A parametric study of graphite dust deposition on high-temperature gas-cooled reactor (HTGR) steam generator tube bundles

Annals of Nuclear Energy 123 (2019) 135–144

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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Review

A parametric study of graphite dust deposition on high-temperature gas-cooled reactor (HTGR) steam generator tube bundles Wei Mingzhe, Zhang Yiyang ⇑, Wu Xinxin, Sun Libin 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, China

a r t i c l e

i n f o

Article history: Received 11 May 2018 Received in revised form 4 September 2018 Accepted 8 September 2018

Keywords: HTGR Steam generator Large eddy simulation Graphite dust Deposition

a b s t r a c t Accumulation of abraded graphite dust is a major concern during a potential accident in hightemperature gas-cooled reactors (HTGR). AVR experiments indicated that it is of great safety interest to develop and benchmark numerical approaches for predicting deposition of dust particles in steam generator (SG) area. The present study analyzes the deposition behavior of graphite dust on one typical part of HTGR SG tube bundles using the open source CFD code OpenFOAM. The effects of Reynolds number, temperature difference and bundle structure parameters are discussed. The results show that the mechanism of thermophoresis dominates the deposition of particles below 2 lm, and the inertia impact becomes important for larger particles. As the flow Reynolds number increases, the impact rate decreases in the thermophoresis-controlled regime but increases in the inertial regime and the peak of deposition rate shifts to the smaller size. The effect of flow Reynolds number can be entirely represented by particle Stokes number in the inertia regime. As for the effect of bundle arrangement parameters, the impact rate first increases and then decreases as the longitudinal spacing increases, and decreases as the transverse spacing increases. Ó 2018 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Geometry description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Flow field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Particle-fluid interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Particle-wall interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Particle deposition on tube bundles: The basic case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Effects of temperature gradient and Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Effect of bundle structure parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Distribution of particle deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. E-mail address: [email protected] (Y. Zhang). https://doi.org/10.1016/j.anucene.2018.09.009 0306-4549/Ó 2018 Elsevier Ltd. All rights reserved.

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M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

