Numerical Analysis of Local Flow andHeat Transfer Characteristic in a Regular Arranged Pebble Bed Reactor

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ADVANCED LEVEL, 30 ECTS

-STOCKHOLM BEIJING, 2018

Numerical Analysis of Local Flow and

Heat Transfer Characteristic in a Regular

Arranged Pebble Bed Reactor

Bai Yuyu

KTH

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Numerical Analysis of Local Flow and

Heat Transfer Characteristic in a Regular

Arranged Pebble Bed Reactor

Bai Yuyu

Thesis Submitted to

Tsinghua University KTH Royal Institute of Technology In partial fulfilment of the requirement

for the degree of Master of Science

In

Nuclear Science and Technology

In partial fulfilment of the requirement for the degree of

Master of Science In

Nuclear Energy Engineering

Co-supervisor: Professor Yang Xingtuan Co-supervisor: Professor Henryk Anglart Institute of Nuclear and New Energy

Technology/107 Laboratory

Alba Nova University Centre/Division of Nuclear Reactor Technology

UNDER THE COOPERATION AGREEMENT ON DUAL MASTER’S DEGREE PROGRAM IN NUCLEAR ENERGY RELATED DISCIPLINES

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Abstrakt

Avhandlingstitel: Numerisk analys av lokal flöde och

värmeöverföringskarak-täristik i en regelbunden arrangerad stenbäddsreaktor

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Abstract

Pebble bed reactor (PBR) has attracted much attention due to its inherent safety characteristics, low power density and high efficiency of energy conversion. Fuel pebbles are arranged randomly in a pebble bed reactor. The helium, which acts as coolant, will flow through the multiple flow channels of a pebble bed reactor and take away the heat. Because of the particularity and complex flow channel of pebble bed reactor, there will be enormous cost and limitations in the experimentation on it. Therefore, more and more researchers use numerical simulation to study the reactor. This paper uses Fluent, which is a professional numerical simulation software, to simulate the gas flow in a pebble bed reactor. In order to simplify the model, many studies assume that the fuel pebbles are regularly arranged. Due to the limitation of time, only four models were selected in this project for study: the body-centred cubic (BCC) model, the face-centred cubic (FCC) model, simple cubic(SC) model, and BFCC model, whose porosity value is between that of FCC and BCC. An appropriate turbulence model is also significant in order to get accurate results. This paper uses k− ϵ turbulence model to simulate the pebble bed and compares the results with the results obtained by the q-DNS method. It was found that both models can get similar calculation results. The contact mode between each of the fuel spheres is also a problem that needs to be addressed. In this paper, the surface contact method and near-miss model are studied. It was found that there are certain differences in the results of different contact methods, but the smaller the artificial gap, the more similar the results obtained with the contact considered. In addition, this paper also discusses the effect of different inlet Reynolds numbers on the flow and heat transfer process within a PBR. It was found that when the inlet Reynolds number is large, not only the surface temperature lower, but the surface temperature distribution is also more uniform. Finally, this paper compares the flow conditions of the four different arrangement of pebbles and the existence of incomplete fuel balls in the flow field. It was found that the arrangement of the fuel balls inside the pebble bed reactor has a significant influence on the flow field and the temperature field. Changes in the flow field structure may cause changes in the entire downstream flow field.

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Nomenclature

Abrreviation

PBR Pebble Bed Reactor

BCC Body-Centered Cubic

FCC Face-Centered Cubic

SC Simple Cubic

BFCC A combination of Body-Centered Cubic and Face-Centered Cubic PBMR Pebble Bed Moderated Reactor

NPP Nuclear Power Plant

EBR-1 Experimental Breeder 1 PWR Ressurized water reactor (BWR Boiling water reactor

URD Advanced Light Water Reactor Utility Requirements Document EURD Euopean Utility Requirements Document for Advanced LightWater

Reactors

PBMR Pebble Bed Moderated Reactor

NPP Nuclear Power Plant

GIF The Generation IV International Forum VHTR Very High Reactor Temperature Reactor CFD Computational Fluid dynamics

RANS Reynolds Average Navier-Stokes DNS Direct numerical simulation LES Large eddy simulation FEM Finite element method

FVM Finite volume method

FDM Finite diﬀerence method

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T Temperature ρ Densityea p Pressure S Source ui Velocity component xi Space interval cp Specific heat ω Vorticity τi j Tensor of stress µ Dynamic viscosity

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Chapter 1 Introduction and Motivation

1.1 Background

Nuclear energy plays an important role in the structure of the world energy. It may meet our country’s energy needs in the future as a green and low-carbon economic energy. In 1951, the world first nuclear power plant the Experimental Breeder 1 (EBR-1) was established in the United States, and people started to use nuclear power to generate electricity. Till now, the history of the development of nuclear power has been almost 70 years. The Chinese government also pay close attention to the use of nuclear energy. As of May 2017, the People’s Republic of China had 37 nuclear reactors operating with a capacity of 32.4 GW and 20 under construction with a capacity of 20.5 GW[1]_{and became} the world’s second largest nuclear-energy-used country. Additional reactors are planned, providing 58 GW of capacity by 2020. Nuclear power plants are also wildly used in other countries. It was operated in 31 countries. The United States is the largest producer of nuclear power, while France has the largest share of electricity generated by nuclear power.

Figure 1.1 Four generation of nuclear reactor

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roughly divided into 4 generations: in the 1950s, the Soviet Union built an experimental nuclear power plant with Capacity of 5 MW, in the same time the United States built the Shippingport Atomic Power Station which is the first full-scale PWR nuclear power plant. The successful establishment of these experimental and prototype nuclear power plant mentioned above makes a lot of sense. It has proved that we can use nuclear energy to generate electricity. And these nuclear power plants are called the first generation of nuclear power reactors.

From then on, more nuclear power units, such as graphite moderated reactor reac-tors(PBMK), pressurized water reactors(PWR), heavy water reactors(HWR) and boiling water reactors(BWR) were built, and nuclear energy went commercial. In the 1970s, sharp price increases of petroleum energy triggered the energy crisis which promotes the rapid development of nuclear power generation. Therefore, during this period of time, more than 400 commercial nuclear power plants rapidly appeared in the world. And these nuclear power units are called second-generation nuclear power units.

The most severe accident of nuclear power plants which called "The Chernobyl nuclear accident" happened on April 26, 1986, and the "Three Mile Island (TMI) accident" happened on March 28, 1979. These two accidents caused not only huge economic losses but also brought great safety problems. The public was triggered to devote much attention to the safety of Nuclear Power Plant (NPP).

As a result, experts and professors worldwide have begun to explore how to prevent and mitigate the severe accidents in nuclear power plants. Then the United States and Europe have promulgated the "Advanced Light Water Reactor Utility Requirements Doc-ument"(URD) and "European Utility Requirements Document for Advanced Light Water Reactors"(EURD) which require the nuclear power plant meet higher standards such as superior thermal eﬃciency, significantly enhanced safety systems (including passive nu-clear safety), and standardized designs for reduced maintenance and capital costs. Usually, we called the reactors which meet the requirements in "Advanced Light Water Reactor Utility Requirements Document"(URD) or "European Utility Requirements Document for Advanced Light Water Reactors"(EURD) the Third Generation Nuclear Power Units.

