Experimental study of flow field characteristics on bed configurations in the pebble bed reactor

Annals of Nuclear Energy 102 (2017) 1–10

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Experimental study of flow field characteristics on bed configurations in the pebble bed reactor Xinlong Jia a,b, Nan Gui a, Xingtuan Yang a, Jiyuan Tu a,b, Haijun Jia a, Shengyao Jiang a,⇑ a Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, PR China b School of Aerospace, Mechanical & Manufacturing Engineering, RMIT University, Melbourne, VIC 3083, Australia

a r t i c l e

i n f o

Article history: Received 25 December 2015 Received in revised form 1 December 2016 Accepted 8 December 2016

Keywords: Pebble bed High temperature gas-cooled reactor Particle Tracking Velocimetry Bed configuration Flow field characteristics

a b s t r a c t The flow field characteristics are of fundamental importance in the design work of the pebble bed high temperature gas cooled reactor (HTGR). The different effects of bed configurations on the flow characteristics of pebble bed are studied through the PTV (Particle Tracking Velocimetry) experiment. Some criteria, e.g. flow uniformity (r) and mass flow level (a), are proposed to estimate vertical velocity field and compare the bed configurations. The distribution of the Dh (angle difference between the individual particle velocity and the velocity vector sum of all particles) is also used to estimate the resultant motion consistency level. Moreover, for each bed configuration, the thickness of displacement is analyzed to measure the effect of the funnel flow zone based on the boundary layer theory. Detailed information shows the quantified characteristics of bed configuration effects on flow uniformity and other characteristics; and the sequence of levels of each estimation criterion is obtained for all bed configurations. In addition, a good design of the pebble bed configuration is suggested and these estimation criteria can be also applied and adopted in testing other geometry designs of pebble bed. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction It is generally known that the high temperature gas-cooled nuclear reactor is considered as one potential solution for the generation IV (the fourth generation) advanced reactor (Magwood IV, 1996). The Institute of Nuclear and New Energy Technology (INET) at Tsinghua University has developed the HTR-PM (High Temperature Reactor – Pebble bed Modules) (Wu et al., 2002; Zhang et al., 2002, 2004), which has been regarded as one of the National Special Grand Science-Technology Projects of China (Zhang and Yu, 2002; Zhang et al., 2006; Sui et al., 2014). The mechanisms and characteristics of the pebble-bed reactor have drawn great attention. Some countries, such as South Africa (Koster et al., 2003) and the United States (Berte, 2004), have developed their own test and demonstration reactors (PBMR, (Pebble Bed Modular Reactor), and MPBR (Modular Pebble Bed Reactor), resp.) and their prototype reactor, for example, the AVR (Arbeitsgemeinschaft Versuchsreaktor) early in Germany, and so forth (Schulten, 1978). The basic mechanism of granular flow has not been fully understood yet, especially this specific pebble flow in the reactor core. ⇑ Corresponding author. E-mail addresses: [email protected] (N. Gui), shengyaojiang@sina. com (S. Jiang). http://dx.doi.org/10.1016/j.anucene.2016.12.009 0306-4549/Ó 2016 Elsevier Ltd. All rights reserved.

However, the flow field characteristic is vital to the efficiency and safety of HTGR. Pebbles’ behavior should be insured to fulfill thermal hydraulic rules and radiation safety requirements (Jiang et al., 2012). In the reactor core of HTR-PM (Auwerda et al., 2010), the pebbles are flowing greatly slowly which are driven by gravity, termed as a quasi-static flow regime. The particles are dumped from the outlet at the bottom, and reloaded at the top of the reactor core, forming a recirculation mode of operation. In this recirculation process, the velocities of particles throughout the bed are varied greatly, relying on the bed configurations, loading method, and etc. In common, particles flow fast in the central part and slowly near the wall. The uniformity of pebble flow is of crucial importance for the performance and safety of reactor operation, which should be focused in reactor core design work (Gui et al., 2014). Although experimental and numerical studies (Li et al., 2005) have been done on the flowing characteristics in silo bed of different geometries, those cases mostly belong to the free outflow and focus on the prediction of outflow rate (Balevicˇius et al., 2011; González-Montellano et al., 2011b; Oldal et al., 2012; Albaraki and Antony, 2014). Other relating studies have also been carried out, contributing to various related aspects of them such as velocity profiles (Choi et al., 2005; Li et al., 2009; Kim et al., 2013), phenomenological analysis (Yang et al., 2012), stagnant

