- Email: [email protected]
Numerical analysis of flow resistance characteristics in an inclined rod bundle channel
Progress in Nuclear Energy 122 (2020) 103247
Contents lists available at ScienceDirect
Progress in Nuclear Energy journal homepage: http://www.elsevier.com/locate/pnucene
Numerical analysis of flow resistance characteristics in an inclined rod bundle channel Yinxing Zhang a, Puzhen Gao a, *, Xiaoqiang He a, Chong Chen b, Qiang Wang b, Chunping Tian c a
College of Nuclear Science and Technology, Harbin Engineering University, Harbin, 150001, PR China China Ship Development and Design Center, Wuhan, 430064, PR China c Wuhan 2nd Ship Design and Research Institute, Wuhan, 430060, PR China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Inclination Rod bundle Friction resistance coefficient
The thermal-hydraulic characteristics of the rod bundle channel under ocean conditions have attracted much attention. In this paper, the three-dimensional CFD numerical simulation software STAR-CCMþ is used to study the frictional resistance characteristics of the rod bundle under non-vertical conditions at inclination angles of 0� , 15� , 30� and 45� , so as to provide a reference for future reactor core design of floating nuclear power plants. Using CFD, the relationship between the frictional resistance coefficient and the Reynolds Number (Re) is ob tained in a wide range of Re, including the laminar regime, the transition regime and the turbulent regime. The predicted correlation results are good and have an estimated error within �10%. The influence of the heating flux and the inlet temperature of the test bundle is considered, and a kinematic viscosity correction coefficient is introduced. Owing to the structural characteristics of the square rod bundle channel, this paper considers two different directions of tilt: rotations about an axis perpendicular to the channel faces and about an axis which passes through opposite corners of the channel. A difference in the frictional resistance coefficient between in clined and vertical conditions only occurs in the laminar flow regime, and there is no frictional resistance co efficient difference in the transition regime or the turbulent regime. Additionally, the frictional resistance coefficient is reduced considerably if the boundary layer is disturbed, due to the reversed fluid flow in the laminar regime.
1. Introduction As the basic structure of the nuclear reactor core, the rod bundle channel has naturally become the research topic of scientists. Since floating nuclear power plants are operating under ocean conditions, the core will inevitably tilt and roll due to the effects of ocean conditions. Such motion will affect nuclear reactors in varying degrees. The purpose of this paper is to investigate the effect of inclined conditions on the flow resistance characteristics of the core structure of a nuclear reactor, the rod bundle channel, through numerical simulations. The flow resistance characteristics in the rod bundle channel are expressed in the form of the frictional resistance coefficient, which can be used as a reference for future research on offshore floating nuclear power plants. Many scholars have studied related aspects of flow characteristics in rod bundles. C. Lifante et al. (Lifante et al., 2014) used ANSYS CFX to perform single-phase numerical research on a 3 � 3 rod bundle. The effects of different geometries are considered in the paper, including the
effects of the inlet vessel and the grid spacer. Their paper paves the way for numerical analysis of multiphase flow in order to accurately predict the boiling water flow in the rod bundle channel under high pressure. Han Wang et al. (Wang et al., 2016) studied the heat transfer charac teristics of supercritical pressurized water in a 2 � 2 rod bundle channel by numerical simulation. They showed that the SSG Reynolds Stress Model (Wang et al., 2016) can make the best prediction for normal and enhanced heat transfer models and described the relationship between heat transfer characteristics and pressure, mass flux and heat flux of supercritical water in the rod bundle channel. K. Zhang et al. (Zhang et al., 2016) conducted experimental and theoretical studies on verti cally upward two-phase flow of air-water across staggered rod bundles with inclination angles from the vertical of 0� , 45� and 90� . They developed pressure drop correlations for single-phase fluids in rod bundle channels at different inclination angles. The two-phase friction multipliers they developed successfully predict the two-phase pressure drop data in inclined rod bundles. Faruk A. Sohag et al. (Sohag et al., 2017) used ANSYS Fluent 16 to conduct a careful study of a vertically
* Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (P. Gao). https://doi.org/10.1016/j.pnucene.2020.103247 Received 28 November 2018; Received in revised form 26 October 2019; Accepted 8 January 2020 Available online 30 January 2020 0149-1970/© 2020 Elsevier Ltd. All rights reserved.
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Nomenclature P D d Re a L n ΔPf l ui u g ct Fi v g0 E ΔP Sh k
keff p T T1 T2
rod pitch, m diameter of the rod, m hydraulic diameter, m Reynolds number rectangular casing width, m second piezometric section length, m number of the rods frictional pressure drop, Pa the distance to the wall, m x-direction velocity component, m/s velocity vector, m/s gravitational acceleration, m/s2 kinematic viscosity correction coefficient external body forces, N/m3 velocity, m/s initial gravitational acceleration, m/s2 empirical constant in turbulence model total pressure drop, Pa the other volumetric heat sources turbulence kinetic energy, m2/s2
effective conductivity, W/m2⋅K system pressure, MPa temperature, K tilt 1 tilt 2
Greek letters δij Kronecker delta tensor ρ density, kg/m3 λ frictional resistance coefficient μ dynamic viscosity, Pa⋅s υ kinematic viscosity, m2/s τij Reynolds stress tensor ω specific dissipation rate, 1/s φ intermittency factor θ inclined angle, � Subscripts i o w f
inlet outlet wall fluid
studied the incompressible, isothermal and fully developed turbulent flow pressure drop in a hexagonal rod bundle channel. The results show that the number of rods in a rod bundle has no measurable effect on the pressure drop coefficient. He (Rehme, 1973) also predicted the hy draulic losses in multirod fuel bundles over a broad range, which is necessary for reactor design calculations. Zhu et al. (Zhu et al., 2018) carried out an experimental study on flow resistance characteristics in a 3 � 3 rod bundle channel under natural circulation rolling motion conditions. They only fitted laminar data on frictional resistance coef ficient and did not consider the influences of heating power on the friction coefficient changes. Due to the limitation of experimental con ditions, they could only obtain a selected ranges of Reynolds number (Re) conditions, consequently the results for the turbulent regime were difficult to achieve, and the laminar regime had few experimental re sults. Based on the above description, it can be seen that there are many researchers working on the thermal-hydraulics characteristics in rod bundle channels, which is of great significance for the nuclear industry. In this paper, the numerical simulation method can overcome experimental limitations and estimate the frictional resistance coeffi cient under all conditions, including the laminar, the transition and the turbulent regimes. We then analyze some conditions that our experi ment cannot achieve, like inclination in different directions. The paper aims to obtain the correlation of frictional resistance coefficient over a broad range of Reynolds numbers under inclined conditions by computational fluid dynamics (CFD). In this study, not only the incli nation angles of 0� , 15� , 30� and 45� were considered, but also the in fluence of two different tilt axes on the frictional resistance coefficient were considered. This paper focuses on the frictional resistance char acteristics of the fluid in the laminar regime. The laminar flow condition are likely to occur in accident conditions of nuclear power plants. Therefore, the paper has important guiding significance for the safety analysis of nuclear power plants. In addition, the change of fluid phys ical properties caused by the heating power in the test section was also studied in detail in order to get a more accurate expression of the fric tional resistance coefficient of the rod bundle channel. It is worth mentioning that the reversed flow phenomenon in laminar flow region under inclined conditions found in this paper occurs only when the outlet pressure is set to a constant value.
heated rod bundle in mixed and forced convection. They used the Shear-Stress Transport (SST) K-Omega model, which is also used in this paper, to study the heat transfer of a vertical rod-bundle with the rela tionship of spacer grid spacing and spacer grid blockage ratio. The simulation results are in good agreement with some notable correlations found in the literature. Deqi Chen et al. (Chen et al., 2016) carried out a numerical study of the thermal-hydraulic characteristics of a 5 � 5 rod bundle in a reactor. Changes in thermal hydraulic behavior downstream of the spacer grid caused by dimple, spring and mixing vane were described in detail, including with and without the spacer grid. The study found that the secondary flow induced by the spacer grid could have a significant impact on heat transfer. C.C. Liu et al. (Liu et al., 2012) evaluated the turbulence model of the fuel rod bundle with a spacer assembly and detailed the differences in wall treatment, mesh refine ment and appropriate definition of boundary conditions that would have a significant impact on the numerical simulation results. Through that study, they were able to determine the best enhanced wall treatment and the most appropriate turbulence models for modeling nuclear reactor coolant systems. F. C. Engel et al. (Engel et al., 1979) tested the pressure drop across wire-wrapped hexagonal rod bundles in laminar, transition, and turbulent parallel flow regions using air, water and sodium. A smooth transition region from laminar to turbulent flow was found to occur over the Reynolds number (Re) range from 400 to 5000. They developed correlations for the frictional resistance coefficient for different flow regimes and also a general laminar flow friction factor correlation with different wire-wrap lead pitch-to-diameter ratios. Mary V. Holloway et al. (Holloway et al., 2008) studied the convective heat transfer in the turbulent regime through rod bundles that can represent nuclear fuel rods used in pressurized water reactors. Using air as the working fluid, they measured the single-phase convective heat transfer coefficients for flow downstream of support grids in a rod bundle and obtained a general correlation to predict the heat transfer development downstream of support grids. E. Bubelis and M. Schikorr (Bubelis and Schikorr, 2008) summarized the existing wire-wrapped fuel bundle friction factor and pressure drop correlations and qualitatively evaluate which of the existing friction factor correlations is the best, which could be reflected in matching the results of the existing large body of experimental data under different coolant conditions. That paper is of great help to scholars looking for a large number of references in the study of rod bundle channels. Klaus Rehme (1972) systematically 2
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
2. Experimental facility In order to ensure the accuracy of numerical simulation, experi mental research should be carried out before the CFD simulation, and the experimental data thus obtained should used to verify the model. We have done such an experimental verification of our model. The experi mental facility shown in Fig. 1 has been described in detail in the Yinxing Zhang’s paper (Zhang et al., 2019). The test section shown in Fig. 2 is a 3 � 3 rod bundle channel made of stainless steel with a rectangular casing. Three pressure ports are located in one side of the rod bundle, as shown in Fig. 3. The length of the two piezometric sections L2 and L are 150 mm and 400 mm, respectively. The frictional pressure drop along the rod bundle channel, which is of most concern in this paper, is measured by the second piezometric section. For more details, please refer to Yinxing Zhang’s paper (Zhang et al., 2019). The uncertainty of the Reynolds number and the friction resistance coefficient are analyzed by the error transfer formula. The calculated uncertainties of the Reynolds number and the frictional resistance co efficient are 2% and 5%, respectively, according to the method of J. P. Holman (Holman and Gajda, 2001). Since this paper focuses on the CFD analysis, the details of the experimental procedure are not described in this paper. The inclination angle of the experimental facility is achieved by a crank rocker mechanism. For the data used for the validation of this paper’s model, the inclination angle is 20� , as determined by the acquisition system.
