Science and Technology of Nuclear Installations

Volume 2014 (2014), Article ID 820481, 10 pages

http://dx.doi.org/10.1155/2014/820481

Research Article

Some Movement Mechanisms and Characteristics in Pebble Bed Reactor

Xingtuan Yang,¹ Yu Li,¹ Nan Gui,¹ Xinlong Jia,¹ Jiyuan Tu,1,2 and Shengyao Jiang¹

¹Institute of Nuclear and New Energy Technology of Tsinghua University and the Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Beijing 100084, China

²School of Aerospace, Mechanical & Manufacturing Engineering, RMIT University, Melbourne, VIC 3083, Australia

Received 16 November 2013; Accepted 4 May 2014; Published 1 June 2014

Academic Editor: Yanping Huang

Copyright © 2014 Xingtuan Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The pebblebed-type high temperature gas-cooled reactor is considered to be one of the promising solutions for generation IV advanced reactors, and the two-region arranged reactor core can enhance its advantages by flattening neutron flux. However, this application is held back by the existence of mixing zone between central and peripheral regions, which results from pebbles’ dispersion motions. In this study, experiments have been carried out to study the dispersion phenomenon, and the variation of dispersion region and radial distribution of pebbles in the specifically shaped flow field are shown. Most importantly, the standard deviation of pebbles’ radial positions in dispersion region, as a quantitative index to describe the size of dispersion region, is gotten through statistical analysis. Besides, discrete element method has been utilized to analyze the parameter influence on dispersion region, and this practice offers some strategies to eliminate or reduce mixing zone in practical reactors.

1. Introduction

The high-temperature gas-cooled reactor (HTGR) [1] is generally recognized as a probable solution for the generation IV advanced reactors [2, 3] for its advantages of security, environmental applicability, high efficiency, and industrial-process heat applied in producing hydrogen. A pebble bed-type reactor core is one of the mainstream types for HTGRs, which has been adopted in many tests or demonstration reactors such as HTR-10 [4–6] in China, PBMR [7, 8] in South Africa, and their prototype reactor known as AVR [9–11] early in Germany. Compared with conventional reactors, a pebble-bed reactor is formed with spherical coated fuel pebbles and graphite pebbles instead of fixed fuel assembles. Pebbles descend along the core under gravity, whose movements are determined by pebble flow. Pebble-bed HTGR runs in a recirculating way, in which fuel pebbles are drained out from discharge hole at the bottom of the core and loaded into the core from the top. When pebble bed reaches equilibrium state, the number of pebbles in the core approximately remains constant.

The two-region arranged reactor core is expected as a promising technique for pebble bed HTGRs [12]. In this concept, the reactor core is divided into two distinct regions, a central column region consisting of graphite pebbles (called graphite region) and an outer annular region consisting of fuel pebbles (called fuel region). In running circumstances, at the top of the core, graphite pebbles are inserted into the core from a single central spot and fuel pebbles are loaded from several positions in the annular periphery of the core. All the pebbles are discharged from the base hole. The two-region arrangement brings numerous advantages. It flattens the neutron flux and consequently allows a significantly higher power output without reducing safety margin. The decay heat also transfers a shorter distance from the core to outside during accidents [13].

It was found that a stable two-region arrangement could be formed under experimental conditions [14, 15]. There is still a crucial issue remained to be verified. As the two-region arrangement formed, a mixing interface appeared between the fuel-pebble region and the graphite-pebble region as a result of the pebble dispersion. Figure 1 depicts the sketch map of the two-region arranged reactor core. The mixing interface is limited within several diameters of pebbles, which approximately has no impact on the stability of two-region arrangement. However, neutron moderation in the mixing interface is enhanced, which leads to more intensive reactions as well as more heat. Obviously, the temperature in mixing interface is higher and the fuel pebbles are easier to be damaged. In other words, in order to meet the requirements of reactor security, the mixing interface should be small enough. To some extent, the feasibility of two-region depends on the size of mixing interface which is directly related to the dispersion of pebble flow.

