Thermal-hydraulic simulations of a wire spacer fuel assembly

Transcription

1 Thermal-hydraulic simulations of a wire spacer fuel assembly S. Rolfo, C. Peniguel, M. Guillaud and D. Laurence Abstract The paper presents refined three-dimensional simulations of the flow and heat transfer in fuel assemblies as found or suggested for liquid-metal coolant fast reactors. The wire spacers, helically wound along the pin axis, generate a strong secondary flow pattern in opposition to smooth pins. The eddy viscosity and second moment turbulence models yield predictions of global friction and heat transfer coefficients that are quite similar, and within the range of experimental correlations available. The four configurations simulated range from a small test rig to the full scale reactor bundle (7, 19, 61 and 271 pins) in order to separate (a) global swirl boundary effects, where helical wires leaning against the casing deflect the flow in unison, from (b) homogeneous flow patterns in the core, where wire helices counteract each other. The 61 and 271 pins simulations show a clear decoupling of (a) and (b), meaning that a tri-periodic DNS or refined LES of a single pin and wire could be quite relevant. Indeed the effect of the variation of the helix pass is also investigated for the 7 pin geometry. Finally the paper considers the variation of the results as function of the meshing and in particular with the level of detail used for the pin-wire connection. 1. Introduction Fast reactors, with liquid metal coolant, have recently received a renewed interest due to their more efficient usage of uranium resources, and they are one of the proposals for the Generation IV reactors. Fuel rods of fast reactors are generally arranged in bundles of a triangular configuration and each pin is wrapped with wire spacer following a helical pattern around the rod axis. The wire maintains the gap between adjacent pins but also reduces vibrations and avoids trapping of the liquid metal coolant (generally sodium). In earlier reactor design stages the effect of the wire was introduced via experimental correlations, which provide the friction factor as a function of geometrical and hydraulic parameters. A first example is provided by Novendstern (1972), where the Blasius formula for pipe flow is corrected taking into account several parameters including the number of pins and the hydraulic diameter of the different types of sub-channels present in the fuel bundle. Another correlation proposed by Rehme (1973) introduces a shape factor, which takes into account the pitchover-diameter ratio P/D and the helix-over-diameter ratio H/D. A milestone Preprint submitted to Nuclear Engineering and Design 12th April 2011

2 in the experimental evaluation of flow inside wire-wrapped fuel bundle is found in Cheng and Todreas (1986), who present two sets of correlations. The first one takes into account several geometrical and hydraulic parameters, making the correlations suitable for many configurations and flow regimes (i.e. laminar or turbulent), but relatively difficult to use. The second one is a simplified version and the two formulations converge toward same values as the Reynolds number increases. Because this work focus on fully turbulent flow with relatively high Reynolds number, only the simplified version is considered for the comparison. A last correlation used herein, is by Bubelis and Schikorr (2008), which is a modified version of an early correlation from Engel et al. (1979). The recent work of Bubelis and Schikorr (2008) presents a very wide and complete evaluation of several correlations, taking into account arrangements with different numbers of pins and different types of coolant. In case of liquid sodium authors validate the correlations against the CFD data of Gajapathy et al. (2007). On the other hand Gajapathy et al. (2007) use Rehme s correlation to validate the numerical results. Now an obvious question arises: which set of data is to be trusted and used as reference? Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), as the one presented by Fischer et al. (2007), could serve as arbitrator. LES and DNS can deliver a large and accurate amount of data (provided that a proper code and mesh are used), whereas detailed results in such a dense packed geometry are difficult to obtain with experimental techniques. Instantaneous flow field and extensive averaged data (as for example the Nusselt number distribution along the fuel rod) will be available, for more rigorous validation of RANS models. Because of the extreme time consumption and cost of LES and DNS (very powerful High Parallel Computing, HPC, facilities are necessary), they are still limited to subsections of the global geometry and moderate Reynolds numbers, whereas usual Reynolds Averaged Navier Stokes (RANS) can be devoted to the study of more industrial cases. An even more difficult task is to find precise and accurate experimental data for heat transfer. Several studies were conducted during the sixties and analytical correlations were devised for the Nusselt number. In recent years those data were revisited in the context of international collaborations for advanced nuclear reactor core configurations (see the reviews of Pfrang and Struwe (2007) and Mikityuk (2009)). Both reviews present several correlations for the Nusselt number as function of the Peclet number and the P/D ratio for both triangular and square arrays. The major problem is that the pins are considered without the wire wrapping typical of more recent and future designs. In particular Pfrang and Struwe (2007) write In most of the available publications on experimental campaigns no hint is given, whether spacers or what type of spacers have been used, and the few works which explicitly mention a support device refer to spacer grids, but not to wire wrapping. Several recent studies based on RANS models (i.e. Raza and Kim (2008)) are generally carried out with commercial unstructured codes, reduced geometries, and representing a limited number of pins (i.e. seven or nineteen) and comparison with well-establish experimental correlations. The present work follows the

