Chapter 1 Environmental Engineering: Air Quality Modeling#
1. Introduction#
Air polluiton#
Air pollutants can exist as gases or particulates, with particulate pollutants further classified by their size and origin. Dust consists of relatively large solid particles generated from mechanical processes such as coal handling or sandblasting. Fumes are extremely fine particles, typically ranging from 0.03 to 0.3 micrometers, often composed of metallic oxides formed through chemical reactions. Mists are liquid droplets between 0.5 and 3 micrometers in size, produced by vapor condensation or chemical processes. Smoke includes solid particles ranging from 0.005 to 1 micrometer, usually resulting from the incomplete combustion of carbon-based materials. Each type of particulate pollutant has distinct physical characteristics and forms through different mechanisms, influencing how it behaves in the atmosphere and impacts human health and the environment.
🌬️ Measurement of Air Quality#
Air quality monitoring is generally categorized into three main types:
Emission Monitoring – Focuses on pollutants released from natural and anthropogenic sources.
Ambient Monitoring – Measures the concentration of toxic and non-toxic contaminants in the surrounding air.
Meteorological Monitoring – Tracks atmospheric conditions that influence the transport and dispersion of air pollutants.
To measure gaseous pollutants such as SOₓ, NOₓ, ozone (O₃), hydrocarbons, and vapors, techniques include grab sampling, absorption in liquids or solids, and freeze-out sampling. For particulate matter (e.g., PM₁₀, PM₂.₅), common collection methods include dust fall jars and electrostatic precipitation.
🛠️ Control of Air Pollutants#
Gaseous Pollutants#
Gaseous air pollutants can be controlled by chemically transforming them, capturing them from the emission stream, or modifying the processes that generate them. The choice of control method depends on the pollutant’s concentration and the flow rate of the emission stream:
Low flow rate & low concentration: Photocatalysis is effective.
High concentration & high flow rate: Catalytic and thermal oxidation are preferred.
Intermediate conditions: Adsorption and absorption methods are commonly used.
High concentration & low flow rate: Membrane separation and condensation are effective.
Low concentration & high flow rate: Biofiltration is suitable.
Adsorbers can achieve removal efficiencies exceeding 95% and are effective for pollutants like SOₓ, H₂S, and NOₓ. However, absorption processes may generate wastewater, potentially shifting the pollution burden from air to water. For organic compounds, incineration methods—including direct combustion, thermal incineration, and catalytic incineration—are widely used.
Particulate Pollutants#
Particulate matter is typically removed using cyclones, electrostatic precipitators, and baghouse filters:
Electrostatic precipitators apply electrical forces to charge particles negatively and attract them to positively charged collection plates. They are highly effective for fine particles.
Fabric filters (baghouses) operate like vacuum filters, drawing dirty gas through filter bags. They are efficient even for submicron-sized particles.
Cyclones are economical and effective for larger particles but have lower efficiency for fine particulates. For smaller particles, baghouses, scrubbers, and electrostatic precipitators offer superior performance.
⚖️ Regulatory Framework#
The Clean Air Act (CAA) [United States Environmental Protection Agency, 1990], enacted in 1970, provides the foundational regulatory framework for controlling air pollution in the United States. It authorizes the EPA to establish National Ambient Air Quality Standards (NAAQS) for six criteria pollutants:
Carbon monoxide (CO)
Lead (Pb)
Nitrogen dioxide (NO₂)
Ozone (O₃)
Particulate matter (PM)
Sulfur dioxide (SO₂)
These standards include:
Primary standards – Protect public health.
Secondary standards – Safeguard public welfare, including visibility and property.
Based on compliance with NAAQS, regions are designated as attainment or nonattainment areas. Each state is required to develop a State Implementation Plan (SIP) outlining how it will achieve and maintain air quality standards.
Additionally, under the Prevention of Significant Deterioration (PSD) program, attainment areas are classified into:
Class I – Allows minimal air quality deterioration (e.g., national parks and wilderness areas).
