Tsuang, B.

Atmospheric Environment 36 (2002) 411–419
Quantiﬁcation on source/receptor relationship of primary
pollutants and secondary aerosols from ground sourcesF
Part I. Theory
Ben-Jei Tsuang*, Chien-Lung Chen, Rong-Chang Pan, Jen-Hui Liu
Department of Environmental Engineering, National Chung-Hsing University, Taichung 40227, Taiwan, ROC
Received 26 January 2001; received in revised form 12 September 2001; accepted 19 September 2001
Abstract
A new algorithm has been derived for trajectory models to determine the transfer coeﬃcient of each source along or
adjacent to a trajectory and to calculate the concentrations of SO2, NOx, sulfate, nitrate, ﬁne particulate matter (PM)
and coarse PM at a receptor. The transfer coeﬃcient tf (s m1) is deﬁned to be the ratio between the contributed
concentration DC (mg m3) to the receptor from a ground source and the emission rate of the source q (mg m2 s1) at a
grid, i.e. tf DC=q: The model is developed by combining with a backward trajectory scheme and a circuit-type’s
parameterization. First, the transfer coeﬃcients of grids along or adjacent a back-trajectory are calculated. Then, the
contributed concentration of each emission grid is determined by multiplying its emission rate with the transfer
coeﬃcient of the grid. Finally, the concentration at the receptor is determined by the summation of all the contributed
concentrations within the domain of simulation. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Transfer coeﬃcient; Trajectory; Contributed concentration; Circuit model; Inverse approach
1. Introduction
High concentrations of particulate matter (PM) have
become a problem for many countries (Fung and Wong,
1995; Elbir et al., 2000). In Taiwan, the number of days
with the pollutants standards sub-index (PSI) (EPA/US,
1982) of PM10 over 100 of a normal station ranges
from around 2–13% yr1 in the past 10 yr. The PM10
limit is 150 mg m3 of its daily mean for PSI over 100.
The annual area average PM10 concentration ranges
from 58 to 94 mg m3 (EPA/ROC, 2000) in the last
decade. The concentration in many regions in Taiwan
still exceeded the annual EPA/ROC standard of
65 mg m3.
PM consists of primary aerosols as well as secondary
aerosols such as sulfate, nitrate and organics (Pandis
*Corresponding author. Tel.: +886-4-22851206; fax: +8864-22862587.
E-mail address: [email protected] (B.-J. Tsuang).
et al., 1992; Chiang, 1993; Fung and Wong, 1995; Tsai
and Cheng, 1999). Major factors that cause the high PM
concentration in the region include emissions, gas and
aerosol chemical reactions, gas-to-particle conversion,
meteorology and the complex terrain. Because PM
results from the interaction of several complex physical
processes and diﬀerent emission sources, an air quality
model is neededFone that can identify possible sources
and determine their contributions to the concentrations
of primary PM secondary sulfate and nitrate at a
receptor. Such a model can serve as a tool for designing
a better abatement strategy.
Back trajectories (e.g., Rolph and Draxler, 1990; Doty
and Perkey, 1993; Seibert, 1993; Stohl et al., 1995; Fast
and Berkowitz, 1997; Lee et al., 1997; Cheng et al., 1998;
Cheng and Lam, 1998) are popular for identifying
possible sources of pollutants measured at a receptor
location qualitatively because they pose smaller computational demands than Eulerian models (e.g., Chang
et al., 1987; Scheﬀe and Morris, 1993; Peters et al., 1995;
1352-2310/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 2 - 2 3 1 0 ( 0 1 ) 0 0 4 9 1 - 5
412
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
Jacobson et al., 1996; Jacobson, 1997a, b; Lurmann
et al., 1997; Nenes et al., 1999). Statistical analysis
methods for large sets of trajectories can further increase
the probability of identifying possible sources (e.g.,
Cheng et al., 1993a, b). However, it has been noted that
while the angular resolution for the methods of
statistical analysis is good, the radial resolution is poor
because of the convergence of all trajectories toward the
receptor (Vasconcelos et al., 1996). Conventionally, to
quantitatively determine the source–receptor relationship for trajectory models, we must couple a trajectory
model with physical-chemical modelsFa so-called
Lagrangian box (column) model, which has aerosol
chemistry, gas-to-particle conversion and diﬀusion
mechanisms (e.g., Eliassen and Saltbones, 1983; Pandis
et al., 1992; Asman and Janssen, 1987; Asman and van
Jaarsveld, 1992; Hertel et al., 1995; Bourque and Arp,
1996). Then, we must perform sensitivity analysis to
determine the source–receptor relationship in the model.
