8.4 Gaussian Plume Modeling, Plume Rise, and Environmental Fate

Gaussian Plume Modeling, Plume Rise, and Environmental Fate

Stability class and plume shape are the qualitative picture. The Gaussian plume equation turns that picture into a predicted downwind concentration, and multimedia partitioning predicts where a chemical ends up once it leaves the air.

1. The Gaussian Plume Dispersion Model

The standard regulatory model for continuous stationary point source emissions assumes steady-state conditions, conservation of mass, and that turbulent diffusion causes pollutant concentrations to follow normal (Gaussian) distributions along both the lateral (y) and vertical (z) axes.

                                  z (Height)
                                  ▲
                                  │      Plume Centerline (z = H)
                                  │     . - - - - - - - - - - - - - -
                                  │   .               / 
                                  │ .  σz            /   Wind (u)
                    Stack (hs) ───┼─────────────────► ─────────►
                                  │                 x (Downwind Distance)
                                 O└────────────────────────►
                                 / ◄────── σy ──────►
                                / 
                               ▼ y (Crosswind Distance)

Coordinate System Conventions

• x-axis: Downwind direction along the mean horizontal wind vector (y = 0).
• y-axis: Crosswind horizontal direction perpendicular to the mean wind vector.
• z-axis: Vertical axis representing elevation above ground level (z = 0 is ground surface).
• H: Effective stack height (H = hs + Δ h, where hs is physical stack height and Δ h is plume rise).

The Full 3D Point Source Gaussian Plume Equation

Accounting for total reflection of the plume at the ground surface (z = 0) using the method of virtual images:

$C(x,y,z; H) = \frac{Q}{2\pi \cdot u \cdot \sigma_y \cdot \sigma_z} \cdot \exp\left(-\frac{y^2}{2\sigma_y^2}\right) \cdot \left[ \exp\left(-\frac{(z - H)^2}{2\sigma_z^2}\right) + \exp\left(-\frac{(z + H)^2}{2\sigma_z^2}\right) \right]$

Where:

• C(x,y,z; H) = Pollutant concentration at receptor coordinate (x,y,z) (g/m³ or µg/m³)
• Q = Continuous mass emission rate from the stack (g/s or mg/s)
• u = Mean wind speed at effective stack height (H) (m/s)
• σy = Lateral (crosswind) dispersion coefficient (standard deviation of plume concentration spread, in meters)
• σz = Vertical dispersion coefficient (standard deviation of plume concentration spread, in meters)
• H = Effective stack height (meters)

Ground-Level Centerline Concentration Equation (y = 0, z = 0)

For an observer on the ground directly along the downwind plume centerline (y = 0, z = 0):

$C(x,0,0; H) = \frac{Q}{\pi \cdot u \cdot \sigma_y \cdot \sigma_z} \cdot \exp\left(-\frac{H^2}{2\sigma_z^2}\right)$

Ground-Level Release Equation (H = 0)

For a ground-level release (such as a surface spill or low fugitive leak where H = 0):

$C(x,0,0; 0) = \frac{Q}{\pi \cdot u \cdot \sigma_y \cdot \sigma_z}$

Key Dispersion Relationships

1. Concentration is directly proportional to emission rate (C ∝ Q): Doubling stack emissions doubles ground-level concentration.
2. Concentration is inversely proportional to wind speed (C ∝ 1/u): Higher wind speeds stretch out the plume volume, increasing dilution.
3. Dispersion coefficients grow with downwind distance: Both σy(x) and σz(x) increase monotonically with downwind distance x, parameterized via empirical power-law equations (chief among them σy = a · x(b) and σz = c · x(d)).
4. Maximum Ground-Level Concentration Distance (xmax): For an elevated source, ground concentration is initially zero at the stack base (x = 0), rises to a peak maximum at xmax (where σz = H / √2 ≈ 0.707 H), and then decays asymptotically at long distances downwind as σy σz expansion dominates.

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2. Effective Stack Height and Briggs Plume Rise (Δ h)

Effluent exiting a stack possesses vertical momentum (exit velocity vs) and thermal buoyancy (Ts > Ta). The plume bends over in the crosswind and rises to an Effective Stack Height (H):

$H = h_s + \Delta h$

Where hs is the physical stack height and Δ h is the plume rise calculated using the Briggs Plume Rise Equations.