Fig. 2. Cross section rod bundle channel structure.
the modeling geometric structure should only be the part between pressure tube 2 and pressure tube 3 in Fig. 3. It is then helpful to match the numerical simulation results with the experimental results and to model with the 3D-CAD Modeler directly in STAR-CCMþ 11.06. In this paper, two kinds of mesh elements were used to generate grids, including a polyhedral mesh and an extruder mesh. The use of the extruder mesh can greatly reduce the amount of mesh points, which has the benefit for saving the calculation cost, under the premise of ensuring the quality of calculation. In addition, the prism layer mesh model is applied to solve boundary conditions on all geometry walls. The ge ometry, including the mesh, is shown in Fig. 4. The surface remesher can be used to improve the overall quality of an existing surface and opti mize it for the volume mesh models. To ensure the accuracy of the numerical calculation, a grid inde pendent test should be performed for the simulation model. The stabi lization of the frictional resistance coefficient (λ), the coolant outlet temperature (To), the total pressure drop (ΔP), and the coolant outlet velocity (vo) were used for calculating convergence. The total pressure
3. Numerical methods 3.1. Modeling and meshing In order to simulate the flow resistance characteristics of 3 � 3 rod bundle channel under tilted conditions, this paper uses the CFD software STAR-CCMþ 11.06 based on the finite volume method to do the nu merical calculation. This article only focuses on the frictional pressure drop in the rod bundle channel, not the influence of the spacer grid on the resistance characteristics in the rod bundle channel. Consequently,
Fig. 1. Thermal-hydraulic experimental facility. 3
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
drop is the direct result of the calculation relevant to the mesh size. It is useful to change the number of layers in the extrusion parameter to refine the mesh. After refinement of the mesh, four different numbers of grids are generated on the geometry model. The number of grids are 346,757; 658,769; 791,047 and 981,486, respectively, correspond to model 1, model 2, model 3 and model 4. The values of λ, To ΔP and vo under different mesh models are shown in Table 1. As can be seen from the table, the relative error of model 3 has reached within 0.1%, to meet the accuracy requirements. In this paper, the model 3 with 791,047 grids was adopted for all following simulations. 3.2. Governing equations and boundary conditions CFD provides a comprehensive range of modeling capabilities for incompressible and compressible, laminar, transitional and turbulent fluid problems. Three-dimensional modeling can be used to study the thermal-hydraulic properties of complex geometric structures. For example, the frictional resistance characteristics of a 3 � 3 rod bundle channel under different inclination conditions are discussed in this paper. The governing equations of coolant properties are written as follows: Table 1 Grid independence test.
Fig. 3. Longitudinal section rod bundle channel structure.
Model
Grid number
λ
To [oC]
ΔP [Pa]
vo[m/s]
#1 #2 #3 #4
346,757 658,769 791,047 981,486
0.036309 0.03585 0.036214 0.036206
28.87317 29.02993 28.87765 28.87803
4266.528 4261.947 4265.591 4265.512
0.550215 0.5502278 0.5502205 0.5502213
Fig. 4. The mesh applied in the geometry mode:(a) cross section, (b) detail of cross section, (c) global. 4
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Continuity equation:
∂ρ v Þ¼0 þ r ⋅ ðρ! ∂t
d¼
(1a)
Momentum equation: �
∂ ∂ ρui uj ¼ ðρui Þ þ ∂t ∂xj where � � �� ∂ui ∂uj τij ¼ μ þ ∂xj ∂xi
Re ¼
∂p ∂τij þ þ ρgi þ Fi ∂xi ∂xj
(2a)
2 ∂ul μ δij 3 ∂xl
(3a)
∂ ∂ ∂ ∂T k ðρEÞ þ ðui ðρE þ pÞÞ ¼ ∂t ∂xi ∂xi eff ∂xi
X � hj0 Jj0 þ uj τij eff
λ¼
(4a)
0
j
SST K-Omega model (Menter and Kuntz, 2004): The transport equations for the kinetic energy k and the specific dissipation rate ω are:
� ∂ ðρωÞ þ r ⋅ ρω v ∂t
vg
��
¼ r ⋅ ½ðμ þ σk μt Þrk� þ Pk
ρβ* ðωk
ω0 k0 Þ þ Sk (5a)
vg
��
¼ r ⋅ ½ðμ þ σ ω μt Þrω� þ Pω
ρβ ω2
�
ω20 þ Sω (6a)
where, vandvg are the mean velocity and the reference frame velocity relative to the laboratory frame, respectively, Pk and Pω are production terms, k0, ω0 are the ambient turbulence values that counteract turbu lence decay, Sk and Sω are the user-specified source terms.
σk ¼ 1
0:15F1 ; σ ω ¼ 0:856
β * ¼ 0:09; β ¼ 0:0828
0:356F1
0:0078F1
(7a) (8a)
where, ! !#! pffiffi 2k k 500υ ; 2 min max ; l CDkω 0:09ωl l2 ω
" F1 ¼ tanh
� 1 CDkω ¼ max rk ⋅ rω; 10
ω
υ
¼
� v 4a2 π D2 n υð4a þ πDnÞ
(2b)
2dΔPf Lρv2
(4b)
As previously stated our modeling validation was under the condi tion of an inclination angle, θ, of 20� . The calculation results obtained are shown in Fig. 5. It can be seen from the figure that SST (Menter) KOmega is most suitable for the application and calculation of the rod bundle channel, which could fit the experiment data best. Menter (1994) suggested using a blending function (which includes functions of wall distance) that would include the cross-diffusion term far from walls, but not near the wall. This approach effectively blends a K-Epsilon model in the far-field with a K-Omega model near the wall and cures the biggest drawback to applying the K-Omega model to practical flow simulations. Menter also introduced a modification to the linear constitutive equa tion and dubbed the model containing this modification the SST (shear-stress transport) K-Omega model. The SST model works best for rod bundle geometries, which is consistent with Faruk A. Sohag’s research (Sohag et al., 2017), where viscous flows are typically resolved and turbulence models are applied throughout the boundary layer. This paper will use this model as a predictor of the frictional resistance co efficient in rod bundle channel. The laminar flow model and the SST (Menter) K-Omega model are used to numerically simulate the same working conditions of the laminar regime. The results show that there is 0.01% deviation between the two models for the pressure drop calcu lation. Thus it is shown that the SST (Menter) K-Omega model can accurately simulate the laminar flow. Therefore, the SST (Menter) K-Omega model is also selected for the laminar regime in this paper. The error bars on experimental data from the calculated un certainties of the Reynolds number and the frictional resistance coeffi cient are 2% and 5%, respectively.
! þ Sh
vd
(1b)
The frictional resistance coefficient, λ, is calculated as follows: � θπ � ΔPf ¼ ΔP ρgL⋅cos (3b) 180
Energy equation:
� ∂ ðρkÞ þ r ⋅ ρk v ∂t
4a2 πD2 n 4a þ πDn
(9a)
� 20
(10a)
For the boundary conditions, the inlet is the velocity inlet, the outlet is a pressure outlet, the constant heat flux is used for the wall of the rod bundle, and the adiabatic boundary condition is used for the outer wall. All wall boundary conditions are set to be no-slip. 3.3. Modeling validation In order to ensure the accuracy of this numerical simulation for the later part of the paper, we present the verification of the numerical models of the rod bundle channel under inclined conditions by comparing the simulations to our experimental results. The experi mental data were obtained from the experimental facility described above. The specific experiment was conducted under the conditions of 0.4 MPa system pressure and 18 kW/m2 test section heating flux. The validation numerical simulation is carried out under the same conditions as the experimental conditions in order to find a suitable physical model for the numerical simulation in this paper. Six models were calculated and analyzed respectively, including SST (Menter) K-Omega, Realizable K-Epsilon, Realizable K-Epsilon Two-Layer, Reynolds Stress Turbulence, Standard (Wilcox) K-Omega, Standard K-Epsilon. The parameters used for the determination of the Reynolds number are as follows:
Fig. 5. Frictional resistance coefficients calculated by CFD and experimental data (p ¼ 0.4Mpa, Q ¼ 18 kW/m2, θ ¼ 20� ). 5
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Todreas, 1986; Sadatomi et al., 1982)and the numerical simulation re sults. The results are most similar to Engel’s research, as they claimed that transition points occur at 400 and 5000 Re. This article will study the relationship between λ and Re on the basis of Engel’s research. According to the CFD simulation results, the relationship between the frictional resistance coefficient λ and Re under vertical condition in the rod bundle channel is fitted with a fixed heating power as for the laminar region λ ¼
88 Re
for the transition region λ ¼
88 ð1 Re
(5b) ϕÞ1=2 þ
0:24 1=2 ϕ Re0:23
(6b)
0:24 Re0:23
(7b)
Re 400 3600
(8b)
for the turbulent region λ ¼ where Fig. 6. Frictional resistance coefficients calculated by CFD and different empirical correlations (p ¼ 1Mpa, Q ¼ 20 kW/m2, Ti ¼ 300 K, θ ¼ 0� ).