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zones (Jia et al., 2014), diffusion and mixing (Choi et al., 2004), as well as numerical simulations (Goda and Ebert, 2005; Gui et al., 2014), etc. However, few of them have carried out experiment investigations on the effect of bed configuration under the recirculation operation, especially with the contraction configuration at the bottom of the pebble bed, as well as the flow uniformity and the very slow region (the funnel flow regime) in such kinds of configurations. Mass flow index (MFI) is introduced as a quantitative way to evaluate the radial flow uniformity (Johanson, 1964; GonzálezMontellano et al., 2011a, 2012), considering a simple index to recognize the mass and funnel flow regime (Langston et al., 1995) in the silo pebble flow. The pebbles move rapidly in the mass flow regime locating in the central part of pebble bed, as while funnel flow stays nearly stagnant near the walls and corners. That is to say, the pebbles firstly loaded will be discharged firstly in the middle region, but lastly in the near-wall region. The occurrence of the two regimes is analogous with the field of boundary layer theory (Schlichting et al., 2000) in fluid mechanics, although the physics of fluids is fundamentally different from the granular media. Nevertheless, the rich body of work devoted to the analysis of the boundary layer characteristics (such as displacement thickness, momentum thickness) and the work relating to the analysis and design of contraction pipes (for instance, the wind tunnel, (Morel, 1975; Doolan and Morgans, 2007)), is similar with the base configuration of the pebble bed. These works provide a suitable framework to be applied to characterize the pebble flow regime and the near-wall flow behaviors. Furthermore, the Relaxation Method (RM) PTV is based on the use of iterative relaxation process technique and it is widely applicable to complex flows with local rotating and shear motions (Ohmi and Li, 2000). In RM, as the reference particle selected by

the quasi-rigidity radius finds many neighbors flowing similarly, the positive estimate of such movement is increased. The most important advantage of this relaxation method is that the examination is on the basis of the probability of particle matching between the first and the second frames, defined for every possible pair of particles inclusive of the probability of those being nomatched (or the probability of the loss of partners). Fortunately, it is suitable to measure the 2-D flow regime of very slow and dense granular flow here since there exist particles which will disappear when they are flowing towards the outlet. Since the frictional and hard-core inelastic interactions between particles in the near-wall pebble flow have common characteristics with the molecular interactions in a fluid (Radjai and Roux, 2002). Motivated this consideration, this study aims to analyze the particle motion in the near-wall region with analogy to the boundary layer theory, and demonstrate some typical results on the bed configuration effect. Besides, the snapshots of the flow regimes and velocity profiles with different bed configurations will be shown. In addition, the whole velocity uniformity will be analyzed with different estimation criteria.

2. Methodology 2.1. Experimental Set-up In this study, a 2-D test facility is designed based on a real pebble-bed reactor at Tsinghua university with the scale of 1:5 (Fig. 1a). The experimental setup consists of several main parts. Firstly, the vessel is made up of Plexiglas which has the dimensions of 800 * 1000 * 120 mm in width, height, and thickness, respectively. About 70000 black glass pebbles with the diameter 12 mm

Fig. 1. Sketch of experiment setup and images. (a) The experiment setup; (b) Different bed base configurations. (c)The experimental images of particles (left inset), the locations of the particles marked by the red crosses (middle inset), and the trajectories of individual particles (right inset).

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X. Jia et al. / Annals of Nuclear Energy 102 (2017) 1–10 Table 1 Characteristic index for each bed base configuration. Categories

