Axisymmetric lattice Boltzmann model for simulating the freezing process of a sessile water droplet with volume change

Droplet freezing not only is of fundamental interest but also plays an important role in numerous natural and industrial processes. However, it is challenging to numerically simulate the droplet freezing process due to its involving a complex three-phase system with dynamic phase change and heat transfer. Here we propose an axisymmetric lattice Boltzmann (LB) model to simulate the freezing process of a sessile water droplet with consideration of droplet volume expansion. Combined with the multiphase flow LB model and the enthalpy thermal LB model, our proposed approach is applied to simulate the sessile water droplet freezing on both hydrophilic and hydrophobic surfaces at a fixed subcooled temperature. Through comparison with the experimental counterpart, the comparison results show that our axisymmetric LB model can satisfactorily describe such sessile droplet freezing processes. Moreover, we use both LB simulations and analytical models to study the effects of contact angle and volume expansion on the freezing time and the cone shape formed on the top of frozen droplets. The analytical models are obtained based on heat transfer and geometric analyses. Additionally, we show analytically and numerically that the freezing front-to-interface angle keeps nearly constant (smaller than 90°).

Figure 1

Schematic of the experimental setup for single water sessile droplet freezing on a subcooled substrate surface.

Figure 2

Schematic of the axisymmetric simulation domain with a water droplet placed on the substrate surface. The boundary conditions used for computations are indicated.

Figure 3

Water droplet profiles with different grid schemes: (a) before freezing; (b) after freezing.

Figure 4

Simulated droplet freezing process on the substrate surface of contact angle $\theta ={90}^{\circ }$ with the ice-water density ratio $\gamma =0.9$: (a) time evolution of the droplet profiles at different freezing stages (time, with the simulated total freezing time as $t^{*}=76$); (b) droplet volume change during the entire freezing process (here $V_0$ is the original liquid droplet volume at $t^{*}=0$); (c) temperature contours at $t^{*}=38$; (d) enthalpy contours and velocity vectors at $t^{*}=38$.

Figure 5

Comparison of water droplet freezing shapes at different times between experiment (left) and simulation (right): (a) substrate contact angle $\theta ={64}^{\circ }$; (b) substrate contact angle $\theta ={98}^{\circ }$.

Figure 6

Comparison of the droplet freezing front height and radii between experimental and simulation results: (a) substrate contact angle $\theta ={64}^{\circ }$; (b) substrate contact angle $\theta ={98}^{\circ }$.

Figure 7

Droplets with the same initial volume on the substrate surface of different contact angle: (a) final frozen droplet shapes; (b) total freezing time versus contact angle with a comparison among the simulations, theoretical predictions, and experiments.

Figure 8

Droplet frozen shapes for different volume expansion coefficient on the substrate surface of contact angle $\theta ={90}^{\circ }$.

Figure 9

Cone angle 2α formed on the frozen droplet: (a) sketch of the tip geometry on the droplet at the last freezing stage with the freezing front-to-interface angle indicated as ${\theta }_u+{\theta }_d$ [10]; (b) cone angle versus volume expansion coefficient; (c) cone angle versus contact angle; (d) time evolution of angles ${\theta }_u$ and ${\theta }_d$ for the case $\theta ={90}^{\circ }$ and $\gamma =0.9$.