Figure 1: Experimental installation at INET of Tsinghua University and sketch of the two-region arranged reactor core.

Dispersion of pebble flow is investigated through experiments and numerical simulations. Some fundamental mechanisms underlying pebble flow remain unknown, and the experimental way is still a principal approach to study it. The phenomenological method [15] is widely accepted to study experimental phenomena of pebble flow, which is an approach to study the dense pebble flow by means of investigating the interface features of different areas composed of differently colored pebbles. Several representative experimental facilities related to their own HTGRs have been built, including the installation developed by INET, Tsinghua University, according to the HTR-PM in China. Modeling of pebble flow which can be called granular flow in a broader term began during 1960s. Over the past 50 years, a number of theoretical approaches have been proposed for granular flow. Models based on continuum assumption are derived from other science branches, which fail to predict the dispersion of pebble flow and can only be applied into two extreme forms: quasistatic flow and rapid flow [16]. Some other models, such as the void model [17, 18] and spot model [19], provide different ideas for dense pebble flow and achieve certain agreements with the process of pebble motion but lead to some unexpected problems. Among these models, discrete element method [20, 21] is recognized to be more appropriate and its qualitative accuracy has been verified. The DEM simulation has been utilized to investigate the influence of some key parameters on pebble flow.

2. Experimental Installation

The experimental installation (Figure 1) is designed on the basis of the two-region pebble bed reactor with the scale of 1 : 5. The 2-dimensional experimental vessel is equivalent to an axial central cut piece of the 3-dimensional cylinder with the same ratio. The geometries of the vessel, including the base angle and the discharge-hole diameter, are supposed to have great effect on pebble flow. Experimental vessels for various combinations of bed and pebble parameters are planned to be set up to determine the parameter influences on the characteristics of pebble flow. Experiments discussed in this paper are for one combination of these parameters. More details can be found in [15]. Table 1 shows the main design parameters.

Table 1: Main parameters of experimental facility.

Experiment vessel

(width × height × thickness)
800 × 1000 × 120 mm;

800 × 2200 × 120 mm (higher vessel)

Base cone angle
30°

Diameter of discharge hole
120 mm

Diameter of particle
12 mm

Number of particle
70,000;

150,000 (higher vessel)

As a 2-dimensional model, the pebble flow in the experimental pebble bed is not fully the same as the one in practical reactor core. However, important geometries and pebble parameters are in accord with similarity theory, including base angle, discharge-hole diameter, and pebble size. Therefore, the studies still have an important practical meaning of guidance, whose conclusions can also be considered as general rules of the pebble flow in a reactor core.

In addition, the DEM methods used for comparative study and its application in pebble flows can be found in recently published papers of our group.

3. Experimental Results

3.1. Experimental Observation

Experiments were designed to investigate the establishment of the two-region arrangement and the mixing interface between the two regions. Experimental vessel was first filled with about 70,000 colorless pebbles to form the initial state of random pebble packing. During the recirculating procedure, black pebbles were loaded from the central inlet tube while colorless pebbles were loaded from the two-side inlet tubes. Meanwhile, pebbles were discharged from the bottom outlet tube at a reasonable rate to keep summation of pebbles constant. After a period of time, black pebbles were expected to form the central region (representing the graphite region) of the two-region arrangement by replacing initial colorless pebbles. Likewise, loaded colorless pebbles were expected to form the side regions (representing the fuel region). Different states of the development of two-region arrangement were recorded by snapshots at intervals. Figure 2 depicts the equilibrium state of two-region arrangement.

(a)

(b)

Figure 2: Equilibrium state of the two-region arrangement after about 7.5 h (a) and the counterpart obtained by DEM simulation (b).

It is shown that two-region arrangement reached an equilibrium state after a period of running. Obviously, there was a quite rough boundary between central region and side region, which is the mixing interface. The maximum size of the mixing interface was about 4 to 5 times of pebble diameter. In addition, the appearance of some scattering pebbles results from the pebbles which are bounced away when they were loaded at the top of vessel, which is not caused by dispersion.

