Investigation of velocity gradient as driving force of flow pulsation in fuel assemblies

Transcription

1 Investigation of velocity gradient as driving force of flow pulsation in fuel assemblies by Patrick F. Everett SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCE AND ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERING AT MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE Massachusetts Institute of Technology. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature of Author: Certified by: Accepted by: Patrick Everett Department of Nuclear Science and Engineering May 18, 2017 Emilio Baglietto Norman C. Rasmussen Associate Professor of Nuclear Science and Engineering Thesis Supervisor Michael Short Assistant Professor of Nuclear Science and Engineering Chairman, NSE Committee for Undergraduate Students

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3 Investigation of velocity gradient as driving force of flow pulsation in fuel assemblies by Patrick F. Everett Submitted to the Department of Nuclear Science and Engineering on May 18, 2017 in partial fulfillment of the requirements for the degree of Bachelor of Science in Nuclear Science and Engineering ABSTRACT The presence of quasi-periodic flow pulsations in fuel assemblies has been observed since the 1960 s but is still not fully understood. Current design and licensing practices for nuclear reactor fuel mostly rely on 1-dimensional subchannel simulation tools, which might not accurately predict the increased subchannel mixing caused by flow pulsations. The present work develops a quantitative relationship between subchannel mixing and the inter-subchannel velocity gradient, shown to be the driving force of flow pulsation. A sensitivity study on rod-bundle geometry, based on an experiment by Bardet and Balaras at George Washington University, was conducted with a URANS method in transient simulations using the commercial software Star-CCM+. A linear relationship was observed between crossflow mixing and v bulk, defined as the di erence in bulk velocities of adjacent subchannels. A threshold value of v bulk was seen close to 0.4 m/s, below which very little crossflow mixing was observed. Using these results, an analytical relationship between inter-subchannel velocity gradient and crossflow mixing could be developed and implemented into subchannel codes for more accurate modeling of flow in a fuel assembly. Thesis Supervisor: Emilio Baglietto Title: Norman C. Rasmussen Associate Professor of Nuclear Science and Engineering 2

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5 Contents 1 Introduction 7 2 Background Fuel Assembly Design Subchannel Analysis History of Flow Pulsation Current Knowledge of Flow Pulsation Modelling of Flow Pulsation GWU Experiment Methods 14 4 Results 18 5 Conclusion 22 4

6 List of Figures 1 Major PWR fuel assembly variants Fuel rod lattice Crossflow in 2-subchannel model Secondary flows in non-circular channels Flow oscillations from Hofmann Formation of vortices in gap region Experimental setup at George Washington University CAD model of computational domain Crossflow plane in computational domain Bu er and adjacent subchannels Crossflow velocity over time for P/D ratio of Crossflow velocity at t=15 seconds for P/D ratio of Average crossflow variance over time for constant bu er ratio Average crossflow variance over time for constant P/D ratio Average crossflow variance versus v bulk for all cases Peak and second peak frequencies versus v bulk for 11 cases

7 List of Tables 1 Geometric properties of base case of simulation Summary of simulation methods Summary of geometric properties of the 12 simulated cases