That is, we vary the emission rates of sources of interest
source by source. Finally, we repeatedly run the model
to quantify the change in the simulated concentration at
the receptor with the new emission proﬁles. We can
thereby sequence the relative importance of each source
quantitatively. Although the use of this model is
possible, it is very time-consuming and tedious. Stohl
(1998) has an excellent review on the development of
trajectory models.
A new trajectory model based on a circuit-type
parameterization (Tsuang and Chao, 1997, 1999) is
derived in this paper, which calculates transfer coeﬃcients based on wet scavenging, dry deposition, vertical
and horizontal diﬀusion, and gas-to-particle conversion
processes. This study deﬁnes terminologies of ‘‘transfer
coeﬃcient’’ and ‘‘transport ratio’’ to quantify the
source–receptor relationship. The transfer coeﬃcient is
the ratio between the contributed concentration from a
ground source and the emission rate of the source (Cass
and McRae, 1983). The transport ratio is the fraction of
a concentration originating at an upwind location and
transporting to the receptor along a trajectory. Assuming a piecewise steady state, using the circuit scheme to
parameterize the advection–diﬀusion equation for a
ground source, the transfer coeﬃcient and the transport
ratio can be easily determined. A detailed derivation of
the transfer coeﬃcient will be presented later. A Circuit
Trajectory transfer-coeﬃcient model (CTx) developed
based on the derived theory will be presented and tested
in a companion paper (Chen et al., 2002).
2. Theory
In the past decade, atmospheric aerosol modeling has
gained increasing attention (Wexler et al., 1994;
Jacobson, 1997a, b; Lurmann et al., 1997; Nenes et al.,
1999; Tsuang and Chao, 1999). For simplicity, we
neglect gas chemistry; assume that gas-to-particle
conversion only takes place in the conversion of gaseous
SO2 to particulate sulfate and in gaseous NOx to
particulate nitrate; assume that the supply of gaseous
ammonia is abundant; and assume that the radii of
particles does not grow. If the supply of gaseous
ammonia is abundant, the converted sulfate and nitrate
will be present in the forms of (NH4)2SO4 and NH4NO3,
respectively (Seinfeld and Pandis, 1998). Under the
above conditions, the transfer coeﬃcients can be
determined easily and the governing equations for
describing the evolution of PM can be written as
qPMi
q
qPMi
q2 PMi
qPMi
¼ Kz
þ Ky
pLPMi
u
qz
qt
qz
qy2
qx
MðNH4 Þ2 SO4 Sul
þ
fi kSO2 SO2
MSO2
MNH4 NO3 Nit
þ
fi kNOx NOx ;
ð1Þ
MNOx
qNOx
q
qNOx
q2 NOx
¼ Kz
þ Ky
qz
qt
qz
qy2
qNOx
pLNOx kNOx NOx ;
u
qx
qSO2 q
qSO2
q2 SO2
¼ Kz
þ Ky
qt
qz
qy2
qz
qSO2
pLSO2 kSO2 SO2 ;
u
qx
ð2Þ
ð3Þ
where coordinate x is along the wind direction and
coordinate y is perpendicular to the wind direction, PMi
is the concentration of suspended PM of size category i
(mg m3), fSul
and fNit
are the fractions of mass in the
i
i
size category i of the converted secondary sulfate and
nitrate, respectively, NOx and SO2 are the concentrations of gaseous SO2 (mg m3) and NOx (mg m3),
respectively, t is time (s), u is average wind speed
(m s1), Kz and Ky are eddy diﬀusivities in the z direction
and y direction, respectively (E10 m2 s1); p is the
proportion of time raining (s s1), L is scavenging
coeﬃcient (s1) which ranges from 0.4 105 s1 to
3 103 s1; kSO2 (%S s1) and kNOx (%N s1) are gas–
particle conversion rates for SO2 to sulfate and NOx to
nitrate, respectively. A review of kSO2 and kNOx can be
found in Seinfeld (1986). Rates of kSO2 range from 0%
to 13%S h1. Rates of kNOx range from 5% to
24%N h1. MðNH4 Þ2 SO4 ; MNH4 NO3 ; MSO2 and MNOx are
molecular weights of (NH4)2SO4, NH4NO3, SO2 and
NOx, respectively. The above equations are derived
from the traditional advection–diﬀusion equations for
air pollutants (e.g., Hanna et al., 1982; Beryland, 1991;
Seinfeld, 1986) by including wet deposition and chemical
reaction terms, but neglecting the diﬀusion term along
the wind direction.