Thus, the existence of mixing interface is evident, and the size of mixing interface is finite. According to the physical calculations for the PBMR reactor, the size of mixing interface should be less than 5.5 times of pebble diameter to meet security requirements [12]. It seems that the mixing interface obtained from the experiment agrees with practical application, but the safety margin is not large enough.

Mixing interface depends on the pebble’s dispersion. In the recirculating mode of the two-region pebble bed reactor, the motion of individual pebble determines the two-region arrangement as well as the mixing interface. For the purpose of exploring the characteristics of pebble motion, statistical investigations have been undertaken.

3.2. Study on Pebble Tracks

The pebble motions were recorded at intervals by tracing the tagged pebbles, when they descended with the overall downward movement of pebble flow. At the beginning, the experimental vessel was filled with colorless pebbles. Then, the pebbles tagged with different color were distributed uniformly along the radial length and pairwise symmetrized against the central axis of the vessel. When the installation ran, positions of each tagged pebble at particular moments were visually detected. In this way, tracks of tagged pebbles were determined and they are found to be generally like “streamline” form, while these tracks are not as smooth as those of common fluids.

The above experimental procedure was repeated for numerous times. Each tagged pebble was placed at the same starting point repeatedly and a bunch of tracks for each tagged pebble were obtained. The motion characteristics of individual pebble are analyzed statistically based on the tracks of tagged pebbles and visual observation. Figure 3 plots all tracks for each tagged pebble.

Figure 3: Descending tracks of tagged pebbles.

As seen in Figure 3, the tracks of tagged pebbles located at the same initial position are different, but the differences between them are small. It is recognized that pebbles in the pebble bed move randomly to a small extent, and tracks of pebbles can only be statistically determined. The movements of pebbles are greatly confined by their neighbors, which makes dispersion among pebbles limited and is totally different from general flows. Therefore, the two-region arrangement is capable to be established without breaking the configuration during running, and consequently the mixing interface can be constrained to a small size.

Some characteristics of pebble dispersion can be concluded preliminarily. Due to randomness of pebble flow, it can be considered that the descending track of a pebble is not a single “stream line” but a statistical “stream tube” consisted of a bunch of “stream lines” starting from the same point. The diameters of “stream tubes,” which indicate the extent of dispersion, are varied in the vertical and horizontal directions. Specifically, dispersion of each pebble increases to the maximum at the midvessel and then decreases according to the specific geometry of bottom vessel. The decrease of dispersion at the bottom of vessel results from the small discharging hole which forces pebbles to move more compactly. In addition, pebbles staying away from the central axis have a tendency of larger dispersion through the visual stream lines in Figure 3.

After adequate experimental observations and statistical analysis, a common view on the behavior of individual pebble has been obtained. Individual pebbles in the same initial position at the top of the vessel move downward within a limited region and rarely exceed the slim region. To put it simply, these pebbles flow in the pebble bed like flowing in a stream tube, which is mentioned above. In other words, pebbles’ random movements have been restricted by their surrounding pebbles, which lead to a finite dispersion constrained in a tube-like region. Next, the distribution of pebbles in the tube-like region is going to be discussed, as the mixing interface in two-region arrangement is actually a tube-like region located at a particular radial position. Features of mixing interface, such as size and shape, are determined by the behaviors of a series of pebbles flowing in a specific tube-like region.

3.3. Radial Distribution of Pebbles in Dispersion Region

After a period of stable running, attentions are paid to the distribution of pebbles within a tube-like region which have same starting-point at a particular height by getting enough pebble tracks. The tube-like region is a dispersion region of a series of pebbles. At a particular height, the dispersion region is divided into several equal intervals, and the number of pebble tracks within each interval is calculated, as shown in Figure 4 (the upper inset). The tagged pebbles are located inside certain intervals according to their initial positions of mass center. For instance, pebble A belongs to interval 6 and pebble B belongs to interval 10.

(a) The way to determine pebbles’ radial intervals

(b) Distribution of tagged pebbles at the height of 204 mm

Figure 4: The method to illustrate radial distribution of pebbles.

