Holland Plume Rise

Holland published this in 1953 from observations at Oak Ridge, and it does in one step what Briggs does in several: it adds a momentum term and a buoyancy term and returns a single final rise with no downwind distance in it at all. The structure is easier to see when the bracket is read as two pieces. A stack 2 m across discharging at 15 m/s into a 5 m/s wind, at 1000 mb, with gas at 400 K against air at 280 K, gives $(T_s-T_a)/T_s = 0.3$, a buoyancy term of $2.68\times10^{-3} \times 1000 \times 2 \times 0.3 = 1.608$, a bracket of $1.5 + 1.608 = 3.108$, and a rise of $(15 \times 2/5) \times 3.108 = 6 \times 3.108 = 18.6$ m.

The 1.5 is the momentum coefficient, and $v_s d/u$ is the momentum length scale, so a cold plume with no buoyancy at all still rises $1.5 v_s d/u$ simply because it was thrown upward. The second term carries the buoyancy, and its constant of $2.68\times10^{-3}$ is dimensional: it is only correct with pressure in MILLIBARS, which is hectopascals, because that is the unit Holland's data was tabulated in. This is the single most common way to get a wrong answer out of this equation. Enter 101325 pascals where 1013 millibars belongs and the buoyancy term is inflated a hundredfold. The solver here converts internally from whatever unit you choose, so the constant always sees millibars.

Compare the two methods on the same stack and the difference is not small. That example has $F = 9.80665 \times 15 \times 4 \times 120/(4 \times 400) = 44.1$ m⁴/s³, which is below Briggs' 55 threshold, so $x^* = 14 \times 44.1^{5/8} = 149$ m and final rise arrives at about 523 m. Briggs then gives $1.6 \times 3.53 \times 523^{2/3}/5 = 73$ m against Holland's 18.6 m. Holland is low by roughly a factor of four here, and that is typical: it was fitted to sources much smaller and cooler than a modern utility boiler, and it is well documented as under-predicting hot buoyant plumes. ASME's guidance is to multiply Holland's result by 1.1 to 1.2 in unstable air and by 0.8 to 0.9 in stable air, which narrows the gap without closing it.

So use it for what it is. It is a screening number, it is conservative in the direction that matters for a permit, it needs no stability class, and it can be done on the back of an envelope in a plant corridor. It is not a regulatory answer, and no agency that requires a dispersion model will accept it in place of the Briggs treatment inside AERMOD. Watch the downwash condition as well: when the exit velocity is less than about 1.5 times the wind speed, the plume is drawn into the low-pressure wake behind the stack itself and any calculated rise is optimistic, whichever equation produced it.