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Two-Phase Flow And Heat Transfer Model

Calibration and Code Validation: A Sub-Cooled Boiling Flow Case Study

Idaho National Laboratory and North Carolina State

University

A. Bui, T.N. Dinh, R.R. Nourgaliev, and R.W. Youngblood

May 12-17, 2013:

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International Topical Meeting on Nuclear Reactor Thermal - Hydraulics, NURETH-15 NURETH15-637 Pisa, Italy, May 12-17, 2013

TWO-PHASE FLOW AND HEAT TRANSFER

MODEL CALIBRATION AND CODE VALIDATION:

A SUBCOOLED BOILING FLOW CASE STUDY

A. Bui1, T.N. Dinh

2, R.R. Nourgaliev

1 and R.W. Youngblood

1

1 Idaho National Laboratory, 2525 N. Fremont Ave, Idaho Falls, ID 83415, USA 2 Department of Nuclear Engineering, North Carolina State University, P.O. Box 7909,

2500 Stinson Dr, Raleigh, NC 27695-7909, USA

[email protected], [email protected], [email protected], [email protected]

ABSTRACT

This paper presents a calibration and validation case study of a (relatively) complex model of subcooled boiling flows which is widely used in nuclear reactor thermal-hydraulics analysis. The model calibration/validation approach constitutes a partial realization of the so-called “total data-model integration” concept where multidimensional data of different types, obtained from different sources and of different qualities are simultaneously used to improve models and model predictions. A statistical Bayesian model analysis framework is used which allows to incorporate information about not only data uncertainty, but also data quality in model calibration and validation. In addition to the calibration of closure model parameters, the proposed approach is shown to be able to reveal and quantify overall model inadequacy, which can be caused by both deficient model form and imprecise closure laws.

In summary, this paper discusses lessons learned from this subcooled boiling case study, the possible extension of the approach to the validation and calibration of more complex models, and the data qualification/characterization requirements in support of the total data-model integration concept realization.

1. INTRODUCTION

Advanced modeling and simulation of nuclear reactor systems increasingly involve high-fidelity/high-resolution multi-physics, multi-scale methods and tools. As noted in [1] and [2], assessment of such modeling tools requires data from different types of experiment, i.e. single-physics, multi-physics, separate-effect and integral-effect tests, and a systematic approach to model-data integration. The current practice of complex model validation and calibration would normally involve tuning or calibration of various sub-/closure-models using separate-effect test (SET) data and whole model validation using integral-effect test data (Figure 1a). The uncertainty in data and model bias can hardly be accounted for in the whole model predictions using such approach. More importantly, the relevance/adequacy of the whole model calibrated/validated based on the experimental data obtained at the conditions far different from the real plant conditions and the values of “extrapolative” model predictions are difficult to judge.

The so-called “total model-data integration” concept of complex model calibration/validation has been introduced in [1] and elaborated in [2] as illustrated in Figure 1b. This approach addresses

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the above-mentioned shortcomings of the traditional approach and, in particular, allows to:

• Account for data uncertainty; • Quantify whole model prediction uncertainty; • Use all available data including SETs and IETs in simultaneous sub- and whole model

calibration/validation; • Incrementally update model when more data become available.

(a) (b)

Figure 1 - “Traditional” approach to complex model calibration and validation (a) versus “total model-data integration” approach (b).

To enable such an approach to complex model calibration/validation, special attention has to be paid to data availability, quality and characterization. As mentioned in [1], data available for nuclear system code assessment are very heterogeneous, i.e. greatly different in their availability at different system levels and quality defined by their relevancy, scalability and uncertainty. For instance, separate-effect test (SET) data used in closure model derivation/calibration are often obtained from small-scale experiments conducted mainly under laboratorial conditions which are far from realistic plant conditions, i.e. low pressure, low and uniform wall heat flux, simple flow geometry, etc. The value of such data is therefore restricted by their relevancy and scalability and this fact should be taken into consideration in the “total model-data integration” approach. A data-value “weight” factor based on key measures, namely RPP or Reactor Prototypicality Parameter and EMU or Experimental Measurement Uncertainty, is introduced [1] and used in the model calibration/validation process. The RPP measures how close the test conditions are to the reactor conditions in the scenarios of interest and the EMU indicates the quality of data in terms of uncertainty.

