Air pollution dispersion modelling_P. Goyal _Centre for Atmospheric Sciences Indian Institute of Technology, Delhi

1. AAiirr PPoolllluuttiioonn DDiissppeerrssiioonn
MMooddeelllliinngg
PP.. GGooyyaall
CCeennttrree ffoorr AAttmmoosspphheerriicc SScciieenncceess
IInnddiiaann IInnssttiittuuttee ooff TTeecchhnnoollooggyy,, DDeellhhii..

2. Introduction
 Air quality modelling is a numerical methodology, based upon
physical principles, for estimating pollutant concentrations in
space and time as a function of the emissions distribution and the
meteorological and geophysical conditions.
 One of the approaches to the mathematical description of
turbulent diffusion is the gradient theory (or K-theory) which
involves the solution of the basic diffusion equation with
appropriate boundary conditions (B.Cs).

3. AAddvveeccttiioonn DDiiffffuussiioonn EEqquuaattiioonn
The advection diffusion equation is: (symbols have the usual meanings)
(1)
Source Sink
K c
ù
é
¶
æ
ö ¶
çè
ù
+ ¶ úû
é
ö
÷ ÷ø
ç çè æ
¶
ù
+ ¶ úû
¶
æ
ö ¶
çè
úû
+ + x y z z
K c
y z
K c
÷ø
x y
u c
v c
x
w c
y
¶
= - ¶
¶
- ¶
¶
- ¶
¶
+ ¶
¶
z x
C
t
êë
÷ø
¶
êë
¶
¶
é
êë
¶
This equation can be obtained by using the principle of conservation of mass
(Wark & Warner, 1981).
Assumptions for Equation (1) :
a) Steady State : This is valid for air quality models in which the emissions are
generally steady and hourly mean concentrations are computed.
b) Advection Terms : We take x axis along the wind direction and assuming
horizontal homogeneity in the wind speed, we can ignore the lateral advection
term. The assumption is justifiable because we take hourly mean values of the
wind velocity vector for computation of hourly mean concentrations.

4. AAddvveeccttiioonn DDiiffffuussiioonn EEqquuaattiioonn……..
The vertical advection term can be ignored because is at least 2
order lesser than the horizontal components of the wind velocity. We
are not assuming calm wind conditions in which this may not be valid.
c) Eddy Diffusivity : Longitudinal contributions is ignored if we
compare its magnitude with the advection term . This is valid for
plume model in which the time for the plume to reach the receptor is
less than the emission time.
The equation reduces to the following :
W
u ¶
c
¶
x
ö
÷ ÷ø
æ
u C y Z ¶
K C
ç çè
ö
+ ÷ ÷ø
æ
K ¶
C
ç çè
=
¶
¶
2
2
2
2
z
Y
x
¶
¶
(2)

5. AAddvveeccttiioonn DDiiffffuussiioonn EEqquuaattiioonn……..
Boundary conditions
(i) The flux per unit time across a plane normal to the wind condition
(i.e. II yz plane) is equal to the emission rate Q (source located at
(0, 0, h)) :
C 0, y, z =Qd d -
Removal mechanism in this model has been ignored. h is the height
of the source, which is taken zero in the solution.
(ii)
(iii)
(iv)
( ) ( y) ( z h)
u
¥ ¥
ò ò
-¥
=
0
uC(x, y, z)dydz Q
C®0as y®±¥
c
¶ at z
z
=0, =0
¶
C®0as z®¥

6. Solution :
(3)
AAddvveeccttiioonn DDiiffffuussiioonn EEqquuaattiioonn……..
C x y z Q
Where r2 = x2 + y2 + z2
ù
é
ö
æ
ù
é
z
y
u
( ) ú ú
= - +
1 4
s 2 =
K u z
z 2
Notice that there is an inverse decrease of concentration with x along axis
of the plume (y = z = 0).
The scale and which shows that concentration pattern follows normal
distribution law.
2 2
y z
C x y z Q
( , , ) exp 1
= - +
On Substitution in (3), we obtain:
û
ê ê ë
÷ ÷
ø
ç ç
è
ú úû
ê êë
y z
y z
k
k
x
r K K
2 2
2
exp
2
( , , )
p
ù
ö
ú ú
û
é
ê ê
ë
ù
÷ ÷
ø
æ
ç ç
è
ú úû
é
ê êë
y z y z
u
pss s s
2
x
(4)

7. AAddvveeccttiioonn DDiiffffuussiioonn EEqquuaattiioonn……..
If h ≠ 0, i.e. effective stack height is H, then solution (4) with reflection becomes :
(5)
ù
é + - + úû
é - -
ù
é
ö
æ
y z H z H
exp ( )
exp ( )
C x y z Q
= - 2
which is suitable for gaseous release from a stack.
For particulate release, we use the solution without reflection:
(6)
( )
ïþ
ïý ü
ïî ïí ì
úû
êë
ù
êë
÷ ÷ ø
ú úû
ê êë
ç ç
è
2
2
2
2
2
2
2
2
exp
2
, ,
y z y z z
u
p s s s s s
C x y z Q
( )
ïþ
ïý ü
ù
ïî
ïí ì
ù
úû
é - -
y z H
exp ( )
= - 2
êë
ö
ú úû
é
ê êë
÷ ÷
ø
æ
ç ç
è
2
2
2
2
2
exp
2
, ,
y z y z
u
pss s s

