Investigation of the freezing process of water droplets based on average and local initial ice fraction

ABSTRACT

We presented both experimental and numerical studies on the freezing of impacting water droplets on a cold surface at different surface temperatures. The numerical model consists of two parts. The first one is to determine the temperature evolution of the droplet prior to the occurrence of freezing by solving the heat conduction equation, and the second one is to simulate the freezing process via an extended phase change model. Experiments were conducted to observe and record the freezing process. The droplet profile and the propagation of moving water–ice interface during freezing were obtained from image analysis. Based on the numerical pre-recalescence temperature of the droplet, the average initial ice fraction and local initial ice fraction were obtained. Then, both the two kinds of initial ice fractions were used to figure out the difference that they brought to the predicted freezing process. Through a comparison of the experimental observations and the numerical predictions, the freezing process predicted by using local initial ice fraction showed a better agreement with the experiment than using average initial ice fraction.

KEYWORDS:

• Freezing process
• heat transfer
• water droplet
• water–ice interface
• local initial ice fraction

Nomenclature

$c$	=	specific heat (J/kg·K)
$D_0$	=	initial droplet diameter (m)
$D_h$	=	horizontal width of the droplet (m)
$D_v$	=	vertical length of the droplet (m)
$f_a$	=	average initial ice fraction
$f_l$	=	local initial ice fraction
$h$	=	enthalpy per unit volume (J/kg)
$h_c$	=	heat transfer coefficient (W/m2·K)
$h_s$	=	specific sensible enthalpy (J/kg)
$L$	=	latent heat of phase change (J/kg)
$Oh$	=	Ohnesorge number
$Re$	=	Reynolds number
$t$	=	time (s)
$T$	=	temperature (°C)
$T_e$	=	equilibrium freezing temperature (°C)
$T_l$	=	lower bound of the phase change temperature range (°C)
$T_r$	=	droplet temperature at pre-recalescence instant
$T_{r,a}$	=	average temperature of the entire droplet at pre-recalescence instant (°C)
$T_u$	=	upper bound of the phase change temperature range (°C)
$T_w$	=	surface temperature (°C)
$U_0$	=	impact velocity of the water droplet (m/s)
$V$	=	volume of the droplet during the freezing process (m3)
$V_0$	=	initial volume of a liquid water droplet (m3)
$We$	=	Weber number

Greek Symbols

$\lambda$	=	thermal conductivity (W/m·K)
$\gamma$	=	liquid volume fraction
$\rho$	=	density (kg/m3)
$\sigma$	=	surface tension coefficient (N/m)
$\mu$	=	dynamic viscosity (Pa$\cdot$s)
${\theta }_e$	=	equilibrium contact angle (°)

Subscripts

$i$	=	ice
$m$	=	water–ice mixture
$w$	=	water

Additional information

Funding

This work was supported by the National Key R&D Program of China under Grant No. 2017YFC0803300; National Natural Science Foundation of China under Grant No. U1633203; and R&D project of Civil Aviation Administration of China under Grant No. 20160103.