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former reactors because of fewer nuclear waste and its ability to prevent nuclear prolifer-ation. The representative fourth generation reactors include Supercritical Water-Cooled Reactor, Molten Salt Reactor, Gas-cooled Fast Reactor, and Sodium-cooled Fast Reactor, Lead-cooled Fast Reactor and Very High Reactor Temperature Reactor (VHTR). The Very High-Temperature Gas-cooled Reactor has good safety features. The fuel of it is recyclable and produces very little nuclear waste. The system is easy to modularize and this specialty gave this type of reactor its economic competitive edge. The Pebble Bed Reactor is categoried to be one type of the VHTR reactor.

Figure 1.2 pebble bed reactor scheme

The pebble bed reactor is a graphite-moderated, gas-cooled nuclear reactor which is shown in Figure 1.2. And nuclear fuel of the pebble bed reactor is spherical fuel elements. These fuel elements are made of pyrolytic graphite, about 60mm in diameter, and contain thousands of micro-fuel particles which are called TRISO particles. TRISO particles consist of a fissile material surrounded by a coated ceramic layer of silicon carbide.

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Figure 1.3 TRISO fuel

of the process, the fuel elements can be taken out from the reducer at the bottom of the reactor to measure the amount of radiation. If the amount of radiation is too low, then it is determined that it has been completely processed and taken out as nuclear waste and replaced with a new fuel ball. If it is within the normal range, it will be reloaded into the core to continue work so as to avoid waste of cost. One of its greatest advantages is its extremely high thermal eﬃciency. The temperature of the helium flow into the steam generator is about 725◦C in a pebble bed reactor. The thermal eﬃciency can even higher than that of modern fossil fuel power stations. Another advantage is the pebble bed reactor is safer.

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coeﬃcient is large, which contributes to reactor stability. Second, as compared to light-water reactors, once the coolant system is depressurized, since it does not have the phase change problem of the coolant, so it will not cause a lot of energy release due to spray. In addition, a large amount of graphite can provide considerable thermal capacity. Therefore, once the cooling capacity is lost, the temperature of the core rises slowly, and there is enough time to take remedial measures. The density of helium is not suﬃcient even under pressure unless it is in a forced circulation system. So it is designed to let the forced circulation system be ready at any time to ensure that the exhaust heat can be removed after shutdown. And even if the coolant is completely lost, the expected result is not so serious like it in light water reactors. However, in the extreme cases of interruption of the cooling medium, as temperature increases, the neutron absorption rate of uranium-238 will increase, which will reduce the nuclear fission of U-235 to stop the reaction gradually. And on the other hand, the dense pyrolytic carbon and silicon carbide on the fuel elements can limit the escape of fission products in the extremely high temperature. In addition, the number of fuel balls can be adjusted to control the volume of the entire core. So this kind of nuclear reactor can be designed as a tiny reactor as a reactor that directly serves as a vehicle power source.

Rudolf proposed the pebble bed reactor concept in 1950. Later, Germany established the first prototype pebble bed reactor in West Germany, which is a 15MW test reactor used to develop and test a variety of fuels and machinery. In 2004, Tsinghua University built a 10MW prototype reactor HTR-10. The experimental reactor has approximately 27,000 fuel balls and uses helium as a coolant to promote power generation. Huaneng’s(a huge energy company in China) first PBR power plant with a capacity of 195MW was built in Weihai, Shandong Province. The project will be completed by the end of 2017. South African energy company ESKOM is one of the leaders in pebble bed reactor technology. They plan to cooperate with British nuclear fuel and Exelon to build a pebble bed reactor power station, which the main function is an increase of power generation at peak hours, also available for desalination. But the plan is delayed due to the environmental problems finally.

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1.2 Literature Review

1.2.1 Research Status of Experiments About PBR

The fuel balls in the pebble bed reactor are full of cores, closely arranged. Its physical model is very similar to that of a porous medium containing a strong internal heat source. The coolant flows in its complex flow path to take away fission-generated heat. Because the pores formed by the accumulation of fuel balls in the pebble bed are irregular, fluids are continuously mixed and flow separated in a pebble bed with so many spherical surfaces. This process involves a variety of heat transfer mechanisms, making the problem extremely complicated. And because experimenting directly with real pebble bed reactors requires significant costs and is limited by measurement technology and safety, experiments become diﬃcult to achieve. Therefore, most experimental studies considered the pebble bed reactors as a packed pebble bed, the flow path as a porous medium flow path and set up a small test stand to carry out experiments. Porous media refers to solids with many pore channels inside. Chemical packed beds, soils, and porous cores of thermal tubes can all be regarded as porous media. Many pebble bed reactors where fuel balls accumulate form complex pore channels, and thus can also be seen as porous media.

Packed pebbles are wildly used in engineering applications. Many researchers have conducted detailed studies on the flow and heat transfer characters in packed pebble beds. In fact, the pebble bed reactor model is geometrically similar to packed pebble bed reactor. However, most of the studies focus on how to enhance heat transfer for solid-filled particles without internal heat sources.

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in the water-filled flow channels packed with solid particles, the heat transfer coefficient increased by 5 to 10 times and in an air-filled flow channel packed with solid particles, the heat transfer coefficient can increase 3 to 30 times. Huang and Zhao[9,10] _{studied the} effect of solid particles with different thermal conductivity on the heat transfer coefficient. The results show that the greater the thermal conductivity of the filled solid particles, the higher the heat transfer efficiency. Zhang Zhijun et al.[11,12] _{combed the concluded} the experimental results of heat transfer between two parallel plates with solid particles and drawn the following conclusions: When the distance between the plates is constant, the Nusselt number change monotonically with the diameter of the solid particles. The smaller the particle size, the higher the heat exchange efficiency. Ichimiya et al.[13]_studied the local flow and heat exchange characters in porous media. It was found that when the thermal conductivity of the porous media particles is high, there is an optimal Darcy number, which can enhance the convective heat transfer without significantly increasing the flow resistance, thereby obtaining a higher heat exchange efficiency. In short, fill the solids into the flow channel can indeed considerably enhance the heat transfer. And heat transfer coefficient has a more significant correlation with the diameter of the filled solid particles and their thermal conductivity. But the above studies are only qualitative analysis and do not give some quantitative heat transfer standards. Other scholars have proposed some similar heat transfer criteria for pebble bed reactors. Wakao and Kagei[14]_combed and summarized the previous experimental results: For porous media, when the Reynolds number is range from 20 to 10000, and when the flowing medium is air or water, the following heat exchange relationship exists:

Nu = 2 + 1.1Re0.6Pr1/3 (1-1)

Whitaker[15]_{derived heat transfer criteria for a single pipe, multiple pipes, flat plates,} a single fuel sphere, and the entire pebble bed. The scope of this rule is to use air as the heat transfer medium, with a Reynolds number of 22 to 8000.

