Plume Rise Formulas
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Unstable or Neutral - Crossover Between Momentum and Buoyancy
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Plume Rise When Schulman and Scire Buiding Downwash is selected
The plume height is used in the calculation of the Vertical Term described in Section 1.1.6. The Briggs plume rise equations are discussed below. The description follows Appendix B of the Addendum to the MPTER User's Guide (Chico and Catalano, 1986) for plumes unaffected by building wakes. The distance dependent momentum plume rise equations, as described in (Bowers, et al., 1979), are used to determine if the plume is affected by the wake region for building downwash calculations. These plume rise calculations for wake determination are made assuming no stack-tip downwash for both the Huber-Snyder and the Schulman-Scire methods. When the model executes the building downwash methods of Schulman and Scire, the reduced plume rise suggestions of Schulman and Scire (1980) are used, as described in Section 1.1.4.11.
In order to consider stack-tip downwash, modification of the physical stack height is performed following Briggs (1974, p. 4). The modified physical stack height hs´ is found from:
where hs is physical stack height (m), vs is stack gas exit velocity (m/s), and ds is inside stack top diameter (m). This hs´ is used throughout the remainder of the plume height computation. If stack tip downwash is not considered, hs´ = hs in the following equations.
For most plume rise situations, the value of the Briggs buoyancy flux parameter, Fb (m4/s3), is needed. The following equation is equivalent to Equation (12), (Briggs, 1975, p. 63):
where )T = Ts - Ta, Ts is stack gas temperature (K), and Ta is ambient air temperature (K).
For determining plume rise due to the momentum of the plume, the momentum flux parameter, Fm (m4/s2), is calculated based on the following formula:
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, ()T)c, is determined by setting Briggs' (1969, p. 59) Equation 5.2 equal to the combination of Briggs' (1971, p. 1031) Equations 6 and 7, and solving for )T, as follows:
for Fb <
55,
and for Fb $ 55,
If the difference between stack gas and ambient temperature, )T, exceeds or equals ()T)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 )T exceeds ()T)c as determined above, buoyancy is assumed to dominate. The distance to final rise, xf, is determined from the equivalent of Equation (7), (Briggs, 1971, p. 1031), and the distance to final rise 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:
and for Fb $ 55:
The final effective plume height, he (m), is determined from the equivalent of the combination of Equations (6) and (7) (Briggs, 1971, p. 1031):
for Fb < 55:
and for Fb $ 55:
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 )T is less than ()T)c from Equation (1-10) or (1-11), the assumption is also made that the plume rise is dominated by momentum. The plume height is calculated from Equation (5.2) (Briggs, 1969, p. 59):
Briggs (1969, p. 59) suggests that this equation is most applicable when vs/us is greater than 4.
For stable situations, the stability parameter, s, is calculated from the Equation (Briggs, 1971, p. 1031):
As a default approximation, for stability class E (or 5) M2/Mz is taken as 0.020 K/m, and for class F (or 6), M2/Mz 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 crossover temperature difference, ()T)c , is determined by setting Briggs' (1975, p. 96) Equation 59 equal to Briggs' (1969, p. 59) Equation 4.28, and solving for )T, as follows:
If the difference between stack gas and ambient temperature, )T, exceeds or equals ()T)c, plume rise is assumed to be buoyancy dominated, otherwise plume rise is assumed to be momentum dominated.
For situations where )T exceeds ()T)c as determined above, buoyancy is assumed to dominate. The distance to final rise, xf, is determined by the equivalent of a combination of Equations (48) and (59) in Briggs, (1975), p. 96:
The plume height, he, is determined by the equivalent of Equation (59) (Briggs, 1975, p. 96):
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 )T is less than ()T)c as determined by Equation (1-18), the assumption is also made that the plume rise is dominated by momentum. The plume height is calculated from Equation 4.28 of Briggs ((1969), p. 59):
The equation for unstable-neutral momentum rise (1-16) is also evaluated. The lower result of these two equations is used as the resulting plume height, since stable plume rise should not exceed unstable-neutral plume rise.