Arc shape with finite radius

Arc shape with infinite radius

R1

R2

R3

Characteristic parameters

pffiffiffi R1 ¼ es = 3

R2 = es

pffiffiffi R1 ¼ es = 7

Total mass flow level hin r(Dh) Average of d⁄ Standard deviation of d⁄

0.5407 77.6° 18.0° 9.01 7.29

0.6585 73.4° 12.2° 5.94 4.80

0.6391 75.5° 17.2° 5.78 4.40

are filled (Yang et al., 2012). Secondly, three inlet tubes are set on the vessel top through which the pebbles can fall into the bed. Finally, the discharge hole is in the bottom of the set-up, with 120 mm in diameter and 200 mm in length. The operation method can be described as follows. (Yang et al., 2009; Jiang et al., 2012): 1). The experiment begins with the random packing by prefilling the black pebbles in the vessel. 2). The pebble bed is fed with 14, 122 and 14 pebbles per minute from the left, center, right inlet tubes, respectively. The pebbles from the central inlet tube represent the graphite moderator pebbles while the ones from the two side inlet tubes stand for the fuel pebbles in the real pebble bed reactor. 3). Meanwhile, pebbles are discharged from the outlet hole with the rate of 150 pebbles per minute. Thus, the total number of the pebbles in the vessel remains constant. This recirculation mode is different from the previous experiments (Yang et al., 2012) where the drainage pebbles flow freely. In order to show the effect of bed configurations on pebble flow, especially on the contraction configuration at the bottom, some arc-shaped configurations are employed. As shown in Fig. 1b, i.e. (1) a straight line with a length es (base angle 30o) or equivalently named as an arc shape (named R1;30 ) of infinite radius R1 ; (2) a pffiffiffi straight line with length 6es =2 (base angle 45°, named R1;45 ); pffiffiffi (3) a straight line with length 3es (base angle 60o, named pffiffiffi R1;60 ); (4) an arc shape (named R1 ¼ 3es =3) for the circumcircle of the equilateral triangle which has an edge of the inclined straight line es; (5) an arc shape with radius R2 equal to the edge of the equilateral triangle, i.e. R2 = es; (6) an arc shape (name R3) on a circle with radius R3; The distance between the centers of circle R1 and R2 is R1, equal to the distance between the centers of R2 and R3. It is noticed that the centers of these circles (R1, R2, and R3) are on the perpendicular bisector of the inclined edge (R1;30 ). The radii of series of circles with centers on the perpendicular bisector are pffiffiffi within the range of ½ 6es =2; 1Þ. Thus, R1, R2, R3 and R1;30 are typical representatives of this series. In addition, the three bed configurations with straight lines (R1;30 , R1;45 and R1;60 ) take the main representatives of the series with infinite radius. The six cases of the bed configurations share same horizontal span in the pebble pffiffiffi bed, equal to 3es =2 . For clarity, all the circumstances of distinct bed configurations are listed in Table 1.

R1,30°

R1,45°

R1,60°

R=1 Angle = 30° 0.6369 75.6° 13.5° 5.96 4.52

R=1 Angle = 45° 0.7217 73.8° 12.6° 3.52 3.18

R=1 Angle = 60° 1 66.4° 9.56° 0 0

enty thousand pebbles in the vessel, the rate of taking one image per second is acceptable (Choi et al., 2005). The trajectories of individual pebbles are focused in Particle Tracking Velocimetry (PTV) method, while the PIV (particle image velocimetry) method can only obtain the velocity filed without the movement information of a given pebble (Hassan and Dominguez-Ontiveros, 2008). The images are preprocessed to reduce the background and the noise, which includes convolving the image with a Gaussian filter then an average filter. The circular shaped filter has been added to the algorithm to improve the particle tracking as for the circular shape of the glass particles. In PTV, the relaxation method (RM) (Baek and Lee, 1996) is used. In this algorithm, the location of the pebble is identified with the dynamic threshold binarization (DTB), where the threshold level is detected particle by particle. The DTB is better than the single-threshold binarization (STB) using a uniform threshold level to detect particles. (Ohmi and Li, 2000). 2.3. Boundary layer theory In fluid mechanics, the boundary layer is a layer of fluid in the vicinity of a wall surface where the influence of viscosity is significant. The boundary layer thickness (d) is the distance across a boundary layer from the wall to a point where the flow velocity has essentially reached the point of 99% of the free stream velocity (Fig. 2a). This distance is defined normal to the wall (Schlichting et al., 2000). The displacement thickness d⁄ is employed to measure and characterize the mass flow effect by the boundary layer. It is the distance by which a surface would have to be moved in the direction perpendicular to its normal vector away from the reference plane in an inviscid fluid stream of velocity U to give the same flow rate as occurs between the surface and the reference plane in a real fluid (see Fig. 2b). The definition for incompressible flow can be based on volumetric flow rate (White, 2003):

d ¼

Z 0

d

  uðyÞ dy 1 U

ð1Þ

where U is not a constant. We choose the component U in the tangential direction (y-direction in Fig. 2c) of the wall surface at the outer of the boundary layer to calculate the displacement thickness. 3. Results and discussions

2.2. Particle Tracking technique

3.1. Snapshots of velocity profiles

The polished stainless black glasses balls are chosen to achieve the necessary resolution in measuring the particle trajectories (Garcimartín et al., 2011). Under convenient illumination, each particle reflects a small, bright and well-defined spot, which allowed for a very precise position measurement (Fig. 1c). In the images, every pebble diameter occupies about d = 50 pixels. As the flow rate is about 2.5 particles per second within almost sev-

As aforementioned, the main focus of this paper is to explore the flow field characteristics under the six types of bed configurations. The transverse velocity is an indicative of the character of horizontal dispersion in pebble flow (Fig. 3). The horizontal velocities in the central and near-wall region are small while they are larger in the middle of the two regions on the same heights. This is based

