Master Thesis

Investigation of Spent Nuclear Fuel Pool Coolability

Fredrik Nimander

Supervisor: Henryk Anglart

Division of Nuclear Reactor Technology Royal Institute of Technology Stockholm, Sweden, August 2011

TRITA-FYS 2011:51 ISSN 0280-316X ISRN KTH/FYS/–11:51–SE ii ABSTRACT

The natural catastrophe at Fukushima Dai-ichi 2011 enlightened the nuclear community. This master thesis reveals the non-negligible risks regarding the short term storage of spent nuclear fuel. The thesis has also investigated the possibility of using natural circulation of air in a passive safety system to cool the spent nuclear fuel pools. The results where conclusive: The temperature difference between the heated air and ambient air is far too low for natural circulation of air to remove any significant amount of heat from the spent nuclear fuel pool in a worst case scenario. Air, as with any gas, has too low density and a specific heat too low to be able to remove the heat generated by spent nuclear fuel shortly after it has been removed from the reactor core. The author does not deny the possibility of slightly prolonging the boiling with other designs. The author does however suggest other possibilities to prolong cooling with the conclusion that large enough spent fuel pools would constitute the simplest solution.

iii iv Abstract ACKNOWLEDGEMENTS

The author would like to thank Henryk Anglart for his help and supervision of the master thesis. He allowed the author to complete the master thesis during the summer of 2011 even though it is standard to write a thesis during a normal semester. The author also wants to thank PHD-student Roman Thiele for his help with OpenFoam. This software package is extremely useful and valuable, but it has a rather steep learning curve when getting started since the documentation is very general rather than detailed. Without the help of Roman there might not have been enough time to learn how to use this software package as a tool for the work that needed to be done. A special thanks goes to Nils-Gunnar Ohlson, teacher at KTH, for his help with finding the appropriate information on how to calculate the structural integrity of the copper wall designed for the passive safety system that was investigated for this thesis.

v vi Acknowledgements NOMENCLATURE

Symbol Dimensions Description Latin Symbols −1 −1 Cp Jkg K Specific heat E J Energy Esat J Energy required to saturate pool Evap J Energy reguired to evaporate water Gr - Grashof number h Wm−2K−1 Heat transfer coefficient k Wm−1K−1 Thermal conductivity Nu - Nusselt number P r - Prandtl number q00 Wm −2 Heat flux Q Js−1 Heat −1 Q0 Js Thermal Power of reactor during operation Ra - Rayleigh number s Pa Stress top s Continuous perational time of fuel in reactor ts s Time since reactor shutdown tsat s Time until spent fuel pool becomes saturated T K Temperature Ti K Initial temperature Tsat K Saturation temperature T∞ K Bulk temperature V m3 Volume 3 Vaf m Volume of water above fuel assemblies Greek Symbols α m2s−1 Thermal diffusivity β - Dimensionless structure coefficient λ Wm−1 Thermal conductivity µ cm2g−1 Gamma attenuation coefficient ν m2s−1 kinematic viscosity ρ kgm−3 Density Subscripts af Above fuel i Initial op Operation sat Saturation vap Vaporization

vii viii Acknowledgements CONTENTS

Abstract iii

Acknowledgements v

1 Introduction 1 1.1 Background ...... 1 1.2 Thesis objectives ...... 2

2 March 2011 events at Fukushima Dai-ichi 3 2.1 The Natural Disaster ...... 3 2.2 Event sequence at Fukushima Dai-ichi ...... 4

3 Spent nuclear fuel pools, spent fuel and Decay heat 7 3.1 Spent Nuclear Fuel Pools ...... 7 3.2 Spent Nuclear Fuel ...... 8 3.2.1 Radiotoxicity ...... 8 3.3 Boil off times ...... 10 3.3.1 Boiling off ...... 10 3.3.2 Governing equations ...... 11 3.4 Results ...... 11 3.4.1 Case 1 ...... 12 3.4.2 Case 2 ...... 12 3.4.3 Case 3 ...... 12 3.5 Conclusions ...... 12

4 The possibility of air cooled passive safety systems 15 4.1 Design requirements ...... 15 4.2 Design Suggestion ...... 15 4.2.1 Copper wall design ...... 16 4.2.2 Wall strength and thickness ...... 16 4.3 Theory of heat transfer and governing equations ...... 18

ix x CONTENTS

4.3.1 Convection ...... 18 4.3.2 Conduction ...... 19 4.3.3 Heat transfer ...... 19 4.3.4 Air duct mass flow ...... 20 4.4 Results and feasibility ...... 20 4.4.1 Structual Integrity of the wall ...... 20 4.4.2 Heat transfer ...... 20 4.4.3 Verification with OpenFoam ...... 22 4.5 Conclusions ...... 22

5 Alternative design suggestions and emergency prepardness 25 5.1 Closed System ...... 25 5.2 Prolonging Designs ...... 25 5.2.1 Larger pools ...... 25 5.2.2 Implementing other types of existing passive cooling systems ...... 25 5.2.3 Enclosed pool ...... 26 5.2.4 Multiple pool heat sink ...... 26 5.2.5 Lessons from Fukushima Dai-ichi ...... 26

6 Discussion and Conclusion 27

Appendix A Verification with openFOAM 31 A.1 OpenFoam framewrok ...... 31 A.2 CFD...... 31 A.2.1 Boussinesq approximation ...... 32 A.2.2 Newtonian Fluid ...... 32 A.2.3 Mesh ...... 32 A.2.4 Boundary conditions ...... 33 A.2.5 Accuracy of the results ...... 33 A.3 Post processing ...... 34

Appendix B Heat loss calculations 35

Appendix C Boil off times 37

Appendix D Passive safety system calculations 39

Appendix E OpenFoam Mesh 41 * LIST OF FIGURES

2.1 Picture showing the epicenter location of Japan March 2011 earthquake...... 3 2.2 Picture showing units 1,2,3, and 4 starting from the right...... 4

3.1 Total radiotoxic inventory in spent nuclear fuel as a funciton of time[13]...... 9 3.2 The time dependency of decay heat along with some of the largest contributing isotopes[13]. .9

3.3 Thermal power after shut down for different burnup times top...... 10

4.1 Passive safety system that was investigated...... 16 4.2 Passive safety system that was investigated...... 17 4.3 Copper wall design...... 17 4.4 Stress distribution in copper wall...... 21 4.5 Temperature distribution through copper wall [3] ...... 21 4.6 Pressure distribution...... 22 4.7 Temperature distribution...... 23 4.8 Flow velocities...... 23

5.1 Alternative safety system design ...... 26

A.1 The data for the physical properties were extracted form the red line at the top of the duct. . 34

xi xii List of Figures LIST OF TABLES

3.1 Relative power of fuel for different burnup times given in %...... 10 3.2 Variables used for boil-off calculations...... 12 3.3 Case 1 results...... 12 3.4 Case 2 results...... 12 3.5 Case 3 results...... 12 3.6 Case 3 corrected results...... 13

4.1 Temperature distribution for a full core unloading, case 1...... 21

A.1 OpenFoam bondary conditions ...... 33

xiii xiv List of Tables CHAPTER 1

INTRODUCTION

1.1 Background

The events that occurred in Japan in March of 2011 showed us that there might be parts of the nuclear industry that has a lack of proper safety margins. The nuclear reactors that managed to withstand such a powerful earthquake and an unprecedented tsunami belongs to the second generation of reactors that were built in the 1970’s. This freak of nature catastrophe had a very low probability of occurring but came very close to having disastrous consequences. Even though the design of these reactors belong to the history books there are still some similarities in the design between the oldest generation of reactors and the modern reactors that are being built and designed today, generation three and generation three plus. The Three Mile Island(TMI) accident in 1979 was a milestone withinin the nuclear industry that will not be forgotten. It showed us the importance of safety systems and control systems in a nuclear power plant. Maybe most importantly it showed us that the human factor can play a vital role in safety. Attention was given both to the possible direct actions of a single human being but also the behavior of an organization. Modern reactors incorporate safety systems that can even counter faulty human behavior. Even though TMI did not cause any dangerous release of radiotoxic material, it suffered a severe meltdown from a to that day unpredicted sequence of events. The mindset at that time was that large break LOCAs would constitute the worst-case scenario and therefore no studies had been made on the behaviour of a reactor during small break LOCAs. This changed the way we think about safety and it gave rise to deterministic and probabilistic safety calculations. In 1986, when Chernobyl suffered a massive power surge leading to an enormous release of radiotoxic materials it became clear that we had not yet learned enough. The Chernobyl accident was caused by faulty operation in a forbidden operational zone. The control rods also had a design flaw that largely contributed to the accident. After Chernobyl computers became part of the world. Thanks to them we can make almost any type of calculation, making it possible for us to run simulations that could calculate even some of the most unpredictable chains of events. From the Chernobyl accident we did learn how important it is to protect every power plant from design base accidents and ensuring passive and redundant safety systems. 25 years after the Chernobyl accident Mother Nature gave Japan all her wrath with an enormous earthquake that even slowed down the earths rotation and a following tsunami of mythological proportions. The nuclear power plants at Fukushima Dai-ichi came disturbingly close to a second nuclear catastrophe (only counting Chernobyl as a catastrophe) that could even have become worse than Chernobyl. Due to these events design flaws have come in to light once again, showing us that some of our predictions might underestimate the probability of certain chains of events. Risk is defined as the probability times the consequence and it is therefore of the uttermost importance that we learn from these events in order to ensure that this was the last accident within the nuclear industry.

