Algorithmic augmentation in the pseudopotential-based lattice Boltzmann method for simulating the pool boiling phenomenon with high-density ratio

The pseudopotential-based lattice Boltzmann method (LBM), despite enormous potential in facilitating natural development and migration of interfaces during multiphase simulation, remains restricted to low-density ratios, owing to inherent thermodynamic inconsistency. The present paper focuses on augmenting the basic algorithm by enhancing the isotropy of the discrete equation and thermodynamic consistency of the overall formulation, to expedite simulation of pool boiling at higher-density ratios. Accordingly, modification is suggested in the discrete form of the updated interparticle interaction term, by expanding the discretization to the eighth order. The proposed amendment is successful in substantially reducing the spurious velocity in the vicinity of a static droplet, while allowing stable simulation at a much higher-density ratio under identical conditions, which is a noteworthy improvement over existing Single Relaxation Time (SRT)-LBM algorithms. Various pool boiling scenarios have been explored for a reduced temperature of 0.75, which itself is significantly lower than reported in comparable literature, in both rectangular and cylindrical domains, and also with micro- and distributed heaters. All three regimes of pool boiling have aptly been captured with both plain and structured heaters, allowing the development of the boiling curve. The predicted value of critical heat flux for the plain heater agrees with Zuber correlation within 10%, illustrating both quantitative and qualitative capability of the proposed algorithm.

Figure 1

Vectorial representation of the D2Q9 lattice adopted in the present paper; here ${\mathbf{c}}_i$ is the unit vector in the $i\text{th}$-velocity direction and $w_i$ is the corresponding weight factor.

Figure 2

Appearance of spurious velocity currents around a static liquid droplet being stabilized in a periodic domain at $T_R=0.73$, following (a) discretization scheme A and (b) discretization scheme B, and (c) variation of maximum false velocity with density ratio following both schemes. Proposed scheme B offers substantial suppression in spurious velocity, while also enabling simulation at much higher-density ratios.

Figure 3

Two layers of neighboring nodes considered during the proposed discretization scheme. Layer 1 involves the eight immediate neighbors and layer 2 includes the 16 next neighbors, resulting in a 25-point discretization scheme.

Figure 4

Qualitative validation of the proposed algorithm: (a) The pressure differential across the interface of a static droplet exhibits a linear relationship with the reciprocal of droplet radii for three different saturation temperatures; (b) liquid and vapor coexisting densities predicted with scheme B referred to earlier demonstrate excellent conformity with the Maxwell area construction rule [37] at significantly lower reduced temperatures.

Figure 5

Schematics of the geometric configurations considered for the present paper: (a) an open rectangular domain with a micro- or distributed heater at the bottom surface; (b) an open fluid-filled cylinder housing a noncoaxial rotating solid cylinder, which embeds a microheater at the top surface; and (c) a structured heater composed of four identical columns mounted along the bottom surface of an open rectangular domain.

Figure 6

Numerical characterization nucleate boiling from a microheater in an open rectangular domain: (a) snapshots of one bubble ebullition cycle presenting temperature contours, velocity vectors, and bubble contour (thick black line); (b) validation of Fritz's correlation [45] to prove that the bubble departure diameter is inversely proportional to $g$; and (c) temporal variation in wall heat flux for two different wall superheat values, demonstrating periodic dewetting and rewetting of the heated surface.

Figure 7

Snapshots of one bubble ebullition cycle from a microheater in an open cylindrical domain presenting velocity vectors and phase contour; here red and blue colors, respectively, symbolize liquid and vapor phases.

Figure 8

Characterization of nucleate boiling from a distributed plain heater in an open rectangular domain with $\mathrm{\Delta }{T}_{\text{sup}}=0.21$: (a) snapshots of bubble nucleation and growth from discrete nucleation sites, where existence of the superheated liquid layer and bubble-induced liquid motion can clearly be seen, and (b) temporal variations in wall heat flux and vapor area fraction, showing considerable rise in heat transmission following departure of a bubble.

Figure 9

Characterization of transition boiling from a distributed plain heater in an open rectangular domain with $\mathrm{\Delta }{T}_{\text{sup}}=0.27$: (a) snapshots of bubble nucleation from an unstable vapor film, the expanse of which keeps on changing with time and the superheated liquid layer of which is also unstable, and (b) temporal variations in wall heat flux and vapor area fraction, signifying the quite chaotic nature of the phenomenon.

Figure 10

Characterization of film boiling from a distributed plain heater in an open rectangular domain with $\mathrm{\Delta }{T}_{\text{sup}}=0.31$: (a) snapshots demonstrating the formation of a stable vapor film over the entire heated surface and occasional vapor release following interfacial instabilities and (b) temporal variations in wall heat flux and vapor area fraction, substantiating the stable nature of the film with the area fraction being near constant around unity.

Figure 11

Snapshots of pool boiling from a distributed plain heater in an open rectangular domain during (a) the nucleate regime with $\mathrm{\Delta }{T}_{\text{sup}}=0.21$, (b) transition boiling with $\mathrm{\Delta }{T}_{\text{sup}}=0.27$, and (c) film boiling with $\mathrm{\Delta }{T}_{\text{sup}}=0.31$. Here red and blue colors, respectively, symbolize liquid and vapor phases. Nucleation can clearly be seen to initiate from the corners of the columnar structures.

Figure 12

Boiling curves with plain and structured distributed heaters in an open rectangular domain. The transition regime with the structured heater is noticeably shorter, with CHF getting shifted towards the right and the Leidenfrost point shifted towards the left. The heat flux level with the structured heater is also consistently higher than that with the plain heater, yielding a greater CHF value.

Figure 13

The static contact angle inversely varies with the false wall density and approaches the theoretical extremes of $0^{\circ }$ and ${180}^{\circ }$ on assuming the limiting density values corresponding to vapor and liquid, respectively, as shown within the insets.

Figure 14

Effect of contact angle on the nature of bubble nucleation demonstrated for (a) $\theta =39.{22}^{\circ }$ and (b) $\theta =56.{97}^{\circ }$, with red and blue colors, respectively, symbolizing liquid and vapor phases; (c) temporal variations in wall heat flux and vapor area fraction for $\mathrm{\Delta }{T}_{\text{sup}}=0.25$; and (d) boiling curves for the same contact angles. Fluid with higher contact angle clearly shows a greater inclination towards stable film formation, resulting in a substantial reduction in CHF under identical operating conditions.