The Pennsylvania State University

The Graduate School

College of Engineering

OPTIMIZATION OF COUPLED MULTIPHYSICS METHODOLOGY FOR SAFETY

ANALYSIS OF PEBBLE BED MODULAR REACTOR

A Dissertation in

Nuclear Engineering

by

Peter Tshepo Mkhabela

© 2010 Peter Tshepo Mkhabela

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2010

ii

The dissertation of Peter Tshepo Mkhabela was reviewed and approved* by the following:

Kostadin N. Ivanov

Distinguished Professor of Nuclear Engineering

Thesis Advisor Co-Chair of Committee

Maria N. Avramova Assistant Professor of Nuclear Engineering

Co-Chair of Committee

Robert M. Edwards Professor of Nuclear Engineering

Yousry Y. Azmy Professor of Nuclear Engineering

Frederik Reitsma Pebble Bed Modular Reactor (Pty) Ltd

Special Member

Ntate D. Kgwadi North-West University

Special Member

Michael Adewumi

Professor of Petroleum and Natural Gas Engineering

Arthur Motta

Professor of Nuclear Engineering and Material Science and Engineering

Chair of Nuclear Engineering Program

*Signatures are on file in the Graduate School

iii

ABSTRACT

The research conducted within the framework of this PhD thesis is devoted to the high-

fidelity multi-physics (based on neutronics/thermal-hydraulics coupling) analysis of Pebble Bed

Modular Reactor (PBMR), which is a High Temperature Reactor (HTR). The Next Generation

Nuclear Plant (NGNP) will be a HTR design. The core design and safety analysis methods are

considerably less developed and mature for HTR analysis than those currently used for Light

Water Reactors (LWRs). Compared to LWRs, the HTR transient analysis is more demanding

since it requires proper treatment of both slower and much longer transients (of time scale in

hours and days) and fast and short transients (of time scale in minutes and seconds). There is

limited operation and experimental data available for HTRs for validation of coupled multi-

physics methodologies.

This PhD work developed and verified reliable high fidelity coupled multi-physics models

subsequently implemented in robust, efficient, and accurate computational tools to analyse the

neutronics and thermal-hydraulic behaviour for design optimization and safety evaluation of

PBMR concept The study provided a contribution to a greater accuracy of neutronics

calculations by including the feedback from thermal hydraulics driven temperature calculation

and various multi-physics effects that can influence it. Consideration of the feedback due to the

influence of leakage was taken into account by development and implementation of improved

buckling feedback models. Modifications were made in the calculation procedure to ensure

that the xenon depletion models were accurate for proper interpolation from cross section

tables.

To achieve this, the NEM/THERMIX coupled code system was developed to create the

system that is efficient and stable over the duration of transient calculations that last over

several tens of hours.

Another achievement of the PhD thesis was development and demonstration of full-

physics, three-dimensional safety analysis methodology for the PBMR to provide reference

solutions. Investigation of different aspects of the coupled methodology and development of

iv

efficient kinetics treatment for the PBMR were carried out, which accounts for all feedback

phenomena in an efficient manner. The OECD/NEA PBMR-400 coupled code benchmark was

used as a test matrix for the proposed investigations.

The integrated thermal-hydraulics and neutronics (multi-physics) methods were extended

to enable modeling of a wider range of transients pertinent to the PBMR. First, the effect of the

spatial mapping schemes (spatial coupling) was studied and quantified for different types of

transients, which resulted in implementation of improved mapping methodology based on user

defined criteria. The second aspect that was studied and optimized is the temporal coupling

and meshing schemes between the neutronics and thermal-hydraulics time step selection

algorithms. The coupled code convergence was achieved supplemented by application of

methods to accelerate it. Finally, the modeling of all feedback phenomena in PBMRs was

investigated and a novel treatment of cross-section dependencies was introduced for

improving the representation of cross-section variations.

The added benefit was that in the process of studying and improving the coupled multi-

physics methodology more insight was gained into the physics and dynamics of PBMR, which

will help also to optimize the PBMR design and improve its safety. One unique contribution of

the PhD research is the investigation of the importance of the correct representation of the

three-dimensional (3-D) effects in the PBMR analysis. The performed studies demonstrated

that explicit 3-D modeling of control rod movement is superior and removes the errors

associated with the grey curtain (2-D homogenized) approximation.

v

TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................................VIII

LIST OF TABLES ......................................................................................................................XI

ACKNOWLEDGEMENTS ........................................................................................................XII

CHAPTER 1 INTRODUCTION ...................................................................................... 1

1.1 Problem identification ...................................................................................................... 1 1.2 Literature review ............................................................................................................. 1

1.2.1 History of gas-cooled reactors ............................................................................ 1 1.2.2 Current developments in gas-cooled reactors ..................................................... 2 1.2.3 Neutron kinetics methods ................................................................................... 5 1.2.4 Cross section generation and modeling .............................................................. 9

1.3 Objectives and layout of the thesis ................................................................................ 12

CHAPTER 2 ENERGY GROUP SENSITIVITY STUDIES ........................................... 13

2.1 Introduction ................................................................................................................... 13 2.2 Procedure ..................................................................................................................... 18

2.2.1 NEM diffusion solution ...................................................................................... 18 2.2.2 MCNP probabilistic transport solution ............................................................... 21

2.3 Results ......................................................................................................................... 26 2.3.1 Results at 300K ................................................................................................ 28 2.3.2 Results at 1000K .............................................................................................. 29

2.4 Conclusion .................................................................................................................... 30

CHAPTER 3 DESCRIPTION OF THE NEM/THERMIX CODE SYSTEM .................... 31

3.1 Description of NEM ....................................................................................................... 31 3.1.1 Cylindrical geometry ......................................................................................... 31 3.1.2 Steady state solution procedure ....................................................................... 34 3.1.3 Description of the transient solution .................................................................. 35

3.2 Description of THERMIX-DIREKT ................................................................................. 37 3.3 Description of coupling scheme..................................................................................... 41 3.4 Description of coupling scheme..................................................................................... 41 3.5 Status of verification...................................................................................................... 45 3.6 Conclusions .................................................................................................................. 47

CHAPTER 4 MULTIPHYSICS CODE DEVELOPMENT AND OPTIMIZATION........... 48

4.1 Introduction ................................................................................................................... 48 4.2 Thermal-hydraulic modeling .......................................................................................... 48

4.2.1 Calculation of the kernel temperature ............................................................... 48 4.2.2 NEM/THERMIX kernel model ........................................................................... 53 4.2.3 New NEM fuel kernel temperature model.......................................................... 56 4.2.4 TINTE kernel model ......................................................................................... 58

4.3 Optimizing temporal coupling schemes ......................................................................... 59 4.4 Improved and efficient feedback modeling..................................................................... 61

4.4.1 Thermal and fast buckling calculation ............................................................... 61 4.5 Xenon and Iodine models ............................................................................................. 68

vi

4.5.1 Final 5-D steady problem convergence ............................................................. 71 4.6 Transient cross section modeling .................................................................................. 72 4.7 Control rod movement modeling ................................................................................... 74

4.7.1 Linear Rod Motion Model (ROMO) ................................................................... 75 4.7.2 Volume-weighting method ................................................................................ 78 4.7.3 Flux-volume weighting Method ......................................................................... 78

4.8 Decay heat calculation .................................................................................................. 85 4.9 Conclusions .................................................................................................................. 88

CHAPTER 5 NEA/OECD PBMR-400 COUPLED CODE BENCHMARK .................... 89

5.1 Introduction ................................................................................................................... 89 5.2 Steady State Cases ...................................................................................................... 91

5.2.1 Exercise 1 (Case S-1): Neutronics solution with fixed cross sections................. 91 5.2.2 Exercise 2 (Case S-2): Thermal hydraulic solution with given power ................. 92 5.2.3 Exercise 3 (Case S-3): Combined neutronics thermal hydraulics calculation ..... 93

5.3 Comparison of steady state results ............................................................................... 93 5.4 Transient cases .......................................................................................................... 100

5.4.1 DLOFC........................................................................................................... 100 5.4.2 Control rod ejection (CRE).............................................................................. 102 5.4.3 Control rod withdrawal (CRW) ........................................................................ 108

5.5 Conclusions ................................................................................................................ 110

CHAPTER 6 THREE-DIMENSIONAL SPATIAL MODELS ....................................... 111

6.1 Introduction ................................................................................................................. 111 6.2 Optimizing spatial coupling schemes ........................................................................... 112 6.3 Reference neutronic solution ....................................................................................... 114

6.3.1 DORT Model of PBMR-268 ............................................................................ 115 6.3.2 TORT Model of PBMR-268 ............................................................................. 116

6.4 Results for 3-D spatial modeling .................................................................................. 119 6.5 PBMR-400 Steady state 3-D modeling with NEM ........................................................ 126 6.6 3-D spatial modeling of PBMR-400 with TORT-TDS .................................................... 131

6.6.1 The LMW 3-D transient problem without feedback .......................................... 132 6.7 Reference thermal-hydraulic models ........................................................................... 136 6.8 ATTICA3D model ........................................................................................................ 138

6.8.1 Description of ATTICA3D ............................................................................... 138 6.8.2 Results of ATTICA3D modeling ...................................................................... 140

6.9 Conclusions ................................................................................................................ 141

CHAPTER 7 CFD TRANSIENT MODELING OF PBMR-400 .................................... 142

7.1 Description of CFD model ........................................................................................... 142 7.2 PHOENICS calculations.............................................................................................. 143 7.3 DASPK solution .......................................................................................................... 145

7.3.1 Results and Discussion of CFD modeling ....................................................... 145 7.3.2 Transient calculations ..................................................................................... 150

7.4 The 1-D Model PLOFC and DLOFC results ................................................................. 156 7.5 Conclusion .................................................................................................................. 158

CHAPTER 8 CONTRIBUTIONS AND FUTURE WORK ........................................... 159

8.1 Contributions .............................................................................................................. 159 8.2 Future work ................................................................................................................ 162

REFERENCES ....................................................................................................................... 163

vii

APPENDIX A .......................................................................................................................... 167

APPENDIX B .......................................................................................................................... 171

viii

LIST OF FIGURES

Figure 1: PBMR spherical fuel element [41] .............................................................................................. 5

Figure 2: Comparison of cross section representation in PBMR modeling .............................................. 11

Figure 3: Methodology for the energy group sensitivity studies ............................................................... 19

Figure 4: PBMR-400 model in NEM and MCNP ..................................................................................... 21

Figure 5: TRISO particle for PBMR-400 reactor ...................................................................................... 22

Figure 6: Pebble bed fuel sphere with homogeneous distribution of TRISO particles .............................. 23

Figure 7: Lattice of TRISO particles in graphite matrix ............................................................................ 24

Figure 8: Axial cross sectional view of PBMR-400 reactor in MCNP model ............................................. 25

Figure 9: Horizontal cross sectional view of PBMR-400 reactor in MCNP model ..................................... 25

Figure 10: Comparison of K-Effective for 300K and 1000K with upscattering correction .......................... 26

Figure 11: Flux ratio for MCNP to NEM .................................................................................................. 27

Figure 12: Radial flux distribution at 300K .............................................................................................. 28

Figure 13: Axial flux distribution at 300K ................................................................................................. 29

Figure 14: Axial flux distribution at 1000K ............................................................................................... 29

Figure 15: Radial flux distribution at 1000K ............................................................................................ 30

Figure 16: Old transient Xenon model .................................................................................................... 44

Figure 17: Old NEM/THERMIX coupling scheme.................................................................................... 45

Figure 18: Comparison of axial power profiles for steady state ............................................................... 46

Figure 19: Comparison of radial power profiles for steady state .............................................................. 47

Figure 21: Convergence of temperature after modifications .................................................................... 49

Figure 20: Temperature transfer to NEM ................................................................................................ 49

Figure 22: Power during the control rod ejection transient [23] ................................................................ 51

Figure 23: Representative micro-system for shell calculation [45] ........................................................... 52

Figure 24: Simplified kernel heat transfer model ..................................................................................... 57

Figure 25: Convergence of the buckling distribution ............................................................................... 64

Figure 26: Simplified representation of spectral zones of PBR ................................................................ 65

Figure 27: New Xenon model ................................................................................................................. 69

Figure 28: Xenon changes in response to power changes [46] ............................................................... 70

Figure 30: Steady state convergence of 3-D map of Xenon number densities ......................................... 71

Figure 29: Xenon model processing for cross section interpolation ......................................................... 71

Figure 31: The k-effective convergence .................................................................................................. 72

Figure 32: Partially rodded nodes ........................................................................................................... 75

Figure 33: Simulation of control rod movement ....................................................................................... 76

Figure 34: Material homogenisation approach ........................................................................................ 79

Figure 35: Flux estimation for partially rodded nodes .............................................................................. 80

Figure 36: Flowchart for the control rod model........................................................................................ 82

Figure 37: Index of rod position .............................................................................................................. 83

Figure 38: Tracking of rod tip during DLOFC transient ............................................................................ 83

Figure 39: Results testing for the flux approximation .............................................................................. 84

Figure 40: Cusping effects during rod movement ................................................................................... 85

Figure 41: Decay heat behaviour (% of fission power) ............................................................................ 86

Figure 42: Log interpolation data points and time step error estimation ................................................... 87

Figure 43: New NEM/THERMIX feedback model.................................................................................... 88

Figure 44: PBMR-400MWth reactor ....................................................................................................... 90

Figure 45: Neutronic model for case S1 ................................................................................................. 92

Figure 46: Comparison of k-eff for OECD PBMR-400 Case S1 ............................................................... 94

Figure 47: Axial power distribution in the PBMR-400 reactor .................................................................. 95

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Figure 48: Radial power distribution for PBMR-400 reactor .................................................................... 96

Figure 49: Radial thermal flux distribution in PBMR-400 reactor ............................................................. 97

Figure 50: Comparison of outlet temperature for OECD PBMR-400 exercise 2 ....................................... 98

Figure 51: NEM axial power distribution for cases S1 and S3 ................................................................. 99

Figure 52: DALTON axial power distribution for cases S1 and S3 ........................................................... 99

Figure 53: Power during reactor scram................................................................................................. 101

Figure 54: Maximum temperatures reached during LOFC accidents ..................................................... 102

Figure 55: Power evolution for combinations of multiple time step sizes during the CRE transient ........ 103

Figure 56: Power evolution for single rod ejection for Case 5c .............................................................. 104

Figure 57: Power for two rods CRE ...................................................................................................... 105

Figure 58: Maximum temperature for two rods CRE ............................................................................. 105

Figure 59: Case 5b with all rods ejected ............................................................................................... 106

Figure 60: Maximum temperature for all rods CRE without kernel model .............................................. 106

Figure 61: Sensitivity on number of rods ejected .................................................................................. 107

Figure 62: Effect of fuel kernel model for CRE transient ....................................................................... 107

Figure 63: Maximum temperature for all rods CRE with the kernel model ............................................. 108

Figure 64: Case 5a control rod withdrawal transient ............................................................................. 109

Figure 65: Fuel temperature during CRW transient ............................................................................... 109

Figure 66: Reduction of 3-D model to 2-D ............................................................................................ 113

Figure 67: Neutronic model for PBMR-268 ........................................................................................... 116

Figure 68: Actual size of rods in 3-D model .......................................................................................... 117

Figure 69: Top view of the PBMR-400 reactor ...................................................................................... 117

Figure 70: Control rod equivalent volume ............................................................................................. 118

Figure 71: K-effective for 3-D modeling of PBMR-268 reactor............................................................... 119

Figure 72: 3-D view of thermal flux at top of PBMR-268 ....................................................................... 120

Figure 73: Azimuthal flux distribution at the top of the reactor ............................................................... 120

Figure 74: Middle core thermal flux distribution for PBMR-268 .............................................................. 121

Figure 75: Azimuthal flux distribution in the middle of the reactor .......................................................... 121

Figure 76: Thermal flux distribution at the bottom of PBMR-268 reactor................................................ 122

Figure 77: Thermal flux distribution at the bottom of PBMR-268 reactor................................................ 122

Figure 78: Top flux distribution PBMR-268 ........................................................................................... 123

Figure 79: Top azimuthal flux distribution comparison .......................................................................... 124

Figure 80: Middle of core flux distribution for PBMR-268 ...................................................................... 124

Figure 81: Flux comparison PBMR-268 ................................................................................................ 125

Figure 82: Bottom of core flux distribution for PBMR-268...................................................................... 125

Figure 83: Comparison of 10 sectors azimuthal flux distribution ............................................................ 126

Figure 84: PBMR-400 flux distribution at the top with ORO ................................................................... 127

Figure 85: PBMR-400 flux distribution at the top with ARI ..................................................................... 128

Figure 86: PBMR-400 flux distribution at rods with ORO....................................................................... 128

Figure 87: PBMR-400 Flux distribution at rods with ARI ....................................................................... 129

Figure 88: PBMR-400 Flux distribution bottom core with ORO .............................................................. 129

Figure 89: PBMR-400 Flux distribution bottom core with ARI ................................................................ 130

Figure 90: Middle PBMR-400 core ARI................................................................................................. 130

Figure 91: Middle of PBMR-400 core ORO........................................................................................... 131

Figure 92: PBMR-400 model using TORT-TDS .................................................................................... 132

Figure 93: Source convergence for the LMW problem .......................................................................... 133

Figure 94: Power for the LMW benchmark using TORT-TDS ................................................................ 134

Figure 95: LMW comparison of results ................................................................................................. 135

Figure 96: Reactor period for the LMW problem ................................................................................... 135

Figure 97: Comparison of axial thermal fluxes for Case S-3 PBMR-400................................................ 138

Figure 98: Aspects of transport, reaction and phase change in porous media ....................................... 139

x

Figure 99: Aspect of transport, reaction and phase change at pore level .............................................. 140

Figure 100: PHOENICS model of PBMR-400 ....................................................................................... 144

Figure 101: Velocity vectors in steady state PBMR-400 reactor ............................................................ 146

Figure 102: Recirculation at the bottom of the PBMR reactor................................................................ 147

Figure 103: Velocity vectors for steady state flow at the top of PBMR reactor ....................................... 148

Figure 104: Steady state temperature distribution for PBMR-400.......................................................... 149

Figure 105: HTR-10 Maximum temperatures as function of emissivity .................................................. 150

Figure 106: Recirculation at the top of the reactor ................................................................................ 151

Figure 107: Temperature evolution during PLOFC transient in PBMR reactor ....................................... 152

Figure 108: Temperature variation with emissivity at 8hrs for PLOFC ................................................... 153

Figure 109: Variation of temperature with emissivity at 20hrs ............................................................... 153

Figure 110: Variation of maximum temperature with emissivity at 40hrs ............................................... 154

Figure 111: Axial temperature profile at 8hrs ........................................................................................ 154

Figure 112: Heat transferred by radiation ............................................................................................. 155

Figure 113: Radial temperature distribution during PLOFC transient..................................................... 155

Figure 114: Heat transferred by convection during PLOFC ................................................................... 156

Figure 115: Emissivity sensitivity of PBMR surfaces during PLOFC transient ....................................... 156

Figure 116: Radial temperature distribution at different stages of the PLOFC transient ......................... 157

xi

LIST OF TABLES

Table 1: Historical HTRs that have been operated.................................................................................... 2

Table 2: Energy group structures for Xe oscillation studies by Yamasita ................................................. 16

Table 3: Energy group structure used at Fort St. Vrain ........................................................................... 17

Table 4 Thermal cut-off values for cross section generation in COMBINE .............................................. 19

Table 5: Reactor material compositions .................................................................................................. 21

Table 6: Thermal conductivity of fuel kernel layers ................................................................................. 54

Table 7: Specific heat capacity of fuel kernel layers ................................................................................ 54

Table 8: Temperature representation in the fuel element ........................................................................ 55

Table 9: Suggested convergence criteria ............................................................................................... 60

Table 10: Suggested convergence criteria and step sizes for transient cases ........................................ 60

Table 11: 5-dimensional cross section table ........................................................................................... 73

Table 12: Major design and operating characteristics of the PBMR-400 reactor ...................................... 91

Table 13: Comparison of k-eff with and without feedback ....................................................................... 98

Table 14: PBMR-268 axial slices for flux profile .................................................................................... 123

Table 15: Description of levels for 3-D flux distribution .......................................................................... 127

Table 16: Parameter for LMW transient calculation .............................................................................. 134

Table 17: Eigenvalue comparison for case S-3 PBMR-400 ................................................................... 137

Table 18: Exercise 1 DLOFC without scram ......................................................................................... 171

Table 19: Exercise 2 DLOFC with scram .............................................................................................. 172

Table 20: Exercise 3 PLOFC with scram .............................................................................................. 174

Table 21: Exercise 4a load follow without control rod movement .......................................................... 175

Table 22: Exercise 4b load follow with control rod movement ............................................................... 178

xii

Acknowledgements

First of all, I would like to express profound appreciation and deep regards for my Ph.D.

adviser, Prof. Kostadin N. Ivanov, for his vision, professional expertise, and continued

guidance.

I thank Dr Armin Seubert, Dr Andreas Pautz and Mr Antonio Sureda Sureda from GRS

mbH in Germany for their technical support.

I would like to thank Profs. Robert M. Edwards, Maria N. Avramova, Yousry Y. Azmy,

Michael Adewumi and Ntate D. Kgwadi as well as Mr. Frederik Reitsma for their time and effort

in reviewing this work and serving on my doctoral committee.

I thank Messrs. A.J. van der Merwe, Z. Mbambo, Jeff Victor and Dr Alex Tsela of PBMR

(Pty) Ltd for affording me the contract to support my studies.

I want to thank the Penn State University and Department of Mechanical and Nuclear

Engineering for giving me this opportunity.

Lastly, I want to acknowledge my friends and family for being supportive and for having

faith in me.

1

Chapter 1 Introduction

1.1 Problem identification

The Next Generation Nuclear Plant (NGNP) will be a High Temperature (HTR) design. For

HTRs core design and safety analysis methods are considerably less developed and mature

than those currently used for Light Water Reactors (LWRs).

The continued development of the NGNP requires verification of HTR design and its safety

features with reliable, and high fidelity coupled multi-physics models within the framework of

robust, efficient, and accurate code systems. While the coupled three-dimensional (3-D)

neutron kinetics/thermal-hydraulics methodology has been extensively researched and

established for LWR applications, there is a limited experience for HTRs in this area.

Compared to LWRs, HTR transient analysis can be more complex because it is required that

the safety analysis accounts for slower and much longer transients, as well as for fast and

short transients. High fidelity kinetics methods are important for core transients involving

significant variations of the flux shape and these methods have not been systematically

applied to HTRs. These facts motivate establishing consistent, sophisticated and efficient

coupled methodologies for the HTR.

1.2 Literature review

1.2.1 History of gas-cooled reactors

Conventional nuclear reactors have limitations on the outlet temperature, which also limit

the thermal efficiency of these reactors. Other industries could take advantage of the higher

outlet temperature provided by high temperature reactors that could be used as the source of

heat. The HTR, which are graphite-moderated, helium/CO2-cooled and use graphite as

reflector material, are the most appropriate candidates with outlet temperatures ranging

between 750°C to 950°C and with the inlet temperature of about 350°C. Although the high

temperatures of these reactors limit the use of metallic cladding material, fission product

2

retention takes place at the microscopic level of the coated particle that could take a

BISO(double coating) or TRISO (triple coating) configuration using coating material such as

SiC.

Earlier development of the HTR technology was with the Dragon project which was

collaboration between different countries. Other historical HTR developments are summarized

in Table 1.

Table 1: Historical HTRs that have been operated

Plant Thermal

Power (MW)

Electrical

Power (MW)

Fuel Element Site Operation

AVR 46 15 Pebble-shaped fuel

element

Germany 1965-1988

DRAGON 20 Tubular fuel element Great

Britain

1966-1975

Peach

Bottom

115 40 Tubular fuel element USA 1965-1988

THTR 750 308 Pebble-shaped fuel

element

Germany 1985-1988

Fort St.

Vrain

852 342 Block Type fuel

element

USA 1976-1989

1.2.2 Current developments in gas-cooled reactors

Recently Gas-Cooled Reactor (GCR) systems were investigated by the Gas-Cooled

Reactor Technical Working Group (GCR-TWG) to fulfill the goals for sustainability, safety and

reliability, and economics for Generation IV nuclear energy systems. Twenty-one concept

descriptions were evaluated based on the public response to the U.S. Department of Energy

request for information [1] and nineteen of the twenty-one concepts considered are grouped

into the following four concept sets, representing the common capabilities and attributes

among the concepts:

• Modular Pebble Bed Reactor Systems (PBRs)

• Prismatic Fuel Modular Reactor Systems (PMRs)

3

• Very-High-Temperature Reactor Systems (VHTRs)

• Gas-Cooled Fast Reactor Systems (GFRs).

Other efforts for exploring the resurgence of the HTRs are taking place internationally and

have sparked research and development in the area of fuel design and management, reactor

design and operation, as well as safety analysis. Research initiatives are under way with larger

research initiatives undertaken under the EURATOM project.

In China, the modular HTR-10 was designed based on the AVR design technology using

spherical fuel [2]. Its construction was started in 1995 and began operation in 2002. It was

constructed to acquire the know-how of the design, construction and operation of HTRs; to

demonstrate the inherent safety features of modular HTRs and to test the electricity and heat

co-generation technology. Post Irradiation Examination (PIE) for the spherical pebble fuel for

the HTR-10 was recently conducted in Russia [3]. It was shown that the fabricated fuel

elements were suitable for use in the HTR-10 reactor.

In another development, China Institute of Nuclear Energy Technology (INET) was

investigating the Ordered Bed Modular Reactor (OBMR). In this design, the annular reactor

core is filled with an ordered bed of fuel spheres that are packed in a rhombohedral geometry

[4]. The unit cell layer is formed by four spheres lying at the corners of a square and the

individual spheres in subsequent layers fill the cusps formed by them. This reactor is said to

have the most of advantages from both the pebble bed reactor and block type reactor and

decreases core pressure drop.

In Japan, the operating 30 MW High Temperature Test Reactor (HTTR) plant achieved its

criticality in 1998. It achieved the full power of 30MW and reactor outlet coolant temperature of

about 850°C on 7 December 2001. The reactor outlet coolant temperature of 950°C was

reached after several operation cycles on 19 April 2004 [5]. This reactor uses block type

tubular fuel elements and it was constructed for testing of intermediate heat exchangers.

4

Other developments have been reported in the IAEA coordinated research meeting (CRM)

held in Vienna in 2002 on the status of development in the HTR technology by various

countries for the Coordinated Research Project (CRP-5), with the final report on the

benchmark being published in the TECDOC1382 [6].

While the traditional fuel design of HTR fuel is of the TRISO configuration with SiC and

uranium dioxide (UO2) (see Figure 1), there are developments looking into using alternative

kernel composition (UCO) as well as alternative coating layer (ZrC). UCO would help in

decreasing the CO pressure build-up in the particle and ZrC remains more stable at higher

temperature than SiC; thus providing increased margins in accident conditions. This work is

undertaken in the RAPHAEL (ReActor for Process Heat And ELectricity) project that was

established in April 2005 as part of EURATOM’s 6th Framework Programme [7].

It has been experimentally proven that the release of fission products from the TRISO-

coated particles become significant when the temperatures exceed 1600°C. This was tested in

the heat-up experiment of irradiated pebbles to measure release of fission products as a

function of time and temperature under simulated accident conditions. The information thus

obtained was required for the determination of the consequences of accident conditions in an

HTR and the suitability of the fuel. The heat-up experiment was made with the KÜFA device,

which was used in the 1990s at Julich Research Centre FZJ in Germany.

5

Figure 1: PBMR spherical fuel element [41]

The HTR transient analysis is conducted using deterministic methods by employing

computer codes designed to model various accident conditions. The computer codes used for

designing the reactor must be verified and validated for the assurance of safety of operation

under normal operations and accident conditions. To test the performance of different

methodologies adopted by various types of computer codes, there are efforts by various

organizations to engage in comparative analyses using a set of defined design and operating

parameters as it will be shown with the OECD PBMR-400 benchmark in the work presented in

this PhD thesis.

1.2.3 Neutron kinetics methods

There are different approaches to the solution of the time-dependent neutron diffusion

equation for reactor physics calculations. The ultimate goal is to determine the reactivity and

power density distribution of the reactor so that various safety calculations can be conducted.

Three well-known approaches based on flux factorization have been investigated in the past,

which include the Point Reactor Model (PRM), Adiabatic Model (AM) and Improved Quasistatic

Model (IQSM). The IQSM has been shown to be the most accurate of the three options in the

6

analysis of reactivity accidents as compared with the adiabatic and the point kinetic models for

the PBMR type reactor [8].

In the PRM the reactivity is a function of time only since it is assumed that the flux

distribution in the reactor remains constant during the transient. In heterogeneous systems,

this assumption would be flawed since the reactivity of the reactor also depends on the spatial

flux distribution itself. The flux factorization utilized in PRM and IQSM methods assumes slow-

varying change of flux shape. The best approach is to introduce complete finite difference (or

higher order) formulations of flux derivatives and treat the space and time dependence directly

and simultaneously [9]. These formulations are the basis for the NEM code that is being used

in the proposed work described in this PhD thesis.

The neutron energy distribution depends on the spatial temperature distribution, which

depends on the heat/power distribution. This dependence on temperature arises from the

dependence of the macroscopic cross sections that depend on the number densities of

materials in the core, which are used to calculate the power distribution. Hence the

temperature feedback is important in the determination of the power distribution in a reactor

and power is important for the determination of the correct temperature distribution. The

challenge is to develop models that account for temperature and power feedback effects in the

safety analysis. Hence the coupling of neutronic and thermal hydraulic computer codes has

become necessary, resulting in multi-physics tools for reactor dynamics simulation.

