1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN IN –
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH 0976
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 6, September – October 2013, pp. 269-277
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
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IJARET
©IAEME
TIME – DEPENDENT TWO DIMENSIONAL MATHEMATICAL MODEL OF
AIR POLLUTION DUE TO AREA SOURCE WITH SETTLING VELOCITY
AND TRANSFORMATION PROCESSES OF PRIMARY AND SECONDARY
POLLUTANTS
Dr. Lakshminarayanachari.K
Associate Professor, Department of Mathematics, Sai Vidya Institute of Technology,
Bangalore -560 064, INDIA.
ABSTRACT
A comprehensive time dependent two dimensional advection-diffusion numerical model for
primary pollutants emitted and converted into secondary pollutants for an urban area is presented.
This model takes into account the realistic form of variable wind velocity and eddy diffusivity
profiles. In this model we consider that the secondary pollutants are formed by means of first order
chemical conversion of primary pollutants. This assumption results in a coupled system of partial
differential equations of primary and secondary pollutants. This intricate coupled system of mixed
initial boundary value problem is solved by using Crank-Nicolson implicit finite difference
technique. The effect of time dependent emission of pollutants and the effect of various
meteorological parameters on the dispersion of pollutants on concentration contour are analysed
extensively.
Keywords: Primary and secondary pollutant, Crank-Nicolson method, Dry deposition, Gravitational
settling.
1. INTRODUCTION
The dispersion of atmospheric contaminant has become a global problem in the recent years
due to rapid industrialization and urbanization. The toxic gases and small particles could accumulate
in large quantities over urban areas, under certain meteorological conditions. This is one of the
serious health hazards in many of the cities in the world. An acute exposure to the elevated levels of
particulate air pollution has been associated with the cases of increased cardiopulmonary mortality,
hospitalization for respiratory diseases, exacerbation of asthma, decline in lung function, and
restricted life activity. Small deficits in lung function, higher risk of chronic respiratory disease and
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2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
increased mortality have also been associated with chronic exposure to respirable particulate air
pollution [1] Epidemiological studies have demonstrated a consistent increased risk for
cardiovascular functions in relation to both short- and long-term exposure to the present-day
concentrations of ambient particulate matter [2]. Exposure to the fine airborne particulate matter is
associated with cardiovascular functions and mortality in older and cardiac patients [3]. Volatile
organic compounds (VOCs), which are molecules typically containing 1–18 carbon atoms that
readily volatilize from the solid or liquid state, are considered a major source of indoor air pollution
and have been associated with various adverse health effects including infection and irritation of
respiratory tract, irritation to eyes, allergic skin reaction, bronchitis, and dyspnea [4: 5: 6].
A two dimensional advection-diffusion mathematical model of primary and secondary
pollutants of an area source with chemical reaction and gravitational settling is presented [7].
Rudraiah et al. have studied the atmospheric diffusion model of secondary pollutants with settling
[8]. Khan and Venkatachalappa et al. have presented a time dependent mathematical model of an air
pollutant with instantaneous and delayed removal [9: 10]. The above models are analytical in nature
with simple form of wind velocity and eddy diffusivity under restrictive assumptions.
Lakshminarayanachari et al. have studied the mathematical model with chemically reactive
pollutants without considering the settling velocity on secondary pollutants [11]. Mathematical
models are important tools and can play a crucial role in the methodology developed to predict air
quality. A numerical model for primary and secondary pollutants in the atmosphere with more
realistic wind velocity and eddy diffusivity profiles by considering the various removal mechanisms
such as dry deposition, wet deposition and gravitational settling velocity is presented.
2. MODEL DEVELOPMENT
The physical problem consists of an area source which is spread out over the surface of the
city with finite down wind and infinite cross wind dimensions. We assume that the pollutants are
emitted at a constant rate from uniformly distributed area source. The major source being vehicular
exhausts due to traffic flow and all other minor sources are aggregated. The vertical height extends
up to mixing height 624 meters above which pollutants do not rise due to the temperature profile of
the atmosphere. We have considered the source region within the urban centre (0 ≤ x ≤ l) which
extends up to l = 6 km from the origin and source free region beyond l (l ≤ x ≤ X0). We compute the
concentration distribution till the desired down wind distance X0 = 12 km i.e. 0 ≤ x ≤ X0. We have
considered two layers, surface layer and planetary boundary layer to evaluate the wind velocity as
accurate as possible. The physical description of the model is shown schematically in figure 1.
Fig 1. Physical Layout of the Model
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3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
2.1 PRIMARY POLLUTANT
The basic governing equation of primary pollutant
∂C p
∂t
+U ( z)
∂C p
∂x
=
∂C p 
∂ 
 Kz ( z)
 − k + K wp C p ,
∂z 
∂z 
(
)
(1)
where C p = C p ( x, z , t ) is the ambient mean concentration of pollutant species, U is the mean wind
speed in x-direction, K z is the turbulent eddy diffusivity in z-direction, k wp is the first order
rainout/washout coefficient of primary pollutant C p and k is the first order chemical reaction rate
coefficient of primary pollutant C p .
We assume that the region of interest is free from pollution at the beginning of the emission. Thus
the initial conditions are
Cp = 0 at t = 0,
0 ≤ x ≤ X0 and 0 ≤ z ≤ H,
(2)
Cp = 0 at x = 0,
0 ≤ z ≤ H and ∀ t > 0.
(3)
The air pollutants are being emitted at a steady rate from the ground level and are removed from the
atmosphere by ground absorption
Kz
∂C p
∂z
= VdpC p − Q at
= VdpC p
Kz
∂C p
∂z
= 0 at
at

