chapter 5 concentration models: diffusion model. diffusion model using the gaussian plume idea....

CHAPTER 5

Concentration Models: Diffusion Model

Diffusion model

• Using the Gaussian plume idea.• Consideration:– The point source is the chimney or

smoke stack.– One need to measure

concentration downwind form the point source

Plume of contaminated air

• The Gaussian Plume.Physical stack height = hThe plume rise = hEffective stack height, H = h + h

Figure A

Plume of contaminated air

• The Gaussian Plume.

Assumptions:• Wind blows in the x direction, with

velocity, u and emission rate, Q, andit is independent of time, location or elevation.

Figure A

• Through material balance around a cube of space near the center of the plume, and considering the dispersion due to turbulent mixing:

z

x

y

Diffusion Model – Gaussian Plume

• Gaussian puff, 3D spreading• Applicable to an instantaneous shot-

term release of pollutants from the chimney shown in previous figure, i.e. at x = y = 0 and z = H

z

2

y

2

x

2

2/1zyx

2/3 KHz

Ky

Kx

t41

expKKKt8

tQc

where• K = turbulent dispersion coefficient• x = the distance upwind or downwind from the

center of the moving puff• t = time since release• t = time duration of release

Diffusion Model – Gaussian Plume

• Gaussian plume, 2D spreading• Applicable to steady-state release of

plume.• Assume negligible net transfer of material

in the x direction

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z

2

y

2

2/1zy

2/3 KHz

Ky

t41

expKKt4

u/Qc

• The above equation is generally used by making the following substitutions:

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ux

t

5.0Kxu

5.0K

2zz

2yy

Where:y = horizontal dispersion coefficient z = vertical dispersion coefficient

Diffusion Model – Gaussian Plume

• Making the substitutions, we find:

• Basic 2D Gaussian Plume equation

2

z

2

2y

2

zy 2σHz

2σy

expσuσ 2π

Qc

2

z

2

2y

2

zy 2σHz

exp 2σy

expσuσ 2π

Qc

Example 5:

• A factory emits 20 g/s of SO2 at height H. The wind speed is 3 m/s. At a distance of 1 km downwind, the values of σy and σz are 30 and 20 m, respectively.

What are the SO2 concentrations at the centerline of the plume, and at a point 60 meters to the side and 20 meters below the centerline?

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Solution

• At centreline, y = 0 and z = H (refer Fig. A). Thus, at centreline:

• At the point away from the centreline,

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33 mg

1770mg

00177.0m20m30s/m32

s/g20c

3

22

mg

145m20m20

21

m30m60

21

expm20m30s/m32

s/g20c

Diffusion Model – Gaussian Plume

• The basic Gaussian plume equation predicts a plume that is symmetrical with respect to y and with respect to z.

• Different values of σy and σz mean that spreading in the vertical and horizontal directions is not equal.

• To find the approximated values for σ y and σ z ,

Diffusion Model – Gaussian Plume

Surface Wind Speed

(at 10 m), m/s

Day Night

Incoming Solar radiation

Thinly overcast or 4/8 cloud

Clear or 3/8 cloud

StrongModerat

eSlight

0 – 2 A A – B B – –

2 – 3 A – B B C E F

3 – 5 B B – C D D E

5 – 6 C C – D D D D

6 C D D D D

Note: The neutral class D should be assumed for overcast conditions during day or night

• Horizontal dispersion coefficient

Horizontal dispersion coefficient, y, as a function of downwind distance from the source for various stability categories

• Vertical dispersion coefficient

Vertical dispersion coefficient, z, as a function of downwind distance from the source for various stability categories

Diffusion Model – Gaussian Plume

Some modifications The effect of the ground

• The ground damps out vertical dispersion and vertical spreading terminates at ground level.

• Commonly assumed that any pollutants that would have carried below z = 0 if the ground were not there; are ‘reflected’ upward as if the ground is a mirror

Diffusion Model – Gaussian Plume

Some modifications • Therefore:

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2

z

2

z

2

yzy

Hz5.0exp

Hz5.0exp

y5.0exp

u2Q

c

Example 6:

• If z = 10 m, repeat the calculation in Example 5 for the cases where H = 20 m and where H = 30 m.

Solution:

• H = 20 m

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3

22

2

mg

289

202010

5.0exp20

20105.0exp

3060

5.0exp203032

20c

Solution:

• H = 30 m

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3

22

2

mg

178

203010

5.0exp20

30105.0exp

3060

5.0exp203032

20c

Example 7

A large, poorly controlled copper smelter has a stack 150 m high and a plume rise of 75 m. It is currently emitting 1000 g/s SO2. Estimate the ground level concentration of SO2 from this source at a distance 5 km directly downwind when the wind speed is 3 m/s and the stability class is C.

Solution

• Q = 1000 g/s• u = 3 m/s• y = 438 m – from Figure 1

• z = 264 m – from Figure 2

• y = h + h = 225 m

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2

z

2

2y

2

zy σ2Hz

expσ2y

exp σ σ u2

Qc

Diffusion Model – Gaussian Plume Ground level concentration,

simplified• At ground level, z = 0.• Substituting into the previous equation:

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2

z

2

yzy

H5.0exp

y5.0exp

uQ

c

Diffusion Model – Gaussian Plume Ground level concentration,

simplified• At y = 0 and z = 0

correspond to the line on the ground directly under the centerline of the plume

• Rearrange:

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2

zzy

H5.0exp

1Qcu

2

zzy

H5.0exp

uQ

c

Diffusion Model – Gaussian Plume Ground level concentration,

simplified• We can plot a graph of cu/Q vs. distance x.

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2

zzy

H5.0exp

1Qcu

Ground-level , directly under the plume centreline, as a function of downwind distance from the source an effective stack height, H, in meters, for stability Class C only

Example 8

A plant is emitting 750 g/s of particulates. The stack height is 100 m and the plume rise is 50 m. The wind speed is 7 m/s and the stability category is C.a) What is the maximum estimated

ground-level concentration ? b) How far downwind it does occur?

Plume Rise

• Figure below shows the plume rising a distance h, called the plume rise, above the top of the stack before leveling out.

Plume Rise

• Plumes rise buoyantly because they are hotter than the surrounding air and also because they exit the stack with a vertical velocity that carries them upward.

Plume Rise

• They stop rising because:(i) they mix with surrounding air(ii) they lose velocity(iii) they cool by mixing

• To estimate h, Holland’s formula is:

where h = plume rise, mVs = stack exit velocity, m/sD = stack diameter, mu = wind speed, m/sP = pressure, mbarTs = stack gas temperature, KTa = atmospheric temperature, K

s

as3s

TTT

PD10x68.25.1uDV

h

Plume Rise

• Estimate the plume rise for a 3 m diameter stack whose exit gas has a velocity of 20 m/s when the wind velocity is 2 m/s, the pressure is 1 atm, and the stack and surrounding temperatures are 100oC and 15oC, respectively.

• Solution:

m

x x x x

101373

288373310131068.25.1

2320

h 3

Example

End of Lecture