ϕ ¼ intermittencyfactor ¼
Based on the relational expression above, the frictional resistance coefficient calculated at different Reynolds numbers are compared with experimental data as shown in Fig. 7. The frictional resistance coeffi cient calculated from CFD simulation are shown at the same time. As can be seen from this figure, the difference among them is within �10%, which satisfies the prediction accuracy requirement. The experimental data under inclined condition in Fig. 7 is the same as that in Fig. 5. It can be seen that our prediction formula can not only predict the frictional resistance coefficient under vertical conditions, but also predict the λ of the rod bundle channel under inclined conditions. However, our pre diction formula only applies to the transition and the turbulent regime. For the laminar regime, it is not applicable under inclined conditions as will be shown in detail in Section 4.2. In this paper, the effects of fluid properties on the frictional resis tance of the rod bundles were investigated by adjusting the heating flux of the test section to be 20, 100, 200 kW/m2. The numerical simulation results are shown in Fig. 8. From this figure, it can be seen that as the heat flux of the rod bundle increases, the frictional resistance coefficient gradually decreases. This is because the average temperature of the fluid goes up as the heating flux increases. As a result, the viscosity of the fluid is reduced, resulting in a reduced resistance pressure drop. From the figure we also notice that the Reynolds number range is different for different heating flux. The greater the heating flux, the greater the starting point of the Reynolds number value in Fig. 8. This is taken into account to ensure that the rod bundle channel is filled with single-phase
4. Results and discussion 4.1. Vertical condition Before studying the hydraulic characteristics of the rod bundle channel under inclined conditions, the vertical condition was investi gated as a special tilt (θ ¼ 0� ). In order to study the characteristics of frictional resistance coefficients as a function of the Reynolds number in the rod bundle channel, the method of adjusting Re was performed by changing the inlet velocity of the test section. Fig. 6 shows the variation of the frictional resistance coefficients with Re. In the simulation pro cess, the inlet temperature of 300 K was kept constant, the system pressure was 1 MPa, the heat flux was 20 kW/m2. It can be seen from the figure that the frictional resistance gradually decreases with the increase of the Reynolds number. It is noteworthy that transition points occur at Re ¼ 400 and Re ¼ 4000 in the graph, which is a little different from Engel’s (Engel et al., 1979) study. From the data in Fig. 6, it could be proposed that for the laminar-transition Reynolds number, 400, and the transition-turbulent Reynolds number, 4000. The laminar, transition and turbulent regions are then defined as Re < 400, 400 < Re < 4000, and Re > 4000, respectively. From this figure, it can be seen that there are some differences between other scholars’ research (Cheng and
Fig. 7. Comparison among predicted frictional resistance coefficient, CFD re sults and experimental results (p ¼ 0.4Mpa, Q ¼ 18 kW/m2, θ ¼ 20� ).
Fig. 8. Frictional resistance coefficients with different test section heating powers (p ¼ 1Mpa, Ti ¼ 300 K, θ ¼ 0� ). 6
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
fluid. It can also be seen from the figure that when Re > 20,000, the change of heat flux does not affect the frictional resistance characteris tics. It can be considered that the fluid temperatures at different heat flux are substantially the same when the fluid velocity is quite large, resulting in the same fluid viscosity and the same resistance pressure drop. In order to quantitatively express the influence of the fluid prop erties on the frictional resistance characteristics in the rod bundle channel, the kinematic viscosity correction coefficient (Sieder and Tate, 1936) is introduced on the basis of the previously obtained formulae (5), (6), and (7) to reflect the influence of the fluid viscosity on the resistance characteristics. The expression is as follows: � �0:3 υw ct ¼ 1:05 (9b)
υf
where, υw and υf are the kinematic viscosity calculated by means of the average wall temperature of the rod bundle and the average fluid tem perature, respectively. The correlation of the frictional resistance coefficient in the rod bundle channel under vertical conditions can be summarized as � �0:3 92 υw for the laminar region λ ¼ (10b) Re υf � for the transition region λ ¼
for the turbulent region λ ¼
92 ð1 Re
ϕÞ1=2 þ
� �0:3 0:25 υw Re0:23 υf
0:25 1=2 ϕ Re0:23
�� �0:3
υw υf
Fig. 10. Frictional resistance coefficient changes with inlet temperatures at different mass flow rate (p ¼ 1Mpa, Q ¼ 20 kW/m2, θ ¼ 0� ).
(11) (12)
Again, the simulation results are compared with the results of the prediction formula above. The results are shown in Fig. 9. It can be seen from the figure that the formula has a good prediction with an uncer tainty within �10%. The simulation conditions of the previous study are all constant inlet temperatures. However, inlet temperature is also an important control parameter during experimental research. To study the effect of inlet temperature on the frictional resistance coefficient, the mass flow rates studied in this paper are 0.04, 0.4 and 0.2 kg/s, respectively. The inlet temperature is studied at 280–400 K, the heat flux is 20 kW/m2 and the system pressure is 1 MPa. The simulation results are shown in Fig. 10. It can be seen from the figure that the inlet temperature at high mass flow has no influence on the frictional resistance coefficient, and the frictional resistance coefficient decreases with increasing inlet temper ature at low mass flow. When the mass flow is low, especially in the
Fig. 11. Relationship between predicted frictional resistance coefficient and CFD results with different mass flow and inlet temperature (p ¼ 1Mpa, Q ¼ 20 kW/m2, θ ¼ 0� ).
laminar regime, the higher the temperature, the lower the viscosity of the fluid at the same heating flux, resulting in the smaller the resistance coefficient. At high mass flow rates, especially in fully developed tur bulent regime, the main determinant of pressure drop in high-speed flowing fluids is the velocity, at which point the effect of viscosity is not significant. It has been verified that the characteristics of the fric tional resistance in the rod bundle channel under the condition of var iable inlet temperature can still be expressed by formulae (10), (11), and (12), and the uncertainty is within �10%, as shown in Fig. 11. It can be seen that formulae (10), (11), and (12) in this paper can be applied to the prediction of frictional resistance coefficient, not only for a wide range of Reynolds number, but also for different heating power and coolant average temperature. 4.2. Inclined condition In order to study the influence of inclination on the frictional resis tance characteristics in the rod bundle channel, the changing of the di rection of gravity in the CFD simulation must be considered. The specific method is as follows: the direction of gravity in the CFD simulator is
Fig. 9. Relationship between predicted frictional resistance coefficient and CFD results with different test section heating power (p ¼ 1Mpa, Ti ¼ 300 K, θ ¼ 0� ). 7
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 12. Our definitions of tilted 1 and 2, wherein the tilting is a rotation about the y-axis, as shown. Table 2 The simulation inclined angles and Reynolds numbers for the studies in this paper. T1 represents tilted 1 and T2 represents tilted 2. θ/�
0
tilt style Re
changed for the tilts as shown below: g0 cos θÞ
30
45
T1
T2
T1
T2
T1
T2
312 349 387 426 500 649 797 1527 3699 4046 13016 21647 43231 72017
314 350 387 426 500 649 797 1527 3699 4053 13016 21647 43231 72017
309 350 385 425 500 648 796 1526 3699 4046 13016 21647 43231 72017
313 348 386 423 500 648 795 1526 3698 4053 13016 21647 43231 72017
309 351 385 425 499 648 795 1525 3699 4046 13016 21647 43231 72017
312 348 386 423 499 648 794 1525 3698 4053 13016 21647 43231 72017
were the same as for the vertical orientation above. The results of the calculation under tilted 1 and 2 conditions are shown in Fig. 13. The simulated inclination angle and Reynolds number are shown in Table 2. It can be seen from the figure that the inclination angles have no influence on frictional resistance coefficient in the transition regime and turbulent regime, but there is a significant affect in the laminar regime. The frictional resistance coefficient in the laminar flow region under the inclined condition is much smaller than that in the vertical condition. It can also be noticed that the frictional resistance coefficient under the titled 2 condition is slightly smaller than that under the titled 1 condi tion, and the difference of the coefficient between tilted 1 and 2 will be smaller as the inclination angle increases. After tilting, the fluid in the rod bundle channel generates a radial velocity, which means the velocity has components in the x and y di rections, under the influence of gravity and the constraint of the chan nel. This can be seen from the distributions of the radial velocity at the
Fig. 13. Frictional resistance coefficient under tilted 1 (T1) and 2 (T2) condi tion (p ¼ 1Mpa, Q ¼ 20 kW/m2, Ti ¼ 300 K).