Buoyancy Flux Parameter (Fb)

Thermal buoyancy dominates when stack gas exit temperature (Ts) significantly exceeds ambient air temperature (Ta):

$F_b = g \cdot v_s \cdot d^2 \cdot \left(\frac{T_s - T_a}{4 \cdot T_s}\right)$

Where:

• g = 9.81 m/s²
• vs = Stack gas exit velocity (m/s)
• d = Stack inside exit diameter (meters)
• Ts, Ta = Absolute temperatures of stack gas and ambient air, respectively (Kelvin)

Briggs Final Plume Rise in Neutral and Unstable Conditions (Classes A–D)

For buoyant plumes with Fb < 55 m⁴/s³, final plume rise occurs at downwind distance xf = 49 · Fb(5/8):

$\Delta h = 21.42 \cdot \frac{F_b^{3/4}}{u}$

For buoyant plumes with Fb ≥ 55 m⁴/s³, final plume rise occurs at downwind distance xf = 119 · Fb(2/5):

$\Delta h = 38.71 \cdot \frac{F_b^{3/5}}{u}$

Briggs Plume Rise in Stable Conditions (Classes E and F)

In stable atmospheres, vertical rise is arrested by the ambient thermal inversion gradient (s is the atmospheric stability parameter: s = g/Ta dθ/dz):

$\Delta h = 2.6 \cdot \left(\frac{F_b}{u \cdot s}\right)^{1/3}$

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3. Chemical Environmental Fate and Multimedia Partitioning

Once an organic chemical enters the environment, its distribution between air, water, soil, sediment, and biological tissue is dictated by fundamental physicochemical thermodynamic partition coefficients.

                                  ┌─────────────────────────────┐
                                  │      AIR (Atmosphere)       │
                                  └──────────────┬──────────────┘
                                                 │ ▲
                        Henry's Law Constant (H) │ │ Volatilization
                                                 ▼ │
                                  ┌─────────────────────────────┐
                                  │    WATER (Aqueous Phase)    │
                                  └──────────────┬──────────────┘
                                                 │ ▲
                          Octanol-Water (Kow)   │ │ Soil Adsorption (Koc / Kd)
                          Bioconcentration (BCF) │ │
                                                 ▼ │
                     ┌───────────────────────────┴───────────────────────────┐
                     ▼                                                       ▼
     ┌───────────────────────────────┐                       ┌───────────────────────────────┐
     │   BIOTA (Lipid / Tissue)      │                       │   SOIL / SEDIMENT (Organics)  │
     └───────────────────────────────┘                       └───────────────────────────────┘

Octanol-Water Partition Coefficient (Kow / log Kow)

Kow is the ratio of chemical concentration in n-octanol (surrogate for biological lipid) to its concentration in pure water at equilibrium:

$K_{ow} = \frac{C_{\text{octanol}}}{C_{\text{water}}}$

• log Kow < 1: Highly hydrophilic (water-soluble); low lipophilicity; rapid excretion; negligible bioaccumulation.
• log Kow = 1 - 4: Moderately lipophilic; intermediate bioaccumulation.
• log Kow > 4 - 5: Highly hydrophobic and lipophilic; strong affinity for adipose tissue; high potential for bioaccumulation and biomagnification up the trophic food chain (e.g., DDT log Kow ≈ 6.9, PCBs log Kow ≈ 6-8, Dioxins log Kow ≈ 6.8).

Bioconcentration Factor (BCF) and Bioaccumulation

• Bioconcentration Factor (BCF): The ratio of chemical concentration in an aquatic organism (Cbiota, in mg/kg) to the concentration in ambient water (Cwater, in mg/L) resulting strictly from direct gill/integument exposure:

$\text{BCF} = \frac{C_{\text{biota}}}{C_{\text{water}}}$

• Mackay Empirical Estimation: For non-metabolized hydrophobic neutral organics:

$\log \text{BCF} \approx 0.85 \cdot \log K_{ow} - 0.70 \quad \text{or} \quad \text{BCF} \approx 0.048 \cdot K_{ow}$

Soil Organic Carbon-Water Partition Coefficient (Koc and Kd)

Governs the chemical sorption onto organic matter in soils and aquifer sediments:

• Kd (Soil Distribution Coefficient): Kd = Csoil/Caqueous (in L/kg).
• Koc (Normalized to Soil Organic Carbon): Koc = Kd/foc, where foc is the mass fraction of organic carbon in the soil.
• Groundwater Mobility Scale:
  • Koc < 50 mL/g : Very High Mobility (readily leaches into groundwater aquifers; e.g., vinyl chloride, MTBE).
  • Koc = 50 - 500 mL/g : Moderate Mobility (e.g., benzene, TCE).
  • Koc > 5,000 - 10,000 mL/g : Immobile (strongly bound to topsoil; e.g., PAHs, PCBs, heavy petroleum fractions).

Henry's Law Constant (H / KH)

Measures the equilibrium partitioning between the gaseous phase (air) and the aqueous phase (water):

$H = \frac{P_{\text{vapor}}}{C_{\text{aqueous}}} \quad (\text{expressed in } \text{atm}\cdot\text{m}^3/\text{mol})$

• Dimensionless Henry's Law Constant (H'):

$H' = \frac{C_{\text{air}}}{C_{\text{water}}} = \frac{H}{R \cdot T}$

Where R = 8.2057 × 10⁻⁵ atm·m³/(mol·K) and T is temperature in Kelvin.

