i 
 
REACTOR CORE CONVERSION STUDIES OF GHANA 
RESEARCH REACTOR – 1 AND PROPOSAL FOR ADDITION OF 
SAFETY ROD  
 
 
BY 
 
HENRY CECIL ODOI 
(10235606) 
M Phil Nuclear Engineering – 2008  
(University of Ghana, Legon) 
B Sc Physics – 2004 
 (Kwame Nkrumah University of Science and Technology, Kumasi) 
 
 
 
 
This Dissertation Submitted to University Of Ghana, Graduate School of 
Nuclear and Allied Sciences, Department of Nuclear Engineering, in Partial 
Fulfillment of the Requirement for the Award of PhD Nuclear Engineering 
 
June 2014 
 
 
 
 
 
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Declaration  
 “This dissertation is the result of research work undertaken by Henry Cecil Odoi in the 
Department of Nuclear Engineering, School of Nuclear and Allied Sciences (SNAS) of 
University of Ghana (UG), under the supervision of Prof. Emeritus Edward H. K. 
Akaho and Prof. Sunday A. Jonah” 
 
 
 
 
 
Sign: ........................................... 
Date: ........................................... 
Henry Cecil Odoi 
      (Student) 
 
Sign: ........................................... 
Date: ..........................................                        
Prof. Emeritus Edward H. K. Akaho 
            (Supervisor)                                      
 
Sign: ........................................... 
Date: ............................................                    
Prof. Sunday A. Jonah 
        (Supervisor)                                      
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Dedication 
To the trio: my mother, my wife and my son. 
To the Lord Almighty, and mother Ghana. 
With Love. 
 
 
 
 
 
 
 
 
 
 
 
 
 
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ACKNOWLEDGEMENTS 
I am grateful to the Almighty God for his protection and guidance. 
I do thank all the lecturers of School of Nuclear and Allied Sciences for their selfless 
gestures. 
Further appreciation goes to my entire family for their invaluable support; spiritually, 
mentally and financially. 
I greatly appreciate the advice and encouragement of my supervisors; Prof. Emeritus E. 
H. K. Akaho and Prof. S. A. Jonah during the undertaking of this project.   
I do acknowledge with gratitude the financial support of IAEA for the award of 
Research Contract Number 13946 and also the Fellowship in Nigeria – GHA/09003.  
I also appreciate the fellowships sponsored by US Department of Energy in 2010 and 
2011, and I am thankful to the staff of Nuclear Engineering Division of Argonne 
National Laboratory, especially, J. Roglans-Ribas, J.E. Matos, J.R. Liaw, M. 
Kalimullah, J. Morman and S.C. Mo for hosting fellowships, and the importance they 
assigned to this project.  
 
 
 
 
 
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Table of Contents 
Title page …………………………………………………………………….………………………… i 
Declaration ……………………………………………………………..……..………………………… ii 
Dedication ……………………………………………………………..……..………………………… iii 
Acknowledgements ……………………………………………………………..……..……………………… iv 
Table of contents  ……………………………………………………………..……..………………………… v  
List of tables ……………………………………………………………..……..………………………… viii 
List of figures ……………………………………………………………..……..………………………… x 
Nomenclature ……………………………………………………………..……..………………………… xiii 
Abstract ……………………………………………………………..……..………………………… xvii 
1. INTRODUCTION ...................................................................................................... 1 
1.1 Reactor Facility Changes .................................................................................... 7 
1.2 Operation License and Procedural Changes ........................................................ 7 
1.3 Characteristics of Reference GHARR-1 LEU Core ............................................ 8 
1.4 Outline of Thesis ............................................................................................... 11 
2 LITERATURE REVIEW ......................................................................................... 13 
2.1 General .............................................................................................................. 13 
2.1.1 Similarity with Other Converted Facilities ..................................... 13 
2.1.2 Advocates of Core Conversion ........................................................ 14 
2.2 GHARR-1 Reactivity Management .................................................................. 18 
2.2.1 Reactivity worth of top beryllium reflector plates .......................... 20 
2.2.2 Comparison of the HEU and LEU fuel characteristics ................... 21 
2.2.3 Xenon poisoning calculations ......................................................... 21 
2.3 Central control rod of GHARR-1 ...................................................................... 23 
2.3.1 Control rod drive mechanism .......................................................... 24 
2.3.2 Measurement of the central control rod worth ................................ 27 
3 THEORETICAL ANALYSIS .................................................................................. 28 
3.1 NUCLEAR REACTOR THEORY ................................................................... 28 
3.2 Solution of the Neutron Transport Equation ..................................................... 30 
3.2.1 Diffusion Theory Approximation .................................................... 31 
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3.2.2 Multigroup Calculations .................................................................. 34 
3.3 Generation of Group Constants ......................................................................... 36 
3.4 Few Group (Macro-group) Constants ............................................................... 38 
3.5 Monte Carlo Techniques ................................................................................... 39 
3.5.1 General Overview ............................................................................ 39 
3.5.2 Criticality Calculations .................................................................... 42 
3.5.3 Neutron Cross section Data Libraries ............................................. 44 
3.5.4 Thermal Scattering Matrix         ................................................ 45 
3.5.5 Mathematical Basis for Monte Carlo Simulation ............................ 46 
3.5.6 Generation of Random Variable ...................................................... 49 
3.5.7 Variance Reduction ......................................................................... 57 
3.5.8 Variance Reduction Techniques in Monte Carlo ............................ 59 
3.5.9 Monte Carlo Simulations ................................................................. 61 
3.5.10 Monte Carlo Estimates .................................................................... 67 
3.6 Burnup ............................................................................................................... 71 
3.6.1 BURNUP ANALYSIS .................................................................... 75 
3.6.2 DESCRIPTION OF REBUS ........................................................... 76 
3.7 Analyses of MNSR Thermal Hydraulics .......................................................... 78 
3.7.1 Heat transfer process of MNSR ....................................................... 78 
3.7.2 Thermal – Hydraulic Characteristics of MNSR .............................. 84 
3.7.3 Heat Transfer Calculations of the Steady State ............................... 85 
3.7.4 Transient Thermal hydraulic calculation ......................................... 91 
4 METHODOLOGY ................................................................................................. 102 
4.1 Monte Carlo Model for GHARR-1 ................................................................. 103 
4.1.1 Auxiliary shutdown system ........................................................... 106 
4.2 REBUS Model for GHARR-1......................................................................... 112 
4.3 Thermal Hydraulic Analyses ........................................................................... 114 
4.3.1 Steady State ................................................................................... 115 
4.3.2 Transients Analysis ....................................................................... 118 
5 Results and Discussion ........................................................................................... 122 
5.1 Criticality Results ............................................................................................ 122 
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5.1.1 Integral and Differential Control Rod Worth ................................ 124 
5.1.2 Flux Distributions .......................................................................... 126 
5.1.3 Reactivity Coefficients .................................................................. 131 
5.1.4 Worth of the top beryllium reflector ............................................. 134 
5.2 ANALYSIS FOR AUXILIARY CONTROL ROD ........................................ 135 
5.3 CORE LIFETIME ANALYSIS ...................................................................... 137 
5.4 THERMAL HYDRAULICS ANALYSES ..................................................... 138 
5.4.1 Steady State ................................................................................... 138 
5.4.2 Transient ........................................................................................ 141 
6 CONCLUSION AND RECOMMENDATION ..................................................... 146 
6.1 Conclusion ....................................................................................................... 146 
6.2 Recommendation ............................................................................................. 148 
References ...................................................................................................................... 149 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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List of Tables 
Table 1.1 Control Rod and Guide Tube Design Parameters in MNSR Models ................ 9 
Table 1.2  Comparison of Reference GHARR-1 HEU and LEU Fuel Uranium Isotopes . 9 
Table 1.3  Comparison of Key Parameters for Reference GHARR-1 HEU and LEU 
Cores ................................................................................................................ 10 
Table 2.1 Lattice positions for the fuel cage .................................................................... 19 
Table 2.2 Characteristics of the GHARR-1 fuel element & dummy ............................... 21 
Table 2.3 Equilibrium xenon reactivity as a function of core average flux ..................... 22 
Table 4.1 Dimension of cadmium in capsule (average along each side) ....................... 108 
Table 4.2 Uncertainties Included in the Six Hot Channel Factors ................................. 117 
Table 5.1 Comparison of Reactivities for various cores. ............................................... 122 
Table 5.2: Comparison of Criticality Results for HEU and LEU .................................. 123 
Table 5.3: Differential Reactivity, mk ........................................................................... 125 
Table 5.4:  Peak Flux in the Inner Irradiation Channels (n/cm2s) .................................. 130 
Table 5.5: Fuel Temperature Coefficient ....................................................................... 132 
Table 5.6: Moderator temperature coefficient (partial).................................................. 133 
Table 5.7: Moderator void coefficient ........................................................................... 133 
Table 5.8: Moderator Temperature coefficient .............................................................. 134 
Table 5.9. Auxiliary rod worth, mk for unmodified irradiation site (inner) .................. 136 
Table 5.10. Auxiliary rod worth, mk for modified irradiation site (inner) .................... 136 
Table 5.11: Xenon Worth and Reactivity Change Rates ............................................... 137 
Table 5.12: Comparison of HEU and LEU steady-state parameters using PLTEMP/ANL
 .................................................................................................................... 139 
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Table 5.13: Comparison of computed coolant outlet temperatures at various powers and 
inlet temperatures ......................................................................................... 140 
Table 5.14: Effect of Inlet Temperature on Temperature Difference at Nominal 
Operating Power for the HEU and LEU Cores ............................................ 141 
Table 5.15: Peak power, peak fuel temperature and peak clad temperature for various 
reactivity insertions ...................................................................................... 145 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
  
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List of Figures 
Figure 2.1  Schematic of Reactor Pool (Cross sectional plane through centres of reactor            
pool and reactor vessel) .................................................................................. 23 
Figure 3.1 :  Partial distribution of neutron energies into G groups ................................ 35 
Figure 3.2 Monte Carlo random works for single particle .............................................. 63 
Figure 3.3 Power iteration for Monte Carlo criticality calculation of the mean value of  
keff ................................................................................................................... 63 
Figure 3.4  Decision points in generating the history of a single neutron in a Monte Carlo 
calculation ...................................................................................................... 67 
Figure 3.5 The most important actinide nuclides and their relations, via a-decay, ß-decay 
and neutron capture reactions......................................................................... 73 
Figure 3.6: Schematics of Reactor Core [55] ................................................................... 79 
Figure 3.7: Heat transfer and transport process ............................................................... 83 
Figure 3.8: Fuel element .................................................................................................. 85 
Figure 3.9: Temperature distribution in fuel and coolant ................................................ 90 
Figure 3.10: Water temperature variation at core inlet (T1) ............................................. 99 
Figure 3.11: Variation of temperature difference between core inlet and outlet. .......... 100 
Figure 3.12: Variation of coolant velocity in reactor core. ............................................ 100 
Figure 4.1: MCNP plot of the core configuration and associated experimental channels
 ...................................................................................................................... 105 
Figure 4.2: MCNP plot of vertical cross-section of GHARR-1 reactor (in full power 
mode)............................................................................................................ 106 
Figure 4.3: Cadmium Capsule........................................................................................ 107 
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Figure 4.4: Cadmium foil ............................................................................................... 107 
Figure 4.5: Measurement of Cadmium (Side)................................................................ 107 
Figure 4.6: Measurement of Thickness (a) .................................................................... 108 
Figure 4.7: Measurement of Thickness (b) .................................................................... 108 
Figure 4.8: Cadmium foil in irradiation channel (Visual editor of MCNP)................... 109 
Figure 4.9 Three concentric foils in unmodified irradiation site (CFU) ........................ 110 
Figure 4.10. Solid auxiliary rod (SR) ............................................................................. 110 
Figure 4.11 Foil with a similar arrangement to the cadmium capsule (F2) ................... 111 
Figure 4.12 Three concentric foils in modified irradiation site (CFM) ......................... 111 
Figure 4.13 Solid auxiliary rod (SR2) ............................................................................ 111 
Figure 4.14  RZ model of the reactor core (figure is not drawn to scale); ..................... 114 
Figure 5.1: The Integral Control Rod Curve. ................................................................. 125 
Figure 5.2: Comparison of average flux distribution in inner irradiation channel at 30 
kW. ............................................................................................................... 127 
Figure 5.3:  Comparison of Flux Distribution in Fission Chamber at 30 kW. ............... 127 
Figure 5.4: Comparison of average flux distribution in outer irradiation channel at 30 
kW. ............................................................................................................... 128 
Figure 5.5: Comparison of average flux distribution in inner irradiation channel at 
nominal powers. ........................................................................................... 128 
Figure 5.6: Comparison of average flux distribution in fission chamber at nominal 
powers. ......................................................................................................... 129 
Figure 5.7: Comparison of average flux distribution in outer irradiation channel at 
nominal powers. ........................................................................................... 130 
Figure 5.8: Peak Power Pin Axial Profile (21 Segments) .............................................. 131 
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Figure 5.9: Reactivity worths of top Be shims for HEU and LEU cores ....................... 135 
Figure 5.10: Differential Reactivity worths of top Be shims for HEU and LEU cores . 135 
Figure 5.11:Time vs. Power  for a 3.8 mk Reactivity Insertion with HEU and LEU Fuel
 ...................................................................................................................... 142 
Figure 5.12: Fuel Temperature comparison of HEU and LEU cores for 3.8 mk reactivity 
transient. ....................................................................................................... 142 
Figure 5.13: Clap Surface Temperature comparison of HEU and LEU cores for 3.8 mk 
reactivity transient. ....................................................................................... 143 
Figure 5.14: Reactor power for reactivity insertions of 3.8 mk, 6 mk and 8 mk for LEU.
 ...................................................................................................................... 143 
Figure 5.15: Fuel Temperature for reactivity insertions of 3.8 mk, 6 mk and 8 mk for 
LEU .............................................................................................................. 144 
Figure 5.16: Clad Temperature for reactivity insertions of 3.8 mk, 6 mk and 8 mk for 
LEU .............................................................................................................. 144 
 
 
 
 
 
 
 
 
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NOMENCLATURE 
Roman Letters 
Af Cross sectional area of flow channel (m
2) 
C Specific heat capacity of fuel meat (J/kg°C) 
D Diffusion constant 
De Equivalent diameter of channel (m) 
E Energy of particle 
f Friction factor 
h Heat transfer coefficient (w/m
2°C) 
J Neutron current density 
keff Multiplication factor 
N Atomic density 
p Probability of the outcome of a particular event 
P Pressure (Pa) 
Q Heat  
r Particle (quantity) distribution in space 
R Sequence of numbers 
S Number of particles emitted from a Source term 
t Time in seconds 
U Coolant velocity (m/s) 
x, y, z General variables 
 
 
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Greek Letters 
Ω Direction in plane, spherical or cylindrical geometry 
Σ Macroscopic cross-section 
  Spatial gradient 
  Particle Flux (n/cm
2s) 
  Angle (°) 
ν Average number of fission neutrons released 
  Track trajectory length, (cm) 
ᵪ fraction of fission neutrons 
       Scattering matrix 
Abbreviations 
ANL Argonne National Laboratory  
BNCT Boron Neutron Capture Therapy  
CANDU Canadian Deuterium Uranium Reactor 
CDF Cumulative Distribution Function 
CHF Critical Heat Flux 
CHFR Critical Heat Flux Ratio 
CIAE China Institute of Atomic Energy 
DNBR Departure from nucleate boiling ration 
ENDF Evaluated Nuclear Data File 
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FOM Figure of Merit 
GAEC Ghana Atomic Energy Commission 
GHARR-1 Ghana Research Reactor–1 
GTRI Global Threat Reduction Initiative 
HEU  Highly Enriched Uranium 
IAEA International Atomic Energy Agency 
IHNI In-Hospital Neutron Irradiator 
INL Idaho National Laboratory 
INSARR Integrated Safety Assessment for Research Reactors 
LEU Low Enriched Uranium 
MCNP Monte Carlo N- Particle Code 
MNSR Miniature Neutron Source Reactor 
NNRI National Nuclear Research Institute 
NNSA National Nuclear Security Administration  
ONBR Onset of Nuclear Boiling Ratio 
PARET Programme for Analyses of Reactor Transients 
PDF Probability Density Function 
PLTEMP Plate Temperature Analysis 
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PSA Project Supply Agreement 
RERTR Reduced Enrichment for Research and Test Reactors 
RRDB Research Reactors Database 
SAR Safety Analysis Report 
SLOWPOKE Safe Low Power Critical (K) Experiments 
REBUS Reactor Burn-up Systems 
U-Al4  Uranium-Aluminium 
UO2  Uranium Oxide  
XSDIR Cross-Section Directorate 
Zirc-4 Zircaloy-4 
 
 
 
 
 
 
 
 
 
 
 
 
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ABSTRACT 
The inclusion of an additional safety rod in conjunction with a core conversion 
study of Ghana Research Reactor-1 (GHARR-1) was carried out using neutronics, 
thermal hydraulics and burnup codes. The study is based on a recommendation by 
Integrated Safety Assessment for Research Reactors (INSARR) mission to 
incorporate a safety rod to the reactor safety system as well as the need to replace 
the reactor fuel with LEU. Conversion from one fuel type to another requires a 
complete re-evaluation of the safety analysis. Changes to the reactivity worth, 
shutdown margin, power density and material properties must be taken into account, 
and appropriate modifications made.  Neutronics analysis including burnup was 
studied followed by thermal hydraulics analyses which comprise steady state and 
transients. Four computer codes were used for the analysis; MCNP, REBUS, 
PLTEMP and PARET. The neutronics analysis revealed that the LEU core must be 
operated at 34 kW in order to attain the flux of 1.0E12 n/cm2.s as the nominal flux 
of the HEU core. The auxiliary safety rod placed at a modified irradiation site gives 
a better worth than the cadmium capsules. For core excess reactivity of 4 mk, 348 
fuel pins would be appropriate for the GHARR-1 LEU core. Results indicate that 
flux level of 1.0E12 n/cm2.s in the inner irradiation channels will not be 
compromised, if the power of the LEU core is increased to 34 kW. The GHARR-1 
core using LEU-UO2-12.5% fuel can be operated for 23 shim cycles, with cycle 
length 2.5 years, for over 57 years at the 17 kW power level.  All 23 LEU cycles 
meet the ~ 4.0 mk excess reactivity required at the beginning of cycle. For 
comparison, the MNSR HEU reference core can also be operated for 23 shim cycles, 
but with a cycle length of 2.0 years for just over 46 years at the 15.0 kW power 
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level. It is observed that the GHARR-1 core with LEU UO2 fuel enriched to 12.5% 
and a power level of 34 kW can be operated ~25% longer than the current HEU core 
operated at 30 kW. Based on the results presented in this report, it is concluded that 
the conversion of the GHARR-1 to LEU core is not likely to compromise safety nor 
increase the frequency/severity of any of the postulated design basis accidents 
identified in the current approved SAR.   
 