Class II – Permits moderate deterioration (most U.S. regions).
Class III – Would allow greater deterioration, but no areas have been designated as Class III.
Pollution control requirements for stationary sources vary by area designation and source type:
New or modified sources in attainment areas must implement Best Available Control Technology (BACT), determined case-by-case based on effectiveness, cost, and environmental impact.
New or modified sources in nonattainment areas must meet the Lowest Achievable Emission Rate (LAER), the most stringent standard, regardless of cost.
Existing sources in attainment areas typically apply technologically achievable controls, often defined by general performance standards.
Existing sources in nonattainment areas must implement Reasonably Available Control Technology (RACT), which balances effectiveness with economic and technical feasibility.
These tiered requirements ensure that air quality is protected while accounting for regional conditions and the nature of each emission source.
Gaussian Plume Dispersion Model – Interactive Visualization#
The Gaussian Plume Model is a widely used mathematical model that describes how air pollutants disperse from a point source, such as a smokestack, under steady-state conditions. It assumes that the concentration of a pollutant in the atmosphere follows a normal (Gaussian) distribution in the crosswind and vertical directions, resulting in a “plume” of pollution downwind of the source [Masters and Ela, 2008].
The general equation for pollutant concentration ( C(x, y, z) ) at location ( (x, y, z) ) is:
Where:
\(( C \)): concentration [µg/m³]
\(( Q \)): emission rate [g/s]
\(( u \)): wind speed at stack height [m/s]
\(( H \)): effective stack height [m]
\(( \sigma_y, \sigma_z \)): lateral and vertical dispersion coefficients [m]
\(( x \)): downwind distance [m]
\(( y \)): crosswind offset [m]
\(( z \)): height above ground [m]
✅ Ground Reflection#
The plume reflects off the ground due to the no-flux boundary condition.
This is modeled by the image source at ( -H ), contributing the second exponential term in the equation.
✅ Thermal Inversion Reflection#
A thermal inversion layer acts like a lid, trapping pollutants below.
Reflection from this layer can be modeled by adding additional image sources above the inversion height.
🧠 Conceptual Insight#
Reflections increase concentration near the ground.
Inversion layers suppress vertical mixing, leading to higher ground-level concentrations.
The Gaussian model assumes steady-state, constant wind, and uniform terrain.
❓ Quiz Questions#
What causes the plume to reflect off the ground?
How does a thermal inversion layer affect pollutant dispersion?
Why is the second exponential term in the Gaussian equation important?
What assumptions does the Gaussian plume model make?
How would you modify the model for complex terrain or variable wind?
Estimating Dispersion Coefficients#
The coefficients ( \sigma_y ) and ( \sigma_z ) vary with distance and atmospheric stability. The following empirical formulas are adapted from Pasquill-Gifford curves.
