Plume Models: Multiple Sources and Receptors

Lecture 20

October 20, 2023

  Activating project at `~/Teaching/BEE4750/fall2023/slides`

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 Distribution

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 150 $\mu\text{g/m}^3$):

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

Multiple Point Source Example

Our goal:

Minimize cost of removing SO₂ from the plume sources to ensure all receptors are not exposed beyond the 150 $\mu\text{g/m}^3$ standard.

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:

\[\text{Exp}_j \leq .00015 \text{g/m}^3\]

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

Developing Constraints

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

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

Developing Constraints

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

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

\[ \begin{align} \text{Exp}_j &= \sum_i Q_i t_i(x_j, y_j) \\ &= (1-R_i)E_it_i(x_j, y_j) \leq 0.00015\ \text{g/m}^3 \end{align} \]

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.

Dispersion Coefficients

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

# delta_x, delta_y 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/(2 * pi * u * σy * σz)
        tf = tf_coef * exp((-0.5 * (Δy / σy)^2) + (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)

2.521069616495523e-6

Objective

What is our objective?

Minimize cost:

\[ \begin{align} \min_{R_i} & \sum_i RemCost_i \times (E_i \times R_i) \\[0.5em] &= (0.20) (34560) R_1 + (0.45)(172800) R_2 + (0.60)(103680) R_3 \\[0.5em] &= 6912R_1 + 77760R_2 + 62208R_3 \end{align} \]

Final Problem

\[ \begin{align} \min_{R_i} \quad & \sum_i RemCost_i \times (E_i \times R_i) & \\ \text{subject to:} & \\[0.5em] & \sum_{i=1}^3 E_i (1-R_i)t_{ij} \leq 0.00015 & \forall j \in 1:5 \\[0.5em] & R_i \geq 0 & \forall i \in 1:3 \\[0.5em] & R_i \leq 1 & \forall i \in 1:3 \end{align} \]

Solution

Source	SO2 Emissions (g/m^3)	Removal Percentage (%)
1	400	85.0
2	2000	100.0
3	1200	75.0

Does this make sense?

Untreated Exposure

In the absence of our treatment plan (exposure in $\mu$g/m$^3$):

Source	R2	R4	R5
1	1008.0	51.0	5.0
2	0.0	278.0	1000.0
3	0.0	58.0	603.0

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 ~85% to comply at receptor 5, but then would also need to reduce source 3’s emissions by 100%.

This plan involves eliminating all of source 2’s emissions, but only 75% of source 3’s.

Since removal at source 2 ($0.45/kg) is cheaper than source 3 ($0.60/kg), eliminating all of source 2’s is cheaper, at a cost of $124,416.

Treated Exposure

Source	R2	R4	R5
1	150.0	8.0	1.0
2	0.0	0.0	0.0
3	0.0	14.0	149.0

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: Mixed Integer Linear Programming

Wednesday/Friday: Applications (network models, unit commitment)

Assessments

HW4: Due next Friday (10/27) at 9pm.

Regulatory Report: Due next Friday also.

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)