1 2 Introduction

1.2 Thesis objectives

This MSc.Thesis recognizes that spent fuel pools used at some power plants today might have design flaws. These pools are today dependent on electricity to ensure cooling and the circulation of water with the use of pumps. Accident scenarios where power is lost can be postulated with far less serious initiating events than a groundbreaking earthquake or a mythologically sized tsunami. This thesis will investigate not only how long these pools remain safe without a power supply but will also investigate a simple design for a passive cooling system that does not require electricity. The thesis uses conservative calculations as well Computation Fluid Dynamics in order to try and verify the calculations made. The initial plan was to investigate the possible accident scenarios for all of the Swedish reactors but unfortu- nately only Ringhals 1 data were made available to this author during the course of this work and this power plant will therefore serve as a real life example. The scenarios investigated are however generall and could be applied to all spent nuclear fuel pools with some modifications. Calculations made throughout this thesis uses conservative simple correlations for heat and mass transfer. The reader should therfore be familiar with with heat transfer and how a nuclear power plant is designed. Chapter 2 describes the events that occured in Japan 2011. In Chapter 3 the spent nuclear fuel is investigated and the decay heat which it generates. The problem of storing it in spent nuclear pool with active safety systems is brought into light as well as a possible worst case accident scenario. Chapter 4 investigates the possbility och passive cooling using air as the heat transport medium. In the first appendix the Author describes the OpenFoam software that as been used to validate the results in Chapter 4. In Chapter 5 the author suggest possible systems that might investigated for the purpose of passive safety. CHAPTER 2

MARCH 2011 EVENTS AT FUKUSHIMA DAI-ICHI

The events that occurred in Japan March 2011 serves as the background for the work done in this thesis. The author therefore wanted to describe what happened at the Fukushima Dai-ichi nuclear power plants.

2.1 The Natural Disaster

Figure 2.1: Picture showing the epicenter location of Japan March 2011 earthquake.

Earthquakes are not something unusual in Japan. History has taught the Japanese not to underestimate the forces of nature but on 11 March 2011 it became painfully clear that nature can and will be underestimated. An earthquake of magnitude 9 occurred in the ocean east of Japan. Nuclear power plants and other utilities that handle radioactive material are highly protected against earthquakes and they are supposed to auto- matically shut down whenever the ground acceleration reaches certain magnitudes. However, the earthquake was extremely powerful and a tsunami shortly followed the earthquake. This tsunami reached as high as 38.9 meters in some coastal areas and around 14 meters at the Fukushima Daiichi power plant. Except for the consequences at the Fukushima Dai-ichi nuclear power plant 15391 people lost their lives and 8171 people

3 4 March 2011 events at Fukushima Dai-ichi are unaccounted for. It must be mentioned than none has yet died from the effects of exposure to radiation. Towns where washed away and much of the east coast infrastructure was destroyed. In other words the 11 March 2011 was a tragedy in itself but it could have become much worse [1].

2.2 Event sequence at Fukushima Dai-ichi

The power plants at Fukushima Dai-ichi were successfully shutdown after the earthquake as with many other power plants. The external power was lost but all back up diesel generators were put in to use and everything seemed under control. The ground acceleration did not exceed the standard design basis at units 1,4 and 6. Thereby showing that the risk of earthquakes had been accounted for. But, this earthquake was large enough to exceed the design basis at units 2,3 and 5. However, 46 minutes after the accident a tsunami struck the power plants reaching a maximum height of 14 meters. The tsunami exceeded the design basis at all plants thereby reaching deep into the facilities. All backup power was lost except for one diesel generator and outside help was at this point very far away. Units 5 and 6 shared the working diesel generator power. This was in a sense unlucky since the fuel hade been removed from the reactor cores at unit 4,5 and 6 and power was urgently needed at the other units. All instrumentation and control systems were therefore lost at units 1,2,3 and 4. Communications systems were also affected. The personnel at all units were temporarily evacuated due to aftershocks and further tsunami warnings. When they returned they had to secure all nuclear facilities along with fuel storage facilities without any instrumentation and reduced communication possibilities. Reactors 5 and 6 remained in cold shutdown, which is a safe condition. Cooling was however lost in units 1, 2 and 3 and coolability of the spent fuel pools was lost at unit 4 [1].

Figure 2.2: Picture showing units 1,2,3, and 4 starting from the right.

As is known and will be shown in the next chapter the loss of coolability can in some facilities quite rapidly become a problem. The residual decay heat within the reactors and the spent fuel pools, which can be in the order of MW, caused the water to start boiling within the reactor cores. Indications have shown that the water also started boiling in the spent fuel pools at unit 4. When the fuel became hot enough hydrogen production started as a chemical reaction between the fuel cladding and the cooling water. After the earthquake some cooling was available. For example a gravity driven isolation condenser (IC) was available for some time at unit 1 and in units 2 and 3 a reactor core isolation cooling system (RCIC) was available. The RCIC uses produced steam to drive a pump that could inject water in the core. The RCIC in unit three failed approximately 19.5 hours after the accident and a high-pressure injection system (HPCI) was available for another 14 hours. When the IC failed in unit 1 and the RCIC in unit 2 and 3 as well as the HPCI in unit 3 failed other modes of cooling had to be established. Fire trucks and low pressure pumps were therefore used[1]. In the afternoon on 12 March 2011 the first of three hydrogen explosions occurred at unit 1. When venting of the hydrogen was later confirmed workers established means to inject borated seawater into the core. On 25 March 2011 this this was discontinued as they managed to establish means to insert fresh water that continues to this day. At units 2 and 3 fire trucks were used to try and supply the cores with borated seawater. However, steam was produced which was bleeding into the suppression pools through safety relief valves. This meant that Event sequence at Fukushima Dai-ichi 5 the temperature increased in the suppression pools and pressure rose in the containments making it difficult to refill the system with low-pressure pumps. Pressure had to be lowered. At unit 2 workers managed to open relief valves that operated on pressurized air on 13 March 2011. A second hydrogen explosion that occurred at unit 3 would render these valves inoperable. The safety relief valves of the containment that remained needed both DC power and pressurized nitrogen to be realigned. This was not possible at unit 2 since the nitrogen pressure was to low. The second hydrogen explosion occurred on 14 March 2011 in unit 3 further damaging the plant and the valves using pressurized air. The nitrogen pressure at unit three was high enough and DC power was established with car batteries after the explosions. Once the pressure was reduced injection of seawater was possible. Fresh water injection was established at unit 3 shortly after unit 2 on 25 March 2011 [1]. On 15 March 2011 the third hydrogen explosion occurred in unit 4. Indications hade shown that the fuel was never uncovered prior to the explosion leading to the theory that hydrogen had leaked from unit 3 into unit 4 since they share venting lines leading to the exhaust stack. The core melt down sequences has not yet been confirmed but Tepco have been running simulations showing that damage to the unit 1 core started only 4 hours after the shutdown. Damage and partial meltdowns to units 1, 2 and 3 has been confirmed. All reactors remain shutdown to this day. This is the case for many other power plants in Japan since safety requirements have been heightened and public support has been lowered[1]. 6 March 2011 events at Fukushima Dai-ichi CHAPTER 3

SPENT NUCLEAR FUEL POOLS, SPENT FUEL AND DECAY HEAT

Nuclear power constitutes a great alternative for electricity production. Its effective, relatively cheap and great for the environment compared with fossil fuels such as coal, gas and oil. Simplified one can say that there are only two drawbacks to nuclear power. The first one is the safety related to the operation of the nuclear power plant, even during normal operation, but especially during unforeseeable chains of events. The second drawback is of course the radiotoxic waste that is produced, which can be very harmful to humans for a very long time after it’s been used. There is an ongoing debate on the long-term safety and handling of nuclear waste but this thesis will only concern itself with the short-term safety issues that are related to the storage of spent nuclear fuel in the reactor buildings. These issues are clearly worth investigating in the aftermath of the events in Japan 2011.

3.1 Spent Nuclear Fuel Pools

Spent nuclear fuel pools serve as short-term fuel storage before the fuel is shipped away for intermediate storage or reprocessing. That is at least the original purpose for them. In some countries there is no plan for the handling of the nuclear waste and spent fuel has therefore been accumulating in the reactor pools. In Sweden the fuel is shipped off to an intermediate storage (CLAB). The spent nuclear pools are designed to safely cool the fuel as well as protect the workers from the radiation. There are also regulations for dealing with the criticality issue of the fuel, but that was not a topic for the work done in this thesis. For radiation safety reasons minimum water coverage of three meters is required to be present at all times in the pools. Most pools like the ones at Ringhals 1 has a water coverage of approximately ten meters during normal operation to increase the safety margins[12]. A design flaw today is that the piping is installed only three meters above the fuel in order to ensure that even with a leak the radiation protection level of water is kept at three meters. But as will be shown this can gravely diminish the safety margins during certain postulated events. There might be other designs present elsewhere where the pipes are elevated. The CLAB intermediate storage facility in Sweden has this type of design and the author therefore assume that this might be a safety standard at least for Swedish nuclear facilities. The safety issue that this thesis concerns itself with is what would happen if the possibility to refill and cool the spent fuel pools were lost. The decay heat may then at some points in time be sufficiently large enough to evaporate or even too boil off all of the protective water. If this were to happen radioactive nuclides might be released into the reactor building and to the environment. When the fuel pins get hot enough hydrogen production starts due to a thermochemical reaction between the Zircaloy cladding that encapsulates the fuel and the water surrounding it. This build up of hydrogen could lead to a hydrogen explosion that could further damage the power plant leading to even more dangerous accidents[16]. At Fukushima Dai-ichi unit 1,3 and 4 three such hydrogen explosions occurred. These hydrogen build-ups were however caused by the heating of the rods within the reactor core rather than the rods in the spent fuel pools[4]. There are a number of scenarios that can be postulated under which the backup electricity supply to a nuclear power plant might be lost. As will be shown there are few high probability scenarios where the time to boil off is short enough to cause any real concern. But suppose that the entire core has been unloaded into the spent fuel pools as was the situation at Fukushima Dai-ichi unit 4 and suppose that a leak in the

7 8 Spent nuclear fuel pools, spent fuel and Decay heat piping occurs so that the water level is not 10 m above the fuel but rather the minimum 3 m. In this type of scenario there might be cause for concern.

