### x - Ari Rabl

```Atmospheric models for damage costs
Ari Rabl, ARMINES/Ecole des Mines de Paris, November 2013
There are many different models for atmospheric dispersion and chemistry, with different
objectives: e.g.
microscale models (street canyons),
local models (up to tens of km),
regional models (hundreds to thousands of km),
short term models for episodes,
long term models for long term (annual) averages.
For damage costs of air pollution, note that the dose-response functions for health (dominant
impact) are linear  only the long term average concentration matters
For agricultural crops and buildings they are nonlinear, but can be characterized in terms of
seasonal or annual averages  only the long term average concentration is needed
Dispersion of most air pollutants is significant up to hundreds or thousands of km
 need local + regional models for long term average concentrations
(they tend to be more accurate than models for episodes)
1
Dispersion of
Air Pollutants
Depends on
meteorological
conditions:
wind speed and
atmospheric stability
rate, see diagrams at
left)
2
Gaussian plume model
for atmospheric dispersion
(in local range < ~50 km)
3
Gaussian plume model, concentration c at point (x,y,z)
Underlying hypothesis: fluid with random fluctuations around a dominant
direction of motion (x-direction)
c=concentration, kg/m3
Q=emission rate, kg/s
v= wind speed, m/s,
in x-direction
y=horizontal plume width
z=vertical plume width
he=effective emission height
Source at x=0,y=0
é
ù
é
ù
2
2
ê 1 æ(z-he)ö ú
ê 1æyö ú
Q
c(x,y,z) = 2 π sy sz v expêë- 2 çèsy÷ø úû expêë- 2 çè sz ÷ø úû
Plume width parameters y and z increase with x
4
Gaussian plume width parameters
There are several models for estimating y and z as a function of
downwind distance x,
for example the Brookhaven model
s y =a y × x
by
s z =a z × x
bz
where
To use model one needs data for wind speed and direction,
and for atmospheric stability (Pasquill class);
the latter depends on solar radiation and on wind speed.
5
Gaussian plume
with reflection
terms
When plume hits
ground or top of
mixing layer, it is
reflected
6
Gaussian plume with reflection terms, cont’d
for 1.08 < sz/H (this is the limit of large distances)
2 π sz
replace S(z) ®
H
this corresponds to uniform vertical mixing
7
Effect of stack parameters
Plume rise:
fairly complex, depends on velocity and temperature of flue gas, as well as
on ambient atmospheric conditions
8
Effect of stack parameters, examples
Influence of Emission Source Parameters and Meteorological Data on
Damage Estimates. The Source is Located in a Suburb of Paris.
Normalized Damage
3
Reference State
Stack height = 100 m
Exit temp = 473 K
Exit speed = 10 m/s
Exit diameter = 2 m
Meteo data are
average values for
the period 1987-92.
Stack height
2,5
2
Exit temperature
Exit diameter
1,5
1
Exit speed
Weather data
0,5
0
0
0,5
1
1,5
2
Normalized Parameter
2,5
9
Removal of pollutants from atmosphere
Mechanisms for removal of pollutants from atmosphere:
1) Dry deposition
(uptake at the earth's surface by soil, water or vegetation)
2) Wet deposition
(absorption into droplets followed by droplet removal by
precipitation)
3) Transformation
chemical transformation SO2 NH4)2SO4).
They can be characterized in terms of deposition velocities,
(also known as depletion or removal velocities)
vdep = rate at which pollutant is deposited on ground, m/s
(obvious intuitive interpretation for deposition)
vdep depends on pollutant
determines range of analysis: the smaller vdep the farther the pollutant travels)
Typical values 0.2 to 2 cm/s for PM, SO2 and NOx
Gaussian plume model can be adapted to include
removal of pollutants
10
Regional Dispersion, a simple model
Far from source gaussian plume with reflections implies vertically uniform concentrations
Therefore consider line source for regional dispersion
(point source and line source produce same concentration at large r)
Assume wind speed is always = v, uniform in all directions f
the pollutant spreads over an area that is proportional to r
11
Simple model for regional dispersion, cont’d
Consider mass balance as pollution moves from r to r+r, if uniformity in all directions
mass flow v c(r) H r f across shaded surface at r
= mass flow v c(r+r) H (r+r) f across shaded surface at r+r
+ mass vdep c(r +r/2) r (r+r/2) f deposited on ground between r and r+r
Taylor expand c(r+r) = c(r) + c’(r) r and neglect higher order terms
 Differential equation c’(r) = - ( + 1/r) c(r)
with  = vdep/(v H)
Solution c(r) = c0 exp(- r)/r with constant c0 to be determined
12
Simple model for regional dispersion, cont’d
Determination of c0 by considering integral of flux v c(r) over cylinder of height H
and radius r in limit of r 0
This integral must equal to emission rate Q [in kg/s].
