Uncertainty analysis of sub-channel code calculated ONB wall superheat in rod bundle experiments using the GRS methodology

Progress in Nuclear Energy 65 (2013) 42e49

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Uncertainty analysis of sub-channel code calculated ONB wall superheat in rod bundle experiments using the GRS methodology Robert K. Salko*, Maria N. Avramova** The Pennsylvania State University, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 December 2012 Received in revised form 1 February 2013 Accepted 7 February 2013

Rod bundle experiments were performed for prototypical PWR operating conditions in the project “New Experimental Studies of Thermal-Hydraulics of Rod Bundles (NESTOR)”. The intent of the project was to improve the understanding of the Axial Offset Anomaly (AOA) through improved modeling of Onset of Nucleate Boiling (ONB) (EPRI, 2008) using sub-channel codes. Skewing of the axial power profile (AOA) is most likely driven by the deposition of boron in the crud layer on nuclear fuel rods, which is caused by boiling on the fuel rod surface (EPRI, 2008). VIPRE-I (Srikantiah, 1992), a sub-channel code, was chosen for the analysis of NESTOR tests and for which uncertainty analysis was performed. NESTOR experimental results were used to optimize grid-loss coefficients, friction-loss coefficients, and a single-phase heat transfer model in the code. By modeling NESTOR ONB tests, the VIPRE-I calculated wall superheat was determined at the experimental ONB locations. This calculated ONB wall superheat could be used as a criterion in VIPRE-I for the prediction of ONB; however, it is important to quantify the uncertainty of this calculated ONB wall superheat in order to know the accuracy of such a criterion. The VIPRE-I model optimization process, however, was a complicated one and involved interaction of both experimental and code modeling uncertainties. The propagation of these uncertainties was treated using the Gesellschaft für Anlagen und Reaktorsicherheit (GRS) methodology; a process which is detailed in this paper. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: VIPRE-I SUSA NESTOR ONB

1. Introduction 1.1. Code modeling uncertainty A thermal-hydraulic simulation code like VIPRE-I describes a physical system using (Strydom, 2010):
A mathematical model comprised of governing equations as well as sub-grid equations that capture complex physical processes like convective heat transfer and turbulent exchange
A numerical technique to solve the analytical mathematical model
Physical parameters that describe the system being modeled (e.g. boundary conditions and geometry) Code modeling uncertainty is the inability of the code to capture the behavior of a system with complete accuracy. This uncertainty

* Corresponding author. 18 Reber Building, University Park, PA 16802, USA. Tel.: þ1 570 972 0988. ** Corresponding author. Tel.: þ1 814 865 0043. E-mail address: [email protected] (R.K. Salko). 0149-1970/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pnucene.2013.02.003

is introduced by each of the three aforementioned aspects of code modeling. For example, physical models e like convective heat transfer correlations e have a specified level of uncertainty owing to uncertainties in the experiments they were developed from and their inherent inability to perfectly capture the physics of the phenomenon. Numerical techniques, which must be used for solving the governing equations, introduce truncation error. The system that is being modeled e the NESTOR tests, in this case e has uncertainties associated with the operating conditions and geometry. The manner in which these uncertainties interact and propagate through the solution will be complex and convoluted. One such method that has been developed to characterize this propagation is the GRS methodology. 1.2. GRS methodology The GRS methodology uses a statistical type of approach to track the propagation of code input uncertainties and quantify their effect on some output Figure of Merit (FOM). The System for Uncertainty and Sensitivity Analysis (SUSA) was the actual program used for performing the analysis using the GRS methodology. A benefit in this methodology is that it requires no modification to the

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49

Nomenclature G

btm DPform DPfriciton DZ d _ m

r s A a b C Ct d,e,f Drod Dc Dh F f Fmod fmod G hdb hmod i k kMVG,mod kSSG,mod L m,n n P p Pout

average mass flux between adjacent sub-channels turbulent mixing coefficient form pressure loss friction pressure loss axial unit length parameter uncertainty coolant mass flow rate coolant density standard deviation sub-channel cross-sectional area probability FOM will fall between tolerance limits confidence in probability, a test section channel width turbulent momentum factor correlating coefficients in the GSCE rod diameter predicted/measured quantity difference parameter sub-channel hydraulic diameter radial peaking factor friction loss coefficient radial power factor modification parameter friction factor modification parameter coolant mass flux single phase HTC obtained using DittuseBoelter single phase HTC using optimized model index of sub-channel grid loss coefficient MVG form loss modification parameter SSG form loss modification parameter rod heated length correlating coefficients in heat transfer model number of calculations required for probability, a, and confidence, b pressure rod pitch outlet pressure