1. Introduction As an important option for the fourth generation advanced nuclear reactor, the high temperature gas-cooled reactor (HTGR) is promising to produce electricity or process heat with high efficiencies and unique safety features (J. Kupitz1, 2014; Nerac and GIF, 2002). A typical HTGR uses helium gas as the coolant, and the moderator function is performed by carbon in the form of graphite. The fuel is embedded in the graphite moderator and able to withstand high temperatures up to 1600 °C (Engle et al., 1974; Marsden, 2000). Various experimental and prototype pebble-bed reactors have been successfully operated worldwide. Apart from those, two full-scale HTGRs, the two-module plant HTR-PM, are currently under construction in China (Zhang et al., 2009). While the efficient operation has been demonstrated, the dust generation, transport, deposition and resuspension in the primary circuit of these reactors are identified as one of the foremost concerns for HTGRs (Humrickhouse, 2011; Kissane, 2009; Xu et al., 2017a). The graphite dusts are mainly generated from the partiallygraphitized matrix material of the fuel pebbles due to abrasion in the core and fuel handling system. In a 400 MW thermal pebblebed HTGR, the abrasion-generated graphite dust is anticipated to be in the range 30–100 kg/year (Kissane et al., 2012). The dust is essentially composed of carbonaceous compounds, fission and activation products (Mayer et al., 2010; Metcalfe and Mills, 2015). Graphite dust has been a highly-concerned problem for pebble bed reactors 7since 1960s (IAEA, 1997; Zhang and Yu, 2002), because the transport of fission products (FPs) is closely coupled with dust particles in the primary circuit. Especially in the steam generator (SG) tube bundles, a large fraction of free FPs (e.g. Cs137 and Sr-90) condenses on the surface of either air-borne or deposited particles, and thus stays and accumulates. The experiments in AVR showed that the specific activity of graphite dusts sampled on SG tubes was much higher than other places e.g. cold gas zone or fuel element surface (Rainer, 2008). These dust particles are very likely to re-suspend and be discharged in a depressurization or water-ingress accident (Xu et al., 2017a). Moreover, large deposition area, strong turbulence and high temperature gradient make it even easier for particles to deposit on the SG tube bundles region. The deposited graphite dust not only complicates system maintenance in the tube bundles region, but also reduces the heat transfer rate (Lind et al., 2010). Therefore, the deposition of graphite dust on SG tubes needs special attention. The particle deposition in a tube bundle cross-flow is generally influenced by both impact rate and sticking efficiency (Li et al., 2011). There are several mechanisms for particle impact, including inertia impact, inception, thermophoresis and diffusion etc. (Friedlander, 2000; Guha, 2008). The contribution of each mechanism largely depends on the various parameters, including flow dynamic parameters e.g. Reynolds number and structure parameter e.g. tube spacing. On the other way, the sticking efficiency is determined by the particle–wall interaction after impact, which requires the understanding of the dynamic impact process (Chen et al., 2015; Zhang et al., 2015). Particularly for the issue of aerosol deposition on nuclear reactors SG tubes, the research is still quite limited and far from fully understood. Several papers have been published regarding the behavior of dust in HTGRs. Some studies (Hiruta et al., 2013; Luo et al., 2017; Troy et al., 2015; Xiaowei et al., 2005) give the quantity and size distribution of the graphite particle dust based on various estimated distribution functions. Some studies form INET (Xie et al., 2017; Xie et al., 2013; Xu et al., 2017b) measure and estimate the concentration of the radioactive dust in the primary loop of HTR-10. Some studies (Gutti and Loyalka, 2009; Nguyen and Loyalka, 2015; Sun et al., 2018) concentrate on the aerodynamic characteristics of irregular

graphite particles. Some studies (Kissane, 2009; Lind et al., 2010; Rainer, 2008) discussed the transport of the radioactive dust and some others (Barth et al., 2014; Peng et al., 2016; Peng et al., 2013a; Peng et al., 2013b; Wei et al., 2018) investigate the deposition and resuspension behavior of dust in HTGRs. However, there is still very limited study on the parametric influence of the graphite dust deposition on the primary side of SG tube bundles even though the results have directly significant importance to reduce particle deposition and give guidance to SG maintenance work. Compared to traditional experiments, numerical simulations have the advantage of repeatability, less effort and the ability to view the flow field. Computational fluid dynamics (CFD) methods have been widely used to predict the transport and deposition behavior of small particles in various environments (Bouris et al., 2001; Dehbi and Martin, 2011; Greifzu et al., 2016; Lu et al., 2015; Lu and Lu, 2015) instead of experiments to get consistent results. The investigation of parametric influence can be easily performed by switching the related parameter and comparing the results. With the rapid development of CFD tools, the open source CFD codes such as OpenFOAM are more and more popular because it is economically feasible and easier to implement self-defined functions. In summary, the deposition behavior of graphite dust on HTGR SG tube bundles is a complex phenomenon taking particle-vortex interaction and particle–wall interaction in consideration. The present study analyzes the parametric influence of flow dynamic parameters and bundle structure parameters on the deposition behavior of one typical part of HTGR SG tube bundles with the open source CFD code OpenFOAM. 2. Methodology 2.1. Geometry description Fig. 1 shows the two-dimensional axisymmetric geometry model used in this study. The steam generator of HTR-PM is consisted of 19 identical components, each containing five layers of helical tubes (Li et al., 2014; Ma et al., 2014). Here we select a part of 7 rows to represent the whole tube bundles. The tubes, with an outer diameter of 19 mm, are arranged in an in-line form with normalized transverse and longitudinal spacing ST/D and SL/D to be 1.58 and 1.32 respectively. The distances between the wall and the centers of wall adjacent column tubes are H = ST/2. The computational area is 175 mm
150 mm. Gas flow enters the tube bundles from the left. Periodically fully developed boundary condition is applied at the inlet and outlet. The fluid phase is helium with constant properties. The operating pressure is 7 MPa. The mass

Fig. 1. The geometry of the computational domain.

