University of Ghana http://ugspace.ug.edu.gh COMPUTER SIMULATION OF THERMAL-HYDRAULICS OF MNSR FUEL- CHANNEL ASSEMBLY USING LabVIEW By LEONARD ALELE GADRI (10363740) BSc. Mathematics (KNUST), 1990 This thesis is submitted to the University of Ghana, Legon in partial fulfillment of the requirement for the award of MPhil Computational Nuclear Sciences and Engineering Degree JULY, 2013 University of Ghana http://ugspace.ug.edu.gh DECLARATION I hereby declare that with the exception of references to other people work which have duly been acknowledged, this compilation is the result of my own research work and no part of it has been presented for another degree in this university or elsewhere. …………………………………….. LEONARD ALELE GADRI (Candidate) Date………………………………… I hereby declare that the preparation of this project was supervised in accordance with guidance of the supervision of Thesis work laid down by the University of Ghana. ……………………………. …………………. …………………………… Prof. EDWARD AKAHO NANA (Prof.) A. AYENSU GYEABOUR I. (PRINCIPAL SUPERVISOR) (Co-SUPERVISOR) Date……………………… Date……………………… ii University of Ghana http://ugspace.ug.edu.gh DEDICATION To my children James Nutifafah, Belinda, Brenda, Yayra and Makafui for their support and prayers. iii University of Ghana http://ugspace.ug.edu.gh ACKNOWLEDGEMENT I thank Jehovah for His favor, love and mercy which have enabled me to come out with this work. My sincere thanks go to all the Lecturers in the Department of Nuclear Engineering, more especially to my supervisors Nana (Prof.) A. Ayensu Gyeabour I and Prof. E. H. K. Akaho whose efforts and support have brought this research work to fruition. I dearly recall my course mates, particularly, Mr. Justice Darko, Mr. Robert Nunoo and Mr. Joseph Djangmah for their advice and support. I deem it fit to put on record the advice of my predecessors: Mr. Alex Akoto Danso, Miss Rita Appiah and Mr. Emmanuel Proven Adzri towards the preparation of this work. iv University of Ghana http://ugspace.ug.edu.gh TABLE OF CONTENTS DECLARATION ............................................................................................................... ii DEDICATION .................................................................................................................. iii ACKNOWLEDGEMENT ................................................................................................ iv TABLE OF CONTENTS ................................................................................................... v LIST OF TABLES .......................................................................................................... viii LIST OF FIGURES .......................................................................................................... ix LIST OF SYMBOLS AND CONSTANTS ...................................................................... xi LIST OF ABBREVIATIONS ......................................................................................... xiii ABSTRACT .................................................................................................................... xiv CHAPTER ONE ................................................................................................................ 1 INTRODUCTION ............................................................................................................. 1 1.1 Background of the research ..................................................................................... 1 1.2. Research Problem Statement. ................................................................................. 3 1.3. Justification of the Research Project ....................................................................... 3 1.4. Research Aims and Objectives. .............................................................................. 4 1.5. Scope of Research. .................................................................................................. 5 1.6. Organization of the Thesis. ..................................................................................... 5 CHAPTER TWO ............................................................................................................... 6 LITERATURE REVIEW .................................................................................................. 6 2.1 Generation of heat in nuclear reactors ..................................................................... 6 2.1.1. Thermal energy release during fission ............................................................. 6 2.1.2 Analysis of energy converted to heat. ............................................................... 7 2.1.3 Heat distribution among the different reactor components............................. 8 2.1.4 Volumetric heat generation rate in reactor fuel ................................................ 8 2.1. 5 Thermal Energy Terminologies ....................................................................... 9 v University of Ghana http://ugspace.ug.edu.gh 2.1.6 Spatial volumetric heat generation distribution in the reactor core .................. 9 2.1.7 Determination of the maximum neutron flux. ................................................ 12 2.1.8 Maximum and Average Linear Power of the Fuel Channel ......................... 15 2.2 Thermal-hydraulics of a fuel channel .................................................................... 17 2.3. Gap conductance between the fuel and cladding. ............................................... 19 2.3.1 Open gap. ........................................................................................................ 20 2.3.2. Closed gap. ..................................................................................................... 22 CHAPTER THREE ......................................................................................................... 28 RESEARCH METHODOLOGY..................................................................................... 28 3.1. Overview. .............................................................................................................. 28 3.2 Analytical equations of temperature distribution in the reactor core components28 3.2.1. Coolant temperature along the fuel channel in axial direction. ...................... 28 3.2.2. Temperature of the cladding. ..................................................................... 30 3.2.3. Temperature profile of fuel element .............................................................. 32 3.2.4. Temperature profile in the fuel center .......................................................... 33 3.3. Temperature profile in the radial direction. ...................................................... 35 3.3.1. Fuel temperature profile in the radial direction ............................................ 36 3.3.2. The temperature profile of the cladding in the radial direction. ................. 37 3.3.3. The temperature profile of the coolant in the radial direction ...................... 37 3.4 Summary of equations of the temperature distribution for simulation and visualization ................................................................................................................. 37 3.4.1 Analytical equations of temperature distribution in the axial direction adopted for the simulation and visualization .......................................................... 37 3.4.2 Analytical equations of temperature distribution in the radial direction adopted for the simulation and visualization ........................................................................ 38 3.5. Thermal-hydraulic parameters of MNSR Channel Assembly. ............................ 38 vi University of Ghana http://ugspace.ug.edu.gh 3.6. Computer simulation of axial and radial temperature profile using LabVIEW development platform. ................................................................................................. 40 3.6.1. Simulation of axial temperature profile ........................................................ 42 3.6.2. Simulation of radial temperature profile ..................................................... 43 3.7. Validation by the MATLAB and Excel software programs ................................. 45 3.7.1. MATLAB codes for axial temperature profile ........................................... 45 3.7.2. MATLAB codes for radial temperature profile ............................................ 48 3.7.3 Excel spreadsheet for axial temperature profile ............................................. 49 3.7.4. Excel code for the axial temperature distribution ......................................... 50 3.7.5. Excel spreadsheet for radial temperature profile .......................................... 51 3.7.6. Excel code for the radial temperature distribution .................................... 52 CHAPTER FOUR ............................................................................................................ 53 RESULTS AND DISCUSSIONS .................................................................................... 53 4.1. LabVIEW Simulation of the Mathematical models of temperature distribution in the coolant, outer surface of cladding, fuel surface and fuel center. ........................... 53 4.2. Temperature profile of the coolant, cladding and fuel element. .......................... 55 4.3. The plots of temperature profile of the fuel-channel elements in the axial direction. ..................................................................................................................................... 64 4.4. The plots of temperature profile of the fuel-channel elements in the radial direction ....................................................................................................................... 68 4.5. Discussions ........................................................................................................... 71 4.5.1. Temperature distribution in the axial direction ............................................. 72 4.5.2. Temperature distribution in the radial direction ........................................... 73 CHAPTER FIVE ............................................................................................................. 74 CONCLUSIONS AND RECOMMENDATIONS .......................................................... 74 5.1. Conclusions ......................................................................................................... 74 5.2. Recommendations ............................................................................................... 77 REFERENCES ................................................................................................................ 79 vii University of Ghana http://ugspace.ug.edu.gh LIST OF TABLES Table Caption Page 2.1 Emitted and recoverable energies for fission of U-235 7 2.2 Distribution of heat released among different components in MNSR 8 3.1 Thermal-hydraulic parameters of the MNSR Channel Assembly 39 4.1 The axial temperature distribution in the coolant (T) , outer surface of the cladding ( ), fuel surface ( ) and fuel center ( ) using MATLAB 57 4.2 Excel calculation of axial temperature distribution in the fuel channel using spreadsheet 59 4.3 Location of maximum temperature in the components of the fuel channel 61 4.4 The radial temperature distribution in the fuel using MATLAB 61 4.5 The radial temperature distribution in the sheath/cladding using MATLAB 62 4.6 The radial temperature distribution in the coolant using MATLAB 62 4.7 Radial temperature distribution in the fuel, clad and coolant using spreadsheet 63 viii University of Ghana http://ugspace.ug.edu.gh LIST OF FIGURES Figure Caption Page 2.1 Radial and axial thermal neutron distribution obtained with eqn. (2.4) and with two neutron energy groups. 