In Figure 3, it is easy to find that most tracks lie near the center of the dispersing region, and track numbers near the border of dispersion region are much less. Figure 4 (the lower inset) also demonstrates the phenomenon that most tagged pebbles are concentrated in the central intervals. All the tagged pebbles start from the radial position of 0, and their radial positions are recorded at the height of 204 mm to form the radial distribution.

In order to study the distribution at a particular height statistically, the probability density of radial distribution is defined as quantity fraction of pebbles in each interval divided by the length of interval (5 mm). In this way, the curve of probability density of radial distribution at the height of 204 mm (Figure 4) is gotten and shown in Figure 5. Several features of the curve should be noticed. The curve has a decent symmetry whose shape is high in the middle and low in both sides, which looks quite similar to the Gaussian distribution.

Figure 5: Gaussian fitting curve on probability density of radial distribution.

The relevance of the probability density curve to Gaussian distribution has been investigated through curve fitting based on the experimental data. The obtained fitting curve agrees with those data points on probability density to a large extent, as is shown in Figure 5. The radial position of symmetry axis of the Gaussian fitting curve, 0.4 mm, is close to the mean value of radial positions of these data points, −0.2 mm. Therefore, at least the radial distribution at the height of 204 mm of such pebbles which start from the specific radial position, 0 mm, accords with Gaussian distribution.

According to the uniform characters of pebble flow in experiments, it is supposed that the radial distributions located at different heights of pebbles starting from different radial positions accord with Gaussian distribution, which has been justified by the coincidences between experimental data and fitting curves (Figure 6). Therefore, in the entire pebble flow field, it can be generally recognized that radial distribution of pebbles which have the same starting point agrees with Gaussian distribution at a particular height in dispersion region.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 6: Coincidences between probability density points and fitting curve at different heights and radial positions.

If a random variable obeys the Gaussian distribution whose expectation and deviation are and , the probability density of accords with the following formula:

where determines the location of Gaussian curve and determines the range of distribution. The characteristics of Gaussian distribution define that the area surrounded by the curve and horizontal axis is 1. When the selected region ranges from to by the sides of mean value , the surrounded area is 0.9545.

When the law of Gaussian distribution is applied to the dispersion region, a statistical conclusion can be obtained. It is not difficult to find agreements between Figures 3 and 6. The slim part of dispersion region in Figure 3 corresponds with the “tall” and “thin” distribution in Figure 6, and similarly the bulky part of dispersion region relates to the “dumpy” distribution. A small standard deviation means a concentrative distribution, which represents a small dispersion region. Therefore, one of the parameters in Gaussian distribution, namely, the standard deviation , indicates the size of dispersion region.

We investigate standard deviations of radial positions at different heights and radial positions, in order to find the variation trend of standard deviations along the descending path. Figure 7 shows tracks starting from different radial positions. The tracks and variation trends of standard deviations are got through calculating the mean value of radial coordinates at different heights. Actually, there are essential resemblances between Figures 3 and 7, and identical conclusions can also be drawn from Figure 7. Four curves in Figure 7 reveal that the standard deviations increase to the maximum at the height of about 300 mm and then begin to decrease, which is like the variation trend of sizes of dispersion regions. Besides, the curve of id3 has relatively large values, for pebbles represented by id3 stay far away from the central axis, which can also be observed in Figure 3 that pebbles in side region have larger dispersion.

Figure 7: The variation trend of standard deviations of tracks.

The value of standard deviation is in proportion to the size of dispersion region. Virtually, the concept of dispersion region is also statistical, since it is only realistic for the region to comprise a majority of pebbles instead of all of them. For instance, the dispersion region can comprise 95.45% pebbles if radiuses of this region are set to be 2 times of standard deviations at different heights. To some extent, it is acceptable to regard 2 or 3 times of standard deviation as radius of dispersion region in engineering. In conclusion, quantity and variation trend of standard deviation display the features of dispersion region well, and it is applicable to treat standard deviation as a quantitative index of dispersion region.