existing code if the uncertain parameters can be captured by input parameters to the code. Further, the methodology can be used for any type of computational tool. The principle of the methodology is to quantify the uncertainties of selected code input uncertainties. The user is not limited in the number of input uncertainties they can choose to investigate. This set of input uncertainties is used to create some number of code input cases containing random variations of the selected input parameters. Their randomization will be dependent on their uncertainty and probability distribution function. These cases are then run and their outputs are used to set tolerance bands that some selected FOM will fall between. It is the number of input cases generated that will determine the probability that the FOM will fall between the tolerance bands to a given confidence. The number of cases required for a specified probability and confidence is determined using Wilks’ formula (Wilks, 1941), which is shown for two-sided statistical tolerance intervals in Equation (1).

ð1  an Þ  nð1  aÞan1 > ¼ b

(1)

In Equation (1), b is the confidence that the code result for the FOM will fall between the stated tolerance bands with probability, a. The term, n, is the number of code runs required to achieve the stated probability and confidence. As an example, for a standard

43

r Rdb S Tc,ONB Tc Texp,wo Ti,m,EOHL Tin Tout Tsat

rod radius ratio of experimental HTC to calculated HTC gap width calculated ONB wall superheat calculated rod surface temperature experimental rod surface temperature measured temperature at EOHL for sub-channel i experimental inlet temperature experimental outlet temperature local fluid saturation temperature y coolant velocity 0 linear heat rate W 0 lateral cross-flow due to turbulence w total rod bundle power WT Xi,c,EOHL predicted, non-dimensional temperature at EOHL measured non-dimensional temperature for subXi,m channel i z axial elevation in rod bundle distance to the nearest upstream grid Zg AOA Axial Offset Anomaly CC Pearson’s productemoment correlation coefficient EOHL End of Heated Length FOM Figure of Merit GRS Gesellschaft für Anlagen und Reaktorsicherheit GSCE Grid Spacer Cooling Enhancement HTC Heat Transfer Coefficient HTCmod HTC modification parameter LDV Laser Doppler Velocimetry MVG Mixing Vane Grid NESTOR New Experimental Studies of Thermal-Hydraulics of Rod Bundles ONB Onset of Nucleate Boiling PDF Probability Distribution Function PWR Pressurized Water Reactor RMS Root-Mean Square SSG Simple Support Grid SUSA System for Uncertainty and Sensitivity Analysis

95% confidence and 95% probability, 93 code runs would be required, which is the number of code runs that were performed for this study. Specific steps for using the GRS methodology for performing code uncertainty analysis were outlined by Salah et al. (2006): 1. Identification of all relevant sources of uncertainty, represented by uncertain parameters 2. Definition of uncertainty ranges for the identified parameters (e.g. minimum and maximum values) as well as the uncertain parameter probability distribution 3. Generation of a random sample of size N from their probability distributions 4. Execution of the computer code with the generated sample of input values 5. Derivation of the quantitative uncertainty of the code predictions by specifying the statistical tolerance limits of the FOM 6. Computation of the quantitative sensitivity measures to rank the importance of the individual uncertain source parameters Benefits of the GRS methodology include being able to quantify code uncertainty in a straightforward manner, with resulting tolerance band confidence and probability only being dependent on the number of code runs. The methodology is, however,

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dependent on the user’s selection of uncertain input parameters and their associated uncertainty ranges and probability distributions. This aspect has an element of subjectivity and can be a source of error in the uncertainty analysis. 1.3. Application to NESTOR test data analysis The purpose of the NESTOR tests was to develop a wall superheat criterion based model for the accurate prediction of ONB in PWR cores. It was envisioned that an improved understanding of ONB in PWR cores could facilitate the development of an AOA model since AOA is likely caused by the boiling-induced deposition of boron into the fuel rod crud layer (EPRI, 2008). The tests were performed in heated and unheated configurations of 5  5 rod bundle facilities. The rod bundles utilized both Simple Support Grids (SSG) and Mixing Vane Grids (MVG). High-fidelity local transverse and axial velocity profile measurements were taken via Laser Doppler Velocimetry (LDV) in the unheated configuration along with grid span pressure drop measurements. High-fidelity rod surface temperature measurements were also taken in the heated configuration via thermocouple probes that were able to move axially and azimuthally around the inside surfaces of the heater rods. Pressure drop measurements from unheated tests were used to optimize the friction and grid pressure drop models in VIPRE-I. Velocity measurements and sub-channel temperature measurements at the test section outlet were used to optimize the turbulent mixing model in VIPRE-I. Single-phase heated experiments were used to develop a dedicated single-phase heat transfer model. All test analysis was performed on the central heater rod, dubbed Rod 5 throughout this paper, and on the four inner-type sub-channels surrounding the rod. Further information relating to the NESTOR program and the modeling optimization process is available (see (Péturaud et al., 2011; Salko, 2010)). Tests were also run with operating conditions that caused ONB to occur in the bundle. VIPRE-I was then used, with all models optimized to the NESTOR configuration, to determine the calculated wall superheat at the experimental ONB locations. The calculated wall superheat results at experimental locations could be utilized for tuning the wall superheat ONB criterion in VIPRE-I for rod bundle geometry. Such a modification, however, would require a definite statement of the uncertainty in the calculated rod surface temperature, which is the primary component in the calculated ONB wall superheat as shown in Equation (2). Therefore, the purpose of this uncertainty analysis is to quantify the impact of the experimental (boundary conditions and test section geometry) and code modeling uncertainties on the calculated rod surface temperature.