Interest is growing in the improvement of capabilities of different computer codes for safety

analysis of HTRs. The neutronic calculation without feedback could lead to incorrect

representation of the behavior of the reactor. Hence the neutronic codes are coupled with

thermal hydraulic codes in safety calculations. Several developments are currently under way

in coupling computer codes in the area of HTRs analysis.

One of the developments is with the HELIOS/MASTER code that is being developed at the

Korean Atomic Energy Research Institute (KAERI) [10]. The code has been modified for

7

analysis of HTRs of both pebble bed and prismatic fuel by introducing the treatment of double

heterogeneity, thermal up-scattering and the effects of spectrum shift.

The VSOP code has been coupled with a CFD code (Flownex) to perform heat source

calculations that were mapped into CFD code. The CFD is used in the design of the PBMR

reactor unit to calculate temperature fields and gradients, pressure drops and flow distributions

for the PBMR [11]. Empirical models were used to model the detailed heat transfer

phenomena and provide detailed results to structural analysis codes.

The CATHARE code was initially developed for the thermal hydraulic analysis of the

French Pressurized Water Reactors. This code has been adapted to evaluate gas cooled

reactor for various transient situations and has been applied in the analysis of the direct

Brayton cycle in the SALSA project [12]

The Time Dependent Neutronic and Temperature (TINTE) code was developed to analyze

the nuclear and transient behavior of high temperature reactors with full neutron, temperature,

and Xenon feedback effects taken into account in two-dimensional r-z geometry. The code

was developed by Kernforschungsanlage (KFA), today Forschungzentrum Jülich (FZJ) in

Germany. The main time-dependent calculation components mentioned here are:

• The neutron flux

• The nuclear heat generations source distribution

• The heat transport from the kernel to the fuel sphere surface

• Global temperature distribution

• The coolant gas flow distribution

• Convection and its feedback on the circulation

• The gas mixing effects including corrosion between gases and solid structures.

The code incorporates numerous material property correlations for graphite and other core

structure materials including the temperature and fast fluence dependence of the thermal

8

conductivity. More details about further coding information can be obtained from the work of

Gerwin 1987 and 1989 [13].

Most of the codes used in HTR safety analysis use one or other form of THERMIX [14] as

thermal hydraulic code because it was one of the initial thermal hydraulic codes designed

specifically for HTRs in Germany. A variety of neutronics calculation codes have been coupled

with this code and few that have been identified are mentioned below.

The commonly investigated cases for the HTR are the loss of forced cooling under

depressurized and pressurized conditions. The equivalence of the LOCA accident in an HTR is

often referred to as depressurized loss of forced cooling (DLOFC) [15] and Depressurized

Conduction Cooling (DCC) [16, 17]. To demonstrate the inherent safety of these types of

reactors, the codes have to demonstrate the capability of conduction, convection and radiative

heat transfer. The main goal is to demonstrate by calculation that the temperature limit of

1600°C would not be exceeded during a severe accident conditions for HTR.

PEBBED, a three-dimensional core simulator code was developed at the Idaho National

Laboratory (INL) specifically for pebble-bed reactor design and depletion analysis, was

recently coupled with THERMIX to determine the maximum temperature of the RPV during a

depressurize conduction cool down accident in a PBMR reactor [18].

The Control Rod Ejection accident in the PBMR-400 was analyzed using the U.S. NRC

neutronics code PARCS coupled to THERMIX-DIREKT. The calculated results were analyzed

using the “Nordheim Fuchs” linear feedback model for one of the cases of the OECD PBMR-

400 Coupled Neutronics and Thermal Hydraulics Transient Benchmark Problem [19].

Coupled steady state and transient calculations for the PBMR-400 design have been

conducted with the diffusion code DALTON [20].

9

Another code system comprised of KIND coupled with THERMIX to model fast transients of

HTR and ZIRKUS with THERMIX for steady state calculations has been developed at IKE in

Germany. This code system includes introduction of LabView for real-time intervention [21].

The NEM code, developed at PSU, has been coupled with THERMIX for analysis of PBMR

reactor in the framework of the OECD PBMR-268 benchmark [22]. The optimisation,

improvements and further development of this code system will be discussed later in this

thesis.

DORT-TD, a numerically optimized transient extension of the well known steady-state SN

code DORT, was developed at the Technical University of Munich, Germany in 2001. In

developing the time-dependent features of DORT-TD, the usage of a fully implicit time

discretization scheme was favored, which required the extension of DORT’s steady-state

formulation by a “time-like” source term, precursor contributions and some modifications to

total cross section and fission spectra. It is now possible e.g. to take into account delayed

neutron spectra different from the prompt fission spectrum as well as to have spatially varying

neutron group velocities [23]. DORT-TD code has been coupled with THERMIX-DIREKT for

analysis of the PBMR-268 and PBMR-400 transients that involve control rod movement.

1.2.4 Cross section generation and modeling

Few-group homogenized cross sections are typically pre-calculated and then stored in

some form or another to be reconstructed during most reactor cycle depletion and transient

analyses. This approach has to be followed since the detailed fine-group transport solutions

needed to calculate the few-group homogenized cross sections are typically very expensive

and therefore on-line calculations will not be practical. Furthermore historical experience has

shown that a fairly accurate representation of the cross sections can be found without too

much effort. More recently the need for increased accuracy and more complex fuel designs

required enhancements in the cross section models to include environment effects, history

effects, the inclusion of cross-term dependency of the state parameters and some additional

base parameters such as the spectral index.

10

Traditionally the most commonly used representations were linear interpolation in tables or

the use of some functional representation such as polynomials of varying order. The aim of

these approaches is to obtain accurate few-group cross section data at the material specific

conditions or state parameters in a computational efficient way. Typically the selected method

or the level of sophistication of the model needed is found as a balance between accuracy, the

complexity of the method and the calculation efficiency.

A derivative program of TINTE code called MGT has been developed with different options

such as: multi-group time-dependent neutron diffusion calculations with cross section

representations as the on-line spectrum calculations, tabulated cross-sections using linear

interpolation and the polynomial cross-section expansions. These options were evaluated for

the OECD PBMR-400 benchmark transient cases such as the loss of flow (case 3) and case 5

where different control rod or control banks were withdrawn to investigate the effect of cross

section representation.

The power excursions for the CRW and CRE cases are shown in Figure 2 for the four

different cross section representations. Significant differences were noticed between the

results although the general behavior and magnitude of the excursions were similar. Part of the

differences was attributed to different spectrum region definitions; especially the mesh on

which the spectrum analysis in the reflector regions was performed had a huge effect. The

results for the polynomial representation were significantly lower than for the other approaches

in the CRE case. In this extreme case the linear extrapolation used for the table

representations was more accurate than the extrapolation of the polynomial function, which

was possibly used beyond its range of application.

11

Figure 2: Comparison of cross section representation in PBMR modeling

These comparisons between the different cross section representation methodologies

provided a first estimation of the uncertainties that can be introduced due to the approximate

representations relative to the use of the on-line fine-group spectrum evaluations. The

availability of an efficient and accurate cross section representation will be required in future

because the on-line spectrum calculation is only practical if a very simple fine-group solution

method is used. The use of more advanced transport models and solution methods will be too

expensive. Ideally the same cross section representation should also be used in the steady-

state core analysis code [24]

The HTGR coordinated projects have investigated different benchmark cases using various

HTR reactors. Different core physics methods ranging from Monte Carlo, transport and

diffusion have been used in the investigations. The challenges identified in these studies are

double heterogeneity, library data and streaming of neutrons. In the thermal hydraulic

12

investigations, the methods include finite volume, finite element and CFD. For all these

methods there are still challenges in the geometry, and validity of correlations [25].

1.3 Objectives and layout of the thesis

The objectives of the PhD work reported in this thesis are as follow:

a) Development and verification of reliable high fidelity coupled multi-physics models

and their subsequent implementation in robust, efficient, and accurate computational

tools for design and safety analysis of the Pebble Bed Modular Reactor (PBMR);

b) Development and demonstration of a PBMR full-physics, three-dimensional safety

analysis methodology for reference solutions;

c) Optimization of different aspects of the coupled multi-physics methodology and

development of efficient kinetics treatment for the PBMR, which accounts for all

feedback phenomena in a sophisticated and efficient manner.

d) Utilization of the OECD/NEA PBMR-400 coupled code benchmark as a test matrix

for the performed investigations.

Chapter 2 introduces energy group studies that were conducted to justify the optimized

group structure for 2-group deterministic core calculation. The energy group structure finds its

application in the NEM/THERMIX code system presented in Chapter 3. The developments and

optimization work are presented in Chapter 4 followed by Chapter 5 where the models are

verified and validated. The reference models for the calculations are conducted in Chapter 6

and Chapter 7 followed by a summary of the contributions of this PhD research and the

envisioned future work in Chapter 8.

13

Chapter 2 Energy group sensitivity studies

2.1 Introduction

The work described in this chapter was conducted by the author as a part of his PhD

research since the OECD PBMR-400 benchmark, which is used as a test framework for the

developments of this thesis, was defined in two energy groups. Thus, the two-energy group

structure had to be optimized for HTR core analysis. The two-group optimization was also a

part of a broader sensitivity study on the few-group energy group structure for HTR core

analysis performed at PSU in collaboration with Idaho National Laboratory (INL). It is known

that the few-group energy structure for reactor analysis provides approximation for the physical

phenomena occurring in the reactor. This is due to the discretization of the continuous energy

spectrum and lumping together the cross-section regions into groups such as the two-group

structure adopted in this study. It was also postulated that the actual cut-off energy for the

thermal energy range could have an effect on the outcome of the diffusion solution i.e. flux, k-

effective, reaction rates and power. Hence, sensitivity study on the energy group structure

optimization was performed first for analysis of the reactor physics.

Probabilistic transport codes utilize statistical approach in obtaining the solution of neutron

problem and it allows continuous energy spectrum calculation. This enables the calculation to

capture all reactions (scattering, capture, fissions, etc.) in transport calculation whereas

diffusion codes require the discretization into few- or multi-group structure. Hence, MCNP was

identified as a transport code that could provide a good reference solution of the reactor

physics problem with all physical phenomena accounted for.

Nuclear reactions between nuclei and neutron are divided into potential and resonance

reactions. Potential scattering occurs when the neutrons are deflected from the nuclei by the

potential field of the nuclei. Resonance reactions occur when a compound nucleus is formed in

an excited state energy composed of the binding energy and neutron kinetic energy. When the

excited state energy corresponds to quantum states of the nucleus, cross sections as function

of enrgy of incoming neutron present sharp peaks around these energies called resonances.

The compound nucleus will decay by neutron emission (scattering), fission and gamma

(capture) reactions with different probabilities.

14

The probabilities of these decay modes are given by:

Γ

Γ=

∑i

i

i

i

λ

λ

2.1

If resonances are well separated, the cross sections are defined by the Breit-Wigner

formula of equation 2.2.

( )( ) 22

0

2

4

1Γ+−

ΓΓ=

EE

gE xn�πσ

2.2

At low energies the v

1 behavior of cross sections is obtained when the Enn 0Γ=Γ is

substituted into Breit-Wigner formula.

Reactor physics calculations require data on the neutron-induced reactions covering the

range of energies of the interest in the calculation for all the materials present in the system.

This data is obtained from different laboratories in the form of Evaluated Nuclear Data Files

(ENDF). The recommended reference data set is the ENDF/B and contains evaluated data set

for each material.

The neutron cross sections are represented by a series of tabulated values and a

method of interpolating between input values. Processing codes are needed to process the

cross section data from the ENDF/B libraries so that the information could be applied to

specific neutron calculation codes. For this purpose we utilize COMBINE Code Version 6

developed at INL [36].

Neutron calculation codes are designed to solve the multi-group diffusion equation of

the form represented in equation 2.3.

( ) ( ) ( ) ( ) ( )∑∑==

→ +Σ+Σ=Σ+∇−N

k

ikfkk

eff

iN

k

kiksitiirSr

krrrD

11

,0

2 φυχ

φφφ 2.3

In these calculations the reaction rates are represented by ( )riti

φΣ where the energy

dependence of the cross section determines the number of reactions occurring per unit volume

in a certain energy range. Hence accurate energy representation of cross sections is required.

15

This is attained by subdividing the neutron energy spectrum into manageable energy group

structures. For solving the diffusion equation, the neutron diffusion code NEM was used.

For a detailed spectrum calculation, high number of energy groups is required with only

a rough approximation on the spatial dependence. A library of fine structure (40-200 groups) is

commonly used by codes including all the cross sections, transfer matrices, resonance

parameters, fission spectra, etc. For few group calculations, usually the thermal group’s cut-off

boundary is where the probability of neutron gaining energy in a collision is very small. For

High Temperature Reactors (HTRs) this boundary is between 2 and 4eV [37]. Spectrum

calculations have to be performed to ensure proper treatment of neutrons slowing down in a

heterogeneous reactor system in the presence of anisotropic scattering and leakage. In HTRs

the fundamental buckling is used in the approximation of the leakage because of the limited

anisotropy of graphite scattering. Once the spectrum calculations are performed, average

constants for few group reactor calculations are produced. These constants are averaged

according to the structure of the energy groups.

The microscopic cross section for energy group I can be expressed as:

( ) ( )

( )∫

∫

−

−=i

i

i

i

E

E

E

E

i

dEE

dEEE

1

1

''

φ

φσσ

2.4

Other constant like the transfer coefficients, Diffusion coefficients are obtained in a

Spectrum calculations are performed for heterogeneous systems having the core and reflector

regions. These calculations are conducted separately for the two regions and coupled to

account for the energy dependent buckling. Although the HTRs are more homogeneous as

compared to other reactor types, heterogeneity exists in HTR. This is encountered on the

microscopic level (fuel kernel) and macroscopic level (fuel pebble) where the term ‘double

heterogeneity’ arises. However, spectrum calculations of HTRs adopt the cell homogenization

approach in the determination of group constants. This is done by ensuring the conservation of

reaction rates in the homogenized cell as compared to the real cell using disadvantage factors

(self-shielding factors). The self-shielding factors, referred to as the Dancoff factors, can be

obtained by using different methods described in references [38] and [39].

16

Space-dependent spectrum calculations in heterogeneous cells could be done using a cell-

transport calculation with a high number of groups. This is adopted by the codes such as

MICROX (200 groups) and THERMOS. In MICROX the leakage is treated with the diffusion

formulation of B1 equations that are transformed into multi-group diffusion equation. The

transport cross section is obtained by weighting the scattering matrices on the reference

current spectrum. Another code that is used in the spectrum calculations is the WIMS code,

which includes 69 group libraries using the MULTICELL approximation. The advantage of this

approach is that very large meshes can be used where the spectrum is constant allowing finer

meshes where the spectral variations are large.

Yamasita et al.[40] modeled Xe oscillations and investigated the effect of the energy

group structures on the oscillations. In that study, the diffusion code CITATION and the burnup

code DELIGHT were used. DELIGHT uses the Garrison-Roos model to perform depletion

calculations and the energy group structures adopted 1, 2, 4, 6, 7 and 8 groups with

boundaries as shown in Table 2.

Table 2: Energy group structures for Xe oscillation studies by Yamasita

Upper

bound

1 2 4 6 7 8

10MeV 1 1 1 1 1 1

183keV 2 2 2

961eV 2 3 3 3

2.38eV 2 3 4 4 4

0.65eV 5 5 5

0.35 6 6

0.105eV 4 6 7 7

0.055eV 8

It was shown that the Xenon absorption cross section was higher below 0.65eV when

the 8 energy-group structure was adopted (5% Σabs,fuel). The concentration of Xenon decreased

as the finer energy group structure (7 and 8) was adopted since the absorption cross sections

17

in the energy below 0.65eV was higher. This was a demonstration that the energy-group

structure had significant effect on the reaction rates calculations.

The number of energy groups could be chosen to be as low as possible to limit the

computer time. The effect of the reduction of groups may not be evident in the calculation of

the k-effective value but in the power distribution in the regions with strong spatial dependence

on the neutron spectrum (e.g. core-reflector boundaries). The choice depends on the type of

phenomena to be separated and the types of nuclear reactions. The partitioning could be used

to distinguish between range of fission source spectrum, a range of unresolved and resolved

resonances, a thermal energy range, etc. For shielding calculations, a high number of fast

groups may be required. The data on Table 3 and Table 4 show energy group structures

adopted in the Fort St. Vrain and THTR calculations respectively.

Table 3: Energy group structure used at Fort St. Vrain

Energy

Boundaries(eV)

Upper Lower 9 groups 7 groups 4 groups

1.50E+07 1.83E+05 1 1 1

1.83E+05 961 2 2 2

961.000 17.6 3

17.600 3.93 4 3 3

3.930 2.38 5

2.380 0.414 6 4 4

0.414 0.1 7 5

0.100 0.04 8 6

0.040 0 9 7

18

Table 3: Energy group structure used for the THTR

Energy Group Structure for THTR Calculations

Energy

Boundaries(eV) Number of Groups

Upper Lower

7

groups

6

groups

4

groups

2

groups

1.00E+07 6.74E+04 1 1 1 1

6.74E+04 748.5 2

748.500 17.6 3 2 2

17.600 1.9 4 3 3 2

1.900 0.37 5 4 4

0.370 0.03 6 5

0.030 0.0025 7 6

2.2 Procedure

2.2.1 NEM diffusion solution

The energy group structure considered for the cross section generation for this problem

was determined by the fixed thermal energy cutoff points in COMBINE. The energy range in

COMBINE is from 0.001 eV to 16.905 MeV spanned by 166 discrete groups. The equations

solved for the energy-dependent fast and thermal neutron spectra are the B-1 and B-3

approximations of the transport equation using two neutron spectrum codes INCITE (thermal

spectrum) and PHROG (fast spectrum). The thermal cutoff points in the code are shown in

Table 4. The sensitivity in the group structure was brought about by the difference in the

treatment of the solution for the cross section generation in the thermal spectrum. The

methodology for performing energy group sensitivity studies is depicted in Figure 3.

19

Table 4 Thermal cut-off values for cross section generation in COMBINE

Lower Energy (eV) Upper Energy (MeV)

0.414eV

16.905

0.532eV

0.683eV

0.876eV

1.125eV

1.44eV

1.86eV

2.38eV

Cross sections were generated at each temperature for all compositions in the reactor

model and used to define materials characteristic of the PBMR reactor. These cross sections

were used in the diffusion equation solved by NEM.

Figure 3: Methodology for the energy group sensitivity studies

20

The diffusion solution for the steady state problem was solved using the NEM code from

the Pennsylvania State University (PSU). NEM approximates node-averaged fluxes and other

group constants using the transverse-integration technique on the nodal equation. The solution

from the code includes the fluxes, power distributions and effective multiplication of the reactor,

which are utilized and presented in this study.

The reactor model adopted in this study is shown in Figure 4 with the ‘void’ replaced by

graphite material from the original model described in the OECD PBMR-400 benchmark

specifications

21

Figure 4: PBMR-400 model in NEM and MCNP

Table 5 shows material compositions provided in the benchmark specifications [41].

Table 5: Reactor material compositions

Isotope

Graphite

(reflector)

all regions

RCS/RSS Core Barrel

C 8.925E-02 8.925E-02 0.0

B-10 1.0E-09 6.0E-06 0.0

Fe (Nat) 0.0 0.0 5.810E-02

Cu (Nat) 0.0 0.0 3.861E-04

Co-59 0.0 0.0 1.544E-04

Si 0.0 0.0 2.488E-04

Ni (Nat) 0.0 0.0 7.996E-03

Mo (Nat) 0.0 0.0 1.733E-03

Mn – 55 0.0 0.0 1.278E-03

Cr (Nat) 0.0 0.0 1.590E-02

2.2.2 MCNP probabilistic transport solution

Explicit modeling of the kernels and the pebbles in the pebble bed reactor were

conducted by Karriem et al. [42] using the MCNP model of the ASTRA facility. In that study, it

was found that the modeling of the individual coatings had no significant influence on the k-

effective. The smearing of the layers of the coating to one has yielded significant reduction in

the calculation time by about 30%. Another model by Colac and Seker [43] was used to model

the HTR-10 using MCNP with explicit modeling of the heterogeneity of the reactor. Although

22

higher computational expense is encountered, there is a capability of modeling the double

heterogeneity of PBMR reactor using MCNP, which is described in the MCNP manual.

Figure 5: TRISO particle for PBMR-400 reactor

Options available in the MCNP solution for the problem allow using heterogeneous or

homogeneous compositions of the reactor material. Although the heterogeneous model for the

reactor would yield more accurate results for the problem, it was realized that the time required

for the solution was enormous. Hence the homogeneous core model was adopted. However,

the fuel TRISO particle (see Figure 5) and the fuel pebble with homogenously distributed

TRISO particles (see Figure 6 and Figure 7) were modeled with MCNP. The input decks for

MCNP for the TRISO particle and the pebble are attached to this thesis as Appendix A.

23

Figure 6: Pebble bed fuel sphere with homogeneous distribution of TRISO

particles

To obtain the unit volume of combined TRISO particles and graphite matrix, the volume

of the inner fuel region of the pebble was divided by the number of TRISO particles. It was

estimated that the pebble would contain 15000 particles. Since this volume would be spherical,

it was converted into a cube by taking the cube root of the spherical volume to determine the

length of the side of the cube. Then the TRISO particles could be inserted into these cubes as

shown in Figure 6.

24

Figure 7: Lattice of TRISO particles in graphite matrix

In Figure 7 the TRISO particles are arranged in square lattice, hence making the

arrangement of the particles in the fuel sphere to be fixed. This is not practical since the

process of binding fuel particles can only be produced in a random arrangement. Conducting

calculations at this scale has already proved to be computationally expensive and can be

imagined how the full core calculations involving this setup would entail.

The final reference model for MCNP that was consistent with the diffusion model is

shown in Figure 8 and Figure 9. The void above the reactor core and next to the core barrel

was removed and replaced with graphite reflector material. The material in the core is

homogenized for simplification of the geometry and calculation in MCNP, hence the double

heterogeneity is not modeled. The control rods are inserted about 350 cm from the top of the

reactor, hence 150 cm into the core.

25

Figure 8: Axial cross sectional view of PBMR-400 reactor in MCNP model

For the heterogeneous model, many parameters were determined using the benchmark

specifications [41].

Figure 9: Horizontal cross sectional view of PBMR-400 reactor in MCNP model

26

2.3 Results

The reactivity comparison was done based on the k-effective (in pcm) difference of the

diffusion solution from the MCNP reference solution. The results indicate a strong sensitivity of

the neutronic solution to the neutrons energy group structure for 2-group calculation. The k-

effective (in pcm) difference is calculated as shown below where the MCNP result serves as a

reference value:

( ) 510eff MCNP i

k k k∆ = − ×

Figure 10: Comparison of K-Effective for 300K and 1000K with upscattering correction

The results for the k-effective indicate that the diffusion solution is closer for the 2.38eV at

1000K temperature than at 300K temperature for the 2-group calculation as shown in Figure

10. Please note that for the energy cut-off point of 0.414 ev no result was produced with the

NEM/COMBINE methodology. Although the calculation in NEM could be performed with the

27

full scattering matrix, the removal cross section in NEM was corrected for up-scattering. This

was conducted by the following procedure:

2Re 1 2 2 1

1

moval

φ

φ→ →Σ = Σ − Σ

The upscattering would be considered for groups that have a subsequent energy level to

ensure accounting for upscattering in the removal cross section. Since 0.414eV cut-off energy

was the lowest cut-off point in COMBINE multi-group structure, upscattering below this group

was not available. Hence this case could not produce any results for comparison.

0.13 0.11 0.11 0.11 0.12 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.13 0.14 0.14 0.14 0.15 0.16 0.18 0.22 0.29 0.39 0.62 0.82 0.85 0.77 0.64 0.64 0.54 0.51 0.48 0.55 0.58 0.53 0.62 1.22

0.33 0.34 0.32 0.32 0.33 0.33 0.33 0.34 0.35 0.35 0.35 0.36 0.36 0.37 0.37 0.38 0.39 0.40 0.41 0.44 0.47 0.52 0.63 0.79 1.08 1.85 2.45 2.27 1.91 1.88 1.90 1.75 1.64 1.58 1.62 1.62 1.37 1.55 2.33

0.52 0.51 0.50 0.49 0.51 0.52 0.54 0.54 0.53 0.53 0.55 0.56 0.58 0.59 0.59 0.59 0.62 0.64 0.65 0.69 0.73 0.79 0.94 1.17 1.63 2.85 3.92 3.86 3.33 3.20 2.99 2.87 2.50 2.37 2.41 2.49 2.42 2.29 3.80

0.68 0.69 0.64 0.65 0.65 0.66 0.69 0.71 0.73 0.73 0.73 0.75 0.74 0.75 0.75 0.75 0.78 0.81 0.83 0.88 0.94 0.99 1.13 1.40 2.01 3.31 4.94 4.72 4.25 3.93 3.44 3.14 3.01 2.74 2.84 2.80 2.69 2.76 4.64

0.74 0.78 0.83 0.79 0.77 0.80 0.81 0.81 0.84 0.85 0.85 0.86 0.85 0.85 0.86 0.90 0.91 0.91 0.95 1.01 1.08 1.15 1.25 1.54 2.15 3.48 4.64 4.72 4.34 3.77 3.38 3.31 3.01 2.98 2.61 2.48 2.75 2.66 4.31

0.90 0.88 0.89 0.89 0.88 0.90 0.90 0.92 0.92 0.95 0.97 0.95 0.94 0.93 0.95 0.98 0.98 1.00 1.02 1.07 1.13 1.22 1.36 1.59 2.19 3.58 4.90 4.59 4.19 3.78 3.51 3.03 2.76 2.65 2.63 2.64 2.45 2.71 4.80

1.03 1.01 0.97 0.95 0.97 0.97 1.00 0.98 0.98 1.00 1.02 1.01 1.00 0.98 1.01 1.03 1.03 1.04 1.04 1.09 1.15 1.23 1.37 1.63 2.17 3.36 4.41 3.93 3.73 3.46 3.29 2.90 2.60 2.39 2.41 2.38 2.38 2.36 4.09

1.03 1.03 1.02 1.00 0.99 1.03 1.03 1.02 1.03 1.04 1.04 1.05 1.05 1.04 1.05 1.05 1.03 1.04 1.05 1.11 1.14 1.20 1.29 1.49 1.91 3.00 3.88 3.47 3.17 2.89 2.72 2.44 2.31 2.16 2.17 2.19 2.20 2.28 3.65

1.09 1.10 1.09 1.04 1.02 1.03 1.03 1.04 1.05 1.05 1.07 1.06 1.06 1.05 1.06 1.05 1.05 1.07 1.08 1.08 1.14 1.18 1.23 1.37 1.69 2.45 2.91 2.77 2.51 2.49 2.38 2.13 2.04 2.04 2.09 2.16 2.26 2.57 4.55

1.24 1.12 1.08 1.04 1.04 1.05 1.07 1.06 1.06 1.07 1.08 1.05 1.05 1.05 1.05 1.06 1.06 1.08 1.08 1.09 1.12 1.15 1.19 1.28 1.41 1.73 2.04 2.04 2.02 2.00 1.98 1.84 1.80 1.82 1.87 2.00 2.27 2.50 4.33

1.06 1.05 1.07 1.08 1.09 1.11 1.09 1.07 1.09 1.08 1.07 1.05 1.05 1.05 1.05 1.05 1.06 1.07 1.08 1.09 1.12 1.11 1.14 1.19 1.30 1.47 1.70 1.70 1.72 1.74 1.72 1.74 1.77 1.79 1.79 1.89 2.06 2.39 4.06

1.04 1.03 1.07 1.10 1.08 1.08 1.08 1.08 1.08 1.08 1.05 1.06 1.07 1.06 1.05 1.04 1.05 1.06 1.06 1.07 1.10 1.10 1.13 1.15 1.24 1.54 1.76 1.68 1.67 1.63 1.64 1.69 1.66 1.73 1.76 1.81 1.97 2.27 3.82

1.05 1.06 1.07 1.08 1.07 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.06 1.06 1.06 1.07 1.08 1.10 1.11 1.12 1.13 1.15 1.20 1.33 1.41 1.40 1.39 1.40 1.42 1.42 1.45 1.49 1.52 1.57 1.67 2.01 3.45

1.04 1.04 1.05 1.06 1.06 1.06 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.07 1.09 1.11 1.11 1.11 1.13 1.13 1.14 1.16 1.17 1.19 1.21 1.26 1.30 1.29 1.30 1.34 1.34 1.34 1.39 1.38 1.42 1.47 1.60 1.88 3.15

1.02 1.02 1.04 1.04 1.04 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.05 1.06 1.09 1.10 1.11 1.11 1.12 1.13 1.14 1.16 1.17 1.18 1.21 1.25 1.29 1.28 1.30 1.29 1.28 1.32 1.34 1.36 1.41 1.46 1.56 1.84 3.01

1.02 1.04 1.05 1.06 1.06 1.05 1.05 1.04 1.04 1.05 1.04 1.05 1.05 1.06 1.08 1.10 1.11 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.20 1.25 1.28 1.28 1.27 1.29 1.28 1.30 1.30 1.32 1.37 1.43 1.55 1.83 3.04

1.04 1.05 1.05 1.05 1.06 1.06 1.05 1.04 1.04 1.04 1.03 1.03 1.04 1.05 1.07 1.09 1.10 1.11 1.12 1.13 1.13 1.14 1.15 1.16 1.18 1.23 1.27 1.25 1.25 1.25 1.24 1.25 1.26 1.28 1.31 1.39 1.48 1.72 2.82