11 2.2 Fuel channel with single rod 15 2.3 Temperature profile in the region and temperature jump 21 2.4 Closed fuel-cladding gap 23 2.5 Four basic arithmetic operation functions that were used in the LabVIEW simulation 25 2.6 The For Loop programming structure in LabVIEW 26 2.7 Mathematical functions in LabVIEW 27 3.1 Fuel channel 30 3.2 Definition sketch that is used in the determination of radial temperature of radial temperature in a fuel –sheath-coolant of nuclear recover channel 36 3.3 Bundle function used in LabVIEW simulation 40 3.4 Build array used in LabVIEW simulation 41 3.5 Front panel view developed in LabVIEW to display the axial temperature profile 42 3.6 Block diagram developed in the LabVIEW to display the axial temperature profile 43 3.7 Front panel view developed in LabVIEW to display the radial temperature profiles 44 3.8 Block diagram developed in the LabVIEW to display the radial temperature profile 45 ix University of Ghana http://ugspace.ug.edu.gh 4.1 Plot of temperature in degrees Celsius against axial height in m with = 30 kW 53 4.2 Plot of temperature profile in the reactor fuel-channel elements for values of (i) 5kW (ii) 10kW (iii) 15 kW and (iv) 25 kW. 54 4.3. Plots of temperature in degrees celcius against radial distance in mm 55 4.4 The axial temperature distribution in the coolant 65 4.5 The axial temperature distribution in the outer surface of the cladding 66 4.6 The axial temperature distribution in the fuel surface 67 4.7 The axial temperature distribution in the fuel center 67 4.8. Combined temperature distribution in the fuel channel (coolant, outer surface of cladding, fuel surface and fuel center) 68 4.9 Radial temperature distribution in the fuel 69 4.10 Radial temperature distribution in the sheath/cladding 69 4.11 Radial temperature distribution in the coolant 70 4.12 Combined temperature distribution in the fuel, cladding and coolant. 71 5.1 The axial temperature distribution in channel (1) Coolant, (2). Outer surface of cladding, (3) Fuel surface and (4) Fuel centerline 74 5.2 Radial temperature profile across a fuel rod and coolant channel. 76 x University of Ghana http://ugspace.ug.edu.gh LIST OF SYMBOLS AND CONSTANTS Symbol Meaning Units (E) Fission cross-section as a function of neutron Energy. barns Φ(r,z) Neutron flux as a function of neutron Energy neutrons/ Number of fissile atoms per unit volume atoms/ (r) Volumetric heat generation MW/ (E) Macroscopic fission cross-section barns Zero order Bessel function of first kind First order Bessel function of first kind Maximum heat generation rate within reactor kW Total power of the reactor kW Total fuel Channel Power kW Mean free path in the gas m Total conductance W/ ℃ Heat flux kW/ ℃ Q Power density W/ Heat transfer W/ ℃ Radius of fuel m Radius of cladding m z Axial height m μ Viscosity of the gas Ns/ V Volume of the reactor core Gap conductance W/ ℃ Mass flow rate kg/s T Temperature (or temperature of the coolant) ℃ xi University of Ghana http://ugspace.ug.edu.gh τ time s Convective heat transfer W/ ℃ C Specific heat J/kg Specific mass of fuel kg/ Specific mass of cladding kg/ σ Stefan-Boltzman constant Temperature of fuel center line ℃ Temperature of fuel surface in axial (or in the fuel in the radial) directions ℃ Temperature at outer surface of fuel ℃ Temperature in the cladding in the radial direction ℃ Temperature in the coolant in the radial direction ℃ Thermal conductivity of the fuel W/m ℃ Thermal conductivity of the of the cladding W/m ℃ xii University of Ghana http://ugspace.ug.edu.gh LIST OF ABBREVIATIONS AP 1000 Westinghouse AP 1000 (Advanced Passive) Nuclear Reactor GHARR-1 Ghana’s Research Reactor 1 LabVIEW Laboratory Virtual Instrumentation Engineering Workbench MATLAB Matrix Laboratory MNSR Miniature Neutron Source Reactor PC Personal Computer xiii University of Ghana http://ugspace.ug.edu.gh ABSTRACT A LabVIEW simulator of thermal hydraulics has been developed to demonstrate the temperature profile of coolant flow in the reactor core during normal operation. The simulator could equally be used for any transient behavior of the reactor. Heat generation, transfer and the associated temperature profile in the fuel-channel elements viz: the coolant, cladding and fuel were studied and the corresponding analytical temperature equations in the axial and radial directions for the coolant, outer surface of the cladding, fuel surface and fuel center were obtained for the simulation using LabVIEW.. Tables of values for the equations were constructed by MATLAB and Excel software programs. Plots of the equations with LabVIEW were verified and validated with the graphs drawn by the MATLAB. In this thesis, an analysis of the effects of the coolant inlet temperature of 24.5 ℃ and exit temperature of 70.0 ℃ on the temperature distribution in fuel- channel elements of the reactor core of cylindrical geometry was carried out. Other parameters, including the total fuel channel power, mass flow rate and convective heat transfer coefficient were varied to study the effects on the temperature profile. The analytical temperature equations in the fuel channel elements of the reactor core were obtained. MATLAB and Excel software were used to construct data for the equations. The plots by MATLAB were used to benchmark the LabVIEW simulation. Excellent agreement was obtained between the MATLAB plots and the LabVIEW simulation results with an error margin of 0.001. The analysis of the results by comparing gradients of inlet temperature, total reactor channel power and mass flow indicated that inlet temperature gradient is one of the key parameters in determining the temperature profile in the MNSR core. xiv University of Ghana http://ugspace.ug.edu.gh CHAPTER ONE INTRODUCTION 1.1 Background of the research Nuclear power reactor produces nuclear energy in a controlled manner. Nuclear reactors are used for either research or power production. A power reactor is designed to produce heat for use in driving steam turbines to generate electricity. A research reactor is designed to produce beams of radiation for experimental applications. The heat produced is a waste product and is dissipated as proficiently as possible by the coolant. Since the nuclear energy renaissance, there has been a mounting interest for nuclear professionals which in turn has given rise to the creation of nuclear science and engineering education programs at universities in a number of countries, including Ghana. The increase in the number of students undergoing nuclear degrees at the various levels need deeper understanding of the phenomena in connection with nuclear fission by way of simulation in other to assist in their education and training endeavors. A reactor is an enormously complex machine. A number of things can go wrong with the reactor undetected. One malfunction will lead to another and then to a series of others until the core of the reactor begins to melt. A highly trained nuclear engineer may not know how to respond by way of putting the situation under control. [1] In nuclear reactors, the fission is the origin of the thermal energy generation and much smaller source comes from non-fission process due to neutron capture in the fuel, 1 University of Ghana http://ugspace.ug.edu.gh coolant, moderator and structural materials. The coolant affects significantly, the operating temperature and pressure, the size of the core, and methods of fuel handling. The thermal energy released in the fuel rods is transferred by heat conduction to the surface of the rod and then by convection to the coolant which circulates around the rods. In the process, the coolant transports the thermal energy released by fission in the reactor to external heat exchangers called steam generators of nuclear reactors. Research reactor has received much attention in recent years due to its redesign and installation in a number of countries which wanted to supplement their power supply by nuclear power in future. Research reactor has many uses and building research reactor has also been investigated as a potential start of building power reactors. Thermal-hydraulics are computed on the basis of a one or multidimensional core description. Lumped parameters models for thermal-hydraulics were generally considered only for qualitative studies of reactor dynamics. In these simplified models temperature effects were accounted for through a single effective temperature for the whole core. The temperature effects were described by considering four distinct temperature regions, corresponding respectively to the coolant, outer surface of cladding, fuel surface and fuel center. This enables reactivity feedback to include all the major contributions, namely, moderator temperature effects, cladding temperature effects and fuel temperature effects. In line with the point model concept, coolant, clad and fuel temperatures were assumed to be functions separable in space and time. The space dependence was postulated to be the static distribution corresponding to a one-dimensional (in the axial direction) core. 2 University of Ghana http://ugspace.ug.edu.gh 1.2. Research Problem Statement Computer simulation of the temperature profile in the coolant, clad and the fuel of an MNSR was undertaken in order to examine and visualize the thermal-hydraulics of the nuclear reactor by LabVIEW development platform, which was later validated by MATLAB and Excel programs. Nuclear reactor is in fact one of the most sophisticated and complex systems man has ever made which is associated with very high risks due to nuclear fission. Considering the immense uses and benefits of nuclear reactors which outweighed the risks, it has become very essential to have in depth understanding of the behavior of the reactor core during operation and of the technological solutions inherent in the thermal design of nuclear reactors. 1.3. Justification of the Research Project A number of countries including Ghana, in their energy policy plan to adopt nuclear energy has taken giant steps towards operating research reactors, as a starting point in acquiring nuclear power plants. The GHARR-1 MNSR research reactor has been well managed and researches have been conducted to acquire knowledge and monitor the operations. Critical research has been on thermal-hydraulics, both engineering and analytical and numerical computations. The present research dealt with computer simulations of thermal-hydraulics which would serve as teaching and learning simulator. Apart from this research being used as a teaching and learning aid, more can be learnt of the temperature distribution in the reactor core which can: (a) keep alive the proposal by the government of Ghana to install nuclear power plant by the year 2020. 3 University of Ghana http://ugspace.ug.edu.gh (b) bring about the awareness of technological solutions of nuclear reactor challenges to ward off fear. (c) make this civilized idea of going nuclear by 2020 a reality and give it further impetus. Besides, though extensive research has been carried out on the temperature profile in the reactor core, investigation revealed that LabVIEW has not been applied to simulate the temperature distribution in the components of the reactor core. 1.4. Research Aims and Objectives. The main aim of this research was to simulate the mathematical models for the temperature distribution in the components of the MNSR fuel channel assembly by using LabVIEW development platform. The objectives were to: (a) find out, whether or not the proposed temperature changes in the fuel-channel elements will put the operation of the reactor outside the operational limits and conditions. (b) state the rate of heat generation and removal in relation to possibility of overheating or insufficient cooling. (c) predict the behavior of the research reactor. (f) train reactor operators and students in nuclear reactor systems using this research simulator for teaching and learning methodology. 