EVALUATION OF THE INITIAL CRITICAL CONFIGURATION OF THE ... · graphite-reflected, HTGRs, HTR-10, Pebble-bed reactor, test reactor, uranium dioxide SUMMARY INFORMATION 1.0 DETAILED

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Gas Cooled (Thermal) Reactor – GCR

HTR10-GCR-RESR-001 CRIT-REAC

EVALUATION OF THE INITIAL CRITICAL CONFIGURATION OF THE HTR-10 PEBBLE-BED REACTOR

Evaluators

William K. Terry Leland M. Montierth

Soon Sam Kim Joshua J. Cogliati

Abderrafi M. Ougouag Idaho National Laboratory

Yuliang Sun

Institute of Nuclear and New Energy Technology

Internal Reviewers J. Blair Briggs Soon Sam Kim

Leland M. Montierth Hans D. Gougar

Idaho National Laboratory

Xingqing Jing Shuren Xi

Zhengpei Luo Institute of Nuclear and New Energy Technology

Independent Reviewer

Virginia F. Dean Under subcontract to the Idaho National Laboratory

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Evaluator and Reviewer Contributions

This evaluation has been made by a group of scientists at the Idaho National Laboratory (INL). The principal evaluator and author of this report is Dr. William K. Terry. He is responsible for the content, including any errors that may remain. He is grateful for the contributions of the other evaluators and the reviewers, some of whom had both roles. Mr. J. Blair Briggs and Dr. Soon Sam Kim were the primary internal reviewers. Mr. Briggs, who leads the IRPhEP, was a constant source of guidance on the philosophy and substance of IRPhEP evaluations. He also provided many helpful comments on specific aspects of the report. Dr. Kim performed the discrete ordinates (Sn) analysis with the code TWODANT and reviewed the evaluation except for the portions dealing with his own work. Dr. Kim also performed some preliminary Monte Carlo analysis with MCNP and reviewed the detailed MCNP models constructed by Dr. Leland M. Montierth. Dr. Montierth, in turn, reviewed the Sn work of Dr. Kim. Mr. Joshua J. Cogliati and Dr. Abderrafi Ougouag performed the analysis of the upper surface shape of the pebble bed with their code PEBBLES. Dr. Hans D. Gougar performed a review of the original draft. Dr. Virginia Dean was a strict independent reviewer, and her observations greatly improved the quality of the final report. The information on which this INL evaluation is based comes primarily from IAEA-TECDOC-1382, an International Atomic Energy Agency (IAEA) publication. This IAEA publication is the outcome of an IAEA Coordinated Research Project (CRP) entitled “Evaluation of HTGR Performance”. Dr. Yuliang Sun of the Institute of Nuclear and New Energy Technology (INET), which designed, built and operates the HTR-10 test reactor, is the principal author of the chapters in IAEA-TECDOC-1382 on the HTR-10 core physics calculation performed by INET. Upon completion of this INL evaluation report in its draft status, Dr. Sun performed a thorough review and provided recommendations for revision, focusing on the first two sections of the evaluation report. His colleagues Prof. Xingqing Jing, Prof. Shuren Xi, and Prof. Zhengpei Luo joined the review.

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Status of Compilation / Evaluation / Peer Review

Section 1 Compiled Independent Review

Working Group Review

Approved

1.0 DETAILED DESCRIPTION YES YES YES YES 1.1 Description of the Critical and / or Subcritical Configuration

YES YES YES YES

1.2 Description of Buckling and Extrapolation Length Measurements

N/A N/A N/A N/A

1.3 Description of Spectral Characteristics Measurements

N/A N/A N/A N/A

1.4 Description of Reactivity Effects Measurements

YES NO NO NO

1.5 Description of Reactivity Coefficient Measurements

N/A N/A N/A N/A

1.6 Description of Kinetics Measurements N/A N/A N/A N/A 1.7 Description of Reaction-Rate Distribution Measurements

N/A N/A N/A N/A

1.8 Description of Power Distribution Measurements

N/A N/A N/A N/A

1.9 Description of Isotopic Measurements N/A N/A N/A N/A 1.10 Description of Other Miscellaneous Types of Measurements

N/A N/A N/A N/A

Section 2 Evaluated Independent Review


Approved

2.0 EVALUATION OF EXPERIMENTAL DATA

YES YES YES YES

2.1 Evaluation of Critical and / or Subcritical Configuration Data

YES YES YES YES

2.2 Evaluation of Buckling and Extrapolation Length Data

N/A N/A N/A N/A

2.3 Evaluation of Spectral Characteristics Data

N/A N/A N/A N/A

2.4 Evaluation of Reactivity Effects Data NO NO NO NO 2.5 Evaluation of Reactivity Coefficient Data N/A N/A N/A N/A 2.6 Evaluation of Kinetics Measurements Data

N/A N/A N/A N/A

2.7 Evaluation of Reaction Rate Distributions

N/A N/A N/A N/A

2.8 Evaluation of Power Distribution Data N/A N/A N/A N/A 2.9 Evaluation of Isotopic Measurements N/A N/A N/A N/A 2.10 Evaluation of Other Miscellaneous Types of Measurements

N/A N/A N/A N/A

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Section 3 Complied Independent Review


Approved

3.0 BENCHMARK SPECIFICATIONS YES YES YES YES

3.1 Benchmark-Model Specifications for Critical and / or Subcritical Measurements

YES YES YES YES

3.2 Benchmark-Model Specifications for Buckling and Extrapolation Length Measurements

N/A N/A N/A N/A

3.3 Benchmark-Model Specifications for Spectral Characteristics Measurements

N/A N/A N/A N/A

3.4 Benchmark-Model Specifications for Reactivity Effects Measurements

NO NO NO NO

3.5 Benchmark-Model Specifications for Reactivity Coefficient Measurements

N/A N/A N/A N/A

3.6 Benchmark-Model Specifications for Kinetics Measurements

N/A N/A N/A N/A

3.7 Benchmark-Model Specifications for Reaction-Rate Distribution Measurements

N/A N/A N/A N/A

3.8 Benchmark-Model Specifications for Power Distribution Measurements

N/A N/A N/A N/A

3.9 Benchmark-Model Specifications for Isotopic Measurements

N/A N/A N/A N/A

3.10 Benchmark-Model Specifications of Other Miscellaneous Types of Measurements

N/A N/A N/A N/A



Approved

4.0 RESULTS OF SAMPLE CALCULATIONS

YES YES YES YES

4.1 Results of Calculations of the Critical or Subcritical Configurations

YES YES YES YES

4.2 Results of Buckling and Extrapolation Length Calculations

N/A N/A N/A N/A

4.3 Results of Spectral Characteristics Calculations

N/A N/A N/A N/A

4.4 Results of Reactivity Effect Calculations NO NO NO NO 4.5 Results of Reactivity Coefficient Calculations

N/A N/A N/A N/A

4.6 Results of Kinetics Parameter Calculations

N/A N/A N/A N/A

4.7 Results of Reaction-Rate Distribution Calculations

N/A N/A N/A N/A

4.8 Results of Power Distribution Calculations

N/A N/A N/A N/A

4.9 Results of Isotopic Calculations N/A N/A N/A N/A 4.10 Results of Calculations of Other Miscellaneous Types of Measurements

N/A N/A N/A N/A



Approved

5.0 REFERENCES YES YES NO NO Appendix A: Computer Codes, Cross Sections, and Typical Input Listings

YES YES YES YES

Appendix B: Calculation of Base-case Atomic Number Densities in the Core Region

YES YES YES YES

Appendix C: Angle of Repose for Criticality Considerations in Pebble Bed Reactors

YES YES NO NO

Appendix D: Nuclear Constants YES YES YES YES

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Revision: 0 Page 1 of 82 Date: March 31, 2007

EVALUATION OF THE INITIAL CRITICAL CONFIGURATION OF THE HTR-10 PEBBLE-BED REACTOR

IDENTIFICATION NUMBER: HTR10-GCR-RESR-001 CRIT-REAC KEY WORDS: benchmark evaluation, Chinese reactor, criticality, graphite-moderated,

graphite-reflected, HTGRs, HTR-10, Pebble-bed reactor, test reactor, uranium dioxide

SUMMARY INFORMATION 1.0 DETAILED DESCRIPTION The HTR-10 is a small (10 MWt) pebble-bed test reactor intended to develop pebble-bed reactor (PBR) technology in China. “HTR” denotes a high-temperature reactor, in particular a high-temperature gas-cooled reactor (HTGR). It is used to test and develop fuel, verify PBR safety features, demonstrate combined electricity production and co-generation of heat, and provide experience in PBR design, construction, and operation. Table 1.1 gives major design features of the reactor, and Figure 1.1 illustrates the reactor layout. The parameters in the table characterize the design of the reactor under full power with the equilibrium core, and not the initial criticality experiment evaluated here. The reactor was not at power during the subject experiment, and some key features, such as the properties of the gas in the void spaces, were different in the experiment from the specifications in Table 1.1, as explained in detail below.

Table 1.1. Reactor Design Features (from Reference 1).

Reactor thermal power MW 10 Primary coolant pressure MPa 3.0 Reactor core diameter cm 180 Average core height cm 197 Average coolant temperature at reactor outlet °C 700 Average coolant temperature at reactor inlet °C 250 Coolant mass flow rate at full power kg/s 4.3 Main steam pressure at steam generator outlet MPa 4.0 Main steam temperature at steam generator outlet °C 440 Feed water temperature °C 104 Main steam flow rate t/hr 12.5 Number of control rods in side reflector 10 Number of absorber ball units in side reflector 7 Nuclear fuel UO2 Heavy metal loading per fuel element g 5 Enrichment of fresh fuel element % 17 Number of fuel elements in equilibrium core 27,000 Fuel loading mode multi-pass

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Figure 1.1. Layout of HTR-10 (from Reference 1).

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As a test reactor intended to demonstrate all the technologies associated with PBRs, the HTR-10 contains a complete cooling system that supplies hot helium to a steam generator for generation of electricity and supply of district heating. The primary coolant loop contains three pressure vessels: the reactor vessel, the steam generator vessel, and the connecting vessel. The initial criticality experiment evaluated in this report was performed under air rather than helium, so that the void spaces in the reactor were filled with ambient air in the experiment. The HTR-10 first became critical on 1 December 2000. Primarily based on published information, this evaluation assesses, in the IRPhEP framework, whether the HTR-10 initial criticality experiment can serve as a benchmark for reactor physics codes. It is concluded that the experiment is an acceptable benchmark. In the initial critical condition, all control rods were withdrawn. Measurements of control rod worth were also made on the cold clean core. Data from the control rod worth experiments are provided in Section 1.4; however, they have not yet been evaluated in the IRPhEP framework. 1.1 Description of the Critical and / or Subcritical Configuration 1.1.1 Overview of Experiment The HTR-10 is located at the Institute of Nuclear and New Energy Technology (INET), a research institute of Tsinghua University in Beijing. The HTR-10 project was approved by the Chinese State Council in March 1992, ground was broken in 1994, and construction was completed in 2000. Initial criticality was achieved on 1 December 2000. The experiment was performed by the INET. In advance of the actual experiment, relevant data were provided by INET in an IAEA Coordinated Research Project (CRP) for interested participating groups to use in benchmarking calculations with various computer codes. The pre-experiment benchmark problem specifications differed in a few respects from the experiment as it was ultimately run. Some compositions provided in the benchmark specifications were different from the as-built compositions, the benchmark and as-run temperatures were different, and the void spaces in the actual experiment were filled with air instead of helium. The INET group performed calculations for both the pre-experiment specifications and the as-run configuration. Specifications and results for both the pre-experiment benchmark problem and the as-run configuration are given in Reference 1. The uncertainties in the specifications of the initial criticality experiment are shown in Section 2 to lead to a total uncertainty in calculated values of keff of slightly less than 0.004. The experiment is judged acceptable as an IRPhEP benchmark. 1.1.2 Geometry of the Experimental Configuration and Measurement Procedure Like any PBR, the HTR-10 is fueled by billiard-ball-size spheres containing TRISO-coated fuel particles embedded in a matrix. These fuel spheres are the “pebbles” for which the PBR is named, and they are denoted as “pebbles” in the remainder of this document. The terms “particles,” “coated particles,” and “TRISO-coated particles,” depending on the context, are used for the fuel microspheres in the fuel zone of the fuel pebbles. The individual components (e.g., the kernels) of the coated particles are denoted by name. During reactor operation, the pebbles are introduced at the top and slowly flow downwards through the core region. As shown in Figure 1.1, the HTR-10 core is a cylindrical cavity above a conical zone (the “conus”) that funnels pebbles into a discharge tube. The core, conus, and discharge tube are surrounded by graphite blocks (the reflector) and boronated carbon blocks, some of which are penetrated by various borings for coolant flow, control rods, shutdown absorber balls, and other purposes. In the initial critical experiment, the voids in the core and reflector were filled by ambient air rather than the helium with which the reactor is cooled while in operation, and the conus and discharge tube were filled

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with “dummy” pebbles – spheres the same size as fuel pebbles but made only of graphite. In the cylindrical core, a mixture of fuel pebbles and dummy pebbles in the ratio of 57:43 was added until the core became critical. The packing fraction in the core was assumed to be 61%, the commonly accepted average value based on empirical data.a During this initial fuel loading, pebbles were not removed at the bottom, so the pebbles remained stationary after they settled into position in the core. A small Am-Be source (4.4x107 neutrons/s) was provided to assist startup, and the neutron flux was tracked by three neutron counters in the side reflector. Figure 1.2 shows the reciprocal neutron multiplication No/N, obtained from the counting rate N, as a function of the number of pebbles in the core; this method is explained by Glasstone and Sesonske.b Criticality was achieved when 16,890 pebbles (9627 fuel pebbles and 7263 dummy pebbles) had been loaded. This number of pebbles is equivalent to a level core height of 123.06 cm, as one can readily confirm from the pebble diameter, the core diameter, and the packing fraction.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

5000 7000 9000 11000 13000 15000 17000Number of loaded balls

N0/N

Counter 1Counter 2Counter 3

Figure 1.2. Experimental Determination of Critical Core Loading (from Reference 1).

Dimensions of the initial critical experiment were provided by the INET in Reference 1. Figure 1.3 shows the construction of the fuel pebble and the TRISO fuel particles embedded in the fuel zone matrix of the fuel pebble. The kernel of the particle is composed of the fissile material, UO2. A porous carbon buffer zone, designed to provide space for fission-product gases, immediately surrounds the kernel. Next, a dense carbon zone – the inner pyrolytic carbon (IPyC) layer – provides a diffusion barrier for the fission products. This is followed by a layer of silicon carbide (SiC), which supplies most of the fuel particle’s strength. Finally, an outer pyrolytic carbon (OPyC) layer protects the SiC layer and provides a further impediment to diffusion. Table 1.2 shows the dimensions of the various components of the fuel pebble and fuel particle illustrated in Figure 1.3. Table 1.3 shows the nominal dimensions of key components in the core and reflector, together with tolerance limits, observed ranges, and standard deviations where they are available in References 1 and 2. The nominal dimensions are taken or derived from Reference 1, and the uncertainties in the fuel pebble and coated particle dimensions are taken from Reference 2. Reference 2 presents data obtained by 1997.

a R. F. Benenati and C. B. Brosilow, “Void Fraction Distribution in Beds of Spheres,” A. I. Ch. E. Journal 8, No. 3, pp. 359-361 (1962). b Samuel Glasstone and Alexander Sesonske, Nuclear Reactor Engineering, Van Nostrand Reinhold (1981), pp. 190-191.

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Figure 1.3. Illustration of Fuel Pebbles and TRISO Coated Particles.

Table 1.2. Dimensions of Pebbles and Fuel Particles ( from Reference 1).

Diameter of fuel and dummy (no-fuel) pebbles 6.0 cm Diameter of fuel zone 5.0 cm Volume fraction of pebbles in the core 0.61 Radius of the kernel (mm) 0.25 Coating layer materials (starting from kernel) Buffer/PyC/SiC/PyC Coating layer thicknesses (mm), respectively 0.09/0.04/0.035/0.04

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The reflector region comprises numerous graphite blocks, surrounded by boronated carbon bricks at the periphery. Graphite blocks adjacent to the hot gas duct contain borings filled with boronated carbon to protect the duct from neutron irradiation damage. The variations in the volume-averaged compositions of the blocks arise both from different concentrations of (natural) boron equivalent and from different sizes and placements of borings. Figure 1.4 shows a coarse representation of the different regions of the reactor. The reflector is contained in a metallic vessel (labeled the “core barrel” in Figure 1.1) that is supported on the SA516-70 steel pressure vessel. This pressure vessel is 4.2 m in inside diameter and 11.1 m high; it weighs approximately 142 T. The pressure vessel is supported by four brackets. Not all dimensions are given in Reference 1 (e.g., the thickness of the pressure vessel, the dimensions of the “core barrel,” and the detailed shapes of some of the reflector blocks). The details of components of complex shape are proprietary, but the INET has provided average atomic number densities in Reference 1 over regions containing such components, as far as the outer surfaces of the boronated carbon bricks. In operation, the reactor coolant enters the side of the reactor in an annular channel around the hot gas exit duct. The inlet channel separates into twenty vertical channels in the reflector, equally spaced azimuthally, in which the coolant flows upwards past the core cavity. Then the coolant turns horizontally, flows radially inward, and next turns downward to enter the core. After flowing downward through the core, it enters borings at the conus and passes into a hot gas plenum under the reflector, whence it exits through the hot gas exit duct. Explicit information on these flow channels is given in Reference 1 only for the hot gas duct and the vertical channels in the reflector; see Table 1.3. The 100 cm length of the hot gas duct given in the table applies only to the portion of the hot gas duct in the side reflector and not to the external portion; in fact, this dimension is the reflector thickness (90 cm ≤ r ≤ 190 cm). The average atomic number densities given in Section 3 (from Reference 1) for regions that contain coolant channels account for the presence of these channels. Figure 1.5 shows a horizontal cross section at an elevation within the core, displaying the borings (channels) for coolant flow, control rods, experiments, and shutdown absorber balls. The small absorber ball channels extend from the top of the reflector to the bottom; from the top of the reflector to the top of the core cavity (z=130 cm) and below the bottom of the conus (z=388.764 cm), they are cylindrical in cross section; between these levels they are oval, as shown in Figure 1.5. The twenty borings near the inner wall of the side reflector, for control rods, shutdown absorber balls, and irradiation measurements, are also equally spaced azimuthally.

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Table 1.3. Nominal Values and Variations in Dimensions.

Nominal Design Observed Standard Deviation Item Value Tolerances Range (design/observed) Core height 123.06 cm n/a n/a n/a Diameter of core cavity 180 cm n/a n/a n/a Height of core cavity (a) 221.818 cm n/a n/a n/a Height of conus (a) 36.946 cm n/a n/a n/a Dimensions of graphite blocks Various n/a n/a n/a Outer diameter of graphite reflector 380 cm n/a n/a n/a Overall height of graphite reflector 610 cm n/a n/a n/a Diameter of cold coolant flow channels 80 mm n/a n/a n/a Radial location of cold coolant flow channels 1446 mm n/a n/a n/a Height of cold coolant flow channels 5050 mm n/a n/a n/a Diameter of control rod and irradiation channels 130 mm n/a n/a n/a Height of control rod and irradiation channels 4500 mm n/a n/a n/a Radial location of control rod and irradiation channels 1021 mm n/a n/a n/a Diameter of round part of KLAK channels(b) 60 mm n/a n/a n/a Width of oval part of KLAK channels 60 mm n/a n/a n/a Length (in plane of cross section) of oval part of KLAK channels 160 mm n/a n/a n/a Diameter of hot gas duct 300 mm n/a n/a n/a Position of hot gas duct axis below top of reflector 4800 mm n/a n/a n/a Length of hot gas duct (see text) 1000 mm n/a n/a n/a Radius of fuel discharge tube 250 mm n/a n/a n/a Length of fuel discharge tube (a) 2212.36 mm n/a n/a n/a Diameter of fuel pebble 6.0 cm 59.8-60 mm 59.8-60 mm n/a Diameter of kernel 0.500 mm 500 μm 501 μm 25/10.2 (μm)

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Table 1.3 (cont’d.). Nominal Values and Variations in Dimensions.

Nominal Design Observed Standard Deviation Thickness of buffer layer 0.09 mm 90 μm 90.2 μm 18/4.4 (μm) Thickness of IPyC layer 0.04 mm 40±10 μm 39.6±2.8 μm n/a Thickness of SiC layer 0.035 mm 35±4 μm 35±2.6 μm n/a Thickness of OPyC layer 0.04 mm 40 ±10 μm 42±3.0 μm n/a Packing fraction 0.61 n/a n/a n/a (a) Some quantities are calculated from data given in Reference 1. (b) The HTR-10 uses two independent shutdown systems – control rods and small absorber balls that fall by gravity into channels near the core

boundary. These small absorber balls are called KLAK from the German “kleine aufsauger Kugeln” (“small absorber balls”). The KLAK channels are cylindrical above and below the core level, but oval beside the core; see Figure 1.5 and associated text.

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Figure 1.4. Regions of HTR-10 Initial Configuration (dimensions are in cm).

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Figure 1.5. Cross Section of Reactor in Core Region. 1.1.3 Material Data The nominal data for the fuel and dummy pebble materials, as given in Reference 1 for the actual as-built configuration, are given in Table 1.4. The only materials reported in Reference 1 in the reflector are carbon, natural boron, and the cooling fluid. Impurities in graphite are treated by giving an overall representative natural boron content that is determined by experiment. This is believed reasonable from the viewpoint of neutronics criticality evaluations. The relative boron fraction is higher in the graphite blocks adjacent to the hot gas duct, where boron has been added to protect the duct from neutron damage, and it is much higher in the boronated carbon bricks. The zone-averaged graphite density is tabulated in Reference 1 for 82 different defined annular regions in the reflector; this density varies, because many of the zones have voids for various purposes. Since the control rods were withdrawn in the initial criticality experiment, and since no shutdown absorber balls had been introduced into the reflector, all the voids were assumed to be filled with ambient air. The “core barrel” is made of a metallic material not specified in Reference 1. The pressure vessel itself is primarily made of SA516-70 steel. (In Reference 1, compositions were given only for the regions the INET group expected to have significant effects on keff; these regions did not extend beyond the boronated carbon bricks. The steel specification was given for the pressure vessel as general information.) Although the HTR-10 is cooled by helium in operation, in the initial criticality experiment the voids in the reactor were filled with “moist” aira at a total pressure of 0.1013 MPa and a temperature of 15 °C.

a The word “moist” was used by the INET in Reference 1. It is taken to mean saturated, as the water vapor density specified at the nominal temperature of 27 °C (see Section 1.1.4) is in fact the saturation density.

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Table 1.4. Nominal Material Properties of Fuel and Dummy Pebbles as Built (from Reference 1).

