Biomedical Engineering Reference
In-Depth Information
The differentials in the conduction terms on opposing bound-
ary faces are approximated via a Taylor series expansion with all
higher order terms dropped:
T
∂
T
< 0
∂
r
= −
∂
∂
Q
x
QQ
−
x
dx
x
x
+
dx
Heat flux
= −
∂
∂
Q
y
x
QQ
−
dy
.
y
y
+
dy
(1.7)
= −
∂
∂
Q
z
r
QQ
−
x
dz
z
z
+
dzx
FIGURE 1.2
A positive flow of heat along a coordinate occurs by
application of a negative gradient in temperature along the direction
of flow.
When Fourier's law is substituted for the heat flows, the
boundary interactions are written in terms of the temperature
gradients
to be accounted for in applying conservation of energy to this
elemental volume.
The time rate of change of energy stored in the elemental sys-
tem is described as
∂
∂
∂
∂
T
x
QQ
−
=
dx dy dz
⋅
⋅
x
x
+
dx
x
∂
∂
∂
∂
T
y
QQ
−
=
dx dy dz
⋅
⋅
.
y
y
+
dy
y
(1.8)
dT
dt
=ρ
Ec
dx dy dz
.
⋅
⋅
(1. 5)
st
p
∂
∂
∂
∂
T
z
QQ
−
=
dx dy dz
⋅
⋅
z
z
+
dzx
z
The individual conduction exchanges across the system bound-
ary are written in terms of the Fourier law:
The constitutive Equations 1.5 and 1.8 may be substituted in
the conservation of energy Equation 1.1 for the limited bound-
ary interactions assumed in this analysis, noting that each
resulting term contains the system volume,
dx
⋅
dy
⋅
dz
, which can
be divided out:
∂
∂
T
x
Qk dy dz
=− ⋅
⋅
x
∂
∂
T
y
Qk dx dz
=− ⋅
⋅
.
y
(1.6)
∂
∂
T
t
∂
∂
∂
∂
T
x
+
∂
∂
∂
∂
T
y
+
∂
∂
∂
∂
T
z
ρ
c
=
k
k
k
.
∂
∂
T
z
(1.9)
p
Qk dx dy
=− ⋅
⋅
x
y
x
z
Q
z+dz
Q
y+dy
z
y
x
Q
me t
dz
Q
x+dx
Q
x
E
st
Q
y
dy
dx
Q
z
FIGURE 1.3.
A small interior elemental system for analysis of heat conduction consisting of differential lengths
dx
,
dy
, and
dz
in Cartesian coor-
dinates as identified within a larger overall system.