4 University of Ghana http://ugspace.ug.edu.gh 1.5. Scope of Research. The research reactor type whose fuel rods are placed in channels which run through the reactor core is considered. The coolant circulates around the rods. The reactor core is cylindrical in shape and surrounded by lateral and axial reflectors. The reflectors prevent the leaking neutrons from escaping but return them to the reactor core. To enable the prediction of the behavior of reactor core it requires a sound knowledge of the temperature distribution in the reactor fuel channel assembly during the steady state and transient conditions. Consequently, the temperature distributions in the radial and axial directions in the cylindrical fuel element are considered but for steady state conditions only. 1.6. Organization of the Thesis. The thesis is organized in the following way. The first Chapter presents the background, justification and the objectives of the research. Chapter 2 describes the literature review. Chapter 3 illustrates the research methodology. Detailed results and discussions are dealt with in chapter 4. Finally, the conclusions and recommendations are summarized in chapter 5. 5 University of Ghana http://ugspace.ug.edu.gh CHAPTER TWO LITERATURE REVIEW A large number of researches have been carried out on the thermal-hydraulics and heat generation studies of nuclear reactor cores. 2.1 Generation of heat in nuclear reactors 2.1.1. Thermal energy release during fission The thermal energy released during fission is mainly due to fission process and much smaller amount due to non-fission neutron capture in the fuel, moderator, coolant and structural materials [2] A total of about 207 Mev of energy is emitted during fission. The distribution of this energy among different components of a neutron-induced fission of U-235 is shown in Table 2.1 which also includes the energy converted into heat or the recoverable energy. Apparently, the emitted energy and the recoverable energy are not the same. This is due to the fact that for the total emitted energy during the fission process, the kinetic energy of the fission fragments, neutrons, gammas and betas are converted into heat in the reactor core and shielding materials while the neutrinos which accompany β-decay escape from the reactor virtually without any interaction . 6 University of Ghana http://ugspace.ug.edu.gh 235 Table 2.1. Emitted and recoverable energies for fission of U [2]. Components Emitted Energy Energy Converted to heat or recovered energy Mev Mev % of total Fission fragments 168 168 84 Neutrons 5 5 2.5 Prompt rays 7 7 3.5 Delayed radiations: Β-rays 8 8 4 γ-rays 7 7 3.5 Neutrinos 12 - - Capture γ-rays - 5 2.5 TOTAL 207 200 100 Moreover, a new item which consists of capture γ-rays has been created. The capture γ-rays or radioactive capture reactions account for (v-1) neutrons per fission (v is the average number of neutrons emitted per fission; v is approximately equal to 2.5 for U- 235) which are absorbed in the reactor without producing any fission reaction. About 3 to 12 Mev of capture γ-radiation is produced per fission. An average value of 5 Mev is included in Table 2.1 and all this γ-ray is recoverable for the reactor system. 2.1.2 Analysis of energy converted to heat. Nearly 84 % and 4 % of the energy converted to heat are due to the fission fragments and to the β-rays, respectively. The very short range of fission fragments: mm from the fission site and β-rays less than 1mm ensures that this heat release takes place within the fuel elements. The energy release during neutron thermalization takes place in the moderator. The energy release due to neutron capture occurs largely 7 University of Ghana http://ugspace.ug.edu.gh within the fuel and a small portion of this energy is released in the structural material, in the moderator and in the coolant as shown in Table 2. 1. 2.1.3 Heat distribution among the different reactor components In line with the thermal design of nuclear reactors, it can be assumed that 94 % of the fission energy release takes place in the in the fuel, 5.2 %, 0.6 % and 0.3 % in the moderator, coolant and shielding respectively of an MNSR as illustrated in Table 2.2 Table 2.2 Distribution of heat release among different components in a research reactor [2] Component Released Thermal Energy, % Energy Released in fuel 93.9 Energy Released in coolant 0.6 Energy released in moderator 5.2 Energy released in the Shielding 0.3 Total 100.0 2.1.4 Volumetric heat generation rate in reactor fuel Given fission cross-section as a function of the neutron energy, , the neutron flux as a function of the neutron energy and space ф( ) and the number of fissile atoms per unit of volume , the volumetric heat generation rate in the reactor fuel can be expressed as [3] (2.1) 8 University of Ghana http://ugspace.ug.edu.gh where is the recovered fission energy (200 Mev) and is the part of this energy deposited in the fuel. Substituting for which is the macroscopic fission cross-section of the fuel, eqn. 2. 1 can be written as, (2.2) In thermal reactors most of the fission are induced by thermal neutrons. Thus is simply equal to the macroscopic fission cross-section of thermal neutrons and ф( ) is represented by ф( ) which is the thermal neutron flux. Therefore eqn. (2.2) becomes (2.3) 2.1. 5 Thermal Energy Terminologies The thermal energy terminologies used in the Thesis are defined below [4]: (a) Power density of the core is the thermal energy released per unit volume of the core, kW/ . (b) Specific power of the fuel describes the thermal energy released per unit mass of the fuel, kW/kg or MW/ton, and (c) Power density of the fuel also is used to describe the thermal energy released per unit volume of the fuel, MW/ . 2.1.6 Spatial volumetric heat generation distribution in the reactor core Equation (2.3) shows that the distribution of heat generation rate in the reactor is determined by the distribution of the neutron flux throughout the region of the core 9 University of Ghana http://ugspace.ug.edu.gh which contains the fuel. Since an accurate neutron flux density distribution determination is a complicated process, only cylindrical reactors with axial and radial symmetry are considered. In addition, the assumption is that the fuel elements have a constant enrichment and are uniformly distributed throughout the core. This assumption allows the treatment of the reactor core as homogeneous reactor and determines the overall features of the neutron flux distribution. However, homogeneous reactor approach can still be used to get an idea on the overall neutron flux pattern. Using a one group neutron approach, the thermal neutron flux distribution in the core region of a homogeneous reactor with radial and axial reflectors can be approximated by: . (2.4) where is the flux at the center of the reactor, is zero-order Bessel function of the first kind, r and z are the radial and axial coordinates and with reference to eqn. (2.1) vary between O and R, and 1/2H and -1/2H, respectively. R and H are the radius and the height of the reactor core and and are the effective radius and height of the core including an allowance for the reflector. R and H are respectively smaller than . 10 University of Ghana http://ugspace.ug.edu.gh Fig 2.1 Radial and axial thermal neutron distribution obtained with eqn. (2.4) and with two neutron energy groups. [4]. The average flux in the core region is given by: (2.5) Substituting ф in equation (2.4) into eqn. (2.5) and integrating yields [5]: (2.6) where is the first order Bessel function . Equation (2.3) indicates that the heat generation rate or the power density of the core is proportional to the thermal neutron flux. For that reason equation (2.6) also gives the ratio of the heat generation rate where is the maximum heat generation rate within the reactor. In the case of bare reactor, where R= and H= the reciprocal of the ratio given by eqn. (2 .6) becomes 11 University of Ghana http://ugspace.ug.edu.gh Therefore for a reactor equipped with radial and axial reflectors, it can be assumed that In this light the value of the ratio is in the region of 2.35. For a uniform radial neutron flux, the same ratio has the value of 1.57. These simple calculations show that the maximum neutron flux or heat generation rate in the core may be substantially higher than the average neutron flux or heat generation rate. In reactor design, a high ratio is not desirable. In a given reactor, it may be possible to increase the power output by decreasing this ratio. In other words, the power output of the reactor can be increased by a more uniform distribution of the heat generation rate. This is usually referred to as flattening of the heat generation distribution or of the neutron flux [5]. 2.1.7 Determination of the maximum neutron flux. The maximum thermal neutron flux for a homogeneous reactor consisting of axial and radial reflectors is determined by using equations (2.3) and (2.4). Using these two equations the total power of the reactor can be written as [4], (2.7) 12 University of Ghana http://ugspace.ug.edu.gh where is the macroscopic fission cross-section of the thermal neutrons. As the reactor is homogeneous, the assumption is that all the recovered fission energy, , is deposited in the core. In this case α ≈ 1. Considering the fact that (2.8) then equation (2.7) can be integrated to obtain (2.9) Making the subject, eqn. (2.9) becomes (2.10) By making the assumption for R/ =H/ = 0.83, eqn. (2.10) now becomes (2.11) where V is the volume of the core. Thus a reactor with axial and radial reflectors, we have =2.35 and the total power of the reactor from eqn. (2.10) can be written as: (2.12) Therefore, Substituting eqn. (2.11) into eqn. (2.4) the distribution of the thermal neutron flux becomes, (2.13) 13 University of Ghana http://ugspace.ug.edu.gh Equation (2.13) can also be used to approximate the flux in a reactor which consists of fuel assemblies of n rods, provided that the value of is computed for an equivalent homogeneous mixture. Suppose that there are N fuel assemblies in the reactor core and each assembly consists of n rods of fuel radius and length H. Also taking as the macroscopic cross-section of the fuel, the total fission cross-section of the entire core is: while the average value of in the core is given by: . (2.14) Substituting equation (2.14) into (2.13) yields; . (2.15) The distribution of the heat sources in the reactor is obtained by introducing equation (2.15) into eqn. (2.3) and taking as being equivalent to ; therefor . (2.16) Equation (2.16) is the global distribution of the heat sources or power density in the reactor core. The dependence of on the variable r gives the variation of the power density from rod assembly to rod assembly in the core but not across any individual assembly nor across the fuel rod shown in figure 2.2 14 University of Ghana http://ugspace.ug.edu.gh Fig 2.2 Fuel channel with single rod Assume as being constant in the fuel assembly and in the fuel rod even though this assumption is arguable for fuel assemblies any errors in the heat transfer calculations especially for small diameter natural uranium or low enrichment fuel rods used in most power reactors and MNSRs are not significant [11]. 