3 same path, but increasing the number of pins up to 271 of the real geometry. Indeed this work tries to establish the influence on the results of some important parameters, such as turbulence modelling, geometry meshing and takes also into account some preliminary tests on heat transfer. The heat transfer analysis employs simplified boundary conditions for the temperature (i.e. A Dirichlet constant wall temperature and a Neumann constant wall heat flux), whereas the real situation at the fluid/solid interface is far more complicated: the fuel is located only inside the pin and there is no energy deposit inside the wire (in this study the same boundary condition is applied over all the pin surface including the wire). 2. Numerical methods In this study the open source CFD software Code_Saturne (Archambeau et al., 2004; Fournier et al., 2011), developed by EDF R&D, is used. The RANS equations are discretised using a cell centred co-located finite volume approach. Velocity and pressure coupling is insured by a prediction/correction method with a SIMPLEC algorithm and the Poisson equation is solved with a multigrid algorithm. The Rhie and Chow (1983) interpolation in the correction step is also used in order to stabilise the solution. The code handles completely unstructured grids, including polyhedral cells and embedded refinements. For more details on all the above numerical methods see e.g. Ferziger and Peric (1997) 2.1. Governing equations for periodic flows The major interest of this work is toward a fully developed turbulent flow, which is reached after several helices. In order to avoid the usage of a very long geometry a periodic procedure has been set up with a beneficial reduction of the domain to only one helix. In a fully developed turbulent flow, pressure and temperature can be decomposed as: P = P βx 3 (1) T = T +αx 3 (2) where x 3 is the periodic direction, β = p/ x 3 is the pressure drop in the stream-wise direction and α the slope of the bulk temperature increase which can be evaluated from a heat balance and reads: α = T B x 3 = q WS W,h ṁc p L 3 (3) where T B is the bulk temperature, q W the desired wall heat flux, S W,h the heated wall surface, ṁ the mass flow rate, c p the specific heat capacity and L 3 the length in the periodic directionx 3 (which corresponds also to the streamwise direction). The terms βx 3 and αx 3 take into account the global pressure drop

4 and the rise in enthalpy respectively, whereas the functions P and T repeat them-self at every wire angle position. Consequently the RANS equations read: 2 P x 2 i = ρ U i x j U j x i (4) DρU i Dt Dρ T Dt = + [ ( Ui µ + U ] j ) ρu i u j x j x j x i P x i +βδ i,3 (5) [ = x j µ Pr T ] ρu j θ ραu j δ i,3 (6) x j with µ the kinematic viscosity, Pr the Prandtl number, u i u j the Reynolds stress tensor, u j θ the turbulent heat flux and δ the Kronecker s delta. The pressure drop β is adjusted at every time step in order to ensure the desired mass flow rate. 3. Turbulence modelling In this investigation two different turbulence models have been used: the standard k ε eddy viscosity model (EVM) of (Jones and Launder, 1972) and a second moment closure (SMC), denoted by ε in the graphs. In the first case two additional transport equations, one for the turbulent kinetic energy k and the second for the rate of dissipation ε, are introduced and read: Dρk Dt = P ρε+ x j Dρε Dt = C ε ε1 k P ρc ε2 [( µ+ µ t ε 2 k + x j σ k ) ] k [( µ+ µ t σ ε x j ) ] ε wherep is the production that contains also the buoyant term, σ k andσ ε are the turbulent Prandtl numbers for the turbulent kinetic energy and the dissipation, C ε1 and C ε2 two constants. The turbulent viscosity ν t, which relates stresses to rate of strain, is evaluated from k and ε as: k 2 µ t = ρc µ (9) ε with C µ = 0.09, that has been tuned mainly on log layer. The unknown Reynolds stresses of Eq. (5) are modelled as: [ Ui ρu i u j = µ t + U ] j 2 x j x i 3 ρkδ ij (10) x j (7) (8)