3.2 Spent Nuclear Fuel

The most common reactors today are light water reactors. And the nuclear fuel that is most common consists of natural or enriched Uranium. This fuel is practically entirely safe before it has been irradiated in the reactor core. Within the reactor core highly radioactive nuclides are created through the fission process of heavy nuclides as well as through the neutron capture process in many nuclides. The spent nuclear fuel is transferred to the spent nuclear fuel pools after it’s been irradiated in the reactor core. This fuel is therefore highly radioactive. But radio toxicity is not the only problem with the spent nuclear fuel. Heat generation is the natural consequence of radioactive decay. This decay heat constitutes almost 7% of the total heat generation within a operating reactor core. The half lives of some isotopes is very short and therefore the decay heat decreases quite rapidly as soon as the reactor has been shut down. There are however longer lived fission products like 137Cs that has a half life of 30 years along with actinides that can be as long lived as 105 years. Radiotoxicity and decay heat will therefore be a concern even for intermediate and long term storage as well as for the short term storage[16].

3.2.1 Radiotoxicity

When heavy metal nuclear fuels are irradiated by neutrons in a reactor core there are several processes oc- curring. The main process that occurs is the fission process that generates the bulk heat within a reactor. The fission products that are produced are often radioactive and decay by means of β and γ decay. During normal operation these decays constitutes a non-negligible amount of the total heat. Neutron capture is an- other process where particles may absorb an extra neutron. This process gives us several different radioactive isotopes that can decay by the same means as the fission product but some of the heavy metal isotopes might also decay by the emission of α particles. These two processes are the main processes that occur along with elastic and non-elastic scattering that might excite nuclei to higher energy states that most often decay by the emission of γ radiation. There is also a mode of decayed competing with β decay called electron capture. When the reactor is shut down this radioactive decay continues. All generated radioactive isotopes have different half lives, spanning from very short almost unmeasurable times to very long geological time-spans. The radiotoxicity of spent nuclear fuels is therefore strongly coupled to the composition of the fuel after it has been irradiated. The typical radiotoxic profile of spent light water reactor fuel can be seen in figure 3.1. The issue with radiotoxicity is important for safety reason, were the water to boil off, but it is also important to make sure that the design of a spent fuel pool safely protects the workers during normal conditions. This will play an important part in the suggested passive safety system design in Chapter 4. The storage of nuclear fuel within the power plant should be limited to a maximum of a few years and during this period of time it’s the fission products that will generate the bulk of the decay heat. The first decades of decay heat are dominated by decay of 90Sr and 137Cs. In the first year however we need to account for all fission products[13]. Since the decay heat is in effect a result of the beta and accompanied gamma decay it will decrease in proportion to the half lives of the different isotopes. It is therefore ensured that the maximum decay heat will be generated directly after shutdown of the reactor and that we will have a quite rapid decrease within the first 100 days after shutdown[13]. For this thesis a worst case scenario will be investigated for the decay heat and boil off times. The decay heat is therefore investigates during a short period of time after operation. The typical total heat production per metric ton heavy metal (MTHM) can be seen in figure 3.2. The decay heat that is being generated by the fuel assemblies from one core can be related to their initial power Q0 by equation (3.1) where ts[s] is the time since shutdown of the reactor and top[s] is the operational time of the fuel in the reactor[5].This equation as been determined for 4.5% enrichment of 235U in the fuel. The amount of fission products goes up with enrichment and can therefore safely be used for Swedish reactors that usually utilizes a lower enrichment. Since the amount of fission products would saturate when the burnup time of the fuel goes towards infinity we could assume a long burnup for the calculations in order to ensure a worst-case scenario. Within the first minute, delayed neutrons does play a role in the heat generation but we can assume that no one removes the fuel assemblies within a minute of shutdown, nor within the first hours or even days. According to [11], an assumption that fuel is placed in the spent fuel after 7 days is a valid assumption. Spent Nuclear Fuel 9

Figure 3.1: Total radiotoxic inventory in spent nuclear fuel as a funciton of time[13].

Figure 3.2: The time dependency of decay heat along with some of the largest contributing isotopes[13].

Q(ts) −0.2 −0.2 = 0.065[ts − (ts + top) ] (3.1) Q0

Figure 3.3 has been generated in Matlab using equation 3.1 and it shows the relative decay heat generation in fuel as function of the total burnup time. Starting from the bottom the first three lines corresponds to 1, 2 and respectively 3 years of burnup and the fourth line corresponds to the solution for 1000 Years of burnup. Due to this saturation behavior of the burnup we can use say 5 years of continuous burnup and 10 Spent nuclear fuel pools, spent fuel and Decay heat be quite confident the that the amount of heat production represents a realistic value for any thermal light water reactor with an enrichment of 4.5% 235U or less.

Figure 3.3: Thermal power after shut down for different burnup times top.

In table 3.1 we can see the fraction of heat generation in the fuel relative to the operational power at different points in time after shutdown for different operational times(burnup times).

Table 3.1: Relative power of fuel for different burnup times given in %.

ts top = 1y top = 10y 1 week 0.25 0.32 1 month 0.14 0.21 3 months 0.075 0.14

Equation 3.1 is generated for a fuel enrichment of 4.5% 235U. This means that the Swedish reactors will have a decay heat generation lower than the prediction of this equation which will ensure a worst-case scenario prediction. In the data received from Ringhals they postulate a worst-case scenario where the entire core is unloaded giving a thermal load of 7.245MW one week after shutdown. Equation (3.1) with a burnup of 5 years predicts a thermal load of 7.614 MW[12].

3.3 Boil off times

3.3.1 Boiling off

Throughout this chapter it’s assumed that the spent fuel pool will boil at some points. These calculations are made conservative assuming that radiation and evaporation heat loss is negligible for the short term storage. This will not be the case for lower thermal loads nor for large enough pools [17]. Some simple calculations can be made using the equations in example 14.13 in [3]. The calculations shows that for a temperature difference of 80 degrees celsius between the pool water and the air the total heat loss for radiation, convection and evaporation adds up to 55.3 kW. Comparing this to the heat loads of the spent fuel which is in the order of a couple MW certainly validates that a worst case scenario calculation can neglect these heat loss effects. The Matlab code can be found in appendix B. Results 11

3.3.2 Governing equations

To calculate the time it will take until the fuel assemblies are uncovered the process is divided into two steps. The first step is the process when the pool water saturates. Until the water has been saturated vapor will condense soon after its been created before it reaches the surface of the pool and therefore no mass will be lost from the so called subcooled pool boiling[3]. The second step is when the pool has been saturated. When the pool is already saturated we can assume that all heat generated from the radioactive decay goes undivided to the phase change process to ensure a worst-case scenario. In a best-estimate calculation one would have to account for heat losses to the surrounding enviroment.

Step 1: Pool saturation

This first step is very simple since we know how much energy we transfer from the fuel to the water each second thanks to equation (3.1). If we know the volume of the pool, the density of the water, initial temperature of the pool, the saturation temperature and the amount of energy required to increase the temperature of the water by one degree(specific heat Cp) we can easily find out how much time it takes to saturate the entire pool. Equation (3.2) gives us the total amount of energy required to saturate the pool. The volume occupied by the fuel assemblies has not been accounted for.

Esat = ρV Cp(Tsat − Ti) (3.2)

Equation (3.3) then gives us the time until the pool has become saturated. The fuel decay heat is given in W [Js−1].

Esat tsat = (3.3) Qfuel

Step 2: Boil off

When the pool is saturated it can be assumed that all decay heat goes too the phase change of the water in order to keep the calculations conservative. Some heat will be lost to the the surroundings and even a small fraction will be lost through heat radiation. Since these energy losses are small and very extensive to evaluate they have been ignored for the insurance of the worst case scenario. The amount of energy required to boil off all the water above the fuel is then given by equation (3.4).

EBoil = ρVaf Evap (3.4)

The critical time when enough energy has been produced to boil off the water above the can be found by integrating equation (3.1) from the time where cooling is lost tstart to the the critical time tc that we are looking for. This gives us equation (3.5) which can easily be solved in Mathematica with the numerical solver NSolve() where we set E to EBoil and Q0 is the initial thermal power of the reactor.

Z tc −0.2 −0.2 E = 0.065Q0[ts − (ts + top) ]dts (3.5) tstart

3.4 Results

At Fukushima Dai-ichi unit 4 the entire core had been unloaded in november and therefore this is assumed as a certainly plausible scenario. A full core unloading correspond to an initial thermal power 2500MW for Ringhals 1. Since the thesis is trying to evaluate a worst-case scenario three different cases have been postulated. In case 1 the normal case of unloading a third of the core with a full pool is investigated. In the second case a scenario close to the events at Fukushima Dai-ichi unit 4 where a full core has been unloaded is investigated. And in the third case a worst case scenario is investigated. In all three cases geometries and thermal power corresponds to Ringhals 1 since this is the plant for which the actual geometry data was available. In table 3.2 the variables used for the three different cases can be seen. For all three cases the boil off times have been calculated for three individual scenarios where the cooling is lost after respectively 7, 30 and 100 days. The code used for these calculations can be found in Appendix C. 12 Spent nuclear fuel pools, spent fuel and Decay heat

Table 3.2: Variables used for boil-off calculations.

Const Case 1 Case 2 Case 3 3 Vsat[m ] 995.5 995.5 511.875 3 Vaf [m ] 648.375 648.375 204.75 Tsat[K] 373.15 373.15 373.15 Tin[K] 313.15 313.15 313.15 −1 −1 Cp(Tin)[kJkg K ] 4.205 4.205 4.205 −3 ρ(Tin)[kgm ] 992.2 992.2 992.2 −1 Q0[MW s ] 833.333 2500 2500

3.4.1 Case 1

This is the normal case when the pool is filled, cooled to 313.15 K and only a third of the core has been unloaded. This should constitute the normal circumstance when refueling a reactor core. The results of these calculations can be seen in table 3.3.

Table 3.3: Case 1 results.

Days after shutdown 7 30 100 Days until uncovered fuel 8.77 13.07 20.38

3.4.2 Case 2

This case does not constitute the worst case scenario, rather a quite possible scenario. Had disaster struck Fukushima Daiich shortly after the entire core had been unloaded the scenario would look close to whats simulated in case 2. In table 3.4 the time until the fuel would be uncovered after loss of cooling has been calculated.

Table 3.4: Case 2 results.