8 1 Introduction The success of nuclear power as a carbon-free electricity source depends heavily on its economic standing among electricity sources. Uncertainties associated with the design and with the measurement of operational parameters lead to a reduction in maximum power output of the plant in order to meet the required safety margins. As a result, much of the work in this industry is aimed at reducing uncertainties through improved modeling, resulting in greater power-plant e ciency. The design and licensing processes for nuclear reactor fuel primarily utilize 1-dimensional subchannel simulation tools, while leveraging the high resolution of computational fluid dynamics (CFD) as a support to evaluate the local e ect of grid spacers and confirm the accuracy of 1-D tools. While this approach has robustly supported the industry and delivered highly reliable fuel, complex unsteady flow phenomena such as flow pulsations across subchannels require further study [1]. Periodic flow pulsations in the transverse direction across adjacent subchannels have been observed in several experiments dating back to the 1960s [1]. As most previous work has studied tight-lattice rod bundles, it has not been shown that these pulsations occur in commercial pressurized water reactors (PWR), but this possibility has not been ruled out [2]. Turbulent mixing across subchannels is increased by these flow pulsations, and heat and momentum transfer have been shown to be greatly enhanced; as a result, the temperature and velocity distributions are di erent than expected [3, 4]. At high enough transverse flow velocities, it is possible that fuel rods undergo flow-induced vibration and could even lead to anticipated fuel failure [5]. An understanding of this phenomenon, and whether or not it might occur in existing PWRs, is necessary for a reactor operator to ensure safe operation of a reactor within its safety limits. Further, engineers of next-generation reactors must be able to predict and model these pulsations to create safe, economic reactor designs. This research was primarily inspired by the recent work by Philippe Bardet and Elias Bardaras at George Washington University (GWU). In this experiment, a partial PWR assembly was constructed to study the e ects of mechanical vibration caused by seismic activity on flow behavior. Contrary to expectations, periodic flow pulsations were observed, first computationally by Balaras and later in the test assembly while operating without mechanical vibration (Bardet, Balaras, Personal communications). As this experiment points to the possibility that these pulsations do occur in existing PWRs, careful study is needed to determine if this is a result of the experimental apparatus, namely a large bu er region surrounding the fuel assembly, or indeed a major discovery. While these pulsations have been observed in several experiments, there has not been a quantitative study on the relationship between flow pulsation and the axial velocity gradient in a fuel assembly. An investigation of the size of the bu er region surrounding the fuel assembly is necessary to understand the results of the GWU experiment. In this experiment, a model of the GWU apparatus was created using the commercially available Star-CCM+ software, and the bu er region size and rod spacing were varied while keeping 7

9 the Reynolds number constant. An unsteady Reynolds Averaged Navier-Stokes (URANS) method, based on the work of Baglietto (2006), was used in transient simulations, and converged statistics were accumulated for the mixing characteristics [6]. A quantitative relationship between inter-subchannel velocity gradients and crossflow mixing was found, deepening the understanding of this phenomenon. 2 Background 2.1 Fuel Assembly Design In a reactor, fuel rods are arranged in a lattice to form a fuel assembly. Rod spacing is constrained by both neutronics and heat transfer requirements. The spacing of fuel rods is typically defined by the pitch-to-diameter (P/D) ratio, shown in Figure 2. For existing PWRs, the P/D ratio is about 1.33; for fast reactors requiring no neutron moderation, the P/D ratio can be as low as 1.1 [7]. A single PWR fuel assembly contains fuel rods, typically in a 17x17 lattice [7]. The two major variants of PWR fuel assemblies are the Westinghouse-CE and Westinghouse, shown in Figure 1. (a) Westinghouse-CE fuel assembly (b) Westinghouse fuel assembly Figure 1: Two major variants of PWR fuel assemblies: Westinghouse-CE (left) and Westinghouse (right). On left, fuel rods are shown in orange and guide thimbles in white. On right, fuel rods are shown in red and guide thimbles in blue [8]. 2.2 Subchannel Analysis Because of the importance of accurately predicting temperature and velocity distributions within a fuel assembly, the development of thermal-hydraulic models has been a focus of the 8

10 nuclear industry since the 1950s. Recent improvements in computational fluid dynamics (CFD) are promising, but full-core simulation using CFD remains costly. In practice, the conservation equations of mass, momentum, and energy are solved using subchannel analysis. Subchannel analysis ignores the detailed distribution of velocity and temperature within a single subchannel, which is defined as the control volumes between fuel rods [9]. The averaged mass flow rate and fluid temperature are calculated within each subchannel by subchannel codes. Figure 2 shows a fuel assembly with interior, edge, and corner subchannels highlighted [10]. Figure 2: A 4x4 square lattice of fuel rods. Highlighted are interior, edge, and corner subchannels. The diameter (D) of fuel rods and pitch (P) between rods are used to define the rod bundle s pitch-to-diameter ratio (P/D) [10]. For the majority of subchannel analysis, subchannels are assumed to be individual, parallel channels. In reality, adjacent subchannels are open to interaction through the gap between neighboring fuel rods. This crossflow between subchannels must be accounted for in subchannel codes. The turbulent interactions between subchannels are described by a mixing coe cient, a parameter in subchannel codes. For a simplified model of 2 subchannels, shown in Figure 3, the crossflow from subchannel n to m is shown as W mn [11]. 9