2.1.8 Maximum and Average Linear Power of the Fuel Channel Fuel channel which consists of cylindrical fuel rods inserted in a circular cross-section channel is considered. To obtain the highest possible heat transfer from the reactor and achieving the maximum fuel temperature below the permissible value, the heat transfer should be increased. For each fuel rod, the heat transfer surface may be increased either by dividing it into a number of small diameter rods or by adding fins on the cladding of the rod. 15 University of Ghana http://ugspace.ug.edu.gh Considering the neutron flux distribution given by equation (2.16), that the maximum power density is at the center of the core where r=0 and z =0, and both functions in eqn. (2.16) are unity and the maximum value is; . Hence the power density distribution throughout the reactor can be stated as; (2.17) From equation (2.17), the power density distribution in a fuel channel located at a given radial distance is given by; (2.18) where Equation (2.18) can be expressed conveniently as: (2.19) Using equation (2.19) the average power density in a fuel channel can be written as: (2.20) The liner power distribution along the channel is obtained by multiplying equation (2.20) by the cross sectional area of the fuel; (2.21) where n is the number of the rods in the fuel assembly. By integrating eqn. (2.21) along the channel gives the total fuel channel power as; (2.22) By taking (2.23) Equation (2.22) then becomes (2.24) 16 University of Ghana http://ugspace.ug.edu.gh 2.2 Thermal-hydraulics of a fuel channel The thermal-hydraulics equations consist of the energy conservation equations applied to the coolant, the cladding and to the fuel with appropriate boundary conditions. To determine the coolant energy conservation equation, a single fuel channel is considered as shown in Fig. 2.2 with control volume laterally limited by the channel wall and the fuel, and axially by the planes z and z+ dz. The analysis on a single fuel rod can be easily extended to a multi-rod fuel bundle provided that a strong mixing exists between the laterally interconnected sub-channels bounded by the fuel rods or by the fuel rods and the channel wall. The application of energy conservation principle to the chosen control volume gives equation of the form, (2.25) or (2.26) for energy and mass conservation where is heat on the fuel element surface, kW/ C, A is channel flow area ( ), s is heated perimeter (m), ρ is fluid specific mass ( kg/ ), u is specific internal energy ( kJ/kg), h is specific enthalpy ( kJ/kg), τ is time ( s). 17 University of Ghana http://ugspace.ug.edu.gh Assuming that the heat conduction in the axial and angular directions are negligible compared to that in the radial direction, the following conduction equations holds for the fuel element [6]. (2.27) and for the cladding: (2.28) where: is conductivity of the fuel, usually a strong function of the temperature, . conductivity of the cladding, it may be taken as constant ( ) specific mass ( f-fuel, c-cladding ) (kg/ ) c is specific heat (J/kg) is power density (W/ ) The control variables and boundary conditions for eqns. (2.27) and (2.28) are: 1. Inlet mass flow rate (kg/s) 2. Inlet temperature (°C) 3. Inlet pressure ( Pa or MPa) 4. Heat convection at the fuel element surface: (2.29) where is convective heat transfer ( ), is the temperature on outer surface of cladding (°C), T is temperature of the coolant (°C), is radius of the fuel element (m). 18 University of Ghana http://ugspace.ug.edu.gh Heat transfer at the fuel-cladding interface: = ( ( ) - ( ) (2.30) where: gap conductance, W/ C, temperature at the surface of the fuel, °C temperature at the inner surface of the cladding, °C 6. Continuity of the heat fluxes at the fuel-cladding interface: , . (2.31) 2.3. Gap conductance between the fuel and cladding. In manufacturing the fuel rods, a gap of about 0.08 mm is created between the outer surface of the fuel and the inner surface of the cladding in order to insert easily the fuel pellets into the cladding tubes. In some reactors graphite powder is used to facilitate the insertion of fuel pellets. The cladding tubes are filled at atmospheric pressure with an inlet gas such as helium to avoid corrosion and assure a reasonable initial heat transfer [20]. Since the gap is so small, the convection currents cannot develop in the gas. The heat transfer in the gap region is by conduction through the filling gas. Because of the swelling and the thermal expansion of the fuel, the gap region closes and the direct contact between the surfaces at several discrete points take place. Consequently, the heat transfer by conduction at these points should be taken into account when modeling the heat transfer through the gap region [7]. The gap heat transfer are usually expressed in terms of a gap heat transfer coefficient or conductance, . Taking the temperature on the fuel surface and on the inner 19 University of Ghana http://ugspace.ug.edu.gh surface of the cladding as and respectively, then the linear heat flux across the gap is: = 2 . (2.32) The gap heat transfer coefficient (gap conductance) has been considered for two general cases of open and closed gaps [8]. 2.3.1 Open gap. If the fuel and cladding is not in physical contact, which is true for fresh fuel or fuel operating at very low linear power rate, the fuel stands freely within the cladding. In this case the heat transfer mechanisms are conduction through the filling gas and radiation. If the space between the fuel and the cladding is larger than the mean free path of the atoms at the prevailing temperature and pressure, the gap conductance, considering also the heat exchange by radiation between the exposed surfaces, is given by [9] + (2.33) is the thermal conductivity of the gas, is the gap thickness, σ is the Stefan-Boltzman constant, and are the surface emissivities of the fuel and cladding respectively. In cases of small gaps where the temperature gradient is also sustained, a steep change in the gas temperature is observed in the region close to the solid surfaces. The variation of the gas temperature in the gap region is a steep change that takes place within a mean free path from the walls and is called the temperature jump denoted by distances and in figure (2.4) [10]. 20 University of Ghana http://ugspace.ug.edu.gh Fig 2.3 Temperature profile in the region and temperature jump [10]. The extrapolation of the temperature profile in the bulk of the gas intersects the fuel and cladding temperature in the solids at distances and respectively. These distances are called temperature jump distances and should be added to the actual gap thickness in order to predict the correct fuel –cladding temperature difference by using the Fourier conduction law. The conductance for narrow gaps of few mean free path is then given by [9] = + (2.34) [10] gives the temperature jump distance as: = 2( )( )( ) . (2.35) where is mean free path in the gas, г is the radial distance change and is the ratio of for the gas, is specific heat at constant pressure, μ is velocity of the gas and k 21 University of Ghana http://ugspace.ug.edu.gh the conductivity of the gas. α is the thermal accommodation coefficient of the gas in contact with a solid surface, α = (2.36) is the temperature of molecules that strike the solid, is the temperature of the solid and is the temperature of the reflected molecules. The mean free path is given by: (2. 37) where T is the absolute temperature in Kelvin, p is gas pressure in bar and is the property of the gas that depends on the molecular or atomic diameter [10] 2.3.2. Closed gap. In practice, because of the fuel swelling, the differential expansion of the fuel and the pressure exerted by the coolant on the outside surface of the cladding, the gap tends to close. As a result the gap reduces and because of the roughness of the fuel and cladding surfaces, solid-to-solid contact between them will be established. Under this condition, heat transfer in the gap region occurs through the points of solid contact and across the discontinuous gas gap between these points of contact. Figure (2.4) shows how the closed gap may look like. 22 University of Ghana http://ugspace.ug.edu.gh Fig. 2.4. Closed fuel-cladding gap. The different components of conductance in a closed gap are [20] (a). Gas conductance through discontinuous gap, . (b). Conduction through the solid-to-solid contact points . (c). Radiation through the discontinuous gap According to the work of Cetinkale and Fisherden [8] the following relationship for the conductance due to solid-to-solid contact is established. (2.38) where A dimensional constant, (=10 ). conductivity of the fuel, . conductivity of the cladding, surface contact pressure, . Meyer hardness number of softer material . root- mean-squre of contact material surface rughness and is given by: m 23 University of Ghana http://ugspace.ug.edu.gh where are the surface roughness of the interface material, m. Ross and Stoute [6] assumed that the thickness of a closed gap is related to the surface roughness as: (2.39) Where C is a constant which depends on the interface pressure. The proposed correction for this constant is: (2.40) where is the pressure at the interface in N/ . Substituting eqn. (2.39) into eqn. (2.34) and neglecting the radiation term yields; (2.41) The total conductance, of a closed gap is obtained by adding eqns. (2.38) and (2.41) and considering heat exchanges by radiation between the fuel and cladding: + + (2.42) 2.4. LabVIEW Simulation The cornerstone of modern engineering studies and practices is the investigation of the engineering system process by simulation. LabVIEW was used to simulate the temperature distribution in the reactor fuel- channel elements based on analytical mathematical models derived from the generation of heat in nuclear reactor fuel. 24 University of Ghana http://ugspace.ug.edu.gh 2.4.1. General Procedure From the front panel window, a numeric control button was created to input the power to generate the axial temperature profile. The control was linked directly in the block diagram to provide the necessary data for computation of the temperature values. (i) (ii) (iii) (iv) Fig 2.5 Four basic arithmetic operator functions used in the LabVIEW simulation Figure 2.5 shows the four basic arithmetic operator functions that were used in developing the governing relations. These functions accepted only two varying data types and produced a single output. Figures 2.5 (i), (ii), (iii) and (iv) represents the multiplication, division, addition and subtraction functions respectively. 25 University of Ghana http://ugspace.ug.edu.gh 2.4.2. For Loop Programming Structure Fig 2.6 The For Loop programming structure in LabVIEW Figure 2.6 shows the For loop programming structure used to generate the range of values representing the distance along the fuel element. The structure executed a sub-diagram n times, where n is the value wired to the count (N) terminal. The iteration (i) terminal produced the current loop iteration count which ranges from 0 to n-1. For the simulation, an axial length of 0.23 m of fuel element was divided into two where the ranges -0.115 m to 0.0 m represented the lower half and 0.0 m to 0.115 m the upper half. 26 University of Ghana http://ugspace.ug.edu.gh (i) (ii) (iii) Fig 2.7 Mathematical functions in LabVIEW The LabVIEW functions for sine, cosine and natural logarithm were also employed and functions accepted only one input to produce a single output. The input for the functions was either a 1-D array of values or a single data type value. Figure 2.7 (i), (ii) and (iii) represent the sine, cosine and natural logarithmic functions used in Lab VIEW. 27 University of Ghana http://ugspace.ug.edu.gh CHAPTER THREE RESEARCH METHODOLOGY 3.1. Overview There has been limited research work carried out on computer simulation of the temperature distribution in the MNSR reactor core. Numerical experimentation work was conducted to model and simulate the governing equations which relate to temperature distribution in the components of the reactor core by LabVIEW in order to improve on the existing knowledge and to advance the understanding of behavior of the reactor core due to temperature changes (steady state and transient, etc). 