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wall region is the smallest which provides less space for particles close to the wall to move towards the wall. The particles in the top part exhibit the plug flow behavior where the difference between the vertical velocity magnitude in the central and nearwall region is small. Based on the detailed velocity field data discussed above, the flow channel boundary, which is defined as the interface between mass and funnel flow (Johanson, 1964; Watson and Rotter, 1996). A ratio of critical velocity (RCV) is introduced as a criterion to delineate the boundary between the ‘‘flowing” particles from the ‘‘stationary” ones: RCV ¼ v 0 =v 00 . The v0 denotes the vertical velocity of a particle at a given position, v0 0 is the vertical velocity of the particle at the central position of the corresponding level that stands for the highest vertical velocity at the same level. Generally, the values of RCV < 0.3 are indicative of funnel flow while those of RCV > 0.3 are typical mass flow. The overall flow pattern for the silo is determined by the change in RCV over the height of the silo. The RCV line is depicted (Fig. 3) except for the case with the base angle equal to 60°, which means that no particles belong to the funnel flow regime. In addition, the cases with different bed configurations will be discussed by comparing the velocity profiles. For the R1;30 configuration, the pebbles move nearly uniformly in the central part of the bed, except the near-wall regions. Some pebbles are stagnated near the wall, moving fairly slowly and then stay around the bottom corner of the bed. As mentioned above, the formation of stagnant zones will not benefit the real reactors, although the pebbles may be discharged from the outlet after a sufficiently long time at last. By comparison, the stagnant zone is smaller for the R1;45 configuration, and it even disappears for the R1;60 configuration. The case of the R1 configuration seems to be much worse than the situation of R1;30 . A larger stagnant region consisting of stagnant pebbles is noticed clearly around the corner. The main reason for this would be that a flatter bottom promotes the formation of the stagnant region. When the pebbles reached the flat bottom, the driven force of gravity was resisted absolutely and the particles have to flow horizontally until they arrived at the outlet region. The velocity profile of R3 configuration is quite similar to that of the R1;30 configuration since the arc curve in R3 configuration is quite near the straight line edge in R1;30 configuration. The R2 case appears to be better than R1 and R3 situations.

Fig. 2. The boundary layer (a), displacement thickness illustration (b) and global and local coordinate system, vector diagram (c).

on the fact that particles tend to drift horizontally towards a zone with faster downward flows because they are likely to get more space to move in the transverse direction. A largely uniform transverse velocity profile with smaller magnitude (less than 0.01d/s) is established across the radius of the silo above h = 30d. Below this height, the velocity profile is not uniform and become increasingly concentrated towards the outlet region on lower heights. The largest vertical velocity component v0 (approximately 0.0158d/s) is found along the centerline on the level of h = 0d under all types of bed configurations. In this experiment, for the sake of visualization, seven different zones have been identified in the pebble bed during recirculation depending on the ratio a of local actual vertical velocity component v to v0. A central rapidly flowing pipe in the lower part with a velocity larger than 0.0048 (a = 30%) was surrounded by a slowly flowing zone and a nearly stagnant zone near the silo wall. The stagnant zone herein is defined by the zones with velocities less than 0.0008d/s (a = 5%) located in the lower corner of the pebble bed. The vertical velocity in near-

3.2. Flow uniformity in radial direction In the previous section, we employed contour map to highlight the vertical movement of the pebble flow. In this section, the features of the pebbles on the same levels are used to reflect the radial flow uniformity. Horizontal layer-like regions with 4d height (6872d, 28-32d, etc.) are selected, and then we calculate the vertical velocity of all pebbles within each region during all the experiment time. For example, in a radially-uniform flow layer, it takes almost the same time for the particles on the same level to flow out of this layer, which shows mass flow pattern. However, in a flow field with poor radial uniformity, the pebbles in the central part will flow out of the layer earlier than the particles in the near-wall region on the same height as shown in Fig. 3. That is, the particles near the wall move down with the flow pattern of a first-in and last-out type. Standard deviation r of all vertical velocities within the same layer can reflect the extent of dispersion in the vertical direction, which would be a good indication of radial uniformity. In the mass flow region, the r can remain smaller values even at lower part of the experiment vessel. In contrast, the magnitude of r will rise greatly in the non-uniform field, especially in the lower section which presents significant vertical dispersion.

X. Jia et al. / Annals of Nuclear Energy 102 (2017) 1–10

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Fig. 3. Flow profiles in the pebble beds with different bed configurations. The transverse velocity profiles are given in the left half. The zones characterized by different vertical velocity ranges are shown in the right half. The line RCV = 0.3 is plotted (if exists) in each case. (a) R1;30 configuration. (b) R1;45 configuration. (c) R1;60 configuration. (d) R1 configuration. (e) R2 configuration. (f) R3 configuration.