In the “total model-data integration” approach, data obtained from the so-called “computer experiments” are equally important as data obtained from physical experiments. This recognizes the fact that many computer models are based on verifiable and universal first-principal physical laws which are less restricted by the scalability issue. For severe accident scenarios where extraordinary plant conditions (high temperature, high energy release, shock pressure, etc.) take place, “computer experiments” are probably the only ways to obtain insights into what happens in the considered system.

To deal with the heterogeneity of data (that in a broader sense may also include expert opinions)

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a strategy for data validation and homogenization is needed which, for instance, involves [3]:

• Faulty data detection: identification of doubtful or missing data; • Data correction: modification of doubtful data or estimation of missing data –

interpolation, data smoothing, data mining, data reconciliation, etc.; • Data quality quantification: UQ, scalability assessment, etc.

Data homogenization is used to remove the factors which cause data variations/biases which are not related to the physics of the considered phenomenon. These factors include, for instance, instrumental inaccuracies, variations in data acquisition and processing methods/procedures, changes in experimental settings and conditions, etc. If computer experiments are used for data generation, grid resolution/dependency would be the factor which needs to be addressed.

To be able to assimilate/integrate multiple heterogeneous datasets obtained at different scales and conditions in the whole model calibration/validation a significant burden is placed on the model-data integration framework. As described in [2], the model-data integration method based on Bayesian inference is found to be best suited for this purpose due to its capabilities to:

• account for uncertainty in observed data or take into consideration the “weight” or “values” of data given their uncertainty;

• quantify prediction uncertainty; • exploit the results of past validations/calibrations (in construction of more informative

priors for analysis) in sequential model updating; • handle “missing” data and allow the validation of unobserved quantity predictions; • handle multiple (multiphysics) coupled models using Bayesian influence networks.

This paper presents a partial realization of the above-described strategy for data characterization and integration and its application in the calibration of a typical, though simplified, example of multiphysics model – a subcooled boiling flow model – which is widely used in reactor thermal hydraulics analysis.

2. SUBCOOLED BOILING TWO-PHASE FLOW DATA AND MODELS

As illustrated in Figure 2, subcooled boiling flows involve a wide range of physics occurring at different scales. Modeling of subcooled boiling flows is normally based on a mix of different modeling approaches. At the macroscopic level, fluid and vapor transports and associated heat transfer in confined geometries are described using continuum mechanics and conservation laws which, in essential, are represented by partial differential equations (PDEs) formulating the variations of flow characteristics, such as pressure, temperature, velocity, and phase distribution, in time and space.

The simplified one-dimensional (1D) drift-flux model [4] is used in the present case study for simulation of heat-mass transports by subcooled boiling flows. This model can be derived from the two-fluid 6-equation model with the relative velocity between phases defined analytically. As a result, the total number of equations reduces by one and the resulted equation system arguably becomes more well-posed and easier to be solved.

At the micro-/meso-scopic scales where many important physical phenomena and interactions take place and where the continuum models are unable to resolve due to grid and/or physics restrictions, a less precise and more empirical approach to modeling is usually employed. The so-called constitutive models are derived based on particular sets of data and analytically define the

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local dynamics and interactions, i.e. condensation rate, bubble growth and detachment rates, etc., based on local flow and boundary characteristics. Due to the empirical nature of these constitutive models, their applications to the conditions significantly different from the conditions of the experiments which produce the data used for their derivation are very questionable and the model calibration and validation efforts mostly focus on improvement/verification of these constitutive models. A list of the major constitutive models used in the present subcooled boiling flow model is given in the following Table 1

Figure 2 - Subcooled boiling flow models and data.

Table 1 - Subcooled boiling flow constitutive models.

