Critical heat flux is a major thermal limit in water-cooled nuclear reactors because heat-transfer deterioration can produce a rapid rise in fuel-cladding temperature and potentially damage the cladding. Under non-uniform axial heating, the local margin is not determined solely by pressure, mass flux, geometry, or equilibrium quality. It also depends on where heat has been deposited earlier along the channel, which alters the evolving two-phase flow and thermodynamic state. Engineering methods commonly begin with a critical heat-flux value for equivalent uniform heating and then apply a correction representing the axial power history. The study keeps that structure, using a frozen neural network trained on a large uniform-heating database as the baseline and training a compact transformer only to estimate the additional non-uniform-heating correction.

The experimental basis was a benchmark-ready database derived from non-uniform critical-heat-flux measurements compiled in a Korea Atomic Energy Research Institute report. Its 888 experiments were divided among inlet-peaked, middle-peaked, outlet-peaked, and spike-like axial power profiles. Before modeling, the researchers excluded 39 cases with two-phase inlet conditions because the frozen uniform-heating backbone produced unreliable predictions for them. They also excluded 123 cases that required the backbone to extrapolate beyond its diameter, mass-flux, or equilibrium-quality ranges. The resulting 726 experiments were randomly divided at the experiment level into training, validation, and test sets at 70%, 15%, and 15%. These exclusions define the model’s applicability domain and are not intrinsic limitations of the transformer architecture. A separate leave-one-profile-family-out evaluation tested what happened when an entire heating-profile family was withheld from training and validation.

For each axial location, the transformer received the normalized axial position, local equilibrium quality, and normalized axial heat-flux shape as a sequence token. Average heat flux and mass flux were also included, while mass flux was treated as a global operating-condition variable because it was conserved along the steady, constant-area test channels. The model used one encoder layer, a 32-dimensional embedding, two attention heads, a 64-unit feed-forward layer, GELU activation, and 0.10 dropout. No index-based positional encoding was added because physical axial position was already represented explicitly. A causal attention mask prevented downstream locations from influencing the correction at an upstream location, allowing the correction to depend on the current and accessible upstream history without using information from the future. After attention, a two-layer feed-forward head predicted an unconstrained value that was mapped to a positive correction factor through a bounded exponential transformation.

Training targeted the experiment-level minimum departure from nucleate boiling ratio, the minimum local thermal margin that determines the limiting critical-heat-flux condition. The objective penalized squared deviation of that minimum from a value of one while regularizing corrections away from unity along the sequence. This differs from pointwise training on the local critical heat flux, because the experimental data provide an experiment-level critical condition rather than separate local targets at every axial position. The researchers stated that alternative objectives, including logarithmic percentage-error losses, were not evaluated. Predicted correction factors ranged from 0.354 to 2.948 across all locations in the primary test set and from 0.814 to 2.493 at the transformer-selected limiting locations, so the numerical bounds were not approached closely. The authors also emphasized that the broad bounds were safeguards against extreme training values, not mechanistic limits on physical critical heat flux.

On the primary random split, the frozen uniform-heating neural network without any non-uniform correction substantially overpredicted the limiting critical heat flux. Its prediction-to-measurement mean was 1.247, its standard deviation was 0.388, and its mean absolute percentage error was 0.313. Applying the classical correction reduced the standard deviation to 0.143 and the mean absolute percentage error to 0.122. The Tong F-factor correction also improved on the uncorrected neural network but was less accurate on this test set, with a standard deviation of 0.246 and mean absolute percentage error of 0.178. Look-up-table predictions had still greater scatter, although they were evaluated on 108 of the 109 test cases because one middle-peaked case selected a limiting location outside the usable quality range. On the common 108-case subset, the transformer retained lower standard deviation and error than the classical correction, so the conclusion did not depend on the excluded look-up-table case. Paired bootstrap comparisons using 200,000 experiment-level resamples produced 95% confidence intervals below zero for differences in error and scatter, although no multiple-comparison adjustment was applied.

Results differed by axial profile. The transformer had lower scatter and mean absolute percentage error for the inlet-peaked and middle-peaked cases, which made up most of the test set. For 15 outlet-peaked cases, its standard deviation was close to that of the classical correction, while its error and mean were slightly better. Only one spike-like case appeared in the primary test set, so its result could not support a profile-level conclusion. The leave-one-family-out analysis revealed the model’s most important limitation. It retained lower scatter and error when the middle-peaked family was hidden, but performed worse than the classical correction when either the inlet-peaked or outlet-peaked family was held out. For the unseen inlet-peaked family, transformer scatter and mean absolute percentage error were 0.332 and 0.349, versus 0.235 and 0.200 for the classical correction. The corresponding held-out outlet-peaked values were 0.250 and 0.224 for the transformer and 0.182 and 0.152 for the baseline. Generalization to an entirely unseen profile family was therefore uneven.