Fuel Pebble Density of graphite in matrix and outer shell 1.73 g/cm3 Heavy metal (uranium) loading (weight) per pebble 5.0 g Uranium enrichment (235U/total U by weight) 17 % Equivalent natural boron content of impurities in uranium 4 ppm Equivalent natural boron content of impurities in graphite 1.3 ppm Volume fraction of pebbles in the core 0.61 UO2 density(g/cm3) 10.4 Coating layer materials (starting from kernel) Buffer/PyC/SiC/PyC Coating layer densities (g/cm3), respectively 1.1/1.9/3.18/1.9 Average number of fuel particles per pebble 8335 Volume fraction of fuel particles in pebble fuel zone(a) 5.0248 % Dummy Pebble Density of graphite in dummy pebbles 1.84 g/cm3 (b) Equivalent natural boron content of impurities in graphite in dummy pebbles

0.125 ppm(b)

(a) This quantity is derived from the dimensions of the pebbles and particles and the number of particles per pebble.

(b) These are the actual as-built values, corrected from the initial specifications (see p. 248 of Reference 1).

Table 1.5 presents the composition of most of the SA516-70 steel pressure vessel. Table 1.6 specifies the compositions of all the remaining components in the reactor except the “core barrel” and the structural components. The compositions of these components are not given in Reference 1. Tolerance limits and observed ranges are shown where they are publicly available. The nominal compositions are taken from Reference 1, and the uncertainties are taken from Reference 2. As noted above for the dimensions, Reference 2 presents data obtained by 1997. 1.1.4 Temperature Information The initial criticality experiment was run with a clean cold core. In the benchmark specifications provided in advance of the experiment, the temperature was given as 20 °C, and the composition of the air occupying the void spaces in the reactor was specified at 27 °C, but the actual temperature on the day of the experiment was 15 °C. 1.1.5 Additional Information Relevant to Critical and Subcritical Measurements

No additional information relevant to critical and subcritical measurements is given in Reference 1.

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Table 1.5. Composition of Pressure Vessel (SA516-70 steel, more than 4 in. thick(a)).

Constituent (except iron) Concentration (percent) Carbon <0.31 Manganese 0.79-1.30 Phosphorus <0.035 Sulfur <0.035 Silicon 0.13-0.45 (a) ASTM International, Designation: A516/A516M-05, Standard Specification for Pressure Vessel

Plates, Carbon Steel, for Moderate- and Lower-Temperature Service, July 2005.

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Table 1.6. Nominal Values and Variations in Compositions.

Item Nominal Value(a)

Design Tolerances(b) Observed Range(b) Standard Deviation(b)

Uranium fuel loading (g/fuel pebble) 5.0 g 5.0±0.1 g 4.95-5.05 g (c) n/a Density of graphite in matrix and outer shell of fuel pebble 1.73 g/cm3 1.75±0.02 g/cm3 1.73 g/cm3 n/a Total ash in fuel element 0 ≤300.0 ppm 130-190 ppm n/a Lithium in fuel element 0 ≤0.3 ppm 0.007-0.023 ppm n/a Boron in fuel element 1.3 ppm ≤3.0 ppm 0.15 ppm n/a Ratio of oxygen to uranium in kernel 2 <2.01 n/a n/a Density of kernel 10.4 g/cm3 >10.4 g/cm3 10.83 g/cm3 n/a Density of buffer layer 1.1 g/cm3 ≤1.1 g/cm3 1.02 g/cm3 0.03 g/cm3 observed Density of IPyC layer 1.9 g/cm3 1.9±0.1 g/cm3 1.86±0.06 g/cm3 n/a Density of SiC layer 3.18 g/cm3 ≥3.18 g/cm3 3.21±0.02 g/cm3 n/a Density of OPyC layer 1.9 g/cm3 1.9±0.1 g/cm3 1.87±0.02 g/cm3 n/a Density of reflector graphite 1.76 g/cm3 n/a n/a n/a Density of boron equivalent in natural graphite reflector elements 4.8366 ppm n/a n/a n/a Density of boronated carbon brick including B4C 1.59 g/cm3 n/a n/a n/a Weight ratio of B4C in boronated carbon brick 5% n/a n/a n/a Density of coolant 1.149e-3 g/cm3 n/a n/a n/a

Composition of coolant

78.084% N2, 20.9476% O2, 0.934% Ar (by volume)(d) n/a n/a n/a

Moisture content of coolant 2.57E-5 g/cm3 n/a n/a n/a (a) From Reference 1 (b) From Reference 2 (c) Mark F. Bryan, Evaluation of NP-MHTGR Performance Test Fuel Quality Control Data, Draft EGG-NPR-10130, February 1992 (d) http://www.physlink.com/Reference/AirComposition.cfm, citing the CRC Handbook of Chemistry and Physics, David R. Lide, Editor-in-Chief, 1997 Edition

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1.2 Description of Buckling and Extrapolation Length Measurements Buckling and extrapolation length measurements were not reported in Reference 1. 1.3 Description of Spectral Characteristics Measurements Spectral characteristics measurements were not reported in Reference 1. 1.4 Description of Reactivity Effects Measurements With a slight excess reactivity (i.e., at a total loading of 17,000 pebbles, or 110 more than the critical count), a control rod worth calibration experiment was carried out, as described in Reference 1. One “typical” control rod (control rod S3, identified in Figure 1.5) was inserted through a distance of 223 cm. The worth of the control rod was measured as a function of rod position. The measured integral worth of the rod is 1.4693%.a 1.4.1 Overview of Experiment The control rod worth measurement was carried out under the same conditions as the initial criticality experiment, except for the control rod insertion and the slightly increased number of pebbles. The temperature was 15 °C and the void spaces were filled with air. The general comments on the HTR-10 in Section 1.1.1 apply. 1.4.2 Geometry of the Experimental Configuration and Measurement Procedure The same general description of the reactor given in Section 1.1.2 applies to the control rod worth measurement as well. The locations of the control rods are shown in Figure 1.5. The active absorbing medium in the control rods is B4C, boron carbide. The boron carbide is contained in five ring segments located between inner and outer sleeves of stainless steel. These segments are connected by tubular iron joints. The upper and lower ends are capped with iron. Table 1.7 presents the dimensions of the components of the control rod. The radial and axial coordinates in the table are local coordinates. That is, they are relative to the centerline of the control rod and the lower end of the control rod. When the control rod is fully withdrawn, its lower end is at an axial position of z=119.2 cm in reactor coordinates. (In reactor coordinates, as shown in Figure 1.4, z=0 is at the top of the reflector.) When it is fully inserted, its lower end is at an axial position of z=394.2 cm in reactor coordinates. In the control rod worth experiment, the initial position of the lower end of the control rod was at z=171.2 cm. This position is essentially the expected location of the equivalent level height of the equilibrium core (i.e., the steady-state core that would develop after extended operation at constant power). There is a radial gap of 0.5 mm between the control elements and each sleeve. Reactivity measurements were based on the neutron counters located in the side reflector. As the control rod was inserted, the reactivity was measured at discrete positions until the rod was fully inserted. The curve of reactivity versus control rod position, replotted from Reference 1, is given in Figure 1.6.

a At one place, Reference 1 also gives the number of 1.4369%, which is a typing error.

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-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 50 100 150 200 250

Control rod lower end displacement (cm)

Rea

ctiv

ity

Figure 1.6. Reactivity Versus Control Rod Position.

Table 1.7. Nominal Control Rod Dimensions.

Component Rinner (mm) Router (mm) Z lower (mm) Zupper (mm) Lower end cap 27.5 55.0 0.0 45.0 Outer sleeve segment 1 53.0 55.0 45.0 532.0 Inner sleeve segment 1 27.5 29.5 45.0 532.0 Control segment 1 30.0 52.5 45.0 532.0 Joint 1 27.5 55.0 532.0 568.0 Outer sleeve segment 2 53.0 55.0 568.0 1055.0 Inner sleeve segment 2 27.5 29.5 568.0 1055.0 Control segment 2 30.0 52.5 568.0 1055.0 Joint 2 27.5 55.0 1055.0 1091.0 Outer sleeve segment 3 53.0 55.0 1091.0 1578.0 Inner sleeve segment 3 27.5 29.5 1091.0 1578.0 Control segment 3 30.0 52.5 1091.0 1578.0 Joint 3 27.5 55.0 1578.0 1614.0 Outer sleeve segment 4 53.0 55.0 1614.0 2101.0 Inner sleeve segment 4 27.5 29.5 1614.0 2101.0 Control segment 4 30.0 52.5 1614.0 2101.0 Joint 4 27.5 55.0 2101.0 2137.0 Outer sleeve segment 5 53.0 55.0 2137.0 2624.0 Inner sleeve segment 5 27.5 29.5 2137.0 2624.0 Control segment 5 30.0 52.5 2137.0 2624.0 Upper end cap 27.5 55.0 2624.0 2647.0

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1.4.3 Material Data Table 1.8 presents the compositions of the control rod components.

Table 1.8. Compositions of Control Rod Components.

Component Composition Density Control elements B4C 1.7 g/cm3

Stainless steel Cr-18%, Fe-68.1%, Ni-10%, Si-1%, Mn-2%, C-0.1%, Ti-0.8%

7.9 g/cm3

Joints and end caps(a) Fe (atom density) 0.04 atoms/barn-cm (a) It is suggested in Reference 1 that the joints and end caps be represented simply as iron; the

experimenters do not provide details of the composition. 1.4.4 Temperature Information The control-rod worth experiment was run with a clean cold core. In the benchmark specifications provided in advance of the experiment, the temperature was given as 20 °C, and the composition of the air occupying the void spaces in the reactor was specified at 27 °C, but the actual temperature on the day of the experiment was 15 °C.

1.4.5 Additional Information Relevant to Reactivity Effects Measurements No additional information relevant to reactivity effects measurements is given in Reference 1. 1.5 Description of Reactivity Coefficient Measurements Reactivity coefficient measurements were not reported in Reference 1. 1.6 Description of Kinetics Measurements Kinetics measurements were not reported in Reference 1. 1.7 Description of Reaction-Rate Distribution Measurements Reaction-rate distribution measurements were not reported in Reference 1. 1.8 Description of Power Distribution Measurements Power distribution measurements were not reported in Reference 1.

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1.9 Description of Isotopic Measurements Isotopic measurements were not reported in Reference 1. 1.10 Description of Other Miscellaneous Types of Measurements Other miscellaneous types of measurements were not reported in Reference 1.

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2.0 EVALUATION OF EXPERIMENTAL DATA The overall uncertainty in the calculated value of keff, which is a function of multiple input parameters, is given bya

1

2 2,

1 1 1( ) ( ) 2 ( )( )

N N N

c eff i i j i ji i j i

u k k k k r−

= = = +

= Δ + Δ Δ∑ ∑∑ (2.1)

In Equation 2.1, Δki is the change in keff when parameter i is changed by the increment σi, the standard deviation in the parameter, and ri,j is the correlation coefficient for parameters i and j. This expression for the uncertainty in keff can be evaluated accurately when the standard deviations in the parameters on which keff is dependent are available. References 1 and 2 are the principal sources of information for this evaluation of the HTR-10 initial criticality experiment. These references do not contain a complete set of data on the uncertainties or tolerances of all relevant parameters. All further information on the experiments is either proprietary to the INET or unavailable for other reasons. Therefore, most of the data presented in this report comes from published sources, primarily these two references. Uncertainty information on the boron content of the fuel matrix and the reflector graphite is given by the INET author in the course of this evaluation. The following list summarizes the unknown quantities relevant to this evaluation:

• The ranges of variation and standard deviations of almost all dimensions and compositions • The actual impurity content of all components (as opposed to the boron equivalent) • Detailed shapes of the graphite blocks in the reflector • Source data for the compositions of the graphite blocks (as opposed to calculated atomic number

densities provided in Reference 1). The remainder of this introduction addresses how this evaluation addresses the lack of required uncertainty data, and it also describes possible inaccuracies introduced by the methods of calculation. Data on dimensions and materials are given in Table 1.4. Most of the data are given only as nominal values, with no information provided about tolerances, observed variations, or standard deviations. Where standard deviations are available, they are used for calculating the Δki. Where observed ranges are given, but not standard deviations, the limiting values of the observed ranges are usually applied, and plausible distribution functions are assumed for finding the Δki. Where only tolerances are given, their limiting values are used, along with plausible distribution functions. Where no guidance is given on the variability of a parameter, engineering judgment is used to select a range of variation that will produce the largest reasonable uncertainty in keff. The bounding values in this range are then applied in the uncertainty analysis. If the overall uncertainty in keff predicted by this approach is small enough that the experiment can be judged an acceptable benchmark, one can be confident that the real experiment is actually even better. All uncertainties are adjusted to values of one standard deviation (1σ). No information is available on correlations among parameters, so all parameters and their uncertainties are assumed to be uncorrelated. One category of unknown dimensional uncertainties includes the dimensions of the graphite blocks from which the reflector is constructed. In typical milling operations, tolerances on the order of tenths of millimeters are common. Cumulative uncertainties from stacking a set of ten or so blocks milled with such tolerances would be on the order of millimeters or less. For the purpose of obtaining the largest plausible effect on keff, a tolerance of ±1 cm is usually assumed for dimensions specifying the positioning of graphite

a International Handbook of Evaluated Criticality Safety Benchmark Experiments, NEA/NSC/DOC(95)03/I-VIII, OECD-NEA, “ICSBEP Guide to the Expression of Uncertainties, ”Revision 1, p. 29, September 30, 2004.

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reflector elements. Because borings have to line up much more precisely than that, a tolerance of 0.25 cm is usually assumed for the locations and radii of the various channels in the core. Usually, no information is publicly available about the distribution function of the deviation of a parameter from its nominal value. In most cases, it is reasonably assumed that the most relevant quantity is uniformly distributed. For example, if the change in keff from its nominal value is dependent on the change in the volume of a spatial region, then it is assumed that the deviation of the volume of that region from its nominal value is uniformly distributed. The following sections discuss the calculation of the effects of uncertainties in the parameters listed in Table 1.4. The values of keff are computed in the as-run critical configuration and in the configuration with each parameter assigned its maximum variation (or its standard deviation when available), one parameter at a time. The bases for the choices of the parameter values are discussed. Only a few parameters are important enough that reasonable variations in them produce relatively large variations in keff, and these parameters are discussed at greater length. The calculations of keff performed for the uncertainty analysis described in this section applied the diffusion model discussed in Section 4. For clarity in discussions of the details of the individual calculations, reference is often made to region numbers illustrated in Figure 4.1.a in Section 4.1.2, which illustrates the different material zones defined for the Sn and diffusion models. The value of keff computed for a given configuration depends on the microscopic cross sections used in the calculation. The calculated microscopic cross sections are somewhat sensitive to the fast and thermal buckling values used in the input to COMBINE, the cross section processing codea

used in this evaluation. The buckling values were found by a buckling search. In most cases, variations in the problem specifications were small enough that the same cross sections were used for both the critical baseline configuration and the perturbed configuration. However, in some cases, new cross sections were calculated for a perturbed case. The baseline value of keff for the simplified benchmark model (see Section 3) is 1.02310 with PEBBED, the code used for the uncertainty analysis; however, for computational efficiency a somewhat looser mesh was used in the uncertainty analysis, with less accurate values of keff (1.03257 for the base case) but much faster calculation times. The looser mesh has almost no effect on the uncertainties. The baseline value of keff computed for this evaluation is farther from 1.0 than is ordinarily acceptable in a benchmark calculation. PEBBED,b the code used to calculate keff, has been verified extensively,c but its results are naturally dependent on the quality of the cross sections supplied to it. Calculation of cross sections in graphite systems presents problems that have not yet been resolved. Among the more difficult cross sections to evaluate are the transport cross sections, which are used to define the diffusion coefficient. In discussing the transport cross section, the COMBINE manual states, “…the transport cross section for any ‘finite system’ based on the transport equation cannot be expressed in terms of elementary cross sections alone, but requires the linearly anisotropic neutron currents also, which result from the finite boundary conditions.” In COMBINE, the finite boundary conditions are represented

a Robert A. Grimesey, David W. Nigg, and Richard L. Curtis, COMBINE/PC – A Portable ENDF/B Version 5 Neutron Spectrum and Cross-Section Generation Program, EGG-2589, Rev. 1, Idaho National Engineering Laboratory, February 1991. b W. K. Terry, H. D. Gougar, and A. M. Ougouag, “Direct Deterministic Method for Neutronics Analysis and Computation of Asymptotic Burnup Distribution in a Recirculating Pebble-Bed Reactor,” Annals of Nuclear Energy 29 (2002), pp. 1345-1364. c Hans D. Gougar, Abderrafi M. Ougouag, and William K. Terry, “Validation of the Neutronic Solver within the PEBBED Code for Pebble Bed Reactor Design,” proc. Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, Palais des Papes, Avignon, France, September 12-15, 2005, American Nuclear Society, LaGrange Park, IL (2005).

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by buckling values, which were found by a search as explained above. The value of keff was found to be sensitive to the buckling values and consequently to the transport cross sections. Therefore, these parameters are considered to be likely causes for the deviation of the baseline value of keff from the correct value of 1.0. Another possible source of inaccuracy in the PEBBED results is the use of only two spectral zones in the model – one in the core, and the other in the reflector. This might be inadequate, even in the HTR-10 initial startup core, which is uniform in composition. Again, however, this representation introduces in the results a constant bias that should not affect the uncertainties calculated when dimensional or compositional parameters are varied. Another minor source of inaccuracy in the PEBBED results is the assumption that the core density is uniform. Of course, the pebble packing fraction is only about 61%, so on the scale of a pebble diameter the density is highly nonuniform, but on the scale of the core dimensions, this nonuniformity is not expected to make a significant difference. However, spheres in a container are not distributed uniformly throughout. It has been observed experimentallya that the packing fraction varies from zero at the edges through several damped oscillations to approach an asymptotic value of about 61% over a distance of about five sphere diameters. The effects of these inhomogeneities have been studied for a simple hypothetical PBR reactor configuration.b It was found that the difference in the calculated value of keff was about 5x10-4 for that reactor. The packing fraction itself is uncertain, but the uncertainty in this parameter is addressed in the analysis. 2.1 Evaluation of Critical and / or Subcritical Configuration Data The following sections contain evaluations of the critical configuration measurement described in Section 1.1. The diffusion code PEBBEDc was generally used to calculate keff in the nominal and variational cases. However, the Monte Carlo code MCNPd was used to calculate keff in the nominal and variational cases for the shape of the upper core surface. A summary of the uncertainty calculations, showing the nominal and bounding values used for each varied parameter, is given in Table 2.1. In many cases, the assumptions on the uncertainties in experimental parameters are very crude. However, the only important parameters are those whose uncertainties lead to uncertainties of more than 10-3 in keff. When reasonable bounding assumptions lead to smaller uncertainties than that in keff, little is to be gained by trying to be more precise. In all cases where tolerances or observed variations apply to large numbers of objects (pebbles and fuel particles or portions thereof), both deterministic uncertainties (applying to all the objects equally) and random uncertainties (different from one object to the next) will occur. For the fuel particles and their subregions especially, the random uncertainties are extremely small (the tolerance limit for the random uncertainty divided by the square root of the number of fuel particles in the core, √(4751x16,890) = 8957.9 (see Appendix B)). For uncertainties in properties of pebbles, the random uncertainty is equal to the tolerance limit for the random component divided by the square root of the number of pebbles, or 129.96. In all cases, division by such large numbers would make the random a R. F. Benenati and C. B. Brosilow, “Void Fraction Distribution in Beds of Spheres,” A. I. Ch. E. Journal 8, No. 3, pp. 359-361 (1962). b W. K. Terry, A. M. Ougouag, Farzad Rahnema, and Michael Scott McKinley, “Effects of Spatial Variations in Packing Fraction on Reactor Physics Parameters in Pebble-bed Reactors,” American Nuclear Society Mathematics & Computation Division Conference, Gatlinburg, Tennessee, April 6-11, 2003. c W. K. Terry, H. D. Gougar, and A. M. Ougouag, “Direct Deterministic Method for Neutronics Analysis and Computation of Asymptotic Burnup Distribution in a Recirculating Pebble-Bed Reactor,” Annals of Nuclear Energy 29 (2002), pp. 1345-1364. d J. F. Briesmeister, Ed., “MCNP – A General Monte Carlo N-Particle Transport Code – Version 4B,” LA-12625-M (March 1997).

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component of the uncertainty negligible. However, since no information is available about how the uncertainties are divided between the deterministic and random components, it is assumed throughout that the uncertainties are all deterministic. This assumption leads to substantial overprediction of the effect on keff, in keeping with the philosophy discussed above for dealing with unknown parameter distributions. It is also noted that assuming a uniform distribution of a parameter between its limits leads to overprediction of the effect on keff. When a variable x is judged to be distributed uniformly between the bounding values μ-a and μ+a, the standard deviation in x can be shown to be a/√3.a Then the uncertainty in a linear function f(x) is taken as ⎟f(μ+a/√3) –f(μ)⎟. When a variable is judged to be distributed uniformly between the bounding values μ and μ+a, then the mean is actually μ+a/2, and the standard deviation in x from the mean is a/(2√3). Then the uncertainty in a linear function f(x) is taken as ⎟f(μ +(a/2)+a/(2√3))-f(μ+a/2)⎟ = ⎟ f(μ+a((1/2)+1/(2√3)))-f(μ+a/2) ⎟ = ⎟((1/2)+1/(2√3)-1/2)(f(μ+a)-f(μ))⎟ = ⎟(f(μ+a)-f(μ))/(2√3)⎟. When a variable x is most likely to have a value c in the interval a<x<b, then the distribution function may be taken as triangular, and the variance in the triangular distribution is given byb

2 2 22

18a b c ab ac bcσ + + − − −= ,

and the uncertainty in a linear function f(x) in the interval is

( ) ( )f c f cσ± − ,

where the choice of the plus or minus sign depends on the specific circumstances. These observations are used repeatedly in the following analysis. 2.1.1 Core Height and Core Diameter The core height is a calculated parameter, based on the number of pebbles contained in the core. As discussed above, the upper core surface is actually a mound, but it is modeled as a plane at an elevation consistent with the core volume and a cylindrical core shape. The total number of pebbles (fuel and dummy) in the cylindrical core region at criticality was given as 16,890, and no information is given in Reference 1 on the accuracy of the count. The core height was computed from the number of pebbles and the nominal core diameter at an assumed packing fraction of 0.61. The number of pebbles in the core is

2p c c

pp

f R HN

Vπ

= , (2.2)

where fp =packing fraction, Rc = core radius,

a International Handbook of Evaluated Criticality Safety Benchmark Experiments NEA/NSC/DOC(95)03/I-VIII, OECD-NEA, “ICSBEP Guide to the Expression of Uncertainties,” Revision 1, p. 29, September 30, 2004. b M. Evans, Nicholas Hastings, and Brian Peacock, Statistical Distributions, 3rd Edition, John Wiley and Sons, New York, 2000, pp. 187-188.