Days after shutdown 7 30 100 Days until uncovered fuel 2.68 4.17 6.61

3.4.3 Case 3

In this scenario a leak has occurred prior to the loss of cooling and the water level is only the minimum 3 m above the fuel. In addition an entire core has been unloaded. Table 3.5 shows the boil off times for case 3.

Table 3.5: Case 3 results.

Days after shutdown 7 30 100 Days until uncovered fuel 0.9 1.42 2.27

3.5 Conclusions

For normal circumstances where only a third of the core is unloaded to a filled spent fuel pools there is no reason for concern. However if procedure requires or does not prohibit the removal of an entire core, severe situations might occur. It is quite clear that routines for unloading the fuel at plant with spent fuel pools with the current design must guarantee that this type of situation does not occur. If there are situations where the entire core must be unloaded, procedures or design has to be changed to accommodate the heat load. Extra back up systems might be installed in existing reactor while passive safety system might be a Conclusions 13 requirement for the next generation of nuclear reactors. The author recognizes that the data for Ringhals 1 might not constitute a standard for which all nuclear power plants can be measured against. Situations for loss of cooling in a power plant spent fuel pool has prior been investigated at least for normal circumstances. One such investigation for the Ignalina NPP showed that the time until the fuel became uncovered would be 12.5 days for a total thermal load of 4.253MW and a water volume more than five times larger (5070 m3) then the one at Ringhals (995 m3)[9]. The thermal load used corresponds the thermal load after 41 days for Ringhals 1. Using the model used in this thesis for the same thermal load and water volumes as the article results in 11.4 days for the top fuel assemblies (complicated geometry of Ignalina spent fuel pool). In the article a Relap 5 best estimate code was used rather than a worst-case scenario approximation. The difference in the results are probably due to the fact that the Relap code is a best estimate code and does account for the thermal losses through the concrete walls as well as to the building surrounding the pools. The difference due to these effects can then be estimated to 8.77% assuming this effect is close to linear. Then the a best estimate for case three can be seen in table 3.6. When the values have been corrected with an increase of 8.77% one would still draw the same conclusions.

Table 3.6: Case 3 corrected results.

Days after shutdown 7 30 100 Days until uncovered fuel worst case 0.9 1.42 2.27 Days until uncovered fuel best estimate 0.98 1.54 2.47 14 Spent nuclear fuel pools, spent fuel and Decay heat CHAPTER 4

THE POSSIBILITY OF AIR COOLED PASSIVE SAFETY SYSTEMS

A passive safety system could be one that either prolongs the boling process or one that makes sure boiling never occurs. The first type of system that comes to mind is the one that prolongs the time for boiling so that the worst case scenario will have the same safety margins as the existing systems has today for a normal scenario. Such a system might be quite complicated since it might have to account for boiling, vapor and rising pressures. Therefore the author decided to investigate the possibility of a system that ensure that boiling never occurs. Other design suggestion are brought up in Chapter 5.

4.1 Design requirements

As was shown in Chapter 3 a realistic situation might occur where the fuel might be uncovered rapidly. The probability for such a series of events is not necessarily negligible. Whether or not the the probability of such a chain of events is higher than the regulations allow is not under investigation in this thesis. Since the system only has to lower the probability of boil off, a system should be designed that will weigh in the safety issues but also the economical aspects as with all engineering problems. For this thesis it was investigated whether an open passive cooling system could be built using air as the heat transfer medium. The feasibility of such a system is investigated in this chapter. The author investigated a passive safety system that would remove enough energy to ensure that boiling never occurs. In other words a system that would keep the pool temperature below the saturation temperature Tsat. Due to the temperature difference of the pool and the surrounding air, evaporation of the water would still occur [3]. To counteract the loss of water through this mechanism a system is suggested that encloses the spent fuel pool and retains the water. The type of machinery that would enclose the pool when not being filled or emptied is not part of this thesis. It is however possible to imagine that such a system need not be very complicated since it would not have to handle pressures above normal. A simple hydraulic thin folding lid with a seal could be one solution. If the pool is enclosed and if the temperature is kept below the boiling point any vapor would then condense and return to the pool. In this way the water level can be kept nearly constant except for the height increase due to the change in density when the water heats up. Whether or not the height would remain nearly constant could be investigated in a small scale lab where one could compare the water height in a enclosed aquarium with a heat source with the water heights in such an aquarium where one of the walls is replaced by a heat exchanging metal wall.

4.2 Design Suggestion

In figure 4.1 and 4.2 a graphic interpretation of the suggested system can be seen. The dark grey area is the concrete fuel pool. The red part will be any type of doors that can seal the pool whenever fuel is not being inserted or removed from the pool. A lid that can be moved would allow for the use of the same type of storage system that are used today to move the fuel in and out of the pools under water. The light grey represents the air ducts that will transport air from the bottom of the reactor building into the building and up via a heat exchanging copper wall to a stack. The air duct bottom part would probably be a concrete

15 16 The possibility of air cooled passive safety systems design feature of the power plant while the top would be made out of metal. The air duct has two inlets for the purpose of redundant safety were anything to happen to the outer walls on one side of the building. The orange shows the upper part of the pool wall that will be replaced by a copper wall. As can be seen in figure 4.2 the pool has thick concrete walls that serves as radiation protection surrounding the pool and air duct. Most spent fuel pools today are lined with stainless steel and so would this one be except for the copper wall.

The initial idea of this design is a simple heat exchanger where heat is transported through the wall by means of convection and conduction. Since both of these heat transport phenomena are strongly dependent on the thermal conductivity of the wall material used the necessity for a metal is quite obvious. Copper was chosen since the author believes that it has the highest thermal conductivity of the metals that can realistically be used for this type of application.

Figure 4.1: Passive safety system that was investigated.

4.2.1 Copper wall design

Initially a plane wall was designed. The feasibility of this type of heat exchanger is strongly dependent on the outdoor temperature since it’s a passive system and initial results showed it might not be efficient enough during the hottest summers in Sweden. Also the initial wall design was tall and narrow but since the height of the wall defines the characteristic length of the wall for convection a broader but shallower design might be preferred. And finally the wall side have been designed with triangular teeth shown in figure 4.3. If designed with many short teeth(1 cm or shorter) the conduction conditions could be considered nearly unchanged while the area for heat convection may be doubled using in the shape of equilateral triangles.

4.2.2 Wall strength and thickness

The feasibility of such a design would first of all depend on the structural strength of the copper wall. The design of the system would also have to ensure radiation protection work workers in the plant. Design Suggestion 17

Figure 4.2: Passive safety system that was investigated.

Figure 4.3: Copper wall design.

Radioactive Particles

Radioactive particles will give the minimum thickness of the copper wall. α, β and γ particles are all created in the decay of radioactive particles and can be harmful to health and cannot be allowed out of the power plant nor be allowed to cause any harm to the workers. 3 meters of water is enough to contain these particles but how about copper? Decay particles such as α and β are easily stopped but γ photons are what we would need to concern ourselves with. Energy from radiation is measured in the unit Grey [Gy] and is calculated into Sievert through a radiation weighting factor that takes into the the damage of the particle in question. For gamma rays the weighting factor is equal to one for all energies. Equation 4.1 can be used to calculate the attenuation in a material as a function of distance where µ is the attenuation coefficient that is dependent on the absorption coefficient of the material. Since the duct is surrounded by the same amount of concrete as the pool the radial distribution from the pool can be considered safe. Depending on the final design, the upper part of the duct has to be investigated. If this part of the duct were to be designed with concrete as well the problem might easily be solved. However, concrete structures constitutes the largest cost in a nuclear power plant and cost saving designs should be investigated if the passive design turned out to be feasible. 18 The possibility of air cooled passive safety systems

I = e−µx (4.1) I0

Structural strength

The copper wall could be built very thick from the structural integrity and the radiation protection perspec- tive. There are however two limiting factors, the cost and the heat transfer. As can be seen in equation 4.12 a thinner wall will give a higher heat flux through the wall [3]. The second limiting factor of cost is always present since the largest costs of nuclear power are the construction costs of the nuclear power plant itself. According to [8] the maximum stress for a flat plate with three supported edges (here: bottom and sides) and with a third free edge(here: top) can be calculated with equation (4.2) where w is the hydrostatic stress at the bottom of the wall where its the highest, t is the thickness of the wall, b is the wall height and β is determined by tables in [8] A standard norm for the allowed maximum stress is given by equation (4.3) which gives a large margin to plastic deformation of the wall.

ωb2 s = β (4.2) max t2 σ s = s (4.3) max 1.5

Combining these equations we get equation (4.4) that gives us the minimum allowed thickness of the wall. s 1.5ωb2 t = β (4.4) σs

4.3 Theory of heat transfer and governing equations

Heat transfer is defined as the natural redistribution of thermal energy due to a temperature gradient in a medium. Since there will be a temperature difference between the spent fuel pool water and the air in the air duct thermal energy will travel from the pool to the wall by natural convection, trough the wall by thermal conduction and then to the air in the duct by natural convection [7]. Because the air will be heated the density will be lowered and the air will rise through the duct due to the buoyancy effect caused by a density difference. The duct must therefore be designed to allow for enough air to travel through the duct by itself to cool the wall enough to cause a heat gradient large enough to keep the pool below it’s saturation temperature Tsat. If enough buoyancy can be achieved an entirely passive safer system could be designed.

4.3.1 Convection

Convection is a transport phenomena caused by the bulk motion of the medium particles. The heat fluxed is defined according to Newtons law of cooling (4.5) where h is the heat transfer coefficient which is strongly dependent on the surface and the flow conditions [3]. The heat transfer coefficient can be calculated with equation (4.6) where Nu is the Nusselt number, λ is the thermal conductivity of the solid material and L is the characteristic length of the surface.

