Biomedical Engineering Reference
In-Depth Information
The external power is the sum of the power of the body forces (per unit volume)
P
V
with the body force vector k and the power of the surface forces P
A
with the
stress vector t
n
to be exerted to the body surface
P
V
:
¼
Z
V
P
A
:
¼
Z
A
P
¼
P
V
þ
P
A
with
k
vdV
;
t
n
vdA
:
ð
3
:
133
Þ
The time rate of change by heat transfer is the sum of the heat transfer to the
body surface A due to the heat flow vector q and the heat transfer to the volume
V by radiation heat r.
Q
¼
Z
A
n
qdA
þ
Z
V
qrdV
:
ð
3
:
134
Þ
The time derivations needed in (
3.130
) can be obtained using (
3.118
) and
(
3.131
)
2
as follows
dV
¼
Z
Z
edm
¼
Z
qedV
¼
Z
E
¼
d
dt
q
d
dt
1
2
v
v
q
1
2
ð
v
v
þ
v
v
Þ
dV
m
V
V
V
¼
Z
V
qv
vdV
ð
3
:
135
Þ
Z
udm
¼
Z
V
U
¼
d
dt
q udV
:
ð
3
:
136
Þ
m
Substitution (
3.133
)to(
3.136
)in(
3.130
) leads to the first law of thermody-
namics in global form (for the entire body)
Z
qv
vdV
þ
Z
q udV
¼
Z
k
vdV
þ
Z
Z
þ
Z
t
n
vdA
n
qdA
qrdV
:
V
V
V
A
A
V
|{z}
I
1
|{z}
I
2
ð
3
:
137
Þ
To generate the local balance of (
3.137
), both surface integrals I
1
and I
2
on the
right-hand side are transferred in volume integrals using the G
AUSS
' integral the-
orem as well as (
3.93
) and I
1
:
¼
R
A
t
n
vdA
¼
R
A
n
S
vdA
¼
R
V
rð
S
v
Þ
dV and
I
2
:
¼
R
A
n
qdA
¼
R
V
r
qdV . The underlined integral term in the integral I
1
can
further be transformed using the ''product rule'' and the definitions of the velocity
gradient
L :
¼
v
r
in
r
S
ð Þ¼r ð Þ
v
þ
S
v
ðÞ
|{z}
L
rð Þ
v
þ
S
L
such that substitution in (
3.137
) leads to the following form (the term in paren-
theses represents the balance of linear momentum (
3.120
) and thus vanishes):