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Hc = core height, and Vp = pebble volume. Then, in part, keff is a function of the packing fraction, core radius, core height, and pebble volume. The dependencies of keff on pebble volume and packing fraction are treated in subsequent sections. In this Section, pebble volume and packing fraction are held constant. It is convenient to treat core radius and core height together; it is reasonable to treat them as independent (uncorrelated) variables. No estimate is given on the uncertainty in the number of pebbles in the core, although the method of counting, based on the use of filled packages containing a known number of pebbles each, suggests that the count is likely to be exact. Nevertheless, an uncertainty estimate is made for the effects of a counting error. Engineering judgment suggests that a counting uncertainty of more than 1% would be very poor, while an accuracy of ±0.1% (at one standard deviation) might be acceptable. The latter figure corresponds to a miscount of about 17 pebbles in the core. PEBBED calculations (based on changes of 1 cm in radius and height) show that a change in the core radius leads to a change in keff of 1.10643x10-5 per pebble, while a change in core height leads to a change in keff of 2.16393x10-5 per pebble. (This difference in sensitivity is due to the relatively short and wide configuration of the core at initial criticality.) Then for a miscount of 17 pebbles, Δkcore radius = 1.9x10-4 and Δkcore height = 3.7x10-4. 2.1.2 Height of Core Cavity This parameter is the height of the cavity in which the core resides, which includes the void space above the core, not the height of the core itself. The effect on keff of variations in this parameter is assessed by raising the datum plane at the top of the core cavity. No other computational cells are affected by this change. No guidance is given on the possible variation in core cavity height, but this parameter depends on the precision with which the graphite components of the reflector are machined and stacked. As noted above, a very large value of ±1 cm is assumed for most graphite dimensional uncertainties. If the top of the core cavity is raised by 1 cm, keff changes by -4.2x10-4. For lack of specific knowledge about how uncertainties in core cavity height might be distributed, it is assumed that the distribution is uniform. Then the standard deviation in core height uncertainties and the resulting change in keff are the limiting values divided by √3:a Δkcore cavity = -2.4x10-4. 2.1.3 Height of Conus An uncertainty in the height of the conus is represented by moving the datum plane at the top of the conus upwards by 2.7355 cm, an increment that fits conveniently with the PEBBED spatial mesh, and by moving the datum plane at the top of the core by an equal amount. Thus, the active core height does not change. The shift roughly amounts to one pebble radius, which corresponds to a miscount of about 375 pebbles, an unlikely amount. For this assumed uncertainty, the change in keff is found to be 1.05x10-3. Assuming a uniform distribution function for uncertainties in conus height, one divides this number by √3 to find (to the nearest significant figure) Δkconus = 6.1x10-4.

a International Handbook of Evaluated Criticality Safety Benchmark Experiments NEA/NSC/DOC(95)03/I-VIII, OECD-NEA, “ICSBEP Guide to the Expression of Uncertainties,” Revision 1, p. 29, September 30, 2004.

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2.1.4 Dimensions of Graphite Blocks As discussed in Section 2, tolerances of about 0.1 mm are assumed for milling the graphite blocks. Such tolerances appear reasonable in view of the necessity of coolant, control rod, and shutdown absorber ball channels to line up. Except in the outer periphery of the reflector, where boronated carbon bricks are used to reduce neutron leakage into the surroundings, the only differences among adjacent blocks are in their void fraction; uncertainties in void fraction are accounted for below where the sizes and locations of the different borings are varied. Uncertainties in the inner and outer reflector diameters and the inner and outer reflector axial dimensions are also treated in other sections. The importance of neutrons is low in the outer periphery. Therefore, it is not necessary to analyze the effects of uncertainties in the internal boundaries between reflector blocks. Figure 1.5 appears to show that the reflector is composed of blocks shaped in the form of annular sectors, between which the boundaries are radial planes. It has not been confirmed that this is correct, but if it is there is a potential for gaps to develop along these planes, allowing increased neutron leakage. It was assumed that a wedge-shaped gap could develop between two blocks, 1 cm wide at the reflector outer boundary. (This gap represents the total of all the gaps between blocks all the way around the reflector.) The MCNP model described in Section 3.1 was used to find that the effect of this gap is to decrease keff by 5.6x10-4. A cumulative gap width of zero is unlikely, and a cumulative gap width of 1 cm may be considered excessive because of neutron streaming concerns. If the mean is taken as 0.5 cm, and gap widths are assumed to be distributed uniformly in the interval from zero to 1 cm, and the difference between the nominal value of keff and the value corresponding to a gap is considered linear in the interval, then the standard deviation of the change in keff from the nominal value is 5.6x10-4 / (2√3) = 1.6x10-4. Δkeff/gaps = 1.6x10-4. 2.1.5 Outer Diameter of Graphite Reflector The outer diameter of the reflector could differ from the nominal value by an accumulation of off-nominal radial dimensions in graphite blocks at several radial locations or by off-nominal radial dimensions at a single radial location. Two cases of the latter possibility were explored. First, the outermost zone was increased 1 cm in radius; this led to an increase in keff of 1.8x10-4. Second, with the outermost zone again at its nominal thickness, the next-to-outermost zone was increased 1 cm in radius; this led to a decrease in keff of 1.4x10-4. The outermost zone consists of boronated carbon bricks, installed for the purpose of capturing neutrons that would otherwise escape into the surrounding pressure vessel. Even though boron has a high absorption cross section for thermal neutrons, increasing the thickness of the boronated carbon bricks provides more scattering sites from which neutrons can be scattered back towards the core. In other words, nothing is as good an absorber as a vacuum boundary condition, so replacing vacuum with anything increases keff. However, increasing the thickness of the region closer to the core than the boronated carbon layer (while keeping the thickness of the boronated layer constant) is found to decrease keff. It is surmised that this additional layer of low-boron graphite softens the neutron energy spectrum slightly, which increases the absorption of neutrons in the boronated carbon bricks and decreases the number of neutrons that can be reflected back into the core.

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The larger effect is taken as the uncertainty in keff resulting from differences in the reflector outer diameter from the nominal value. The change in keff is proportional to the change in the cross-sectional area of the outermost zone. If the uncertainty in the cross-sectional area is uniformly distributed, a reasonable assumption, then the uncertainty in keff associated with uncertainty in reflector outside diameter is 1.8x10-4/√3, or (to the nearest significant figure) Δkrefl O.D. = 1.0x10-4 . 2.1.6 Height of Graphite Reflector In this case, because of the large height of the reflector, a greater uncertainty was assumed in the height: ±1%, which is a dimensional uncertainty of ±6.1 cm. Increasing the axial thickness of the first axial region increases keff by 1.0x10-5. Increasing the axial thickness of the last region by 6.1 cm causes no change in keff. Whatever distribution function applies, the uncertainty at the standard deviation is smaller, but when it is rounded to the nearest significant figure the value returns to 1x10-5. Δkrefl ht = 1x10-5 . 2.1.7 Diameter of Cold Coolant Flow Channels The nominal diameter of the coolant flow channels is 8.0 cm; tolerance information is not given in Reference 1. A very large tolerance of ±0.5 cm is assumed; more realistic tolerances would be of the order of a fraction of a millimeter. The coolant flow channels occupy portions of Regions 58-65 in Figure 4.1.a. From Table 3.9, one can infer the void fraction in each region (Regions 22, etc., have no voids) and the part of the void space taken by the flow channels in each region. Then a reduction in the flow channel diameter can be used to find the new void fraction and the new nuclide number densities in each region. It is found that a simultaneous reduction of 0.5 cm in all the flow channel diameters produces an increase in keff of 2x10-5. If the uncertainty in cross-sectional area is assumed to be distributed uniformly, then the uncertainty in keff is 2x10-5/√3, or, to the nearest significant figure, ΔkCoolant flow dia = 1x10-5. 2.1.8 Radial Location of Cold Coolant Flow Channels The coolant flow channels occupy the entire radial span of Regions 58-65 of Figure 4.1.a; i.e., r = 140.6-148.6 cm. No tolerance information is given in Reference 1. Therefore, the radial location of the channels can be adjusted by moving the inner and outer boundaries of these regions by equal amounts. The flow channels must line up well in axially adjacent blocks in order for the coolant to flow smoothly, so the tolerances in radial location must be quite small. For the maximum plausible effect on keff, it is assumed that the tolerance is ±0.25 cm. When the model regions are shifted outwards, the cross-sectional area of the regions increases, but the cross sectional area of the flow channels does not increase. This requires a correction in the number densities of the materials in the regions. When these corrections are made, it is found that an outward shift of 0.25 cm in the radial location of the flow channels produces no change in keff. ΔkCoolant rad loc = 0.0.

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2.1.9 Height of Cold Coolant Flow Channels Reference 1 states that the cold coolant flow channels extend to the bottom of the reflector. Thus, their height cannot be changed by adjustments at the bottom. Changes in their height are accounted for here by extending them at the top. It was assumed for modeling convenience that the cold coolant flow channels were extended through the entire axial span of Region 57 of Figure 4.1.a, an increase of 10 cm. This is an implausibly large increase. Nevertheless, this adjustment caused no change at all in keff: ΔkCoolant chan ht = 0.0 2.1.10 Diameter of Control Rod and Irradiation Channels The control rod and irradiation channels are located at the same radial position and have the same diameter and axial extent. They occupy the entire radial span of Regions 27-31, 41, 42, and 82 of Figure 4.1.a, or r = 95.6-108.6 cm. No tolerance range is given in Reference 1. The diameter is assumed to be 0.5 cm smaller than nominal, which is an unrealistically large tolerance. The same approach is applied as to the coolant flow channels, discussed in Section 2.1.7 above. The change in keff is 6.1x10-4, and the uncertainty is 6.1x10-4/√3: ΔkC R dia = 3.5x10-4 . 2.1.11 Height of Control Rod and Irradiation Channels The control rod and irradiation channels extend from the top of the reflector through Region 42 of Figure 4.1.a. The region below Region 31 is Region 43, which is 15 cm in height. Because neutrons are more important near the core than in the coolant flow channel regions, it is excessively coarse to assume a perturbation in the height of the control rod and irradiation channels that extends the entire length of Region 43. It is assumed instead, still generously, that the perturbed channels extend 2 cm into Region 43. This perturbation does not change keff. ΔkC R ht. = 0.0 2.1.12 Radial Location of Control Rod and Irradiation Channels No tolerances are given in Reference 1 on the location of the control rod and irradiation channels. If the inner and outer boundaries of the annulus containing these channels are moved outwards by 0.25 cm, the change in keff is 1.5x10-4. Assuming a uniform uncertainty distribution in the radial location of the channels in a range of ±0.25 cm, and correcting the number densities for expanding the region without expanding the voids, one finds the uncertainty in keff to be ΔkC R loc = 9x10-5 . 2.1.13 Dimensions of KLAK Channels The HTR-10 has two independent shutdown systems: the normal control rods and a supply of small spherical absorbers, often called “KLAK” from the German acronym for small absorber balls (“kleine absorber Kugeln”), that can be dropped into seven channels located in the same radial regions of the model as the control rod and irradiation channels. The KLAK channels run the entire length of the model. In the core and conus axial range, the channel cross sections are ovals 6 cm wide and 16 cm long, with the long sides oriented in the circumferential direction of the reactor (see Figure 1.5). Above and below that axial range, the KLAK channels are cylindrical, with a diameter of 6 cm. No tolerance ranges are given in Reference 1.

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The channels occupy a 6 cm radial zone in model regions 13 cm in radial extent. The KLAK channels can move in a range of 7 cm radially in these regions without changing the region densities. Such large perturbations in the KLAK channel locations are implausible. However, the KLAK channels effectively shifted outwards with the control rod and irradiation channels when the zone containing all of them was moved in Section 2.1.12. The results of that calculation and the effects of perturbations in the size of the channels indicate that the effects of mislocations of the KLAK channels are probably very small. For the KLAK channels, an uncertainty of 10% (in standard deviation) was assumed in the cross-sectional area for all regions. This corresponds to about 5% in linear dimensions (√1.1 = 1.04881), or about 1.5 mm in radius in the cylindrical sections and 8 mm in the long direction in the oval section. The uncertainties in keff were computed from such increases in the sizes of the three sections of the KLAK channels. Upper cylindrical section: ΔkKLAK upper = 0.0 Oval section: ΔkKLAK middle = -2.8x10-4. Lower cylindrical section: ΔkKLAK lower = 0.0. 2.1.14 Dimensions of Hot Gas Duct The hot gas duct extends from r=90 cm to the outer edge of the reflector, in Regions 26, 44, 53, 62, 70, and 77 of Figure 4.1.a in an axial zone from z=465 cm to z=495 cm. The nominal diameter of the duct, 30 cm, spans this axial range. No tolerance ranges are given in Reference 1. Because the duct is large, a large perturbation of ±1 cm was assumed in the diameter. The same approach described above for the cold coolant flow channels was applied to the duct diameter. The changed duct diameter had no effect on keff. The effect of a perturbation in the length of the duct was examined by assuming the duct to extend through the radial extent of Region 15, which is adjacent to Region 26. Region 15 has a high void fraction in its nominal configuration, but its void fraction is reduced further by the introduction of an additional void volume corresponding to the hot air duct. This change extends the length of the duct by 19.25 cm, which is completely unrealistic. However, the perturbation caused no change in keff, so more realistic perturbations would also have no effect. A shift in the axial location of the duct could be accomplished by changing the locations of the datum planes at z=465 cm and z=495 cm, but this change would affect the nuclide number densities in other regions besides those containing the duct. An axial shift could also be represented by calculating perturbed nuclide number densities in regions adjacent to the nominal location of the duct, into which the duct would move in the perturbation, but this approach would alter the nuclide number densities throughout the adjacent regions, which would give distorted neutron flux distributions. Because of the lack of influence the other changes in duct dimensions have on keff, it is simply assumed that realistic shifts in hot gas duct axial position also have no effect. Δkhot gas duct = 0.0 2.1.15 Radius of Fuel Discharge Tube The fuel discharge tube occupies Regions 6, 7, and 81 in Figure 4.1.a. Its nominal radius is 25 cm; no tolerance range is given in Reference 1. The uncertainty in discharge tube radius is modeled by moving the radial boundary of Regions 6, 7, and 81 outwards by 0.25 cm. There is no effect on keff. Δkdischarge radius = 0.0.

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2.1.16 Height of Fuel Discharge Tube The fuel discharge tube extends to the bottom of the model. The effect of increasing the height of the model has already been examined. A perturbation in the fuel discharge tube height is redundant. Δkdischarge height = 0.0 2.1.17 Diameter of Fuel Pebble The HTR-10 fuel pebble is a sphere 6 cm in diameter, consisting of an inner fuel zone 5 cm in diameter and an outer unfueled layer. As stated in Reference 1, the inner fuel zone consists of a graphite matrix containing an average of 8335 TRISO coated fuel particles (this number can be verified from the nominal uranium loading and the properties of the fuel kernels). The outer layer contains only graphite. The core also contains dummy graphite pebbles, which are also 6 cm in diameter; the ratio of fuel pebbles to dummy pebbles is 57:43. The tolerance range on fuel pebble diameter is 5.98-6.0 cm; this is also the observed range. The uncertainty in keff resulting from uncertainty in fuel pebble diameter is evaluated by reducing the pebble diameter to 5.98 cm. There are several ways to do this, but the greatest changes in average nuclide densities in the core region are produced by keeping the diameter of the fuel zone constant and reducing the thickness of the outer graphite layer. For example, reducing the diameter of the fuel zone and keeping the outer graphite layer thickness constant would shrink the fuel zone volume almost in proportion to the shrinkage in total pebble volume; then the average nuclide densities in the core would be almost unaffected and keff would hardly change at all. The average densities are further changed by assuming that the dummy pebbles are also reduced to 5.98 cm in diameter. The changes in fuel zone and pebble dimensions are not assumed to change the core height; thus, more pebbles are contained in the core in the perturbed condition. (The effect of changing core height is assessed separately in Section 2.1.1.) The result of making those changes is to decrease keff by 2.12x10-3. As noted in Section 2.1, the uncertainty within the tolerance range presumably comprises a deterministic component and a random component. However, it is not well known how the uncertainty is apportioned between the deterministic and random components. The approach that will give the maximum effect on keff is to assume that the uncertainty is all deterministic. In this case, it is assumed that the change in keff is proportional to the volume change, rather than the diameter change. The volume of the nominal pebble sphere is (4π/3)63 cm3 = 904.779 cm3, and the volume of the smallest allowable sphere is (4π/3)5.983 cm3 = 895.761 cm3. If the nominal value is assumed to be the most likely, a triangular distribution is appropriate. The standard deviation in volume is found to be 2.12556 cm3, and the standard deviation in the change in keff from nominal is found to be ΔkFPdia = -5.0x10-4. The result obtained by assuming a uniform probability distribution for volumes in the same range is not significantly different from this result. It merits comment that keff is reduced by packing more fuel pebbles, which each contain the same amount of fuel as the nominal pebbles, into the same volume of core. Evidently the HTR-10 is undermoderated, so that the loss of moderation caused by reducing the amount of graphite in the core overshadows the increase

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in the macroscopic fission cross section obtained by increasing the amount of fuel. A study of the effect of changing the relative amounts of fuel and moderator in PBRs has found that the usual formula of 5 cm for the fuel zone diameter and 6 cm for the fuel pebble diameter does not provide optimal moderation, at least for some fuel particle packing fractions.a 2.1.18 Diameter of Kernel The nominal kernel diameter is 0.5 mm, and the measured standard deviation is 0.0102 mm. The 1σ kernel is 6.2457% larger in volume than the nominal kernel. However, there is a tolerance in pebble fuel loading of 5.0±0.1 g/pebble. This is only a variation of ±2%. Therefore, if all of the kernels were allowed to reach their 1σ diameter, the limit on fuel loading would be exceeded. Hence, the limit on fuel loading (see Section 2.1.23) bounds the effects of uncertainties in the fuel kernel diameter, and the fuel kernel diameter is not used in the uncertainty analysis. 2.1.19 Thickness of Buffer Layer The buffer layer is a relatively porous layer of carbon immediately surrounding the kernel. Its porosity is intended to provide room for fission product gases without excessive pressure buildup. In the HTR-10 fuel, the buffer layer thickness is nominally 0.09 mm, and the experimentally observed standard deviation is ±0.0044 mm. In the uncertainty analysis, it was assumed that all the buffer layers in the core were increased in thickness by one standard deviation from the nominal value, and that all the other layers in the fuel particles kept their original volumes. The new core average nuclide densities were computed, and keff was found to decline by -3.0x10-5. It is assumed that the uncertainty in buffer volume is uniformly distributed; then the uncertainty in keff is -3.0x10-5/√3, or, to the nearest significant figure, Δkbuffer thickness = -2x10-5 . The reason for the decrease in keff is that the expanded buffer layer displaces matrix material, since the diameter of the fuel zone is held constant. Then there is a decrease in carbon density in the core, which reduces moderation. 2.1.20 Thickness of IPyC Layer Outside the buffer layer, the TRISO particle consists of three hard and dense layers that provide a miniature containment vessel for fission products. The first of these is the inner pyrolytic carbon (IPyC) layer. Its nominal thickness is 0.04 mm, its tolerance range is 0.03-0.05 mm, and its observed range of variation is 0.0368-0.0424 mm. For maximum effect on keff, it was assumed that all of the IPyC layers in the core were expanded to their maximum allowable thickness, while all the other layers retained their nominal volumes. The density of the pyrolytic carbon layers is greater than that of the matrix material, so expansion of the layers increases the carbon density in the core and enhances moderation. Therefore, keff increases slightly; the change is 3x10-5. If the volume uncertainty is uniformly distributed, then to the nearest significant figure, ΔkIPyC thickness = 2x10-5. 2.1.21 Thickness of SiC Layer The second hard and dense layer outside of the buffer zone is the silicon carbide layer. This is the strongest

a Abderrafi M. Ougouag, Hans D. Gougar, William K. Terry, Ramatsemela Mphahlele, and Kostadin N. Ivanov, “Optimal Moderation in the Pebble-Bed Reactor for Enhanced Passive Safety and Improved Fuel Utilization,” proc. PHYSOR 2004, The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments, American Nuclear Society, Chicago, IL, April 25-29, 2004.

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layer in the fuel particle and the principal containment shell. It has a nominal thickness of 0.035 mm, a tolerance range of 0.031-0.039 mm, and an observed range of 0.0324-0.0376 mm. Standard deviations are not given. First, it was assumed that all of the SiC layers in the core were expanded to their maximum allowable thickness, while all the other layers retained their nominal volumes. This perturbation produced a decrease in keff of -3.4x10-4. In cases like the IPyC layer, where even perturbations up to the maximum tolerance value produce only very small changes in keff, it is inconsequential to use the maximum allowable perturbation to compute the uncertainty in keff. However, when such a perturbation produces a larger change in keff, it is more reasonable to assume the maximum observed perturbation. The computed change in keff and the fractional volume change were used to calculate

32.70499 10effdkx

dV−= − . (2.3)

Then an increase of the SiC thickness to its maximum observed value gives a change in keff of -2.2x10-4. If the change in SiC volume is assumed to be distributed uniformly, then ΔkSiC thickness = -1.3x10-4 . 2.1.22 Thickness of OPyC Layer The final hard and dense layer in the TRISO particle, and its outermost shell, is the outer pyrolytic carbon (OPyC) layer. This shell has a nominal thickness of 0.04 mm, a tolerance range of 0.03-0.05 mm, and an observed range of 0.039-0.045 mm. It was assumed that all the OPyC layers in the core were increased to their maximum allowable values while all the other layers retained their nominal volumes. This assumption led to an increase of 5x10-5 in keff. If the change in the OPyC volume is distributed uniformly, ΔkOPyC thickness = 3x10-5 . 2.1.23 Uranium Fuel Loading The nominal uranium loading in the HTR-10 fuel is 5 g per pebble. The tolerance range is 4.9-5.1 g. No observed range is given for the HTR-10 fuel, but experiments in the U.S.a with the same tolerance range found an observed range of 4.95-5.05 g. The same range is assumed for the HTR-10 pebbles. In the calculation, the fuel loading was increased by changing the uranium and oxygen number densities appropriately. The tolerance range was used to estimate the derivative

0.179eff

U

dkdm

= , (2.4)

where Udm is the change in the normalized uranium mass in the pebble.

a Mark F. Bryan, Evaluation of NP-MHTGR Performance Test Fuel Quality Control Data, Draft EGG-NPR-10130, February 1992

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The observed variation in Um is ±1%, so the associated variation in keff is 1.79x10-3. If the possible deviation of the uranium loading from the nominal value is distributed uniformly within the observed range, the uncertainty in keff is 1.79x10-3/√3, or ΔkU loading = 1.03x10-3 . 2.1.24 Density of Graphite Matrix in Fuel Pebble The density of the graphite matrix in the fuel pebble is nominally 1.73 g/cm3. The tolerance range is 1.73-1.77 g/cm3, and only the nominal value was observed. It is not plausible to assume that there is no uncertainty in the graphite density, but the observation of only one value suggests that a uniform distribution is not appropriate. For this case, a triangular distribution was assumed, with the mode (i.e., the most frequently observed value) equal to the nominal value and the distribution function declining to zero at the maximum tolerance value. As explained in Section 2.1, the variance in the triangular distribution is given bya

2 2 22

18a b c ab ac bcσ + + − − −= , (2.5)

where a and b are the limits of the interval on which the distribution function is defined, and c is the mode. This formula was applied to the deviation of the graphite density from 1.73 g/cm3, so that a=0, b=0.04, and c=0. Then σ2 = 8.88889x10-5 and σ = 0.009428. The uncertainty is found by computing keff at one standard deviation of the graphite density away from its nominal value. Thus, the value of the graphite matrix density at which keff is to be evaluated is 1.739428 g/cm3. The boron content of the graphite matrix is assumed to remain at 1.3 ppm. The fuel matrix graphite is only one component of the carbon in the core region. All the other carbon components are assumed to be unaffected. When the new graphite and boron densities are used in PEBBED, the associated uncertainty in keff is found to be Δkfuel graphite density = 1.13x10-3. 2.1.25 Total Ash in Fuel Element The fuel specifications permit small amounts of unspecified impurities. Without knowing what they are, it is impossible to assess their effects quantitatively. The tolerance limits given on the boron content of the fuel and reflector are actually limits on equivalent boron content, which includes the unspecified impurities. Because different impurities have different resonance absorption behavior, boron is not truly equivalent to other impurities. However, treating the impurities as boron equivalents is the best available option. The effects of uncertainties in the boron content are treated in other sections.

a M. Evans, Nicholas Hastings, and Brian Peacock, Statistical Distributions, 3rd Edition, John Wiley and Sons, New York, 2000, pp. 187-188.