3.4. Experiment in Heightened Vessel

Inherent safety of HTGR in china is insured by its fuel characteristics and passive safety system, which is attributed to the specific design of small radius of reactor core to some extent. Small radius allows the decay heat to transfer from the inner core to the outside more easily because of a shorter distance, so that passive approaches like radiation heat transfer and heat convection can meet the requirements of heat-transfer capacity during accidents. On the other hand, in terms of the total power, such design restricts the output due to the size of the core. Thus, enlarging the height of the core is the adoptive way, which can drive up the total power and maintain the passive safety capacity at the same time.

As mentioned in the section of experimental installation, the height of the experimental vessel can be heightened to a height of 2.2 m. Similar experiments have been conducted in the heightened vessel so as to study whether there would be differences or not.

Figure 8 shows the tracks in the heightened vessel. The tracks in the heightened vessel are also smoothly varied and symmetric which is similar to those in 1 m vessel. As a result of the cone-shaped bottom and the small discharging hole, tracks bend towards the axis and flow more compactly. There is a region in the upper vessel where the tracks are uniformly straight, and these tracks do not bend until they reach the bottom vessel. In terms of tracks, there is not an essential link between the upper and bottom regions. The flow field below 1 m in heightened vessel shares many features with that in standard vessel.

Figure 8: The standard deviations of tracks in the heightened vessel.

The standard deviation in heightened vessel depicted in Figure 8 reveals more characteristics. The curve of id4 is based on relatively limited data, and it perhaps cannot represent the physical law well. Compared with the results in the last section shown in Figure 7, several differences are concluded. In general, pebble flow in heightened vessel has larger dispersion regions, because standard deviations in Figure 8 reach maximums above 20 mm, whereas they are about 12 mm in Figure 7. What is more, apart from id4, curves do not split until they reach the depth of 1000 mm, which refers to the uniform flow region in upper vessel. Of course, several laws are justified again. The standard deviations increase to maximums at the middle vessel and then decrease. Pebbles staying far from axis have larger dispersion, which can be illustrated by the curve of id3. At last, from the perspective of engineering, the most important issue that we should be concerned about is that heightening reactor core will bring larger dispersion region, which is not expected.

4. Discussions

The main factors that determine the phenomenon of dispersion still remain unknown. It is generally accepted that the qualitative accuracy of DEM has been verified. DEM simulation has been adopted to discuss principal parameters’ influence on dispersion. The standard deviation at a particular height, as a quantitative index to show the size of the dispersion region, will be investigated with the change of other parameters.

4.1. Radial Velocity

It is natural to associate dispersion with radial velocity which can be detected in DEM simulation. Figure 9 illustrates variation trend of standard deviation as well as radial velocity both in 1 m vessel and heightened vessel. The unit of radial velocity (d/s) means diameter per second and the data were drown from those pebbles starting at half-radius position. In Figure 9, the size of dispersion region does not correlate well with radial velocity, since the standard deviation has a sharp increase whereas the radial velocity remains stable at a low level in the upper vessel. The overall correlations have been proved weak, with the Pearson correlation coefficients of 0.0823 and 0.19725, respectively.

(a) Simulation in 1-meter vessel

(b) Simulation in heightened vessel

Figure 9: DEM simulations on standard deviation and radial velocity in 1-meter and heightened vessels.

In Figure 10, which set radial velocity as abscissa axis, correlation between these two parameters is illustrated more obviously. When radial velocity is less than 0.005 (d/s), standard deviation has a wide range and is not affected by radial velocity. Due to the majority of spots standing at low velocity, the overall correlation is not apparent. However, when radial velocity is larger than 0.005 (d/s), a much more significant correlation appears, with Pearson correlation coefficients of −0.9618 in 1 m vessel and −0.9648 in heightened vessel, respectively. It means that there is a critical value of radial velocity, above which the size of dispersion region drops with the rise of radial velocity.

(a) Data from 1-meter vessel

(b) Data from heightened vessel

Figure 10: Correlation between standard deviation and radial velocity in 1-meter and heightened vessels.