Nu = (0.5Re1/2+ 0.2Re2/3)Pr1/3 (1-2)

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Reynolds number of 10 to 5000 and a heat exchange medium of air:

Nu = 0.23Re0.7pr1/3 (1-3)

Nie et al.[17]_{summarized the heat transfer relationships for porous media that are suitable} for Reynolds numbers from 5 to 280 and that the heat exchange medium is air:

Nu= 0.419Re0.8572Pr1/3 (1-4)

However, the above convective heat transfer criterion relations are all derived on the premise that no internal heat source particles are contained. The medium used is mostly air, and its applicability is limited. The actual pebble bed reactor is based on helium as the heat exchange medium and contains an internal heat source. It’s a complex conjugate heat exchange problem. Some scholars have also conducted relevant experimental research. The most common way to obtain internal heat source is to add an electrical resistance heater wire. This method is to wrap the resistance wire around the local part that needs to be heated. The heating power can be adjusted by changing the voltage across the two ends. Due to its simple structure, stable heating power and controllability, it is widely used in the experiments. But this heating method can only be used for local heating. And the internal flow path of porous media is generally too complex to place much resistance wire in case of aﬀecting the flow field and aﬀecting the accuracy of the final result. Zhai

Figure 1.4 Experimental schematic of heating with resistance wire

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influencing factors of heat transfer in the pebble channel. Their experimental equipment is a 160-mm-diameter steel tube filled with 31.8-mm diameter metal balls in a regular delta arrangement.1.4 A total of 18 layers are filled, but heated metal balls are only placed in the middle of layers 4, 10, and 16 layers. The rest of the metal balls are not heated. The metal ball near the wall surface is cut to eliminate the wall effect as much as possible. The wall effect refers to the fact that relatively large voids are formed in the vicinity of the contact between the spherical particles and the wall surface so that the flow resistance at this point is greatly reduced. Therefore, most of the cooling fluid will pass through this large gap, thereby reducing the heat exchange efficiency. Their research finally got the following conclusions:

1. The flow field in the pebble bed is very uneven. And the healing power of the metal ball is large; it intensifies the inhomogeneity of the temperature field which can en-hance lateral natural convection and increase the heat transfer coeﬃcient. When the heating power is low, the heat transfer coeﬃcient does not change significantly with heating power and pressure but continues to decrease as the inlet water temperature increases.

2. The dimensionless criterion correlation of the local heat transfer coeﬃcient of the pebble bed with the internal heat source is summarized and proved by the data. The error between the experimental value and the calculated heat transfer coeﬃcient is within 10%. The research in Zhai’s work is more comprehensive. The way in which the resistance wire is embedded in the metal ball can avoid changing the flow field as much as possible.

However, there are still some problems. First, the resistance wire cannot be com-pletely and evenly embedded inside the ball. This leads to a non-uniform heating power of the individual fuel balls. In addition, the limitation of the number of heated metal pellets can cause the uneven heating power of the entire pebble bed. But in a real reactor core heating power is uniform. Finally, deliberately eliminating the wall eﬀect is not in line with the actual situation and may aﬀect the accuracy of the experimental results.

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can heat the entire pebble bed reactor uniformly at the same time. An electromagnetic induction heating method was used for the experimental study for the first time in Wu’s[21] work. Naik et al.[22]_{used an electromagnetic induction heater with an alternating magnetic} field frequency of 450 kHz and power of 10 kW and 25 kW, respectively, to study the effect of inlet speed, pipe diameter and filling ball diameter on temperature rise and pressure drop in the pebble bed. Miyazaki et al.[23–25]_{investigated the effect of pressure drop and} ball diameter on heat transfer in a packed bed, where the metal ball is electromagnetic induction heated. Catton et al.[26] _{used electromagnetic induction heating to study the} effect of pressure drop on the dry-up of a packed pebble bed, and derived the laws of the heating power distribution. Their experimental results were similar to those model of Lipinski[27]_{. Athens and Berthoud}[28] _{explored how the internal heat transfer of the} pebble bed change with ball diameter and height of filled balls, pebble bed height, and other factors. Schafer et al.[29,30] _{used electromagnetic induction heating to perform} experiments on a packed pebble bed with a strong internal heat source. The experiment filled cylindrical barrels with metal balls of 6 mm or 3 mm in diameter. The filled height is 640 mm, with water as the heat transfer medium. The top of the water is about 310 mm higher than the top of the metal ball (see Figure 1.5). It mainly explores the influences of the ball diameter and the height of the pebble bed on the critical heat flux and the pressure drop. Their experimental results show that the results when using metal balls with 6mm diameter are in good agreement with the Tung et al.[31] _{model, and the results with the} 3mm diameter are very different from their models.

Weisberger et al.[32–34] _{filled metal balls in an inclined bottom square sink and used} the electromagnetic induction heating method to investigate the two-phase boiling heat transfer in the pebble bed .

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Figure 1.5 Experimental setup

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problems progressed slowly. In view of these problems and diﬃculties in the experiments, numerical simulation methods have become an increasingly important tool for the study of pebble bed reactors.

1.2.2 Research Status of Numerical Simulation of PBR

The method of numerical simulation has advantages over the experiment. For ex-ample, the cost and time for setting up the experimental bench can be saved. The heating power and other parameters can also be set as actual parameters. Not only does it cir-cumvent the difficulty of data measurement in the experiment, but it also gets richer data. Therefore, many scholars have used CFD methods to study spherical particle-packed beds such as pebble bed reactors. Guo Xueyan et al.[35]_{used a finite volume method based} on the Chimera mesh for packed pebble bed reactors. In the simulation, the spherical particles were all of the same diameters. The results showed that the flow channels in the packed pebble bed reactor are complicated and the flow has strong non-uniformity, wall effect, and channelling phenomenon. As the ratio of the diameter of the fixed bed to the diameter of the fuel ball increases, the flow inhomogeneity in the packed bed will be improved, and the wall effect and channelling effect will be significantly reduced, which in turn will make heat transfer processes and chemical reactions more uniform.

It is necessary to use precise mesh and appropriate calculation methods to simulate the entire pebble bed and to obtain accurate and reliable data in local areas. However, most simulation studies now are only based on using coarser unstructured mesh for the entire pebble bed and using the RANS method to solve problems and obtaining steady results for analysis. These results often do not reflect details of local flow fields and temperature fields. Therefore, in order to obtain more accurate local flow field and temperature field information, many researchers have simplified the reactor model due to the limitation of computational resources. And Ferng and Lin[36]_{believe that the randomly} arranged pebble bed should actually be randomly composed of three arrangements, which are Simple Cubic(SC), Body-Centered Cubic(BCC), and Face-Centered Cubic(FCC), as shown in Figure 1.6.