• Volatilization Screening Criteria:
  • H > 10⁻³ atm·m³/mol : Rapid volatilization from surface waters (e.g., VOCs, TCE, tetrachloroethylene).
  • H = 10⁻⁵ - 10⁻³ atm·m³/mol : Moderate volatilization.
  • H < 10⁻⁵ atm·m³/mol : Low volatilization; chemical remains predominantly in the water column (e.g., phenol, alcohols).

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4. Worked Step-by-Step Calculation Examples

Worked Example 7.3: Gaussian Plume Ground-Level Centerline Concentration

Problem: A chemical plant discharges sulfur dioxide (SO2) continuously from a stack at a mass emission rate of Q = 50.0 g/s (50,000 mg/s). The effective stack height is H = 60.0 meters, and the mean wind speed at stack height is u = 4.0 m/s. At a downwind community receptor located on the centerline (y = 0, z = 0) at x = 1.5 km under Pasquill Stability Class C (Slightly Unstable), atmospheric dispersion coefficients are determined to be:

• σy = 150.0 meters
• σz = 85.0 meters

Calculate the steady-state ground-level centerline concentration of SO2 at the receptor in micrograms per cubic meter (µg/m³).

Solution Steps:

1. State the Ground-Level Centerline Gaussian Equation: $C(x,0,0; H) = \frac{Q}{\pi \cdot u \cdot \sigma_y \cdot \sigma_z} \cdot \exp\left(-\frac{H^2}{2\sigma_z^2}\right)$

2. Calculate the Pre-Exponential Dispersion Dilution Factor: $\text{Denominator} = \pi \times u \times \sigma_y \times \sigma_z$ $\text{Denominator} = \pi \times 4.0\text{ m/s} \times 150.0\text{ m} \times 85.0\text{ m} = 160,221.2\text{ m}^3/\text{s}$ $\frac{Q}{\pi \cdot u \cdot \sigma_y \cdot \sigma_z} = \frac{50.0\text{ g/s}}{160,221.2\text{ m}^3/\text{s}} = 3.1207 \times 10^{-4}\text{ g/m}^3 = 312.07\ \mu\text{g/m}^3$

3. Calculate the Exponential Elevation Decay Term: $\frac{H^2}{2\sigma_z^2} = \frac{(60.0\text{ m})^2}{2 \times (85.0\text{ m})^2} = \frac{3,600}{2 \times 7,225} = \frac{3,600}{14,450} = 0.249135$ $\exp(-0.249135) = 0.77947$

4. Multiply the Factors to Compute Ground-Level Concentration (C): $C = 312.07\ \mu\text{g/m}^3 \times 0.77947 = 243.25\ \mu\text{g/m}^3 \approx 243.3\ \mu\text{g/m}^3$

Result: The predicted steady-state ground-level SO2 concentration is 243.3 µg/m³ (0.243 mg/m³).

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Worked Example 7.4: Environmental Partitioning (Kd) and Fish Bioconcentration (BCF)

Problem: An industrial solvent released into a river basin has an octanol-water partition coefficient of log Kow = 3.50 (Kow = 10(3.50) = 3,162.3). Environmental testing indicates that riverbed sediment has a mass fraction of organic carbon of foc = 0.015 (1.5% organic carbon). The aqueous concentration of the solvent in the river water is measured at Cw = 0.050 mg/L (50 µg/L).

1. Estimate the soil organic carbon-water partition coefficient (Koc) using the empirical relation: log Koc = log Kow - 0.21.
2. Calculate the sediment-water distribution coefficient (Kd) in L/kg.
3. Estimate the Bioconcentration Factor (BCF) using BCF = 0.048 · Kow, and predict the steady-state concentration of the contaminant in fish tissue (Cfish) in mg/kg.

Solution Steps:

1. Calculate Koc: $\log K_{oc} = 3.50 - 0.21 = 3.29$ $K_{oc} = 10^{3.29} = 1,949.8\text{ L/kg} \approx 1,950\text{ L/kg}$

2. Calculate Sediment Distribution Coefficient (Kd): $K_d = K_{oc} \times f_{oc} = 1,949.8\text{ L/kg} \times 0.015 = 29.25\text{ L/kg}$

3. Calculate Bioconcentration Factor (BCF) and Tissue Concentration: $\text{BCF} = 0.048 \times 3,162.3 = 151.79\text{ L/kg} \approx 151.8\text{ L/kg}$ $C_{\text{fish}} = \text{BCF} \times C_w = 151.79\text{ L/kg} \times 0.050\text{ mg/L} = 7.589\text{ mg/kg} \approx 7.59\text{ mg/kg}$

Result: The sediment distribution coefficient is 29.25 L/kg, the estimated BCF is 151.8, and the predicted fish tissue burden is 7.59 mg/kg.