137

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144 Table 1 Material properties. Notation

Physical quantity

Value

Etb Egh

Young’s modulus of the tube material, GPa Young’s modulus of graphite, GPa Poisson’s ratio of the tube material Poisson’s ratio of graphite graphite density, kg/m3

157.6 8.0 0.31 0.16 1781

surface free energy of the tube material, J/m2 surface free energy of graphite (Abrahamson, 1973), J/m2

0.700 0.095

mtb mgh qgh ctb cgh

2013; Rollet-Miet et al., 1999). Above all, LES is an advanced modeling methodology which results in a more realistic particle deposition predictions than lower order modeling approach RANS (Jayaraju et al., 2008) and needs far less computation resources than DNS. Therefore, we opt here the LES approach as a good compromise between accuracy and practicality. The LES (Pope, 2000), is a time-dependent, spatial-filtering simulation technique which aims at properly resolving large coherent and generally anisotropic structures which drive particle motion towards the wall. The remaining unresolved fraction of turbulent kinetic energy (TKE) is modeled using so called sub-grid scale (SGS) models. Within the LES framework, the dynamics of the continuous phase is described by the space-filtered equations for the conservation mass, energy and momentum subject to the Boussinesq approximation:

8  @ui > ¼0 > @x > > < i   2 @u u @s @ui ui @p þ @xi j ¼  q1 @x þ m @x@ @x  @xij @t j i j j j > > > > @T  @hj 2 : þ u  r T ¼ a r T  i @t @x

ð1Þ

j



ui is the filtered velocity in i direction, p the filtered pressure, and T the filtered temperature. The residual SGS shear stress tensor and heat flux vector are given respectively by:  

sij ¼ ui uj  ui uj Fig. 2. Structured grid distribution.

ð2Þ

 

hij ¼ ui T  ui T

ð3Þ

flow rate at the inlet is set to be 5.074 kg/s, with a Reynolds number of 10,500. The upstream bulk temperature is 1023 K. Constant temperature and non-slip wall boundary conditions are set at the tubes surface. The material properties are listed in Table 1. The sufficient resolution of the near-wall turbulence is important for accurate prediction of particle deposition, especially in the area very close to the wall e.g. yþ < 4 (Lecrivain and Hampel, 2012; Tian and Ahmadi, 2007). Structured meshes are used to discretize the domain and the boundary layer is fully resolved. An example of generated fully hexahedral mesh for the tube bundle is presented in Fig. 2. Overall meshes are designed in according to Best Practices Guidelines (BPG) (Mahaffy et al., 2007), such that numerical diffusion is reduced with accuracy and convergence enhanced. For each simulation, a hierarchy of grids is constructed with coarse, medium and fine grid resolution. Grid sensitive quantities such as the velocity magnitude, temperature and deposition rates are checked to ensure that the results are grid-independent, as shown in Fig. 3. The profiles are extracted from the upper surface of the tube in row 3 column 3 and along its wake (Y/D = 0.5, X/D = 2.63  3.95, the origin of coordinate system is at the center of the first tube in the center row). The results indicate that the grid independence can be ensured.

The remaining unresolved fraction of TKE is modeled using the Smagorinsky model.

