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Plume Rise Formulas
The plume height is used in the calculation of the Vertical Term. The distance dependent momentum plume rise equations are used to determine if the plume is affected by the wake region for building downwash calculations.
Stack-tip Downwash.
In order to consider stack-tip downwash, modification of the physical stack height is performed. The modified physical stack height hs is found from:
hs’=hs+2ds[(vs/us)-1.5] for vs<1.5us (5)
or
hs’=hs for vs> o =1.5us (6)
where hs is physical stack height (m), vs is stack gas exit velocity (m/s), and ds is stack top diameter (m). If stack tip downwash is not considered, hs’= hs in the following equations.
Buoyancy and Momentum Fluxes.
For most plume rise situations, the value of the Briggs buoyancy flux parameter, Fb (m4/s3), is needed
Fb=gvsds2(DT/4Ts) (7)
where DT = Ts - Ta, Ts is stack gas temperature (K), and Ta is ambient air temperature (K).
For determining plume rise, the momentum flux parameter, Fm (m4/s2), is calculated based on the following formula:
Fm=gvs2ds2(Ta/4Ts) (8)
Unstable or Neutral - Crossover Between Momentum and Buoyancy.
For cases with stack gas temperature greater than or equal to ambient temperature, it must be determined whether the plume rise is dominated by momentum or buoyancy. The crossover temperature difference, (DT)c, is determined as follows:
for Fb < 55,
(DT)c=0.0297 Ts(vs/ds2)1/3 (9)
and for Fb >= 55,
(DT)c=0.00575 Ts(vs2/ds)1/3 (10)
If DT, exceeds or equals (DT)c, plume rise is assumed to be buoyancy dominated, otherwise plume rise is assumed to be momentum dominated.
Unstable or Neutral - Buoyancy Rise.
For situations where DT exceeds (DT)c as determined above, buoyancy is assumed to dominate. The distance to final rise, xf, is assumed to be 3.5x*, where x* is the distance at which atmospheric turbulence begins to dominate entrainment. The value of xf is calculated as follows:
for Fb < 55:
xf=49Fb5/8 (11)
and for Fb >= 55:
xf=119Fb2/5 (12)
The final effective plume height, he (m), is determined as
for Fb < 55:
he=hs+(21.425 Fb3/4/us) (13)
and for Fb = 55:
he=hs+(38.71 Fb3/5/us) (14)
Unstable or Neutral - Momentum Rise.
For situations where the stack gas temperature is less than or equal to the ambient air temperature, the assumption is made that the plume rise is dominated by momentum. If DT is less than (DT)c, the assumption is also made that the plume rise is dominated by momentum. The plume height is calculated as:
he=hs+3ds(vs/us) (15)
Briggs suggests that this equation is most applicable when vs/us is greater than 4.
Stability Parameter.
For stable situations, the stability parameter, s, is calculated:
s=g[(dT/dz)/Ta] (16)
As a default approximation, for stability class E (or 5) dT/dz is taken as 0.020 K/m, and for class F (or 6), dT/dz is taken as 0.035 K/m.
Stable - Crossover Between Momentum and Buoyancy.
For cases with stack gas temperature greater than or equal to ambient temperature, it must be determined whether the plume rise is dominated by momentum or buoyancy. The (DT)c is determined and solving for DT, as follows:
(DT)c=0.019582 Ts vs s1/2 (17)
If the difference between DT exceeds or equals (DT)c, plume rise is assumed to be buoyancy dominated, otherwise plume rise is assumed to be momentum dominated.
Stable - Buoyancy Rise.
For situations where DT exceeds (DT)c as determined above, buoyancy is assumed to dominate. The distance xf is determined by
xf=2.0715 us s-1/2 (18)
The plume height, he, is determined by
he=hs+2.6 [Fb/(uss)]1/3 (19)
Stable - Momentum Rise.
Where the stack gas temperature is less than or equal to the ambient air temperature, the assumption is made that the plume rise is dominated by momentum. Then,
he=hs+1.5[Fm/(uss1/2)]1/3 (20)
The equation for unstable-neutral momentum rise is also evaluated. The lower result of these two equations is used as the resulting plume height.
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