3.2 Analytical equations of temperature distribution in the reactor core components Four analytical equations of the temperature distribution in the coolant, outer surface of the cladding, fuel surface and fuel center in the axial direction and three in the coolant, cladding and fuel in the radial direction were derived for simulation. 3.2.1. Coolant temperature along the fuel channel in axial direction. Assuming steady state conditions, equation (2.26) can be expressed as: (.3.1) or (3.2). where is the mass flow rate and is the linear heat flux or power density whose variation along the channel is given by eqn. (2.21). Substituting in eqn. (3.2) into 28 University of Ghana http://ugspace.ug.edu.gh eqn. (2.21) and integrating the resulting equation between the inlet of the channel located at z = -½H and a given location z, illustrated by Fig. (3.1), gives; (3.3) or (3.4) By substituting in equation (2.22) into equation (3.2) yields: . (3.5) Taking equation (3.5) can be expressed in the temperature form as: (3.6) or recalling that the total channel power is given by: (3.7) The coolant temperature can be expressed as: (3.8) where and are the coolant enthalpy and temperature at the channel exit respectively. 29 University of Ghana http://ugspace.ug.edu.gh Fig. 3.1 Fuel channel 3.2.2. Temperature of the cladding. The temperature of the outside surface of the cladding is obtained by using Newton cooling law [6]: (3.9) where is the outside temperature of the cladding, is the transfer coefficient and s is the heated perimeter. Substituting eqns. (3.8) and (2.21) into eqn. (3.9) and taking into account eqns. (2.23) and (2.24) yields : = + sin2β + cos2 (3.10) or = + (sin2β + cos2β (3.11) 30 University of Ghana http://ugspace.ug.edu.gh where (3.12) by substituting for the expression obtained in eqn. (3.7). Taking derivative of equation (3.11), equating to zero and solving yields: (3.13) It follows that the maximum temperature is then located at: with the value of = + [sin (arctan + cos (arctan ] (3.14) The maximum cladding temperature should be less than the maximum allowable temperature with a reasonable safety margin. As an example, the maximum allowable temperature for Zircaloy-4 cladding ranges from [6]. The temperature distribution in the cladding in the radial direction at a given axial location is determined by using eqn. (2.28). Considering steady state condition, this equation becomes with (3.15) where heat conduction in the angular and axial directions are neglected compared to that in the radial direction. Integrating eqn. (3.15) yields; (3.16) To determine the constants A and B, take the inner surface of the cladding subject to a constant linear heat flux, whereas the outer surface is kept at the temperature thus the boundary conditions can be expressed as: 31 University of Ghana http://ugspace.ug.edu.gh r = T = The constants A and B are evaluated as (3.17) (3.18) Therefore the variation of the temperature through the cladding is given by: with (3.19) Finally, the inner surface temperature of the cladding is obtained by setting (3.20) 3.2.3. Temperature profile of fuel element Combining eqns. (3.20) and (2.32), an expression for the surface temperature of the fuel is obtained as; (3.21) The variation of is given by eqn. (3.11). Replacing this equation into the above equation and taking into consideration equations (2.21) and (2.22) we obtain : = + [sinβ + ( + In + ) cos2β (3.22) By letting = + In + ) (3.23) Equation (3.22) can be expressed as: (3.24) 32 University of Ghana http://ugspace.ug.edu.gh Differentiating equation (3.24) with respect to z to obtain the location of the maximum temperature as: (3.25) (3.26) Therefore maximum temperature is: (3.27) 3.2.4. Temperature profile in the fuel center To determine the temperature distribution in the fuel and analyze the variation of fuel centerline temperature along the channel, the following assumptions were made: (a) The neutron flux within the fuel pellet and heat generation rate are uniform. (b) No angular variation in convective heat transfer coefficient and in the gap conductance no significant angular temperature gradient exists in the fuel. (c) The axial conduction of heat is small compared to that in radial direction which is confirmed by length to diameter ratio being higher than 20. (d) The steady state conditions prevail, [11, 21 ]. Based on the above conditions equation (2.27) can be written as : (3.28) Integrating equation (3.28) gives; (3.29) Assuming that the fuel has an internal cavity of radius which is not cooled and there is no heat flux at , a corresponding boundary condition is written as: 33 University of Ghana http://ugspace.ug.edu.gh (3.30) Therefore the constant A can be determined: (3.31) Integrating eqn. (3.29) from r to which is outside the fuel diameter gives: [ - ( In( ] (3.32) In the case of a fuel pellet where = 0 the eqn. (3.32) becomes; (3.33) A relationship between the temperature at the center , at the surface and the radius of the fuel can be obtained by substituting in eqn. (3.33) r =0; (3.34) Equation (3.34) is equivalent to (3.35) where is the linear power or heat flux which is given by . For a constant conductivity equation (3.33) can be integrated to obtain the radial temperature distribution in the fuel; [ ] (3.36) The fuel center temperature of the fuel is obtained by putting r=0; (3.37) Using equations (2.21), (2.23), (2.24) and (3.24) the variation of the fuel center temperature along the channel is obtained: 34 University of Ghana http://ugspace.ug.edu.gh (3.38) where (3.39) The maximum centerline temperature is located at: (3.40) and the maximum value is: + (3.41) 3.3. Temperature profile in the radial direction. The appropriate steady-state heat transfer relations needed in determining the radial temperature profile of nuclear reactor channel involves conservation of energy in a cylindrical differential volume element in the fuel : (3.42) and the Fourier’s law of heat conduction (3.43) where T(r, z) – local medium temperature °C k – thermal conductivity of the medium, W/ – vector heat flux, W/ – power density, W/ The power density is zero everywhere except in the nuclear fuel region. In the fuel region it is defined in terms of the fission process: (3.44) 35 University of Ghana http://ugspace.ug.edu.gh Figure 2.6 illustrates the material composition of the fuel rod, viz fuel, sheath- cladding, coolant cell. The thermal conductivities in each material have been determined. We indicate the local temperature at specific radial coordinates [12]. Fig. 3.2. Definition sketch that is used in the determination of radial temperature profile in a fuel – sheath –coolant of nuclear reactor channel. 3.3.1. Fuel temperature profile in the radial direction Using equations (3.42) and (3.43) yields; (3.45) Restricting the analysis to a specific z-coordinate and taking the thermal conductivity, , and nuclear power density, , as constants permits the integration of eqn. (3.45) twice to obtain the radial fuel temperature, in terms of two constants of integration. These constants are determined by imposing the conditions that is finite in the fuel and equal to , temperature at the fuel sheath interface [20]. 36 University of Ghana http://ugspace.ug.edu.gh Thus (3.46) a quadratic function in r. 3.3.2. The temperature profile of the cladding in the radial direction. The heat flux in the radial direction is, however, linear in r; (3.47) Since no energy is generated in the sheath, is zero in the sheath and we write eqn. (3.42) as: (3.48) and finally obtain temperature profile in the sheath: . (3.49) 3.3.3. The temperature profile of the coolant in the radial direction The mean temperature in the coolant, , may be represented with the aid of Newton’s law of cooling by (3.50) where h is the film coefficient of heat transfer. Therefore, we write (3.51) 3.4 Summary of equations of the temperature distribution for simulation and visualization 3.4.1 Analytical equations of temperature distribution in the axial direction adopted for the simulation and visualization 1. Coolant: 37 University of Ghana http://ugspace.ug.edu.gh 2. Outer surface of the cladding: we can call the underlined expression: 3. Fuel surface: = + [sinβ + ( + In + ) cos2β alternatively + In + ) = 4. Fuel center: [ where 3.4.2 Analytical equations of temperature distribution in the radial direction adopted for the simulation and visualization 5. Fuel: 6. Cladding (sheath): 7. Coolant: = 3.5. Thermal-hydraulic parameters of MNSR Channel Assembly. The thermal-hydraulic parameters used to simulate the mathematical model of temperature variation equations and fuel flow in the channel are given in 38 University of Ghana http://ugspace.ug.edu.gh Table 3.1.Thermal-hydraulic parameters of the MNSR Channel Assembly[ 4, 13, 18] - Total Channel Power range = 0 – 30 kW. - mass flow rate = 0.448 kg/s - specific heat at constant pressure = - inlet temperature = C. 1.4717 kJ/kgK z- axial axis range = 0 – 0. 23 m. - exit temperature = C. R- radius of reactor core = 0.115 m. H- core height = 0.23 m. - core height including the reflector = - core radius including the reflector = 0.276 m. 0.138 m. s- heated perimeter =0.391 m. - heat transfer coefficient = 15 W/ K. - radius of the fuel = 0.00215 m - radius of the cladding = 0.00275 m - thermal conductivity of the fuel, W/ C – thermal conductivity of the cladding, = 0.15. W/ C =15. - surface contact pressure, N/ =1.96 Psi. - conductivity of the pure gases = 5 , m- temperature jump distances for W/mK. gas, helium = 10x -root-mean square of contact material -temperature at the inner surface of roughness = 0.37. cladding= 86 . - surface roughness of the interface -temperature at the outer surface of material, m = 0.37 each. cladding = 30 ℃. Q – power density = 30 W/ q – heat flux. = 1628 W/ Other constants derived for the thermal-hydraulic parameters are: 1. 2. total conductance of a closed gap : = 466K = 193 Substituting these constants into the equations of the temperature distributions in the axial direction, the equations simplifyied to: (3.52) 39 University of Ghana http://ugspace.ug.edu.gh (3.53) 3.6. Computer simulation of axial and radial temperature profile using LabVIEW development platform. . To plot the governing equations (3.6), (3.10), (3.22) and (3.38) on a single graph, each data set for the relation was bundled before finally being used in the graph. The bundle function accepted a number of inputs of varying data types which produced a single output for onward processing. For the axial temperature profile, two 1-D array data type was used as input for each of the bundle functions. The main function of the bundle was to assemble a cluster from individual elements. Figure 3.3 depicts the bundle function used for the computer simulation. Fig 3.3 Bundle function used in LabVIEW simulation 40 University of Ghana http://ugspace.ug.edu.gh Fig 3.4 Build Array used in LabVIEW simulation After the governing relations has been transformed using the various functions and structures the values from the bundle function for each of the governing relation were passed onto a single function known as build array function which function concatenated multiple arrays or appended elements to an n-dimensional array. The function assembled all the results of the governing relations for the axial temperature profiles for the coolant, outer surface of the cladding, fuel surface and fuel center into a single container and then passed the output for further processing to the multiple plot graphs. Figure 3.4 shows the build array function in LabVIEW. 