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Figure 1.6 Diﬀerent arrangements

Figure 1.7 Contact models

between the fuel spheres is a detail that needs to be considered. The general contact point processing methods are: near-miss, point contact and surface contact as shown in Figure 1.7, the near-miss model refers to artificially leaving a gap between the fuel balls, thereby avoiding the diﬃculty of mesh generation near the contact points and reducing the number of mesh. The point contact model is that a ball touches another ball at one point. And the surface contact is to consider the fact that the two spherical fuel balls come into contact with each other due to the slight deformation of the fuel ball.

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in Figure 1-7, where the contact point is not well performed due to the mesh adhesion, causing the mesh to distort, and aﬀecting the calculation results. Theoretically, it requires infinite encryption to avoid this problem. In the same year, Lee et al.[38]_{conducted detailed} studies on fuel spheres using diﬀerent contact methods. The results show that the results of point contact methods are similar to those of near-miss model. However, the paper still uses unstructured mesh and standard k-ϵ model to calculate.

In 2009, Kim et al.[39]_{] used unstructured mesh, artificial gap contact methods, and} standard k-ϵ models to investigate simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) aligned simplified pebble bed reactor. The results show that the simplified reactor core with the simple cubic (SC) arrangement has the largest porosity and the simplest flow path, resulting in the fluid does not fully transfer heat with the surface of the fuel ball, so the surface temperature of the fuel is the highest. In 2013, Pavlidis and Lathouwers[40]_{used a large-eddy simulation (LES) to study the pebble bed} using an unstructured self-adaptive mesh. Artificial gaps model were used between the fuel balls. But this method introduced the deformation of the mesh; it is diﬃcult to judge if this work is valuable. But at least it shows that many researchers propose solutions to the mesh generation method for the pebble bed reactor. In 2010, Wu et al.[21] _used a standard k-ϵ model to compare the results of modelling of a ball-ball array with an unstructured mesh and with body-centred cubic (BCC) arrangement and using a porous medium hypothesis. Similar results were obtained. However the hypothetical approach of porous media can save computational cost, but it can not capture the details of the flow channel while using actual pebble model can capture some of the flow details, but the computational cost is also greatly increased. Nijemeisland and Dixon[41]_{studied the} near-miss model and point contact model. They concluded that when the clearance is 1/100 or less of the diameter of the fuel sphere, the result will be consistent with the results of the point contact method. In 2012, Li et al.[42]_{compared the eﬀect of fuel sphere diameter on} temperature field. The paper still uses unstructured mesh, standard k-ϵ turbulence model, and artificial gaps between fuel spheres. 2013 Ferng and Lin[36]3_{used RSM to compare} the results of body-centred cubic (BCC) and face-centred cubic (FCC) aligned pebble bed models with unstructured mesh.

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FCC model in detail using the method of the polyhedral mesh using q-DNS(quasi-direct numerical simulation) to simulate the simplified reactor. The results show that even though the arrangement of the fuel balls is regular, the flow field and the temperature field show an asymmetrical distribution, indicating that the presence of a large number of vortices makes the flow field and temperature field extremely complex.The contact model is a near-miss model. Although a large number of the polyhedral mesh are used and are encrypted in gaps, the quality of the mesh does not meet the requirements of the DNS, so it’s called q-DNS. This shows that it is challenging to divide the structured fuel mesh model in SC, BCC, and FCC simplified PBR.

In summary, the main diﬃculties in studying the flow field and temperature field in the pebble bed reactor are as follows:

1) Overall pebble bed and local pebble bed model The whole pebble bed simulation can certainly obtain more reasonable local details in diﬀerent areas such as the core, near wall, etc. But the premise is that a vast number of the mesh are needed, which is impos-sible for the existing calculation conditions. However, numerical simulation of regularly arranged ball arrays can greatly reduce the number of mesh, but the boundary condi-tions around them will inevitably need to be treated as symmetric boundary condicondi-tions or periodic boundary conditions, which may have some impact on the results. Therefore, trade-oﬀs and choices need to be made on the basis of specific research problems. This paper mainly studies the local flow field in the pebble bed reactor core. Considering comprehensively, the local pebble-bed model is more appropriate.

2) Mesh Generation Most of the researchers chose unstructured mesh to reduce the diﬃculty of mesh generation. To get reliable results, the number of unstructured mesh needed is enormous. In particular, where the fuel ball contacts, the mesh needs to be encrypted. Many scholars are also trying diﬀerent mesh generation methods including adaptive unstructured mesh and polyhedral mesh. Because of the complex flow path in the pebble bed reactor, the quality requirements for the mesh are also very high. The author spent a lot of time to study various mesh division software to find the impact on the mesh quality and finally got a higher quality mesh.

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they calculated using the RANS method. The RANS method is of course highly eﬃcient. However, if you want to study the local flow field and temperature field, the internal velocity and temperature fluctuations are also vital data. But the RANS method based on the time average derivation and seemed impossible to get these details accurately.

4) Treatment of contact models The basic processing methods include near-miss model, point contact and surface contact. The choice of contact method not only has an eﬀect on the diﬃculty of grid generation but also has a great influence on the calculation result. This article has carried on the computational analysis and the detailed comparison to near-miss model and contact model.

5) Conjugate or non-conjugate heat transfer Conjugate heat exchange means that there is a body mesh inside the fuel ball. The fuel ball generates heat as a body heat source. Instead of conjugate heat exchange, it removes the fuel ball’s internal mesh to reduce the number of meshes, thereby greatly reducing the calculation cost and converting the thermal power to fuel ball spherical surface boundary heat flux. Many studies have adopted non-conjugate heat exchange methods to reduce costs. This article conducts detailed studies using conjugate heat exchange method.

1.3 Purpose and Objective

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1.4 Contents of the Research

Numerical simulation of the entire pebble bed reactor requires a large amount of grid and a long iteration time. It is not realistic for the existing calculation conditions, so this paper simplifies the real pebble bed reactor into a regularly arranged ball array. In the real pebble reactor, the fuel elements are arranged randomly. It is considered as a combination of diﬀerent structured arrangements, including SC, BCC, FCC and other kinds of arrangements. We simulated four types of the arrangements, including SC, BCC, FCC and a combination structure of BCC and FCC(BFCC). Then we observed some flow patterns and velocities of the fluid domain or in some selected planes. It is easier to draw some regular conclusions when studying the arrangement of rules. The content of this article includes the following aspects:

1) Explore the eﬀects of diﬀerent contact modes between fuel spheres such as near-miss model, point contact model and surface contact model on the calculation results and the reason.

2) Comparing the results of diﬀerent turbulence calculation methods, the turbulence calculation method selected is the standard k-ϵ model. The results are compared to q-DNS calculation method on the local flow field and temperature field of the pebble bed reactor. 3) The results of conjugate heat exchange (body heat source) and non-conjugate heat transfer (surface heat source) are compared to investigate the diﬀerence in the spherical temperature distribution of the fuel sphere and its reason.