5 In the second moment closure a transport equation for each component of the Reynolds Stress Tensor is solved: Dρu i u j = P ij +D ij +Φ ij ρε ij (11) Dt where P ij is the production, D ij the turbulent diffusion and ε ij the dissipation. The SSG model (Speziale et al., 1991) is used for pressure strain rate correlation Φ ij. The heat transfer modelling employs a simple gradient diffusion hypothesis and the turbulent heat fluxes read: ρu j θ = µ t Pr t T x j (12) with Pr t the turbulent Prandtl number. These models are nowadays very standard. Full equations and constants are recalled in Archambeau et al. (2004) for the Code_Saturne framework, and validation cases are available on-line as well as the source code Wall modelling The wall modelling is based on the classical log law (Von Kármán, 1930): u + = 1 κ lny+ +B (13) with κ = 0.41 being the von Kármán constant and B = 5.2. u + = U/u τ and y + = ρyu/µ are the dimensionless velocity and distance from the wall, with y the distance to the wall and u τ being friction velocity. In this work a modified version, called scalable wall functions, proposed by Grotjans and Menter (1998) is used, circumventing the complex viscous sub-layer effects by instead acting as if the wall was pushed back when y + is initially too small. This fixture removes the requirement to place the non-dimensional wall distance y + above the buffer layer (y + 30), and it has been found of benefit in industrial applications, where the geometry is relatively complex and characterised by a large variation of the dimensionless wall distance. In the case of a scalar with a Dirichlet boundary condition (BC) Code_Saturne employs a three-scale model, which reduces to two-scale in case of very low Prandtl number, as it is in the present case (Arpaci and Larsen, 1984). The expression of the dimensionless temperature reads: T + = Pr y + if y y 0 + T + = Pr ( ) t y + κ ln +Pr y 0 + if y > y 0 + y + 0 (14) 1 Code_Saturne and its validation test cases can be downloaded from:

6 with T + = (T W T)/T τ and y 0 + = Prt /κpr. In the present case y since Pr = Even at the highest Reynolds number y + < 200, consequently the linear approximation of Eq. (14) is always used. When the BC is of a Neumann type, the heat flux q W is directly inserted into the balance in the boundary cell and the wall temperature is evaluated from: ( µ q W = c p Pr + µ ) t TW T I (15) Pr t I F with I being the projection of the cell centre on the normal of the wall face passing through the wall face centre F and with I F being the distance. 4. Test case set-up The test case considered herein is constituted by a fuel rod bundle arranged into a triangular array with a pitch-to-diameter ratio P/D = 1.1, encapsulated into a hexagonal casing with a wall-to-diameter ratio W/D = 1.1. The diameter of the wrapped wire is equal to 95% of the minimum gap between two adjacent rods. For the wire helical pattern two wire-helix-step to fuel-rod-diameter ratio H/D = 21 and 16.8 are considered (only for the 7 pin geometry). The real industrial fuel bundle is composed of 271 pins, but in this study reduced geometries with 7, 19 and 61 pins are also simulated in order to study the influence of numerical boundaries on the bundle s core flow-pattern. Since the flow is considered periodic in the streamwise direction (see Sec. 2.1) only one helix pass H is employed and periodic boundary conditions are used between inlet and outlet. Periodic boundary conditions are generally used for refined turbulent LES and DNS simulations in order to reduce the domain size. Periodic boundaries are treated as fully implicit, i.e. identical to inner cells, as needed to allow unsteady flow reversal in LES mode, or, as herein in RANS mode, fully developed turbulent quantities, instead of ad-hoc user defined inlet condition. Despite the easy geometrical definition, where only five parameters define the entire geometry, the meshing stage is relatively difficult because of the singularities introduced every time the wire on each pin comes almost in contact with the surrounding pins. Many attempts with several different commercial mesh generators have been tested, in particular using an unstructured approach, resulting in too large meshes (order of few millions of cells for only 7 pins) and unsatisfactory mesh quality (high level of skewing and insufficient number of points between two adjacent pins). As a consequence a specific procedure has been developed, which consists of a block-structured configuration where the wire is inserted through a deformation of the mesh. Moreover the methodology allows a high control of the mesh in particular in the pin normal direction, where at least six cells are used between two adjacent pins. Several ways to handle the connection between the wire and the pin have been investigated (see Fig. 1) and their influence on the hydrodynamic field is presented in Sec. 8. A complete description of the mesh procedure and all previous attempts is reported in (Péniguel et al., 2009). The resulting mesh sizes for the different configurations

