Where ReD is Reynolds number for external airflow, Pr is Prandtl’s number and μ is dynamic viscosity. Note that Prandtl’s number for air at -20°C is 0.75. Considering ice growth, a quasi-steady process, by energy balance at water-ice interface Eq. (3.23) can be obtained in this case as well. Due to the radial ice growth, the velocity of freezing front is written as Eq. (3.30).
According to energy balance Eq. (3.31) could be obtained as,
Where Ai is ice-water interface area and Ao is ice-air interface area. By substituting Eq. (3.23) and (3.30) in Eq. (3.31), the following equation is obtained,
(3.32)
Now, we can find an equation for ice growth rate. Through solution of heat equation in spherical coordinates, we have,
(3.33)
T(r = ri) = Tf and T(r = r0) = Ts are ice-water and ice-air interface temperatures, respectively. The boundary conditions are written as,
By applying boundary conditions on Eq. (3.34), we have,
We define,
Thus,
(3.35)
(3.36)
The surface temperature is as follows:
where
Thus,
And
Where Biot number is defined as:
By writing the energy balance at the ice-water interface one has:
(3.38)
By substitution of θs from Eq. (3.37) in Eq. (3.39):
And by simplification of this equation, one finds
Through integration of both sides:
And we have the initial condition for ice growth as:
Thus,
Through Eq. (3.40), the radius of ice as a function of time, i.e. ice growth rate, in high air flow condition is obtained. Irajizad et al. [5] plotted Eq. (3.40) for different air velocities and temperatures. As shown in Figure 3.9, the initial rate of ice growth is low. However, the rate of ice growth increases as the ice growth proceeds.
A
