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Experimental and numerical study on helium flow characteristics in randomly packed pebble bed
Annals of Nuclear Energy 128 (2019) 268–277
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Experimental and numerical study on helium flow characteristics in randomly packed pebble bed Zhenxing Wu, Yingwei Wu ⇑, Chenglong Wang ⇑, Simiao Tang, Di Liu, Suizheng Qiu, G.H. Su, Wenxi Tian Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China Shanxi Key Lab. of Advanced Nuclear Energy and Technology, Xi’an Jiaotong University, China
a r t i c l e
i n f o
Article history: Received 19 March 2018 Received in revised form 10 January 2019 Accepted 11 January 2019
Keywords: Helium flow characteristics Spherical pebble bed Experimental study Numerical study
a b s t r a c t The high temperature gas cooled reactor (HTGR) with a packed pebble bed core has been one of the fourth generation reactors. The helium cooled pebble bed (HCPB) blanket packed spherical pebbles of lithium orthosilicate (Li4SiO4) has been recommended to Chinese Fusion Engineering Test Reactors (CFETR). Helium flow characteristics in spherical pebble bed are the key issues to design the pumping power. In this work, helium flow characteristics in a rectangular pebble bed channel filled with stainless steel particles whose diameters are 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm, respectively, were investigated experimentally and numerically. The experimental results show that the compressibility cannot be neglected and it has an important influence on flow characteristics in the conditions with the enough large ratio of pressure drop to system pressure. Ergun correlation was compared with experimental values, and Ergun correlation modified with compressibility was deduced. The results show that Ergun correlation modified with compressibility can improve the ability to predict experimental values. The numerical results show that the particle friction factors modified with compressibility (fp*) in a rectangular pebble bed channel randomly filled with spherical particles can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements. This work would provide some supports for the design of the helium supply system of the packed bed reactors. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Spherical pebble bed have been widely used in some reactors, such as high temperature gas cooled reactor (HTGR), pebble bed water cooled reactor (PBWR), and helium cooled pebble bed (HCPB) blanket of CFETR. HTGR with a packed bed core has been one of the fourth generation reactors due to the higher inherent safety. The energy generated in a HTGR core is brought by helium from the core into the heat exchanger. In addition, the helium cooled pebble bed blanket packed spherical pebbles of lithium orthosilicate (Li4SiO4) was proposed for Chinese Fusion Engineering Test Reactors (CFETR) (Li et al., 2015). Lithium orthosilicate (Li4SiO4) pebble bed has been selected as tritium breeder in many test blanket module (TBM) designs. The bred tritium is transported by low pressure helium from pebble bed to the tritium extraction system. Thus, helium pressure drop through the randomly packed pebble bed is a key parameter to design helium supply system. ⇑ Corresponding authors. E-mail addresses: [email protected] (Y. Wu), [email protected] (C. Wang). https://doi.org/10.1016/j.anucene.2019.01.016 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
In recent decades, there were many experimental investigations on fluid flow in the packed bed, but their research subjects were either water or air. The well-known correlation used to predict pressure drop through packed bed was proposed by Ergun (Ergun, 1952) for incompressible flow:
DP lU ð1 eÞ2 qU 2 ð1 eÞ þ 1:75 ¼ 150 2 3 L Dp e3 Dp e
ð1Þ
After Ergun’s equation was proposed, most researchers believed that pressure drop is made up of viscous and kinetic losses. Ergun’s equation has been widely adopted to estimate pressure drop through packed bed for Darcy and Forchheimer flow. The application of Ergun’s equation depended on two constants, which have been discussed and improved by many researchers. Based on Ergun’s pressure drop correlation in the literature (Ergun, 1952), many researchers proposed a modified form of Ergun equation, where the Ergun constants were replaced according to their studies. Macdonald (Macdonald, 1979) proposed a correlation with the Ergun constants of 180 and 1.8 for smooth particles, and 180 and 4.0 for rough particles. Avontuur and Geldart (Avontuur and
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277
269
Nomenclature Dp fp fp* L L1 L2 L3 M p Dp R Re Rep
particle diameter, m particle friction factor, (Dp)(Dp/L)/(qU2)(e3/(1e)) particle friction factor modified with compressibility, (Dp)(Dp/L)/(qU2)(e3/(1-e))(p1 + p0)/(2 p1) bed length, m length of inlet block, m length of packed channel, m length of outlet block, m mass flow rate, kg/s pressure, Pa pressure drop, Pa molar gas constant, J/(molK) Reynolds number, qU Dp/l particle Reynolds number, qU Dp/l/(1e)
Geldart, 1996) proposed the fixed value of 141 and 1.51. Yu (Yu et al., 2002) and Aerov and Tojec (Aerov and Tojec, 1968) also did the same works to obtain different Ergun constants. Ali (Ali et al., 2014) experimentally studied helium flow characteristics in a rectangular pebble bed channel packed particles with mixed diameters. The results showed that Ergun’s equation agrees well with the experimental data. Some works have shown that the tube-to-particle diameter ratio (D/Dp) has an important influence on flow characteristics of the packed bed. The effect of D/Dp has been ‘‘wall effect”. Foumeny (Foumeny et al., 1993) took the effects of confining wall into account and proposed a modified form. The results showed that the published correlations could not be expected to obtain reliable pressure drop, particularly for packed bed with D/Dp less than 10. Mottllet (Montillet et al., 2007) concluded other experimental data and proposed a correlation, where the tube-to-particle diameter ratio was considered. With the development of computer technology, computational fluid dynamics (CFD) have been widely used to simulate fluid flow characteristics of packed pebble bed. Xu (Xu and Jiang, 2008) presented numerical simulations with structured pebble bed models to predict flow resistance characteristics, and to provide more details of flow in microporous media. The results showed that the effects of rarefaction on air flow should be took into account in the conditions when the particle diameters is less than 90 lm. Yang (Yang and Wang, 2010, 2012) carried out experimentally and numerically more comprehensive studies on the flow and heat transfer characteristics of packed bed with SC, BCC, and FCC arrangements. The 1/4 channel model was adopted with air as the working fluid, and pressure drop and heat transfer correlations were obtained. Both the experimental and numerical results showed that the inertial resistance coefficients of three kinds of packed forms are smaller than that of the randomly packed bed. Li (Li et al., 2013) extended the packed bed with BCC arrangements to 11 layers, and compared the resistance characteristics in the single channel, four channels and nine channels models, respectively. The results showed that in the range of Re = 225–3750, the sizes of these bed have little effect on the calculated values of the resistance coefficient. The sizes of these bed can represent the packed bed with BCC arrangements to analyze the flow resistance characteristics, and the axial distribution of pressure showed that the flow is fully developed at the third layer. In the actual packed bed, there were a large number of sharp corners between the ball and the ball, and the ball and the wall. These extremely narrow regions bring great difficulty to the grid generation. Therefore, many researchers adopted different meth-
S T U u
area of the radial cross section, m2 temperature, K superficial mean fluid velocity, m/s flow velocity, m/s
Greek letters e bed porosity l fluid dynamic viscosity, Pas q fluid density, kg/m3 Subscripts 0 outlet block of packed channel 1 inlet block of packed channel
ods to deal with point contact, including the gap, the overlap, the bridge and the cap methods. Bu (Bum and Yang, 2014) studied the effect of different bridge diameters on flow and heat transfer characteristics. The results showed that bridge method is the best way to predict the flow and heat characteristics of packed bed. In addition, the bridge diameters ranging from 16% Dp to 20% Dp are recommended for simulations. In recent decades, many investigations on flow characteristics in the packed bed channel were based on water or air, but few works were based on helium. In addition, the compressibility of the working fluid was neglected in the majority of studies. However, helium is a type of compressible gas, and the compressibility of helium has an important influence on flow characteristics. Taking the effect of compressibility of helium into account, this paper experimentally and numerically studied helium flow characteristics in a rectangular pebble bed channel.