! g ¼ ðg0 sin θ; 0;
279 350 389 427 502 651 799 1528 3698 4052 13016 21647 43231 72017
15
(13)
where, g0 is the gravitational acceleration under vertical condition. However, although the rod bundle channel is symmetrical, with a square cross section, there will be some differences in the flow when the channel is inclined at different directions. This paper selected the two most representative directions. We refer to these directions as “tilt 1” and “tilt 2”, as shown in Fig. 12. Consequently, the STAR-CCMþ was used to numerically simulate the rod bundle channel at 15� , 30� and 45� inclination angles with the two inclination directions described above. The heat flux was 20 kW/m2, the system pressure was 1 MPa, and the inlet temperature was 300 K, which 8
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 14. Radial velocity distributions at the outlet under different conditions.
outlet under vertical and inclined conditions in Fig. 14. In the figures, the velocity direction at higher flow rates are marked in red. At the same time, the heated fluid will undergo thermal stratification, and reversed flow will occur under the action of gravity. The reason why reversed flow happens near the outlet is as follows: The titled heated fluid flows up vertically under the influence of density difference at low Reynolds number. However, the fluid velocity is too small to keep flowing up. The buoyancy is not enough to support the fluid to continue to move up wards and stay due to gravity. Then the reversed flow will occur as shown in Fig. 15 enclosed in red. The reversed flow can inhibit the heat transfer of the fluid in the rod bundle channel, which in turn causes the fluid temperature to decrease and the fluid viscosity in the channel to increase. According to the foregoing description, an increase in viscosity causes an increase in pressure drop. But the actual situation is the opposite, so we can think that reversed flow’s disturbing the boundary layer is the main cause of the reduction of the pressure drop in rod bundle channel. It can be seen from Fig. 14 that the maximum value of
the fluid velocity in the radial direction increases with the inclination angle under the buoyancy force. In the process of the numerical simu lation, the number of reversed flow faces on outlet under different conditions are also indicated in Fig. 14. The higher the maximum fluid velocity in the radial direction, the more severe the reversed flow and the greater the damage to the boundary layer. Fig. 15 shows the reversed flow in the longitudinal section of the full flow channel. The smaller the angle of inclination, the smaller the proportion of radial velocity to the axial velocity, which is considered as the mainstream velocity, resulting in the smaller the damage to the boundary layer, and then the larger the frictional resistance coefficient with the other conditions being the same. The proportion of radial velocity to the axial velocity in the laminar regime is large, and the proportion in the transition regime and turbulent regime is small because of the high mainstream velocity and can be neglected. The reversed flow phenomenon will not occur at high mainstream speed, which is the reason why a tilt has no effect on the frictional resistance coefficient in the transition regime and turbulent 9
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 15. Axial velocity distributions along the full flow channel, at the position shown by the plane at the pink line. The area enclosed in red in the top part of the figure shows the reversed velocity vectors.
numerical simulation. The correlation of the frictional resistance coef ficient under vertical condition was obtained and further research was conducted with different tilt directions and angles. This paper focuses on the analysis of the effect of inclination on frictional resistance charac teristics of the rod bundles under low Reynolds number conditions (laminar flow), which serves as a guide for the occurrence of accident conditions in nuclear power plants. First, the numerical simulation results under vertical conditions were analyzed. The relationship between the frictional resistance coefficient λ and the Reynolds number was obtained in laminar, transition and tur bulent regimes. A kinematic viscosity correction coefficient was then introduced to modify the relationship. Actual experimental data were compared to the predicted results from Equations (10)–(12), and they were in good agreement. The prediction result from the correlation agreed with the CFD simulation values to within �10% uncertainty. We therefore propose our general correlation of the frictional resistance coefficient in future investigations for the inclined rod bundle channel to reduce the cost of experimentation and CFD simulation. Second, for the numerical simulation study under inclined condi tions, this paper considers two different inclination directions - tilt 1 and 2, and three inclination angles of 15� , 30� and 45� . No matter what kind of inclination, there is no influence on the frictional resistance coeffi cient in the transition regime and the turbulent regime, and the fric tional resistance coefficient in those regions can still be predicted by the expression for the non-inclined, i.e. vertical, condition. However, there are some differences between the vertical condition and the inclined conditions in laminar regime. The frictional resistance coefficient will be smaller under the inclined condition than that under the vertical con dition in the laminar regime. The results show that there is not much different between the coefficient under tilted 1 and 2 conditions. The frictional coefficient under titled 2 condition is slightly smaller than that under titled 1 condition, and the difference of the coefficient between tilted 1 and 2 condition will be smaller as the inclination angle increases.
Fig. 16. Axial velocity distributions along the full flow channel, which is shown in Fig. 15. (Re ¼ 648, tilted 2 θ ¼ 45� ).
regime. In order to verify the above statement, Fig. 16 shows the fluid velocity distribution in the transition regime with a low Re ¼ 648 and a large inclination angle θ ¼ 45� in the same full flow channel as that in Fig. 15. It can be seen from the figure that no reversed flow occurs. Without interference to the boundary layer, there is no effect on the frictional resistance coefficient. Under the condition shown in Fig. 16, the fluid is more likely to reverse than other conditions in transition regime and turbulent regime. It can be explained that the fluid in the transition regime and the turbulent regime does not produce reversed flow to interfere with the boundary layer as the condition in Fig. 16 has no reversed flow. Therefore, the frictional resistance coefficient in in clined condition is the same as that in the vertical condition in the transition regime and turbulent regime. The reversed flow phenomenon under low flow rate conditions is difficult to occur during the normal operation of the reactor, but it may occur when the reactor is in an accident condition. Therefore, this paper carried out research in this field in detail. 5. Conclusions In this paper, the frictional resistance characteristics of the rod bundles under vertical and inclined conditions are studied by CFD 10
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
In addition, the greatly reduced frictional resistance coefficient in the laminar regime can be understood as the result of reversed flow, which disturbs the boundary layer in tilted conditions. Any other more complicated channels could be further studied using the research method in this paper. The method will guide the safe development of the nuclear industry by researching the thermal hy draulic characteristics of different channels under different inclined conditions.
Engel, F.C., Markley, R.A., Bishop, A.A., 1979. Laminar, transition, and turbulent parallel flow pressure drop across wire-wrap-spaced rod bundles. Nucl. Sci. Eng. 69 (2), 290–296. Holloway, M.V., Beasley, D.E., Conner, M.E., 2008. Single-phase convective heat transfer in rod bundles. Nucl. Eng. Des. 238 (4), 848–858. Holman, J.P., Gajda, W.J., 2001. Experimental Methods for Engineers. McGraw-Hill, New York, 7. Lifante, C., Krull, B., Frank, T., et al., 2014. 3�3 rod bundle investigations, CFD singlephase numerical simulations. Nucl. Eng. Des. 279, 60–72. Liu, C.C., Ferng, Y.M., Shih, C.K., 2012. CFD evaluation of turbulence models for flow simulation of the fuel rod bundle with a spacer assembly. Appl. Therm. Eng. 40, 389–396. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence modeling for engineering applications. AIAA J. 32 (8), 1598–1605. Menter, F.R., Kuntz, M., 2004. Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow behind Vehicles, pp. 339–352. Rehme, K., 1972. Pressure drop performance of rod bundles in hexagonal arrangements. Int. J. Heat Mass Transf. 15 (12), 2499–2517. Rehme, K., 1973. Pressure drop correlations for fuel element spacers. Nucl. Technol. 17 (1), 15–23. Sadatomi, M., Sato, Y., Saruwatari, S., 1982. Two-phase flow in vertical noncircular channels. Int. J. Multiph. Flow 8 (6), 641–655. Sieder, E.N., Tate, G.E., 1936. Heat transfer and pressure drop of liquids in tubes. Ind. Eng. Chem. 28 (12), 1429–1435. Sohag, F.A., Mohanta, L., Cheung, F.-B., 2017. CFD analyses of mixed and forced convection in a heated vertical rod bundle. Appl. Therm. Eng. 117, 85–93. Wang, H., Bi, Q., Wang, L., 2016. Heat transfer characteristics of supercritical water in a 2�2 rod bundle – numerical simulation and experimental validation. Appl. Therm. Eng. 100, 730–743. Zhang, K., Tian, W.X., Hou, Y.D., et al., 2016. Pressure drop characteristics of vertically upward flow in inclined rod bundles. Exp. Therm. Fluid Sci. 78, 208–219. Zhang, Y., Gao, P., Kinnison, W.W., et al., 2019. Analysis of natural circulation frictional resistance characteristics in a rod bundle channel under rolling motion conditions. Exp. Therm. Fluid Sci. 103, 295–303. Zhu, Z., Wang, J., Yan, C., et al., 2018. Experimental study on natural circulation flow resistance characteristics in a 3�3 rod bundle channel under rolling motion conditions. Prog. Nucl. Energy 107, 116–127.
Acknowledgements The authors wish to thank the LLOYD’S REGISTER FOUNDATION for financial support. The authors would like to thank Professor W. Wayne Kinnison for his help on this article, including language and content guidance. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pnucene.2020.103247. References Bubelis, E., Schikorr, M., 2008. Review and proposal for best fit of wire-wrapped fuel bundle friction factor and pressure drop predictions using various existing correlations. Nucl. Eng. Des. 238 (12), 3299–3320. Chen, D., Xiao, Y., Xie, S., et al., 2016. Thermal–hydraulic performance of a 5�5 rod bundle with spacer grid in a nuclear reactor. Appl. Therm. Eng. 103, 1416–1426. Cheng, S.-K., Todreas, N.E., 1986. Hydrodynamic models and correlations for bare and wire-wrapped hexagonal rod bundles — bundle friction factors, subchannel friction factors and mixing parameters. Nucl. Eng. Des. 92 (2), 227–251.