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CHAPTER ONE 
1. INTRODUCTION 
The purpose of this introduction is to present the historical background of 
Ghana’s first and only research reactor, justify the need for core conversion from HEU 
to LEU and highlight improved safety due to addition of an auxiliary safety rod. 
The Ghana Research Reactor-1 (GHARR-1) is a commercial version of the 
Miniature Neutron Source Reactor (MNSR) and belongs to the class of tank-in-pool type 
reactors [1]. It is under-moderated with an H/U atom ratio of 197. Thermal power is 
rated at 30 kW with a corresponding peak thermal neutron flux is 1.0×1012 n/cm2.s. For 
fresh core, its cold clean excess reactivity is about 4mk. Cooling is achieved by natural 
convection using light water. Presently, the GHARR-1 core consists HEU (U-Al4 
alloyed) fuel elements arranged in ten concentric rings about a central control rod guide 
tube which houses the reactor’s only control rod. The reactor is fuelled with uranium 
aluminium alloy with an enrichment of 90.2 %. The control rod’s reactivity worth is 
about -7mk, providing a core shutdown margin of -3mk of reactivity. The small core has 
a U-235 content of about 1 kg. However, its relatively large negative temperature 
coefficient of reactivity is capable of boosting its inherent safety properties [2]. The 
small size of the core facilitates neutron leakage and escape in both axial and radial 
directions. To minimize such loses and thereby conserve neutron economy, the core is 
heavily reflected respectively on the side and underneath the fuel cage by a thick 
annulus and slab of beryllium material. Adding regulated thickness of beryllium shims 
to the top tray can compensate loss of reactivity due to axial neutron leakage. 
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GHARR-1  was obtained under a Project Supply Agreement (PSA) between the 
International Atomic Energy Agency (IAEA), the China Institute of Atomic Energy 
(CIAE) and the Government of Ghana in 1994 [3]. The reactor was assembled between 
October and December 1994 and it went critical in 17th December, 1994 [4]. 
Subsequently, it was commissioned on 15th March 1995. Since then it has been used for 
neutron activation analysis, experiments and training of human resource in nuclear 
science and technology.  
According to the IAEA Research Reactor Database (RRDB), 735 research 
reactors have been constructed around the world for civilian applications. On the basis 
of the RRDB, 244 research reactors are currently in operation, 148 are shut down and 
306 have been decommissioned, whiles others are under temporary shutdown [5]. One 
type of such research reactors in operation is the Miniature Neutron Source Reactors 
(MNSR).  
MNSR can be classified as low power research reactor. It is a tank-in-pool type 
reactor, water-moderated and cooled. The general principles of design and operation are 
similar to other low power research reactors as SLOWPOKE 1, SLOWPOKE II. 
SLOWPOKE, acronym for Safe LOW POwer Critical (K) Experiments, is a nuclear 
reactor developed by the Atomic Energy of Canada Limited (AECL). SLOWPOKE-2, 
the commercial version of this reactor, is similar to GHARR-1 and is installed in 8 
institutes. The reactors have a combined operating record of over 80 reactor years. 
SLOWPOKE-2 is a small, light-water moderated, tank-in-pool type reactor, 
which has an inherent safety feature based on a large negative temperature coefficient of 
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reactivity. It contains a small critical mass with HEU fuel. All these characteristics are 
associated with GHARR-1. To maintain a steady neutron flux in the reactor for 
experimental purposes, the reactor also employs a cadmium control rod that responds to 
a self-powered neutron flux detector that is located in the beryllium annulus surrounding 
the core. These features provide a safe and cost-effective reactor as it is with GHARR-1. 
All the features described briefly before are inherent to all the 8 SLOWPOKE-2 
installations except the facility at the Royal Military College of Canada. This particular 
SLOWPOKE-2 facility uses low enriched uranium fuel (LEU) of 19.89 % U-235 in the 
core [6]. A heavy water column is installed in the reactor vessel from which a vertical 
neutron beam is installed for neutron radiography studies. 
Several transient experiments were performed for the insertion of large 
reactivities to demonstrate the self-limiting power excursion features of SLOWPOKE 
type reactors. The results of these investigations have been applied to GHARR-1 to 
show that it exhibits similar safety features [2]. 
All MNSR facilities are equipped with a single control rod, which serves to 
compensate for the excess reactivity necessary for long term core operation and also to 
adjust the power level of the reactor in order to bring the core to power, follow load 
demands and shutdown the reactor. Because the movement of the single control rod is 
controlled by mechanical clutches, it is possible for the mechanical systems to 
malfunction and therefore jeopardize its safety function. In the case of malfunction of 
the single control rod such as rod stuck situation, emergency shutdown of the reactor is 
achieved by pumping Cd rabbits and strings into irradiation channels [2]. Pumping of 
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cadmium rabbits may not also be achieved in case of pressure failure. There is a 
possibility of single point failure which could lead to excessive power excursion and 
ultimately to radiation exposure of personnel in the process of inserting cadmium 
strings. Even though the MNSR is inherently safe due to high negative temperature 
coefficient of reactivity, which provides self limiting power excursion characteristics, 
reactor safety experts have recommended for redundancies in design for reliability of 
systems important to safety.   
One of such recommendations was made during an Integrated Safety Assessment 
for Research Reactors (INSARR) Mission in May, 2009 [7]. It was stated in section 3.11 
of the INSARR report that, “The design of the GHARR-1 has been carried out in 
accordance with Chinese standards with some attention being focussed on relevant 
IAEA standards. However, this design is not fully in conformance with the IAEA Safety 
Standards. In particular, the lack of redundancies of safety rods, and neutronic, 
temperature, and water level channels does not provide for protection against single 
failure”.  
INSARR Mission is provided upon request to all Member States of IAEA. It is a 
peer review of the safety of research reactors against IAEA Safety Standards and 
provides recommendations for safety improvements [8]. To date, over 70 INSARR 
missions were implemented in research reactors in different regions of the world. 
INSARR missions cover all safety areas of research reactors. Key review areas include 
design, safety analysis, operational limits and conditions, regulatory supervision, reactor 
operation and maintenance, ageing management, radiation protection and waste 
management, and experiments and modifications. The host organization may request a 
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full-scope INSARR or a review of a specific area(s). INSARR missions are in three 
stages, namely, PRE-INSARR, INSARR and FOLLOW-UP. 
The single control rod does not satisfy the safety requirements of the IAEA 
Standards which are stated below: 
1.  At least one automatic shutdown system shall be incorporated into the design. The 
provision of a second independent shutdown system may be necessary, depending on the 
characteristics of the reactor, and this shall be given due consideration. 
2. The effectiveness, speed of action and shutdown margin of the reactor shutdown 
system shall be such that the specified limits and conditions are met. 
3.  No single failure in the shutdown system shall be capable of preventing the system 
from fulfilling its safety function when required (e.g. with the most reactive shutdown 
rod stuck in the out position). 
4.  One or more manual initiations suitable for emergency shutdown may be necessary 
and this shall be given due consideration. 
Several activities related to normal research reactor operation involve safety 
evaluation. In principle, any activity that may influence neutronic, thermal-hydraulic and 
mechanical properties of the reactor should be supported by safety evaluation. Typical 
activities involving safety evaluation are as follows: 
1. Modifications of reactor components, power upgrading, new 
experimental setups, aging, etc. 
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2. Complete safety evaluation related to major changes in reactor system 
and resulting in safety analysis report modifications is normally performed by 
competent institutions (e.g. reactor manufacturer, national institutions, consultant 
companies and the like). 
The Nuclear Reactors Research Centre (NRRC) of National Nuclear Research 
Institute (NNRI) which operates GHARR-1 is undertaking steps to convert the reactor 
core. This is in response to the global trend in converting research and test reactors from 
the use of High Enriched Uranium (HEU) to Low Enriched Uranium (LEU) in civil 
nuclear application. As part of the Core Conversion of the GHARR-1 to LEU Core, the 
National Nuclear Research Institute (NNRI) – the Operating Organization of the reactor 
– is considering the introduction of a safety rod to augment the safety features of the 
GHARR-1.  
The first objective of this study is to design an LEU core with similar operational 
capabilities as the original HEU core and with acceptable safety margins under both 
normal and anticipated or design basis accident conditions. In order to provide 
comparisons between the proposed LEU core and the initial HEU core, thorough 
analyses must be performed for both cores. It was decided that the proposed LEU core 
consists of UO2   fuel elements clad in Zircaloy-4 alloy, the control element of the 
control rod material will remain unchanged but the diameter of the absorber material 
would increase, leaving the diameter of the control rod guide tube unchanged.  
The second objective is to design an auxiliary shut down rod for emergency 
shutdown of the reactor core thereby augmenting the safety of the GHARR-1 facility. 
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The auxiliary shutdown rod will replace the cadmium capsules in emergency situations. 
As earlier stated, this is in response to the recommendation made by experts during an 
Integrated Safety Assessment for Research Reactors (INSARR) Mission in 2009. 
Two changes that are required for the conversion are Reactor Facility changes 
and Operation License and Procedural Changes. 
1.1 Reactor Facility Changes 
For the Reactor Facility Changes, there is the need 
i. To replace the current UAl4 HEU fuel elements with UO2 LEU fuel 
elements enriched to 12.5 % in U-235. The dimensions of the LEU fuel elements 
are identical to the HEU fuel. In addition, the proposed LEU fuel has obtained 
approval from the China Institute of Atomic Energy, the manufacturers of the 
facility for use in non-power reactors such as the Miniature Neutron Source 
Reactors (MNSR). Section 1.3 provides a detailed description of the proposed 
LEU fuel. 
ii. To improve on the shutdown margin of the control rod by increasing the 
cadmium component.  
iii. To design means of emergency shutdown in case of control rod failure; 
either by an auxiliary safety rod placed in an irradiation channel or displacing 
part of annular beryllium reflector to accommodate the safety rod.   
1.2 Operation License and Procedural Changes 
For the operational license and procedural changes, the GHARR-1 operating 
license   shall have to be amended to allow possession of both the LEU and HEU fuel 
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inventories during conversion as stated in Section 17.3 of the facility’s Safety Analysis 
Report (SAR). Changes to the technical specifications that are significant include: 
1. Changing all references to fuel type from UAl4 in Al matrix to UO2 fuel. 
2. Review of the characteristics of the central control rod. 
3. Review of provision on amount of reactor fuel to be held by NNRI during 
the conversion exercise. 
4. Addition of auxiliary safety rod or displacement of part of the annular 
beryllium reflector. 
1.3  Characteristics of Proposed GHARR-1 LEU Core 
The GHARR-1 LEU core base model was established from the reference HEU 
core base model.  The core layout was kept the same. Only the components within the 
fuel cage (fuel pins, dummy pins, tie rods, and top and bottom grid plates) and the 
control rod design were different between these two reference models. All other core 
and reactor structure components are kept the same. A control rod design with a larger 
outer diameter for the cadmium absorber was also implemented for the LEU core as 
shown in Table 1.1.   
Uranium isotopes used in the GHARR-1 models are compared in Table 1.2.  
Comparison of key core parameters between the HEU and LEU models are shown in 
Table 1.3.   In an actual LEU core, adjustments to the excess reactivity are likely to be 
needed to compensate for differences between the reactor design data and the 
manufacturer’s as-built materials and geometry data. 
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Table 1.1 Control Rod and Guide Tube Design Parameters in MNSR Models 
Parameter (Dimension in mm) HEU LEU 
Outer diameter of cadmium 3.9 4.5 
Thickness of SS cladding 0.5 0.5 
Thickness of water gap 2.0 1.7 
Thickness of guide tube 1.5 1.5 
Material of guide tube LT-21 Zirc-4 
 
Table 1.2  Comparison of Reference GHARR-1 HEU and LEU Fuel Uranium 
Isotopes 
Uranium Isotopic, wt% HEU LEU 
U-235 90.2 12.5 
U-238 8.3 87.65 
U-234 1.0 0.2 
U-236 0.5 0.25 
Total 100.0 100.0 
Some of the most important of these differences are in the as-built U-235, U-234 
and U-236 loadings, and the as-built impurity levels in the fuel meat.  The adjuster rods 
would be made available to make these adjustments, depending on the results of the low 
power experiments that are planned at the CIAE before the LEU core is shipped to the 
customer’s site. 
 
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Table 1.3  Comparison of Key Parameters for Reference GHARR-1 HEU and LEU 
Cores 
Key Parameters HEU LEU 
Fuel Meat U-Al4 UO2 
U-235 Total Core Loading, g ~ 998 ~ 1358 
U-235 Enrichment, wt % 90.2 12.5 
U-234 content, wt% 1.0 0.2 
U-236 content, wt% 0.5 0.25 
Density of Meat, g/cm3 3.456 10.6 
Meat Diameter, mm 4.3 4.3 
Cladding Diameter, mm 5.5 5.5 
Thickness of He Gap, mm None 0.05 
Cladding Material Al-303-1 Zirc-4 
Number of Fuel Rods 344 348 
Material for Grid Plates LT-21 Zirc-4 
Top Shim Tray (not modeled) LT-21 LT-21 
Number of Dummy Elements 6 2 
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Material for Dummy Elements Al-303-1 Zirc-4 
Number of Tie Rods 4 4 
Material for Tie Rods Al-303-1 Zirc-4 
Adjuster Guide Tubes 4 4 
 
Ghana, a member of the IAEA and a signatory to various conventions, is to use 
nuclear science and technology for peaceful applications. The Ghana Atomic Energy 
Commission (GAEC), and for that matter NNRI, is poised to support any advocate or 
initiative that intends to minimize the likelihood of proliferation of nuclear materials. 
Hence the Commission is putting in all efforts to make the reactor core conversion a 
success.  
1.4 Outline of Thesis 
The literature review is presented in Chapter Two with reference made to the 
paradigm shift of research reactors from HEU cores to LEU cores. 
In Chapter Three, nuclear theory and the required reactor physics as well as 
thermal hydraulics analyses are highlighted. 
In Chapter Four, the 3-D Monte Carlo methodology developed for GHARR-1 
reactors using the versatile MCNP transport code and upon which this work is based is 
described. Three other codes used are also reviewed. 
The results of the neutronics and thermal hydraulics analyses performed for the 
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conversion of GHARR-1 from the use of HEU to LEU has been highlighted in Chapter 
Five of this thesis. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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CHAPTER TWO 
2 LITERATURE REVIEW 
2.1  General 
The conversion of MNSRs is being performed as part of the Global Threat 
Reduction Initiative (GTRI) – Conversion Program of the United States Government at 
Argonne National Laboratory and the IAEA’s Coordinated Research Project on 
Conversion of Miniature Neutron Source Research Reactors to Low Enriched Uranium 
[9]. The project started in 2006 and various neutronics, thermal hydraulic and numerical 
computations have been performed. 
The Reduced Enrichment for Research and Test Reactors (RERTR) Program 
develops technology necessary to enable the conversion of civilian facilities using high 
enriched uranium (HEU) to low enriched uranium (LEU) fuels and targets. The RERTR 
Program was initiated by the U.S. Department of Energy in 1978. During the Program's 
existence, over 40 research reactors have been converted from HEU (= or >20% U-235) 
to LEU (< 20% U-235) fuels, and processes have been developed for producing 
radioisotopes with LEU targets. The RERTR Program is managed by the Office of 
Nuclear Material Threat Reduction in the National Nuclear Security Administration 
(NNSA). The main technology components of the program includes the development of 
advanced LEU fuels and design and safety analysis for research reactor conversion. 
2.1.1 Similarity with Other Converted Facilities 
In the framework of the IAEA Coordinated Research Project entitled 
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“Conversion of MNSRs to LEU”, it has been established that it is possible to convert 
GHARR -1 to UO2 LEU fuel [10]. The CIAE has successfully commissioned a variant 
of the MNSR with LEU core, called the In-Hospital Neutron Irradiator (IHNI) designed 
exclusively for the treatment of cancer based on the boron neutron capture therapy 
(BNCT) technique [11].  
Over 60 research reactor facilities worldwide have been converted to LEU since 
the 1980s due to international concerns on nuclear proliferation. In 1985, the first LEU-
fuelled SLOWPOKE-2 reactor was commissioned at the Royal Military College of 
Canada (RMC) in Ontario [12]. The reactor at Ecole Polytechnique, Montreal was 
installed in 1976 and operated with HEU fuel for 21 years [13]. In 1997 the tray of 
beryllium shim-plates was full and there was no remaining excess reactivity. The 
exhausted fuel was removed and replaced with LEU fuel in the same beryllium reflector, 
making it essentially identical to the reactor at RMC. Other conversions include the RPI 
reactor in Lisbon, Portugal in 2007 [14], Kyoto University Research Reactor in 2010 
[15] and the HFETR near Chengdu around 2008 [16]. 
In the U.S., research reactors at Washington State and Oregon State Universities 
were recently converted to use LEU fuel rather than HEU.  Research reactor conversions 
were also completed at Purdue University in September 2007 and at Texas A&M and 
the University of Florida in September 2006. 
2.1.2 Advocates of Core Conversion 
The Department of Energy's National Nuclear Security Administration (NNSA) 
has successfully converted a research reactor at the University of Florida from the use of 
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highly enriched uranium to low enriched uranium [17]. This conversion comes on the 
heels of a reactor conversion at Texas A&M University. The NNSA had converted over 
six U.S. and international research reactors. 
As a part of its nonproliferation mission, NNSA converts research reactors in the 
U.S. and around the world from operating on highly enriched uranium (HEU) to low 
enriched uranium (LEU) fuel. The LEU is not suitable for use in a nuclear weapon and is 
not sought by terrorists or criminals. The conversion is part of the Bush administration's 
efforts to minimize the use of highly enriched uranium in civil applications around the 
world. 
HEU is primarily used in research reactors to produce isotopes for medical 
applications, and early reactor technology used HEU fuel because it was more difficult 
to achieve comparable power levels using LEU. However, modern reactor designs can 
use newer high-density LEU fuels while maintaining comparable power levels, making 
conversion an attractive option for limiting the availability of HEU nuclear material. 
The NNSA worked closely with the Department of Energy's Office of Nuclear 
Energy, the U.S. Nuclear Regulatory Commission, and the University of Florida to 
complete this reactor conversion. On October 20, 2006 Senior Officials from NNSA 
joined others in Gainesville to commemorate the successful fuel conversion of the 
reactor. 
Under the 2005 North American Security and Prosperity Partnership, the United 
States, Mexico, and Canada agreed to convert civil HEU reactors on the North American 
continent to LEU fuel by 2011, where such LEU fuel is available. The University of 
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Florida research reactor is the second of six domestic research reactors the United States 
will convert. Mexico will convert its one research reactor in Mexico City, and Canada 
will convert three research reactors. 
This reactor conversion also supports the 2005 Bratislava Joint Statement on 
Nuclear Security Cooperation issued by President Bush and Russian President Vladimir 
Putin. Under the statement, the United States and Russia agreed to work together to 
convert more than 30 U.S. and Russian-supplied research reactors around the world 
from the use of HEU to LEU. 
GTRI mission includes returning and securing nuclear fuel and radiological 
sources, protecting radiological and nuclear material and converting research reactors 
around the world. Currently, GTRI is working to convert 59 more reactors around the 
world from HEU to LEU by 2014. 
Established by Congress in 2000, NNSA is a semi-autonomous agency within 
the U.S. Department of Energy responsible for enhancing national security through the 
military application of nuclear science.  NNSA maintains and enhances the safety, 
security, reliability, and performance of the U.S. nuclear weapons stockpile without 
nuclear testing; works to reduce global danger from weapons of mass destruction; 
provides the U.S. Navy with safe and effective nuclear propulsion; and responds to 
nuclear and radiological emergencies in the United States and abroad.  
One of the most practical ways to prevent nuclear terrorism is to secure the 
materials necessary for making a weapon of mass destruction [18]. That was the call to 
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action outlined in Prague in 2009, and again at the Nuclear Security Summit in 2010. 
Today, the message is resonating. 
Worldwide, nuclear materials that could be stolen and fashioned into a nuclear 
weapon exist in dozens of nations. However, ongoing efforts led by the National Nuclear 
Security Administration (NNSA) and supported by U.S. national laboratories are 
working to minimize the use of HEU. 
At Idaho National Laboratory (INL), engineers support the NNSA Global Threat 
Reduction Initiative’s (GTRI) Reactor Conversion Program, which converts domestic 
research reactors from operating with highly enriched uranium fuel to using LEU fuel. 
The LEU fuel is less than 20 percent enriched, and cannot be used to make a nuclear 
weapon. 
Engineers at INL integrate conversion activities including performing conversion 
analyses, fuel fabrication, and fuel replacement. During conversions, three parameters 
are considered, including (1) the new fuel will not significantly impact the reactor’s 
performance, (2) no major modification to the reactor is required, and (3) there is no 
significant increase in operational costs. The laboratory is also developing a high-
density, low-enriched uranium fuel that can be used in global high-power reactors. 
 The INL collaborates with the Nuclear Regulatory Commission, national 
laboratories, universities, international partners, and commercial entities for both reactor 
conversions and fuel development. 
Other interesting information includes:  
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• To date, reactors at the University of Florida, Texas A&M, Purdue University, 
Oregon State University, Washington State University, University of Wisconsin, and 
INL have been converted. 
• INL researchers are developing a high-density LEU fuel by leading the design, 
fabrication technology development, irradiation testing, and fuel performance modeling. 
• More than 300 fuel plates have been irradiated at INL to support development 
of a uranium-molybdenum-based alloy fuel design that can be qualified for the high-
power reactors. 
2.2 GHARR-1 Reactivity Management 
The core consists of 344 cylindrical fuel elements, which form a fuel cage. The 
cage is located inside an annular beryllium reflector and rests on a lower beryllium 
reflector plate [2]. The volume of the vessel which accommodates the core is 1.5 m3. 
The fuel elements are all enriched uranium-aluminium (U-Al4) alloy extrusion clad with 
aluminium. They are arranged in 10 multi-concentric circle layers at a pitch distance of 
10.95 mm. The fuel cage consists of 2 grid plates, 4 tie rods and a guide tube for the 
control rod. Screws connect the 2 grid plates and 4 tie rods. The total number of lattice 
positions is 354 and the number of fuel elements is 344. The remaining positions are 
occupied with 6 dummy aluminium elements. 
Ten circles of holes are concentrically arranged in the upper and lower grid 
plates to form a lattice.  
Table 2.1 contains the characteristics of the arrangement for the lattice. 
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The annulus and lower reflectors are spaced to form the lower orifice, which 
controls water flow through the core. The top plate of the core and annulus are spaced to 
form the upper orifice. An aluminium tray holds the upper reflector, which contains 
semicircular beryllium shims which are added approximately once a year to compensate 
for fuel burn-up and samarium (Sm) poisoning. 
The top reflector is of variable thickness and assembled by stacking semi-circular 
plates within an aluminium tray. It is composed of a group of semi-circular beryllium 
shims with internal diameter of 243 mm. Four different thickness sizes are available, 
namely 10 pieces of 1.5 mm, 20 pieces of 3.0 mm, 8 pieces of 6.0 mm and 8 pieces of 
12.0 mm. 
Table 2.1 Lattice positions for the fuel cage 
Circle no. Lattice no. Circle diameter (mm) Rod pitch (mm) 
0 1 0.00 0.00 
1 6 21.90 11.47 
2 12 43.80 11.47 
3 19 67.70 10.86 
4 26 87.60 10.98 
5 32 109.60 10.78 
6 39 131.40 10.58 
7 45 153.30 10.70 
8 52 175.20 10.58 
9 58 197.10 10.68 
10 65 219.00 10.58 
 
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Long-term reactivity control is exercised by periodically increasing the thickness 
of this reflector to compensate for reactivity loss caused by fuel burn-up and samarium 
poison. Under normal operating conditions of the reactor, the top shims need to be added 
less frequently than once every one and half years. The maximum thickness of top shims 
is 109.5 mm for a cold, clean reactor, which is equivalent to 18 mk. In addition to the 
initial excess reactivity of the core, the presence of the shims ensures that the core life of 
the reactor fuel elements shall be longer than 10 years.                                
2.2.1 Reactivity worth of top beryllium reflector plates 
The core excess reactivity gradually decreases as 235U is consumed and poisons 
build up in the fuel. Compensation for this decrease is accomplished by increasing 
periodically the thickness of the top reflector. This is accomplished by adding small 
beryllium shim plates of known reactivity worth. Specially authorized persons using 
appropriate procedures will perform this operation [19]. The typical plot of measured top 
reflector reactivity worth against thickness of reflector plates is shown in the results 
given in Chapter 5 of this thesis. As can be observed from the plot when the core burn-
up proceeds, the top reflector thickness increases correspondingly and the specific worth 
of additional beryllium decreases. The graph is used to select the correct thickness to be 
added to the aluminium tray when the core excess reactivity drops just below 3 mk. 
 
