Cases 1 through 10 above all depend on the specification of a value for

the eddy diffusivity, Kj. In general, Kj changes with position, time, wind

velocity, and prevailing weather conditions. While the eddy diffusivity

approach is useful theoretically, it is not convenient experimentally and

does not provide a useful framework for correlation.

Sutton solved this difficulty by proposing the following definition for

a dispersion coefficient.

(37)

with similar relations given for y and z. The dispersion coefficients,

x, y, and z represent the standard deviations of the concentration in the

downwind, crosswind and vertical (x, y, z) direction, respectively.

nx utC 222

2

1

A plume dispersing in a normal distribution along two axes - distance crosswind and distance vertically

Values for the dispersion coefficients are much easier to obtain

experimentally than eddy diffusivities.

The dispersion coefficients are a function of atmospheric conditions

and the distance downwind from the release. The atmospheric conditions

are classified according to 6 different stability classes shown in Table

2. The stability classes depend on wind speed and quantity of sunlight.

During the day, increased wind speed results in greater atmospheric

stability, while at night the reverse is true. This is due to a change in

vertical temperature profiles from day to night.

The dispersion coefficients, y and z for a continuous source were

developed by Gifford and given in Figures 10 and 11, with the

corresponding correlation given in Table 3. Values for x are not provided

since it is reasonable to assume x = y. The dispersion coefficients y and

z for a puff release are given in Figures 12 and 13. The puff dispersion

coefficients are based on limited data (shown in Table 3) and should not be

considered precise.

The equations for Cases 1 through 10 were rederived by Pasquill using

relations of the form of Equation 37. These equations, along with the

correlation for the dispersion coefficients are known as the Pasquill-

Gifford model.

Table 2 Atmospheric Stability Classes for Use with the Pasquill-Gifford Dispersion Model

Day radiation intensity Night cloud coverWind

speed (m/s) Strong Medium Slight CloudyCalm &

clear

< 2 A A – B B

2 – 3 A – B B C E E

3 – 5 B B – C C D E

5 – 6 C C – D D D D

> 6 C D C D D

Stability class for puff model :

A,B : unstable

C,D : neutral

E,F : stable

Figure 10 Horizontal dispersion coefficient for Pasquill-Gifford plume

model. The dispersion coefficient is a function of distance downwind and

the atmospheric stability class.

Figure 11 Vertical dispersion coefficient for Pasquill-Gifford plume

model. The dispersion coefficient is a function of distance downwind and

the atmospheric stability class.

Figure 12 Horizontal dispersion coefficient for puff model. This data is

based only on the data points shown and should not be considered reliable

at other distances.

Figure 13 Vertical dispersion coefficient for puff model. This data is

based only on the data points shown and should not be considered reliable

at other distances.

Table 3 Equations and data for Pasquill-Gifford Dispersion Coefficients

Equations for continuous plumes

Stability class y (m)

A y = 0.493x0.88

B y = 0.337x0.88

C y = 0.195x0.90

D y = 0.128x0.90

E y = 0.091x0.91

F y = 0.067x0.90

Stabilityclass

x (m) z (m)

A100 – 300

300 – 3000Z = 0.087x0.88

log10z = -1.67 + 0.902 log10x + 0.181(log10x)²

B100 – 500

500 – 2 × 104Z = 0.135x0.95

log10z = -1.25 + 1.09 log10x + 0.0018(log10x)²

C 100 – 105 Z = 0.112x0.91

D100 – 500500 – 105

Z = 0.093x0.85

log10z = -1.22 + 1.08 log10x - 0.061(log10x)²

E100 – 500500 – 105

Z = 0.082x0.82

log10z = -1.19 + 1.04 log10x - 0.070(log10x)²

F100 – 500500 – 105

Z = 0.057x0.80

log10z = -1.91 + 1.37 log10x - 0.119(log10x)²

Data for puff releases

x = 100 m x = 4000 mStabilitycondition

y (m) z (m) y (m) z (m)

Unstable 10 15 300 220

Neutral 4 3.8 120 50

Very stable 1.3 0.75 35 7

This case is identical to Case 7. The solution has a form similar to Equation 33.

(38)

The ground level concentration is given at z = 0.

(39)

2

2

2

22

23

*

2

1exp

2,,,

zyxzyx

m zyutxQtzyxC

2

22

23

*

2

1exp

2,0,,

yxzyx

m yutxQtyxC

The ground level concentration along the x-axis is given at y = z= 0.

(40)

The centre of the cloud is found at coordinates (ut,0,0). The concentration at the centre of this moving cloud is given by

(41)

The total integrated dose, Dtid received by an individual standing at

fixed coordinates (x,y,z) is the time integral of the concentration.

(42) dttzyxCzyxD ,,,,,0tid

2

23

*

2

1exp

2,0,0,

xzyx

m utxQtxC

zyx

mQtutC

23

*

2,0,0,

The total integrated dose at ground level is found by integrating

Equation 39 according to Equation 42. The result is -

(43)

The total integrated dose along the x-axis on the ground is

(44)

Frequently the cloud boundary defined by a fixed concentration is

required. The line connecting points of equal concentration around the

cloud boundary is called an isopleth.