1.03 1.04 1.04 1.03 1.04 1.05 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.05 1.07 1.09 1.10 1.10 1.11 1.12 1.12 1.13 1.14 1.14 1.16 1.21 1.25 1.22 1.21 1.21 1.20 1.20 1.21 1.24 1.26 1.31 1.41 1.66 2.76

1.04 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.03 1.02 1.03 1.03 1.03 1.05 1.07 1.08 1.08 1.09 1.09 1.10 1.11 1.12 1.12 1.12 1.12 1.15 1.17 1.15 1.14 1.14 1.14 1.15 1.16 1.18 1.21 1.26 1.36 1.60 2.65

1.05 1.04 1.04 1.03 1.03 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.04 1.05 1.07 1.07 1.08 1.08 1.08 1.08 1.09 1.09 1.08 1.07 1.06 1.06 1.06 1.06 1.07 1.08 1.09 1.11 1.13 1.16 1.20 1.29 1.53 2.54

1.03 1.03 1.04 1.03 1.03 1.03 1.03 1.02 1.02 1.02 1.01 1.01 1.02 1.03 1.04 1.06 1.06 1.06 1.07 1.07 1.07 1.07 1.06 1.05 1.04 1.03 1.03 1.03 1.04 1.05 1.05 1.06 1.08 1.09 1.12 1.17 1.26 1.48 2.46

1.05 1.04 1.03 1.03 1.02 1.02 1.02 1.02 1.01 1.01 1.00 1.01 1.01 1.02 1.04 1.05 1.06 1.06 1.06 1.06 1.06 1.06 1.06 1.05 1.03 1.02 1.02 1.02 1.03 1.04 1.04 1.05 1.06 1.08 1.11 1.16 1.23 1.46 2.43

1.02 1.01 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.00 1.00 1.01 1.02 1.04 1.05 1.05 1.06 1.06 1.06 1.05 1.05 1.05 1.04 1.02 1.01 1.01 1.01 1.02 1.02 1.03 1.05 1.06 1.08 1.10 1.14 1.22 1.44 2.40

1.00 1.00 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.01 1.03 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.03 1.02 1.01 1.00 1.01 1.01 1.02 1.02 1.04 1.05 1.07 1.10 1.14 1.22 1.43 2.37

1.00 1.00 1.00 1.00 1.01 1.00 1.00 0.99 1.00 0.99 0.99 0.99 0.99 1.01 1.02 1.04 1.04 1.04 1.05 1.04 1.04 1.04 1.04 1.03 1.01 1.00 1.00 1.00 1.01 1.01 1.02 1.03 1.04 1.06 1.09 1.13 1.21 1.43 2.37

0.99 0.99 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.02 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.03 1.02 1.00 0.99 0.99 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.09 1.13 1.21 1.43 2.39

0.97 0.98 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 1.00 1.01 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.00 0.99 0.99 0.99 0.99 1.00 1.01 1.02 1.04 1.05 1.08 1.12 1.20 1.42 2.36

0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.98 0.98 0.98 0.99 1.01 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.01 1.00 0.99 0.98 0.99 0.99 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.20 1.41 2.35

0.99 0.99 0.98 0.98 0.99 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.98 0.99 1.01 1.02 1.03 1.02 1.03 1.03 1.02 1.02 1.02 1.01 0.99 0.98 0.98 0.98 0.99 1.00 1.01 1.02 1.03 1.05 1.08 1.12 1.20 1.41 2.35

0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.98 0.98 0.98 0.97 0.97 0.97 0.98 1.00 1.02 1.02 1.02 1.03 1.02 1.02 1.02 1.02 1.01 0.99 0.98 0.98 0.98 0.99 1.00 1.00 1.01 1.03 1.04 1.07 1.12 1.20 1.41 2.35

0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.98 1.00 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 0.99 0.98 0.98 0.98 0.98 0.99 1.00 1.01 1.02 1.04 1.07 1.11 1.20 1.41 2.34

0.98 0.98 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.97 0.97 0.98 1.00 1.01 1.01 1.01 1.02 1.02 1.02 1.02 1.01 1.00 0.99 0.98 0.97 0.98 0.98 0.99 0.99 1.00 1.02 1.03 1.06 1.11 1.19 1.41 2.32

0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.98 0.99 1.01 1.01 1.01 1.02 1.02 1.01 1.01 1.01 1.00 0.98 0.97 0.97 0.98 0.98 0.99 0.99 1.00 1.02 1.03 1.06 1.11 1.18 1.40 2.32

0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 0.99 0.98 0.97 0.97 0.97 0.97 0.98 0.99 1.00 1.01 1.03 1.06 1.10 1.18 1.40 2.32

0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.96 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 0.99 0.98 0.97 0.96 0.97 0.97 0.98 0.99 1.00 1.01 1.03 1.06 1.10 1.18 1.39 2.31

0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.97 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 0.99 0.98 0.97 0.96 0.97 0.97 0.98 0.98 1.00 1.01 1.03 1.06 1.10 1.18 1.39 2.31

0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.98 0.99 1.00 1.01 1.00 1.01 1.01 1.00 1.01 1.00 0.99 0.97 0.96 0.96 0.97 0.97 0.98 0.98 0.99 1.01 1.03 1.05 1.09 1.17 1.38 2.29

0.99 0.99 0.98 0.98 0.98 0.98 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.00 1.00 1.01 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.97 0.97 0.98 0.99 1.01 1.02 1.05 1.09 1.17 1.39 2.32

0.97 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.97 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.97 0.97 0.98 0.99 1.00 1.02 1.05 1.09 1.17 1.38 2.30

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.97 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.96 0.96 0.96 0.96 0.97 0.98 0.99 1.00 1.02 1.05 1.09 1.17 1.38 2.30

0.96 0.96 0.96 0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.95 0.95 0.95 0.96 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.96 0.96 0.96 0.96 0.97 0.97 0.99 1.00 1.02 1.04 1.09 1.16 1.37 2.27

0.96 0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.95 0.95 0.96 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.96 0.95 0.96 0.96 0.97 0.98 0.99 1.00 1.02 1.04 1.08 1.16 1.37 2.29

0.97 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.95 0.95 0.95 0.95 0.96 0.98 0.99 0.99 0.99 1.00 1.00 0.99 0.99 0.99 0.98 0.96 0.95 0.95 0.96 0.96 0.97 0.97 0.99 1.00 1.01 1.04 1.08 1.16 1.36 2.27

0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.95 0.94 0.94 0.94 0.94 0.96 0.97 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.96 0.95 0.95 0.95 0.95 0.96 0.97 0.98 0.99 1.01 1.03 1.08 1.15 1.36 2.27

0.96 0.96 0.95 0.95 0.95 0.96 0.95 0.95 0.95 0.94 0.94 0.94 0.94 0.95 0.97 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.98 0.97 0.96 0.95 0.95 0.95 0.95 0.96 0.97 0.98 0.99 1.01 1.03 1.07 1.15 1.36 2.26

0.94 0.95 0.94 0.94 0.95 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.97 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.98 0.97 0.95 0.95 0.94 0.94 0.95 0.96 0.96 0.97 0.99 1.00 1.03 1.07 1.14 1.35 2.24

0.94 0.94 0.94 0.94 0.95 0.94 0.94 0.94 0.94 0.94 0.93 0.94 0.94 0.95 0.97 0.98 0.98 0.98 0.99 0.99 0.98 0.98 0.98 0.97 0.95 0.94 0.94 0.94 0.94 0.95 0.96 0.97 0.98 1.00 1.02 1.06 1.14 1.34 2.24

0.94 0.94 0.94 0.94 0.95 0.95 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.95 0.96 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.97 0.96 0.95 0.94 0.93 0.94 0.94 0.95 0.95 0.96 0.98 0.99 1.02 1.06 1.13 1.33 2.23

0.94 0.95 0.95 0.94 0.95 0.95 0.94 0.94 0.93 0.93 0.93 0.93 0.93 0.94 0.96 0.97 0.97 0.97 0.98 0.98 0.97 0.97 0.97 0.96 0.94 0.93 0.93 0.93 0.94 0.94 0.95 0.96 0.97 0.99 1.02 1.06 1.13 1.34 2.23

0.94 0.95 0.95 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.93 0.94 0.95 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.95 0.94 0.93 0.93 0.93 0.93 0.94 0.95 0.96 0.97 0.99 1.01 1.05 1.13 1.33 2.21

0.96 0.95 0.95 0.94 0.93 0.93 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.93 0.95 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.95 0.93 0.92 0.92 0.92 0.93 0.93 0.94 0.95 0.96 0.98 1.00 1.04 1.11 1.31 2.19

0.93 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.91 0.91 0.91 0.91 0.91 0.92 0.94 0.95 0.95 0.96 0.96 0.96 0.95 0.96 0.95 0.94 0.93 0.91 0.91 0.92 0.92 0.93 0.93 0.94 0.96 0.97 1.00 1.04 1.11 1.31 2.17

0.93 0.92 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.90 0.90 0.90 0.90 0.92 0.93 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.94 0.92 0.91 0.91 0.91 0.91 0.92 0.93 0.94 0.95 0.96 0.99 1.03 1.10 1.30 2.16

0.90 0.89 0.90 0.90 0.91 0.91 0.90 0.90 0.90 0.89 0.89 0.89 0.89 0.91 0.92 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.91 0.90 0.90 0.90 0.90 0.91 0.92 0.93 0.94 0.96 0.98 1.02 1.09 1.29 2.14

0.88 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.90 0.91 0.92 0.93 0.92 0.93 0.93 0.93 0.93 0.92 0.91 0.90 0.89 0.89 0.89 0.89 0.90 0.91 0.92 0.93 0.94 0.97 1.01 1.08 1.28 2.13

0.88 0.87 0.87 0.88 0.88 0.88 0.88 0.88 0.88 0.87 0.87 0.87 0.87 0.88 0.90 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.90 0.89 0.88 0.88 0.88 0.88 0.89 0.90 0.91 0.92 0.94 0.96 0.99 1.06 1.26 2.09

0.87 0.85 0.86 0.87 0.87 0.87 0.87 0.86 0.86 0.86 0.85 0.85 0.85 0.86 0.88 0.89 0.89 0.89 0.90 0.89 0.89 0.90 0.89 0.89 0.88 0.87 0.86 0.87 0.87 0.88 0.88 0.89 0.91 0.92 0.94 0.98 1.05 1.25 2.09

0.83 0.84 0.85 0.86 0.86 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.85 0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.85 0.86 0.86 0.87 0.87 0.88 0.89 0.90 0.91 0.93 0.97 1.04 1.23 2.03

0.84 0.84 0.85 0.85 0.85 0.84 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.83 0.84 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.85 0.85 0.85 0.86 0.86 0.87 0.88 0.88 0.90 0.92 0.96 1.03 1.22 2.01

0.82 0.82 0.82 0.83 0.83 0.83 0.82 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.83 0.83 0.83 0.84 0.84 0.84 0.84 0.85 0.85 0.86 0.86 0.87 0.89 0.91 0.94 1.02 1.19 1.98

0.81 0.81 0.80 0.80 0.80 0.80 0.81 0.81 0.81 0.80 0.80 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.82 0.82 0.82 0.82 0.82 0.83 0.83 0.82 0.82 0.82 0.83 0.83 0.84 0.85 0.86 0.87 0.89 0.93 1.00 1.19 1.95

0.78 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.79 0.80 0.80 0.79 0.80 0.80 0.80 0.80 0.80 0.80 0.81 0.81 0.82 0.83 0.83 0.85 0.86 0.90 0.97 1.15 1.93

0.73 0.73 0.73 0.72 0.72 0.73 0.73 0.73 0.73 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.77 0.78 0.78 0.79 0.81 0.82 0.86 0.93 1.09 1.82

0.68 0.67 0.66 0.67 0.67 0.67 0.67 0.68 0.68 0.69 0.69 0.69 0.70 0.69 0.70 0.70 0.70 0.70 0.70 0.71 0.72 0.71 0.71 0.71 0.71 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.73 0.75 0.76 0.79 0.85 1.00 1.68

0.60 0.60 0.59 0.59 0.60 0.59 0.60 0.60 0.61 0.62 0.62 0.62 0.62 0.62 0.63 0.63 0.63 0.63 0.64 0.64 0.64 0.64 0.64 0.64 0.65 0.65 0.65 0.64 0.65 0.64 0.65 0.65 0.65 0.66 0.68 0.71 0.76 0.90 1.53

0.49 0.49 0.49 0.50 0.50 0.50 0.51 0.51 0.51 0.52 0.52 0.52 0.52 0.53 0.53 0.53 0.53 0.54 0.54 0.54 0.54 0.54 0.55 0.55 0.55 0.54 0.55 0.54 0.55 0.54 0.55 0.55 0.56 0.56 0.58 0.60 0.65 0.77 1.28

0.36 0.37 0.37 0.37 0.38 0.38 0.38 0.39 0.39 0.39 0.40 0.40 0.40 0.40 0.40 0.41 0.41 0.41 0.42 0.41 0.42 0.42 0.42 0.42 0.42 0.41 0.42 0.41 0.42 0.42 0.42 0.42 0.43 0.43 0.44 0.46 0.50 0.59 0.96

0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.25 0.25 0.25 0.25 0.25 0.25 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.27 0.27 0.27 0.27 0.28 0.29 0.31 0.37 0.61

0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.10 0.10 0.11 0.13 0.45

Figure 11: Flux ratio for MCNP to NEM

28

The result of NEM consisted of the thermal and the fast neutron fluxes whereas the

MCNP only produced one set of total flux. The NEM results were reduced by adding the

thermal and the fast fluxes. The axial flux was obtained by radially averaging the flux

distribution and the radial flux distribution by axially averaging the flux distribution.

The spatial ratio between the flux of NEM and MCNP showed differences in the reflector

material and where the control rods were located as shown in Figure 11. This was a difference

arising from the fact that the diffusion theory approximation is known to be deficient around

strongly absorbing materials and near boundaries with strong material changes.

2.3.1 Results at 300K

Figure 12 and Figure 13 show the results of the 300K case for the total flux distributions. The

ratios discussed above, show closer agreement in the distributions for the 2.38eV case than

those of the lower energy cut-offs.

1.00E+14

1.10E+14

1.20E+14

1.30E+14

1.40E+14

1.50E+14

1.60E+14

1.70E+14

1.80E+14

1.90E+14

100 120 140 160 180 200

Flu

x (

#.c

m-2

.s-1

)

Radial Distance (cm)

MCNP_300K

2.38eV_300K

1.86eV_300K

1.44eV_300K

1.125eV_300K

.876eV_300K

.683eV_300K

0.532eV_300K

0.414eV_300K

Figure 12: Radial flux distribution at 300K

29

0.00E+00

5.00E+13

1.00E+14

1.50E+14

2.00E+14

2.50E+14

3.00E+14

200 400 600 800 1000 1200 1400

Flu

x(#

.cm

-2.s

-1)

Axial Distance (cm)

MCNP_300K

2.38eV_300K

1.86eV_300K

1.44eV_300K

1.125eV_300K

.876eV_300K

.683eV_300K

0.532eV_300K

0.414eV_300K

Figure 13: Axial flux distribution at 300K

2.3.2 Results at 1000K

Analysis results for 1000K cases are shown in Figure 14 and Figure 15. These

comparisons also indicate closer agreement in the distributions for the 2.38eV case than those

of the lower energy cut-offs.

0.00E+00

5.00E+13

1.00E+14

1.50E+14

2.00E+14

2.50E+14

3.00E+14

0.00 500.00 1000.00 1500.00

Flu

x (

#.c

m-2

.s-1

)

Height (cm)

Axial_MCNP_1000K

2.38eV_1000K

1.86eV_1000K

1.44eV_1000K

1.125eV_1000K

0.876eV_1000K

0.6834eV_1000K

0.532eV_1000K

0.414eV_1000K

Figure 14: Axial flux distribution at 1000K

30

0.00E+00

5.00E+13

1.00E+14

1.50E+14

2.00E+14

2.50E+14

100 120 140 160 180 200

Flu

x (

#.c

m-2

.s-1

)

Radial Distance (cm)

MCNP_1000K

2.38eV_1000K

1.86eV_1000K

1.44eV_1000K

1.125eV_1000K

0.876eV_1000K

0.683eV_1000K

0.532eV_1000K

0.414eV_1000K

Figure 15: Radial flux distribution at 1000K

2.4 Conclusion

The results for the cut-off energy of 2.38 eV are closer to the MCNP results in all cases for

the distributions of the flux and the k-effective comparisons. The differences between the

MCNP and NEM/COMBINE for the 1000K case are minimized. These conditions are closer to

the operating conditions of the PBMR at average temperature around the 1000K. Hence, the

use of the thermal energy cut-off of 2.38 eV for this reactor can be considered optimal for two-

group PBMR analysis studies conducted within the framework of this PhD thesis.

Please note that the fine group structure hardwired within INL’s COMBINE-6 code that is

used to generate broad group constants is fixed and has not been optimized for HTR analyses.

If the fine group structure is not sufficiently refined in energy regions of importance, such a

structure may prevent the flexibility needed for more accurate cell-level energy collapsing. The

limitations inherent in the actual fine group structure are being currently addressed explicitly in

a follow-up study performed by PSU and INL.

31

Chapter 3 Description of the NEM/THERMIX code system

3.1 Description of NEM

The Nodal Expansion Method (NEM) code is a 3-D multi-group nodal code used at PSU for

modeling both steady state and transient core conditions. It utilizes a transverse integration

procedure and is based on the partial current formulation of the nodal balance equations. The

code has options for modeling of 3-D Cartesian, cylindrical and hexagonal geometry. The

cylindrical option utilizes fourth-order polynomial expansions of the 1-D transverse-integrated

flux distribution in the R-, Z- and θ-directions. It is important to note that the detailed treatment

of the effects of azimuthally dependent reactor control rods requires a full three-dimensional

representation of the PBMR.

3.1.1 Cylindrical geometry

In this section, the detail of deriving the NEM in 3-D cylindrical geometry is outlined. This

option is used for PBMR core modeling. To begin this derivation, the steady-state diffusion

equation within node l in two energy groups (for sake of simplicity – please note that NEM is a

multi-group code with no limitation on the number of energy groups) is rewritten as:

2,1,),,(

),,(),,(),,(

),,(1

,,(1

2

2

2

2

2

=∈

=+∂

∂−

∂

∂−

∂

∂

∂

∂−

gVzr

zrQzrAzrz

D

zrr

Dzrr

rrr

D

l

l

g

l

g

l

g

l

g

l

g

l

g

l

g

l

g

l

g

θ

θθφθφ

θφθ

θφ

3.1

where

32

( )

1 1 ,12 1

2 2

1 2 2

2 12 1

2 2

12

1

1( , , ) ( , , )

( , , ) ( , , )

1

2

1 2

l l l l

a s f

l l

a

l l l

f

l l l

l

out in

ag

Ak

A

Q r z r zk

Q r z r z

V R R z Volume

absorptionXS

ScatteringXS

υ

θ ν φ θ

θ φ θ

θ

= Σ + Σ − Σ

= Σ

= Σ

= Σ

= − ∆ ∆ ≡

Σ =

Σ = →

where Rin and Rout are inner and outer radius of node l respectively.

In cylindrical geometry, the Fick’s law may be used to express the three components of the

current vector as:

),,(),,( zrr

Dzrj l

g

l

g

l

grθφθ

∂

∂−= 3.2

),,(),,(

),,(),,(

zrz

Dzrj

zrDzrj

l

g

l

g

l

gz

l

g

l

g

l

g

θφθ

θφθ

θθ

∂

∂−=

∂

∂−=

3.3

From these relationships, Equation 3.1 can be written as:

( ) ( ) ( )

,),,(

),,(),,(

),,(1

),,(1

),,(1

l

l

g

l

g

l

g

l

gz

l

g

l

gr

Vzr

zrQzrA

zrrjzr

zrrjr

zrrjrr

∈

=+

∂

∂+

∂

∂+

∂

∂

θ

θθφ

θθθ

θ θ

3.4

If we assume the coordinate origin to be at the center of the cell l, Equation 3.4 can be

integrated over the node to obtain the nodal balance equation in cylindrical geometry:

( ) ( )

l

g

l

g

l

g

l

gz

l

gz

l

g

l

g

l

gr

l

gr

QA

JJz

JJR

Jr

RJr

RrR

=+

−∆

+−∆

+

∆−−

∆+

∆−+−+−+

φ

θθθ

11

22

1

3.5

where

33

R= radius from the coordinate centerline to a point midway between Rin and Rout

dzrdrrdzrV

z

z

l

g

rR

rR

l

l

g θθφφθ

θ∫∫∫

∆

∆−

∆

∆−

∆+

∆−

=2/

2/

2/

2/

2/

2/

),,1

= node volume average flux

dzrdrrdzrQV

Q

z

z

l

g

rR

rR

l

l

g θθθ

θ∫∫∫

∆

∆−

∆

∆−

∆+

∆−

=2/

2/

2/

2/

2/

2/

),,1

= node volume average source

/ 2 / 2 / 2

/ 2 / 2 / 2

1 1 1{ ( , , )}

2 2

R r z

l l l

gr gr grl

R r z

r rR J R J rj r z rdrd dz

R r V r r

θ

θ

θ θ+∆ ∆ ∆

+ −

−∆ −∆ −∆

∆ ∆ ∂ + − − =

∆ ∂ ∫ ∫ ∫

≡±l

grJ average r-directed net current on node faces 2

rR

∆±

( ) =−∆

−+l

g

l

gJJ

Rθθ

θ

1dzrdrdzrj

rV

z

z

l

g

rR

rR

lθθ

θθ

θ

θ∫∫∫

∆

∆−

∆

∆−

∆+

∆−∂

∂2/

2/

2/

2/

2/

2/

)},,({11

≡±l

gJ θ average θ-directed net current on node faces 2

θ∆±

( ) =−∆

−+l

gz

l

gzJJ

z

1dzrdrdzrj

zrV

z

z

l

gz

rR

rR

lθθ

θ

θ∫∫∫

∆

∆−

∆

∆−

∆+

∆−∂

∂2/

2/

2/

2/

2/

2/

)},,({11

≡±l

gzJ average z-directed net current on node faces 2

z∆±

For central nodes, i.e. where Rin =0 and R =2

z∆ one can see that l

gzJ − drops out of the

nodal balance equation. In order to develop a relationship between the node average flux and

the face average net currents in Equation 2.5, one must use transverse integration method in

directions perpendicular to the direction of interest.

For the r-direction the transverse integrated diffusion equation becomes:

( ) )(1

)(1

)()()(1

rLz

rLr

rQrArrjdr

d

r

l

gx

l

g

l

gr

l

gr

l

g

l

gr∆

−∆

−=+ θθ

φ 3.6

The theta-direction transverse integrated diffusion equation is written as:

( ) )(1

)(1

)()()(1

θθθ

θθφθθ

θθl

gx

l

g

l

gr

l

gr

l

g

l

gL

zL

rQAj

d

d

R ∆−

∆−=+ 3.7

and for the z-direction we have:

( ) )(1

)(1

)()()( zLr

zLR

zQzAzrjdz

d l

gx

l

g

l

gz

l

gz

l

g

l

gz∆

−∆

−=+ θθ

φ 3.8

From Equations 3.6, 3.7 and 3.8:

34

θθθ

θ

θ

dzdzrjz

rLl

gz

z

z

l

gz ),,(1

)(

2/

2/

2/

2/

∫∫∆

∆−

∆

∆−∂

∂

∆= = r-direction transverse leakage

dzrdrzrjzrR

Ll

gz

z

z

rR

rR

l

gz ),,(1

)(

2/

2/

2/

2/

θθ ∫∫∆

∆−

∆+

∆−∂

∂

∆= = z-directed transverse leakage

rdrdzrjzrr

zLl

g

z

z

rR

rR

l

g θθθθ ),,(11

)(

2/

2/

2/

2/

∫∫∆

∆−

∆+

∆−∂

∂

∆= = theta direction transverse leakage

Finally, NEM requires that the one-dimensional fluxes in Equations 3.6, 3.7 and 3.8 be

expanded in a series of polynomials, i.e.

)()(1

rfar n

N

n

l

grn

l

g

l

gr ∑=

+= φφ 3.9

)()(1

θφθφ θθ n

N

n

l

ng

l

g

l

g fa∑=

+= 3.10

)()(1

zfar n

N

n

l

gzn

l

g

l

gz ∑=

+= φφ 3.11

In the current version of the NEM code at PSU the cylindrical option utilizes fourth-order

polynomial expansions of the 1-D transverse-integrated flux distribution in the R-, Z- and θ-

directions. Implementing these expansions one can solve the resulting equations by deriving

and solving the response matrix equations and simply calculating the transverse leakage

moments. To speed up the convergence of the outer or source iterations, coarse mesh

rebalancing and asymptotic extrapolation can be used.

3.1.2 Steady state solution procedure

The multi-group equations in NEM are solved by inner/outer iteration multi-group diffusion

theory method. In order to invert the diffusion removal matrix, inner iterations/multiple sweeps

are performed through the mesh with a known internal source. Outer iterations or source

iterations are then performed around the inner iterations to calculate the correct eigenvalue

and the space and energy dependent neutron source distribution. The solution is tested for

35

convergence and whether to proceed to the next outer iteration. The production source for

node l is calculated as:

2

1

ll l

fg g

g

S rdrd dzυ φ θ=

= Σ∑ 3.12

Then these sources are summed over the entire system to get the global source ( totalS ).

Next the a new value of the eigenvalue keff is calculated using

n n 1

1

n

total

eff eff n

total

SK K

S

−

−= ⋅ 3.13

3.1.3 Description of the transient solution

Two energy groups, three-dimensional (3D) transient neutron diffusion and precursors

equations for node l, which has constant neutron properties, can be written as:

( )2

1 1 1 1 2 1 1

1

2 2

1

1( , ) ( , ) (1 ) ( , )

(1 ) ( , ) ( , )

l l l l l l l

a a f

Il l l

f i i

i

r t D r t r tV t

r t C r t

φ φ β ν φ

β ν φ λ=

∂− ∇ + Σ + Σ − − Σ

∂

= − Σ +∑

),(),(),(),(1

112222

2

22

2

trtrtrDtrtV

llll

a

lll φφφφ Σ=Σ+∇−∂

∂

( ) IitrCtrtrtrCt

l

ii

ll

f

ll

fi

l

i,1),(),(),(),( 2211 =−Σ+Σ=

∂

∂λφνφνβ

3.14

where

gV is the energy group g neutron velocity;

),( trl

gφ is the time- and space-dependent neutron flux in group g;

l

gD is the group g diffusion coefficient;

ν is the average number of neutrons produced by a fission event;

l

iC is the time- and space-dependent group i delayed-neutron precursor concentration;

ν is the fraction of delayed neutron in group i;

∑ ==

I

i i1ββ is the total fraction of delayed neutrons where

iβ is the fraction of delayed

neutrons in group i and I is the number of delayed-neutron groups;

36

iλ is the decay constant for delayed-neutron precursors in group i;

agΣ is the group g absorption cross-section;

12Σ is the group 1 to 2 scattering cross-section; and

fgΣ is group g fission cross-section.

First-order finite difference expression for the time derivatives of the fluxes is:

t

rrtr

t

oldl

g

l

gl

g∆

−=

∂

∂ )()(),(

,φφφ

3.15

Delayed neutron source contribution to the fast group equation is given by:

( )

1 1 2 2

( , ) ( , )

( ( , ) ( ( , ))

i

newnew

i

old

tl new l old

i i

tt tl l l l

i f ft

C r t C r t e

r t r t e dt

λ

λβ ν φ ν φ

− ∆

− −

=

+ Σ + Σ∫

3.16

It is assumed that the fission source varies linearly between time steps:

)()()()(

),(,,

,,

rttt

rrtr

oldl

g

oldl

fg

old

oldl

g

oldl

fg

l

g

l

fgl

g

l

fg φνφνφν

φν Σ+−∆

Σ−Σ=Σ

3.17

Then

( )

( )

−

∆

−−Σ+Σ+

∆

−−Σ+Σ+=

∆−∆−

∆−∆−

t

i

toldloldl

f

oldloldl

f

i

i

i

tll

f

ll

f

i

itoldl

i

newl

i

i

i

i

i

et

err

t

erretrCtrC

λλ

λλ

λφνφν

λ

β

λφνφν

λ

β

11)()(

11)()(),(),(

,

2

,

2

,

1

,

1

2211

3.18

Finally

2,1)()()(2 ==+∇− grQrArD l

g

l

g

l

g

l

g

l

g φφ 3.19

where

∑=

∆−

∆

−−Σ+Σ−−

∆+Σ+Σ=

I

i i

tl

fi

l

f

ll

a

l

t

e

tVA

i

1

11

1

1211

11)1(

1

λνβνβ

λ

tVA l

a

l

∆+Σ=

2

22

1

3.20

37

( )

,,1

1 2 2 2 2

11

, , , ,

1 1 2 2

( ) 1( ) (1 ) ( ) ( ) ( ) 1

1( ) ( )

i

i

i

i

tl old Itl l l l old l li

f i i f

i i i

ttl old l old l old l old

f f

i

r eQ r r C r e r

V t t

er r e

t

λλ

λλ

βφβ ν φ λ ν φ

λ λ

ν φ ν φλ

− ∆− ∆

=

− ∆− ∆

−= + − Σ + + Σ − ∆ ∆

−+ Σ + Σ − ∆

∑

)()(

)( 112

2

,

22 r

tV

rrQ

lloldl

l φφ

Σ+∆

=

a. These equations can be solved in either cylindrical geometry using the NEM spatial

approximations. It is assumed that there is no dependence of the delayed neutrons precursor

yield on the neutron energy.

b. ( )1

dgk fg k g fg

pgk fg k g fg

υ β υ

υ β υ

Σ = Σ

Σ = − Σ

It is a fixed source calculation where the total source magnitude is fixed and no need to

calculate an eigenvalue. The new time flux distribution is calculated by the previous one

applying inner/outer iteration strategy. However, to calculate the two-group flux distribution at

time step one, time zero values for the average fluxes and flux moments have to be already

obtained. The initial time zero fluxes are calculated by performing an initial steady-state

eigenvalue calculation. The initial precursors’ concentrations are calculated from this steady-

state condition by setting

( )

( ))0,()0,()0,(

0),(),(),(),(

2

0,

21

0,

1

22112

rrrC

trCtrtrtrCt

ll

f

ll

f

i

il

i

l

ii

ll

f

ll

fi

l

φνφνλ

β

λφνφνβ

Σ+Σ=

=−Σ+Σ=∂

∂

3.21

The old time fluxes are calculated at the previous time step, while the previous time

precursor concentrations are calculated from one and two time steps previous fluxes.

