Environmental Engineering Reference
In-Depth Information
integrand in the first and the second integrals, we obtain
N
0
erf
x
2
√
Dt
−
N
0
√
π
x
2
√
Dt
2
e
−
η
v
(
x
,
t
)=
−
d
η
=
−
,
x
2
√
Dt
where the
Gauss error function
erf
x
is defined by
x
2
√
π
2
e
−
ξ
erf
x
=
d
ξ
.
0
Finally, the solution of PDS (3.57) is
N
0
1
erf
N
0
erfc
x
2
√
Dt
x
2
√
Dt
N
(
x
,
t
)=
−
=
.
3.6.3 Diffusion from an Instant Plane Source
We still use diffusion of mixed materials into a silicon plate to describe the physical
model. The instant plane refers to the part of the plate surface where a very thin
layer of mixed material is deposited. In a very short time period from the start of
diffusion, all the mixed material on the plate is inside the silicon plate, and there
is no further material being mixed into the plate surface afterwards. Therefore, the
total amount of mixed material inside the silicon plate is constant with respect to
time. As the thickness of the mixed material layer deposited on the plate surface
is infinitesimal, such kind of diffusion is sometimes called the
diffusion from the
infinitesimal layer
. For the one-dimensional case, the particle density
N
(
x
,
t
)
should
then satisfy
⎧
⎨
N
t
=
DN
xx
,
−
∞
<
x
<
+
∞
,
0
<
t
,
⎧
⎨
N
0
, |
x
|≤
h
,
(3.60)
⎩
N
(
x
,
0
)=
ϕ
(
x
)=
0
,
|
x
| >
h
,
⎩
where
h
is a very small positive constant. The total particle number is
Q
=
2
hN
0
.
The solution of PDS (3.60) is, by the result in Section 3.4,
+
∞
h
N
(
x
,
t
)=
ϕ
(
ξ
)
V
(
x
,
ξ
,
t
)
d
ξ
=
N
0
V
(
x
,
ξ
,
t
)
d
ξ
(3.61)
−
∞
−
h
h
Q
2
√
π
1
2
h
e
−
(
x
−
ξ
)
2
Q
2
√
π
Dt
e
−
(
x
−
ξ
)
2
=
4
Dt
d
ξ
=
,
(3.62)
4
Dt
Dt
−
h
¯
−
<
ξ
<
where the mean value theorem of integrals has been used, and
h
h
.
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