00 q = h(Ts − T∞) (4.5)

Nuλ h = (4.6) L

The Nusselt number is the dimensionless heat gradient at the surface and varies with the type of medium and the flow conditions. It will therefore differ on the water-metal side of the heat exchanger from the air-metal side. The Nusselt number is dependent on the Prandtl number (equation 4.7) and the Grashof number(equation (4.8)) which have the same definition for both the air and water side. The difference between the two convection conditions is how the Nusselt number is related to the Grashof number and the Prandtl number [2, 7]. This dependency is often determined empirically. Theory of heat transfer and governing equations 19

ν P r = (4.7) α

gβ(T − T )L3 Gr = s ∞ (4.8) ν2

Nusselt on water-metal side

On the water side of the heat exchanger we have water and we have free convection driven by the boyancy forces of the water caused by a temperature difference due to the heating of the water from the fuel assemblies. The recommended equation for external natural convection is given by equation (4.9) [2, 7] for external laminar natural convection for a vertical wall. Ra is the Rayleigh number which substitutes for the Reynolds number for natural convection it is defined in equation (4.10) [2, 7].

1 0.67Ra 4 Nu = 0.68 + (4.9) 9 4 0.492 16 9 (1 + ( P r ))

Ra = P rGr (4.10)

Nusselt on air-metal side

On the air side of the metal we also have free convection due too buoyancy forces acting on the air in the duct. On this side however we have air but we also have a duct, not an infinite heat sink. In a duct we can not threat the convection boundary conditions as free convection on a vertical plate since the enclosed fluid will behave differently from a an infinite heat sink that we could threat the pool as. Instead we have to use a correlation for an internal free heat convection in a rectangular vertical tall duct given in equation (4.11) [2].

L 1 Nu = 0.364 Ra 4 (4.11) H

4.3.2 Conduction

Conduction is a heat transport phenomena that can be conceived as an effect of particle movement. Energy is transported from the more energetic particles to the lesser energetic ones through interactions between these particles. The rate of heat transfer in one dimension is determined by Fourier’s law given in equation (4.12) where k is the coefficient known as the thermal conductivity of a material [7]. Equation (4.12) gives the heat flux and equation (4.13) gives the total heat rate through a wall of thicknes x.

dT q00 = −k (4.12) x dx

kA∆T Q = (4.13) ∆x

4.3.3 Heat transfer

The total heat transfer can be calculated with equation (4.14) by adding all thermal resistances from the conduction and convection boundaries. For a vertical plate the areas would be the same but due to my design the area for conduction is twice the area of conduction due to the equilateral triangle design of the copper wall teeth.

1 ∆x 1 ∆T = Q( + + ) (4.14) h1A1 kA2 h2A1 20 The possibility of air cooled passive safety systems

4.3.4 Air duct mass flow

The air duct must be designed to allow for an air mass flow large enough to get a heat transfer coefficient h that is large enough to remove all the heat generated by the spent nuclear fuel. The buoyancy forces cause by the temperature difference of the ambient air and the air in the duct can be calculated using equation (4.15). The pressure losses in the air duct can be calculated using equation (4.16) where ξ is the friction loss coefficient and Dh is the hydraulic diameter of the duct.

Pb = gHρairβ∆T (4.15)

2 Hρairv ∆Pch = ξ (4.16) 2Dh

To achieve buoyancy the forces must counteract the pressure losses. Setting these two equation equal to one another we can calculate the flow velocity in the duct (equation (4.17)). With the velocity and temperature difference known we can calculate the heat removed by the air in the duct using equation (4.18). The friction loss coefficient is calculated using a handbook[6]. In this engineering handbook loss coefficents can be looked up for flows in the particualr geometries used along with coefficents for flow area changes. These loss coefficients can the be added up and used in equation (4.17).

s 2P D v = b h (4.17) ξHρair

Qremoved = W Cpair∆T (4.18)

4.4 Results and feasibility

4.4.1 Structual Integrity of the wall

Most non-alloyed coppers have a tensile strength between 200-400 MPa. Pure copper properties are used throughout this thesis. The water pressure and the added atmospheric pressure adds up to about 290kPa at a depth of 9 m. Inserting the values along with the dimensions of the copper wall into equation (4.4) results in a minimum thickness of 14.68 cm and therefore the wall design would have to be 15 cm thick.

Teacher Nils-Gunnar Ohlson at the solid-mechanics department was kind enough to input the values into a program called EMRC by NISA which solves this problem by the Finite Element Method. The model did not account for the walls own weight nor the stresses caused by the edge-walls but these should be negligible for this case since the wall is vertical and the walls that its attached to are 1-1.5 m thick concrete walls. Figure 4.4 shows the stresses in the wall if it is 15 cm thick. The highest stress found is 76.75 MPa and according to equation (4.3) the maximum allowed stress is 133 MPa, in other words a good margin to plastic deformation. Teacher Nils-Gunnar also did the calculations for a wall 10 cm thick and the maximum stress for that case was 170 MPa which is above the allowed value confirming the calculations used by the author.

4.4.2 Heat transfer

The author investigated the the maximum allowed temperature of the air in the duct that would allow for an safety system that would passively remove all heat generated by the spent fuel. Since we know the saturation temperature of water we can say that the maximum allowed temperature of the spent fuel pool water is 95 degrees celsius. And since we know how much heat is produced we also know how much heat that must be transferred to the air in the duct. We can there for calculate the temperature required fore each part of the system each step by step starting with the convection from the water to the wall. In appendix D the Matlab code used for these calculations can be seen. The entire heat transfer could have been calculated by equation (4.14) but had the heat transfer coefficients been know. The author set the maximum temperature for the pool, and since the amount of heat produced was known the temperature distribution could be calculated Results and feasibility 21

Figure 4.4: Stress distribution in copper wall.

Figure 4.5: Temperature distribution through copper wall [3] step by step finally finding out the maximum allowed temperature of the outdoors temperature allowing for full passive safety. The temperature distribution is illustrated in figure 4.5. Calculations were made for what would constitute the worst case scenario corresponding to an accident occurring seven days after a full core unloading. The author calculated the temperature distribution based on the requirement that the spent fuel pool temperature, T∞1 in figure 4.5, must remain at 368.15 K. Results for these calculations can be seen in table 4.1.

Table 4.1: Temperature distribution for a full core unloading, case 1.

T emperature Celsius Bulk Water 95 Wall Water Side 94.41 Wall Air Side 63.8 Bulk Air 37.9

If the bulk air leaving the duct would have an average temperature of 37.9 degrees celsius and the ambient outdoors temperature would bearound 20 degrees celsius the velocity of the the air over the plate would be m 0.7996 s and would remove only about 7.91kW of heat(out of 6.7 MW ). This means that temperature of the wall and air boundary layer in the duct would reach close to the saturation temperature of the water in the spent fuel pool but would still only remove a fraction of the heat needed. With an average temperature 22 The possibility of air cooled passive safety systems

m of 37.9 degrees celsius in the duct the air would reach a flow velocity of 0.8 s .

4.4.3 Verification with OpenFoam

The author has used openFoam as a verification tool. In Appendix A the software package openFoam is presented and explained. Simulations were run to investigate the flow velocities and temperatures within the air duct. The author set the hot wall temperature in OpenFoam to 100 degrees celcius since it was known that only a small amount of heat would be removed and the temperature of the wall would rise to or near the boiling temperature of the pool. These simulations verifies the calculated results. In figure 4.6,4.7 and 4.8 the pressure, temperature and flow velocities caluclated are shown. The author spent a lot of time trying to contruct a three dimensional case but the mesh grew too big and therefore a two dimensionall case was chosen. The author also tried constructing a two dimensional duct representating the air duct but did not manage to apply the correct boundary conditions and the solution became non-physical. The correct result have been obtained for a closed natural circualtion system such as the ones used in [2] when explaining a buoyance driven system with a heat source and a heat sink.

Figure 4.6: Pressure distribution.

The author believs the obtained results represents a physical system since the pressure distribution shows a higher density in the bottom of the system and the lower density air in the top. The author also believs that the system used has a heat sink large enough since the temperature distribution is quite small. The velocities behaves as expected, the air rises in the duct and circulates down i the large ”enviroment”. The average velocity in the top of the duct is 0.4012ms−1 and the average temperature is 310.5492K. For the suggested design this means that the system would remove 3.15kW compared with 7.9 kW that was caclulated by the author.

4.5 Conclusions

The suggested design is not effective enough to remove enough heat in a worst case scenario. The pool would still boil and the reduction in saturation and boil off time would be negligible. At the point of boiling the water level would decrease uncovering the copper wall with time and thereby further decrease its effective- ness. Another problem with the correlations used by the author shown with the OpenFoam calculations is the fact the only a thin layer close to the heated copper wall will reach the ”bulk air temperatures” calcu- lated. Therefore the mass flow would be even lower and the amount of heat removed will be even less than calculations used for internal natural convection. As is seen by these calculations even a perfect heat transfer to air at atmospheric pressure would require an enormous heat gradient in order to achieve the buoyancy Conclusions 23

Figure 4.7: Temperature distribution.

Figure 4.8: Flow velocities. needed to remove enough heat from the spent fuel pool wall. The author has therefore concluded that an air cooled passive safety system can’t feasibily be designed and applied to a spent nuclear fuel pool within any nuclear power plant. 24 The possibility of air cooled passive safety systems CHAPTER 5

ALTERNATIVE DESIGN SUGGESTIONS AND EMERGENCY PREPARDNESS

5.1 Closed System

The first type of passive air-cooled safety system isn’t efficient enough. A different possibility might be closed systems with a heat transfer medium different from air. A quick check through reference tables makes it obvious that the heat capacity along with the low densities of gases at atmospheric pressures is too low. The density for water is however a thousand times higher and the specific heat is four times larger than the specific heat for air. A closed system with a pressurizer ensuring the correct water level might be a possible solution. The water-cooled closed system would work if the system is designed with the outdoors temperature as a heat sink. A large enough heat exchanger must be design to keep the water at a constant temperature but with flow resistance low enough to ensure stable natural buoyancy driven flow. The problem with water might be that the low heat gradient won’t be enough for a stable flow.

