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Progress in Nuclear Energy 65 (2013) 42e49

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Progress in Nuclear Energy

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Uncertainty analysis of sub-channel code calculated ONB wall superheat in rodbundle 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

Article history:Received 10 December 2012Received in revised form1 February 2013Accepted 7 February 2013

Keywords:VIPRE-ISUSANESTORONB

* Corresponding author. 18 Reber Building, UnivTel.: þ1 570 972 0988.** Corresponding author. Tel.: þ1 814 865 0043.

E-mail address: rks171@psu.edu (R.K. Salko).

0149-1970/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.pnucene.2013.02.003

a b s t r a c t

Rod bundle experiments were performed for prototypical PWR operating conditions in the project “NewExperimental Studies of Thermal-Hydraulics of Rod Bundles (NESTOR)”. The intent of the project was toimprove the understanding of the Axial Offset Anomaly (AOA) through improved modeling of Onset ofNucleate Boiling (ONB) (EPRI, 2008) using sub-channel codes. Skewing of the axial power profile (AOA) ismost likely driven by the deposition of boron in the crud layer on nuclear fuel rods, which is caused byboiling on the fuel rod surface (EPRI, 2008).

VIPRE-I (Srikantiah, 1992), a sub-channel code, was chosen for the analysis of NESTOR tests and forwhich uncertainty analysis was performed. NESTOR experimental results were used to optimize grid-losscoefficients, friction-loss coefficients, and a single-phase heat transfer model in the code. By modelingNESTOR ONB tests, the VIPRE-I calculated wall superheat was determined at the experimental ONB lo-cations. This calculated ONB wall superheat could be used as a criterion in VIPRE-I for the prediction ofONB; however, it is important to quantify the uncertainty of this calculated ONB wall superheat in orderto know the accuracy of such a criterion. The VIPRE-I model optimization process, however, was acomplicated one and involved interaction of both experimental and code modeling uncertainties. Thepropagation of these uncertainties was treated using the Gesellschaft für Anlagen und Reaktorsicherheit(GRS) methodology; a process which is detailed in this paper.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Code modeling uncertainty

A thermal-hydraulic simulation code like VIPRE-I describes aphysical system using (Strydom, 2010):

� A mathematical model comprised of governing equations aswell as sub-grid equations that capture complex physical pro-cesses like convective heat transfer and turbulent exchange

� A numerical technique to solve the analytical mathematicalmodel

� Physical parameters that describe the system being modeled(e.g. boundary conditions and geometry)

Codemodeling uncertainty is the inability of the code to capturethe behavior of a system with complete accuracy. This uncertainty

ersity Park, PA 16802, USA.

All rights reserved.

is introduced by each of the three aforementioned aspects of codemodeling. For example, physical models e like convective heattransfer correlations e have a specified level of uncertainty owingto uncertainties in the experiments they were developed from andtheir inherent inability to perfectly capture the physics of thephenomenon. Numerical techniques, which must be used forsolving the governing equations, introduce truncation error. Thesystem that is being modeled e the NESTOR tests, in this case e hasuncertainties associated with the operating conditions and geom-etry. The manner in which these uncertainties interact and prop-agate through the solution will be complex and convoluted. Onesuch method that has been developed to characterize this propa-gation is the GRS methodology.

1.2. GRS methodology

The GRSmethodology uses a statistical type of approach to trackthe propagation of code input uncertainties and quantify their ef-fect on some output Figure of Merit (FOM). The System for Uncer-tainty and Sensitivity Analysis (SUSA) was the actual program usedfor performing the analysis using the GRS methodology. A benefitin this methodology is that it requires no modification to the

Nomenclature

G average mass flux between adjacent sub-channelsbtm turbulent mixing coefficientDPform form pressure lossDPfriciton friction pressure lossDZ axial unit lengthd parameter uncertainty_m coolant mass flow rater coolant densitys standard deviationA sub-channel cross-sectional areaa probability FOM will fall between tolerance limitsb confidence in probability, aC test section channel widthCt turbulent momentum factord,e,f correlating coefficients in the GSCEDrod rod diameterDc predicted/measured quantity difference parameterDh sub-channel hydraulic diameterF radial peaking factorf friction loss coefficientFmod radial power factor modification parameterfmod friction factor modification parameterG coolant mass fluxhdb single phase HTC obtained using DittuseBoelterhmod single phase HTC using optimized modeli index of sub-channelk grid loss coefficientkMVG,mod MVG form loss modification parameterkSSG,mod SSG form loss modification parameterL rod heated lengthm,n correlating coefficients in heat transfer modeln number of calculations required for probability, a, and

