On the prediction of critical heat flux using a physics-informed machine learning-aided framework

Xingang Zhao; Koroush Shirvan; Robert K. Salko; Fengdi Guo

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On the prediction of critical heat flux using a physics-informed machine learning-aided framework

Xingang Zhao, Koroush Shirvan, Robert K. Salko, Fengdi Guo

arXiv:1906.11124v1Jun 26, 2019physics.data-anphysics.app-phphysics.flu-dyn

TL;DR

Robust CHF prediction is hindered by disagreement over the DNB-triggering mechanism and limitations of existing models. The paper combines domain knowledge with ML trained on prediction residuals, finding superior performance across flow conditions and proposing window-type extrapolation mapping for future experiments.

Problem
Accurate, robust CHF prediction remains elusive because there is no general agreement on the mechanism triggering DNB, despite CHF’s importance to thermal-system safety.
Method
A physics-informed hybrid framework uses a domain-knowledge model as a baseline and trains ML to capture residual information between measured and domain-knowledge-predicted CHF.
Results
The hybrid approach significantly outperforms standalone models across a wide range of flow conditions, with reduced scatter and unbiased parametric trends.
Takeaways & Limitations
The framework extends CHF predictive capabilities while supporting flexible applicability and window-type extrapolation mapping for informing future experiments.
Takeaways & Limitations
Standalone ML methods can produce scattered, physically undesired solutions because of their purely data-driven, black-box nature.

Abstract

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The critical heat flux (CHF) corresponding to the departure from nucleate boiling (DNB) crisis is essential to the design and safety of a two-phase flow boiling system. Despite the abundance of predictive tools available to the thermal engineering community, the path for an accurate, robust CHF model remains elusive due to lack of consensus on the DNB triggering mechanism. This work aims to apply a physics-informed, machine learning (ML)-aided hybrid framework to achieve superior predictive capabilities. Such a hybrid approach takes advantage of existing understanding in the field of interest (i.e., domain knowledge) and uses ML to capture undiscovered information from the mismatch between the actual and domain knowledge-predicted target. A detailed case study is carried out with an extensive DNB-specific CHF database to demonstrate (1) the improved performance of the hybrid approach as compared to traditional domain knowledge-based models, and (2) the hybrid model's superior generalization capabilities over standalone ML methods across a wide range of flow conditions. The hybrid framework could also readily extend its applicability domain and complexity on the fly, showing an elevated level of flexibility and robustness. Based on the case study conclusions, the window-type extrapolation mapping methodology is further proposed to better inform high-cost experimental work.

Greek letters

The paper abbreviates several engineering terms and methods, including ANN, API, DNB, EPRI, NN, PWR, ReLU, and rRMSE.

ANN means artificial neural network.
API means application programming interface.
DNB means departure from nucleate boiling.
EPRI means Electric Power Research Institute.
NN means feed-forward neural network, while PWR means pressurized water reactor.
ReLU means rectified linear unit, and rRMSE means relative root-mean-square error.

1. Introduction

DNB-type CHF prediction matters for thermal-system safety, but robust modeling remains difficult because the DNB-triggering mechanism lacks general agreement. Existing domain-knowledge and standalone ML tools each have important limitations, motivating a hybrid framework.

Motivation: DNB crisis sharply deteriorates heat transfer as vapor blankets the heated surface, potentially causing cascading component failures.CHF is a regulatory limit for commercial PWRs worldwide.
Research gap: Accurate, robust CHF prediction remains elusive because researchers lack general agreement on the mechanism triggering DNB.
Existing approaches: Empirical correlations and LUTs are easy to implement but may fail beyond their validity ranges in geometry and operating conditions.
Existing approaches: Physics-based models rely on limited physical understanding and mostly empirical constitutive relations to close conservation equations.
ML limitations: CHF ML research faces limited experimental or high-fidelity numerical data, while prior studies lacked cross-validation and hyperparameter-sensitivity assessments.
ML limitations: Standalone ML tools can produce undesired, unphysical solutions because of their purely data-driven, black-box character.
Proposed direction: The proposed parallel hybrid framework uses domain knowledge as a baseline and ML to learn residual information, while the series approach is not explored because it requires a more extensive database.
Study scope: The work comprehensively evaluates the hybrid framework and proposes window-type extrapolation mapping to inform future high-cost experiments.

2. Methodology and dataset

The methodology combines a fixed domain-knowledge CHF model with ML trained on the residual between measured and domain-knowledge-predicted outputs. The case study uses standard CHF input variables and two prior models: the Groeneveld 2006 LUT and Liu’s physics-driven model.

Hybrid framework: The parallel hybrid framework combines a prior model and ML, with ML learning residuals rather than the final output directly.
Inputs: The CHF input vector comprises pressure, mass flux, local equilibrium quality, equivalent diameter, heated diameter, and heated length.
Training: During training, the residual is measured output minus prior-model prediction, and ML is optimized by comparing predicted and actual residuals through a loss function.
Evaluation: The hybrid output sums the prior-model prediction and ML-predicted residual, then is evaluated against experimental CHF using rRMSE.
Prior models: The study selects the Groeneveld 2006 LUT and Liu model as prior models for the hybrid framework.The Groeneveld LUT is data-driven; Liu’s model is physics-driven and based on liquid sublayer dryout.