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
Following a similar analogy of Tsuang and Chao
(1997) and neglecting the ill-mixed factor (Tsuang and
Chao, 1999), we can derive analytical solutions for PMi,
SO2, NOx, Suli (sulfate), and Niti (nitrate) as
PMi:
PMi ¼ ðPMi0 PMi * Þ
i
V
2Ky
u
þ pL t þ PMi * ;
exp d þ 2 þ
Dx
le
Dy
ð4Þ
Vdi qPMi
Ky
u
PMbi
þ 2 ðPMli þ PMri Þ þ
i
Dx
le V d
Dy
MðNH4 Þ2 SO4 Sul
MNH4 NO3 Nit
þ
fi kSO2 SO2 þ
fi kNOx NOx
MSO2
MNOx
:
PMi * ¼
i
Vd 2Ky
u
þ pL
þ 2þ
Dx
le
Dy
ð5Þ
SO2:
SO2 ¼ ðSO20 SO2 * Þ
Vd 2Ky
u
þ pL þ kSO2 t þ SO2 * ;
exp þ 2þ
Dx
le
Dy
ð6Þ
SO2 *
Ky
Vd qSO2
u
SOb2
þ 2 ðSOl2 þ SOr2 Þ þ
Dx
le Vd
Dy
¼
:
Vd 2Ky
u
þ pL þ kSO2
þ 2þ
Dx
le
Dy
ð7Þ
NO2:
NOx ¼ ðNOx0 NOx * Þ
Vd 2Ky u
exp þ 2 þ þ pL þ kNOx t þ NOx * ;
le Dy Dx
ð8Þ
NOx *
Ky
Vd qNOx
u
NObx
þ 2 ðNOlx þ NOrx Þ þ
Dx
le V d
Dy
¼
: ð9Þ
Vd 2Ky
u
þ pL þ kNOx
þ 2þ
Dx
le
Dy
Sulfate:
Suli ¼ ðSuli0 Suli * Þ
i
V
2Ky
u
þ pL t þ Suli * ;
exp d þ 2 þ
le
Dy
Dx
Suli *
413
Nitrate:
Niti ¼ ðNiti0 Niti * Þ
i
V
2Ky
u
þ pL t þ Niti * ;
exp d þ 2 þ
Dx
le
Dy
Niti *
ð12Þ
MNO3 Nit
Ky
u
Nitbi þ
ðNitli þ Nitri Þ þ
f kNOx NOx
Dy2
MNOx i
Dx
¼
;
Vdi 2Ky
u
þ 2þ
þ pL
le
Dy
Dx
ð13Þ
where subscript * denotes steady-state concentration at
near surface of a trajectory grid (mg m3); subscript 0
denotes its initial concentration (mg m3); superscripts l
and r denote concentrations on the left-hand side and on
the right-hand side of the grid perpendicular to wind
direction, respectively (mg m3); subscript b denotes the
upwind concentration of the grid; q is the emission rate
per unit area of the grid (mg m2 s1); t is time (s); Vd is
the dry deposition velocity (m s1); le is the eﬀective
mixing length (m). The derivation of le can be found in
Tsuang and Chao (1997).
2.1. Piecewise steady state
Under steady state, according to Eqs. (4)–(13), PMi ¼
PMi * ; SO2 ¼ SO2 * ; NOx ¼ NOx * : These steady-state
concentrations can be decomposed into several fractions. The concentration of PMi * ; for example, according to Eq. (5) can be rewritten as (Fig. 1)
Vdi qPMi
Ky
u
þ 2 ðPMli þ PMri Þ þ
PMbi
le Vdi
Dy
Dx
MðNH4 Þ2 SO4 Sul
MNH4 NO3 Nit
þ
fi kSO2 SO2 þ
fi kNOx
MSO2
MNOx
PMi * ¼
Vdi 2Ky
u
þ 2þ
þ pL
le
Dy
Dx
¼ fqPMi þ Xy ðPMli þ PMri Þ þ Xb PMbf
MðNH4 Þ2 SO4 Sul
MNH4 NO3 Nit
þ
fi XSul SO2 þ
fi XNit NOx
MSO2
MNOx
ð14Þ
and,
1
le
ð15Þ
Xy T i
Ky
Dy2
ð16Þ
Xb T i
u
Dx
ð17Þ
fPMi Ti
ð10Þ
MSO2 sul
Ky
u
4
ðSulli þ Sulri Þ þ
f kSO2 SO2
Sulbi þ
2
Dy
MSO2 i
Dx
:
¼
Vdi 2Ky
u
þ pL
þ 2þ
Dx
le
Dy
ð11Þ
XSuli Ti kSO2
ð18Þ
414
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
conversion ratio, which is the ratio of nitrogen element
between the concentration of converted nitrate and the
concentration of NOx within the same grid.