3.2 Description of THERMIX-DIREKT

THERMIX-DIREKT is a two-dimensional code for thermal hydraulic analysis of HTRs. The

code consists of steady-state or transient conduction module (THERMIX) and the quasi-steady

38

state convection module (DIREKT). Both modules in the code use the finite-difference method

with successive point-wise over-relaxation solution technique. THERMIX solves the time-

dependent general heat conduction equation with temperature dependent material properties.

DIREKT solves the steady-state continuity, energy and momentum equations for the core and

adjacent flow regions.

Heat transfer between the spherical fuel element and the coolant gas occurs in the thermal

boundary layer surrounding the fuel element. In this thin layer it is considered that heat

conduction is equal in importance to heat convection, whereas outside this layer the heat

conduction is relatively small.

For a spherical fuel element in a gas stream, the heat transfer Q (in W) from the pebble to

the surrounding gas is calculated by [27]:

( - )k k gQ A T Tα=� 3.22

where Ak is the surface area of fuel element in m2,

Tk, is the average fuel element surface temperature in K,

Tg is the gas temperature in K and

α is the mean heat transfer coefficient of the fuel element surface in Wm-2K-1.

The heat transfer coefficient α is calculated by [27]:

Nu

d

λα = 3.23

where α is in Wm-2K-1

λ is the heat (thermal) conductivity of the gas in Wm-1K-1,

Nu is the Nusselt number (dimensionless) and

d is the outer diameter of the fuel element in m.

The heat conductivity of helium is well studied and is calculated from experimentally

measured parameters [28]:

43 0.71(1.0 2.0 10 ) 32.682 10 (1.0 1.123 10 )PT Pλ−− − × −= × + × 3.24

39

where λ is in Wm-1K-1

P is the pressure in bar and

T is the temperature in K.

The Nusselt number is a dimensionless coefficient of heat transfer and determines the size

of the thermal boundary layer. The Nusselt number measures the enhancement of heat

transfer from a surface which occurs in a “real” situation, compared to the heat transfer that

would be measured if only conduction could occur. Typically it is used to measure the

enhancement of heat transfer when convection takes place. The Nusselt number for spherical

fuel elements in a pebble bed core is given by [27]:

1 3 1 20.36 0.86

1.18 1.07

Pr PrNu 1.27 Re 0.033 Re

ε ε= + 3.25

where ε is the porosity of the bed, i.e. the relation between the volume filled by the gas and the

total volume of the reactor packed with fuel elements. Thus (1- ε) is the sphere-packing factor,

Pr is the Prandtl number (dimensionless) and

Re is the Reynolds number (dimensionless).

The Reynolds number is a non-dimensional parameter that compares the inertia to viscous

forces. If the Reynolds number is low, then viscosity plays an important part in the flow

phenomena. The Reynolds number determines whether the gas flow over the fuel spheres is

laminar or turbulent. The two types of flows, laminar or turbulent, have different heat transfer

mechanisms, and influence the formation and size of the thermal boundary layer. The

Reynolds number is calculated by the following equation:

( )Re

m dA

η=

�

3.26

where m� is the helium mass flow through the core in kgs-1

A is the core (sphere pile) cross section in m2

η is the dynamic viscosity of the gas (helium) in kgm-1s-1, defined in equation 3.28.

40

Prandtl number is the non-dimensional ratio between the product of heat advection and

viscous forces and the product of heat diffusion and inertial forces in a given fluid. Standard

thermo fluids textbooks define Pr as:

Pr PCη

λ= 3.27

where λ is the heat (thermal) conductivity of the gas in Wm-1K-1, defined in equation 3.24,

CP is the specific heat of the gas at constant pressure

(KTA3102.1 gives for Helium CP =5195 Jkg-1K-1) and

η is the dynamic viscosity of the gas (helium) in kgm-1s-1.

The dynamic viscosity η of the coolant gas (Helium) is a function of the temperature and is

given as [28]:

7 0.73.674 10 Tη −= × 3.28

where T is the gas (helium) temperature in K.

The pressure drop through the core is defined by [29]: 2

3

1 1

2

mP H

d A

ε

ρε

− ∆ = Ψ

� 3.29

where H is the height of the reactor core in m,

Ψ is the coefficient of pressure loss defined in equation 3.29 and

ρ is the density of the gas (helium) in kg.m-3.

The density of the helium is defined by [28]:

1.2

48.14

1 0.446

P

T

P

T

ρ

=

+ ×

3.30

where P is the pressure in bar and

T is the temperature in K.

ρ is in kg.m-3

The coefficient of loss of pressure through friction shall be determined in accordance with

the following empirical correlation [29]:

41

0.1

320 6

Re Re

1 1ε ε

Ψ = + − −

3.30

where Re is the Reynolds number defined in equation 3.30.

3.3 Description of coupling scheme

The NEM code has been coupled with THERMIX-DIREKT using a serial integration

approach (see Figure 17). The spatial mesh overlays are exact in r-z geometry and provide a

capability for different spatial meshing in neutronics and thermal-hydraulics models. The

temporal coupling is based on the same time step size used by NEM and THERMIX-DIREKT

with the time step determined by the latter code. During both steady state and at each time

step of transient a coupling iteration loop is performed between neutronics and thermal-

hydraulic calculations upon reaching a defined convergence in temperature distribution. The

cross-section dependencies on feedback parameters are modeled through linear surface

interpolation in multi-dimensional tables.

3.4 Description of coupling scheme

The NEM code has been coupled with THERMIX-DIREKT using a serial integration

approach (see Figure 17). The spatial mesh overlays are exact in r-z geometry and provide a

capability for different spatial meshing in neutronics and thermal-hydraulics models. The

temporal coupling is based on the same time step size used by NEM and THERMIX-DIREKT

with the time step determined by the latter code. During both steady state and at each time

step of transient a coupling iteration loop is performed between neutronics and thermal-

hydraulic calculations upon reaching a defined convergence in temperature distribution. The

cross-section dependencies on feedback parameters are modeled through linear surface

interpolation in multi-dimensional tables.

42

3.4.1.1 Old fuel temperature approximation

The temperature distribution obtained from THERMIX was considered to be the fuel

temperature distribution. This was the only distribution that was transferred to NEM. The idea

was that the solid temperature refers to both the fuel and the moderator temperatures, which

was a good estimation for the homogenized material. It was also known that the actual fuel

temperature is higher than the moderator temperature.

3.4.1.2 Old moderator temperature approximation

An intuitive approach to the estimation of the moderator temperature was adopted. This

was based on the observation that the fuel temperature in the pebble bed was about 50°C

higher than the fuel temperature. Hence the adopted approach for the estimation of moderator

temperature was using equation 3.31:

Tmoderator = Tfuel -50°C 3.31

This approximation was not entirely correct but was sufficient to enable calculations for the

coupled code system since both the fuel and the moderator temperature distributions would be

available when the cross section interpolation routine is called.

3.4.1.3 Old Xenon approximation

To account for Xenon reactivity, node-wise Xenon number densities were calculated

internally in NEM. The equilibrium number densities at steady state were computed using

equation (3.32) for Iodine and (3.33) for Xenon for each neutronic spatial node. The

assumptions were that after a long time of operation, the reactor reaches an equilibrium state

where the xenon and iodine number densities no longer change with time.

( ) 0I f

t

I

I t Iγ φ

λ∞→∞

Σ→ = 3.32

( )( ) 0

0

I X f

t X

X a

X t Xγ γ φ

λ σ φ∞→∞

+ Σ→ =

+ 3.33

43

0φ = Flux at steady state condition

I∞ = Iodine concentration at the beginning of the transient

X ∞ = Xenon concentration at the beginning of the transient

Iλ =Iodine decay constant

Xλ = Xenon decay constant

Iγ = Iodine fission yield

Xγ = Xenon fission yield

For time-dependent calculations the number densities of Xenon and Iodine had to be

tracked. The previous implementation of time-dependent Xenon and Iodine concentrations was

using equations (3.34) and (3.35).

( ) ( )0I f II t I t Iγ φ λ∞ ∞= + ∆ Σ − 3.34

( ) ( )( )0 0

X

X f I X aX t X t I Xγ φ λ λ σ φ∞ ∞ ∞= + ∆ Σ + − + 3.35

The results shown in Figure 16 for these models show that Iodine and Xenon

concentrations could only build-up in a system and are very sensitive to the time step size

changes in calculations. For instance, when the time step is changed at 60 hours, the there

was an abrupt jump in the Xenon number densities as displayed in Figure 16. This anomaly

justified the need for the improvement of the Xenon number density prediction model. Hence

the models in equations 3.34 and 3.35 cannot be used to approximate the correct reactor

conditions where there is a change in power/flux level and time step since they were only

dependent on the steady state flux, 0φ and time step changes.

44

Figure 16: Old transient Xenon model

This shortcoming was identified as one area for improvement since all transient cases that

were studied in this work required realistic models to best estimate reactor conditions. The

improvements implemented in this aspect are discussed in Chapter 4.

The old coupled code system involved NEM starting with initial guess of the temperature

distribution to initiate the calculation. NEM passes the power distribution to THERMIX, which

uses the power distribution to calculate the temperature distribution. As discussed above, the

temperature distribution is send to the cross section interpolation routine where fuel

temperature and the moderator temperature distributions are estimated. The two temperature

distributions and the saved Xenon number densities from the previous coupled iteration NEM

calculation, are used to interpolate the new cross section data using three-dimensional (i.e.

Tfuel, Tmoderator and [Xe]) interpolation routine.

45

Figure 17: Old NEM/THERMIX coupling scheme

The ultimate coupled scheme for the code system was organised as shown in Figure 17

with the 3-dimensional cross-section interpolation scheme. The coupled calculation continues

until a converged solution is reached on the temperature and xenon distribution. This system

was validated as discussed in section 3.5 below with some results shown in Figure 18 and

Figure 19.

3.5 Status of verification

The stand-alone version of the THERMIX-DIREKT code was verified for use in modular

HTR (PBMR) through simple stand-alone cases at PSU [22]. In fact, an extensive verification

(and validation) of this code was performed in Germany many years ago for the AVR reactor.

At PSU, the verification was performed for PBMR applications using the stand-alone thermal-

hydraulic test cases of the PBMR-268 and recently the PBMR-400 international benchmark

efforts between the PBMR Ltd (South Africa), NRG (Netherlands), PSU, PU and INL (USA),

and other organizations.

The verification of the stand-alone NEM has been performed through another benchmark

initiative, which was a joint effort between three institutions namely: Penn State, Purdue

NEM

Cross Section Library

Interpolation THERMIX Temp

Mass Flow Rates

Pressure

Power Density

TModerator

Tfuel

Xe

Flux

46

University (PU) and the PBMR Ltd. Company in South Africa [22]. These institutions defined a

benchmark with three steady-state core physics test problems hereafter referred to as test

cases. The ultimate intention was to develop these test problems into HTGR benchmarks

similar to the well-known IAEA PWR steady-state benchmarks. These test cases were

analyzed with three different codes: VSOP (PBMR Ltd), NEM (Penn State University), and

PARCS (Purdue University). In the framework PBMR-268 benchmark, which was a

comprehensive benchmark exercise aimed at both steady-state and transient analysis of the

PBMR reactor core, NEM was tested further in standalone mode and in coupled mode also. In

this benchmark, the PBMR Ltd still used VSOP for steady state analysis and the TINTE code

for transients, PSU used NEM/THERMIX; NRG used PANTHERMIX, and PU used

PARCS/THERMIX. Figure 18 and Figure 19 show the comparisons between different

participants for axial and radial power distributions for the initial steady state conditions, using

coupled neutronics/thermal-hydraulics models. Transient verification of both code systems was

also performed for different scenarios within the framework of the PBMR-268 benchmark

activities.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 100 200 300 400 500 600 700 800 900

Axial position (cm)

Re

lati

ve

Po

wer

NEM

PANTHERMIX

PARCS

Figure 18: Comparison of axial power profiles for steady state

47

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 20 40 60 80 100 120 140 160 180 200

Radial Position (cm)

Re

lati

ve

Po

we

r

NEM

PANTHERMIX

PARCS

Figure 19: Comparison of radial power profiles for steady state

3.6 Conclusions

In this chapter the basic coupled multi-physics code system NEM/THERMIX, developed

and verified at PSU, was introduced. NEM/THERMIX is the tool selected for performing the

proposed optimization studies of multi-physics coupling methodology since it has

demonstrated a potential for accurate and efficient steady-state and transient solutions. This

sets the platform for enhancements and optimization studies on NEM/THERMIX codes

system. Such studies require the extension of the three-dimensional cross-sectioninterpolation

scheme to five-dimensional scheme in order to model all of the important cross-section

feedback dependencies including spectrum (environment) dependence. It was also learned

that the models for Xenon and Iodine would require improvements to ensure temporal stability

to enable longer transient calculations to be performed in an efficient manner.

48

Chapter 4 Multiphysics code development and optimization

4.1 Introduction

Multi-physics approach deals with coupled fields, such as neutronics and thermal-

hydraulics fields, analysis to determine the combined effects of multiple physical phenomena

on design and safety. In this research the coupled thermal-hydraulics and neutronics methods

were extended to enable modeling of a wider range of transient scenarios pertinent to the

PBMR. As described in the Appendix B, such transient problems are defined within the

framework of the OECD PBMR-400 benchmark. The benchmark is suitable to verify and

validate PBMR transient and safety analyses methods, and the benchmark suite includes

variety of transient scenarios to cover the whole transient range (from slow to fast transients)

and feedback phenomena involved. The accuracy and efficiency of the coupled neutronics /

thermal-hydraulics model depends on many parameters related to the neutronics model,

thermal-hydraulics model and the utilized coupling scheme. In order to evaluate the

importance of each parameter related to the multiphysics coupling sensitivity studies were

performed. The obtained results were utilized further in this research to reduce the coupled

code system dependency on these parameters thus optimizing it.

4.2 Thermal-hydraulic modeling

4.2.1 Calculation of the kernel temperature

The first challenge was the transfer of the temperature distribution itself, which was not

conducted properly into the NEM meshes in the previous version of the code. Since THERMIX

provides the temperature in 2-D maps, there was a need to expand the model to be a full 3-

map in NEM as shown in Figure 25.

49

After the modifications of the temperature transfer were implemented properly the

convergence of the solution was achieved as shown in Figure 21. This figure demonstrates

spatial temperature convergence realized on the r-theta-z spatial mesh – for a given spatial

node as function of coupled iterations.

0

100

200

300

400

500

600

700

800

900

1,000

0 5 10 15 20

Te

mp

era

ture

(°K

)

Iteration

Tfuel1(1,1,1)

Tfuel1(1,2,1)

Figure 21: Convergence of temperature after modifications

TTHERMIX(2-D)

NEM (3-D)

Interpolate

T(r,z)

Tfuel (r,z)

Tfuel1 (r,θ,z)

Figure 20: Temperature transfer to NEM

50

The PBMR feedback effects are affected by the large heat capacity of the core (that results

in very slow overall temperature changes during transients). There was a need to model the

core heterogeneous temperatures variations and studies were performed to evaluate the effect

of heterogeneous vs. homogenous modeling for different transients as discussed later.

The effect of the fuel kernel thermal-hydraulics model was studied and is discussed first.

For fast reactivity transients the thermal conductivity models employed to determine fuel

temperatures during normal operation often does not take the coated particles and its thermal

properties into account. In most cases this omission is acceptable since a rapid heat transfer

takes place from the coated particle to the surrounding graphite matrix. However, if fast power

transients occur the temperature differences between the coated particles and the surrounding

graphite will be large and must be taken into account to be able to predict fast reactivity

transients accurately. A method was developed to introduce the solution of a representative

micro-system representing the coated particles within discretized fuel pebble shells. The

developed time-dependent algorithm to solve for the temperatures in a pebble was introduced

into NEM/THERMIX and results are presented in this thesis for fast reactivity transient cases.

The THERMIX-DIREKT code provides for two methods to calculate temperature of the fuel,

namely, the so-called homogeneous model and the shell-model. In the homogeneous model, it

is assumed that there is a uniform distribution of temperature across the fuel pebble, and

therefore the fuel and moderator are essentially the same, with the exception that for the

moderator, only the temperature of the outer graphite of the pebble is considered. On the other

hand, the shell model divides the fuel pebble into five shells and the temperature of the inner

most shell is regarded as the maximum fuel temperature, while the average over the four inner

shells is regarded as the fuel temperature. The moderator temperature is then computed to be

the average of all five shells. The first model is obviously fraught with very extreme

assumptions and can only be used for overly simplified calculations. For a reactor operating at

a significant level of power, the assumption of uniform temperature is not valid because the

central (fuel) region of the pebble will experience a higher temperature than the surface. The

two most important consequences will be that (a) the maximum fuel temperature will be

underestimated, thus resulting in the neutronics model calculating higher powers and (b) the

temperature at which the cross sections are tabulated i.e. the feedback temperature will in fact

be wrong.

51

Studies have shown that this difference can be as small as 36oC at steady-state conditions.

This temperature difference albeit smaller for steady state calculations suggests that there is a

need for correction introduced in order to accurately model the fuel kernel temperature for fast

reactivity insertion transients where this difference may be larger. The importance of this

correction was demonstrated by showing the difference between using the shell model and the

correction model for fast transients [23], such as control rod ejection (CRE) transient.

Effect of fuel temperature model on CRE power

0

2000

4000

6000

8000

10000

12000

0 0.5 1 1.5 2 2.5 3

Time (s)

Rela

tive P

ow

er

(%)

Doppler Temperature from shell model

Doppler Temperature from shell model + fuelkernel correction

Figure 22: Power during the control rod ejection transient [23]

It can be seen from Figure 22 that there are significant differences in the maximum fuel

temperature peak and consequently in the fission power time evolution depending on which

fuel temperature model was used. When the shell model is used, (i.e. treating the temperature

of the inner shell of the pebble as the fuel without taking into consideration that the kernels are

embedded into a matrix of graphite) this results in very high maximum fuel temperature

because the Doppler feedback to turn back the power is delayed due to the lowered thermal

conductivity of uranium dioxide and the coatings which makes heat transfer even slower in

kernel, thereby causing the fission power to surge by a factor of 120 before returning promptly

to about 600% within 0.5 seconds. On the other hand, using the model that corrects the shell

model to take care of fuel kernels explicitly results in a much reduced power peak and hence a

52

much lower maximum fuel temperature peak. It is thus very important to use the correct fuel

temperature treatment for this transient, especially for safety analysis purposes.

Figure 23: Representative micro-system for shell calculation [45]

The implemented method involves the following steps:

1. Discretization of fuel sphere into shells;

2. Assignment of a homogeneous (independent of r) heat source to each shell within the

inner 2.5 cm of fuel sphere;

53

3. Definition of a representative micro-system for each shell within the inner 2.5 cm of fuel

sphere. Each kernel consists of UO2, four kernel coatings and a part of the graphite

matrix, as shown in Figure 23;

4. Temperature dependent thermal conductivity and heat capacity values for UO2, kernel

coatings and graphite as shown in Table 6 and Table 7 are used;

5. Micro-system assigns the average temperature of the fuel sphere shell (as usually

calculated);

6. Time-dependent discrete mathematical model for heat conduction is solved;

7. The end result is average temperatures in shells of each representative micro-system at

each time point.

4.2.2 NEM/THERMIX kernel model

The moderator and Doppler temperatures are obtained from THERMIX using the

currently implemented procedure for cross section interpolation purposes. The anticipated

changes were in the calculation of heterogeneous temperature in THERMIX. In the current

model the fuel sphere is discretised into a number of shells for solving the differential equation

of heat conduction. Nuclear heat sources are considered to be homogeneously distributed (i.e.

smeared out) throughout the shells, which are inside the inner 2.5 cm of the fuel sphere. This

is clearly not true since the nuclear heat sources are heterogeneously distributed throughout

each of the shells inside the inner 2.5 cm of the fuel sphere. Sources are concentrated in the

small UO2 kernels, which are coated with several layers. Thermal properties of UO2 and

coatings are different from that of the graphite in which they are embedded.

The effect of homogenization approximation of the kernel and graphite are justified

during steady state operation or slow reactivity transients since the temperature difference

between fuel kernels and surrounding graphite matrix is small. However, when fast reactivity

transients occur, there is a strong power surge and the temperature of the UO2 kernels and

coatings rise well above that of the surrounding graphite matrix. This introduces large

differences between homogeneously and heterogeneously distributed sources as well as in the

explicit coated particle models.

54

To give an idea of the differences in the material properties of the coatings and the fuel

kernel in the fuel pebble, some of the properties are given in Table 6 and Table 7.

Table 6: Thermal conductivity of fuel kernel layers

Region Thermal conductivities (W.cm-1.K-1)

UO2 3.7E-02

PyC buffer layer 0.5E-02

Inner / Outer PyC 4.0E-02

SiC layer 16.0E-02

graphite matrix 0.54E-02

Table 7: Specific heat capacity of fuel kernel layers

Region Specific heat capacity (J. g-1. K-1)

UO2 0.04E-01

PyC buffer layer 3.50E-01

Inner / Outer PyC 3.50E-01

SiC layer 0.25E-01

graphite matrix 0.50E-01

The modeling of correct heterogeneous representation of temperature includes the

discretization of fuel sphere into shells and representing temperature in the pebble according

to Table 8.

55

Table 8: Temperature representation in the fuel element

Parameter Description Unit

Average and Maximum

Fuel Temperature

The “fuel temperature” is defined as the average fuel kernel

temperatures of all the fuel spheres present in a single mesh.

(This is the value used for Doppler feedback calculations in

each mesh point).

The maximum value that occurs in the 2D spatial map (the

report mesh) is defined as the “maximum fuel temperature”,

and the average of all the spatial moderator temperatures is

defined as the “average fuel temperature”. These are reported

in the time dependent results list.

oC

Maximum Kernel

Temperature

The maximum kernel temperature of a pebble refers to the

maximum temperature seen by a single kernel, assumed in the

centre of a fuel element (a region the size of a kernel).

The core maximum kernel temperature (as is required in the

time dependent single parameter edit for Cases 5) is the

highest of all the maximum kernel temperatures of pebbles that

occurs in the entire 2D spatial map.

To exclude mesh effects these parameters are to be calculated

in the reported mesh and not in the refined calculational mesh.

Thus values calculated in a refined mesh should first be

averaged per the reported mesh and then the maxima should

be found.

oC

56

Parameter Description Unit

Average and Maximum

Moderator Temperature

The “moderator temperature” is defined as the average

temperatures of all graphite in the fuel spheres present a single

mesh (i.e. in the fuel graphite matrix and the outer fuel free

graphite zone of a sphere). (This is the value used for

moderator temperature feedback calculations in each mesh

point).

The maximum value that occurs in the 2D spatial map (the

report mesh) is defined as the “maximum moderator

temperature”, and the average of all the spatial moderator

temperatures is defined as the “average moderator

temperature”. These are reported in the time dependent results

list.

oC

4.2.3 New NEM fuel kernel temperature model

The correct modeling of kernel temperature was developed using the heat resistance

model and the conductivities as listed in Table 7. Heat transfer in the kernel is through

conduction since the layers are solid as shown in Figure 24. Newton’s Law of cooling applies

since the heating is from the inner part of the fuel particle (kernel) and since the temperature of

the outer layer (the graphite) is known, and it forms the boundary condition for our purpose. It

is also assumed that heat deposited in the graphite is also homogeneous in all layers of the

fuel particle.

The fuel kernel has different layers that have different properties as noted above. To

determine the heat transfer through the fuel layer, the heat resistance through the layers of the

fuel particle had to be conducted. Each material has its own thermal conductivity, which is

named k and the indices of SiC for instance are 1 (on the left) and 2 (on the right), hence SiC

has properties temperature (T12) thermal conductivity k12 and thickness (x1 – x2). The thickness

is determined by the difference between the locations of the boundaries. The thermal

resistance is then converted into a total resistance.

57

0 10 1 0

01

1 21 2 0

12

2 32 3 0

23

3 43 4 0

34

4 54 5 0

45

x xT T q

k

x xT T q

k

x xT T q

k

x xT T q

k

x xT T q

k

−− = −

−− = −

−− = −

−− = −

−− = −

where

ki,i+1 is the thermal conductivity of the material

Ti = temperature at the boundary

xi-xi+1 = the thickness of the layer

Adding the terms on both sides:

Graphite Pyrolytic Carbon

SiC Pyrolytic Carbon

Porous Buffer Carbon

UO2

T0

T34

T45

T12

T23

T01

K23 k12 k0 k01 k34 k45

Figure 24: Simplified kernel heat transfer model

58

0 1 2 3 3 4 4 51 20 5 0

01 12 23 34 45

x x x x x x x xx xT T q

k k k k k

− − − −−− = − + + + +

0 1 2 3 3 4 4 51 25 0 0

01 12 23 34 45

x x x x x x x xx xT T q

k k k k k

− − − −−= + + + + +

Currently the thickness of the layers is fixed as the first approximation. But it is known that

the properties of materials change in response to changes in conditions such as temperature

that cause the expansion and contraction. The radiation effects on the materials can also

change the properties of graphite and result in poor conductivity. In future the effects of these

changes can be taken into account.

4.2.4 TINTE kernel model

An approximation to a complete explicit model is available in the TINTE code system where

the temperature in the UO2 fuel kernel, used for cross section reconstruction, is calculated for

a steady state case as:

mlff TQT +′′′= �'α

where

f'α = User supplied parameter

lQ ′′′� = Local heat production in the mesh (Watt/cc)

Tm = Matrix (graphite) temperature

Note that the heat production is defined as being homogenized over a mesh (fuel sphere

and helium) and include only the locally deposited heat component – in the PBMR-400

benchmark definition this includes all fission heat. This will of course influence the definition of

f'α .

59

In the time dependent case, the fuel temperature at the end of the time step is given in

terms of the steady state values as follows (subscripts 0 and 1 indicate beginning and end of

time step respectively):

( ) ( )∆

−−+−+=

∆−∆−

f

ff

f

fe

AAeATATλ

λλ 1

100011

where

000 ' mlf TQA +′′′= �α

111 ' mlf TQA +′′′≡ �α

( ) fpfUO

U

be

bef

cM

M

SM

V

,'

1

12

αελ

−=

∆ = Time step length (s)

3

3

4bebe

RV π=

beR = Fuel sphere radius (3 cm)

SM = Heavy metal loading per fuel sphere (9 g)

beε = Pebble bed void fraction (0.39)

2UOM = Molar mass of uranium dioxide fuel (270 g/mol)

UM = Molar mass of low enriched uranium metal (238 g/mol)

fpc , = Specific heat capacity for UO2 fuel (0.3 kJ/kg/K)

4.3 Optimizing temporal coupling schemes

The temporal coupling and meshing schemes aspect was studied and optimized between

the neutronics and thermal-hydraulics time step selection algorithms. The temporal coupling

can be either direct (implicit) or iterative (explicit). Implicit coupling requires a single matrix

solution of both fields, while explicit coupling sequentially solves the individual problems,

passing explicit values across the field interfaces and iterating until all solutions converge.

All coupled transient calculations start with a fully converged steady state solution. The

convergence criteria utilized in NEM/THERMIX for coupled steady state solution are shown in

Table 9.

60

Table 9: Suggested convergence criteria

1 k-eff 0.00001

2 Local Fluxes 0.0001

3 Local Temperature °C 0.01

4 Local Flows m/s 0.1

The time step size and total run time information are specified as an input parameter. A

time step size adjustment could be input at certain time points with a certain fraction of the

initial time step size. The temporal discretization of all fields was performed with a theta

method which enables to specify any type of scheme from fully explicit to fully implicit. At each

time step a marching scheme through the calculation fields must be performed until the

specified convergence is achieved.

For fast transient scenarios such as Control Rod Ejection (CRE) or Control Rod Withdrawal

(CRW), the typical time step size is in order of milliseconds to achieve a reasonable accuracy.

On the other hand, some of the anticipated accident scenarios that are investigated for reactor

safety characteristics of the PBMR type reactors require very long run times. For instance, the

“Depressurized Loss Of Forced Cooling without SCRAM” accident is analyzed for about 100

hours in order to capture the effects of re-criticality which occurs around 72 hours. To simulate

such an accident scenario with a reasonable accuracy, a time step size of less than 5 seconds

should be applied, which results in a total CPU time of 7 – 8 days. The optimization of this

aspect enabled the efficient run of such calculation in hours.