11 Figure 3: A simplification of subchannel analysis using a 2-subchannel model. Crossflow from subchannel n to m is shown by W mn [11]. 2.3 History of Flow Pulsation During the 1960s, several experiments detected unusual turbulent intensities and high inter-subchannel mixing rates without a conclusive explanation [12, 13, 14, 15]. It was first explained by Skinner et al. that these results were driven by secondary flows, since the mixing rates were higher than could be described by turbulent di usion alone [16]. Formation of secondary flows in non-circular channels had been described by Nikuradse in 1926, shown in Figure 4 [17]. Figure 4: Secondary flows in non-circular channels as predicted by Nikuradse in It was thought that high inter-subchannel mixing rates in fuel assemblies were driven by these secondary flows [17]. During the following decades, numerous experiments were conducted to confirm that these high mixing rates were caused by secondary flows, as no other explanation had 10

12 gained significant support [18]. It was shown that secondary flow velocities in rod bundles are very small, on the order of 1% of the axial flow velocity, and do not play a major role in transporting momentum or energy through the gap regions [19, 20, 21]. Thus, it can be concluded that the high mixing rates that had been observed were not driven by secondary flows [1, 9]. Instead, it was shown that these turbulent intensities and mixing rates were driven by large-scale, turbulent structures [22, 23, 24]. These structures had first been observed by Hofmann in 1964 using a 7-rod bundle with an aluminum powder as tracer, shown in Figure 5, though his results were not reported externally [15]. The presence of these turbulent structures was confirmed by numerous experiments in the following years [22, 23, 25]. It is now generally accepted that the high inter-subchannel mixing rates observed in rod bundles is driven by these quasi-periodic turbulent structures, often referred to as flow pulsations [1]. Figure 5: The first (unreported) observation of quasi-periodic turbulent structures by Hofmann in Aluminum powder was used as a tracer to visualize flow oscillations [15]. 2.4 Current Knowledge of Flow Pulsation Flow pulsations in rod bundles originate from a velocity gradient across adjacent subchannels. Flow in the open subchannel region has a higher velocity than flow in the narrow gap region. As a result of this velocity di erence, vortices form in the mixing layer. In a rod bundle, two large channels are connected by a single narrow gap; thus, vortices are 11

13 formed on both sides of the gap. These vortices stabilize and move across the gap while being transported by the main flow in the axial direction [26]. It has been observed that the pair of vortices across a gap region are out of phase with each other by 180 and rotate in opposite directions, shown in Figure 6 [25, 27]. Figure 6: The vortices formed in a narrow gap region as a result of the gradient in velocity profile. Vortices are separated by a distance /2 and rotate in opposite directions across the gap [1]. As these vortices move across the gap region, there is a significant amount of crossflow between subchannels, as can be observed in the transverse velocity profile. The velocity, wall shear stress, and pressure signals each exhibit quasi-periodic behavior; this is the inspiration for the term flow pulsation [25, 28, 29]. It is worth noting that Meyer prefers the term vortex train, as he points out that the fuel assembly geometry is just a special case of a narrow channel connected to a larger channel where this phenomenon occurs. Spectral analysis of these physical properties indicate that flow pulsations have a characteristic frequency [23, 29]. Experiments by Hooper, Hooper and Rehme, and Moller have shown that the characteristic frequency increases with increasing Reynolds number [25, 28, 29]. The Strouhal number, a non-dimensional frequency often used in fluid mechanics, is defined as follows: Str = fd u where D is the rod diameter and u is the flow velocity. It was shown by Moller that the Strouhal number is independent of Reynolds number and inversely proportional to the gap width [26]. Since the velocity gradient that drives flow pulsation is heavily influenced by the rodbundle geometry, many experiments on this topic are studies of the influence of the P/D ratio. It has been shown that tight-lattice fuel assemblies, with small P/D ratio and narrow gap regions, are more prone to flow pulsation [29, 30]. It is suggested that existing PWRs, 12

