Introduction

Nuclear fusion is one of the most sustainable sources of future energy. Thermonuclear fusion is the process of atomic nuclei combining or fusing using high temperatures to drive them close enough together for this to become possible. Such temperatures cause the matter to become a plasma and, if confined, fusion reactions can occur due to collisions with extreme thermal kinetic energies of the particles. Tokamak as one of typical thermonuclear fusion reactor types is a device which uses a powerful magnetic field to confine plasma in the shape of a torus. Divertor is a special device located at the bottom of a vacuum vessel in most tokamaks. Its purpose is to collect and remove impurities from the plasma. In a divertor, intense heat loads of 10–20 MW/m2, which is ten times the heat load experienced by a spacecraft re-entering Earth’s atmosphere, appear as a result of the high-energy plasma particles striking the vertical targets in the divertor. The extreme thermal loads, therefore, lead to considerable stress and deformation in the structure and material of the divertor, emphasizing the necessity for improved cooling channel design [1].

Various cooling channel design concepts to sustain such high heat loads have been proposed, including swirl tube, screw tubes and hypervapotron [2,3,4,5]. The viability of the hypervapotron design has been effectively demonstrated over a period more than 20 years at the Joint European Torus [6]. This achievement has prompted the consideration of utilizing this technology as an alternative cooling approach at the ITER facility. As a result, the progression of hypervapotron structures is highly significant for advancing fusion technology.

The hypervapotron design concept was introduced by Beurtheret [7], which includes fins positioned transversely to the fluid flow as shown in Fig. 1. The hypervapotron cooling channel maintains a stable boiling regime even under high heat flux conditions, and the fin structure enables the discharge of vapor formed within the fin slot into the bulk fluid. Ongoing research aims to explain the mechanisms behind the significant enhancement in heat transfer performance due to the arrangement of fin structures. Cattadori et al. [8] conducted a study in which visualization experiments were carried out on the hypervapotron channel to analyze the vapotron effect phenomenon. This phenomenon describes a cyclic process in which a phase change causes the internal slots of the fin to fill with steam. The steam is then rapidly condensed by subcooled bulk liquid, resulting in the cyclic injection of cold liquid back into the fin interiors. The visualization results confirmed that the vapotron effect occurs simultaneously across all fin slots, but only at lower flow rates. Falter and Thompson [9] conducted an experimental study on the heat transfer performance of the hypervapotron cooling channel at varying flow velocity. The operating mode of the hypervapotron was identified by measuring the frequency of the noise generated at low flow velocity, which was low frequency. This was similar to the frequency calculated by considering the volume and heat transfer of the fin slot. However, at high flow velocity, the frequency of the noise increased significantly. The fluid exchange between the bulk liquid and the fin slots, as well as the probability of vortex formation within the slots, increases with the velocity of the bulk liquid. In this situation, subcooled liquid enters the fin slots, and the nucleated bubbles are expected to collapse before being expelled into the bulk liquid stream. As a result, the boiling that occurs in the fin slots is similar to typical subcooled flow boiling. The mechanisms that contribute to the enhancement of hypervapotron cooling performance can be identified as the vapotron effect, which occurs at lower velocities, and the increase in heat transfer area coupled with the promotion of turbulence through vortex flow formation by fin structures at higher velocities. This approach highlights the complex relationship between structural design and fluid dynamics in enhancing thermal performance in hypervapotron cooling channels.

Fig. 1
figure 1

Configuration of a typical hypervapotron cooling channel

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Experimental and computational studies have continuously been conducted to enhance the performance of the hypervapotron cooling channel. Altmann et al. [10] investigated the hypervapotron design with a particular focus on the performance improvement achieved by incorporating a slot between the fins and the central web. The modified design demonstrated a significant reduction in temperature, up to 100 K at a heat flux of 10 MW/m2, effectively relocating the peak temperature zone to above the center of the fins. Ciric et al. [11] conducted fatigue tests on three design variants: the standard JET, the new Box Scraper, and the Box Scraper prototype, measuring the maximum temperature rise at peak power density to validate the prototype design. Escourbiac et al. [12] conducted experimental optimization of the hypervapotron channel by fabricating channels with widths of 27, 40, and 50 mm and conducting Critical Heat Flux (CHF) measurement tests. The experiments showed that while the 27 mm design could manage a heat removal of 28 MW/m2 at a flow velocity of 5 m/s and inlet subcooling of 120 K. However, temperatures became unsustainable at heat flux levels above 20 MW/m2. It was observed that an increasing the width of the hypervapotron resulted in a reduction of CHF by more than 30%. This may be due to the enhanced fluid ingress from the side slots into the fin interiors. Experimental studies, such as the critical heat flux measurement and fatigue testing, have primarily focused on verifying the cooling performance and determining the CHF of hypervapotron. However, due to the experimental nature of these studies, visualization of internal fluid flow within metal cooling channels under high heat flux conditions has been notably limited. This has made analyzing the impact of design modifications on cooling performance challenging.