All Conditions - Distance Less Than Distance to Final Rise.
Where gradual rise is to be estimated for unstable, neutral, or stable conditions, if the distance downwind from source to receptor, x, is less than the distance to final rise, the equivalent of Equation 2 of Briggs ((1972), p. 1030) is used to determine plume height:
*** MISSING EQUATION ***
This height will be used only for buoyancy dominated conditions; should it exceed the final rise for the appropriate condition, the final rise is substituted instead.
For momentum dominated conditions, the following equations (Bowers, et al, 1979) are used to calculate a distance dependent momentum plume rise:
where x is the downwind distance (meters), with a maximum value defined by xmax as follows:
where x is the downwind distance (meters), with a maximum value defined by xmax as follows:
The jet entrainment coefficient, $j, is given by,
As with the buoyant gradual rise, if the distance-dependent momentum rise exceeds the final rise for the appropriate condition, then the final rise is substituted instead.
Calculating the plume height for wake effects determination
The building downwash algorithms in the ISC models always require the calculation of a distance dependent momentum plume rise. When building downwash is being simulated, the equations described above are used to calculate a distance dependent momentum plume rise at a distance of two building heights downwind from the leeward edge of the building. However, stack-tip downwash is not used when performing this calculation (i.e. hs´ = hs). This wake plume height is compared to the wake height based on the good engineering practice (GEP) formula to determine whether the building wake effects apply to the plume for that hour.
The procedures used to account for the effects of building downwash are discussed more fully in Section 1.1.5.3. The plume rise calculations used with the Schulman-Scire algorithm are discussed in Section 1.1.4.11.
Plume Rise When Schulman and Scire Building Downwash is Selected
The Schulman-Scire downwash algorithms are used by the ISC models when the stack height is less than the building height plus one half of the lesser of the building height or width. When these criteria are met, the ISC models estimate plume rise during building downwash conditions following the suggestion of Scire and Schulman (1980). The plume rise during building downwash conditions is reduced due to the initial dilution of the plume with ambient air.
The plume rise is estimated as follows. The initial dimensions of the downwashed plume are approximated by a line source of length Ly and depth 2Ro where:
LB equals the minimum of hb and hw, where hb is the building height and hw the projected (crosswind) building width. A is a linear decay factor and is discussed in more detail in Section 1.1.5.3.2. If there is no enhancement of Fy or if the enhanced Fy is less than the enhanced Fz, the initial plume will be represented by a circle of radius Ro. The FUNC { SQRT 2} factor converts the Gaussian Fz to an equivalent uniform circular distribution and FUNC { SQRT {2B}} converts Fy to an equivalent uniform rectangular distribution. Both Fy and Fz are evaluated at x = 3LB, and are taken as the larger of the building enhanced sigmas and the sigmas obtained from the curves (see Section 1.1.5.3). The value of Fz used in the calculation of Ly also includes the linear decay term, A.
The rise of a downwashed finite line source was solved in the BLP model (Scire and Schulman, 1980). The neutral distance-dependent rise (Z) is given by:
The stable distance-dependent rise is calculated by:
with a maximum stable buoyant rise given by:
Fb= buoyancy flux term (Equation 1-8) (m4/s3)
Fm= momentum flux term (Equation 1-9) (m4/s2)
x = downwind distance (m)
us= wind speed at release height (m/s)
vs= stack exit velocity (m/s)
ds= stack diameter (m)
$= entrainment coefficient (=0.6)
$j= jet entrainment coefficient
FUNC {~=~{1 over 3} + {{u_s} over {v_s}}}
FUNC {~=~ g {{M theta}/Mz } over {T_a}}
The larger of momentum and buoyant rise, determined separately by alternately setting Fb or Fm = 0 and solving for Z, is selected for plume height calculations for Schulman-Scire downwash. In the ISC models, Z is determined by solving the cubic equation using Newton's method.
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