Look-up table (LUT)

The Groeneveld 2006 LUT is a normalized CHF database for a vertical 8 mm water-cooled round tube, adjusted for different conditions using correction factors. It offers broad coverage and low computational cost but applies to both DNB and dryout.

Definition: The Groeneveld 2006 LUT is a normalized data bank for a vertical 8 mm water-cooled round tube.
Coverage: Its database contains over 30,000 data points covering 0.1 ≤ P ≤ 21 MPa, 0 < G ≤ 8,000 kg/m2-s, and -0.50 < x_e ≤ 0.90.
Corrections: Correction factors adjust table values for different diameter, geometry, and heat-flux-distribution conditions.
Corrections: Hydraulic diameter is recommended for diameter correction in noncircular channels because it better describes vapor formation and development in subcooled and low-quality flow.
Practical use: The LUT applies to both DNB and dryout scenarios, is simple to use, and has very low computational cost.

Mechanistic Liu model

The Liu model uses liquid sublayer dryout theory, where DNB follows complete evaporation of a superheated liquid layer beneath a vapor blanket. Its mechanistic formulation requires estimating liquid-sublayer thickness, vapor-blanket velocity, and vapor-blanket length, but existing closures largely address round tubes and performance degrades at low subcooling.

Mechanism: The model assumes DNB begins when a superheated liquid sublayer beneath a vapor blanket completely evaporates.The vapor blanket forms through coalescence of small bubbles rising near the heated wall.
Formulation: A heat-balance equation expresses CHF using liquid density, latent heat, liquid-sublayer thickness, vapor-blanket velocity, and vapor-blanket length.The model’s key task is determining the three vapor-blanket and liquid-sublayer parameters.
Mechanism: The Liu model analyzes instabilities at both the liquid-sublayer/vapor-blanket and vapor-blanket/bulk-region interfaces.These interfaces are part of the model’s liquid-sublayer dryout framework.
Scope: Although the mechanism is geometry-insensitive, existing mechanistic closures, including Liu’s, predominantly address round tubes.This creates a geometry-related scope constraint in the available closure relations.
Performance: For 2,482 tube data, Liu et al. reported favorable validation against earlier models, but the data-driven LUT had better statistics and Liu’s model was deficient at low subcooling.The comparison concerns the model’s reported validation matrix and its stated low-subcooling weakness.
ML comparison: Feed-forward neural networks and random forests are the machine-learning methods used alongside mechanistic modeling in the case study.The NN is described as effective for nonlinear relationships, while RF is characterized as fast and requiring little tuning.

Feed-forward neural network (NN)

A feed-forward neural network uses fully connected input, hidden, and output layers to learn nonlinear mappings through activation functions. Its weights and biases are trained by backpropagation with gradient descent, while regularization helps reduce overfitting.

Architecture: A feed-forward neural network contains at least three fully connected layers: input, hidden, and output.The supplied example is a 3/5/6/2 architecture.
Architecture: Activation functions between layers enable the network to perform nonlinear mappings.This capability supports the network’s use for complex input–output relationships.
Training: Weights and biases are initialized from a uniform distribution and iteratively updated during training by backpropagation with gradient descent.The update procedure adjusts network parameters during training.
Regularization: Dropout and early stopping can be applied to reduce test error by limiting overfitting.These are presented as regularization techniques for improving generalization.

Random forest (RF)

The random forest is a tree-based ensemble that aggregates predictions from multiple decision trees built using randomly selected observations and features. In this study, it predicts CHF from flow, geometry, and thermodynamic inputs evaluated with cross-validation.

RF method: Random forest aggregates multiple decision trees trained with randomly selected observations and features, then averages their predictions.Bootstrap aggregation is described as improving accuracy, preventing overfitting, and reducing variance.
Dataset: The CHF dataset contains 1,865 test cases spanning broad flow conditions and DNB-specific characteristics.The heaters include round tubes, annuli, and one-side-heated plates, with equilibrium quality ≤ 0.2.
Features: Inputs are pressure, mass flux, local or exit equilibrium quality, channel equivalent diameter, heated diameter, and heated length; the target is CHF.The axial power profile is uniform, and no geometry indicator is included in the baseline feature vector.
Dataset: The experimental CHF data have estimated uncertainties of 5–15%.The uncertainty estimate applies to the compiled experimental database.
Features: Feature engineering transforms raw data using prior expertise, while RF generally performs well with raw inputs and NN requires feature scaling.The NN inputs are standardized using arithmetic mean and standard deviation.
Validation: Both NN and RF are trained and validated for out-of-sample prediction using shuffled 10-fold cross-validation.Each fold uses nine subsamples for training and one for validation; a separate test fold is omitted because the dataset is relatively small.