Dispersion Coefficient Formulas#
Stability Class |
Description |
( a ) |
( b ) |
( c ) |
( d ) |
|---|---|---|---|---|---|
A |
Very unstable |
0.22 |
0.0001 |
0.20 |
0.0001 |
B |
Moderately unstable |
0.16 |
0.0001 |
0.12 |
0.0001 |
C |
Slightly unstable |
0.11 |
0.0001 |
0.08 |
0.0001 |
D |
Neutral |
0.08 |
0.0001 |
0.06 |
0.0001 |
E |
Slightly stable |
0.06 |
0.0001 |
0.03 |
0.0001 |
F |
Very stable |
0.04 |
0.0001 |
0.016 |
0.0001 |
Effective Stack Height#
The effective stack height ( H ) is given by:
Where:
( h ): physical stack height [m]
( \Delta h ): plume rise [m] (due to buoyancy and/or momentum)
Approximate Buoyant Plume Rise (Stable Conditions)#
Where:
\(( F \)): buoyancy flux [m⁴/s³]
\(( g \)): gravitational acceleration ≈ 9.81 m/s²
\(( V \)): stack gas exit volume flow rate [m³/s]
\(( \Delta T \)): stack–ambient temperature difference [K]
\(( T_a \)): ambient temperature [K]
\(( u \)): wind speed [m/s]
Notes#
Use stability classes (A–F) based on surface conditions, solar radiation, and wind speed
The ground reflection term accounts for pollutant dispersion near the surface
The model assumes steady-state conditions (constant wind and emissions)
This notebook provides an interactive visualization of the ground-level concentration of pollutants based on:
📍 Source emission rate (Q)
🌬️ Wind speed (u)
📏 Effective stack height (H)
🌡️ Atmospheric stability class (A–F)
Key Features#
Simulate how different environmental and source parameters affect pollutant dispersion
Visualize the concentration distribution at ground level as a 2D contour plot
Explore six atmospheric stability classes (A = very unstable, F = very stable)
Adjustable inputs with interactive sliders for real-time exploration
Applications#
Air quality impact assessments for industrial stacks
Regulatory modeling and compliance
Environmental engineering education and plume behavior analysis
Foundational Literature#
[Masters and Ela, 2008] presents the Gaussian plume model as a core tool for estimating air pollutant concentrations from point sources, such as smokestacks. It derives the model from the advection-diffusion equation, assuming steady-state flow and uniform turbulence, and defines key parameters such as emission rate, wind speed, and dispersion coefficients. The model is applied in air quality assessments and regulatory analysis, with attention to limitations such as terrain effects and atmospheric stability, using Pasquill-Gifford classes. While [Masters and Ela, 2008] provides a solid introductory overview, the {cite}’venkatram2003gaussian’ offer deeper mathematical derivations, regulatory context, and advanced modeling techniques—ideal for graduate-level study, research, or regulatory modeling.
2. Simulation#
🌬️ Gaussian Plume Air Quality Model – Summary in Markdown#
This Python-based model simulates pollutant dispersion from a point source (e.g., smokestack) using the Gaussian plume equation. It visualizes how wind speed, atmospheric stability, stack height, and inversion layer influence concentration at various downwind distances and heights.
🔧 Model Components#
Component |
Description |
|---|---|
|
Horizontal & vertical dispersion using empirical formulas |
|
Calculates pollutant concentration with reflections |
|
Rotates coordinates to align plume with wind direction |
|
Interactive 2D heatmap showing concentration distribution |
|
Sliders and dropdowns to adjust model inputs |
💡 Inputs and Controls#
Parameter |
Description |
|---|---|
|
Emission rate (µg/s) |
|
Wind speed (m/s) |
|
Stack height (m) |
|
Atmospheric stability class (A–F) |
|
Inversion layer height (m) |
|
Wind direction (degrees) |
📈 What It Shows#
Heatmap of pollutant concentration vs. distance and height
Plume widening or narrowing with stability class
Inversion effects when vertical mixing is restricted
Wind direction adjusts plume orientation
🧠 How to Interpret Results#
Feature |
Meaning |
|---|---|
Bright plume core |
High concentration near the stack and ground level |
Wider dispersion |
Occurs in unstable (A–C) conditions with higher turbulence |
Inversion line |
Limits vertical mixing; pollutants may pool below it |
Wind rotation |
Plume shifts with prevailing wind direction |
🧪 Equation Used (Simplified)#
Optional terms account for reflection from the inversion layer.
This model offers a practical and interactive way to explore how atmospheric conditions affect pollution dispersion — valuable for air quality engineers, planners, and educators.