Tc;ONB ¼ Tc  Tsat

(2)

Table 1 Code input uncertain parameters. Uncertain input parameter

Symbol

Parameter range (2s uncertainty)

Test section channel width Rod pitch Rod radius Radial peaking factor Heated length Axial elevation in rod bundle Inlet temperature Absolute pressure Differential pressure Mass flow rate Test section power Experimental rod surface temperature

dC dp dr dF/F dL/L dz dTin dP/P dP/P _ m _ dm= dWT/WT dTexp,wo

0.15 mm 0.14 mm 10 mm 0.2% 0.07% 1 mm 0.2 K 0.5% 0.1% 1% 0.1% 2.0 K

information on sub-channel cross-sectional area and wetted perimeter. The uncertainty in parameters like rod radius, r, and rod pitch, p, affect these terms. Equation (3), for example, was used to calculate a normal channel flow area.

Ainner ¼ p2  pr 2

Considering that the dimensional uncertainties are independent and random, the uncertainty in sub-channel flow area can be calculated as follows, with the d terms representing the parameter uncertainty.

dAinner

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 vAinner vAinner dp þ dr ¼ vp vr

(4)

This lead to a sub-channel area 2s uncertainty of 2  106 m2 for the inner-type sub-channel. The inner-type channel is depicted in Fig. 1 as Sub-channel numbers 4, 5, and 6. Uncertainties of the side (channels 2 and 3) and corner (channel 1) type channels were also calculated and used in the analysis, though the rod surface temperature uncertainty analysis was only performed for the central rod. Another example involves the test section power. VIPRE-I takes power input in units of kW/ft, which must be calculated from the supplied test section power, WT, and the total length of the 25 heater rods, L. Since the section length also had uncertainty, it was necessary to calculate the propagation of uncertainty into the linear 0 heat rate, W .

W0 ¼

WT 25L

(5)

Similarly, the inlet flow boundary condition was given to VIPRE-I in terms of mass flux, whereas uncertainty was given in terms of mass flow rate. Calculation of the mass flux term involved three uncertain parameters, as shown below:

2. Identification of uncertainty sources The VIPRE-I input uncertain parameters needed to be listed and quantified for the GRS methodology. These uncertain parameters include both measured uncertain values (e.g. inlet mass flux and temperature) as well as calculated parameters (e.g. friction and grid loss coefficients). Test operating condition uncertainty and geometry uncertainty was obtained from NESTOR documentation (EPRI, 2009, 2010; Peturaud and Decossin, 2001) and is repeated here in Table 1. The uncertain parameters of Table 1 weren’t all directly used in writing the VIPRE-I input deck. For example, the input deck takes

(3)

Fig. 1. One-quarter symmetric section of the NESTOR test bundle.

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49

G ¼

_ m C 2  25*pr 2

(6)

These calculated uncertain input parameters are summarized in Table 2. The terms A, Pw, PH, S, and L represent the sub-channel cross-sectional area, wetted perimeter, heated perimeter, gap width, and gap length, respectively. The subscripts, “I”, “S”, and “C”, represent the inner-type, side-type, and corner-type channel configuration, respectively. The subscripts, CeS, SeS, SeI, and IeI, represent the type of gap e possible connections include cornere side, sideeinner, sideeside, and innereinner. The aforementioned uncertain parameters were also used in the calibration of VIPRE-I models which include the turbulent mixing model parameters (btm and Ct), the friction and grid loss coefficients (f and k), and the single-phase heat transfer coefficient model. Each of these error sources is discussed separately in the following subsections. 2.1. Turbulent mixing model VIPRE-I uses a simple turbulent diffusion model, as shown in Equation (7).