Table 10 provides the convergence criteria and time step sizes utilized in NEM/THERMIX

for transient simulations.

Table 10: Suggested convergence criteria and step sizes for transient cases

1 Temperature iterations °C 0.2

2 k-eff 1.0E-05

Step sizes

61

3 Case 1 – 3 (during heat-up and cool-down) sec 60

4 Recriticality phase of Case 1 sec 2

5 Case 5 sec 0.1

The true coupled code convergence and application of innovative methods to accelerate

it and solve the coupled field problem simultaneously was another important issue, which was

addressed in this research through the development of reference solutions as discussed in

Chapter 6.

4.4 Improved and efficient feedback modeling

4.4.1 Thermal and fast buckling calculation

Finally, the modeling of all feedback phenomena in HTRs was investigated and novel

treatment was introduced for improving the representation of cross-section variations. The

dependence on fast / thermal leakage (buckling) may introduce problems. The leakage

dependence parameterization approach was thoroughly evaluated and compared to the

directly calculated cross section data sets. Attention was given to the restrictive fast and

thermal buckling input values yielding physical cross sections results to prevent excessive

extrapolation with potential negative cross sections.

As it has been mentioned before when generating PBMR cross-section libraries it is

very important to model the spectrum dependence. As a consequence the cross-sections

should be represented as a function of some spectral parameter as for example the leakage

term - L or buckling – B2. Leakage of neutrons from/to the adjacent spectrum zones is included

by buckling terms, which are generated from the diffusion calculation over the whole reactor.

From the 3-D NEM diffusion core simulator calculation the leakage terms LSI are calculated for

each spectrum zones S and for each coarse energy group I (normally four). The leakage terms

can be transformed into bucklings:

62

SSISI

SI

SIVD

LB

•Φ•=2

4.1

The leakage can be also transferred into the albedo at the surface of the spatial cell (the

group indices are omitted for sake of simplicity):

+

−=

J

Ja

4.2

This is the ratio of the partial current J- of neutrons entering the cell divided by J+ leaving

the cell.

The partial currents are given by:

00

2

1

4JJ −

Φ=−

4.3

00

2

1

4JJ +

Φ=+

4.4

in which J0 = J+ - J- is the net current leaving the cell per cm2.

The net current of the cell is equal to the ratio of the leakage LC of the cell per surface SC .

C

C

S

LJ =0

4.5

The leakage of the cell is equal to the LS of the whole spectrum zone divided by the number

of cells in the spectrum zone, as given by the ratio of the volumes VS/VC of the spectrum zone

and cell.

S

CSC

V

VLL •=

4.6

Further the neutron flux Φ0 at the surface of each cell is equivalent to the average neutron

flux ΦS of the whole spectrum zone. As a result the albedo is calculated as:

C

C

SS

S

C

C

SS

S

S

V

V

LS

V

V

L

A

••Φ

+

••Φ

−=

21

21

4.7

63

In the case of a spherical cell of the pebble bed the VC includes the volume of the void per

cell, but SC is just the surface of the cell.

The calculation of buckling is based on the transverse buckling equivalence method [30].

The approximation of the transverse leakage is given as:

gu g

S

L J dS= ∫� 4.8

The surface integral is performed on the surface orthogonal to the collapsed directions,

where u is our direction of interest ( , ,r z θ ).

out in

in out

J J J

J J J

+

−

= −

= − 4.9

The total leakage is:

( ) ( ) ( )g r r r r z z zL J S J S J J S J J Sθ θ θ+ + − − + − + −= − + − + − 4.10

The buckling is then given as

12

1

G

g

g

g G

g g

g

L

B

D Vφ

=

=

=

∑

∑

4.11

In Figure 25 it is shown that the two-group buckling converges in a given spatial node (as

illustration of convergence in all nodes) for the 3-D calculations.

64

-4.00E-03

-3.50E-03

-3.00E-03

-2.50E-03

-2.00E-03

-1.50E-03

-1.00E-03

-5.00E-04

0.00E+00

0 5 10 15 20

Bu

cklin

g

Iteration

Buckling(1,1,1,1)

Buckling(1,2,1,1)

Figure 25: Convergence of the buckling distribution

From the core diffusion calculation the leakage term (and thus the albedo and buckling) for

the spectrum feedback is obtained. The core thermal-hydraulic model provides the

corresponding temperatures of the fuel, moderator and coolant averaged over the volumes of

the reactor spectrum zones, being ready for further neutronics evaluation.

In the improved in this study the PSU base feedback model the cross-sections are

calculated for each spectral zone (see Figure 26) and for each coarse energy group (normally

four). The basic unit of material composition is a batch. In each layer a number of batches with

different irradiation ages can reside. These are mixed and put together to form a layer. These

layers present partial volumes of the reactor core, which provide the distribution of materials

(or cross sections) for the flux calculation. All the batches within a layer are exposed to the

same local flux. Number of batches can be placed together to form a spectral zone. Spectral

calculations are based on the averaged atom densities of all the batches within this zone and

therefore provide the broad group cross sections for the respective batches.

65

Figure 26: Simplified representation of spectral zones of PBR

In the void above the pebble bed calculations are performed by employing adapted

(direction-dependent) diffusion coefficients. Since a diffusion coefficient for each coordinate

direction is provided in the PSU model the different streaming effects (depending on the

relative size and shape of the void) can be adjusted.

The major PBR reactivity feedback parameters are fuel temperature and moderator

(graphite) temperature, and to the less extent the coolant (helium) temperature. The spatial

variation of these parameters requires a proper modeling of local cross-section dependencies

on these feedback parameters for a multi-dimensional PBR core analysis. Entire ranges of

changes of the major feedback parameters are covered using multi-dimensional table

interpolation thus avoiding simplified polynomial fitting with possibilities for extrapolation.

Unloaded fuel elements

StorageBoxes

Batch 1,2 3

Fresh FuelElements

66

Special software package is developed that reads the tables and interpolates in them to

obtain macroscopic cross-section values for each spectral zone (core region) and for each

broad energy group. The testing of the functionality of the package was performed successfully

in the framework of the NEM/THERMIX-DIREKT code. By using this method there is no

chance that the calculation of the cross section can be outside of the bounds set by the user

and to involve extrapolation procedure. In addition, this approach also helps to improve the

accuracy of modeling the cross section variations by getting around user-calculated

coefficients that could contain errors and by treating explicitly the cross terms of cross-section

dependencies (which is especially important for transient simulations). This method takes into

account the non-linear thermal hydraulic feedback parameter phenomena that are critical for

accurate prediction of the cross section behavior.

The developed feedback model has three options: the base one using linear surface

interpolation routine, the second using a higher order interpolation and the third using Besier

splines (B-spline) surface interpolation routine.

For the base feedback model it is necessary to tabulate a sufficient number of points to

achieve the degree of accuracy that is desired. Since linear interpolation is used and the

behavior of the cross sections is not linear with respect to temperature, many points are

necessary. In addition temperature varies significantly and non-linearly across the core in both

the radial and axial directions. When more points are added, the table size increases

disproportionately. To keep the number of cross-section generation calculations to a minimum,

and also the total amount of data that has to be tabulated while maintaining a particular degree

of accuracy, a higher order interpolation scheme has been implemented to interpolate the

tables. The use of a higher order interpolation over linear interpolation has the advantage of

yielding the same degree of accuracy with fewer points.

The utilized higher order interpolation scheme is based on quintic spline routine. It uses a

fifth order polynomial to fit the data. This routine calculates the coefficients at a set of knots

(independent points at which dependent points are known) and these coefficients are then

used for the interpolation. A front-end routine is needed to supply the routine with the required

data and a back end routine is needed to do the interpolation using the coefficients.

67

The curve fitting subroutine calculates the coefficients of a piece wise natural quintic spline

with knots and uses the method of least resistance. The term spline was adopted from a tool

that draftsmen used to create a smooth curve through a set of points. The method of least

resistance ensures that the curve passes through each point with the least amount of tension

on the curve. This is important to prevent the formation of oscillations between the points

defining the curve (knots). This curve fitting method can only be used for a strictly increasing or

decreasing sequence of knots. The knots must be formed such that the fifth power of X(I+1)-

X(I) can be formed without overflow or underflow of the exponents. This does not create a

problem with the parameters used in this application. The equations used are:

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )IYPIBPICPIDPIEPIFxS +×+×+×+×+×= 4.12

or

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )( )1

11111

++×

+−×++×+−×++×+−=

IYQ

IBQICQIDQIEQIFxS

4.13

where :

S(x) = Interpolated spline value at point x.

F(I) , E(I) , D(I), C(I), B(I) = Calculated spline coefficients at point I.

P = x – x(I)

Q = x(I+1) – x

When cross terms are present the additive effect is less accurate when compared to actual

cross section data calculated by the spectral codes. This is due to interdependence (non-

linearity) of the cross section behavior on the parameters. Such non-linearity can be explained

with the spectral shifts in the neutron behavior based on these parameters that cannot be

accurately described by simply adding up the individual effects to obtain a total representative

cross section.

More sophisticated interpolation routines have been utilized to help model these

interactions. Multi-dimensional surface interpolation schemes such as Besier splines (B-spline)

surface interpolation have been shown to be able to handle varying shapes and to be very

stable. They also have the advantage of being local, which means that the basis function is

only affected by the neighboring nodes. The number of neighboring nodes that is needed is

68

based on the smoothness of the curve that is desired, which means that fewer points have to

be evaluated to obtain the same information thus reducing the computational effort that is

required. Another advantage to B-splines is that they are an approximation not an

interpolation. This means that the curve or surface generated only has the requirement that it

comes close to the point and is not forced through the point. It is well known that any computer

code can produce varying results and when spectral codes are pushed to bounds that are not

well defined, such as points at the extreme limits of the off-nominal condition range; they can

produce results that do not correspond to a smooth curve.

The three interpolation methods – linear surface interpolation, higher order (quintic splines)

interpolation, and Bezier surface interpolation method – have been implemented in the

NEM/THERMIX feedback model. The test results demonstrated the superiority of the Bezier

surface interpolation method in terms of an optimal combination of accuracy and efficiency for

steady state and transient calculations.

4.5 Xenon and Iodine models

The previously implemented time-dependent Xenon and Iodine models for transient

calculation have been improved using equations 4.14 and 4.15. The improvement in the

representation of Xenon is shown in Figure 27.

( ) 0I II ft t

I

I t I e eλ λ

γ φ

λ− −

∞

Σ= = 4.14

( ) ( )

( )( )0 0

0

X X I

X X I

t t tI

I X

I X f I ft t t

X

X a I X

IX t X e e e

e e e

λ λ λ

λ λ λ

λ

λ λ

γ γ φ γ φ

λ σ φ λ λ

− − −∞∞

− − −

= + −−

+ Σ Σ = + −

+ −

4.15

However, these equations assume that the reactor shutdown occurs at the beginning of the

transient. Hence the conditions are valid only if the flux disappears promptly after initiation of

the transient and these models still need further improvements.

69

Figure 27: New Xenon model

These Xenon and Iodine models have to be modified to enable the calculation that reflects

response to changes in power as function of time appropriately as represented in Figure 28.

These changes have been implemented by the introduction of analytical models represented

by equations (4.16) for Iodine and (4.17) for Xenon.

( ) 0 1 0

1

1 II f t

I

I t eλ

γ φ φ φ

λ φ−

Σ −= −

4.16

( )( ) ( )

( )( )1

1 1 0

1 1 1

1

1

1X

X a

XX aI

tI X f X

X X

X a X a

XttX aI

X

I X X a I

X t e

e e

λ σ φ

λ σ φλ

γ γ φ φ φ λ

λ σ φ φ λ σ φ

λ σ φγ

γ γ λ σ φ λ

− +

− +−

+ Σ −= −

+ +

+ + − + + −

4.17

70

Figure 28: Xenon changes in response to power changes [46]

The Xenon interpolation routine had challenges in a sense that the units of interpolation

table were stated in barn-1.cm-1 whereas NEM returns the Xenon atom concentration in

atoms.cm-3. These challenges have been addressed in this study by modifying the feedback

model in NEM/THERMIX as shown in Figure 29. The steady state convergence of Xenon

number density in two selected spatial nodes, as illustration of convergence in all spatial

nodes, when using the improved Iodine and Xenon models in NEM/THERMIX is depicted in

Figure 30.

71

0.00E+00

2.00E+15

4.00E+15

6.00E+15

8.00E+15

1.00E+16

1.20E+16

1.40E+16

1.60E+16

1.80E+16

0 5 10 15 20

Nu

mb

er

De

nsi

ty (

ato

ms.

cm3)

Iteration

xennold(17,1,19)

xennold(12,3,19)

Figure 30: Steady state convergence of 3-D map of Xenon number densities

4.5.1 Final 5-D steady problem convergence

The overall solution convergence of the coupled calculations with all of the implemented

feedback modeling enhancements in NEM/THERMIX as part of this PhD research are shown

[Xe] barn-1.cm-1

x

aσ barn

LIBRARY x

aσ cm2

[Xe] atoms.cm-3

BBSTEA

*1024

barn.cm-2 *10-24

cm2.barn-1

BBTRANS

Figure 29: Xenon model processing for cross section interpolation

72

in Figure 31 on the example of the multiplication factor convergence as a function of the

number of coupled iterations.

0.98500

0.99000

0.99500

1.00000

1.00500

1.01000

1.01500

1.02000

1.02500

0 5 10 15 20

K-E

ffe

ctiv

e

Iteration

Figure 31: The k-effective convergence

4.6 Transient cross section modeling

The previous modeling of instantaneous cross-sections dependencies (important for

transient analysis) was limited in accuracy and applicability for cross section data sets with

three-dimensional tables. The new model performs direct five-dimensional table interpolation

to account for most feedback phenomena. A feedback model has been developed to account

for the feedback effects of temperature and spectrum and was implemented into NEM. The

five parameters of interest in the model are fuel temperature (Tf), moderator temperature (Tm),

fast buckling ( 2

1Β ), thermal buckling ( 2

2Β ) and Xenon concentration ([ ]Xe ).

The ultimate advantage was to extend the range of applicability and eliminate the

possibility of negative cross sections by introducing, in addition to the interpolation between

points, extrapolation beyond prescribed points in the tables; thus obtaining improved efficiency

in the transient analysis calculation. Two software packages have been developed as part of

this new feedback model. The first package automatically generates a macroscopic cross

section library for a given PBMR core model using MICROX-2. This library contains sets of

73

macroscopic 5-D cross section tables for each composition. Users can specify the ranges of

the feedback parameters as well as the number of reference points themselves. Once this

information is selected it is stored at the beginning of the cross section tables. This information

is read by NEM together with the reference cross section values. This library contains tables

for transport, absorption, fission, production and scattering cross sections. The second

software package reads the tables and interpolates in them to obtain the macroscopic cross

section values for each spectral zone (core region) and for each broad energy group. This

package interacts with NEM in the following way: first, the cross section library is read once at

the beginning of the calculation process and stored in the NEM arrays. During the calculation

process for each spatial node of the NEM core model, five parameters representative of this

node are passed to the feedback module. Using these values, five-dimensional tables are then

interpolated for the appropriate macroscopic cross section values. The updated macroscopic

cross sections are passed back to NEM to perform core calculations, pass the power

distribution to THERMIX-DIREKT and getting the relevant thermal-hydraulics data in turn, and

this calculation loop continues. The layout of a typical cross section table for PBMR with

parameters Tf, Tm, 2

1Β , 2

2Β and [Xe] (the fuel temperature, moderator temperature, fast buckling,

thermal buckling and Xenon number density respectively) is shown in Table 11.

Table 11: 5-dimensional cross section table

Tf1 Tf2 Tf3 Tf4 Tm1

Tm2 Tm3 Tm4 Tm5 Tm6

Tm7 Bf1 Bf2 Bf3 Bt1

Bt2 Bt3 X1 X2 X3

Σ1 Σ2 Σ3 Σ4 Σ5

Σ6 Σ7 Σ8 Σ9 Σ10

Σ11 … … … …

Σ756

74

where:

• Tf is the fuel temperature

• Tm is the moderator temperature

• Bf is the fast buckling

• Bm is the thermal buckling

• X is the Xenon number density

• Σ is the macroscopic cross section

The layout of cross section tables is as follows:

• Σ1 is a function of (Tf1,Tm1,Bf1,Bt1,X1)

• Σ2 is a function of (Tf2, Tm1,Bf1,Bt1,X1)

• Σ3 is a function of (Tf3, Tm1,Bf1,Bt1,X1)

• Σ4 is a function of (Tf4, Tm1,Bf1,Bt1,X1)

• Σ5 is a function of (Tf1, Tm2,Bf1,Bt1,X1)

• …

• Σ253 is a function of (Tf1,Tm1,Bf1,Bt1,X2)

• …

• Σ756 is a function of (Tf4,Tm7,Bf3,Bt3,X3)

4.7 Control rod movement modeling

The reactor designer has to specify the Operating Technical Specifications (OTS) for the

reactor design. Within this section the limiting conditions of operation have to be detailed. This

entails the level of safe operation dictated by the performance of the equipment. The control

rods form an important part of this aspect in the justification of the control of reactivity. The

demonstration of this safety aspect involves the best estimate deterministic modeling of the

75

reactivity insertion/withdrawal during operation. In this section the movement of the control

rods modeling is demonstrated by adopting different approaches for the estimation of reactivity

change.

The challenge in the modeling of control rod movement is in the estimation of reaction rates

in partially rodded nodes during a numerical procedure of blending rodded and unrodded

cross-section values. The setup for the system is depicted in Figure 33.

NEM processing that includes the control rod movement is shown in Figure 32.

4.7.1 Linear Rod Motion Model (ROMO)

The control rod movement has been implemented to introduce the capability of modeling

transients that involve shutdown of the reactor or control rod insertion, rod withdrawal and

control rod ejection.

The control rods positioned in the side reflector cannot be represented explicitly in two-

dimensional axi-symmetric geometry. A number of models are commonly used to overcome

σF

VF ϕg

dzc, Vc, σC

dzF, VF, σF

Zin(k)

Partially Rodded Material (PRM)

Unrodded Material (URM)

k

Rodded Material (RM)

Zin(k+1)

ktop

kbot

Z

r

Figure 32: Partially rodded nodes

76

this limitation. The ‘grey’ curtain model is adopted. This approximation models the 24 control

rods as a ring or curtain of absorber material (for all azimuthal meshes-symmetry in 2-D) by

defining a material with an effective absorber boron concentration that conserves the reactivity

effect of the control rods. This method can be used with great success to conserve the

reactivity effect of the control rods. This model is easily implemented by means of overlaid

cross-section sets. Typically cross-section sets are defined as follows: one for the control rod

fully inserted and another for the control rod withdrawn from a given material mesh as shown

in Figure 33.

Control rod withdrawnfrom mesh

Control rod fullyInserted in mesh

etcrffa ;;;; ΣΣΣΣ ν etcrffa ;;;; ΣΣΣΣ νCross-section set A Cross-section set B

Figure 33: Simulation of control rod movement

When continuous control rod movements need to be simulated, the same principle can be

used by adjusting the neutron poison concentration in the given axial mesh where the control

rods are partially inserted into. A typical overlaid cross-section value is calculated by blending

unrodded and rodded cross-section values. In the case of Figure 33 this is ( )BA

Σ+Σ−=Σ ςς1 .

The correct choice of the parameter ς ( 10 ≤≤ ς ) is essential for modeling small control rod

movements because the parameter is not linearly dependent on control rod position. If a

simple linear relation or volume weighting is used the reactivity effect of the partial insertion will

be overestimated since the self-shielding effect cannot be modeled correctly and the diluted

boron have a relatively too large absorption effect. This leads to the so-called “cusping effect”

where the rate of reactivity insertion as a function of rod movement is not smooth function. This

problem is solved if flux volume weighting is used to obtain the effective boron concentration in

the partially rodded mesh but an axial flux shape is then needed in advance to get the correct

77

weighting. This axial flux shape within the mesh has to be calculated explicitly or estimated

from the available flux information. Typically this information is not available in many codes. If

correct flux-volume-weighting can be performed this is the preferred method to apply.

A simple approximate method to obtain a smooth reactivity insertion for the partially rodded

meshes is given below and is based on the TINTE implementation developed at FZJ, the so-

called ‘Linear Rod Motion Model’ (ROMO). This model uses a special exponential interpolation

algorithm for the absorber concentration in the partially rodded mesh (node). The calculation of

the variable absorber concentration c* as a function of the insertion length l of the rods in an

annular region with partially inserted rods is given by:

( )( )

LlcS

SL

l

lc ≤≤∗−

−

= 0,1ln

1ln*

2.44

c absorber concentration in the rod region when rod bank is fully inserted in this

region

c*(l) absorber concentration in the rod region when rod bank is partially inserted in

this region over the length l , according to the linear rod motion model

l insertion length of the rod bank in the rod region

L length (axial height) of the rod region

S interpolation factor of the special exponential interpolation scheme of the linear

rod motion model, also called “absorber blackness value” because it

characterizes the neutron absorber poison strength; it is always: 0 < S < 1.

The cross sections of the partially rodded meshes can thus be mixed in this way to reduce

the cusping effects. The interpolation factor S is in general dependent on the mesh size and

the blackness of the grey curtain and can be found by inspection of performing several k-

effective calculations at various control rod positions and adjusting the value until a smooth

curve, per mesh is obtained, resulting in potentially different S factors per mesh. This treatment

is important for the control rod withdrawal transient. For the 50 cm meshes applicable in the

benchmark a starting guess of 0.9 can be used. If the mixing of the two sets of cross sections

by volume weighting is applied on a very fine sub-mesh (< 5 cm) and not on the material mesh

defined (50cm), the cusping effect will still appear for each of these sub-meshes when they are

78

partially rodded but the overall effect is much reduced and should not introduce major

variations in the overall results.

4.7.2 Volume-weighting method

The properties of the new material FM are given by:

( ) ( )M F

M F

M F

g V VFM

g

g V V

E dV E dV dE

dV dV dE

σ σ

σ

+

=

+

∫ ∫ ∫

∫ ∫ ∫

This method has limitations since it does not provide for the preservation of reaction

rates and leads to cusping effect.

4.7.3 Flux-volume weighting Method

The control rod methodology for flux-volume-weighting approach was developed and

implemented in NEM/THERMIX as part of this PhD study. This involves the mixing of materials

from the rodded region and the unrodded regions in a sophisticated manner to ensure that the

cusping effects are eliminated and that the reaction rates are preserved.

The method involves the use of the Equivalence Theory in homogenization of materials

(e.g. M and F in Figure 34) in a cell of interest.

79

The properties of the new material FM are given by:

( ) ( ) ( ) ( )

( ) ( )

, ,

, ,

M F

M F

M F

g V VFM

g

g V V

E dV r E E dV r E dE

dE dV r E dV r E

σ φ σ φ

σ

φ φ

+

=

+

∫ ∫ ∫

∫ ∫ ∫

( ) ( )

( ) ( )

M F

M F

M F

FM V Vg

V V

dV r dV r

dV r dV r

σ φ σ φ

σφ φ

+

=+

∫ ∫

∫ ∫

Homogenisation of the cross section FC

gσ for cell k in Figure 38, in the partially rodded

area is given by:

( ) ( )

( ) ( )

C F

C F

C F

g C g F

FC V Vg

g C g F

V V

r dV r dV

r dV r dV

σ φ σ φ

σφ φ

+

=+

∫ ∫

∫ ∫

C C C F F F

g g g gFC

g C C F F

g g

V V

V V

σ φ σ φσ

φ φ

+=

+

Figure 34: Material homogenisation approach

VF, σF

VM, σM

80

4.7.3.1 Linear interpolation of the flux

The flux in the partially rodded nodes will also depend on the flux of the neighbors (see

Figure 35). Linear interpolation will be used to obtain the approximated flux:

1

2

k ka

z zz −−

=

2 1

2

k kb

z zz + +−

=

( ) ( ) ( ) ( )a b a

a b a

z z z z

z z z z

φ φ φ φ− −=

− −

( ) ( ) ( ) ( )ag g b g a g a

b a

z zz z z z

z zφ φ φ φ

− = ⋅ − + −

Figure 35: Flux estimation for partially rodded nodes

Φg(Z1

)

Φg(Z)

Zk-

Φg(Z0)

ΦgF

ΦgC

k-1=ka

Za Zmin= Zk Zk+2

Zb

dZF

Zk+1=Zmax

Zc

z

K+1=kb

dZC

81

( )F

g g Fzφ φ=

( )C

g g Czφ φ=

where

( )1 1

2 2

F

F k k kz z z z z dz= − + = +

( )1

1 1

2 2

C

C kz z z z z dz+= − + = +

4.7.3.2 Implementation and testing for control rod movement

The algorithm for the control rod movement was implemented as shown in Figure 42

82

The most important thing to record during calculation procedure is the location of the

control rod tips. First the location is tracked by the location index of the node corresponding to

the distance of insertion. In Figure 37, the indices of the location of the rods are shown with ka

indicating the node below and kb indicating the node above the partially rodded node.

Change Material in kCR MixCR

Identify and index affected control rod locations setCR_NEM (x, y)

Calculate ZCR(*) MoveCR

Is the rod tip covering the next boundary?

Check against zdim(k) to index the z location of control rod

Mix materials mixCR

Interpolate cross sections XSTAB

Solve diffusion equation

Figure 36: Flowchart for the control rod model

83

16

16.5

17

17.5

18

18.5

19

19.5

20

20.5

21

0 10 20 30 40 50 60

No

de

Nu

mb

er

Time(sec)

k_a

k

k_b

Figure 37: Index of rod position

0

200

400

600

800

1000

1200

1400

10 11 12 13 14 15 16 17 18

Dis

tan

ce (

cm)

Time(sec)

Figure 38: Tracking of rod tip during DLOFC transient

Figure 38 shows the actual distance of insertion during the reactor scram that lasts 3

seconds for the DLOFC transient

84

0.00E+00

2.00E+13

4.00E+13

6.00E+13

8.00E+13

1.00E+14

1.20E+14

1.40E+14

0 10 20 30 40 50 60

Flu

x (

#.c

m-2

.s-1

)

Time(sec)

phi_a

phi_C

phi_k

phi_F

phi_b

Figure 39: Results testing for the flux approximation

To test the impletentation of flux interpolation model, the control rod simulation was

conducted by withdrawing the rods from the original operation location for 20 seconds from 30

to 50 seconds. The flux interpolation showed that the flux could be tracked at every time step

during the calculation as shown in Figure 39.

Figure 40 illustrtaes that the Flux-Volume-Weighting (FVW) method reduces the

cusping effects as comapred to the Volume-Weighting (VW) approach.

2.50E+13

2.55E+13

2.60E+13

2.65E+13

2.70E+13

2.75E+13

2.80E+13

2.85E+13

2.90E+13

20 40 60 80 100 120

Flu

x(#

.cm

-2.s

-1)

Time(sec)

VW

FVW

85

Figure 40: Cusping effects during rod movement

It must be emphasised that the time step size selection for the analysis of the fast and slow

transients is important. This is not just the consideration for stability of the numerical solution,

but also for the dynamics of the problem that is being investigated. For instance if the time step

size will allow the tip of the rod to skip axial nodes during the movement, this can result in the

material map not being updated for the nodes that were not traversed by the rod tip – see

Figure 40. Hence there would be traces of rodded material trailing in the preceding nodes

during the movement.

This has posed additional limitations to the time step size selection in the current code. The

requirement for the proper modeling in this instance is that the time step has to be selected

considering the speed of the rod and the smallest axial node size in mind. Hence the product

of the speed and the time step size (in seconds) should not exceed the size of the smallest

node (in cm).

4.8 Decay heat calculation

The decay heat source is only of importance in certain transient cases, typically where the

fission power is reduced to zero during the event as shown in Figure 41. For the steady-state

cases the decay heat is assumed to be part of the energy released per fission, which is

assumed to be all deposited locally, i.e. where the fission took place.

86

0%

1%

10%

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Decay time (seconds)

De

ca

y h

ea

t (%

of

tota

l p

ow

er)

Decay Heat (%)

Figure 41: Decay heat behaviour (% of fission power)

The decay heat value for each material mesh in the core is derived making use of the

relative core average decay heat behaviour (values provided as determined from the DIN

25485 standard) and the material mesh power. This implies that the decay heat is directly

related to the steady-state power produced in the mesh prior to the start of the transient. No

history effects or power excursions after the start of the transient (t=0) are taken into account.

The decay heat as calculated for the equilibrium core during DLOFC transient is shown

in Figure 41 for a period of just over 100 hours (365000 seconds). NEM reads and linearly

interpolates within the set of data points to obtain total decay heat relative to the total power at

any given time step and re-distributes it spatially according to the initial steady state power

distribution.

As an alternative a reduced set of data points can also be utilised for interpolation and

that will lead to an acceptable error. A linear interpolation of the log of time and log of the

decay heat is used. The maximum error is -1.5% but this only applies to the first half second.

87

After 100 seconds the time-integrated heat error is already smaller than -0.2%

(underestimated) while over the 100 hour total period the error is only -0.026%. For the

transient cases where this decay heat data is used the effect of these errors will be

insignificant. This is illustrated in Figure 42. The data points (25 in total) is shown with the

“LOG interpolated decay heat” data that falls on top of the reference “Decay Heat” set. The

small differences are shown as the “LOG Interpolation error” and are also expressed as a

percentage difference.