import numpy as np
import plotly.graph_objects as go
from ipywidgets import interact, FloatSlider, Dropdown
# --- Dispersion functions ---
def sigma_y(x, stability='D'):
a = {'A': 0.22, 'B': 0.16, 'C': 0.11, 'D': 0.08, 'E': 0.06, 'F': 0.04}
b = {k: 0.0001 for k in a}
return a[stability] * x * (1 + b[stability] * x) ** -0.5
def sigma_z(x, stability='D'):
a = {'A': 0.20, 'B': 0.12, 'C': 0.08, 'D': 0.06, 'E': 0.03, 'F': 0.016}
b = {k: 0.0001 for k in a}
return a[stability] * x * (1 + b[stability] * x) ** -0.5
# --- Gaussian plume with reflection ---
def gaussian_plume(x, z, Q, u, H, stability, z_inv=None):
sy = sigma_y(x, stability)
sz = sigma_z(x, stability)
term1 = Q / (2 * np.pi * u * sy * sz)
base = np.exp(- (z - H)**2 / (2 * sz**2)) + np.exp(- (z + H)**2 / (2 * sz**2))
if z_inv and z_inv > 0:
base += np.exp(- (2*z_inv - z - H)**2 / (2 * sz**2))
base += np.exp(- (2*z_inv + z - H)**2 / (2 * sz**2))
return term1 * base
# --- Coordinate rotation for wind direction ---
def rotate_coords(x, y, angle_deg):
theta = np.radians(angle_deg)
x_rot = x * np.cos(theta) + y * np.sin(theta)
return x_rot
# --- Plotting function with wind direction ---
def plot_plotly(Q, u, H, stability, z_inv, wind_dir):
x_vals = np.linspace(-500, 2500, 150)
z_vals = np.linspace(0, 300, 100)
x_grid, z_grid = np.meshgrid(x_vals, z_vals)
# Rotate x according to wind direction (assume y = 0 slice)
x_rot = rotate_coords(x_grid, np.zeros_like(x_grid), -wind_dir)
x_rot = np.maximum(x_rot, 1.0) # prevent division by zero
C = gaussian_plume(x_rot, z_grid, Q, u, H, stability, z_inv)
fig = go.Figure(data=go.Heatmap(
x=x_vals,
y=z_vals,
z=C,
colorscale='Viridis',
zmin=0,
zmax=0.12, # Adjust this upper limit as needed
colorbar=dict(title='Concentration (µg/m³)'),
hovertemplate='x: %{x:.0f} m<br>z: %{y:.0f} m<br>Conc: %{z:.2f} µg/m³<extra></extra>'
))
fig.update_layout(
title=f"Gaussian Plume with Wind Direction {wind_dir}°<br>Stability: {stability} | Inversion: {z_inv or '∞'} m",
xaxis_title='Projected Downwind Distance (m)',
yaxis_title='Height z (m)',
height=500
)
if z_inv and z_inv > 0:
fig.add_shape(type="line", x0=x_vals[0], x1=x_vals[-1], y0=z_inv, y1=z_inv,
line=dict(color="red", dash="dash"), name="Inversion")
fig.show()
# --- Widgets with wind direction added ---
interact(
plot_plotly,
Q=FloatSlider(value=100, min=10, max=5000, step=10, description="Emission Q"),
u=FloatSlider(value=5.0, min=0.5, max=50.0, step=0.5, description="Wind Speed"),
H=FloatSlider(value=50, min=0, max=150, step=5, description="Stack Height"),
stability=Dropdown(options=['A', 'B', 'C', 'D', 'E', 'F'], value='D', description="Stability"),
z_inv=FloatSlider(value=0, min=0, max=300, step=10, description="Inversion Height"),
wind_dir=FloatSlider(value=0, min=0, max=360, step=5, description="Wind Dir (°)")
)
<function __main__.plot_plotly(Q, u, H, stability, z_inv, wind_dir)>
3. Self-Assessment#
Objective: Gaussian Plume Receptor Concentration Estimator#
The main objective of this code is to provide an interactive tool for estimating air pollutant concentration at any specified receptor point ((x, y, z)), using the Gaussian plume dispersion model with both ground and inversion layer reflections.