14 with a P/D ratio close to 1.33, do not experience flow pulsation in significant amounts [2]. Experimental testing through a variety of rod-bundle geometries is challenging and costly, so recent research has been largely computational. 2.5 Modelling of Flow Pulsation The focus of recent work on this topic has been improving models of these flow pulsations in CFD. Steady-state CFD methods are unable to capture the contribution to turbulent mixing by large-scale coherent structures [31]. Unsteady CFD is typically performed using large eddy simulation (LES) or direct numerical simulation (DNS), but these methods are computationally expensive [31]. It was thought that these structures could not be captured by an unsteady Reynolds-averaged Navier-Stokes (URANS) approach until Merzari and Baglietto in 2009 [31]. Further comparison between LES and URANS methods in modelling a tight-lattice rod bundle showed good agreement between the two, though the LES approach was able to resolve smaller structures [5]. From spectral analysis, it can be observed that the energy of high-frequency turbulent structures is much smaller in URANS than LES; thus, the URANS approach is applicable when lower frequency structures are relevant, as is the case for flow pulsation [5]. 2.6 GWU Experiment Recently, an experiment by Philippe Bardet and Elias Bardaras at George Washington University demonstrated the presence of flow pulsation in a 6x6 lattice PWR assembly. Surrounding the assembly on two sides was a large bu er region, as shown in Figure 7. The purpose of this experiment was to evaluate the e ect of rod vibrations caused by seismic loads. Flow pulsation was not expected to occur in this configuration but was observed computationally by Balaras. Physical testing with no mechanical vibrations confirmed the presence of flow pulsation in this geometry. The present research further develops this experiment through a sensitivity study of the rod-bundle geometry. 13

15 (a) Top view (b) Side view Figure 7: The experimental setup at George Washington University. The 6x6 square lattice of fuel rods, held in place by the spacer grid, can be seen from the top view, surrounded on top and bottom by large bu er regions. Spacer grid can be seen from side view. 3 Methods The experimental setup from GWU was modeled using the commercial software Star- CCM+. Taking advantage of the symmetrical flow distribution, the computational domain was reduced from a 6x6 lattice to a single row of subchannels covering half of the rod bundle, roughly a 3x1 lattice, as shown in Figure 8. 14

16 (a) Top view (b) Isometric view Figure 8: CAD model of the computational domain used in the present research. Labeled surfaces include top (in purple); right (symmetry plane); wall; front (in yellow); and bottom. Periodic boundary conditions were employed on the front and back sides of the model, and the right surface of the model was defined as a symmetry plane. With these boundary conditions, the reduced computational domain represented a rod-bundle lattice of 6 rods in the x-direction, repeating infinitely in the y-direction. The geometry of the model is characterized by the diameter of the rods, the pitch of the rods, the length of the bu er region, and the height of the computational domain. These properties are summarized for the base case in Table 1. Table 1: A summary of the geometric properties of the base case, with P/D ratio of Property Diameter of fuel rods Pitch of fuel rods Length of bu er region Height of computational domain Value mm mm mm 200 mm For numerical solution, a segregated flow solver based on the SIMPLE algorithm applied on collocated variables with Rhie-Chow interpolation was used. All convective terms 15

17 were approximated with a non-oscillatory second-order scheme using the Venkatakrishnan reconstruction gradient limiting. The solution methods are summarized in Table 2. Table 2: A summary of solution methods used in Star-CCM+. Property Value Solver Implicit unsteady based on SIMPLE algorithm Temporal discretization Second-order Spatial discretization Second-order Turbulence model Anisotropic k-epsilon [6] Time-step size s Total solution time 15 s The geometry of the computational domain was varied using two di erent methods: 1. Changing the pitch while maintaining a constant diameter, thus changing the P/D ratio; and 2. Changing the size of the bu er region while maintaining the P/D ratio, expressed as a change in the bu er length ratio, defined by the base case bu er length. During this sensitivity analysis, the Reynolds number of the water was kept constant at This was done by changing the mass flow rate through the region. The 12 cases are summarized in Table 3, highlighting the P/D ratio, bu er length ratio, mass flow rate, and hydraulic diameter. Table 3: The geometric properties of the 12 simulated cases. The P/D ratio was varied from 1.1 to 1.56 and the bu er ratio was varied from 0.5 to 1.5 for a constant P/D of P/D ratio Bu er ratio ṁ (kg/s) D h (mm)