w0 ¼ btm SG

(7) 0

The amount of mixing, w , caused by turbulence is expressed as a product of three terms: (1) the width of the gap, S, (2) the average of 0 the axial mass fluxes, G, in the two adjacent sub-channels that w is being evaluated for and, (3) a coefficient, btm, that captures the 0 strength of the mixing. The term, w , when calculated, can be used in each governing equation to determine the quantity of mass, momentum, and energy transferred through the gaps due to turbulent exchange. The btm term is left for the user to choose. An empirical model, such as that given by Rogers and Rosehart (1972), could be utilized to estimate the mixing coefficient term. However, detailed axial and lateral velocity measurements as well as (End of Heated Length) EOHL temperature measurements were available in the single phase tests of NESTOR. It was possible to use this information to optimize certain terms, such as the mixing coefficient, for the NESTOR geometry and operating conditions. The turbulent mixing coefficient was optimized by minimizing the difference between predicted and measured, non-dimensionalized, EOHL temperatures through variation of the btm parameter. The measured, non-dimensional EOHL temperature was defined as Table 2 Uncertainty of VIPRE-I input parameters. Input parameter

Uncertainty (2s)

AI AC AS Pw,I Pw,C Pw,S PH,I PH,C PH,S SCS SSS SSI SII LCS LSI LSS LII Linear Heat Rate

4  106 m2 2.5  106 m2 2.6  106 m2 6  105 m 6  104 m 1.4  104 m 6  105 m 1.6  105 m 3  105 m 3  104 m 3  104 m 1.4  104 m 1.4  104 m 8  105 m 8  105 m 1.4  104 m 1.4  104 m kW 1:6  104 ft kg 23 2 s$m

Inlet Mass Flux

45

shown in Equation (8), where the numerator is the difference between the measured sub-channel temperature and the experimental outlet temperature and the denominator is the difference between the experimental outlet and inlet temperatures. A VIPREI-predicted, non-dimensional temperature was similarly defined by using calculated temperature in place of measured temperature at EOHL.

Xi;m ¼

Ti;m;EOHL  Tout Tout  Tin

(8)

A difference between predicted and measured, nondimensional temperature was calculated for each sub-channel in the model. Then, the RMS of these differences was taken to create a single parameter, shown in Equation (9), that quantified the difference between predicted and measured EOHL temperatures for a given VIPRE-I run (using a single b value). The reader is guided towards (Salko, 2010) for further information on the turbulent mixing parameter optimization procedure.

 2 Dc ¼ Si Xi;c;EOHL  Xi;m;EOHL

(9)

The optimization was done so for the SSG and MVG configurations. An optimum btm value was found for the SSG configuration, leading to a minimum Dc parameter, but no such value was found for the MVG configuration, which was the focus of this uncertainty analysis. Rather, it was found that it was necessary to raise the mixing coefficient to very high values to best match experimental results. It was assumed that this was due to the directed cross-flow effects, brought on by the presence of MVGs, which acted to flatten the lateral temperature profile in the experiments. As Fig. 2 demonstrates, the magnitude of the Dc parameter decreased exponentially with increasing btm for all MVG tests. For this reason, a large mixing coefficient of 0.3 was chosen for future modeling of NESTOR tests using VIPRE-I. With no optimum btm value available from the analysis, it was also not possible to quantify an uncertainty of the mixing coefficient used and, therefore, the turbulent mixing parameter was left out from the uncertainty analysis. It is evident from Fig. 2 that the predicted temperatures had a low sensitivity to the mixing coefficient as it was increased past 0.1. Sensitivity studies on predicted velocity and temperature throughout the model confirmed this fact (Salko, 2010). 2.2. Friction- and grid-loss correlation uncertainty The friction- and grid-loss correlations were calculated using pressure drop measurements from unheated tests. Tests were

Fig. 2. Sensitivity of non-dimensional, predicted temperature at EOHL with respect to b for MVG configuration tests.

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R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49

performed over a wide range of Reynolds numbers. The friction loss coefficients were obtained by using pressure measurements taken over a bare section of the bundle (no grids). The coefficients were back-calculated from Equation (10). The grid form loss coefficients were obtained for both SSG and MVG types using pressure drop measurements over those respective grids. The form loss coefficients were back-calculated from Equation (11).

