Gaussian Plume Dispersion Calculator

Introduction to Gaussian plume dispersion from elevated stacks

When a chimney, flare, vent or process stack releases pollution into a moving atmosphere, the exhaust is stretched into a long ribbon by the mean wind and simultaneously torn apart by turbulent eddies. The Gaussian plume model is the classical steady-state description of that process: it assumes the time-averaged concentration profile across the plume is normally distributed in the crosswind and vertical directions, with standard deviations that grow with travel distance. Nearly every regulatory screening tool in use today, from the United States Environmental Protection Agency SCREEN3 and ISC3 models to the plume component of AERMOD, is built on this same skeleton.

This calculator implements that skeleton faithfully rather than approximately. The dispersion coefficients are not invented constants: rural runs use the Pasquill-Gifford curve fits published in Tables 1-1 and 1-2 of the EPA ISC3 User's Guide, and urban runs use the Briggs formulas fitted to the McElroy-Pooler St. Louis tracer data in Tables 1-3 and 1-4 of the same guide. The optional plume rise module reproduces the Briggs buoyancy and momentum equations coded into ISC3, including stack-tip downwash. As a result the numbers this page returns are directly comparable with a hand calculation done from EPA guidance.

A Gaussian plume estimate answers a very specific question: if the emission rate, wind speed and stability stay constant for about an hour, what average concentration would a receptor at a given downwind distance, crosswind offset and height experience? That is exactly the question a permit engineer asks when siting a monitor, an emergency planner asks when drawing a first cordon, and a student asks when learning why a taller stack lowers a nearby impact but shifts the maximum further out.

How to use the stack dispersion calculator on this page

Work through the form from the top. Enter the emission rate of the single pollutant you care about and choose the unit that matches your permit or emissions inventory; the calculator converts grams per second, kilograms per hour, pounds per hour and short tons per year internally. Enter the wind speed at release height, not the airport ten-metre value, unless you have already applied a power-law profile. Choose rural or urban dispersion to match land use within a few kilometres of the source, then pick the Pasquill stability class that matches the hour you are modelling.

You can either type an effective stack height directly, or tick the plume-rise box and supply physical stack height, inside top diameter, exit velocity, stack gas temperature and ambient temperature. In the second case the calculator applies the Briggs equations and reports the plume rise it computed, so you can see how much lift the buoyancy is giving you. Finally place the receptor: downwind distance along the wind vector, crosswind offset perpendicular to it, and receptor height above ground, which is conventionally 1.5 to 2 metres for a breathing-zone estimate.

• Emission rate scales the answer linearly. Doubling it doubles every concentration on the page.
• Wind speed appears in the denominator, so a light wind concentrates and a strong wind dilutes, all else equal.
• Stability class changes both spread parameters at once and is usually the single most influential input.
• Effective height pushes the plume away from ground receptors; the ground-level maximum falls roughly with the square of effective height and moves further downwind.
• Crosswind offset only ever reduces the concentration relative to the plume centreline.
• Receptor height matters most between the ground and the plume centreline; a rooftop monitor can read several times a sidewalk monitor.

After you calculate, the page draws a downwind profile chart, lists the ground-level maximum and the distance at which it occurs, and shows the same receptor evaluated under all six stability classes so you can see the sensitivity immediately. A copy button and a CSV download let you move the whole scenario into a report or spreadsheet.

Formula: the Gaussian point-source equation with ground reflection

For a continuous point source of strength released at effective height into a wind of speed , the time-averaged concentration at downwind distance , crosswind offset and height is

Formula: C = Q / (2 π σ_y σ_z u) e^-y^2/(2σ_y^2) e^-(z-H)^2/(2σ_z^2) + e^-(z+H)^2/(2σ_z^2)

The leading fraction is pure mass conservation: the emitted mass per second is smeared over an elliptical cross-section of area travelling at speed . The first exponential is the crosswind Gaussian, equal to one on the centreline. The bracketed pair is the vertical structure: the first term is the real plume centred on , and the second is an image source placed at . Adding the image enforces a zero-flux boundary at the ground, which is what physically happens when a non-depositing gas hits a surface it cannot pass through and is reflected upward. Omitting that term, a common error in simplified implementations, understates ground-level concentration by up to a factor of two.

On the plume centreline at ground level the expression collapses to a form that is easy to sanity-check by hand:

Formula: C_gl = Q / (π σ_y σ_z u) e^-H^2/(2σ_z^2)

Differentiating that expression shows the classical screening result that, when is roughly constant, the ground-level maximum occurs where . This calculator does not rely on that approximation; it locates the maximum by scanning the profile numerically, which stays correct even where the curve fits change slope.

Dispersion coefficient formulas taken from EPA ISC3

Everything hinges on and . For rural terrain the EPA fits the Pasquill-Gifford curves with

Formula: σ_y = 465.11628 ⁢ x ⁢ tan ⁡ (0.017453293 ⁢ [c - d ⁢ ln ⁡ x])

Formula: σ_z = a ⁢ x^b

with in kilometres and the coefficients , , and tabulated by stability class. Note that the vertical fit is piecewise: neutral class D alone uses six different coefficient pairs between 0.3 km and beyond 30 km. For urban terrain the Briggs fits to the McElroy-Pooler data are single closed-form expressions, reproduced below exactly as EPA publishes them, with in metres.

