Gaussian Plume Dispersion Calculator

How Pollutants Spread Downwind

The atmosphere is constantly in motion, stirring pollutants released from chimneys, flares, vents, or industrial stacks into complex three-dimensional patterns. Environmental scientists and regulators often rely on the Gaussian plume model to estimate how a continuous point source disperses as the wind carries it downwind. This approach assumes that turbulent eddies mix the plume into a bell-shaped, or Gaussian, distribution in both the vertical and crosswind directions. Although the model simplifies reality, it provides a useful first approximation and forms the backbone of many air-quality screening methods around the world.

This calculator is designed for exactly that first-pass estimate. You enter the source strength, the wind speed, the effective stack height, and the receptor location. The tool then estimates the concentration at one point in space, which makes it handy for screening studies, classroom problems, and quick scenario comparisons. It is especially helpful for seeing how strongly the answer depends on stability class. A stable night can keep the plume narrow and concentrated, while a sunny unstable afternoon can spread the same emission much more widely.

In plain language, each input answers a different part of the dispersion story. The emission rate Q says how much pollutant leaves the source each second. Wind speed U controls how quickly the air carries and dilutes that pollutant. Effective stack height H combines physical stack height with any plume rise caused by buoyancy or momentum. Downwind distance x locates the receptor farther from the source. Crosswind distance y measures how far off the plume centerline the receptor sits, and receptor height z lets you compare ground-level and elevated points. Together, those values define one observation point in the plume field.

The core equation computes concentration $C$ at a point located downwind distance $x$, crosswind offset $y$, and height $z$ from a stack of effective height $H$ emitting continuously at rate $Q$. The basic form 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)

$$C=\frac{Q}{2\pi {\sigma }_y{\sigma }_{z}U}{e}^{-\frac{y^2}{2{{\sigma }_y}^2}}\left(e^{-\frac{{z-H}^2}{2{{\sigma }_z}^2}}+e^{-\frac{{z+H}^2}{2{{\sigma }_z}^2}}\right)$$

Here ${\sigma }_y$ and ${\sigma }_z$ are the standard deviations of the plume in the crosswind and vertical directions, so they describe how wide and how tall the plume becomes as it travels. The mirrored exponential term accounts for reflection of the plume off the ground, which is why ground-level receptors can still see notable concentration even when the stack is tall. Because the equation assumes steady wind and steady emissions, it is best viewed as a screening tool rather than a full atmospheric simulation.

What each input means in practice

If you are new to dispersion modeling, the easiest way to use the calculator is to imagine a specific release and a specific monitoring point. Start with the source itself. If a stack emits 100 grams of pollutant per second, then Q = 100 g/s. Next decide how quickly the air is moving past the source. If the local wind is 5 meters per second, then U = 5 m/s. Then choose the source height that the plume effectively behaves as if it were released from. That is the effective stack height H, not always just the physical stack top. Finally, define the receptor position: downwind distance x, crosswind offset y, and receptor height z.

• Emission rate Q increases concentration almost proportionally. Double the emission rate and, all else equal, the predicted concentration roughly doubles.
• Wind speed U dilutes the plume. Faster winds carry the same mass through a larger volume of air per second, reducing concentration.
• Effective stack height H lifts the plume away from ground receptors. A taller effective release point often lowers nearby ground-level impacts, though the maximum can occur farther downwind.
• Downwind distance x affects both travel and mixing. As distance grows, the plume usually dilutes, but ground-level concentration may rise at first if the plume needs time to mix down from a high stack.
• Crosswind distance y moves the receptor left or right of the plume centerline. The highest concentration usually occurs near $y=0$.
• Receptor height z matters when you compare sidewalk, rooftop, or elevated platform locations. Ground-level studies often use about 1.5 to 2 meters.

When you interpret the output, remember the reported value is a modeled concentration at one receptor location. It is not automatically a regulatory compliance result or a human exposure estimate. Real assessments usually compare modeled values against ambient standards, short-term averaging requirements, existing background pollution, and more detailed meteorology. Still, the number is highly informative because it reveals how sensitive the scenario is to wind and atmospheric stability.

Determining Dispersion Coefficients

Values of ${\sigma }_y$ and ${\sigma }_z$ depend on atmospheric stability, a measure of how readily air parcels mix vertically. Meteorologists classify conditions into six Pasquill stability categories from A (very unstable) to F (very stable). Bright sunny days with light winds often fall into class A, while clear nights with gentle breezes can produce class F. Turbulent unstable air promotes rapid dispersion, which means larger sigma values and lower centerline concentration. Stable air does the opposite: it keeps the plume tight and can create higher concentrations closer to the centerline.

For this simplified calculator, the spread parameters are approximated with linear relations of the form shown below. These are convenient screening formulas rather than a replacement for a full site-specific dispersion model.

Formula: σ_y = a x, σ_z = b x

$${\sigma }_y=ax,{\sigma }_z=bx$$

For example, at 1,000 meters downwind on a neutral day (class D), the crosswind spread is ${\sigma }_y=0.08x=80$ m, while the vertical spread is ${\sigma }_z=0.06x=60$ m. Plugging those values into the main equation shows how dilution develops with distance. If you change only the stability class to F, the smaller sigma values compress the plume and can noticeably raise the predicted concentration along the centerline.

Worked example and how to read the result

Consider a smokestack releasing 100 g/s of sulfur dioxide at an effective height of 50 m with a wind speed of 5 m/s. A receptor located 1 km downwind at ground level directly under the plume centerline experiences a concentration computed by the calculator under neutral conditions. That estimate comes from substituting the dispersion coefficients for class D and setting $y=0$, which maximizes the crosswind Gaussian term. If you move the receptor 200 meters off the centerline, the exponential crosswind factor reduces the concentration substantially. In a more stable atmosphere, the same emission can produce higher centerline values because the plume remains narrower for a longer distance.

The result box also reports ${\sigma }_y$ and ${\sigma }_z$, which are worth paying attention to instead of looking only at the final concentration. Those sigma values explain why the answer changes. If wind speed stays the same but sigma values get larger, the plume is spreading more, so centerline concentration tends to drop. If sigma values are small, the plume is tightly focused and even moderate emissions may produce notable concentrations near the centerline. That makes the output especially useful for comparing scenarios rather than treating a single number as absolute truth.

Assumptions, limits, and when to use a more advanced model

The Gaussian plume model is intentionally simple. It assumes flat terrain, steady-state wind, uniform meteorological conditions, and a pollutant that behaves conservatively over the modeled distance. It does not explicitly include building downwash, shoreline effects, terrain channeling, strong chemistry, wet scavenging, dry deposition, intermittent emissions, or rapid weather shifts. If your site has hills, nearby buildings, or short release durations, the real plume may bend, split, or pulse in ways this calculator cannot represent.

That limitation does not make the calculator unhelpful. In fact, this simplified model remains a valuable teaching tool and a fast screening method. Students can build intuition about dilution, emergency planners can estimate where a centerline hotspot may occur, and facility staff can compare how sensitive a source is to stable versus unstable conditions. Regulatory models such as AERMOD and ADMS add more physics on top of the same basic idea, so understanding the simple Gaussian form is a strong foundation for deeper air-quality work.

Finally, remember that dispersion is not the same thing as pollution control. Lower concentration at one location may simply mean the pollutant has been spread over a larger area. Real air-quality improvement usually comes from reducing emissions at the source through process changes, fuel switching, controls, or cleaner technologies. This calculator helps you see what the atmosphere does with an emission once it is released, but it also highlights why preventing the emission is usually the most effective strategy.