Enlarging radial velocity seems to help in dispersion control as shown in the above figures, if this rule can be applied in practice. Nevertheless, the pebble flow in experimental vessel as well as in practical reactor belongs to quasistatic flow, so velocity of most pebbles is under the critical value (appropriate 0.005 d/s). Only in bottom vessel larger radial velocity can be realized, and the size of dispersion region declines to some extent. Additionally, we cannot focus on the growth of radial velocity to announce that it directly leads to the decrease of dispersion region, because radial velocity rises at the result of specific bottom geometry which may also exerts impact on dispersion.

4.2. Base Cone Angle and Friction Coefficient

The vessel geometry impacts the overall pebble flow dramatically. To say specifically, the base cone angle in bottom vessel plays an important role in developing the main features of pebble flow, including the dispersion region. The base cone angle of experimental vessel can be adjusted from 30 to 60 degrees, and corresponding DEM simulations are carried out as well.

As shown in Figure 11, dispersion regions in vessels whose cone angles are 30 and 45 degrees are nearly matched, which means that this change has little impact on the size of dispersion region. However, with angle changing from 45 to 60 degrees, the maximum size of dispersion region has a significant drop. Therefore, geometry influences the phenomenon in a nonlinear way. Perhaps there is also a particular angle or a specific geometry that can minimize the dispersion. Moreover, the standard deviation does not show differences until they reach the middle vessel, which means the bottom geometry fails to influence flow pattern in the upper vessel. This enables simplified researches of heightened vessel, and we can concentrate on the bottom flow field on some occasions.

Figure 11: Simulation on different base cone angles.

Friction coefficient is determined by physical property of individual pebble. Apart from gravity, the friction is an important force of driving or resisting. It is natural to suppose that a larger friction restricts the behavior of pebbles. Then, the dispersion would be reduced to some extent. In fact, the friction resulting from surrounding pebbles also acts as driving force which may push pebbles away from its original path and bring about more obvious dispersion. As a result, we believe that there is a competitive mechanism between driving and resisting effects produced by friction. This hypothesis is demonstrated by Figure 12. It is hard to predict variation trend of dispersion by changing the friction coefficient, because the relation between them is not consistent in different friction scopes. Bottom pebbles in the heightened vessel suffer more pressure from above pebbles and competition between the driving and resistance forces is more intense. As a consequence, the size of dispersion approximately remains stable when friction coefficient changes from 0.2 to 0.4, for resistance force as well as driving force increases to some extent.

(a) Simulation in 1-meter vessel

(b) Simulation in heightened vessel

Figure 12: Simulations on different friction coefficients in 1-meter and heightened vessels.

5. Summary

Through the above analysis, we can summarize the following.

In our experiment, the stable two-region arrangement in pebble bed reactor can be established with a definite and finite mixing interface. The dispersion happened in mixing interface results from pebbles’ random motion behavior. Pebbles with the same starting point flow downward in a “tube-like” region and rarely get out statistically. This region is called the dispersion region which directly relates to the size of mixing interface. The side or peripheral dispersion regions have larger diameters compared with central ones, and, for each region, it usually reaches the maximum size in the middle of vessel. In dispersion region, the radial distribution of pebbles at a particular height accords with Gaussian distribution and the standard deviation determines the size of dispersion as a quantitative index.

In the heightened vessel, apart from the uniform flow field in the upper vessel, pebble flow has a more obvious dispersion. There is a critical value of radial velocity (approximately 0.005 diameter/s) that determines the influence on dispersion. The rise of radial velocity larger than the critical value can reduce the size of dispersion linearly.

Enlarging base cone angle helps in the control of dispersion. The impact of friction on dispersion region is complicated, because friction is related to driving as well as resistance force of dispersion and the competitive mechanism between them remains unknown.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful for the support of this research by the National Natural Science Foundations of China (Grant nos. 11072131 and 51106180) and the research funds of Tsinghua University (no. 2010Z02275).

