Air Quality & Dispersion formulas — 19 solvers

Barometric Pressure with Altitude

P = P_0 \, e^{-Mgz/RT}

Air Quality & DispersionAtmospheric pressure at a height above a reference level, for an isothermal layer. The exponential decay that gives the atmosphere a scale height of about 8.4 km.

Briggs Buoyancy Flux

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

Air Quality & DispersionThe 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}

Air Quality & DispersionHow 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.

Effective Stack Height

H = h_s + \Delta h

Air Quality & DispersionThe 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.

Emission Correction to Reference Oxygen

C_{corr} = C_{meas} \, \frac{20.9 - O_{2,ref}}{20.9 - O_{2,meas}}

Air Quality & DispersionRestates a measured stack concentration at the reference oxygen a limit is written at, so that adding dilution air can no longer make an emission look cleaner than it is. The US EPA convention, built on 20.9 % oxygen in ambient air.

Emission Rate from Stack Concentration

E = C \, Q_v

Air Quality & DispersionThe mass of pollutant leaving a stack per unit time, from the measured concentration and the volumetric flow. This is the number a permit limit is written against and the source strength every dispersion model asks for.

Environmental Lapse Rate

\Gamma = \frac{T_1 - T_2}{z_2 - z_1}

Air Quality & DispersionHow fast the air cools with height, from two temperatures at two altitudes. Comparing it against the dry adiabatic rate of 9.8 °C/km is what decides whether the atmosphere is stable or unstable.

Excess Air from Flue Gas Oxygen

EA = \frac{O_2}{20.9 - O_2}

Air Quality & DispersionHVAC & HydronicsHow much air beyond stoichiometric is passing through a burner, read straight off the oxygen in the flue gas. The single most useful number a combustion analyser gives you, because everything about efficiency follows from it.

Gaussian Plume Ground-Level Concentration

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

Air Quality & DispersionGround-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.

Good Engineering Practice Stack Height

H_{GEP} = h_b + 1.5 L

Air Quality & DispersionThe stack height that just clears the turbulent wake of a nearby building, by the US EPA formula: the building height plus one and a half times its lesser dimension. It is a ceiling on credit, not a permit to build tall.

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)

Air Quality & DispersionPlume 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.

Isokinetic Sampling Rate

Q_n = v_s A_n

Air Quality & DispersionThe flow a sampling train must draw so that gas enters the nozzle at exactly the stack velocity. Get this wrong and the particulate result is biased in a direction you can predict from the sign of the error.

Maximum Ground-Level Concentration

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

Air Quality & DispersionThe 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.

Particulate Collection Efficiency

\eta = \frac{C_{in} - C_{out}}{C_{in}}

Air Quality & DispersionThe fraction of the dust entering a control device that does not leave it, from inlet and outlet loadings. The number on every baghouse, cyclone and precipitator datasheet, and the one most often quoted with too many nines.

Pasquill–Gifford Dispersion Coefficient

\sigma = a \, x^{b}

Air Quality & DispersionThe 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.

ppm to mg/m³ Conversion

C = \frac{ppm \cdot M}{24.45}

Air Quality & DispersionConverts a gas concentration by volume into one by mass, using the molar volume of an ideal gas at 25 °C and 101.325 kPa. The bridge between an analyser that reads in ppm and a limit written in mg/m³.

Stack Draft Pressure (Chimney Effect)

\Delta p = h \, g \, (\rho_a - \rho_s)

Air Quality & DispersionHVAC & HydronicsThe pressure a chimney generates on its own, from the height of the column and the density difference between cold outside air and hot flue gas. The same equation explains why a tall building's lobby doors are hard to open in January.

Stack Exit Velocity

v_s = \frac{4 Q_v}{\pi d^2}

Air Quality & DispersionHVAC & HydronicsThe speed exhaust leaves a round stack, from the volumetric flow and the inside diameter. Every plume-rise calculation starts here, and so does the check against stack-tip downwash.

Wind Speed at Height (Power Law)

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

Air Quality & DispersionExtrapolates 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.