7 are: 550,000 cells for the 7 pins, 2.8 millions for the 19 pins, 4.8 millions for the 61 pins and 19.5 millions for the 271 pins. This last case is investigated using only the k ε model. Base Blended Triangular Figure 1: Cross-section of the three different meshes employed. 5. Flow description In the absence of spacer-wire (the smooth configuration ) six weak secondary vortices, with an average intensity of about 0.4% of the bulk velocity, are generated by the Reynolds stress anisotropy in every triangular sub-channel following the symmetry lines (Vonka, 1988; Ninokata et al., 2009). In addition, when the geometry is very tight, large coherent structures appear between the sub-channels. This flow pulsations phenomenon was investigated both experimentally (Krauss and Meyer, 1998) and numerically (Chang and Tavoularis, 2007; Rolfo et al., 2010). The addition of the wire obviously tends to induce a local swirl around each pin, preventing the secondary and coherent motions of the smooth configuration, but this spin is countered by the spin of the six neighbour pins, as they all have same parity: all wires rotate counter-clockwise as we move downstream as well as in all figures herein (i.e. all cross sections are seen from the leeward side). The presence of the casing then destroys the symmetry of the flow field in the cross-plane section and introduces a complicated flow pattern. Two different areas can be consequently identified: an outer zone, composed mainly by edge and corner sub-channels, and an inner zone made of the internal sub-channels unaffected by the casing boundary (see Fig. 2). All figures correspond to the

8 k ε model results, unless explicitly mentioned. The outer zone is distinguished by a global counter-clockwise rotation following the wire path (see Figs. 2, 3 and 4), and by a global swirling motion in the cross section (see Fig. 4 on the left). In Fig. 3, at = 180 o the northwest wall casing is the one that has just been swept over, and this is where we find the highest axial velocity as the streamlines are forced to converge by the wire. The effect is seen to fade away as we move clockwise to the northeast, east and southeast casing walls. The maximum acceleration against the casing wall reaches 35% of the bulk velocity, whereas in the inner region it is limited to 10%. The global swirling motion is not uniform across all edge and corner subchannels: where the axial velocity is higher the intensity of the global swirling is lower and with a presence of recirculation areas; where the axial velocity is lower the global swirling is stronger and cuts across the sub-channel without generating any recirculation. In the inner area the secondary motion follows the wire rotation and right behind the wire an apparent separation bubble is formed, followed by a stream of high secondary motion for almost 180 o. This large secondary motion provides efficient momentum transfer between the narrow gap region, which separates two adjacent pins, and the wider channel in the middle of three pins. The result is a fairly uniform distribution of the axial velocity in the inner area of the bundle. Note that the apparent separation bubble has negligible effect on the friction because the axial velocity is still very high (it is only visible because of pseudo-streamlines generated by the projection velocity vectors into the cross section plane). As the number of pin increases (compare Fig. 4 with 5) the main flow features remain unchanged; the difference between inner and outer areas become more visible and it is clear that the global swirling motion affects mainly the most external and the first inner circle of channels. The change in turbulence model does not alter the flow features and indeed the apex values of the stream-wise velocity (see Fig 6) and secondary motion intensity are very similar. The difference is mainly concentrated at the wall, where the two models give a different estimation of the friction coefficient c f as it can be seen in Figs. 8, 9 and 10, which report the c f along a line and two cross sections for the centre tube of the seven pins geometry. Values given by the ε model are generally higher with respect to the k ε model, but profiles have similar trends. As expected the highest value of the c f is located on the top of the wire, which comes along with a very rapid variation of the profile.