Model description Reference Comments

Drift velocity [4] Flow regime dependent

Mixture-wall friction factor [4] Flow regime dependent

Wall boiling – heat flux partitioning [5] [6]

Wall boiling - nucleation density [7] [6] [8] Nucleation boiling mode

Wall boiling - bubble detachment frequency

[7] [9] Nucleation boiling mode

Wall boiling- bubble detachment size [7] [10] Nucleation boiling mode

Bulk flow condensation [7] [11] [12]

Flow regime transition model [13] [7]

The flow transition model is based on empirically derived flow regime maps which correlate the factors such as the local void fraction or quality, flow direction (upward, downward, horizontal),

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heat flux, temperature, etc. in determination of the local configuration of vapor and liquid [7]. Since many other constitutive models are flow regime-dependent (see above table), invalidity/uncertainty in determination of local flow regime contributes greatly to the incorrectness and uncertainty of overall model predictions.

For subcooled boiling flows, SET and IET data have been obtained for both constitutive model derivation and whole model validation and some are listed as follows,

From IET measurements:

• Void temporal and spatial (axial and radial) distributions [14]; • (Axial) interfacial area distribution [15] [16]; • Flow field and temperature distributions;

From SET measurements: [6] [10] [9]

• Onsets of Nucleate Boiling (ONB) and Significant Void (OSV); • Nucleation site density; • Bubble growth rate; • Bubble detachment frequency and diameter; • Bubble Sauter diameter.

Table 2 - Incomplete list of boiling and two-phase flow experiments which deem relevant to calibration/validation of models of subcooled flow boiling.

Authors (year) Phenomena investigated Scalability

Bartel et al. (2001) [15] Interfacial area distribution 1 (1 atm)

Garnier et al. (2001) [17] Local measurements 1 (R12)

Kang et al. (2002) [18] Vapor phase measurements 1 (R113)

Warrier et al. (2002) [19] Interfacial heat transfer 1 (low P)

Roy et al. (1997) [20] Turbulence, void fraction 1 (R133)

Chen et al. (2003) Bubble coalescence 1 (1 atm)

Yeoh et al. (2004) [21] Bubble departure, bubbly flow 1 (1-2 atm)

Okawa et al. (2005a,b) Bubble slide 1 (1 atm)

Mauruset al. (2006) Bubble; boundary layer 1 (horiz.)

Bang et al. (2004) Visual bubble 1 (R134a)

Situ et al. (2004) [22] Bubble dynamics 1 (1 atm)

Ünal (1976) [23] Bubble growth 3 (full P)

Chang et al. (2002) Wall bubble 1 (R134a)

Basu et al. (2005) [24] Wall heat partitioning 1 (low P)

Basu et al. (2002) [6] Boiling onset, nucleation site density 1 (low P)

Hibiki& Ishii (2003) [8] Nucleation site 1 (1 am)

Theofanous et al. (2001) Nucleation on different surfaces 1 (1 atm, PB)

Dinh et al. (2004) Nucleation 1 (1 atm, PB)

* PB – pool boiling

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As shown in Table 2, the available experimental data were obtained mostly for low pressure condition which renders poor scalability of the constitutive models derived from those data. Additionally, successful validation of a flow model based on those data does not warrant equal successful application of the model in predictions of prototypic reactor conditions where much higher pressure is present. To address this pressure scalability issue, a data-relevancy weight factor can be derived based on the ratio [pe / ps] where pe is the pressure under which a particular dataset has been obtained and ps is the coolant (water) pressure of the simulated system.

3. BOILING TWO-PHASE FLOW MODEL CALIBRATION

The LANL Gaussian Process Models for Simulation Analysis (GPMSA) toolbox [25] is used in this paper to calibrate the subcooled boiling model described above. Using GPMSA, the calibration process is conducted in the following steps:

1. Construction of a model surrogate using a process convolution technique [26] based on Principal Component Analysis (PCA) and Gaussian processes (GPs);

2. Conducting Bayesian calibration via Markov chain Monte Carlo (MCMC) sampling.

Construction of subcooled boiling flow model surrogate is based on 200 simulation runs with the inputs varying in the ranges listed in Table 3. The pipe diameter and flow rate have been kept unchanged at 24 mm and 890 kg/(m2s), respectively. The input ranges have been chosen based on the conditions of a subset of experiment data obtained by Bartolomej et al. [14] which are to be used in the calibration (see Table 4). In this preliminary study, only two parameters are chosen for calibration: one is related to the subcooled boiling suppression and the other to the bulk flow condensation. The inputs are sampled using the Latin hypercube sampling (LHS) method.

