A mathematical model for predicting the densification and growth of frost on a flat plate

The widely used correlation of Hayashi et al. expresses frost density dependent on frost surface temperature only and considerably overpredicts frost density at relatively high frost surface temperatures close to the melting temperature.

“Note that the widely used correlation of Hayashi et al. [24] expresses frost density dependent on frost surface temperature only, and is known to considerably overpredict the frost density at relatively high frost surface temperatures close to the melting temperature [14].”

Tao et al. (1993) stated that the effective diffusivity can actually be higher than the diffusivity of water vapor in the air for the special case of frost, which means diffusion resistance factor more than 1.

“On the other hand, Tao et al. (1993) stated that the effective diffusivity can actually be higher than the diffusivity of water vapor in the air for the special case of frost, which means diffusion resistance factor more than 1.”

The Tao et al. numerical model assumed that the gradient of frost density at the frost surface is zero without justification.

“The Tao et al. [10] numerical model assumed that the gradient of the frost density at the frost surface is zero. They gave no justification for this assumption.”

Tao et al. [10] and Le Gall and Grillot [6] used a saturation model and adjusted the tortuosity factor to force agreement of their model predictions to experimental data, resulting in a tortuosity factor greater than 1.0.

“Tao et al. [10] and Le Gall and Grillot [6] used a saturation model and compared their predicted results to their experimental data. They adjusted the tortuosity factor to force agreement of their model predictions to the experimental data. This resulted in a tortuosity factor greater than 1.0”

The melted water penetrated the frost layer and froze, forming a new, considerably thicker, ice layer.

“The melted water penetrated the frost layer and froze, thus forming a new, considerably thicker, ice layer [11,12].”

The DWC period is determined by ambient and surface conditions including surrounding air temperature or relative humidity, natural or forced convection or air flowing speed, cold surface temperature, and surface roughness value.

“This period was determined by the ambient and surface conditions, including the surrounding air temperature or the relative humidity (RH), the natural or forced convection or the air flowing speed, the cold surface temperature, and the surface roughness value.”

Under natural convection, when the cold plate temperature decreases from ambient to a steady-state low temperature, the frost formation deviates from the three-period process defined by Hayashi et al., mainly during the CG period.

“However, when the frost growth periods followed in a typical frost formation process under natural convection conditions, during which the temperature of the cold plate decreased from an initial ambient temperature to a previously set steady-state low temperature, the frost formation would deviate from the abovedefined three-period formation process, mainly during the CG period.”

As vapor-liquid and liquid-solid phase change occurs, droplets merge and solidify, and the diameters of the solid droplets increase significantly.

“As the vapor-liquid and liquid-solid phase change occurs, the droplets merge and solidify; the diameters of the solid droplets increased significantly.”

The STG period starts from the critical time to a transitional time tt, when a relatively uniform porous layer of frost formed.

“The second period, namely the STG period, started from the critical time to a transitional time, t t , when a relatively uniform porous layer of frost formed.”

After reaching the critical time tc, all coalescent droplets turned into ice particles or crystals.

“After reaching the critical time, t c , all coalescent droplets turned into ice particles or crystals.”

During the FLG period, frost branches formed at the top of the ice crystals, grew in three dimensions, and connected to neighboring frost branches, forming a flat frost layer.

“The entire process of the frost branch formation was referred to as the FLG period when frost branches would form at the top of the ice crystals. These frost branches grew in three dimensions and connected to the neighboring frost branches, thus forming a flat frost layer [9,10].”

During the DWC period, condensing droplets in a subcooling state formed on the cold surface; droplet nucleation occurred first, and coalescence of the droplets occurred immediately after nucleation.

“During the initial cooling period, namely the DWC period, the condensing droplets in a subcooling state formed on the cold surface. As shown in Fig. 2, the droplet nucleation occurs first, as well as the coalescence of the droplets occurred immediately after the nucleation.”

Tao et al. divided the entire frost formation process under natural convection into three periods: dropwise condensation (DWC), solidified liquid tip growth (STG), and densification and bulk growth (DBG).

“In this study, the entire frost formation process, namely the frost nucleation and growth process, was also divided into three periods according to the growth timeline, namely (1) the dropwise condensation (DWC) period, (2) the solidified liquid tip growth (STG) period, and (3) the densification and bulk growth (DBG) period [4].”

For the first frosting type, before the frost layer growth stage, there are two stages: a dropwise condensation stage and a solidification and tip-growth stage.

“Tao [13] focused on the initial stage of frosting and found that there existed two stages before entering the frost layer growth stage. The first was a dropwisecondensation stage where water droplets nucleated and merged on the cold surface. The second was the solidification and tip-growth stage where coalescent droplets solidified, and crystals grew on their tips in one dimension.”

An effective water diffusion coefficient was significantly larger than the standard water diffusion coefficient.

“Tao et al. [72] indicated that an effective water diffusion coefficient was significantly larger than the standard water diffusion coefficient.”

The exact value of F was hard to be evaluated but experimentally ranged from 2.2 to 3.5.

“It was also indicated that the exact value of F was hard to be evaluated but experimentally ranged from 2.2 to 3.5.”