41 University of Ghana http://ugspace.ug.edu.gh 3.6.1. Simulation of axial temperature profile Fig 3.5 Front panel view developed in LabVIEW to display the axial temperature profiles The front panel developed in LabVIEW to display the axial temperature profiles is shown in Figure 3.5. The temperature (in degrees celsius) was located on the abscissa of the graph and the ordinate of the graph recorded the axial length of the fuel presented in (m). The legend for the various temperature profiles was shown on the top-right corner of the graph. Finally, the numeric control button, which accepted the value of power in kilowatts was also displayed. 42 University of Ghana http://ugspace.ug.edu.gh Fig 3.6 Block diagram developed in LabVIEW to display the axial temperature profiles The block diagram for the simulation of the axial temperature profile is presented in figure 3.6. The various arithmetic functions and structures were combined so as to depict the governing relations developed for the axial temperature profile. 3.6.2. Simulation of radial temperature profile The various arithmetic operations and structures used in axial simulator were also used in determining the radial temperature profile. 43 University of Ghana http://ugspace.ug.edu.gh Three for loop structures were developed for the simulation and were used to generate the various intervals for the fuel meat, clad and the coolant. The governing equations were then developed using the various LabVIEW functions and structures. Figure 3.7 was the LabVIEW block diagram representation for the relations governing the radial temperature profile. A similar approach as with that of the axial temperature profile was employed by assembling the various governing relations for the fuel meat, cladding and the coolant into three separate bundle functions and then passing them to the build array for graphical processing. The block diagram for the radial profile had temperature in degrees celsius on the ordinate of the graph and the radial distance in millimeters on the abscissa. Fig 3.7 Front panel view developed in LabVIEW to display the radial temperature profiles 44 University of Ghana http://ugspace.ug.edu.gh Fig 3.8 Block diagram developed in LabVIEW to display the radial temperature profiles 3.7. Validation by the MATLAB and Excel software programs For the purpose of verifying and validating the values of temperature distribution in the components by LabVIEW both MATLAB and Excel were used. 3.7.1. MATLAB codes for axial temperature profile Listed below are the relevant thermal-hydraulic parameters and constants used including the resulting MATLAB codes for the temperature profile in the axial direction. % initialization of thermal-hydraulic parameters used in simulating in % the axial direction. z = [-0.115:0.005:0.115]; % range and interval along the z-axis = 24.5; % inlet temperature = 70; % exit temperature pc = 30; % total channel power cp =1.4717; % specific heat at constant pressure 45 University of Ghana http://ugspace.ug.edu.gh m = 0.448; % mass flow rate H =0.23; % core height Beta = 1.31; % constant = H/ (2 ) H = 0.276 ; % height including the reflector. hc = 15; % heat transfer coefficient s = 0.391; % heated perimeter %%***************************************************************** % Computing terms in the main equation MATLAB codes for the equations in the axial direction VAL1 = (pc/2*m*cp) ; VAL2 = sin (2*beta.*z/H)/sin (beta)+1; VAL3 = (Ti+Te)/2; VAL4 = (Te-Ti)/2*sin (beta); VAL5 = sin (2*beta.*z/H); VAL6 = (pc*beta*cos (2*beta.*z/H))/ (hc*s*H*sin (beta)); VAL7 = (1.4581*pc*cos (2*beta.*z/H))/ (H*(Te-Ti)); VAL8 = (2.4720*pc*cos (2*beta.*z/H))/ (H*(Te-Ti)); %-------------------------------------------------------------------------- 1. T = Ti+VAL1*VAL2; 2. = VAL3+VAL4*VAL5+VAL6; 3. = VAL3+VAL4*(VAL5+VAL7); 4. = VAL3+VAL4*(VAL5+VAL8); The plotting routine of the equations 1. figure; plot (T, z, 'm’); xlabel (‘ Temperature ℃.’ ), ylabel (‘Axial Height’); title (‘graph of axial temp. distribution in the coolant’) 2. figure; plot (Tc ,z, 'b-.'); 46 University of Ghana http://ugspace.ug.edu.gh xlabel (‘ Temperature ℃.’ ), ylabel (‘Axial Height’); title (‘graph of temp. distribution in the outer surface of the cladding’); 3. figure; plot (Tf ,z, 'g--'); xlabel (‘ Temperature ℃.’ ), ylabel (‘Axial Height’); title (‘graph of temp. distribution in the fuel surface’); 4. figure; plot (To ,z, 'r-*'); xlabel (‘ Temperature ℃.’ ), ylabel (‘Axial Height’); title (‘graph of temp. distribution in the fuel center’); The combined plot figure; plot (T, z, 'm', Tc, z,' b-.' Tf , z, 'g--', To, z,' r-*'); xlabel (‘ Temperature ( Degrees Celcius)’ ), ylabel (‘Height of Reactor Core (m)’); title (‘combined temp. distribution in the coolant, outer surface of the cladding, fuel surface, fuel center’); %-------------------------------------------------------- %The thermal-hydraulic parameters used in simulating in the radial direction. %ri, rj and rk are distances from fuel element center (m). %-------------------------------------------------------------------------- 47 University of Ghana http://ugspace.ug.edu.gh 3.7.2. MATLAB codes for radial temperature profile The relevant thermal-hydraulic parameters and constants used including the resulting MATLAB codes for the temperature profile in the radial direction were: %The thermal-hydraulic parameters used in simulating in the radial direction. %ri, rj and rk are distances from fuel element centre (mm). ri = [0.0:0.05:2.15]; % range and interval for the fuel meat of radius 2.15 mm; rj = [2.15:0.05:2.75]; % range and interval from the inner to outer surfaces of cladding; rk = [2.75:0.05:2.9]; % width of the coolant channel; h = 15; % heat transfer coefficient; rf = 2.15; % radius of the fuel; rc = 2.75; % width of the cladding; kf = 0.15; % conductivity of the fuel; kc = 15; % conductivity of the cladding; q = 1628; % heat flux; Q = 30; % power density; Ta = 86; % temperature at inner surface of cladding; Tb = 30; % temperature at the outer surface of cladding; % codes for the equations in the radial direction. MATLAB codes for the equations in the radial direction. 1. Tf = Ta+(Q/(4*kf))*(rf.^2-ri.^2); 2. Td = Ta-((rf*q)* reallog (rj/rf)/kc); 3. Th = Tb-(Q*rk.^2)/(2*h* rc); The plotting routine of the equations. 1. figure; plot(ri,Tf,'r'); xlabel(‘Radial distance (mm)’,ylabel(‘Temperature ℃’); title (‘graph of radial temp. distribution in the fuel’); 48 University of Ghana http://ugspace.ug.edu.gh 2. figure; plot(rj,Tc,'g--'); xlabel(‘Radial distance (mm)’,ylabel(‘Temperature ℃’); title (‘graph of radial temp. distribution in the sheath(cladding)’); 3. figure; plot(rk,Th,'b-'); xlabel(‘Radial distance (mm)’,ylabel(‘Temperature ℃’); title (‘graph of radial temp. distribution in the coolant’); The combined plot figure; plot(ri,Tf,'r', rj,Td,'g--', rk,Th,'b-'); xlabel(‘Radial Distance (mm)’,ylabel(‘Temperature (Degrees Celcius)’); title (‘ Combined radial temp. distribution demarcating clearly each region ’); legend (‘Fuel Rod’,’ Cladding’, ‘Coolant’) 3.7.3 Excel spreadsheet for axial temperature profile (1) In cells A1, B1, C1, D1 , E1 , F1, G1, H1 and I1 were entered Te, pc, beta, m, cp, H, te, hc and s respectively. (2) In cells A2, B2, C2, D2, E2, F2, G2, H2, I2 and J2 the following values were entered respectively: 24.5, 30, 1.31, 0.448, 1.4717, 0.23, 70, 15, 0.391 and 0.005 for the step size (3) Name cells A5, B5, C5, D5 and E5 as z, T, respectively. (4) Entered in cell A6 the value -0.115 and Enter pressed. Place the cell highlight in cell A6 and highlight the block of cells A6 to A52 by holding down the mouse button 49 University of Ghana http://ugspace.ug.edu.gh and wiping the highlight down to cell A52. Click the Edit command on the Command bar and point at Fill from the drop-down menu. Select Series from the next drop-down menu and accept the default Step value of 0.005 by clicking OK in the series window. This yielded -0.115, -0.11, -0.105, -0.1, -0.095, …, 0.115. (5) In the cell B6 was entered the formula = $A$2+($B$2/(2*$D$2*$E$2))*((SIN(2* $C$2*A6/$F$2)/SIN($C$2)+1)) (6) The cell highlight was placed in cell B6. The Edit command was clicked and Copy was selected from the drop-down menu. The contents of cell B6 were copied to the clipboard. Now the cell highlight was placed in cell B7 and the block cells from B7 to B52 were highlighted. The Edit command was clicked again but now Paste was selected from the drop- down menu. The result was 24.8825, 25.3376, 25.8637, 26.4592, 27.1221,…, 70.0013. (7) Similarly, in the cells C6, D6 and E6 the under-listed formulas for were entered respectively and the processes in (7) were repeated. 3.7.4. Excel code for the axial temperature distribution T =$A$2+($B$2/(2*$D$2*$E$2))*((SIN(2* $C$2*A6/$F$2)/SIN($C$2)+1)); =(($A$2+$G$2)/2)+($G$2$A$2)*(SIN(2*$C$2*A6/$F$2))/(2*SIN($C$2))+($B$2 *$C$2*COS(2*$C$2*A6/$F$2))/($H$2*$I$2*$F$2*SIN($C$2)); = (($A$2+$G$2)/2)+ (($G$2$A$2)/(2*SIN($C$2)))*(SIN(2*$C$2*A6/$F$2)+((1.4581*$B$2*COS(2*$C$ 2*A6/$F$2)))/($F$2*($G$2-$A$2))); (($A$2+$G$2)/2)+(($G$2$A$2)/(2*SIN($C$2)))*(SIN(2*$C$2*A6/$F$2)+((2.4 72*$B$2*COS(2*$C$2*A6/$F$2)))/($F$2*($G$2-$A$2))); 50 University of Ghana http://ugspace.ug.edu.gh 3.7.5. Excel spreadsheet for radial temperature profile (1) In cells A1, B1, C1, D1 ,E1 and F1 were entered ri, rj, rk, Tf, Td and Th respectively. (2) In cells M2, M3, M4, M5, M6, M7, M8, M9 and M10 were named h, rf, rc, kc, kf, q, Q, Ta and Th respectively. (3) Cells N2 to N10 contained the corresponding values of the hydraulic parameters in (3) above: 15, 2.15, 2.75, 15, 0.15, 1628, 30, 86, and 3.0 (4) In cells A1, B1 and C1 were entered ri, rj and rk. highlight the block of cells A2 to A45 by holding down the mouse button and wiping the highlight down to cell A45. Click the Edit command on the Command bar and point at Fill from the drop-down menu. Select Series from the next drop-down menu and accept the default Step value of 0.05 by clicking OK in the series window. This yielded 0.00, 0.05, 0.10, 0.15, 0.20 ,…, 2.15. (5) Same process was carried out in columns B and C starting with the initial values of 2.15 and 2.75 in cells B2 and C2 respectively. (6) In the cell A45 was entered the formula : =$N$9+ ($N$8/(4*$N$6))*(($N$3^2) - (A45^2)) (7) The cell highlight was placed in cell A45. The Edit command was clicked and Copy was selected from the drop-down menu. The contents of cell A45 were copied to the clipboard. Now the cell highlight was placed in cell A44 and the block cells from A44 to A2 were highlighted. The Edit command was clicked again but now Paste was selected from the drop- down menu. The result was 317.125, 317.000, 316.625, 316.000, 315.125, …, 86.000. (8) Similarly, in the cells B2 and C2 were entered the under-listed formulas for and respectively and the processes in (7) were repeated. 