Where the plume lands

Gaussian plume modelplume riseeffective stack heightground level concentrationdispersion modelling

From a stack to the ground-level concentration at the fence: wind at stack height, plume rise, effective height and the Gaussian equation.

Wind Speed at Height (Power Law)

u_2 = u_1 \left(\frac{z_2}{z_1}\right)^{p}

Extrapolates a wind measurement to another height using the power-law profile. Anemometers sit at 10 m and stacks do not, so this step precedes every plume-rise calculation.

Briggs Buoyancy Flux

F = \frac{g \, v_s \, d^2 (T_s - T_a)}{4 \, T_s}

The buoyancy flux parameter of a hot stack plume, from exit velocity, stack diameter and the temperature difference against ambient. The input to every Briggs plume-rise equation.

Briggs Plume Rise (Neutral and Unstable)

\Delta h = \frac{1.6 \, F^{1/3} x^{2/3}}{u}

How far a buoyant plume climbs above the stack at a given downwind distance, in neutral or unstable air — the Briggs two-thirds law that underlies most regulatory dispersion modelling.

Holland Plume Rise

\Delta h = \frac{v_s d}{u}\left(1.5 + 2.68\times10^{-3} P d \, \frac{T_s - T_a}{T_s}\right)

Plume rise by Holland's 1953 equation, combining momentum and buoyancy in one step. Simpler than Briggs and generally conservative, which is why it survives as a screening estimate.

Effective Stack Height

H = h_s + \Delta h

The height dispersion actually starts from: the physical stack plus the plume rise. Because ground-level concentration falls with the square of this height, plume rise is worth more than steel.

Pasquill–Gifford Dispersion Coefficient

\sigma = a \, x^{b}

The crosswind or vertical spread of a plume at a downwind distance, in the power-law form fitted to the Pasquill stability classes. The width term the Gaussian model needs.

Gaussian Plume Ground-Level Concentration

C = \frac{Q}{\pi \sigma_y \sigma_z u} \, e^{-H^{2}/(2\sigma_z^{2})}

Ground-level concentration directly downwind of an elevated point source, on the plume centreline, with full reflection from the ground. The core equation of regulatory air-quality modelling.

Maximum Ground-Level Concentration

C_{max} = \frac{2Q}{e \pi u H^{2}} \cdot \frac{\sigma_z}{\sigma_y}

The worst ground-level concentration a plume ever produces, wherever downwind it occurs. Falls with the square of effective stack height, which is the whole argument for building tall.

How they fit together

Do the wind first or everything after it is wrong. The wind speed power law shears the anemometer reading, taken at 10 m, up to stack height — and every formula downstream of it wants u at stack height, not at the mast. The exponent is a stability question rather than a terrain question alone: around 0.07 in very unstable air, 0.15 in neutral, and up to 0.55 in a stable night-time inversion over rough ground. Using one exponent all year is the quiet error that makes a model agree with monitors in the afternoon and disagree with them at dawn.

Plume rise is next, and here the set has a fork rather than a step. Briggs and Holland are two routes to the same quantity, not two stages of one calculation, so run one of them. Briggs, reached through buoyancy flux, is the modern and regulatory-standard route, it is the better physics, and it is what you want for a hot buoyant plume from a combustion source — which is most of them. Holland is the older empirical form, it takes the stack conditions directly without a separate flux step, and it survives because it handles momentum-dominated plumes reasonably and is quick to run for a screening estimate. It generally gives a lower rise, so it is the conservative one, which is the other reason it persists. If you compute both and they disagree by a factor of two, that is normal rather than a mistake in your arithmetic. Note also that Briggs' rise grows with downwind distance x until the plume levels off, so the answer depends on where you asked.

Effective stack height is the addition that matters more than any other single number here, because H enters the Gaussian equation inside an exponential and squared. A plume rise of 50 m on a 50 m stack does not add 50% to your dilution — it can cut peak ground-level concentration by a factor of several. This is why a hot fast plume from a short stack can outperform a cold slow one from a tall stack, and why turning down a fan to save energy has been known to create a nuisance at the fence line. Pasquill-Gifford coefficients then give σy and σz at the receptor distance, and they carry the model's real uncertainty: they are curve fits to field trials over open, level, rural ground, and applying them over a city or broken terrain is an approximation everybody makes and nobody defends. Finally, the two Gaussian forms answer different questions and should not be substituted for one another. Ground-level concentration is the value at a specific receptor — a house, a school, a monitor — and maximum ground-level concentration is the worst case wherever it happens to fall, which is the number a permit limit is usually written against. Both assume flat terrain, steady wind and no chemistry, so treat either as an order-of-magnitude screening answer rather than a measurement.