Biomedical Engineering Reference
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the nodal points between
W
and
P
and between
P
and
E
. The diffusive fluxes are
evaluated as
dφ
dx
e
A
E
φ
E
−
;
φ
P
x
E
A
E
=
e
dφ
dx
w
A
W
φ
P
−
φ
W
x
W
A
W
=
(7.31)
w
and the source term is
1
V
S
φ
dV
=
S
φ
(7.32)
V
where
S
φ
is assumed to be constant within
V
which is the finite control volume.
The final form of the discretised equation becomes
φ
E
−
φ
P
x
E
φ
P
−
φ
W
x
W
e
A
E
V
w
A
W
V
−
+
S
φ
=
0
(7.33)
It can be seen that the discretised equation represents the difference between the
diffusive fluxes of
φ
at the east and west faces of the control volume equal to the
generation of
φ
and constitutes a balance equation for
φ
over the control volume.
Equation (7.33) can be re-arranged as
e
A
E
x
E
+
φ
P
e
A
E
x
E
φ
E
+
w
A
W
x
W
φ
W
+
1
V
w
A
W
x
W
1
V
1
V
=
S
φ
(7.34)
which gives the rate of change per unit volume. Multiplying out the volume term
and grouping together the coefficients of
φ
E
,
φ
W
, and
φ
P
as
a
E
,
a
W
and
a
P
,gives
a
P
φ
P
=
a
E
φ
E
+
a
W
φ
W
+
b
(7.35)
Where
e
A
E
x
E
w
A
W
x
W
a
P
=
a
E
+
a
W
;
a
E
=
;
a
W
=
;
b
=
S
φ
V
(7.36)
If the values for the diffusion coefficient,
at the control volume faces,
e
and
w
need
to be known then for a uniform grid, a linear interpolation can be performed and we
get:
P
+
E
P
+
W
e
=
;
and
w
=
2
2
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