9 U 3/U B Figure 2: Comparison between geometries of the dimensionless stream-wise velocity: top left 7pins, bottom left 19 pins, right 61 pins (H/D = 21). The wire location is at = 10 o. U 3/U B Figure 3: Comparison between geometries of the dimensionless stream-wise velocity: top left 7pins, bottom left 19 pins, right 61 pins (H/D = 21). The wire location is at = 180 o.

10 Global swirling motion: strong area: weak area: Wake Recirculation Figure 4: Dimensionless stream-wise velocity U 3 /U B (left) and secondary motion intensity U 2 1 +U2 2 /U B (right) for the 19 pins geometry (H/D = 21). The wire location is at = 180 o. U 3/U B U 2 1 +U 2 2 U B Figure 5: Dimensionless stream-wise velocity U 3 /U B (left) and secondary motion intensity U 2 1 +U2 2 /U B (right) for the 271 pins geometry (H/D = 21). The wire location is at = 180 o.

11 k ε ε U 3/U B Figure 6: Comparison of the dimensionless stream-wise velocity U 3 /U B between k ε (left) and second moment closure ε (right) for the 7 pins geometry (H/D = 21) atre = 25,000. The wire location is at = 180 o. c f 10 3 ε k ε Figure 7: Comparison of the friction coefficient c f at the wall between k ε (left) and second moment closure ε (right) for the central rod of 7 pins geometry (H/D = 21) at Re = 25,000. The square blocks indicate the presence of the wire.

12 9 8 7 cf x 2 4 x 1 k ε ε x 3 /H Figure 8: Friction factor c f along the line x 2 = 0 for the central rod of 7 pins geometry (H/D = 21) at Re = 25,000. The square blocks indicate the presence of the wire k ε ε 8 7 cf θ Figure 9: Friction factor c f along the cross section located at x 3 /H = 0 for the central rod of 7 pins geometry (H/D = 21) at Re = 25,000. The square blocks indicate the presence of the wire. The indicates the narrow gap between two adjacent rods.

13 10 9 k ε ε 8 7 cf θ Figure 10: Friction factor c f along the cross section located at x 3 /H = for the central rod of 7 pins geometry (H/D = 21) at Re = 25,000. The square blocks indicate the presence of the wire. The indicates the narrow gap between two adjacent rods. 6. Thermal field Two types of temperature boundary conditions have been simulated: a Dirichlet uniform wall temperature and a Neumann uniform wall heat flux 2,3. The first case corresponds to a small Biot number, where the heat conduction inside the solid is larger than the heat convection away from the body surface. The second one is the opposite, where the heat conduction in the solid is slower with respect to convection in the fluid. The source term introduced in Eq. (6) is explicit and maintains the bulk temperature constant so the thermal field remains periodic, and α is equal to the constant axial temperature gradient that is subtracted for this purpose. This is equivalent to imposing an axial temperature gradient α in the solids then resetting the inlet and outlet fluid temperatures. Such a linear transformation is permitted, because the mean and fluctuating temperature equations are linear even in a turbulent flow-field (as long as feedback effects on velocities, such as buoyancy, are neglected). Plots report the dimensionless temperature T ad = (T max T)/[(T W ) m T B ] (with T max the maximum T and (T W ) m the average wall temperature over all the heated surface), therefore a minimum of T ad correspond to a maximum of T and viceversa. 2 With the Neumann BC the value imposed at the wall must be equal to the one used to evaluate α (see Eq. (3)) in order to satisfy the heat balance. 3 The external hexagonal casing has an adiabatic BC (q W = 0).

