Biomedical Engineering Reference
In-Depth Information
Table 7.2
The Coefficients at
each node of the control
volumes
Node
a
P
a
E
a
W
b
1
3,000
1,000
0
3,000
+
2,000
T
L
2
2,000
1,000
1,000
3,000
3
2,000
1,000
1,000
3,000
4
3,000
0
1,000
3,000
+
2,000
T
R
The above set of equations yields the steady state temperature distribution for the
given situation. For a one-dimensional steady heat conduction process we obtained
the algebraic equations in matrix form. Because the problem only involves a small
number of nodes, the matrix can be solved easily. Analytically the matrix can be
solved by methods such as Gaussian elimination which is discussed later in Sect. 7.3.
This matrix algorithm will be discussed in the next section.
7.2.4
One-dimensional Steady State Convection-diffusion
Respiratory flows involve moving air (considered as an incompressible fluid) and as
such the effects of convection are important. The convection terms moves the scalar
variable,
φ
through the flow domain in the flow direction determined by the velocity
components while the diffusion process distributes the variable in all directions by its
diffusion coefficient,
and its gradient. If we ignore the source terms and focus on
the steady convection and diffusion of the variable
φ
in one dimension, its equation
is
dφ
dx
d
(
uφ
)
dx
d
dx
=
(7.38)
Note that the convection diffusion equation, Eq. (7.38), is representative of the
momentum equation if
φ
=
u
and
=
ν
, and the energy equation if
φ
=
T
and
=
ν
Pr
.
In this section we demonstrate the FV approach. The 1D mesh for nodal point
P
, and
its neighbouring nodes is given in Fig.
7.17
.
Integration of Eq. (7.38) over the control volume gives
e
dφ
dx
w
dφ
dx
(
uφ
)
e
−
(
uφ
)
w
=
e
−
convective flux
w
diffusive flux
where the two diffusive fluxes are approximated with a linear interpolation as
e
e
=
dφ
dx
φ
E
−
φ
P
x
PE
e
=
D
e
(
φ
E
−
φ
P
);
w
dφ
dx
w
φ
P
−
φ
W
x
WP
w
=
=
D
w
(
φ
P
−
φ
W
)
where
D
e
=
e
/x
PE
and
D
w
=
w
/x
WP
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