Hence
ì
c 0 exp(-b r) ü
Q =lim {2π rHv c(r) } = lim í2π rH v
ý
r®0
r®0 î
þ
r
Q=c0 2π v H
Therefore final result
Q exp(-b r)
c(r)=
2π vH
r
with
v dep
b=
vH
This model can readily be generalized
(i) To case where wind speeds in each direction are variable with a distribution f(v(f), f)
2p
¥
with normalization
1=
df
f (v(f ),f ) dv
ò
0
ò
0
(ii) To case where trajectories of puffs meander instead of being straight lines: then exp(- r) is replaced by
exp(- t(r)) where t(r) = transit time to r;
13
all else remains the same.
Impact vs cutoff rmax
Total impact I = integral of  sER c(r)
with  = receptor density and sER = slope of exposure-response function
Simple case:  and sER independent of r and f
∞
Q exp(-b r)
with c(r)=
2π vH
r
ER
0
I= r s
ò
I= r sER
I=
2π r c(r)dr
Q
vH
r sER Q
I(rmax )= I
ò
0
0
exp(-b r)dr
with
b=
v dep
vH
v dep
If cutoff rmax for integral
rmax
ò
∞
exp(-b r)dr
I(rmax )=I[1-exp(-b rmax )]
Range 1/ = v H/vdep = 800 km
for
mixing layer height H = 800 m
wind speed v = 10 m/s
depletion velocity vdep = 0.01 m/s
14
Chemical Reactions
Primary pollutants (emitted)  secondary pollutants
aerosol formation from NO, SO2 and NH3 emissions.
SO2
OH
NH3
H 2SO4
H 2O 2
Sulfate
aerosol
Note: NH3 background,
mostly from agriculture
Emission
Dry deposition
Wet deposition
O3
O3
NO
NO2
OH
hn
Emission
Dry deposition
Aerosol
HNO3
NH3
Wet deposition
Nitrate
aerosol
15
UWM: a simple model for damage costs
Product of a few factors (dose-response function, receptor density,
unit cost, depletion velocity of pollutant, …),
Exact for uniform distribution of sources or of receptors
UWM (“Uniform World Model”) for inhalation
• verified by comparison with about 100 site-specific calculations by
EcoSense software (EU, Eastern Europe, China, Brazil, Thailand,
…);
• recommended for typical values for emissions from tall stacks,
more than about 50 m (for specific sites the agreement is usually within a
factor of two to three; but for ground level emissions the damage of primary
pollutants is much larger: apply correction factors).
UWM for ingestion is even closer to exact calculation, because food is
transported over large distances average over all the areas where the food is
produced  effective distributions even more uniform.
