Formula & Calculator
Gaussian Plume Concentration
Ground-level concentration downwind of a continuous point source under steady wind.
Interpretation
C(x,y,0) = Q/(π u σ_y σ_z) · exp(−y²/2σ_y²). Models downwind concentration of a continuous point source. Used in air quality dispersion analysis and regulatory permitting.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| C | Concentration | g/m³ |
| Q | Emission rate | g/s |
| u | Wind speed | m/s |
| σ_y,σ_z | Dispersion coefficients | m |
What it means
The Gaussian plume model is the most widely used method for predicting the downwind concentration of pollutants emitted from a continuous point source (e.g., a smokestack) under steady‑state conditions. The formula gives the ground‑level concentration (z=0) at a downwind distance x and crosswind distance y. It depends on the emission rate Q, wind speed u, and the dispersion parameters σ_y and σ_z, which vary with atmospheric stability and distance. The equation assumes a Gaussian distribution of the plume in the horizontal and vertical directions. This model is used for regulatory compliance, environmental impact assessments, and stack design. Example: For a stack emitting 100 g/s of SO₂ at a wind speed of 5 m/s, with σ_y=20 m and σ_z=10 m at a given x, the concentration at the plume centreline (y=0) is C = 100/(π × 5 × 20 × 10) = 100/(3141.6) ≈ 0.032 g/m³. This allows engineers to determine if ground‑level concentrations meet air quality standards.
Worked example
Gaussian Plume Concentration – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Q | 100 g/s |
| u | 2 m/s |
| σ_y | 20 m |
| σ_z | 10 m |
| y | 0 m |
| Parameter | Value |
|---|---|
| Q | 100 g/s |
| u | 2 m/s |
| σ_y | 20 m |
| σ_z | 10 m |
| y | 10 m |
Common mistakes
- Unit of C: Concentration (e.g., µg/m³, mg/m³) – ensure Q has matching units (e.g., µg/s, mg/s).
- Dispersion coefficients σ_y and σ_z: Depend on atmospheric stability class and downwind distance – use appropriate Pasquill‑Gifford curves or calculations.
- Wind speed u: The average wind speed at release height – in m/s.
- Ground‑level concentration: The formula given is for z=0 (ground level) – for elevated releases, additional terms (reflection) are needed.
- Assumptions: Steady‑state, continuous point source, constant wind speed, flat terrain, no deposition.
Applications
The Gaussian plume concentration equation estimates the ground‑level concentration of a pollutant released from a point source, such as a smokestack, under steady‑state atmospheric conditions. It accounts for emission rate (Q), wind speed (u), and atmospheric dispersion parameters (σ_y, σ_z) that depend on atmospheric stability. This model is widely used in air quality management to predict the impact of industrial emissions, to set permit limits, and to assess compliance with ambient air quality standards. Environmental engineers use it to design stacks of adequate height, to evaluate the effectiveness of emission controls, and to conduct risk assessments for accidental releases. While simplified, the Gaussian plume remains a foundational tool in regulatory air pollution modelling.
- Air quality impact assessment for industrial facilities
- Design of stack height and emission controls
- Evaluation of compliance with National Ambient Air Quality Standards (NAAQS)
- Risk assessment for hazardous air pollutants (HAPs)
- Emergency response planning for accidental releases
Frequently Asked Questions
The Gaussian plume model predicts the downwind ground‑level concentration of a pollutant emitted from a continuous point source (e.g., a smokestack) under steady wind conditions. The equation is C(x,y,0) = (Q / (π·u·σ_y·σ_z)) · exp(–y²/(2σ_y²)). It assumes a Gaussian (normal) distribution of the plume in the horizontal and vertical directions.
- Q – emission rate (mass/time, e.g., g/s).
- u – average wind speed (m/s) at the effective stack height.
- σ_y – horizontal dispersion coefficient (m) depending on downwind distance and atmospheric stability.
- σ_z – vertical dispersion coefficient (m).
- y – crosswind distance from the plume centreline (m).
They are empirical functions of downwind distance (x) and atmospheric stability class (Pasquill‑Gifford stability classes A‑F). For a given stability and distance, σ_y and σ_z are obtained from standard curves or formulas. They increase with distance as the plume spreads.
- Steady‑state (constant emissions and meteorology).
- Uniform wind speed and direction.
- Continuous point source.
- No deposition, chemical transformation, or terrain effects.
- Complete reflection at the ground (z=0).
- Gaussian distribution applies.
The exponential term exp(–y²/(2σ_y²)) describes the horizontal spread of the plume. It is maximum at y=0 (centreline) and decreases as y increases. This reflects that most of the plume mass is near the centreline.
Stability classes range from A (very unstable, strong dispersion) to F (stable, weak dispersion). Under unstable conditions (A, B), σ_y and σ_z are large, so the plume spreads widely and ground‑level concentrations are lower. Under stable conditions (E, F), the plume stays narrow, leading to higher concentrations downwind.
A taller stack releases the plume higher, reducing ground‑level concentrations (since the plume must travel further downward to reach ground level). The effective stack height includes plume rise (due to momentum and buoyancy). The Gaussian model can be extended to include stack height by using z = H and applying reflection.
- Assumes flat, uniform terrain; complex terrain (hills, buildings) invalidates it.
- Does not account for chemical reactions or transformations.
- Not valid for very short time periods (hours).
- Assumes steady wind; not suitable for calm or variable conditions.
- Limited for near‑source (within a few hundred metres) due to plume rise assumptions.
For a stack of height H, the ground‑level concentration at y=0 is C = (Q / (π·u·σ_y·σ_z)) · exp(–H²/(2σ_z²)) (assuming reflection at ground). If the plume is reflected, multiply by a factor that accounts for the image source. More complete equations include a summation term for reflections.
More advanced models include:
- Lagrangian puff models (for non‑steady conditions).
- Computational Fluid Dynamics (CFD) for complex geometries.
- Eulerian grid models (e.g., CMAQ, CALPUFF) for regional scales.
- Dispersion models that account for chemical reactions and deposition (e.g., AERMOD).