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Chapter 2 Mathematical Model

2.1 Development and Application of Computational Fluid Dynamics

Newton discovered the basic law of motion of macroscopic objects in 1687, which laid the foundation for the study of fluid motion. Until the early 1950s, experimental research and theoretical research were the main methods for exploring the law of fluid motion. But with the development of research, their advantages and diﬃculties have also been gradually recognized by people respective. Although the experiment can obtain reliable observation results, the cost of its equipment is generally higher, the duration is longer, and the research scope is more limited. Theoretical research can give analytical solutions to certain problems within a certain scope of application, but it is generally limited to some simpler flow problems. In fact, all flow problems follow the three basic laws: conservation of mass, conservation of momentum, and conservation of energy. The equations derived from these basic laws, coupled with the associated state equations, constitutive equations, and boundary conditions, can describe fluids reasonably. These equations are often called the governing equations. However, the flow of fluid is actually very complicated. Its governing equations are usually a highly complex and highly nonlinear equation set. Therefore, it is almost impossible to obtain an analytical solution. However, with the dramatic increase in computer capabilities, using numerical methods to solve the flow problem becomes more and more popular, and forming a new discipline - Computational Fluid Dynamics(CFD), which based on fluid mechanics, heat transfer theory, thermodynamics, computational graphics, etc.

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Figure 2.1 flowchart of CFD procress

information of the flow field distribution in the entire calculation area is obtained. The results are processed by computer graphics technology and finally visually represented on the computer graphic interface. Through the numerical simulation method, the distribu-tion of pressure, velocity, temperature, and even chemical components at various places under various complicated conditions can be obtained, as well as the variation of these quantities over time. This technical method can save a lot of experimental costs and time. And detailed phenomena such as flow, heat transfer, and mass transfer that are diﬃcult to observe in experiments can be obtained. CFD is also called numerical simulation, numerical emulation or numerical calculation. The flow chart of CFD work procress is shown in Figure 2.1.

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in academic research and engineering fields. At present, scholars at home and abroad have made great progress in the application of CFD technology to pebble bed reactors. CFD has become an indispensable tool for pebble bed reactor research.

2.2 Basic Mathematical Models

In this paper, we use a widely used CFD numerical simulation software, Fluent and simulates a regularly arranged pebble bed reactor. As with all flow and heat transfer problems, it is necessary to solve mass conservation equations, momentum conservation equations, and energy conservation equations also known as Navier-Stokes equations.

The He gas is flowing through a segment of the pebble-packed reactor core, and transfer heat with the sphere fuel elements. In this kind of flow and heat transfer process, the continuity, momentum and energy equation always need to be solved. When comes to radiation, We can simply ignore it because of the transparency of the coolant and the small temperature diﬀerence between neighbouring fuel elements as shown in Sobes’s[48] work. The following set of equations are the Governing equations solved by FLUENT 15.0 in this process. Continuity equation is as follows:

∂ρ ∂t + ∂ ∂xi (ρui) = 0 (2-1) Momentum equation ∂ ∂t(ρui) + ∂ ∂xj (ρuiuj) = − ∂p ∂xi ( µ∂ui ∂xj − ρu′ iu′j ) + Si (2-2) Energy equation ∂ ∂t(ρE) + ∂ ∂xi (ui(ρE + p)) = ∂ ∂xi ( λ∂T ∂xi + uiτi j ) + Q (2-3) 2.3 Turbulence Modelling

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The standard turbulence model(k − ϵ) The standard turbulence model is the most widely used two-equation turbulence model. Transport equations of k andϵ need to be introduced. k is turbulent energy andϵ is turbulent dissipation rate, which is defined as :

ε = µ_ρ( ∂ui′ ∂xk ) ( ∂u′ j ∂xk ) (2-4)

The turbulent viscosityµt can be described with k andϵ as

µt = ρCµ k2

ε (2-5)

The transport equations of k andϵ are ∂(ρk) ∂t + ∂(ρku∂xii) = ∂ ∂xj [( µ + µt σk ) ∂k ∂xj ] + Gk + Gb− ρε − YM + Sk (2-6) ∂(ρk) ∂t + ∂(ρku∂xii) = ∂ ∂xj [( µ + µt σε ) ∂ε ∂xj ] + Clεε_k(Gk + C3εGb) − C2ερε 2 k + Sε (2-7)

Some useful parameters in the equations are given as follow

Gk = µi ( ∂ui ∂xj + ∂uj ∂xi ) ∂ui ∂xj (2-8) Gb = βgi_Prµt t ∂T ∂xi (2-9) β = −1 ρ∂T∂ρ (2-10) YM = 2ρεM_t2 (2-11)

In the formula, Gk is the turbulent kinetic energy which is caused by the average velocity gradient; Gb is the turbulent kinetic energy caused by the buoyancy; YM is the influence of the expansion diﬀusion in the compressible flow on the overall dissipation rate; C1ϵ,C2ϵ, and C3ϵ are constant values, which are respectively 1.44, 1.92, 0.09; σk andσ_ϵ are the Prandtl numbers of Turbulence, which are respectively 1.0 and 1.3; Skand S_ϵ are user-defined source terms.

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2.4 Parameters and Criteria

2.4.1 Vorticity

In fluid mechanics, vorticity is introduced to determine the centre of rotation of a vortex. The definition of the vorticity is as follow:

ω = ▽ × u = (∂w_∂y − ∂v_∂z,∂u_∂z − ∂w_∂x,∂v_∂x − ∂u_∂y) (2-12) In term of mathematics, the vorticity of the velocity field is equivalent to 2 times the angular velocity of the velocity field. The vorticity reflects the intensity of vortex of the physical micelles.

2.4.2 The Synergy Angle

The synergy Angle is defined as the angle between velocity and the temperature gradient. We can see from the energy equation that The angle between temperature gradient and speed has an important eﬀect on heat transfer. The smaller the angle between them, the better the surface heat transfer of the fuel ball, so it is meaningful to analysis this parameter.

2.4.3 Drag Coeﬃcient(Cd)

In fluid dynamics, the drag coeﬃcient is a dimensionless quantity which used to describe the drag force of an object in a fluid. The drag force mainly contains skin friction resistance and form drag. Sometimes lift-induced drag and interference drag also play a big role. The magnitude of Cd comprehensively expresses the influence of angle, shape, surface smoothness, Reynolds number, etc. So it’s also an important parameter to describe the flows in the pebble bed.

2.4.4 Nusselt Number(Nu)

It’s a dimensionless number that named after Wilhelm Nusselt. The definition is

Nu= h L/λ =

hL

λ (2-13)

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a fluid where both conduction and convection dominate. This kind of problem is called conjugate heat transfer problem. And in a PBR, the pebbles contain a heat source. So it’s a conjugate heat transfer problem at the pebble surface. In a particular problem，L and λ is constant, and Nu number can represent the intensity of convection heat transfer. 2.5 Chapter Summary

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Chapter 3 Influence of Diﬀerent Contact Model

3.1 Model Descriptions and Numerical Treatment

3.1.1 Basic Physical Model

Figure 3.1 Geometry and mesh of FCC arranged simplified PBR

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parameters are shown in Table 3-1 and Table 3-2.