Table 3 - Variation ranges of input parameters.

Input p,

MPa

Inlet Tsub,

K

Heat flux,

kW/m2

Condensation

parameter, p1

Subcooled boiling

suppression

parameter, p2

Minimum 1 20 360 0.0 0.01 Maximum 5 55 800 0.2 0.5

Table 4 - Experimental conditions [14]. Experime

nt p, MPa

Inlet

Tsub, K

Heat flux,

kW/m2

1 1.5 20.9 380 2 1.5 49.2 790 3 3.0 23.9 380 4 3.0 47.0 790 5 4.5 26.4 380 6 4.5 52.4 790

Since information about measurement error is not available from the published source, it is artificially introduced and, for the case study purpose, all data are assumed to have a standard deviation of 0.02.

The predictions of steady-state 1D axial distributions of void fraction, fluid temperature, pressure and mixture velocity are obtained at 35 spatial locations. Given the high dimensionality of the outputs, the principal component analysis (PCA) technique [27] is needed for dimensionality

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reduction. The results of PCA are shown in Figures 3-4, indicating that up to 98.3% of variance can be captured by the first 5 principal components (PCs).

Figure 3 - Accumulative percentage of variances “explained” by principal components (PCs).

Figure 4 - Variations captured by PCs 1-5 – Note the increasingly smaller variances being

captured by higher PCs.

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The relationship between observed data D and model prediction (or response) � can be expressed as [28]

� = ��, �� + �� + �� (1)

with �� being a model discrepancy/bias function, � indicating the observation error/uncertainty and x denoting the vector of known variable inputs, i.e. pressure, inlet fluid subcooling and wall heat flux in this study.

A multivariate emulator which approximates the real model response can be constructed as follows [26]

��, �� = ∑ �� , �� + � (2)

where { ki } is a collection of orthogonal basis vectors obtained as a result of the PCA,

��, �� are the weights defined over input space, � is an error term, and �� is the number

of PCs used to capture all significant variances.

Each weight �� is assumed to be a zero-mean Gaussian process and expressed as ��, �� ~

� [0, cωi ( ��, ��, ��′, �′�)] where the covariance function cω is defined as [26]

Cωi ( ��, ��, ��′, �′�� = �� ∏ ��,!

"#�$��%&'(

∏ ��,)*�+"#�,��-

& '(�.)��

�+!�� (3)

with the control hyperparameters �� and �� to be defined (together with �) in the

calibration process.

Both and � can be assumed to be normally distributed with � having a zero mean, i.e. (�) ~ � [mδ (�),cδ ( �, �′)] and �� ~ �[0,c� ( �, �′)] with m and c being the mean and covariance functions, respectively. The covariance functions of and and � are defined in the

similar manner as for weight �� .

The 1D model bias/discrepancy �� is constructed in this study as a linear combination of 13 normal kernels placed at equidistant locations along the pipe.

In the Bayesian calibration step, the posterior probabilistic distributions of the considered parameters are sampled with MCMC. Following the Bayes theorem, the posterior distributions of parameters θθθθ are computed as follows:

/��|�� ∝ ℒ��|��/�� (4)

where /�� is a prior probabilistic distribution of θθθθ and ℒ��|�� is the “likelihood” of the observation D given specific values of θθθθ.

Given the above assumptions about the distributions of data, observation error, and model discrepancy, the ‘likelihood’ function is also a normal probability density function defined as

ℒ��|�� ∝ exp 6− �8 �� − ��9:�� − ��; (5)

with V being a matrix which combines the variance and covariance matrices of D, M, and �

[29].

As can be seen in its formulation, the likelihood function is where comparison of model predictions and data occurs. When there are multiple observed datasets, a joint likelihood function can be derived which is simply a multiplication of the individual likelihood functions, i.e.

