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Algorithms II · atmospheric pollution

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 h_s is found from:

h_s’=h_s+2d_s[(v_s/u_s)-1.5]   for  v_s<1.5u_s      (5)

or

h_s’=h_s                   for v_s> o =1.5u_s   (6)

where h_s is physical stack height (m), v_s is stack gas exit velocity (m/s), and d_s is stack top diameter (m). If stack tip downwash is not considered, h_s’= h_s in the following equations.

Buoyancy and Momentum Fluxes.

For most plume rise situations, the value of the Briggs buoyancy flux parameter, F_b (m⁴/s³), is needed

F_b=gv_sd_s²(DT/4T_s)    (7)

where DT = T_s  - T_a, T_s is stack gas temperature (K), and T_a is ambient air temperature (K).

For determining plume rise, the momentum flux parameter, F_m (m⁴/s²), is calculated based on the following formula:

F_m=gv_s²d_s²(T_a/4T_s)    (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 F_b < 55,

(DT)_c=0.0297 T_s(v_s/d_s²)^1/3      (9)

and for F_b >= 55,

(DT)_c=0.00575 T_s(v_s²/d_s)^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, x_f, is assumed to be 3.5x^*, where x^* is the distance at which atmospheric turbulence begins to dominate entrainment. The value of x_f is calculated as follows:

for F_b < 55:

x_f=49F_b^5/8      (11)

and for F_b >= 55:

x_f=119F_b^2/5     (12)

The final effective plume height, h_e (m), is determined as

for F_b < 55:

h_e=h_s+(21.425 F_b^3/4/u_s)    (13)

and for F_b = 55:

h_e=h_s+(38.71 F_b^3/5/u_s)          (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:

h_e=h_s+3d_s(v_s/u_s)     (15)

Briggs suggests that this equation is most applicable when v_s/u_s is greater than 4.

Stability Parameter.

For stable situations, the stability parameter, s, is calculated:

s=g[(dT/dz)/T_a]         (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 T_s v_s s^1/2      (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 x_f is determined by

x_f=2.0715 u_s s^-1/2        (18)

The plume height, h_e, is determined by

h_e=h_s+2.6 [F_b/(u_ss)]^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,

h_e=h_s+1.5[F_m/(u_ss^1/2)]^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.