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2.1.26 Lithium in Fuel Element The nominal fuel element contains no lithium, but a trace amount of up to 0.3 ppm is permitted, and concentrations of 0.007-0.023 ppm are observed. The maximum allowable amount of lithium was assumed, not only in the fuel elements but also in the dummy pebbles. A new cross section set was generated, and the revised multiplication constant was computed. The change in keff is 1.7x10-5. The deviation of the lithium concentration from zero to the maximum allowable value is assumed to be uniform. Then the uncertainty in keff is 1.7x10-5/2√3, or, to the nearest significant figure, ΔkLi = 1x10-5 . 2.1.27 Boron in Fuel Element The nominal boron concentration in the fuel graphite is 1.3 ppm, while the tolerance limit is 3.0 ppm. The only observed value reported in Reference 2 for the boron in the fuel element graphite matrix is 0.15 ppm. However, measurements performed by the INET found a standard deviation of ±10% in the fuel graphite boron concentration. A calculation was performed for the limiting concentration of 3.0 ppm in all components of the fuel element except the kernels, for which a value of 4 ppm was used as reported in Reference 1; this calculation gave a change in keff of -7.43x10-3 from the nominal value. This change leads to the derivative

35.68176 10eff

B

B

dkxdn

n

−= − .

When Δ nB /nB = 0.1, Δ keff = -5.7x10-4. Because this is an actual standard deviation, no further adjustment is needed. 2.1.28 Density of Reflector Graphite The nominal graphite density in the reflector material is 1.76 g/cm3; no information is given in Reference 1 on tolerances, observed ranges, or standard deviations. It is assumed that the total tolerance range is the same as for the graphite matrix in the fuel pebbles, i.e., 0.04 g/cm3, but that the nominal value is centered in the range. So the assumed tolerance range becomes 1.76±0.02 g/cm3. Furthermore, it is assumed that the deviation from the nominal value is distributed uniformly within this range. The boron content is assumed to vary in proportion to the carbon density. With these assumptions, when keff is evaluated at a density of 1.78 g/cm3, the change in keff becomes 1.09x10-3, and the uncertainty is found to be 1.09x10-3√3, or Δkreflector graphite = 6.3x10-4. 2.1.29 Density of Boron in Reflector Graphite Here, the boron is assumed to vary independently from the carbon density. The nominal boron concentration is 4.8366 ppm by weight; no information is given in Reference 1 on tolerances, observed ranges, or standard deviations. However, by measuring the diffusion length in more than 30 T of the reflector graphite, the INET found that the equivalent boron concentration in the reflector graphite has a standard deviation of ±2%. This value applies to the graphite in the reflector, and not to boronated carbon bricks or other structures in which the boron concentration is elevated above the natural level.

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The change in keff was calculated for a 10% increase in boron density from the nominal value. The result was -5.00x10-3. The derivative is thus

25 10eff

B

B

dkxdn

n

−= − .

For a 2% change, the standard deviation is Δkreflector boron = -1.00x10-3. 2.1.30 Ratio of Oxygen to Uranium in Kernel The nominal value is the stoichiometric ratio of 2.0; the tolerance limit is 2.01. It is assumed that the distribution is uniform from the nominal value to the limit. The multiplication factor is found to change by 1x10-5, so that the uncertainty is 1x10-5/2√3, or, to the nearest significant figure, ΔkO/U = 1x10-5. If the nominal ratio were considered to be the most probable, the triangular distribution function would apply, and the uncertainty would be even smaller. 2.1.31 Density of Kernel The nominal kernel density is 10.4 g/cm3, which is also the lower tolerance limit. The observed value is given in Reference 2 as 10.83 g/cm3. The observed value is considerably higher than the nominal value, so new cross sections were computed for this uncertainty calculation. Then keff was found for a kernel density of 10.83 g/cm3. The change in keff was found to be 5.48x10-3. This is a very large change in keff, but the difference in kernel density from the nominal value is also quite large, namely 4.13%. There is a tolerance range on fuel loading per pebble, as discussed in Section 2.1.24 above, of 4.9-5.1 g, or ±2%. Therefore, if all kernel densities were equal to the value reported in Reference 2, and if the number of kernels were unchanged, the pebble would exceed its tolerance on fuel loading. The fuel loading tolerance limit is bounding on the average kernel density, and the effects of variations in kernel density within the fuel loading tolerance limit are included in the uncertainty reported in Section 2.1.23. 2.1.32 Density of Buffer Layer The nominal value of the density of the buffer layer is 1.1 g/cm3, the tolerance range is 0-1.1 g/cm3, the observed value given in Reference 2 is 1.02 g/cm3, and the standard deviation is 0.03 g/cm3. The boron concentration is assumed to remain at 1.3 ppm. A deviation of 0.1 g/cm3 was used to calculate dkeff/dρbuffer ,but the uncertainty was evaluated at the standard deviation; to the nearest significant figure, this is Δkbuffer density = -3x10-5. 2.1.33 Density of IPyC Layer The nominal IPyC density is 1.9 g/cm3, the tolerance range is 1.8-2.0 g/cm3, and the observed range is 1.8-1.92 g/cm3. No standard deviation is given. The change in keff from the nominal value was computed from the tolerance limit of 2.0 g/cm3, with the boron concentration unchanged at 1.3 ppm. The change in keff was found to be 6x10-5; if the probability distribution is uniform, then to the nearest significant figure, ΔkIPyC density = 3x10-5.

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2.1.34 Density of SiC Layer The nominal SiC density is 3.18 g/cm3, the tolerance range is ≥3.18 g/cm3, the observed range is 3.19-3.23 g/cm3, and no standard deviation is given. For maximum plausible effect on keff, the change in keff was computed for the upper tolerance limit and found to be -2x10-5. If the distribution in the range from 3.18-3.23 g/cm3 is uniform, then to the nearest significant figure, ΔkSiC density = 1x10-5. 2.1.35 Density of OPyC Layer The nominal OPyC density is 1.9 g/cm3, the tolerance range is 1.8-2.0 g/cm3, and the observed range is 1.85-1.89 g/cm3. The derivative dkeff/dρOPyC was calculated at the tolerance limit of 2.0 g/cm3, but the distribution was assumed uniform between 1.85g/cm3 and 1.9 g/cm3 for simplicity; then the change in keff is -4.5x10-5 and the uncertainty is that value divided by √3; to the nearest significant figure, this is ΔkOPyC density = 3x10-5. 2.1.36 Composition of Coolant The coolant in actual reactor operation is helium, but in the initial criticality experiment the voids in the reactor were occupied by ambient air, which was specified as “moist” air at a total pressure of 0.1013 MPa and a temperature of 15 °C. The specified pressure of 0.1013 MPa is standard sea level atmospheric pressure, 29.92 in Hg. The term “moist” is taken to mean saturated, because the water vapor density given in Reference 1 does actually correspond to saturation at the temperature of 27 °C that was given in the pre-experiment benchmark specifications. Therefore, the baseline composition is saturated air at 15 °C and 0.1013 MPa. There are two potential causes of uncertainty in the specification of the coolant. First, the air was probably not fully saturated. Second, the air pressure was probably not standard sea level pressure. The second possibility is treated in the next section. A check of weather data in Beijing on the Internet shows a wide variability in the relative humidity, from low humidity (e.g., around 20%) to nearly saturated. The experiment was performed in December, when the humidity is relatively low. The lowest possible value of relative humidity is zero. The change in keff from saturated air to totally dry air was found to be -3x10-5. Assuming a uniform distribution between the extremes of humidity, one finds Δkwater vapor = -3x10-5/(2√3), or, to the nearest significant figure, Δkwater vapor = -1x10-5. 2.1.37 Air Pressure Standard sea level pressure is 29.92 in. Hg (0.1013 MPa). Sea level pressure typically varies about ±1 in. Hg. While larger variations are not uncommon, they represent more extreme weather than normal. Atmospheric pressure also varies with elevation above sea level; the elevation at Beijing is given on the Internet variously from 35 to 55 m. This is sufficiently close to sea level not to affect the standard atmospheric pressure significantly. A cursory search of the Internet failed to reveal a standard deviation for atmospheric pressure, so 1 in. Hg was taken as a generous estimate. With the components of dry air made

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denser by a factor of 30.92/29.92, the change in keff was found to be -3.0x10-4. This is a small change compared to some of the other uncertainties, and not worth trying to refine further. Δkair pressure = -3.0x10-4. 2.1.38 Boron in Kernels and Dummy Pebbles Reference 1 gives the boron concentration in the kernels as 4 ppm and the boron concentration in the dummy pebbles as 0.125 ppm. No information is given on tolerances, observed ranges, or standard deviations. No information was given on these boron concentrations in Reference 2. It is implied that these concentrations are confidently known values. In the specifications for the international benchmarking exercise that was performed before the initial criticality experiment (cf. Reference 1), the boron concentration in the dummy pebbles was given as 1.3 ppm, but Reference 1 states that the actual value was 0.125 ppm instead of the previously stated value of 1.3 ppm. This implies that the corrected value is known accurately. It seems unlikely that whatever tolerance range applies to this parameter is as large as those that apply to the reflector and fuel pebble matrix graphite compositions. It is customary in such circumstances to assume an uncertainty of one-half the last digit, i.e, ±0.5x0.001=±0.0005, or ±0.4%. This is too small to merit analysis. The argument to support confidence in the value of 4 ppm for the kernel boron concentration is less compelling, but in the absence of specific data on the variability of this parameter, the given value is taken as a measurement with the customary uncertainty of one-half the last digit: ±0.5x1=±0.5. This tolerance was applied to the kernel graphite, with a resulting change in keff of -7x10-5. Assuming a uniform uncertainty distribution, one finds Δkkernel boron = -4x10-5. 2.1.39 Packing Fraction The average packing fraction is assumed to be 0.61, as has been observed experimentally.a But other packing fractions are theoretically possible, and a numerical study has shown that shifts in packing fraction, as might happen in an earthquake, can induce substantial changes (of the order of 1%) in keff.b Evidently an increase in packing fraction causes a reduction in leakage even though the total mass of the core remains constant. The statistical variation in packing fraction in a bed of spheres is not known. It is unlikely that the packing fraction will exceed 0.64, which corresponds to the maximally random jammed state.c This is the condition where the pebbles are as closely packed as possible while remaining randomly arranged. Closer packing requires organized lattice structure. In a perfectly cylindrical vessel, the theoretical packing fraction is zero at the walls and fluctuates in space over several sphere diameters until it approaches its asymptotic value. In the HTR-10, the vessel walls have a pattern of indentations, like the surface of a golf ball but on a larger scale; this feature decreases the magnitude of the spatial fluctuations and produces a more uniform packing fraction.

a R. F. Benenati and C. B. Brosilow, “Void Fraction Distribution in Beds of Spheres,” A. I. Ch. E. Journal 8, No. 3, pp. 359-361 (1962). b A. M. Ougouag and W. K. Terry, “A Preliminary Study of the Effect of Shifts in Packing Fraction on k-effective in Pebble-Bed Reactors,” American Nuclear Society Mathematics & Computation Division Conference, Salt Lake City, Utah, September 9-13, 2001. c S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is Random Close Packing of Spheres Well Defined?” Phys. Rev. Lett. 84, p. 2064, 6 March 2000.

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The INL has been engaged in studies of pebble packing using its PEBBLES code,a a discrete-elements-method code. Such codes simultaneously solve the equations of motion of all the pebbles in the system. It has been found that the (average) packing fraction in a pebble bed depends on the coefficient of friction at the pebble contact surfaces and on the method by which the pebbles are introduced. Higher coefficients of friction correspond to lower packing fractions, and vice versa. Pebble beds assembled gently have lower packing fractions than pebble beds assembled more roughly (as by dropping pebbles from a meter or more above the bed surface). Strong agitation causes settling to higher packing fractions. Sample calculations by PEBBLES of realistic models of the HTR-10 (i.e., coefficients of friction in the range of 0.25-0.4 and introduction of pebbles by dropping from the top of the core cavity) produce packing fraction values close to 0.61. Values of packing fraction greater than 0.62 require unrealistically low values of the coefficient of friction (i.e., <0.1) or agitation more vigorous than one would see in a strong earthquake. Values of packing fraction less than 0.60 require unrealistically gentle assembly of the pebble bed.b In this analysis, it is assumed that the packing fraction varies uniformly in the range from 0.60-0.62. This assumption is believed to be bounding. When the inventory of core constituents remains constant, a change in packing fraction entails a change in core height. This change is uncorrelated with the change analyzed in Section 2.1.1, because there the change in core height was assumed to result from an inaccurate pebble count. The change in keff from an increase in packing fraction to 0.62 is found to be 3.3x10-3, when the core height changes as needed to keep the core material inventories constant. For a uniform distribution, the corresponding uncertainty in keff is Δkpacking fraction = 1.90x10-3. 2.1.40 Boron in Boronated Carbon Bricks If the boron content in the boronated carbon bricks is changed from the nominal value (the atomic number density associated with a 5% B4C component of a total density of 1.59 g/cm3) by 10%, keff changes by 3.3x10-4. The boron content of these bricks is controlled by a manufacturing process, unlike the natural boron content of most of the reflector elements. Thus, a tolerance of ±10% would be very lax, and the use of such tolerances will substantially overpredict the effect on keff. The uncertainty associated with this tolerance is 3.3x10-4/√3, or ΔkBCBricks = 1.9x10-4

. 2.1.41 Shape of Upper Surface When granular materials are dropped from a single point into a container, a mound builds up under the drop point. The shape of the mound can be determined by an analysis in which the equations of motion of all of the individual grains are solved simultaneously as the grains are dropped into place. A computational technique called the discrete element method (DEM) can perform such an analysis. The INL has developed the PEBBLES code based on the DEM. A discussion of the code and the PEBBLES analysis for the HTR-10 is contained in Appendix C. The analysis finds that the mound is a cone with a mean angle of 19.5° from the horizontal and limits of 17° and 22° from the horizontal.

a Abderrafi M. Ougouag and Joshua J. Cogliati, “Methods for Modeling the Packing of Fuel Elements In Pebble Bed Reactors,” Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, Palais des Papes, Avignon, France, September 12-15, 2005, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2005). b Joshua J. Cogliati, personal communication, June 7, 2006.

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The high-fidelity MCNP model was used to compute keff for the three values of the cone angle. The case with the 19.5° angle is the base case, or the high-fidelity experiment benchmark, with keff = 1.01190±0.00021. With the 22° angle, keff = 1.00844±0.00021, and with the 17° angle, keff = 1.01560±0.00022. The deviations in the two cases are -3.46x10-3 and +3.70x10-3, respectively. The difference in magnitude of these two cases is considered meaningless, and the larger deviation is assumed to apply to both sides of the mean. If the distribution of angles between the observed limits is assumed to be uniform, the uncertainty is 3.7x10-3/√3, or Δkcone angle = 2.14x10-3. This effect is surprisingly large. In fact, it is an order of magnitude higher than the effect of changes in the shape of the upper surface investigated independently by the INET. The reasons for the discrepancy have not been identified, so the larger effect is being used in this analysis. If this estimate is indeed an order of magnitude too high, the overall uncertainty presented in Section 2.1.44 will be reduced to 3.01x10-3, which is somewhat lower but still in the same general range. 2.1.42 Effect of Steel Pressure Vessel and “Core Barrel” Reference 1 does not specify the thickness of the stainless steel pressure vessel or the component labeled “core barrel” in Figure 1.1, because it was believed that any materials outside the boronated carbon bricks would have negligible effects on criticality. However, as general information, Reference 1 does state the pressure vessel material (SA516-70 steel), inside diameter (4.2 m), height (11.1 m), and weight (142 T). A rough estimate of the thickness is made by using the density of iron (7.86 g/cm3); it is found that the thickness is approximately 10 cm. Adding this component to the detailed MCNP model results in a difference in keff of -4.3x10-4 from the baseline case. This is a minor effect; thus, more accurate analysis is not warranted. Furthermore, it is close to the statistical uncertainty in the Monte Carlo calculations (±0.00021). If the uncertainty in the steel thickness is distributed uniformly from zero to 10 cm, the uncertainty in keff is -4.3x10-4/2√3, or Δkvessel thickness = -1.2x10-4. Because of the roughness of the estimate, this may also be taken to account for the “core barrel.” 2.1.43 Arrangement of Fuel Particles in Pebble The TRISO fuel particles in the fuel pebbles are actually distributed randomly. In most of the MCNP calculations, they were modeled as being arranged in a cubic lattice. An MCNP analysis was performed in which the fuel particles were distributed randomly. The difference in keff between the two calculations was 6x10-4. If this difference is taken as a measure of the variation in keff for differences in fuel particle packing arrangement, and if it is assumed that keff varies uniformly in this range as the fuel particle packing arrangement is varied, then the uncertainty in keff is 6x10-4/√3, or Δkfuel particle packing = 3.5x10-4. 2.1.44 Overall Uncertainty in keff When Equation 2.1 is applied to all the uncertainties calculated in Sections 2.1.1-2.1.42, assuming that all the uncertainties are uncorrelated (a reasonable and pragmatically necessary assumption), the result is uc = 3.69x10-3, or about 0.4%. This number was obtained by consistently making assumptions on the variability of parameters on which keff may depend that are expected to overpredict the variation in keff, often greatly. The parameters that could have the largest effect are the boron-equivalent density in the reflector, the graphite matrix density,

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the fuel loading, and the pebble diameter. For the purposes of the IRPhEP, an uncertainty of 1% is considered to be the maximum tolerable value. Therefore, the initial criticality measurement in HTR-10 is judged to be an acceptable benchmark. Table 2.1 summarizes the results of the uncertainty calculations. 2.2 Evaluation of Buckling and Extrapolation Length Data Buckling and extrapolation length measurements were not reported in Reference 1. 2.3 Evaluation of Spectral Characteristics Data Spectral characteristics measurements were not reported in Reference 1. 2.4 Evaluation of Reactivity Effects Data The control rod worth measurements have not yet been evaluated in the IRPhEP framework. The experimental data are available, as cited in Section 1.4. 2.5 Evaluation of Reactivity Coefficient Data Reactivity coefficient measurements were not reported in Reference 1. 2.6 Evaluation of Kinetics Measurements Data Kinetics measurements were not reported in Reference 1. 2.7 Evaluation of Reaction-Rate Distributions Reaction-rate measurements were not reported in Reference 1.

2.8 Evaluation of Power Distribution Data Power distribution measurements were not reported in Reference 1.

2.9 Evaluation of Isotopic Measurements

Isotopic measurements were not reported in Reference 1.

2.10 Evaluation of Other Miscellaneous Types of Measurements Other miscellaneous types of measurements were not reported in Reference 1.

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Table 2.1. Individual and Total Uncertainties (shaded entries are dominant).

Item Nominal & Bounding Values Uncertainty in keff (Δki)

(absolute value) Core radius 90 cm, +17 pebbles (see text) 1.9e-4 Core height 123.06 cm, +17 pebbles (see text) 3.7e-4 Height of core cavity 221.818 cm, 222.818 cm 2.4e-4 Height of conus 36.946 cm, 39.6815 cm 6.1e-4 Dimensions of graphite blocks No gaps, gap 1 cm wide at outside of reflector 1.6e-4 Outer diameter of graphite reflector 380 cm, 382 cm 1.0e-4 Height of graphite reflector 610 cm, 616.1 cm 1e-5 Diameter of cold coolant flow channels 8.0 cm, 8.5 cm 1e-5 Radial location of cold coolant flow channels 144.6 cm, 144.85 cm 0 Height of cold coolant flow channels 405 cm, 415 cm 0 Diameter of control rod and irradiation channels 13 cm, 12.5 cm 3.5e-4 Height of control rod and irradiation channels 450 cm, 452 cm 0 Radial location of control rod and irradiation channels 102.1 cm, 102.35 cm 9e-5 Diameter of KLAK channels (upper) 6 cm, 6.2929 cm 0 Dimensions of KLAK channels (middle) Area=88.2743 cm2, 97.1017 cm2 2.8e-4 Diameter of KLAK channels (lower) 6 cm, 6.2929 cm 0 Dimensions of hot gas duct D=30 cm, 31 cm; L=100 cm, 119.25 cm 0 Radius of fuel discharge tube 25 cm, 25.25 cm 0 Height of fuel discharge tube 610 cm, 616.1 cm 0 Diameter of fuel pebble 6.0 cm, 5.98 cm 5.0e-4 Diameter of kernel NA § 2.1.18 (bounded by fuel loading limit) NA § 2.1.18 Thickness of buffer layer 0.009 cm, .00944 cm 2e-5 Thickness of IPyC layer 0.004 cm, 0.005 cm 2e-5 Thickness of SiC layer 0.0035 cm, 0.00376 cm 1.3e-4 Thickness of OPyC layer 0.004 cm, 0.005 cm 3e-5 Uranium fuel loading 5 g/pebble, 5.05 g/pebble 1.03e-3 Density of graphite matrix in fuel pebble 1.73 g/cm3, 1.77 g/cm3 1.13e-3 Total ash in fuel element NA § 2.1.25 NA § 2.1.25 Lithium in fuel element 0, 0.3 ppm 1e-5 Boron in fuel element 1.3 ppm, 1.43 ppm 5.7e-4

Density of graphite matrix in reflector 1.76 g/cm3, 1.78 g/cm3 6.3e-4 Density of boron in reflector graphite 4.8366 ppm, 4.93333 ppm 1.00e-3 Ratio of O to U in kernel 2.0, 2.01 1e-5 Density of kernel NA § 2.1.31 (bounded by fuel loading limit) NA § 2.1.31 Density of buffer 1.1 g/cm3, 1.07 g/cm3 3e-5 Density of IPyC layer 1.9 g/cm3, 2.0 g/cm3 3e-5 Density of SiC layer 3.18 g/cm3, 3.23 g/cm3 1e-5 Density of OPyC layer 1.9 g/cm3, 2.0 g/cm3 3e-5 Composition of coolant (saturated vs. dry air) Saturated, dry 1e-5 Air pressure 0.1013 MPa, 0.104686 MPa 3e-4 Boron in kernels 4 ppm, 4.5 ppm 4e-5 Boron in dummy pebbles 0.125 ppm, 0.1255 ppm NA § 2.1.38 Boron in boronated carbon bricks 3.46349e-3 atoms/barn-cm, 3.80984e-3 atoms/barn-cm 1.9e-4 Pebble packing fraction 0.61, 0.62 1.9e-3 Angle of upper-surface cone from horizontal 19.5°, 17°, 22° 2.14e-3 Thickness of pressure vessel and “core barrel” 0, 10 cm 1.2e-4 Arrangement of fuel particles in pebble Regular lattice, random 3.5e-4 Total (root mean square) 3.69e-3 (NA = not applicable – see discussion in indicated subsections.)

