Improvement of two-phase closure models in CTF using Bayesian inference☆
Introduction
CTF (developed from legacy subchannel code COBRA-TF) is a thermal hydraulics code based on the two-fluid, three-field model (i.e., liquid film, dispersed droplets, and dispersed bubble/continuous vapor) jointly developed by Oak Ridge National Laboratory and North Carolina State University for light-water reactor (LWR) core analysis (Salko et al., 2019a). Qualification of the code (Salko et al., 2019b) revealed that CTF has a tendency to over-predict the void fraction, particularly at low void fraction corresponding to bubbly slug flow regimes. A sensitivity and uncertainty quantification study (Porter et al., 2018) demonstrated that the interfacial drag model is one of the most important closure models in correctly predicting void fraction. Another study (Zhao et al., 2019) expanded on the CTF two-phase validation matrix, showing that the subcooled boiling model used in CTF also has a significant impact on the quality of void fraction predictions. The legacy wall boiling model was thus updated and the approach used by TRACE was implemented into CTF and tested for the CTF validation matrix (Kumar and Salko, 2020). The legacy interfacial drag model was also updated to a drift-flux model approach for defining the slip between vapor and liquid phases in the small bubble and slug bubble flow regimes. Concomitantly with the over-prediction in void fraction, the two-phase pressure drop was also found to be substantially over-predicted (Zhao et al., 2019). To address this over-prediction, the annular flow regime transition was updated by implementing a new flow regime map, and the treatment of the unstable film interfacial drag was revised. Additional details on these improvements can be found in other work by Salko and others (Salko et al., 2020, Kumar et al., 2021), which summarize the various two-phase closure model improvements.
Physics-informed machine learning (ML) approaches can be used to tune these new models to achieve improved prediction of key output parameters such as void fraction, two-phase pressure drop, and wall temperature. Various ML approaches have been used to improve the physical models in CTF. A data-driven, supervised Hi2Lo mapping that derived training data from detailed STAR-CCM computations was developed to capture important statistical properties of the flow field that are unresolved by the closure models in CTF, such as the turbulent kinetic energy (TKE) (Gurecky, 2019). These models strongly influence the CRUD growth rates, which are determined based on CTF rod temperature distributions. This statistical downscaling method was based on importance sampling, multivariate copula, and gradient boosting. The turbulent mixing coefficient in CTF was calibrated with experimental data using a Bayesian inference-based approach using Markov chain Monte Carlo (MCMC), as well as Hi2Lo mapping using high-resolution STAR-CCM simulations (Gordon et al., 2019). A hybrid of prior/domain knowledge methods (both data driven and mechanistic) and neural network-based ML approaches was shown to improve the critical heat flux (CHF) model prediction in CTF, in comparison to either standalone approach (Zhao et al., 2020). A physics-discovered, data-driven methodology was used to modify the functional form of the existing closure models in CTF using experimental void fraction data from BWR full bundle tests (BFBT) (select assemblies) and PWR subchannel and bundle tests (PSBT) (all bundles) (Borowiec et al., 2021). The code prediction was shown to be improved for both void fraction and two-phase pressure drop predictions for the validation dataset, highlighting the effectiveness of an alternative approach to ML techniques.
The principal advantage of using a Bayesian inference approach in this regard is that it provides both the optimal parameter values and the uncertainty associated with each free parameter, given the experimental data. Even for cases in which a traditional optimization approach encounters difficulties, such as when the root-mean-square error (RMSE) response surface is non-smooth near the optimal value, a numerical approach that uses MCMC might still provide accurate mean and higher moments of the parameter distributions at the expense of thousands of model evaluations. For quickly executing models, this computational expense is not a concern. Using the Bayesian Inference approach, selected parameters in two-phase closure models can be calibrated to experimental data. The calibrated models can then be tested on previously unseen datasets to demonstrate the effectiveness of the approach.
