Biomedical Engineering Reference
In-Depth Information
where
D
h
is the distance between the two heaters,
T
h
is the heating period,
k
is the thermal conductivity
of the capillary material, and
q
0
0
is the heat flux inside the capillary wall. Introducing the dimensionless
variables
t
*
¼ t
/
T
h
,
x
*
¼ x
/
D
h
, and
q
¼ qk
=ðq
0
0
$
D
h
Þ
and solving the corresponding dimensionless
energy equation result in the dimensionless temperature
[44]
:
f
N
1
l
n
F
2
n
exp
ðl
n
Þ
F
1
n
exp
ð
2
l
n
Þ
1
q
ðx
;
exp
ðl
n
x
Þ
t
Þ¼<
n¼N
F
2
n
exp
ðl
n
Þ
F
1
n
exp
ð
2
l
n
Þ
1
F
1
n
exp
ðl
n
x
Þ
g
nt
Þ
þ
exp
ði
2
p
(6.32)
where
<
denotes the real part of a complex number,
i
is the imaginary unit, and
(
F
1
n
¼
0
:
5
n ¼
0
:
F
2
n
¼
0
:
5
1
n
ns
0
:
F
1
n
¼ F
2
n
¼
n
½
1
ð
1
Þ
2
p
q
b
2
l
n
¼
þ i
2
p
nh
(6.33)
2
¼
2
hD
h
R
o
½kðR
o
R
i
Þ
and
h ¼ D
h
with
b
=
ðaT
h
Þ
.
Figure 6.36
shows the temperature distribution
as a function of time.
Since the surface tension
s
of the liquid depends on the temperature:
s
lg
ðqÞ¼s
lg
0
þ gðq q
0
Þ
(6.34)
where
s
lg0
is the surface tension at the reference temperature
q
0
and the temperature field is a function
of the position
x
, the surface tension is also a function of the position:
s
lq
ðqÞ¼f ½qðxÞ ¼ gðxÞ¼s
lg
ðxÞ:
(6.35)
The velocity
u
can be then determined through the force balance equation
[44]
:
8
v
R
2
u þ
rR
i
L
plug
s
lg
ðx þ LÞ
cos
f
a
s
lg
ðxÞ
cos
f
r
¼
0
d
u
d
t
þ
2
(6.36)
where
L
plug
is the length of the plug,
R
i
is the inner radius of the capillary, and
n
is the kinetic viscosity
of the plug liquid. The receding and advancing contact angles are denoted by
f
r
and
f
a
, respectively.
The three terms discussed above represent the acceleration, the friction, and the thermocapillary force,
respectively. The solution for the plug velocity is:
B
A
½
1
exp
ðAtÞ
u ¼
(6.37)
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