51 University of Ghana http://ugspace.ug.edu.gh 3.7.6. Excel code for the radial temperature distribution = $N$9+ ($N$8 / (4*$N$6))*(($N$3^2) - (A2^2)); = ($N$9) - ((($N$3*$N$7) / $N$5)*L N (B2 / $N$3)); = ($N$10) - (($N$8 / (2*$N$2*$N$4))*(C2^2)); Excel has been used to calculate and analyze the temperature values at discrete points in each of the components of the reactor core. Tables 4.2 and 4.7 This was done to ascertain the phenomenon and structural behavior of the reactor core under normal operation using different procedures. In table 4.2 the location of maximum temperatures in the four components in the axial direction were marked with asterisk. This is displayed in table 4.3 52 University of Ghana http://ugspace.ug.edu.gh CHAPTER FOUR RESULTS AND DISCUSSIONS 4.1. LabVIEW Simulation of the Mathematical models of temperature distribution in the coolant, outer surface of cladding, fuel surface and fuel center. Shown in figures 4.1 and 4.3 are the plots of the equations in the axial and radial directions using LabVIEW for the given sets of initial conditions. The combined plots of temperature profile in the rector components in the axial direction using LabVIEW and MATLAB were very identical. The LabVIEW and MATLAB plots of temperature profile in the components in the radial direction were also similar. Compare figures 4.1 with 4.8 for axial plots and 4.3 with 4.12 for the radial plots. LabVIEW Simulation (Axial direction) FRONT PANEL Fig.4.1. Plot of temperature in degrees celsius against axial height in m. with = 30 kW 53 University of Ghana http://ugspace.ug.edu.gh Figure 4.2 are the graphs that were the visual simulation of the temperature profile in the four components in the axial direction with varying values of (i) 5 kW, (ii) 10 kW, (iii) 15 kW and (iv) 25 kW Axial Temperature Profile at Power=5kW Axial Temperature Profile at Power=10kW Axial Temperature Profile at Power=15kW Axial Temperature Profile at Power=25kW Fig 4.2 Plot of temperature profile in the reactor components for values of (i) 5 kW (ii) 10 kW (iii) 15 kW and (iv) 25 kW 54 University of Ghana http://ugspace.ug.edu.gh LabVIEW simulation (radial direction) FRONT PANEL Fig.4.3 Plots of temperature in degrees celcius against radial distance in mm 4.2. Temperature profile of the coolant, cladding and fuel element. The axial temperature distribution of the coolant, outer surface of the cladding, surface of the fuel and fuel center were obtained using eqns. (3.6), (3.10), (3.22) and (3.38) respectively and the corresponding results presented in Table 4.1 55 University of Ghana http://ugspace.ug.edu.gh The radial temperature distribution in the fuel, cladding and the coolant were also constructed using eqns. (3.46), (3.49) and (3.51) and the data are shown in Tables 4.4, 4.5 and 4.6 respectively. Table 4.1 Axial temperature distribution in the coolant (T), outer surface of the cladding ( ), fuel surface ( ) and fuel center ( ) using MATLAB for: T = , and z = 0.0 to 0.115 m , and z = 0.0 to 0.115m = + [sinβ + ( + In + ) cos2β , and z = 0.0 to 0.115 m = + [sinβ + ( + In + ) cos2β ], and z = 0.0 to 0.115 m 56 University of Ghana http://ugspace.ug.edu.gh Table 4.1 The axial temperature distribution in the coolant, outer surface of cladding, fuel surface and fuel centerline using MATLAB Height of Temperature ℃ reactor core(m) Z T -0.115 24.500 33.788 49.703 66.177 -0.110 24.883 35.791 55.075 75.036 -0.105 25.338 37.830 60.422 83.805 -0.100 25.864 39.901 65.726 92.456 -0.095 26.459 41.996 70.970 100.960 -0.090 27.122 44.107 76.137 109.289 -0.085 27.850 46.229 81.210 117.418 -0.080 28.642 48.354 86.174 125.319 -0.075 29.493 50.476 91.011 132.967 -0.070 30.402 52.586 95.706 140.336 -0.065 31.366 54.68 100.244 147.404 -0.060 32.381 56.750 104.610 154.147 -0.055 33.444 58.789 108.790 160.544 -0.050 34.553 60.790 112.770 166.573 -0.045 35.702 62.747 116.538 172.214 -0.040 36.889 64.654 120.081 177.451 -0.035 38.109 66.505 123.389 182.266 -0.030 39.360 68.293 126.449 186.642 -0.025 40.635 70.013 129.252 190.567 -0.020 41.932 71.660 131.790 194.026 0.015 43.247 73.227 134.052 197.010 --0.010 44.574 74.710 136.034 199.508 -0.005 45.910 76.103 137.728 201.512 0.000 47.251 77.403 139.128 203.016 57 University of Ghana http://ugspace.ug.edu.gh 0.005 48.591 78.606 140.230 204.015 0.010 49.927 79.83 141.031 204.505 0.015 51.255 81.589 141.528 204.485 0.020 52.569 82.364 141.718 203.955 0.025 53.866 83.026 141.603 202.918 0.030 55.142 83.572 141.181 201.375 0.035 56.392 84.000 140.455 199.332 0.040 57.612 84.308 139.427 196.796 0.045 58.799 84.497 138.099 193.775 0.050 59.949 84.564 136.477 190.279 0.055 61.057 84.511 134.566 186.320 0.060 62.120 84.337 132.371 181.909 0.065 63.136 84.042 129.900 177.061 0.070 64.099 83.629 127.162 171.792 0.075 65.008 83.097 124.164 166.120 0.080 65.860 82.449 120.916 160.062 0.085 66.651 81.687 117.430 153.638 0.090 67.379 80.813 113.716 146.869 0.095 68.042 79.830 109.787 139.777 0.100 68.638 78.742 105.655 132.384 0.105 69.164 77.551 101.333 124.716 0.110 69.619 76.263 96.836 116.797 0.115 70.001 75.025 92.178 108.652 58 University of Ghana http://ugspace.ug.edu.gh Table 4.2. Excel calculation of axial temperature distribution in the fuel channel using spreadsheet. pc Beta m cp H hc s 24.5 30 1.31 0.448 1.4717 0.23 70 15 0.391 z T -0.115 24.5 32.27506 49.87802 67.5248 -0.11 24.88252 34.30342 55.63264 77.01495 -0.105 25.33758 36.37377 61.36007 86.40858 -0.1 25.86372 38.4794 67.04175 95.67521 -0.095 26.45921 40.61346 72.65924 104.7848 -0.09 27.12213 42.76906 78.19432 113.7078 -0.085 27.85034 44.93918 83.62904 122.4152 -0.08 28.64146 47.1168 88.94578 130.8789 -0.075 29.49294 49.29485 94.12729 139.0713 -0.07 30.402 51.46627 99.15678 146.966 -0.065 31.36571 53.62402 104.0179 154.5373 -0.06 32.38094 55.76109 108.6949 161.7606 -0.055 33.44439 57.87056 113.1727 168.6125 -0.05 34.55262 59.94558 117.4367 175.0709 -0.045 35.70203 61.97944 121.473 181.1147 -0.04 36.88889 63.96552 125.2686 186.7243 -0.035 38.10936 65.89739 128.8112 191.8816 -0.03 39.35948 67.76878 132.0893 196.5699 -0.025 40.63518 69.57363 135.0922 200.7738 -0.02 41.93235 71.30608 137.8102 204.4799 -0.015 43.24676 72.96051 140.2345 207.676 -0.01 44.57416 74.53156 142.3573 210.3519 -0.005 45.91023 76.01413 144.1716 212.4988 0 47.25066 77.40341 145.6716 214.1098 0.005 48.59108 78.6949 146.8524 215.1796 0.01 49.92716 79.88441 147.7102 215.7048 0.015 51.25456 80.96808 148.2421 215.6836 59 University of Ghana http://ugspace.ug.edu.gh 0.02 52.56897 81.94239 148.4465 215.1162 0.025 53.86613 82.8042 148.3227 214.0044 0.03 55.14184 83.55069 147.8712 212.3518 0.035 56.39195 84.17945 147.0933 210.1637 0.04 57.61242 84.68845 145.9915 207.4472 0.045 58.79929 85.07603 144.5696 204.2112 0.05 59.9487 85.34093 142.832 200.4662 0.055 61.05693 85.4823 140.7844 196.2243 0.06 62.12038 85.49967* 138.4335 191.4992 0.065 63.1356 85.39299 135.7869 186.3062 0.07 64.09931 85.16261 132.8531 180.6623 0.075 65.00838 84.80927 129.6417 174.5858 0.08 65.85986 84.33412 126.1631 168.0962 0.085 66.65098 83.7387 122.4286 161.2147 0.09 67.37918 83.02494 118.4502 153.9637 0.095 68.04211 82.19516 114.2409 146.3665 0.1 68.6376 81.25204 109.8144 138.4479 0.105 69.16373 80.19865 105.185 130.2335 0.11 69.61879 79.0384 100.3676 121.7499 0.115 70.00132* 77.77506 95.37802 113.0248 60 University of Ghana http://ugspace.ug.edu.gh Table 4.3. Locations of maximum temperatures in the components of the fuel rod COMPONENT AXIAL TEMPERATURE ( ) HEIGHT(Z) Coolant 0.115m 70.00 Surface of clad. 0.06m 85.50 Fuel surface 0.02m 148.45 Fuel center 0.01m 215.70 Table 4.4 Radial temperature distribution in the fuel using MATLAB , (for r = 0.00 to 2.15 mm). r 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 T 317.13 371.00 316.63 316.00 315.13 314.00 312.63 311.00 r 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 T 309.13 307.00 304.63 302.00 299.13 296.00 292.63 289.00 r 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 T 2.85.13 281.00 276.63 272.00 267.13 262.00 256.63 251.00 r 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 T 245.13 239.00 232.63 226.00 219.13 212.00 204.63 197.00 r 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 T 189.13 181.00 172.63 164.00 155.13 146.00 136.63 127.00 r 2.00 2.05 2.10 2.15 T 117.13 107.00 96.63 86.00 61 University of Ghana http://ugspace.ug.edu.gh Table 4.5 The radial temperature distribution in the sheath (cladding) using MATLAB = - In ( ) , (for r = 2.15 to 2.75 mm) r 2.15 2.20 2.25 2.30 2.35 2.40 2.45 T 86.00 80.64 75.39 70.26 65.24 60.33 55.52 r 2.50 2.55 2.60 2.65 2.70 2.75 T 50.81 46.19 41.65 37.21 32.85 28.57 Table 4.6 The radial temperature distribution in the coolant using MATLAB Coolant: = - , (for r = 2.75 to 2.90 mm) r 2.75 2.80 2.85 2.90 T 27.2500 27.1491 27.0464 26.9418 Table 4.3 was teased out from the structured excel Table 4.2 to highlight on the positions of maximum temperatures in each component in the axial direction and it showed the order in which the maximum temperatures were attained in the four components. Temperature distribution in the radial direction of the fuel, clad and coolant by the excel program is shown in Table 4.7 62 University of Ghana http://ugspace.ug.edu.gh Table 4.7. Radial temperature distribution in the fuel, clad and coolant using the spreadsheet rf rd rh 0.00 2.15 2.75 317.1250 86.0000 27.2500 0.05 2.20 2.80 317.0000 80.6355 27.1491 0.10 2.25 2.85 316.6250 75.3915 27.0464 0.15 2.30 2.90 316.0000 70.2628 26.9418 0.20 2.35 2.95 315.1250 65.2444 26.8355 0.25 2.40 3.00 314.0000 60.3317 26.7273 0.30 2.45 312.6250 55.5202 0.35 2.50 311.0000 50.8060 0.40 2.55 309.1250 46.1851 0.45 2.60 307.0000 41.6540 0.50 2.65 304.6250 37.2091 0.55 2.70 302.0000 32.8474 0.60 2.75 299.1250 28.5657 0.65 296.0000 0.70 292.6250 0.75 289.0000 0.80 285.1250 0.85 281.0000 0.90 276.6250 0.95 272.0000 1.00 267.1250 1.05 262.0000 1.10 256.6250 1.15 251.0000 1.20 245.1250 1.25 239.0000 1.30 232.6250 1.35 226.0000 1.40 219.1250 63 University of Ghana http://ugspace.ug.edu.gh 1.45 212.0000 1.50 204.6250 1.55 197.0000 1.60 189.1250 1.65 181.0000 1.70 172.6250 1.75 164.0000 1.80 155.1250 1.85 146.0000 1.90 136.6250 1.95 127.0000 2.00 117.1250 2.05 107.0000 2.10 96.6250 2.15 86.0000 4.3. The plots of temperature profile of the fuel-channel elements in the axial direction. A plot by MATLAB of the axial temperature distribution in the coolant, cladding, fuel surface and fuel center using Table 4.1 is illustrated by Figures 4.4, 4.5, 4.6, 4.7 and 4.8. Figure 4.4 is a plot of temperature profile of the coolant in the axial direction. The trend approximated to a straight line graph of positive gradient. The only variations were identified at the inlet region were the rise in temperature was more rapid and towards the exit where a more rapid reduction in temperature was experienced. The increase in temperature was steady between these two regions. 64 University of Ghana http://ugspace.ug.edu.gh Fig.4.4. Axial temperature distribution in the coolant Figure 4.5 shows the trend of temperature profile in the outer surface of the cladding. The graph depicted a steady rise in temperature until the maximum point was reached. Thereafter there was a rapid decrease in temperature to the reactor exit. The shape of the graph is hyperbolic. 65 University of Ghana http://ugspace.ug.edu.gh Fig.4.5 Axial temperature distribution in the outer surface of the cladding Figure 4.6 is the plot of temperature profile in the surface of the fuel. The shape was hyperbolic and the trend was also similar to that of the temperature profile in the outer surface of the cladding. The temperature profile at the fuel center is shown in figure 4.7. Although the shape and the trend were similar to the temperature profiles in the outer surface of the cladding and surface of the fuel, its hyperbolic shape was much more symmetrical. 66 University of Ghana http://ugspace.ug.edu.gh Fig 4.6 The axial temperature distribution in the fuel surface Fig.4.7. The axial temperature distribution along the fuel center 67 University of Ghana http://ugspace.ug.edu.gh Figure 4.8 is the combined plots of the temperature profile in all the four components in the axial direction for comparison and analysis. 0.15 0.1 0.05 0 -0.05 -0.1 coolant outer surface of cladding -0.15 Fuel surface Fuel centerline -0.2 0 50 100 150 200 250 Temperature (Degrees Celcius) Fig.4.8.Combined temperature distribution in the axial direction in a fuel channel (coolant, outer surface of the cladding, fuel surface and fuel center). 