As depicted in Fig. 4, there are similar tendencies among various bed configurations on the values of r in the horizontal-stripe layers on several levels. The middle part of the layer belonging to dominating central flow move down faster in the lower section of the setup, while the side part has to move slower by overcoming additional friction force against the wall. Consequently, near-wall particles move down slowly and form structured packing and stagnant or dead region. This leads to the loss of uniformity and rising of r for the pebble flow in lower layers. The upper bed gets a less significant vertical dispersion than that in the bottom, which is caused by the more uniform distribution of velocity along the radius. For the layers with high levels, vertical velocities of the particles increase slowly with reduction of height because of the con-

stant width of the vessel. However, in the wedged part of the pebble bed, particle velocity in the middle region becomes much larger than the upper bed. As a result, the growth rate of r is larger than the upper bed. Nevertheless, there seem to be some differences among various bed configurations on the values of r. The R1;60 bed configuration presents the smallest profile of r in horizontal-stripe layers, from 0.00101d/s at h = 70d to 0.00287d/s at h = 8d, while the R1 bed configuration keeps its value largest in each layer and experiences a significant raise from 0.0016 d/s to 0.0043 d/s. It is noticed that the R3 and R1;30 bed configurations are quite similar and so are R2 and R1;45 bed configurations. Both of them indicate the following sequence of levels of radial uniformity quality:

R1;60 > ðR2 or R1;45 Þ > ðR3 or R1;30 Þ > R1

ð2Þ

From the mass flow zones depicted in Fig. 3, we found that the whole pebble flow of the bed belongs to the mass flow pattern field for the R1;60 situation with higher radial uniformity quality. By comparison, the particles in R1 situation with flow stagnations prove a poor uniformity of radial flow, and formulate a smaller mass flow zone. A main reason for that is the larger loss of flow movement in the middle and side part within the same layer. In a certain sense, the stagnant region propagates upwards along the wall and induces a typical funnel flow. Good radial flow uniformity is a significant characteristic of mass flow pattern. 3.3. Mass flow level

Fig. 4. The standard deviation of vertical velocity in each horizontal-stripe layer at different levels for all bed configurations.

To some extent, the r is also one point to estimate the percentage of particles under the mass flow pattern within the mixedpattern flow field. However, the quantitative proportion of the mass flow zone is significantly necessary when estimating the flow patterns in the real pebble bed reactor design. In a mixed-pattern flow field, particles belonging to mass flow pattern always occupy

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the middle part of the pebble bed, leaving the funnel-flow particles in the near-wall region. The mass flow level can be expressed as the ratio (a) of the number of mass-flow particles to the total number of particles in a given horizontal-stripe layer. As aforementioned, the line is plotted in Fig. 3 to identify the mass flow region in the middle part where RCV > 0.3 at each height. By comparison, the relative size (Lm) of mass flow region is also defined as the ratio of the width of mass flow zone to the width of the experiment vessel at a certain height. Several horizontal-stripe layers are chosen to calculate the mass flow level during the experiment time of 3300 s. Fig. 5 shows the dependence of the mass flow level (a) and relative size (Lm) of mass flow region on the height. On one hand, a and Lm hold almost the same value at each level. For the R1;60 bed configuration, all particles in the vessel flow with the mass flow pattern. That is, the number of mass-flow particles is equal to the total number of particles in a horizontal-stripe layer. The width of the mass flow region and experiment vessel are also equal in size. There is no doubt that a and Lm keep exactly the same value of 1 for the situation R1;60 . On the other hand, the Lm shows a little smaller than a in the cases of mixed-pattern flow apart from R1;60 bed configuration. The void fraction in the middle part is always larger than that in the side part when pebbles keep flowing, which has been studied with simulation by our group (Yang et al., 2014). Thus, there are fewer particles per unit area for the central region than that for the near-wall region, which makes the a smaller. However, the difference between a and Lm is little since the void fraction does not change much along the radius at the same height when the speed of pebbles is small enough. The mass flow level can also be studied by focusing on the relation between Lm and height instead. At each height in the cylinder section of the vessel, there are not large differences of Lm among the bed configurations except R1;60 . This difference usually keeps around 10%. For instance, Lm arranges from 80% for R1 to 90% for R2 at the height of 75d. In the cylinder section, Lm experiences a significant decrease from 90% at h = 75d to 40% at h = 15d on average except the situation R1;45 and R1;60 . The poor flow uniformity in the radial direction makes the mass flow region narrower in the lower part of the cylinder section. In the wedged section, it presents a rising profile of Lm from 55% at 30d to 95% at 1d for the R1;45 case, and from 40% at 15d to 80% at 1d for the R1;30 case. The main reason for this is that the contraction of the bed base keeps the width of the vessel