 
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2.2.2 Comparison of the HEU and LEU fuel characteristics  
For better fuel characteristics, UO2 is the preferred option for the conversion.    
Comparison of some characteristics of the two fuels in question is shown in the Table 
2.2. 
Table 2.2 Characteristics of the GHARR-1 fuel element & dummy  
          Item  HEU LEU 
Fuel/Cladding material  U-Al alloy/Al alloy 
(LT-21) 
UO2 / Zirc-4  
Uranium wt. (%) / U-235 Enrichment (%)  27.5/90.2 12.5 
Density (g/cm3) 3.456  10.6 
Diameter (mm)/Length (mm) 4.3/230.0  4.3/230.0 
Uranium loading of each rod (g) 2.88   
No. of fuel rod positions/fuel rod loading 350/344 350/348 
No. of dummy elements/thickness (mm) 6/0.6  2/0.6 
Dummy element material  Al   Zirc-4 
 
2.2.3 Xenon poisoning calculations 
Xenon poisoning is small for low power reactors such as GHARR-1. The 
equilibrium xenon poisoning has been determined to be 5.2 mk [20]. Since the core 
excess reactivity is only 4 mk, the operation of the reactor is strongly influenced by 
xenon. This means that the reactor can only be operated intermittently. After operating 
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the reactor for a period of 2.5 hours at the rated power, the reactor will be shut down 
because of the xenon poisoning and can re-start only after xenon decay. Under the 
conditions of maximum flux of 1 ×1012 n/cm2.s in the inner experimental tube, a cold 
excess reactivity of 4 mk and initial core temperature of 15.8 °C, the maximum operable 
time is about 7.25 hours per day [21]. By lowering the operating power, the maximum 
operating time can be increased. The variation in the equilibrium xenon poisoning as the 
average core flux changes can be seen in Table 2.3. 
Table 2.3 Equilibrium xenon reactivity as a function of core average flux 
Core average flux, 1012 n/cm2.s 1.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 
Equilibrium xenon, mk 0.6 2.2 2.7 3.25 3.8 4.2 4.7 5.0 
The trend of fuel management for seventeen years of successful operations: 3 
mm thick of beryllium shim have been added to the shim tray on three different 
occasions. These were added in February, 2002; April, 2009 and January, 2011. All 
three were added when the reactivity had fallen below 2.5 mk. Regulatory rods are 
sometimes used to fine-tune the reactivity. 
Wet Storage of Spent Fuel could be done in the pool. For this reason, the reactor 
vessel is not placed at the centre of the pool but rather placed in order to make room for 
the spent fuel as show in Figure 2.1. In the figure shown, the cross sectional plane cuts 
through the centre of the reactor pool as well as the centre of the reactor vessel. The 
code used for the burnup analyses is described in Section 3.7.2 of this thesis. 
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Figure 2.1  Schematic of Reactor Pool (Cross sectional plane through centres of reactor            
pool and reactor vessel)  
2.3 Central control rod of GHARR-1 
The reactor assembly consists of the reactor core, beryllium (Be) reflector, small 
fission chambers for detecting neutron fluxes, one central control rod made of cadmium 
(Cd) and its drive mechanism 
The reactor is designed to have self-limiting power excursion characteristics. 
Only 1 control rod is at the centre of the reactor core. A fail-safe principle is adopted in 
the design of the reactor control system. A single cadmium rod is used for regulating the 
power level, compensating for fuel consumption, startup and shutdown of the reactor. 
The control rod drive mechanism is mounted on the top plate of the container.   
The control range of the automatic control system is designed to cover 108 -
1012n/cm2 .s. The control accuracy is ±1% of the maximum value of the range. The 
design control rod worth of the reactor is 6.8 mk and the shutdown margin is 3.0 mk for 
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maintaining the reactor in safe shutdown conditions. The total cold excess reactivity to 
be compensated is about 4.0 mk by the control rod. 
As stated earlier, there is only one central rod for GHARR-1 serving as shim, 
regulation as well as safety rod. The rod is installed to the control rod mechanism. The 
total cold excess reactivity to be compensated is about 3.5 - 4.0 mk. The central control 
rod has a worth of   at least 6.8 mk. The shutdown margin is 3 mk. Thus, it can be used 
to maintain the reactor in a safe shutdown condition.  
The control rod has a total length of 440 mm and its diameter is 5 mm. The 
length of the control rod absorber is 226 mm and its diameter is 3.9 mm. The thickness 
of the stainless steel cladding is 0.5 mm. Results of zero-power critical experiments have 
indicated that the movement of the central control rod will have no effect on the flux 
distribution at the positions of both inner and outer irradiation sites. This characteristic is 
very beneficial for neutron activation analysis and production of short-lived 
radioisotopes. 
2.3.1 Control rod drive mechanism  
The control rod drive mechanism is mounted on the top cover plate of the reactor 
vessel. It consists of a small AC servomotor, an electromagnetic clutch, a gearbox, a 
winch drum linked with the steel rope and an autosyn.  
The servomotor and electromagnet are energized when the power is applied to 
the motor and to the ‘Protection Ready’ on the control console. For this case the 
electromagnetic clutch is closed. When receiving the signal for control rod withdrawal 
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or insertion, the motor will rotate clockwise or anti-clockwise; this rotation drives the 
gear and this operates the winch drum. The drive mechanism moves the control rod up 
or down at a speed of 8.7 mm/s. For off-normal conditions when the power supply to the 
clutch is cut off, the clutch immediately opens and the control rod, weighing 1 kg, will 
fall rapidly into the core. By so doing, fast reactor shutdown is achieved. There are limit 
switches provided in the control rod drive mechanism to limit movement of the control 
rod. The switches are connected to a circuit on the main control console, which indicates 
the rod position either on the top or bottom of the core through the display lights. The 
autosyn of the drive mechanism is connected to the rod position indicator on the main 
control console to show the exact position of the control rod in the core. Procedures for 
the maintenance of the control rod drive mechanism are given in reference [22].   
 The control for the GHARR-1 reactor is achieved mainly by the withdrawal and 
insertion of the control rod located in the centre of the reactor core. This control rod is 
used both as safety and regulating rod. The main parameters are: 
i. The control rod with its balance weight is 1 kg, 
ii. Working distance of the rod is 230 mm, 
iii. Manual rod up and down speed ~9 mm/s, 
iv. Quick rod dropping time for the whole distance <11 s, 
v. Position indicating error ± 1 mm. 
The drive mechanism consists of three parts: 
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i. Drive mechanism - It consists of AC servo-motor, electromagnetic clutch, 
compact gearbox, multi-turn potentiometer, water seal tube, guide tube cover, etc. 
ii. Control Rod Position Indicator - The remote indication is achieved owing to the 
characteristics of the synchronisation between the sending autosyn and the receiving 
autosyn. The position indicator is installed in the central part of the vertical panel on the 
control console.  
iii. Control Rod - The control rod consists of a 3.9 mm diameter cadmium rod with 
stainless steel cladding. Its weight, including the balance is 1 kg.  It is installed in the 
central lattice hole in the active area and hanged by a wire rope. It is connected to a 
winch of the drive mechanism through the water seal tube, the position limit tube and 
the guide tube. Its worth is  7mk. 
If the drive mechanism of the control rod is out of order and the control rod 
cannot be inserted into the core for reactor shutdown, the irradiation system described in 
Section 10.3.1 may be used to shut down the reactor. To do this, 4 cadmium rabbits are 
inserted into the inner irradiation sites to provide negative reactivity worth greater than 4 
mk. If the transfer system is operating, the rabbits can be inserted pneumatically. If the 
transfer system is inoperable, the sample rabbits can be inserted manually by opening 
the plastic cover of the reactor, disconnecting the plastic tubing from the top of the 
aluminium rabbit tube, and inserting a rabbit or a string of rabbits into the tube. 
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2.3.2 Measurement of the central control rod worth 
In order to determine the shutdown margin of GHARR-1, the worth of the 
central control rod of the reactor was measured by means of the in-hour method [23]. 
The overall travel length of the control rod in the reactor core is 230 mm and its total 
worth is at least 6.8 mk. When the central control rod is totally inserted into the core, the 
reactor shutdown margin is therefore at least 2.8 mk. 
In conclusion, the Operating Organization of the Ghana first research reactor has 
decided to undertake the necessary studies, first by characterizing the HEU Core and 
subsequently, seek the optimum LEU Core characteristics that can replace the current 
HEU core.  
The theoretical analysis of this thesis is presented next in Chapter Three. 
 
 
 
 
 
 
 
 
 
 
 
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CHAPTER THREE 
3 THEORETICAL ANALYSIS 
The technical and the design specification of reactor core determine the 
behaviour of the neutrons in the reactor and hence the associated criticality conditions 
[24]. This is an essential requirement of reactor design. In particular, the design of a 
nuclear reactor is based on the interaction of several variables including fuel type, the 
moderator and reflector, core geometry and composition, radiation shielding, thermal 
hydraulics, criticality and reactor safety.  
In the next and subsequent sections, a brief review is made of the basic nuclear 
theory and the required reactor physics as well as thermal hydraulics analyses.  
3.1 NUCLEAR REACTOR THEORY  
In a steady state radiation field, the net rate at which particles are lost in an 
element must exactly be balanced by the rate of secondary or source particles which are 
introduced into the volume. The linear particle transport equation or linearized 
Boltzmann transport equation is used to determine the particle distribution in space, r 
direction Ω and energy, E in plane, spherical or cylindrical geometry. The equation is 
the basic physical model to describe the transport of uncharged particles such as 
neutrons and photos (γ - rays, light, etc.) 
The transport equation can be derived by simple balancing of the various 
mechanisms by which particles can be gained of lost from an element in a phase-space, 
(r, Ω, t) i.e. 
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time rate of change due to change due
particle density physical leakages to collision sources
change
       
                
                                 
(3.1)                                                  
which can be expressed mathematically as: 
' ' ' ' ' '
0 4
1
( , , , ) . ( , , , ) ( , , , )
( , , ) ( , , , ) ( , , , ) ( , , , )t s
r E t r E t S r E t
t
r E t r E t dE d r E E t r E t

 

 


      

        
                          (3.2)  
where               
' '( , , , )r E t  = the expected number of particles per unit time at r with energies in 
unit energy about E and moving in a unit angle about Ω,  
( , , , )S r E t = the number of source particles emitted in a volume dV travelling in a 
cone of direction dΩ about Ω with energies between E and dE  
Σs(r, E΄→E, Ω΄
 → Ω) = macroscopic scattering cross-section which is the probability 
that the particles at position r with energy E΄ travelling in the direction Ω΄ scatter into dE 
about E and into the cone of direction dΩ about Ω. 
The corresponding steady state equation is      
  ' ' ' ' ' ' '
' '
. ( , , ) ( , , ) ( , , )
( , , )
t sr E dE r E E r E d dE
S r E
          

      (3.3)                                                      
The equation is solved for shielding problems, including deep penetration and neutron-
gamma coupled problems as well as multiplying media problems in the reactor theory, 
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including criticality search and others. 
For neutrons besides a scattering reaction, a collision of neutrons with a host 
medium can generate other neutrons by fission (multiplying medium). Equation 3.3 is 
written to describe the system containing fissile nuclides which can be regarded as sub-
critical, critical or supercritical. The criticality problem is approached by introducing the 
effective multiplication factor to balance the fission source with other terms in the 
transport equation such that neglecting external source of Equation 3.3 becomes:   
' ' ' ' ' ' '
0
4
' ' ' ' ' '
0
4
. ( , , ) ( , , )
1 ( )
( ) ( , ) ( , , )
4
t s
f
r E E d dE r E d dE
E
v E r E r E d dE
k


  





          
  
 
 
    (3.4)          
The neutron transport equation is a linearized version of the Boltzmann equation based 
on neutron conservation and vector calculus. The “integro-differential” transport 
equation is linear with seven independent variables [25].  
3.2 Solution of the Neutron Transport Equation  
Generally, the particle transport equation is difficult to solve analytically except 
in simple cases with numerical methods. The transport equation is thus solved 
analytically for limited cases only. For estimation of particle flux it is approximated by 
solving it numerically using various techniques [25].  
Most reactor studies assume that neutrons diffuse from regions of high to low 
density, akin to the diffusion of heat from high to low temperature, or as gas molecules 
diffuse to reduce spatial variations in concentration. However, while diffusion of heat 
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31 
and gases is quite accurately modeled, the treatment of neutron transport as a diffusion 
process is quite limited.  In conventional diffusion, particle trajectories are irregular, 
random, and zigzag-like; yet, in neutron transport, low reaction cross-sections (~1 barn, 
10-24 cm2), lead to long mean free paths, with dimensions characteristic to the transport 
medium (inhomogenieties). Thus, a more accurate description of neutron transport is 
required. 
Neutron diffusion equation follows from integrating out the angular dependence 
of the neutron flux, among other approximations. Recent advances in computational 
power and methods provide an enormous opportunity for improved and more accurate 
computational solutions to the neutron transport/diffusion equations and problems. 
The two general approaches to solving the neutron particle transport include                                      
- Deterministic methods (Finite Difference, Finite Element, Discrete Ordinates): These 
methods include discretization of variables, replacement of the mathematical equation 
with coupled linearized algebraic equations and solution of the system of equations with 
numerical methods. 
- Stochastic methods (Monte Carlo): Stochastic methods are statistical and describe a 
physical system by its corresponding probability density functions and also utilize 
numerical methodologies to simulate the probability density functions.   
3.2.1 Diffusion Theory Approximation 
As per the previous discussion, diffusion theory approximation and Monte Carlo 
approaches of the neutron transport theory are used in reactor analysis and design. In 
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32 
particular, diffusion theory approximates the transport theory and provides an exact 
solution and analysis. In this section, the diffusion equation for describing the behaviour 
of neutrons in a reactor system is presented.   
The neutron conservation equation indicates that the net rate of change of the 
neutron density, n, with time t in a given nuclear system is balanced by the rate of 
neutron production from sources, loss rates by absorption and leakage per unit volume, 
i.e.,  
 
i.e., n Sources Absorption Leakaget
        
Mathematically, the above can be expressed as  
a
n S Leakaget 
   
               
(3.5)
            
 
where 
a   is the absorption term and a  is the macroscopic cross-section for neutron 
absorption. 
Calculation of the neutron leakage term is based on the use of a physics 
parameter called the neutron current density, J. In particular, the neutron current density 
is defined as the defined as the net number of neutrons crossing a unit surface area in a 
given direction per unit time. In a system defined in a 3-D configuration in x, y, and z-
directions, the neutron current density will have corresponding Jx,  Jy , and Jz 
components. Then the rate of loss of neutrons per unit volume in the system in the x, y 
and z-directions can be expressed as  
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x-direction 
xJ
z
 
,   y-direction yJ
z
    
and  z-direction 
zJ
z
 
 
Thus, the net rate of loss (or leakage) of neutron per unit volume 
 
yx zJJ J
x y z
     
divJ  .J           (3.6) 
Using the above definition, Equation 3.5 reduces to   
       , , , . ,an r t S r t r t J r tt 
   
                  
(3.7)
 
The diffusion equation is derived via Fick’s Law of Diffusion. Accordingly, the net rate 
of neutron flow in a given direction is proportional to the (negative) spatial gradient of 
the neutron flux. Mathematically, 
J D                          (3.8) 
 where D is the diffusion constant  
Equation 3.7 can then be re-written, using Equation 3.8 as   
       , , , . ,an r t S r t r t D r tt  
       
 
    
     . , , ,aD r t r t S r t                    (3.9) 
                                  2. , ,D r t D r t                             (3.10) 
                           
where 2 represents the Laplacian Operator and 
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2 2 2
2
2 2 2x y z
          
 
The diffusion equation, (Equation 3.9) can then be commonly written in the form 
       2, , , ,an r t D r t r t S r tt  
    
  
                   (3.11)
 
For steady state conditions, 
0nt
 
 
The diffusion equation in steady state condition becomes  
     2 0aD r r S r                    (3.12) 
In the absence of a primary (or independent) source, the external source, S, is given by 
aS k                                   (3.13) 
where k is the infinite neutron multiplication factor.  
3.2.2 Multigroup Calculations 
The energy of neutrons produced in nuclear fission range up to about 10 MeV. In 
a nuclear reactor, these neutrons are slowed down by scattering collisions with atomic 
nuclei until they are thermalized. In the thermal energy region (below about 1 eV), the 
neutrons exchange energy with the moderator atoms, so that up-scattering as well as 
down-scattering can occur. Consequently, the neutron energies in a reactor core range 
from about 10 MeV to 0.001 eV or less. 
Reactor analysis is performed in favour of reactor design, neutronics calculations 
and other important particle transport activities in nuclear reactors. Since the neutron 
energy varies, it is impractical in reactor analysis to treat the neutron energy as a 
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35 
continuous variable. Consequently, the energy range of interest is divided into a finite 
number of discrete energy groups as shown below in Figure 3.1. 
 
Figure 3.1 :  Partial distribution of neutron energies into G groups  
where  E0 > E1 > E2 > - - - > Eg-1 > Eg > - - - > EG. 
The larger the number of energy groups, the more closely will the group energy 
distribution approach the actual neutron energy spectrum. However, increasing the 
number of groups also increases the complexity of the computation. Hence, the number 
of energy groups selected depends on circumstances and represents a compromise 
between accuracy on one hand, and the complexity of the calculations on the other hand 
[26]. 
To set up the multigroup equations for a critical reactor, three group constants 
are first be defined. These are 
 Σfg  = group-average macroscopic fission cross-section for group g 
υg = average number of fission neutrons released as the result of nuclear 
fissions induced by neutrons in the gth group: and  
 χg  = fraction of fission neutrons that are emitted with energies in the  
  gth group.  
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The number of fissions per cm3/s in the hth group can now be written as 
fh h  where φh 
the flux in the hth group. As a result of these fissions, υh Σfhφh neurons are released. The 
total number of neutrons emitted as the result of fissions in all groups is then              
                                             
h hf hn    
                                        (3.14) 
where the sum is carried over all groups. If the fraction χg of these neutrons appears in 
the gth group, it follows that the source term for the gth multigroup equation is  
    
1
N
g g h fh h
k
S   

       (3.15) 
and the gth group equation in a set of N multigroup equations is 
1
2
1 1 1
0
gN N
g g ag g g h g h g h g h fh h
h g h h
D       

 
   
               (3.16) 
For a typical reactor, which has several regions of differing material properties, 
there is a set of equations like those represented by Equation 3.16 for each region. In this 
case, it is necessary to solve the equations in each region and satisfy boundary 
conditions at every interface as well as at the reactor surface.  
3.3 Generation of Group Constants 
Before an attempt is made to solve Equation 3.16, the flux weighted quantities, 
Dg, Σtg, Σg’→g and υg Σfg known as group constants must be determined. The group 
macroscopic cross-section is expressed as  
 
g g
g
= (E) (E)dE   
                                (3.17)a     
 or  
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37 
           
g
g g
1= (E) (E)dE 
                    (3.17)b  
where  g  is the group flux and defined by  
      
g
= (E)dE g 
                                           (3.18) 
and  (E)   is the neutron flux per unit energy interval at energy E within the group and 
the integration is carried over the energy range of the group g, i.e. from Eg-1 to Eg. The 
evaluation of the group cross-sections requires knowledge of Σ and of   as functions of 
the neutron energy for all nuclides present.  
A simpler procedure is based on the assumption that, even though the number of 
microgroups is large over a small range,  (E) may be taken as constant and Equation 
3.18 then reduces to               
         E  ≈ (constant)  Eg      (3.19)                                                                                                                  
where 
1g g gE E E   and Equation 3.18 becomes  
                                            1 ( ) ( )g
g g
E E E dEE   
                                 (3.20) 
thus Σg  can be evaluated. In general, the group constants that Equation 3.20 is applicable 
are for absorption, scattering and fission.   
The group diffusion coefficient reduces to the form 
    1 ( )g
g g
D D E dE 
1 ( )
g g
D E dEE  
                             (3.21)  
assuming  (E) is taken as constant over the group energy range.  
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38 
In calculating Dg, the quantity 1
3 t o sD E   
 is invoked to express D(E) in 
terms of cross-sections as functions of the neutron energy. The transfer (scattering) 
cross-section, Σsg→g1 is defined by 
   
1 1
1 1 11 ( ) ( )ssg g g gg
E E E dEdE    
   (3.22) 
where                                                                                                                      .                                                 
Σs (E
1 → E) is the differential scattering cross-section from energy E1 to E.                 
The group fission constant υg Σfg  is defined by 
                                 
1
1 ( ) ( ) ( )g fg f
g g
E E E dE    
                                     (3.23) 
                                            
1
1 ( ) ( ) ( )f
g g
E E E dEE    
                                         (3.24)                
3.4 Few Group (Macro-group) Constants 
Many group constants are collapsed into few group (macro-group) constants for 
determining the special distribution of the neutron flux. For water-moderated reactors, 
four groups are commonly used. These are: 
1.  Fast region – from 10 MeV to 0.05 MeV. The fission neutrons possess energies 
in this group and both elastic and inelastic scattering occur. 
2. Slowing-down region – from 50 keV to 500 eV. In this range slowing down is by 
elastic scattering and there may be absorption in unresolved resonance. 
3. Resonance region – from 500 eV to 0.625 eV. Slowing down occurs by elastic 
scattering and there is absorption in resolved resonances. 
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4. Thermal region – below 0.625 eV. Both down-scattering and up-scattering can 
occur in addition to absorption. 
For some reactor analysis, 3 or 2 neutron energy groups are sufficient. For 
graphite-moderated reactors, more than four groups may be needed. For Ghana MNSR 
which is both water and beryllium reflected, four groups are adequate [25]. For the 
MNSR MCNP model, three groups are considered for the neutronic analysis. 
3.5 Monte Carlo Techniques 
Stochastic Monte Carlo methods are extensively used in nuclear applications 
such as radiation shielding, radiation transport, neutronics, reactor physics, design and 
analysis. Monte Carlo methods refer to statistical methodology wherein the expected 
characteristics of particles (e.g. particle flux) are estimated by sampling a large number 
of individual particle histories whose trajectories are simulated by a digital computer. In 
some cases, there are equations that adequately describe the behaviour of such systems 
and that can be solved either analytically or numerically.  
3.5.1 General Overview  
The basic advantage of Monte Carlo techniques over the deterministic 
techniques (e.g., numerical solution of the Boltzmann transport equation) is that Monte 
Carlo methods more accurately represent the geometry and nuclear data than do 
deterministic techniques. Deterministic methods require reasonably simple geometries 
for the numerical techniques to work and use multigroup group approximations to 
continuous energy neutron cross-section data [27]. 
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Additionally, Monte Carlo techniques can handle complex geometries and 
continuous cross-section data, as well as the simple geometries and multigroup data. In 
many cases, the geometry of a system is more complex than a cylinder or a stack of 
cubes; it often includes both cylindrical and planar surfaces. For these situations, Monte 
Carlo is a better technique as it statistically evaluates the system with few 
approximations rather than trying for a numerical approximation to the analytic 
description.  
The disadvantages of the Monte Carlo approach are that it is statistical in nature 
and does not provide an exact solution to the problem. All results represent estimates 
with associated uncertainties. Also, Monte Carlo techniques can be quite time 
consuming on a computer if very small uncertainties are required. The relationship 
between Monte Carlo and deterministic techniques can best be summarized as: 
deterministic techniques provide an exact solution to an approximation of the problem 
while Monte Carlo techniques provide an approximate solution to an exact 
representation of the problem. 
Monte Carlo methods can be used to duplicate theoretically a statistical process 
and is particularly useful for complicated geometries and complex problems that cannot 
be modeled with deterministic codes. The individual probabilistic events constitute 
processes and are simulated sequentially. The probability distributions governing these 
events are statistically sampled to describe the total phenomenon.  
In particle transport, the Monte Carlo technique is pre-eminently realistic and 
consists of actually following each of many particles from a source throughout its life to 
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its death in some terminal category such as absorption and leakage. Probability 
distributions are randomly sampled using transport data to determine the outcome at 
each step of its life. 
In a neutron field such as in the core of a nuclear reactor, when a neutron 
traverses a material, it interacts with the constituent atoms of that material. The neutron 
gets scattered or absorbed depending on the cross-sections of the material. These 
processes occur statistically in nature with the probability of occurrence determined by 
its cross-section. Whereas the distance of travel by one neutron particle cannot be 
predicted exactly in the material before interacting; the distribution of flight distances 
that a large number of those particles will have prior to the first interaction, can 
however, be predicted via the Monte Carlo approach.  
Using “random” numbers, a statistical history for the life of each particle (a 
random walk analysis) can be generated by a computer. That is, an individual particle 
may experience many scattering interactions before finally being absorbed or leaked 
from the system. Random numbers (a set of numbers which have no pattern and are 
sampled uniformly between zero and one) are used at each interaction to determine 
which process (absorption, fission, elastic scattering, etc.) occurs, how much energy is 
lost, what is the new direction of the particle (for scattering), or how many neutrons are 
released in a fission event.  
The life of a particle begins at birth, either from an external neutron source or 
from a fission event, and ends with absorption or with a scattering event that moves the 
neutron outside the assembly. The events that occur during a particle’s life are tabulated 
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as the particle’s histories. Because a single particle is usually not representative of the 
total system, a number of histories must be evaluated to accurately describe what occurs. 
3.5.2 Criticality Calculations 
Nuclear criticality is an important activity required in reactor analysis, reactor 
design, reactor operation and safety. Hence, the effective multiplication factor in Monte 
Carlo criticality applications is of primary interest. In these calculations, a group of 
neutron histories is often referred to as a keff cycle with the multiplication factor of the 
assembly defined by the ratio of the number of neutrons generated at the end of the keff 
cycle to the number of neutrons whose histories are evaluated in this cycle. The 
expected value of the multiplication factor is then estimated by averaging over the 
events in the keff cycle.  
The relative error in the estimate of the effective multiplication factor will 
usually decrease as the number of keff cycles increases. Thus, numerous criticality cycles 
are necessary to arrive at a good estimate. In addition, the first few cycles are inaccurate 
because the spatial neutron source has not converged. Because the distribution of source 
(fission) neutrons in a system is dependent on the Eigen-value of the system and on its 
geometry, it takes a number of inactive cycles for the Monte Carlo spatial neutron 
distribution to approach the converged distribution. For this reason, the first few cycles 
are ignored in the final estimate of keff.. The estimates of keff from the remaining cycles 
are averaged to obtain a mean value for the effective multiplication factor.  
In the MCNP Monte Carlo, if G criticality generations are evaluated and the first 
D cycles are discarded, then the estimated effective multiplication factor of the system is 
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given by 
  1
1 G
i
i D
k kG D  
  
              (3.25) 
where k is the estimated system multiplication factor and ki is the multiplication factor 
determined from the ith cycle. The repeatability of the estimate (i.e., if the same 
calculation is performed with different random numbers, how much different will the 
estimate of k be) is determined from the estimated standard deviation of the mean. The 
standard deviation of the mean is calculated using the standard deviation, σ, of the 
distribution of k-values. 
   