PM i b
Xb
Xy
PM i l
Xy
PMi*
XNit
PM i r
Eqs. (16)–(19) introduces several dimensionless ratios.
In order to understand their magnitude, a scale analysis
is conducted as follows. For a time step (i.e. Dt or Dx=u)
of 1 h, Vd is 6.6E3 m s1, a mean value for ﬁne particle
(Tsuang and Chao, 1999), le is set to be 60 m (Tsuang
and Chao, 1997), Ky is 10 m2 s1, L is 1.5E4 s1, a
mean value for particle (McMahon and Denison, 1979),
kSO2 is 5%S h1 and kNOx is 3%N h1 (Kuo et al. 1996)
and Dy is 10 km; in rainless conditions (p ¼ 0) Xb
is determined to be 72%, XSul is 3.6%, XNit is 2.1%,
and Xy is 0.026%; in rainy conditions (p ¼ 1) Xy is
determined to be 0.018%, Xb is 52%, XSul is 2.6%, and
XNit is 1.5%. Hence we can conclude that
Xy 5XSul XNit oXb o1 under both rainless and rainy
conditions and horizontal diﬀusive ratio Xy normally is
of three orders of magnitude less than the advective ratio
Xb while Dt ¼ 1 h and Dy ¼ 10 km.
XSul
f
q
NOx
2.2. Scale analysis
SO2
Fig. 1. The diagram of PMi* associated with emission, transformation and transport.
XNiti Ti kNOx
ð19Þ
V i 2Ky
1
u
þ pL;
dþ 2þ
Ti
Dx
le
Dy
ð20Þ
where f is deﬁned to be the local transfer coeﬃcient
(s m1) which converts emission into concentration
within the same grid; Xy (dimensionless) is deﬁned to
be the horizontal diﬀusive ratio, which is the ratio
between the concentration diﬀused from an adjacent
neighborhood grid in the y-direction to the grid and the
concentration of the neighborhood grid; Xb (dimensionless) is deﬁned to be the advective ratio, which is the
ratio between the concentration advected from the
upwind grid into the grid and the concentration of
the upwind grid; XSul (dimensionless) is deﬁned to be
the local SO2-sulfate conversion ratio, which is the ratio
of sulfur between the concentration of converted sulfate
and the concentration of SO2 within the same grid; XNit
(dimensionless) is deﬁned to be the local NOx-nitrate
PM il ( 0)
f0
q0 l
Xb0
X y0
PM ic (0)
receptor
PM il (1)
f1
X y1
q1l
PM ic (1)
Xb0 f
1
q 1c
f0
q0c
unit
capacity
2.3. Trajectory linkage
We can extend a single grid (Fig. 1) to linked grids by
trajectories as shown in Fig. 2 under the piecewise steady
state. The piecewise steady state means that each
trajectory grid along a trajectory is under steady state
while pollutants arrive at that particular grid but unlike
Gaussian-type plume model meteorological conditions
are not horizontally homogeneous among grids. Fig. 2
shows a schematic diagram of the developed trajectory
model. There are three back trajectories: central left and
right. Most Lagrangian box (or column) models
consider only a single central trajectory. That is
conventionally only pollutants emitted along the central
PM il (k )
Xb1
PM il ( 2)
f2
left
trajectory
X y2
q 2l
PM ic (2)
X b1
f2
fk
q kl
PM ic (k )
central
trajectory
fk
q kc
q 2c
right
trajectory
Fig. 2. Diagram of the developed trajectory model, where ‘‘left’’ or ‘‘right’’ trajectory is according to the receptor’s point of view.