confidence, bP pressurep rod pitchPout outlet pressure

r rod radiusRdb ratio of experimental HTC to calculated HTCS gap widthTc,ONB calculated ONB wall superheatTc calculated rod surface temperatureTexp,wo experimental rod surface temperatureTi,m,EOHL measured temperature at EOHL for sub-channel iTin experimental inlet temperatureTout experimental outlet temperatureTsat local fluid saturation temperaturey coolant velocityW

0linear heat rate

w0

lateral cross-flow due to turbulenceWT total rod bundle powerXi,c,EOHL predicted, non-dimensional temperature at EOHLXi,m measured non-dimensional temperature for sub-

channel iz axial elevation in rod bundleZg distance to the nearest upstream gridAOA Axial Offset AnomalyCC Pearson’s productemoment correlation coefficientEOHL End of Heated LengthFOM Figure of MeritGRS Gesellschaft für Anlagen und ReaktorsicherheitGSCE Grid Spacer Cooling EnhancementHTC Heat Transfer CoefficientHTCmod HTC modification parameterLDV Laser Doppler VelocimetryMVG Mixing Vane GridNESTOR New Experimental Studies of Thermal-Hydraulics of

Rod BundlesONB Onset of Nucleate BoilingPDF Probability Distribution FunctionPWR Pressurized Water ReactorRMS Root-Mean SquareSSG Simple Support GridSUSA System for Uncertainty and Sensitivity Analysis

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49 43

existing code if the uncertain parameters can be captured by inputparameters to the code. Further, the methodology can be used forany type of computational tool.

The principle of the methodology is to quantify the un-certainties of selected code input uncertainties. The user is notlimited in the number of input uncertainties they can choose toinvestigate. This set of input uncertainties is used to create somenumber of code input cases containing random variations of theselected input parameters. Their randomization will be dependenton their uncertainty and probability distribution function. Thesecases are then run and their outputs are used to set tolerance bandsthat some selected FOM will fall between. It is the number of inputcases generated that will determine the probability that the FOMwill fall between the tolerance bands to a given confidence. Thenumber of cases required for a specified probability and confidenceis determined using Wilks’ formula (Wilks, 1941), which is shownfor two-sided statistical tolerance intervals in Equation (1).

ð1� anÞ � nð1� aÞan�1 >¼ b (1)

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

95% confidence and 95% probability, 93 code runs would berequired, which is the number of code runs that were performed forthis study.

Specific steps for using the GRS methodology for performingcode uncertainty analysis were outlined by Salah et al. (2006):

1. Identification of all relevant sources of uncertainty, representedby uncertain parameters

2. Definition of uncertainty ranges for the identified parameters(e.g. minimum and maximum values) as well as the uncertainparameter probability distribution

3. Generation of a random sample of size N from their probabilitydistributions

4. Execution of the computer code with the generated sample ofinput values

5. Derivation of the quantitative uncertainty of the code pre-dictions by specifying the statistical tolerance limits of the FOM

6. Computation of the quantitative sensitivity measures to rankthe importance of the individual uncertain source parameters

Benefits of the GRS methodology include being able to quantifycode uncertainty in a straightforward manner, with resultingtolerance band confidence and probability only being dependenton the number of code runs. The methodology is, however,

Table 1Code input uncertain parameters.

Uncertain input parameter Symbol Parameter range(�2s uncertainty)

Test section channel width dC 0.15 mmRod pitch dp 0.14 mmRod radius dr 10 mmRadial peaking factor dF/F 0.2%Heated length dL/L 0.07%Axial elevation in rod bundle dz 1 mmInlet temperature dTin 0.2 KAbsolute pressure dP/P 0.5%Differential pressure dP/P 0.1%Mass flow rate d _m= _m 1%Test section power dWT/WT 0.1%Experimental rod surface temperature dTexp,wo 2.0 K

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

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e4944

dependent on the user’s selection of uncertain input parametersand their associated uncertainty ranges and probability distribu-tions. This aspect has an element of subjectivity and can be a sourceof error in the uncertainty analysis.