3 Results and discussion

The case study finds that hybrid models combining prior domain-knowledge predictions with ML outperform standalone models and generalize more reliably across interpolation and extrapolation conditions. The approach also reduces scattered or physically undesirable predictions and supports targeted experimental planning.

Standalone ML configuration: 600–800 epochs were sufficient for convergence, while dropout did not further reduce validation error and overfitting was not considered a concern.The validation error never increased with the number of epochs.
Standalone ML configuration: The best-estimate standalone NN used a 6/50/50/50/1 architecture, Adam with learning rate = 0.001, ReLU activation, and no regularization.Standalone RF used 100 trees, 50%-70% features per tree, and no regularization.
Hybrid performance: Hybrid models clearly outperformed standalone ML and domain-knowledge models, with much smaller rRMSE values and faster-rising cumulative data-fraction curves.The comparison used 10-fold cross-validation for all ML results.
Hybrid performance: The hybrid framework guarantees predictions at least as accurate as the standalone prior model for interpolation or generalization, regardless of ML-method complexity.A prior model can be data-driven or physics-driven, allowing ML to learn from residual information while retaining the prior model’s baseline.
Generalization: Hybrid models corrected scattered and biased trends across pressure, mass flux, exit equilibrium quality, tube diameter, and tube length-to-diameter ratio, confirming enhanced generalization.The same conclusion held across the tested combinations of ML and prior models.
Extrapolation: For extrapolation, standalone RF performed poorly, whereas hybrid models consistently improved; LUT was more effective than Liu when training data were relatively small.Hybrid models were at least as accurate as standalone domain-knowledge counterparts, but extrapolation beyond a prior model’s validity range remained constrained.
Extrapolation: Window-type mapping identifies pressure regions where existing data suffice and flags 8 MPa as requiring new measurements because existing subsets cannot accurately predict DNB trends and values.Existing coverage from 4–14 MPa or 16–19 MPa was deemed sufficient for pressure extrapolation.

4 Conclusions

The physics-informed hybrid framework significantly outperforms standalone models across diverse flow conditions, while offering stronger extrapolation, extensibility, and experimental-planning support. Its domain knowledge and ML components can be expanded with model and data complexity without extra training or validation burden.

The hybrid approach significantly outperforms standalone models across a wide range of flow conditions, with reduced scatter and unbiased parametric trends.It uses prior domain knowledge as a foundation and ML to capture information in the mismatch between actual and domain-knowledge-predicted targets.
The framework supports on-the-fly extension of its applicability domain and complexity, including additional geometries, flow characteristics, and surface characteristics.Potential extensions include rod bundles, nanofluids, oxidation, wickability, and wettability features.
The hybrid ML structure can be simpler than its standalone counterpart, reducing computational cost for large datasets, while cross-validation reduces bias through more randomly generated subsets.Cross-validation results compare closely with conventional train–test splitting.
The hybrid framework provides more robust extrapolation than standalone ML methods, especially when the training dataset is relatively small.The choice of domain-knowledge model matters more for extrapolation than interpolation because it affects the ML-aided component.
Window-type extrapolation mapping leverages existing data to inform whether new measurements are needed for a targeted thermal-system investigation.The methodology is intended to help guide future experiments and is proposed for applications where domain knowledge and experimental or high-fidelity numerical data are available.

Appendix A. Standalone ML vs. LUT: summary

Standalone ML methods generally match or outperform LUT across the combined CHF dataset, but performance depends on geometry coverage and validation design. LUT notably under-predicts one-side heated plate CHF, while ML extrapolation is not guaranteed without representative training data.

Overall comparison: Standalone NN and RF are compared with the established LUT across 1,865 tube, annulus, and one-side heated plate data points.The dataset contains 1,439 tube, 378 annulus, and 48 plate measurements.
Sensitivity and validation: Standalone ML training errors can be very small, so validation performance is emphasized because the objective is performance on unseen data.The reported training rRMSE is 7.2% with NN and 5.2% with RF.
Overall comparison: RF performs similarly to LUT on tube data, reduces errors on annulus data, and significantly outperforms the table method on plate data.LUT performs marginally better than RF on tube data, while RF performs better for annulus and plate geometries.
Geometry-specific results: About 90% of predicted plate data fall within ±20% uncertainty for both ML methods, which agree closely with each other and measurements.The plate dataset is small, containing only 48 collected points.
Geometry-specific results: LUT dramatically under-predicts one-side heated plate CHF, potentially because it was developed for tubes and may require plate-specific correction factors.The smaller heated length of the plate heater is also identified as a possible explanation.
Extrapolation boundary: Standalone ML does not guarantee improved extrapolation when plate data are absent from training.The study expects similarly poor performance if no plate data are included during training.
Sensitivity and validation: Ten-fold versus five-fold cross-validation has negligible effect, while adding a geometry indicator contributes little and would require more training data.The validation errors across all-data and geometry-specific cross-validation are generally similar, except for plate predictions with NN.