ME 433 Professor John M. Cimbala Lecture 16 Today, we will: • Modify the Gaussian solution for a buoyant plume, and do some examples/applications • Compare a ground that absorbs the pollutant vs. one that does not absorb (reflects) the

pollutant

Gaussian plume model (steady with no buoyancy and no ground effect):

U

z

x

Side view, origin at the source at the exit of the stack

hs

j ,sm

We had: 2 2

, 1exp2 2

j sj

y z y z

m y zcUπ σ σ σ σ

= − +

, and we use empirical relations for the

dispersion coefficients: by axσ = , d

z cx fσ = + , x in units of km and σy and σz in units of m. Coordinate transformation to make z = 0 on the ground:

We simply use z – hs in place of z in the above equation! We get: 2 2

, 1exp2 2

j s sj

y z y z

m y z hcUπ σ σ σ σ

− = − +

U

z

x

hs

j ,sm

Side view, origin at ground, directly under the stack and under the source

Modification for a buoyant plume:

• Most plumes from smoke stacks are hot, and rise before they level off. • Notation: Let δh = the additional elevation (above the stack exit) due to plume buoyancy. • Thus, the total plume height, measured from the ground (once the plume levels off), is the

sum of stack height hs and this additional buoyancy height δh. We define H = hs + δh, where H is the effective stack height of the plume, accounting for buoyancy, as sketched.

hs z

j ,sm

U

x

Side view, origin at ground, under the apparent source

Actual plume centerline

Gaussian model plume centerline

δh

H

Apparent source

Coordinate transformation to keep z = 0 on the ground, and move the source up as shown:

We simply use z – H in place of z – hs in the above equation! We get:

Modified solution for a buoyant plume: 2 2

, 1exp2 2

j sj

y z y z

m y z HcUπ σ σ σ σ

− = − +

Note: The above equation is for the case of a ground that absorbs the pollutant. We call this an absorbing ground.

z x

j ,sm

U

H

Example: Atmospheric stability classification, Martin Model Given: It is a clear summer day at noon. The average wind speed is 6 m/s. To do:

(a) What stability class (A, B, C, D…) would we use for the Gaussian plume model? (b) What value of constant c should we use at x = 2.0 km to determine the dispersion

coefficient?

Solution: Dispersion coefficients: Tables scanned from Cooper, C. D. and Alley, F. C. Air Pollution Control: A Design Approach, Edition 4, Waveland Press, Inc., Long Grove, IL, 2011, pp. 662-663.

Ground effects (reflecting vs. absorbing ground): Summary of Gaussian plume model: (where sH h hδ= + = effective stack height) With ground absorption,

2 2, 1exp

2 2j s

jy z y z

m y z HcUπ σ σ σ σ

− = − +

(4)

With ground reflection, 2 22 2

, 1 1exp exp2 2 2

j sj

y z y z y z

m y z H y z HcUπ σ σ σ σ σ σ

− + = − + + − +

(5)

Example: Gaussian plume dispersion coefficients Given: A plant emits air pollution from a stack under the following conditions:

z x

j ,sm

U

H

• daytime with moderate solar radiation (summer day with a few clouds) • average wind speed = 4.0 m/s

To do: (a) What stability class is this? (b) What value of constant d should we use at x = 2.0 km to determine the dispersion

coefficient?

Solution: (a) Use Table 1 to determine the stability class (see table below).

(Continued) Example: Gaussian plume dispersion coefficients Given: A plant emits air pollution from a stack under the following conditions:

z x

j ,sm

U

H

• daytime with moderate solar radiation (summer day with a few clouds) • average wind speed = 4.0 m/s

To do: (a) What stability class is this? (b) What value of constant d should we use at x = 2.0 km to determine the dispersion

coefficient?

Solution: (b) Use Table 2 to determine constant d at x = 2.0 km (see table below).