Briggs Buoyancy Flux

Buoyancy flux is the single number that says how hard a plume wants to rise. It is not a measured quantity, it is a constructed one, and its units of m⁴/s³ are the giveaway: this is a volume flow multiplied by a reduced gravity. Written out, $F = g \, v_s r^2 (T_s - T_a)/T_s$ with the stack RADIUS, which becomes the familiar $g v_s d^2 (T_s-T_a)/(4T_s)$ once you substitute $r = d/2$. A stack 2 m across, discharging at 20 m/s and 400 K into air at 280 K, has $F = 9.80665 \times 20 \times 4 \times 120/(4 \times 400) = 94144/1600 = 58.8$ m⁴/s³, which is a mid-sized industrial boiler.

The temperature ratio is doing something specific and it is worth seeing why. At constant pressure the ideal gas law makes density inversely proportional to absolute temperature, so the fractional density deficit of the hot gas relative to ambient is $(\rho_a - \rho_s)/\rho_a = (T_s - T_a)/T_s$. That fraction multiplied by g is the reduced gravity acting on the plume, which is why the temperature difference has to sit over an absolute temperature and never over a Celsius one. Which absolute temperature belongs in the denominator is a genuine small ambiguity in the literature: Briggs and the EPA formulations use $T_s$, several textbooks use $T_a$, and for a stack at 400 K against air at 280 K the two differ by 43 percent in the denominator but only about 12 percent in the final plume rise, since F enters the rise as a cube root. Pick one, say which one, and do not mix sources.

The value of F sorts a stack into a regime. Briggs' own break point is 55 m⁴/s³, below which the distance to final rise is $x^* = 14F^{5/8}$ and above which it is $34F^{2/5}$. Small F also means the plume may be momentum-dominated rather than buoyancy-dominated, in which case the rise comes from a different equation entirely and this parameter is the wrong tool. And F can be negative. A wet scrubber leaves its exhaust saturated and close to ambient temperature, sometimes below it, and a negative buoyancy flux means a plume that sinks out of the stack rather than rising from it. That is precisely why scrubbed stacks so often carry a reheat burner: the fuel is buying stack height that steel would cost far more to provide.

The errors here are mechanical rather than conceptual. Using the stack cross-sectional AREA where the equation wants the diameter squared costs a factor of $\pi/4$. Using Celsius anywhere in the expression is fatal. And the exit velocity has to be the ACTUAL velocity at stack conditions, not one back-calculated from a flow rate quoted at standard conditions: a flue gas at 400 K occupies about 1.4 times the volume it does at 293 K, so using the standard-condition figure understates both the velocity and the flux by that much.