14 This standard is widely used in order to have temperature trends similar to the ones of the velocity (i.e. zero at the wall, maximum far from the wall). In order to keep a visual correspondence between hot and cold regions, the usual colour agreement is reversed and therefore red stands for low values of T ad and blue the opposite. Fig. 11 reports the two different situations at the wall. In the case of a constant wall temperature (left part of the figure) the heat flux largely varies across the pin surface, with a maximum in the wall region facing the wider channel in the middle of three pins, and a minimum in the gap region between two pins, where the wire-spacer is almost in contact with the adjacent element. High values mean an efficient heat transfer between solid and fluid and, as expected, this is reached where the stream-wise velocity is higher. It has to be noticed that the average value of the heat flux is equal to the one imposed through α (see Eq. (3)), as required by the heat balance. The wall temperature for the Neumann case has a different behaviour with a minimum located around the wire-spacer, and a local maximum in the wider section of the sub-channel. In the near wall region the turbulent contribution to the overall heat transfer is negligible because of the low Prandtl number (see Sec. 3.1) and, away from the wall, the flow features, captured by the two turbulence models, are very similar. As a consequence the predictions given by the two turbulence models are very similar in both trends and values, and only the fields given by the k ε model are plotted The thermal field, characterised by Dirichlet BC, has also an inner and an outer areas as for the hydrodynamic field (see Fig. 12). The inner area has relatively uniform distribution of the temperature, with a local maximum in the centres of each sub-channel. The outer zone has a hot part located where the wire is almost in contact with the external case and this hot spot rotates anticlockwise following the wire pattern. The dimensionless temperature reaches its maximum in the wide gaps between pins and casing where high axial velocity brings or has brought cooling. As the number of pins is increased the thermal field preserves the same characteristics, with just an enlargement of the hot inner area. On the contrary in the case of Neumann BC the temperature field (see Fig. 13) displays a minimum value around the central pins and gradually increases toward the external casing as a consequence of the higher flow-rates in these peripheral sub-channels. As the geometry enlarges, increasing the number of pins, the temperature difference, between centre and edge area, increases. The rotational behaviour, proper of the stream-wise velocity, is still present in both temperature fields but it is less clear in the Neumann BC case because of the doubling of the temperature scale.

15 q W/(q W) m T ad Figure 11: Dimensionless wall heat flux q W /(q W ) m for the Dirichlet constant wall temperature (left) and dimensionless wall temperature T ad = (T max T W )/ [ (T W ) m T B ] (right) for the Neumann constant wall heat flux. Contours are plotted on the central rod of the 7 pins geometry (H/D = 21) at Pe = 125. In case of T ad blue correspond low values of T W and red to high T W. Figure 12: Comparison between geometries of the dimensionless temperature T ad = (T max T)/ [ (T W ) m T B ] using a Dirichlet boundary condition: top left 7pins, bottom left 19 pins, right 61 pins (H/D = 21) at Pe = 125. The wire location is at = 180 o.

16 Figure 13: Comparison between geometries of the dimensionless temperature T ad = (T max T)/ [ (T W ) m T B ] using a Dirichlet boundary condition: top left 7pins, bottom left 19 pins, right 61 pins (H/D = 21) at Pe = 125. The wire location is at = 180 o. 7. Profiles A quantitative evaluation of the results can be obtained with the comparison of the numerical results with experimental correlations for the friction factor f and the Nusselt number Nu. The friction factor f is defined as: ( ) 1 L 3 f = p 2 ρu2 B (16) where D h is the hydraulic diameter, U B the bulk velocity and p the pressure loss. The pressure drop can be evaluated in two different ways: (MOD1) computing the elementary pressure and viscous shear stress on each fluid-structure boundary cell, projecting it along the x 3 axis and integrating over the whole domain. Or (MOD2) from the pressure drop β (or rather driving force) defined in Eq. (1) and re-evaluated at each time-step in order to keep the mass flow rate constant. In an ideal finite volume simulation these two methods should lead to rigorously equal results. For the distorted meshes used herein both are nevertheless reported as an accuracy check. The friction coefficient profiles as function of Reynolds number are reported in Figs 14, 15, 16, 17 and 18 for different H/D ratios and numbers of pins. The CFD predictions are in the range given by several experimental correlations and they seem to have a good correspondence with the one of Cheng and Todreas (1986) for all geometrical configurations. For high Reynolds numbers the two methods of computing the friction coefficient are in close agreement as expected. On the other hand there are some small discrepancies at the lower Reynolds D h

