A plume leaves a stack hot and moving, and it keeps climbing. The metres it gains are added to the stack's own height and dispersion starts from the sum — and because ground-level concentration falls with the SQUARE of that sum, plume rise is the cheapest height in the industry.

The modern route is Briggs, in two steps. First the buoyancy flux $F = \dfrac{g\,v_s\,d^{2}(T_s - T_a)}{4\,T_s}$ — F equals g v-s d squared, T-s minus T-a, over four T-s. Subscript s is the stack gas and subscript a the ambient air, both temperatures absolute, in kelvin; $v_s$ is the exit velocity in m/s and $d$ the inside diameter in metres. $F$ comes out in m⁴/s³ and is not a length, a speed, or anything you can picture — it is the plume's lifting power in one number. Note that the difference on top would survive a Celsius slip and the $T_s$ underneath would not, which is exactly why that slip is so quiet.

Then the two-thirds law, $\Delta h = \dfrac{1.6\,F^{1/3} x^{2/3}}{u}$ — delta-h equals one point six, F to the one-third, x to the two-thirds, over u. $x$ is the downwind distance in metres, $u$ the wind at stack top in m/s, and $\Delta h$ the rise in metres. Read the exponents: buoyancy enters WEAKLY at one-third, distance strongly at two-thirds, and the wind divides — which is why the calm hours give the worst ground-level numbers, not the windy ones.

Be honest about what that equation is: transitional rise, valid until the plume levels off. Briggs puts final rise near $3.5x^*$, with $x^*$ about $14F^{5/8}$ below F = 55 m⁴/s³ and $34F^{2/5}$ above it — past that distance the plume has stopped climbing and this form over-predicts. A paper that asks for the value at 1 km wants the value at 1 km; a real assessment caps it.

Holland's 1953 equation is the screening alternative, doing momentum and buoyancy in one pass: $\Delta h = \dfrac{v_s d}{u}\left(1.5 + 2.68\times10^{-3} P d\,\dfrac{T_s - T_a}{T_s}\right)$. The 1.5 is the momentum term, which a cold plume still gets; the second term is buoyancy. $P$ is atmospheric pressure and Holland's constant was written for millibars — feed it pascals and the plume goes to the stratosphere. Holland is generally conservative, which is its whole value: if the screening number already clears the limit, the fuller treatment will only be kinder.

Either way it ends the same: $H = h_s + \Delta h$, the effective stack height, where $h_s$ is the steel and $H$ is what every dispersion equation actually uses.