Input ablations showed that the transformer relied on more than a single thermal variable. Removing global mass flux increased prediction scatter from 0.115 to 0.150, indicating that channel-level mass flux supplies important scaling information. Removing the normalized heat-flux profile raised scatter to 0.144, making the model comparable to the classical correction in that measure. Removing equilibrium-quality history produced the largest mean absolute percentage error among the ablations, 0.115, and shifted the prediction-to-measurement mean below unity. The complete model therefore provided the best balance of scatter, bias, and absolute error in the reported test set. The model also transferred between axial resolutions without retraining: a model trained at 200 locations was evaluated at 40, 100, 200, and 400 locations, and its prediction-to-measurement standard deviation remained between 0.114 and 0.115. The study did not investigate alternative global-conditioning strategies, and the results did not establish that the particular conditioning strategy used was optimal.

Robustness testing also exposed variation associated with model initialization. Across four training seeds, the transformer produced a lower mean absolute percentage error than the classical correction in every run. It reduced prediction scatter in three runs, but one seed yielded a standard deviation of 0.164, above the classical correction’s 0.143, although that seed still had a lower mean absolute percentage error of 0.115. The researchers said that engineering use would require further characterization or mitigation of this stochastic variability. Ensembling was suggested as a possible strategy but was not tested. Attention diagnostics at selected limiting locations showed that the model generally distributed attention over multiple accessible upstream positions rather than depending on one point. Weighting changed with the power-profile shape, including concentrations nearer earlier high-power regions for some profiles and closer to the downstream high-power region in an outlet-peaked case. These visualizations were treated as qualitative representations of learned information flow, not direct proof of physical causality or a complete account of any prediction.

Overall, the study shows that a small causal transformer can learn useful sequence-dependent corrections for critical heat flux under non-uniform axial heating while preserving a familiar engineering formulation. Its strongest reported advantage was narrower prediction error on the primary benchmark, with additional support from ablations, resolution-transfer tests, seed checks, and paired bootstrap analysis. Its weaknesses were equally concrete: the non-uniform database was small and imbalanced, unseen profile families produced inconsistent results, and the baseline was specific to one frozen uniform-heating neural network. The study did not compare alternative baseline predictors or establish an improved allowable design margin, because the statistical treatment of prediction errors and other uncertainties needed for such a limit was not developed. The authors characterized the result as a data-driven correction-factor model rather than a mechanistic model of critical heat flux.

Subject of Research: Machine-learning correction of critical heat-flux predictions under non-uniform axial reactor heating

Article Title: A transformer-based correction factor for critical heat flux prediction under non-uniform axial heating

Article References: Bourisaw, R., & Iskhakov, A. S. (2026). A transformer-based correction factor for critical heat flux prediction under non-uniform axial heating. Frontiers in Mechanical Engineering, 12, Article 1980739. https://doi.org/10.3389/fmech.2026.1980739

Image Credits: AI Generated

DOI: 10.3389/fmech.2026.1980739

Keywords: critical heat flux, non-uniform axial heating, causal transformer, neural networks, nuclear reactors, two-phase flow, minimum DNBR, model robustness, correction factor, transformer-based, correction, factor

The significance of the result lies in the tension at the heart of chip cooling. Water is, in many respects, the ideal coolant: it has a high specific heat capacity and excellent thermal conductivity, and embedded microfluidic systems that etch cooling channels directly into the silicon substrate have used it to dissipate heat fluxes as high as 3,000 watts per square centimeter. But water conducts electricity. In a device where coolant channels run mere micrometers from active circuitry, any leak, condensation or ionic contamination risks short circuits and catastrophic failure. Dielectric fluids such as HFE-7100, the fluorinated engineered fluid used in this study, are electrically insulating and therefore inherently safe to place in intimate contact with live electronics. The price of that safety has traditionally been poorer thermal performance, because dielectric fluids have lower specific heat and thermal conductivity than water.

The Peking University group’s answer is a geometric one. Their device, which they call a horizontal-up manifold microchannel cooler, or HUMMC, reorganizes the way liquid enters and vapor leaves the cooling structure. The manifold features horizontal inlets that facilitate liquid replenishment across the heated area, paired with vertical outlets that accelerate the evacuation of vapor. This separation of duties is crucial in two-phase cooling, where the coolant boils as it absorbs heat and the resulting vapor must be removed quickly to make room for fresh liquid. When vapor lingers over hot surfaces, it forms insulating films that choke heat transfer and can trigger the dreaded critical heat flux condition, the point at which the surface temperature surges uncontrollably because liquid can no longer reach it.

Manifold microchannel architectures are not new. In a conventional design, coolant is distributed from above through an array of alternating inlets and outlets, forcing it to flow short distances laterally through fine microchannels etched into the substrate before exiting. This shortens the flow path dramatically compared with a single long channel and boosts heat transfer coefficients substantially. The team’s earlier work, published in 2025, pushed jet-enhanced manifold microchannels cooled with water to an extraordinary 3,000 watts per square centimeter, a level comparable to the heat flux at the surface of the sun’s photosphere and far beyond what any conventional heat sink can manage. The challenge was to bring that class of performance to a dielectric fluid, whose lower heat capacity, lower thermal conductivity and poorer wetting characteristics make flow boiling far harder to sustain at extreme fluxes.