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3.0 BENCHMARK SPECIFICATIONS 3.1 Benchmark-Model Specifications for Critical and / or Subcritical Measurements This section contains benchmark model specifications for the critical configuration described in Section 1.1. Two models are presented, a detailed model that preserves the geometric details in the core and the reflector as closely as possible, and a simplified model in which the upper surface of the core is approximated as a horizontal plane and the reflector is made azimuthally symmetric. 3.1.1 Description of the Benchmark Model Simplifications A model may be defined as the geometric and compositional description of a system from which the input for a computer code is derived. Ideally, the model is an exact representation of the actual system, and the input parameters can be derived from the model without approximations. However, this ideal is never realized for a complicated system like a nuclear reactor. Approximations are inevitably required to adapt the representation of the real system to the requirements of computer codes. The high-fidelity model is the closest approximation to the actual HTR-10 reactor that is practical based on available published information; this model is suitable for codes that use combinatorial geometry. The simplified model makes some concessions to the requirements of codes with less powerful geometric modeling capabilities. The only materials present in the benchmark models are graphite (with varying boron concentrations), fuel, and coolant. The reactor vessel and other structural components are omitted. Code calculations show that these components are all located in regions of very small neutron importance, so their omission is reasonable. (Perusal of the flux output file from the code PEBBED, which is discussed in Section 4, shows that the thermal neutron flux in the boronated carbon bricks is between four and eleven orders of magnitude lower than that in the core, as depicted in Figure 3.1; the other neutron energy groups show similar disparities.) The channels for coolant flow, control rods, and KLAK are merely borings in the reflector, so there are no structural materials associated with them. The graphite and boron number densities for the reflector regions were supplied in Reference 1 by the HTR-10 team. As detailed experimental data (e.g., component drawings) are not provided in Reference 1, and the data that are provided are based on detailed drawings not available to the public, the provided data are presumed superior to anything that could be inferred from sources such as Figure 1.1. The two benchmark models are described in the following sections. Further requirements of specific codes are met by further simplifications as described in Section 4. High-fidelity Model - The foundational model of the reactor is a geometric cell definition provided by the HTR-10 research group in Reference 1, based on the compositions of the various graphite structures in the reflector. This cell definition is illustrated in Section 3.1.2 below.

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Figure 3.1. Thermal Neutron Flux in HTR-10 Nominal Case.

(the origin is at the top center of the model) The modeling of the reflector regions is straightforward. It is in the core and conus regions that difficulties arise. It is well known that the double heterogeneity of the pebble-bed reactor must be taken into account for accurate analysis of PBRs. This accounting may be accomplished by the use of properly calculated Dancoff factors in deterministic codes,a by explicit modeling of individual fuel pebbles and TRISO fuel particles in Monte Carlo codes,b or by a technique called the sub-group method.c However, there are a huge number of fuel particles in a PBR, and a daunting number of pebbles. Furthermore, the fuel particles and pebbles are not organized in regular lattices, and it is important to model the random placement of the pebbles.b A model has been constructed which addresses these difficulties. It explicitly specifies all the pebbles and all the TRISO particles in the core. The TRISO particles can either be placed in a simple cubic lattice, or they can be randomly placed within the fuel pebble with each particle explicitly modeled. Unless special care is taken when specifying the lattice arrangement, the TRISO particles that intercept the outer surface of the fuel zone in the fuel pebble are typically represented as partial spheres, making it (practically) impossible to place the exact number of particles inside the fuel pebbles. When the position of each TRISO particle is randomly chosen, the exact number of particles is placed in each fuel pebble, the only restriction being that the positioning of the particles in every fuel pebble is identical. In preliminary calculations, it was found that the values of keff for both methods of positioning the TRISO particles, even when using a J. Kloosterman and A. M. Ougouag, “Computation of Dancoff Factors for Fuel Elements Incorporating Randomly Packed TRISO Particles,” INEEL/EXT-05-02593, Idaho National Engineering and Environmental Laboratory, January 2005. b Üner Çolak and Volkan Seker, Monte Carlo Criticality Calculations for a Pebble Bed Reactor with MCNP, Nucl. Sci. Eng. 149, 1-7 (2005). c T. D. Newton, “The Development of Modern Design and Reference Core Neutronics Methods for the PBMR,” ENC 2002 International Nuclear Conference, Lille, France, October 7-9 2002.

050

100150

200

0200

400600

8000

2

4

6

8

x 106

PEBBED Flux (n/cm^2/s) for Group 6

R(cm)Z(cm)

Peak Flux = 6.4100e+006

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partial spheres, are not significantly different (less than 6x10-4), which is consistent with previously reported results.a Therefore, in the high-fidelity model the cubic lattice is used for the TRISO particle positions. The pebbles are explicitly modeled using a technique that arranges each layer of pebbles in a hexagonal array, with each subsequent layer appropriately shifted to maximize the pebble packing fraction. The spacing of the hexagonal array and layers is characterized by hexagonal and axial pitches. The location of each pebble can then be randomly displaced to represent some degree of randomness in the packing of the pebbles. The maximum magnitude of the displacement can be chosen to range from zero (no randomness) up to larger values that result in intersecting pebbles. Intersecting pebbles are removed. Many different pebble configurations are possible if different pitches and maximum displacement magnitudes are chosen. If the total number of pebbles remaining exceeds 16,890, then the excess pebbles are randomly deleted. Typically, the pitches and maximum displacement magnitude are chosen to minimize the number randomly deleted. A successful configuration produces the nominal packing fraction of 0.61. The final step is to assign randomly which pebbles are fuel pebbles, with the remainder being dummy pebbles. The shape of the upper surface of the core is not known. However, the HTR-10 fuel is introduced by dropping from a single point above the center of the core. Spheres dropped in that manner pile up in a mound below the drop point, and it is possible to analyze the accumulation of spheres in the mound. The shape of the mound was determined by running the PEBBLES code,b which simultaneously solves the equations of motion for all the pebbles in the core. This analysis is discussed further in Appendix C. The process of dropping the pebbles into the core was followed until a stable shape was attained for the mound. The model was not axisymmetric, so a range of variation of the cone angle could be assessed by measuring the cone angle at various locations around the base of the cone. The angle was found to vary between 17° and 22° from the horizontal. The high-fidelity model is taken as the model with a cone angle of 19.5°. Sensitivity of keff to this angle is considered in Section 2. The principal departures of the high-fidelity model from reality are that the pebble packing in the model is still not truly random and that the desired packing fraction is still achieved by an artificial choice of pebble spacing. In fact, in an actual operating pebble-bed core, the arrangement of pebbles – i.e., the distribution of pebbles with different burnup histories and the local proportion and relative position of fuel and dummy pebbles – is constantly changing, so that it is not only impossible to represent it exactly in a static model but even meaningless to do so. In the HTR-10 initial core, the random pebble arrangement that actually existed was only one of an infinite number of possible arrangements, and it is not known which one it was. The chosen method of representing the random pebble packing is the best currently achievable since it exactly represents the true number of pebbles with the fuel pebbles quasi-randomly distributed. Simplified Model - As noted above, the principal differences between the high-fidelity model and the simplified model are that the simplified model represents the upper surface of the core as a horizontal plane and the radial zones containing borings are azimuthally homogenized. The total number of pebbles in the core remains the same in both models.

a Üner Çolak and Volkan Seker, Monte Carlo Criticality Calculations for a Pebble Bed Reactor with MCNP, Nucl. Sci. Eng. 149, 1-7 (2005). b Abderrafi M. Ougouag, Joshua J. Cogliati, and Jan-Leen Kloosterman, “Methods for Modeling the Packing of Fuel Elements in Pebble Bed Reactors,” proc. Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, American Nuclear Society, Avignon, France, September 12-15, 2005.

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3.1.2 Dimensions In both the high-fidelity and simplified models, the pebbles and the fuel particles (and all of the layers thereof) are represented explicitly. Both models contain 16,890 pebbles (9627 fuel and 7263 dummy). Each fuel pebble contains 8335 coated fuel particles. The dimensions relevant to the fuel and dummy pebbles are given in Table 3.1 and shown in Figure 3.2.

Table 3.1. Dimensions of Pebbles and Fuel Particles.

Diameter of fuel and dummy (no-fuel) pebbles 6.0 cm Diameter of fuel zone in fuel pebble 5.0 cm Volumetric filling fraction of pebbles in the core 0.61 Radius of the kernel of fuel particle (cm) 0.025 Fuel particle coating layer materials (starting from kernel) Buffer/PyC/SiC/PyC Fuel particle coating layer thicknesses(cm) 0.009/0.004/0.0035/0.004

Dimensions of High-fidelity Model - The high-fidelity and simplified models differ in their representation of the borings and ducts in the reflector and in their representation of the upper surface of the core. These are represented explicitly in the high-fidelity model. The dimensions of the borings and ducts are presented in Table 3.2. The geometric cell definition provided by the HTR-10 research group in Reference 1 is shown in Figure 3.3. The borings are shown explicitly in Figure 3.4. The numerical labeling of a few of the regions in Figure 3.3 is different from that in Reference 1; however, the correspondence between zones and compositions is preserved.

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Figure 3.2.a. Fuel and Dummy Pebbles.

Figure 3.2.b. TRISO Fuel Particle. The conical upper surface of the pebble bed is shown in Figure 3.3. The cone extends from its peak on the core axis to the radial boundary at the inner surface of the core. It makes an angle of 19.5° above the horizontal. It is idealized as a perfect cone, although in reality its surface is bumpy, with extra or missing pebbles in places. It is not reasonable to try to model these, as the buildup of the cone is a stochastic process and will never progress in the same way twice, and the exact arrangement of the pebbles in the HTR-10 initial core is not known. In the model, there are exactly 16,890 pebbles in the core (9627 fuel and 7263 dummy), as in the actual reactor.

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Figure 3.3. Zones of HTR-10 for High-fidelity Model. (dimensions are in cm)

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Figure 3.4. Cross Section of Reactor in Core Region. (used in high-fidelity model)

Table 3.2. Dimensions of Borings and Ducts in HTR-10. (see Figure 3.3 for axis definitions)

Region Diameter

(cm) Coordinate of Center

(cm) Channel Length Range

(cm) Twenty coolant flow channels(a)

8.0 144.6 radius 105.0≤z≤610.0

Thirteen control rod/irradiation channels(a)

13.0 102.1 radius 0≤z≤-450.0

Seven KLAK channels, sections with circular cross section(a)

6.0 98.6 radius 0≤z<-130.0 and 388.764<z≤-610.0

Seven KLAK channels, sections with oval cross section(a)

See Figure 3.4.

98.6 radius (the long axis of the cross section is

circumferential)

130.0≤z≤-388.764

Hot gas duct 30.0 z=480.0 90.0≤r≤190.0 Fuel discharge tube 50.0 r=0 388.764≤z≤610.0 (a) The twenty coolant flow channels are equally spaced azimuthally, as are the twenty channels for

control rods, irradiation experiments, and KLAK. The inner borings align radially with the outer borings, as shown in Figure 3.4.

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Dimensions of Simplified Model - The geometry of the simplified model is illustrated in Figures 3.5 and 3.6. 3.1.3 Material Data Compositions of High-fidelity Model Zones - The properties of the constituent materials are given in Tables 3.3 and 3.4. The atomic number densities provided in Reference 1 are given in Table 3.5 for the geometric cells defined in Figure 3.5. The calculated number densities of the reflector and core constituents are given in Tables 3.6 and 3.7, respectively. (The values used for Avogadro’s number and atomic weights are given in Appendix D.) It is worth noting that the graphite densities in Table 3.4 can be back-calculated from the atomic number densities in Table 3.5 for simple regions where voids are either absent or well defined. All voids are filled with saturated air at 15 °C. The number densities of the constituents of the air at the experiment conditions are given in Table 3.8.

Table 3.3. Nominal Material Properties of Fuel and Dummy Pebbles as Built (from Reference 1).

Density of graphite in matrix and fuel-pebble outer shell 1.73 g/cm3 Uranium mass per pebble 5.0 g 235U enrichment 17 wt.% Boron content in uranium 4 ppm Boron content in graphite (assumed in particle coatings) 1.3 ppm Volumetric filling fraction of pebbles in the core 0.61 Kernel UO2 density (g/cm3) 10.4 Particle coating layer materials (starting from kernel) Buffer/PyC/SiC/PyC Particle coating layer density(g/cm3) 1.1/1.9/3.18/1.9 Density of graphite in dummy pebbles 1.84 g/cm3 Boron content in dummy-pebble graphite 0.125 ppm

Table 3.4. Nominal Properties of Reflector Materials (from Reference 1).

Density of reflector graphite 1.76 g/cm3 Boron in reflector graphite 4.8366 ppm Density of boronated carbon bricks including B4C 1.59g/cm3 Weight percent of B4C in boronated carbon bricks 5%

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Figure 3.5. Zones of HTR-10 for Simplified Model. (dimensions are in cm)

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Figure 3.6. Cross Section of Reactor in Core Region for Simplified Model. The boron densities in Table 3.5 are not consistent with the given boron weight fraction of 4.8366 ppm in the reflector if the standard atomic weight of natural boron (10.811 g/molea) is used for converting weight fraction to atom fraction; a value of 4.8326 ppm is found instead. It can be shown that the stated boron weight fraction is consistent with a boron atomic weight of 10.820 g/mole, which corresponds to an isotopic composition of 19.0% 10B and 81.0% 11B, rather than the standard values of 19.9% 10B and 80.1% 11B. Natural boron varies considerably in its isotopic distributiona (from 19.1% to 20.3% 10B), but 19.0% is out of the observed range. References 1 and 2 make no mention of the isotopic composition. Also, the given boron concentrations are actually boron equivalent. Finally, the difference between 4.8326 ppm and 4.8366 ppm is only 0.004 ppm, or 0.083%. Based on the sensitivity of keff to boron concentration in the reflector found in Section 2.1.29, the effect on keff of using one of these values or another is only about 4x10-5, which is negligible compared to the effects of the assumed uncertainty in the boron concentration. In this evaluation, the standard isotopic composition of boron is assumed. A few of the reflector components (regions 10-14, 16, 24, 28, 29, 51, 68, and 82) have boron densities two to three orders of magnitude greater than 4.83 ppm, but still only in the range of tenths or hundredths of a percent. No comment is made in Reference 1 about the graphite in these regions. The extra space taken by the boron in these regions is estimated by assuming that the boron atoms occupy additional volume in the material. This assumption reduces the void fraction estimate by 0.35% or less. Also, some of the compositions in Table 3.5 indicate carbon with no boron (regions 18, 47, 56, 65, 73, and 81).

a Chart of the Nuclides, Thirteenth Edition, General Electric Company, 1984,

NEA/NSC/DOC(2006)1




Table 3.5.a. Compositions of Zones in Reflector (high-fidelity model) (Reference 1, with revisions(a)).

(a) The numbers of regions 48 and 66 are reversed from those of Reference 1. Also, the graphite and boron concentrations in the dummy balls have been corrected to the as-run compositions.

No. of Zone

Carbon Density (atoms/barn-cm)

Natural Boron Density (atoms/barn-cm )

Description

83 0.851047E-01 0.456926E-06 Bottom reflector with hot coolant flow borings 1 0.729410E-01 0.329811E-02 Boronated carbon bricks 2 0.851462E-01 0.457148E-06 Top graphite reflector 3 0.145350E-01 0.780384E-07 Cold coolant chamber 4 0.802916E-01 0.431084E-06 Top reflector 5 Top core cavity 7 0.572501E-01 0.277884E-08 Dummy pebbles, simplified as graphite of lower

density 8 0.781408E-01 0.419537E-06 Bottom reflector structures 9 0.823751E-01 0.442271E-06 Bottom reflector structures

10 0.843647E-01 0.298504E-03 Bottom reflector structures 11 0.817101E-01 0.156416E-03 Bottom reflector structures 12 0.850790E-01 0.209092E-03 Bottom reflector structures 13 0.819167E-01 0.358529E-04 Bottom reflector structures 14 0.541118E-01 0.577456E-04 Bottom reflector structures 15 0.332110E-01 0.178309E-06 Bottom reflector structures 16 0.881811E-01 0.358866E-04 Bottom reflector structures

17,27,46,55,64,72,74,75,76,77,78,79

0.765984E-01 0.346349E-02 Boronated carbon bricks

18,47,56,65,73

0.797184E-01 0.000000E+00 Carbon bricks

19 0.761157E-01 0.344166E-02 Boronated carbon bricks 20 0.878374E-01 0.471597E-06 Graphite reflector structure 21 0.579696E-01 0.311238E-06 Graphite reflector structure

22,23,25,26,28,30,31,41,43,44,45,48,49,50,52,53,54,58,59,61,62,63,67,69,70,71,80,

82

0.882418E-01 0.473769E-06 Graphite reflector structure

24,51,68 0.879541E-01 0.168369E-03 Graphite reflector structure 29 0.524843E-01 1.819690E-05 Graphite reflector structure 42 0.879637E-01 0.162903E-03 Graphite reflector structure 66 0.582699E-01 0.312850E-06 Graphite reflector structure 57 0.728262E-01 0.391003E-06 Graphite reflector structure 60 0.879538E-01 0.168369E-03 Graphite reflector, cold coolant flow region 81 0.847872E-01 0.000000E+00 Dummy pebbles, but artificially taken as carbon

bricks

NEA/NSC/DOC(2006)1




Table 3.5.b. Compositions of Zones in Reflector (high-fidelity model) – Coolant Components.

No. of Zone (a) Coolant Volume

Fraction (b) N O (air) Ar H (water) O (water)

83 3.5551E-02 1.38993E-06 3.72877E-07 8.31317E-09 3.05038E-08 1.52519E-081 4.7748E-02 1.86679E-06 5.00805E-07 1.11653E-08 4.09691E-08 2.04846E-082 3.5081E-02 1.37155E-06 3.67947E-07 8.20327E-09 3.01005E-08 1.50502E-083 8.3528E-01 3.26568E-05 8.76083E-06 1.95320E-07 7.16694E-07 3.58347E-074 9.0096E-02 3.52247E-06 9.44972E-07 2.10679E-08 7.73049E-08 3.86524E-085 1.0000E+00 3.90968E-05 1.04885E-05 2.33838E-07 8.58028E-07 4.29014E-077 3.9000E-01 1.52478E-05 4.09052E-06 9.11968E-08 3.34631E-07 1.67315E-078 1.1447E-01 4.47541E-06 1.20062E-06 2.67674E-08 9.82185E-08 4.91092E-089 6.6484E-02 2.59931E-06 6.97317E-07 1.55465E-08 5.70451E-08 2.85226E-08

10* 4.0560E-02 1.58577E-06 4.25414E-07 9.48447E-09 3.48016E-08 1.74008E-0811* 7.2253E-02 2.82486E-06 7.57826E-07 1.68955E-08 6.19951E-08 3.09975E-0812* 3.3478E-02 1.30888E-06 3.51134E-07 7.82843E-09 2.87251E-08 1.43625E-0813* 7.1278E-02 2.78674E-06 7.47599E-07 1.66675E-08 6.11585E-08 3.05793E-0814* 3.8613E-01 1.50964E-05 4.04992E-06 9.02919E-08 3.31310E-07 1.65655E-0715 6.2364E-01 2.43823E-05 6.54105E-06 1.45831E-07 5.35101E-07 2.67550E-0716* 2.8700E-04 1.12208E-08 3.01020E-09 6.71115E-11 2.46254E-10 1.23127E-10

17,27,46,55,64, 72,74,75,76,77, 78,79 0 0 0 0 0 0

18,47,56,65,73 0 0 0 0 0 0

19 6.3017E-03 2.4638E-07 6.6095E-08 1.4736E-09 5.40704E-09 2.70352E-0920 4.5829E-03 1.7918E-07 4.8068E-08 1.0717E-09 3.93226E-09 1.96613E-0921 3.4306E-01 1.3413E-05 3.5982E-06 8.0220E-08 2.94355E-07 1.47178E-07

22,23,25,26,28*,30,31,41,43,44,45,48,49,50, 52,53,54,58,59,61,62, 63,67,69,70,71,80,82*

0 0 0 0 0 0

24*,51*,68* 1.3580E-03 5.3093E-08 1.4243E-08 3.1755E-10 1.16520E-09 5.82601E-1029* 2.3064E-01 9.0173E-06 2.4191E-06 5.3932E-08 1.97896E-07 9.89478E-0842 4.0522E-01 1.5843E-05 4.2501E-06 9.4756E-08 3.47690E-07 1.73845E-0766 3.1516E-03 1.2322E-07 3.3056E-08 7.3696E-10 2.70416E-09 1.35208E-0957 3.3966E-01 1.3280E-05 3.5625E-06 7.9425E-08 2.91438E-07 1.45719E-0760 1.7470E-01 6.8302E-06 1.8323E-06 4.0851E-08 1.49897E-07 7.49487E-0881 3.9000E-01 1.5248E-05 4.0905E-06 9.1197E-08 3.34631E-07 1.67315E-07

(a) In addition, there are explicitly modeled borings containing coolant in zones in bold font. (Compare positions of borings given in Table 3.2 to zones shown in Figure 3.2.)

(b) Coolant volume fraction is calculated using the formula 1 – NC/Ng, where NC is the carbon atom density of the particular zone and Ng is the carbon atom density of reflector graphite or of the boronated carbon bricks as given in Table 3.6, depending on the zone description. However, where boron is more than a trace of a region not designated as a boronated carbon brick (these regions are denoted by *), the formula is modified to 1 – (NC+NB)/Ng, where NB is the boron density (see text just prior to Table 3.5.a).

NEA/NSC/DOC(2006)1




Table 3.6. Number Densities in Solid Reflector Components.

Nuclide Atomic Number Density (atoms/barn-cm)

Graphite at 1.76 g/cm3 8.82418E-2 10B at 4.8366 ppm total boron concentration 9.42800E-8 11B at 4.8366 ppm total boron concentration 3.79489E-7 Carbon in boronated carbon bricks 7.65984E-2 10B in boronated carbon bricks 6.89235E-4 11B in boronated carbon bricks 2.77426E-3

Table 3.7. Number Densities in Core Components.