Table 3.1 Boundry conditions

Location Type value

Inlet Velocity inlet 0.6m/s Temperature 523K Outlet Pressure outlet 0Pa

Temperature 600K

Boundry Symmetry

-Pebble surface Wall

-Table 3.2 Parameters

Parameter Helium Graphite

Density(kg/m3₎ _5.36 ₁₇₅₀

Specific heat (J/kg · K) 5441.6 1690 Thermal conductivity (W/m · K) 0.3047 25

Dynamic viscosity(kg/m · s) 3.69×10−5

-The selection of the conversion range in this paper is based on the parameter in the q-DNS simulation performed by Shams et al.[47]_{The inlet velocity is 0.3 to 1.2m/s in this} paper. Four main surfaces along x- and y-axis are symmetry boundaries, which means that the velocity and scalar variable gradients are set 0 in this kind of plane. The inlet is located at z = 0.2m, and He is injected from the inlet at a special velocity with a fixed temperature of 523◦C According to Ferng and Lin[36]_{, the outlet is located at -1.0 m, which} is far away from the last layer of pebbles to prevent the reversed flow during the simulation. Besides, a near wall arrangement was adopted at the wall boundary. In addition, in the fueled region, a constant energy source 6.23MW/m3_{is set as fission source.}

3.1.2 Mesh Model

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Figure 3.2 Local mesh near contact area

is the spacing between the fuel balls used in previous works such as Shams, and the 0.06 mm is based on the Nijemeisland et al.’s[41] _{work who believed that when the gaps} between the fuel balls is less than or equal to 1% of the diameter of the fuel ball, the results are consistent with the point contact model. Therefore, in this paper, there are three examples for investigating the influence of the fuel sphere contact method. They are the body-centred cubic (BCC) fuel arranged model with 5 mm and 0.6 mm artificial gaps, and surface contact respectively. In fact, due to the successive decrease of the porosity of the three case, the inlet cross-sectional area will be slightly diﬀerent. However, the change is small and can be basically ignored. So this article uses the same inlet speed for the above three calculations.

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3.1.3 Calculation Method and Verification

In this chapter, the k − ϵ model is adopted. Considering the limited computing resources, the physical properties of the gas are set to constant. And the eﬀects of gravity and heat radiation was not taken into account. The detailed calculation method is introduced in Chapter 2, which will not be repeated here. In order to verify the applicability of the calculation method and the grid to the numerical simulation of the pebble bed reactor, the same calculation method and mesh are used here to compare with Shams et al.’s FCC model. As shown in Figure 3.3, the average speed of the centre position

Figure 3.3 Location of selected lines

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with a large curvature, resulting in some deviation in the velocity distribution in the fluid domain. And DNS-method is more accurate in predicting vortices. These vortices have great interactions with each other, which makes momentum and energy transfer between more fully and velocity distribution more uniform.

Figure 3.4 Temperature of center line with diﬀerent calculate method

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3.2 Results and Discussion

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In the following article, they are called plane 1, plane 2, plane 3, plane4 and plane5 respectively. Line 1, line 2, line 3, and line 4 on the centre of the 4th layer of the fuel ball are also extracted. It should be noted that the line 2 passes through the contact points of the two fuel balls and is in a special position where the result may be influenced because of the symmetry of the geometric model and the momentum on both sides of the contact point may cancel out each other, and some special details may not be observed. Therefore, the line 4 is taken on the adjacent side to compare the area better where the fluid passes through the contact point and the flow separation and vortices have a great influence. The streamlines near the center fuel ball on the 4th layer is shown in Figure

Figure 3.7 Streamlines near the center fuel ball

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two pebbles. This is due to in the body-centred cubic (BCC) arrangement pebble beds, such area can form a flow channel with low flow resistance from top to bottom. Relatively speaking, the flow resistance in this channel is the smallest, while in the other regions, the flow resistance is larger due to the obstruction of the fuel ball. So most of the fluid tends to flow to the area with small flow resistance which causes a large flow velocity.

The arrangement of ball arrays is becoming less and less compact in three cases, and the ratio of flow resistance in other areas to the flow channel is decreasing. The resistance distribution is more and more even. So it can be seen that the velocity of the surface contact case along the line 3 is the largest. The flow velocity distribution in the model is more inhomogeneous. And the uneven flow rate will inevitably cause uneven heat transfer because most of the coolant passes along the direction of the flow channel along the side of the fuel sphere and between the plane 2 and plane 3, which of course can improve the surface heat transfer conditions in this area. But in fact, the highest temperature on the spherical surface of the fuel sphere is not in this area but near the contact area of the fuel balls. Therefore, most of the cooling fluid passes through this channel, which results in failure to suﬃciently exchange heat with the fuel sphere, and the heat exchange eﬃciency is low. Therefore, the uneven flow rate is not conducive to the heat transfer of the fuel sphere in the BCC array.

Figure 3.8 show the average temperature distribution over five sections (surface contact, 0.6mm gap, 5mm gap from left to right respectively). It can be found that in the surface contact case, the temperature of the fuel ball among three cases is higher and most uneven. This is due to the fact that the surface-contacting BCC arrangement of the fuel ball array is more compact. The fuel balls are closer together, and the heat flux between them is larger. And on the other hand, such a compact structure further exacerbates the uneven distribution of the flow velocity, which indirectly also exacerbates the uneven temperature distribution.

Figure 3.9 shows the time-averaged temperature distribution of the spherical fuel ball at the 4th layer. It represents the top, bottom, side and 45-degree sides of the fuel ball from top to bottom. From left to right, the results are surface contact, 0.6mm gaps, and 5mm gaps. It can be seen that the highest time-averaged temperature is at the contact point or near the “artificial gap”, and the lowest temperature appears at the position along the upper hemisphere of line 7 due to the higher fluid velocity and better heat transfer.

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Figure 3.8 Temperature distribution in five planes

in selected pebbles. The blue line in the figure shows the results of the case with a 3mm artificial gap; the red line shows the results of the case with a 0.6mm artificial gap; the black line shows the results of surface contact gap. First of all, it can be observed that as the gap becomes smaller and smaller, the average sphere temperature distribution of the fuel sphere becomes closer to the surface contact. Therefore, it can be considered that when the spacing is small enough, the near-miss model results are similar to those of the contact. The main difference between the temperature of the artificial gap and the surface contact at the contact area. It is conceivable because no matter how small the artificial gap is, there will be fluid passing through it. Convection heat transfer dominates. But when the fuel ball is in contact, the heat transfer near the contact point is the effect of both heat conduction and convection heat transfer between different fuel spheres. And heat conduction dominates. The main heat transfer mechanism has changed, so in actuality, there will be some local hot spots at the contact point.

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Figure 3.9 Temperature distribution of fuel ball surface at the 4th layer in Figure 3.12.