🔧 What the Code Does#
📥 Accepts user inputs for:
Receptor location:
( x ): Downwind distance (m)
( y ): Crosswind offset (m)
( z ): Height above ground (m)
Emission rate ( Q ) [g/s]
Wind speed ( u ) [m/s]
Stack height ( H ) [m]
Stability class (A–F)
Optional inversion layer height ( z_{\text{inv}} )
📈 Calculates concentration using the Gaussian plume equation:
Incorporates horizontal and vertical dispersion using stability-dependent functions ( \sigma_y(x), \sigma_z(x) )
Includes both ground reflection and optional inversion reflection
🧮 Core equation: $\( C(x, y, z) = \frac{Q}{2\pi u \sigma_y \sigma_z} \cdot \exp\left(-\frac{y^2}{2\sigma_y^2}\right) \cdot \left[ \exp\left(-\frac{(z - H)^2}{2\sigma_z^2}\right) + \text{(reflections)} \right] \)$
Interactive interface with
ipywidgetsin a Jupyter Notebook:Users input values via sliders and text boxes
Concentration is calculated and displayed in nicely formatted Markdown output
Why It’s Useful#
This tool helps students, engineers, and environmental analysts:
Explore the influence of stack and meteorological parameters
Visualize how concentration varies with location
Understand plume reflection mechanisms
Get hands-on with a model used in regulatory and industrial settings
Want to enhance it further with a visual heatmap or batch receptor logging?
Let’s build that into the next version together 🚀
4. Simulation#
🌬️ Gaussian Plume Receptor Point Query – Summary#
This is an interactive Jupyter-based tool that estimates pollutant concentration at a specific receptor location using a Gaussian plume dispersion model. It includes:
Input widgets for plume parameters and receptor coordinates
A button-triggered calculation using the dispersion equation
Markdown-based result display with formatted output
🧪 How It Works#
Inputs via Widgets:
(x, y, z): receptor location (meters)Q: emission rate (g/s)u: wind speed (m/s)H: stack height (m)stability: atmospheric turbulence class (A–F)z_inv: height of inversion layer
Concentration Calculation:
Uses horizontal (
σ_y) and vertical (σ_z) dispersion formulasApplies Gaussian plume equation with reflections for ground and inversion
Computes concentration at user-defined point
Output:
After clicking “Get Concentration”, displays:
📍 Receptor at (x=..., y=..., z=...): **Estimated Concentration:** `... µg/m³`
🔍 How to Interpret Results#
Output Element |
Interpretation |
|---|---|
|
Receptor coordinates downwind and vertically from source |
|
Estimated pollutant level at the receptor |
|
Indicates far distance, poor mixing, or dispersion limits |
|
Suggests proximity to plume center and limited dispersion |
This tool is ideal for evaluating exposure risk, site-specific concentration, and regulatory compliance scenarios.
import ipywidgets as widgets
from IPython.display import display, Markdown, clear_output
# Compute concentration at specified (x, y, z) using existing dispersion model
def concentration_query(x, y, z, Q, u, H, stability, z_inv):
sy = sigma_y(x, stability)
sz = sigma_z(x, stability)
if sy <= 0 or sz <= 0:
return 0.0
part1 = Q / (2 * np.pi * u * sy * sz)
part2 = np.exp(- y ** 2 / (2 * sy ** 2))
part3 = (
np.exp(- (z - H)**2 / (2 * sz**2)) +
np.exp(- (z + H)**2 / (2 * sz**2))
)
if z_inv and z_inv > 0:
part3 += np.exp(- (2 * z_inv - z - H)**2 / (2 * sz**2))
part3 += np.exp(- (2 * z_inv + z - H)**2 / (2 * sz**2))
return part1 * part2 * part3
# UI elements
x_input = widgets.BoundedFloatText(value=500, min=1, max=10000, step=10, description='x (m):')
y_input = widgets.BoundedFloatText(value=0, min=-1000, max=1000, step=10, description='y (m):')
z_input = widgets.BoundedFloatText(value=1, min=0, max=500, step=1, description='z (m):')
Q_input = widgets.FloatSlider(value=100, min=10, max=5000, step=10, description='Q (g/s)')
u_input = widgets.FloatSlider(value=5.0, min=0.5, max=50, step=0.5, description='Wind (m/s)')
H_input = widgets.FloatSlider(value=50, min=0, max=150, step=5, description='Stack H')