2.2. Flow field simulation

The accurate LES flow field is used to compute particle paths and fates in the tube bundle area by way of an Euler-Lagrange approach (Maxey and Riley, 1983). When turbulence is present in the flow, the computation of particle dispersion becomes significantly more involved because of the random velocity fluctuations that preclude the deterministic computation of particle trajectories. Particles are assumed to interact with the filtered fluid velocity field only, and hence the effects of sub-grid scales on particle transport are neglected. Particle-particle interaction is not considered (one-way coupling) since the particle volume fraction is far below 1%, which indicates that the dispersed phase is dilute enough not to affect the continuous flow field (Elghobashi, 1994). Then particle trajectories are calculated by Discrete Phase Method (DPM) (Friedlander, 2000), which is naturally deduced from New-

Computational Fluid Dynamics modeling has emerged as a useful tool in simulating the particle transport and deposition phenomena. Complex particle–wall and particle-vortex interactions are the main concern of these simulations (Falkovich et al., 2001). There are three main approaches to numerically solve Navier-Stokes equation in turbulent flows: Reynolds-averaged Navier–Stokes (RANS), Large eddy simulation (LES), and Direct numerical simulation (DNS). DNS is most accurate but too expensive for the Reynolds number and computational domain in this study (Elghobashi, 1994). RANS is least consuming but lack of ability to resolve different scales of vortices, which has been proven to be important for evaluation of particle deposition (Dehbi et al.,



1 3

1 3

sij  skk dij ¼ 2lsgs Sij  Skk dij



ð4Þ



hij ¼ 

lsgs @ T Prsgs q@xj

1=2 lsgs ¼ C k ksgs D

Above,

ð5Þ ð6Þ

lsgs is the sub-grid viscosity, and Prsgs is the sub-grid

Prandtl number equal to 0.85. ksgs is the SGS turbulent kinetic energy. D is the filter size computed from the third root of the cell volume. The numerical method for the flow field is validated by comparing to a similar experimental case (Konstantinidis et al., 2000), including both time-averaged velocities and fluctuations. The detailed comparison can be referred to our recent publication (Wei et al., 2018). 2.3. Particle-fluid interaction

20

Coarse Medium Fine

15

10

5

0 3.2

3.5

3.8

4.1

4.4

4.7

RSME velocity magnitude (m/s)

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

Mean velocity magnitude (m/s)

138

20

Coarse Medium Fine

15

10

5

0 3.2

3.5

3.8

X/D

(a) Mean velocity magnitude

4.7

25

Coarse Medium Fine

1050

RMSE temperature (K)

Mean temperature (K)

4.4

(b) RMSE velocity magnitude

1070

1030 1010 990 970

Coarse Medium Fine

20 15 10 5 0

950 3.2

3.5

3.8

4.1

4.4

3.2

4.7

3.5

3.8

4.1

4.4

4.7

X/D

X/D

(c) Mean temperature

(d) RMSE temperature 10%

100%

Coarse Medium Fine

Coarse Medium Fine

5%

Deposion rate

Impact rate

4.1

X/D

10%

1%

1% 0.5%

0.1%

0.1% 0

1

2

3

4

5

6

0

Particle diameter ( m)

1

2

3

4

5

6

Particle diameter ( m)

(e) Impact rate

(f) Deposition rate

Fig. 3. Grid sensitivity for basic case, profiles of selected variables.

ton’s second law, allowing one to include all the relevant forces which are significant. In this situation, the forces acting on the particle are drag and thermophoresis. Other forces (Brownian diffusion, lift, etc.) are neglected. The vector force balance on a spherical particle is given by:

!   ! dup ! ! ¼ FD u  u p þ F T dt

ð7Þ

The first term on the right is the drag force for unit particle mass for ! ! spherical particles. u p is the particle velocity and u is the fluid ! velocity. F T is the thermophoretic force per unit mass, and F D is

the drag force per unit mass defined based on the work of (Sommerfeld et al., 1992). Considering the slip boundary condition in the transition and continuum regimes, F D is expressed as

FD ¼

C D Rep 24sp

The particle relaxation time

sp ¼

qp d2p C c 18l

ð8Þ

sp is defined as: ð9Þ

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

where l is the gas molecular viscosity, q the gas density, qp the particle density, dp the particle diameter, C c the Cunningham slip correction factor. The particle Reynolds number is expressed as:





! qdp ! u  u p Rep ¼ l

ð10Þ

The drag coefficient is computed as (Morsi and Alexander, 1972)

C D ¼ a1 þ

a2 a3 þ Rep Rep 2

ð11Þ

The a’s are constants which apply to wide ranges of particle Reynolds number (Feng and Bolotnov, 2018). The particle Stokes number is always introduced to describe the particle inertia effect, which is defined as the ratio of the particle relaxation time to a typical fluid flow time:

Stk ¼

sp D=u1

¼

2 d dp u1

q

18lD

ð12Þ

The thermophoretic effect on particles is calculated as (Talbot et al., 1980):

! rT F T ¼ DT mp T DT ¼

7:02pdp l2 ðK þ 2:18K n Þ qð1 þ 3:42K n Þð 1 þ 2K þ 4:36K n Þ

ð13Þ

In the above, DT is the thermophoretic coefficient. K n ¼ 2k=dp is the Knudsen number, k the mean free path of the fluid, K ¼ k=kp , k ¼ 15lR=4 the fluid thermal conductivity based on translational energy only, kp the particle thermal conductivity, T the local fluid temperature. The trajectory of the particle is obtained by the time integration of Eq. (7), the solution of which determines particle trajectories and velocities:

up ¼

dx dt

capture velocity can be described as a function of the properties of particle and wall, including Young’s modulus, Poisson’s ratio, surface energy, particle size and density. To the best of our knowledge, there is no published experimental data on the critical capture velocity of nuclear graphite dust. In this work we choose Kim and Dunn’s model to estimate the critical capture velocity (Brach and Dunn, 1995; Kim and Dunn, 2007, 2008), whose validity has been demonstrated for stainless steel, glass and polystyrene micro-sized particles. Based on Hertz contact theory, Kim and Dunn added a simplified analysis of energy dispassion to calculate the restitution coefficient, which is a ratio of reflective velocity to incident velocity,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u !15   35 u 4 4 5 r t e ¼ 1  2C R f 0 V i 5 3p 4K m2p

ð15Þ

To achieve accurate trajectory computations, the time step for the integration of the ordinary differential equations must be of the order of the particle relaxation time and hence are smaller than the fluid time steps. In this case, particles are injected linear from the inlet boundary and then tracked until one through leaving the computational domain. Given that the stochastic particle tracking is a Monte Carlo process, the number of particles injected has to be large enough for the sample size not to affect the results. This number was varied depending on cases (particle inertia) from 1e6 to 1e9, until results became independent of sample sizes. The diameter range of the graphite particles was chosen to be 0.1–7 lm according to the HTGR situation (Xiaowei et al., 2005). 2.4. Particle-wall interaction After hitting the wall, the particles may deposit, rebound or even break up. The outcome is determined by the energy balance of particle–wall interaction in the impact. A parameter named critical capture velocity is often used in aerosol science to quantify adhesion/rebound behavior of particle–wall impacts. If the normal impacting velocity exceeds the critical capture velocity, the particle will rebound, otherwise deposition occurs. The critical capture velocity, which actually represents the probability of adhesion, is mainly determined by two factors: i) the damping of particle kinetic energy; ii) the surface adhesion energy that turns to stick the particle. Hence, from a view of energy balance, the critical

ð16Þ

where C R is a modification coefficient for surface roughness (Cheng  1  et al., 2010), f 0 ¼ 9Krx2A =2p 3 , K ¼ 4E1 E2 =3 E1 þ E2  E1 v 22  E2 v 21 is system rigidity, E and v are Young’s modulus and Poisson’s ratio pffiffiffiffiffiffiffiffiffiffi respectively, xA ¼ 2 c1 c2 is reduced surface energy with c1 and c2 are surface energies of particle and wall respectively. Set the restitution coefficient to be zero and the critical capture velocity can be calculated,