5.2 Prolonging Designs

5.2.1 Larger pools

The simplest solution to the problem of spent fuel pool boil off is of course to design the pools large enough so that the boil off times for a worst case scenario are within an acceptable limit. On could reason the water from the other pools could be used to enlarge the total amount of water but since we are investigating a worst-case scenario we can presume that the other pools are emptied for some maintenance right after the core has been refueled. Presuming that there is a fixed limit to the shortest allowed time for possible boil off determined by the authorities, and then the pool would have to be built to uphold these conditions.

5.2.2 Implementing other types of existing passive cooling systems

There are several system used today to prevent pressure build-ups and ensure cooling in current and future reactor designs. One such system is the gravity driven passive cooling system utilizing the pressure head of an elevated reserve pool to inject water in the spent fuel pool could be a simple solution that might save space. This can be designed with passive pressure valve so that injection of water starts when the water level of the spent fuel pool reaches a certain level. Another benefit to this system is that the water in the reserve pool could be saturated with boron to further ensure any criticality issues that might occur during accident scenario such as those at Fukushima Dai-ichi. This could also be combined with a system that condense as recycles the water that evaporates from the spent fuel pool.

25 26 Alternative design suggestions and emergency prepardness

5.2.3 Enclosed pool

Having some type of sealing or lid made from a highly conductive material such as aluminum that could close off the pool would prolong the time until boil off since the water would remain in the pool as Vapor and vapor would condense and return to the pool. And due to the high thermal conduction of aluminum or copper some heat would be transferred to the reactor hall. This type of system might lack in efficiency due to the same reasons, as the design in Chapter 2, the convective properties of air during free circulation is rather ineffective. The pool could however be built to withstand large vapor pressures.

5.2.4 Multiple pool heat sink

As was mentioned in Chapter 2 passive systems that prolongs the evaporation and boil off times could be investigated. One such system might utilize the rest of the water in the reactor hall without a connected passaged way. Figure 5.1 represents this design. Here as in the design suggested in Chapter 2 one of the concrete walls would replaced by a copper wall that in turn could be coupled with rerouted water from the other pools in the reactor hall. The orange wall is the wall that would exchange heat with the rest of the water in the reactor hall thereby prolonging the time until boil off. The green circle only symbolizes the pool that is on top of the reactor lid. This type design is rather easy to implement.

Figure 5.1: Alternative safety system design

5.2.5 Lessons from Fukushima Dai-ichi

In the event that any severe accident occurs external action might be vital to ensure the safety of the power plant. At Fukushima Dai-ichi fire trucks and make up fire trucks with low pressure pumps where used to pump water into the reactor buildings [1]. Ensuring that trucks fitted with water and pumps are close enough for a quick response but at such a safe distance from the power plant to not be affected by the same external events as the power plant might be a recommendation for some utilities since it increases the redundancy of possible safety actions that can be taken. Mobile power, compressed air and water supply that is safely located is the second and third lesson learned in IAEAs expert fact finding report made after the events in Japan[1]. NISA has required that all Japanese power plants implement these lessons without delay. CHAPTER 6

DISCUSSION AND CONCLUSION

Since the author was limited to the data from Ringhals 1 the assumptions made in Chapter 2 might not be relevant for other nuclear power plants. It might also be the case that routines at Ringhals are such that the event scenarios suggested within this thesis are not valid. With that said the author believes that there might be some need for evaluation of the probabilities for possible chains of events that might lead to a worst-case scenario. The case might be such that the probability for a pipe leak that would lead to ”case 3”-scenario is far to low for any real concern. If such investigations would show non-negligible risk there might be a need for revisions. One conclusion can be drawn from the work and research done for the possibility of an air-cooled passive safety system. A passive safety system that mitigates the consequences of a worst-case scenario is not feasible with gases at atmospheric pressures. The efficiency would be so low that the size and implementation of such a system would not be feasible. Several soccer fields of wall area would be needed to remove all the heat or even prolong the boil off times using natural circulation. There are prolonging systems that might and should implemented if the conditions are such that there are accident event sequences that might lead to same conditions as in Case 3 in Chapter 2. The author mentions several times that other nuclear power plants might have larger spent nuclear fuel pools and therefore larger safety margins. But there might also be nuclear power plants where the spent nuclear fuel pools contains even smaller quantities of water and therefore inhabiting smaller safety margins. The possibility of using personnel during accident scenario should not be underestimated since the prepared- ness of the personnel can severely affect the outcome of a severe accident. For examples if valves are realigned directly when external power is lost to prepare for external refilling of water crucial time can be saved in the event that small incidents becomes accidents. Today internal events are very well understood and power plants are inherently safe. External events such as natural disasters and possible human threats are however hard to predict. This area has to be well understood and governments and utilities should take this into account when building, revising and updating security measurements. With that said the author still believes in the inherent safety of modern nuclear power plants. Nuclear power will have to be part of the future at least as a intermittent solution to the environment issues the world is facing today. The author believes that limiting the nuclear industry is a risk greater than expanding it since a larger industry gives more funding and incentives for safety research and improvements.

27 28 Discussion and Conclusion BIBLIOGRAPHY

[1] International Atomic Energy Agency. Iaea international fact finding expert mission of the fukushima daiichi npp accident following the great east japan earthquake and tsunami, 2011. Published on www.IAEA.org. [2] Adrian Bejan. Convection heat transfer (John Wiley and Sons New York, Incorporated, 1995, 2. ed). ISBN 0471579726. [3] Yunus. A. Cengel. Heat Transfer a Practical Approach ed. 2 (McGraw-Hill Education, 2002). ISBN 9780070634534. [4] Biello David. Partial meltdowns led to hydrogen explosions at fukushima nuclear power plant. http://www.scientificamerican.com/article.cfm?id=partial-meltdowns-hydrogen-explosions-at- fukushima-nuclear-power-plant, 2011. Scientific American article. [5] Samuel Glasstone. Nuclear Reactor Engineering (Van Nostrand Reinhold New York, Incorporated, 1981, 3. ed). ISBN 0442200579. [6] I. E. Idelchik. Flow resistance, a design guide for engineers (Hemisphere Publishing Incorporated, 1989). ISBN 0891164359. [7] Frank P. Incropera. Fundamentals of heat and mass transfer (John Wiley and Sons New York, Incor- porated, 2002, 5. ed). ISBN 0471386502. [8] Raymond J.Roark. Formulas for stress and strain (McGraw Hill, 1965, 4th:ed). [9] A. Kaliatka, V. Ognerubova, and V. Vileiniskisa. Analysis of the processes in spent fuel pools of ignalina npp in case of loss of heat removal. Nuclear Engineering and Design, 240, 2010. doi:10.1016/j.nucengdes. 2009.12.026. [10] Roman Thiele PHD KTH. Interview, based on openfoam course. [11] Nilsson Markus. Okg, 2011. Email with Oskarshamn employee. [12] Henrik Nylen. Vattenfall, 2011. Spentfuelstorage.doc regarding Ringhals 1, consider as quote. [13] Massachusetts Institute of Technology. The Future of Nuclear Power an interdisciplinariy MIT study (2003). ISBN 0615124208. [14] OpenFoam. Openfoam users guide. Http://www.openfoam.com/docs/user/. [15] OpenFoam. Openfoam users guide tutorials. Http://www.openfoam.com/docs/user/cavity.php. [16] Bengt Pershagen. Light Water Reactor Safety 2.ed (KTH, 1996). [17] M.E. Weech and Y.J. Lee. Heat transfer in spent fuel storage. Nuclear Engineering and Design, 67, 1982. doi:10.1016/0029-5493(82)90066-8. [18] David C. Wilcox. Turbulence Modeling for CFD (2006). ISBN 9781928729082.

29 30 Bibliography APPENDIX A

VERIFICATION WITH OPENFOAM

The feasibility of the suggested passive safety system is largely dependent on the the conditions in the air duct. The velocity and distribution largely effects the convective heat transfer. An experimental verification of the results is of course out of the question but a numerical verification of the results using a computational fluid dynamics(CFD) code is possible. In the effort to support the results of the previous chapters openFOAM has been used.

A.1 OpenFoam framewrok

OpenFoam is an open source computational fluid dynamics(CFD) software package. CFD codes such as OpenFoam has been developed for the purpose of theoretical verification of dynamical fluid systems since experimental validation is very complicated and expensive. Researchers can now first do the veification on their computer and since the OpenFoam code is open source it can be changed and improved upon by the user. OpenFoam accounts for all physical properties and can therefore compute anything from fluid flows, chemical reactions, turbulence, heat transfer and even electromagnetics by the addition of Maxwell’s equations. OpenFoam contains many standard solvers that are used for different purposes. This thesis has used the‘‘buoyantBoussinesqSimpleFoam’’ and the ‘‘buoyantBoussinesqPimpleFoam’’ solver that is used for buoyant turbulent flow with for a incompressible fluid. The next section will motivate the choice of solver [14].

A.2 CFD

Air flowing through a duct is constituted by real particles moving around individually with respect to each other but they also have collective flow properties. A macroscopic system would be rather impossible to describe by describing each one of these particles with computers used today. The fluid is therefore modeled as continuum that tries to approximate the fluid structure. This continuum is based on the fundamental principles conserving mass, momentum and energy. In CFD the momentum equation(Newtons second law of motion) is replace by the Navier-Stokes equation that relates the stresses applied to the surface of the fluid to the viscous properties of the fluid. The Navier-Stokes equation can be seen in equation (A.1).

δu g ∇P + u · ∇u = − + ν∇2u (A.1) δt ρ ρ

A computation fluid dynamics software such as OpenFoam solves the mass, momentum and energy equa- tions through an iterative process. These equation are however very difficult or impossible to solve for more complex problems since the introduction of the convective acceleration(second term on LHS of equationA.1 makes it a non-linear equation. The convective acceleration is the change in velocity related to the position of the fluid.The non-linearity arises from the turbulence within the fluid and its impact on the stresses act-

31 32 Verification with openFOAM ing on the fluid. A model for the turbulence is therefore needed to solve the problem for turbulent conditions.