1.3. Application to NESTOR test data analysis

The purpose of the NESTOR tests was to develop a wall super-heat criterion based model for the accurate prediction of ONB inPWR cores. It was envisioned that an improved understanding ofONB in PWR cores could facilitate the development of an AOAmodel since AOA is likely caused by the boiling-induced depositionof boron into the fuel rod crud layer (EPRI, 2008). The tests wereperformed in heated and unheated configurations of 5 � 5 rodbundle facilities. The rod bundles utilized both Simple SupportGrids (SSG) and Mixing Vane Grids (MVG). High-fidelity localtransverse and axial velocity profile measurements were taken viaLaser Doppler Velocimetry (LDV) in the unheated configurationalong with grid span pressure drop measurements. High-fidelityrod surface temperature measurements were also taken in theheated configuration via thermocouple probes that were able tomove axially and azimuthally around the inside surfaces of theheater rods.

Pressure drop measurements from unheated tests were used tooptimize the friction and grid pressure drop models in VIPRE-I.Velocity measurements and sub-channel temperature measure-ments at the test section outlet were used to optimize the turbulentmixing model in VIPRE-I. Single-phase heated experiments wereused to develop a dedicated single-phase heat transfer model. Alltest analysis was performed on the central heater rod, dubbed Rod5 throughout this paper, and on the four inner-type sub-channelssurrounding the rod. Further information relating to the NESTORprogram and the modeling optimization process is available (see(Péturaud et al., 2011; Salko, 2010)).

Tests were also run with operating conditions that caused ONBto occur in the bundle. VIPRE-I was then used, with all modelsoptimized to the NESTOR configuration, to determine the calcu-lated wall superheat at the experimental ONB locations. Thecalculated wall superheat results at experimental locations couldbe utilized for tuning the wall superheat ONB criterion in VIPRE-Ifor rod bundle geometry. Such a modification, however, wouldrequire a definite statement of the uncertainty in the calculated rodsurface temperature, which is the primary component in thecalculated ONB wall superheat as shown in Equation (2). Therefore,the purpose of this uncertainty analysis is to quantify the impact ofthe experimental (boundary conditions and test section geometry)and code modeling uncertainties on the calculated rod surfacetemperature.

Tc;ONB ¼ Tc � Tsat (2)

2. Identification of uncertainty sources

The VIPRE-I input uncertain parameters needed to be listed andquantified for the GRS methodology. These uncertain parametersinclude both measured uncertain values (e.g. inlet mass flux andtemperature) as well as calculated parameters (e.g. friction and gridloss coefficients). Test operating condition uncertainty and geom-etry uncertainty was obtained from NESTOR documentation (EPRI,2009, 2010; Peturaud and Decossin, 2001) and is repeated here inTable 1.

The uncertain parameters of Table 1 weren’t all directly used inwriting the VIPRE-I input deck. For example, the input deck takes

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

Ainner ¼ p2 � pr2 (3)

Considering that the dimensional uncertainties are independentand random, the uncertainty in sub-channel flow area can becalculated as follows, with the d terms representing the parameteruncertainty.

dAinner ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�vAinnervp

dp�2

þ�vAinner

vrdr�2

s(4)

This lead to a sub-channel area 2s uncertainty of 2�10�6 m2 forthe inner-type sub-channel. The inner-type channel is depicted inFig. 1 as Sub-channel numbers 4, 5, and 6. Uncertainties of the side(channels 2 and 3) and corner (channel 1) type channels were alsocalculated and used in the analysis, though the rod surface tem-perature uncertainty analysis was only performed for the centralrod.

Another example involves the test section power. VIPRE-I takespower input in units of kW/ft, which must be calculated from thesupplied test section power, WT, and the total length of the 25heater rods, L. Since the section length also had uncertainty, it wasnecessary to calculate the propagation of uncertainty into the linearheat rate, W

0.