2. Experimental study on flow characteristics in pebble bed 2.1. Experimental system and data processing The experimental loop is designed as shown in Fig. 1, including four helium tanks connected with parallel method, pressure reducing valve, a rectangular experimental section filled with spherical pebbles, vacuum pump and data acquisition system. The experimental section with the dimensions of 20 mm 20 mm 500 mm is designed. Five kinds of stainless steel particles with the diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are filled into the test section, respectively. Five pressure ports are fixed in 50 mm, 150 mm, 250 mm, 350 mm, and 450 mm along the flow direction, respectively. This experimental system is an open loop, and the helium is not reused during the experiment process. The helium cylinder provides driving force for the flow of helium in the loop. The helium is passed through the pressure reducing valve and the flow meter into the experimental section, and finally released into the atmosphere to complete the entire flow process. Before the start of the experiment, the air in the experiment system must be exhausted by the vacuum pump to ensure that only pure helium is present in the experimental system during the experiment process. In this experiment, the gas heater is not used, and may be used in the future to study the helium flow characteristics at high temperatures. The inlet velocity of experimental section cannot be directly measured, and the mass flow rate is converted according to the
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Data acquisition system
Atmosphere T2
T3
P2
Vacuum pump
P3
Check valve Atmosphere
Test section T1
Gas heater
P1
Gas flowmeter Helium reducing valve
Helium tank
Fig. 1. Experimental apparatus.
flow meter. The NIST software is used here. In the experimental data processing, the influence of the inlet block and the outlet block needs to be considered. Therefore, only the pressure drop between 50 and 450 mm in the experimental section is selected as the data processing object. In the experimental process, the superficial mean fluid velocity, pressure drop along the main flow direction, bed porosity are obtained for data processing. The form of Ergun correlation is well-known. To simplify this form, some non-dimensional parameters are defined as follows. The particle Reynolds number with the porosity is defined as follow:
Rep ¼
qUDp lð1 eÞ
ð2Þ
The particle friction factor with the porosity is defined as follow:
fp ¼
Dp Dp 2
qU L
e3
1e
ð3Þ
where, e is the porosity; Dp is the particle diameter; q is the fluid density; L is the test section length; U is the superficial mean fluid velocity. Helium is a type of compressible gas and the compressibility of helium should be considered. Li (Li and Zhang, 2003) has given theoretical derivation to modify the compressibility of gas. In order to establish a reasonable model, some assumptions are made. 1) 2) 3) 4)
Steady flow; Flow pressure drop satisfies the form of Ergun’s equation; Isothermal flow; Ideal gas; 2
dp lð1 eÞ qð1 eÞ 2 ¼ A UB 3 U 2 3 dx e Dp e Dp
M ¼ qUS p ¼ qRT
)U¼
MRT pS
ð4Þ
ð5Þ
" # dp 1 lð1 eÞ2 ð1 eÞ ¼ p1 U 1 A þ Bq1 U 1 3 dx p e Dp e3 D2p
ð6Þ
where, A and B are the Ergun constants, respectively; S is area of the radial cross section; R is the molar gas constant; M is the mass flow; p1 is the pressure of inlet; U1 is the velocity of inlet; q1 is the density of inlet. For a steady flow, M is a constant. Boundary condition:
x ¼ 0; p ¼ p1
ð7Þ
x ¼ L; p ¼ p0
The pressure drop between inlet and outlet of the experimental section can be obtained. Where, p0 is the outlet pressure, and p1 is the inlet pressure.
"
p20
¼
p21
#
lð1 eÞ2 ð1 eÞ 2p1 U 1 A þ Bq 1 U 1 3 L e Dp e3 D2p
p21 p20 lð1 eÞ2 ð1 eÞ U1 þ B 3 q U2 ¼A 2p1 L e Dp 1 1 e3 D2p
ð8Þ
ð9Þ
Thus, a new non-dimensional parameter is defined, which is known as the particle friction factor modified with compressibility.
f p ¼
Dp D p 2
qU L
e3
1e
p1 þ p0 p þ p0 ¼ fp 1 2p1 2p1
ð10Þ
The well-known correlation used to predict pressure drop through a packed bed was proposed by Ergun for incompressible flow. Thus, Ergun equation should be modified with compressibility for a compressible gas. The modified Ergun equation is defined as follow:
" # DP 2p1 lU ð1 eÞ2 qU 2 ð1 eÞ ¼ 150 þ 1:75 L Dp e3 p1 þ p0 D2p e3
ð11Þ
The uncertainty analysis is performed on the experimental results through the approach of Kline (Kline and McClintock, 1953). The uncertainties of the measuring parameters are listed in Table 1.
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277 Table 1 The uncertainties of measuring parameters. Measuring parameters
Range
Uncertainty(%)
Particle diameter Mass flow rate Absolute pressure Differential pressure
0.5–2 mm 0–0.0002 kg/s 0–200 kPa 0–6 kPa 0–60 kPa 0–200 kPa 0–10 kg
±0.5% ±0.75% ±1.0% ±0.7%
Mass
±1.0%
In the method, a dependent variable, named R, which has independent linear parameters, like R = R (u1, u2. . .un). Thus, the uncertainty of R can be calculated by Eq(1). Moreover, for all the parameters which is collected by system of data acquisition, then the total uncertainties of the experimental results are listed in Table 2.
dR ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 du1 du2 dun þ þ ::: þ u1 u2 un
ð1Þ
2.2. Results and discussion Porosity is a key parameter and has an important influence on the flow characteristics in the pebble bed. In this paper, the porosity measurement is done by weighing. Firstly, the spherical particles with different diameters are randomly sampled, respectively. Secondly, the number of samples and the total mass of the samples are recorded to obtain the density of the spherical particles with different diameters, respectively. Finally, the spherical particles are packed into the experimental section. The weights of the experimental sections before and after packing are recorded to obtain the total volume of the packed particles, and the porosity is obtained in each channel. The particle parameters are listed in Table 3. For the randomly packed bed, the tube-to-particle diameter ratio has a certain relationship with the porosity. In this study, the porosity tendency with D/Dp was compared with the existing experimental investigations (Foumeny et al., 1993) (Fand and Thinakaran, 1990; Beavers et al., 1973; Chu and Ng, 1989; Sato et al., 1974; Zou and Yu, 1995). Fig. 2 shows that with the increase of D/Dp, the porosity decreases. In addition, the porosity tendency agrees with the existing investigations, which verify the reliability of the measurement method. Superficial velocity and particle diameter have an important influence on flow characteristics through spherical pebble bed. In this study, the effects of these parameters on low pressure helium flow characteristics were experimentally investigated. Five rectangular pebble bed channels are filled with particles, whose diameters are 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm, respectively. As shown in Fig. 3, the relationship between pressure drop per meter and the particle Reynolds numbers (Rep) are described. The results show that pressure drop per meter is a function of the particle Reynolds numbers (Rep), and that pressure drop increases with the increase of the particle Reynolds numbers (Rep), and that decreases with the increase of particle diameter. As shown in Figs. 3 and 4, the experimental values are compared with the values calculated by Ergun equation and Ergun equation
Table 2 The uncertainties of experimental results. Parameters
Particle Reynolds number (Ref)
Particle friction factor (f)
Uncertainty
1.95%
2.07%
271
modified with compressibility, respectively. The results show that Ergun equation can agree well with the experimental values in the region with low particle Reynolds numbers (Rep), and show that it cannot agree well with the experimental values in the region with large particle Reynolds numbers (Rep). The deviation of prediction increases with the increase of the particle Reynolds numbers (Rep). Ergun’s correlation were obtained based on water, which can be considered as an incompressible fluid. For compressible gases, Ergun correlation is not suitable when the ratio of pressure drop to system pressure is large enough. In condition with low particle Reynolds numbers, the ratio of pressure drop to system pressure is so small that the compressibility can be neglected. In condition with large particle Reynolds numbers, the ratio of pressure drop to system pressure is so large that the compressibility can’t be neglected. As shown in Fig. 4, Ergun equation modified with compressibility can improve the ability to predict experimental values. Fig. 5 shows the effects of the compressibility on the particle friction factor with the particle diameters of 0.5 mm and 2.0 mm, respectively. The results show that the compressibility has some impact on the particle friction factor with the particle diameter of 0.5 mm, and that the effects of the compressibility on the particle friction factor with the particle diameter of 2.0 mm can be neglected. The reason is that pressure drop with the particle diameter of 0.5 mm is greater than that of 2.0 mm. It is concluded that the greater pressure drop is, the greater the effects of compressibility on the particle friction factor are. It is easy to be understood that the density decreases with the decrease of pressure for a compressible gas. As shown in Fig. 6, pressure decreases along the helium flow direction, leading to the decrease of the density of helium. According to the conservation of mass, the velocity increases with the decrease of the density, which lead to the increase of pressure drop. Compared with the formula (3) and formula (10), fp and fp* can be obtained. Fig. 7 shows the relative error between fp and fp* with the particle diameters of 0.5 mm and 2.0 mm, respectively. The results show that the relative error increases with the Rep. The reason is that the greater the Rep is, and the greater the ratio of pressure drop to system pressure is. It is obvious that the relative error is close to 20% with the particle diameter of 0.5 mm, and the relative error is less than 2% with the particle diameter of 2.0 mm. Under the condition with the same Rep, the smaller Dp is, and the greater pressure drop is. The greater pressure drop means that the ratio of pressure drop to system pressure is larger, and that the effect of compressibility is larger. Thus, the relative error of the particle friction factors before and after considering the compressibility is larger when Dp is smaller.