Numerical studies have emerged as a promising alternative for in-depth analysis of hypervapotron cooling channel due to the challenges associated with conducting experimental studies. Pascal-Ribot et al. [13] verified the possibility of modeling the flow and conjugate heat transfer within a hypervapotron using 3D simulations, using SYRTHES code [14] for solid heat conduction and NEPTUNE CFD code [15] for two-phase flow thermal hydraulics through by coupling. Milnes et al. [16] carried out a systematic analysis using an Ansys CFX-based CFD model to model the hypervapotron cooling channel. They performed a series of sensitivity analyses for turbulence models and grid sizes, and adapted the RPI wall boiling model, thus laying the foundation for CFD modeling techniques for accurate fluid flow and heat transfer analysis in the hypervapotron. Savoldi et al. [17] investigated the effect of aspect ratio on heat transfer performance in hypervapotron fin-like square cavities. The authors compared steady-state and transient simulations and discovered that the maximum temperature was significantly underestimated in steady-state results, while transient state results slightly overestimated it. Youchison et al. [18] investigated the influence of hypervapotron cooling performance on 36 mm, 50 mm, and 70 mm channel width using STAR-CCM+. They found only small differences in maximum surface temperature but large differences in the distribution of void fraction. In the 36 mm channel, bubbles were concentrated toward the side slot, while in the 50 mm and 70 mm channels, a higher void fraction appeared in the center of the channels. Wider channels may prevent vapor form escaping from the center to the side slot. Domalapally and Dellabiancia [19] performed an optimization of a three-dimensional hypervapotron cooling channel design using STAR-CCM+ under DEMO conditions. A systematic analysis of the main design factors of the Hypervapotron cooling channel was carried out, including the thickness of the front wall, the channel height, the channel width and the thickness of the side channel. However, only a single-phase flow analysis was carried out, which did not consider the effect of phase change on heat transfer performance.

The preceding literature review showed that experimental and numerical studies on the design of hypervapotron have been carried out continuously, but the experimental studies only have focused on the validation of the cooling channel performance and the measurement and improvement of the CHF. In addition, CFD models have been developed to evaluate the performance of the hypervapotron cooling channel, but a universally applicable analysis methodology for various hypervapotron geometries has not yet been established. This study aims at numerically analyzing the influence of geometrical design factors on the internal flow pattern and cooling performance of hypervapotron channels using CFD. A numerical methodology is developed to accurately model the vortex flow within the fin slots, which is one of the main mechanisms used to improve heat transfer performance in hypervapotron. The effects of phase change phenomena on the internal flow pattern are also investigated, which have not been precisely addressed in the previous literature. Based on this, a suitable CFD method for hypervapotron simulation is established and a comparative analysis of different 3D hypervapotron cooling channel designs is performed.

CFD Methodology and Case Setup

The CFD simulation was conducted using the open-source CFD package OpenFOAM [20]. The ‘multiphaseEulerFoam’ which is based on a multi-fluid Eulerian model that adequately accounts for conjugate heat transfer between heated walls and fluid, boiling, and condensation of bubbles in subcooled liquid, among diverse physical phenomena occurring within a hypervapotron cooling channel. Detailed information on this solver can be found in relevant literatures [21, 22], the interphase models and the wall boiling model which are crucial for modeling hypervapotron described in following sections.

