2. ENVIRONMENTAL ChEMISTRy & TEChNOLOGy 2.1. Lectures

Chem. Listy, 102, s265–s1311 (2008) Environmental Chemistry & Technology
2
∂C
∂ C
ux = Dy
2 z
∂x
∂y
( x)
+ D ( x)
2
∂ C ∂C
+ W 2
∂z
∂z
Here u x is a wind speed in the x direction; all other variables
are the same as in equation (2).
To be completed, the PDE (3) needs to have specified
boundary conditions. The first condition follows from an assumption
of continuous point source with constant strength
located in (0,0,h) coordinates:
The next two conditions follow from the natural assumption
that pollutant concentration approaches zero far from the
source in the lateral y directions and high above the source:
( )
(3)
C x, +∞ , z = 0 .
(3b)
( )
C x, −∞ , z = 0 .
(3c)
C( x, y , +∞ ) = 0 .
(3d)
The last boundary condition is that pollutant deposition
onto the ground occurs at a rate proportional to local air
concentration 3 (for simplicity, the flat ground is taken into
account only):
⎡ ∂C
⎤
⎢
D W C
⎥
vC
⎣ ⎦
( ∞ ) + = [ ] = 0
z z
∂z
z=
0
.
(3a)
(3e)
The deposition velocity v depends on many factors such
as type and size of pollutant particles, the roughness of terrain
and its other surface properties and the meteorological
conditions.
A n a l y t i c a l m o d e l
The analytical solution of equation (3) with respect to
boundary conditions (3a-e) is derived in (refs. 3,5,6 ), and it is
done by expression (4):
(4)
s387
The standard deviations of plume width and height, σ y
and σ z , are defined in terms of their respective diffusion coefficients
3,6 :
The error function erfc (ξ) is defined in (ref. 4 ) as:
n u m e r i c a l M o d e l
In our model we have decided to use the well know
method of lines, which was frequently used for solving different
kinds of PDEs by classical analog and hybrid computers.
In order to perform this method the given PDE (3) must be
transformed into the system of ordinary differential equations
(ODEs). To do that the discretization of all variables except
one must be done. In our case we have discretized y (points
j, step Δy) and z (points k, step Δz) variables and we have
let the x variable continuous (because of the assumed wind
direction).
The obtained system of ODEs is as follows:
( , , )
dC x j k
∂x
=
( ) ( , + 1, ) − 2 ( , , ) + ( , −1,
)
Dy x C x j k C x j k C x j k
⋅ +
u y
2
x
∆
( ) ( , , + 1) − 2 ( , , ) + ( , , −1)
D x C x j k C x j k C x j k
+
z
⋅ +
u z
2
x
∆
( , , + 1 ) − ( , , −1)
.
W C x j k C x j k
+ ⋅
ux 2∆z
The system of ODEs (6) is solved inside the
interval along the y direction, and inside the along the
z direction, where N j and N k are natural numbers. Thus the
total number of ODEs is (2N j + 1) × (N k + 1) and all equations
can be solved for current time point in parallel.
The boundary condition (3a) is transformed to new initial
conditions (7a):
C
C
( 0,
0,
H )
Q
=
u ∆y∆z
x
( 0,
j,
k)
= 0 otherwise .
(7)
(7a)
And new boundary conditions (7b–e) for the ODEs system
(7) with respect to (3b–e) are:
(5)
(6)