18 Simulations of these 12 cases were conducted, and several flow properties were monitored to analyze the influence of flow pulsation. The majority of these properties were measured on the crossflow plane, defined as the vertical plane in the center of the gap region closest to the bu er region, as shown in Figure 9. Figure 9: The location of the crossflow plane in the computational domain. The plane is located in the center of the gap region between fuel rods. As described in Section 2, flow pulsation in rod bundles is driven by the velocity gradient between subchannels. To analyze this relationship, the bulk velocity in the bu er subchannel and the adjacent subchannel were measured throughout the simulation. These subchannels are divided by the centerline of the fuel rods in the x-direction, as shown in Figure 10. Figure 10: A top view of the computational domain, highlighting the relevant subchannels. Bu er and adjacent subchannels are divided by the crossflow plane. 17

19 4 Results The crossflow velocity, defined as the flow velocity in the x-direction, was monitored at 10 equally spaced points along the centerline of the crossflow plane. For each case, the crossflow velocity took several seconds to reach a maximum amplitude. The crossflow velocity at the center of the gap region at a height of 0.11 m from t=13 seconds to t=15 seconds for 3 cases are shown in Figure 11. Figure 11a shows the base case, with a P/D ratio of 1.33 and a bu er ratio of 1; Figures 11b and 11c show a bu er ratio of 0.5 and 1.5, respectively. (a) Bu er ratio = 1 (b) Bu er ratio = 0.5 (c) Bu er ratio = 1.5 Figure 11: The crossflow velocity as a function of time for 3 cases. P/D ratio was 1.33, and the bu er ratio was varied from 0.5 to 1.5. It can be observed that the amplitude of the crossflow oscillations is greater for the larger bu er region, as is the trend for all cases of P/D ratio of The crossflow velocity across the center plane of the model is shown for these same cases at t=15 seconds in Figure 12. The base case of P/D ratio of 1.33 and bu er region is shown in Figure 12a; Figures 12b and 12c show the bu er ratio of 0.5 and 1.5, respectively. 18

20 (a) Bu er ratio = 1 (b) Bu er ratio = 0.5 (c) Bu er ratio = 1.5 Figure 12: The crossflow velocity at t=15 seconds for 3 cases. P/D ratio was 1.33, and the bu er ratio was varied from 0.5 to 1.5. Because the crossflow velocity through the gap region is periodic, it is di cult to quantify the amount of crossflow using this parameter. Instead, the variance of the transverse velocity was calculated throughout the model and averaged across the crossflow plane. This quantity, referred to as the average crossflow variance, takes into account both the amplitude and the unsteadiness of flow pulsations. The average crossflow variance versus time was plotted for both methods of geometry adjustments; Figures 13 and 14 show these results for changes in the P/D and bu er ratio, respectively. Figure 13: The average crossflow variance over time for 6 cases of constant bu er ratio of 1. The P/D ratio was varied from 1.1 (blue) to 1.56 (pink). 19

21 Figure 14: The average crossflow variance over time for 7 cases of constant P/D ratio of The bu er ratio was varied from 0.5 (pink) to 1.5 (purple). It can be observed that the average crossflow variance is highly dependent on geometry, as expected [1, 12, 24]. Cases with tight-lattice rod bundles, expressed by a small P/D ratio, were prone to crossflow significantly more than loosely packed rod bundles. Further, crossflow variance was larger in cases with a larger bu er ratio; the amount of crossflow increased with increasing bu er ratio. As it is expected that the velocity gradient is the driving force of this crossflow, the di erence in v bulk between the bu er subchannel and adjacent subchannel is defined as v bulk. The average crossflow variance versus v bulk for both geometry adjustment methods is shown in Figure 15, with P/D changes shown as circles and bu er ratio changes shown as squares, and fit with a linear regression, shown in blue. 20

