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drop and the tube as between such two surfaces that one is completely enclosed
by the other. This case is equivalent to thermal radiation between two parallel flat
surfaces. Assuming both the water surface and the copper surface are not transparent
for the thermal radiation, and thus only emission and reflection exist, the heat
flux density (in W m −2 ) results in
4 4
T1 T2
˙q = C 12 − , (3.5)
100 100
where T 1 and T 2 are taken at 273 K and 77 K respectively for water and copper
surfaces. The coefficient C 12 is defined as
C 12 =
C S
1
+ 1 − 1 , (3.6)
ɛ 1 ɛ 2
where C S is 5.67 J s −1 m −2 K −4 , a scaled coefficient derived from Stefan-Boltzmann
constant for the convenience of arithmetic. ɛ 1 and ɛ 2 are emissivity of water and
copper, taken as 0.96 and 0.049 respectively [139]. The emissivity of metal is in
general very low because of the high reflectivity.
Substituting these values into Eq. 3.5 and Eq. 3.6, the heat flux density of radiation
is 15.3 W m −2 . This value is negligible compared to heat transfer by forced convection:
6 W m −2 K −1 to 260 W m −2 K −1 at temperature differences of over 150 K
as computed below. Therefore only the forced convection is considered in the
temperature calculation.
The heat transfer by forced convection is governed by
˙Q = h conv A(T − T ∞ ) = cm∆T , (3.7)
∆t
where h conv is the heat transfer coefficient, A is the surface area, A = πd 2 , m is the
mass of the drop, c is the specific heat of water at the temperature T . The initial
condition is:
T(t = 0) = 0. (3.8)
The heat transfer coefficient h conv is calculated by the Nusselt number:
Nu = h conv d
k air
. (3.9)
3.1. Supercooling Method 51