415
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
trajectory will be included in the calculation of
concentration at the receptor. However if there is a
source lying along the right or the left trajectories with
the emission rate of few orders of magnitude higher than
those along the central trajectory the contribution from
the source can have the same magnitude of impact on
the concentration of the receptor. Therefore three back
trajectories are designed in this study.
In order to quantify the source/receptor relation we
introduced two terms: transport ratio Pj ðkÞ (dimensionless) and transfer coeﬃcient tf (s m1). The transport
ratio Pj ðkÞ is the fraction of a concentration originating
at a grid at time k along the trajectory j and transporting
to the receptor at time 0. The transfer coeﬃcient tf
(s m1) is deﬁned as the ratio between the contributed
concentration DC (mg m3) to the receptor from a
ground source and the emission rate of the source q
(mg m2 s1) ðtf DC=qÞ: According to Eq. (14) the
local transfer coeﬃcient is equal to f : Hence the transfer
coeﬃcient tjf ðkÞ at grid k upwind along the trajectory j
can be determined by the multiplication of the local
transfer coeﬃcient f at grid k and the transport ratio Pj
at the grid as
tjf ðkÞ ¼ Pj ðkÞf ðkÞ;
ð21Þ
where the superscript j denotes the trajectory (c for
central l for left and r for right trajectories, respectively).
The transport ratios Pj ðkÞ of a grid k along the central
trajectory the left trajectory and the right trajectory are
determined as follows:
(1) Central trajectory (j=c). According to Eq. (14) we
can determine the transport ratio Pc ðkÞ along the central
trajectory for transporting PMi SO2 and NOx as (Fig. 3)
Pci ðkÞ
¼
Xbi ðk
1ÞPci ðk
1Þ þ
OðXy2 Þ
ð22Þ
and
Pci ð0Þ
¼ 1;
ð23Þ
where the superscript c denotes the central trajectory
and index i represents pollutant PMi SO2 and NOx.
Note that OðXy2 Þ is the truncation error term for
calculating Pc ðkÞ which neglects that a pollutant can
diﬀuse from the central trajectory to an adjacent
trajectory and then diﬀuse back to the central trajectory
or diﬀuses back and forth several times. Luckily OðXy2 Þ
is very small according to the previous scale analysis
(E107–108 for Dt ¼ 1 h and Dy ¼ 10 km).
For pollutants involving phase transformation such as
sulfate from SO2 and nitrate from NO2 we have to
consider two routes. First the gaseous pollutants transform to particulate then advect to the downwind grid.
Secondly the gaseous pollutants advect to the downwind
grid then transform to particles during the route to the
receptor. As a result the transport ratios for formation
of sulfate and nitrate can be written as
j=c
phase
transformation
k
grid
XSul(k)
XNit(k)
Concentration
Sulfate(k)
Nitrate(k)
SO2(k)
NO2(k)
PMi(k)
Xb(k-1)
phase
transformation
k-1
grid
Concentration
Sulfate(k-1)
Nitrate(k-1)
XSul(k-1) SO2(k-1)
XNit(k-1) NO2(k-1)
central
trajectory
PMi(k-1)
c
Π (k-1)
c
Π (k)
receptor
Fig. 3. Relationship of the concentration-transfer ratio Pc ðkÞ
along the central trajectory line.
PcSuli ðkÞ ¼ XbSO2 ðk 1ÞPcSuli ðk 1Þ
þ XSuli ðkÞPcPMi ðkÞ þ OðXy2 Þ
ð24Þ
PcNiti ðkÞ ¼ XbNOx ðk 1ÞPcNiti ðk 1Þ
þ XNiti ðkÞPcPMi ðkÞ þ OðXy2 Þ
ð25Þ
and
PcSuli ð0Þ ¼ XSuli ð0Þ
ð26aÞ
PcNiti ð0Þ ¼ XNiti ð0Þ;
ð26bÞ
where the ﬁrst terms on the right hand side of Eqs. (24)–
(25) represent the gaseous pollutants transported to the
downwind grid in a gaseous phase and then transformed
to particles afterwards. The second terms represents the
gaseous pollutants’ transformation to particulate ﬁrst
and then transport to the receptor.