2
1
1
1
G
i
i D
k kG D  
   
       (3.26) 
For a valid Monte Carlo calculation, the range k  - σ to k  + σ should include the precise 
keff.  
The literature review on the associated nuclear reactor physics has been briefly 
limited to particle transport theory and the approximated diffusion theory as well as a 
discussion on the differences between deterministic and Monte Carlo approaches to the 
transport theory. As mentioned earlier, the MCNP Monte Carlo code was utilized in this 
work and the review is thus somewhat skewed towards Monte Carlo techniques. For 
purposes of brevity, the presentation made in this thesis about the MCNP code is 
however limited. Details of the physics, theory and mathematical models upon which the 
MCNP code is built can be obtained in the literature [27]. 
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3.5.3 Neutron Cross section Data Libraries  
The basis for the simulation of all fundamental process in MCNP, are the cross 
section and other nuclear data for neutrons and correlated secondary particles, which are 
organized in data libraries. Neutron-induced cross sections depend primarily on the 
nucleus with which the neutron is interacting, incident neutron energy and temperature 
of the medium. Cross section values are derived from experimental data and theoretical 
models if experimental measurement is not possible. Cross sections are summarized and 
organized in the Evaluated Nuclear Data Files (ENDF) [28]. To utilize the evaluated 
data libraries with neutronics codes such as the MCNP, the data is adequately formatted 
to suit the requirement of the code and the problem to be solved. For MCNP this task is 
done by processing code NJOY [29]. The processed nuclear data libraries retain as much 
detail from the original evaluations as is feasible to faithfully reproduce the evaluator’s 
intent. 
MCNP uses continuous-energy nuclear and atomic data libraries. Nuclear data 
tables exist for neutron interactions, neutron-induced photons, photon interactions, 
neutron dosimetry or activation, thermal particle scattering        etc.  In the case of 
neutron interaction, continuous-energy and discrete reaction data are available, the latter 
based on multi-group cross section. In addition to cross section data, the MCNP libraries 
also include angular distributions for scattering, fission yields, Q-values, etc. 
The calculations performed in this work are based on the cross section data 
distributed with the code and thermal particle treatment for hydrogen in light water. 
Each data table available to MCNP is listed on a directory file, XSDIR.  Specific data 
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tables may be selected through unique identifiers for each table, called ZAIDs. These 
identifiers generally contain the atomic number Z, mass number A, and library indicator 
ID. 
Since GHARR-1 is operated at relatively low pressure and temperature, nuclear 
data evaluated for a temperature of 300K are used in all simulations. Cross sections are 
required for all materials used in the model of the reactor. The materials include: 
structural materials, the coolant and the moderator and nuclides built-up in the fuel. 
Other sources of nuclear data apart from the Evaluated Nuclear Data File are 
Advanced Computational Technology Initiative (ACTI) [30],the Evaluated Nuclear Data 
Library (ENDL) [31], Evaluated Photon Data Library (EPDL) [32], the Activation 
Library (ACTL) [33] compilations from Livermore, and evaluations from the Nuclear 
Physics (T–16) Group [34] [35] [36] at Los Alamos. 
3.5.4 Thermal Scattering Matrix         
For low energies, the scattering behavior of neutrons depends on binding effects 
in the material and the possibility to excite vibrational modes, if the target nucleus is a 
low mass number (A) material embedded in a molecule or lattice. In this case, 
incoherent inelastic scattering strongly contributes to the thermal scattering cross 
sections of neutrons described as [29]: 
            
 
   
  
 
 
     
 
 
                  (3.27) 
where   and   are the energies of the incident and the scattered neutron respectively,   
is the cosine of the scattering angle in the laboratory system, and   is the integral cross 
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46 
section of the bound nucleus for the limit of low neutron energies at    . The 
scattering matrix S      is a function of the momentum transfer  and energy transfer , 
which are both defined as dimensionless quantities. 
   
           
   
    and     
    
  
                 (3.28)
      
For gaseous and unbound nuclei, MCNP explicitly calculates the scattering 
matrix        with the analytical expression. 
        
 
    
     
     
  
                   (3.29)                  
In all other occasions, i.e. for specified material compositions, the values of 
S      are tabulated in additional libraries and can be requested by the code. Since 
incoherent inelastic scattering is most relevant for materials with low mass number 
nuclei, MCNP provides tabulated data for hydrogen in light water, deuterium in heavy 
water, beryllium or graphite [37]. 
3.5.5 Mathematical Basis for Monte Carlo Simulation 
3.5.5.1 Generation of Random Numbers 
A computer is a precise and deterministic machine, and hence, the output of a 
computer is predictable and not truly “random”. The word “random” represents the 
output of a truly random physical process such as the decay of radioactive nuclei. In 
radioactive decay process, we cannot predict which nucleus is going to decay and when. 
However, we can say that after time  , how many nuclei would have decayed using 
probabilistic models. Computer generated sequences are “pseudo-random” number [38]. 
It is preferable to have a random number sequence that is reproducible. This way it is 
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47 
possible to reproduce a calculation. If every calculation uses different numbers, it is hard 
to make a distinction between a new result of the calculation and a change in result due 
to the new set of random numbers. 
A random number with a uniform probability distribution in the unit interval 
      is generated and any number in the unit interval generated is equally likely to be 
picked. Most commonly used methods for generating sequences   of numbers from a 
uniform distribution are based upon a mixed congruential relationship [39] [40]  given 
by Halton [41] and Marseguerra [42]: 
                              (3.30) 
The new number      is the remainder, modulo a positive integer  , of an 
affine transform of the old   , with a non-negative integer coefficient   and  . The 
sequence obtained is made up of numbers      and is periodic with period    . 
The sequences generated with Equation 3.30 are called pseudorandom numbers and are 
reproducible. The constants  ,   and  may be selected so that (Bart, 2009): 
i. The sequence pass statistical test for randomness 
ii. The period   is very large. 
In MCNP the period is typically 246 [43] [44]. In other words, 246, random 
numbers can be generated before the numbers start to repeat themselves. 
The numbers generated in Equation 3.30 is smaller than   so that when divided 
by  (   
    
  ), they must all lie in the interval        i.e Uniform PDF on      . 
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48 
3.5.5.2   Probability Density Function 
A Probability Density Function (PDF) is the mathematical expression of a 
stochastic physical law. In Monte Carlo, PDF is the input in which the underlying 
physical process is described [42]. 
Let   be a random variable (a variable that takes on a particular values with a 
frequency that is determined by some underlying probability distribution) that takes the 
value in the interval            . The likely hood of obtaining   in the interval is 
the ratio     of the number   of values   that fall within the interval and the total 
number   of generated   values, in the limit as   . A PDF,      is the probability 
that the outcome of a particular event will occur in    about   for continuous and 
discrete distribution. 
For continuous distribution, we can define 
                              (3.31)
    
On physical grounds,      must be everywhere non-negative and be normalized to a 
total probability of 1, i.e., 
                       and             
 
  
                    (3.32)
  
A function that satisfies Equation 3.32 is interpreted as PDF. The likely hood of 
obtaining   in the interval       is 
                   
 
 
       (3.33) 
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The Cumulative Distribution Function (CDF) for the continuous distribution is defined 
by 
          
 
 
                                  (3.34) 
where      is non-negative and monotonically increasing from  to  and it is 
continuous and differentiable at will 
              
     
  
                                   (3.35) 
The probability of having   in the interval       is 
     
 
 
                                    (3.36)
   
 For discrete distribution, the PDF of a random variable   is 
                                            (3.37) 
Where         is the Dirac Delta function       must be everywhere non-negative 
and be normalized to  . 
                        or                               (3.38)
   
The discrete CDF is 
                                                         (3.39) 
The uniform distribution is frequently used in Monte Carlo simulation since sampling 
from the distribution allows obtaining random variable obeying any other distribution. 
3.5.6 Generation of Random Variable  
In this section we discuss various methods for generating non-uniform random 
variables from a prescribed distribution. We will consider the inverse-transform method 
and the acceptance-rejection method which are of great importance to this work. 
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3.5.6.1 Sampling by Inversion of CDFs from Continuous Distribution 
Let          be a random variable with CDF         and PDF       namely [42]: 
        
 
  
                              (3.40) 
Where, 
         ,        ,    
 
  
            (3.41)
   
Since       is a non-decreasing function, for any         its inverse is defined as 
  
                                               (3.42)
  
We can obtain values         starting from values   sampled from the uniform 
distribution        . Since both       and       are increasing function of      , for 
each         there exists only one value 
    
               (3.43)
    
The   values computed are actually sampled from       i.e.        : for any 
        there is one   value such that        . We then have  
                 
                           (3.44) 
Applying the function       to the right hand side of 2.74, we get  
                                 (3.45) 
Also applying the function       to both sides of 2.75we have 
                  
 
  
                        (3.46) 
Equation 3.46 gives the fundamental relationship of the inverse transform method since 
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for any   value sampled from the uniform distribution         it gives the 
corresponding   value, which is sampled from the       distribution. 
A random variable   is said to be uniformly distributed in the interval       if 
the PDF and CDF is respectively given by 
   
 
   
for          
   
   
 for       
        for                          and     for                  (3.47)
    for            for     
Substituting into Equation 3.46 and solving with respect to   gives  
                                (3.48) 
Thus, to generate a random variable   with cumulative distribution function  , draw 
         and set     
     . In practical terms,   is generated from the cumulative 
distribution in the the following steps: 
i) Generate   from        
ii) Return           
 
3.5.6.2 Sampling by Direct Inversion of CDFs from Discrete Distribution 
Let  be a random variable that can only have discrete values   ,  
where                   with probability [42] 
                  ,              (3.49) 
Ordering the     sequence so that        , the cumulative distribution is 
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                      (3.50)  
where,      . The normalization condition of the cumulative distribution now is                        
                     (3.51) 
Following the scheme of the inversion method, in the discrete case, given a value 
  sampled from the uniform distribution, the probability that   falls within the interval 
           is 
                              
  
    
            (3.52) 
that is, for any         , we get the random variable      where   is the index for 
which          . 
Practically,   is sampled from the cumulative distribution    in the following 
steps: 
i) sampling an          
ii) find the smallest positive integer,  ,  such that     
iii) return to (the required value)     
3.5.6.3 Sampling of a Direction   from an Isotropic Distribution 
Consider sampling of a direction   from an isotropic distribution in 3-dimension. 
In polar co-ordinates the direction   is identified by         between   and the   axis 
and by the angle          between the projection of   on the    plane and the   
axis. We can now write [42]  
                        (3.53)
    
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where,       . The probability distribution function of the isotropic distribution is 
given by 
                   
    
  
                      (3.54) 
where, 
      
 
 
    and          
 
  
                        (3.55) 
The probability distribution function is the product of the two uniform 
probability distribution functions      and      . If    and    are two random variable 
       we have  
        
  
  
     
   
 
 and        
 
  
      
   
  
    (3.56) 
and solving Equation 3.56 we have   
          and              (3.57) 
     
Practically, the direction cosines      of  are obtained in the following steps: 
i) sampling of two random variables              
ii) computation of                     
iii) direction cosines: 
                        , 
                                 , 
        
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3.5.6.4 Acceptance-Rejection Method  
The inverse-transform and composition methods are direct methods in the sense 
that they deal directly with the CDF of the random variable to be generated. The 
acceptance-rejection method is an indirect method due to Stan Ulam and John von 
Neumann [45]. It can be applied when the Inverse transform method fail or turn out to 
be computationally inefficient. The rejection technique [46] [47] has the advantage that 
the sampling procedure can be carried out using only the PDF of the random variable. In 
the simplest form, a constant     and another density function      are assigned to 
     in such a way that: 
                                                      (3.58) 
for all values of  . First, a value for   is sampled from the distribution     . Another 
random variable  , is then sampled on the unit interval and the value of   is accepted as 
a sample from      if: 
          
    
     
                                                           (3.59) 
If the inequality does not hold, the value is discarded and the procedure is 
repeated from the beginning. There are certain properties that are desired from     . 
The function should be chosen in such a way that the sampling can be performed using 
the inversion method (or some other efficient technique). Another important aspect is 
that the difference between the functions       and      should be small, or more 
precisely, the ratio of the integrals: 
  
       
 
  
        
 
  
         (3.60) 
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should be close to unity as possible. The ratio is called the efficiency of the algorithm. If 
the efficiency is low, computing time is wasted as the re-sampling loop has to be 
repeated over and over again before finding an acceptable value for the variable. 
Sampling procedure: 
i) Sample   from     :           
ii) Test:               
If true accept  . 
If false reject   and try again. 
3.5.6.5 Exponential Distribution 
A system for which the probability of future events in time or space depends 
only on the present state of the system and not on the way this state was reached i.e. the 
future of the system does not depend on its past history is called Markovian system. A 
Markovian system is homogeneous when the events occur with constant probability 
density    . Starting from a given point - in time or space – the interval distribution for 
the occurrence of the next event is exponential with decay constant    . 
A random variable         is said to be exponentially distributed if its 
cumulative distribution function and its probability distribution function are respectively 
[42] 
           
      and         
 
 
         for        
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                                                                                        (3.61) 
Substituting into Equation 3.46 and solving for   gives 
                    
Since the random variable     is distributed as   we can write 
                                    (3.62) 
The algorithm for the generation of an exponential random variable is 
i) Generate          
ii) Return          as a random variable from        ) 
3.5.6.6 Multinomial Distribution 
Consider a random game such as throwing a dice and neutron nucleus interaction 
(scattering, absorption, fission, etc) once it is known that the interaction has occurred. 
Such a random game can have only   possible values, the probability of      one 
being   .  The game is simulated by first dividing the interval       in   successive 
subintervals of amplitudes            and then performing a large number of trials in 
each of which a random variable          is thrown on the interval      . Each time a 
point falls on the      subinterval, we say a     event out of the   possible ones 
has occurred. The probability of such event obeys the multinomial distribution. 
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3.5.7 Variance Reduction 
3.5.7.1 Tally Variance 
Two basic principles of statistics are used to determine the accuracy of 
calculation. They are the laws of large numbers and central limit theory [48]. When we 
run a random-walk simulation, the     history contributes a score    to the tally. If the 
particle (or its daughters) never reaches the tally region, then   =0, whereas, if reaches 
tally without interaction, the score   generally is very large. The probability any history 
will contribute a score between   and      is denoted by a probability distribution 
function        .In MCNP simulation, we seek the mean score (or expected value) of  , 
given by 
                         (3.63) 
Unfortunately, we do not know     . Instead, we estimate     by the mean   of the 
scores of N particles, i.e.  
       
 
             (3.64) 
As    , the law of large numbers guarantees that       , provided     is finite. 
 The variation in the different scores    is measured by the standard deviation of 
the population (histories), which for large   
     
 
 
    
 
      
              (3.65) 
where 
       
 
 
   
  
                   (3.66) 
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The estimated variance of the mean    is then 
    
  
 
 
           (3.67) 
The central limit theory states that if we repeated the simulation a large number of times 
(each with   histories) the variation of the means   from each simulation will be 
distributed normally about the true mean    and have a variance    
 . It is this uncertainty 
or variance we are trying to reduce in MCNP simulations, i.e. for a fixed number of 
particles, we seek an estimate   which has the least uncertainty or minimum    . 
3.5.7.2 Relative Error and Figure of Merit 
In any variance reduction method, we change the simulation and hence change 
the underlying distribution      so that it produces fewer zero-score histories and 
becomes more concentrated about its mean   . By making      more concentrated 
about its mean (which remains the same as the analog PDF), the variance of the mean   
  
will be less than that of the analog PDF, i.e., the estimate of the mean will be more 
precise. 
For each tally, the MCNP calculates several other statistics apart from sample 
mean   . One of such statistics and the most important is the relative error   (tells us the 
fraction of the error compared to the value) defined as 
                     (3.68) 
It can be seen that we want to make   as small as possible. For a reliable tally 
confidence interval, R should be less than 0.1. Combining Equations 3.67 and 3.68, we 
have 
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                (3.69) 
Another important statistics generated by MCNP is the figure of merit (FOM) defined as 
      
 
   
                     (3.70) 
where  is the simulation time or CPU time, which is proportional to   the number of 
histories run. We can see in Equation 3.70 that     is independent of  . For different 
simulations of the same problem, the simulation with the largest     is preferred since 
it requires the least time or produces the smallest error. 
3.5.8 Variance Reduction Techniques in Monte Carlo  
Methods for reducing the variance in the estimated solution to reduce the 
computational time for Monte Carlo simulation is called variance reduction technique. 
Alternative to large number of particles and long computation is to, reduce the variance 
   by more efficient random walk designed to reduce the spread of the tally results 
about the mean. There will be some histories which will not contribute to the tally 
because of absorption or motion in the wrong direction. The computer time spent on 
these zero contribution histories can be reduced by implicit capture or population control 
of low weight and out of range particles. 
3.5.8.1 Implicit Capture 
Particles which are tracked but are absorbed before they reach the tally surface 
represent wasted computing effort. This Analog capture is replaced with implicit 
capture. In Analog simulation the particle is killed at collision site, with a 
probability
  
   . However in Implicit capture, the particle is allowed to continue on its 
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trajectory as if no interaction had occurred with the particle but the particles weight    is 
reduced by 
               
  
          (3.71) 
By so doing, no particles are lost due to absorption, but absorption effects are properly 
for. 
3.5.8.2 Population Control 
Population control makes sure that there are enough particles in the interesting 
region. These are methods used by the MCNP to boost the particle population in the 
region of interest at the expense of other region. They include: 
3.5.8.2.1 Geometry Splitting by Cell Importance 
To track neutron in the reactor core, it is divided into several thicknesses of 
approximately one mean free path and have increasing importance within each 
thickness. The neutron splits into two at each surface within the material and each new 
neutron has half the weight of original neutron. The paths of the separate neutrons are 
followed and their scores are added to that of the original neutron. Here the weight of the 
neutron is conserved and this way there are more different scores in the interesting 
region or tally side thereby generating lower variance in the region. However, the 
calculation time is higher. This technique is used in the region of interest. 
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3.5.8.2.2 Russian Roulette 
In Russian roulette, neutrons with low weight are terminated, below a threshold 
    with a known probability because they will not contribute a lot anymore. There is a 
probability   that the particle terminates; if the particle survives the weight will become: 
     
  
 
         (3.72) 
Defining   by a survival weight    we have: 
    
  
  
          (3.73) 
Combining Equations 3.72 and 3.73 the weight of the surviving particle is  . By setting 
the threshold and survival weight different in the different areas, the number of neutrons 
can be controlled. This way the result is unbiased, the calculation time will decrease, but 
the variance will increase. This technique is therefore used in less interesting part of the 
core. Splitting and Russian roulette methods can also used with particle energy. 
3.5.9 Monte Carlo Simulations 
The nature of Monte Carlo simulation is shown in Figure 3.2. A source particle 
is introducedA source particle is introduced with phase space coordinates (   ) which 
may be sampled randomly according to PDFs representing the spatial, directional and 
energy distributions of source particles in the specific physical problem considered.  
In the transport portion of the analysis, the distance to the particle’s next 
collision is sampled randomly from the PDF which describes the random-walk of 
particles in a background medium. Geometric information describing the material and 
region boundaries, usually in the form of first or second degree surface equations, is then 
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analyzed to determine whether the sampled distance to collision is less than distance to a 
boundary. If less, the collision did not occur, and the collision analysis proceeds by 
sampling the particle’s exit energy and direction from the appropriate PDFs. Production 
of secondary particles such as (   ) or (   ) reactions is also determined by sampling 
from the appropriate PDFs. The Monte Carlo analysis alternates between transport and 
collision analysis until the particle and its progeny have been killed by absorption or 
leakage from the system. Another source particle is then introduced and followed 
throughout its history, and so on. Typical problems can involve the processing of up to 
several million particle histories to achieve accurate scores. 
For reactor calculations, iterative procedure is used to obtain the eigenvalue [49] 
[50] shown in Figure 3.3. An assumed spatial distribution of fission site is used to 
perform the initial iteration (generation 0). New fission sites recorded during the 
random-walk analysis are then used to provide estimates of both the eigenvalue and the 
source distribution to be used for the next generation. Additional generations are then 
analyzed as needed to converge the eigen function. 
 