235U in kernel 3.99198E-3 238U in kernel 1.92441E-2 16O in kernel 4.64720E-2 10B in kernel 4.06384E-7 11B in kernel 1.63575E-6 Carbon in buffer 5.51511E-2 10B in buffer 1.58513E-8 11B in buffer 6.38035E-8 Carbon in IPyC and OPyC 9.52610E-2 10B in IPyC and OPyC 2.73795E-8 11B in IPyC and OPyC 1.10206E-7 Carbon in SiC 4.77597E-2 Si (natural) in SiC 4.77597E-2 Carbon in fuel matrix 8.67377E-2 10B in fuel matrix 2.49298E-8 11B in fuel matrix 1.00345E-7 Carbon in dummy pebbles 9.22528E-2 10B in dummy pebbles 2.54951E-9 11B in dummy pebbles 1.02621E-8

NEA/NSC/DOC(2006)1




Table 3.8. Number Densities of Air Constituents in Experiment Conditions (15°C and 0.1013 MPa).

Nuclide Volume Percent in Dry Air(a)(b)

Atomic Number Density (atoms/barn-cm)

Nitrogen in dry air 78.084 3.95901E-5 Oxygen in dry air 20.9476 1.06209E-5 Argon in dry air 0.934 2.36778E-7 Hydrogen in vapor at saturation 8.58028E-7 Oxygen in vapor at saturation 4.29014E-7 (a) http://www.physlink.com/Reference/AirComposition.cfm, citing the CRC Handbook of Chemistry

and Physics, David R. Lide, Editor-in-Chief, 1997 Edition. (b) The remaining constituents of dry air are ignored. Compositions of Simplified Model - Since the simplified reflector model is two-dimensional, it is axisymmetric, as shown in Figure 3.6. The average number densities in the regions with borings are then reduced by homogenizing the solid material and the voids. Table 3.9 gives the new atomic number densities, as provided in Reference 1, in all the reflector zones, including the homogenized ones. Only the total boron densities were specified in Reference 1; these were converted to 10B and 11B densities by multiplying by the natural abundance (0.199 for 10B and 0.801 for 11B; cf. the Chart of the Nuclides.a) The atomic number densities of the gaseous constituents (see Table 3.8) were calculated from the stated temperature and total pressure, 15 °C and 0.1013 MPa, and the assumption that the air is saturated. The compositions listed in Tables 3.3, 3.4, 3.6, 3.7, and 3.8 also apply to the simplified model. Table 3.3 presents the compositions of the pebble constituents. Table 3.4 specifies the nominal compositions of the reflector graphite and boronated carbon bricks. Table 3.6 presents the calculated atomic number densities in the solid reflector components. Table 3.7 presents similar results for the core constituents. Table 3.8 presents number densities for the air.

a Chart of the Nuclides, Thirteenth Edition, General Electric Company, 1984

NEA/NSC/DOC(2006)1




Table 3.9.a. Compositions of Zones in Reflector – Simplified Model (Reference 1, w. revisions(a)).

(a) The number densities in the dummy pebbles have been corrected to the as-built values.

No. of Zone

Carbon Density ( atoms/barn-cm )

Natural Boron Density (atoms/barn-cm )

Description

83-90 0.851047E-01 0.456926E-06 Bottom reflector with hot coolant flow borings 1 0.729410E-01 0.329811E-02 Boronated carbon bricks 2 0.851462E-01 0.457148E-06 Top graphite reflector 3 0.145350E-01 0.780384E-07 Cold coolant chamber 4 0.802916E-01 0.431084E-06 Top reflector 5 Top core cavity

6,7,91-98 0.572501E-01 0.277884E-08 Dummy pebbles, simplified as graphite of lower density 8 0.781408E-01 0.419537E-06 Bottom reflector structures 9 0.823751E-01 0.442271E-06 Bottom reflector structures

10 0.843647E-01 0.298504E-03 Bottom reflector structures 11 0.817101E-01 0.156416E-03 Bottom reflector structures 12 0.850790E-01 0.209092E-03 Bottom reflector structures 13 0.819167E-01 0.358529E-04 Bottom reflector structures 14 0.541118E-01 0.577456E-04 Bottom reflector structures 15 0.332110E-01 0.178309E-06 Bottom reflector structures 16 0.881811E-01 0.358866E-04 Bottom reflector structures

17,55,72,74,75,76,

78,79

0.765984E-01 0.346349E-02 Boronated carbon bricks

18,56,73 0.797184E-01 0.000000E+00 Carbon bricks 19 0.761157E-01 0.344166E-02 Boronated carbon bricks 20 0.878374E-01 0.471597E-06 Graphite reflector structure 21 0.579696E-01 0.311238E-06 Graphite reflector structure

22,23,25,48, 49,

50,52,54, 67,69,71,

80


24*,51*,68*


26 0.846754E-01 0.454621E-06 Graphite reflector structure 27 0.589319E-01 0.266468E-02 Boronated carbon bricks

28*,82* 0.678899E-01 1.400000E-05 Graphite reflector structure 29 0.403794E-01 1.400000E-05 Graphite reflector structure

30,41 0.678899E-01 0.364500E-06 Graphite reflector structure 31 0.634459E-01 0.340640E-06 Graphite reflector, control rod borings region 42 0.676758E-01 0.125331E-03 Graphite reflector structure

43,45 0.861476E-01 0.462525E-06 Graphite reflector structure 44 0.829066E-01 0.445124E-06 Graphite reflector structure 46 0.747805E-01 0.338129E-02 Boronated carbon bricks 47 0.778265E-01 0.000000E+00 Carbon bricks 66 0.582699E-01 0.312850E-06 Graphite reflector structure 53 0.855860E-01 0.459510E-06 Graphite reflector structure 57 0.728262E-01 0.391003E-06 Graphite reflector structure

58,59,61,63

0.760368E-01 0.408240E-06 Graphite reflector, cold coolant flow region

60 0.757889E-01 0.145082E-03 Graphite reflector, cold coolant flow region 62 0.737484E-01 0.395954E-06 Graphite reflector, cold coolant flow region 64 0.660039E-01 0.298444E-02 Boronated carbon bricks 65 0.686924E-01 0.000000E+00 Carbon bricks 70 0.861500E-01 0.462538E-06 Graphite reflector structure 77 0.749927E-01 0.339088E-02 Boronated carbon bricks 81 0.847872E-01 0.000000E+00 Dummy pebbles, but artificially taken as carbon bricks

NEA/NSC/DOC(2006)1




Table 3.9.b. Compositions of Zones in Reflector (simplified model) – Coolant Components.

No. of Zone (a) Coolant fraction (b) N O (air) Ar H (water) O (water)

83-90 3.5551E-02 1.38993E-06 3.72877E-07 8.31317E-09 3.05038E-08 1.52519E-08

1 4.7748E-02 1.86679E-06 5.00805E-07 1.11653E-08 4.09691E-08 2.04846E-08

2 3.5081E-02 1.37155E-06 3.67947E-07 8.20327E-09 3.01005E-08 1.50502E-08

3 8.3528E-01 3.26568E-05 8.76083E-06 1.95320E-07 7.16694E-07 3.58347E-07

4 9.0096E-02 3.52247E-06 9.44972E-07 2.10679E-08 7.73049E-08 3.86524E-08

5 1.0000E+00 3.90968E-05 1.04885E-05 2.33838E-07 8.58028E-07 4.29014E-07

6,7,91-97 3.9000E-01 1.52478E-05 4.09052E-06 9.11968E-08 3.34631E-07 1.67315E-07

8 1.1447E-01 4.47541E-06 1.20062E-06 2.67674E-08 9.82185E-08 4.91092E-08

9 6.6484E-02 2.59931E-06 6.97317E-07 1.55465E-08 5.70451E-08 2.85226E-08

10* 4.0560E-02 1.58577E-06 4.25414E-07 9.48447E-09 3.48016E-08 1.74008E-08

11* 7.2253E-02 2.82486E-06 7.57826E-07 1.68955E-08 6.19951E-08 3.09975E-08

12* 3.3478E-02 1.30888E-06 3.51134E-07 7.82843E-09 2.87251E-08 1.43625E-08

13* 7.1278E-02 2.78674E-06 7.47599E-07 1.66675E-08 6.11585E-08 3.05793E-08

14* 3.8613E-01 1.50964E-05 4.04992E-06 9.02919E-08 3.31310E-07 1.65655E-07

15 6.2364E-01 2.43823E-05 6.54105E-06 1.45831E-07 5.35101E-07 2.67550E-07

16* 2.8700E-04 1.12208E-08 3.01020E-09 6.71115E-11 2.46254E-10 1.23127E-10

17,55,72,74,75, 76,78,79 0 0 0 0 0 0

18,56,73 0 0 0 0 0 0

19 6.3017E-03 2.4638E-07 6.6095E-08 1.4736E-09 5.40704E-09 2.70352E-09

20 4.5829E-03 1.7918E-07 4.8068E-08 1.0717E-09 3.93226E-09 1.96613E-09

21 3.4306E-01 1.3413E-05 3.5982E-06 8.0220E-08 2.94355E-07 1.47178E-07 22,23,25,48,49, 50,52,54,67,69,

71,80 0 0 0 0 0 0

24,51,68* 1.3580E-03 5.3093E-08 1.4243E-08 3.1755E-10 1.16520E-09 5.82601E-10

26 4.0416E-02 1.5801E-06 4.2390E-07 9.4508E-09 3.46781E-08 1.73390E-08

27 2.3064E-01 9.0173E-06 2.4191E-06 5.3932E-08 1.97896E-07 9.89478E-08

28,82* 2.3048E-01 9.0110E-06 2.4174E-06 5.3895E-08 1.97758E-07 9.88791E-08

29* 2.3064E-01 9.0173E-06 2.4191E-06 5.3932E-08 1.97896E-07 9.89478E-08

30,41 2.1920E+00 8.5700E-05 2.2991E-05 5.1257E-07 1.88080E-06 9.40399E-07

31 2.8100E-01 1.0986E-05 2.9473E-06 6.5708E-08 2.41106E-07 1.20553E-07

42 4.0522E-01 1.5843E-05 4.2501E-06 9.4756E-08 3.47690E-07 1.73845E-07

43,45 2.3734E-02 9.2792E-07 2.4893E-07 5.5499E-09 2.03644E-08 1.01822E-08

44 6.0461E-02 2.3638E-06 6.3415E-07 1.4138E-08 5.18772E-08 2.59386E-08

46 2.3733E-02 9.2788E-07 2.4892E-07 5.5497E-09 2.03636E-08 1.01818E-08

47 2.3732E-02 9.27845E-07 2.48913E-07 5.54944E-09 2.03627E-08 1.01814E-08

66 3.1516E-03 1.2322E-07 3.3056E-08 7.3696E-10 2.70416E-09 1.35208E-09

53 3.0097E-02 1.1767E-06 3.1567E-07 7.0378E-09 2.58241E-08 1.29120E-08

57 3.3966E-01 1.3280E-05 3.5625E-06 7.9425E-08 2.91438E-07 1.45719E-07

58,59,61,63 1.3831E-01 5.4075E-06 1.4507E-06 3.2342E-08 1.18674E-07 5.93369E-08

60 1.7470E-01 6.8302E-06 1.8323E-06 4.0851E-08 1.49897E-07 7.49487E-08

62 1.6425E-01 6.4216E-06 1.7227E-06 3.8408E-08 1.40931E-07 7.04655E-08

64 1.3831E-01 5.4075E-06 1.4507E-06 3.2342E-08 1.18674E-07 5.93369E-08

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Table 3.9.b (cont’d.). Compositions of Zones in Reflector (simplified model) – Coolant Components.

No. of Zone (a) Coolant fraction (b) N O (air) Ar H (water) O (water)

65 1.0321E-01 4.0352E-06 1.0825E-06 2.4134E-08 8.85571E-08 4.42785E-08

70 2.3705E-02 9.2679E-07 2.4863E-07 5.5431E-09 2.03396E-08 1.01698E-08

77 1.0963E-02 4.2862E-07 1.1499E-07 2.5636E-09 9.40656E-09 4.70328E-09

81 3.9000E-01 1.5248E-05 4.0905E-06 9.1197E-08 3.34631E-07 1.67315E-07 (a) See Footnote ‘(b)’ to Table 3.5.b.

The graphite density in Regions 22, 23, 25, 48, 49, 50, 52, 54, 67, 69, 71, and 80 of Figure 3.5 is solid density (at 1.76 g/cm3, the specified density of the reflector graphite); the coolant volumes are determined in the other regions from the fractions of the solid density that the graphite in them occupies. In regions where the void fraction is easily inferred, such as Regions 58, 59, 61, and 63 (which are solid except for the coolant flow channel borings), the number densities in Table 3.9 are readily confirmed. The data in Regions 17, 55, 72, 74, 75, 76, 78, and 79 of Figure 3.5 are solid boronated carbon bricks; the number densities in Tables 3.5 and 3.9 can also be shown to agree with the data on the boronated carbon bricks in Table 1.5. The carbon and boron atomic number densities are also tabulated in Reference 1 for the dummy pebbles, but the tabulated densities had to be corrected for the as-built composition (1.84 g/cm3 instead of 1.73 g/cm3 for carbon and 0.125 ppm instead of 1.3 ppm for boron). 3.1.4 Temperature Data The temperature of the entire system was a uniform 15 °C, as stated in Reference 1.

3.1.5 Experimental and Benchmark-Model keff and / or Subcritical Parameters The value of keff is not explicitly given in Reference 1. Instead, the number of pebbles in the core cavity at criticality is given (9627 fuel pebbles and 7263 dummy pebbles, for a total of 16,890 pebbles). It is implied that keff is exactly equal to 1 when this number of pebbles is loaded into the core cavity. This number of pebbles is equivalent to a level core height of 123.06 cm at the assumed packing fraction of 0.61. The effects of uncertainty in the core height specification, which is equivalent to an uncertainty in the number of pebbles in the core, are discussed in Section 2.1.1. Because the simplifications made in constructing the high-fidelity model, discussed in Section 3.1.1, were judged to have no significant effect on keff, the expected calculated result for the high-fidelity benchmark model also is 1. The high-fidelity model was analyzed by the code MCNP, as discussed in Section 4.1.1. The value of keff obtained in the base case, in which the cone angle of the upper surface is 19.5°, is (keff)high-fidelity/MCNP = 1.01190±0.00021. The simplified model was also analyzed by MCNP, as discussed in Section 4.1.1. The value of keff obtained for this model is (keff)simplified/MCNP = 1.02500±0.00021. Therefore, the difference, Δkeff = 1.02500 - 1.01190 = 0.01310 ± 0.00030, is the correction to the expected keff for the simplified model. These results are summarized in Table 3.10. The uncertainty is the result obtained in Section 2.

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Table 3.10. Benchmark-Model keff Values and Uncertainties.

Model Expected keff Δkeff Uncertainty (1σ) High-fidelity model 1.0000 0.0037 Simplified model 1.0131 0.0037

3.2 Benchmark-Model Specifications for Buckling and Extrapolation-Length

Measurements Buckling and extrapolation length measurements were not reported in Reference 1. 3.3 Benchmark-Model Specifications for Spectral Characteristics Measurements Spectral characteristics measurements were not reported in Reference 1. 3.4 Benchmark-Model Specifications for Reactivity Effects Measurements The control rod worth measurements are not evaluated in this report. 3.5 Benchmark-Model Specifications for Reactivity Coefficient Measurements Reactivity coefficient measurements were not reported in Reference 1. 3.6 Benchmark-Model Specifications for Kinetics Measurements Kinetics measurements were not reported in Reference 1. 3.7 Benchmark-Model Specifications for Reaction-Rate Distribution Measurements Reaction-rate distribution measurements were not reported in Reference 1. 3.8 Benchmark-Model Specifications for Power Distribution Measurements Power distribution measurements were not reported in Reference 1. 3.9 Benchmark-Model Specifications for Isotopic Measurements Isotopic measurements were not reported in Reference 1. 3.10 Benchmark-Model Specifications for Other Miscellaneous Types of Measurements Other miscellaneous types of measurements were not reported in Reference 1.

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4.0 RESULTS OF SAMPLE CALCULATIONS This section presents results of calculations for the high-fidelity and simplified models obtained by the INL research group. The code MCNPa was used for both models. The Sn code TWODANTb and the diffusion code PEBBEDc were also used to analyze the simplified model. Auxiliary codes were used for such purposes as computing multigroup cross sections for TWODANT and PEBBED. Principal results are given in Table 4.1 for both the high-fidelity and simplified models. Detailed discussions are provided in the following sections explaining the codes and further simplifications made for the Sn and diffusion codes.

Table 4.1. Sample Calculated Results of keff.

Case keff 100(C-E)/E High-fidelity model Expected benchmark value (experimental results) 1.00000 0.000 Monte Carlo result 1.01190±0.00021 1.190 Simplified model Expected benchmark value 1.0131± 0.000297 0.000 Monte Carlo result 1.02500±0.00021 1.175 Discrete ordinates result (30-group) 1.0144 0.128 Discrete ordinates result (B-3, S-16, 6-group) 1.02023617 0.704 Discrete ordinates result (P-1, S-8, 6-group) 1.02028653 0.709 Diffusion result 1.02310 0.987 4.1 Results of Calculations of the Critical or Subcritical Configurations 4.1.1 High-fidelity Model The Monte Carlo code MCNP was used to analyze the high-fidelity model. MCNP is one of the most widely used Monte Carlo codes in reactor physics today. It applies a continuous-energy cross section library obtained from the ENDF/B data files,d it uses combinatorial geometry to construct very complex configurations from combinations of spheres, cylinders, planes, cones, and surfaces of revolution, and it contains lattice and repeating-structure options to simplify input for large arrays of objects. These techniques were used to represent the randomly arranged pebbles and the idealized array of fuel particles as explained in Section 3. MCNP also contains powerful variance-reduction techniques to accelerate execution times in complex problems.

a J. F. Briesmeister, Ed., “MCNP – A General Monte Carlo N-Particle Transport Code – Version 4B,” LA-12625-M (March 1997). b R. E. Alcouffe, F. W. Brinkley, Jr., D. R. Marr and R. D. O'Dell, `Ùser's Guide for TWODANT: A Code Package for Two-Dimensional Diffusion-Accelerated, Neutral-Particle Transport,'' Los Alamos National Laboratory report LA-10049-M, Revision 1 (1984). c W. K. Terry, H. D. Gougar, and A. M. Ougouag, “Direct Deterministic Method for Neutronics Analysis and Computation of Asymptotic Burnup Distribution in a Recirculating Pebble-Bed Reactor,” Annals of Nuclear Energy 29 (2002), pp. 1345-1364. d R. Kinsey, Data Formats and Procedures for the Evaluated Nuclear Data File ENDF, Brookhaven National Laboratory, BNL-NCS-50496, 1979.

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The high-fidelity benchmark model includes a conical mound at the top of the core, making an angle of 19.5° from horizontal. The value of keff obtained from this model, within the 1σ statistical uncertainty inherent in the Monte Carlo method, is (keff)high-fidelity(base)/MCNP =1.01190±0.00021. Input listing are provided in Appendix A. 4.1.2 Simplified Model MCNP Calculations - MCNP was also used to calculate keff for the simplified model. This calculation reveals the bias resulting from homogenizing the reflector azimuthally and flattening the top of the core (while preserving the total number of pebbles present. The result of the MCNP calculation for the simplified model is (keff)simplified/MCNP = 1.02500±0.00021. The bias is 1.02500-1.01190±√(2x0.000212) = 0.01310±0.00030. TWODANT Calculations - Each of the approximations required by PEBBED introduces biases in the calculated value of keff. So that these biases could be assessed individually, the simplifications were made sequentially. The Sn transport model provides an intermediate level of fidelity between the Monte Carlo models of MCNP and the diffusion model of PEBBED. To implement the Sn model, the code TWODANTa was chosen. In the Sn model, additional simplifications are made in the geometry and the physics, but the more accurate transport treatment is retained in such regions as the void space above the core. First, as a two-dimensional (r-z) model, it cannot represent the sloping surface of the conus; therefore, it uses a stair-step approximation. It cannot represent the pebbles explicitly, so the core is represented as a homogeneous cylinder. The geometry of the Sn model is illustrated in Figure 4.1. The definition of zones in Figure 4.1 is mostly as specified by the HTR-10 research group in its invitation to participants in its international benchmark project. This same definition is used for the diffusion model discussed in Section 4.1.2, “Diffusion Model.” However, for compatibility with input requirements of the diffusion code, the numbering of some zones in the reflector, the conus, and the bottom reflector section with hot coolant flow borings was changed slightly from the numbering assigned by the HTR-10 modeling group, but the correspondence between zones and compositions is preserved. Figure 4.1 shows the numbering used in this evaluation report. The only change in numbering that affects the fidelity of the model is the stair-step approximation of the slanted surface of the conus, as shown in Figure 4.1.b. This is a necessary approximation for the use of any code in cylindrical geometry; some participants in the pre-experiment benchmark program applied the same approach. The stair-steps are small, so that the effects of the approximation should be negligible.

a R. E. Alcouffe, F. W. Brinkley, Jr., D. R. Marr and R. D. O'Dell, `Ùser's Guide for TWODANT: A Code Package for Two-Dimensional Diffusion-Accelerated, Neutral-Particle Transport,'' Los Alamos National Laboratory report LA-10049-M, Revision 1 (1984).

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Figure 4.1.a. Zones of HTR-10 for Sn and Diffusion Models. (dimensions are in cm)

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Figure 4.1.b. Detail in Conus Region. (dimensions are in cm)

As shown in Figure 4.1, the model regions are numbered from 1 to 99. In the model of the initial critical configuration, the control rod region in Figure 4.1, Region 31, was originally divided into Regions 31-40 in Reference 1. This consolidation leaves a gap from 32-40. (Dividing the control rod region axially into several model regions allowed partial insertion of control rods to be represented.) Because the cross sections of the same nuclides will be different in the core and the reflector, which are treated as different spectral zones in the model, nuclides that appear in both zones are treated separately in the two zones, with different material numbers. The number densities in the core were calculated from the specifications given in Reference 1. First, a typical cell was defined as a homogeneous sphere comprising 57% of a fuel pebble, 43% of a dummy pebble, and the coolant volume associated with one pebble, based on a packing fraction of 61% as stated in Reference 1. The volume of the cell was found to be 185.405 cm3. Table 3.7 specifies the compositions of all the components in the core region, and Table 3.8 gives the composition of the air in the core. From these data, one can calculate the average number densities of all the nuclides in the core. Because this calculation is tedious, it is sketched in Appendix B. The results of these calculations appear in Table 4.2.

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Table 4.2. Homogenized Atomic Number Densities in the Model.