4.4. The plots of temperature profile of the fuel-channel elements in the radial direction A plot by MATLAB of radial temperature distribution in the fuel, cladding and coolant using tables 4.4, 4.5 and 4.6. Figure 4.9 is the plot of temperature profile in the fuel at the radial direction. The trend was parabolic while the temperature axis acted as the line of symmetry. Figure 4.10 shows the temperature profile in the clad. The profile followed a straight line graph of negative gradient descent. 68 Height of Reactor Core (m) University of Ghana http://ugspace.ug.edu.gh Fig. 4.9 Radial temperature distribution in the fuel Fig 4.10 .Radial temperature distribution in the sheath (cladding) 69 University of Ghana http://ugspace.ug.edu.gh The plot of temperature profile in the coolant in the radial direction is shown in figure 4.11. The trend was approximately a horizontal straight line. This trend described very small changes in temperature. Fig.4.11. Radial temperature distribution in the coolant Figure 4.12 is the combined plot of the temperature profile in the three components in the radial direction for the purposes of comparison and analysis. 70 University of Ghana http://ugspace.ug.edu.gh 350 Fuel Rod Cladding Coolant 300 250 200 150 100 50 0 0 0.5 1 1.5 2 2.5 3 Radial Distance (mm) Fig.4.12. Combined radial temperature distribution in the fuel, cladding and coolant 4.5. Discussions The maximum cladding temperature should be less than the maximum allowable temperature with a good safety margin. The maximum allowable temperature for zircaloy -4 cladding ranges from 380 ℃ to 400 ℃ [5]. By theory, since the location of the maximum temperature is given by an equation of the form: = ɑ arctan (b) where b = γ or or and increased as b increased and the maximum temperature drifted further downstream of the centerline of the channel. 71 Temperature (Degrees Celcius) University of Ghana http://ugspace.ug.edu.gh Thus the location of the maximum temperature in the components was related to the thermal resistance and it shifted further downstream of the channel center as the resistance decreased. 4.5.1. Temperature distribution in the axial direction The maximum temperature recorded at maximum channel power =30 kW in the axial direction for fuel surface was 215 ℃. This occurred in the fuel center and was within the specification. The coolant temperature increased continuously from the inlet of 24.5 ℃ to the exit value of 70.0 ℃. However the increase in temperature decreased as the exit point was approached a phenomenon which underpinned a safety condition and indicated effective cooling by natural condition. Furthermore, a more efficient heat transfer was observed in the outer surface of the cladding, fuel surface and fuel center, where the temperature increased continuously along the channel in the region below the mid-point and reached a maximum value in each component in the upper half of the fuel channel in the order as shown in Table 4.3. The temperature of the fuel center, the hottest part of the reactor, reached maximum point first then decreased in temperature after this point and was fastest considering the other two components, a design in support of safety condition. The next maximum point was in the fuel which also began to decrease at a faster rate thereafter. Finally, the temperature in the surface of cladding attained its maximum and decreased very quickly to the exit. The data in table 4.2 and figures 4.5, 4.6 and 4.7 were in support of these conclusions. 72 University of Ghana http://ugspace.ug.edu.gh 4.5.2. Temperature distribution in the radial direction (i) Radial temperature distribution in the fuel The temperature increased continuously from the fuel surface to the fuel center. For a fuel surface temperature of 86.00 ℃, the temperature along the center could increase to 317.13 ℃ an increase of 231.13 ℃ at a full operational power of 30 kW. However, the increase in temperature was not a steady one but decreased continuously between successive discrete points to the fuel center. Considering a space step size of 0.05 mm for the range (0.05 ≤ r ≤ 2.15 mm) on the radial distance axis, the decrease in temperature in degrees Celsius followed the sequence: 0.13, 0.37, 0.63, 0.87, 1.13, 1.37, 1.63, 1.87, 2.13, . . . , 9.37, 9.63, 9.87, 10.13, 10.37, 10.63. Thus the temperature gradient was steep close to the fuel surface and gradual as the fuel center was approached. (ii) Radial temperature distribution in the sheath (cladding) A decrease in temperature from the inner to the outer surface of the cladding was obtained. The decrease increased towards the outer surface. The decrease in temperature in degrees Celsius for the clad thickness r (2.15 mm ≤ r ≤ 2.75 mm) followed the sequence described by: 4.28, 4.36, 4.44, 4.54, 4.62, 4.71, 4.81, 4.91, 5.02, 5.13, 5.25, 5.36. (iii) Radial temperature distribution in the coolant In the coolant, the radial temperature decreased as expected as the coolant flowed away from the outer surface of the cladding. The decrease in temperature corrected to one decimal place was approximately 0.1 ℃, a constant, as shown by the sequence: 0.1009, 0.1027, 0.1046, 0.1063, 0.1082 73 University of Ghana http://ugspace.ug.edu.gh CHAPTER FIVE CONCLUSIONS AND RECOMMENDATIONS 5.1. Conclusions The temperature profile in the channel of fuel assembly of MNSR has been simulated using LabVIEW software and the temperature distribution was validated by MATLAB and Excel software programs developed . The temperature profile in the radial direction revealed that the coolant temperature increased continuously from the inlet value of 24.5℃ to the exit value of 70.0 ℃. In the outer surface of the cladding, the surface of the fuel and the fuel center components, the temperatures rose along the channel and reached maximum values above the centerline of the fuel channel as shown in Figure 5.1 [17]. Fig 5.1 The axial temperature distribution in the fuel channel: 1. Coolant 2. Outer surface of the cladding 3. Fuel surface and 74 University of Ghana http://ugspace.ug.edu.gh 4. Fuel center. On reaching the maximum temperatures, the temperatures in each of the other three zones continued to decrease to the exit of the fuel channel because the linear power after the maximum point began to drop more rapidly than the increase of coolant temperature. Moving away from fuel center the maximum temperature shifted towards the exit of fuel channel (fig 5.1), mainly due to thermal resistance between the outside surface of the cladding and the coolant, in the cladding, in the gap region and in the fuel which decreased as one moved away from the fuel center. In a good agreement, the LabVIEW simulation revealed that increasing the convective heat transfer coefficient, , or decreasing the coolant mass flow rate, , shifted the point of maximum cladding temperature further away from the fuel channel mid- plane or centerline. A detailed knowledge of the temperatures of the cladding and fuel center was important since together with critical heat flux, the temperature imposed limitations on the maximum allowable heat generation in the fuel element. The changes in the temperature distribution in the radial direction were more significant than in the axial direction. Although the coolant temperature was 28℃, the maximum temperature at the fuel center exceeded 317 ℃ in the radial direction. 75 University of Ghana http://ugspace.ug.edu.gh Figure 5.2 Radial Temperature Profile across a Fuel Rod and Coolant Channel. The temperature distributions were skewed by the changing capacity of the coolant to remove the heat energy. Since the coolant increased in temperature as it flowed up the channel, the fuel cladding and the fuel temperatures were higher in the upper axial region of the core. A radial temperature profile across a reactor core (assuming all channel coolant flows are equal) followed the radial power distribution. The areas with the highest heat generation rate (power) produced the most heat and had the highest temperatures. A radial temperature profile for an individual fuel rod and coolant channel is shown in figure 5.2 [19]. 76 University of Ghana http://ugspace.ug.edu.gh The shapes of the temperature profiles were dependent upon the heat transfer coefficient of the various materials involved. The analysis on a single fuel rod can be easily extended to a multi-rod fuel bundle once there is a strong mixing existing between the laterally interconnected sub-channels bounded by the fuel rods or by the fuel rods and the channel wall. However, the position of the fuel rod in the entire fuel channel assembly will vary temperature values. The closer the fuel rod is to the center of the core the higher the temperature value. LabVIEW has been used as a development environment and as generic graphical user interface because LabVIEW is based on a graphical language which is user friendly, can provide excellent graphics capabilities and in addition has the ability to make all results available and in real time. 5.2. Recommendations The thesis presented an effective and a high quality LabVIEW simulations to execute on PC, the temperature profile in the channels of MNSR core so as to provide immediate access to knowledge on the temperature distribution in reactor core to nuclear personnel (mostly students) for familiarization, design evaluation, training and early guide to commissioning and operating procedures. It is recommended that countries which are either aware of nuclear power or nuclear power ready in sharing this experience with interested parties, should develop various levels of training programs using LabVIEW and other simulation tools. In the process of working together, the goals of self-reliance and the transfer of necessary nuclear reactor knowledge will be obtained which will help to fulfill the objectives of the nuclear human resource development initiatives. 77 University of Ghana http://ugspace.ug.edu.gh The simulation was carried out on nuclear reactor of cylindrical geometry. It can be demonstrated on other geometries to compare the efficiency of heat transfer. Future work will focus on including additional interactive features for operator actions as well as extending the development to AP 1000 reactor design. 78 University of Ghana http://ugspace.ug.edu.gh REFERENCES [1] Fenech, H (Editor), Heat Transfer and fluid flow in Nuclear Systems, Pergamon Press, University of California , USA ,1981. [2] Tong L.S., Heat Transfer in water cooled nuclear reactors, Nuclear Eng. and Design, Vol. 6, 1967. [3] Galliland, E. R. The Science and Engineering of Nuclear Reactors, Academic Press, San Diego, CA, USA, 1949 [4] Kampf, H and Karsten, G., Table of Parameters, Nuclear Application Vol. 9, 1970. [5] Ogawa ,S.Y., Lees, E.A. and Lyons, M.F., Power Reactor High Perfomance U Program Fuel Design Summary and Program Status, GEAP-5591 Pergamon Press Inc New York,1968. [6] Ross, A.M. and Stoute, R.L., Heat Transfer Coefficient Between U02 and Zircaloy-2, AECL-1552, John Wiley & Sons Inc. New York, 1962. [7] Lyons, F.M. et al. UOz P'owder and Pellet Thermal Conductivity During Irradiation, GEAP-SIOO-l, Harwood Academic Publishers, New York, 1966 [8] Cetinkale, T.K. and Fishenden , M., Thermal Conductance of Metal Surfaces in Contact, Proc. General Discussion on Heat Transfer, Institute of Mechanical Engineers, London. UK, 1951. [9] Godfrey, T.G., Fulkerson, W., Kollie, T.G., Moore, J.P and McElroy, D.L., Thermal Conductvity of Uranium Dioxide and Armco Iron by an Improved Radial Flow Technique, ORNL -3556. 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