References

A. J. Goodjohn, “Summary of Gas-Cooled reactor programs,” Energy, vol. 16, no. 1-2, pp. 79–106, 1991.

W. Magwood, “Roadmap to the Next Generation Nuclear Power Systems: A Vision for a Powerful Future, Nuclear News,” American Nuclear Society, 2000

J. Ryskamp, “Next Generation Nuclear Plant: High-Level Functions and Requirements,” Tech. Rep. INEEL/EXT-03-01163, Idaho National Engineering and Environmental Laboratory, 2003.

D. Z. Wang and Y. Y. Lu, “Roles and prospect of nuclear power in China’s energy supply strategy,” Nuclear Engineering and Design, vol. 218, pp. 3–12, 2002.

Y. H. Xu and Y. L. Sun, “Status of the HTR programme in China, IAEA TCM on High Temperature Gas Cooled Reactor Application and Future Prospects ECN,” Petten, The Netherlands, 1997

Z. Y. Zhang, Z. X. Wu, Y. H. Xu, Y. L. Sun, and F. Li, “Design of Chinese modular high-temperature gas-cooled reactor HTR-PM,” in Proceedings of the 2nd International Topical Meeting on High Temperature Reactor Technology, Beijing, China, 2004.

A. Koster, H. D. Matzner, and D. R. Nicholsi, “PBMR design for the future,” Nuclear Engineering and Design, vol. 222, no. 2-3, pp. 231–245, 2003.

W. K. Terry, H. D. Gougar, and A. M. Ougouag, “Novel method for the deterministic design, analysis, and optimization of the in-core fuel cycle of a recirculating pebble-bed reactor,” Annals of Nuclear Energy, vol. 29, pp. 1345–1364, 2001.

R. Schulten, “Pebble bed HTRs,” Annals of Nuclear Energy, vol. 5, no. 8–10, pp. 357–374, 1978.

H. Frewer, W. Keller, and R. Pruschek, “The modular high-temperature reactor,” Nuclear Science and Engineering, vol. 90, no. 4, pp. 411–426, 1985.

G. H. Lohnert and H. Reutler, “The modular HTR—a new design of high temperature pebble-bed reactor,” Nuclear Energy, vol. 22, no. 3, pp. 197–201, 1982.

International Atomic Energy Agency, “Current status and future development of modular high temperature gas cooled reactor technology,” IAEA TEDOC-1223, Vienna, Austria, 2001.

Pebble Bed Modular Reactor (Pty) Limited, “PBMR safety analysis report,” Document 001929-207/4, 2002.

X. T. Yang, W. P. Hu, and S. Y. Jiang, “Experimental Investigation on feasibility of two-region-designed pebble-bed high-temperature gas-cooled reactory,” Journal of Nuclear Science and Technology, vol. 46, no. 4, pp. 374–381, 2009.

X. T. Yang, W. P. Hu, S. Y. Jiang, K. K. L. Wong, and J. Y. Tu, “Mechanism analysis of quasi-static dense pebble flow in pebble bed reactor using phenomenological approach,” Nuclear Engineering and Design, vol. 250, pp. 247–259, 2012.

C. S. Campbell, “Granular material flows—an overview,” Powder Technology, vol. 162, no. 3, pp. 208–229, 2006.

W. W. Mullins, “Stochastic theory of particle flow under gravity,” Journal of Applied Physics, vol. 43, no. 2, pp. 665–678, 1972.

H. Caram and D. C. Hong, “Random-walk approach to granular flows,” Physical Review Letters, vol. 67, no. 7, pp. 828–831, 1991.

M. Z. Bazant, “The spot model for random-packing dynamics,” Mechanics of Materials, vol. 38, no. 8–10, pp. 717–731, 2006.

N. Gui, X. T. Yang, J. Y. Tu, and S. Y. Jiang, “A simple geometrical model for analyzing particle dispersion in a gravity-driven monolayer granular bed,” Powder Technology, vol. 254, pp. 432–438, 2014.

N. Gui, X. T. Yang, J. Y. Tu, and S. Y. Jiang, “Effect of bed configuration on pebble flow uniformity and stagnation in the pebble bed reactor,” Nuclear Engineering and Design, vol. 270, pp. 295–301, 20142014.