3. Numerical simulation of helium flow in pebble bed 3.1. Physical model As is shown in Fig. 8, three kinds of structured packed cells, including simple cubic (SC-1), body center cubic (BCC-1) and face center cubic arrangements (FCC-1), are selected as representative forms to investigate helium flow resistance of the randomly packed bed. A full-scale simulation of the entire packed bed with all the particles is very difficult because of the huge calculations. Therefore, based on three types of structured packed cells, different packed channels are selected as computational domain of fluid flow. As shown in Fig. 9, the packed channels are connected by 8 packed cells. As shown in Fig. 10, the inlet and outlet blocks are considered in order to guarantee fully developed flow. The length of the inlet block was set to 4 Dp and the length of outlet block was set to 7 Dp in this paper. As shown in Fig. 10, a typical point modification with bridges treatment is selected. According to the
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Table 3 Particle parameters. Particle diameter Dp(mm)
0.5
0.8
1.0
1.5
2.0
Porosity e (%)
0.3698
0.3811
0.3735
0.3711
0.3768
160
0.60 Sato Zou and Yu Beaver Foumeny Fand and Thinkaran Chu Current work
0.54 0.51 0.48
120 100
0.45
80 60
0.42
40
0.39
20
0.36
0
0
5
10
15
20
25
30
35
Experimetal Value(Dp=0.5mm) Modified Ergun Equation(Dp=0.5mm) Experimetal Value(Dp=0.8mm) Modified Ergun Equation(Dp=0.8mm) Experimetal Value(Dp=1.0mm) Modified Ergun Equation(Dp=1.0mm) Experimetal Value(Dp=1.5mm) Modified Ergun Equation(Dp=1.5mm) Experimetal Value(Dp=2.0mm) Modified Ergun Equation(Dp=2.0mm)
140
/L(KPa)
0.57
40
0
5
10 15 20 25 30 35 40 45 50 55 60
Rep
D/Dp Fig. 2. Comparison of the current work and the existing works in the tendency of porosity.
160
Continuity equation: Experimetal Value(Dp=0.5mm) Ergun Equation(Dp=0.5mm) Experimetal Value(Dp=0.8mm) Ergun Equation(Dp=0.8mm) Experimetal Value(Dp=1.0mm) Ergun Equation(Dp=1.0mm) Experimetal Value(Dp=1.5mm) Ergun Equation(Dp=1.5mm) Experimetal Value(Dp=2.0mm) Ergun Equation(Dp=2.0mm)
140 120
/L(KPa)
Fig. 4. Comparison of experimental values and the values calculated by modified Ergun correlations with different particle diameters.
100 80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
50
55
60
Rep Fig. 3. Comparison of experimental values and the values calculated by Ergun correlations with different particle diameters.
r! u ¼0
ð12Þ
Momentum equation:
! ! q ! u r u ¼ rp þ r lr u
ð13Þ
Where, u is the flow velocity; p is the pressure; l is the viscosity. q is the density. In the present calculations, the laminar model is adopted according the experimental results. FLUENT code adopts the SIMPLE velocity-pressure coupling algorithm, standard discretization for the pressure and second-order discretization for advection terms. Symmetric boundary conditions are adopted on the boundaries of the fluid flow. The inlet and outlet of computational domain are set as velocity inlet boundary and pressure outlet boundary, respectively. In Fluent code, the governing equations are discretized using the finite volume method. To obtain more accurate results, the convergence criterion is set to be 10-5 for all the variables. 3.3. Grid independence test
paper of Bu (Bum and Yang, 2014), the bridge diameters ranging from 16% Dp to 20% Dp were recommended for simulation. As shown in Fig. 11 16% Dp is selected as bridge diameter in this paper. Five kinds of particles with the diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are investigated, and the geometrical parameters are listed in Table 4, respectively.
The grid independence is tested to obtain higher order accuracy results. As shown in Table 5, the computational results are compared with different number of grid. The Richardson extrapolation approach (Roache, 1997) is adopted to test grid independence. The Richardson extrapolation is defined as follow:
/R ¼ /F þ 3.2. Governing equations and computational method The flow process through the structured packed bed is treated to be three-dimensional, laminar and incompressible, because the calculation region is short and the Reynolds number is very small.
/F /c r 2g 1
ð14Þ
where, / is the pressure drop; and Subscripts of F and C is the fine grid and the coarse grid, respectively; rg = (NF/NC); N is the number of grid. To order to ensure both the economical efficiency and the accuracy, the grid numbers with 0.94, 1.79, and 3.15 million are selected, respectively.
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70 160
60
Experimental Value fp
140
Experimental Value fp*
120
40
fp
100
fp
Experimental Value fp Experimental Value fp*
50
80
30
60
20
40
10 20
0
0 0
2
4
6
8
10
Rep
12
0
14
5
10
(a) Dp=0.5mm
15
20
Rep
25
30
35
(b) Dp=2.0mm
Fig. 5. Comparison of fp and fp* with the particle diameters of 0.5 mm and 2.0 mm.
3.4. Results and discussion 3.4.1. The effects of particle diameter on pressure drop As a geometric parameter of packed bed, particle diameter has an important influence on flow characteristics. In this paper, the packed form with SC-1 arrangements is used as an analysis object to investigate the effects of particle diameter on pressure drop. The calculated results with the different particle diameters are described in Fig. 12. It is clear that pressure drop per meter decreases with the increase of particle diameter. In addition, the effect of particle diameter on pressure drop is nonlinear.
70 60
Dp=0.5mm Dp=0.8mm Dp=1.0mm Dp=1.5mm Dp=2.0mm
50
kPa
40 30 20
3.4.2. The effects of packed forms on pressure drop and the particle fiction factor As shown in Fig. 13, the effects of packed forms on pressure drop are described. The results show that pressure drop with FCC-1 arrangements is the largest and pressure drop with SC-1 arrangements is the smallest. Velocity distributions of three packed forms with the same inlet Reynolds numbers (Re) are described in Fig. 14. The results show that the maximum velocity in the channel with FCC-1 arrangements is about 1.37 m/s, and show that the maximum velocity in the channel with SC-1 arrangements is about 0.96 m/s. The porosities with SC-1, BCC-1, and
10 0 100
150
200
250
300
350
400
Channel length mm Fig. 6. Effect of the channel length on the pressure drop under Rep = 12.5.
20
15
the relative error(%)
the relative error(%)
2.0
10
5
0
1.5
1.0
0.5
0.0
0
2
4
6
8
Rep
10
(a) Dp=0.5mm
12
14
0
5
10
15
20
25
Rep
(b) Dp=2.0mm
Fig. 7. The relative error between fp and fp* with the particle diameters of 0.5 mm and 2.0 mm.
30
35
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50 SC-1(Dp=0.5mm) SC-1(Dp=0.8mm) SC-1(Dp=1.0mm) SC-1(Dp=1.5mm) SC-1(Dp=2.0mm)
(a) SC-1
(b)BCC-1
p/L(kPa·m -1)
40
(c)FCC-1
Fig. 8. Different packed cells of spheres.
30
20
10
0 (a) The channel with SC-1 arrangements
0
10
20
30
40
Rep (b) The channel with BCC-1 arrangements
Fig. 12. The effects of particle diameter on pressure drop.