z = 0, 0 < x ≤ l 
 ∀t > 0 ,
z = 0, l < x ≤ X 0 

z = H , 0 < x ≤ X 0 , ∀t > 0 ,
(4)
(5)
where Q is the emission rate of primary pollutant species, l is the source length in the downwind
direction, Vdp is the dry deposition velocity, X0 is the length of desired domain of interest in the wind
direction and H is the mixing height.
2.2 SECONDARY POLLUTANT
The basic governing equation for the secondary pollutant Cs is
∂Cs
∂C
∂C 
∂C s
∂ 
+ U ( z ) s =  K z ( z ) s  + Ws
+ Vg kCs − K ws Cs .
∂t
∂x
∂z 
∂z 
∂z
(6)
The appropriate initial and boundary conditions on Cs are :
Cs = 0 at t = 0, for 0 ≤ x ≤ X0 and 0 ≤ z ≤ H,
(7)
Cs = 0 at x = 0, for 0 ≤ z ≤ H
(8)
and ∀t > 0.
Since there is no direct source for secondary pollutants, we have
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4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
Kz
∂C s
+ Ws Cs = Vds Cs
∂z
Kz
∂Cs
= 0 at
∂z
z = 0, 0 ≤ x ≤ X 0 , ∀t > 0 ,
at
z = H , ∀t > 0 ,
(9)
(10)
where Kws is the first order wet deposition coefficient of secondary pollutants, Vg is the mass ratio of
secondary particulate species to the primary gaseous species which is being converted, Ws is the
gravitational settling velocity and H is mixing height.
3. METEOROLOGICAL PARAMETERS
In order to solve the equations (1) and (6), it is essential to know the profiles of eddy
diffusivity and wind speed for various atmospheric stability conditions. It is assumed that the surface
layer terminates at z = 0.1k (U∗ f ) for neutral stability condition, where k is the Karman’s Constant
≈ 0.4, f is the Coriolis parameter and U* is the friction velocity.
For stable case, the surface layer extends up to z = 6L, where L is the Monin-Obukhov
stability length parameter. The following wind velocity profiles are used.
In the surface layer, logarithmic profiles are used for neutral stability with z < 0.1k (U∗ f ) , ie
U=
U∗  Z + Z0 
ln 
 (within surface layer ) .
k
 Z0 
For stable flow with 0 <
For stable flow with 1 <
z
< 1,
L
z
< 6,
L
U=
U=
(11)
U∗   Z + Z0
l n 
K   Z0

U∗
K
 α 
 + z .
 L 

  Z + Z0 

l n 
 + 5.2  .
  Z0 



(12)
(13)
In the planetary boundary layer, above the surface layer, power law scheme has been used
p
 Z − Z SL 
U = U g − U SL 
 + U SL ,
 Z m − Z SL 
(
)
(14)
where Ug is the geostrophic wind, ZSL is the top of the surface layer, USL is wind at ZSL, Zm is the
mixing height H and p is an exponent which depends up on the atmospheric stability.
0.2

We have used, p = 0.35
0.5

for neutral condition
for slightly stable flow
for stable flow .
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5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
The following eddy – diffusivity profiles are used for the entire boundary layer (surface layer
and planetary boundary layer)
K z = 0.4 U*Ze−4 Z / H , for neutral case
(15)