W 0 ¼ WT

25L(5)

Similarly, the inlet flowboundary conditionwas given to VIPRE-Iin terms of mass flux, whereas uncertainty was given in terms ofmass flow rate. Calculation of the mass flux term involved threeuncertain parameters, as shown below:

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49 45

G ¼ _m2 2 (6)

C � 25*pr

These calculated uncertain input parameters are summarized inTable 2. The terms A, Pw, PH, S, and L represent the sub-channelcross-sectional area, wetted perimeter, heated perimeter, gapwidth, and gap length, respectively. The subscripts, “I”, “S”, and “C”,represent the inner-type, side-type, and corner-type channelconfiguration, respectively. The subscripts, CeS, SeS, SeI, and IeI,represent the type of gap e possible connections include cornereside, sideeinner, sideeside, and innereinner.

The aforementioned uncertain parameters were also used in thecalibration of VIPRE-I models which include the turbulent mixingmodel parameters (btm and Ct), the friction and grid loss coefficients(f and k), and the single-phase heat transfer coefficient model. Eachof these error sources is discussed separately in the following sub-sections.

2.1. Turbulent mixing model

VIPRE-I uses a simple turbulent diffusion model, as shown inEquation (7).

w0 ¼ btmSG (7)

The amount of mixing,w0, caused by turbulence is expressed as a

product of three terms: (1) thewidth of the gap, S, (2) the average ofthe axial mass fluxes, G, in the two adjacent sub-channels thatw

0is

being evaluated for and, (3) a coefficient, btm, that captures thestrength of the mixing. The term, w

0, when calculated, can be used

in each governing equation to determine the quantity of mass,momentum, and energy transferred through the gaps due to tur-bulent 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 wellas (End of Heated Length) EOHL temperature measurements wereavailable in the single phase tests of NESTOR. It was possible to usethis information to optimize certain terms, such as the mixing co-efficient, for the NESTOR geometry and operating conditions. Theturbulent mixing coefficient was optimized by minimizing thedifference between predicted and measured, non-dimensionalized,EOHL temperatures through variation of the btm parameter. Themeasured, non-dimensional EOHL temperature was defined as

Table 2Uncertainty of VIPRE-I input parameters.

Input parameter Uncertainty (�2s)

AI 4 � 10�6 m2

AC 2.5 � 10�6 m2

AS 2.6 � 10�6 m2

Pw,I 6 � 10�5 mPw,C 6 � 10�4 mPw,S 1.4 � 10�4 mPH,I 6 � 10�5 mPH,C 1.6 � 10�5 mPH,S 3 � 10�5 mSC�S 3 � 10�4 mSS�S 3 � 10�4 mSS�I 1.4 � 10�4 mSI�I 1.4 � 10�4 mLC�S 8 � 10�5 mLS�I 8 � 10�5 mLS�S 1.4 � 10�4 mLI�I 1.4 � 10�4 mLinear Heat Rate

1:6� 10�4kWft

Inlet Mass Flux23

kgs$m2

shown in Equation (8), where the numerator is the difference be-tween the measured sub-channel temperature and the experi-mental outlet temperature and the denominator is the differencebetween the experimental outlet and inlet temperatures. A VIPRE-I-predicted, non-dimensional temperature was similarly defined byusing calculated temperature in place of measured temperature atEOHL.

Xi;m ¼ Ti;m;EOHL � ToutTout � Tin

(8)

A difference between predicted and measured, non-dimensional temperature was calculated for each sub-channel inthe model. Then, the RMS of these differences was taken to create asingle parameter, shown in Equation (9), that quantified the dif-ference between predicted and measured EOHL temperatures for agiven VIPRE-I run (using a single b value). The reader is guidedtowards (Salko, 2010) for further information on the turbulentmixing parameter optimization procedure.

Dc ¼ Si�Xi;c;EOHL � Xi;m;EOHL

�2 (9)

The optimization was done so for the SSG and MVG configura-tions. An optimum btm value was found for the SSG configuration,leading to a minimum Dc parameter, but no such value was foundfor the MVG configuration, which was the focus of this uncertaintyanalysis. Rather, it was found that it was necessary to raise themixing coefficient to very high values to best match experimentalresults. It was assumed that this was due to the directed cross-floweffects, brought on by the presence of MVGs, which acted to flattenthe lateral temperature profile in the experiments. As Fig. 2 dem-onstrates, the magnitude of the Dc parameter decreased exponen-tially with increasing btm for all MVG tests.

