Biomedical Engineering Reference
In-Depth Information
treat the mass transfer for an evaporating droplet on a rigorous mathematical basis.
This problem can indeed be mapped into the calculation of the electrostatic poten-
tial generated by a conductive surface obtained by the intersection of two eccentric
spheres forming a biconvex lens. The analogy with droplet evaporation provides
here the following equation for the mass loss [39, 70, 71]:
d
m
d
t
=−
DR(c(T
s
)
−
Hc(T
∞
))φ(θ),
(9)
where
m
is the droplet's mass,
c(T)
is the vapor water concentration at temper-
ature
T
,
D
water vapour diffusion coefficient in air and
φ(θ)
is a dimensionless
function of the contact angle reflecting geometrical effects. All other notations have
been defined. The main advantage of this equation is to bring back the time depen-
dence in the model and, in doing so, helps address the problem of water circulation
in deposited and heated droplets. This approach is justified by the fact that tem-
perature differences inside the droplet and along its interface are all small for the
initial conditions that will be considered in the following. It is hence reasonable
to assume that Eq. (9) also holds here, even if it has been essentially developed
in the literature for isothermal situations. The small temperature gradients that the
system will develop will only slightly perturb the vapor water concentration in the
neighborhood of the interface. But overall thermocapillary phenomena will ensue.
Surface tension temperature dependence in the temperature range of the present
section (30
◦
C
70
◦
C) is described by equation:
σ
T
s
=
σ
0
+
β
×
(T
−
T
0
),
(10)
10
−
3
10
−
4
where
σ
0
=
72
×
N/m and
β
≈−
1
.
6
×
N/(m K) are respectively the
surface tension at temperature
T
0
=
298 K and the temperature coefficient of sur-
face tension that is supposed to be temperature independent. Temperature gradients
along the interface are key here and this is where special numerical efforts have to
be devoted in particular for an accurate computation of
v
∗
and the precise integra-
tion of Eqs (6) and (7) in the computational domain describing the water droplet
(see Fig. 4). Droplet evaporation is always accompanied by a temperature decrease
due to evaporative cooling [7, 31]. This effect has been accounted for in the liter-
ature and is of first importance in isothermal systems where there is no heat input
to balance the corresponding energy loss. It might also become important in the
neighborhood of the contact line due to large mass flux. Considering the small size
of the droplets and the relative large difference
T
s
−
T
∞
(itisalwayslargerthan
10
◦
C), evaporative cooling is assumed to be compensated rapidly and at all times
by the important substrate incoming heat flow.
As discussed before, the numerical method that is used to simulate Eqs (6)-(8)
is based on stationary hypotheses and hence treats evaporating droplets as coupled
thereafter subsystems. No free interfaces have to be accounted for in this approach.
The copper/water and copper/air interfaces are fixed in time whereas the water/air
evolves according to the mass evolution given in Eq. (9). For each time step, spe-
cific finite elements calculation stages are performed for their description. The first
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