11
Contents lists available at ScienceDirect
Progress in Nuclear Energy journal homepage: http://www.elsevier.com/locate/pnucene
Numerical analysis of flow resistance characteristics in an inclined rod bundle channel Yinxing Zhang a, Puzhen Gao a, *, Xiaoqiang He a, Chong Chen b, Qiang Wang b, Chunping Tian c a
College of Nuclear Science and Technology, Harbin Engineering University, Harbin, 150001, PR China China Ship Development and Design Center, Wuhan, 430064, PR China c Wuhan 2nd Ship Design and Research Institute, Wuhan, 430060, PR China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Inclination Rod bundle Friction resistance coefficient
The thermal-hydraulic characteristics of the rod bundle channel under ocean conditions have attracted much attention. In this paper, the three-dimensional CFD numerical simulation software STAR-CCMþ is used to study the frictional resistance characteristics of the rod bundle under non-vertical conditions at inclination angles of 0� , 15� , 30� and 45� , so as to provide a reference for future reactor core design of floating nuclear power plants. Using CFD, the relationship between the frictional resistance coefficient and the Reynolds Number (Re) is ob tained in a wide range of Re, including the laminar regime, the transition regime and the turbulent regime. The predicted correlation results are good and have an estimated error within �10%. The influence of the heating flux and the inlet temperature of the test bundle is considered, and a kinematic viscosity correction coefficient is introduced. Owing to the structural characteristics of the square rod bundle channel, this paper considers two different directions of tilt: rotations about an axis perpendicular to the channel faces and about an axis which passes through opposite corners of the channel. A difference in the frictional resistance coefficient between in clined and vertical conditions only occurs in the laminar flow regime, and there is no frictional resistance co efficient difference in the transition regime or the turbulent regime. Additionally, the frictional resistance coefficient is reduced considerably if the boundary layer is disturbed, due to the reversed fluid flow in the laminar regime.
1. Introduction As the basic structure of the nuclear reactor core, the rod bundle channel has naturally become the research topic of scientists. Since floating nuclear power plants are operating under ocean conditions, the core will inevitably tilt and roll due to the effects of ocean conditions. Such motion will affect nuclear reactors in varying degrees. The purpose of this paper is to investigate the effect of inclined conditions on the flow resistance characteristics of the core structure of a nuclear reactor, the rod bundle channel, through numerical simulations. The flow resistance characteristics in the rod bundle channel are expressed in the form of the frictional resistance coefficient, which can be used as a reference for future research on offshore floating nuclear power plants. Many scholars have studied related aspects of flow characteristics in rod bundles. C. Lifante et al. (Lifante et al., 2014) used ANSYS CFX to perform single-phase numerical research on a 3 � 3 rod bundle. The effects of different geometries are considered in the paper, including the
effects of the inlet vessel and the grid spacer. Their paper paves the way for numerical analysis of multiphase flow in order to accurately predict the boiling water flow in the rod bundle channel under high pressure. Han Wang et al. (Wang et al., 2016) studied the heat transfer charac teristics of supercritical pressurized water in a 2 � 2 rod bundle channel by numerical simulation. They showed that the SSG Reynolds Stress Model (Wang et al., 2016) can make the best prediction for normal and enhanced heat transfer models and described the relationship between heat transfer characteristics and pressure, mass flux and heat flux of supercritical water in the rod bundle channel. K. Zhang et al. (Zhang et al., 2016) conducted experimental and theoretical studies on verti cally upward two-phase flow of air-water across staggered rod bundles with inclination angles from the vertical of 0� , 45� and 90� . They developed pressure drop correlations for single-phase fluids in rod bundle channels at different inclination angles. The two-phase friction multipliers they developed successfully predict the two-phase pressure drop data in inclined rod bundles. Faruk A. Sohag et al. (Sohag et al., 2017) used ANSYS Fluent 16 to conduct a careful study of a vertically
* Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (P. Gao). https://doi.org/10.1016/j.pnucene.2020.103247 Received 28 November 2018; Received in revised form 26 October 2019; Accepted 8 January 2020 Available online 30 January 2020 0149-1970/© 2020 Elsevier Ltd. All rights reserved.
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Nomenclature P D d Re a L n ΔPf l ui u g ct Fi v g0 E ΔP Sh k
keff p T T1 T2
rod pitch, m diameter of the rod, m hydraulic diameter, m Reynolds number rectangular casing width, m second piezometric section length, m number of the rods frictional pressure drop, Pa the distance to the wall, m x-direction velocity component, m/s velocity vector, m/s gravitational acceleration, m/s2 kinematic viscosity correction coefficient external body forces, N/m3 velocity, m/s initial gravitational acceleration, m/s2 empirical constant in turbulence model total pressure drop, Pa the other volumetric heat sources turbulence kinetic energy, m2/s2
effective conductivity, W/m2⋅K system pressure, MPa temperature, K tilt 1 tilt 2
Greek letters δij Kronecker delta tensor ρ density, kg/m3 λ frictional resistance coefficient μ dynamic viscosity, Pa⋅s υ kinematic viscosity, m2/s τij Reynolds stress tensor ω specific dissipation rate, 1/s φ intermittency factor θ inclined angle, � Subscripts i o w f
inlet outlet wall fluid
studied the incompressible, isothermal and fully developed turbulent flow pressure drop in a hexagonal rod bundle channel. The results show that the number of rods in a rod bundle has no measurable effect on the pressure drop coefficient. He (Rehme, 1973) also predicted the hy draulic losses in multirod fuel bundles over a broad range, which is necessary for reactor design calculations. Zhu et al. (Zhu et al., 2018) carried out an experimental study on flow resistance characteristics in a 3 � 3 rod bundle channel under natural circulation rolling motion conditions. They only fitted laminar data on frictional resistance coef ficient and did not consider the influences of heating power on the friction coefficient changes. Due to the limitation of experimental con ditions, they could only obtain a selected ranges of Reynolds number (Re) conditions, consequently the results for the turbulent regime were difficult to achieve, and the laminar regime had few experimental re sults. Based on the above description, it can be seen that there are many researchers working on the thermal-hydraulics characteristics in rod bundle channels, which is of great significance for the nuclear industry. In this paper, the numerical simulation method can overcome experimental limitations and estimate the frictional resistance coeffi cient under all conditions, including the laminar, the transition and the turbulent regimes. We then analyze some conditions that our experi ment cannot achieve, like inclination in different directions. The paper aims to obtain the correlation of frictional resistance coefficient over a broad range of Reynolds numbers under inclined conditions by computational fluid dynamics (CFD). In this study, not only the incli nation angles of 0� , 15� , 30� and 45� were considered, but also the in fluence of two different tilt axes on the frictional resistance coefficient were considered. This paper focuses on the frictional resistance char acteristics of the fluid in the laminar regime. The laminar flow condition are likely to occur in accident conditions of nuclear power plants. Therefore, the paper has important guiding significance for the safety analysis of nuclear power plants. In addition, the change of fluid phys ical properties caused by the heating power in the test section was also studied in detail in order to get a more accurate expression of the fric tional resistance coefficient of the rod bundle channel. It is worth mentioning that the reversed flow phenomenon in laminar flow region under inclined conditions found in this paper occurs only when the outlet pressure is set to a constant value.
heated rod bundle in mixed and forced convection. They used the Shear-Stress Transport (SST) K-Omega model, which is also used in this paper, to study the heat transfer of a vertical rod-bundle with the rela tionship of spacer grid spacing and spacer grid blockage ratio. The simulation results are in good agreement with some notable correlations found in the literature. Deqi Chen et al. (Chen et al., 2016) carried out a numerical study of the thermal-hydraulic characteristics of a 5 � 5 rod bundle in a reactor. Changes in thermal hydraulic behavior downstream of the spacer grid caused by dimple, spring and mixing vane were described in detail, including with and without the spacer grid. The study found that the secondary flow induced by the spacer grid could have a significant impact on heat transfer. C.C. Liu et al. (Liu et al., 2012) evaluated the turbulence model of the fuel rod bundle with a spacer assembly and detailed the differences in wall treatment, mesh refine ment and appropriate definition of boundary conditions that would have a significant impact on the numerical simulation results. Through that study, they were able to determine the best enhanced wall treatment and the most appropriate turbulence models for modeling nuclear reactor coolant systems. F. C. Engel et al. (Engel et al., 1979) tested the pressure drop across wire-wrapped hexagonal rod bundles in laminar, transition, and turbulent parallel flow regions using air, water and sodium. A smooth transition region from laminar to turbulent flow was found to occur over the Reynolds number (Re) range from 400 to 5000. They developed correlations for the frictional resistance coefficient for different flow regimes and also a general laminar flow friction factor correlation with different wire-wrap lead pitch-to-diameter ratios. Mary V. Holloway et al. (Holloway et al., 2008) studied the convective heat transfer in the turbulent regime through rod bundles that can represent nuclear fuel rods used in pressurized water reactors. Using air as the working fluid, they measured the single-phase convective heat transfer coefficients for flow downstream of support grids in a rod bundle and obtained a general correlation to predict the heat transfer development downstream of support grids. E. Bubelis and M. Schikorr (Bubelis and Schikorr, 2008) summarized the existing wire-wrapped fuel bundle friction factor and pressure drop correlations and qualitatively evaluate which of the existing friction factor correlations is the best, which could be reflected in matching the results of the existing large body of experimental data under different coolant conditions. That paper is of great help to scholars looking for a large number of references in the study of rod bundle channels. Klaus Rehme (1972) systematically 2
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
2. Experimental facility In order to ensure the accuracy of numerical simulation, experi mental research should be carried out before the CFD simulation, and the experimental data thus obtained should used to verify the model. We have done such an experimental verification of our model. The experi mental facility shown in Fig. 1 has been described in detail in the Yinxing Zhang’s paper (Zhang et al., 2019). The test section shown in Fig. 2 is a 3 � 3 rod bundle channel made of stainless steel with a rectangular casing. Three pressure ports are located in one side of the rod bundle, as shown in Fig. 3. The length of the two piezometric sections L2 and L are 150 mm and 400 mm, respectively. The frictional pressure drop along the rod bundle channel, which is of most concern in this paper, is measured by the second piezometric section. For more details, please refer to Yinxing Zhang’s paper (Zhang et al., 2019). The uncertainty of the Reynolds number and the friction resistance coefficient are analyzed by the error transfer formula. The calculated uncertainties of the Reynolds number and the frictional resistance co efficient are 2% and 5%, respectively, according to the method of J. P. Holman (Holman and Gajda, 2001). Since this paper focuses on the CFD analysis, the details of the experimental procedure are not described in this paper. The inclination angle of the experimental facility is achieved by a crank rocker mechanism. For the data used for the validation of this paper’s model, the inclination angle is 20� , as determined by the acquisition system.