60
(c) The channel with FCC-1 arrangements
SC-1 BCC-1 FCC-1
Inlet
Outlet
Inlet Section
Outlet Section
Heium
p0
T1,U1,p1 L1
L2
p/L(kPa · s-1)
Fig. 9. Different packed pebble channels.
40
20
L3
Fig. 10. The geometric diagram of the packed channels.
0 0
5
10
15
20
25
30
35
40
Rep 0.16Dp
Fig. 13. The effect of packed forms on pressure drop.
Fig. 11. A typical point modifications with bridges treatment.
Table 4 Geometrical parameters. Packed form
SC-1
BCC-1
FCC-1
Porosity(e)
0.476
0.320
0.259
FCC-1 arrangements are 0.476, 0.320, and 0.259, respectively. According to Ergun equation, it is concluded that pressure drop is negatively correlated with the porosity. The smaller porosity causes the narrower flow area, and results in greater velocity in bed. Thus, pressure drop with FCC-1 arrangements is the largest and pressure drop with SC-1 arrangements is the smallest. The relationship of the particle friction factors (fp*) and the particle Reynolds numbers (Rep) are shown in Fig. 15. The results show that the variation of the particle friction factors of the packed forms
Table 5 Mesh parameters. The number of grid (SC-1) SC-1(Dp/Pa) Richardson extrapolation (SC-1) The relative error The number of grid (BCC-1) BCC-1(Dp/Pa) Richardson extrapolation (BCC-1) The relative error The number of grid (FCC-1) FCC-1(Dp/Pa) Richardson extrapolation (FCC-1) The relative error
522,119 11.32 12.34 8.26% 502,130 61.48 72.27 14.93% 1,510,155 209.78 247.97 15.40%
722,665 11.29
941,561 11.95
2,124,244 12.19
8.51% 1,079,336 67.97
3.16% 1,787,706 69.70
1.21% 4,899,842 71.93
5.95% 2,027,138 228.74
3.56% 3,157,087 235.48
0.47% 6,003,352 244.51
7.75%
5.03%
1.39%
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277
275
(a)SC-1
(b)BCC-1
(c)FCC-1 Fig. 14. Velocity distributions in different packed bed with the same inlet Re.
Since the packed form with SC-1 arrangements is the simplest, the packed form with SC-1 arrangements is selected to calculate the particle friction factors in next calculation.
300 250
Numerical values with SC-1 Numerical values with BCC-1 Numerical values with FCC-1
fp *
200 150 100 50 0 0
2
4
6
Rep
8
10
12
14
Fig. 15. The effects of packed forms on the particle friction factor.
with SC-1, BCC-1, and FCC-1 arrangements are similar with the particle Reynolds numbers. The differences between the numerical results with SC-1, BCC-1, and FCC-1 arrangements are less than 10% so that any packed form may be used to calculate the particle friction factor. The channels with SC-1, BCC-1, and FCC-1 arrangements have different velocity and pressure drop distributions at the same inlet velocity, which is mainly due to the difference of porosity. In this paper, Rep is the particle Reynolds number, which is a dimensionless number considering both porosity and flow velocity. Moreover, the particle friction factors also consider the effect of porosity. Therefore, the curve tendency of Rep and fp is basically the same.
3.4.3. The prediction to the particle friction factor in the experiment The actual packed form is random in this experiment. The difference between the randomly packed bed and the structured packed bed is mainly porosity. To obtain a new dimensionless parameter including porosity, the particle friction factor modified by compressibility (fp*) is defined in formula of (10) according to Ergun equation. The effects of velocity and particle diameter on the particle friction factors modified with compressibility (fp*) are investigated. Fig. 16 compares the calculated results with experimental results. As shown in Fig. 16, five types of particle diameters with 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are compared, respectively. It is obvious that the calculated results of all particle diameters agree well with experimental results. Thus, it is concluded that the particle friction factors modified with compressibility in this experiment with randomly packed form can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements.
4. Conclusions In this paper, the low pressure helium flow characteristics in a rectangular pebble bed channel filled with stainless steel particles with 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm were investigated experimentally. Three-dimensional numerical studies are carried out using three structured packed bed of spheres to predict flow performances of the experiment with randomly packed bed. Based on the studies, some findings have been obtained as follows:
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120
200
100
Experimental values Numerical values
Experimental values Numerical values
80
fp*
fp *
150
100
60 40
50
20 0
0 0
2
4
6
Rep
8
10
12
0
14
2
4
6
8
10
12
14
16
18
Rep
(a) 0.5mm
(b) 0.8mm 60
80
Experimental values Numerical values 40
fp *
fp*
60
Experimental values Numerical values
40
20
20
0
0
0
5
10
15
20
25
0
Rep
5
10
15
20
25
30
Rep
(c) 1.0mm
(d) 1.5mm
70 60
Experimental values Numerical values
50
fp*
40 30 20 10 0 0
5
10
15
20
25
30
35
40
Rep
(e) 2.0mm Fig. 16. Comparison of the experimental values and the numerical values with different particle diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm.
(1) The experimental results show that Ergun equation can agree well with the experimental results in the region with low particle Reynolds numbers, and show a bad prediction in the region with large particle Reynolds numbers. The
deviation of prediction increases with the increase of the particle Reynolds numbers. In addition, Ergun equation modified with compressibility can improve the ability to predict experimental values.
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277
(2) It can be concluded that the compressibility has an important influence on flow characteristics in the condition with the large ratio of pressure drop to system pressure. In addition, the effects of the compressibility increase with the particle Reynolds numbers (Rep). The relative error between fp and fp* is close to 20% with the particle diameter of 0.5 mm, and the relative error is less than 2% with the particle diameter of 2.0 mm. (3) The numerical results with the same Reynolds numbers, show that the pressure drop with FCC-1 arrangement is the largest, and that with SC-1 arrangement is the smallest, and that the maximum velocity with FCC-1 is greater than that with SC-1 arrangements. Furthermore, the numerical results show that the particle friction factor modified with compressibility (fp*) in this experiment with randomly packed form can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements. (4) This work provides an approach for the flow characteristics simulation of the randomly packed bed.
Acknowledgement This work was supported by a grant from the evaluation and research on the thermal-hydraulic design and safety analysis of Chinese fusion engineering test reactor (CFETR) test blanket module (No. 2014 GB114000). References Aerov, M.E., Tojec, O.M., 1968. Hydraulic and Thermal Basis on the performance of Apparatus with Stationary and Boiling Granular Layer. Himia Press, Leningrad. Ali, A., Frederik, A., Lorenzo, V.B., Georg, S., 2014. Measurements of the purge helium pressure drop across pebble bed packed with lithium orthosilicate and glass pebbles. Fusion Eng. Des. 89, 1459–1463. Avontuur, P.P.C., Geldart, D., 1996. A quality assessment of the Ergun equation, in: The 1996 IChemE Research Event/Second European Conference for Young Researchers, pp. 994–996.