2 V c ¼ 42C R f 0

ð14Þ

139

!15 354  35 4 5 r 5 3p 4K m2p



ð17Þ

When the particle center of mass happens to be at a distance less than the particle radius, the particle is considered to impact the wall. In the CFD numerical study, statistical analysis can be conducted on the impact and deposition rate of particles. The impact rate is defined as the ratio of impact particle number to incoming particle number. The deposition rate is defined as the ratio of deposited particle number to incoming particle number. 3. Results and discussions 3.1. Particle deposition on tube bundles: The basic case Fig. 4 shows the instantaneous and time-averaged velocity field. From the instantaneous velocity field, we can see that the fluid in the tube bundle sways heavily as it passes the tubes, which shows the intrinsic unsteadiness of the cross flow over tube bundles. The transient vortex structure around the bundles can be clearly described by LES. As the mean velocity field shows, the velocity of the flow between the central tubes is larger due to the wall effect. The mean maximum velocity is 4 times the mean average velocity at the inlet, which indicates that in these regions particles may obtain a higher velocity magnitude. Fig. 5 presents the critical capture velocity, impact rate and deposition rate of dust particles. The critical capture velocity is calculated by Kim and Dunn’s formula, i.e. Eq. (17) and with material properties listed in Table 1. The critical capture velocity drops rapidly as particle size increases, which indicates that rebound is much more likely to occur for larger particles. For particles larger than 1 lm, the critical capture velocity is already below 0.5 m/s, which is in the same order of particle velocity in boundary layers. On the other hand, the wall impact rate of graphite particles, which is a result of particle–fluid interaction, increases with particle size. For particles with larger inertia and thus larger Stokes number, the velocity fluctuations due to vortices are further enlarged and thus take more particles into the boundary layers. The impact rate of 1 mm particles is only 0.3%, while the impact rate of 6 mm particles reaches 35%. The particle deposition rate is determined jointly by several parameters: the impact rate, the critical capture velocity

140

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

while for large particles it is limited by the low sticking efficiency due to both low critical capture velocity and high impact velocity. 3.2. Effects of temperature gradient and Reynolds number For the conditions in this study, the dominating mechanisms for particle deposition are inertia impaction (mainly from turbulent eddies) and thermophoresis (Guha, 2008). Therefore we investigate the effects of temperature gradient and Reynolds number in this section. Fig. 6 shows the particle impact and deposition rate for different temperature difference between the tube wall and main flow. We can see that the effect of thermophoresis is dominant for particles below 2 lm. The impact rate almost scales proportionally with the temperature difference, which indicates that nearly all the particle impact comes from the thermophoresis. For particles above 4 lm, the contribution of inertia impact overwhelms thermophoresis due to larger Stokes number. When particle rebound is concerned, the effect of thermophoresis is even more important, as shown in Fig. 6(b). Because the impacting velocity of inertia-induced impacts is generally larger than thermophoresis-induced impacts, which more likely leads to rebound. Fig. 7 shows the impact and deposition rate of graphite particles for different Reynolds number. The trend is opposite for large and small particles with a transition point at 2 lm. For particles below

(a) Instantaneous velocity field

100%

Impact rate

10%

1%

(b) Averaged velocity field

T=1K T=35K T=70K T=140K

0.1%

Fig. 4. Instantaneous and averaged velocity field.

0.01%

Critical capture velocity Collision rate Deposition rate

3.0

Rate

10% 2.0 1% 1.0

0.1%

0.0 0

1

2

3

4

5

0

6

Particle diameter ( m) Fig. 5. Curve of critical capture velocity varying with particle diameter, and the calculated impact and deposition rate.