Interphase Models

The interphase momentum transfer model includes both drag and non-drag forces, including lift, wall lubrication, turbulent dispersion, and virtual mass forces. The drag model of Schiller and Naumann [23] was used for momentum transfer due to drag. The lift coefficient was derived from the model of Tomiyama et al. [24], while the wall lubrication and turbulent dispersion forces modeled according to Antal et al. [25] and Lopez de Bertodano et al. [26], respectively. The characteristic physical phenomenon occurs in the hypervapotron cooling channel where vapor is intensively generated due to boiling inside the fins and suddenly condensed due to direct contact with subcooled bulk liquid. The thermal phase change model in the solver uses the two-resistance approach to calculate the interfacial heat transfer, as described below:

$${q}_{\text{int}.l}^{{\prime}{\prime}}={K}_{\text{int}.l}\left({T}_{\text{int}}-{T}_{l}\right)+{\left(\frac{d{m}_{l}}{dt}\right)}_{\text{int}}{H}_{\text{int}.l}$$
(1)
$${q}_{\text{int}.v}^{{\prime}{\prime}}={K}_{\text{int}.v}\left({T}_{\text{int}}-{T}_{v}\right)+{\left(\frac{d{m}_{v}}{dt}\right)}_{\text{int}}{H}_{\text{int}.v}$$
(2)
$${\left(\frac{dm}{dt}\right)}_{\text{int}}=-{\left(\frac{d{m}_{v}}{dt}\right)}_{\text{int}}={\left(\frac{d{m}_{l}}{dt}\right)}_{\text{int}}$$
(3)

where \({h}_{\text{int}.l}\) and \({h}_{\text{int}.v}\) represent enthalpies of the liquid and vapor phases, respectively. \({T}_{\text{int}}\) is the temperature at the interface between the phases. If the temperature of the interface is set at the saturation temperature and the balance of heat through the interface is maintained, the mass transfer resulting from the phase change can be calculated as

$${\left(\frac{d{m}_{l}}{dt}\right)}_{\text{int}}=\frac{{K}_{\text{int}.l}\left({T}_{\text{sat}}-{T}_{l}\right)+{K}_{\text{int}.v}\left({T}_{\text{sat}}-{T}_{v}\right)}{{H}_{\text{int},2}-{H}_{\text{int},1}}$$
(4)

The heat transfer coefficients (\({K}_{l}, {K}_{v}\)) are determined from the heat transfer models. The Ranz-Marshall model [27] is widely used, but employing a constant Nusselt number of 2 has been found to enhance the model's robustness, as noted in [16]. Consequently, the heat transfer coefficient was modeled using the constant Nusselt number.

Once the mass flux due to phase change has been calculated, the interfacial temperature is accordingly updated:

$${T}_{\text{int}}=\frac{{K}_{\text{int}.l}{T}_{l}+{K}_{\text{int}.v}{T}_{v}+{\left(\frac{d{m}_{1}}{dt}\right)}_{\text{int}}\left({H}_{\text{int},2}-{H}_{\text{int},1}\right)}{{K}_{\text{int}.l}+{K}_{\text{int}.v}}$$
(5)

Wall Boiling Model

In OpenFOAM, the wall boiling model is applied as thermal wall function. The effective thermal diffusivity of the wall is set based on the heat flux expected from the wall boiling model. This approach can incorporate a wall heat flux partitioning function to determine the effective liquid phase fraction for the boiling wall thermal diffusivity term based on the liquid phase volume fraction [22, 28].

$${D}_{T,l}=\frac{{f}_{\text{liquid}}}{\text{max}\left({\alpha }_{l}, {10}^{-8}\right)}{D}_{T,l,\text{init}}$$
(6)
$${D}_{T,v}=\frac{1-{f}_{\text{liquid}}}{\text{max}\left({\alpha }_{l}, {10}^{-8}\right)}{D}_{T,v,\text{init}}$$
(7)

The Lavieville model was chosen for the wall heat flux partitioning function [29], with the critical void fraction defined as \({\alpha }_{\text{crit}}=0.2\).

$$\begin{array}{cc}{\alpha }_{l}\ge {\alpha }_{crit}& {f}_{\text{liquid}}=1-\frac{1}{2}{e}^{-20\left({\alpha }_{l}-{\alpha }_{\text{crit}}\right)}\\ {\alpha }_{l}<{\alpha }_{crit}& {f}_{\text{liquid}}=\frac{1}{2}{\left(\frac{{\alpha }_{l}}{{\alpha }_{\text{crit}}}\right)}^{20{\alpha }_{\text{crit}}}\end{array}$$
(8)