The horizontal-up configuration addresses the specific failure modes of dielectric flow boiling. Because the inlets run horizontally, liquid is delivered in a way that maintains a robust supply to the heated walls even as vigorous boiling generates vapor. The vertical outlets, meanwhile, give buoyant vapor a direct escape route, preventing it from accumulating in the channels and disrupting the liquid film. The researchers compared their HUMMC design against a standard parallel manifold microchannel configuration, referred to as SPMC in their thermo-hydraulic performance comparisons, and found the horizontal-up arrangement delivered markedly higher critical heat flux along with operational stability at the highest cooling loads. The combination of rapid replenishment and rapid venting effectively delays the onset of dryout, the local exhaustion of liquid that caps the performance of most two-phase coolers.

Two-phase cooling itself deserves explanation, because it is the physical engine behind these numbers. Unlike single-phase cooling, where the coolant simply warms up as it flows, two-phase cooling exploits the latent heat of vaporization. When the coolant boils, each kilogram of fluid absorbs far more energy than it could by a temperature rise alone, and the boiling process itself sustains high heat transfer coefficients at the wall. The trade-off is complexity: flow boiling in microchannels is prone to instabilities, pressure fluctuations and premature dryout, all of which worsen as the fluid’s surface tension rises and its latent heat falls, exactly the regime of dielectric fluids. Engineering the channel geometry to manage the two-phase flow, rather than merely to carry liquid, is therefore the central design problem, and it is precisely where the horizontal-up manifold makes its contribution.

The fabrication approach is equally important for the technology’s prospects. The cooler is built using microelectromechanical systems technology, the same suite of processes that underpins standard semiconductor manufacturing. That means the cooling structures can, in principle, be directly integrated with chip fabrication and scaled across wafers, rather than bolted on as an afterthought. Embedded cooling of this kind places the microchannels within millimeters, or even micrometers, of the transistor hot spots, slashing the thermal resistance between the heat source and the coolant. This co-design of electronics and microfluidics echoes a broader trend in the field: earlier landmark work demonstrated monolithic microfluidic cooling integrated into gallium nitride power electronics, and researchers have explored exotic substrates such as diamond and boron arsenide, materials with exceptional thermal conductivity, to spread heat before it reaches the coolant.

The motivation for such extreme measures is easy to find in the trajectory of the industry. Data centers powering artificial intelligence workloads are consuming electricity at a pace that strains grids, and a growing fraction of that energy goes into cooling. Wide-bandgap and ultra-wide-bandgap power devices, including gallium nitride transistors, concentrate heat into ever smaller footprints, and their reliability and lifetime degrade rapidly as channel temperatures climb. Defense electronics, radar and laser systems face similar constraints. Agencies such as the United States Defense Advanced Research Projects Agency have run programs explicitly targeting device-scale heat removal, recognizing that thermal limits now gate performance as surely as transistor scaling once did. A cooler that handles kilowatt-per-square-centimeter fluxes with an electrically inert fluid addresses all of these use cases at once.

The dielectric route also opens possibilities that water cannot offer. Because HFE-7100 and similar engineered fluids are insulating, they can be used in immersion cooling schemes, where entire boards are bathed in liquid, or in embedded systems where coolant contacts or nearly contacts live circuitry without elaborate sealing and isolation. The fluid’s low boiling point suits it to two-phase operation at temperatures compatible with silicon devices. The trade-off remains cost and thermophysical performance, which is why the demonstration of 1,570 watts per square centimeter with a dielectric fluid is more than an incremental record. It shows that with the right manifold geometry, the safety advantage of dielectric coolants no longer has to be purchased with a threefold penalty in heat flux capability.

There are, of course, steps between a laboratory demonstration and deployment in a server rack or a radar array. The researchers report operational stability at their highest cooling performance, and their comparison against representative two-phase cooling technologies, from spray cooling to pool boiling on enhanced surfaces to other embedded manifold designs, places the HUMMC at the top of the dielectric-cooled field. Scaling the approach to full-size chips, integrating the fluid delivery plumbing with real packages, and proving long-term reliability with fluorinated coolants are the natural next challenges. But the core insight, that separating horizontal liquid replenishment from vertical vapor evacuation unlocks dielectric two-phase cooling at fluxes once thought reachable only with water, gives chip designers a new tool at exactly the moment thermal constraints are tightening around the industry. As computing demand accelerates, the machines of the future may well be cooled from the inside out, by insulating liquids flowing through silicon labyrinths etched alongside the circuits they protect.

Subject of Research: Embedded two-phase microfluidic cooling of high-power chips using a dielectric fluid in a horizontal-up manifold architecture

Article Title: Horizontal-up microfluidics for embedded chip cooling with a dielectric fluid

Article References: Xiao, W., Wu, Z., Jiang, Z., Wang, Y., Wang, W., & Song, B. (2026). Horizontal-up microfluidics for embedded chip cooling with a dielectric fluid. Nature Electronics. https://doi.org/10.1038/s41928-026-01724-y

Image Credits: AI Generated

DOI: 10.1038/s41928-026-01724-y

Keywords: microfluidics, chip cooling, thermal management, dielectric fluid, HFE-7100, two-phase cooling, manifold microchannels, critical heat flux, embedded cooling, MEMS, Nature Electronics, electronics cooling