17 numbers due to post-processing error on the shear stress when the log-law of the wall is applied at the lower end of its range. As already observed in Sec. 5 the ε model gives higher estimations of the friction factor f and in particular at the low Reynolds number, whereas results substantially converge toward same values as the Re increases. Both correlations of Cheng and Todreas (1986) and Rehme (1973) give an increase of the friction factor of about 8.5% from the high to the low H/D (H/D = 21.8 in Fig. 19, H/D = 17.8 in Fig. 20). The numerical results predict also this jump, but the increment is only of the order of 5.5%. Indeed ε gives a closer agreement with the correlation of Cheng and Todreas (1986) for the smaller H/D, whereas the k ε has a closer agreement for the higher one. As the number of fuel pins increases the experimental correlations present two different scenarios: in Cheng and Todreas (1986) f is only function of P/D and H/D and therefore the friction factor is constant; on the other hand Rehme (1973) predicts an increase of about25% moving from 7 to 271 pins. CFD results display a trend similar to the first situation, since the k ε model predicts an increment in the range 0 3% from 7 to 271 pins whereas the ε a larger increase of 6 8% from 7 to 61 pins. f (Novendstern, 1972) (Rehme, 1973) (Bubelis and Schikorr, 2008) 0.05 (Cheng and Todreas, 1986) k ε MOD k ε MOD2 R MOD2 ij e3 10e3 50e3 Re Figure 14: Friction factor for the 7 pin geometry (H/D = 21).

18 f (Novendstern, 1972) (Rehme, 1973) (Bubelis and Schikorr, 2008) 0.05 (Cheng and Todreas, 1986) k ε MOD k ε MOD2 MOD e3 10e3 50e3 Re Figure 15: Friction factor for the 7 pin geometry (H/D = 16.8). f (Novendstern, 1972) (Rehme, 1973) (Bubelis and Schikorr, 2008) 0.05 (Cheng and Todreas, 1986) k ε MOD k ε MOD2 MOD e3 10e3 50e3 Re Figure 16: Friction factor for the 19 pin geometry (H/D = 21).

19 f (Novendstern, 1972) (Rehme, 1973) (Bubelis and Schikorr, 2008) 0.05 (Cheng and Todreas, 1986) k ε MOD k ε MOD2 MOD e3 10e3 50e3 Re Figure 17: Friction factor for the 61 pin geometry (H/D = 21) (Novendstern, 1972) (Rehme, 1973) (Bubelis and Schikorr, 2008) 0.05 (Cheng and Todreas, 1986) k ε MOD k ε MOD2 f e3 10e3 50e3 Re Figure 18: Friction factor for the 271 pin geometry (H/D = 21). The evaluation of the global heat transfer is estimated throughout the Nus-

20 selt number that reads: Nu = q w (T W ) m T B D c p µ/pr (17) The Nusselt number profiles for the different configurations are reported from Fig. 19 to 22. Graphs also report the experimental correlations of Mikityuk (2009) that is designed for the smooth configuration, as described in the introductory Sec. 1, and it is only included for reference. As the number of pins increases the Nusselt profiles, for the imposed wall temperature, remains substantially unchanged ranging from 15 at Pe = 50 to 23 at Pe = 250. On the other hand profiles for the constant wall heat flux Nu decreases with the enlargement of the geometry (3 Nu 15 for the 7 pins, 1 Nu 5 for the 61 pins). This is not surprising since, as described in Sec. 6 and clear from Fig. 13, the average temperature difference between (T W ) m and T B increases with the growth of the number of pins. Indeed the rate of growth in the Nusselt number tends to become less steep with the enlargement of the geometry. It is interesting to notice that all CFD profiles have a faster rate of increase with respect to the experimental correlation, which are, in the analysed range, relatively constant. The variation of the turbulence model has almost a negligible influence on the heat transfer results, the only small exception is the evaluation of the Nusselt, at high Peclet number, for the 7 pin at the higher H/D when a constant wall heat flux BC is used (see Fig. 19). Mikityuk k ε T w 20 k ε q w T w q w Nu Pe Figure 19: Nusselt number for the 7 pin geometry (H/D = 21).

21 25 Mikityuk k ε T w k ε q w 20 T w q w Nu Pe Figure 20: Nusselt number for the 7 pin geometry (H/D = 16.8). Mikityuk k ε T w 20 k ε q w T w q w Nu Pe Figure 21: Nusselt number for the 19 pin geometry (H/D = 21).