The generation of heat from a heated surface typically involves heat transfer from that surface to the liquid. To accurately model this process, most CFD software employs the wall boiling model. Among these, the RPI wall boiling model, proposed by Kurul and Podowski [30], is notably prevalent. This model delineates the heat flux emanating from the wall into three distinct mechanisms of heat transfer, which include single-phase convection in the liquid phase, evaporative heat transfer from nucleate bubbles, and quenching heat transfer.

$${q}_{w}^{{\prime}{\prime}}={q}_{c}^{{\prime}{\prime}}+{q}_{e}^{{\prime}{\prime}}+{q}_{q}^{{\prime}{\prime}}$$
(9)

The convective heat flux is given by the formula:

$${q}_{c}^{{\prime}{\prime}}={h}_{c}{A}_{1}\left({T}_{w}-{T}_{l}\right)$$
(10)

where \({h}_{c}\) represents the convective heat transfer coefficient and \({A}_{1}\) represents the area fraction unaffected by boiling.

The evaporative heat flux is determined by the evaporation mass flux and the enthalpy difference between the vapor and liquid phases. The evaporation mass flux (γ) depends on the bubble’s departure diameter, frequency, and nucleation site density.

$${q}_{e}^{{\prime}{\prime}}=\gamma \left({H}_{v.\text{sat}}-{H}_{l}\right)$$
(11)
$$\gamma =\frac{\pi {D}_{\text{dep}}^{3}}{6}{\rho }_{v}fN$$
(12)

The quenching heat flux is formulated as Del Valle and Kenning [31].

$${q}_{q}^{{\prime}{\prime}}={h}_{q}{A}_{2}\left({T}_{w}-{T}_{l}\right)$$
(13)

Here, \({h}_{q}\)​ is the quenching heat transfer coefficient, and \({t}_{w}\)​ denotes the bubble waiting time related to the bubble departure frequency:

$${h}_{q}=2{k}_{l}f\sqrt{\frac{{t}_{w}{C}_{p.l}{\rho }_{l}}{\pi {k}_{l}}}$$
(14)
$${t}_{w}=\frac{0.8}{f}$$
(15)

The total area fraction is the sum of the area fractions affected (\({A}_{2}\)) and unaffected (\({A}_{1}\)) by boiling, which equals unity. \({A}_{2}\)​ is computed from the bubble departure diameter, nucleation site density, and the wetted area fraction as influenced by the heat flux partitioning function mentioned previously. To minimize grid dependency, the Jacob number calculation uses the liquid temperature at a wall distance of y+ = 250.

$${A}_{1}+{A}_{2}=1$$
(16)
$${A}_{2}=\text{min}\left(1,\frac{{A}_{l}\pi {D}_{dep}^{2}N}{4}\right)$$
(17)
$${A}_{l}={f}_{\text{liquid}}4.8{e}^{-\frac{Ja}{80}}$$
(18)
$$Ja=\frac{{\rho }_{l}{C}_{p.l}\left({T}_{\text{sat}}-{T}_{l.{y}^{+}=250}\right)}{{\rho }_{v}\left({H}_{v.\text{sat}}-{H}_{l}\right)}$$
(19)

The wall boiling model necessitates several parameters, including nucleation site density, bubble departure diameter, and bubble departure frequency, all crucial for determining the contribution of single-phase and two-phase heat transfer.

The bubble departure diameter is calculated using the Tolubinsky and Kostanchuk model [32]:

$${D}_{\text{dep}}=\text{max}\left[\text{min}\left({D}_{\text{max}}, {D}_{\text{ref}}{e}^{-\frac{\Delta {T}_{\text{sup}}}{\Delta {T}_{\text{ref}}}}\right), {D}_{\text{min}}\right]$$
(20)

Bubble departure frequency follows the Cole model [33]

$$f=\sqrt{\frac{4g\left({\rho }_{l}-{\rho }_{v}\right)}{3{\rho }_{l}{D}_{\text{dep}}}}$$
(21)

The model proposed by Lemmert and Chawla [34] is employed for nucleation site density:

$$N={C}_{n}{N}_{\text{ref}}{\left(\frac{\Delta {T}_{\text{sup}}}{\Delta {T}_{\text{ref}}}\right)}^{n}$$
(22)