 
 
 
 
 
 
 
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Figure 3.2 Monte Carlo random works for single particle 
 
 
Figure 3.3 Power iteration for Monte Carlo criticality calculation of the mean value of  
keff 
For each trial, the neutron typically goes through the following steps: 
Transport 
*sample distance to collision track to collision 
point 
*split/Russian roulette 
Tally, bias, … 
 
Collision 
*sample exit group/energy 
*sample exit direction 
Tally, bias, … 
*secondary particles 
 
Source (   ) 
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i) Determination of the source parameters       .  
We start with a fission neutron source so that the direction  and the energy   may 
be sampled from an isotropic distribution as in Equation 3.57 and from the Watt 
spectrum, respectively. For multiplying systems such as nuclear reactors, the 
iterated fission source methodology which closely follows the              
method isadopted. The first iteration starts by assuming a suitably large number   of 
fission locations distributed throughout the reactor core. A guess of the 
multiplication factor of the system say  is made and we assume that the mean 
number of neutron released in fission, instead of having the true value   is     
    . Then the Monte Carlo simulation is carried out: the      fission neutrons 
emitted is followed one after the other and the    locations of the fissions neutrons 
they induce are recorded. The   locations are better distributedwithin the core than 
   since they are due to neutrons which follow real physics processes. In order to 
proceed to the second iteration we update   and    . This is done as the reactor were 
critical by requiring that at each iteration the number of fission neutrons should 
remain equal to the initial value      . Generally, if    is the locations at the      
iteration (by following the fates of the neutrons of the preceding generation), then 
we have 
                            
from which  
           
  
  
             
  
  
    
  
  
                             ( 3.74) 
ii) Determination of the next collision point.  
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From the knowledge of the neutron position  and energy   the code looks up in its 
files the region, say   , where the neutron is and the total macroscopic cross section 
   pertaining to that region. The next collision point is then sampled from the 
exponential distribution on the straight line starting at   in the direction . If    were 
infinite in extent the neutron would travel a distance given by the Equation3.62 
namely, 
      
   
            (3.75) 
where,          and   
 
  
. If   is less than the geometric chord    from   to the 
   boundary along , the collision actually occurs at the point      which is 
within   . If      the neutron, going along , enters the next region, say   , 
whose total cross section at the energy   is   . At the point     on the boundary 
between    and    the residual way is 
      
   
                   (3.76) 
and everything happens as   were the distance to be traveled from the point 
   within   . More formally if         are successive regions along and 
       ,         are the corresponding optical lengths (macroscopic cross 
section times geometrical chord), the neutron will collide in that region    for which 
              
 
   
   
                 (3.77) 
where by definition, the first sum vanishes if    . 
iii) Determination of the kind of collision: 
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 Assume that the total cross section    ofthe      region where the collision occurs 
is made up of   components namely                 relating to the various nuclides 
in the medium and to the various reactions. Then the particular      reaction 
occurs with discrete probability           and the selection of the collision as 
mentioned insection 2.10.6 
iv) Parameters of the emerging neutron 
Assume that at point   in zone   a neutron ofenergy   and direction   has an 
interaction with a given nucleus of mass A. If the collision is an absorption, the 
neutron disappears and we start a new trial that is another fission neutron among 
the      of the present      generation, begins to be processed.If the collision is 
fission the point   is recorded as a fission location for the            generation 
to be used as in 1 above and again, another trial is started.Finally if the collision is 
elastic scattering the code looks up the angular distribution of the involved nucleus 
at the neutron energy    in its files and samples a scattering angle   .  Knowledge 
of this angle allows computing the energy of the scattered neutron from the 
kinematics of the elastic collision. To obtain the new direction we have still to 
sample the azimuthal angle   around    from the uniform distribution in       . By 
so doing the new direction has been found with respect to the previous direction    
so that we have to reorient the axes to get the new direction  with respect to the 
coordinate system.  
As illustrated in Figure 3.2, Monte Carlo code is basically a collection of random 
decision points shown in Figure 3.4, with relatively simple arithmetic in between. The 
physics of a problem is contained in the PDFs used for the random sampling of the 
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collision kernel and the transport kernel, problem geometry is involved in the surface 
equations utilized for particle tracking in the transport kernel and results are obtained by 
tallying the quantities of interest. 
 
 
 
 
 Material Boundary Crossing      Collision 
 
 
Crossing Boundary of New Material     Collision                    Absorption 
   Scattering   Fission    
         
 
 
Figure 3.4  Decision points in generating the history of a single neutron in a Monte Carlo 
calculation 
3.5.10 Monte Carlo Estimates  
Now the particles are simulated and behave the same as the physical particles and it is 
time to extract results from the simulation. The quantities that need to be measured have 
to be estimated by estimators. To estimate the flux, we may need three different 
Initial velocity and position of neutron (   )  
   and    assumed from initial conditions  
Length of free flight ( ) 
   determined from material properties. 
Length of free flight in new material,  . 
    determined from properties of new 
material. 
Type of collision   
   determined by known branching ratios 
 
Chain 
terminated 
Number and velocities of new neutron 
    
      
  
             determined from fission cross 
section. 
 
New velocity    
    determined from 
scattering cross sections 
and incoming velocity. 
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estimators namely [44]: collision estimator, an absorption estimator and a track length 
estimator.    
An estimator is a specific function of the random samples, of a random variable, 
that statistically represents a true unknown mean.  If   is a random variable, with an 
associated distribution and an unknown mean, then the function                         
is an estimator of the unknown mean.  The set                        consists of   
independent random samples selected from the probability density distribution of . A 
good estimator should be unbiased, consistent, and efficient.  An estimator is unbiased if 
its expectation value equals the true mean,  
                             (3.78) 
for all  .  An estimator is consistent if it approaches, in a statistical sense, the truemean, 
as   gets larger. An efficient estimator is the one, among a group of unbiased estimators, 
which produces the minimum variance for a Mathematical expression for each of the 
three estimators of     . 
3.5.10.1 Collision Estimator  
The collision estimate for     for any active cycle is given by:   
      
  
 
 
     
         
       
             (3.79) 
where   is summed over all collisions in a cycle where fission is possible; 
  is summed over all nuclides of the material involved in the     collion; 
    is the total microscopic fission cross section;  
    is the microscopic fission cross section;  
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    is the average number of prompt or total neutrons produced per fission by the 
collision nuclide at the incident energy;  
   is the atomic fraction for nuclide k;  
  is the nominal source size for a cycle; and  
   is the weight of particle entering collision.    
Because     represents the number of neutrons entering the     collision:  
      
         
       
                 (3.80)
  
is the expected number of neutrons to be produced from all fission processes in the 
collision.  Thus      is the mean number of fission neutrons produced per cycle.   
3.5.10.2 Absorption Estimator  
The absorption estimator for     for any active cycle is made when a neutron 
interacts with a fissionable nuclide.  For implicit capture which is a default in MCNP5 it 
is given by:   
      
  
 
 
     
   
       
            (3.81) 
where   is summed over all collisions in which fission is possible and  
   
                                (3.82) 
is the weight absorbed in the implicit capture.  The difference between the implicit 
absorption estimator and the collision estimator is that only the nuclide involved in the 
collision is used for the absorption      estimate rather than an average of all nuclides in 
the material for the collision      estimator. The absorption estimate also differs from 
the collision estimate in that the collision estimate is based upon the expected value at 
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each collision, whereas the absorption estimate is based upon the events actually 
sampled at a collision.  Thus all collisions will contribute to the collision estimate     
  
of and by the probability of fission in the material.  Contribution to the absorption 
estimate will only be made for the nuclide sampled.    
3.5.10.3 Track Length Estimators  
The track length estimator of      is accumulated every time the neutron 
traverses a distance d in a fissionable material cell:   
      
  
 
 
                                 (3.83) 
Where    is summed over all neutron trajectories,  
  is the atomic density in the cell, and   
  is the trajectory track length from the last event. 
 Because             is the expected number of fission neutrons produced 
along trajectory  ,     
  is a third estimate of the mean number of fission neutrons 
produced in a cycle per nominal fission source neutron. The track length estimator tends 
to display the lowest variance for optically thin fuel cells (e.g. plates) and fast systems 
where large cross–section variations because of resonances may cause high variances in 
the other two estimators.  
The methodology adopted for performing this work using the versatile MCNP particle 
transport code is presented next in Chapter 4.  
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3.6 Burnup 
The initial composition of a fuel element will depend on the source of the fuel. 
For reactors operating on the uranium cycle, fuel developed directly from natural 
uranium will contain a mixture of U-234, U-235 and U-238, with the fissile U-235 
content varying from 0.72% (for natural uranium) to more than 90%, depending on the 
enrichment. Recycled fuel from reprocessing plants will also contain the various 
isotopes produced in the transmutation-decay process of uranium [51]. Presently, a 
typical MNSR uses enriched uranium fuel of about 90 % of U235. 
 During the operation of a nuclear reactor a number of changes occur in the 
composition of the fuel. Various fuel nuclei are transmuted by neutron capture and 
subsequent decay. For a uranium-fueled reactor, this process produces a variety of 
transuranic elements in the actinide series of the periodic table. For a thorium fueled 
reactor, a number of uranium isotopes are produced. The fission event destroys a fissile 
nucleus, of course, and in the process produces two intermediate mass fission products. 
The fission products tend to be neutron-rich and subsequently decay by beta or neutron 
emission (usually accompanied by gamma emission) and undergo neutron capture to be 
transmuted into a heavier isotope, which itself undergoes radioactive decay and neutron 
transmutation, and so on. The fissile nuclei also undergo neutron transmutation via 
radiative capture followed by decay or further transmutation [51]. Figure 3.5 shows a 
simplified diagram of transmutation of nuclei in the uranium series. Actinides of interest 
are marked with thick borders.  
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The buildup of actinides in the U-238 transmutation-decay process introduces a fuel 
reactivity penalty because some of actinides act primarily as parasitic absorbers. While 
Pu-239 and Pu-241 are fissionable in a thermal reactor, and Pu-240 transmutes into Pu-
241(fissile), Pu-242 transmutes into Pu-243 with a rather small cross section and thus 
build up in time. Pu-243 decays immediately towards Am-243(non-fissile). Pu-242 is 
effectively a parasitic absorber. The Am243 also accumulates and acts primarily as a 
parasitic absorber. Whereas the Am-243, which is produced by the decay of Pu-243, can 
be separated readily, it is quite impossible to separate the different plutonium isotopes 
from each other by chemical means, so the negative Pu-242 reactivity effect is 
exacerbated if the plutonium is recycled with uranium [51]. 
 
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Figure 3.5 The most important actinide nuclides and their relations, via a-decay, ß-decay 
and neutron capture reactions  
 
 
 
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In nuclear power technology, burn-up is a measure of the number of fuel atoms 
that underwent fission. It is normally quoted in megawatt–days per metric ton of heavy 
metal (MWd/THM), or its equivalent gigawatt–days/T (GWd/tHM). The unit GWd/tHM 
is the (average) thermal output, multiplied by the time of operation, and divided by the 
mass of heavy involved. This gives a rough measure of the number of nuclear fission 
events that have taken place within the fuel. 
 The actual fuel may be uranium, plutonium, or a mixture of these or of either or 
both of these with thorium. This fuel content is often referred to as the heavy metal (the 
only nuclei which can fission to produce energy) to distinguish it from other elements 
present in the fuel, such as oxygen of an oxide fuel or those used for cladding. The 
heavy metal is typically present as either metal or oxide, but other compounds such as 
carbides or other salts are possible. In a power station, high fuel burn-up is desirable for:  
 Reducing downtime for refueling   
 Reducing the number of fresh nuclear fuel elements required and spent nuclear 
fuel elements generated while producing a given amount of energy  
 Reducing the potential for diversion of plutonium from spent fuel for use in 
nuclear weapons, for short BU, the plutonium produced is essentially Pu-239 
which highly fissile. Alpha decay of this isotope also leads to the formation of U 
235.  
It is also desirable that burn-up should be as uniform as possible both within 
individual fuel elements and from one element to another within a fuel charge [52]. In 
reactors with online refueling (as in the case of CANDU reactor), fuel elements can be 
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75 
repositioned during operation to help achieve this. In reactors without this facility, fine 
positioning of control rods to balance reactivity within the core, and repositioning of 
remaining fuel during shutdowns in which only part of the fuel charge is replaced may 
be used, this type of fuel management is incorporated in this work with four 
fractionating cycles for the study of mono-recycling of the americium. 
3.6.1 BURNUP ANALYSIS 
One of the most important and difficult problems in reactor design is the 
prediction of the properties of a reactor core throughout its life. How these properties 
change in time must be known to ensure that the reactor will operate safely throughout 
the lifetime of the core. The changing reactor properties also determine the length of 
time that a given reactor core will remain critical, producing power at the desired level. 
This is an important practical consideration also for waste and spent-fuel 
characterization, as well as natural resource consumption. It obviously makes little sense 
for a public utility to purchase a reactor that must be shut down every month for 
refueling, if it can obtain one that needs to be refueled every 2 years, except for power 
reactors that can be fueled online such as the CANDU system. However, it makes 
equally little sense to buy a core that has fuel enough to remain critical for 20 years 
when radiation damage is known to compromise the fuel elements in 3 or 4 years [24]. 
In a typical UOX fueled thermal reactor, as energy is removed from a reactor, the 
fuel composition changes as a result of loss of fissionable material due to fission, 
buildup and decay of fission products, and transmutation of other reactor materials due 
to neutron capture. These composition changes occur over a relatively long period of 
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time [53]. All fission products absorb neutrons to some extent, so they are known as 
reactor poisons. Most fission product poisons simply build up slowly as the fuel burns 
up and are accounted for as a long-term reactivity. The neutron absorbing fission 
products such as Xe-135 have particular operational importance. Their concentrations 
can change quickly, producing major changes in neutron absorption on a relatively short 
time scale. Each arises from the decay of a precursor fission product, which controls 
their production rate, but, because they have large absorption cross-sections, their 
removal changes quickly with changes in flux [54]. Notwithstanding, U-235 in a UOX 
fuel fissions into other fission products and minor actinides while the fertile isotopes U-
238 undergoes neutron capture to build up fissile isotope Pu-239. All these phenomenon 
and transmutations tend to modify the core parameters during the core-lifetime of the 
fuel.  
Computations of changing reactor properties are known as burnup calculations or 
lifetime calculations [24].  Reactor Burnup Systems (REBUS) is a code that is used for 
such calculations. 
3.6.2 DESCRIPTION OF REBUS 
The REBUS-PC code has evolved away from its roots as a design tool for fast 
reactors to encompass the needs of thermal reactor design and analysis. Since its 
conception in the 1960’s, REBUS (the REactor BUrnup System) has always been a 
general-purpose tool.  
REBUS designed for the analysis of research reactor fuel cycles. Optionally, 
reprocessing may be included in the specification of the external fuel cycle and 
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discharged fuel may be recycled back into the reactor. For non-equilibrium cases, the 
initial composition of the reactor core may be explicitly specified or the core may be 
loaded from external feeds and discharged fuel may be recycled back into the reactor as 
in equilibrium problems. 
Four types of search procedures may be carried out in order to satisfy user-
supplied constraints: 
1) adjustment of the reactor burn cycle time to achieve a specified discharge burnup,  
2) adjustment of the fresh fuel enrichment to achieve a specified multiplication constant 
at a specified point during the burn cycle,  
3) adjustment of the control poison density to maintain a specified value of the 
multiplication constant throughout the reactor burn cycle, and 4) adjustment of the 
reactor burn cycle time to achieve a specified value of the multiplication constant at the 
end of the burn step. 
REBUS-PC will handle both equilibrium and non-equilibrium problems using a 
number of different core geometries including triangular and hexagonal mesh. The 
neutronics solution may be obtained using finite difference or nodal diffusion-theory 
methods. Other features include: fully automatic restart capability, no restrictions on 
number of neutron energy groups, and general external cycle with no restrictions on 
number of external feeds, reprocessing plants, etc. Fuel management is completely 
general for non-equilibrium problems.  
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Microscopic cross sections are permitted to vary as a function of the atom 
density of various reference isotopes in the problem as appropriate for thermal reactor 
systems. The previous capability where neutron capture and fission processes were fitted 
to low-order polynomials as functions of burnup is retained. In addition, the user now 
may select cubic spline interpolation for (n,gamma), (n,fission), (n,alpha), (n,p), (n,d), 
and (n,t) reactions as functions of burnup. The user may specify control rod positions at 
each time node in the problem. Output edits have been extensively revised and better 
organized for use in a PC environment. A number of ASCII format datasets containing 
various types of summary results are available for use in tailoring reports with the aid of 
auxiliary PC software such as spreadsheet or word processor programs.   
3.7 Analyses of MNSR Thermal Hydraulics     
3.7.1 Heat transfer process of MNSR 
The thermal hydraulic design of the reactor is closely related to the neutronic 
calculations and the structural design [2]. The geometrical dimensions of the fuel 
element, fuel element temperatures, coolant pressure, temperatures and velocities are 
analytically shortened to satisfy the requirements of reactor safety during all operational 
states.  
The core of the MNSR is immersed into the lower part of the reactor tank, which 
is filled with coolant water [55]. The tank is immersed in a quite large water pool. The 
heat generated from the nuclear fuel is carried out of the core by coolant through heat 
conduction, heat convection and heat transport. 
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The heat generated by the nuclear fuel is first transferred to the outer surface of 
cladding through the heat conduction of fuel pellets and cladding, and then the heat is 
transferred to the coolant water by the natural convection. The density of the heated 
water decreases, and the buoyancy force causes coolant water to flow up, thus the heat is 
carried away from the reactor, which is called the transport process.  
 
 
 
   Fuel Element 
 
 
Dimensions in millimeters 
 
 
 
Figure 3.6: Schematics of Reactor Core [56]  
Core Fuel element 
1 – Elevation line of core outlet 1. Top end plug  
2 – Fuel element 2. Cladding 
3 – Beryllium reflector 3. Meat 
4 – Downcomer  4. Lower end plug 
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5 – Reactor vessel  
6 – Pool water   
  
T1 – Inlet temperature  
T2 – Outlet temperature  
T3 – Temperature above Core  
T4 – Temperature at downcomer region  
To – Pool temperature  
The core region of GHARR-1 is located 4.7 m under water close to the bottom of 
a watertight reactor vessel. The quantity of water is 1.5 m3 in the vessel, which serves 
the purpose of radiation shielding, moderation and as primary heat transfer medium. In 
addition, heat can be extracted from the water in the vessel by means of a water-cooling 
coil located near the top of the vessel. The water-filled reactor vessel is in turn immersed 
in a water-filled pool of 30 m3. Cold water is drawn through the inlet orifice by natural 
convection. The water flows past the hot fuel elements and comes out through the core 
outlet orifice. The hot water rises to mix with the large volume of water in the reactor 
vessel and to the cooling coil. Heat passes through the walls of the container to the pool 
water. A diagrammatic representation of the heat transfer mechanism is represented in 
Figure 3.6. 
The core inlet flow orifice impedes the natural circulation of water through the 
core. Its area is fixed during assembly of the reactor and it is deliberately chosen such 
that the highest power achieved during the design basis self-limiting power excursion 
can cause no damage to the core or present any hazard to staff about the reactor. 
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The GHARR-1 reactor has a small core. It can be shown that the coolant flow in 
the core is at the transient phase from laminar flow to turbulent flow. The flow transition 
will occur when there is an increase in power. The closer to the upper part of elements, 
the stronger the turbulence becomes. The buoyancy force in natural circulation must 
overcome the friction. Calculations show that the friction resistance is small, about 10% 
of the total resistance. Meanwhile, the inlet resistance is about 70% of the resistance and 
thus has a great effect on the state of flow. An appropriate choice for the inlet flow 
orifice is a very important factor in the thermal-hydraulic design. As inlet orifice gets 
smaller, the flow velocity and in turn, the turbulence in the core will increase. As a 
result, the heat transfer from fuel element to coolant will be improved. However, a 
smaller inlet orifice will cause an increase in resistance and a decrease in flow rate 
resulting in a rapid increase of temperature. An appropriate temperate rise is an essential 
condition. 
The water in the reactor is not pressurized and it relies upon natural convection. 
The problems of de-pressurization or coolant flow pump failure are not posed. A water-
cooled coil of limited heat removal capacity removes the thermal energy generated in the 
core eventually. Due to the fact that the reactor possesses limited excess reactivity and 
reactivity feedback characteristics, any significant deterioration in heat removal 
capability will eventually result in an automatic decrease in reactor power to match up 
with the new heat removal capacity. 
The temperature in an operating reactor varies from point to point within the 
system.  As a consequence, there is always one fuel rod (usually one near the centre of 
the reactor) that at some point along its length is hotter than all the rest. This maximum 
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fuel temperature is determined by the power level of the reactor, the design of the 
coolant system, and the nature of the fuel [24]. One major design of a reactor coolant 
system is to provide for the removal of heat produced at the desired power level while 
ensuring that the maximum fuel temperature is always below this predetermined value. 
This in tend ensures good safety margin. Under sub-cooled flow boiling conditions, the 
boiling crisis is often called departure from nucleate boiling (DNB) and the heat flux at 
which the boiling crisis occurs is named the critical heat flux (CHF) [57]. In general, 
thermal performance improvements are highly desirable and it is therefore needed to 
predict CHF accurately at the earliest stages of a new product design. In the case of a 
nuclear reactor core, CHF margin gain (e.g. using improved fuel assembly design) can 
allow power uprate and enhanced operating flexibility [58]. Most metal-finishing 
operation score tiny grooves on the surface, but they also typically involve some 
chattering or bouncing action, which hammers small holes into the surface. When a 
surface is wetted, liquid is prevented by surface tension from entering these holes, so 
small gas or vapour pockets are formed. These little pockets are sites at which bubble 
nucleation occurs [24]. The ONB is not a limiting criterion in the design of a fuel 
element. However, it is a heat transfer regime which should be identified for proper 
hydraulic and heat transfer considerations, i.e., single-phase flow versus two-phase flow. 
For reactor design purposes, acceptable data on burnout heat flux are needed 
since departure from nucleate boiling (DNB) is potentially a limiting design constraint. 
Optimization of core cooling against other neutronic, economic, and materials 
constraints can best be accomplished by judicious use of standard, experimentally-
deduced DNB correlations [59]. The parameter most used to evaluate margin to failure 
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by boiling crisis is the critical heat flux ratio (CHFR), or departure from nucleate boiling 
ratio (DNBR). This is the ratio of critical heat flux calculated by the correlation to the 
most limiting heat flux condition in the reactor [53]  
The heat generated from the nuclear fuel is carried out of the core by coolant 
through heat conduction, heat convection and heat transport. The heat generated by the 
nuclear fuel is first transferred to the outer surface of cladding through the heat 
conduction of fuel pellets and cladding, and then the heat is transferred to the coolant 
water by the natural convection. The density of the heated water decreases, and the 
buoyancy force causes coolant water to flow up, thus the heat is carried away from the 
reactor, which is called the transport process [60].  The progression is depicted in Figure 
3.7. 
 