Any property of a turbulent fluid can be described as an average value with a fluctuating part, in other words as a stochastic variable. These stochastic variables are averaged over a time interval representing the time of fluctuation. Applying this time averaged variable results in the Reynolds Averaged Navier-Stokes equation, equation (A.2) [18], for the momentum equation. This makes it possible to solve the non-averaged Navies-Stokes equations for turbulent fluids. However, expressing the momentum equation for an incom- pressible fluid means that the velocity of the fluidis described by introducing its fluctuating velocity. The non vanishing fluctuating part, last term on RHS of equation (A.2), of the velocity means that we have more variables than equations and we need to add another equation relating the fluctuating part of the velocity to the viscous properties in order to solve the system of equations.

δui δui δP δ  0 0  ρ + ρuj = − + 2µSji − uiuj (A.2) δt δxj δxi δxj

The Boussinesq assumption, equation (A.3) says that the effective viscosity can be calculated by the sum of the laminar viscosity with the turbulence viscosity. The Menter k-ω-SST model relates the turbulent viscosity with the kinematic turbulence energy k and the specific dissipation ω with equations (A.4). k and ω are related to the turbulent velocities by a two equation system. The system of equations is then closed, making them solvable by replacing the non-linear term with equation (A.5).

νeff = ν + νt (A.3)

k ν = (A.4) t ω

    0 0 δu¯i δu¯j 2 δu¯k −uiuj = νt + − k + νt δij (A.5) δxj δxi 3 δxk

A.2.1 Boussinesq approximation

The Boussinesq assumption used in the previous section is not the Boussinesq approximation that is referenced to in the solver name “buoyantBoussinesqSimpleFoam/PimpleFoam”. The Boussinesq approxima- tion is used for heat transfer solvers where the temperature change of the fluid is rather small. The density in the approximation can then be approximated from a reference temperature and the thermal expansion coefficient, see equation (A.6).

ρ(T ) = ρ(Tref )(1 + β(Tref )∆T ) (A.6)

A.2.2 Newtonian Fluid

The definition of a newtonian fluid is a fluid which has a linear dependence between the stress and the strain. This is the case for fluids such as water and air. The proportionality constant that defines this dependence is the viscosity µ. Therefore, a Newtonian fluid description of the air flowing the duct was used.

A.2.3 Mesh

When using OpenFoam one would first define the geometry which in itself can be difficult when you do not have any experience. The mesh must have cells small enough so allowing for the solution to converge if infact your simulation will end in a steady state solution. The mesh can however not be to small, partly because of the runtime of the simulation that you want as short as possible and because the k- and ω-wallfunctions are not accurate enough for all cell sizes. The k-ω model was chosen since it is less sensitive to the wall cell size than the k- model. With some experience a best guess could be used for approximating the amount of cells needed. However for someone that is unexperienced with the code such as the author, an iterative CFD 33 method was used to determine the amount of cells needed. After a lot of trial and error the mesh defined in appendix E was used.

A.2.4 Boundary conditions

In order to to run OpenFoam calculations the user must define a set of initial and boundary conditions in the ‘‘0"-file. Different models require different sets of initial conditions. The boundary conditions used in the simulations run by the author can be seen in Table A.1. First the initial temperatures and initial velocities are set accordingly. The pressure initial conditions are set in two seperate files. The ‘‘p’’ file contains the intitial pressure conditions which are calculated in this case since we use the Boussinesq approximation. In the ‘‘p rgh’’-file the pressure is set to ‘‘calculated’’ as ρ times g time h. For the ‘‘omega’’ (specific turbulence kinetic energy dissipation rate), ‘‘nut’’ (turbulent viscosity), ‘‘kappat’’ (turbulent conductance energy) and ‘‘k’’(turbulence kinetic energy)-files wall functions are set. The ref- erence are set to standard values since the values are later calculated. The wall functions are used to keep a high resolution in the wall area to avoid inaccurate predictions in the velocity profile. The problem that the models have is the predictions made by ‘the law of the wall”. The law of the wall has been determined empirically and gives a logaritmic velocity distribution close to the wall. The omega model is the only model that gives acceptable values to these predictions without the addition of dampening factors. This is what gives the k-ω model an advantage in a large model with large cells over the k- model. Initial values are given for the wall funcitons but these are used only for the post processing tool ‘ParaView” [18].

Table A.1: OpenFoam bondary conditions

Properties Value Unit Hot Wall Temperature 373.15 K Wall Temperatures 293.15 K Initial velocities 0 m s−1 Pressure “calculated” Pa Prgh ‘buoyantPressure” Pa Omega “omegaWallFunction” s−1 Nut “nutkWallFunction” m2s−1 kappat “kappatJayatillekeWallFunction” m2s−1 k “kqRWallFunction” m2s−2

A.2.5 Accuracy of the results

When a suitable model has been chosen an iterative method solves the governing equations. For each step a residual can be calculated, a measure of how much the solution has changed from one step to another. The optimal criterion would be if this residual reached zero but this will not happen in practice. A ”small enough” residual could be criterion enough to establish that the solution is converging, but how small depends on each individual problem. A normalized residual is therefore calculated by OpenFOAM removing the individuality of the problem. A general criterion of a residual which is of the order of 10−6 is therefore an acceptable con- vergence criterion for any simulations. This is established due to the fact that the computer is only accurate to the power 10−8. The author first tried to solve the problem using buoyantBoussinesqSimpleFoam but the time needed for convergence with the computer available would be enormous[10].

The solver was then changed to buoyantBoussinesqPimpleFoam which is a non-steady state solver. The PimpleFoam gives the solution over a time interval. When running a non-stead-state solver using openFoam convergence of the result canon be controlled in the same manner as for stead state solvers. In order to prevent inherent errors propagating in the solution the Courant number is introduced. This can be interpreted as how far each ”particle” moves in the mesh with each iteration. To ensure non-propagating errors the Courant number limit it must be set to be lower than 1 in the control dict file. This basically means that each time step is kept short enough so that the fluid does not travel further than on cell during one time step [15]. 34 Verification with openFOAM

A.3 Post processing

There are several methods that can be used to interpret the results of the computations. Graphically, paraFoam can be used to see the distributions of the physical properties. Data can also be exported in files that can be used with Matlab for example. To calculate the average physical properties for Chapter 4 a line was defined at the upper boundary of the duct for which all needed physical properties where extracted from the output files using the command sample. The line from which the data was extracted can be seen in figure A.1

Figure A.1: The data for the physical properties were extracted form the red line at the top of the duct. APPENDIX B

HEAT LOSS CALCULATIONS

1 %Fredrik Nimander KTH thesis 2 %Radiation, convection and evaporation heat loss form hot bath 3 %Exampel 14−13 Cengel Heat transfer 2:ed 4 5 6 % Radiation 7 8 epsilon = 0.95;%Emissivity 9 As = 10.5*6.5;%Surface area 10 sigma = 5.67*10ˆ−8;%StefanBoltzman constant 11 Ts = 373;%Saturation Temperature 12 T = 298;%Ambient Temperature 13 14 Qrad = epsilon*As*sigma*(Tsˆ4 −Tˆ4);%Stefan −Boltzma radiation law 15 16 17 %Convection 18 19 Pv = 1.65*10ˆ3;%Vapour pressure 20 rhos = 0.942*323/373;%Suuface air vapor mixture density 21 rhoinf = 1.0684;%Ambient air vapor mixture density 22 rhoave = (rhos + rhoinf)/2;%Average density 23 Lc = As / (2 *(10.5 +6.5));%Characteristic lenght 24 nu = 1.848 *10ˆ5;%kinetic viscosity 25 k = 0.02644;%Water heat conductivity 26 27 Gr = 9.81 *(rhoinf − rhos) *Lcˆ3 / rhoave/ nuˆ2;%Grashof number 28 Pr = 0.7261;%Prandtl number 29 30 Nu = 0.15 * (Gr*Pr)ˆ(1/3);%Nusselt Number 31 32 hc = Nu *k /Lc;%Heat transfer coefficient 33 34 Qcon = hc * As * (Ts − T);%Convective heat 35 36 % Evaporation 37 hfg = 2383*10ˆ3;%Latent heat 38 Dab = 3*10ˆ−5;%Diffusion coefficient 39 Sc = 0.616;%Schmidt number 40 sh = 76.2%Sherwood Number 41 hm = sh *Dab/Lc;%Mass transfer coefficient 42 rhovs = 0.0828;%Vapor surface density 43 rhovinf = 0.012;%Vapor ambient density 44 45 mv = hm *As *(rhovs −rhovinf);%Evaporation rate 46 47 Qev = mv *hfg;%Evaporation Heat 48 49 %Qtot 50 51 Qtot = Qrad + Qcon + Qev%Total heat

35 36 Heat loss calculations APPENDIX C

BOIL OFF TIMES

Fredrik Nimander KTH M.Sc thesis Boil off times Geometry and power data for Ringhals Physical properties found in Wolfram Alpha

Clear[“Global`*”]

Cp = 4.205 ∗ 10∧3; (*WolframAlphaforTi = 313.15K*)

Tsat = 373.15; (*Wolfram Alpha Boiling Temperature*)

Ti = 313.15; (*Should be below 333.15 but assuming more realistic 40*) h = 9.5; (*WaterHeightabovecore, 9.5m = max forRinghals*)

V1 = 5 ∗ 10.5 ∗ 6.5 ∗ (h + 4.5); (*Entire water volume*)

ρ = 992.2; (*Water density at 313.15K*) m = V1 ∗ ρ;

Esat = m ∗ Cp ∗ (Tsat − Ti); (*Amount of energy required to saturate pool*) q0 = 2500 ∗ 10∧6; (*Thermal energy of Ringhals reactor*) top = 3600 ∗ 24 ∗ 365 ∗ 5; (*Operationaltime[s]*) ta = 41 ∗ 24 ∗ 3600; (*Timesinceshutdown[[[s]]*)]

(*Saturation Time*) q[t ]:=q0 ∗ 0.065 ∗ (t∧ − 0.2 − (t + top)∧ − 0.2);

Eq[tc ] = Integrate[q[t], {t, ta, tc}];

T = NSolve[Esat==Eq[tc], tc, Reals];

Tc = tc/.T[[[1][[1][1]]]];;;

