.3, text(string(receptors[i]), :black, 14, :center))    
end
plot!(size=(700, 550))
plot!([(0, 7.75), (10, 7.75)], arrow=true, color=:blue, linewidth=2, label="")
annotate!(4.5, 8, text("Wind", :left, 18, :blue))

Figure 1: Results of Generating Capacity Expansion Example

Modeling Considerations

Need to know relationship between source emissions ($Q_i$) and receptor exposure.

Receptors are only affected by upwind sources.
Individual plume model gives us concentrations. How to combine?

\[ C_\text{total} = \frac{M_1 + M_2 + M_3}{V} = \frac{M_1}{V} + \frac{M_2}{V} + \frac{M_3}{V} = C_1 + C_2 + C_3 \]

Decision Variables

What is the key set of decision variables?

Fraction of SO₂ removed at source $i$: $R_i$.

Alternatively, can reframe as level of emissions: \[Q_i = (1-R_i) \times E_i,\] where $E_i$ is the emissions level (given in problem).

Constraints

What is our main constraint?

Need to ensure compliance with the air quality standard:

\[\text{Exp}_j \leq .013 \text{g/m}^3\]

This means that we need to express $\text{Exp}_j$ as a linear function of the $R_i$.

Developing Constraints

How do we relate the exposure level to $Q_i$?

Write $C_i(x,y) = Q_it_i(x,y)$, where the $t_i$ is the transmission factor from the Gaussian dispersion model:

\[t_i(x,y) = \frac{1}{\pi u \sigma_y \sigma_z} \exp\left(\frac{-y^2}{2\sigma_y^2}\right)\exp\left(\frac{H^2}{2 \sigma_z^2}\right)\]

Developing Constraints

This lets us write the exposure constraints as a linear function of $R_i$:

For a receptor $j$ (with fixed location $(x_j, y_j)$),

\[ \begin{align} \text{Exp}_j &= \sum_i Q_i t_i(x_j, y_j) \\ &= \sum_i (1-R_i)E_it_i(x_j, y_j) \leq 0.013\ \text{g/m}^3 \end{align} \]

Write $t_{ij} = t_i(x_j, y_j)$.

Dispersion Spread

However, we still need to do this analysis given a particular atmospheric stability class (or can test across all). Let’s assume we’re in stability class C.

Source: https://courses.washington.edu/cee490/DISPCOEF4WP.htm

Dispersion Spread

Using the equations:

\[\begin{align} \sigma_y &= ax^{0.894} \\[0.5em] \sigma_z &= cx^d + f, \end{align}\]

we have \[\sigma_y = 104x^{0.894}, \qquad \sigma_z = 61x^{0.911}.\]

Calculating Transmission Factors

# Δx, Δy should be in m
function transmission_factor(Δx, Δy, u, H)
    if Δx <= 0 # check if source is upwind of receptor
        tf = 0.0 # ensure this is a Float
    else
        σy = 104 * (Δx / 1000)^0.894
        σz = 61 * (Δx / 1000)^0.911
        tf_coef = 1/(pi * u * σy * σz)
        tf = tf_coef * exp(-0.5 * (Δy / σy)^2) * exp(-0.5 * (H / σz)^2)
    end
    return tf
end

Calculating Transmission Factors

For example, from Source 1 to Receptor 1:

transmission_factor(1000, 5500, 1.5, 50)

0.0

Or from Source 1 to Receptor 2:

transmission_factor(3000, 0, 1.5, 50)

4.4002801438721065e-6

Objective

What is our objective?

Minimize cost:

\[ \begin{align} \min_{R_i} & \sum_i RemCost_i \times (E_i \times R_i) \\[0.5em] &= 2R_1 + 11.25R_2 + 10.8R_3 \end{align} \]

Final Problem

\[ \begin{align} \min_{R_i} \quad & \sum_i 2R_1 + 11.25R_2 + 10.8R_3 & \\ \text{subject to:} & \\[0.5em] & \sum_{i=1}^3 E_i (1-R_i)t_{ij} \leq 0.013 & \forall j \in 1:5 \\[0.5em] & R_i \geq 0 & \forall i \in 1:3 \\[0.5em] & R_i \leq 1 & \forall i \in 1:3 \end{align} \]

Solution

Source	SO2 Emissions (kg/s)	Removal Percentage (%)
1	10000	70.0
2	25000	26.0
3	18000	100.0

Does this make sense?

Untreated Exposure

In the absence of our treatment plan (exposure in $\text{mg/m}^3$):

Source	R2	R4	R5
1	44.0	2.0	0.0
2	0.0	3.0	17.0
3	0.0	2.0	18.0

Need to reduce Source 1’s emissions to bring Receptor 2’s exposure down. What about Sources 2 and 3?

Why This Plan Makes Sense

We could reduce emissions from source 2 by 100% and from source 3 by ~28% to comply at receptor 5.

But removal at source 2 is more expensive per percentage removed than source 3.

Treated Exposure

Source	R2	R4	R5
1	13.0	1.0	0.0
2	0.0	3.0	13.0
3	0.0	0.0	0.0

Key Takeaways

Gaussian plume models can be useful for modeling continuous point source emissions.
Can turn multi-source and receptor problems into an LP by linearizing Gaussian dispersion model when locations are fixed.

Upcoming Schedule

Next Classes

Monday: Fixed Costs and Mixed Integer Programs

Wednesday and Next Week: Applications (network models, unit commitment)

Assessments

HW4: Due tomorrow (10/31) at 9pm.

Project Proposal: Also due tomorrow.

Variable	Meaning
\(C\)	Concentration (g/m\(^3\))
\(Q\)	Emissions Rate (g/s)
\(H\)	Effective Source Height (m)
\(u\)	Wind Speed (m/s)
\(y, z\)	Crosswind, Vertical Distance (m)