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Beavers, G.S., Sparrow, E.M., Rodenz, D.E., 1973. Influence of Bed Size on the Flow Characteristics and Porosity of Randomly Packed Bed of Spheres. J. Appl. Mech. 40 (3), 655. Bum, S., Yang, J., et al., 2014. On contact point modifications for forced convective heat transfer analysis in a structured packed bed of spheres. Nuclear. Eng. Des. 270, 21–33. Chu, C.F., Ng, K.M., 1989. Flow in packed tubes with a small tube to particle diameter ratio. AIChE J., 148–158 Ergun, S., 1952. Fluid flow through packed columns. Chem. Eng. Pro. 48, 89–94. Fand, R.M., Thinakaran, R., 1990. The influence of the wall on flow through pipes packed with spheres [J]. J. Fluids Eng.-Trans. ASME 112 (1), 84–88. Foumeny, E.A., Benyahia, F., Castro, J.A.A., Moallemi, H.A., Roshani, S., 1993. Correlations of Pressure Drop in Packed Bed Taking into Account the Effect of Confining Wall. Int. J. Heat Mass Transf. 36 (2), 536–540. Kline, S.J., McClintock, F.A., 1953. Describing uncertainties in single-sample experiments. Mech. Eng. 75, 3–8. Li, M., Chen, H., Zhou, G., Liu, Q., Wang, S., Lv, Z., et al., 2015. Preliminary structural design and thermo-mechanical analysis of helium cooled solid breeder blanket for Chinese Fusion Engineering Test Reactor. Fusion Eng. Des. 91, 39–43. Li, J., Song, X., Lu, J., Li, Z., 2013. Scaling effect on flow characteristics of a structured packed bed using CFD method. Nucl. Power Eng. 34, 25–29. Li, H., Zhang, X., 2003. Analysis of non-darcy steady compressible gas flow in porous media and its application. Chinese J. Appl. Mech. 20, 1–4. Macdonald, I.F., EI-Sayed, M.S., Dullien, F.A.L., 1979. Flow through porous media— the Ergun equation revisited. Ind. Eng. Chem. Fundam. 18, 199–208. Montillet, A., Akkari, E., Comiti, J., 2007. About a correlating equation for predicting pressure drops through packed bed of spheres in a large range of Reynolds numbers. Chem. Eng. Process. Process Intensif. 46, 329–333. Roache, P.J., 1997. Quantification of uncertainty in computational fluid dynamics. Annual Rev. Fluid Mech. 29, 123–160. Sato, Y., Hirose, T., Takahashi, F., Toda, M., 1974. Pressure loss and liquid holdup in packed bed reactor with cocurrent gas-liquid down flow. J. Chem. Eng. Japan 6 (2), 147–152. Xu, R., Jiang, P., 2008. Numerical simulation of fluid flow in microporous media. Int. J. Heat Fluid Flow 29, 1447–1455. Yang, J., Wang, Q., et al., 2010. Computational study of forced convective heat transfer in structured packed bed with spherical or ellipsoidal particles. Chem. Eng. Sci. 65, 726–738. Yang, J., Wang, J., et al., 2012. Experimental analysis of forced convective heat transfer in novel structured packed bed of particles. Chem. Eng. Sci. 71, 126– 137. Yu, J., Zhang, M., Fan, W., Zhou, Y., Zhao, G., 2002. Study on performance of the ball packed-bed regenerator: experiments and simulation. Appl. Therm. Eng. 22, 641–651. Zou, R.P., Yu, A.B., 1995. The packing of spheres in a cylindrical container: the thickness effect. Chem. Eng. Sci. 50 (9), 1504–1507.
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Experimental and numerical study on helium flow characteristics in randomly packed pebble bed Zhenxing Wu, Yingwei Wu ⇑, Chenglong Wang ⇑, Simiao Tang, Di Liu, Suizheng Qiu, G.H. Su, Wenxi Tian Department of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China Shanxi Key Lab. of Advanced Nuclear Energy and Technology, Xi’an Jiaotong University, China
a r t i c l e
i n f o
Article history: Received 19 March 2018 Received in revised form 10 January 2019 Accepted 11 January 2019
Keywords: Helium flow characteristics Spherical pebble bed Experimental study Numerical study
a b s t r a c t The high temperature gas cooled reactor (HTGR) with a packed pebble bed core has been one of the fourth generation reactors. The helium cooled pebble bed (HCPB) blanket packed spherical pebbles of lithium orthosilicate (Li4SiO4) has been recommended to Chinese Fusion Engineering Test Reactors (CFETR). Helium flow characteristics in spherical pebble bed are the key issues to design the pumping power. In this work, helium flow characteristics in a rectangular pebble bed channel filled with stainless steel particles whose diameters are 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm, respectively, were investigated experimentally and numerically. The experimental results show that the compressibility cannot be neglected and it has an important influence on flow characteristics in the conditions with the enough large ratio of pressure drop to system pressure. Ergun correlation was compared with experimental values, and Ergun correlation modified with compressibility was deduced. The results show that Ergun correlation modified with compressibility can improve the ability to predict experimental values. The numerical results show that the particle friction factors modified with compressibility (fp*) in a rectangular pebble bed channel randomly filled with spherical particles can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements. This work would provide some supports for the design of the helium supply system of the packed bed reactors. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Spherical pebble bed have been widely used in some reactors, such as high temperature gas cooled reactor (HTGR), pebble bed water cooled reactor (PBWR), and helium cooled pebble bed (HCPB) blanket of CFETR. HTGR with a packed bed core has been one of the fourth generation reactors due to the higher inherent safety. The energy generated in a HTGR core is brought by helium from the core into the heat exchanger. In addition, the helium cooled pebble bed blanket packed spherical pebbles of lithium orthosilicate (Li4SiO4) was proposed for Chinese Fusion Engineering Test Reactors (CFETR) (Li et al., 2015). Lithium orthosilicate (Li4SiO4) pebble bed has been selected as tritium breeder in many test blanket module (TBM) designs. The bred tritium is transported by low pressure helium from pebble bed to the tritium extraction system. Thus, helium pressure drop through the randomly packed pebble bed is a key parameter to design helium supply system. ⇑ Corresponding authors. E-mail addresses: [email protected] (Y. Wu), [email protected] (C. Wang). https://doi.org/10.1016/j.anucene.2019.01.016 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
In recent decades, there were many experimental investigations on fluid flow in the packed bed, but their research subjects were either water or air. The well-known correlation used to predict pressure drop through packed bed was proposed by Ergun (Ergun, 1952) for incompressible flow:
DP lU ð1 eÞ2 qU 2 ð1 eÞ þ 1:75 ¼ 150 2 3 L Dp e3 Dp e
ð1Þ
After Ergun’s equation was proposed, most researchers believed that pressure drop is made up of viscous and kinetic losses. Ergun’s equation has been widely adopted to estimate pressure drop through packed bed for Darcy and Forchheimer flow. The application of Ergun’s equation depended on two constants, which have been discussed and improved by many researchers. Based on Ergun’s pressure drop correlation in the literature (Ergun, 1952), many researchers proposed a modified form of Ergun equation, where the Ergun constants were replaced according to their studies. Macdonald (Macdonald, 1979) proposed a correlation with the Ergun constants of 180 and 1.8 for smooth particles, and 180 and 4.0 for rough particles. Avontuur and Geldart (Avontuur and
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277
269
Nomenclature Dp fp fp* L L1 L2 L3 M p Dp R Re Rep
particle diameter, m particle friction factor, (Dp)(Dp/L)/(qU2)(e3/(1e)) particle friction factor modified with compressibility, (Dp)(Dp/L)/(qU2)(e3/(1-e))(p1 + p0)/(2 p1) bed length, m length of inlet block, m length of packed channel, m length of outlet block, m mass flow rate, kg/s pressure, Pa pressure drop, Pa molar gas constant, J/(molK) Reynolds number, qU Dp/l particle Reynolds number, qU Dp/l/(1e)
Geldart, 1996) proposed the fixed value of 141 and 1.51. Yu (Yu et al., 2002) and Aerov and Tojec (Aerov and Tojec, 1968) also did the same works to obtain different Ergun constants. Ali (Ali et al., 2014) experimentally studied helium flow characteristics in a rectangular pebble bed channel packed particles with mixed diameters. The results showed that Ergun’s equation agrees well with the experimental data. Some works have shown that the tube-to-particle diameter ratio (D/Dp) has an important influence on flow characteristics of the packed bed. The effect of D/Dp has been ‘‘wall effect”. Foumeny (Foumeny et al., 1993) took the effects of confining wall into account and proposed a modified form. The results showed that the published correlations could not be expected to obtain reliable pressure drop, particularly for packed bed with D/Dp less than 10. Mottllet (Montillet et al., 2007) concluded other experimental data and proposed a correlation, where the tube-to-particle diameter ratio was considered. With the development of computer technology, computational fluid dynamics (CFD) have been widely used to simulate fluid flow characteristics of packed pebble bed. Xu (Xu and Jiang, 2008) presented numerical simulations with structured pebble bed models to predict flow resistance characteristics, and to provide more details of flow in microporous media. The results showed that the effects of rarefaction on air flow should be took into account in the conditions when the particle diameters is less than 90 lm. Yang (Yang and Wang, 2010, 2012) carried out experimentally and numerically more comprehensive studies on the flow and heat transfer characteristics of packed bed with SC, BCC, and FCC arrangements. The 1/4 channel model was adopted with air as the working fluid, and pressure drop and heat transfer correlations were obtained. Both the experimental and numerical results showed that the inertial resistance coefficients of three kinds of packed forms are smaller than that of the randomly packed bed. Li (Li et al., 2013) extended the packed bed with BCC arrangements to 11 layers, and compared the resistance characteristics in the single channel, four channels and nine channels models, respectively. The results showed that in the range of Re = 225–3750, the sizes of these bed have little effect on the calculated values of the resistance coefficient. The sizes of these bed can represent the packed bed with BCC arrangements to analyze the flow resistance characteristics, and the axial distribution of pressure showed that the flow is fully developed at the third layer. In the actual packed bed, there were a large number of sharp corners between the ball and the ball, and the ball and the wall. These extremely narrow regions bring great difficulty to the grid generation. Therefore, many researchers adopted different meth-
S T U u
area of the radial cross section, m2 temperature, K superficial mean fluid velocity, m/s flow velocity, m/s
Greek letters e bed porosity l fluid dynamic viscosity, Pas q fluid density, kg/m3 Subscripts 0 outlet block of packed channel 1 inlet block of packed channel
ods to deal with point contact, including the gap, the overlap, the bridge and the cap methods. Bu (Bum and Yang, 2014) studied the effect of different bridge diameters on flow and heat transfer characteristics. The results showed that bridge method is the best way to predict the flow and heat characteristics of packed bed. In addition, the bridge diameters ranging from 16% Dp to 20% Dp are recommended for simulations. In recent decades, many investigations on flow characteristics in the packed bed channel were based on water or air, but few works were based on helium. In addition, the compressibility of the working fluid was neglected in the majority of studies. However, helium is a type of compressible gas, and the compressibility of helium has an important influence on flow characteristics. Taking the effect of compressibility of helium into account, this paper experimentally and numerically studied helium flow characteristics in a rectangular pebble bed channel.
