Gaussian Plume Model: Pollutant Dispersion Fundamentals

A widely used mathematical framework for Gaussian plume model assumes that the concentration distribution of pollutants in cross-sections perpendicular to the wind direction follows a normal distribution in both vertical and horizontal directions. The model is applied under specific assumptions:

1. Point source emissions with low vertical velocities.
2. Emissions at temperature ($T$) equal to or slightly above ambient air.
3. Stable atmospheric conditions, at least over one-hour intervals.
4. Flat and uniform soil with little surface roughness.

Hypotheses for the Model

• Wind speed and direction: constants.
• Conditions of the parameters that characterize the dispersive properties of the atmosphere: constants.
• Soil: uniform and flat.

General Diffusion Equation

The general diffusion equation describes the atmospheric concentration $X_v(x, y, z, t)$:

$$\text{Equation (1)} \quad K_x, K_y, K_z$$

Where $K_x, K_y, K_z$ are the turbulent diffusion coefficients, $u$ is the mean wind velocity, and $S$ is the source term.

Continuous Point Emission Solution

For a continuous point emission of rate $Q$ (g/s or Bq/m$^3$) from a stack of effective height $H$, the solution for steady-state condition is:

$$\text{Equation (2)} \quad X$$

Where $X$ is the concentration (g/m$^3$ or Bq/m$^3$), $\sigma_y$ and $\sigma_z$ are the dispersion coefficients in the horizontal and vertical directions, respectively.

Ground Level Concentration

For ground level concentration ($z=0$):

$$\text{Equation (3)} \quad X$$

This formulation highlights how plume dispersion is influenced by emission rate, wind speed, atmospheric stability, and effective stack height. The model parameters $\sigma_y$ and $\sigma_z$ are derived from empirical correlations based on the Pasquill-Gifford stability classes.

Worst-Case Concentration

Finally, for $z=0$ and $y=0$ (worst condition, when the plume touches the ground) if the emission is at $H$:

$$\text{Equation (4)} \quad X$$

Dispersion Coefficients Parameters

The parameters of the model are the dispersion coefficients. The Pasquill-Gifford equation is:

$$\text{Equation (5)} \quad \sigma_j = f(\text{distance}, \text{stability class})$$

• For $j=y$: $B_y = 0.9031$ and $C_y = 0$.
• For $j=z$: $B_z, A_z, C_z$ are not equal to 0.

Atmospheric Stability Influence

The atmospheric stability is one of the most influential factors in the atmospheric dispersion. The atmospheric stability depends on the difference of temperature of a parcel of air and its surroundings. According to these differences, different types of stabilities will be produced. Slightly unstable conditions lead to greater dispersion.