8. Puff Solution
Once the standard deviation (σ) of the distribution of material in a puff is known,
the concentration (c) of the material can be calculated by the Gaussian formula :
(7)
ö
Q r
ç çè æ
2
C p
( ) ÷ ÷ø
= exp
- 2
3 2
2p 2 s 3
s
where Qp = emissions (in mass per second)
Diffusion is assumed to be isotropic;
r is the radial distance from the puff centre.
When Diffusion is not Isotropic
The concentration, C of a pollutant at x, y, z from an instantaneous puff released
with an effective emission height, H, is given by the following equation :
ïý ü
ïî
ïí ì
ù
úû
æ - = - 2
é + - + úû
C x y z H Q x ut y z H z H
êë
ù
é - -
exp ( )
êë
ù
ú úû
é
ê êë
ö
÷ ÷
æ
-
ù
é
ö
æ
ö
exp 1
3 2
p s s s s s s s
( ) ïþ
ø
ç ç
è
ú ú
û
ê ê
ë
÷ ÷
ø
ç ç
è
÷ ÷ø
ç çè
2
2
2
2
2 2
2
exp ( )
2
2
exp
2
2
( , , , )
x y z z
x y z
(8)

9. Puff Solution…..
Diffusion parameters are functions of travel time t rather than of downwind
distance.
Following the puff and assuming σx equals σy expressed as σr where
r = (x-ut)2+y2, the puff equation can be written as follows :
(9)
ïý ü
ïî
ïí ì
ù
úû
é + - + úû
2
2
C r Z H Q r z H z H
exp ( )
exp ( )
= - 2
êë
ù
é - -
êë
ù
ú úû
é
ê êë
ö
÷ ÷ø
ç çè æ
2
exp
3 2
2
2
p s s 2
s 2
s s
ïþ
( 2
) 2
2
( , , )
r z z
r z
When σz becomes larger than eight tenths of the mixed layer depth, L, the puff is
assumed to be well mixed and the concentration equation is expressed as
(10)
Where
2
ù
é
ö
æ
C r Z H Q z
÷ ÷ø
ç çè
= - s
3 >
( , , ) 2
p s s
( )
r for L
L
r
2
r
.8
2
exp
2
2
ú úû
ê êë
( ) 2 4 3
F buoyancy flux parameter g T T S S
= = - S a
d V m s-
T
4
S
b
Dq
Dz
g = 9.8 m s-2 and the potential temperature gradient is expressed in
0°K / 100 m

10. Line Source
In some situations, such as a series of industries located along a river or
harbour, or heavy traffic along a straight stretch of highway, the pollution
problem may be modelled as a continuous emitting infinite line source. When
the wind direction is normal to the line of emission, the ground level
concentration downwind is given by :
(11)
¥
ïý ü
ïî
ïí ì
þ ý ü
2
2
h y dy
C Q
= - 2
u
ò
-
p ss s s
2
exp
¥ - ïþ
î í ì
2
2
exp
y Z Z y
Obtained from (3). Here Q is the source strength in gm/s. m. This is reduced
to :
(12)
C x Q
( )
( ) þ ý ü
î í ì
2
h
= exp
- 2
1 2
p s s
2
2
,0
Z
Z
u
Notice that there is no dependence of the solution (12) on y because the
concentration should be uniform in the y – direction at a given x – direction.

11. Line Source…
If the line source is small and perpendicular to the wind direction, then it is
convenient to define the x-axis in the direction of the wind and also passing
through the sampling point downwind. The ends of the line source then are at
two positions in the crosswind direction, y1 and y2, where y1 is less than y2.
The concentration along the x-axis at the ground level is given by the
expression :
(13)
C x Q
( )
p
2
ò þ ý ü
2
h ( p )dp
î í ì
= -
1
1 exp .5
2 1
p s 2
s p
exp
( ) u
Z p
( )
Z
1
2
2
2
2
2
,0,0
Where p1 = y1/σy and p2 = y2/ σz. Once the limits of integration are established,
the value of the integral may be determined from standard statistical tables.

12. Area Source
The GLC at a fixed receptor due to all area sources upwind of the receptor
is given by (using Gaussian formulation):
æ
-
Q y
exp
pss
s
y ò ò
Thus the GLC at a receptor is:
(14)
(15)
zs p ò ¥
C x Q
From (14) and (15)
(16)
= Q C = C 1-b 1-b
+1 p S åi
If (mixing height)
(17)
ö
dy dz
u
C
y z
¥ ¥
-¥
÷ ÷
ø
ç ç
è
=
0
2
2
2
( ) dx
u
=
0
2
b
z s =ax
[ X - X ] 2
a(1- b)
i i
i
i
i u
C = C = Q [ - ] 2 1 p S å +
i
i i
i
mix
i
i
X X
uh
z mix s >h