DPfriction ¼ DPform ¼

f rv2 DZ 2Dh

krv2 2

(10)

(11)

The friction loss coefficients were plotted against bundle Re and a curve fit was applied to capture the friction factor dependence on Re. The distribution of the actual friction loss coefficients about the correlation values was found to be normal. The standard deviation of the normalized data (i.e. measured over predicted friction factors) was 0.006. Fig. 3 shows “measured” friction factors, extracted from the bare bundle region pressure drop measurements, plotted against the calculated friction factors using the developed f(Re) correlation. Associated 2s uncertainty is shown by the green bands. It was assumed that 3s covered the range of all f values when entering the uncertainty into SUSA. The correlation was multiplied by a modification factor, determined by SUSA based on the distribution and uncertainty of the f(Re) correlation, for the VIPRE-I uncertainty analysis runs. Reynolds-dependent correlations were also developed for the form losses associated with the two grid types, similar to what was done for the friction losses. Fig. 4 shows the measured SSG form losses plotted against the values calculated by the developed kSSG(Re) correlation. The associated 2s uncertainty is shown by the green bands. Likewise, Fig. 5 shows the measured and predicted MVG form losses and associated uncertainty. The 3s uncertainty bands were used to define the max/min range of the form loss values in SUSA, as was done for the friction loss coefficients. It was discovered that the MVG data followed a normal distribution about the correlation value; however, the SSG data did not, and so it’s distribution was assumed uniform. 2.3. Single-phase heat transfer model uncertainty Rod surface temperatures were obtained from the NESTOR tests using sliding and rotating thermocouple mechanisms which probed the interior of the heater tubes. Outer surface temperatures were obtained using the 1-D conduction equation. Using

Fig. 3. Measured friction factor coefficients plotted against calculated values with associated 2s uncertainty.

Fig. 4. Measured SSG form loss coefficients plotted against calculated values with associated 2s uncertainty.

experimental heat flux, the experimental rod surface temperatures, and the VIPRE-I calculated sub-channel temperatures, corresponding experimental Heat Transfer Coefficients (HTC) were obtained. Comparing the experimental HTCs to VIPRE-I calculated HTCs using DittuseBoelter, it was found there was a slight slope in the ratio of experimental to calculated HTC with respect to Reynolds number. This was corrected by adjusting the Reynolds number exponent as well as the leading coefficient of the Dittuse Boelter correlation. Furthermore, there was an enhancement of heat transfer downstream of the grids that was captured by creating a dedicated Grid Spacer Cooling Enhancement (GSCE) correlation, similar to the approach of (Yao et al., 1982). The form of the resulting single-phase heat transfer model is shown in Equations (12) and (13).

  GSCEðzÞ ¼ dexp eZg þ f

(12)

hmod ¼ GSCEðzÞmhdb Ren

(13)

The correlating coefficients are denoted with d, e, and f for the GSCE model. The correlating coefficients are denoted with m and n for the optimized heat transfer coefficient model. The Dittuse Boelter calculated HTC is represented by hdb and it is modified by the m coefficient and the n exponent on Re number. The GSCE was captured by fitting an exponential curve fit to the ratio of experimental-to-calculated HTCs (Rdb) with respect to distance from the grid. Two correlations were developed e one for the wake

Fig. 5. Measured MVG form loss coefficients plotted against calculated values with associated 2s uncertainty.

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49

of the MVG and one for the wake of the SSG. The curve fit was applied to the circumferentially-averaged values. The Re exponent modification was determined using a plot of the local ratios of Rdb with respect to local Reynolds number (calculated by VIPRE-I). Further details on the HTC model development can be found in Péturaud et al. (2011), Salko (2010). The model was developed from 13 single-phase heated test cases with varying operating conditions. The model was then added in VIPRE-I and the 13 single-phase cases were re-run. The uncertainty was characterized by calculating the standard deviation of the circumferentially-averaged Rdb values for the 13 singlephase cases with the model applied. Since the ONB test cases were run on the same geometry as the single-phase cases, it was assumed that this dedicated heat transfer model would be applicable to the ONB test case analysis. The average Rdb standard deviation was 0.036. The distribution of the Rdb terms about the mean was found to be normal.

47

There were a total of 13 NESTOR MVG single-phase tests performed. The uncertainty analysis was performed for a single test, which most closely represented the prototypical operating conditions of a PWR. Because information on the PDF shape of the experimental measurement values was not available, it was necessary to assume all shapes to be uniform. 3.1. Generation of random uncertain parameters After all uncertain parameters were defined in detail, they were input into SUSA. The standard deviation, PDF shape and maximum/ minimum values of the uncertain parameters were input into the code for normal distributions. The maximum and minimum values in the range were used for uniform distributions. SUSA then generated 93 sets of input values using simple random sampling. Each set out of the 93 included the 27 uncertain parameters of Table 3. The uncertain parameters, however, were randomized according to the shape and range of their PDF.