Fig. 5. The mass flow level a (dots) for several horizontal-stripe layers and the relative size Lm (lines) of mass flow region at each height for various bed configurations.

declining more greatly than the width of mass flow region. For R2 and R3, the Lm goes up after dropping from h = 15d to h = 5d. Particularly, there is a continuous fall of Lm from 30% at h = 15d to 5% at h = 1d for R1 bed configuration since that the vessel width seems not to show apparent decreasing from h = 15d to h = 1d. In conclusion, the total mass flow level can be calculated by average all Lm at each height. Table 1 shows the total mass flow level for all bed configurations. All the values indicate the following sequence of total mass level:

R1;60 > R1;45 > R2 > ðR3 or R1;30 Þ > R1

ð3Þ

The combination (r  a) of mass flow level and the flow uniformity in the radial direction can be used to estimate the flow pattern and flow uniformity quantitatively in the pebble flow. 3.4. Resultant movement estimation Although vertical velocity field was discussed in the previous section, the resultant velocity field is also significantly in need when estimating the whole flow field in the real pebble bed reactor. As is known, in the pebble bed, directions of resultant velocity vector for pebbles with different locations are various and their distribution can be regarded as a good indicator to estimate the resultant movement for the whole pebble flow. To estimate the relation between vertical and transverse movement and flow characteristics of the resultant flow, the tilted (or inclined) angle hin is defined as the average of the angles between the resultant velocities and the radial directions for all particles in the vessel during the experiment time firstly. The tangent value of hin can be simply calculated as the ratio of the vertical to transverse average displacement magnitude of the whole particles as follows:

hin ¼ arctan

  Sy Sx

ð4Þ

where sx (sy ) is the average of transverse (vertical) displacement magnitude of all particles. From depicted in Table 1, growing up of base angle (from 30° to 60°) indicates the dropping of hin (from 75.6° to 66.4°) for the arc shapes with infinite radius. As shown in Fig. 3, the R1;60 bed configuration presents a larger transverse velocity field than other situations. Thus the average transverse displacement magnitude sx for all particles remains greater for R1;60 situation. Furthermore, there is higher flow uniformity and mass flow level in the radial direction for the case with steeper base bottom. A larger mass flow zone indicates more particles flow with a relatively higher vertical velocity. Consequently, particles tend to drift horizontally towards a zone with faster downward flows because they are likely to get more space to move in the transverse direction. Meanwhile, particles flow in the pebble bed follows approximately the equation of continuity. That is, the vertical mass flux in one horizontal-stripe layer is equal to that in another layer statistically. All bed configurations keep the same recirculation flow rate (150/min) and the same pile height of the pebbles in the bed, which all make the differences of the sy among these cases small. In the arctangent function, the tilt angle hin of the resultant movement monotonically decreases with the reduction of the ratio of sy to sx . Thus a steeper bottom usually means a smaller tilted angle. Moreover, there are smaller differences of hin among the arc shapes with finite radius for the similar transverse velocity profile as shown in Fig. 3. R1, R2 and R3 situation all display smaller transverse dispersion in the base corner of the pebble bed. Specifically, the R3 bed configuration presents similar tilted angle (75.5°) with the R1;30 case because of the slight base distinction between the two situations. As aforementioned, higher mass level and flow uni-

X. Jia et al. / Annals of Nuclear Energy 102 (2017) 1–10

formity will make a smaller tilted angle. That is why the hin of the R1 case is little larger than that of the R2 situation. The velocity vectors of all particles in the pebble bed are collected during the experiment time of 3300 s. The angle between velocity vector and the radial direction for the particle i at time t can be expressed as hi (t), then the difference between hi(t) and hin is:

Dhi ðtÞ ¼ hi ðtÞ  hin

ð5Þ

More than 6,000,000 individual velocity vectors are enough to calculate statistically the probability distribution of Dh (Fig. 6) and standard deviations r (Dh) (Table 1) of Dh for each bed configuration. From Fig. 6, the distribution of Dh indicates some similar characteristics for all bed configurations. Firstly, it is noticed that the number of particles with a positive value of Dh accounts for less than 50% of the total number of particles in the pebble bed. That is, there are more than 50% of particles whose velocity vector angle h is less than hin. The vertical velocity of particles with larger h is significantly larger than those with smaller h. As a result, the ratio of Sy to Sx becomes larger with making the hin greater than the average of the h of all particles. So velocity vectors of particles with larger h contribute more to calculating the hin. Moreover, the Dh presents a left-skewed distribution profile, namely the tail on the left side of the probability density function is longer or fatter than the right side. The positive Dh displays a concentrated distribution profile arranging from 0° to 20°, while the negative Dh presents a dispersed distribution. For instance, the Dh reaches 50° which means the particles move faster in transverse than vertical direction for the R1 bed configuration. However, there are also some differences of the probability density functions (PDFs) of Dh among all bed configurations. All cases show different amount of variation or dispersion of the set of the

