Hybrid lattice-Boltzmann–finite-difference approach for the simulation of micro-phase-change-material slurry in convective flow

In this paper, we present a hybrid numerical scheme that couples the lattice Boltzmann method (LBM) with the finite difference method (FDM) to model micro-phase-change-material (MPCM) suspensions in a minichannel. Within this framework, the LBM was employed to solve the continuity, momentum, and energy equations for the fluid domain, while a Lagrangian scheme replicates the motion of MPCM particles. The LBM is coupled with an FDM solver which operates under the lumped capacitance assumption to address the phase-change phenomena within the microparticles. This hybrid coupling eliminates the necessity for any specific treatment in handling phase transitions and tracking phase interfaces. The proposed method is first evaluated on classic particle cases, demonstrating its ability to achieve four-way coupling. Furthermore, the current model effectively adapted viscosity changes when integrating the microparticles, obviating the need for homogenous viscosity models. Subsequently, the potential of this approach is demonstrated by examining the influence of the near-wall thermal interaction of MPCM particles considering three scenarios based on particle density: light $({\rho }_p<{\rho }_f)$, neutrally buoyant $({\rho }_p\approx {\rho }_f)$, and dense $({\rho }_p>{\rho }_f)$ microparticles. The hybrid approach further revealed insights into the impact of the volume fraction on the heat transfer coefficient as well as on the overall heat transfer coefficient and performance index from a Lagrangian perspective.

Figure 1

(a) $D2Q9$ and (b) $D3Q19$ models.

Figure 2

The algorithmic framework of the current numerical model.

Figure 3

Schematic for the single-particle sedimentation problem.

Figure 4

Streamwise particle velocity vs time in the current model compared with the experimental setup of ten Cate et al. [56] for four cases.

Figure 5

Normalized vertical particle position plotted against time for the current model along with the findings of the experimental setup by ten Cate et al. [56] for four cases.

Figure 6

Schematic for the elliptical particle sedimentation.

Figure 7

(a) Particle lateral position and (b) particle orientation plotted against the normalized settling position for the current model along with the findings of Xia et al. [58] and Chen et al. [57].

Figure 8

The schematic on the left-hand side represents the two-particle interaction problem. P1 and P2 denote the leading and trailing particles, respectively. Contours on the right-hand side of the figure show the drafting-kissing-tumbling timeframes.

Figure 9

Particle lateral position over time: a comparative analysis with literature (Wang et al. [59], Nie et al. [60], Krause et al. [26], and Feng and Michaelides [61]). Filled symbols denote the leading particle, and unfilled symbols represent the trailing particle.

Figure 10

Validation setup for particle-wall contact model.

Figure 11

Validation of particle rebound height for discrete contact model [53] and the model of Crowe et al. [52] with experimental work [63].

Figure 12

Comparison between the hybrid numerical method with the closed form solution.

Figure 13

The fully developed velocity profile across the height of the channel obtained using the analytical solution against the current model.

Figure 14

The distribution of the local Nusselt number across the normalized local length of the channel is obtained using both the analytical solution and the current model.

Figure 15

Visualization of the computational mesh generated for the model.

Figure 16

A depiction of the setup for obtaining the viscosity of the slurry.

Figure 17

Ratio of dynamic viscosity to that of the base fluid for varying volume fractions.

Figure 18

An illustration of the current model setup with the applied boundary conditions. Note: The particles are not drawn to scale.

Figure 19

Distinctive sedimentation patterns exhibited by relatively light, neutrally buoyant, and heavy micro-phase-change-material (MPCM) particles.

Figure 20

A diagram illustrating the different behaviors of heavy [red (dark gray)], neutrally buoyant [black], and light [blue (light gray)] micro-phase-change-material (MPCM) particles for different volume fractions at various points of the normalized local position.

Figure 21

Normalized temperature contour for the neutrally buoyant micro-phase-change-material (MPCM) particles. (a) Normalized temperature for the full channel at $\mathrm{R}{\mathrm{e}}_{Dh}=200$. (b) A segment of the channel for the volume fraction of 0.1%. (c) A segment of the channel for the volume fraction of 0.8%.

Figure 22

Normalized temperature contours for various conditions: (a) Heavy micro-phase-change-material (MPCM) particles in the full channel at $\mathrm{R}{\mathrm{e}}_{Dh}=200$. (b) A 0.1% volume fraction of heavy MPCM particles in a channel segment. (c) Light MPCM particles in the full channel at $\mathrm{R}{\mathrm{e}}_{Dh}=200$. (d) A 0.8% volume fraction of light MPCM particles in a channel segment.

Figure 23

Comparative analysis of temperature rate change in light and dense micro-phase-change-material (MPCM) particles.

Figure 24

The distribution of local Nusselt numbers for (a) upper local Nusselt number ${\mathrm{Nu}}_x^{\mathrm{T}}$ and (b) bottom local Nusselt number ${\mathrm{Nu}}_x^{\mathrm{B}}$.

Figure 25

A compilation of local Nusselt number trends is presented at the entrance region, portraying the relationship with nondimensional local position and volume fraction, characterized by Peclet number.

Figure 26

A compilation of local Nusselt number trends is presented at the fully developed region, portraying the relationship with nondimensional local position across various flow regimes (a) $\mathrm{Re}=50$, (b) $\mathrm{Re}=100$, (c) $\mathrm{Re}=200$, and (d) $\mathrm{Re}=300$.

Figure 27

Comparison between the average Nusselt number and Reynolds number for different volume fractions.

Figure 28

An examination of the relationship between the performance index and Reynolds number is conducted across various volume fractions and base fluid configurations.