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Figure 3.10 Temperature distribution in line 1

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Figure 3.14 Distribution of the cosine value of the field synergy angle

cosine value of the field synergy angle in seleted lines. The result of the synergy can tell us that in the case with surface contact method, the angle between the velocity and the temperature gradient is large which means the heat tranfer condition is terrible. Especially in the vicinity of the contact area there is a huge diﬀerence between two method. This is due to that the fluid is blocked near the contact area in the surface contact model. In other regions, the fuel sphere whith a small gap and the fuel sphere contact with each other can get more consistent results with an error less than 1%.

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Figure 3.15 Distribution of the cosine value of the field synergy angle in line 1

Figure 3.16 Distribution of the cosine value of the field synergy angle in line 2

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Figure 3.17 Distribution of the cosine value of of the field synergy angle in line 3 periphery of the centre of rotation. In the area immediately below the contact surface, the trajectory of the fluid rotation path is just along the normal direction of the spherical surface of the fuel ball. Therefore, the angle between the average velocity and the average temperature gradient is smaller, and the heat transfer is better. And as for return fluid, the periphery of the centre of rotation is increasing, and the outermost part of the fluid impinges on the surface of the fuel sphere. At this area, the average velocity is opposite to the direction normal to the outer surface of the spherical surface of the fuel ball. Therefore, there will be a strip area on two sides of the fuel sphere, in which the velocity and the average temperature gradient form a large angle, and the heat transfer is poor.

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Figure 3.18 Isosurface of vorticity

Figure 3.19 Local streamlines of surface contact model 3.3 Chapter Summary

Based on the comparison and analysis of the contact modes between fuel spheres in this chapter, the following conclusions can be drawn:

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In the pebble bed reactors, it is obvious that at the contact point heat conduction is dominate, so the numerical simulation of the pebble bed reactor using the artificial gap treatment method will cause some errors.

2. The ball arrays arranged in body-centred cubic (BCC) with a 5mm gap and 0.6mm gap, and surface contact method respectively. As the structure becomes more and more compact, the ratio of the flow resistance between the upper and lower fuel spheres and the flow resistance in the side grows larger and larger, and the flow velocity becomes more and more inhomogeneous, thereby making the heat transfer uneven and unfavourable for heat exchange.

3. The vortex structure in the pebble bed has a large difference with the artificial gap and surface contact. As the artificial gap becomes smaller and smaller, the flow field and the temperature field will become closer and closer to each other. But the difference in average temperature at the contact point (or close to the artificial gap) is still large. The vortex structures of the two are quite different, and U-shaped horseshoe vortex structures are presented in four directions in the area between the upper and lower fuel spheres using surface contact results.

4. For the low-velocity area below the contact point or contact surface, the vortex has an important influence on the heat transfer there. The two vortices at the contact point (or the contact surface) interact with each other and bring out the heat generated near contact point along the outer normal line of the fuel ball sphere (or the contact surface) to the fuel ball side high-speed area. During the process, some fluid of the partial rotation area hits the spherical surface of the fuel sphere, which has some negative influence on the area below the contact point (or the contact surface). However, the advantages of the vortex heat exchange mechanism outweigh the disadvantages, and it is beneficial to the heat transfer in the pebble bed, while the eddy heat transfer intensity of the surface contact is the highest.

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Chapter 4 Influence of Diﬀerent Reynolds Number

4.1.1 Basic Physical Model

The calculation model used in this chapter is the same as that described in Chapter 3, Section 3.1. Due to space limitations, this chapter only applies the k- turbulence model for the body-centered cubic (BCC)-arranged model with 0.6mm gaps between fuel spheres.

4.1.2 Mesh Model

This chapter uses the standard turbulence model for the simulation calculation of the flow in a BCC arranged pebble bed reactor. Considering the limited computational resources, this chapter does not consider the eﬀects of gravity and heat radiation. The gas properties of helium are set as constants. The specific algorithms for the turbulence model and mesh model are described in Chapter 2.3, Section 2.3, and are not repeated here.

4.2 Results and Discussion

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Figure 4.1 Temperature distribution in selected fuel pebble

diﬀerence. When the inlet Reynolds number becomes 10458, the temperature of the fuel sphere is more uniform throughout the sphere. The reason that the spherical temperature distribution of the fuel ball becomes uniform is the heat flux distribution on the spherical surface of the fuel ball to become more and more uniform as the inlet Reynolds number increases. And this is due to the uneven distribution of heat flux distribution.

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Figure 4.3 Temperature distribution in plane 4 and plane 5

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that the intensity of the vortex is increasing, and this is the fundamental reason why the heat transfer of the fuel ball becomes even. The intensity of the vortex becomes larger and the disturbance in fluid become stronger which makes the velocity distribution more uniform, thus making the heat flux in the fuel ball surface more evenly distributed, indirectly causing the temperature distribution on the spherical surface of the fuel ball to become uniform. Therefore, increasing the inlet Reynolds number not only improves the heat transfer around the spherical surface of the fuel ball but also enhances the uniformity of the fluid velocity within the pebble bed due to the increased disturbance, thereby improving the uniformity of the spherical temperature distribution of the fuel sphere.

Figure 4.7 shows the mean time-averaged temperature of each layer in the fluid domain. Since the contact time between helium and the fuel ball with an inlet velocity of 0.3 m/s is the longest and the heat transfer is more adequate, the temperature is highest, and rises most fast. In order to compare the heat transfer of three diﬀerent inlet Reynolds numbers, it is more appropriate to select the amount of heat absorbed per second of helium as the comparison parameter. The amount of heat absorbed per second of helium is defined as:

Heat = ρvinletSinletcp(Tposition− Tinlet) (4-1)

Where Vinlet is the inlet velocity, Sinlet is the inlet cross-sectional area, ρhelium is the helium density, and C pheliumis helium Specific heat capacity, Tposition represents the average temperature of each layer, and Tinlet represents the average temperature of the inlet. As shown in Figure 4.8, it is clear that helium at 3m/s will bring more heat per second and the heat exchange efficiency will be higher. However, according to the conservation of energy, this value should be equal. The reason why this results occur is that the outlet temperatures are different due to different flow rates. And the fluid properties are not constant at different temperatures, thus resulting in this calculation results.

4.3 Chapter Summary

Based on the comparison and analysis of diﬀerent inlet velocity simulation results obtained by using simplified BCC pebble bed in this chapter, the following conclusions can be drawn:

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Figure 4.6 Vorticity distribution in plane 4 and plane 5

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Figure 4.8 Heat absorption per second each layer

ball is significantly improved, and the surface temperature of the fuel ball is significantly reduced. This is due to the increase in the coeﬃcient of convective heat transfer at the high Reynolds number and the ability to carry more amount of heat at the same time. This makes the heat exchange process enhance, and the surface temperature of the fuel ball reduce.

2. Due to the large number of spherical surfaces in the pebble bed, as the inlet Reynolds number increases, the increase in the flow velocity results in increased fluid disturbances in the flow path and increased vortex strength, especially in such low velocity areas between the upper and lower fuel spheres, making the fluid along the The flow velocity and heat distribution perpendicular to the cross-section of the flow direction are more uniform, so that the spherical temperature distribution of the fuel ball is more uniform.