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

7%

0 10 20 30 40 50 60 70 80 90 100 110

Decay time (hours)

De

cay

he

at

(% o

f to

tal

po

we

r)

Decay Heat (%)

Data Points

LOG Interpolated decay heat

LOG Interpolation error

Figure 42: Log interpolation data points and time step error estimation

The final feedback model developed in this PhD research for NEM/THERMIX coupled

calculations is shown in Figure 43. The feedback model was improved and accounts for five

cross-sectional dependencies of fuel temperature, moderator temperature, Xenon number

densities, fast and thermal buckling.

88

4.9 Conclusions

This chapter summarized the completed optimization studies of the coupled multi-physics

methodology for PBMR safety analysis. Such studies were designed to quantify the effects of

thermal-hydraulic modeling, coupling spatial and temporal schemes and feedback modeling for

steady state and transient simulations. The NEM/THERMIX code system is now ready to

analyze any of the problems within the OECD benchmark framework.

Figure 43: New NEM/THERMIX feedback model

NEM

Cross Section Library

Interpolation THERMIX Temperature

Mass Flow Rates

Pressure

Power Density

TModerator

Tfuel

Xe

Flux

Buckling 2

Buckling 1

89

Chapter 5 NEA/OECD PBMR-400 Coupled Code Benchmark

5.1 Introduction

The Nuclear Energy Agency (NEA) of the Organization for Economic Cooperation and

Development (OECD) has accepted, through the Nuclear Science Committee (NSC), the

inclusion in its program, the Pebble-Bed Modular Reactor (PBMR) coupled neutronics/thermal

hydraulics transient benchmark problem.

The benchmark is complementary to other ongoing or planned efforts in the reactor

physics community. The PBMR-268 benchmark problem [22]; initiated by PBMR Pty Ltd, PSU

and NRG, served as the predecessor to this effort. The work was concluded and future efforts

were focused on this benchmark. The PBMR-400 MW core design was also a test case in the

IAEA CRP-5 (TECDOC2 in preparation) but important differences exist between the test case

definitions and approaches. The OECD benchmark includes additional steady-state and

transient cases including reactivity insertion transients that are not included in the CRP5 effort.

Furthermore it makes use of a common set of cross sections (to eliminate uncertainties

between different codes) and includes specific simplifications to the design to limit the need for

participants to introduce approximations in their models which could be inconsistent.

The scope of the benchmark is to establish a well-defined problem, based on a common

given set of cross sections and to compare methods and tools in core simulation and thermal

hydraulics analysis with a specific focus on transient events through a set of multi-dimensional

computational test problems. In addition the benchmark exercise has the following objectives:

• establish a standard benchmark for coupled codes (neutronics/thermal-hydraulics) for

PBMR design;

• code-to-code comparison using a common cross section library, which is important for

verification and validation; and

• obtain a detailed understanding of the events and the processes.

90

Figure 44: PBMR-400MWth reactor

The reference design for the PBMR-400 benchmark problem is derived from the PBMR-

400MW NPP design described in Table 12. Several simplifications were made to the design as

shown in Figure 44 in order to limit the need for any further approximations to a minimum.

During this process care has been taken to ensure that all the important characteristics of the

reactor design were preserved. This ensures that the results from the benchmark will be

91

representative of the actual design’s characteristics. This benchmark is used in this PhD

studies as a test set for the developed optimized coupled code models.

Table 12: Major design and operating characteristics of the PBMR-400 reactor

PBMR Characteristic Value

Installed thermal capacity 400 MW(t)

Installed electric capacity 165MW(e)

Load following capability 100-40-100%

Availability ≥ 95%

Core configuration Vertical with fixed centre graphite reflector

Fuel TRISO ceramic coated U-235 in graphite

spheres

Primary coolant Helium

Primary coolant pressure 9Mpa

Moderator Graphite

Core outlet temperature 900°C.

Core inlet temperature 500°C.

Cycle type Direct

Number of circuits 1

Cycle efficiency ≥ 41%

Emergency planning zone 400 meters

5.2 Steady State Cases

The steady state benchmark calculation cases include 3 exercises:

5.2.1 Exercise 1 (Case S-1): Neutronics solution with fixed cross sections

An equilibrium core is utilized with the reactor operational state achieved after a

considerable time of operating at a specific set of conditions. Operating conditions are defined

to be at full power and with the control rods inserted 2.0 m below the bottom of the top reflector

(therefore 1.5 m alongside the pebble-bed). Once equilibrium is reached no significant

changes can be observed in the properties of the core. For example the k-eff, power profile,

temperatures and isotopic concentration distribution do no longer change.

92

Cross sections, which were generated by making use of the isotopic distribution calculated

in VSOP99, were provided. Hence no state parameter dependence and thermal-hydraulic

feedback modeling were required. All participants used this common set of cross sections to

facilitate better and well-defined comparisons as well as to allow broader participation in the

benchmark.

There are 190 different material sets for the core at equilibrium, which were arranged in the

2-D model as shown in Figure 45.

Figure 45: Neutronic model for case S1

5.2.2 Exercise 2 (Case S-2): Thermal hydraulic solution with given power

This exercise makes use of the thermal hydraulic properties and model description and the

following conditions:

o The provided power/heat source density given with the values corresponding to core

regions 1 – 110 as used in Exercise 1 (Case S1).

0 10 41 73.6 80.55 92.05 100 117 134 151 168 185 192.95 204.45 211.4 225 243.6 260.6 275 287.5 292.5

-200 10 31 32.6 6.95 11.5 7.95 17 17 17 17 17 7.95 11.5 6.95 13.6 18.6 17 14.4 12.5 5

-150 50 133 133 133 133 155 116 113 113 113 113 113 135 164 144 144 152 152 152 189 190

-100 50 133 133 133 133 155 116 113 113 113 113 113 135 164 144 144 152 152 152 189 190

-50 50 133 133 133 133 155 116 112 112 112 112 112 135 164 144 144 152 152 152 189 190

0 50 133 133 133 133 155 116 111 111 111 111 111 135 165 144 144 152 152 152 189 190

50 50 134 134 134 125 156 117 1 23 45 67 89 136 166 145 145 153 153 153 189 190

100 50 134 134 134 125 156 117 2 24 46 68 90 136 167 145 145 153 153 153 189 190

150 50 134 134 134 126 157 118 3 25 47 69 91 137 168 146 146 153 153 153 189 190

200 50 134 134 134 126 157 118 4 26 48 70 92 137 169 146 146 153 153 153 189 190

250 50 134 134 134 126 157 118 5 27 49 71 93 137 170 146 146 153 153 153 189 190

300 50 134 134 134 127 158 119 6 28 50 72 94 138 171 147 147 153 153 153 189 190

350 50 134 134 134 127 158 119 7 29 51 73 95 138 172 147 147 153 153 153 189 190

400 50 134 134 134 127 158 119 8 30 52 74 96 138 173 147 147 153 153 153 189 190

450 50 134 134 134 127 158 119 9 31 53 75 97 138 174 147 147 153 153 153 189 190

500 50 134 134 134 128 159 120 10 32 54 76 98 139 175 148 148 153 153 153 189 190

550 50 134 134 134 128 159 120 11 33 55 77 99 139 176 148 148 153 153 153 189 190

600 50 134 134 134 128 159 120 12 34 56 78 100 139 177 148 148 153 153 153 189 190

650 50 134 134 134 128 159 120 13 35 57 79 101 139 178 148 148 153 153 153 189 190

700 50 134 134 134 129 160 121 14 36 58 80 102 140 179 149 149 153 153 153 189 190

750 50 134 134 134 129 160 121 15 37 59 81 103 140 180 149 149 153 153 153 189 190

800 50 134 134 134 129 160 121 16 38 60 82 104 140 181 149 149 153 153 153 189 190

850 50 134 134 134 129 160 121 17 39 61 83 105 140 182 149 149 153 153 153 189 190

900 50 134 134 134 130 161 122 18 40 62 84 106 141 183 150 150 153 153 153 189 190

950 50 134 134 134 130 161 122 19 41 63 85 107 141 184 150 150 153 153 153 189 190

1000 50 134 134 134 130 161 122 20 42 64 86 108 141 185 150 150 153 153 153 189 190

1050 50 134 134 134 131 162 123 21 43 65 87 109 142 186 151 151 153 153 153 189 190

1100 50 134 134 134 131 162 123 22 44 66 88 110 142 187 151 151 153 153 153 189 190

1150 50 132 132 132 132 163 124 114 114 114 114 114 143 188 151 151 154 154 154 189 190

1200 50 132 132 132 132 163 124 115 115 115 115 115 143 188 151 151 154 154 154 189 190

1250 50 132 132 132 132 163 124 115 115 115 115 115 143 188 151 151 154 154 154 189 190

93

o Calculate the temperatures distribution, outlet temperature, pressure drop over the core

and heat loss to the constant temperature boundary.

o Assume fresh fuel Zehner-Schlünder pebble bed effective thermal conductivities

(resultant values to be provided to be used as input if required).

5.2.3 Exercise 3 (Case S-3): Combined neutronics thermal hydraulics calculation

This exercise represents the equilibrium cycle steady-state conditions and makes use of

the state-parameter dependent cross section library and thus is a coupled neutronics/thermal-

hydraulics calculation. This case also represents the starting conditions for the transient

events.

5.3 Comparison of steady state results

The k-effective results for Case S1 are shown in Figure 46 with the average value of

1.00437 shown with the black line. The codes were in reasonable agreement in the prediction

of the multiplication factor for this case.

94

1.00437 + 22 pcm

1.00370

1.00380

1.00390

1.00400

1.00410

1.00420

1.00430

1.00440

1.00450

1.00460

1.00470

K-E

ffe

ctiv

e

Figure 46: Comparison of k-eff for OECD PBMR-400 Case S1

The power distribution reported by participants was in 2-D maps. The data was reduced by

radially averaging for axial power distribution and axially averaging for radial distribution. From

the axial and radial power distributions shown in Figure 47 and Figure 48, it can be noted that

different codes predicted different maximum power densities. The power at the top of the

reactor was also different as result of challenges in modeling the helium space at the top of the

pebble bed reactor. From the visual inspection of the data, it was evident that the power and

the flux distributions reported by all participants were in agreement. The main difference was at

the top of the reactor where cavities are encountered. This was an indication that some

spatially optimized calculation is still required.

95

0

1

2

3

4

5

6

7

8

9

10

0 200 400 600 800 1000 1200

PO

WE

R D

EN

SIT

Y(W

.cm

-3)

AXIAL POSITION(cm)

DALTON

DORT-TD Diff.

CAPP (IC FEM)

CAPP (CMFDM)

TOPS

TINTE

NEM

PARCS

BOLD VENTUREMGRAC

CITATION

PEBBED_FD

Figure 47: Axial power distribution in the PBMR-400 reactor

96

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

100 120 140 160 180

PO

WE

R D

EN

SIT

Y(W

/cm

-3)

RADIAL POSITION(CM)

DALTON

DORT-TD Diff.

CAPP (IC FEM)

CAPP (CMFDM)

TOPS

TINTE

NEM

PARCS

BOLD VENTURE

MGRAC

CITATION

PEBBED_FD

PEBBED_ND

Figure 48: Radial power distribution for PBMR-400 reactor

The flux distributions also displayed evident differences especially in the reflector and

the helium regions of the reactor as shown in Figure 49.

97

0

2E+13

4E+13

6E+13

8E+13

1E+14

1.2E+14

1.4E+14

1.6E+145

25.5

57.3

77.0

75

86.3

96.0

25

108.

5

125.

5

142.

5

159.

5

176.

5

188.

975

198.

7

207.

925

218.

2

234.

3

252.

1

267.

8

281.

25 290

FL

UX

(n

/cm

2/s

)

RADIAL POSITION (cm)

DALTON

DORT-TD Diffusion

CAPP (IC FEM)

CAPP-CMFDM

TOPS

TINTE

NEM

PARCS

BOLD VENTURE

OSCAR-4

CITATION

PEBBED_FD

PEBBED_ND

Figure 49: Radial thermal flux distribution in PBMR-400 reactor

The high Helium outlet temperature of about 900C was demonstrated by the results of

Case S2 as shown in Figure 50.

98

896.5

897.0

897.5

898.0

898.5

899.0

899.5

900.0T

EM

PE

RA

TU

RE

(°

C)

Figure 50: Comparison of outlet temperature for OECD PBMR-400 exercise 2

One can notice the differences introduced by modeling the reactor with and without

feedback in Table 13 (i.e. between exercise S1 and S3).

Table 13: Comparison of k-eff with and without feedback

Code K-eff_S1 K-eff_S3

NEM 1.00045 1.09060

PARCS 0.99283 1.04099

DALTON-Fine Mesh 0.99928 1.00476

The change in power distribution can also be demonstrated by the results of NEM

(Figure 51) and DALTON (Figure 52).

99

Comparison of axial power distribution

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 200 400 600 800 1000 1200 1400

Axial distance from top to battom(cm)

Po

wer

den

sit

y (

W/c

m^

3) Axial_Power_S3

Axial_Power_S1

Figure 51: NEM axial power distribution for cases S1 and S3

Figure 52: DALTON axial power distribution for cases S1 and S3

100

5.4 Transient cases

Six transient cases, covering the range from slow to fast neutronic transients, as well as

feedback effects from thermal-hydraulic parameters and fission products, are defined in the

transient part of the benchmark study. The cases are given below and their detailed

descriptions are given in Appendix A (Table 18 through Table 22):

• De-pressurized loss of forced cooling (DLOFC) without scram-case 1,

• DLOFC with scram-case 2,

• Pressurized loss of forced cooling (PLOFC) with scram -case 3,

• Load follow 100%-40%-100% -case 4,

• Reactivity insertion by Control rod ejection (CRE) and Control Rod Withdrawal (CRW) -

case 5, and

• Cold Helium injection into the core inlet plenum -case 6.

The focus of the PhD work was on the analysis of cases 1,2,3 and 5.

5.4.1 DLOFC

The DLOFC and PLOFC cases involve the control rod insertion over 3 seconds to scram

the reactor from 13 to 16 seconds. The modeling of control rods adopted the generally

accepted grey curtain approach where the rods move as one continuous sector of absorber

material. In this PhD study flexibility was introduced to enable the movement of control rods

independent of each other based on individual location in the reflector material. In NEM, one-

twelfth symmetry of the core consisting of 3 sectors of 10° each, is usually utilized. This implies

that the grey curtain will have three sectors of absorber material moving simultaneosly.

To check the implemented flexibility of the movement of the rods, the scram was conducted

with only parts of the control rods moving at their actual location in the reactor. Since there are

three sectors, sensitivity study was conducted by moving 1, 2 and then all (3) rods into the

RCS channels. Taking into account the one-twelfth core symmetry these movemnets

coresspond to insertion of 12, 24 and 36 rods respectively. The reactivity effect on power

101

during the time period of scram is shown in Figure 53. When the number of rods inserted is

increased, the negative reactivity increases; thus causing the power to decrease more for

larger number of rods than in the fewer cases.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

13.0 13.5 14.0 14.5 15.0 15.5 16.0

Re

lati

ve

Po

we

r

Time(sec)

One_Rod

Two_Rods

All_Rods

Figure 53: Power during reactor scram

The implementation of the conrol rod movement has enabled the NEM/THERMIX analysis

of the cases like Loss Of Coolant Accidents (LOFC), which include DLOFC and PLOFC

(shown in Figure 54) with scram, which were previoulsy unattainable due to the absence ot a

robust control rod model. The cases with scram (WS) exibit lower maximum fuel tempertaures

than the cases without scram (WOS).

102

Figure 54: Maximum temperatures reached during LOFC accidents

5.4.2 Control rod ejection (CRE)

In this case it was assumed that the control rods are ejected from the reactor over a period

of 0.1 seconds. This provided another test scenario, which introduces fast reactivity changes in

a short period of time. The case description is similar to the pulsing in the operation of the

TRIGA reactor in the following manner:

• Large reactivity insertion

• Prompt critical transient

• Power increases

• Fuel temperature increases

• Reactivity decreases due to negative reactivity temperature effect (feedback reactivity)

• Power decreases and temperature decreasing

• Power stabilising at a lower level higher than the initial power level

103

Figure 55: Power evolution for combinations of multiple time step sizes during the CRE

transient

The challenge was imposed by the global code system time-control. The time step

combinations in the calculation introduced some errors since thermal hydraulic time steps

should be compatible with the neutronics time steps in terms of size and stability limitations in

both numerical solutions. Since the time step is controlled by THERMIX, it is allowed for the

user to select multiple time step sizes during different intervals. This added flexibility helps to

meet the requirement to refine time step or to make it coarser in order to obtain an optimal

balance in terms of accuracy and efficiency. The utilization of very different time step sizes

introduced unstable results as seen in Figure 55. (please explain what mean the two cases).

This observation prompted performing sensitivity studies on time-step size and explicit iterative

temporal coupling in NEM/THERMIX for different transients. Time step selection algorithms

were developed for neutronics and thermal-hydraulics models supplemented by automatic

meshing schemes between them.

After this implementation, the results improved dramatically as shown in CRE cases shown

below in Figure 56 through Figure 63. The observations in this cases are in agreement with the

postulations described above.

104

Figure 56: Power evolution for single rod ejection for Case 5c

Sinsitivity studies on the CRE transients were on the time step sizes (see Figure 56 where

0.001, 0.002 and 0.003 are in seconds time step sizes used during CRE simulations), and also

on the number of rods ejected. As expected the more rods were ejected, the more the

reactivity insertion was. Hence the negative feedback reactivity would be more. This is

illustrated by the comparison of maximum power attained after ejection of the rods, which

increases with the number of rods ejected in Figure 56, Figure 57 and Figure 59 and is also

summarised in Figure 61.

105

Figure 57: Power for two rods CRE

Figure 58: Maximum temperature for two rods CRE

106

Figure 59: Case 5b with all rods ejected

Figure 60: Maximum temperature for all rods CRE without kernel model

Fewer rods take longer to insert reactivity and this is also accompanied by slower

negative reactivity feedback.

107

Figure 61: Sensitivity on number of rods ejected

Figure 62 shows that the correct prediction of the kernel temperature would result in

stronger negative feedback reactivity than when the fuel kernel temperatures are

homogenised.

Figure 62: Effect of fuel kernel model for CRE transient

108

Figure 63: Maximum temperature for all rods CRE with the kernel model

The maximum temperature predictions for the kernel model were as much as 400°C higher

than in the homogeneous case as shown in Figure 63.

The performed studies suggest that all the rods have to be ejected to have the effect of

power excursion that is significantly high. This situation has to be closely investigated for its

frequency using relevant PRA techniques since in the deterministic analysis it is assumed that

it will occur. The ejection of a single control rod did not introduce significant reactivity as

compared to the ejection of all rods. This transient scenario has no measured results to

compare against since it is impractical to conduct an experiment of this type. The best way to

account for such scenario is to have the knowledge of the control rod worths and design

control rod drive mechnisms with the defence-in-depth consideration to avoid such an

occurrence.

5.4.3 Control rod withdrawal (CRW)

The control rod withdrawal is modeled by the withdrawal of rods at 1 cm/s for 200 seconds.

This would withdraw the control rods 200 cm from the initial insertion level after the transient.

As discussed before, the slower control rod movement is a challenge for estimation of the

109

reactivity insertion that is linear with the linear movement of the rods (cusping effect). Figure 64

shows the deficiency of the volume-weighting (VW) and the superiority of the flux-volume-

weighting (FVW) approach in the elimination of the cusping problem. Using higher order of flux

aproximation in the FVW appraoch would further improve the results.

Figure 64: Case 5a control rod withdrawal transient

Figure 65: Fuel temperature during CRW transient

110

Figure 65 shows the effect of the cusping on the temperature during the rod withdrawal.

This type of transient is longer than the CRE case since the control rod insertion is slower.

CRWs probably will happen many times during the reactor operation. CRE while having much

larger safety consequences than CRW is in orders of maginute less probable than CRW to

occur.

5.5 Conclusions

In this chapter the OECD PBMR-400 coupled code benchmark was introduced. This

benchmark was selected as reference test framework for the optimization studies since it

includes a wide spectrum of transient scenarios ranging from longer and slow transients to

shorter and faster transients. This benchmark helped to demonstrate the effectiveness and

efficiency of the optimization developments described in Chapter 4. This also helped to explain

certain requirements for improvement and guidelines for performing high-fidelity multi-physics

simulations. It was also demonstrated the importance of the interplay between temporal and

spatial effects in the modeling of the PBMR reactor design. The use of the grey curtain

assumption was also demonstrated to be a limitation in terms of the proper analysis of the rod

ejection transient. Hence a full spatially coupled 3-D model has to be used to analyze such

physical phenomena as spatial flux re-distribution during the transient. This modeling aspect is

the focus of Chapter 6 where the 3-D models are discussed and analyzed.

111

Chapter 6 Three-Dimensional Spatial Models

6.1 Introduction

Coarse mesh methods are motivated by the fact that in some instances a reactor may be

adequately described by a model consisting of homogeneous regions that are relatively large.

Coarse mesh methods are able to use mesh sizes, which are much larger than the utilized by

finite difference methods, because they use higher order approximations to the spatial

variations of the unknowns within a mesh cell. As representative of coarse mesh methods,

nodal methods utilize relatively large computational mesh cells to solve multi-dimensional

reactor problems, and use significantly less computer resources than the fine-mesh finite

difference methods. Early nodal methods required a variety of schemes to deal with face-

averaged partial currents and the node-averaged fluxes. The coupling parameters for a node

are defined as the ratios of the face-averaged out-going partial currents to the node-averaged

flux. The homogenized parameters are usually computed by weighting the spatially dependent

cross-sections with the flux solution obtained in an assembly calculation with zero net current

boundary conditions. These parameters are computed using a reference fine-mesh calculation.

While these methods work well in situations in which the conditions analyzed using the nodal

method closely resemble the reference condition at which the coefficients were computed, they

often break down when the difference between the analyzed and reference conditions

becomes large.

Transverse-integrated nodal methods assume that either nodes are truly uniform

throughout their entire volume, or that they may be adequately represented using node-

averaged values of the cross-sections and diffusion coefficients. This assumption of uniformity

of intranodal composition does not apply to most reactor calculations that employ assembly or

quarter-assembly sized nodes. These issues are addressed by advanced nodal

homogenization schemes yielding equivalent diffusion theory parameters that allow

transverse-integrated nodal codes to compute node-averaged quantities agreeing closely with

the results of fine-mesh calculations in which the heterogeneity within the node is explicitly

represented.

112

In order to perform optimization studies of coupled multi-physics code, it is necessary in

addition to reference test problems to have also reference solutions in terms of 3-D higher

order approximation such as transport solution for the neutronics field implicitly coupled to a 3-

D thermal-hydraulic model in Cylindrical geometry. Such developments of the 3-D neutronic

models for diffusion and the transport codes are discussed in this chapter.

6.2 Optimizing spatial coupling schemes

The solution of the neutron balance equation requires the availability of the cross section

data which strongly depend on state feedback parameters as described in the previous

sections. On the other hand, the power distribution must be known to solve the reactor thermal

hydraulic fields. In order to perform a complete calculation of a design or safety analysis

problem, the two calculations must be coupled with accurate and efficient methods.

In this study, multi-dimensional coupling schemes, the 2-D dimensional thermal hydraulic

calculation (THERMIX-DIREKT) that is coupled to 3-D neutronics solution (NEM), are studied.

The effects of thermal feedback required during the flow of calculations in steady state and

transient problems was studied by investigating the effects of mesh structure on the neutronics

calculation to accurately predict the flux distribution.

Currently, the spatial mapping from neutronic to thermal-hydraulic mesh and from

thermal-hydraulic to neutronic mesh (spatial coupling schemes) is done to pass parameters

during the coupled calculations. After each thermal-hydraulics call the cross sections are

recalculated for the current conditions. A converged coupled neutronic and thermal-hydraulic

steady state solution is obtained before any time-dependent simulation in order to initialize the

transient problem.

The reference solution consisting of porous media thermal-hydraulic model and neutron

transport model was developed to demonstrate the effect of spatial mapping schemes on the

distribution of flux. Thus, the effects of mapping methodology were studied and quantified.

113

In Figure 66, a 3-D sector (in R-Θ-Z geometry) of a PBMR core in NEM modeling is

shown in cylindrical geometry, which usually adopts 30o sector of symmetry in θ-direction used

for PBMR calculations. This constitutes a 1/12th sector of symmetry of the full core model. The

control rod channels shown here are usually modeled as continuous absorber material since it

was the best way to represent the control rods when the 3-D model was not available.

However, the actual material representation would be to tap into the 3-D capability of NEM and

generate a full 3-D model that accurately represents the reactor configuration.

Horizontalinlet slots

Control rodchannels

Centralreflector

Pebble bed

Verticalriserchannels

Core barrelannulus

Core barrel

Gas inletmanifold

Corestructures

Figure 66: Reduction of 3-D model to 2-D

114

6.3 Reference neutronic solution

This PhD study utilizes the 3-D time-dependent discrete ordinates multi-group transport

code TORT-TD, coupled to a 3-D thermal-hydraulic module described in next section, as

reference coupled 3D transport neutronics/thermal-hydraulics solution to verify the optimization

studies performed in this research.

TORT-TD is the 3-D counterpart to the 2-D code DORT-TD that has been introduced some

years ago [32]. The TORT-TD code was transferred to PSU as a part of PSU/GRS

cooperation. TORT solves the stationary multi-group transport equation using the discrete

ordinates or SN theory with quadrature order N. Anisotropic scattering is treated in terms of a

Legendre Pn cross section expansion where n denotes the scattering order. For coupled

neutronic/thermal-hydraulic calculations, TORT-TD has also been coupled with the GRS

thermal-hydraulic system code ATHLET [32]. The time-dependent code TORT-TD is based on

the steady-state 3D transport code TORT [32] from the DOORS package, which has been

developed at ORNL.

In the work of B. Tyobeka [23] the 3-D neutron transport SN code TORT, was used with the

cross sections generated from MICROX-2 to perform control rod worth calculations with control

rods accurately and explicitly modeled in three-dimensions. The main objective of these

studies was to obtain an optimum control rod representation in TORT, and use the differential

control rod worth curve resulting from this configuration to adjust the 2-D DORT-TD control rod

approximation so that accurate transient analysis can be performed with the developed at PSU

coupled code DORT-TD/THERMIX. The DORT-TD/THERMIX code system models in 2-D

geometry both the neutronics and thermal-hydraulics phenomena in the PBMR core.

The work performed for this thesis was conducted with the various neutronic codes. These

codes include transport codes (DORT, TORT and TORT-TDS), which results were compared

to the obtained results of the diffusion code (NEM). This was very extensive and

comprehensive work as far as the spatial and temporal modeling is concerned. The modeling

involved moving from 2-D to 3-D geometry, which helped to assess spatial effects for the

115

DORT to TORT transition as well as the temporal effects for the TORT to TORT-TDS

transition. Later the comparisons were performed on the thermal hydraulic side of analysis

from 2-D THERMIX-DIREKT to 3-D Porous Media Code. The ultimate coupled analysis

involved higher order full 3-D modeling of the HTR both on the neutronic and the thermal

hydraulic aspects.

6.3.1 DORT Model of PBMR-268

DORT is a 2-D neutron transport code developed at ORNL. Its limitations in the modeling of

the PBMR reactor problem were in the modeling of the control rod movement, which required a

full 3-D model either for the control movement in symmetry or single rod movement. The code

has performed very well in the steady state 2-D calculations and the models have been used

as the basis for the development of the 3-D model for TORT model. Cases that were

considered were N1 and N2 as described below for the PBMR-268 benchmark.

6.3.1.1 CASE N-1: Fresh fuel and cold conditions

• All fuel is fresh (9 grams HM and 8 w/o enriched): Use ND-set1

• Cold conditions (300K) for all materials

• Use own cross sections

6.3.1.2 CASE N-2: Equilibrium cycle with given number densities

• The equilibrium uranium, graphite and structural number densities are used (no fission

products or higher elements).

• Constant temperature conditions (600K and 900K) for all materials.

• Use own cross sections.

As it was required that the cross sections be generated the MICROX code for cross section

generation was used. For the studies conducted in this work, the PBMR-268 benchmark with

fixed cross sections for cases N1 and N2 were generated in 4 neutron energy groups. These

cross sections were used in the TORT model of the same test problem.

116

0 78.6 110.9 135.7 156.6 175 181 194 215 235 250 275 287 292 300 317 417 418

-231 78.6 32.3 24.8 20.9 18.4 6 13 21 20 15 25 12 5 8 17 100 1

-230 1 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 18-200 30 13 13 13 13 13 13 13 13 13 13 13 13 12 15 16 17 18-175 25 7 7 7 7 7 7 7 7 7 7 7 11 12 15 16 17 18-125 50 9 9 9 9 9 9 9 9 9 3 7 11 12 15 16 17 18-75 50 2 2 2 2 2 3 3 3 6 3 7 11 12 15 16 17 18-25 50 2 2 2 2 2 3 3 3 6 3 7 11 12 15 16 17 180 25 1 1 1 1 1 3 5 3 6 3 7 11 12 15 16 17 18

50 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18100 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18150 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18200 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18250 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18300 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18350 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18400 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18450 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18500 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18550 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18600 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18650 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18700 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18750 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18800 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18850 50 a b c d e 3 5 3 6 3 7 11 12 15 16 17 18900 50 4 4 4 4 4 3 3 3 6 3 7 11 12 15 16 17 18950 50 4 4 4 4 4 3 3 3 6 3 7 11 12 15 16 17 181000 50 4 4 4 4 4 3 3 3 8 3 7 11 12 15 16 17 181050 50 4 4 4 4 4 3 3 3 3 3 7 11 12 15 16 17 181100 50 10 10 10 10 10 3 3 3 3 3 7 11 12 15 16 17 181125 25 7 7 7 7 7 7 7 7 7 7 7 11 12 15 16 17 181155 30 14 14 14 14 14 14 14 14 14 14 14 14 12 15 16 17 181156 1 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18

Figure 67: Neutronic model for PBMR-268

Figure 67 shows the PBMR-268 model on which cases N1 and N2 were based upon. It

should be noted that in this configuration material number 5 is the control material. This means

that the control rods are inserted to the bottom of the core in this problem.