2. Experimental study on flow characteristics in pebble bed 2.1. Experimental system and data processing The experimental loop is designed as shown in Fig. 1, including four helium tanks connected with parallel method, pressure reducing valve, a rectangular experimental section filled with spherical pebbles, vacuum pump and data acquisition system. The experimental section with the dimensions of 20 mm 20 mm 500 mm is designed. Five kinds of stainless steel particles with the diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are filled into the test section, respectively. Five pressure ports are fixed in 50 mm, 150 mm, 250 mm, 350 mm, and 450 mm along the flow direction, respectively. This experimental system is an open loop, and the helium is not reused during the experiment process. The helium cylinder provides driving force for the flow of helium in the loop. The helium is passed through the pressure reducing valve and the flow meter into the experimental section, and finally released into the atmosphere to complete the entire flow process. Before the start of the experiment, the air in the experiment system must be exhausted by the vacuum pump to ensure that only pure helium is present in the experimental system during the experiment process. In this experiment, the gas heater is not used, and may be used in the future to study the helium flow characteristics at high temperatures. The inlet velocity of experimental section cannot be directly measured, and the mass flow rate is converted according to the
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Data acquisition system
Atmosphere T2
T3
P2
Vacuum pump
P3
Check valve Atmosphere
Test section T1
Gas heater
P1
Gas flowmeter Helium reducing valve
Helium tank
Fig. 1. Experimental apparatus.
flow meter. The NIST software is used here. In the experimental data processing, the influence of the inlet block and the outlet block needs to be considered. Therefore, only the pressure drop between 50 and 450 mm in the experimental section is selected as the data processing object. In the experimental process, the superficial mean fluid velocity, pressure drop along the main flow direction, bed porosity are obtained for data processing. The form of Ergun correlation is well-known. To simplify this form, some non-dimensional parameters are defined as follows. The particle Reynolds number with the porosity is defined as follow:
Rep ¼
qUDp lð1 eÞ
ð2Þ
The particle friction factor with the porosity is defined as follow:
fp ¼
Dp Dp 2
qU L
e3
1e
ð3Þ
where, e is the porosity; Dp is the particle diameter; q is the fluid density; L is the test section length; U is the superficial mean fluid velocity. Helium is a type of compressible gas and the compressibility of helium should be considered. Li (Li and Zhang, 2003) has given theoretical derivation to modify the compressibility of gas. In order to establish a reasonable model, some assumptions are made. 1) 2) 3) 4)
Steady flow; Flow pressure drop satisfies the form of Ergun’s equation; Isothermal flow; Ideal gas; 2
dp lð1 eÞ qð1 eÞ 2 ¼ A UB 3 U 2 3 dx e Dp e Dp
M ¼ qUS p ¼ qRT
)U¼
MRT pS
ð4Þ
ð5Þ
" # dp 1 lð1 eÞ2 ð1 eÞ ¼ p1 U 1 A þ Bq1 U 1 3 dx p e Dp e3 D2p
ð6Þ
where, A and B are the Ergun constants, respectively; S is area of the radial cross section; R is the molar gas constant; M is the mass flow; p1 is the pressure of inlet; U1 is the velocity of inlet; q1 is the density of inlet. For a steady flow, M is a constant. Boundary condition:
x ¼ 0; p ¼ p1
ð7Þ
x ¼ L; p ¼ p0
The pressure drop between inlet and outlet of the experimental section can be obtained. Where, p0 is the outlet pressure, and p1 is the inlet pressure.
"
p20
¼
p21
#
lð1 eÞ2 ð1 eÞ 2p1 U 1 A þ Bq 1 U 1 3 L e Dp e3 D2p
p21 p20 lð1 eÞ2 ð1 eÞ U1 þ B 3 q U2 ¼A 2p1 L e Dp 1 1 e3 D2p
ð8Þ
ð9Þ
Thus, a new non-dimensional parameter is defined, which is known as the particle friction factor modified with compressibility.
f p ¼
Dp D p 2
qU L
e3
1e
p1 þ p0 p þ p0 ¼ fp 1 2p1 2p1
ð10Þ
The well-known correlation used to predict pressure drop through a packed bed was proposed by Ergun for incompressible flow. Thus, Ergun equation should be modified with compressibility for a compressible gas. The modified Ergun equation is defined as follow:
" # DP 2p1 lU ð1 eÞ2 qU 2 ð1 eÞ ¼ 150 þ 1:75 L Dp e3 p1 þ p0 D2p e3
ð11Þ
The uncertainty analysis is performed on the experimental results through the approach of Kline (Kline and McClintock, 1953). The uncertainties of the measuring parameters are listed in Table 1.
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277 Table 1 The uncertainties of measuring parameters. Measuring parameters
Range
Uncertainty(%)
Particle diameter Mass flow rate Absolute pressure Differential pressure
0.5–2 mm 0–0.0002 kg/s 0–200 kPa 0–6 kPa 0–60 kPa 0–200 kPa 0–10 kg
±0.5% ±0.75% ±1.0% ±0.7%
Mass
±1.0%
In the method, a dependent variable, named R, which has independent linear parameters, like R = R (u1, u2. . .un). Thus, the uncertainty of R can be calculated by Eq(1). Moreover, for all the parameters which is collected by system of data acquisition, then the total uncertainties of the experimental results are listed in Table 2.