ℒ��|�� = ∏ ℒ��|�� (6)

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where �� and �� being the subsets of the full data space � and parameter space �, respectively.

The weighting factor of dataset �� calculated based on the relevancy, scalability and uncertainty criteria can be used here to express the variance of �� in terms of a common variance determined for all datasets.

The results of GPMSA calibration of the subcooled flow model are presented in Figures 5-8. The traces of MCMC draws in Figure 6 show stabilized distributions of parameters after 1800 draws, indicating convergence of the chains. Posterior distributions of the two considered parameters are displayed in Figure 5, showing relatively spread variations of parameters from their mean values given the small number of datasets used for calibration and (assumed) measurement error. It is worth mentioning that both parameters are responsible for the change of vapor phase and their calibration using only void fraction distribution response would be handicapped by the non-identifiability problem (see discussion in [2]). One possible solution to this issue is based on the use of other responses, e.g. fluid temperature, in the calibration process [30].

Figure 7 show a potential problem with the emulator constructed on the basis the chosen 5 PCs with the parameters �� of the second and third inputs (inlet fluid subcooling and wall heat flux) approaching zero for PCs 4 and 5. In such a case, the predictions by the emulator are questionable given its incapability to find any smooth trend in the data. The �� value of the condensation parameter is seen to be close to 1 in Figure 7, indicating a relatively small sensitivity of the response to this parameter.

The model discrepancy is presented in Figure 8, which also shows the comparison of calibrated predictions and experimental data. The discrepancy is an indication of model inadequacy which may be caused by general model form (equations) and/or constitutive model inadequacy.

Figure 5 - Histogram of draws from posterior distributions of boiling suppression (red) and

condensation (blue) parameters shown on the standardized range [0-1].

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Figure 6 - Traces of MCMC draws of parameters.

Figure 7 - Boxplots of the marginal posterior distributions of <=>.

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Figure 8 - Comparison of data (circles) and predictions (lines) obtained with the calibrated parameter values (left panel), estimated model discrepancy (center panel), and calibrated

predictions made by taking into account model discrepancy (right panel).

4. DISCUSSIONS AND CONCLUSIONS

This paper presents a preliminary realization of the “total model-data integration” concept which is based on the Bayesian inference and a mechanism to accommodate multiple data-streams and models. For efficient statistical analysis, multivariate models with multiple responses are approximated using a convolution process method in which surrogate models are constructed on the basis of multiple GP kernels. MCMC is then applied to sample the posterior probabilistic distributions of the calibrated parameters.

A simplified model of two-phase flows with subcooled boiling has been used as a target for concept testing. The model featuring both first-principle continuum descriptions of vapor and liquid transports and the constitutive closure laws approximating the local thermal and mechanistic interactions is an example of more complex multiphysics models which are used in nuclear engineering analysis.

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Multiple sets of measurements of 1D void fraction distribution in vertical pipes are used for model calibration. Since data uncertainty and measurement error are missing, this information is artificially introduced.

A demonstrative calibration of the model conducted using the LANL GPMSA toolbox shows the capabilities and potentials of the chosen model-data integration framework in

• Integration of multiple datasets of multidimensional (including spatial and temporal) data;

• Integration of data of different qualities via data weighting; • Propagation of data uncertainty to parameter probabilistic distributions; • Identification of the sensitivity of model responses to various parameters (and

corresponding constitutive models); • Identification of the overall model inadequacy.

In the context of total model-data integration, the capability of integration of data obtained at different physical scales and having different quality – a function of relevancy, scalability and uncertainty – constitutes a foundation for the proposed framework. However, data characterization, validation and homogenization are also needed to quantify data information value and establish weight factor based on data quality and relevancy to applications at hand. Therefore, a systematic approach to data characterization which is the subject of an ongoing study constitutes another foundation for the “total data-model integration” framework, where multiple model responses are taken into consideration and a variety of data obtained at different physical scales are simultaneously used in the calibration process.

5. ACKNOWLEDGEMENT

This research has been supported by the Consortium for Advanced Simulation of Light Water Reactors (http://www.casl.org), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725.

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