 −bη
kU ∗ Z
U
 Z  −1/ 2
KZ = 
and µ = ∗ . (16)
 e , for stable case b = 0.91, η =   µ
fL
L
 0.74 + 4.7 Z / L 
U* is the friction velocity and f is the Coriolis parameter (= 10-4).
4. NUMERICAL SOLUTION
We have used Crank-Nicolson implicit finite difference method for the solution of the
equations (1) and (6).
The derivatives are replaced by the arithmetic average of its finite difference approximations
th
at the n and ( n + 1)
can be written as
th
time steps. Then equation (1) at the grid points (i, j ) and time step n + 1 2
1
n
n + 1
n+
n
n + 1


C 
C 
∂C


p
2+ 1 U ( z) ∂Cp + U ( z) ∂Cp

 = 1  ∂  K ( z) ∂ p  + ∂  K ( z) ∂ p 
z


 2  ∂z  z
2
∂t ij
∂x ij
∂x ij
∂z 
z
∂z 
ij ∂ 
ij 



−
1
n
n
( k + kwp ) (Cpij + Cpij+1 ) , i = 1,2,.....,
2
j = 1,2,.....
(17)
On simplifying, equation (17) can be written as
n
n+
n+
n+
n
n
n
n
Aj C pi+−1ij + B j C pij11 + D j C pij1 + E j C pij1 1 = Fj C pi −ij + G j C pij −1 + M j C pij + N j C pij +1 ,
−
+
(18)
for each i = 2,3,4,….. i max l …… i max X 0 , j=2,3,4,……jmax-1 and n=0,1,2,3,……, .
and i maxl and imaxX0 are the i values at x = l and X0 respectively and jmax is the value of j at z = H.
The finite difference equations for the secondary pollutant Cs obtained from the partial differential
equation (6) can be written as
n+1
n+1
n+1
n+1
n
n
n
ACs i−1j +BCs ij−1 +DCs ij +ECs ij+1 = FCsni−1j +GCs ij−1 +MjCs ij +NjCs i j+1 +VgkCsnij
j
j
j
j
j
j
, (19)
The above system of equations has a tridiagonal structure and is solved by Thomas Algorithm [12].
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6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
5. RESULTS AND DISCUSSIONS
A numerical model to study the effects of primary and secondary pollutant concentration
horizontally and vertically is developed. The results of this model have been illustrated graphically in
figures 2 - 7 to analyse the dispersion of air pollutants in the urban area downwind and vertical
direction for stable and neutral conditions of atmosphere.
In figure 2 the ground level concentration at various heights for primary and secondary
pollutants of stable case is studied. For removal mechanisms Ws=0 and Vd=0 for stable and neutral
cases are analysed. The primary pollutant concentration ‫ ܥ‬ሺܺ, 2ሻ attains maximum value of 180. As
height increases i.e. ‫ ܥ‬ሺܺ, 4ሻ and ‫ ܥ‬ሺܺ, 6ሻ the primary pollutant concentration decreases up to 7kms
and there is no substantial decrease with height thereafter. This is because we have considered
source region up to 6kms and no source beyond 6 to 12 kms. The same effect is observed for
secondary pollutants but the concentration of secondary pollutants is maximum at the end of source
region and outside source region. The ground level concentration of secondary pollutants increases
up to 6 kms and remains constant thereafter in source free region. As height increases the
concentration of secondary pollutants decreases.
In figure 3 the concentration at various distances for primary and secondary pollutants of
stable case is studied. As distance increases the primary pollutant concentration increases in the
source region and is maximum at the end of source region and is rapidly decreases thereafter. The
concentration is zero around 55mts height for stable case because we have considered the sources at
the ground level. The secondary pollutant concentration increases with downwind distance and is
maximum in the source free region. The concentration of secondary pollutants is zero about 70mts
height.
In figure 4 the Ground level concentration at various heights of primary and secondary
pollutants for neutral case is obtained. It is observed that the pollutant concentration is 55 at the end
of source region. The concentration is low in neutral atmosphere when compared to stable case.
This shows that neutral atmosphere enhances vertical diffusion.
In figure 5 the same effect is observed but the concentration of primary and secondary
pollutants are zero at the heights 350 and 400 respectively. This shows that neutral condition
enhances vertical diffusion and the concentration is low in downwind distance when compared to
stable condition.
0.06
200
Primary pollutants
180
ws=0
Vd=0
C(X,2)
Ws=0
Vd=0
0.05
