Institut za rudarstvo i metalurgiju Bor

z and r – cylindrical coordinates
ρ′ - oxygen disappearance rate as the
result of reactions
Considering that the rate of combustion
on the wall of coal seam in the channel, at
temperature higher than 1000 o C, is sufficiently
high, the diffusion of oxygen has a
decisive role in this process, where its
concentration on the wall is approaching to
zero:
C r ≈ 0
(2)
In such case, a stationary turbulent
flux, disregarding the longitudinal diffusion
– along the channel, equation becomes
as follows:
r ∂C 1 ∂ ∂C
u = DT
r
(3)
∂z
r ∂r
∂r
Where:
DT – coefficient of turbulent diffusion
Differential equation (3) with the
Dirichlet’s homogenous boundary condition
(2) is resolved using the finite-difference
method. This method is generally based on
the approximation of differential operators in
differential equations to the corresponding
difference operators. Previously, the calculation
domain is divided into a certain grid in
such a way that it would approximate, the
most accurately, to the boundaries of domain
and include a minimum mistake into the
configuration of such domain. The above
replacement of differential operators by corresponding
operators is made as follows:
⎛ ∂C
⎞ C1
− C2
⎛ ∂C
⎞ C3
− C4
⎜ ⎟ ≈ ; ⎜ ⎟ ≈
⎝ ∂x
⎠0
2h
⎝ ∂y
⎠ 2h
⎛ 2
∂ C ⎞ C1
+ C2
− 2C
⎜ ⎟
o
⎜
≈
2
2
x
⎟
⎝ ∂ ⎠o
h
⎛ 2
∂ C ⎞ C3
+ C4
− 2C
⎜ ⎟
o
⎜
≈
y2
⎟
h2
⎝ ∂ ⎠o
0
;
Here, „O“ is a midpoint where the approximation
of operators is made, which is
surrounded by „1“ and „2“ points at x-axis,
that is, by „3“ and „4“ points at the y-axis,
while „h“ step is the grid. The procedure,
that is presented by the equations for „O“
point, is further written for each point of the
grid of the whole calculation domain. According
to the described algorithm of the
finite-difference method, resolving the differential
equation (3) becomes the resolving
systems of algebra equations.
After numerical resolution of this system
of equations (by the Newton-Raphson’s
method), the values of concentration in node
points of the grid are obtained, i.e. change in
concentration of oxygen at length of channel
C = C(
z ) in a form of discrete values. At
that, the discretization is made using a
square-shape grid, the h step.
Similar differential equations may be
also written for the other basic gas components,
such as hydrogen H2, carbon monoxide
CO, carbon dioxide CO2 and methane
CH4, whereby their characteristic parameters
are taken into account, and then
the equations should be resolved according
to the same algorithm as for oxygen O2.
As the result of numerical solutions,
according to the described algorithm, Figure
3 shows curves of changes in concentration
of particular gas components at
channel length for one average stone coal.
In Figure 3:
C - initial concentration of gas components
o
Cz v - average concentration of gas components
in the channel section at distance
„z“ from the channel inlet
No 1, 2010. 133
MINING ENGINEERING