Biomedical Engineering Reference
In-Depth Information
T
C
(e.g., the temperature where gas and liquid phase are indiscernible), Eötvös and
later Katayama and Guggenheim [4] have worked out the semi-empirical relation
n
γ γ
æ
T
T
ö
*
=
-
(3.3)
1
ç
÷
è
ø
C
where
g
*
is a constant depending on the liquid and
n
is an empirical factor, which
value is 11/9 for organic liquids. Equation (3.3) produces very good results for
organic liquids. If the temperature variation is not very important, and taking into
account that the exponent
n
is close to 1, a good approximation of the Guggenheim-
Katayama formula is the linear approximation
(
)
*
γ γ
=
1
+
α
T
(3.4)
where
a
is a constant. It is often easier and more practical to use a measured ref-
erence value (
g
0
,
T
0
) and consider a linear change of the surface tension with the
temperature
(
)
)
(
1
(3.5)
γ γ
=
+
β
T T
-
0
0
The coefficient
b
can be obtained by remarking that
g
= 0 when
T
=
T
c
:
b
=
-1(
T
c
/
T
0
). Relations (3.4) and (3.5) are shown in Figure 3.4. The value of the refer-
ence surface tension
g
0
is linked to
g
*
by the relation
g
0
=
g
*
(
T
c
/
T
0
)/
T
c
.
Typical values of surface tensions and their temperature coefficients
a
are given
in Table 3.1.
The value of the surface tension decreases with temperature. This property is
at the origin of a phenomenon called the
Marangoni convection
or
thermocapillary
instabilities
(Figure 3.5). Suppose that an interface is locally heated (for example,
by radiation) and locally cooled (for example, by conduction). The value of the
surface tension is smaller in the heated area than in the cooled area. A gradient of
surface tension is then induced at the interface between the cooler interface and the
warmer interface. This imbalance creates tangential forces on the interface, pushing
the fluid from the warm region (smaller value of the surface tension) towards the
Figure 3.4
Representation of the relations (3.4) and (3.5).