dR ¼ R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 du1 du2 dun þ þ ::: þ u1 u2 un
ð1Þ
2.2. Results and discussion Porosity is a key parameter and has an important influence on the flow characteristics in the pebble bed. In this paper, the porosity measurement is done by weighing. Firstly, the spherical particles with different diameters are randomly sampled, respectively. Secondly, the number of samples and the total mass of the samples are recorded to obtain the density of the spherical particles with different diameters, respectively. Finally, the spherical particles are packed into the experimental section. The weights of the experimental sections before and after packing are recorded to obtain the total volume of the packed particles, and the porosity is obtained in each channel. The particle parameters are listed in Table 3. For the randomly packed bed, the tube-to-particle diameter ratio has a certain relationship with the porosity. In this study, the porosity tendency with D/Dp was compared with the existing experimental investigations (Foumeny et al., 1993) (Fand and Thinakaran, 1990; Beavers et al., 1973; Chu and Ng, 1989; Sato et al., 1974; Zou and Yu, 1995). Fig. 2 shows that with the increase of D/Dp, the porosity decreases. In addition, the porosity tendency agrees with the existing investigations, which verify the reliability of the measurement method. Superficial velocity and particle diameter have an important influence on flow characteristics through spherical pebble bed. In this study, the effects of these parameters on low pressure helium flow characteristics were experimentally investigated. Five rectangular pebble bed channels are filled with particles, whose diameters are 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm, respectively. As shown in Fig. 3, the relationship between pressure drop per meter and the particle Reynolds numbers (Rep) are described. The results show that pressure drop per meter is a function of the particle Reynolds numbers (Rep), and that pressure drop increases with the increase of the particle Reynolds numbers (Rep), and that decreases with the increase of particle diameter. As shown in Figs. 3 and 4, the experimental values are compared with the values calculated by Ergun equation and Ergun equation
Table 2 The uncertainties of experimental results. Parameters
Particle Reynolds number (Ref)
Particle friction factor (f)
Uncertainty
1.95%
2.07%
271
modified with compressibility, respectively. The results show that Ergun equation can agree well with the experimental values in the region with low particle Reynolds numbers (Rep), and show that it cannot agree well with the experimental values in the region with large particle Reynolds numbers (Rep). The deviation of prediction increases with the increase of the particle Reynolds numbers (Rep). Ergun’s correlation were obtained based on water, which can be considered as an incompressible fluid. For compressible gases, Ergun correlation is not suitable when the ratio of pressure drop to system pressure is large enough. In condition with low particle Reynolds numbers, the ratio of pressure drop to system pressure is so small that the compressibility can be neglected. In condition with large particle Reynolds numbers, the ratio of pressure drop to system pressure is so large that the compressibility can’t be neglected. As shown in Fig. 4, Ergun equation modified with compressibility can improve the ability to predict experimental values. Fig. 5 shows the effects of the compressibility on the particle friction factor with the particle diameters of 0.5 mm and 2.0 mm, respectively. The results show that the compressibility has some impact on the particle friction factor with the particle diameter of 0.5 mm, and that the effects of the compressibility on the particle friction factor with the particle diameter of 2.0 mm can be neglected. The reason is that pressure drop with the particle diameter of 0.5 mm is greater than that of 2.0 mm. It is concluded that the greater pressure drop is, the greater the effects of compressibility on the particle friction factor are. It is easy to be understood that the density decreases with the decrease of pressure for a compressible gas. As shown in Fig. 6, pressure decreases along the helium flow direction, leading to the decrease of the density of helium. According to the conservation of mass, the velocity increases with the decrease of the density, which lead to the increase of pressure drop. Compared with the formula (3) and formula (10), fp and fp* can be obtained. Fig. 7 shows the relative error between fp and fp* with the particle diameters of 0.5 mm and 2.0 mm, respectively. The results show that the relative error increases with the Rep. The reason is that the greater the Rep is, and the greater the ratio of pressure drop to system pressure is. It is obvious that the relative error is close to 20% with the particle diameter of 0.5 mm, and the relative error is less than 2% with the particle diameter of 2.0 mm. Under the condition with the same Rep, the smaller Dp is, and the greater pressure drop is. The greater pressure drop means that the ratio of pressure drop to system pressure is larger, and that the effect of compressibility is larger. Thus, the relative error of the particle friction factors before and after considering the compressibility is larger when Dp is smaller.
3. Numerical simulation of helium flow in pebble bed 3.1. Physical model As is shown in Fig. 8, three kinds of structured packed cells, including simple cubic (SC-1), body center cubic (BCC-1) and face center cubic arrangements (FCC-1), are selected as representative forms to investigate helium flow resistance of the randomly packed bed. A full-scale simulation of the entire packed bed with all the particles is very difficult because of the huge calculations. Therefore, based on three types of structured packed cells, different packed channels are selected as computational domain of fluid flow. As shown in Fig. 9, the packed channels are connected by 8 packed cells. As shown in Fig. 10, the inlet and outlet blocks are considered in order to guarantee fully developed flow. The length of the inlet block was set to 4 Dp and the length of outlet block was set to 7 Dp in this paper. As shown in Fig. 10, a typical point modification with bridges treatment is selected. According to the
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Table 3 Particle parameters. Particle diameter Dp(mm)
0.5
0.8
1.0
1.5
2.0
Porosity e (%)
0.3698
0.3811
0.3735
0.3711
0.3768
160
0.60 Sato Zou and Yu Beaver Foumeny Fand and Thinkaran Chu Current work
0.54 0.51 0.48
120 100
0.45
80 60
0.42
40
0.39
20
0.36
0
0
5
10
15
20
25
30
35
Experimetal Value(Dp=0.5mm) Modified Ergun Equation(Dp=0.5mm) Experimetal Value(Dp=0.8mm) Modified Ergun Equation(Dp=0.8mm) Experimetal Value(Dp=1.0mm) Modified Ergun Equation(Dp=1.0mm) Experimetal Value(Dp=1.5mm) Modified Ergun Equation(Dp=1.5mm) Experimetal Value(Dp=2.0mm) Modified Ergun Equation(Dp=2.0mm)
140
/L(KPa)
0.57
40
0
5
10 15 20 25 30 35 40 45 50 55 60
Rep
D/Dp Fig. 2. Comparison of the current work and the existing works in the tendency of porosity.
160
Continuity equation: Experimetal Value(Dp=0.5mm) Ergun Equation(Dp=0.5mm) Experimetal Value(Dp=0.8mm) Ergun Equation(Dp=0.8mm) Experimetal Value(Dp=1.0mm) Ergun Equation(Dp=1.0mm) Experimetal Value(Dp=1.5mm) Ergun Equation(Dp=1.5mm) Experimetal Value(Dp=2.0mm) Ergun Equation(Dp=2.0mm)
140 120
/L(KPa)
Fig. 4. Comparison of experimental values and the values calculated by modified Ergun correlations with different particle diameters.
100 80 60 40 20 0 0
5
10
15
20
25
30
35
40
45
50
55
60
Rep Fig. 3. Comparison of experimental values and the values calculated by Ergun correlations with different particle diameters.
r! u ¼0
ð12Þ
Momentum equation:
! ! q ! u r u ¼ rp þ r lr u
ð13Þ
Where, u is the flow velocity; p is the pressure; l is the viscosity. q is the density. In the present calculations, the laminar model is adopted according the experimental results. FLUENT code adopts the SIMPLE velocity-pressure coupling algorithm, standard discretization for the pressure and second-order discretization for advection terms. Symmetric boundary conditions are adopted on the boundaries of the fluid flow. The inlet and outlet of computational domain are set as velocity inlet boundary and pressure outlet boundary, respectively. In Fluent code, the governing equations are discretized using the finite volume method. To obtain more accurate results, the convergence criterion is set to be 10-5 for all the variables. 3.3. Grid independence test
paper of Bu (Bum and Yang, 2014), the bridge diameters ranging from 16% Dp to 20% Dp were recommended for simulation. As shown in Fig. 11 16% Dp is selected as bridge diameter in this paper. Five kinds of particles with the diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are investigated, and the geometrical parameters are listed in Table 4, respectively.
The grid independence is tested to obtain higher order accuracy results. As shown in Table 5, the computational results are compared with different number of grid. The Richardson extrapolation approach (Roache, 1997) is adopted to test grid independence. The Richardson extrapolation is defined as follow:
/R ¼ /F þ 3.2. Governing equations and computational method The flow process through the structured packed bed is treated to be three-dimensional, laminar and incompressible, because the calculation region is short and the Reynolds number is very small.
/F /c r 2g 1
ð14Þ
where, / is the pressure drop; and Subscripts of F and C is the fine grid and the coarse grid, respectively; rg = (NF/NC); N is the number of grid. To order to ensure both the economical efficiency and the accuracy, the grid numbers with 0.94, 1.79, and 3.15 million are selected, respectively.
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70 160
60
Experimental Value fp
140
Experimental Value fp*
120
40
fp
100
fp
Experimental Value fp Experimental Value fp*
50
80
30
60
20
40
10 20
0
0 0
2
4
6
8
10
Rep
12
0
14
5
10
(a) Dp=0.5mm
15
20
Rep
25
30
35
(b) Dp=2.0mm
Fig. 5. Comparison of fp and fp* with the particle diameters of 0.5 mm and 2.0 mm.
3.4. Results and discussion 3.4.1. The effects of particle diameter on pressure drop As a geometric parameter of packed bed, particle diameter has an important influence on flow characteristics. In this paper, the packed form with SC-1 arrangements is used as an analysis object to investigate the effects of particle diameter on pressure drop. The calculated results with the different particle diameters are described in Fig. 12. It is clear that pressure drop per meter decreases with the increase of particle diameter. In addition, the effect of particle diameter on pressure drop is nonlinear.