For this reason, a large mixing coefficient of 0.3 was chosen forfuture modeling of NESTOR tests using VIPRE-I. With no optimumbtm value available from the analysis, it was also not possible toquantify an uncertainty of the mixing coefficient used and, there-fore, the turbulent mixing parameter was left out from the uncer-tainty analysis. It is evident from Fig. 2 that the predictedtemperatures had a lowsensitivity to themixing coefficient as itwasincreased past 0.1. Sensitivity studies on predicted velocity andtemperature throughout the model confirmed this fact (Salko,2010).

2.2. Friction- and grid-loss correlation uncertainty

The friction- and grid-loss correlations were calculated usingpressure drop measurements from unheated tests. Tests were

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

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

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e4946

performed over awide range of Reynolds numbers. The friction losscoefficients were obtained by using pressure measurements takenover a bare section of the bundle (no grids). The coefficients wereback-calculated from Equation (10). The grid form loss coefficientswere obtained for both SSG and MVG types using pressure dropmeasurements over those respective grids. The form loss co-efficients were back-calculated from Equation (11).

DPfriction ¼ f rv2DZ2Dh

(10)

DPform ¼ krv2

2(11)

The friction loss coefficients were plotted against bundle Re anda curve fit was applied to capture the friction factor dependence onRe. The distribution of the actual friction loss coefficients about thecorrelation values was found to be normal. The standard deviationof the normalized data (i.e. measured over predicted friction fac-tors) was �0.006. Fig. 3 shows “measured” friction factors, extrac-ted from the bare bundle region pressure drop measurements,plotted against the calculated friction factors using the developedf(Re) correlation. Associated�2s uncertainty is shown by the greenbands. It was assumed that 3s covered the range of all f valueswhen entering the uncertainty into SUSA. The correlation wasmultiplied by a modification factor, determined by SUSA based onthe distribution and uncertainty of the f(Re) correlation, for theVIPRE-I uncertainty analysis runs.

Reynolds-dependent correlations were also developed for theform losses associated with the two grid types, similar to what wasdone for the friction losses. Fig. 4 shows the measured SSG formlosses plotted against the values calculated by the developedkSSG(Re) correlation. The associated �2s uncertainty is shown bythe green bands. Likewise, Fig. 5 shows themeasured and predictedMVG form losses and associated uncertainty. The 3s uncertaintybands were used to define the max/min range of the form lossvalues in SUSA, as was done for the friction loss coefficients. It wasdiscovered that the MVG data followed a normal distribution aboutthe correlation value; however, the SSG data did not, and so it’sdistribution was assumed uniform.

2.3. Single-phase heat transfer model uncertainty

Rod surface temperatures were obtained from the NESTOR testsusing sliding and rotating thermocouple mechanisms which pro-bed the interior of the heater tubes. Outer surface temperatureswere obtained using the 1-D conduction equation. Using

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

experimental heat flux, the experimental rod surface temperatures,and the VIPRE-I calculated sub-channel temperatures, corre-sponding experimental Heat Transfer Coefficients (HTC) were ob-tained. Comparing the experimental HTCs to VIPRE-I calculatedHTCs using DittuseBoelter, it was found there was a slight slope inthe ratio of experimental to calculated HTC with respect to Rey-nolds number. This was corrected by adjusting the Reynoldsnumber exponent as well as the leading coefficient of the DittuseBoelter correlation. Furthermore, there was an enhancement ofheat transfer downstream of the grids that was captured bycreating a dedicated Grid Spacer Cooling Enhancement (GSCE)correlation, similar to the approach of (Yao et al., 1982). The form ofthe resulting single-phase heat transfer model is shown in Equa-tions (12) and (13).

GSCEðzÞ ¼ dexp�eZg

�þ f (12)

hmod ¼ GSCEðzÞmhdbRen (13)

The correlating coefficients are denoted with d, e, and f for theGSCE model. The correlating coefficients are denoted with m and nfor the optimized heat transfer coefficient model. The DittuseBoelter calculated HTC is represented by hdb and it is modified bythe m coefficient and the n exponent on Re number. The GSCE wascaptured by fitting an exponential curve fit to the ratio ofexperimental-to-calculated HTCs (Rdb) with respect to distancefrom the grid. Two correlations were developed e one for the wake

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

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49 47

of the MVG and one for the wake of the SSG. The curve fit wasapplied to the circumferentially-averaged values. The Re exponentmodification was determined using a plot of the local ratios of Rdbwith respect to local Reynolds number (calculated by VIPRE-I).Further details on the HTC model development can be found inPéturaud et al. (2011), Salko (2010).