2

2*

tid 2

1exp0,,

yzy

m y

u

QyxD

u

QxD

zy

m

*

tid 0,0,

This case is identical to Case 9. The solution has a form similar to

Equation 35.

(46)

The ground level concentration is given at z = 0.

(47)

2

2

2

2

2

1exp,,

zyzy

zy

u

QzyxC

2

2

1exp0,,

yzy

y

u

QyxC

The concentration along the centreline of the plume directly

downwind is given at y = z= 0.

(48)

The isopleths are found using a procedure identical to the isopleth

procedure used for Case 1.

For continuous ground level releases the maximum concentration

occurs at the release point.

u

QxC

zy0,0,

This case is identical to Case 10. The solution has a form similar to

Equation 36.

(49)

2

2

2

1exp

2

1exp

2

1exp

2,,

z

r

z

r

yzy

m

HzHz

y

u

QzyxC

The ground level concentration is found by setting z = 0.

(50)

The ground centreline concentrations are found by setting y = z= 0.

(51)

22

2

1

2

1exp

20,,

z

r

yzy

m Hy

u

QyxC

2

2

1exp0,0,

z

r

zy

m H

u

QxC

The maximum ground level concentration along the x-axis, <C>max,

is found using.

(52)

The distance downwind at which the maximum ground level

concentration occurs is found from

(53)

The procedure for finding the maximum concentration and the

downwind distance is to use Equation 53 to determine the distance

followed by Equation 52 to determine the maximum concentration.

y

z

r

m

uHe

QC

2max

2

2r

z

H

For this case the centre of the puff is found at x = ut. The average

concentration is given by

(54)

22

2

23

2

1exp

2

1exp

2

1exp

2,,,

z

r

z

r

yzyx

m

HzHz

yQtzyxC

The time dependence is achieved through the dispersion coefficients,

since their values change as the puff moves downwind from the release

point. If wind is absent (u = 0), Equation 54 will not predict the correct

result.

At ground level, z = 0, and the concentration is computed using

(55)

22

23

*

2

1

2

1exp

2,0,,

z

r

yzyx

m HyQtyxC

The concentration along the ground at the centreline is given at any

y = z = 0,

(56)

The total integrated dose at ground level is found by application of

Equation 42 to Equation 55. The result is

(57)

2

23

*

2

1exp

2,0,0,

z

r

zyx

m HQtxC

22*

tid 2

1

2

1exp0,,

z

r

yzy

m Hy

u

QyxD

For this case, the result is obtained using a transformation of

coordinates similar to the transformation used for Case 7. The result is

(58)

where t is the time since the release of the puff.

22

2

1

2

1exp

)

P(,,,

56 through 54 Equations system,

coordinate moving with equations uff

z

r

y

Hy

tzyxC

The plume model describes the steady state behaviour of material

ejected from a continuous source. The puff model is not steady-state and

follows the cloud of material as it moves with the wind. As a result, only

the puff model is capable of providing a time dependence for the release.

The puff model is also used for continuous releases by representing the

release as a succession of puffs. For leaks from pipes and vessels, if tp is

the time to form one puff, then the number of puffs formed, n, is given by

(59)

pt

tn

where t is the duration of the spill. The time to form one puff, tp, is

determined by defining an effective leak height, Heff. Then,

(60)

where u is the wind speed. Empirical results show that the best Heff to

use is

(61)

For a continuous leak,

(62)

u

Ht p

eff

5.1leak ofheight eff H

pmm tQQ *

and for instantaneous release divided into a number of smaller puffs,

(63)

where (Qm*)total is the release amount.

This approach works for liquid spills, but not for vapor releases. For

vapor releases a single puff is suggested.

The puff model is also used to represent changes in wind speed and

direction.

n

QQ m

mtotal

**

On an overcast day, a stack with an effective height of 60 meters is

releasing sulfur dioxide at the rate of 80 grams per second. The wind speed

is 6 meters per second. The stack is located in rural area .Determine:

a. The mean concentration of SO2 on the ground 500 meters downwind.

b. The mean concentration on the ground 500 meters downwind and 50

meters crosswind.

c. The location and value of the maximum mean concentration on ground

level directly downwind.

a. This is a continuous release. The ground concentration directly

downwind is given by Equation 51.

(51)

From Table 2, the stability class is D. the dispersion coefficients are

obtained from Figures 10 and 11. The resulting values are y = 39 meters

and z = 22.7 meters. Substituting into Equation 51

2

2

1exp0,0,

z

r

zy

m H

u

QxC

34

2

mgm1045.1

m 22.7

m 60

2

1exp

sm6m 22.7m 3914.3

sgm800,0,m 500

C

b. The mean concentration 50 meters crosswind is found using Equation

50 and setting y = 50. The results from part a are applied directly,

35

234

2

mgm1037.6

m 39

m 50

2

1expmgm1045.1

2

1exp0,0,m 005m,0 m,50 500

y

yCC

c. The location of the maximum concentration is found from Equation

53,

From Figure 11, the dispersion coefficient has this value at x = 1200 m. At

x = 1200 m, from Figure 10, y = 88 m. The maximum concentration is

determined using Equation 52,

m 42.42

m 60

2 r

z

H

34

2

2max

mgm104.18

m 88

m 42.4

m 60sm 63.142.72

sgm 802

2

y

z

r

m

uHe

QC