Figure 3.7: Heat transfer and transport process 
In the early stages of the reactor start-up, the water temperature difference 
between the upper part of the tank and the pool is not large; the heat transferred is little, 
so that the heat released from the fuel elements cannot be all transported out. Therefore 
the water temperature in the tank increases continuously, the natural circulation is 
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intensified and the flow velocity of the water in the core increases accordingly. So the 
heat transferred to the pool water increases continuously. After some time a steady 
natural circulations can be established to a large extent.  
After this, the water flow and heat transport in reactor are steady state conditions. 
The time of reactor startup is usually several hours, during which the heat transport and 
flow in reactor are the transient process, and the thermal-hydraulic parameters are under 
transient variation. The purpose of transient calculation is to predict fuel element 
temperature variation with time at reactor inlet and outlet, predict fuel element 
temperature and the relation between reactor power and inlet – outlet temperature 
difference. 
3.7.2 Thermal – Hydraulic Characteristics of MNSR 
Characteristics of MNSR considered for thermal-hydraulics [2]: 
i. Low linear power density of fuel element: about 3.8 W/cm, close to that of 
powers at the period of residual heat removal. 
ii. U-Al alloy is adopted as fuel meat for the highly enriched fuel, but a UO2 is 
proposed for the expected low enriched fuel. 
iii. Natural convection is adopted for cooling. The reactor core is immersed in a 
quite large reactor tank, in which the water possesses a considerable heat 
capacity. The natural circulation cooling of MNSR possesses a very good 
inherent safety. It is also economical since there is no pumping device which 
might in turn need power to operate. 
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3.7.3 Heat Transfer Calculations of the Steady State 
3.7.3.1  Heat conduction in fuel pellets 
 
Figure 3.8: Fuel element 
2
2
1 0v
u
qd T dT
dr r dr K  
                                                  (3.84) 
       
Where  
vq   – heat production rate, W/m
3 
   
uK  – thermal conductivity of pellets, W/m.°C      
When qv is distributed uniformly, solving Equation 3.83 to get 
2
4 2 4
u
v u l
o u
u u u
q r qqrT T K K K   
                                             (3.85) 
Where 
oT  – temperature at the centre of the pellets, 
           
uT  – temperature at the surface of pellets, 
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ur  – radius of pellets; see Figure 3.8. 
                 q  – Surface heat flux, 
      
lq –  linear power density 
 
lq   2  ur q    2ur vq                                     (3.86) 
 Heat conduction of cladding 
                   
c
dTQ K F dr 
   Fourier law [61]              (3.87)    
 Solving Equation 3.87, one can obtain 
                  
ln ln2 2
c l c
u c
c u c u
r q rQT T K L r K r   
                 (3.88) 
Where Q – total heat conducted through the cladding (w), 
          F – area perpendicular to conduction direction 
          L – length of pellets (m). 
3.7.3.2 Heat Convection between cladding outer surface and coolant 
The convection heat transfer mode is comprised of two mechanisms. In addition 
to energy transfer due to random molecular motion, energy is also transferred by the 
bulk, or macroscopic, motion of the fluid. This fluid motion is associated with the fact 
that, at any instant, large numbers of molecules are moving collectively or as aggregates. 
Such motion, in the presence of temperature gradient, contributes to heat transfer [62]. 
Let us consider the convection heat which occurs between coolant and clad surface when 
the two are at different temperature.  
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2 ( )l c c fq r h T T   Newton’s Law of Cooling        (3.89) 
 Or         
 ( )
2
l
c f
c
q zT T r h 
                                                              (3.90) 
Where 
fT
– mixed temperature of coolant (°C) 
               h  – heat transfer coefficient (W/m
2.°C) 
Given 
lq , fT
 and h , one can get the temperature at the outer surface of cladding by 
means of Equation 3.90. Here the key problem is how to calculate h , which is related to 
the velocity of coolant, the thermal properties of coolant and the geometries of flow 
channel etc.  
i. Forced convection heat transfer in a circular channel 
           0.8 0.40.023Nu Re Pr            Dittus – Boelter  [61]      (3.91) 
Where  
f
hDeNu K
, 
f
f
UDeRe  
 and  
( ) fCpPr K

 
De  – equivalent hydraulic diameter (m) 
fK
 – thermal conductivity of coolant (W/m.°C) 
f  – density of coolant (Kg/m
3)  
U   – coolant dynamic viscosity (Pa.S) 
f  – coolant flow velocity (J/Kg.°C) 
Cp  – specific heat capacity for coolant (J/Kg.°C) 
ii. Heat transfer coefficient for rod bundles 
Weisman suggested the following 
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0.8 1/3Nu CRe Pr                                                             (3.92) 
Where C  – constant determined by lattice arrangement. 
For square lattice, when1.1 / 1.3P d  , 
   0.042( / ) 0.024C P d  
For triangular lattice, when 1.1 / 1.5P d   
   0.026( / ) 0.006C P d  
P  – distance between adjacent fuel rods, (m) 
d  – diameter of fuel bundle (m) 
iii. Natural convection heat transfer 
Natural convection refers to the fluid flow which is caused by density gradient 
within the fluid. The density gradient is usually produced by temperature difference in 
the fluid. Therefore, the intensity of natural convection depends on temperature 
difference. The heat transfer coefficient of the natural convection is often expressed in 
the form of the following correlation: 
( )nmNu c GrPr  
Gr  is the Grashof number; 3 2( ) /c f fGr g De T T   
g – acceleration due to gravity (m/s) 
  – volumetric expansion coefficient (1/°C) 
De – characteristic dimension (m) 
f – kinematic viscosity of coolant (m
2/s) 
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m – means 1/ 2( )m c fT T T  as the temperature calculation the physical properties, c and 
n mainly depend on the geometry and orientation of the solid, heat flux direction, and 
the range of . .Gr Pr   
On the full scale simulation facility of the MNSR at China Institute of Atomic 
Energy, the natural heat transfer coefficient was measured and the following correlations 
were obtained: 
0.250.68( )Nu GrPr , when 6( ) 6GrPr E  
1/30.174( )Nu GrPr , when 6( ) 6GrPr E  
3.7.3.3 Heat transport in the reactor 
When coolant is flowing through reactor core, the heat released from fuel 
elements is carried away from the reactor core. The process is called heat transport. For 
steady state conditions, the heat balance equation is  
( ) [ ( ) ]p f finQ z WC T z T 
                                     (3.93) 
Where 
( )Q z  – heat power transported by coolant from inlet of core to axial position z , (W) 
W  – mass flow of coolant (Kg/s) 
pC
 – specific heat of coolant (J/Kg°C) 
From Equation 3.93, coolant temperature distribution may be obtained:  
 ( ) 1( ) ( )
in in
in
z
f f f l
p p z
Q zT z T T q z dzWC WC    
                                         (3.94) 
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Where  
lq  – linear power density at position z, (W/m), 
infT
 – inlet temperature of coolant, (°C) 
Given
infT
, w, and 
lq (z), one can calculate coolant temperature distribution along the 
axial direction for Equation 3.94, 
infT
, maximum cladding surface temperature 
maxcT , 
position , maximum centre temperature of fuel pellet position 
0maxT , position 0maxZ
shown in Figure 3.9. 
 
Figure 3.9: Temperature distribution in fuel and coolant 
Coolant temperature gains maximum value at the exit, the maximum temperature 
at the outer surface of cladding and the maximum centre temperature of the pellet occur 
all between the middle and exit of the flow channel, their values must be less than their 
limiting values [26]. 
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For the HEU reactor at a power of 30 Kw, the calculated maximum fuel temperature 
(
0maxT ) is about 70 ° C, much less than the melting point of U-Al alloy which is 640 ° 
C. 
3.7.4 Transient Thermal hydraulic calculation 
Under transient conditions, the heat generated by the fuel meat Qu is not equal to 
the heat transferred from the outer surface of cladding to the coolant Qf, as part of Qu is 
transferred to the coolant through the cladding, another of Qu is used to raise the 
temperature of fuel elements. The temperatures of fuel and coolant and the velocity of 
coolant all vary with time. Generally speaking, partial differential equations of heat 
conduction within fuel element and hydro-dynamical partial differential equations are 
simultaneously solved. 
3.7.4.1 The partial differential equations of heat conduction within fuel and 
cladding under transient conditions 
Fuel: 
 1( , ) [ ( ) ( , )] ( , )u u vC T r t Ku T r T r t q r tt r r r
     
              (3.95) 
Cladding:  
1( , ) [ ( ) ( , )]c c cC T r t K T r T r tt r r r
    
                              (3.96) 
Boundary conditions: 
i) 
( , ) | ( )cc r r c fK T r t h T Tr 
  
     
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ii) 
( , ) | ( , ) |u uu r r c r rK T r t K T r tr r 
  
 
iii) 
( , ) | 0ur rT r tr 
 
     
                   i), ii) and iii)         (3.97)   
3.7.4.2  The hydro-dynamical partial differential equations of coolant flowing in 
reactor core 
i) Mass conservation equation;   
                             
( ) 0f fUt z
    
                                               (3.98) 
ii) Momentum Conservation equation;  
                  
21 ( )2f f f fe
P U U fU U gz t z D z   
           
      (3.99) 
iii) Energy conservation equation;  
                   
( )f f hpf f pf f c f
f
T T PC C U h T Tt z A 
    
                               (3.100) 
iv) State equation of coolant;  
                              ( , )f ff P T                                                                        (3.101) 
where   
f  – coolant density (kg/m
3),  
              U   – coolant velocity (m/s) 
               
pfC – specific heat capacity of coolant (J/Kg °C) 
               
fT
– coolant temperature (°C)  
               
hP  – heated perimeter (m) 
              
fA
  – cross sectional area of flow channel (m2) 
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               h    – heat transfer coefficient (w/m
2°C) 
                
cT  – temperature at outer surface of cladding (°C) 
               P   – coolant pressure (Pa) 
               f   – friction factor  
               
eD – equivalent diameter of channel (m) 
                  – form resistance coefficient 
The Equations 3.97 to 3.101 stated above include five known variables; T 
(including Tu and Tc), U , 
fT
, P   and 
f . These equations are simultaneously solved by 
the finite difference to obtain the distributions of these five variables and variations with 
time. 
3.7.4.3  Simplified method of integrated parameters  
According to the characteristics of MNSR, a method of integrated parameters is 
adopted: taking the reactor core as a calculated unit, and assuming that [55]: 
i. Uniform power distribution of fuel element, that is ( )vq f t  
ii. Water temperature 
fT
 varies linearly along axial direction  z, and 
taking average water temperature at reactor core 
1 2
1 ( )2fT T T 
  as 
reference variable, ( )fT f t  
iii. Uniform coolant velocity U  in the reactor core, that is, 
0, ( )U U f tz
  
; 
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iv. The fuel meat possesses high thermal conductivity, a small contact 
resistance between the meat and cladding, low power density of fuel 
and small diameter of fuel element, hence the temperature difference 
between the centre and cladding surface approximately the same, 
o cT T   and ( )cT f t  
As a result, the partial differential equations of heat conduction within fuel and 
cladding as shown in equations 3.95 and 3.96, and boundary condition in equation 3.97i 
are approximately expressed as in the following equation: 
 
( ) ( ) [ ( ) ( )]cu u c c u s c fdTG C G C Q t F h T t T tdt   
                    (3.102)    
Where 
uG  –  total mass of fuel meat in core (kg)  
        
uC  – specific heat capacity of fuel meat (J/kg°C) 
      
cG  – total mass of cladding in the core (kg) 
               
cC  – specific heat capacity of cladding (J/kg°C)  
uQ  – reactor power (W) 
sF  – total surface area of fuel elements in core (m
2) 
            
cT  – average temperature of fuel elements (°C) 
            
fT  – average temperature of coolant (°C)                     
            h  – heat transfer coefficient (W/m
2°C) 
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According to the assumptions mentioned above, the energy equation of coolant 
in core can become: 
0 0 0
( )H H Hf f hpf f pf f c f
f
dT T PC dz C U dz h T T dzdt z A 
    
                          (3.103) 
The integrated equation, 3.103, may be written as 
2 1( ) ( )f spf f c f pf f
f
dT FHC h T T C U T Tdt A    
                             (3.104) 
Where 
fA
  – total cross section area of flow channel in core (m2) 
 
0
1 H
f f dzH  
                                                                             (3.105) 
               H  – height of core (cm)                                               
               
1T   – coolant temperature at the inlet of core (°C)                                                
               
2T   – coolant temperature at the outlet of core (°C)   
               
2 12 fT T T                            (3.106) 
If the flow resistance in the downcomer is neglected and coolant velocity in the 
downcomer is approximately equal to coolant velocity in the core U, the integrated 
momentum equation of coolant in the reactor (a cycle path) can be written as [55]      
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1 1 1
1
2
0 0 0
0 0 0
0 0 0
1
0 ( )
2
H H H
f f f
e
H H H
f f f
H H H
f f f
dU f
dz U dz g dz
dt D z
dU dU
dz g dz dz
dt dt
dU
g dz dz g dz
dt
  
  
  

  

   
   
 
  
  
  
          (3.107)                                                                                     
The above equation may be written as 
1 2 1 3 1 4
2 2
4 1 3 2
( )
1
( ) ( )
2 2
f f f f
f f f f f f
e
dU
H H H H
dt
fH
gH gH U U
D
   
     
   
     
                   (3.108)                                               
Where 
1H  equal temperature height above the core (m) 
              
2f  – coolant density at the outlet of core (kg/m
3) 
            
3f  – coolant density above the downcomer (kg/m
3) 
 
4f  – coolant density in the downcomer (kg/m
3) 
Since 
3 4 1f f f    , equation 3.107 can be simplified  
2
1 1 1 1 22( ) ( ) ( ) ( )2 2f f f f f fe
dU fHh H gH gH Udt D     
      
          (3.109) 
Where  
1
64
Re, ff 
 – coolant density at the inlet of core (kg/m3) 
If the reactor power Qu(t) and the inlet coolant temperature T1(t) are known, Equations 
3.102, 3.104, 3.106 and 3.109 are simultaneously solved, and we obtain Tc (t), T f(t), 
T2(t) and U(t). However, the inlet coolant temperature T1(t) is unknown during the 
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reactor start-up. In order to calculate T1(t), it is necessary to supply some energy balance 
equations in upper part of reactor tank, in annular downcomer and outside water pool.  
Bounded by the elevation of the outlet of the reactor core, the water in the reactor 
tank is divided into upper and lower parts, their energy equations are as follows: 
 33 2 2 3 3 0pf up pf f fdTG C A C U T T Qdt    
                          (3.110) 
 44 3 3 4 4 4pf dow pf f f BdTG C A C U T T Q Qdt     
                       (3.111) 
0
0 0 4pf
dTG C Q Qdt  
                                                      (3.112) 
Where 
0Q  and 4Q  – heat transferred to pool, calculated with natural convection 
correlation; 
BQ – heat transferred from beryllium to downcomer. 
Experiments show that the temperature at downcomer does not change much, 
hence T4 = T1. Thus Equation 3.111 is rewritten as 
 44 3 3 1 1 4pf dow pf f f BdTG C A C U T T Q Qdt     
                    (3.113) 
Equations 3.110, 3.112 and 3.113 include three unknown temperatures T0, T1 and T3. The 
pool water, which weighs Go, has a large heat capacity and may be considered as 
isothermal to the ambient, in other words, the heat (Q0 + Q4) is all absorbed by water in 
the pool. 
Solving equations 3.102, 3.104, 3.106, 3.109, 3.110 and 3.112, we can get the cladding 
temperature Tc, the average water temperature of the reactor core
fT , the water 
temperature at the core inlet and outlet T1 and T2, the water velocity in core U, the water 
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temperature in the upper part of the reactor tank T3 and the water temperature in the 
pool, To.  
Figures 3.7, 3.8 and 3.9 show the water temperature at core inlet, T1, water temperature 
difference between core inlet and outlet, and the water velocity in core U, they vary with 
time (minute), the reactor power Qu as parameter.  
During the early period of start-up, the temperature difference is small between 
primary water (water in reactor tank) and the pool water, therefore the heat is less 
transferred to the pool, and most is used to raise the temperature of the primary water 
and the fuel elements, thus the average and the inlet temperature of the reactor rise more 
quickly [55]. With the increasing of primary water temperature, the temperature 
difference for heat transfer increases, and more heat is transferred to the pool, so that the 
temperature increase of the primary water is reduced, and after three hours, the condition 
becomes steady. 
In several minutes, the reactor power reaches the rated value and the temperature 
difference between outlet and inlet reaches maximum value as shown in Figure 3.11. 
After that, the temperature of the primary water increases continuously (Figure 3.10) and 
the buoyancy force increases and the water flow velocity in the reactor core increases 
slightly (Figure 3.12), thus the temperature difference between outlet and inlet decreases 
slightly (Figure 3.11). 
Arranging the calculated results, the correlation between reactor power and inlet-
outlet temperature difference is realized [55]. 
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1(0.59 0.0019 )2.674 0.352 2 1(5.725 147.6 ) Tin uT T H T Q                       
(3.114) 
Where 
inH is the height of the inlet channel (mm), the size of H affects the form 
of resistance at the core inlet, and the form of resistance makes up the greater part of 
total resistance. So that, the 
inH  is smaller, the form resistance is larger, and the coolant 
velocity in the core is smaller. As a result, the inlet-outlet temperature difference 
increases. 
During reactor operation, the temperature difference between core outlet and 
inlet can be measured, and the reactor power can be calculated with the correlation (29) 
 
  
 
Figure 3.10: Water temperature variation at core inlet (T1) 
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Figure 3.11: Variation of temperature difference between core inlet and outlet. 
 
Figure 3.12: Variation of coolant velocity in reactor core.  
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The methodology engaged in undertaking the studies of this work is provided in Chapter 
Four. Firstly, the neutronics aspect including burnup is analysed using neutronics codes; 
the MCNP and the REBUS codes. Secondly, the thermal hydraulics analysis which 
comprises of the Steady State as well as the Transients is analysed using the PLTEMP 
and PARET codes. 
 
 
 
 
 
 
 
 
 
 
 
 
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CHAPTER FOUR 
4 METHODOLOGY 
MNSR reactors are simple in design and structure. As per all types of reactor, 
neutronics analysis of MNSRs can be done with reactor codes based on both 
deterministic and Monte Carlo approaches as described in Chapter 3. In the past, 
deterministic methods and codes have been employed for reactor analysis of the 
GHARR-1 [63]. 
In recent times, however, the Monte Carlo approach to reactor analysis has been 
included. In particular, multipurpose Monte Carlo particle transport codes generally 
have the capability to model and treat different complicated geometries in 3-D and also 
simulate the transport behavior of different particles and nuclear interaction processes. 
Good and accurate modeling of the different zones and diverse geometries of the MNSR 
reactor is important for realizing good neutronics, particle transport simulation, and 
physics analysis. For these reasons, the versatile and widely utilized MCNP code 
particle transport code was employed to develop a 3-D Monte Carlo model for MNSR 
for particle transport simulation and neutronic analysis of MNSR reactors [25]. 
In this Chapter, the 3-D Monte Carlo developed for MNSR reactors using the 
versatile MCNP transport code and upon which this work is derived is described. The 
methodology adopted in the modeling and neutronic simulation relevant to this work is 
also presented.   
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4.1 Monte Carlo Model for GHARR-1 
The Monte Carlo method has been in use for almost sixty years to solve radiation 
transport problems in high energy physics, nuclear reactor analysis, radiation shielding, 
medical imaging, etc [64]. Individual particles histories are simulated using random 
numbers, highly accurate representations of particle interaction probabilities, and exact 
models of three dimensional problem geometry. Monte Carlo methods are sometimes the 
only viably methods for analyzing complex, demanding particle transport problems. 
  This work is based on the Monte Carlo model for MNSR reactors modified at 
Argonne National Laboratory in the USA using the MCNP transport code. The model 
was developed following the approaches used in the 3-D combinatorial, generalized 
geometry and continuous energy methods applied in MCNP. Thus different geometries 
in planar, conical, spherical and cylindrical configurations of the various zones, sections 
and materials such as the fuel assembly, control systems, reflectors, experimental 
channels, shim tray, reactor vessel, reactor pool and other structural components were 
modeled accordingly using available design data. The centre of the GHARR-1 core 
assembly which has a cylindrical configuration with 10 fuel lattices concentrically 
arranged about the central control rod guide was chosen as the geometrical centre of the 
Monte Carlo model.  
Typically, MNSR reactors are designed to have different core loadings. This 
means that the fuel loading of a typical MNSR facility could vary or differ from other 
operating MNSR facilities. Taking cognizance of this variation, the GHARR-1 Monte 
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Carlo model provided for the modeling and simulation of MNSR reactors with varying 
core configurations ranging from 342 – 350 fuel lattices.  
The MCNP5 [27] transport code was used to perform the Monte Carlo 
calculations. Nuclear data for fissile and non-fissile isotopes associated with materials 
(fuel and clad, coolant, moderator, control rod and clad, reflectors, 5 structural 
components) of the physical model was chosen from ENDF/B-VI nuclear data libraries. 
The special S(   ) scattering feature was applied in the nuclear model to treat thermal 
scattering in beryllium and hydrogen in light water for the reflector material and water 
regions respectively of the GHARR-1 Monte Carlo model. Neutronics analyses were 
performed using a condensed 3-group neutron energy structure: up 0.625eV for thermal 
neutrons, < 8.21 eV for epithermal neutrons and up to 20 MeV for fast neutrons.  
All this were done to establish the deck for the HEU core and after it been 
ascertained that results compare very well with experimental data, the necessary 
modifications were made to acquire the LEU model for the core conversion exercise.  
MCNP5 permits a very general, detailed representation of 3D geometry. 
Volumes in space called cells are defined in terms of their bounding surfaces. Surface 
types include planes, quadrics, and tori (aligned with a major axis). Alternatively, users 
may specify macro-bodies (e.g., sphere, finite cylinder, ellipsoid, etc.), which are 
converted internally into individual surfaces. Boolean operators (union, intersection) and 
surface-senses (inside, outside) may be used with the surface specifications to precisely 
define the volume enclosed by a cell. Cells may also include the complement of another 
cell and may be composed of non-convex or disjoint volumes. The MCNP input file 
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describes the problem geometry, specifies the materials and source, and defines the 
results you desire from the calculation. The geometry is constructed by defining cells 
that are bounded by one or more surfaces. Cells can be filled with a material or be void.  
 An MCNP input file has three major sections: cell cards, surface cards, and data 
cards. A one-line title card precedes the cell card section and MCNP uses a blank line 
delimiter to denote separation between the three different sections. The word “card” 
describes a single line of input of up to 80 characters. A section consists of one or more 
cards.  
The MCNP plots of the GHARR-1 core configuration and the vertical cross-
section are shown in Figure 4.1 and Figure 4.2 respectively.  
 