(2) Adjacent trajectories ( j ¼l, r). Along the left (right)
trajectory similarly transport ratios can be written as
(Fig. 4)
Pri ðkÞ ¼ Pli ðkÞ ¼ Xyi ðkÞ Pci ðkÞ
þ Xbi ðk 1ÞPli ðk 1Þ þ OðXy3 Þ
ð27Þ
and
Pri ð0Þ ¼ Pli ð0Þ ¼ Xyi ð0Þ;
ð28Þ
where the superscripts r and l denote the right and the
left back trajectories, respectively, and index i represents
pollutant PMi, SO2 and NOx. The ﬁrst term on the right
416
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
j=c
j=l
Concentration
Sulfate( k)
Nitrate(k)
phase
transformation
k
grid
SO2(k)
NO2 (k)
XSul(k)
X Nit (k)
left
trajectory
phase
transformation
k-1
grid
PMi (k)
phase
transformation
Xy(k)
SO2 (k)
NO2 (k)
XSul(k)
X Nit (k)
PMi (k)
central
trajectory
Xb(k-1)
Concentration
Sulfate( k-1)
Nitrate(k-1)
XSul(k-1) SO2 (k-1)
XNit (k-1) NO2(k-1)
Concentration
Sulfate( k)
Nitrate(k)
Πl(k)
Π c(k)
PM i(k-1)
Πl(k-1)
receptor
Fig. 4. Relationship of the concentration-transfer ratio Pl ðkÞ along the left trajectory line.
hand side of Eq. (27) represents the pollutants diﬀusing
to the adjacent grid along the central trajectory and then
advecting to the receptor. The second term represents
the pollutants advecting one grid downward along the
left (right) trajectory ﬁrst then travel to the receptor.
Note that the ratios of Xy Xb XSuli and XNiti along the
left and right trajectories are assumed equal to those of
the central trajectory. Similarly the gaseous SO2 and
NOx in an adjacent trajectory transforming to ﬁne
particle while it arrives at the receptor have three paths
and can be written as
PrSuli ðkÞ ¼ PlSuli ðkÞ ¼ XySO2 ðkÞPcSuli ðkÞ
þ XSuli ðkÞPlPMi ðkÞ
þ XbSO2 ðk 1ÞPlSuli ðk 1Þ þ OðXy3 Þ
PrNiti ðkÞ
j¼l;c;r k¼0
¼ PlNiti ðkÞ
þ
þ
¼ XyNOx ðkÞPcNiti ðkÞ
XNiti ðkÞPlPMi ðkÞ
XbNOx ðk 1ÞPlNiti ðk 1Þ
ð29Þ
þ
þ
OðXy3 Þ
ð30Þ
and
PrSuli ð0Þ ¼ PlSuli ð0Þ ¼ XySO2 ð0ÞPcSuli ð0Þ
þ XSuli ð0ÞPlPMi ð0Þ
PrNiti ð0Þ
ð31aÞ
¼ PlNiti ð0Þ
þ
¼ XyNOx ð0ÞPcNiti ð0Þ
XNiti ð0ÞPlPMi ð0Þ;
gaseous phase ﬁrst and then transforming to particles
afterwards. The second terms represent the gaseous
pollutants transforming to particulate ﬁrst and then
transporting to the receptor. The third terms represent
the gaseous pollutants advecting to the downwind grid
along the left (right) trajectory in a gaseous phase ﬁrst
and then transforming to particle afterwards.
(3) Concentration at the receptor. Finally the steadystate concentration at the receptor C * ð0Þ under the
assumption of piecewise steady state along trajectories
can be determined by the summation of all the
contributed concentrations from the sources along the
three trajectories as
"
n
X X
PMi * ð0Þ ¼
PjPMi ðkÞfPMi ðkÞqjPMi ðkÞ
MðNH4 ÞSO4 Sul j
fi PSuli ðkÞfSO2 ðkÞqjSO2 ðkÞ
MSO2
MNH4 NO3 Nit j
fi PNiti ðkÞfNOx ðkÞqjNOx ðkÞ
þ
MNOx
#
þ PcPMi ðn þ 1ÞPMci ðn þ 1Þ þ OðXy2 Þ
"
n
X X
tjf PMi ðkÞqjPMi ðkÞ
¼
j¼l;c;r k¼0
MðNH4 Þ2 SO4 Sul j
fi tf Suli ðkÞqjSO2 ðkÞ
MSO2
#
MNH4 NO3 Nit j
j
fi tf Niti ðkÞqNOx ðkÞ
þ
MNOx
þ
ð31bÞ
where the ﬁrst term on the right hand side of the
Eqs. (29)–(30) represent the gaseous pollutants diﬀusing
to the adjacent grid along the central trajectory in a
þ PcPMi ðn þ 1ÞPMci ðn þ 1Þ þ OðXy2 Þ;
ð32Þ
417
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
NOx * ð0Þ ¼
n
X X
of k upwind to a receptor. The tables show that (1) PcPMc
and PcPMf decay exponentially with travel time where
PMc and PMf denote particle in coarse radii (2.5–10 mm)
and in ﬁne radii (0–2.5 mm) respectively. (2) Transport
ratios P are very sensitive to dry deposition velocity.