22 Mikityuk k ε T w 20 k ε q w T w q w 15 Nu Pe Figure 22: Nusselt number for the 61 pin geometry (H/D = 21). 8. Effect of the mesh configuration As already mentioned in Sec. 4 the wire introduces many singularities in the geometry, which makes the meshing phase rather challenging. The influence of the geometrical representation of the rod/wire connection on the flow field characteristics, on the friction factor and on the Nusselt number should be examined. Three different types of meshing have been applied, ranging from a very smooth transition, where the wire has a large blending with the attached pin, to a very detailed discretization of the singularity between wire and pin, obtained with an unstructured mesh of the area (see Fig. 1). The main flow features are captured with all types of meshes employed as it can be seen from Fig. 23. In the case of the hydrodynamic field, the evaluation of the friction factor confirms the very limited influence on the results of a very fine representation of this geometrical detail: in fact the maximum difference of f, with respect to the value given by the experimental correlation of Cheng and Todreas (1986), is below 4%, (see Table 1). On the other hand, in the case of the heat transfer, the meshing of the geometry has a larger influence. The variation of the Nusselt number is of the order of 10 15% with respect to the value given by the most detailed geometrical representation. This is not surprising because the area around the wire is the one that is characterised by the highest wall temperature (imposed q W ) and wall heat fluxes (imposed T W ).

23 U 3 /U B Figure 23: Dimensionless velocity U 3 /U B in the cross-section of the three different meshes employed. Mesh type f (MOD1) [%] f (MOD2) [%] Base Base ( ) Blended Triangular Table 1: Comparison of f for different mesh types at Re = 25,000 and 7 pins geometry. The turbulence model used, if not specified otherwise, is k ε. The % difference is evaluated with respect to the value given by the correlation of Cheng and Todreas (1986) (f = ). Mesh type Nu (T w ) [%] Nu (q w ) [%] Base Base ( ) Blended Triangular Table 2: Comparison of Nusselt number for different mesh types at Re = 25,000 and 7 pins geometry. The turbulence model used, if not specified otherwise, is k ε. The % difference is evaluated with respect to the value given by the triangular mesh, which is supposed to be the most accurate discretisation).

24 9. Conclusions CFD simulations of thermal-hydraulic features of wire-wrapped fuel bundles show that a variation of number of fuel pins (from 7, to 19, 61 and 271) does not have a large influence on the main flow features. The flow presents a very strong secondary motion and it can be divided into an inner area and an outer area, which is mainly composed by edge and corner channels. Simulations show a strong secondary swirling motion at the outer edge of the bundle where wires and external boundary combine forces to deflect the mean flow. This is also visible from rotation of the maximum axial velocity. With the larger bundles (61 and 271 pins), the global swirl stays limited to the edge region, with an increasingly large homogeneous core. The eddy viscosity and second moment closure turbulence models are in good agreement with experimental global correlations, in particular that of Cheng and Todreas (1986). The SMC global friction factor is higher than that given by the EVM but the difference is less than 5% except at low Reynolds number (but down-to the wall models should then be used). Local friction coefficient profiles also show that the SMC generally gives higher value with respect to the EVM. The heat transfer effects would merit finer investigations, possibly by DNS or LES, since no precise experimental data is available (i.e. the experimental correlations found in literature are for smooth walls). In the case of constant wall temperature the Nusselt number profiles are greater than those obtained with a prescribed heat fluxes, and profiles are very similar as the geometry enlarges. On the contrary, in the case of a constant heat flux, as the number of pins increases, the Nusselt number drops and the temperature filed in the cross section is less homogeneous. The heat transfer is insensitive to the choice of turbulence model, as expected when the Peclet number is small, however the conjugate heat transfer cases are now being computed. Since there is a serious lack of experimental data, and we have seen that the core of the bundle is homogeneous, independent of boundary, refined LES and DNS calculations with only one pin and tri-periodicity are relevant, and ongoing to provide valuable local heat transfer data for down to the wall RANS model, or hybrid RANS-LES approach validation, which can then be used at higher Re numbers. Acknowledgements Authors express their thanks to D. Monfort, Y. Fournier and J. C. Uribe for advices in Code_Saturne usage. S. Rolfo is grateful to the UK Engineering and Physical Sciences Research Council for the partial funding under grant EP/H010998/1 Novel hybrid LES-RANS schemes for simulating physically and geometrically complex turbulent flows.

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