70 60
Dp=0.5mm Dp=0.8mm Dp=1.0mm Dp=1.5mm Dp=2.0mm
50
kPa
40 30 20
3.4.2. The effects of packed forms on pressure drop and the particle fiction factor As shown in Fig. 13, the effects of packed forms on pressure drop are described. The results show that pressure drop with FCC-1 arrangements is the largest and pressure drop with SC-1 arrangements is the smallest. Velocity distributions of three packed forms with the same inlet Reynolds numbers (Re) are described in Fig. 14. The results show that the maximum velocity in the channel with FCC-1 arrangements is about 1.37 m/s, and show that the maximum velocity in the channel with SC-1 arrangements is about 0.96 m/s. The porosities with SC-1, BCC-1, and
10 0 100
150
200
250
300
350
400
Channel length mm Fig. 6. Effect of the channel length on the pressure drop under Rep = 12.5.
20
15
the relative error(%)
the relative error(%)
2.0
10
5
0
1.5
1.0
0.5
0.0
0
2
4
6
8
Rep
10
(a) Dp=0.5mm
12
14
0
5
10
15
20
25
Rep
(b) Dp=2.0mm
Fig. 7. The relative error between fp and fp* with the particle diameters of 0.5 mm and 2.0 mm.
30
35
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50 SC-1(Dp=0.5mm) SC-1(Dp=0.8mm) SC-1(Dp=1.0mm) SC-1(Dp=1.5mm) SC-1(Dp=2.0mm)
(a) SC-1
(b)BCC-1
p/L(kPa·m -1)
40
(c)FCC-1
Fig. 8. Different packed cells of spheres.
30
20
10
0 (a) The channel with SC-1 arrangements
0
10
20
30
40
Rep (b) The channel with BCC-1 arrangements
Fig. 12. The effects of particle diameter on pressure drop.
60
(c) The channel with FCC-1 arrangements
SC-1 BCC-1 FCC-1
Inlet
Outlet
Inlet Section
Outlet Section
Heium
p0
T1,U1,p1 L1
L2
p/L(kPa · s-1)
Fig. 9. Different packed pebble channels.
40
20
L3
Fig. 10. The geometric diagram of the packed channels.
0 0
5
10
15
20
25
30
35
40
Rep 0.16Dp
Fig. 13. The effect of packed forms on pressure drop.
Fig. 11. A typical point modifications with bridges treatment.
Table 4 Geometrical parameters. Packed form
SC-1
BCC-1
FCC-1
Porosity(e)
0.476
0.320
0.259
FCC-1 arrangements are 0.476, 0.320, and 0.259, respectively. According to Ergun equation, it is concluded that pressure drop is negatively correlated with the porosity. The smaller porosity causes the narrower flow area, and results in greater velocity in bed. Thus, pressure drop with FCC-1 arrangements is the largest and pressure drop with SC-1 arrangements is the smallest. The relationship of the particle friction factors (fp*) and the particle Reynolds numbers (Rep) are shown in Fig. 15. The results show that the variation of the particle friction factors of the packed forms
Table 5 Mesh parameters. The number of grid (SC-1) SC-1(Dp/Pa) Richardson extrapolation (SC-1) The relative error The number of grid (BCC-1) BCC-1(Dp/Pa) Richardson extrapolation (BCC-1) The relative error The number of grid (FCC-1) FCC-1(Dp/Pa) Richardson extrapolation (FCC-1) The relative error
522,119 11.32 12.34 8.26% 502,130 61.48 72.27 14.93% 1,510,155 209.78 247.97 15.40%
722,665 11.29
941,561 11.95
2,124,244 12.19
8.51% 1,079,336 67.97
3.16% 1,787,706 69.70
1.21% 4,899,842 71.93
5.95% 2,027,138 228.74
3.56% 3,157,087 235.48
0.47% 6,003,352 244.51
7.75%
5.03%
1.39%
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(a)SC-1
(b)BCC-1
(c)FCC-1 Fig. 14. Velocity distributions in different packed bed with the same inlet Re.
Since the packed form with SC-1 arrangements is the simplest, the packed form with SC-1 arrangements is selected to calculate the particle friction factors in next calculation.
300 250
Numerical values with SC-1 Numerical values with BCC-1 Numerical values with FCC-1
fp *
200 150 100 50 0 0
2
4
6
Rep
8
10
12
14
Fig. 15. The effects of packed forms on the particle friction factor.
with SC-1, BCC-1, and FCC-1 arrangements are similar with the particle Reynolds numbers. The differences between the numerical results with SC-1, BCC-1, and FCC-1 arrangements are less than 10% so that any packed form may be used to calculate the particle friction factor. The channels with SC-1, BCC-1, and FCC-1 arrangements have different velocity and pressure drop distributions at the same inlet velocity, which is mainly due to the difference of porosity. In this paper, Rep is the particle Reynolds number, which is a dimensionless number considering both porosity and flow velocity. Moreover, the particle friction factors also consider the effect of porosity. Therefore, the curve tendency of Rep and fp is basically the same.
3.4.3. The prediction to the particle friction factor in the experiment The actual packed form is random in this experiment. The difference between the randomly packed bed and the structured packed bed is mainly porosity. To obtain a new dimensionless parameter including porosity, the particle friction factor modified by compressibility (fp*) is defined in formula of (10) according to Ergun equation. The effects of velocity and particle diameter on the particle friction factors modified with compressibility (fp*) are investigated. Fig. 16 compares the calculated results with experimental results. As shown in Fig. 16, five types of particle diameters with 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm, and 2.0 mm are compared, respectively. It is obvious that the calculated results of all particle diameters agree well with experimental results. Thus, it is concluded that the particle friction factors modified with compressibility in this experiment with randomly packed form can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements.
4. Conclusions In this paper, the low pressure helium flow characteristics in a rectangular pebble bed channel filled with stainless steel particles with 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm were investigated experimentally. Three-dimensional numerical studies are carried out using three structured packed bed of spheres to predict flow performances of the experiment with randomly packed bed. Based on the studies, some findings have been obtained as follows:
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120
200
100
Experimental values Numerical values
Experimental values Numerical values
80
fp*
fp *
150
100
60 40
50
20 0
0 0
2
4
6
Rep
8
10
12
0
14
2
4
6
8
10
12
14
16
18
Rep
(a) 0.5mm
(b) 0.8mm 60
80
Experimental values Numerical values 40
fp *
fp*
60
Experimental values Numerical values
40
20
20
0
0
0
5
10
15
20
25
0
Rep
5
10
15
20
25
30
Rep
(c) 1.0mm
(d) 1.5mm
70 60
Experimental values Numerical values
50
fp*
40 30 20 10 0 0
5
10
15
20
25
30
35
40
Rep
(e) 2.0mm Fig. 16. Comparison of the experimental values and the numerical values with different particle diameters of 0.5 mm, 0.8 mm, 1.0 mm, 1.5 mm and 2.0 mm.
(1) The experimental results show that Ergun equation can agree well with the experimental results in the region with low particle Reynolds numbers, and show a bad prediction in the region with large particle Reynolds numbers. The
deviation of prediction increases with the increase of the particle Reynolds numbers. In addition, Ergun equation modified with compressibility can improve the ability to predict experimental values.
Z. Wu et al. / Annals of Nuclear Energy 128 (2019) 268–277
(2) It can be concluded that the compressibility has an important influence on flow characteristics in the condition with the large ratio of pressure drop to system pressure. In addition, the effects of the compressibility increase with the particle Reynolds numbers (Rep). The relative error between fp and fp* is close to 20% with the particle diameter of 0.5 mm, and the relative error is less than 2% with the particle diameter of 2.0 mm. (3) The numerical results with the same Reynolds numbers, show that the pressure drop with FCC-1 arrangement is the largest, and that with SC-1 arrangement is the smallest, and that the maximum velocity with FCC-1 is greater than that with SC-1 arrangements. Furthermore, the numerical results show that the particle friction factor modified with compressibility (fp*) in this experiment with randomly packed form can be predicted by using the models with SC-1, BCC-1, and FCC-1 arrangements. (4) This work provides an approach for the flow characteristics simulation of the randomly packed bed.
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