Decreasing dry deposition velocity from 0.66 to
0.1 cm s1 increases PcPMf from 1.1E7 to 5.9% for a
travel time of 48 h in comparison between Table 1 and 2.
(3) According to Table 1 under rainless conditions the
highest values of PcSulf and PcNitf occur at locations
where pollutants will arrive at the receptor 2 h later. This
means that emissions of SO2 and NOx located at
locations of a travel time of 2 h cause the highest
concentration levels of sulfate and nitrate at the receptor
while dry deposition velocity is 0.66 cm s1. Moreover
the highest transport ratio of PcNitf occurs further
upwind from at 2 to 12 h upwind to the receptor and
the magnitude increases from 3.3% to 18% while the dry
deposition velocity decreases from 0.66 to 0.1 cm s1
(Table 1 and 2). A similar conclusion can be made for
PcSulf : (4) PcPMf which has a travel time of 24 h, is close to
PlPMf which has a travel time of 0 h (Table 1). This
means that emissions at adjacent grids near the receptor
are equally important to the sources located at 24 h
upwind along the central trajectory. (5) Under rainy
conditions (Table 3), transport ratios P decrease
signiﬁcantly. For example, PcPMf decreases from 1.81%
for locations of the traveling time of 12 h under rainless
PjNOx ðkÞfNOx ðkÞqjNOx ðkÞ
j¼l;c;r k¼0
þ PcNOx ðn þ 1ÞNOcx ðn þ 1Þ þ OðXy2 Þ
n
X X
¼
tjf NOx ðkÞqjNOx ðkÞ þ PcNOx ðn þ 1Þ
j¼l;c;r k¼0
NOcx ðn þ 1Þ þ OðXy2 Þ;
SO2 * ð0Þ ¼
n
X X
ð33Þ
PjSO2 ðkÞfSO2 ðkÞqjSO2 ðkÞ
j¼l;c;r k¼0
þ PcSO2 ðn þ 1ÞSOc2 ðn þ 1Þ þ OðXy2 Þ
¼
n
X X
ð34Þ
tjf SO2 ðkÞqjSO2 ðkÞ þ PcSO2 ðn þ 1ÞSOc2 ðn þ 1Þ
j¼l;c;r k¼0
þ OðXy2 Þ:
In addition, the instantaneous concentration at the
receptor Cð0Þ under the assumption of piecewise steady
state along trajectories can be determined according to
Eqs. (4)–(13).
3. Transport ratios
Table 1–3 show typical values of transport ratios Pj ðkÞ of concentrations at locations of the travelling time
Table 1
Typical transport ratios Pj ðkÞ under rainless conditions, where Vd for particle in coarse radii (2.5–10 mm) (denoted as PMc ) is set to be
a
5E2 m s1, Vd for particle in ﬁne radii (0–2.5 mm) (denoted as PMf) is 6.6E3 m s1
k (h)
0
1
2
3
6
12
24
36
48
PcPMc
PcPMf
PcSulf
PcNitf
PlPMf
100.00%
100.00%
3.58%
2.15%
2.58E04
25.00%
71.60%
5.13%
3.08%
3.69E04
6.25%
51.26%
5.51%
3.30%
3.96E04
1.56%
36.70%
5.26%
3.15%
3.78E04
0.02%
13.47%
3.38%
2.03%
2.43E04
0.00%
1.81%
8.44E03
5.07E03
6.08E05
0.00%
3.29E04
2.95E04
1.77E04
2.12E06
0.00%
5.97E06
7.91E06
4.75E06
5.69E08
0.00%
1.08E07
1.90E07
1.14E07
1.37E09
a
le is 60 m, Ky is 10 m2 s1, L is 1.5E4 s1, kSO2 is 5% h1, kNOx is 3% h1, Dy 10 km, and k is the travel time (h) to a receptor, where
the maximum values are in bold font.