Case Setup

In all the simulations, the working fluid was water with the system pressure of 500 kPa, correlating to a saturation temperature of 424.98 K. The inlet subcooling, denoted as (ΔTsub = Tsat – Tin), was specified at 20 K, and the mass flux was set at 1000 kg/m2s. The material properties and the fluid saturation temperature were obtained from the NIST chemistry web book [35]. The k–ω SST model was used as the turbulence model based on the investigation of turbulence models for simulating the internal flow of the cavity [16, 37]. The Reynolds number for the primary flow was calculated to be approximately 55,000, while for the secondary flow within the fin slot, it was around 10,000. These values confirm the turbulent flow conditions, thereby justifying the selection of the k–ω SST model. According to previous studies ([16, 37]), the k–ω SST model provided the most accurate results compared to experimental results and implicit LES methods. Furthermore, wall functions were applied, which are effective across different y+ ranges, encompassing both the viscous sublayer and logarithmic regions. This ensures that the simulations remain valid under various grid resolutions. The simulation performed a transient analysis and most of the cases converged around 3 s, so the end time of the simulation was set to 5 s. Due to the characteristics of transient analysis, there may be fluctuations in the value over time. Therefore, the temperature data of the heated wall from 4 to 5 s was used to judge the convergence of each analysis result. Maximum, minimum, mean, median, and quartile of temperature changes over time were examined. Under relatively high heat flux conditions of 4 MW/m2, the maximum temperature change at a local point in a single time step remained below 0.5 K for the hypervapotron design with 6 mm fin height and side slots. It was judged based on these results that the transient simulations were reasonably converged.

Results and Discussion

Modeling of Vortex Flow Patterns Inside Fin Slots

The design of typical hypervapotron cooling channel is shown in Fig. 1. To accurately evaluate a hypervapotron cooling channel, it is essential to simulate the internal flow in the fin slot shown in Fig. 1c. A set of grid sensitivity analysis for a two-dimensional unit cell was conducted to obtain reliable simulation results for the vortex flow formed inside the hypervapotron cooling channel. The geometric structure of the computational domain is depicted in Fig. 2. The fin slot is 3 mm wide and 6 mm high, resulting in an aspect ratio of 2 for the 2D hypervapotron unit. Heat conduction from the heated surface to the lower part of solid through the side wall (Fig. 1d) cannot be modeled due to the characteristics of two-dimensional simulation. Therefore, the lower part of solid was excluded from the computational domain as shown in Fig. 2.

Fig. 2
figure 2

Computational domain and boundary conditions for modeling the internal flow of a 2D hypervapotron fin slot

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To minimize the influence of the grid, the uniform hexahedral grid was employed. The simulation was performed with varying grid sizes in the range of 750 μm to 25 μm, as shown in Fig. 3a–f. Figure 4 illustrates the velocity field in the fluid domain obtained for single-phase flow with conjugate heat transfer analysis. In this analysis using grid sizes of 750 μm and 500 μm, the center of vortex flow was observed relatively lower than the others, and the velocity of the formed vortex flow was relatively low compared to the other results. This suggests that the grid sizes of 750 μm and 500 μm are excessively large for accurately simulating fluid inflow from the main flow into the fin interior. In contrast, the analyses conducted with the grid sizes of 250 μm, 100 μm, and 50 μm exhibited relatively fewer variations in vortex flow within the fin interior, but finer flow patterns were observed as the grid size decreased. Particularly, in the analysis using the smallest grid size of 25 μm, two independent vortex flows were formed inside the fin structure, with smaller micro vortex flows observed around these two main vortex flows.

Fig. 3
figure 3

Configuration of grid sensitivity tests to analyze vortex flow patterns in a unit cell of side slots

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Fig. 4
figure 4

Streamlines of single-phase liquid flow in a unit cell of side slots

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Heat transfer between fluid and solid inside the hypervapotron channel is one of the crucial phenomena to be addressed in this analysis. Therefore, additional analysis related to this phenomenon was conducted. Figure 5 illustrates temperature contour in 2D hypervapotron unit, and the variation of wall temperature with respect to the position of the heat transfer area. In the analyses using grid sizes of 750 μm and 500 μm, where vortex flow was not adequately formed, the temperature of solid and fluid inside the fin slot was considerably higher compared to other analysis results. Conversely, in the analysis using a grid size of 25 μm, where two vortex flows were formed, the temperature in the region where the primary vortex flow exists was relatively lower. And the temperature of the region for secondary vortex is higher than primary vortex side. In simulations using medium grids (50–250 μm), the temperature at the wall decreased slightly as the grid size decreased. Finally, at a grid size of 25 μm, where two vortices were formed, the temperature varied depending on the location of the vortices. The temperature was relatively lower in the region where the primary vortex was present, while the temperature was higher in the region where the secondary vortex was present. The difference in temperature depending on the location of the vortex can be seen as a result of poor fluid exchange between the two vortices. But overall, it did not show significant differences compared to the results for 50–250 μm.