Fig. 2. Cross section rod bundle channel structure.
the modeling geometric structure should only be the part between pressure tube 2 and pressure tube 3 in Fig. 3. It is then helpful to match the numerical simulation results with the experimental results and to model with the 3D-CAD Modeler directly in STAR-CCMþ 11.06. In this paper, two kinds of mesh elements were used to generate grids, including a polyhedral mesh and an extruder mesh. The use of the extruder mesh can greatly reduce the amount of mesh points, which has the benefit for saving the calculation cost, under the premise of ensuring the quality of calculation. In addition, the prism layer mesh model is applied to solve boundary conditions on all geometry walls. The ge ometry, including the mesh, is shown in Fig. 4. The surface remesher can be used to improve the overall quality of an existing surface and opti mize it for the volume mesh models. To ensure the accuracy of the numerical calculation, a grid inde pendent test should be performed for the simulation model. The stabi lization of the frictional resistance coefficient (λ), the coolant outlet temperature (To), the total pressure drop (ΔP), and the coolant outlet velocity (vo) were used for calculating convergence. The total pressure
3. Numerical methods 3.1. Modeling and meshing In order to simulate the flow resistance characteristics of 3 � 3 rod bundle channel under tilted conditions, this paper uses the CFD software STAR-CCMþ 11.06 based on the finite volume method to do the nu merical calculation. This article only focuses on the frictional pressure drop in the rod bundle channel, not the influence of the spacer grid on the resistance characteristics in the rod bundle channel. Consequently,
Fig. 1. Thermal-hydraulic experimental facility. 3
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
drop is the direct result of the calculation relevant to the mesh size. It is useful to change the number of layers in the extrusion parameter to refine the mesh. After refinement of the mesh, four different numbers of grids are generated on the geometry model. The number of grids are 346,757; 658,769; 791,047 and 981,486, respectively, correspond to model 1, model 2, model 3 and model 4. The values of λ, To ΔP and vo under different mesh models are shown in Table 1. As can be seen from the table, the relative error of model 3 has reached within 0.1%, to meet the accuracy requirements. In this paper, the model 3 with 791,047 grids was adopted for all following simulations. 3.2. Governing equations and boundary conditions CFD provides a comprehensive range of modeling capabilities for incompressible and compressible, laminar, transitional and turbulent fluid problems. Three-dimensional modeling can be used to study the thermal-hydraulic properties of complex geometric structures. For example, the frictional resistance characteristics of a 3 � 3 rod bundle channel under different inclination conditions are discussed in this paper. The governing equations of coolant properties are written as follows: Table 1 Grid independence test.
Fig. 3. Longitudinal section rod bundle channel structure.
Model
Grid number
λ
To [oC]
ΔP [Pa]
vo[m/s]
#1 #2 #3 #4
346,757 658,769 791,047 981,486
0.036309 0.03585 0.036214 0.036206
28.87317 29.02993 28.87765 28.87803
4266.528 4261.947 4265.591 4265.512
0.550215 0.5502278 0.5502205 0.5502213
Fig. 4. The mesh applied in the geometry mode:(a) cross section, (b) detail of cross section, (c) global. 4
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Continuity equation:
∂ρ v Þ¼0 þ r ⋅ ðρ! ∂t
d¼
(1a)
Momentum equation: �
∂ ∂ ρui uj ¼ ðρui Þ þ ∂t ∂xj where � � �� ∂ui ∂uj τij ¼ μ þ ∂xj ∂xi
Re ¼
∂p ∂τij þ þ ρgi þ Fi ∂xi ∂xj
(2a)
2 ∂ul μ δij 3 ∂xl
(3a)
∂ ∂ ∂ ∂T k ðρEÞ þ ðui ðρE þ pÞÞ ¼ ∂t ∂xi ∂xi eff ∂xi
X � hj0 Jj0 þ uj τij eff
λ¼
(4a)
0
j
SST K-Omega model (Menter and Kuntz, 2004): The transport equations for the kinetic energy k and the specific dissipation rate ω are:
� ∂ ðρωÞ þ r ⋅ ρω v ∂t
vg
��
¼ r ⋅ ½ðμ þ σk μt Þrk� þ Pk
ρβ* ðωk
ω0 k0 Þ þ Sk (5a)
vg
��
¼ r ⋅ ½ðμ þ σ ω μt Þrω� þ Pω
ρβ ω2
�
ω20 þ Sω (6a)
where, vandvg are the mean velocity and the reference frame velocity relative to the laboratory frame, respectively, Pk and Pω are production terms, k0, ω0 are the ambient turbulence values that counteract turbu lence decay, Sk and Sω are the user-specified source terms.
σk ¼ 1
0:15F1 ; σ ω ¼ 0:856
β * ¼ 0:09; β ¼ 0:0828
0:356F1
0:0078F1
(7a) (8a)
where, ! !#! pffiffi 2k k 500υ ; 2 min max ; l CDkω 0:09ωl l2 ω
" F1 ¼ tanh
� 1 CDkω ¼ max rk ⋅ rω; 10
ω
υ
¼
� v 4a2 π D2 n υð4a þ πDnÞ
(2b)
2dΔPf Lρv2
(4b)
As previously stated our modeling validation was under the condi tion of an inclination angle, θ, of 20� . The calculation results obtained are shown in Fig. 5. It can be seen from the figure that SST (Menter) KOmega is most suitable for the application and calculation of the rod bundle channel, which could fit the experiment data best. Menter (1994) suggested using a blending function (which includes functions of wall distance) that would include the cross-diffusion term far from walls, but not near the wall. This approach effectively blends a K-Epsilon model in the far-field with a K-Omega model near the wall and cures the biggest drawback to applying the K-Omega model to practical flow simulations. Menter also introduced a modification to the linear constitutive equa tion and dubbed the model containing this modification the SST (shear-stress transport) K-Omega model. The SST model works best for rod bundle geometries, which is consistent with Faruk A. Sohag’s research (Sohag et al., 2017), where viscous flows are typically resolved and turbulence models are applied throughout the boundary layer. This paper will use this model as a predictor of the frictional resistance co efficient in rod bundle channel. The laminar flow model and the SST (Menter) K-Omega model are used to numerically simulate the same working conditions of the laminar regime. The results show that there is 0.01% deviation between the two models for the pressure drop calcu lation. Thus it is shown that the SST (Menter) K-Omega model can accurately simulate the laminar flow. Therefore, the SST (Menter) K-Omega model is also selected for the laminar regime in this paper. The error bars on experimental data from the calculated un certainties of the Reynolds number and the frictional resistance coeffi cient are 2% and 5%, respectively.
! þ Sh
vd
(1b)
The frictional resistance coefficient, λ, is calculated as follows: � θπ � ΔPf ¼ ΔP ρgL⋅cos (3b) 180
Energy equation:
� ∂ ðρkÞ þ r ⋅ ρk v ∂t
4a2 πD2 n 4a þ πDn
(9a)
� 20
(10a)
For the boundary conditions, the inlet is the velocity inlet, the outlet is a pressure outlet, the constant heat flux is used for the wall of the rod bundle, and the adiabatic boundary condition is used for the outer wall. All wall boundary conditions are set to be no-slip. 3.3. Modeling validation In order to ensure the accuracy of this numerical simulation for the later part of the paper, we present the verification of the numerical models of the rod bundle channel under inclined conditions by comparing the simulations to our experimental results. The experi mental data were obtained from the experimental facility described above. The specific experiment was conducted under the conditions of 0.4 MPa system pressure and 18 kW/m2 test section heating flux. The validation numerical simulation is carried out under the same conditions as the experimental conditions in order to find a suitable physical model for the numerical simulation in this paper. Six models were calculated and analyzed respectively, including SST (Menter) K-Omega, Realizable K-Epsilon, Realizable K-Epsilon Two-Layer, Reynolds Stress Turbulence, Standard (Wilcox) K-Omega, Standard K-Epsilon. The parameters used for the determination of the Reynolds number are as follows:
Fig. 5. Frictional resistance coefficients calculated by CFD and experimental data (p ¼ 0.4Mpa, Q ¼ 18 kW/m2, θ ¼ 20� ). 5
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Todreas, 1986; Sadatomi et al., 1982)and the numerical simulation re sults. The results are most similar to Engel’s research, as they claimed that transition points occur at 400 and 5000 Re. This article will study the relationship between λ and Re on the basis of Engel’s research. According to the CFD simulation results, the relationship between the frictional resistance coefficient λ and Re under vertical condition in the rod bundle channel is fitted with a fixed heating power as for the laminar region λ ¼
88 Re
for the transition region λ ¼
88 ð1 Re
(5b) ϕÞ1=2 þ
0:24 1=2 ϕ Re0:23
(6b)
0:24 Re0:23
(7b)
Re 400 3600
(8b)
for the turbulent region λ ¼ where Fig. 6. Frictional resistance coefficients calculated by CFD and different empirical correlations (p ¼ 1Mpa, Q ¼ 20 kW/m2, Ti ¼ 300 K, θ ¼ 0� ).