stab_input = Dropdown(options=['A', 'B', 'C', 'D', 'E', 'F'], value='D', description='Stability')
zinv_input = widgets.FloatSlider(value=0, min=0, max=300, step=10, description='Inversion (m)')
# Action button
button = widgets.Button(description='Get Concentration')
output = widgets.Output()
def on_button_click(b):
with output:
clear_output()
x = x_input.value
y = y_input.value
z = z_input.value
Q = Q_input.value
u = u_input.value
H = H_input.value
stab = stab_input.value
z_inv = zinv_input.value
C = concentration_query(x, y, z, Q, u, H, stab, z_inv)
display(Markdown(
f"### 📍 Receptor at (x={x} m, y={y} m, z={z} m):\n"
f"**Estimated Concentration:** `{C:.2f} µg/m³`"
))
button.on_click(on_button_click)
# Display form
display(widgets.VBox([
widgets.HTML(value="<h3>🎯 Query Pollutant Concentration at a Receptor Point</h3>"),
widgets.HBox([x_input, y_input, z_input]),
Q_input, u_input, H_input, stab_input, zinv_input,
button, output
]))
5. Self-Assessment#
Gaussian Plume Dispersion Model – Learning Module#
This module supports critical thinking, conceptual understanding, and assessment for students or professionals using the Gaussian plume tool. It includes conceptual questions, reflective prompts, and a quiz to reinforce dispersion modeling fundamentals.
Conceptual Questions#
Why do we use different dispersion coefficients (σy and σz) for the horizontal and vertical directions in the Gaussian plume model?
What physical factors influence the values of σy and σz at a given downwind distance x?
How does wind speed (u) affect the shape and dilution of the plume?
What is the purpose of including reflection terms from the ground and inversion layer in the Gaussian plume equation?
How does atmospheric stability (Classes A–F) influence pollutant dispersion, and why does a more stable atmosphere result in a narrower plume?
Why is effective stack height (H) more than just the physical stack height, and how can plume rise be estimated?
Reflective Prompts#
When switching from stability class D to class F in your simulation, what changes do you observe in ground-level concentration? Why does this happen?
Rotate the wind direction by 90°. How does this alter the shape of the heatmap? What implications does this have for nearby receptors?
If a receptor point lies above the inversion layer, how does that affect the estimated concentration? What are the consequences for exposure modeling?
Which parameters most influence the location and magnitude of the peak concentration? How could this inform industrial design or siting decisions?
Without reducing emissions (Q), what other strategies could reduce the maximum concentration at ground level?
✅ Quiz Questions (Multiple Choice)#
Q1. What do σy and σz represent in the Gaussian plume model?
A. Stack height and plume temperature
B. Wind direction and terrain roughness
C. Crosswind and vertical dispersion
D. Wind speed and turbulence intensity
✅ Answer: C
Q2. Which condition leads to the greatest spread of a Gaussian plume?
A. Stability Class F
B. Stability Class A with strong solar radiation
C. Calm wind and tall stack
D. Heavy rain and high humidity
✅ Answer: B
Q3. Increasing the wind speed (u) typically causes the peak concentration to:
A. Increase
B. Decrease
C. Stay the same
D. Occur closer to the stack
✅ Answer: B
Q4. Why are reflection terms included in the vertical concentration expression?
A. To account for wind shear
B. To model pollutant absorption into soil
C. To simulate vertical boundary effects from the ground and atmosphere
D. To add time dependence to the model
✅ Answer: C
Q5. Increasing the stack height (H) will:
A. Increase ground-level concentrations
B. Move the concentration peak further downwind and upward
C. Have no effect on the plume shape
D. Increase horizontal dispersion
✅ Answer: B
References#
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