2

3

4

5

6

5

6

100% T=1K T=35K T=70K T=140K

10%

1%

0.1%

0.01% 0

and instantaneous particle velocity upon impact. As the result shows, deposition rate first increases, reaches a maximum at about 5 mm, and then drops as the particle size further increases because the rebound becomes important. The deposition rate of small particles is mainly limited by the low impact rate due to small inertia,

1

Particle diameter ( m) (a) Impact rate

Deposition rate

4.0

Critical capture velocity (m/s)

100%

1

2

3

4

Particle diameter ( m) (b) Deposition rate Fig. 6. Impact and deposition as per each particle diameter with series of temperature difference between upstream flow and tube surface.

141

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

100%

Re=5271 Re=10542 Re=15812 Re=21083

10%

Impact rate

Impact rate

100%

1%

10%

1%

Re=5271 Re=10542 Re=15812 Re=21083

0.1%

0.1% 0

1

2

3

4

5

6

0.00

Particle diameter ( m) (a) Impact rate

0.10

0.15

0.20

0.25

100%

100% Re=5271 Re=10542 Re=15812 Re=21083

10%

Deposition rate

Deposition rate

0.05

Square root Stokes number (a)

1%

Re=5271 Re=10542 Re=15812 Re=21083

10%

1%

0.1%

0.1% 0

1

2

3

4

5

6

Particle diameter ( m) (b) Deposition rate

0.00

0.05

0.10

0.15

0.20

0.25

Square root Stokes number (b)

Fig. 7. Impact and deposition as per each particle diameter with series of Reynolds number.

Fig. 8. Impact and deposition as per each square root Stokes number with series of Reynolds number.

2 lm, the impact rate decreases with Reynolds number. This is because larger Reynolds number results in shorter residence time, which decreases the particle deposition in the thermophoresisdominated regime. For larger particles above 2 lm, the trend is changed to the opposite way that the impact rate increases with Reynolds number. As the contribution of inertia impact becomes important, larger Reynolds number leads to higher impact rate due to higher fluctuating velocity. Fig. 7(b) shows that when the particle rebound is concerned, the peak of deposition rate shifts to smaller size for larger Reynolds number. The reason is that the probability of rebound increases with particle fluctuating velocity, which makes the limitation of sticking efficiency more effective. To further illustrate the effect of Reynolds number on particle deposition, we introduce particle Stokes number here. Fig. 8 shows the impact and deposition rate as per each square root Stokes number with different Reynolds number. The reason for choosing square root Stokes number is that it is proportional to the particle diameter. The results show that in this study, for particle square root Stokes number more than 0.15, the impact rates of the particle dust of different Reynolds number fall into the same curve, which indicates that the effect of Reynolds number can be essentially represented by Stokes number, or in another way, totally inertial. For smaller particles whose square root Stokes number is less than

0.05, the impact of the particle dust is then in the thermophoretic-moderated regime. In this regime, the impact rate increases as the particle Stokes number increases and decreases as the flow Reynolds number increases. The square root Stokes number range between 0.05 and 0.15 can be regarded as the transition region. As shown in Fig. 8(b), the deposition rates of different Reynolds number do not coincide. At the same Stokes number, particles with larger Reynolds number result in less deposition rate. This is mainly caused by the lower adhesion efficiency for larger Reynolds number. However the peak is always at around 0.13 of square root of Stokes number for different Reynolds number. 3.3. Effect of bundle structure parameters The geometry parameters determine the tube arrangements, and thus have a complex effect on the vortex structure and particle paths. Two of the most important parameters to design a heat transfer tube bundle (normalized longitudinal spacing and normalized transverse spacing) are taken into consideration here. Longitudinal spacing mainly determines the large vortex structure that carries particles and hits the latter rows with a certain vortex development distance. This mechanism has main effect on the particle inertial deposition. Impact rate with different longitudinal

M. Wei et al. / Annals of Nuclear Energy 123 (2019) 135–144

2 m

3 m

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2 m

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100%

SL/D

10%

4 m

1.4

1.6

1.8

2.0

ST/D (b) Deposition rate Fig. 10. Effect of normalized transverse spacing on normalized impact and deposition rate.