The model was developed from 13 single-phase heated testcases with varying operating conditions. The model was thenadded in VIPRE-I and the 13 single-phase cases were re-run. Theuncertainty was characterized by calculating the standard devia-tion of the circumferentially-averaged Rdb values for the 13 single-phase cases with the model applied. Since the ONB test cases wererun on the same geometry as the single-phase cases, it wasassumed that this dedicated heat transfer model would be appli-cable to the ONB test case analysis. The average Rdb standard de-viationwas 0.036. The distribution of the Rdb terms about the meanwas found to be normal.

3. Definition of uncertain parameter properties

After identifying sources of error, the second step in the GRSmethodology was to determine the behavior and magnitude ofthese uncertainties. While some of this has been previously dis-cussed, this section presents a summary of all considered uncertaininput parameters. These are the parameters that were used inSUSA.

Table 3 provides the summary of uncertain parameters alongwith their probability distribution function (PDF) shape, their B/Evalue, their range of values (minimum and maximum), and theirstandard deviation. The table also gives a numerical label to eachparameter, which is used in further figures. The range of values wasassumed to be correctly represented by 3 standard deviations fromthe B/E value in either direction.

Table 3Summary of SUSA input uncertainties.

Label Uncertain parameter PDF shape Best estim

1 AI (m) Uniform 8.8 � 10�

2 AC (m) Uniform 4.4 � 10�

3 AS (m) Uniform 6.3 � 10�

4 Pw,I (m) Uniform 29.85 � 15 Pw,C (m) Uniform 23.2 � 106 Pw,S (m) Uniform 27.5 � 107 PH,I (m) Uniform 29.85 � 18 PH,C (m) Uniform 7.461 � 19 PH,S (m) Uniform 14.92 � 110 SC�S (m) Uniform 3.1 � 10�

11 SS�S (m) Uniform 3.1 � 10�

12 SS�I (m) Uniform 3.1 � 10�

13 SI�I (m) Uniform 3.1 � 10�

14 LC�S (m) Uniform 10.23 � 115 LS�I (m) Uniform 10.23 � 116 LS�S (m) Uniform 12.60 � 117 LI�I (m) Uniform 12.60 � 118 Drod (m) Uniform 9.50 � 1019 Tin (�C) Uniform 270.6

20 W 0�kWft

�Uniform 6.7933

21 G�

kgm2$s

�Uniform 3560.

22 Fmod Radial power factormodification parameter

Uniform 1.3032

23 Pout (bar) Uniform 155.824 fmod Normal 1.00025 kSSG,mod Uniform 1.0026 kMVG,mod Normal 1.0027 HTCmod Normal 1.0

There were a total of 13 NESTOR MVG single-phase tests per-formed. The uncertainty analysis was performed for a single test,which most closely represented the prototypical operating condi-tions of a PWR. Because information on the PDF shape of theexperimental measurement values was not available, it wasnecessary to assume all shapes to be uniform.

3.1. Generation of random uncertain parameters

After all uncertain parameters were defined in detail, they wereinput into SUSA. The standard deviation, PDF shape and maximum/minimum values of the uncertain parameters were input into thecode for normal distributions. The maximum and minimum valuesin the range were used for uniform distributions. SUSA thengenerated 93 sets of input values using simple random sampling.Each set out of the 93 included the 27 uncertain parameters ofTable 3. The uncertain parameters, however, were randomized ac-cording to the shape and range of their PDF.

4. Quantitative uncertainty statements

After performing the VIPRE-I runs using the variations in the 27uncertain input parameters, SUSA was used to produce summarystatistics for the results. Each axial level had 93 VIPRE-I calculatedrod surface temperatures. For each axial level (39 in total), SUSAproduced the mean rod surface temperature and the two-sidedtolerance limits. Two sided tolerance limits were calculated for acoverage of 95% and a confidence of 95%.