Figure 4.1: MCNP plot of the core configuration and associated experimental channels 
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Figure 4.2: MCNP plot of vertical cross-section of GHARR-1 reactor (in full power 
mode). 
4.1.1 Auxiliary shutdown system 
As stated earlier, the auxiliary shutdown system is a completely independent 
method of shutting down the reactor in the event of a failure of the control system. This 
system is not automatically actuated at the onset of an abnormal condition. Auxiliary 
shutdown is accomplished by the manual insertion of cadmium capsules into the inner 
irradiation sites. A safer means of inserting the cadmium needs to be designed, as per the 
objective of this study. Furthermore, the shutdown margin for the CR of the LEU core is 
below that of the HEU, and an introduction of an auxiliary rod can increase the shut 
down margin of the LEU core. To achieve this, the constituents of the cadmium capsules 
were studied and their reactivity worth measured using the MCNP code. In the course of 
this, it appeared as though the surface area exposed to the neutrons were a factor to the 
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reactivity worth of the cadmium. Hence further studies were carried out to ascertain this 
fact. Results of this analysis are shown in chapter five. The figures below depict the 
nature and dimension of the cadmium and capsule. 
 
Figure 4.3: Cadmium Capsule 
 
Figure 4.4: Cadmium foil 
 
Figure 4.5: Measurement of Cadmium (Side) 
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Figure 4.6: Measurement of Thickness (a) 
 
 
Figure 4.7: Measurement of Thickness (b) 
 
Table 4.1 Dimension of cadmium in capsule (average along each side)  
Parameter Dimension (cm) 
Length  ~ 4.60 
Width  ~ 4.54 
Thickness ~ 0.05 
First of all, the cadmium capsules which are sent into the reactor core during 
control rod failure were analyzed. The cadmium capsule is made up flat cadmium piece 
with dimensions shown in Table 4.1. The width of each cadmium piece was translated 
into the circumference and the length became the height of a cylinder in the MCNP 
modeling for this analysis.  The depth of the irradiation site in the beryllium reflector is 
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18.75 cm which accommodates the cadmium capsules. This is depicted by the MCNP 
visual editor as shown in Figure 4.8 below: 
Figure 4.8: Cadmium foil in irradiation channel (Visual editor of MCNP) 
For this analysis, six various shapes of cadmium were designed with the MCNP 
code. Three of these designs are such that the irradiation site had to be gouged further 
down by 4.475 cm. This is to make the rod longer for a better reactivity worth of the 
auxiliary rod. The hollow foil is investigated first, followed by a three layer foil arranged 
in concentric circles and then finally a solid rod. Results for auxiliary rod analysis are 
presented in section 5.2. Images from the MCNP visual editor are shown in Figure 4.9 to 
Figure 4.13 below for the arrangements considered with the dimensions for the various 
safety rods are shown in Table 4.2. 
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Figure 4.9 Three concentric foils in unmodified irradiation site (CFU) 
 
Figure 4.10. Solid auxiliary rod (SR) 
Table 4.2 Dimensions of various safety rods 
Safety Rod Thickness (cm) Height (cm) Radius (cm) 
F1 0.50 18.70 0.78 
CFU 0.50 18.70 0.78 
SR -- 18.70 0.78 
F2 0.50 23.12 0.78 
CFM 0.50 23.12 0.78 
SR2 -- 23.12 0.78 
 
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Figure 4.11 Foil with a similar arrangement to the cadmium capsule (F2) 
 
Figure 4.12 Three concentric foils in modified irradiation site (CFM) 
 
Figure 4.13 Solid auxiliary rod (SR2) 
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4.2 REBUS Model for GHARR-1   
An R-Z diffusion theory model using the REBUS-3 code was developed to 
understand the basic fuel depletion characteristics of the GHARR-1 core. Important fuel 
cycle parameters such as core excess reactivity, xenon worth, rate of fuel depletion, and 
fuel cycle length can be obtained by performing simple reactivity rundown calculations. 
In these calculations, a newly loaded fresh core with control rod fully withdrawn is 
depleted at a fixed power level for a given length of time. Because the REBUS-3 RZ-
model can run very fast, key core parameters can be calculated at many small time steps 
to capture the details of temporal variations during fuel depletion. These details are 
important to help understand the basic fuel depletion characteristics of the GHARR-1 
core under different operational schemes. Using RZ model, the core region is 
homogenized. The control rod is positioned at the centre of the core, followed by its 
clad. This is followed by the first ring of fuel which is homogenized as one entity. 
Subsequently, the moderator /coolant between the first and second ring modeled. The 
other set of fuel elements in each ring and adjoining moderator are all placed in a 
concentric pattern. This results in a ten concentric ring of fuel elements as depicted in 
Figure 4. 14. 
In addition to this, the beryllium reflector is placed around the core. Furthermore, 
the reactor vessel is place at the centre of the pool in the REBUS model (not shown in 
figure). 
The REBUS input deck allows for specification of the various regions. The dimensions 
of the various radii of the components of the reactor are specified and the dimensions of 
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the height (Z axis) are also specified. The components are divided into subsections for 
the sake of analyses. The more nodes you desire, the more the axial sections (e.g. the 
beryllium reflector was divided into 5 sub sections). Mesh intervals are specified in the 
deck. The mesh intervals were chosen with respect to the stability of the regions. In 
areas where there are many variations, a finer mesh is used in order to capture every 
detail. The homogenization of every fuel ring is very important. The constituents of the 
reactor components are specified. The code allows for the user to end the run after the 
neutronic calculation or continue doing burnup calculation. In reactors where fuel 
management is practiced, you are able to assign the homogenized fuel to a particular fuel 
management path. The input also contains the number of days the reactor is operated, 
the power at which the reactor is operated, the beginning of a particular stage of 
operation and the end of that particular stage of operation. The deck also specifies when 
the neutronic calculation should be done. In order words, it divides the number of hours 
for which the reactor runs into sub intervals based on the run time of the reactor. 
The total reactor burn cycle time is divided into subintervals depending on the 
analysis. An explicit burnup is performed in each region of the reactor over each of these 
subintervals using the average reaction rates over the subinterval. These average reaction 
rates are based on fluxes obtained from an explicit 3-dimensional diffusion theory 
neutronics solution computed at both the beginning and end of the subinterval. The 
transmutation equations are solved by the matrix exponential technique. The isotopes 
considered in the burnup equations, as well as their transmutation reactions, were 
specified.      
 
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Figure 4.14  RZ model of the reactor core (figure is not drawn to scale); 
A – Vertical cross section;                 B – Horizontal cross section 
4.3 Thermal Hydraulic Analyses   
Both the steady state and transients were analyzed for the two cores under the 
thermal hydraulics. PLTEMP/ANL and PARET code were used for the steady state and 
transient analyses respectively. Some of values obtained were compared with those 
stated in current HEU SAR. 
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4.3.1 Steady State 
Four input data files were used in the PLTEMP/ANL V4.1 code to calculate the 
safety margins in the steady-state operation of GHARR-1 with HEU core. In addition, an 
input file giving the axial power shape of the fuel pin modeled (the average power pin or 
the peak power pin in the HEU core) was also used with the four input data files. 
Another set of four similar input data files were used to calculate steady-state safety 
margins of GHARR-1 with LEU core at both 30 kW and 34 kW. In addition, an input 
file giving the axial power shape of the fuel pin modeled (the average power pin or the 
peak power pin in the LEU core) was also used with each set of the four input data files; 
this is required by the PLTEMP/ANL V4.1. 
One set of input models one (average fuel pin) of the 344 or 348 fuel pins in the 
HEU or LEU core respectively with a reactor power of 15 kW and a coolant inlet 
temperature of 24.5 °C. The pin is modeled as a solid rod of radius 2.15 mm in a 0.6 mm 
thick cladding, without any gap resistance in the case of HEU core. This input data file 
was used to calibrate the hydraulic resistance in the PLTEMP/ANL model to reproduce 
an experimentally measured coolant temperature rise of 13 °C (from 24.5 °C to 37.5 °C).  
Another input data file uses the above determined value of the hydraulic 
resistance coefficient, and models one (average fuel pin) of the pins in the HEU and 
LEU cores respectively when the reactor is operating at the nominal reactor power of 30 
kW.  The purpose of this input data file is to raise and adjust the coolant channel inlet 
temperature so that the coolant exit temperature is 70 °C. The next input data file uses 
the above adjusted values of the hydraulic resistance coefficient and the channel inlet 
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temperature, and models the peak power pin of the core, with six hot channel factors 
(HCF). The purpose of this input data file is to determine the maximum allowed 
operating reactor power with all hot channel factors applied. 
The final set of input data files is identical to the third set of input data files 
except that five of the hot channel factors have been set to 1.0 in order to calculate the 
maximum allowed reactor power without hot channel factors. The hot channel factor for 
power was kept unchanged at its actual value because the ratio of the peak pin to the 
average pin power is certain. Using this input data file, the pin power was raised and 
adjusted so that the minimum ONBR on the cladding outer surface is exactly 1.0. The 
minimum ONBR occurs in axial node 10. When this minimum ONBR is 1.0, the pin 
power multiplied by number of pins gives the maximum allowed operating reactor 
power of the core without hot channel factors. 
Six hot channel factors (defined below) are used in the PLTEMP/ANL V4.1 code 
to calculate research reactor safety margins. These factors are different in natural 
circulation flow from those in forced flow. The basic reason for this is that in natural 
circulation the coolant flow is induced by the power produced in the pin (thus softening 
the effect of pin power on inlet-to-outlet coolant temperature rise) whereas it is not so in 
forced flow. In forced flow, the pressure drop induces the coolant flow [65]. The hot 
channel factors for forced flow over research reactor fuel plates have been formulated 
earlier [66]. The equations relating these factors to manufacturing tolerances and other 
uncertainties for natural circulation over fuel pins are derived in Section 2. Table 4.3 
shows the type of uncertainties included in each of the six hot channel factors. The 
uncertainties of pool water level and pin heated length are not included.  
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Table 4.3 Uncertainties Included in the Six Hot Channel Factors  
No.  Uncertainty Type FPOWER FFLOW FNUSLT FBULK FFILM FFLUX 
1 Neutronics calculation 
of  power density in a 
pin, u1 
   X X X 
2 U-235 mass per pin, u2    X X X 
3 UO2 pellet radius, u3     X X 
4 U enrichment in a 
pellet, u10 
    X X 
5 Fuel pin radius, u12    X X X 
6 Fuel pin pitch, u13    X X  
7 Flow redistribution 
among channels, u6 
   X   
8 Reactor power 
measurement 
uncertainty, u7 
 
X 
     
9 Flow uncertainty due 
to uncertainty in 
friction factor, u8 
  
X 
    
10 Heat transfer 
coefficient uncertainty 
due to uncertainty in 
Nu number correlation, 
u9 
   
X 
   
*1 – 8 are for local or random uncertainties whiles 9 – 11 represent system-wide uncertainties 
System-wide or Global Hot Channel Factors: 
FFLOW =  a factor to account for the uncertainty in total reactor flow 
FPOWER =  a factor to account for the uncertainty in total reactor power    
FNUSLT = a factor to account for the uncertainty in Nusselt number correlation 
Local Hot Channel Factors: 
FBULK = a hot channel factor for local bulk coolant temperature rise  
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FFILM = a hot channel factor for local temperature rise across the coolant film  
FFLUX = a hot channel factor for local heat flux from cladding surface 
4.3.2 Transients Analysis 
PARET: “Program for the Analysis of REactor Transients” code was developed 
for testing methods and models and for subsequent applications in the analysis of 
transient behaviour in research reactors [67]. The code was originally developed for the 
analysis of the SPERT-III experiments for temperatures and pressures typical of power 
reactors [68]. Subsequently, the code has been modified to address some reactor 
thermal–hydraulic analysis including a selection of flow instability, departure from 
nucleate boiling, single and two-phase heat transfer correlations, and flow rates 
encountered in research reactors. Essentially, the code provides a coupled thermal–
hydraulic and point kinetics capability with continuous reactivity feedback, and an 
optional voiding model which estimates the voiding produced by subcooled boiling [69]. 
For PARET applications, the reactor core can be represented by one to four regions. 
Each region may have different power generation, coolant mass flow rate, and hydraulic 
parameters as represented in a single fuel pin with its associated coolant channel. The 
heat transfer in each fuel element is computed on the basis of one-dimensional 
conduction solution, providing for a maximum of 21 axial segments. The 
hydrodynamics solution is also one-dimensional for each of the two channels at each 
time node [70]. The heat transfer could take place by natural or forced convection, 
nucleate, transition, or stable film boiling, and the coolant could range  from subcooled 
liquid through the two-phase regime, and up to and including super-heated steam and 
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allows for coolant flow reversal. The code has been used for transient analysis of 
GHARR-1 [71]. 
4.3.2.1 PARET Thermal Hydraulic Model 
MNSRs transients have been studied by the RERTR program for quite some time now 
[72]. The methodology for thermal-hydraulic investigation has been enhanced to account 
for the transient characteristics of these reactors. MNSR transient analyses based upon 
extensions to modeling capabilities of the PARET/ANL code have been addressed [73]. 
The PARET/ANL version 7.3 is intended for the analyses of research and test reactors 
using plate-type (flat) or pin-type fuel elements. The code has been used for many 
research reactor conversion studies in support of reactor licensing. It is essential, when 
simulating many-hour transients, to account for the primary coolant heatup and 
temperature feedback effects. Results will be shown for transient analysis against 
experiment, obtained in Ghana MNSR.   
The water volume that is actively in the natural convection cooling in the core 
thank (primary loop) is treated as passing through an inlet pipe section, the core, an 
outlet pipe section and a lump volume. The pipe sections are assumed to be 10 % of the 
active primary loop volume each, and are subdivided into 40 nodes. Coolant is tracked 
through these nodes and the lump based upon volumetric flow rate discharged from the 
core. By “tracking”, it is meant that the coolant in each node is subjected to heat and 
mass balance equations which are integrated over a time step typically much longer than 
that needed for the reactor kinetics calculations. A one second step size was used in this 
work. Heat is exchanged through a tank surface area between the primary loop and the 
pool, using a specified heat-transfer coefficient.  This heat transfer coefficient can be 
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obtained by other means, such as from a RELAP model, or simply fitted to the 
experimental results described below. The pool acts as a secondary loop. Pool heat up is 
modeled as a lump receiving heat from the primary loop through the tank surface, and at 
the same time being cooled by air flow over the pool surface causing evaporation. The 
pool air temperature, humidity and velocity control the evaporation rate of the water at 
the pool surface and affect its bulk temperature. These parameters can be specified along 
with the pool surface area, volume and initial temperature. Since the pool temperature 
changes only a few degrees over many hours, it is not necessary to model it in greater 
detail. 
Evaporative cooling from the pool surface is computed using an ASHARE model 
[74]. The evaporation rate is   
p w aw ( / ) A(p p )(0.089 0.0782V/Y kg s   
                        (4.1) 
where  
 A = area of pool surface, m
2 
 
wp = saturation vapour pressure taken at surface water temperature, kPa 
ap =saturation vapour pressure taken at surface air temperature, kPa 
V = air velocity over surface water, m/s 
Y = latent heat required to change liquid to vapour at surface water temperature, kJ/Kg 
The original PARET model for natural convection was based on fitting SPERT 
III transient tests in the “E-core” under no forced flow conditions [75]. These tests used 
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pin-type fuel clad in stainless steel. The coolant passed up through the core components 
and then reversed direction to flow downward through the thermal shields surrounding 
the core. This flow path is quite similar to the GHARR-1. But the active height of the 
SPERT III E-core fuel was 97.3 cm, which is substantially longer than the 23 cm active 
height of GHARR-1 fuel. In addition, the coolant passages were partially isolated in a 
square lattice, with a hydraulic diameter of about 11.9 mm, compared to an open rod 
lattice of 2.25 for a typical MNSR. The pitch/diameter ratios were 1.258 and 2.0 for the 
E-core and MNSR respectively. These differences have large imparts on the 
development of the natural convection flow during reactor operation as they affect the 
driving head, the heat transfer coefficient and the friction losses.  
 
 
 
 
 
 
 
 
 
 
 
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CHAPTER FIVE 
5   Results and Discussion  
The results show that throughout the lifetime of the proposed LEU core: 
1. The shutdown margin meets Operational Limits and Conditions (also referred to 
as Technical Specification Limits). 
2. Reactivity coefficients meet required limits and are comparable to the existing 
HEU core. 
3. Fuel integrity is maintained under all operating conditions. 
4. There will be no tradeoff in the thermal neutron fluxes in the experimental 
channels. This will be achieved by increasing the power of the LEU core by 13 %. 
5.1 Criticality Results 
The core excess reactivity calculated for the LEU UO2 fuel with 344 fuel pins 
was below the 3 mk which is insufficient for the design of MNSR core. An insufficient 
excess reactivity implies the core will become subcritical after relatively short operation.  
Hence the number of pins was increased to 348 to achieve the design reactivity of 
MNSR which is between 3. 5 mk and 4.0 mk. Results are presented in Table 5.1.  
Table 5.1 Comparison of Reactivities for various cores 
Fuel / No. of Pins Multiplication factor, keff Reactivity, mk 
HEU; 344 pins  1.00375 ± 0.00005  3.74±0.05  
LEU; 344 pins  1.00289 ± 0.00006  2.88±0.05  
LEU; 348 pins  1.00385 ± 0.00004  3.84±0.04  
 
 
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A comparison of the criticality data for the HEU and LEU cores are shown in  
Table 5.2. The Multiplication factors, keff, and of course the reactivities are quite 
comparable and also compare well with values stated in the HEU SAR. The delayed 
neutron fractions for the two cores as estimated by Monte Carlo N Particle Code are 3.3 
% and 3.9 % higher than MNSR manufacturer’s quoted value of 0.00808 [20] 
respectively. Nevertheless, the two compares well with the delay neutron fraction of 
0.00837 reported for NIRR-1 [76].  
Table 5.2: Comparison of Criticality Results for HEU and LEU 
Criticality Result HEU SAR     HEU        Sigma        LEU      Sigma 
Keff – Control rod 
completely 
withdrawn 
- 1.00375 0.00005 1.00385 0.00004 
Keff - Control rod 
fully inserted 
- 0.99680 0.00004 0.99714 0.00004 
Core excess 
reactivity,  mk 
4.0 3.74 0.05 3.84 0.04 
Delayed neutron 
fraction (  eff),10
3 
8.5 8.347 0.0641 8.395 0.0566 
Prompt Neutron 
lifetime (Λ), s 
8.52×10-5 8.46×10-5 0.06×10-5 7.39×10-5 0.06×10-5 
Control rod worth, 
mk 
6.80 6.95 0.018 6.74 0.017 
Shutdown margin, 
mk 
3.0 3.21 0.012 2.87 0.011 
 
The design control rod worth of the reactor is 6.8 mk and the shutdown margin is 
3.0 mk for maintaining the reactor in safe shutdown conditions. The total cold excess 
reactivity to be compensated is about 4.0 mk by the control rod [2]. The Monte Carlo 
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calculation of the control rod worth is about 10.5 % more for the HEU core. Both the 
HEU and LEU cores have shutdown margin close to 3 mk.  
5.1.1 Integral and Differential Control Rod Worth 
The exact effect of control rods on reactivity can be determined experimentally. 
For example, a control rod can be withdrawn in small increments, such as 1 cm, and the 
change in reactivity can be determined following each increment of withdrawal. By 
plotting the resulting reactivity versus the rod position, a graph obtained for both cores 
are shown in Figure 5.1. The graph depicts integral control rod worth over the full range 
of withdrawal. The integral control rod worth is the total reactivity worth of the rod at 
that particular degree of withdrawal and is usually defined to be the greatest when the 
rod is fully withdrawn.  
Differential control rod worth is the reactivity change per unit movement of a rod 
and is normally expressed as ρ/cm or ΔK/K per cm. The graph for the Differential 
control rod worth is shown in Table 5.3 for both HEU and LEU cores. The integral rod 
worth at a given withdrawal is merely the summation of the entire differential rod worth 
up to that point of withdrawal. It is also the area under the differential rod worth curve at 
any given withdrawal position. The highest differential control rod worth occurred 
below the centre of the core. 
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Figure 5.1: The Integral Control Rod Curve. 
 