Fig. 9. Effect of normalized longitudinal spacing on impact and deposition rate.

3.4. Distribution of particle deposition spacing of small particles varies less than that of large particles (see Fig. 9). As the longitudinal spacing increases, the impact rate first increases and then decreases. In this case, impact rates reach its maximum value at the same longitudinal spacing, which is 2.37. In this distance, the vortex will drive the most particles and hit the tube surface, and the value will truly change with different fluid Reynolds number. The deposition rate as a function of longitudinal spacing behaves differently from impact rate. As longitudinal spacing increases, particles will hit the tube surface with higher velocity, thus resulting in a bit lower deposition rate. As for the transverse spacing, normalized impact rate and deposition rate are employed here to count the different deposition surface ratio effect. The transverse spacing is normalized to keep the particle injecting concentration the same. Calculated normalized deposition rate is equal to the deposition rate times the ratio of transverse spacing to basic case transverse spacing, which is 1.58. Fig. 10 shows that the normalized impact rate decrease with the transverse spacing and the normalized deposition rate increases first and then decreases with the transverse spacing. As the particle size increases, the transverse spacing where normalized deposition rate reaches maximum increase, which shows a similar situation as that of Reynold number.

Fig. 11 presents the distribution of the deposited particles on different tube rows and columns. It should be noted that a periodic boundary condition is applied at the inlet and outlet. The deposited masses on different columns are generally similar, while the deposited mass on the back rows is smaller than that of the front rows. This could be attributed to two reasons. First, the particle concentration slightly decreases due to deposition. Second, the decrease of helium temperature reduces the temperature gradient of the back rows, which reduces the thermophoresis deposition. When the flow gets developed as periodic boundary conditions involved, the inertia impaction driven by vortex is generally even at each column, which gives a way to predict the whole particle deposition results. Fig. 12 presents a typical angular distribution of one certain tube. More deposition occurs at 30 degree or 60 degree, while less deposition on the 0 degree and almost none on the 90 degree. Vortex shedding makes the peak at 30 degree and 60 degree, driving more particles hitting the surface on these positions, illustrating the accumulation of inertia particles. The amounts of particles deposit on the front and back side are almost the same due to thermophoresis and vortex shedding. The angular distribution of larger particles is more non-uniform, especially showing a deeper valley at 0 degree.

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1 m 4 m

106

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Fig. 11. Distribution of the deposited particles on different rows and columns.

(b) Deposition behavior 4. Conclusion Fig. 12. Typical angle distribution of one certain tube (Example, tube 4–3).

In this study, we investigate the deposition of graphite dust particles on one typical part of HTGR SG tube bundles. Large eddy simulation is used to present particle-vortex interaction and critical capture velocity model to present particle–wall interaction. The effects of various parameters including Reynolds number, temperature difference and tube spacing on particle deposition rate are discussed. The main conclusions are drawn as below. 1) The deposition rate first increases then decreases with particle size. The peak is around 5 lm. For small particles the deposition rate is limited by the impact rate while the sticking efficiency becomes the key factor for large particles. 2) For particles below 2 lm, the deposition is dominated by the thermphoresis mechanism. As particles size increases, the inertia impact due to turbulent eddies becomes more important. In the thermophoresis-dominated regime, the deposition rate decreases with flow Reynolds number due to short residence time. In the inertia-dominated regime, the impact rate increases with Reynolds number and the effect of flow Reynolds number can be entirely represented by particle Stokes number. 3) As the longitudinal spacing increases, the impact rate first increases and then decreases while the deposition rate decreases. Particles will hit the tube surface with higher velocity, thus resulting in a bit lower deposition rate. 4) The distribution of particle deposition on different rows and columns of tubes is generally uniform. For a certain tube, the angular distribution of deposition shows a preference

around 60 degree. The angular distribution of larger particles is more non-uniform, especially showing a deeper valley around 0 and 90°.

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