Two features of that table are worth internalising. First, urban for classes A and B grows faster than linearly because the exponent is positive one half, reflecting vigorous convective mixing over hot city surfaces. Second, urban classes E and F collapse onto a single curve, which is EPA's way of saying that a city rarely reaches true rural stability at night. Any implementation that simply multiplies distance by a constant, as many simplified web calculators do, throws away both effects and can be wrong by tens of percent even inside the validated distance range.

Effective height, plume rise and why the stack top is not the release point

Hot exhaust is buoyant and fast exhaust has momentum, so a plume keeps climbing after it leaves the stack until turbulence dilutes its excess buoyancy. The Briggs treatment coded into ISC3 begins with the buoyancy flux parameter

Formula: F_b = (g ⁢ v_s ⁢ d_s^2 ⁢ (T_s - T_a)) / (4 T_s)

and, for unstable or neutral hours dominated by buoyancy, gives the final effective height as

Formula: H = h_s^′ + (21.425 ⁢ F_b^3/4) / u, F_b < 55

switching to a form once the buoyancy flux reaches 55 m⁴/s³. In stable classes E and F the rise is capped by the ambient potential temperature gradient through the stability parameter , and the calculator uses the EPA default gradients of 0.020 K/m for class E and 0.035 K/m for class F. Cool or slow exhausts fall back to the momentum forms, and a stack whose exit velocity is below 1.5 times the wind speed is penalised by stack-tip downwash before any rise is added.

Worked example: a 100 g/s sulphur dioxide stack in neutral rural air

Take a power station stack emitting g/s of sulphur dioxide with an effective height of 50 m into a 5 m/s wind, in rural terrain under neutral class D, and place a person 1000 m directly downwind at a breathing height of 1.5 m on the plume centreline.

The rural fits give m and m at that distance. The leading fraction is g/m³. The crosswind exponential is one on the centreline. The real-source vertical term is and the image-source term is , summing to 0.595. Multiplying through gives g/m³, that is about 866 micrograms per cubic metre.

Two details make this example instructive. First, the ground reflection contributes almost half of the total, so dropping it would have given 465 instead of 866 micrograms per cubic metre. Second, a numerical scan shows that this receptor sits essentially at the ground-level maximum for these conditions: the peak is 866 micrograms per cubic metre at roughly 1.0 km. Move the same receptor to 3 km and the concentration falls to a few hundred; move it to 300 m and it collapses towards zero because the plume has not yet mixed down to the surface.

Now switch only the terrain to urban. The Briggs urban curves give a much larger vertical spread of about 123 m at 1 km, the plume is already well mixed through the surface layer, and the concentration drops to roughly 353 micrograms per cubic metre. Switch instead to rural stability class F and the plume is so tight vertically that only about 23 micrograms per cubic metre reach the ground at 1 km, but the maximum for that class occurs far further downwind. That single pair of comparisons is the whole argument for choosing stability and land use carefully before quoting any number.

Reading the result and the downwind profile chart

The result panel reports the concentration in micrograms, milligrams and grams per cubic metre, the two spread parameters, the effective release height actually used, and the ground-level maximum together with the distance at which it occurs. That maximum is usually the number a screening study cares about, because a receptor can always be placed there later. The stability comparison table repeats the calculation at your receptor for all six classes so you can see the spread of plausible answers before committing to one.

The chart plots concentration against downwind distance on a logarithmic distance axis from 100 m to 10 km, drawing the centreline profile and, when you have entered a crosswind offset, the offset profile as well. The rising limb on the left is the plume mixing down towards the surface; the falling limb on the right is straightforward dilution. Marking your receptor on that curve tells you at a glance whether you are sampling the near field, the peak, or the far tail, and how sensitive your answer is to a small error in distance.

Limitations, assumptions and when to move to a refined model

Every Gaussian plume result rests on assumptions that are easy to state and easy to violate. The model assumes steady, horizontally uniform wind and turbulence for the whole averaging period; flat terrain with no channelling, no hills and no shoreline; a conservative pollutant with no chemistry, no dry deposition and no wet scavenging; total reflection at the ground; a continuous release rather than a puff; and no building downwash. It also assumes the wind is strong enough for the concept of a mean transport direction to make sense, which is why EPA models refuse to run below about 1 m/s at stack height.

The curve fits themselves carry a validity range of roughly 100 m to 10 km downwind, and even inside that band the Pasquill-Gifford sigma values were derived from short-range tracer experiments over flat, open, low-roughness ground. Applying them over forest, complex terrain or an industrial site with large structures will bias the answer, usually towards over-prediction near the source. The calculator also ignores the mixing height, so it will over-predict at long range on a convective afternoon when the plume is actually trapped and well mixed below an inversion.

Treat this page as a screening and teaching instrument. If the screening estimate approaches a threshold you care about, the next step is a refined model such as AERMOD driven by real hourly meteorology and site-specific surface characteristics, run according to 40 CFR Part 51 Appendix W, the EPA Guideline on Air Quality Models. And remember the wider point that dispersion is not abatement: pushing a plume higher spreads the same mass over a larger area rather than removing it, so emission reduction at source remains the only intervention that actually lowers total exposure.