Fig. 6 shows the mean, center-rod surface temperatures withrespect to axial location in the bundle. It also gives the 95/95tolerance bands determined by SUSA. In the figure, there are threediscontinuities in the temperature profile. These are due to thepresence of two SSGs near the 2.8 m and 3.4 m locations and one

ate Min Max �1s5 8.2 � 10�5 9.4 � 10�5 0.2 � 10�5

5 4.0 � 10�5 4.8 � 10�5 0.13 � 10�5

5 5.9 � 10�5 6.7 � 10�5 0.13 � 10�5

0�3 29.76 � 10�3 29.94 � 10�3 0.03 � 10�3

�3 22.3 � 10�3 24.1 � 10�3 0.3 � 10�3

�3 27.29 � 10�3 27.71 � 10�3 0.07 � 10�3

0�3 29.76 � 10�3 29.94 � 10�3 0.03 � 10�3

0�3 7.437 � 10�3 7.485 � 10�3 0.008 � 10�3

0�3 14.88 � 10�3 14.97 � 10�3 0.015 � 10�3

3 2.65 � 10�3 3.55 � 10�3 0.153 2.65 � 10�3 3.55 � 10�3 0.153 2.89 � 10�3 3.31 � 10�3 0.073 2.89 � 10�3 3.31 � 10�3 0.070�3 10.11 � 10�3 10.35 � 10�3 0.04 � 10�3

0�3 10.11 � 10�3 10.35 � 10�3 0.04 � 10�3

0�3 12.39 � 10�3 12.81 � 10�3 0.07 � 10�3

0�3 12.39 � 10�3 12.81 � 10�3 0.07 � 10�3

�3 9.47 � 10�3 9.53 � 10�3 10 � 10�6

270.3 270.9 0.1

6.7928 6.7938 0.0017

3491. 3629. 23

1.299 1.3071 0.0013

154.6 157.0 0.40.9830 1.018 0.0060.76 1.24 0.080.964 1.036 0.0120.892 1.108 0.036

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

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

Rank Uncertain parameter

1 Single-phase HTC coefficient2 Inner channel area3 Inlet mass flux4 Side channel area5 Inner channel heated perimeter

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

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e4948

MVG near the 3 m location. The presence of these grids increasemixing and cause drops in rod surface temperature.

The statistics were not the same for each axial level, but insteadtended to become more uncertain as EOHL was approached. Fig. 7presents the 2s uncertainty of the Rod 5 surface temperature withrespect to axial location. The maximum uncertainty, �2.6 �C,occurred at EOHL.

Another benefit of SUSA is that it can quantify the impact of eachuncertain parameter on the uncertainty of the FOM. SUSA does thisby calculating the Pearson’s productemoment correlation coeffi-cient (CC) for each uncertain parameter. The closer the magnitudeof the CC is to 1, themore the uncertain parameter impacts the FOMe the sign of the coefficient shows whether a change in the un-certain parameter will have either a positive or negative impact onthe FOM. ForN¼ 93 runs, parameters with CC < 0:2 can be deemedinsignificant in terms of effects on FOM uncertainty (Langenbuchet al., 2005).

The value of the CC was dependent on the axial location in thebundle. Fig. 8 shows the average, maximum, and minimum CCvalues for each of the uncertain parameters included in this study.The parameter number (defined in Table 3) is shown on the y-axisof the figure and the CC value is shown on the x-axis. Values ofCC ¼ �0.2 and 0.2 are marked off in the figure with two vertical,dashed lines. Parameters that fell within these vertical lines couldbe deemed insignificant sources of error with regards to the FOMuncertainty. Uncertain parameters that had CCs falling outside ofthis range are labeled in the figure.

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

Table 4 presents the rank of importance of each of the uncertaininput parameters with respect to calculated, central-rod surfacetemperature. The ranks in Table 4 correspond to the mean CC ofeach parameter, since there is some shifting in importance withrespect to axial location. Note that the parameters ranked 6e27 hadCC absolute values less than 0.2 which means they had an insig-nificant contribution to the uncertainty of the calculation of rodsurface temperature. The uncertainty in the single-phase heattransfer model, the inner sub-channel flow area, the inlet mass fluxand, to smaller extents, the side sub-channel flow area and innersub-channel heated perimeter had the largest impact on the un-certainty in rod surface temperature.