Most policy applications need typical values
(people tend to use site specific results as if they were typical
 precisely wrong rather than approximately right)
16
UWM: derivation
Total impact I = integral of  sER c(x) over all receptor sites x = (x,y)
I =sER
with
òò r(x)c(x) dxdy
c(x) = c(x,Q) = concentration at surface due to emission Q Q
(x) = density of receptors (e.g. population)
sER = slope of exposure-response function
Total depletion flux (due to deposition and/or transformation)
F(x) = Fdry(x) + Fwet(x) + Ftrans(x)
Define depletion velocity vdep(x) = F(x)/c(x) [units of m/s]
Replace c(x) in integral by F(x)/k(x)
If world were uniform with
uniform density of receptors  and uniform depletion velocity vdep
then
I=(sER r /vdep ) òò F(x) dxdy
By conservation of mass
òò F(x) dxdy =Q
 “Uniform World Model” (UWM) for damage
Iuni =sER r Q/vdep
17
UWM: example
18
UWM and Site Dependence, example
dependence on site and on height of source for a primary pollutant:
impact I from SO2 emissions with linear exposure-response function, for five sites in
France, in units of Iuni for uniform world model (the nearest big city, 25 to 50 km away, is
indicated in parentheses). The scale on the right indicates YOLL/yr (mortality) from a plant
with emission 1000 ton/yr. Plume rise for typical incinerator conditions is accounted for.
19
Validation of UWM, for primary pollutants
Comparison with detailed model (EcoSense = official model of ExternE)
100
Damage costs in €
2000
per kg
UWM
10
1
Factor of two
0.1
Northern Europe
Central Europe
Sourthern Europe
Southeast Asia
USA
South America
0.01
0.01
0.1
1
10
100
Detailed model
20
UWM for secondary pollutants
Same approach: add a subscript 2 to indicate that concentration, dose-response
function and damage refer to the secondary pollutant
D2 = sER2 óõdx óõdy r(x) c2(x)
Replace c2(x) by depletion flux F2(x) and depletion velocity v2(x)
c2(x) = F2(x)/v2(x)
In a uniform world with v2(x) = v2,uni and r(x) = runi
D2 =
D2 =
P sER2 r uni
v 2uni
ò dx ò dy F (x)
2
P sER2 r Q 2
v2
because surface integral of depletion flux F2(x) equals the total quantity of
secondary pollutant Q2 that has been created
Q2 = ó
õdx ó
õdy
F2(x)
21
UWM for secondary pollutants, cont’d
Let us relate Q2 to the emission Q1 of the primary pollutant:
define a creation flux F1-2(x) as mass of secondary pollutant created per s and per
m2 of horizontal surface
F1-2(x) = v1-2(x) c1(x)
where v1-2(x) is a factor defined as local ratio of F1-2(x) and c1(x).
Integral over the creation flux F1-2(x) is also equal to the total quantity of
the secondary pollutant
Q2 = ó
õdx ó
õdy
F1-2(x) = ó
õdx ó
õdy v1-2(x) F1(x)/v1(x)
If uniform atmosphere with v1-2(x) = v1-2 and v1(x) = v1 independent of x
Q2 =
v1-2
v1
! dx ! dy F1 (x) =
v1-2
Q1
v1
Therefore UWM for secondary pollutants
PsER2 r Q1
D2 =
v 2eff
with
v 2 v1
v 2eff =
v1-2
22
Dependence site and on stack height
Strong variation for primary pollutants
but little variation for secondary pollutants,
because created far from source (hence less sensitive to local detail)
23
Correction factors for UWM
for dependence on site and on stack height
No variation with site for CO2 (long time constants, globally dispersing)
Example: the cost/kg of PM2.5 emitted by a car in Paris is about 15 times Duni.
24
Parameters for UWM
Population density and depletion velocities, in cm/s,
selected data for several regions.
From Rabl, Spadaro and Holland [2013]
Region
r
PM2.5
PM10
SO2
NOx
Sulfates Nitrates
112
0.57
0.86
0.88
1.36
1.85
1.00
Austria
110
0.56
0.84
0.85
1.19
1.95
1.03
France
105
0.45
0.68
0.73
1.47
1.73
0.71
Germany
152
0.52
0.78
0.73
1.01
1.94
0.83
Italy
150
0.71
1.07
0.99
1.38
1.86
1.04
Poland
Spain
97
55
0.57
0.50
0.86
0.75
0.90
0.80
0.96
2.16
2.00
1.65
1.23
0.91
Sweden
75
0.86
1.29
1.27
1.83
2.05
1.26
UK
122
0.59
0.89
0.94
1.18
2.03
1.28
0.37
0.55
0.83
0.40
1.96
0.99
persons/km2
EU-27
USA
25
```