ϕ ¼ intermittencyfactor ¼
Based on the relational expression above, the frictional resistance coefficient calculated at different Reynolds numbers are compared with experimental data as shown in Fig. 7. The frictional resistance coeffi cient calculated from CFD simulation are shown at the same time. As can be seen from this figure, the difference among them is within �10%, which satisfies the prediction accuracy requirement. The experimental data under inclined condition in Fig. 7 is the same as that in Fig. 5. It can be seen that our prediction formula can not only predict the frictional resistance coefficient under vertical conditions, but also predict the λ of the rod bundle channel under inclined conditions. However, our pre diction formula only applies to the transition and the turbulent regime. For the laminar regime, it is not applicable under inclined conditions as will be shown in detail in Section 4.2. In this paper, the effects of fluid properties on the frictional resis tance of the rod bundles were investigated by adjusting the heating flux of the test section to be 20, 100, 200 kW/m2. The numerical simulation results are shown in Fig. 8. From this figure, it can be seen that as the heat flux of the rod bundle increases, the frictional resistance coefficient gradually decreases. This is because the average temperature of the fluid goes up as the heating flux increases. As a result, the viscosity of the fluid is reduced, resulting in a reduced resistance pressure drop. From the figure we also notice that the Reynolds number range is different for different heating flux. The greater the heating flux, the greater the starting point of the Reynolds number value in Fig. 8. This is taken into account to ensure that the rod bundle channel is filled with single-phase
4. Results and discussion 4.1. Vertical condition Before studying the hydraulic characteristics of the rod bundle channel under inclined conditions, the vertical condition was investi gated as a special tilt (θ ¼ 0� ). In order to study the characteristics of frictional resistance coefficients as a function of the Reynolds number in the rod bundle channel, the method of adjusting Re was performed by changing the inlet velocity of the test section. Fig. 6 shows the variation of the frictional resistance coefficients with Re. In the simulation pro cess, the inlet temperature of 300 K was kept constant, the system pressure was 1 MPa, the heat flux was 20 kW/m2. It can be seen from the figure that the frictional resistance gradually decreases with the increase of the Reynolds number. It is noteworthy that transition points occur at Re ¼ 400 and Re ¼ 4000 in the graph, which is a little different from Engel’s (Engel et al., 1979) study. From the data in Fig. 6, it could be proposed that for the laminar-transition Reynolds number, 400, and the transition-turbulent Reynolds number, 4000. The laminar, transition and turbulent regions are then defined as Re < 400, 400 < Re < 4000, and Re > 4000, respectively. From this figure, it can be seen that there are some differences between other scholars’ research (Cheng and
Fig. 7. Comparison among predicted frictional resistance coefficient, CFD re sults and experimental results (p ¼ 0.4Mpa, Q ¼ 18 kW/m2, θ ¼ 20� ).
Fig. 8. Frictional resistance coefficients with different test section heating powers (p ¼ 1Mpa, Ti ¼ 300 K, θ ¼ 0� ). 6
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
fluid. It can also be seen from the figure that when Re > 20,000, the change of heat flux does not affect the frictional resistance characteris tics. It can be considered that the fluid temperatures at different heat flux are substantially the same when the fluid velocity is quite large, resulting in the same fluid viscosity and the same resistance pressure drop. In order to quantitatively express the influence of the fluid prop erties on the frictional resistance characteristics in the rod bundle channel, the kinematic viscosity correction coefficient (Sieder and Tate, 1936) is introduced on the basis of the previously obtained formulae (5), (6), and (7) to reflect the influence of the fluid viscosity on the resistance characteristics. The expression is as follows: � �0:3 υw ct ¼ 1:05 (9b)
υf
where, υw and υf are the kinematic viscosity calculated by means of the average wall temperature of the rod bundle and the average fluid tem perature, respectively. The correlation of the frictional resistance coefficient in the rod bundle channel under vertical conditions can be summarized as � �0:3 92 υw for the laminar region λ ¼ (10b) Re υf � for the transition region λ ¼
for the turbulent region λ ¼
92 ð1 Re
ϕÞ1=2 þ
� �0:3 0:25 υw Re0:23 υf
0:25 1=2 ϕ Re0:23
�� �0:3
υw υf
Fig. 10. Frictional resistance coefficient changes with inlet temperatures at different mass flow rate (p ¼ 1Mpa, Q ¼ 20 kW/m2, θ ¼ 0� ).
(11) (12)
Again, the simulation results are compared with the results of the prediction formula above. The results are shown in Fig. 9. It can be seen from the figure that the formula has a good prediction with an uncer tainty within �10%. The simulation conditions of the previous study are all constant inlet temperatures. However, inlet temperature is also an important control parameter during experimental research. To study the effect of inlet temperature on the frictional resistance coefficient, the mass flow rates studied in this paper are 0.04, 0.4 and 0.2 kg/s, respectively. The inlet temperature is studied at 280–400 K, the heat flux is 20 kW/m2 and the system pressure is 1 MPa. The simulation results are shown in Fig. 10. It can be seen from the figure that the inlet temperature at high mass flow has no influence on the frictional resistance coefficient, and the frictional resistance coefficient decreases with increasing inlet temper ature at low mass flow. When the mass flow is low, especially in the
Fig. 11. Relationship between predicted frictional resistance coefficient and CFD results with different mass flow and inlet temperature (p ¼ 1Mpa, Q ¼ 20 kW/m2, θ ¼ 0� ).
laminar regime, the higher the temperature, the lower the viscosity of the fluid at the same heating flux, resulting in the smaller the resistance coefficient. At high mass flow rates, especially in fully developed tur bulent regime, the main determinant of pressure drop in high-speed flowing fluids is the velocity, at which point the effect of viscosity is not significant. It has been verified that the characteristics of the fric tional resistance in the rod bundle channel under the condition of var iable inlet temperature can still be expressed by formulae (10), (11), and (12), and the uncertainty is within �10%, as shown in Fig. 11. It can be seen that formulae (10), (11), and (12) in this paper can be applied to the prediction of frictional resistance coefficient, not only for a wide range of Reynolds number, but also for different heating power and coolant average temperature. 4.2. Inclined condition In order to study the influence of inclination on the frictional resis tance characteristics in the rod bundle channel, the changing of the di rection of gravity in the CFD simulation must be considered. The specific method is as follows: the direction of gravity in the CFD simulator is
Fig. 9. Relationship between predicted frictional resistance coefficient and CFD results with different test section heating power (p ¼ 1Mpa, Ti ¼ 300 K, θ ¼ 0� ). 7
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 12. Our definitions of tilted 1 and 2, wherein the tilting is a rotation about the y-axis, as shown. Table 2 The simulation inclined angles and Reynolds numbers for the studies in this paper. T1 represents tilted 1 and T2 represents tilted 2. θ/�
0
tilt style Re
changed for the tilts as shown below: g0 cos θÞ
30
45
T1
T2
T1
T2
T1
T2
312 349 387 426 500 649 797 1527 3699 4046 13016 21647 43231 72017
314 350 387 426 500 649 797 1527 3699 4053 13016 21647 43231 72017
309 350 385 425 500 648 796 1526 3699 4046 13016 21647 43231 72017
313 348 386 423 500 648 795 1526 3698 4053 13016 21647 43231 72017
309 351 385 425 499 648 795 1525 3699 4046 13016 21647 43231 72017
312 348 386 423 499 648 794 1525 3698 4053 13016 21647 43231 72017
were the same as for the vertical orientation above. The results of the calculation under tilted 1 and 2 conditions are shown in Fig. 13. The simulated inclination angle and Reynolds number are shown in Table 2. It can be seen from the figure that the inclination angles have no influence on frictional resistance coefficient in the transition regime and turbulent regime, but there is a significant affect in the laminar regime. The frictional resistance coefficient in the laminar flow region under the inclined condition is much smaller than that in the vertical condition. It can also be noticed that the frictional resistance coefficient under the titled 2 condition is slightly smaller than that under the titled 1 condi tion, and the difference of the coefficient between tilted 1 and 2 will be smaller as the inclination angle increases. After tilting, the fluid in the rod bundle channel generates a radial velocity, which means the velocity has components in the x and y di rections, under the influence of gravity and the constraint of the chan nel. This can be seen from the distributions of the radial velocity at the
Fig. 13. Frictional resistance coefficient under tilted 1 (T1) and 2 (T2) condi tion (p ¼ 1Mpa, Q ¼ 20 kW/m2, Ti ¼ 300 K).