Secondary pollutants
160
120
0.04
Concentration
Concentration
C(X,2)
C(X,4)
140
C(X,6)
100
80
60
40
C(X,4)
0.03
C(X,6)
0.02
0.01
20
0
0.00
0
2000
4000
6000
8000
10000
12000
0
Distance X
2000
4000
6000
8000
10000
12000
Distance X
Fig. 2. Ground level concentrations at various heights for primary and secondary Pollutants
(stable case)
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7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
In figure 6 the ground level concentration at various heights for primary and secondary
pollutants with removal mechanisms Ws=0.001 and Vd=0.01 for stable case is analysed. From the
graph we notice that the concentration of primary and secondary pollutants decreases as height
increases along downwind distance.
250
0.060
Primary pollutants
Ws=0
Vd=0
Secondary pollutants
0.055
Ws=0
Vd=0
0.050
200
0.045
C(6000,Y)
150
Concentration
Concentration
0.040
C(3000,Y)
100
C(6000,Y)
0.035
0.030
C(9000,Y)
0.025
0.020
0.015
50
C(3000,Y)
0.010
C(9000,Y)
0.005
0
0.000
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
Height Y
50
60
70
80
90
100
Height Y
Fig. 3. Concentrations at various distances for primary and secondary Pollutants (stable case)
60
Primary pollutants
Ws=0
Vd=0
C(X,2)
50
C(X,4)
Ws=0
Vd=0
0.008
C(X,6)
Concentration
Concentration
40
Secondary pollutants
0.010
30
20
C(X,2)
C(X,4)
C(X,6)
0.006
0.004
0.002
10
0.000
0
0
2000
4000
6000
8000
10000
0
12000
2000
4000
6000
8000
10000
12000
Distance X
Distance X
Fig. 4. Ground level concentrations at various heights for primary and secondary Pollutants
(neutral case)
In figure 7 the concentration at various distances of primary and secondary pollutants with
removal mechanisms Ws=0.0001 and Vd=0.01 for neutral case is studied. It is observed that the
concentration of primary and secondary pollutants is high in the lower heights and as move upwards
the concentration decreases and is zero around 350 mts in neutral case.
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8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
70
Primary pollutants
60
Secondary pollutants
0.010
W s=0
Vd=0
C(6000,Y)
W s=0
Vd=0
0.009
C(6000,Y)
0.008
50
Concentration
Concentration
0.007
40
C(3000,Y)
30
20
0.006
0.005
C(3000,Y)
0.004
0.003
C(9000,Y)
0.002
C(9000,Y)
10
0.001
0.000
0
0
50
100
150
200
250
300
0
350
100
200
300
400
500
Height Y
Height Y
Fig.5. Concentrations at various distances for primary and secondary pollutants (neutral case)
60
Prim ary pollutant
S econdary pollutants
W s=0.001
Vd=0.01
C(X,2)
50
0.005
W s=0.001
V d=0.01
C (X ,2)
C (X ,4)
Concentration
40
Concentration
0.004
C(X,4)
C(X,6)
30
20
C (X ,6)
0.003
0.002
0.001
10
0.000
0
0
2000
4000
6000
8000
10000
0
12000
2000
4000
6000
8000
10000
12000
D istance X
Distance X
Fig. 6. Ground level concentration at various heights for primary and secondary pollutants
(stable case) with removal mechanisms Ws=0.001 and Vd=0.01
Primary pollutants
35
25
W s=0.001
Vd=0.01
C (6000,Y)
0.0030
0.0025
Concentration
30
Concentration
secondary pollutants
0.0035
Ws=0.001
Vd=0.01
20
15
C(80,Y)
10
C (9000,Y)
0.0020
0.0015
C (3000,Y)
0.0010
C(40,Y)
C(120,Y)
5
0.0005
0
0.0000
0
50
100
150
200
250
300
0
350
50
100
150
200
250
300
350
400
450
H eight Y
Height Y
Fig.7. Concentrations at various distances for primary and secondary pollutants with removal
mechanisms Ws=0.001 and Vd=0.01 (neutral case)
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9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME
6. CONCLUSIONS
A numerical model for the computation of the ambient air concentration emitted from an
urban area source undergoing various removal mechanisms and transformation process is presented.
The concentration of primary pollutants and secondary pollutants attains peak value at the downwind
end of the source region. The concentration of primary and secondary pollutants is less in magnitude
for neutral atmosphere when compared to stable condition. The neutral atmospheric condition
enhances vertical diffusion carrying the pollutant concentration to greater heights and thus the
concentration is less at the surface region. Hence neutral case is favorable condition in air pollution
point of view.
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