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PDFs of Dh. For example, the PDF for the R1;60 and R2 cases illustrates closer to the zero point than other bed configurations. The more concentrated distribution profile indicates smaller deviation of directions between the movement of individual particles and the resultant movement through the whole flow field. To quantify this deviation, the standard deviation (r(Dh)) of the PDF data is calculated and shown in Table 1. A smaller standard deviation usually suggests the better motion consistency for the pebble flow. The sequence of the resultant motion consistency level is shown as follows:

R1;60 > ðR1;45 or R2 Þ > R1;30 > R3 > R1

ð6Þ

3.5. Thickness of displacement In the previous section, the resultant movement is estimated and analyzed by studying the characteristics of the angles. In this section, the pebble flow will be measured by the thickness of displacement (d⁄) with analogy of the boundary layer theory in fluid mechanics. For simplification, the extremely slow pebble flow is considered as a laminar flow. As is known, the traditional boundary layer thickness is the distance from the wall to a point where the flow velocity has essentially reached the point of 99% ‘free stream’ velocity. However, the pebble flow is very slow and confined by the friction force from neighbors and walls, and the traditional layer thickness of pebble flow is so larger that the so-called ‘freestream’ zone (where velocity is larger than 99% of U) is much smaller. Instead, considering both transverse and vertical movement for different bed configurations, the funnel flow zone can be referred to the boundary zone. At each height, the velocity of the outer boundary of the mass flow zone will be regarded as the U while calculating the thickness of displacement in the boundary funnel flow zone. A smaller d⁄ is required in the real pebble bed reactor

Fig. 6. The probability distribution of the Dh for each bed configuration.

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design to make the pebbles in the corners spend shorter time flowing out of the vessel. In this way, the d⁄ of R1;60 bed configuration will be equal to zero for that there is no funnel flow zone in this situation. The formation of the boundary layer concerns may terms, for example, the bed configuration, the friction and so on. The pebbles in the lower corner of the pebble bed flow very slow, nearly stagnantly, which is called the stagnant zone. The pebbles between the stagnant zone and the central zone are easier to move under a more inclined base slope, because a large slope of inclination provides larger driving force along the wall. Therefore, the enlarged base cone angle greatly decreases the scale of stagnant zone, up to completely disappear, just as presented in the case of 60° cone angle. Due to the existence of stagnant zone, the flow field becomes non-uniform. Pebbles will form rugged and lumpy borders and hinder the flowing of their neighboring pebbles greatly. Meanwhile, such a resistance will transmit within a certain range toward the bed center and finally produce higher non-uniformity of velocity distribution. In other words, the velocity distribution will be more uniform if without the stagnant zone. Because of the non-uniformity, the averaged vertical velocity decreases continuously from the central zone to the near-wall zone. In this way, the enlarged base cone angle greatly decreases the scale of stagnant zone, even up to completely disappearance just as presented in the case of 60° cone angle. In other words, the enlarged base cone angle increases the uniformity of the vertical velocity throughout the whole flow field. Thus, the mass flow level of the case 60° is a = 1, which means the vertical velocities in pebble bed are larger than 0.3 * v0 0 (the vertical velocity of the particle at the central position that stands for the highest vertical velocity of the bed). In this way, the boundary layer with smaller velocity almost does not exist in the 60° case according to the definition of the boundary layer in this work. The thickness of displacement for each bed configurations is shown in Fig. 7a. The thickness of most curves presents the dropping tendency in the lower part after a slow increasing when the height decreases. As is well known, in fluid mechanics, the thickness of the boundary layer becomes thicker along the flow direction. As a result, the thickness of displacement in the flow will be larger when the boundary layer thickness increases. With the pebbles moving down towards the outlet, the increasing of d⁄ in the higher part seems to be caused by the influence of ‘viscosity’ as in fluid mechanics. The ‘viscosity’ in the pebble flow is more complicated than fluids because of many-body interactions, nonthermal fluctuations and other reasons (Liu and Nagel, 1998; Corwin et al., 2005). However, from the point of the statistics, the pebble flow and fluid flow are both similar in the way using the displacement thickness to estimate the boundary layer effect. On the one hand, the d⁄ begins a sharp dropping at the height 30d and 15d for the case R1;45 and R1;30 , respectively. This dropping of d⁄ starts in the transition region from the cylinder to the wedged section of the vessel. Then it can be concluded that the contraction design of the arc shape with finite radius contributes a lot to shortening the thickness of displacement in the lower wedged section and makes the difference of d⁄ before and after the sharp dropping large. On the other hand, for the bed configuration R1 and R2, there is a smaller difference of d⁄ between the cylinder and wedged section. One reason is that the smooth transition region makes the streamlines of the funnel flow zone smoother. Thus the designs of arc shapes with finite radius will be better if the deviation of the thic; R1;60 kness of displacement at each height is considered. Finally, there is no doubt that the situation R1;60 does best with zero d⁄ since there is no so-called boundary layer in this case. On the contrary, the R1 configuration is the worst and the d⁄ presents an increasing profile which is larger than other