Fig. 5
figure 5

Grid sensitivity test for temperature contour and wall temperature profile along the heated surface for single-phase flow simulation

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Through the grid sensitivity analysis for two-dimensional single-phase flow with conjugate heat transfer, it was confirmed that the analysis using a grid size of 25 μm most accurately simulated the vortex flow inside the fin. The secondary vortex flow was formed at the position adjacent to the heated wall. According to the previous study by Milnes [16], an in-depth analysis of the cavity height (or aspect ratio) confirmed that the geometric structure with an aspect ratio of 2, the same as the one used in this study, forms two vortex patterns, which is the same as the internal flow patterns observed in this study. Therefore, a 25 μm grid is considered sufficient for analyzing the internal flow patterns. The simulation predicted that the wall temperature would be highest at this location, causing bubbles to form at the top of the fin slot. It is important to assess how the behavior of the bubble formed is affected by the vortex flow. In this regard, further two-dimensional conjugate heat transfer analysis with the phase change model was carried out for all the grid sizes. The geometry and boundary conditions used in this simulation remained same, but only the phase change heat transfer model was newly applied.

The velocity field in the computational domain was selected as the qualitative analysis, and for the phase change simulation cases, the volume fraction of vapor also was examined. Comparative analysis of the simulation results focused only on the analysis results using a grid size of 25 μm, where secondary vortex flow was formed. The results obtained from transient simulations were presented in Fig. 6. It was observed that there was no significant difference between the single-phase and two-phase simulation until 0.4 s before boiling began in the fin slot. Additionally, the results of both simulations confirmed the formation of primary vortex flow and secondary vortex flow. However, from the moment when boiling started (1 s), the deformation of the secondary vortex flow was observed as the volume fraction of vapor at the top of the fin slot. At 1.2 s, the size of the secondary vortex flow decreased by half, and then this vortex flow dissipated at 1.3 s. These results show that the flows of two vortices merged as the formed vapor began to move in the fin slot. Under conditions of bubble formation, the influence of secondary vortex flow seen in single-phase flow analysis results may be small or absent. Therefore, it is necessary to determine whether it is effective to keep the grid size at 25 μm to accurately represent the secondary vortex flow within the fin slot under hypervapotron flow conditions with phase change.

Fig. 6
figure 6

Velocity field of single-phase and two-phase and volume fraction field of vapor

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A further heat transfer analysis was performed by investigating the temperature distribution in the solid part using the same approach as for the single-phase flow simulation conditions (Fig. 7). From the analyses, it was observed that the temperature distribution converges within a certain range as the grid size decreases. However, the simulation results for the finest grid size of 25 μm showed that the temperature distribution tended to increase, contrary to the above trend. According to the literature, in simulations applying the wall boiling model, using grid sizes below a certain level can lead to rather unphysical results, and it is recommended to keep the minimum y+ value above 40 [38,39,40]. This recommendation may be reasonable because wall boiling model use a logarithmic wall function to estimate parameters such as wall temperature near the wall based on a y+ value of 250, so the grid nearest the wall must be located within the logarithmic region to ensure the validity of these estimates.

Fig. 7
figure 7

Grid sensitivity test for temperature contour and wall temperature profile along the heated surface for two-phase flow simulation

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To determine the appropriate grid size for this analysis, the relationship between the y+ value and grid size was thoroughly analyzed, considering the operating conditions and geometric structure. This analysis ensured that the minimum y+value was set to at least 30, thus maintaining accuracy within the logarithmic region (30 < y+ < 300). Based on the results, a grid size of 250 μm, which corresponds to a y+ value of approximately 40, was selected. This corresponds to a physically valid range of y+ values. As illustrated in Fig. 8, it was confirmed that this grid size provides reliable results within the range proposed in the literature.