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Chapter 5 Influence of Diﬀerent Arrangement of Pebbles

As mentioned above, in a real pebble bed reactor, the fuel pebble are packed randomly in the reactor vessel. But when investigating the local characteristics, it challenging to build the randomly packed geometry model, so most of the researchers use the regular packed arrangement as an alternative. In the above studies, BCC model is used. In this chapter, FCC, SC, and BFCC(a combination structure of BCC and FCC) are also studied. The geometry schematic is shown in Figure 5.1.

Figure 5.1 Geometry model of 4 arrangement

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Table 5.1 Typical mesh number for diﬀerent pebble arrangement pebble

arrangement SC BCC BFCC FCC

mesh number 2230040 533762 578743 998298

5.2 Result and Discussion

5.2.1 Four diﬀerent pebble arrangement

Figure 5.2 Streamlines of 4 arrangement

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Figure 5.3 Secondary flow of 4 arrangement

index shows that the SC alignment fluid domain has the smallest average velocity. And as the porosity get higher, the maximum velocity magnitude is also getting higher.

Figure 5.3 shows the secondary flow of each arrangement. And the color index represents the velocity magnitude. As can be seen from the color, the smaller the porosity, the greater the secondary flow velocity. From the secondary flow in the selected plane, it can be concluded that in each plot, vortices are identified in the pore between pebbles. Similarly, the smaller the porosity, the stronger the vortices. It clearly shows in the BCC and BFCC case, the vortices are not symmetry although the boundary is set to symmetry. This also revealed that transient calculation is important in this kind of problems.

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a different scale of vortices inside the pebble bed, and the vortex generation and shedding process may induce alternating forces and the fuel ball may flow by the vortex. As a result, vibration and noise occur, and resonance and sonic boom may occur severely. Therefore, this part mainly studies the change of drag coefficient under a different arrangement.

For each arrangement, 34 set of data is obtained with diﬀerent Reinlet number. The range of Reinlet is from 2000 to 7000. The drag coeﬃcient(Cd) was changed along with the Reinlet. It is observed that there is a functional relationship between the two. If the simulation data are used to fit correlations. The simulation results and predicted curves are shown in Figure 5.4 to Figure 5.7. The predicted correlations are as follows:

Cd = 0.02633 − 3.62 × 10−6(Re)2+ 2.92 × 10−10(Re)3 (5-1) Cd = 0.05471 − 1.13 × 10−5(Re)2+ 1.053 × 10−9(Re)3 (5-2) Cd = 0.147 − 6.67 × 10−5(Re)2+ 8.15 × 10−9(Re)3 (5-3) Cd = 0.158 − 4.89 × 10−5(Re)2+ 5.02 × 10−9(Re)3 (5-4)

Figure 5.4 Simulated value and fitted curve with SC arrangement

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Figure 5.5 Simulated value and fitted curve with BCC arrangement

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Figure 5.7 Simulated value and fitted curve with FCC arrangement

extremely complex flow channel. Besides Re_[inlet], the arrangement also affects drag coefficient. As can be seen in four figures, the pebble beds with a compact arrangement have a higher drag coefficient with the same Re_[inlet]; this may because of a different form resistance.

5.3 Chapter Summary

The content of this chapter compares the diﬀerence of velocity field and temperature field with the diﬀerent arrangement of fuel spheres. The relationship between Cd and Re in a certain range of four cases was obtained through simulation data. This study has shown that:

1. In the case where the fuel ball is arranged more closely, and the porosity is small, the internal velocity of the flow field is relatively large. The reason is that when the fluid of the same mass flow rate flows through a tighter structure, the flowing cross-section is smaller and therefore the velocity is greater.

2. When the fluid flows through the pebble bed, due to the complicated flow channels, many vortices are formed. The secondary flow velocity inside the more closely arranged pebble bed is larger, and the formed vortex intensity is greater, so it will inevitably bring about better Heat transfer eﬀect.

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Reynolds number in a certain range. Under the same inlet Reynolds number, the structure becomes more compact as the porosity increases. The resistance coeﬃcient becomes larger.

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Chapter 6 Influence of Incomplete Pebbles in PBR

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6.2 Result and Discussion

Figure 6.2 Streamlines in PBR with incomplete pebbles

Figure 6.2 shows streamlines in five different situations. The color of the streamline indicates the velocity magnitude. When there is an incomplete fuel ball in the flow field, the structure of the flow channel will change, and this change will certainly have an impact on the flow field. In the first case, the blockage of the fluid will form a local high-pressure zone and will cause partial fluid form reverse flow, affecting the upstream flow field. The direction of the fluid also changes and no longer flows along the surface of the fuel ball. And The heat transfer of the incomplete fuel ball completely depends on the vortex. In the second case, the fluid flows along the surface of the pebble and is separated at the edge. A low-velocity and high-pressure zone are formed at the rear, which is not conducive to heat exchange. In the latter three cases, the flow channel becomes no longer symmetrical. The fluid may flow through the two sides of the fuel ball at different speeds, resulting in lateral forces. At the same time, different flow rates on two sides of the incomplete fuel ball can also cause uneven heat transfer.

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Figure 6.3 Velocity vector in a selected vertical plane

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direction caused by the change of flow channel will also affect the angle between velocity and temperature gradient, further affecting heat transfer. Figure 6.4 shows the distribution of the secondary flow of each structure on the selected plane. Several pairs of vortices with different sizes and directions appear on the selected plane.

Figure 6.5 Temperature distribution on selected pebble surface

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6.3 Chapter Summary

The chapter focus on the flow field when there is an incomplete fuel ball inside the pebble bed. The following conclusions can be obtained:

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Chapter 7 Conclusions and Future Work

7.1 Conclusions

The main purpose of this thesis is to investigate the local thermal-hydraulic charac-teristics of PBR. Fluent software are used to simulate a segment of pebble bed reactor. Helium is used as the coolant. The different contact model is used between adjacent pebbles for four different arrangements of the mesh model. The influences of varying contact model, different structure of pebbles, different inlet velocity on flow and heat transfer characteristic are analyzed in this article. Some conclusion can be obtained as follows

1. It is essential to select the appropriate contact model during the analysis of the local heat transfer flow of the PBR. This paper compares the heat transfer and flows condition when contact points are treated by surface contact and near-miss model. It was found that diﬀerent models used diﬀerent heat transfer mechanisms at the point of contact. The former mainly rely on heat conduction, and the latter mostly rely on convection heat transfer. For practical pebble bed reactors, apparently at the contact point, it should also be dominated by heat conduction. But with the near-miss method, the smaller the gap, the results are more similar to the results of contact model.

2. The increase in the flow rate results in a significant improvement in the heat transfer of the spherical surface of the fuel sphere, which not only significantly reduces the spherical temperature of the fuel ball, but also enhances fluid disturbances, especially in low speed areas between the upper and lower fuel sphere, and promotes momentum and heat in the area. Making the spherical temperature distribution of the fuel ball more uniform.