Core (average values over whole core) 235U 6.69520E-6 238U 3.22755E-5 16O in kernels(a) 7.79414E-5 Natural Si 8.51054E-5 Graphite 5.40964E-2 10B 9.91914E-9 11B 3.99258E-8 Nitrogen 1.52478E-5 16O in dry air 4.09052E-6 40Ar(b) 9.11929E-8 Hydrogen 3.34631E-7 16O from water vapor 1.67316E-7

Reflector(b) Graphite at 1.76 g/cm3 8.82418E-2 10B at 4.8366 ppm total boron concentration 9.42800E-8 11B at 4.8366 ppm total boron concentration 3.79489E-7 (a) Minor constituents of natural oxygen and argon are neglected. (b) The number densities of graphite and boron in each reflector region are given

in Reference 1; the number densities for the coolant constituents are computed based on the region void fraction, which is calculated from the region-averaged graphite density.

Cross sections for TWODANT were computed by the INL’s COMBINE code,a so named because it combines the PHROGb and INCITEc fast and thermal spectrum codes. COMBINE uses a 166-group cross section data base derived from the Evaluated Nuclear Data Files (ENDF/B),d Version 6, over the energy range from 0.001 eV to 16.905 MeV. COMBINE solves the B-1 or B-3 approximations to the neutron transport equation, at the user’s choice. The P-1 approximation may be chosen as a special case of the B-1 solution. For the TWODANT calculations, both P-1 and B-3 cross sections were derived.

a Robert A. Grimesey, David W. Nigg, and Richard L. Curtis, COMBINE/PC – A Portable ENDF/B Version 5 Neutron Spectrum and Cross-Section Generation Program, EGG-2589, Rev. 1, Idaho National Engineering Laboratory, February 1991. b R. L. Curtis, et al., PHROG – A FORTRAN-IV Program to Generate Fast Neutron Spectra and Average Multigroup Constants, Idaho Nuclear Corp., IN-1435, April 1971. c R. L. Curtis and R. A. Grimesey, INCITE: A FORTRAN-IV Program to Generate Thermal Neutron Spectra and Multigroup Constants Using Arbitrary Scattering Kernels, Idaho Nuclear Corporation, IN-1062, 1967. d R. Kinsey, Data Formats and Procedures for the Evaluated Nuclear Data File ENDF, Brookhaven National Laboratory, BNL-NCS-50496, 1979.

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The COMBINE calculations in the core and the reflector produce cross sections in the two spectral zones defined in the Sn model – i.e., the core spectral zone and the reflector spectral zone. In the core, COMBINE was applied to the unit cell model shown in Figure 4.2. This unit cell consists of a fuel pebble surrounded by the corresponding quantities of coolant and dummy-pebble graphite in a homogeneous mixture. The fuel pebble consists of a homogenized fuel zone and a pure graphite zone.

Figure 4.2. COMBINE Unit Cell Model in Core. A separate unit cell model was developed for the reflector. In the reflector, the only materials present are carbon, boron, and coolant. A tiny amount of 235U was added to provide a neutron spectrum. The proportions of carbon and boron are the same except in the carbon bricks (with no boron) and the boronated carbon bricks used in the outer region where the neutron importance is low, and in a few regions where the boron concentration is significantly higher, but still small, as described in Section 3.1.3, “Compositions of High-fidelity Model Zones.” Therefore, the atomic number densities used in the COMBINE reflector model were obtained by homogenizing two of the regions shown in Figure 4.1, Regions 22 and 31. The COMBINE input file for the as-run critical configuration is contained in Appendix A. COMBINE has been improved numerous times since it was first written. The version used for this evaluation is COMBINE-6.02.a The reference given in Footnote ‘(a)’ is an abbreviated report, mostly containing revised material library labels and revised input instructions. The report by Grimesey et al. is still the proper source for detailed theoretical elucidation of COMBINE.b

The cross section energy group structure is given in Appendix A. Also, of course, in an Sn model the scattering angle is discretized: TWODANT was run with two different choices of expansion order, angular resolution, and energy group structure: P-1, S-8, and six energy groups; and B-3, S-16, and six energy groups.

a W. Y. Yoon letter to D. W. Nigg, “COMBINE-6 CYCLE 1,” WYY-01-94, January 17, 1994. b Robert A. Grimesey, David W. Nigg, and Richard L. Curtis, COMBINE/PC – A Portable ENDF/B Version 5 Neutron Spectrum and Cross-Section Generation Program, EGG-2589, Rev. 1, Idaho National Engineering Laboratory, February 1991.

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For the P-1, S-8, 6-group TWODANT model, (keff)TWODANT/P-1,S-8,6-gp=1.02028653. For the B-3, S-16, 6-group TWODANT model, (keff)TWODANT/B-3,S-16,6-gp=1.02023617. A 30-group TWODANT model was also attempted. Although it did not converge to a value of keff that met the specified convergence criteria, for practical purposes the “almost-converged” result from this model is (keff)TWODANT/30-gp=1.0144. The base-case TWODANT input file is also included in Appendix A. Diffusion Model - PEBBED is a reactor physics and fuel management code developed by the INL specifically for PBRs. PEBBED obtains simultaneous solutions of the neutron diffusion equation and the nuclide depletion/production equations in the equilibrium core directly, without following the evolution of the nuclide number densities and the neutron flux in time. PEBBED accounts explicitly for the motion of the fuel and treats arbitrarily specified pebble recirculation patterns. In the HTR-10 startup core, the fuel was not moving; PEBBED also treats this special case. PEBBED has the capability to perform full r-θ-z analysis, but this option runs slowly, and for computational efficiency the uncertainty analysis was done in r-z geometry. In the r-z option, PEBBED input must be based on the simplified (azimuthally symmetric) model. Furthermore, PEBBED cannot represent the pebbles explicitly, nor can it represent the slanted surface of the conus. Finally, PEBBED is a multigroup code. For these reasons, the geometry and compositions for PEBBED are the same as those used for TWODANT. Most of the regions in Figure 4.1 were subdivided into several computational mesh cells in the PEBBED model. The computational mesh is presented in Tables 4.3 and 4.4, rounded to the nearest millimeter. The meshing was initially selected on the basis of prior experience with spatial convergence in verification and validation studies of PEBBED, and then refined until further division made the calculation impractically slow. By this point, only a small change in keff (0.1%) resulted from the last halving of the mesh intervals. The PEBBED input deck for the nominal configuration is presented in Appendix A.

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Table 4.3. Radial Mesh Points (cm) in Base-case PEBBED Model (mesh point 1 is at r=0). pt. # r pt. # r pt. # r pt. # r pt. # r pt. # r pt. # r pt. # r 2 1.2 3 2.5 4 3.8 5 5.0 6 6.2 7 7.5 8 8.8 9 10.0 10 11.2 11 12.5 12 13.8 13 15.0 14 16.2 15 17.5 16 18.8 17 20.0 18 21.2 19 22.5 20 23.8 21 25.0 22 26.0 23 27.1 24 28.1 25 29.2 26 30.2 27 31.3 28 32.3 29 33.4 30 34.4 31 35.5 32 36.5 33 37.6 34 38.6 35 39.7 36 40.7 37 41.8 38 42.7 39 43.6 40 44.5 41 45.4 42 46.3 43 47.2 44 48.1 45 49.0 46 49.9 47 50.8 48 51.7 49 52.6 50 53.5 51 54.4 52 55.3 53 56.2 54 57.2 55 58.1 56 59.0 57 59.9 58 60.8 59 61.7 60 62.6 61 63.5 62 64.4 63 65.3 64 66.2 65 67.1 66 68.0 67 68.9 68 69.8 69 70.8 70 72.0 71 73.2 72 74.4 73 75.6 74 76.8 75 78.0 76 79.2 77 80.4 78 81.6 79 82.8 80 84.0 81 85.2 82 86.4 83 87.6 84 88.8 85 90.0 86 90.7 87 91.4 88 92.1 89 92.8 90 93.5 91 94.2 92 94.9 93 95.6 94 96.4 95 97.2 96 98.0 97 98.8 98 99.7 99 100.5 100 101.3 101 102.1 102 102.9 103 103.7 104 104.5 105 105.3 106 106.2 107 107.0 108 107.8 109 108.6 110 109.6 111 110.6 112 111.6 113 112.6 114 113.6 115 114.6 116 115.6 117 116.6 118 117.6 119 118.6 120 119.6 121 120.6 122 121.6 123 122.6 124 123.6 125 124.6 126 125.6 127 126.6 128 127.6 129 128.6 130 129.6 131 130.6 132 131.6 133 132.6 134 133.6 135 134.6 136 135.6 137 136.6 138 137.6 139 138.6 140 139.6 141 140.6 142 141.6 143 142.6 144 143.6 145 144.6 146 145.6 147 146.6 148 147.6 149 148.6 150 149.8 151 151.0 152 152.2 153 153.4 154 154.6 155 155.8 156 157.0 157 158.2 158 159.4 159 160.6 160 161.8 161 163.0 162 164.2 163 165.4 164 166.6 165 167.8 166 169.2 167 170.6 168 172.0 169 173.3 170 174.7 171 176.1 172 177.5 173 178.9 174 180.3 175 181.7 176 183.1 177 184.4 178 185.8 179 187.2 180 188.6 181 190.0 Table 4.4. Axial Mesh Points (cm) in Base-case PEBBED Model (mesh point 1 is at z=0). pt. # z pt. # z pt. # z pt. # z pt. # z pt. # z pt. # z pt. # z 2 1.0 3 2.0 4 3.0 5 4.0 6 5.0 7 6.0 8 7.0 9 8.0 10 9.0 11 10.0 12 11.0 13 12.0 14 13.0 15 14.0 16 15.0 17 16.0 18 17.0 19 18.0 20 19.0 21 20.0 22 21.0 23 22.0 24 23.0 25 24.0 26 25.0 27 26.0 28 27.0 29 28.0 30 29.0 31 30.0 32 31.0 33 32.0 34 33.0 35 34.0 36 35.0 37 36.0 38 37.0 39 38.0 40 39.0 41 40.0 42 41.4 43 42.8 44 44.1 45 45.5 46 46.9 47 48.2 48 49.6 49 51.0 50 52.4 51 53.8 52 55.1 53 56.5 54 57.9 55 59.2 56 60.6 57 62.0 58 63.4 59 64.8 60 66.1 61 67.5 62 68.9 63 70.2 64 71.6 65 73.0 66 74.4 67 75.8 68 77.1 69 78.5 70 79.9 71 81.2 72 82.6 73 84.0 74 85.4 75 86.8 76 88.1 77 89.5 78 90.9 79 92.2 80 93.6 81 95.0 82 96.2 83 97.5 84 98.8 85 100.0 86 101.2 87 102.5 88 103.8 89 105.0 90 106.2 91 107.4 92 108.6 93 109.9 94 111.1 95 112.3 96 113.5 97 114.7 98 115.7 99 116.6 100 117.6 101 118.5 102 119.5 103 120.4 104 121.4 105 122.4 106 123.3 107 124.3 108 125.2 109 126.2 110 127.1 111 128.1 112 129.0 113 130.0 114 131.4 115 132.7 116 134.1 117 135.5 118 136.9 119 138.2 120 139.6 121 141.0 122 142.3 123 143.7 124 145.1 125 146.5 126 147.8 127 149.2 128 150.6 129 151.9 130 153.3 131 154.7 132 156.1 133 157.4 134 158.8 135 160.2 136 161.5 137 162.9 138 164.3 139 165.7 140 167.0 141 168.4 142 169.8 143 171.1 144 172.5 145 173.9 146 175.3 147 176.6 148 178.0 149 179.4 150 180.8 151 182.1 152 183.5 153 184.9 154 186.2 155 187.6 156 189.0 157 190.4 158 191.7 159 193.1 160 194.5 161 195.8

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Table 4.4 (cont’d.). Axial Mesh Points (cm) in Base-case PEBBED Model (mesh point 1 is at z=0). 162 197.2 163 198.6 164 200.0 165 201.3 166 202.7 167 204.1 168 205.4 169 206.8 170 208.2 171 209.6 172 210.9 173 212.3 174 213.7 175 215.0 176 216.4 177 217.8 178 219.2 179 220.5 180 221.9 181 223.3 182 224.6 183 226.0 184 227.4 185 228.8 186 229.8 187 230.8 188 231.8 189 232.9 190 233.9 191 234.9 192 235.9 193 237.0 194 238.0 195 239.0 196 240.0 197 241.1 198 242.1 199 243.1 200 244.1 201 245.2 202 246.2 203 247.2 204 248.2 205 249.3 206 250.3 207 251.3 208 252.3 209 253.4 210 254.4 211 255.4 212 256.4 213 257.5 214 258.5 215 259.5 216 260.5 217 261.6 218 262.6 219 263.6 220 264.7 221 265.7 222 266.7 223 267.7 224 268.8 225 269.8 226 270.8 227 271.8 228 272.9 229 273.9 230 274.9 231 275.9 232 277.0 233 278.0 234 279.0 235 280.0 236 281.1 237 282.1 238 283.1 239 284.1 240 285.2 241 286.2 242 287.2 243 288.2 244 289.3 245 290.3 246 291.3 247 292.3 248 293.4 249 294.4 250 295.4 251 296.4 252 297.5 253 298.5 254 299.5 255 300.5 256 301.6 257 302.6 258 303.6 259 304.6 260 305.7 261 306.7 262 307.7 263 308.7 264 309.8 265 310.8 266 311.8 267 312.9 268 313.9 269 314.9 270 315.9 271 317.0 272 318.0 273 319.0 274 320.0 275 321.1 276 322.1 277 323.1 278 324.1 279 325.2 280 326.2 281 327.2 282 328.2 283 329.3 284 330.3 285 331.3 286 332.3 287 333.4 288 334.4 289 335.4 290 336.4 291 337.5 292 338.5 293 339.5 294 340.5 295 341.6 296 342.6 297 343.6 298 344.6 299 345.7 300 346.7 301 347.7 302 348.7 303 349.8 304 350.8 305 351.8 306 352.5 307 353.2 308 353.9 309 354.6 310 355.2 311 355.9 312 356.6 313 357.3 314 358.0 315 358.7 316 359.3 317 360.0 318 360.7 319 361.4 320 362.1 321 362.8 322 363.8 323 364.8 324 365.9 325 366.9 326 367.9 327 368.9 328 370.0 329 371.0 330 372.0 331 373.1 332 374.1 333 375.1 334 376.2 335 377.2 336 378.2 337 379.2 338 380.4 339 381.6 340 382.8 341 384.0 342 385.2 343 386.4 344 387.6 345 388.8 346 389.6 347 390.4 348 391.2 349 392.1 350 392.9 351 393.7 352 394.6 353 395.4 354 396.2 355 397.0 356 397.9 357 398.7 358 399.5 359 400.3 360 401.2 361 402.0 362 403.0 363 404.0 364 405.0 365 406.0 366 407.0 367 408.0 368 409.0 369 410.0 370 411.0 371 412.0 372 413.0 373 414.0 374 415.0 375 416.0 376 417.0 377 418.0 378 419.0 379 420.0 380 421.0 381 422.0 382 423.0 383 424.0 384 425.0 385 426.0 386 427.0 387 428.0 388 429.0 389 430.0 390 431.3 391 432.5 392 433.8 393 435.0 394 436.3 395 437.5 396 438.8 397 440.0 398 441.3 399 442.5 400 443.8 401 445.0 402 446.3 403 447.5 404 448.8 405 450.0 406 450.9 407 451.9 408 452.8 409 453.8 410 454.7 411 455.6 412 456.6 413 457.5 414 458.4 415 459.4 416 460.3 417 461.3 418 462.2 419 463.1 420 464.1 421 465.0 422 465.9 423 466.9 424 467.8 425 468.8 426 469.7 427 470.6 428 471.6 429 472.5 430 473.4 431 474.4 432 475.3 433 476.3 434 477.2 435 478.1 436 479.1 437 480.0 438 480.9 439 481.9 440 482.8 441 483.8 442 484.7 443 485.6 444 486.6 445 487.5 446 488.4 447 489.4 448 490.3 449 491.3 450 492.2 451 493.1 452 494.1 453 495.0 454 495.9 455 496.9 456 497.8 457 498.8 458 499.7 459 500.6 460 501.6 461 502.5 462 503.4 463 504.4 464 505.3 465 506.3 466 507.2 467 508.1 468 509.1 469 510.0 470 510.9 471 511.9 472 512.8 473 513.8 474 514.7 475 515.6 476 516.6 477 517.5 478 518.4 479 519.4 480 520.3 481 521.3 482 522.2 483 523.1 484 524.1 485 525.0 486 525.9 487 526.9 488 527.8 489 528.8 490 529.7 491 530.6 492 531.6 493 532.5 494 533.4 495 534.4 496 535.3 497 536.3 498 537.2 499 538.1 500 539.1 501 540.0 502 541.0 503 542.0 504 543.0 505 544.0 506 545.0 507 546.0 508 547.0 509 548.0 510 549.0 511 550.0 512 551.0 513 552.0 514 553.0 515 554.0 516 555.0 517 556.0 518 557.0 519 558.0 520 559.0 521 560.0 522 561.0 523 562.0 524 563.0 525 564.0 526 565.0 527 566.0 528 567.0 529 568.0 530 569.0 531 570.0 532 571.0 533 572.0 534 573.0 535 574.0 536 575.0 537 576.0 538 577.0 539 578.0 540 579.0 541 580.0 542 581.0 543 582.0 544 583.0 545 584.0 546 585.0 547 586.0 548 587.0 549 588.0 550 589.0 551 590.0 552 591.0 553 592.0 554 593.0 555 594.0 556 595.0 557 596.0 558 597.0 559 598.0 560 599.0 561 600.0 562 601.0 563 602.0 564 603.0 565 604.0 566 605.0 567 606.0 568 607.0 569 608.0 570 609.0 571 610.0

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The zone-averaged atomic number densities of all nuclides in all zones in the model are given in the base-case PEBBED input in Appendix A. Cards 2xy00 give truncated nuclide labels that are sufficient to identify the nuclide assigned the number xy, while cards 3uv00 et seq. identify material uv and specify the nuclide number densities of nuclides xy in that material in a manner obvious by inspection. For the diffusion model, the calculated value of keff is 1.02310. The deviation from 1.0 is believed to be due to inaccuracies in the cross sections and in the diffusion approximation. A variation of 10% in the diffusion coefficients can cause an error of almost 100% in reactivity. This would be enough to change keff to 1.0. Another recent study of the HTR-10 also found values of keff slightly higher (about 1%) for a diffusion model than for a Monte Carlo model or the experimental results.a The values of keff and the biases introduced by each successive approximation in the various models are tabulated in Table 4.5.

Table 4.5. Summary of Biases from Each Level of Simplification in Models.

Model keff Incremental Bias 1. High-fidelity model 1.01190±0.00021 NA 2. Model 1 plus flat upper core surface and azimuthal symmetry (“simplified” Monte Carlo model)

1.02500±0.00021 +0.0131

3. Model 2 but discrete ordinates (30-group) 1.0144 -0.0106 4. Model 2 but discrete ordinates (B-3, S-16, 6-gp.) 1.02023617 +0.00583617 5. Model 4 but lower-order expansion (P-1, S-8, 6-gp.) 1.02028653 +0.00005036 6. Model 5 plus diffusion 1.02310 +0.00281347 4.2 Results of Buckling and Extrapolation Length Calculations Buckling and extrapolation length measurements were not reported in Reference 1. 4.3 Results of Spectral-Characteristics Calculations Spectral characteristics measurements were not reported in Reference 1. 4.4 Results of Reactivity-Effects Calculations The control-rod worth experiments were not modeled in this evaluation. 4.5 Results of Reactivity Coefficient Calculations Reactivity coefficient measurements were not reported in Reference 1.

a Üner Çolak and Volkan Seker, Monte Carlo Criticality Calculations for a Pebble Bed Reactor with MCNP, Nucl. Sci. Eng. 149, 1-7 (2005).

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4.6 Results of Kinetics Parameter Calculations Kinetics measurements were not reported in Reference 1. 4.7 Results of Reaction-Rate Distribution Calculations Reaction-rate distribution measurements were not reported in Reference 1.

4.8 Results of Power Distribution Calculations Power distribution measurements were not reported in Reference 1.

4.9 Results of Isotopic Calculations Isotopic measurements were not reported in Reference 1. 4.10 Results of Calculations for Other Miscellaneous Types of Measurements Other miscellaneous types of measurements were not reported in Reference 1.

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5.0 REFERENCES 1. “Evaluation of high temperature gas cooled reactor performance: Benchmark analysis related to initial

testing of the HTTR and HTR-10,” IAEA-TECDOC-1382, International Atomic Energy Agency, Vienna, November 2003.

2. “Fuel performance and fission product behavior in gas cooled reactors,” IAEA-TECDOC-978,

International Atomic Energy Agency, Vienna, November, 1997, pp. 11-13.

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APPENDIX A: COMPUTER CODES, CROSS SECTIONS, AND TYPICAL INPUT LISTINGS The codes used in this evaluation are MCNP, TWODANT, PEBBED4, COMBINE-6, PEBDAN, and PEBBLES. Detailed information on these codes is given in this appendix. A.1 Name(s) of code system(s) used.

MCNP

TWODANT PEBBED4

COMBINE-6 version 2

PEBDAN

PEBBLES

A.2 Bibliographic references for the codes used.

J. F. Briesmeister, Ed., “MCNP – A General Monte Carlo N-Particle Transport Code – Version 4B,” LA-12625-M (March 1997). R. E. Alcouffe, F. W. Brinkley, Jr., D. R. Marr and R. D. O'Dell, `Ùser's Guide for TWODANT: A Code Package for Two-Dimensional Diffusion-Accelerated, Neutral-Particle Transport,'' Los Alamos National Laboratory report LA-10049-M, Revision 1 (1984). H. D. Gougar, W. K. Terry, and A. M. Ougouag, PEBBED V.4.3 Manual (DRAFT), Idaho National Engineering and Environmental Laboratory. To be published. Robert A. Grimesey, David W. Nigg, and Richard L. Curtis, COMBINE/PC – A Portable ENDF/B Version 5 Neutron Spectrum and Cross-Section Generation Program, EGG-2589, Rev. 1, Idaho National Engineering Laboratory, February 1991. W. Y. Yoon letter to D. W. Nigg, “COMBINE-6 CYCLE 1,” WYY-01-94, January 17, 1994. J. Kloosterman and A. M. Ougouag, “Computation of Dancoff Factors for Fuel Elements Incorporating Randomly Packed TRISO Particles,” INEEL/EXT-05-02593, Idaho National Engineering and Environmental Laboratory, January 2005. R. Kinsey, Data Formats and Procedures for the Evaluated Nuclear Data File ENDF, Brookhaven National Laboratory, BNL-NCS-50496, 1979. Abderrafi M. Ougouag and Joshua J. Cogliati, Methods For Modeling The Packing of Fuel Elements in Pebble Bed Reactors, Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, Palais des Papes, Avignon, France, September 12-15, 2005, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2005).

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A.3 Origin of cross-section data. MCNP uses cross sections from ENDFB-VI COMBINE uses cross sections from ENDF/B-VI. A.4 Spectral calculations and data reduction methods used.