Fig. 8
figure 8

Relation between the values of y+ and the thickness of first layer grid (P = 500 kPa, G = 1000 kg/m2s, Aflow = 21 × 10 mm2)

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Through the analysis performed above, it was concluded that using a grid size of 250 μm is optimal for future 3D CFD analyses of the hypervapotron. This decision was made considering that using excessively fine grid sizes would result in significant computational costs and may be impractical. Utilizing a grid size of 250 μm allows for effective simulation of complex internal flow patterns and heat transfer phenomena while also considering computational costs, thereby providing a rational setting for analysis conditions.

Effects of Side Slot on Cooling Performance

A typical hypervapotron design consists of a flat channel structure with fin structures arranged perpendicular to the flow direction, as shown in Fig. 9. Two designs for the arrangement of fin structures have been proposed: one where the fin structures directly contact a sidewall (Fig. 9b), and another where a side slot (SS) are created between the fin structures and the sidewall to allow additional fluid passage through the slot (Fig. 9c). To evaluate the influence of a side slot on the internal flow and cooling performance of the hypervapotron, three geometric structures were analyzed: a flat channel, a hypervapotron channel without a side slot, and a hypervapotron channel with a side slot. The simulation conditions were kept consistent with 2D simulation, and the effects of varying heat fluxes on the heated wall, ranging from 1 to 4 MW/m2, were also analyzed. Considering the symmetry of cooling channel design, only a half of the domain was used in this simulation as shown in Fig. 9.

Fig. 9
figure 9

Computational domain of a flat channel, b hypervapotron channel without a side slot, c hypervapotron channel with a side slot

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Figure 10 illustrates the temperature distribution which was obtained from the extraction line indicated in Fig. 9a. As shown in Fig. 10a, under the 1 MW/m2 heat flux condition, the wall temperature was predicted to be highest for the flat channel design, intermediate for the channel without side slot, and low for the channel with side slot. At the sides, the temperature difference between the flat channel and the hypervapotron channel without side slots decreased. However, as the heat flux increased, the wall temperature of the hypervapotron channel without side slot increased, becoming similar to that of the flat channel at the 2 MW/m2 condition (Fig. 10b) and exceeding that of the flat channel at 3 MW/m2 (Fig. 10c). In contrast, the hypervapotron channel with side slot predicted lower wall temperatures than the flat channel at all heat flux conditions. The temperature difference between the flat channel and the hypervapotron channel with side slot remained relatively constant with location.

Fig. 10
figure 10

Effect of side slot on heated surface temperature under varying heat flux conditions

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To evaluate the effect of side slot on wall temperature, internal flow patterns and bubble behavior of each design were examined under the condition of 3 MW/m2 in detail. Figure 11a shows the flow path inside the fin slot before boiling occurs. In the hypervapotron design without side slots, the vortex flow was formed in the direction of the bulk flow. On the other hand, in the hypervapotron design with side slot, it can be seen that the internal flow pattern was formed in a diagonal direction by the influence of the bulk flow and the side slot. After the onset of boiling, the hypervapotron design without side slot formed a simple up and down flow, resulting in most of the area inside the fin slot being filled with vapor, as shown in Fig. 11c, d. The most characteristic feature of the hypervapotron design with side slot was the diagonal flow inside the fin slot, which was maintained even after boiling started. As shown in the cross-section of the fin slot in Fig. 11c, liquid could be seen entering the center, and the distribution of liquid inside the fin slot was predicted to be relatively more. After analyzing the internal flow pattern and bubble behavior, it was found that the effective removal of bubbles inside the fin slot has a dominant effect on the heat transfer performance. As shown in Fig. 11, in the hypervapotron channel without side slot, most of the area was filled with vapor, preventing the liquid from entering properly, which is believed to be the reason for the lower heat transfer performance compared to the flat channel design shown in Fig. 10

Fig. 11
figure 11

Effect of side slot of hypervapotron channel on flow pattern and liquid distribution for Hfin = 6 mm at \({q}_{w}^{{\prime}{\prime}}\) = 3 MW/m2

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Effects of Fin Height on Cooling Performance

In the design with side slots, additional vortex flow is generated from the side, leading to enhanced cooling performance. Therefore, the cooling performance can vary depending on the extent to which additional vortex flow is formed from the side. The cause of vortex flow formation lies in the drag generated by the velocity difference between the fluid inside the fin and the fluid passing through the bulk flow or the side slot where they contact. Hence, additional analysis was conducted on the area where the two fluids contact, the height of the fin, to analyze its impact on cooling performance. For the analysis, fin heights of 2, 4, and 6 mm were set and each design is shown in Fig. 12.

