Environmental Engineering & Air Quality Modeling
Gaussian Air Plume Ground Concentration Formula
Calculate steady-state downwind ground-level centerline pollutant concentration C(x, 0, 0) = (Q / (PI()*u*sigma_y*sigma_z)) * EXP(-H^2 / (2*sigma_z^2)) with ground eddy reflection.
Never drop the factor of 2 from ground reflection: 1 / (PI * u * sigma_y * sigma_z) vs 1 / (2 * PI * ...)!
In free unbounded 3D space, a continuous point source emitting Q micrograms/sec into wind speed u (m/s) with crosswind and vertical Pasquill-Gifford dispersion standard deviations (sigma_y, sigma_z in meters) has centerline denominator 2 * PI() * u * sigma_y * sigma_z. In atmospheric dispersion over an impermeable ground surface (z = 0) from Effective Stack Height H = h_stack + Delta_h (m), perfect ground reflection adds an identical mirror image source at z = -H — DOUBLING ground concentration so the 2 cancels out: Ground-Level Plume Centerline Concentration is C(x, 0, 0) = (Q / (PI() * u * sigma_y * sigma_z)) * EXP(-(H^2) / (2 * (sigma_z^2))) (ug/m^3)!
Verify on Emission_Rate Q = 50,000,000 ug/s (= 50.0 g/s SO2, B2), Wind_Speed u = 5.0 m/s (B3), Crosswind_Dispersion sigma_y = 80.0 m (B4), Vertical_Dispersion sigma_z = 50.0 m (B5), and Effective_Stack_Height H = 50.0 m (B6): Ground-Release Pre-Factor (H = 0) is 50,000,000 / (PI() * 5 * 80 * 50) = 795.77 ug/m^3; Elevated Stack Attenuation at H = sigma_z = 50 m is EXP(-2500 / 5000) = EXP(-0.5) = 0.6065; giving Ground-Level Centerline Concentration 795.7747 * 0.60653 = 482.66 ug/m^3 (whereas forgetting ground reflection would underreport ground exposure by 50% at 241.33 ug/m^3!)!
| Gaussian Plume Metric (Q=50 g/s, u=5 m/s, sigma_y=80 m, sigma_z=50 m, H=50 m) | Formula | Verified Output |
|---|---|---|
| Ground-Level Release Ceiling (H = 0) | Q / (PI() * u * sigma_y * sigma_z) | 795.77 ug / m^3 |
| Stack Elevation Attenuation Factor | EXP(-(H^2) / (2 * sigma_z^2)) | 0.6065 (EXP(-0.5)) |
| Reflected Ground Centerline C(x, 0, 0) | Ceiling * EXP(-H^2 / (2*sigma_z^2)) | 482.66 ug / m^3 |
| Crosswind Off-Axis C(x, y = 80 m, 0) | C(x, 0, 0) * EXP(-y^2 / (2*sigma_y^2)) | 292.75 ug / m^3 |
Reflected Ground-Level Plume Centerline Concentration C(x, 0, 0) Formula
Validate Q >= 0, wind speed u > 0 (Gaussian plume equations break down in calm winds u -> 0, where puff models are required!), sigma_y > 0, sigma_z > 0, and H >= 0.
=LET(q_ug, B2, u_wn, B3, s_y, B4, s_z, B5, h_ef, B6, IF(OR(q_ug < 0, u_wn <= 0, s_y <= 0, s_z <= 0, h_ef < 0), "Invalid Plume Input (Require u > 0)", ROUND((q_ug / (PI() * u_wn * s_y * s_z)) * EXP(-(h_ef ^ 2) / (2 * (s_z ^ 2))), 2)))Off-Centerline Lateral Ground Concentration C(x, y, 0) Formula
Multiply centerline C(x, 0, 0) by lateral Gaussian decay EXP(-(y^2) / (2 * sigma_y^2)) at crosswind offset y (B7).
=LET(q_ug, B2, u_wn, B3, s_y, B4, s_z, B5, h_ef, B6, y_of, B7, IF(OR(q_ug < 0, u_wn <= 0, s_y <= 0, s_z <= 0, h_ef < 0), "Invalid Plume Input (Require u > 0)", ROUND((q_ug / (PI() * u_wn * s_y * s_z)) * EXP(-(h_ef ^ 2) / (2 * (s_z ^ 2))) * EXP(-(y_of ^ 2) / (2 * (s_y ^ 2))), 2)))Try the example
Use the original values or change them to test your understanding.