3. Definition of uncertain parameter properties 4. Quantitative uncertainty statements After identifying sources of error, the second step in the GRS methodology was to determine the behavior and magnitude of these uncertainties. While some of this has been previously discussed, this section presents a summary of all considered uncertain input parameters. These are the parameters that were used in SUSA. Table 3 provides the summary of uncertain parameters along with their probability distribution function (PDF) shape, their B/E value, their range of values (minimum and maximum), and their standard deviation. The table also gives a numerical label to each parameter, which is used in further figures. The range of values was assumed to be correctly represented by 3 standard deviations from the B/E value in either direction.

After performing the VIPRE-I runs using the variations in the 27 uncertain input parameters, SUSA was used to produce summary statistics for the results. Each axial level had 93 VIPRE-I calculated rod surface temperatures. For each axial level (39 in total), SUSA produced the mean rod surface temperature and the two-sided tolerance limits. Two sided tolerance limits were calculated for a coverage of 95% and a confidence of 95%. Fig. 6 shows the mean, center-rod surface temperatures with respect to axial location in the bundle. It also gives the 95/95 tolerance bands determined by SUSA. In the figure, there are three discontinuities in the temperature profile. These are due to the presence of two SSGs near the 2.8 m and 3.4 m locations and one

Table 3 Summary of SUSA input uncertainties. Label

Uncertain parameter

PDF shape

Best estimate

Min

Max

1s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

AI (m) AC (m) AS (m) Pw,I (m) Pw,C (m) Pw,S (m) PH,I (m) PH,C (m) PH,S (m) SCS (m) SSS (m) SSI (m) SII (m) LCS (m) LSI (m) LSS (m) LII (m) Drod (m) Tin ( C)   kW W0 ft   kg G m2 $s Fmod Radial power factor modification parameter Pout (bar) fmod kSSG,mod kMVG,mod HTCmod

Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform

8.8  105 4.4  105 6.3  105 29.85  103 23.2  103 27.5  103 29.85  103 7.461  103 14.92  103 3.1  103 3.1  103 3.1  103 3.1  103 10.23  103 10.23  103 12.60  103 12.60  103 9.50  103 270.6

8.2  105 4.0  105 5.9  105 29.76  103 22.3  103 27.29  103 29.76  103 7.437  103 14.88  103 2.65  103 2.65  103 2.89  103 2.89  103 10.11  103 10.11  103 12.39  103 12.39  103 9.47  103 270.3

9.4  105 4.8  105 6.7  105 29.94  103 24.1  103 27.71  103 29.94  103 7.485  103 14.97  103 3.55  103 3.55  103 3.31  103 3.31  103 10.35  103 10.35  103 12.81  103 12.81  103 9.53  103 270.9

0.2  105 0.13  105 0.13  105 0.03  103 0.3  103 0.07  103 0.03  103 0.008  103 0.015  103 0.15 0.15 0.07 0.07 0.04  103 0.04  103 0.07  103 0.07  103 10  106 0.1

Uniform

6.7933

6.7928

6.7938

0.0017

Uniform

3560.

3491.

3629.

23

Uniform

1.3032

1.299

1.3071

0.0013

Uniform Normal Uniform Normal Normal

155.8 1.000 1.00 1.00 1.0

154.6 0.9830 0.76 0.964 0.892

157.0 1.018 1.24 1.036 1.108

0.4 0.006 0.08 0.012 0.036

20 21 22 23 24 25 26 27

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Fig. 6. VIPRE-I calculated Rod 5 surface temperatures and associated 95/95 uncertainty bands over measurement length.

MVG near the 3 m location. The presence of these grids increase mixing and cause drops in rod surface temperature. The statistics were not the same for each axial level, but instead tended to become more uncertain as EOHL was approached. Fig. 7 presents the 2s uncertainty of the Rod 5 surface temperature with respect to axial location. The maximum uncertainty, 2.6  C, occurred at EOHL. Another benefit of SUSA is that it can quantify the impact of each uncertain parameter on the uncertainty of the FOM. SUSA does this by calculating the Pearson’s productemoment correlation coefficient (CC) for each uncertain parameter. The closer the magnitude of the CC is to 1, the more the uncertain parameter impacts the FOM e the sign of the coefficient shows whether a change in the uncertain parameter will have either a positive or negative impact on the FOM. For N ¼ 93 runs, parameters with CC < 0:2 can be deemed insignificant in terms of effects on FOM uncertainty (Langenbuch et al., 2005). The value of the CC was dependent on the axial location in the bundle. Fig. 8 shows the average, maximum, and minimum CC values for each of the uncertain parameters included in this study. The parameter number (defined in Table 3) is shown on the y-axis of the figure and the CC value is shown on the x-axis. Values of CC ¼ 0.2 and 0.2 are marked off in the figure with two vertical, dashed lines. Parameters that fell within these vertical lines could be deemed insignificant sources of error with regards to the FOM uncertainty. Uncertain parameters that had CCs falling outside of this range are labeled in the figure.