Table 5.3: Differential Reactivity, mk  
Rod Position, cm HEU LEU 
-10 0.3220 0.8754 
-9 0.3519 0.0804 
-6 0.8135 0.8135 
-3 1.0425 1.2228 
-2 0.5005 0.3403 
0 0.7897 0.9195 
2 0.8383 0.7583 
3 0.2692 0.3190 
6 1.0056 1.07499 
9 0.7553 0.49689 
10 0.1291 0.20855 
12.4 0.4664 
0.47635 
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The midpoint of the control rod’s position is designated as 0 cm which happens to be the 
arbitrary geometric centre of the MCNP input deck for the core. 
5.1.2 Flux Distributions  
Measurement of neutron flux and neutron energy spectrum parameters in the 
inner irradiation sites can be utilised to determine linearity, repeatability and stability of 
the neutron measurement system, which includes detectors and secondary instrument. 
The LB1120 miniature fission chamber is employed as a neutron detector for the reactor. 
It has a small size and can be put into the side annulus. In the linear range of this 
detector the absolute neutron flux over 4-5 decades could be measured with both gold 
and manganese foils [2].  
The average flux distributions in the inner irradiation channels, outer irradiation 
channels and that of the fission chambers are shown in Figure 5.2, Figure 5.3 and Figure 
5.4 respectively. The centre of the core is equidistant from the inner irradiation channels 
and the fission chamber which houses the device used in measuring the neutron flux 
experimentally. The various graphs follow the same pattern and also depict the reduction 
in the thermal neutron flux of the LEU core at 30 kW. Both at the inner and outer 
irradiation channels it was observed that the HEU and LEU epithermal fluxes are 
comparable at all points on the graph and the same is observed for the fast fluxes of both 
cores. But as stated earlier, the thermal flux of the LEU core is about thirteen per cent 
less than that of the HEU at the various points. 
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Figure 5.2: Comparison of average flux distribution in inner irradiation channel at 30 
kW. 
 
 
 
Figure 5.3:  Comparison of Flux Distribution in Fission Chamber at 30 kW. 
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Figure 5.4: Comparison of average flux distribution in outer irradiation channel at 30 
kW. 
 
Figure 5.5: Comparison of average flux distribution in inner irradiation channel at 
nominal powers. 
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In order not to compromise the thermal neutron flux especially in the inner 
irradiation channel, the power of the LEU core is increased by 13 % to compensate for 
the fall in flux at 30 kW. The power for LEU core was increased to 34 kW based on the 
average ratio of the thermal neutron flux in the inner irradiation channel at 30 kW of the 
LEU core to that of the HEU core. This is to normalize the thermal neutron flux ratio in 
the inner irradiation channels to unity. So the two profiles of the thermal flux are almost 
completely superimposed on the other as observed in Figure 5.5 above. The effects of 
the increase in power of the LEU core on the neutron fluxes in the other locations are 
shown in Figure 5.6 and Figure 5.7.  
 
Figure 5.6: Comparison of average flux distribution in fission chamber at nominal 
powers. 
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Figure 5.7: Comparison of average flux distribution in outer irradiation channel at 
nominal powers. 
The peak fluxes in the inner irradiation channels are shown in Table 5.4. The 
decreases in the peak fluxes as a result of the core conversion are in the range of 10 % to 
13 % with an average of about 11 %. This supports the increase in power of the LEU 
core by about 13 % to compensate for the decrease in neutron flux estimated.  
Table 5.4:  Peak Flux in the Inner Irradiation Channels (n/cm2s) 
Channels 
(MCNP) HEU  30 kW (n/cm2s) LEU  30 kW (n/cm2s) LEU  34 kW (n/cm2s) 
Cell 971 (1.220±0.0018)E+12  (1.087± 0.0017)E+12 (1.223± 0.0017)E+12 
Cell 933 (1.231± 0.0018)E+12  (1.091± 0.0018)E+12 (1.228± 0.0017)E+12 
Cell 935 (1.217± 0.0018)E+12 (1.100± 0.0018)E+12 (1.238± 0.0018)E+12 
Cell 937 (1.253± 0.0018)E+12 (1.098± 0.0018)E+12 (1.236± 0.0018)E+12 
Cell 939 (1.221± 0.0018)E+12 (1.097± 0.0018)E+12 (1.235± 0.0018)E+12 
Average (1.228± 0.0006)E+12 (1.095± 0.0018)E+12  (1.232± 0.0018)E+12 
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The MCNP5/MCNPX code is capable of computing the axial power profiles of 
the fuel pins in the core. Comparison of peak power profile for the two cores is shown in 
Figure 5.8; the axial power profile of the LEU core at 34 kW is also included. The axial 
power profiles are important for thermal hydraulic analyses, and thermal hydraulic codes 
such as PARET and PLTEMP require both peak and average power profile for 
computation of safety margins, transients, etc. 
 
Figure 5.8: Peak Power Pin Axial Profile (21 Segments) 
 
5.1.3 Reactivity Coefficients 
Reactivity changes are due to changes in the physical properties of the materials 
in the reactor. Reactivity coefficients are useful in quantifying the reactivity change that 
will occur due to the change in a physical property such as the temperature of the 
moderator [77].  The temperature coefficient can conveniently be considered to consist 
of three partial contributions: nuclear temperature coefficient, density temperature 
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coefficient and volume temperature coefficient [78]. Some reactivity coefficients 
evaluated for the core conversion are discussed next.  
The fuel temperature coefficient for both cores at various temperatures is shown in  
Table 5.5. That for the LEU fuel is consistent since the coefficients computed are all 
negative and hence makes the core more inherently stable than the HEU core. 
Table 5.5: Fuel Temperature Coefficient 
Temperature (˚C) HEU  (mk/ ˚C) LEU  (mk/˚C) 
20 - - 
126.85 (1.02±0.001)*10-3 (-6.69±0.005)*10-3 
226.85 (4.80±0.020)*10-4 (-9.57±0.004)*10-3 
326.85 (-9.71±0.002)*10-5 (-10.2±0.003)*10-3 
526.85 (-9.79±0.002)*10-5 (-9.92±0.002)*10-3 
Average 0.000326682 (-9.08±0.002)*10-3 
The moderator (water) temperature coefficient was computed in three ways. 
First, only the temperature card was varied and the results are shown in Table 5.6.  
Secondly, the density of the moderator changes by the introduction of void in 
reactor moderator. As such the density of the water in the coolant cell is changed 
accordingly without the change in temperature. The density was decreased from 0.99825 
g/cm3 to 0.95838 g/cm3 corresponds to temperature increase in the range of 20 ˚C to 100 
˚C respectively, and the void coefficient of reactivity for the HEU and LEU cores in this 
range are show in Table 5.7 
 
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Table 5.6: Moderator temperature coefficient (partial) 
Temperature (˚C) HEU  (mk/ ˚C) LEU  (mk/˚C) 
20 - - 
30 (-6.75±0.049)*10-2 (-3.37±0.026)*10-2 
32 (-4.96±0.030)*10-2 (-3.89±0.025)*10-2 
40 (-6.10±0.052)*10-2 (-3.77±0.015)*10-2 
50 (-1.88±0.008)*10-2 (-3.74±0.010)*10-2 
60 (-6.14±0.017)*10-2 (-3.48±0.007)*10-2 
70 (-5.99±0.013)*10-2 (-3.64±0.006)*10-2 
100 (-6.35±0.007)*10-2 (-3.95±0.004)*10-2 
Average (-5.45±0.020)*10-2 (-3.69±0.008)*10-2 
 
 
Table 5.7: Moderator void coefficient 
Temperature (˚C) HEU  (mk/ ˚C) LEU  (mk/˚C) 
20 0 0 
25 (-1.27±0.021)*10-1 (-1.03±0.017)*10-1 
30 (-1.01±0.009)*10-1 (-0.88±0.007)*10-1 
50 (-1.15±0.003)*10-1 (-1.22±0.003)*10-1 
60 (-1.28±0.003)*10-1 (-1.34±0.003)*10-1 
100 (-1.77±0.002)*10-1 (-1.86±0.002)*10-1 
Average (-1.30±0.008)*10-1 (-1.27±0.007)*10-1 
 
Finally, the temperature and density were both varied simultaneously, as shown in 
the  
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Table 5.8. This is the most viable result amongst the three. 
 
 
Table 5.8: Moderator Temperature coefficient 
Temperature (°C) HEU (mk/°C) LEU (mk/°C) 
32 0 0 
50 -0.15883 -0.18768 
60 -0.18148 -0.20405 
70 -0.39561 -0.40581 
100 -0.24293 -0.25832 
Average -0.24471 -0.26397 
The three means of computing moderator coefficient for both the HEU and LEU show 
comparable results. Water was used as moderator in all cases.  
5.1.4 Worth of the top beryllium reflector 
The purpose of calculating the worths of beryllium shims was to determine the 
reactivity increase by adding different thickness of shim pieces that will compensate the 
reactivity losses due to the burn-up of U-235 and Sm-149 poisoning. The total thickness 
available is 109.5 mm, which corresponds to a total reactivity worth of about 18 mk. 
Figure 5.9 and Figure 5.10 shows the comparison of experimental and calculated shim 
worth of HEU with that of calculated LEU. The deferential reactivity inserted by the 
shim decreases with the increasing thickness of Be shim. 
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Figure 5.9: Reactivity worths of top Be shims for HEU and LEU cores 
 
Figure 5.10: Differential Reactivity worths of top Be shims for HEU and LEU cores 
 
5.2 ANALYSIS FOR AUXILIARY SAFETY ROD FOR LEU CORE 
A similar work has been done for the GHARR-1 HEU Core [79]. The worth of 
the central control rod for the LEU is 6.74 mk (Table 5.2). The control rod worth value 
is achieved because it is placed at the centre of the core. It is impractical to get a similar 
result for a cadmium safety rod placed anywhere else in the surroundings of the core. 
After a couple of trials, the best results for cadmium in a single irradiation channel are 
shown in Table 5.9 and Table 5.10. 
 
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Table 5.9. Auxiliary rod worth, mk for unmodified irradiation site (inner) 
Rod design Multiplication Factor Worth of cadmium capsule 
Reference deck 1.00385 0 
Single layer foil 1.00099 2.85 
Three concentric foil 1.00098 2.86 
Solid rod 1.00092 2.92 
  
The single layer foil is tantamount to the set of cadmium capsules sent into the 
outer irradiation channel when it is necessary. The concentric foil and the solid rod with 
the same height do not give any better worth. Hence the designs with modified 
irradiation sites are more considerable.  
Table 5.10. Auxiliary rod worth, mk for modified irradiation site (inner) 
Rod design Multiplication Factor Worth of cadmium capsule 
Reference deck 1.00385 0 
Single layer foil  1.00042 3.42 
Three concentric foil 1.00031 3.53 
Solid rod 1.00027 3.57 
 
The results in Table 5.10 give cadmium worth higher than that of the cadmium 
capsules, with the best scenario given by the solid cadmium rod. This auxiliary safety 
rod, when fixed in one of the inner irradiation channels, can also be used to increase the 
shutdown margin of the reactor core when not in operation; this is due to the reduction 
in the shutdown margin of the LEU core. 
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5.3 CORE LIFETIME ANALYSIS 
R-Z diffusion theory model using the REBUS-3 code was developed to identify 
the basic fuel depletion characteristics of the GHARR-1 core. Various parts of the core 
were homogenized. Xenon worth, rate of fuel depletion and fuel cycle length were 
obtained by performing simple reactivity rundown calculations. Newly loaded fresh core 
with control rod fully withdrawn is depleted at a fixed power level for a given length of 
time. Cross-sections were generated using WIMS/ANL code. Results of burn-up 
analyses are shown in the Table 5.11. 
The GHARR-1 core has a very small window of excess reactivity (only ~ 1.7 
mk) for operation (between 4.0 mk and 2.3 mk). Based on the above reactivity rundown 
study, it is found that the equilibrium xenon worth is high as compared to the allowed 
excess reactivity window for typical reactor operation. Consequently, GHARR-1 cannot 
be operated continuously to allow the xenon build up to its equilibrium level. As per the 
Table 5.11: Xenon Worth and Reactivity Change Rates 
      Parameters HEU LEU 
15 kW 30 kW 17kW 34 kW 
Equilibrium Xe Worth 
(mk)  
2.013 3.827 1.617 3.099 
Fuel depletion (mk/day)  0.0250 0.0489 0.0198 0.0389 
Hours to operate before 
adding shim 
1631.8 835.1 2059.5 1049.9 
Core Lifetime (years) 46  57  
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design, GHARR-1 can only operate for a very short operational time, no more than ~ 4.5 
hours per day at half power or no more than ~ 2.5 hours per day at full power to avoid 
excessive xenon buildup.    
5.4 THERMAL HYDRAULICS ANALYSES 
5.4.1 Steady State 
The ANL/PLTEMP Code was employed in the Steady State analysis of the two 
cores. The reactor power at minimum onset of nucleate boiling ratio, ONBR=1 without 
hot channel factors is 65.72 kW for the HEU and 67.75 kW for the LEU, furthermore the 
reactor power at ONBR=1 with all six hot channel factors are 51.6 kW and 53 kW for 
the HEU and LEU core respectively. The minimum departure from nucleate boiling ratio 
with all six hot channel factors is 8.9 for the HEU and 8.5 for the LEU core.  
The onset of nucleate boiling ratio (ONBR) and departure from nucleate boiling 
ratio (DNBR) computed so far indicate there is no boiling in both cores. And these 
indicate the limits of operating power for both the HEU and LEU cores. This is the true 
allowed power in the sense that there is no allowance in it for any error in the power 
measuring instrument. This maximum allowed operating power assumes that the power 
measuring instrument is perfect without any measuring error. The results also indicated 
good safety margins so far as the boiling point of the coolant and the melting points of 
both the fuel and clad are concerned. 
Thermal hydraulic parameters obtained from further studies undertaken on both 
the HEU and LEU cores at nominal reactor powers are show in Table 5.12 and Table 
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5.13. The results of the calculations for the clad surface and coolant temperatures using 
an inlet temperature of 30 °C and a coolant pressure of 1 bar are also shown. 
Table 5.12: Comparison of HEU and LEU steady-state parameters using PLTEMP/ANL  
Parameter HEU – 344 rods LEU – 348 rods LEU – 348 rods 
Power (kW) 30.0 30.0 34.0 
Core Flow Rate (Kg/S) 1.1E-3 1.1E-3 1.2E-3 
Peak Fuel Temp. (°C) 104   135 142 
Max. Clad Surface Temp. (°C) 77.3 95.0 98.3 
Max. Coolant Temp. (°C) 53.1 53.4 57.1 
For the LEU core the nominal power is raised to 34 kW in order to meet the flux 
level of 1×1012 n/cm2.s. Hence the computations, using PLTEMP, were done for the 
LEU core at this power and the steady-state parameters were also compared with those 
of HEU and LEU at 30 kW in table 2.12. 
 The melting point of UAl4 and UO2 fuels are 650 and 2800 °C respectively, and 
that of Al and Zirc-4 clad are 600 and 1850 °C respectively. The peak fuel temperature 
for the LEU core is higher than that of the HEU by a factor of 1.37; similarly, the factor 
by which the melting points of the respective fuels are increased is 4.3. The peak clad 
surface temperature is increased by a factor of 1.272 but the melting point of Zirc-4 for 
the LEU is higher than that of Al for the HEU core by a factor of 3.08. These give wider 
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safety margins. Besides, the core is located at a depth of 6 m thus increasing the boiling 
point of water at that pressure to about 113 °C. 
The computer code was used to calculate the thermal-hydraulics parameters of 
the HEU reactor core, which consists of 344 fuel elements generating a total power of 30 
kW. This was repeated for the LEU reactor core with 348 fuel elements. Computed 
outlet temperatures using inlet temperatures at different operating power levels were 
checked against measured values. It can be seen in Table 5.13 that the results measured 
and calculated for HEU compare favourably well. More inlet temperatures were 
computed for the HEU and LEU cores which are also comparable.  
Table 5.13: Comparison of computed coolant outlet temperatures at various powers and 
inlet temperatures 
Power 
(kW)  
Inlet Temp. 
(ºC)  
        HEU Outlet Temp. (ºC) 
                
LEU Outlet           
Temp.(ºC)  
Measured Calculated   
0.3 
32.0 
35.0 
32.6 
- 
33.2 
36.1 
33.2 
36.1 
3 
32.0 
37.0 
39.0 
35.9   
- 
42.0 
37.0 
41.7 
43.6 
37.0 
41.7 
43.6 
15 
30.0 
37.0 
42.0 
-  
48.5 
- 
43.9 
50.0 
54.6 
44.0 
50.2 
54.7 
30 
30.0 
34.0 
37.0 
42.0 
-  
53.0 
-   
- 
51.4 
54.7 
57.3 
61.6 
51.7 
55.4 
57.5 
61.9 
 
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The safety settings of the reactor ensure that protective action will correct an 
abnormal situation before a safety limit is exceeded [2]. For the HEU, the safety system 
settings for reactor thermal power, P, height of water above the top of the core, H, and 
ΔT are as follows: 
P(max) = 36 kW 
H (min) = 465 cm 
ΔT(max) = 21 ºC 
The effect of inlet temperature on temperature difference, as computed by PLTEMP, for 
both HEU and LEU are shown in Table 5.14. 
Table 5.14: Effect of Inlet Temperature on Temperature Difference at Nominal 
Operating Power for the HEU and LEU Cores 
TIN (ºC) 
                 30 kW                     36 kW 
HEU – ΔT (ºC) LEU – ΔT (ºC) HEU – ΔT  (ºC) LEU – ΔT (ºC) 
10 24.10 29.15 27.00 32.28 
15  21.63 27.16 24.20 30.20 
20 20.20 25.59 22.66 28.54 
30 18.60 23.26 20.97 26.03 
35 18.30 22.37 20.63 25.07 
40 18.03 21.61 20.54 24.24 
 
5.4.2 Transient 
PARET code was utilized for the Transient Analysis in order to compare the 
reactor power, fuel temperature and clad temperature for the two cores. Results are 
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shown in Figure 5.11, Figure 5.12 and Figure 5.13 for reactivity insertion of 3.8 mk 
respectively. 
 
Figure 5.11:Time vs. Power  for a 3.8 mk Reactivity Insertion with HEU and LEU Fuel 
 
 
Figure 5.12: Fuel Temperature comparison of HEU and LEU cores for 3.8 mk reactivity 
transient. 
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Figure 5.13: Clap Surface Temperature comparison of HEU and LEU cores for 3.8 mk 
reactivity transient. 
 
Hypothetical reactivity insertions for 6 mk and 8 mk were also done to show the 
inherent safety margin of the LEU core. The results are shown in the Figure 5.14, Figure 
5.15 and Figure 5.16 for the reactor power, fuel temperature and clad temperature 
respectively. 
 
Figure 5.14: Reactor power for reactivity insertions of 3.8 mk, 6 mk and 8 mk for LEU. 
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Figure 5.15: Fuel Temperature for reactivity insertions of 3.8 mk, 6 mk and 8 mk for 
LEU 
 
 
 Figure 5.16: Clad Temperature for reactivity insertions of 3.8 mk, 6 mk and 8 mk for 
LEU 
The peak temperature for the fuel as shown in  
Table 5.15  is far below it melting point of 2800 °C and that of the clad is also far below 
its melting point of 1850 °C, indicating good safety margins. 
 
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Table 5.15: Peak power, peak fuel temperature and peak clad temperature for various 
reactivity insertions 
Reactivity Insertion, mk Peak power, kW Peak Fuel Temp, °C Peak Clad Temp, °C 
3.8 73.5 136 96.1 
6.0 140 200 122 
8.0 350 254 126 
 
 
 
 
 
 
 
 
 
 
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CHAPTER SIX 
6 CONCLUSION AND RECOMMENDATION 
6.1 Conclusion  
Ghana is committed to ensuring the success of the IAEA-RERTR HEU-LEU 
conversion program and  12.5 % enriched UO2 has been chosen as fuel for LEU Core on 
the basis of feasibility studies performed. To achieve a core excess reactivity of 3.5 – 4.0 
mk, 348 fuel pins would be appropriate for the GHARR-1 LEU core. Results indicate 
that flux distribution in the inner irradiation channels will not be compromised if the 
power of LEU core is increased to 34 kW. The properties of UO2 and Zirc-4 will 
increase the safety of the LEU core due to their higher melting points, relative to the 
HEU core. Zirc-4 will increase core lifetime because it is less corrosive. The GHARR-1 
core using LEU-UO2-12.5% fuel can be operated for 23 shim cycles, with cycle length 
2.5 years, for over 57 years at the 17 kW power level.  All 23 LEU cycles meet the ~ 4.0 
mk excess reactivity required at the beginning of cycle. For comparison, the MNSR 
HEU reference core can also be operated for 23 shim cycles, but with a cycle length of 
2.0 years for just over 46 years at the 15.0 kW power level. It is concluded that the 
GHARR-1 core with LEU UO2 fuel enriched to 12.5% and a power level of 34 kW can 
be operated ~25% longer than the current HEU core operated at 30 kW.  Both cores will 
have the same value of thermal neutron flux in their experimental positions. 
On the various designs for the auxiliary safety rod, the solid cadmium rod placed 
in the modified inner irradiation channel gives the highest negative reactivity worth. 
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This makes it more suitable for the intended purpose. This could also be used alongside 
the original control rod to increase the shut down margin.  
The onset of nucleate boiling ratio (ONBR) and departure from nucleate boiling 
ratio (DNBR) computed so far indicated there is no boiling in both cores. And these 
indicate the limits of operating power for both the HEU and LEU cores. This is the true 
allowed power in the sense that there is no allowance in it for any error in the power 
measuring instrument. The results also indicated good safety margins so far as the 
boiling point of the coolant as well as the melting points of both the fuel and clad are 
concerned. 
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6.2 Recommendation  
It is recommended that the NNRI gets its staff acquainted with the reactor core 
conversion steps by sending staffs to the manufacturer in China during the conversion of 
the MNSR prototype. This should include the performance of the Zero Power Test for 
the LEU core. And thereafter the Institute comes up with the steps specifically for 
GHARR-1 for approval by the Regulatory Board.  
Even though it is not likely that the control rod for the LEU will pose any 
challenges, there is the need for further work to ascertain the velocity of the control 
rod’s upward and downward movements both under normal operations and during 
emergency prior to the substitution by one with new dimension. 
After the conversion from HEU to LEU, a number of experiments are 
recommended to ascertain parameters which can be measured experimentally. The 
auxiliary safety rod needs be installed during this period.  
The core lifetime is over forty years and therefore recommended that the water 
purification activity undertaken weekly at the reactor site should be taken seriously since 
corrosion of reactor components have the potential of determining the actual lifetime of 
the reactor system. 
 
 
 
 
 
 
 
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References  
[1]  Y. W. Yan, "Reactor complex of Miniature Neutron Source Reactor Triaining 
Document," China Institute of Atomic Energy, China, 1993. 
[2]  E. H. K. Akaho, S. Anim-Sampong, G. Emi-Reynolds, D. N. A. Dodoo-Amoo and 
T. B. Maaku, "Safety Analysis Report for Ghana Research Reactor -1," GEAC-
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[3]  International Atomic Energy Agency Information Circular , "Project and Supply 
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[4]  E. H. K. Akaho, B. T. Maaku, S. Anim-Sampong, L. Yizheng and Z. Wuqin, "On-
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