Fig. 12
figure 12

Computational domain of flat channel and hypervapotron channel with side slot, various fin height

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Figure 13 shows the temperature distribution which was obtained from the extraction line shown in Fig. 12a with varying heat fluxes and fin heights. As shown in Fig. 13a, the wall temperature of the flat channel was highest at the heat flux condition of 1 MW/m2 and the wall temperature decreases as the fin height increases. It can be seen that the temperature difference was relatively constant for each cooling channel design. At a heat flux of 2 MW/m2 (Fig. 13b), the wall temperature of the flat channel was similar to that of the 2 mm fin height design, while at 3 MW/m2 (Fig. 13c), the temperature was higher. The 4 mm fin height design had lower wall temperatures than the planar channel up to 3 MW/m2, but at 4 MW/m2 (Fig. 13d) the temperature was higher and the wall temperature was similar to the 2 mm fin height design.

Fig. 13
figure 13

Effect of fin height on heated surface temperature under varying heat flux conditions

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The effect of fin height was analyzed through the internal flow pattern and bubble behavior as shown in Fig. 14, as in the previous analysis method. As shown in Fig. 14a, it can be seen that the internal flow pattern before the onset of boiling was formed diagonally by the side slots. However, as the fin height decreased, the magnitude of the flow formed in the diagonal direction decreased because the influence of the side slot was relatively small. As shown in Fig. 14b, the internal flow pattern in the fin slot at the start of boiling showed a simple up and down flow in the low fin height channel design. As the fin height increased, the degree of diagonal internal flow increased. Similar to the design without side slots, the low fin height design showed that the inside of the fin slot was filled with vapor which prevented the liquid from entering properly (Fig. 14c).

Fig. 14
figure 14

Effect of fin height in hypervapotron channel on flow pattern and liquid distribution for Hfin = 2, 4, 6 mm at \({q}_{w}^{{\prime}{\prime}}\) = 3 MW/m2

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The design with side slot appears to promote liquid inflow compared to other designs, resulting in better cooling performance, through the formation of vortices induced by the velocity difference created at the interface where the fluid inside the fin and the fluid passing through the main stream or side slots meet by drag. Therefore, variables such as the area where the two fluids meet, i.e., the height of the fin, can have a significant impact on cooling performance. In summary, by varying the fin height to 2, 4, and 6 mm, it was found that the higher the fin height, the better the cooling performance due to the amount of liquid inflow. This suggests that with increasing fin height, more liquid can flow into the fin, allowing for more effective cooling.

Conclusions

The numerical analysis of fluid flow and heat transfer in various hypervapotron channel designs using OpenFOAM was carried out to investigate the effects of design variations on the internal flow patterns and cooling performance. The main findings of this study are as follows:

  • The optimal conditions for grid generation were determined through a set of grid sensitivity tests to accurately simulate the vortex flow within the fins. While a grid size of 25 μm appeared to adequately simulate the flow patterns in the single-phase flow analysis, consideration of the phase change revealed different flow patterns, necessitating grid size selection based on y+.

  • The inclusion of side slots significantly benefited the design of hypervapotron cooling channels. They facilitated bidirectional fluid ingress, leading to improved cooling performance. The synergetic effect of side slots on internal flow patterns promoted both vertical and horizontal flows, contributing to more effective heat dissipation. Consequently, hypervapotron channels with side slots demonstrated a superior ability to manage high heat flux conditions by maintaining lower wall temperatures across various thermal loads.

  • Increasing the height of fins in the hypervapotron design significantly enhanced cooling performance by facilitating greater liquid inflow. This improvement was attributed to the augmented vortex flow generated by the interaction between the fluid within the fin slot and the fluid passing through the side slots, leading to a more efficient heat exchange. The analysis highlighted the importance of fin height in optimizing hypervapotron channel performance.

The developed numerical methodology and the obtained results in this study are expected to serve as important reference materials for the practical engineering design and optimal operation of hypervapotron cooling technique. However, a systematic analysis of the impact of design parameters and various operational conditions on the internal flow patterns and bubble dynamics within hypervapotron channels is necessary. Such an analysis will deepen our understanding of the relationships between operating conditions and design parameters, paving the way for further improvements in performance. Additionally, based on this understanding, it is anticipated that guidelines for enhancing cooling performance can be provided, contributing to the advancement of hypervapotron cooling channel design and optimization.