6.3.2 TORT Model of PBMR-268

The 4-group cross sections were modified to be compatible with the input definition for

TORT using the GIP preprocessor. This model required finer meshing than the DORT

counterpart since the 3-D model was adopted and azimuthal dependence was introduced.

Since the control rods were previously modeled as a grey curtain, the goal was to ensure that

the control rods were modeled as close as possible to the dimensions in the specifications for

the design of PBMR-268. This would be achieved by setting the discretization of the azimuthal

dependence and the arrangement of control (absorbing) material region in different positions of

the reflector regions as shown in Figure 68 with the control rods shown in purple.

117

Figure 68: Actual size of rods in 3-D model

The real control rod positions and sizes are shown in the top view of the PBMR in

Figure 69 as the RCS (Reactivity Control System) channels.

Figure 69: Top view of the PBMR-400 reactor

118

Ultimately the azimuthal sector containing the absorbing material would span an

equivalent volume of the real control rod as shown in Figure 70, (i.e. circles and squares have

equivalent volumes to preserve the reaction rates as accurately as possible):

Figure 70: Control rod equivalent volume

This ensures that the effective absorption of the region is similar to that of the actual control

rod. The results of these studies have been compared to those of the diffusion calculations to

evaluate the accuracy of this approach without the requirement of expensive calculations that

are generally performed by the transport methodology. Hence, the NEM diffusion code was

used for comparative studies.

Sensitivity studies were conducted in the form of spatial meshing mainly concentrating on

the azimuthal dimensions. The number of azimuthal sectors was increased gradually to obtain

a full 3-D model with the control rod regions representing as closely as possible the actual

configuration of the PBMR-268 reactor design. Hence, the grey curtain model was replaced

with a realistic model with alternating absorbing and reflector regions in the azimuthal direction.

119

6.4 Results for 3-D spatial modeling

Figure 71 shows k-effective (multiplication factor) for the different meshing in the azimuthal

direction. It was interesting to note that the k-effective value was not increasing when the

number of sectors increased. The postulation was that increasing the sectors results in

increased reflector material, hence fewer absorbing regions, which would imply more neutron

population in the system.

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

0 50 100 150 200 250

K-E

ffe

ctiv

e

Theta (Degrees)

NEM

TORT

Figure 71: K-effective for 3-D modeling of PBMR-268 reactor

The following plots (Figure 73 through Figure 77) show the comparison of the azimuthal

flux distribution between NEM and TORT for different axial positions for PBMR-268. The

spatial visualisation of the flux distributions are shown for the 60° symmetry sector models for

the PBMR-268 reactor. The flux profiles indicate cleary the absorption power of the control

rods in their actual locations in the full 3-D model. Please note that the plots are not drawn to

the same scale so that the detail in the flux distribution is not lost.

120

Figure 72: 3-D view of thermal flux at top of PBMR-268

0.0E+00

1.0E+13

2.0E+13

3.0E+13

4.0E+13

5.0E+13

6.0E+13

7.0E+13

8.0E+13

9.0E+13

0 10 20 30 40 50 60

Flu

x (

#.s

-1.c

m-2

)

Theta(degrees)

TORT

NEM

Figure 73: Azimuthal flux distribution at the top of the reactor

121

Figure 74: Middle core thermal flux distribution for PBMR-268

0.0E+00

2.0E+13

4.0E+13

6.0E+13

8.0E+13

1.0E+14

1.2E+14

1.4E+14

1.6E+14

1.8E+14

0 10 20 30 40 50 60

Flu

x(#

.s-1

.cm

2)

Theta (degrees)

TORT

NEM

Figure 75: Azimuthal flux distribution in the middle of the reactor

122

Figure 76: Thermal flux distribution at the bottom of PBMR-268 reactor

0.0E+00

5.0E+12

1.0E+13

1.5E+13

2.0E+13

2.5E+13

3.0E+13

3.5E+13

0 10 20 30 40 50 60

Flu

x (

#.s

-1.c

m-2

)

Theta (Degrees)

TORT

NEM

Figure 77: Thermal flux distribution at the bottom of PBMR-268 reactor

123

Similarly, flux distributions were obtained for the corresponding 90° comparison plots at

different height of the core shown Figure 78 to Figure 83 which were taken according to Table

14.

Table 14: PBMR-268 axial slices for flux profile

Height (cm) Description

25 top

425 middle

825 bottom

Figure 78: Top flux distribution PBMR-268

124

0.0E+00

1.0E+13

2.0E+13

3.0E+13

4.0E+13

5.0E+13

6.0E+13

7.0E+13

8.0E+13

9.0E+13

0 20 40 60 80 100

Flu

x (

#.s

-1.c

m-2

)

Theta (degrees)

TORT

NEM

Figure 79: Top azimuthal flux distribution comparison

Figure 80: Middle of core flux distribution for PBMR-268

125

0.0E+00

2.0E+13

4.0E+13

6.0E+13

8.0E+13

1.0E+14

1.2E+14

1.4E+14

1.6E+14

0 20 40 60 80 100

Flu

x (

#.s

-1.c

m-2

)

Theta (Degrees)

TORT

NEM

Figure 81: Flux comparison PBMR-268

Figure 82: Bottom of core flux distribution for PBMR-268

126

0.0E+00

5.0E+12

1.0E+13

1.5E+13

2.0E+13

2.5E+13

3.0E+13

3.5E+13

0 20 40 60 80 100

Flu

x (

#.s

-1.c

m-2

)

Theta (degrees)

TORT

NEM

Figure 83: Comparison of 10 sectors azimuthal flux distribution

It was noted that the flux profiles tend to agree at middle of the core, but tend to deviate

from each other at the two extremes.

6.5 PBMR-400 Steady state 3-D modeling with NEM

These cases involved two variations:

• All rods inserted (ARI)

• One rod withdrawn (ORO)

The thermal flux distribution is represented in polar coordinate system to ensure that the 3-

D distribution is of the reactor is viewed. As noted the reactor is modeled with the core from the

centred and the radius of the reactor extending to 292.5 cm at the core barrel. It should be

noted that the rods are during the normal operation of this reactor are inserted 150 cm into the

core. In the case All Rods In (ARI) all rods remain at these positions and the thermal flux is

represented for the half core model (Figure 84 to Figure 91) from the 0 to 180 degrees in the

127

azimuthal direction. The plots were produced using NEM to provide the full 3-D modeling of the

PBMR-400 reactor.

The plots are not drawn to the same scale of colour for the contours to appear at each level

since the flux varies according to axial location. In these graphs the axial layers were selected

according to Table 15.

Table 15: Description of levels for 3-D flux distribution

Height (cm) Description

200 Top core

350 At rods’ position

500 Middle Core

1150 Bottom Core

The model consisted of alternating control rod material and reflector material as close

as possible to the actual PBMR control rod arrangement in the reflector region between 192

cm and 211 cm in the radial direction.

Figure 84: PBMR-400 flux distribution at the top with ORO

128

Figure 85: PBMR-400 flux distribution at the top with ARI

Figure 86: PBMR-400 flux distribution at rods with ORO

129

Figure 87: PBMR-400 Flux distribution at rods with ARI

Figure 88: PBMR-400 Flux distribution bottom core with ORO

130

Figure 89: PBMR-400 Flux distribution bottom core with ARI

Figure 90: Middle PBMR-400 core ARI

131

Figure 91: Middle of PBMR-400 core ORO

The effect of the removal of a single rod can be observed from is removed from the

core. The assumption of the grey curtain approach is definitely not adequate for the modeling

of the PBMR reactor. The flux distribution has to be modelled in full 3-D to see the effects on

the spatial flux distribution.

6.6 3-D spatial modeling of PBMR-400 with TORT-TDS

TORT-TDS is the time-dependent version of TORT, which enables transient analysis for

long and short transients of HTR. This is very crucial for the development of reference solution

for the PBMR-400 benchmark, which consists of 6 transient cases. TORT-TDS has capability

of control rod movement for analysis of the full spectrum of the transients described in this

benchmark. Figure 92 shows the result of the 90° symmetry sector of PBMR-400 reactor.

132

Figure 92: PBMR-400 model using TORT-TDS

6.6.1 The LMW 3-D transient problem without feedback

The LMW LWR benchmark problem is a 3D transient problem without thermal-hydraulic

feedback. This benchmark is useful for the verification of 3D kinetic code and is quoted in

many papers. It is modeled with two neutron energy groups, and six delayed precursor

133

families. In this problem, a small core is composed of 2 types of fuel assemblies, reflectors and

2 groups of control banks. The macroscopic cross sections of each region are given by the

LMW benchmark specifications. This problem simulates the motion of 2 groups of banks, and

the transient is initiated by withdrawing of group bank 1, which is partially inserted, at 3.0 cm/s.

Bank 1 is withdrawn to the top in 26.6 s, and bank 2 is inserted to 120 cm deep over the time

interval 7.5 to 47.5 s. The transient calculation for 60 seconds can be compared with

benchmark results. This was used to verify the performance of TORT-TDS code in the analysis

of transients. It should be noted that this problem is analyzed in Cartesian coordinates system.

The HTR transient analysis requires cylindrical geometry.

6.6.1.1 Results for LMW with TORT-TDS

At the beginning of the calculation a few cycles were required for the fission source to

converge before the transient calculation begins as shown Figure 93.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50

De

lta F

issi

on

so

urc

e

Iteration

Figure 93: Source convergence for the LMW problem

The cross sections for the analysis with TORT-TDS were provided in the LMW benchmark

specification and they were fixed at the shown in Table 16 parameters:

134

Table 16: Parameter for LMW transient calculation

Moderator Density Boron (ppm) Fuel Temperature (K)

711.87 1000.00 900.00

A depletion analysis was not considered in the calculation; hence the burnup was

maintained at 0.00 GWd/tU, which represented a fresh core. The power evolution during this

transient is shown in Figure 94.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60

Re

lati

ve

Po

we

r

Time(sec)

Figure 94: Power for the LMW benchmark using TORT-TDS

The power history of the LMW problem with TORT–TDS has been demonstrated to be

consistent with the ANCK and the CUBBOX analysis [35] as shown in Figure 95. This

benchmark problem has been widely used to demonstrate capability of the codes to analyze

transient problems.

135

Figure 95: LMW comparison of results

Figure 96 shows that the reactor is put in positive period in the first 20 seconds and then

negative period to the end of the transient. This is consistent with the power evolution as

shown in Figure 94.

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

0 10 20 30 40 50 60

Re

act

or

Pe

rio

d

Time (sec)

Figure 96: Reactor period for the LMW problem

136

6.7 Reference thermal-hydraulic models

This thesis utilizes a 3-D porous medium thermal hydraulic model as a reference. There are

several potentially important three-dimensional neutron and temperature field effects in the

PBMR. However, the current version of the THERMIX/DIREKT code is designed for the

calculation of two-dimensional problems. Basically the models used in this code are in

cylindrical geometry, i.e. a mesh is superimposed in the r- and z-directions. The influence of

azimuthal temperature/fluid field variations is not taken into account. The heat conduction and

radiation component is also capable of calculating r-z-mesh using cylindrical coordinates or x-

y-meshes with Cartesian coordinates since the third dimension is ignored. The convection

component has only been designed to stimulate matrix grids in the r- and z-directions. The

computer model is divided into two different regions, which are called “compositions” and can

be described as follows:

• The solid model (THERMIX), a region which has the same material composition

(material values, temperature-dependent heat conduction equations, temperature-

dependent heat capacity equations),

• The fluid model (DIREKT), a region in which the same hydraulic properties (percentage

empty spaces/voids, hydraulic diameter), or the same type of internal geometry (e.g.

pebble-bed, pipe geometry, or two-dimensional voids in which flow occurs), are present.

A mesh system is superimposed on this system of inter-connected “compositions”.

Therefore every “composition”, consists of many or few material meshes depending on the

degree of accuracy that is required. Within such a mesh, constant mean values apply to all the

parameters.

In the fluid model flow generally occurs in the r- and the z-direction in the “compositions”

(with the exception of the solid region where no flow occurs). In the “compositions”, which

define pipes in which flow occurs in only one direction (e.g. the top and bottom reflectors) or

one-dimensional circular areas, flow is only described in the z-direction.

137

The important feedback effects in the PBMR are the temperature effects of the fuel and

moderator, and in this sense the extension to 3-D modeling is important for thermal hydraulic

part of the code. The methodology for adding θ-direction to heat conduction, radiation and

convection component models thus having complete 3-D cylindrical (r-θ-z) geometry has been

developed at Purdue University (PU) [34]. The extended 3-D porous medium thermal hydraulic

model (named AGREE) is part of the PARCS code system and it has been verified by code-to-

code comparison with CAPP/MARS results for the coupled neutronic/thermal-hydraulic

exercise S3 of the OECD PBMR-400 benchmark. Some of the comparison results are shown

in Figure 97.

Table 17: Eigenvalue comparison for case S-3 PBMR-400

CODE keff

CAPP/MARS 0.99270

PARCS 0.99283

PARCS * 0.99282

keff (at zero power)

PARCS * 1.04099

KAERI 1.04090

* Calculation with single diffusion coefficient

138

Axial Thermal Flux

0

5E+13

1E+14

1.5E+14

2E+14

2.5E+14

-150 50 250 450 650 850 1050 1250 1450

Axial Position (cm) (top-to-bottom)

Ne

utr

on

Flu

x(n

/cm

2/s

)

KAERIPARCSPARCS *

Figure 97: Comparison of axial thermal fluxes for Case S-3 PBMR-400

A porous media code for thermal hydraulic analysis has been used to model different

problems of HTR. These problems include those described in the HTR-10 and PBMR-400

benchmark exercises as well as the AVR transient analysis.

A test case was developed to familiarize with the input definition of the code. This

consisted of a reflector, fuel and control region to resemble a general configuration of the HTR.

The main focus of the study was to ensure that the interfacing between a neutronic code was

understood so that coupling is conducted in an efficient manner.

6.8 ATTICA3D model

6.8.1 Description of ATTICA3D

The general description of a heat transfer model in porous media is examined. This code is

based on the differential conservation equations and the associated constitutive equations

required for the analysis of transport in porous media. In the last four decades, the transport in

139

these heterogeneous systems has been addressed in sufficient detail. So far, the heat transfer

in porous media has been treated or at least formulated satisfactorily. The theoretical

treatment is based on the local volume-averaging of the momentum and energy equations with

closure conditions necessary for obtaining solutions. Heat transfer models used in the code

are similar to the ones in the KTA rules given in references [27, 28 and 29].

Examination of transport, reaction, and phase change in porous media relies on the

knowledge we have gained in studying these phenomena in plain media. The presence of a

permeable solid (which is assumed to be rigid and stationary) influences these phenomena

significantly. Due to practical limitations, as a general approach these phenomena are

described at a small length scale which is yet larger than a fraction of the linear dimension of

the pore or the linear dimension of the solid particle (for a particle-based porous medium). This

requires the use of the local volume averaging theories.

Figure 98: Aspects of transport, reaction and phase change in porous media

Figure 98 gives a classification of the transport phenomena in porous media based on

the single- or two-phase flow through the pores.

140

Figure 99: Aspect of transport, reaction and phase change at pore level

Figure 99 renders these phenomena at the pore level. Description of transport of

species, momentum and energy, chemical reactions (endothermic or exothermic) and phase

change (solid/liquid, solid/gas, and liquid/gas) at the differential, local phase-volume level and

the application of the volume averaging theories lead to a relatively accurate and yet solvable

local description. ATTICA3D is focused on the transport phenomena in a single-phase flow leg

of Figure 98 which are more relevant for our application in the modeling of the PBMR reactor.

6.8.2 Results of ATTICA3D modeling

The stand-alone steady state results for ATTICA3D are being developed for the

PBMR400 model for benchmarking the code. These developments will be followed by the

coupled TORT-TDS/ATTICA3D calculations for steady state and transient cases as reference

solutions.

141

6.9 Conclusions

The current analysis of the LMW problem has demonstrated the capability of TORT-TDS to

analyze the transient behavior of HTR cores with control rod movement. These studies

demonstrated the capabilities of TORT-TDS for modeling of sophisticated transient

calculations in Cartesian geometry. The development of the code for cylindrical geometry has

been completed and transferred to PSU recently. This development was concluded to enable

studies on different PBMR-400 benchmark cases. Development of input and problem analysis

has taken place after the extensions were concluded.

Comparative results carried out for the 4-group analysis of the PBMR-268 N2 case were

not conclusive since it was noted that there were convergence issues with TORT calculations.

This would require further investigations. Outstanding issues in NEM were in the modeling of

more than 36 sectors of 10 degrees each, which seemed to be the limitation. The sensitivity

studies on the control rod modeling in 3-D geometry, revealed some features of the boundary

conditions specification, which should be taken care of in the input description of the problem.

This chapter introduced the 3-D neutronics code TORT-TDS and the 3-D porous medium

thermal-hydraulic (ATTICA3D) model. The TORT-TDS/ATTICA3D code system was coupled in

an implicit manner to provide reference solutions for steady state and transient calculations of

PBMR. Such reference solutions provide insight to the importance of coupled 3-D effects in

PBMR analysis.

142

Chapter 7 CFD transient modeling of PBMR-400

7.1 Description of CFD model

Reference studies have been conducted as part of this PhD work to validate a specific

model in the THERMIX thermal-hydraulic code. The model consisted of a radial slice of the

PBMR-400 reactor taken at a random axial position as a representative of all axial layers. This

model consisted of the central reflector, core, side reflector, helium gap and core barrel.

A finite difference formulation of the heat conduction, convection and radiation were used to

model the heat transfer in the reactor. The models include temperature dependence of the

conductivities and heat capacities for materials.

One of the fundamental safety functions that must be fulfilled by the PBMR reactor design

is the heat removal. In the PBMR reactor, heat removal is going to be achieved by conduction

in the solid materials, and convection and radiation in the gaps. Hence, the phenomena

governing these heat removal paths must be examined closely. In the event of loss of forced

cooling (LOFC) accident, the convective heat transfer is lost due to the loss of coolant (helium)

in the system. The reactor is expected to rely on passive heat removal mechanisms by

conduction, natural convection and radiation through the gaps where there is helium and air.

The radiative heat transfer between two surfaces is governed by the following relation:-

( )( )

o

i

o

o

i

oi

i

T

T

TTq

⋅−

+

−=

ε

ε

ε

σ

11

44''�

where ε is the emissivity of the surfaces and it plays an important role in the transfer of heat

during the LOFC. In order to verify the effect on the heat transfer by radiation a specialised

study on the emissivity sensitivity studies for the PBMR reactor was conducted. The

emissivities of reactor materials such as core barrel, side reflector, etc. were provided by the

143

PBMR (Pty) Ltd. company. This also provides an independent check of the results calculated

by STAR-CD and FLUENT for the development of the Safety Analysis Report.

7.2 PHOENICS calculations

This study involved the use of the CFD code PHOENICS (Parabolic Hyperbolic Or Elliptic

Numerical Integration Code Series) for modeling of the PBMR-400 and the HTR-10 reactors.

The purpose of this code is to solve finite-domain basic differential equations including

the mass, momentum and energy equations for steady or unsteady flows and in 2D or 3D

geometries. Any property obeying the balance equation can be represented using PHOENICS.

It simulates how fluids (single- or multi-phase) flow, change in chemical and physical

composition and radiation fluxes.

PHOENICS is arranged in such a way that it contains a pre-processor (Satellite), a solver

(Earth) and post-processing units (VR Viewer, Photon, Autoplot, and Result). This

arrangement is referred to as the planetary arrangement. The output allows the prediction of

temperature, pressure, velocities and the geometry of the system under investigation.

At AMEC NNC the code has been used to model different systems including the

modeling of the HTR-10 and PBMR reactors in 2D and in 3D geometries.

The model of the PBMR-400 reactor was generated using PHOENICS and this model

could be viewed using Photon as shown in Figure 100.

144

Figure 100: PHOENICS model of PBMR-400

Steady state calculations were performed for a wide range of possible emissivities from 0.2

to 0.8 using the HTR-10 model that has been developed for the IAEA CRP-5 benchmark

exercise. The same calculations were performed for the PBMR-400 model with further analysis

of the transient calculation for the Pressurized Loss of Coolant Accident (PLOFC).

145

The PLOFC transient conditions in PHOENICS are achieved by reducing the pressure from

the nominal value of 90 MPa to 60 MPa, and modeling the heat source as an exponential

function of time G(t) from the initial steady state value. This gives the characteristic time

function of ~10 hrs; hence was no neutronic calculation performed during the transient

calculation.

7.3 DASPK solution

A one-dimensional independent analytical model of the transient calculation was developed

by the author of this PhD study using a FORTRAN code and a time-dependent solver

(DASPK). This allowed the comparison of the calculations performed by the 2D and 3D models

in PHOENICS.

This code solves a system of differential/algebraic equations of the form G(t,y,y') = 0 ,

using a combination of Backward Differentiation Formula (BDF) methods and a choice of two

linear system solution methods: direct (dense or band) or Krylov (iterative).

7.3.1 Results and Discussion of CFD modeling

7.3.1.1 Steady state calculations

The results of PHOENICS consist of temperatures, velocities and pressure fields in the

model of the reactor. The velocity vectors for the steady state calculation of the PBMR-400

reactor are shown in Figure 101. Helium is blown into the system through the inlet plenum,

goes up the riser channels, into the top reflector and flows down through the pebble bed, out of

the bottom reflector and leaves the system through the outlet plenum.

146

Figure 101: Velocity vectors in steady state PBMR-400 reactor

The magnified flow at the bottom of the reactor is shown in Figure 102. Helium is fed

through the inlet plenum at a mass flow rate of 192.7kg/s and drops by about 75% when it

flows into the pebble bed due to friction. This flow shows a general downward flow and the

recirculation is observed at the bottom as shown in Figure 102. The flow tends to accelerate at

the exit of the outlet plenum. Hence the larger velocity vectors observed.

147

Figure 102: Recirculation at the bottom of the PBMR reactor

148

Figure 103 shows the magnified top of the reactor and a clearly visible forced flow

pattern under nominal conditions.

Figure 103: Velocity vectors for steady state flow at the top of PBMR reactor

149

Figure 104: Steady state temperature distribution for PBMR-400

150

Figure 104 shows the steady state temperature distribution calculated for the emissivity

of 0.7. This temperature distribution was generally observed in all steady state calculations at

different emissivities, but tended to yield different maximum temperatures. Generally, the

maximum temperature decreased as the emissivities of different materials was increased (see

Figure 105).

500

750

1000

1250

1500

1750

2000

2250

2500

2750

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Te

mp

era

ture

(Ce

lciu

s)

Emissivity

PEBBLE SURFACE

PEBBLE CENTRE

SIDE REFLECTOR

BOTTOM REFLECTOR

Figure 105: HTR-10 Maximum temperatures as function of emissivity

7.3.2 Transient calculations

The transient was initiated by reducing the pressure from 90MPa to 60MPa to simulate

the PLOFC. Figure 106 shows the velocity vectors under the PLOFC conditions with a notable

change in the flow as compared to Figure 103. Recirculation occurs at the top of the plenum

due to helium picking up some energy at the surface of the pebble bed at low velocities. This

hot air then becomes buoyant and rises against the walls. As the air rises, it loses energy

against the walls and tends to flow downwards. This causes the recirculation observed in the

gas plenum.

151

Figure 106: Recirculation at the top of the reactor

In Figure 107 the core starts to heat up due to the loss of convective heat removal from

the system. The temperature rises until it reaches a maximum since the heat source is also

disappearing during this period. When the heat source levels off, the reactor also starts to cool

152

off; hence, the decrease in temperature was observed. The emissivity did not play a bigger

role in the initial stages of this transient but had a significant effect in the heat removal in the

later stages. The maximum temperature during the transient changed by 10.96°C between the

0.8 and 0.2 emissivity cases. This maximum is achieved at about 8hrs as shown in Figure 107.

Figure 108, Figure 109 and Figure 110 show the temperature variation at different stages

during the PLOFC transient.

200

400

600

800

1000

1200

1400

1600

1800

2000

0 10 20 30 40

Te

mp

era

ture

(°C

)

Time(hrs)

PB 0.2

PB 0.3

PB 0.4

PB 0.5

PB 0.6

PB 0.7

PB 0.8

CB 0.2

PV 0.2

PV 0.8

CB 0.8

Figure 107: Temperature evolution during PLOFC transient in PBMR reactor

153

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.2 0.4 0.6 0.8 1

Tem

pera

ture

(°C)

Emissivity

Centre Reflector

Pebble Bed

Core Barrel

Pressure Vessel

Figure 108: Temperature variation with emissivity at 8hrs for PLOFC

200

400

600

800

1000

1200

1400

1600

1800

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

(°C

)

Emissivity

Centre Reflector

Pebble Bed

Core Barrel

Pressure Vessel

Figure 109: Variation of temperature with emissivity at 20hrs

154

400

500

600

700

800

900

1000

1100

1200

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

(°C

)

Emissivity

Centre Reflector

Pebble Bed

Core Barrel

Pressure Vessel

Figure 110: Variation of maximum temperature with emissivity at 40hrs

Although the maximum temperature at the centre of the pebble bed was comparable,

the temperature at the end of the side reflector, varied for the emissivities of 0.2 and 0.8. The

temperature of the 0.2 case was higher than those of the 0.8 case, suggesting that the heat

lost at the surface of the side reflector was higher at 0.8 than at 0.2.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

4 6 8 10 12 14 16 18 20

Tem

per

atu

re (D

eg C

)

Height (m)

e8ty12PB

e2ty12PB

e2ty22SRO

e8ty22SRO

Figure 111: Axial temperature profile at 8hrs

155

Figure 112 confirms that the heat transferred by radiation from the side reflector is

higher for the 0.8 emissivity than for the 0.2 emissivity case at the 8 hours mark during the

PLOFC transient. Heat transfer in the core barrel was shown to be dominated by radiation

rather than conduction.

0

100

200

300

400

500

600

700

800

900

1000

2 4 6 8 10 12 14 16 18 20

Ra

dia

tive

He

at (

W)

Height (m)

e8RAD4SRO

e2RAD4SRO

e8RAD1CBO

e2RAD1CBO

e8DiffusionCBI

e2DiffusionCBI

Figure 112: Heat transferred by radiation

0

200

400

600

800

1000

1200

1400

1600

1800

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Tem

per

atu

re (°

C)

Radial Dsitance (m)

e8tz27-8hrs

e2tz27-8hrs

e8tz27-40hrs

e2tz27-40hrs

e8tz27ss

e2tz27ss

Figure 113: Radial temperature distribution during PLOFC transient

156

Comparing the heat transferred by convection (Figure 114) and radiation (Figure 112) at

the side reflector, it can also be concluded that radiative heat transfer dominates.

-50

0

50

100

150

200

250

300

350

400

4 6 8 10 12 14 16 18 20

CO

NV

EC

TIV

E H

EA

T (W

)

HEIGHT (M)

e8CONV4SRO

e2CONV4SRO

e8CONV4CBI

e8CONV1CBO

e2CONV1CBO

e8CONV1PVI

e2CONV1PVI

e8CONV2PVO

e2CONV2PVO

Figure 114: Heat transferred by convection during PLOFC

7.4 The 1-D Model PLOFC and DLOFC results

500

700

900

1100

1300

1500

1700

1900

2100

0 10 20 30 40 50 60

Tem

pera

ture

(°C

)

Time (hours)

Tem_PB_02

Tem_PB_03

Tem_PB_04

Tem_PB_05

Tem_PB_06

Tem_PB_07

Tem_PB_08

Figure 115: Emissivity sensitivity of PBMR surfaces during PLOFC transient

157

The maximum temperature attained at the maximum time indicates that the radiation is

efficient heat removal since the temperature is lower for this emissivity value, thus

demonstrating the fact that the reactor will require material with higher emissivity value to be

efficient heat removal of the core barrel. This verifies the claim that the reactor will definitely

remove heat automatically when the engineered cooling systems fail, making radiation a

critical inherent safety feature.

The increase in emissivity of surface did not have a significant effect on the maximum

temperature attained during the DLOFC transient since the difference of about 20°C was

attained for the maximum temperature. However at the end of the transient (60 hours) the

temperature difference of about 200°C was attained between the two extreme case of 08 and

0.2 emissivity.

The variation of the temperature distribution during the transient is shown in Figure 116.

The temperature of the core was the highest when the maximum temperature of ~2018°C was

attained at about 10hrs. This analysis also demonstrated that the core will be cooler than the

central reflector which had the highest temperature at the end of the transient.

Figure 116: Radial temperature distribution at different stages of the PLOFC transient

158

7.5 Conclusion

It can be concluded that the emissivity contributed to the temperature of the reactor only

after the maximum temperature of the reactor was reached. Hence, the radiative heat transfer

would not have a significant effect on the maximum temperature of the fuel during the

transient.

159

Chapter 8 CONTRIBUTIONS AND FUTURE WORK

8.1 Contributions

The research conducted within the framework of this PhD thesis is devoted to the high-

fidelity multi-physics (based on neutronics/thermal-hydraulics coupling) analysis of Pebble Bed

Modular Reactor (PBMR), which is a High Temperature Reactor (HTR) design reactor. Two

facts motivated the selection of this PhD research scope:

a) The Next Generation Nuclear Plant (NGNP) will be a HTR design;

b) Core design and safety analysis methods are considerably less developed and mature

for HTR analysis than those currently used for Light Water Reactors (LWRs).