StringForm[“Power at failure [%]”]; q[ta][ta]/q0;

StringForm[“Saturation Time[d]”];

37 38 Boil off times

Tc2 = (((tc/.T[[[1][[1][1]]]])) − tata))/3600/2424;;;

(*Time to boil*)

V2 = 10..5 ∗ 6.5 ∗ h; m2 = ρ ∗ V2;

EE = 2..230 ∗ 10∧6 ∗ m2; (*Totalvaporizationenergy(WolframAlpha)*)

Eq2[te ] = Integrate[q[t], {t, Tc, te}]];];; (*Evaluatedfromtimesinceshutdown + staurationtime*)

T2 = NSolve[EE==Eq2[te], te, Reals];

Te = tete/.T2[[1]];

StringForm[“Evaporation time[d]”];

Te2 = (((((tetete/.T2[[1]]) − (Tc))/3600/24;

StringForm[“Total boil-off time including saturation time[d]”]

Tc2 + Te2 (* Prints the total time until Boil off*) APPENDIX D

PASSIVE SAFETY SYSTEM CALCULATIONS

1 % NaturalCirculation heat removal(case 1) 2 % Fredrik Nimander KTH master thesis 3 % Copper wall with equilateral triangle teeth 4 % Removing all decay heat from spent nuclea fuel 5 % Material properties taken from Wolfram Alpha 6 7 clear global 8 clear all 9 clc 10 11 % Design Height=9m w1= 10.5= > w2 =21 12 13 Q = DHP (2500,5,7)%Use the function DHP to get the decay heat seven days after unloading 14 A = 9*21;%Convection area, height9m, width2 *10.5m 15 Acond = 9*10.5;%Conduction are height9m, width 10.5m 16 g = 9.81;%Gravity constant 17 Thickneswall = 0.15;%Copper Wall thickness 18 19 %Convection water−metal 20 kCuw = 394.9;%Copper Heat conductivity 21 Betaw = 7.236*10ˆ−4;%Thermal expansion coefficient 22 Tinfw = 95 + 273.15;%Pool water bulk temperature 23 Tsgw = 367.5624;%Wall surface temperature(manually itterated). 24 CL = 9;%Characteristic Lenght water side= vertical height of wall=9m. 25 rhow = 962.3;%Density Water 26 nu = 2.99*10ˆ−4;%Dynamic viscosity 27 vw = nu/rhow;%Kinetic viscosity water 28 cp = 4.205*10ˆ3;%Specific heat water 29 k = 0.677;%Thermal conductivity water 30 Prw = nu*cp/k;%Wolfram Alpha 31 Grw = −g*Betaw*(Tsgw−Tinfw)*CLˆ3/vwˆ2;%Grashof number water 32 Raw = Grw*Prw;%Rayleigh number water 33 34 Nuw = 0.68 +0.670*Rawˆ(1/4)/(1 +(0.492/Prw)ˆ(9/16))ˆ(4/9);% Nusselt number, Incropera, Bejan reference 35 hwm = Nuw*kCuw/CL% Heat transfer coefficient water side 36 %Q1= hwm *A*(Tinfw−Tsgw);% Heat rate convection water side 37 TW = Tinfw − Q/hwm/A 38 39 dt=Q/kCuw/Acond*Thickneswall%Temperautre difference needed over Copper. to receive heat rate=Q 40 41 % Convection Air−Metal 42 CLa = 4*10.5*0.5 / (2*(11)); 43 kCuwa = 397.3;%Thermal conductivity copper air outer temperature 44 Betaa = 3.3*10ˆ−3;%Thermal expansion coefficient 45 Tinfa = 311.9709%Bulk air temperature, iterated manually 46 Tsga = TW − dt%Duct copper surface temperature 47 Toutw= Tsga − 273.15;%Outer wall temperature for printing 48 rhoa = 1.053;%Air density 49 nua = 2.02*10ˆ−5;%Dynamic viscosity air 50 va = nua/ rhoa;%Kinematic viscosity air 51 cpa = 1008;%Isobaric specific heat air

39 40 Passive safety system calculations

52 k = 0.0289;%Thermal conductivity air 53 Pra = nua *cpa /k;%Prandtl number air 54 Gra = g*Betaa*(Tsga−Tinfa)*CLaˆ3/vaˆ2;%Grashof number air 55 Raa = Pra*Gra;%Rayleigh number 56 57 Nua = 0.364*0.5*Raaˆ(1/4)/9;%Nusselt number from Bejan 58 59 ham = Nua*kCuwa/CLa;%Heat transfer coefficient air side 60 TWO = Tsga − Q/ham/A;%Bulk Air temperature 61 62 % Temperature difference over entire structure 63 % This is to check 64 r = Thickneswall; 65 DeltaT = Q*(1/ham/A + r/Acond/kCuwa + 1/hwm/A) 66 DeltaTcalc = Tinfw − Tinfa 67 Tinfr = Tinfw −DeltaT −273.15 68 69 % Channel Conditions 70 Th = Tinfa ;%Assuming linear heating we can calculate max inlet temperature 71 Text = 293.15 72 Aduct = 10.5*0.5;%Tube Area 73 H = 9;%Total chimney height 74 Hc = 30; 75 f = 1.8195;%Loss coefficient 76 rhoah = 0.993;%hot air density 77 Betaah = 3.3*10ˆ−3; 78 79 Ph =g*H*rhoah*Betaah*(Th−Text);%Buoyancy Pressure head 80 velocity =(Ph*2*CLa/(f*H*rhoah))ˆ(1/2)%Flow velocity 81 W = velocity*rhoah*Aduct; 82 Q4 = W*cpa*(Th−Text)%heat removed APPENDIX E

OPENFOAM MESH

1 /*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*− C++ −*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*\ 2 | ======| | 3 | \\ / F ield | OpenFOAM: The Open Source CFD Toolbox | 4 | \\ / O peration | Version: 2.0.0 | 5 | \\ / A nd | Web: www.OpenFOAM.com | 6 | \\/ M anipulation | | 7 \*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*/ 8 FoamFile 9 { 10 version 2.0; 11 format ascii; 12 class dictionary; 13 object blockMeshDict; 14 } 15 // ************************************* // 16 17 convertToMeters 1; 18 19 vertices // here alla the corners of the blocks are defined. 20 ( 21 (−5 0 0) //0 block1 22 (0 0 0) 23 (0 11 0) 24 (−5 11 0) 25 (−5 0 0.01) 26 (0 0 0.01) 27 (0 11 0.01) 28 (−5 11 0.01)//7 block 1 29 (0.5 0 0) 30 (0.5 1 0) 31 (0 1 0) 32 (0.5 0 0.01) 33 (0.5 1 0.01) 34 (0 1 0.01)// 13 block 2 35 (0.5 10 0) 36 (0 10 0) 37 (0.5 10 0.01) 38 (0 10 0.01)// 17 block 3 39 (0.5 11 0) 40 (0.5 11 0.01)// 19 block 4 41 (−5 1 0) 42 (−5 1 0.01) 43 (−5 10 0) 44 (−5 10 0.01)// 23 45 (−0.01 10 0) 46 (−0.01 11 0) 47 (−0.01 10 0.01) 48 (−0.01 11 0.01)//27 49 (−0.01 1 0) 50 (−0.01 1 0.01) 51 (−0.01 0 0)

41 42 OpenFoam Mesh

52 (−0.01 0 0.01) 53 54 ); 55 56 blocks //The model is made up of cells with only one cell on the z−axis for 2D case. Here the amount of cells is defined. 57 ( 58 //level 1 59 hex (0 30 28 20 4 31 29 21) (800 30 1) simpleGrading (0.25 0.25 1) 60 hex (20 28 24 22 21 29 26 23) (800 180 1) simpleGrading (0.25 1 1) 61 hex (22 24 25 3 23 26 27 7) (800 30 1) simpleGrading (0.25 4 1) 62 63 //level 2 64 hex (1 8 9 10 5 11 12 13) (250 30 1) simpleGrading (1 0.25 1) 65 66 //level 3 67 hex (10 9 14 15 13 12 16 17) (250 180 1) simpleGrading (1 1 1) 68 69 //level 4 70 hex (15 14 18 2 17 16 19 6) (250 30 1) simpleGrading (1 4 1) 71 //extra 72 hex (24 15 2 25 26 17 6 27) (3 30 1) simpleGrading (1 4 1) 73 hex (30 1 10 28 31 5 13 29) (3 30 1) simpleGrading (1 0.25 1) 74 ); 75 76 edges 77 ( 78 ); 79 80 boundary 81 ( 82 hotWall // The heated wall 83 { 84 type wall; 85 faces 86 ( 87 (9 12 16 14) 88 89 ); 90 } 91 Walls // Walls with normals in x,y−direction 92 { 93 type wall; 94 faces 95 ( 96 (0 4 21 20) 97 (20 21 23 22) 98 (22 23 7 3) 99 (28 29 26 24) 100 (10 13 17 15) 101 (0 4 31 30) 102 (30 31 5 1) 103 (1 5 11 8) 104 (8 11 12 9) 105 (14 16 19 18) 106 (2 6 19 18) 107 (25 27 6 2) 108 (3 7 27 25) 109 (28 29 13 10) 110 (24 26 17 15) 111 ); 112 } 113 114 115 emptyWalls // Walls with normals in the z−direction, "empty" for 2D case 116 { 117 type empty; 118 faces 119 ( 120 (0 30 28 20) 121 (20 28 24 22) 122 (22 24 25 3) 123 (4 31 29 21) 124 (21 29 26 23) 125 (23 26 27 7) 126 (1 8 9 10) 127 (10 9 14 15) 43

128 (15 14 18 2) 129 (5 11 12 13) 130 (13 12 16 17) 131 (17 16 19 6) 132 (24 15 2 25) 133 (30 1 10 28) 134 (31 5 13 29) 135 (26 17 6 27) 136 137 ); 138 } 139 140 ); 141 142 mergePatchPairs 143 ( 144 ); 145 146 // ************************************************************************* // 44 OpenFoam Mesh