! g ¼ ðg0 sin θ; 0;
279 350 389 427 502 651 799 1528 3698 4052 13016 21647 43231 72017
15
(13)
where, g0 is the gravitational acceleration under vertical condition. However, although the rod bundle channel is symmetrical, with a square cross section, there will be some differences in the flow when the channel is inclined at different directions. This paper selected the two most representative directions. We refer to these directions as “tilt 1” and “tilt 2”, as shown in Fig. 12. Consequently, the STAR-CCMþ was used to numerically simulate the rod bundle channel at 15� , 30� and 45� inclination angles with the two inclination directions described above. The heat flux was 20 kW/m2, the system pressure was 1 MPa, and the inlet temperature was 300 K, which 8
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 14. Radial velocity distributions at the outlet under different conditions.
outlet under vertical and inclined conditions in Fig. 14. In the figures, the velocity direction at higher flow rates are marked in red. At the same time, the heated fluid will undergo thermal stratification, and reversed flow will occur under the action of gravity. The reason why reversed flow happens near the outlet is as follows: The titled heated fluid flows up vertically under the influence of density difference at low Reynolds number. However, the fluid velocity is too small to keep flowing up. The buoyancy is not enough to support the fluid to continue to move up wards and stay due to gravity. Then the reversed flow will occur as shown in Fig. 15 enclosed in red. The reversed flow can inhibit the heat transfer of the fluid in the rod bundle channel, which in turn causes the fluid temperature to decrease and the fluid viscosity in the channel to increase. According to the foregoing description, an increase in viscosity causes an increase in pressure drop. But the actual situation is the opposite, so we can think that reversed flow’s disturbing the boundary layer is the main cause of the reduction of the pressure drop in rod bundle channel. It can be seen from Fig. 14 that the maximum value of
the fluid velocity in the radial direction increases with the inclination angle under the buoyancy force. In the process of the numerical simu lation, the number of reversed flow faces on outlet under different conditions are also indicated in Fig. 14. The higher the maximum fluid velocity in the radial direction, the more severe the reversed flow and the greater the damage to the boundary layer. Fig. 15 shows the reversed flow in the longitudinal section of the full flow channel. The smaller the angle of inclination, the smaller the proportion of radial velocity to the axial velocity, which is considered as the mainstream velocity, resulting in the smaller the damage to the boundary layer, and then the larger the frictional resistance coefficient with the other conditions being the same. The proportion of radial velocity to the axial velocity in the laminar regime is large, and the proportion in the transition regime and turbulent regime is small because of the high mainstream velocity and can be neglected. The reversed flow phenomenon will not occur at high mainstream speed, which is the reason why a tilt has no effect on the frictional resistance coefficient in the transition regime and turbulent 9
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
Fig. 15. Axial velocity distributions along the full flow channel, at the position shown by the plane at the pink line. The area enclosed in red in the top part of the figure shows the reversed velocity vectors.
numerical simulation. The correlation of the frictional resistance coef ficient under vertical condition was obtained and further research was conducted with different tilt directions and angles. This paper focuses on the analysis of the effect of inclination on frictional resistance charac teristics of the rod bundles under low Reynolds number conditions (laminar flow), which serves as a guide for the occurrence of accident conditions in nuclear power plants. First, the numerical simulation results under vertical conditions were analyzed. The relationship between the frictional resistance coefficient λ and the Reynolds number was obtained in laminar, transition and tur bulent regimes. A kinematic viscosity correction coefficient was then introduced to modify the relationship. Actual experimental data were compared to the predicted results from Equations (10)–(12), and they were in good agreement. The prediction result from the correlation agreed with the CFD simulation values to within �10% uncertainty. We therefore propose our general correlation of the frictional resistance coefficient in future investigations for the inclined rod bundle channel to reduce the cost of experimentation and CFD simulation. Second, for the numerical simulation study under inclined condi tions, this paper considers two different inclination directions - tilt 1 and 2, and three inclination angles of 15� , 30� and 45� . No matter what kind of inclination, there is no influence on the frictional resistance coeffi cient in the transition regime and the turbulent regime, and the fric tional resistance coefficient in those regions can still be predicted by the expression for the non-inclined, i.e. vertical, condition. However, there are some differences between the vertical condition and the inclined conditions in laminar regime. The frictional resistance coefficient will be smaller under the inclined condition than that under the vertical con dition in the laminar regime. The results show that there is not much different between the coefficient under tilted 1 and 2 conditions. The frictional coefficient under titled 2 condition is slightly smaller than that under titled 1 condition, and the difference of the coefficient between tilted 1 and 2 condition will be smaller as the inclination angle increases.
Fig. 16. Axial velocity distributions along the full flow channel, which is shown in Fig. 15. (Re ¼ 648, tilted 2 θ ¼ 45� ).
regime. In order to verify the above statement, Fig. 16 shows the fluid velocity distribution in the transition regime with a low Re ¼ 648 and a large inclination angle θ ¼ 45� in the same full flow channel as that in Fig. 15. It can be seen from the figure that no reversed flow occurs. Without interference to the boundary layer, there is no effect on the frictional resistance coefficient. Under the condition shown in Fig. 16, the fluid is more likely to reverse than other conditions in transition regime and turbulent regime. It can be explained that the fluid in the transition regime and the turbulent regime does not produce reversed flow to interfere with the boundary layer as the condition in Fig. 16 has no reversed flow. Therefore, the frictional resistance coefficient in in clined condition is the same as that in the vertical condition in the transition regime and turbulent regime. The reversed flow phenomenon under low flow rate conditions is difficult to occur during the normal operation of the reactor, but it may occur when the reactor is in an accident condition. Therefore, this paper carried out research in this field in detail. 5. Conclusions In this paper, the frictional resistance characteristics of the rod bundles under vertical and inclined conditions are studied by CFD 10
Y. Zhang et al.
Progress in Nuclear Energy 122 (2020) 103247
In addition, the greatly reduced frictional resistance coefficient in the laminar regime can be understood as the result of reversed flow, which disturbs the boundary layer in tilted conditions. Any other more complicated channels could be further studied using the research method in this paper. The method will guide the safe development of the nuclear industry by researching the thermal hy draulic characteristics of different channels under different inclined conditions.
Engel, F.C., Markley, R.A., Bishop, A.A., 1979. Laminar, transition, and turbulent parallel flow pressure drop across wire-wrap-spaced rod bundles. Nucl. Sci. Eng. 69 (2), 290–296. Holloway, M.V., Beasley, D.E., Conner, M.E., 2008. Single-phase convective heat transfer in rod bundles. Nucl. Eng. Des. 238 (4), 848–858. Holman, J.P., Gajda, W.J., 2001. Experimental Methods for Engineers. McGraw-Hill, New York, 7. Lifante, C., Krull, B., Frank, T., et al., 2014. 3�3 rod bundle investigations, CFD singlephase numerical simulations. Nucl. Eng. Des. 279, 60–72. Liu, C.C., Ferng, Y.M., Shih, C.K., 2012. CFD evaluation of turbulence models for flow simulation of the fuel rod bundle with a spacer assembly. Appl. Therm. Eng. 40, 389–396. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence modeling for engineering applications. AIAA J. 32 (8), 1598–1605. Menter, F.R., Kuntz, M., 2004. Adaptation of Eddy-Viscosity Turbulence Models to Unsteady Separated Flow behind Vehicles, pp. 339–352. Rehme, K., 1972. Pressure drop performance of rod bundles in hexagonal arrangements. Int. J. Heat Mass Transf. 15 (12), 2499–2517. Rehme, K., 1973. Pressure drop correlations for fuel element spacers. Nucl. Technol. 17 (1), 15–23. Sadatomi, M., Sato, Y., Saruwatari, S., 1982. Two-phase flow in vertical noncircular channels. Int. J. Multiph. Flow 8 (6), 641–655. Sieder, E.N., Tate, G.E., 1936. Heat transfer and pressure drop of liquids in tubes. Ind. Eng. Chem. 28 (12), 1429–1435. Sohag, F.A., Mohanta, L., Cheung, F.-B., 2017. CFD analyses of mixed and forced convection in a heated vertical rod bundle. Appl. Therm. Eng. 117, 85–93. Wang, H., Bi, Q., Wang, L., 2016. Heat transfer characteristics of supercritical water in a 2�2 rod bundle – numerical simulation and experimental validation. Appl. Therm. Eng. 100, 730–743. Zhang, K., Tian, W.X., Hou, Y.D., et al., 2016. Pressure drop characteristics of vertically upward flow in inclined rod bundles. Exp. Therm. Fluid Sci. 78, 208–219. Zhang, Y., Gao, P., Kinnison, W.W., et al., 2019. Analysis of natural circulation frictional resistance characteristics in a rod bundle channel under rolling motion conditions. Exp. Therm. Fluid Sci. 103, 295–303. Zhu, Z., Wang, J., Yan, C., et al., 2018. Experimental study on natural circulation flow resistance characteristics in a 3�3 rod bundle channel under rolling motion conditions. Prog. Nucl. Energy 107, 116–127.
Acknowledgements The authors wish to thank the LLOYD’S REGISTER FOUNDATION for financial support. The authors would like to thank Professor W. Wayne Kinnison for his help on this article, including language and content guidance. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pnucene.2020.103247. References Bubelis, E., Schikorr, M., 2008. Review and proposal for best fit of wire-wrapped fuel bundle friction factor and pressure drop predictions using various existing correlations. Nucl. Eng. Des. 238 (12), 3299–3320. Chen, D., Xiao, Y., Xie, S., et al., 2016. Thermal–hydraulic performance of a 5�5 rod bundle with spacer grid in a nuclear reactor. Appl. Therm. Eng. 103, 1416–1426. Cheng, S.-K., Todreas, N.E., 1986. Hydrodynamic models and correlations for bare and wire-wrapped hexagonal rod bundles — bundle friction factors, subchannel friction factors and mixing parameters. Nucl. Eng. Des. 92 (2), 227–251.
11