MCNP is a very well known Monte Carlo neutral-particle transport code. It contains a continuous-energy ENDF/B-based cross section library, but it can also accept multigroup cross sections provided from an external source. It contains sophisticated options for variance reduction when regions of the problem domain have low importance for the particles in question. It uses combinatorial geometry to construct very accurate models of complex geometries from planes, spheres, cylinders, and surfaces of revolution. It is considered the most accurate possible modeling code for most nuclear systems, although the cost of accuracy is often very long run times (i.e., days on PCs or UNIX workstations). However, because of the random organization of pebble locations, MCNP cannot model PBRs with complete fidelity, even at one instant in time.

TWODANT is a component of the DANTSYS code system, which also includes ONEDANT, TWODANT/GQ, and THREEDANT. TWODANT solves the two-dimensional multigroup transport equation in x-y, r-z, and r-theta geometries. TWODANT uses the discrete ordinates approximation for treating the angular variation of the particle distribution. The diamond difference scheme is used for phase space discretization. Both inner and outer iterations are accelerated using the diffusion synthetic acceleration method. Energy is treated in a multigroup scheme using cross sections from a cross section processing code such as COMBINE. PEBBED solves the neutron diffusion equation in one-, two-, or three-dimensional cylindrical or Cartesian geometry by finite-difference methods or in one- or two-dimensional cylindrical geometry by nodal methods. In PEBBED, the neutron diffusion equation is solved simultaneously with the nuclide depletion/production equations in a flowing core for a user-specified number of nuclides in the steady-state configuration of neutron flux and composition distribution. It does this by an iterative scheme without following the development of the state of the reactor in time. PEBBED also contains thermohydraulics modules for computing temperatures in normal operation and accident scenarios. However, in this evaluation, only the diffusion-theory component was needed since the fuel was all fresh and the core was stationary. Because the two-dimensional nodal option was not yet working when this evaluation was initiated, the finite-difference option was applied. COMBINE solves the B-1 or B-3 approximation to the neutron transport equation, or the P-1 approximation as a special case of the B-1 approximation. At the user’s option, it can perform an ABH thermal calculation and Nordheim, GAM-1, or Bondarenko treatment of the resolved resonance region. It uses the Wigner Rational Approximation in the unresolved resonance region. In this evaluation, the B-1 approximation was used, and the Nordheim treatment of the unresolved resonance region was chosen. The ABH option was not used. The only fissile nuclide present in the fresh HTR-10 core is 235U, so a 235U fission spectrum was used. Self-shielding factors were not specified. COMBINE contains an internal module for computing Dancoff factors to account for shadowing effects in doubly heterogeneous systems like PBRs. However, recent advances in treating such double heterogeneities have led to an improved method of calculating the Dancoff factors. This method is implemented in the PEBDAN code. PEBDAN results were provided as input data to the COMBINE input file.

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PEBBLES uses the discrete elements method to solve simultaneously the equations of motion of all the grains in a granular substance, or, in the case of PBR, all of the pebbles in the core. A longer discussion of PEBBLES appears in Appendix C. A.5 Number of energy groups

COMBINE begins with a 166-group energy structure from ENDF/B-6; in this evaluation, the fine group structure was collapsed to six groups in the following energy ranges: group 1 16.905-0.111 MeV group 2 0.111 MeV – 7100 eV group 3 7100 – 29.0 eV group 4 29.0 – 2.38 eV group 5 2.38 – 0.532 eV group 6 0.532 – 0 eV. The two groups of lowest energy are the thermal groups. A TWODANT calculation was performed with COMBINE cross sections in the following 30-group structure: group 1 16.905-6.07 MeV group 2 6.07-2.23 MeV group 3 2.23-0.821 MeV group 4 0.821-0.302 MeV group 5 0.302-0.111 MeV group 6 0.111 MeV – 6.74x104 eV group 7 6.74x104-4.09x104 eV group 8 4.09x104-2.48x104 eV group 9 2.48x104-1.17x104 eV group 10 1.17x104-7100 eV group 11 7100-2610 eV group 12 2610-748.5 eV group 13 748.5-275.4 eV group 14 275.4-79.9 eV group 15 79.9-29.0 eV group 16 29.0-17.6 eV group 17 17.6-10.08 eV group 18 10.08-6.48 eV group 19 6.48-3.93 eV group 20 3.93-2.38 eV group 21 2.38-1.78 eV group 22 1.78-1.30 eV group 23 1.30-0.97 eV group 24 0.97-0.70 eV group 25 0.70-0.532 eV group 26 0.532-0.140 eV group 27 0.140-0.030 eV group 28 0.030-0.008 eV group 29 0.008-0.002 eV group 30 0.002-0.00 eV

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A.6 Component calculations.

A.6.1 MCNP calculation of keff Type of calculation: full reactor Geometry: combinatorial Theory used: Monte Carlo transport Method used: Monte Carlo transport Calculation characteristics: given detailed description of core and reflector regions, compute keff; continuous-energy cross sections; explicit representation of semi-randomly located pebbles in core region; regular lattice of TRISO fuel particles in pebble fuel zones; reflector regions defined by HTR-10 group. A.6.2 TWODANT calculation of keff Type of calculation: full reactor Geometry: cylindrical Theory used: Sn deterministic transport Method used: discrete ordinates in angular distribution, diamond difference in spatial distribution Calculation characteristics: given detailed description of core and reflector regions, compute keff; six energy groups, two thermal groups, homogenized core region, reflector regions defined by HTR-10 group. A.6.1 COMBINE core spectrum and cross section calculation Type of calculation: Unit cell Geometry: spherical Theory used: transport Method used: B-1 approximation, Nordheim numerical method, Wigner rational approximation Calculation characteristics: spectrum calculation in mixture-pebble region of core, cross section calculation for each nuclide in the core (with separate nuclide labels assigned to oxygen in fuel, dry air, and water vapor), resonance materials are 235U and 238U. A.6.2 COMBINE reflector cross section calculation. Type of calculation: coarse-group cross section calculation Geometry: NA Theory used: transport Method used: B1 approximation Calculation characteristics: given 235U fission spectrum, thermal and fast buckling values, compute group microscopic cross sections in reflector materials; no resonance materials are present A.6.3 PEBDAN calculation of Dancoff factors. Type of calculation: computation of Dancoff factors Geometry: spheres in finite media Theory used: transport Method used: Monte Carlo ray tracing Calculation characteristics: given the fuel particle characteristics and packing fraction, find the intrapebble Dancoff factor; given the pebble characteristics and packing distribution, find the interpebble Dancoff factor

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A.6.4 PEBBED calculation of keff. Type of calculation: full reactor Geometry: cylindrical Theory used: diffusion Method used: finite-difference Calculation characteristics: given detailed description of core and reflector regions, compute keff; six energy groups, two thermal groups, homogenized core region, reflector regions defined by HTR-10 group, two spectral zones A.6.5 PEBBLES calculation of angle of repose of conical upper surface of pebble bed. Type of calculation: positions of all pebbles in the HTR-10 core Geometry: 3-d cylindrical Theory used: discrete elements method Method used: discrete elements method Calculation characteristics: given the dimensions of the pebble-bed cavity and the pebbles, and the coefficient of friction between pebbles, compute the locations of the pebbles in the core, including the pebbles in the cone on top A.7 Other assumptions and characteristics – see detailed description in Section 3.1.

A.8 Typical Input Listings for each code system type. Because of their size, the inputs are included in a separate file. ASCII text inputs are also available in the directory of this evaluation.

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APPENDIX B: CALCULATION OF BASE-CASE ATOMIC NUMBER DENSITIES IN THE CORE REGION

This appendix presents the calculation of the atomic number densities for Region 99 in the PEBBED model, which is the core region, where the mixture of fuel and dummy balls is located. First, some data from Tables 1.2 and 1.4 are repeated: Fuel enrichment = 17% by weight Kernel density = 10.4 g/cm3 Buffer layer density = 1.1 g/cm3 IPyC density = 1.9 g/cm3 SiC density = 3.18 g/cm3 OPyC density = 1.9 g/cm3 Fuel matrix density = 1.73 g/cm3 Dummy matrix density = 1.84 g/cm3

Pebble packing fraction = 61% Fuel loading per fuel pebble = 5 g U Number of kernels in a fuel pebble = 8335 Ratio of fuel pebbles to dummy pebbles: 57:43 Boron concentration in kernel (by weight) = 4 ppm Boron concentration in fuel matrix (by weight) = 1.3 ppm Boron concentration in dummy-pebble matrix (by weight) = 0.125 ppm Kernel radius = 0.025 cm Buffer layer thickness = 0.009 cm IPyC layer thickness = 0.004 cm SiC layer thickness = 0.0035 cm OPyC layer thickness = 0.004 cm The weights of one gram-mole of the nuclides and elements used in the calculations are taken from the Chart of the Nuclides (Reference B1). B.1 Nuclides in Kernel First, the computational cell is defined. It is taken to comprise an average pebble and the volume of coolant associated with it. The volume of a pebble is 4πr3/3=113.097 cm3. Since the packing fraction is 0.61, the total volume of the cell is 113.097 cm3/0.61=185.405 cm3. The average pebble contains 57% of a fuel pebble and 43% of a dummy pebble. The uranium is 17% 235U by weight. Ignore all isotopes except 235U and 238U. Let 235N% and 238N% be the atom fractions of 235U and 238U, respectively, 235A and 238A be the molecular weights of 235U and 238U,

respectively, and A be the average molecular weight of the mixture of the two isotopes. Then

235 2350.17A N A= % and 238 2380.83A N A= % ,

so that 235 235 235

238 238 238

235.0439280.170.83 238.050788

N A N xAA N A N x= =% %

% %,

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or 235

238

0.207439NN

=%

%.

But 235 238 1N N+ =% % . This and the previous equation are solved simultaneously to yield

235 0.171801N =% and 238 0.828199N =% . The mass of uranium in an average UO2 molecule is (0.171801x235.043928 + 0.828199x238.050788)(g/mole) x 1 mole/6.022x1023 molecules = 3.94444x10-22 g. The ratio of fuel pebbles to dummy pebbles is 57:43. As defined above, the computational cell contains 57% of a fuel pebble and 43% of a dummy pebble, plus the volume of coolant associated with one pebble in the core. The number of U atoms, or UO2 molecules, in a computational cell is thus (NU)cell = 0.57x5 g/3.94444x10-22 g/molecule = 7.22536x1021 molecules. Then the atomic number densities are (nU-235)cell = 0.171801 x 7.22536x1021 x 10-24/185.405 atoms/barn-cm = 6.69520x10-6 atoms/barn-cm and, analogously, (nU-238)cell = 3.22755x10-5 atoms/barn-cm and 5( ) 7.79414 10 atoms/b-cmkernels

O celln x −= . The oxygen in the kernel is given a different material label from the oxygen in the water vapor in the PEBBED and COMBINE inputs because the cross sections are different in different materials. Similarly, identical nuclides in the core and reflector are given different labels because their cross sections will be different in the two spectral zones. The boron concentration in the kernel is 4 ppm by weight of uranium. The mass of the average boron atom is (10.811g/mole)/6.022x1023 atoms/mole) = 1.79525x10-23 g/atom. Then the ratio of the number of boron atoms to the number of UO2 molecules is

6 225

23

4 10 3.94444 10 8.78861 101.79525 10

x x x xx

− −−

− =

and the number of atoms of boron in the computational cell from kernels is

5 17( ) 8.78861 10 ( ) 6.35009 10 atomskernelsB cell U cellN x N x−= = .

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B.2 Nuclides in Fuel Particle Layers The number of TRISO particles in the cell is 0.57x8335=4751. The volume of the buffer layer is 4π[(0.025 + 0.009)3 – 0.0253]/3 cm3 = 9.91864x10-5 cm3, so the total volume of buffer zone in the cell is 4751 x 9.91864x10-5 cm3 = 0.471235 cm3. The density of the buffer zone is 1.1 g/cm3, and the boron content is 1.3 ppm (by weight). The masses of the average carbon and boron atoms are 12.011/6.022x1023 g = 1.99452x10-23 g and 10.811/6.022x1023 g = 1.79525x10-23 g, respectively; by number, a boron content of 1.3 ppm by mass is 1.3x10-6 x 12.011/10.811 = 1.44430x10-6 by number. Then (nC)buffer = 1.1 g/cm3 / (1.99452x10-23 + 1.4443x10-6 x 1.79525x10-23) g = 5.51510x1022 atoms/cm3 and the total number of C atoms in the cell from buffer zones is

22 3 3 22( ) 5.51510 10 atoms/ 0.471235 2.59891 10 atomsbufferC cellN x cm x cm x= = .

It is assumed that the graphite in the carbon layers of the TRISO particle have the same boron concentration as the graphite fuel matrix. This assumption can only affect the boron content of the computational cell slightly. The number of boron atoms in the cell from buffer zones is

6 22 16( ) 1.4443 10 2.59891 10 atoms=3.75361x10 atomsbufferB cellN x x x−= .

In similar fashion, the carbon, silicon, and boron contents of the remaining zones are found. The silicon carbide layer is assumed to contain no boron.

22( ) 2.95135 10 atomsIPyCC cellN x=

16( ) 4.26264 10 atomsIPyC

B cellN x=

22( ) 1.57790 10 atomsSiCC cellN x=

22( ) 1.57790 10 atomsSiC

Si cellN x=

22( ) 4.30778 10 atomsOPyCC cellN x=

16( ) 6.22173 10 atomsOPyC

B cellN x= B.3 Nuclides in Fuel Matrix and Dummy-pebble Matrix The total volume of TRISO particles in the cell is 4751x4π(0.0455 cm)3/3 = 1.87460 cm3. The volume of one pebble is 4π(3 cm)3/3 = 113.097 cm3, of which in the computational cell 57% is fuel pebble and 43% is dummy pebble. Then the volume of fuel matrix in the cell is 0.57x113.097 cm3 -1.87460 cm3 = 62.5907 cm3, and the volume of dummy matrix in the cell is 0.43x113.097 cm3 = 48.6317 cm3. From the given densities of the graphite matrix in the fuel and dummy pebbles, 1.73 g/cm3 and 1.84 g/cm3, respectively, and the mass of the average carbon atom, it is found that the atomic number densities of carbon in the fuel and dummy matrix materials are 8.67377x1022 atoms/cm3 and 9.22528x1022 atoms/cm3, respectively. Then the carbon atomic inventories are

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24( ) 5.42897 10 atomsfuel matrixC cellN x=

and

24( ) 4.48641 10 atomsdummy matrixC cellN x= .

In the fuel matrix the boron concentration is 1.3 ppm, but in the dummy matrix it is 0.125 ppm by weight, which is equivalent to 0.138875 ppm by number. These proportions give the boron content of the matrix materials:

18( ) 7.84106 10 atomsfuel matrixB cellN x=

and

17( ) 6.23050 10 atomsdummy matrixB cellN x= .

B.4 Total Silicon, Carbon and Boron Number Densities When all the contributions to the silicon, carbon and boron contents of the cell are summed and divided by the cell volume, and the result is converted to atoms/barn-cm, the resulting densities are

2( ) 5.40964 10 atoms/b-cmC celln x −=

910( ) 9.91914 10 atoms/b-cmB celln x −− =

and

811( ) 3.99258 10 atoms/b-cmB celln x −− = .

B.5 Nuclides in Air The densities of the gaseous constituents are found from the specified conditions and the assumption that the air is saturated with water vapor. At 15 °C (288.15 K), at a relative humidity of 100% the water vapor density is (Reference B2) 1.28343x10-5 g/cm3 and the vapor pressure is 1.70507x10-3 MPa. (Reference B2 gives data in British units; 15 °C = 59 °F, at which the saturation pressure is 0.2473 lbf/in2 (psi) and the saturated vapor specific volume is 1248.1 ft3/lbm. The conversion factors used to obtain metric units are (Reference B3) 6894.7572 N/m2-psi and 16.018463 kg-ft3/lbm-m3) At this mass density, the number density of water vapor molecules is found to be 4.29014x1017 molecules/cm3, and the average number densities of hydrogen and oxygen from vapor in the cell are

7( ) 3.34631 10 atoms/b-cmvaporH celln x −=

and

7( ) 1.67316 10 atoms/b-cmvaporO celln x −= .

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The pressure of the dry air is then 0.1013 MPa – 1.70507x10-3 MPa = 9.95949x10-2 MPa. The ideal gas law (with R = 8.314 Pa-m3/mole-K) (Reference B4) then gives a molecular density of 2.50351x1019 molecules/cm3. The proportions given in Section 3.1.3 lead to the following atomic densities:

5( ) 1.52478 10 atoms/b-cmN celln x −=

6( ) 4.09052 10 atoms/b-cmairO celln x −=

8( ) 9.11929 10 atoms/b-cmAr celln x −=

REFERENCES FOR APPENDIX B

B1. Chart of the Nuclides, Thirteenth Edition, General Electric Company, 1984. B2. Joseph H. Keenan and Frederick G. Keyes, Thermodynamic Properties of Steam, Wiley, New York,

Thirty-fifth Printing, 1963. B3. Donald R. Pitts and Leighton E. Sissom, Heat Transfer, Schaum’s Outline Series in Engineering,

McGraw-Hill, New York, 1977. B4. Samuel Glasstone and Alexander Sesonske, Nuclear Reactor Engineering, Third Edition, Van

Nostrand Reinhold, New York, 1981.

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APPENDIX C: Angle of Repose for Criticality Considerations in Pebble Bed Reactors

Joshua J. Cogliati and Abderrafi M. Ougouag Introduction The pebble-bed reactor (PBR) concept consists of a cylindrical or annular vat within which spherical fuel elements (SFEs) or “pebbles” are randomly packed. The packing distribution and the shape of the pile formed depend on the physics that governs the dropping of the pebbles and their subsequent settling into positions of equilibrium. The shape of the pile may include a cone at the top, such as is observed when pouring granular materials on a flat surface or within a vessel. The angle that the cone makes with the horizontal plane is named the “angle of repose.” The neutronics of any reactor, including pebble-bed reactors, depends on the geometric configuration of the fuel, moderator, and other materials within the reactor core and in its proximity. Therefore, the existence of a fuel cone on top of the reactor, or its absence, and the shape of the cone, may have consequences on the criticality of the reactor. In this section the shape and size of the fuel cone are determined through modeling. In this remainder of this section, the principal phenomena underlying the formation of the angle of repose in a pile are briefly reviewed and the most elementary models that represent them are identified. Then the approach used in this work to model the angle of repose in the HTR-10 reactor is presented. As expected, the angle of repose correlates with the static friction of the materials under consideration. However, high precision in the results, barring unrealistically long computer runs and repetitions thereof, is precluded by the apparently stochastic nature of the pebble settling phenomena and the small size of the sample at hand when modeling the HTR-10 reactor top surface. Modeling Angle of Repose in Discrete Media When particulate matter is poured on a flat surface, a cone forms at the top of the materials that have been deposited. The angle of repose is defined as the angle the cone surface makes with the horizontal plane on which the material is resting. If additional material is added that would increase that angle, some material flows until the cone becomes shallower (i.e., decreases in height) and the angle of repose returns to a value typical of the material under consideration. When the angle the pile makes with the horizontal plane is less than the angle of the repose for the material, then addition of material on the peak of the cone can result in an increase until the typical angle of the repose for the material is reached. The phenomenon is governed in large part by the static friction between the granules that constitute the material. An understanding of these phenomena has been available for a long time. The first discussion on record of the angle of repose for granular matter is due to Coulomb. He rightly attributed the behavior to a balance between friction forces and the component of the weight of sheets of individual grains along the surface of the cone. The reference to Coulomb’s work and to other much more modern models can be found in the book by Duran(C1). For the purpose of the present report, models other than those described in that book are preferred. Here the motion of individual grains, or equivalently, pebbles, is considered. This preferred approach is discussed next. Modeling Angle of Repose with the PEBBLES Code The packing and flow of pebbles in a PBR has been modeled at the Idaho National Laboratory using a deterministic model that incorporates the motion of individual pebbles in response to the resultant of all forces acting upon them. The method has been embodied into the PEBBLES(C2) code akin to discrete element methods. These methods account explicitly for the weight and all the contact forces on individual

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pebbles and compute the motion that results from the resultant force for small time steps. The computation is repeated for all pebbles until the total kinetic energy drops below a preset threshold. The code then allows the tallying of pebble positions and the computation of zone-based averages (e.g., packing density). When a path is provided at the bottom of the vat, the code is capable of predicting flow patterns. The PEBBLES code was used to model the pouring of individual pebbles in the final stages of the filling of a vat with the dimensions and shape of the HTR-10 reactor core cavity. The initial stages of the filling, which do not affect the final presence or absence of a cone or the shape of such a cone, were modeled as an initial dump of 14,000 pebbles into the vat. As the final stages of filling were modeled, a cone was indeed observed to form in the model results, as discussed later. The static friction factor for the pebbles was taken as 0.35,(C3) which is representative of the friction factor of cold (20 ºC) graphite pebbles in air. The case delineated in the previous paragraph defines the behavior of the pebble bed in a deterministic sense. In reality, although the methods used in this work are deterministic, the modeling of a very large number of pebbles that interact then fall in place may result in different configurations each time the process is repeated. That is, the model behaves stochastically in the same sense as an analog Monte Carlo model. Therefore, it should be expected that the results produced by running the PEBBLES code should be regarded as instances of a stochastic system, and the results (i.e., angles of repose) as members of an ensemble. However, the number of pebbles used in the model is large (even in the modeling of the final stages of filling the vat). Therefore, the deviation of the angle of repose from its average value over a very large number of instances is expected to be low. Results Using the method described above, the angle of repose has been computed for the case discussed in the previous sub-section. The angle of repose is shown graphically in Figure C.1.

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Figure C.1. Graphic Representation of Pebble Cone on HTR-10 with 0.35 Friction Coefficient.

(crosses represent centers of pebbles)

The green line has a slope of 19.5° from horizontal. Figure 1 is actually a cross section of a three-dimensional cone that is not exactly symmetric azimuthally. Measuring the angle of repose on similar cross sections at numerous azimuthal positions around the circumference shows angles from 68° to 73°. Conclusions The angle of repose for a pile of graphite pebbles was computed by simulation using a discrete element code. References

C1. Jacques Duran, Sands, powders, and grains: an introduction to the physics of granular materials, Springer-Verlag, New York (2000). C2. Abderrafi M. Ougouag and Joshua J. Cogliati, “Methods for Modeling the Packing of Fuel Elements

In Pebble Bed Reactors,” Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, Palais des Papes, Avignon, France, September 12-15, 2005, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2005).

C3. R. E. Nightingale, Nuclear Graphite, Academic Press, New York (1962).

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APPENDIX D: NUCLEAR CONSTANTS Avogadro's Number: 6.022 x 10

23 atoms/gram-mole

TABLE D.1. Atomic Weights. Nuclide or Isotope Atomic Weight H 1.00794 Li 6.941 B 10.811 10

B 10.0129 11

B 11.0093 C 12.011 O 15.9994 Si 28.0855 235

U 235.043928 238

U 238.050788

"Chart of the Nuclides," Thirteenth Edition, General Electric Company, 1984.

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EVALUATION OF THE INITIAL CRITICAL CONFIGURATION OF THE ... · graphite-reflected, HTGRs, HTR-10, Pebble-bed reactor, test reactor, uranium dioxide SUMMARY INFORMATION 1.0 DETAILED