Table 2
Typical transport ratios Pj ðkÞ under rainless conditions, where Vd for particle in coarse radii (2.5–10 mm) (denoted as PMc ) is set to be
5E2 m s1, Vd for particle in ﬁne radii (0–2.5 mm) (denoted as PMf) is 1.E3 m s1 a
K (h)
0
1
2
3
6
12
24
36
48
PcPMf
PcSulf
PcNitf
PlPMf
100.00%
4.71%
2.83%
3.39E04
94.28%
8.89%
5.33%
6.40E04
88.88%
12.57%
7.54%
9.05E04
83.79%
15.80%
9.48%
1.14E03
70.21%
23.17%
13.90%
1.67E03
49.29%
30.21%
18.12%
2.17E03
24.30%
28.63%
17.18%
2.06E03
11.98%
20.89%
12.53%
1.50E03
5.90%
13.64%
8.18%
9.82E04
a
le is 60 m, Ky is 10 m2 s1, L is 1.5E4 s1, kSO2 is 5% h1, kNOx is 3% h1, Dy 10 km, and k is the travel time (h) to a receptor,
where the maximum values are in bold font.
418
B.-J. Tsuang et al. / Atmospheric Environment 36 (2002) 411–419
Table 3
Typical transport ratios Pj ðkÞ under rainy conditions, where Vd for particle in coarse radii (2.5–10 mm) (denoted as PMc ) is set to be
5E2 m s1, Vd for particle in ﬁne radii (0–2.5 mm) (denoted as PMf) is 6.6E3 m s1 a
K (h)
0
1
2
3
6
12
24
36
48
PcPMf
PcSulf
PcNitf
PlPMf
100.00%
2.58%
1.55%
1.86E04
51.63%
2.67%
1.60%
1.92E04
26.66%
2.06%
1.24%
1.49E04
13.77%
1.42%
8.53E03
1.02E04
1.89%
3.42E03
2.05E03
2.47E05
3.59E04
1.21E04
7.23E05
8.68E07
1.29E07
8.32E08
4.99E08
5.99E10
4.63E11
4.42E11
2.65E11
3.18E13
1.66E14
2.10E14
1.26E14
1.51E16
a
le is 60 m, Ky is 10 m2 s1, L is 1.5E4 s1, kSO2 is 5% h1, kNOx is 3% h1, Dy 10 km, and k is the travel time (h) to a receptor,
where the maximum values are in bold font.
conditions (Table 1) to a value of 0.036% under rainy
conditions (Table 3).
4. Conclusion
The theory introduced here shows a systematic way to
determine transfer coeﬃcients qualitatively and quantitatively in the angular resolution as well as in the radial
resolution. The mechanisms of wet scavenging, dry
deposition, vertical and horizontal diﬀusion, and
gas-to-particle conversion processes have been included
in the determination of transfer coeﬃcients. The
idea of determining transfer coeﬃcient by an air quality
model is important for the design of a better
abatement strategy. Using transfer coeﬃcients has been
used for receptor models to determine the least-cost
strategies for the reduction of PM concentrations at
receptors (Cass and McRae, 1983; Tsuang and Chang,
1997).
Sensitive analysis shows that transport ratios for
primary pollutants decrease exponentially with travel
time. In addition, the value of the ratio is around 1% at
a location 2 h upwind with a dry deposition velocity of
5 cm s1 under rainless conditions. This means that only
1% of the concentration at 2 h upwind travels to the
receptor under the conditions. The distance increases as
the dry deposition velocity decreases. It increases to 12 h
upwind as the deposition velocity decreases to
0.66 cm s1, and more than 48 h upwind as the deposition velocity further decreases to 0.1 cm s1. The ratio
for nitrate converted from NOx has the highest values of
3% at a location 2 h upwind with a dry deposition
velocity of 0.66 cm s1, and of 18% at a location 12 h
upwind as the dry deposition velocity decreases to
0.1 cm s1. The ratios for adjacent grids with 10 km
apart to the central trajectory are o1%. The ratios for
adjacent grids with 20 km apart are o0.01%.
A CTx model, based on the theory has been
developed. Applying the CTx to a study site is presented
in a companion paper. This model is available to the
public from ‘‘http://air701.ev. nchu.edu.tw’’ under the
General Public License Agreements.
Acknowledgements
The author would like to acknowledge the support of
this work by National Science Council, R.O.C., under
contract NSC-88-2211-E005-018 and NSC-89-EPA-Z005-004. Professor Len-Fu W. Chang’s helpful suggestions are also appreciated.
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