The continued development of the NGNP requires verification of HTR design and its safety

features with reliable, and high fidelity coupled multi-physics models within the framework of

robust, efficient, and accurate code systems. As mentioned above while the coupled three-

dimensional (3-D) neutron kinetics/thermal-hydraulics methodologies have been extensively

researched and established for LWR applications, there is a limited experience for HTRs in this

area. Compared to LWRs, the HTR transient analysis is more demanding since it requires

proper treatment of both slower and much longer transients (of time scale in hours and days)

and fast and short transients (of time scale in minutes and seconds). High fidelity multi-physics

methods based on neutronics/thermal-hydraulic coupling are important for core transients

involving significant space/time variations of the flux and feedback parameters shapes and

these methods have not been systematically applied to HTRs. These facts motivated the

establishing of consistent, sophisticated and efficient coupled methodologies for the HTR.

There is limited operation and experimental data available for HTRs for validation of

coupled multi-physics methodologies. In this situation the verification based on code-to-code

comparisons on well-specified international benchmark problems becomes very important. The

160

Nuclear Energy Agency of Organisation for Economic Development (NEA/OECD) PBMR-400

coupled code benchmark is one of these benchmark problems, which have been developed in

last few years and which allow for comprehensive testing and qualification of coupled multi-

physics codes for steady-state and transient analysis of PBMRs. This benchmark has been

used in this PhD thesis as a test framework for the performed development and optimization

studies. In such benchmark problems the establishment of reference 3-D high-order steady-

state and time-dependent solutions is very important. The experience gained in the PBMR

benchmark code-to-code comparisons indicated also the importance of optimization of coupled

multi-physics methodologies for PBMR analysis.

The work conducted within the framework of this PhD thesis contributed towards

addressing the above-described modeling and verification needs of the coupled high-fidelity

multi-physics methodologies for PBMR analysis. This PhD research has led to the

establishment of PBMR analysis reference models based on higher-order complete 3-D

coupled methods. On the neutronics side the reference models are based on the 3-D multi-

group neutron transport discrete-ordinates TORT-TDS code for both steady-state and time-

dependent simulations. Two reference thermal-hydraulic models have been developed. The

first one is based on the Computational Fluid Dynamics (CFD) code PHOENICS (Parabolic

Hyperbolic Or Elliptic Numerical Integration Code Series) and is used to generate reference

solutions for the Depressurised Loss of Forced Cooling (DLOFC) and Pressurised Loss of

Forced Cooling (PLOFC) transients. However, the CFD calculations are stand-alone thermal-

hydraulic calculations (there is no neutronics model associated with the calculations) and can

be used only for verification of design and safety analysis thermal-hydraulic codes such as

THERMIX-DIREKT. The second one is using the porous media based ATTICA3D code, which

is coupled with TORT-TDS code in order to provide reference solutions for coupled steady

state and transient analysis.

This PhD research also resulted in more accurate and efficient tool based on the

NEM/THERMIX code system to analyze the neutronics and thermal-hydraulic behavior for

design optimization and safety evaluation of the PBMR concept. New modeling features and

methods enhancements have been developed and implemented in NEM/THERMIX to enable

161

this code system to analyze the full spectrum of safety-related transients in PBMR.

Furthermore, based on comparative analysis with reference results the NEM/THERMIX code

system has been optimized in terms of different aspects of the coupling methodologies.

Finally, the added benefit of this work is that in the process of studying and improving the

coupled methodology more insight was gained into the physics and dynamics of PBMR, which

will help also to optimize the PBMR design and improve its safety by better prediction of

maximum temperature during transient. This in turn would result in better prediction of flux

distribution to ensure better prediction of radiation damage amongst other things.

One unique contribution of the PhD research is the investigation of the importance of the

correct representation of the 3-D effects in the PBMR analysis. Most of the current coupled

codes analyze PBMRs in 2-D R-Z geometry neglecting the θ-direction of the 3-D cylindrical

geometry. One of the examples for such simplifying modeling is the representation of the

control rod movement as a 2-D grey curtain. The studies performed within the framework of

this thesis demonstrated that explicit 3-D modeling of control rod movement is superior and

removes the errors associated with the grey curtain approximation.

In summary, the contributions of this PhD thesis are as follows:

a) Developing reference neutronics, thermal-hydraulics and coupled models for PBMR-400

steady-state and transient analysis using high-order methodologies: TORT-TDS (for

neutronics analysis), PHOENICS and ATTICA-3D (for thermal-hydraulics analysis) and

TORT-TDS/ATTICA3D (for coupled analysis);

b) Completion of the development and verification of NEM/THERMIX by adding modeling

capabilities to the coupled code (such as control rod movement model, Xenon model,

improved temperature feedback model, spectrum feedback model, and high-order

interpolation of cross-section tables) for performing in accurate and efficient manner the

whole spectra of transients important for the PBMR design and safety analysis;

c) Performing optimization studies of different aspects of the NEM/THERMIX multi-physics

methodology such as spatial coupling, temporal coupling, coupled convergence,

feedback modeling and cross-section representation.

162

d) Finding the optimal cut-off energy for the two-group PBMR analysis based on sensitivity

studies;

8.2 Future work

The optimization of multi-group structure of the microscopic cross-section libraries, used as

input in the lattice physics codes such as COMBINE for few-group cross-section generation

has to be performed by utilizing the more systematic, consistent, and sophisticated energy

group selection methodology called CPXSD (Contribution and Point-wise Cross-Section

Driven) methodology. Following the examination and optimization of the multi-group structure,

the finalization of the few-group structure for HTR core analysis can be accomplished using

again the CPXSD methodology.

The envisioned future work includes performing transient reference calculations with

TORT-TDS/ATTICA for all transient cases of the OECD PBMR-400 benchmark, and further

optimization of the NEM/THERMIX coupled code system for simulation of different PBMR

transients by comparisons with the reference TORT-TDS/ATTICA results.

163

References

1. Generation IV Roadmap; “Description of Candidate Gas-cooled Reactor Systems Report

Issued by the Nuclear Energy Research Advisory Committee and the Generation IV

International Forum”; 2002.

2. Y. Xu; “High Temperature Gas-Cooled Reactor Programme in China”, Institute of Nuclear

Energy Technology (INET).

3. C. Tang et al; “Irradiation Testing of Matrix Material for Spherical HTR-10 Fuel Elements”;

Proceedings HTR2006: 3rd International Topical Meeting on High Temperature Reactor

Technology, South Africa, 2006.

4. J. Tian; “A New Ordered Bed Modular Reactor Concept”; Proceedings HTR2006: 3rd

International Topical Meeting on High Temperature Reactor Technology, South Africa,

2006.

5. N. Fujimoto et al; “Present Status of HTTR Project-- Achievement of 950 ºC of Reactor

Outlet Coolant Temperature”; 2nd International Topical Meeting on High Temperature

Reactor Technology; CHINA, 2004.

6. IAEA-TECDOC-1382; “Evaluation of High Temperature Gas Cooled Reactor Performance:

Benchmark Analysis Related to Initial Testing of the HTTR and HTR-1”, IAEA, VIENNA,

2003.

7. M. Phelip et al; “High-Temperature Reactor Fuel Technology in the European Projects

HTR-F1 and RAPHAEL”; Proceedings HTR2006: 3rd International Topical Meeting on High

Temperature Reactor Technology, South Africa, 2006.

8. P. H. Liem et al; “Neutronic Modeling for Modular Pebble Bed Reactor during Reactivity

Accident”; Journal of Nuclear Science and Technology, 29(8), pp.805-812, 1992.

9. B. R. Bandini; A 3-Dimensional transient neutronic routine for the TRAC-PF1 reactor

thermal-hydraulic computer code, PhD Thesis, Pennsylvania State University, 1990.

10. J. M. Noh et al; “Development of a Computer Code for the Analysis of Prism and Pebble

Type VHTR Cores”.

11. J. E. Hoffmann et al; “Verification and validation of the CFD model of the PBMR reactor”,

Proceedings HTR2006: 3rd International Topical Meeting on High Temperature Reactor

Technology, South Africa, 2006.

164

12. N. TAUVERON et al.; “Preliminary Design of a Small Air Loop for System Analysis”;

Proceedings HTR2006: 3rd

International Topical Meeting on High Temperature Reactor

Technology, South Africa, 2006.

13. H. Gerwin, W. Scherer; E. Teuchet; “The TINTE modular code system for computational

simulation of transient processes in the primary circuit of a pebble-bed High Temperature

Gas-Cooled Reactor”, Nuclear Science and Engineering, 103, p302-312.

14. S. Struth; “Direkt – A Computer Programme for non-steady, two-dimensional simulation of

thermo-hydraulic transients”, Kernforschunsanslage Jülich, JÜL-1702, 1999.

15. E.C. Verkerk, A. I. van Heek; “Dynamics of a Small Direct Cycle Pebble Bed HTR”, IAEA-

TECDOC--1238, pp:78-86.

16. J. Simoneau; “Three Dimensional Simulation of Coupled Convection, Conduction and

Radiative Heat Transfer During Decay Heat Removal in an HTR”; Nuclear Engineering and

Design 237 p1923-1937, 2007.

17. H. Haque, W. Feltes, G. Brinkmann; “Thermal Response of High Temperature Reactor

During Passive Cooldown under Pressurised and Depressurized Conditions”, Nuclear

Engineering and Design 236, 2006.

18. H. Gougar et al.; “Reactor Pressure Vessel Temperature Analysis of Candidate Very High

Temperature Reactor Designs”; Proceedings HTR2006: 3rd

International Topical Meeting

on High Temperature Reactor Technology; 2006.

19. V. Seker, T. J. Downar; “Analysis of a PBMR-400 Control Rod Ejection Accident Using

PARCS-THERMIX and the Nordheim Fuchs Model”; Advances in Reactor Physics Analysis

and Design of High-Temperature Reactors-I, 2004.

20. B. Boer; “Coupled neutronics / thermal hydraulics calculations for High Temperature

Reactors with the DALTON-THERMIX code system”; International Conference on Reactor

Physics, Nuclear Power: A Sustainable Resource, Switzerland, 2008.

21. T. RADEMER et al; “Coupling of Neutronics and Thermal Hydraulics Codes for Simulation

of Transients of Pebble Bed HTR Reactors”; 2nd International Topical Meeting on High

Temperature Reactor Technology; CHINA, 2004.

22. B. Tyobeka; “PBMR coupled neutronics and thermal-hydraulics calculations”, Master

Thesis, Pennsylvania State University, 2004.

165

23. B. Tyobeka, “Advanced multi-dimensional deterministic transport computational capability

for safety analysis of pebble-bed reactors”, PhD Thesis, Pennsylvania State University,

2007.

24. F. Reitsma et al; “Cross section representation options for HTR transient analysis tested on

the OECD PBMR-400MW transient benchmark”; International Conference on Reactor

Physics, Nuclear Power: A Sustainable Resource, Switzerland, 2008.

25. M. Methnani; “Highlights of recent IAEA research activities on High-Temperature Gas-

Cooled Reactors”, International Atomic Energy Agency OECD PBMR Benchmark

Workshop, Paris, 2005.

26. F. Reitsma; “OECD/NEA/NSC: PBMR coupled neutronics/thermal hydraulics benchmark:

the PBMR-400 core design”; 2005.

27. KTA Rule 3102.2: “Reactor Core Design for High-Temperature Gas-Cooled Reactor, Part

2: Heat transfer in spherical fuel elements”; June 1983.

28. KTA Rule 3102.1: “Reactor Core Design for High-Temperature Gas-Cooled Reactor, Part

1: Calculation of the Material Properties of Helium”; June 1978.

29. KTA Rule 3102.3: Reactor core design for High-Temperature Gas-Cooled Reactor, Part 3:

Loss of pressure through friction in pebble bed cores; March 1981.

30. A. D. Osso; “A transverse buckling based method in core neutronics models equivalence”,

Annals of Nuclear Energy, 29, p659-671; 2002.

31. A. Pautz, A. Birkhofer, “DORT-TD: A Transient Neutron Transport Code with Fully Implicit

Time Integration”, Nuclear Science and Engineering, Vol. 145, pp. 299-319, 2003.

32. A. Pautz, A. Birkhofer, “Coupling of time-dependent neutron transport theory with the

thermal hydraulics code ATHLET and Application to the Research Reactor FRM-II”, Nucl.

Sci. Eng.; pp 167-180, 2003.

33. W. A. Rhoades, R. L. Childs, “The TORT Three-dimensional Discrete Ordinates

Neutron/Photon Transport Code”; Nucl. Sci. Eng. 107, 397, 1991.

34. V. Seker; “Multiphysics methods development for high temperature gas cooled reactor

analysis”, PhD Thesis, Purdue University, 2007.

35. S. AOKI, et al. , “The verification of the 3-Dimensional nodal kinetic code ANCK using

transient benchmark Problem”, Journal of Nuclear Science and Technology, Vol. 44, No. 6,

p. 862–868 (2007)

166

36. R.A. Grimesey et al., “COMBINE/PC-A portable ENDF/B version 5 neutron spectrum and

cross section generation program”.

37. L. Massimo, “Physics of High temperature Reactors”, Pergamon Press (1976).

38. E.E. Bende et al., “Analytical calculation of average Dancoff factor for a fuel kernel in a high

temperature pebble bed reactor”, Nuclear Science and Engineering: 133, 147-153 (1999).

39. J. Valko, “Calculation of Dancoff factor for Pebble Bed Reactors”, Nuclear Science and

Engineering: 135, 304-307 (2000)

40. K. Yamasita et al., “Effects of core models and neutron energy group structures on Xenon

oscillation in large graphite-moderated reactors”, Nuclear Science and Technology: 30 [3],

249-260 (March 1993).

41. F. Reitsma, “PBMR coupled neutronics/thermal hydraulics transient benchmark for the

PBMR-400 core”, September 2005.

42. Z. Karriem et al., “MCNP modeling of HTGR pebble-type fuel”,

43. U. Colak and V. Seker, “Monte Carlo criticality calculations for a pebble bed reactor with

MCNP”, Nuclear Science and Engineering: 149, 131-137(2005)

44. MCNP Manual volume 1,

45. O. Ubbink et al., “PBMR fuel kernel model for the prediction of accurate temperature

profiles”, Proceedings of the 16th International Conference on Nuclear Engineering,

ICONE16 , May 11-15, 2008, Orlando, Florida, USA

46. J. Duderstadt and L. Hamilton, “Nuclear Reactor Analysis”, John Wiley and Sons, New

York, (1941)

167

Appendix A

TRISO Particle for PBMR-400 Reactor

c Cell Cards

c TRISO particle is part of universe 1

1 1 6.9663E-02 -1 imp:n=1

2 2 5.2654E-02 1 -2 imp:n=1

3 3 9.5279E-02 2 -3 imp:n=1

4 4 9.5536E-02 3 -4 imp:n=1

5 5 9.5279E-02 4 -5 imp:n=1

c Cube of Graphite outside the TRISO particle

6 0 5 imp:n=1

c 6 6 8.9262E-02 5 u=1 imp:n=1

c 7 0 6 -7 8 -9 10 -11 lat=1 fill=1 u=2 imp:n=1

c Surface Cards

1 so 0.0250 $ Kernel UO2 10.4 density

2 so 0.03450 $ Porous Carbon Buffer C 1.05

3 so 0.03850 $ Inner Pyrolytic Carbon C 1.9

4 so 0.04200 $ SiC Barrier SiC 3.18

5 so 0.04600 $ Outer Pyrolytic Carbon C 1.9

c Graphite Cube outside of the TRISO particle

c c6 px -8.1704E-02

c 7 px 8.1704E-02

c 8 py -8.1704E-02

c 9 py 8.1704E-02

c 10 pz -8.1704E-02

c 11 pz 8.1704E-02

c c Fill the graphite matrix with fuel spheres

c 12 so 2.50 $ Graphite matrix sphere

168

c Data Cards

c Criticality Control card

kcode 5000 1.0 50 250

ksrc 0 0 0

c Materials Card

m1 92238.66c 2.0992E-02

92235.66c 2.2292E-03

8016.66c 4.6442E-02

m2 6012.42c 5.2654E-02

m3 6012.42c 9.5279E-02

m4 14028.66c 4.7768E-02

6012.42c 4.7768E-02

m5 6012.42c 9.5279E-02

c m6 6012.42c 8.9262E-02

169

Fuel Pebble for PBMR-400 Reactor

c Cell Cards

c TRISO particle is part of universe 1

1 1 6.9663E-02 -1 u=1 imp:n=1

2 2 5.2654E-02 1 -2 u=1 imp:n=1

3 3 9.5279E-02 2 -3 u=1 imp:n=1

4 4 9.5536E-02 3 -4 u=1 imp:n=1

5 5 9.5279E-02 4 -5 u=1 imp:n=1

c Cube of Graphite outside the TRISO particle

7 6 8.9262E-020 -6 7 -8 9 -10 11 5 u=1 imp:n=1 $ window for

the lattice

71 0 6:-7:8:-9:10:-11 u=1 imp:n=1 $ void outside

unit cell

8 0 -6 7 -8 9 -10 11 u=2 fill=1 lat=1 imp:n=1 $

infinite lattice

9 0 -12 fill=2 imp:n=1 $ Fill

inside of pebble with lattice

10 6 8.9262E-020 12 -13 imp:n=1 $ graphite layer

outside fuel zone

100 0 13 imp:n=0 $ void outside

the pebble

c Surface Cards

1 so 0.0250 $ Kernel UO2 10.4 density

2 so 0.03450 $ Porous Carbon Buffer C 1.05

3 so 0.03850 $ Inner Pyrolytic Carbon C 1.9

4 so 0.04200 $ SiC Barrier SiC 3.18

5 so 0.04600 $ Outer Pyrolytic Carbon C 1.9

c Graphite Cube outside of the TRISO particle

6 px 8.1704E-02

7 px -8.1704E-02

8 py 8.1704E-02

170

9 py -8.1704E-02

10 pz 8.1704E-02

11 pz -8.1704E-02

c Fill the graphite matrix with fuel spheres

12 so 2.50 $ Graphite matrix sphere

13 so 3.00

c Data Cards

c Criticality Control card

c kcode 5000 1.0 50 250

c ksrc 0 0 0

c Materials Card

m1 92238.66c 2.0992E-02

92235.66c 2.2292E-03

8016.66c 4.6442E-02

m2 6012.42c 5.2654E-02

m3 6012.42c 9.5279E-02

m4 14028.66c 4.7768E-02

6012.42c 4.7768E-02

m5 6012.42c 9.5279E-02

m6 6012.42c 8.9262E-02

171

Appendix B

Table 18: Exercise 1 DLOFC without scram

Time

(seconds) Description

Time-specific Output

Generated

0 Equilibrium steady-state completed. Equilibrium steady-state output

at this time point should be

identical to the values of

Exercise 3 of Steady State, so

a new output set will not be

needed here.

0 Assume t=0 as the time zero for the decay heat

(Normalization to total power during steady-state to

be kept in mind)

0 – 13 A reduction in reactor inlet coolant mass flow from

nominal (192.7 kg/s) to 0.2 kg/s over 13 seconds.

The mass flow ramp is assumed linear. A trickle flow

of 0.2 kg/s should then be assumed to remain after

this step to continue flowing through the reactor with

inlet temperature of 500°C.

None.

0 – 13 A reduction in reactor helium outlet pressure from

nominal (90 bar) to 1 bar over 13 seconds. The

pressure ramp is assumed linear. (Note that all

pressures defined in this benchmark study are

absolute pressure values, and not gauge values).

None.

13 Depressurisation phase completed.

Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density and pebble-bed

effective thermal conductivity.

Single parameter value for axial

power/heat offset.

172

Time

(seconds) Description

Time-specific Output

Generated

13 – 360000 No change in input parameters. Just the defined time

dependent edits.

re-critical Re-critical condition should be reached after some

time (cool down and Xenon decay)

Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power/heat density.

Single parameter value for core

power, axial power offset,

reactivity edit.

~ 360000 Transient case completed.

100 hours or at least 10 hours after re-criticality

Spatial maps of the maximum

kernel and fuel temperature,

moderator temperature, power

density.

Single parameter value for axial

power/heat offset.

Table 19: Exercise 2 DLOFC with scram

Time

(seconds) Description

Time-specific Output

Generated

0 Equilibrium steady-state completed. Equilibrium steady-state output

at this time point should be

identical to the values of

Exercise 3 of Steady State, so a

new output set will not be

needed here.

0 Assume t=0 as the time zero for the decay heat

(Normalization to total power during steady-state to

be kept in mind)

173

Time

(seconds) Description

Time-specific Output

Generated

0 - 13 A reduction in reactor inlet coolant mass flow from

nominal (192.7 kg/s) to 0.0 kg/s over 13 seconds.

The mass flow ramp is assumed linear. No external

flow after this step

None.

0 - 13 A reduction in reactor helium outlet pressure from

nominal (90 bar) to 1 bar over 13 seconds. The

pressure ramp is assumed linear. (Note that all

pressures defined in this benchmark study are

absolute pressure values, and not gauge values).

None.

13 Depressurisation phase completed. Natural

convection must be included that will lead to some

internal mass flow. No external mass flow.

Transient output at this time

point should be similar to the

values in Exercise 1 (tricle flow

the only difference), so a new

output set will not be needed

here.

13 – 16 All control rods are fully inserted over 3 seconds to

SCRAM the reactor.

None.

16 Scram phase completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power/heat density.

Single parameter value for

fission power and axial power /

heat offset.

16 – 180000 No change in input parameters. None.

180000 Transient case completed.

50 hours or at least 5 hours after maximum

temperature has been reached

Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power/heat density.

Single parameter value for axial

power / heat offset.

174

Table 20: Exercise 3 PLOFC with scram

Time

(seconds) Description

Time-specific Output

Generated

0 Equilibrium steady-state completed. Equilibrium steady-state output

at this time point should be

identical to the values of

Exercise 3 of Steady State, so

a new output set will not be

needed here.

0 Assume t=0 as the time zero for the decay heat

(Normalization to total power during steady-state to

be kept in mind)

0 - 13 A reduction in reactor inlet coolant mass flow from

nominal (192.7 kg/s) to 0.0 kg/s over 13 seconds.

The mass flow ramp is assumed linear.

None.

0 - 13 A reduction in reactor helium outlet pressure from

nominal (90 bar) to 60 bar over 13 seconds. The

pressure ramp is assumed linear. CHANGE TO

CONSTANT INVENTORY.

None.

13 Pressure equalization phase completed. Natural

convection must be included that will lead to some

internal mass flow. No external mass flow.

The core helium inventory is to stay unchanged thus

pressure changes due to heat-up and cool-down are

possible. Only helium volumes in the core to be

included (no PCU). If needed hand calculation

estimates using the average helium temperature can

be used to adjust the pressure linearly over time if

this function does not exist in the core.

Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density, relative

pressure, mass flow.

Single parameter value for axial

power offset.

13 – 16 All control rods are fully inserted over 3 seconds to

SCRAM the reactor.

None.

175

Time

(seconds) Description

Time-specific Output

Generated

16 Scram phase completed. None.

16 - 180000 No change in input parameters. None.

TBD Maximum fuel temperature reached Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density relative pressure,

mass flow.

Single parameter value for axial

power offset.

180000 Transient case completed.

50 hours or at least 5 hours after maximum

temperature has been reached

Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density relative pressure,

mass flow.

Single parameter value for axial

power offset.

Table 21: Exercise 4a load follow without control rod movement

Time

(seconds) Description

Time-specific Output

Generated

0 Equilibrium steady-state completed. Equilibrium steady-state output

at this time point should be

identical to the values of

Exercise 3 of Steady State, so a

new output set will not be

needed here.

176

Time

(seconds) Description

Time-specific Output

Generated

0 - 360 A reduction in reactor inlet coolant mass flow from

nominal (192.7 kg/s) to 77 kg/s (40% of nominal)

over 6 minutes. The mass flow ramp is assumed

linear. The reactor outlet pressure is decreased over

the same time from nominal (90 bar) to 40% of the

inventory.

None.

0 - 360 A reduction in reactor power level from nominal

400 MW (100%) to 160 MW (40%) over 6 minutes.

The power ramp is assumed linear. The reactor total

power is thus a fixed target condition.

None.

360 100-40% phase completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solid temperature,

power density.

Single parameter value for axial

power offset.

360 - 10800

(3 hours)

No change in input parameters. Spatial maps of the Xenon

concentration every two hours,

at t = 3600, 7200 and 10800 s.

Also axial power offset values

at these times.

10800 –

11160

An increase in reactor inlet coolant mass flow from

77 kg/s (40% of nominal) back to 192.7 kg/s, again

over 6 minutes. The reactor outlet pressure is

increased linearly back to nominal at the same time.

None.

10800 -

11160

An increase in reactor power level from 160 MW to

400 MW, again over 6 minutes. The reactor total

power is thus a fixed target condition.

None.

177

Time

(seconds) Description

Time-specific Output

Generated

11160 40-100% phase completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solid temperature,

power density, Xenon

concentration.

Single parameter value for axial

power offset.

11160 –

32400

No change in input parameters. Spatial maps of the Xenon

concentration every hours up to

9 hours. Also axial power offset

values at these times.

32400 Transient case completed. Spatial maps of the maximum

kernel and fuel temperature,

maximum and average

moderator temperature, power

density, Xenon concentration.

Single parameter value for axial

power offset.

172800 Optional

The Xenon oscillation behaviour can be studied over

a longer period.

Transient up to for 48 hours (at

t = 43200, 57600, 72000,

86400, 100800, 115200,

129600, 144000, 158400, and

172800 s)

178

Table 22: Exercise 4b load follow with control rod movement

Time

(seconds) Description

Time-specific Output

Generated

0 Equilibrium steady-state completed. Equilibrium steady-state output

at this time point should be

identical to the values of

Exercise 3 of Steady State, so a

new output set will not be

needed here.

0 - 32400 Scenario 2: Activate controller moving control rods

to keep the reactor critical. Control rods move at

1 cm.s-1 and the reactivity band width is 0.1% ∆k.

0 – 360 A reduction in inlet reactor coolant mass flow from

nominal (192.7 kg/s) to 77 kg/s (40% of nominal)

over 8 seconds. The mass flow ramp is assumed

linear. The reactor outlet pressure is decreased over

the same time from nominal (90 bar) to 40% of the

inventory.

None.

0 – 360 The reactor fission power to be calculated. It should

more or less follow the 400 MW (100%) to 160 MW

(40%) ramp.

None.

360 100-40% phase completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density.

Single parameter value for axial

power offset.

360 - 10800

(3 hours)

No change in input parameters. Spatial maps of the Xenon

concentration every two hours,

at t = 3600, 7200 and 10800 s.

Also axial power offset values

at these times.

179

Time

(seconds) Description

Time-specific Output

Generated

10800 –

11160

An increase in reactor inlet coolant mass flow from

77 kg/s (40% of nominal) back to 192.7 kg/s, again

over 8 seconds. The reactor outlet pressure is

increased linearly back to nominal at the same time.

None.

10800 -

11160

The reactor fission power to be calculated. It should

more or less follow an increase from around

160 MW to 400 MW. The total power variation is still

a boundary condition as in Exercise 4a.

None.

11160 40-100% phase completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density, Xenon

concentration.

Single parameter value for axial

power offset.

11160 –

32400

No change in input parameters. Spatial maps of the Xenon

concentration every hours up to

9 hours.. Also axial power offset

values at these times.

32400 Transient case completed. Spatial maps of the maximum

kernel and fuel temperature,

moderator/solids temperature,

power density, Xenon

concentration.

Single parameter value for axial

power offset.

172800 Optional

The Xenon oscillation behaviour can be studied over

a longer period.

Transient up to for 48 hours (at

t = 43200, 57600, 72000,

86400, 100800, 115200,

129600, 144000, 158400, and

172800 s)

180

VITA

Peter received his BS in Education with Chemistry and Physics in 1998 and a M.S. in

Applied Radiation Science and Technology in 2001 from North-West University, South Africa.

He also received his MS in Nuclear Engineering from The Pennsylvania State University in

2006. Some of his publications are:

• J. Ortensi, H. Gougar, P. Mkhabela, J. Han, B. Tyobeka, K. Ivanov, "PBMR-400 Coupled Code Benchmark: Challenges and Successes with NEM-THERMIX," Annual ANS-2006 Meeting, (Contributing author), (Peer Reviewed), 2006.

• P. Mkhabela*, A. Ougouag, K. Ivanov, H. Gougar, J. Han, "Systematic Method of Neutron Energy Group Structure Selection for HTR Reactor," TANSAO 96 (First author), (Peer Reviewed), 2007.

• J. Han*, A. Ougouag, K. Ivanov, H. D. Gougar, P. Mkhabela, "Broad Energy Group Structure Sensitivity Studies for the PBMR", TANSAO 97, pp.511-513, (Contributing author), (Peer Reviewed) 2007.

• P. Mkhabela*, J. Han, K. Ivanov, "DLOFC Transient Analysis for PBMR with NEM/THERMIX", PHYSOR-2008 International Conference, Interlaken, Switzerland, (First author), (Peer Reviewed), September 14-19, 2008.

• P. Mkhabela*, K. Ivanov, “Improvements to the NEM/THERMIX coupled code analysis of High Temperature Reactors”, PHYSOR-2010 International Conference, Pittsburg, USA, (First author), (Peer Reviewed), May 9-14, 2010