A reduction in the uncertainty of these parameters would havethe largest impact on uncertainty-reduction in the rod surfacetemperature prediction. Such reduction, for all of these parameters,would require using more accurate measurement instrumentationbefore and during experimentation. The uncertainty in the devel-oped single-phase heat transfer model could be reduced by

6 Corner channel area7 Inlet temperature8 Sideeside gap width9 Sideeinner gap width10 Sideeside gap length11 Cornereside gap length12 Rod diameter13 Cornereside gap width14 Outlet pressure15 Bundle power16 Innereinner gap length17 Corner channel wetted perimeter18 Bundle peaking factor19 MVG loss coefficient20 Side channel heated perimeter21 Corner channel heated perimeter22 Inner channel wetted perimeter23 SSG loss coefficient24 Side channel wetted perimeter25 Friction loss coefficient26 Sideeinner gap length27 Innereinner gap width

R.K. Salko, M.N. Avramova / Progress in Nuclear Energy 65 (2013) 42e49 49

reducing experimental HTC scatter, since the model was developedfrom the experimental HTCs. The experimental HTC was not adirectly measured value, but, rather, it was calculated from exper-imental rod surface temperaturemeasurements, experimental heatflux, and VIPRE-I calculated sub-channel temperatures. FromTable 1, it is clear that the measured rod surface temperatures had afairly high uncertainty of 2 �C e this value is nearly as large as theSUSA-calculated uncertainty in predicted, rod-surface temperatureof 2.6 �C. The sub-channel temperature uncertainty is comprised ofmeasurement (e.g. mass flux, power, inlet temperature, etc.) andmodeling (e.g. turbulent mixing) uncertainties. As was previouslydiscussed, the turbulent mixing model of VIPRE-I is a simpleturbulent-diffusion model, incapable of capturing the effects ofMVG diversion cross-flow. Improvement in predicted sub-channeltemperatures would likely be realized with implementation of adiversion cross-flow model, like that proposed by Avramova(2007).

5. Conclusions

The purpose of the NESTOR program was to support develop-ment of an AOA model by improving the understanding andmodeling of ONB in rod bundle geometries. The program includedoptimization of sub-channel codes for the NESTOR test geometryand operating conditions using unheated tests and single-phaseheated tests. Subsequent modeling of ONB tests followed utilizingsaid optimizations. ONB prediction may be improved using thecode-calculated ONB wall superheat at experimental locations;however, it was first necessary to quantify the uncertainty in thosecode predictions. The VIPRE-I calculated ONB wall superheat was1.8 �C. The associated maximum 2s uncertainty of the VIPRE-Icalculated rod surface temperature has been estimated, throughthis study, to be 2.6 �C.

The rod surface temperature calculation required a complexsolution of thermal-hydraulic equations in VIPRE-I. It was decidedthat the GRS methodology (utilizing SUSA) would be best fordetermining code-calculated rod surface temperature uncertaintybecause it provided tolerance limits, with a suitable degree ofcertainty in a reasonable number of code runs.

The GRS methodology relies on a brute-force, statisticalapproach to determine uncertainty. The user defines the range andPDF shape of all uncertain input parameters. This step constitutesthe main drawback of the GRS methodology e it is dependent onthe user’s subjective selection of uncertain parameters and theirquantity and behavior. SUSA provides a set of random combinationsof the uncertain input parameters to be used in the sub-channelcode. These sets of combinations produce a distribution of theselected output variable that can be used to quantify how the un-certain input parameters propagate through the code and affectthat selected output variable. Results showed that the uncertaintyin the single-phase heat transfer model was the greatest contrib-utor to this rod surface temperature uncertainty. This model wasdeveloped for the specific geometry of the NESTOR tests, and relied

on experimental rod surface temperature measurements, test sec-tion power, and VIPRE-I predicted sub-channel temperatures.

The 2.6 �C 2s uncertainty leads to a calculated ONB wall su-perheat range of �0.8 �C to 4.4 �C. The calculated ONB wall su-perheats for all MVG ONB tests fell within this range; theywere�0.3 �C to 2.9 �C. Granted, the calculated ONBwall superheatsweren’t always physical at experimental locations as evidenced bythe negative values; rather, they offer insight into the range ofuncertainty between experimental and predicted ONB location. Ifwe consider a prototypical PWR temperature rise to be 10.9 �C/m,then a �2.6 �C uncertainty leads to an ONB axial location uncer-tainty of �24 cm.

Acknowledgments

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

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