Analytical Models for the Dispersion of Pollutants in Low Wind Conditions 161
balance equation. Alternatively, this can be accounted for through the boundary of
the domain. Sharan et al. (1999) proposed the equivalent mathematical formulations
for accounting the source either through advection–diffusion equation or through
one of the boundaries. It has been demonstrated that all such formulations lead to the
same solution. Accordingly, by modifying the boundary condition (Equation 6.3c) to
account for the source, the solution of Equation 6.2 is found to be the same as given
by Equation 6.4 (Yadav et al., 1996; Sharan et al., 2003).
For facilitating the practical application of the analytical solution (Equation 6.4),
the eddy diffusivities K x , K y , and K z are expressed in terms dispersion parameters as
2
1 d ,
, ,
2 d
i
i
K
i xyz
t
σ
=
=
(6.5)
in which parameters σ x , σ y , and σ z are the standard deviations of the concentration
distribution in downwind, crosswind, and vertical directions, respectively. These
dispersion parameters are the functions of stability and downwind distance and are
based on the combination of experimental results and theory (Hanna et al., 1982).
For constant K i ’s (i = x, y, z), Equation 6.5 takes the form:
2
2
,
, ,
2
2
i
i
i
U
K
i x y z
t
x
σ
=
=σ
=
(6.6)
in which t is the travel time in terms of mean wind speed and downwind distance.
The analytical solution (Equation 6.4) can be written in terms of the dispersion
parameters as
(
)
(
)
⎡
⎤
⎛
⎞
⎛
⎞
⎛
⎞
−
+
⎢
⎥
=
−
−
+
−
⎜
⎟
⎜
⎟
⎜
⎟
π σ σ
σ
σ
σ
⎢
⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎣
⎦
2
2
2
2
2
2
( , , )
exp
exp
exp
2
2
2
2
s
s
y z
y
z
z
z H
z H
Q
y
C x y z
U
(6.7)
This is the widely used standard Gaussian plume solution, describing the concentration distribution of a nonreactive pollutant emitted from a continuous point source
located at height H s in an infi nite domain. For a ground-level source, the concentration of pollutant can be estimated by taking the limit H s → 0 in Equation 6.7.
The properties of this solution are discussed in the literature (Seinfeld, 1986;
Arya, 1999).
6.3.2 GAUSSIAN PUFF MODEL
Relaxing the assumptions (a) and (d) in standard Gaussian plume model (Section
6.3.1), the differential Equation 6.2 includes the time-dependent and horizontal
diffusion terms and can be written as
∂
∂
∂
∂
∂
+
=
+
+
+ δ( )δ( )δ( − )δ( )
∂
∂
∂
∂
∂
2
2
2
2
2
2
C
x
x
y
z
s
C
C
C
C
U
K
K
K
q x y z H
t
t
x
y
z
(6.8)
© 2010 by Taylor and Francis Group, LLC
balance equation. Alternatively, this can be accounted for through the boundary of
the domain. Sharan et al. (1999) proposed the equivalent mathematical formulations
for accounting the source either through advection–diffusion equation or through
one of the boundaries. It has been demonstrated that all such formulations lead to the
same solution. Accordingly, by modifying the boundary condition (Equation 6.3c) to
account for the source, the solution of Equation 6.2 is found to be the same as given
by Equation 6.4 (Yadav et al., 1996; Sharan et al., 2003).
For facilitating the practical application of the analytical solution (Equation 6.4),
the eddy diffusivities K x , K y , and K z are expressed in terms dispersion parameters as
2
1 d ,
, ,
2 d
i
i
K
i xyz
t
σ
=
=
(6.5)
in which parameters σ x , σ y , and σ z are the standard deviations of the concentration
distribution in downwind, crosswind, and vertical directions, respectively. These
dispersion parameters are the functions of stability and downwind distance and are
based on the combination of experimental results and theory (Hanna et al., 1982).
For constant K i ’s (i = x, y, z), Equation 6.5 takes the form:
2
2
,
, ,
2
2
i
i
i
U
K
i x y z
t
x
σ
=
=σ
=
(6.6)
in which t is the travel time in terms of mean wind speed and downwind distance.
The analytical solution (Equation 6.4) can be written in terms of the dispersion
parameters as
(
)
(
)
⎡
⎤
⎛
⎞
⎛
⎞
⎛
⎞
−
+
⎢
⎥
=
−
−
+
−
⎜
⎟
⎜
⎟
⎜
⎟
π σ σ
σ
σ
σ
⎢
⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎣
⎦
2
2
2
2
2
2
( , , )
exp
exp
exp
2
2
2
2
s
s
y z
y
z
z
z H
z H
Q
y
C x y z
U
(6.7)
This is the widely used standard Gaussian plume solution, describing the concentration distribution of a nonreactive pollutant emitted from a continuous point source
located at height H s in an infi nite domain. For a ground-level source, the concentration of pollutant can be estimated by taking the limit H s → 0 in Equation 6.7.
The properties of this solution are discussed in the literature (Seinfeld, 1986;
Arya, 1999).
6.3.2 GAUSSIAN PUFF MODEL
Relaxing the assumptions (a) and (d) in standard Gaussian plume model (Section
6.3.1), the differential Equation 6.2 includes the time-dependent and horizontal
diffusion terms and can be written as
∂
∂
∂
∂
∂
+
=
+
+
+ δ( )δ( )δ( − )δ( )
∂
∂
∂
∂
∂
2
2
2
2
2
2
C
x
x
y
z
s
C
C
C
C
U
K
K
K
q x y z H
t
t
x
y
z
(6.8)
© 2010 by Taylor and Francis Group, LLC