Inverse uncertainty quantification (UQ) approaches in a Bayesian framework are being increasingly used for characterizing input parameter (epistemic) uncertainties (Oberkampf et al., 2004) and/or code calibration in “best estimate” thermal-hydraulics (T/H) system codes. A historical perspective of UQ in best estimate T/H codes and the introduction of the best estimate plus uncertainty (BEPU) methodology is provided in the paper by D’Auria et al. (2012). Bayesian calibration was used to calibrate the single-phase friction model using experimental data in the T/H code FLICA4, and an improvement was shown in the code’s predictive capability (Bachoc et al., 2014). The bias between the T/H code and experimental data was modeled as a Gaussian process (GP), and the maximum likelihood method was used to estimate the input model parameters mean and variance. A Bayesian inference-based methodology was formulated to estimate the uncertainty in physical models in T/H system codes, using an efficient MCMC method: the blocked Gibbs model (Damblin and Gaillard, 2020). The approach was applied to the CATHARE T/H code using a GP-based surrogate model and tested on condensation models using an experimental database of a scaled emergency core cooling system. Similarly, an inverse UQ approach based on a Bayesian framework using MCMC was applied on an experimental test facility of a passive safety system in RELAP5-3D. Dimensional reduction of the transient data was undertaken using principal component analysis (PCA) and Kriging metamodeling was used to emulate RELAP5-3D. A modular Bayesian approach was implemented for inverse UQ using MCMC to calibrate and quantify uncertainties in closure models in TRACE, using the BFBT steady-state void fraction dataset (Wu et al., 2018). A fast GP-based surrogate model was used to replace the full TRACE model, providing a significant reduction in the MCMC sampling time. A sequential test source allocation approach was used to efficiently split the experimental tests between inverse UQ and validation datasets. It was found that the posterior distribution of the model parameters was over-fitted when the model discrepancy term was not considered in the likelihood function.
There are three main objectives of this study. First, this work demonstrates that Bayesian inference using MCMC can be used as a calibration tool for two-phase closure model optimization in CTF. Tied to this objective, it is important to show that the tuned models work well beyond the chosen datasets used in the calibration process. Second, this work aims to generate good surrogates that can effectively mimic CTF code responses and serve as workflows for other modeling needs. Currently, surrogate generation is more of an art than an established process, especially for non-linear code responses. Third, this study builds a calibration framework for two-phase closure models using in-house and open-source tools, which can be used for other models in the future, such as the drift-flux model, departure from nucleate boiling, etc.
Section snippets
Calibration parameter selection for legacy models
The calibration parameters were down-selected from works of Zhao et al. (2019) and Wu et al. (2018), who performed a UQ analysis for the TRACE code. The list of parameters (in terms of multiplication factors) selected in that work (Wu et al., 2018), based on a sensitivity analysis, and which concurred with Zhao et al. (2019), in terms of models which require the greatest need for improvement, include the single-phase liquid-to-wall heat transfer coefficient (HTC), subcooled boiling HTC, wall
Bayesian inference
The Bayesian inference approach formulation is presented in the following subsection, followed by the implementation of the MCMC algorithm.
Test selection for calibration
This section describes the selection of the tests used for the calibration activity. Tests were selected from the CTF validation matrix since CTF models and data processing tools were already available. The tests selected for the legacy model calibration are described in the next subsection, followed by the tests selected for the Chisholm model calibration.
Surrogate model development
During the initial simulation of the test sets, it was decided that a CTF surrogate should be developed for each test set. The primary reason to use a surrogate model for replacing full CTF calculations in the calibration is to reduce computation times. To compute the likelihood, the CTF predictions are compared with the experimental data for each case to obtain a residual vector. The output quantities of interest for the surrogate model are the (vapor) void fraction and the two-phase pressure
Legacy models calibration results
The DREAM-based calibration was performed by using 10 chains executed for 20,000 generations, resulting in a total of 200,000 samples. The first 5000 generations were discarded as burn-in generations to allow the chains to explore the parameter space and to converge to the stationary posterior density function. Approximate convergence criteria were used, which involved comparing the distributions for different sample sizes (e.g., 1.5e5 and 2e5 samples). The calibration was considered to be
Conclusions
CTF closure models were improved using a Bayesian calibration process to optimize selected modeling parameters to achieve an improved prediction of existing two-phase experimental data available in the CTF validation matrix. Selected parameters included multipliers on interfacial drag models in three different flow regimes for void fraction improvement and parameters in the two-phase flow multiplier on the wall shear model for two-phase pressure drop improvement. Calibration experiments
CRediT authorship contribution statement
Vineet Kumar: Methodology, Investigation, Writing – original draft. William Gurecky: Methodology, Writing – original draft, Reviewing and editing. Robert Salko: Conceptualization, Methodology, Reviewing and editing. Belgacem Hizoum: Conceptualization, Reviewing and editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This research was supported by the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program.
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Cited by (1)
Verification and validation of the Alternative Nonlinear Two-phase Subchannel (ANTS) code
2022, Nuclear Engineering and DesignCitation Excerpt :Note that ANTS uses the default Chisholm two-phase multiplier model, as described in Kropaczek et al. (2022). A recalibration of the two-phase wall shear model was previously performed for CTF, which is described in Kumar et al. (2022). This work used a Bayesian inference approach to recalibrate the modeling coefficients in the Chisholm model, which was found to produce a better prediction of the Risø data.
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