22 Figure 15: The average crossflow variance versus v bulk for 12 simulated cases. Adjustments to bu er ratio and P/D ratio are shown as squares and circles, respectively. A linear best-fit is shown in blue. Two interesting conclusions can be drawn from these results: The average crossflow variance shows a linear relationship with v bulk, indicating that a larger v bulk will produce a greater amount of crossflow between subchannels; and The average crossflow variance is very small for v bulk < 0.4 m/s, indicating that flow pulsation is not a significant phenomenon for gap regions with small v bulk. Power spectral densities (PSD) of crossflow velocity monitors were computed using the commercial software Star-CCM+. The peak and second peak frequencies for 11 cases were found from this spectral analysis (note: the case of P/D = 1.1 and bu er ratio = 1 was not recorded). The frequencies versus v bulk are shown in Figure

23 Figure 16: The peak and second peak frequencies versus v bulk for 11 simulated cases. The peak frequency, calculated by fast Fourier transform of the crossflow velocity profile, is shown in blue. The second peak frequency is shown in red. Adjustments to the bu er ratio and P/D ratio are shown as squares and cirlces, respectively. Based on the work of Chandra et al. (2010), it should be noted that spectral analysis using URANS cannot be trusted fully without further testing [5]. While URANS was able to capture large coherent structures, it was shown that URANS does not accurately resolve fine turbulent structures as well as LES; thus, high frequency modes are under predicted in URANS [5]. With this consideration, it can be noted that the dominant frequencies in the present work are observed in a greater range for low v bulk than high v bulk, as the driving force of these vortices is stronger at a high v bulk. 5 Conclusion The present research was primarily inspired by the recent work at George Washington University that showed the presence of oscillating crossflow mixing in a partial PWR fuel assembly, previously unexpected in this geometry. A sensitivity study on flow pulsation in fuel assemblies was conducted by varying the rod-bundle geometry. This study was conducted in transient simulations using a URANS method, based on the work of Baglietto (2006), and converged statistics were collected for the mixing characteristics [6]. The rod- 22

24 bundle geometry was varied by changing the rod spacing (P/D ratio) and the size of the bu er region (bu er ratio) from the base case of the GWU experiment. Several interesting conclusions can be made from these results: Flow pulsation is indeed driven by the velocity gradient between adjacent subchannels, as described by Meyer [1]; The average crossflow variance, a measure of the amount of crossflow mixing, shows a linear relationship with v bulk ; The average crossflow variance is very small for v bulk < 0.4 m/s, showing that flow pulsation is not a significant phenomenon for gap regions with small v bulk ; and The dominant frequencies of flow pulsations are seen in a wider range for small v bulk, though spectral analysis using URANS cannot be trusted fully without further study. As 1-dimensional subchannel codes do not accurately account for the subchannel mixing caused by quasi-periodic flow pulsation, further work is needed to improve these models. Based on these results, an analytical relationship between crossflow mixing and intersubchannel velocity gradient could be developed and implemented into subchannel codes. Accurate prediction of flow pulsation in fuel assemblies is necessary to build on the successful design and licensing practices for nuclear reactors. 23

25 References [1] L. Meyer, From discovery to recognition of periodic large scale vortices in rod bundles as source of natural mixing between subchannels a review, Nuclear Engineering and Design, vol. 240, pp , [2] A. Lexmond, R. Mudde, and T. van der Hagen, Visualisation of the vortex street and characterization of the cross flow in the gap between two sub-channels, in NURETH- 11, [3] M. Guellouz and S. Tavoularis, The structure of turbulent flow in a rectangular channel containing a cylindrical rod part 1: Reynolds-averaged measurements, Experimental Thermal and Fluid Science, vol. 23, pp , [4] A. Gosset and S. Tavoularis, Laminar flow instability in a rectangular channel with a cylindrical core, Physics of Fluid, [5] L. Chandra and E. Baglietto, Unsteady RANS and LES analyses of Hooper s hydraulics experiment in a tight lattice bare rod-bundle, in The 8th International Topical Meetingon Nuclear Thermal-Hydraulics, Operation and Safety, (Shanghai), October [6] E. Baglietto, H. Ninokata, and T. Misawa, CFD and DNS methodologies development for fuel bundle simulations, Nuclear Engineering and Design, vol. 236, pp , August [7] N. Todreas and M. Kazimi, Nuclear Systems 1: Thermal Hydraulic Fundamentals. Cambridge, MA: Hemisphere Publishing Corporation, [8] Westinghouse Nuclear, Fuel Products. Operating-Plants/Fuel/Fuel-Products. [9] K. Rehme, The structure of turbulence in rod bundles and the implications on natural mixing between the subchannels, International Journal of Heat and Mass Transfer, vol. 35, no. 2, pp , [10] C. Franco and P. Carajilescov, Experimental analysis of pressure drop and flow redistribution in axial flows in rod bundles, Journal of the Brazilian Society of Mechanical Sciences, vol. 22, no. 4, [11] L. Tong and Y. Tang, Boiling Heat Transfer and Two-Phase Flow. CRC Press, [12] N. Todreas and L. Wilson, Coolant mixing in sodium cooled fash reactor fuel bundles, tech. rep., U.S. Atomic Energy Commission,