Fig. 7. The thickness of displacement of pebble flow for different bed configurations.

bed configurations at all heights. In terms of the average and standard deviation of the d⁄, the R2, R3 situation show the similar values. As aforementioned, this thickness d⁄ is defined normal to the wall surface of the vessel. In the cylinder section of the bed, this thickness vector will parallel to the horizontal direction (x direction in Fig. 2c) in the global coordinate system. However, in the wedged section, there is an angle (a as shown in Fig. 2c) between the thickness vector and the x direction. We can get the component of the thickness vector in the horizontal direction dx(h) at height h by :

dx ðhÞ ¼ d ðhÞ  cosðah Þ

ð7Þ

The dx(h) curve of each bed configuration is illustrated in Fig. 7b. By comparison, it shows the same value of dx(h) and d⁄(h) in the cylinder section, while in the wedged section, dx(h) is smaller than d⁄(h) at each height for the same configuration. In a sense, the dx(h) can be regarded as an index to indicate the extended width of the vessel at each height in order to decrease the boundary layer effect.

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Particularly, the extended width of the vessel at h = 0d (increasing the radius of the outlet tube) is an effective way to make more pebbles move down with the mass-flow pattern throughout the whole flow field (Datta et al., 2008; Rycroft et al., 2013). 4. Summary This study carries out the comparisons of the bed configuration effects on the pebble flow characteristics in a gravity drained pebble bed, which is of fundamental importance in the design work of pebble bed high temperature gas-cooled reactor. The bed configurations are compared and analyzed in several aspects. From the point of vertical velocity, two parameters including the mass flow level (a) and flow uniformity (r) are proposed to estimate the flow pattern and flow uniformity quantitatively in the pebble flow. In light of the resultant movement of the pebble flow, the angle hin of the vector sum (vsum) of the velocities of the whole particles in the pebble bed is introduced and the distribution of the Dh (angle difference between the individual particle velocity and vsum) is also calculated to estimate the resultant motion consistency level. Moreover, for each bed configuration, the thickness of displacement is analyzed to measure the funnel flow zone based on the boundary layer theory. The sequence of levels of each estimation criterion is obtained for all bed configurations. The results demonstrate that the pebble flow under the contraction configuration with a base angle of 60° is the best, for both the flow patterns and field characteristics analyses. A larger base angle always means the more desired flow behavior by comparing bed configuration series with infinite radius. In the series of R1, R2, R3 and R1;30 cases, the R2 bed configuration usually does better because smooth transition region makes the streamlines of the funnel flow zone smoother. Finally, the arc shapes with finite radius (R2, R3) show a smaller difference of d⁄ between the cylinder and wedged section than those with infinite radius (R1;30 ; R1;45 ) for their smoother transition region. In addition, the proposed criteria can be also applied and adopted in testing other geometry designs of pebble bed. In this paper, the R1;60 bed configuration is strongly suggested to be used for pebble bed reactor design. Acknowledgement The authors are grateful for the support of this research by the National Natural Science Foundations of China (Grant No. 51576211), the Science Fund for Creative Research Groups of National Natural Science Foundation of China (Grant No. 51321002), the National High Technology Research and Development Program of China (863) (2014AA052701), and the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD, Grant No. 201438); Xinlong Jia especially thanks CSC (China Scholarship Council) for the sponsorship (No. 201406210102). References Albaraki, S., Antony, S.J., 2014. How does internal angle of hoppers affect granular flow? Experimental studies using digital particle image velocimetry. Powder Technol. 268, 253–260. Auwerda, G., Kloosterman, J., Lathouwers, D., Van der Hagen, T., 2010. Effects of random pebble distribution on the multiplication factor in HTR pebble bed reactors. Ann. Nucl. Energy 37 (8), 1056–1066. Baek, S., Lee, S., 1996. A new two-frame particle tracking algorithm using match probability. Exp. Fluids 22 (1), 23–32. Balevicˇius, R., Kacˇianauskas, R., Mroz, Z., Sielamowicz, I., 2011. Analysis and DEM simulation of granular material flow patterns in hopper models of different shapes. Adv. Powder Technol. 22 (2), 226–235.

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