Fig. 7. Calculated Rod 5 surface temperature 2s uncertainty with respect to axial location.

Fig. 8. Uncertain parameter CC ranges for effect on rod surface temperature prediction.

Table 4 presents the rank of importance of each of the uncertain input parameters with respect to calculated, central-rod surface temperature. The ranks in Table 4 correspond to the mean CC of each parameter, since there is some shifting in importance with respect to axial location. Note that the parameters ranked 6e27 had CC absolute values less than 0.2 which means they had an insignificant contribution to the uncertainty of the calculation of rod surface temperature. The uncertainty in the single-phase heat transfer model, the inner sub-channel flow area, the inlet mass flux and, to smaller extents, the side sub-channel flow area and inner sub-channel heated perimeter had the largest impact on the uncertainty in rod surface temperature. A reduction in the uncertainty of these parameters would have the largest impact on uncertainty-reduction in the rod surface temperature prediction. Such reduction, for all of these parameters, would require using more accurate measurement instrumentation before and during experimentation. The uncertainty in the developed single-phase heat transfer model could be reduced by

Table 4 Rank of uncertain parameter importance with respect to effect on predicted central-rod surface temperature. Rank

Uncertain parameter

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Single-phase HTC coefficient Inner channel area Inlet mass flux Side channel area Inner channel heated perimeter Corner channel area Inlet temperature Sideeside gap width Sideeinner gap width Sideeside gap length Cornereside gap length Rod diameter Cornereside gap width Outlet pressure Bundle power Innereinner gap length Corner channel wetted perimeter Bundle peaking factor MVG loss coefficient Side channel heated perimeter Corner channel heated perimeter Inner channel wetted perimeter SSG loss coefficient Side channel wetted perimeter Friction loss coefficient Sideeinner gap length Innereinner gap width

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49

reducing experimental HTC scatter, since the model was developed from the experimental HTCs. The experimental HTC was not a directly measured value, but, rather, it was calculated from experimental rod surface temperature measurements, experimental heat flux, and VIPRE-I calculated sub-channel temperatures. From Table 1, it is clear that the measured rod surface temperatures had a fairly high uncertainty of 2  C e this value is nearly as large as the SUSA-calculated uncertainty in predicted, rod-surface temperature of 2.6  C. The sub-channel temperature uncertainty is comprised of measurement (e.g. mass flux, power, inlet temperature, etc.) and modeling (e.g. turbulent mixing) uncertainties. As was previously discussed, the turbulent mixing model of VIPRE-I is a simple turbulent-diffusion model, incapable of capturing the effects of MVG diversion cross-flow. Improvement in predicted sub-channel temperatures would likely be realized with implementation of a diversion cross-flow model, like that proposed by Avramova (2007).

on experimental rod surface temperature measurements, test section power, and VIPRE-I predicted sub-channel temperatures. The 2.6  C 2s uncertainty leads to a calculated ONB wall superheat range of 0.8  C to 4.4  C. The calculated ONB wall superheats for all MVG ONB tests fell within this range; they were 0.3  C to 2.9  C. Granted, the calculated ONB wall superheats weren’t always physical at experimental locations as evidenced by the negative values; rather, they offer insight into the range of uncertainty between experimental and predicted ONB location. If we consider a prototypical PWR temperature rise to be 10.9  C/m, then a 2.6  C uncertainty leads to an ONB axial location uncertainty of 24 cm.