26 [13] M. Ibragimov, I. Isupov, L. Kobzar, and V. Subbotin, Calculation of the tangential stresses at the wall of a channel and the velocity distribution in a turbulent flow of liquid, Soviet Atomic Energy, vol. 21, no. 2, [14] D. Coates, Interchannel mixing and cooling temperature distribution in seven and nineteen element fuel bundles, tech. rep., Canadian General Electric Company Ltd., [15] G. Hofmann, Qualitative untersuchung ortlicher warmeubergangszahlen im 7- stabbundel [16] V. Skinner, A. Freeman, and H. Lyall, Gas mixing in rod clusters, International Journal of Heat and Mass Transfer, [17] J. Nikuradse, Untersuchung ueber die geschwindigkeitsverteilung in turbulenten stromungen, tech. rep., VDI-Forschungsheft, [18] F. Tachibana, A. Oyama, M. Akiyama, and S. Kondo, Measurement of heat transfer coe cients for axial air flow through eccentric annulus and seven-rod cluster, Journal of Nuclear Science and Technology, no. 4, pp , [19] B. Kjellstrom, Studies of Turbulent Flow Parallel to a Rod Bundle of Triangular Array. Aktiebolaget Atomenergi, [20] N. Neelen, Modellierung des Impulstransports achsparalleler turbulenter Stromungen. PhD thesis, TU Braunschweig, [21] W. Seale, Turbulent di usion of heat between connected flow passages, Nuclear Engineering and Design, vol. 54, no. 2, [22] D. S. Rowe, Measurement of Turbulent Velocity, Intensity and Scale in Rod Bundle Flow Channels. PhD thesis, Oregon State University, [23] T. V. der Ros and M. Bogaardt, Mass and heat exchange between adjacent channels in liquid-cooled rod bundles, Nuclear Engineering and Design, [24] K. Rehme, The structure of turbulent flow through rod bundles, Nuclear Engineering and Design, vol. 99, no. 1, [25] J. Hooper and K. Rehme, Large-scale structural e ects in developed turbulent flow through closely-spaced rod arrays, Journal of Fluid Mechanics, [26] S. Moller, Single-phase turbulent mixing in rod bundles, Experimental Thermal and Fluid Science, vol. 5, no. 1,

27 [27] T. Krauss and L. Meyer, Characteristics of turbulent velocity and temperature in a wall channel of a heated rod bundle, Experimental Thermal and Fluid Science, vol. 12, no. 1, [28] J. Hooper and K. Rehme, The structure of single-phase turbulent flows through closely spaced rod arrays, tech. rep., Insitut fur Neutronenphysic und Reaktortechnik, [29] S. Moller, On phenomena of turbulent flow through rod bundles, Experimental Thermal and Fluid Science, vol. 4, no. 1, [30] K. Singh and C. St-Pierre, Single phase turbulent mixing in simulated rod bundle geometries, Transactions of the Canadian Society for Mechanical Engineering, vol. 1, no. 2, pp , [31] E. Merzari, A. Khakim, H. Ninokata, and E. Baglietto, Unsteady Reynolds-averaged Navier-Stokes: toward accurate prediction of turbulent mixing phenomena, International Journal of Process Systems Engineering, vol. 1, no. 1,