Analytical Models for the Dispersion of Pollutants in Low Wind Conditions 167
For ground-level source, the concentration distribution is deduced by taking the limit
H s → 0 in Equation 6.21. In low wind conditions, the formula yielding the Gaussian
plume solution (Equation 6.4) is approximate whereas the solution (Equation 6.21)
seems more accurate as it accounts for the downwind diffusion. The solution
(Equation 6.4) can also be deduced by taking the limit K x → 0 (ignoring the downwind diffusion) in Equation 6.21.
We defi ne the ratio R of the solution (Equation 6.4) to the solution (Equation 6.21)
for a ground-level source:
⎡
⎤
⎧
⎫
β
⎛
⎞
=
+
+
− +
⎨
⎬
⎢
⎥
⎜
⎟
⎝
⎠
⎩
⎭
⎣
⎦
1 2
(1 ) exp
(1 )
1
2
2
p
R
p
p
(6.24)
where
⎛
⎞
=
+
β=
⎜
⎟
⎝
⎠
2
2
2
and
x
y
z
x
K y
z
Ux
p
x K
K
K
(6.25)
are dimensionless parameters, which can take only positive values. This ratio (R)
essentially represents the overprediction/underprediction by the Gaussian model in
treating the dispersion in low wind conditions (Sharan et al., 1996b).
Notice that β resembles the well-known Peclet number P e , and it essentially represents the ratio of advective transport to diffusive transport. Physically, p represents
the region of interest relative to plume centerline and its small values, close to zero,
indicates the region in the proximity of plume centerline, whereas the magnitude
of β indicates the atmospheric conditions in terms of the strength of winds. Small
magnitudes of β are indicating the weakening of winds when downwind diffusion
becomes important. Based on p and β, it was suggested (Sharan et al., 1996b) that
Gaussian models lead to an overestimation up to 25%.
For facilitating the practical applications of these constant K models, K i ’s are
expressed in terms of dispersion parameters (σ’s) (Equation 6.5). A number of parameterizations of σ’s are proposed in the literature (Hanna et al., 1982; Yadav, 1995;
Yadav and Sharan, 1996; Arya, 1999; Sharan et al., 2003). For near source dispersion,
σ’s are proportional to time t, whereas far away from the source, these are proportional
to t (Sharan et al., 2003) and correspondingly from Taylor’s hypothesis, the spread
of plume in the crosswind and vertical directions starts off with a linear form of downwind distance (proportional to x) and ultimately tends to a parabolic form. Various
parameterizations available in the literature for the horizontal and vertical plume dispersion can be broadly classifi ed into (1) methods based on the power-law functions
(Briggs, 1973; Green et al., 1980), (2) methods based on the statistical parameters
such as σ θ (standard deviation of horizontal wind direction) and σ Φ (standard deviation of vertical wind direction), (Draxler, 1976; Irwin, 1979; Gryning et al., 1987), and
(3) methods based on the similarity theory (Hanna et al., 1982; Seinfeld, 1986).
6.4.2 VARIABLE K-MODELS
In Section 6.4.1, the concentration in low wind conditions is derived by taking the
eddy diffusivities and mean wind as constants. However, this assumption is no longer
© 2010 by Taylor and Francis Group, LLC
For ground-level source, the concentration distribution is deduced by taking the limit
H s → 0 in Equation 6.21. In low wind conditions, the formula yielding the Gaussian
plume solution (Equation 6.4) is approximate whereas the solution (Equation 6.21)
seems more accurate as it accounts for the downwind diffusion. The solution
(Equation 6.4) can also be deduced by taking the limit K x → 0 (ignoring the downwind diffusion) in Equation 6.21.
We defi ne the ratio R of the solution (Equation 6.4) to the solution (Equation 6.21)
for a ground-level source:
⎡
⎤
⎧
⎫
β
⎛
⎞
=
+
+
− +
⎨
⎬
⎢
⎥
⎜
⎟
⎝
⎠
⎩
⎭
⎣
⎦
1 2
(1 ) exp
(1 )
1
2
2
p
R
p
p
(6.24)
where
⎛
⎞
=
+
β=
⎜
⎟
⎝
⎠
2
2
2
and
x
y
z
x
K y
z
Ux
p
x K
K
K
(6.25)
are dimensionless parameters, which can take only positive values. This ratio (R)
essentially represents the overprediction/underprediction by the Gaussian model in
treating the dispersion in low wind conditions (Sharan et al., 1996b).
Notice that β resembles the well-known Peclet number P e , and it essentially represents the ratio of advective transport to diffusive transport. Physically, p represents
the region of interest relative to plume centerline and its small values, close to zero,
indicates the region in the proximity of plume centerline, whereas the magnitude
of β indicates the atmospheric conditions in terms of the strength of winds. Small
magnitudes of β are indicating the weakening of winds when downwind diffusion
becomes important. Based on p and β, it was suggested (Sharan et al., 1996b) that
Gaussian models lead to an overestimation up to 25%.
For facilitating the practical applications of these constant K models, K i ’s are
expressed in terms of dispersion parameters (σ’s) (Equation 6.5). A number of parameterizations of σ’s are proposed in the literature (Hanna et al., 1982; Yadav, 1995;
Yadav and Sharan, 1996; Arya, 1999; Sharan et al., 2003). For near source dispersion,
σ’s are proportional to time t, whereas far away from the source, these are proportional
to t (Sharan et al., 2003) and correspondingly from Taylor’s hypothesis, the spread
of plume in the crosswind and vertical directions starts off with a linear form of downwind distance (proportional to x) and ultimately tends to a parabolic form. Various
parameterizations available in the literature for the horizontal and vertical plume dispersion can be broadly classifi ed into (1) methods based on the power-law functions
(Briggs, 1973; Green et al., 1980), (2) methods based on the statistical parameters
such as σ θ (standard deviation of horizontal wind direction) and σ Φ (standard deviation of vertical wind direction), (Draxler, 1976; Irwin, 1979; Gryning et al., 1987), and
(3) methods based on the similarity theory (Hanna et al., 1982; Seinfeld, 1986).
6.4.2 VARIABLE K-MODELS
In Section 6.4.1, the concentration in low wind conditions is derived by taking the
eddy diffusivities and mean wind as constants. However, this assumption is no longer
© 2010 by Taylor and Francis Group, LLC