5. Conclusions

References

The purpose of the NESTOR program was to support development of an AOA model by improving the understanding and modeling of ONB in rod bundle geometries. The program included optimization of sub-channel codes for the NESTOR test geometry and operating conditions using unheated tests and single-phase heated tests. Subsequent modeling of ONB tests followed utilizing said optimizations. ONB prediction may be improved using the code-calculated ONB wall superheat at experimental locations; however, it was first necessary to quantify the uncertainty in those code predictions. The VIPRE-I calculated ONB wall superheat was 1.8  C. The associated maximum 2s uncertainty of the VIPRE-I calculated rod surface temperature has been estimated, through this study, to be 2.6  C. The rod surface temperature calculation required a complex solution of thermal-hydraulic equations in VIPRE-I. It was decided that the GRS methodology (utilizing SUSA) would be best for determining code-calculated rod surface temperature uncertainty because it provided tolerance limits, with a suitable degree of certainty in a reasonable number of code runs. The GRS methodology relies on a brute-force, statistical approach to determine uncertainty. The user defines the range and PDF shape of all uncertain input parameters. This step constitutes the main drawback of the GRS methodology e it is dependent on the user’s subjective selection of uncertain parameters and their quantity and behavior. SUSA provides a set of random combinations of the uncertain input parameters to be used in the sub-channel code. These sets of combinations produce a distribution of the selected output variable that can be used to quantify how the uncertain input parameters propagate through the code and affect that selected output variable. Results showed that the uncertainty in the single-phase heat transfer model was the greatest contributor to this rod surface temperature uncertainty. This model was developed for the specific geometry of the NESTOR tests, and relied

Avramova, M., 2007. Development of an Innovative spacer grid model utilizing computational fluid Dynamics within a Subchannel analysis tool. Ph.D. thesis, The Pennsylvania State University. EPRI, Palo Alto, CA, EDF, France, CEA, France, 2008. New Experimental Studies of Thermal-hydraulics of Rod Bundles: Part 1: Data Compilations of MANIVEL and OMEGA Test Results on 5x5 Bundle Equipped with Simple Support Grids. Technical Report. Electric Power Research Institute. EPRI, Palo Alto, CA, EDF, France, CEA, France, 2009. New Experimental Studies of Thermal-hydraulics of Rod Bundles (NESTOR). Part 2: Data Compilation of MANIVEL and OMEGA Test Results on 5x5 Bundle Equipped with Mixing Vane Grids. Technical Report 1019423. Electric Power Research Institute. EPRI, Palo Alto, CA, EDF, France, CEA, France, 2010. New Experimental Studies of Thermal-hydraulics of Rod Bundles (NESTOR). Generic Analysis of OMEGA Data. Technical Report 1021039. Electric Power Research Institute. Langenbuch, S., Krzykacz-Hausmann, B., Schmidt, K.D., Hegyi, G., Keresztúri, A., Kliem, S., Hadek, J., Danilin, S., Nikonov, S., Kuchin, A., Khalimanchuk, V., Hamalainen, A., 2005. Comprehensive uncertainty and sensitivity analysis for coupled code calculations of VVER plant transients. Nuclear Engineering and Design 235, 521e540. Peturaud, P., Decossin, E., 2001. Improved Understanding of Heat Transfer in Rod Bundles. Basic Design of a Proposed Experimental Program. Technical Report HI-84/01/011/A. EDF, France. Péturaud, P., Salko, R., Bergeron, A., Yagnik, S., Avramova, M., 2011. Analysis of single-phase heat transfer and onset of nucleate boiling in a rod bundle with mixing vane grids. In: The 14th International Topical Meeting on Nuclear Reactor Thermal Hydraulics. NURETH-14. Rogers, J., Rosehart, R., 1972. Mixing by turbulent interchange in fuel bundles. Correlations and influences. In: AIChE-ASME Heat Transfer Conference, Denver, Co. Salah, A., Kliem, S., Rohde, U., D’Auria, F., Petruzzi, A., 2006. Uncertainty and sensitivity analyses of the Kozloduy pump trip test using coupled thermalhydraulic 3D kinetics code. Nuclear Engineering and Design 236, 1240e1255. Salko, R.K., 2010. Data analysis and modeling of NESTOR SSG and MVG rod bundle experiments using VIPRE-i for the Assessment of the onset of nucleate boiling criterion. Master’s thesis, The Pennsylvania State University. Srikantiah, G., 1992. Vipre-1: a Reactor Core Thermal-hydraulics Analysis Code for Utility Applications. Nuclear Technology, p. 100. Strydom, G., 2010. Use of SUSA in Uncertainty and Sensitivity Analysis for INL VHTR Coupled Codes. Technical Report. Idaho National Laboratory. Wilks, S., 1941. Determination of sample sizes for setting tolerance limits. Annals of Mathematical Statistics 12, 91e96. Yao, S., Hochreiter, L., Leech, W., 1982. Heat transfer augmentation in rod bundles near grid spacers. Journal of Heat Transfer 104, 76e81.

Acknowledgments The authors would like to thank Dr. Suresh Yagnik for his guidance and leadership during the NESTOR project as well as the Electric Power Research Institute for its funding of the work.