COMPUTATIONAL FLUID DYNAMICS AS A TOOL FOR DERIVING

COMPUTATIONAL FLUID DYNAMICS AS A TOOL FOR DERIVING SUBCHANNEL MODEL PARAMETERS – THE PSBT CASE STUDY R. Puragliesi Paul Scherrer Institut Villigen, Switzerland I. Clifford Paul Scherrer Institut Villigen, Switzerland

R. Mukin Paul Scherrer Institut Villigen, Switzerland H. Ferroukhi Paul Scherrer Institut Villigen, Switzerland

ABSTRACT The development of subchannel models for fuel assemblies and reactor cores requires accurate information on flow distribution, wall friction and loss coefficients in order to accurately predict the pressure, temperature and flow distribution on a subchannel level. This paper discusses the use of Computational Fluid Dynamics (CFD) simulations as a practical tool for characterising inlet velocity boundary conditions, an approximation of wall friction factor and spacer grid pressure loss coefficients, which are of fundamental importance to correctly generate a consistent subchannel model of a given assembly system. The geometry of the simplified PWR assembly presented here is based on the NUPEC PWR subchannel and bundle tests. Comparison of the derived friction factors and grid pressure loss coefficient with published and recommended values are reported. Discrepancies are also explained using additional calculations. A comparison of the overall system pressure drop, by comparing numerical and analytical solutions, and the local axial pressure distribution at subchannel level are presented. To make a one-to-one comparison between CFD and subchannel solutions, volumeaveraging is applied to the CFD results according to the chosen subchannel nodalization. The obtained results show a perfect agreement between the two codes. This outcome reflects the correct approach employed to build two consistent numerical models by properly carrying important information from the high-resolution models (CFD) to the low-resolution models (subchannel code). Furthermore, it has been found that the large discrepancies recorded in the CFD prediction of the grid pressure loss coefficients suggested in the benchmark specifications are mainly because the suggested benchmark values do not take into account the presence of the bounding channel that was present in the experimental facility. INTRODUCTION In very recent years the use of CFD for studying PWR assemblies has become common practice during the design phase to optimize the geometry of spacer grids and mixing vanes in order to enhance the turbulent heat transfer from the

M. Seidl PreussenElektra GmbH Hannover, Germany

fuel rods to the coolant with a minimum increase of the pressure drops and mechanical loads on the solid structures. This tendency is justified by the increasing computing power available at reasonable costs. However, full length bundle CFD simulations still remain very challenging. Several publications related to the PSBT benchmark, which is defined in [1], are available in literature. The majority consider flow in a single subchannel [2][3][4][5][6] while few deal with the full length PSBT fuel bundle [7][8][9]. The focus is mostly on the comparison of volume fraction measurements and temperatures while no information is provided on pressure losses across the spacer grids. The influence of boiling models and turbulence closure models are also investigated. Depending on the quantity considered for the comparison, deviations from the measured values are not negligible (for instance 10-30% in void fraction). Recent work that is not strictly related to PSBT, focusing on validation of single-phase CFD simulations for characterizing spacer grids, can be found in [10][11][12][13][14][15][16][17][18][19]. From these studies it is possible to conclude that single-phase, isothermal and nonisothermal, CFD simulations are reasonably accurate once the proper grid type, mesh size and boundary conditions are selected. Also, the transfer of information between high-resolution models (CFD) to low-resolution models (sub-channel codes) is an area of active research [18][20]. It is in this framework that here we present the results obtained using STAR-CCM+ CFD model for determining input parameters for the sub-channel code COBRA-TF. We have focused our attention in improving the consistency of the two models by properly describing the inlet velocity boundary conditions, the wall friction factor and the pressure loss coefficients of the three different types of grids. The paper is organized as follows: in Section 1 a brief description of the chosen PSBT experiment is given; Section 2 summarizes the numerical techniques used in the two codes; Section 3 presents mesh convergence, sensitivity to two turbulence models and to geometrical details; In Section 4 results relating to the inlet boundary conditions, wall friction factor, pressure loss coefficients, overall system pressure drop and axial local pressure and temperature distributions at sub-

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channel are reported; Concluding remarks and future work are reported in Section 5. 1. DESCRIPTION OF THE SELECTED EXPERIMENT Exercise 4 of Phase I [1][22][23] of the PSBT experiments has been selected for the validation/verification study because it partially complies with the modelling hypothesis that makes use of single-phase flow in the CFD model. However, as reported in the benchmark specifications no pressure data is available for code-to-data comparison. Only code-to-code comparison has been used in the benchmark for the typical central sub-channel over the heated section. The reference value provided for the code-to-code comparison was given for the bundle (type B7) at the rated thermal-hydraulic conditions characterizing the initial phase of the power increase transient (test series 7T). In Table 1 the characteristics of the B7 bundle are reported. The material used for the rods, thimble and grids is Inconel 600. The material employed for the channel is not specified. The cosine power profile applied to the heated rods is approximated by the piecewise constant function of Figure 1. It describes the axial distribution of the relative axial power coefficients which are defined as the ratios of the local power to the rod average power and they are provided in the benchmark specifications. Along the heated length of the PSBT B7 5x5 rods assembly, 1 spacer grid without mixing vanes (NMV), 7 grids with mixing vanes (MV) and 8 simple spacer grids (SSG) are placed. The 7 MV grids are placed following an alternating pattern with orientation 0, + , 0, + , 0, + , … along the axial direction as suggested in [19]. The central rod is replaced by a thimble tube and the whole bundle is surrounded by an enclosing channel. A schematic of the full length PSBT facility is shown in Figure 2. The initial rated conditions of PSBT bundle test 7T (transient with power increase) are reported in Table 2. Finally, the reference pressure drop value for code-to-code comparison reported in the PSBT benchmark specifications is p = 156.9064 [kPa]. 2. NUMERICAL MODELS 2.1. CFD Model STAR-CCM+ version v10.04.009 has been used. The flow to be simulated has been considered incompressible, Newtonian, turbulent and, as a first approximation, is assumed to be a single-phase liquid with variable properties. No conjugate heat transfer problem has been solved; instead the heat flux has been imposed on the outer surfaces of the rods. Two turbulent closure models will be compared in Sec. 3, k- SST and k-. As recommended in [24] all-y+ treatment is employed at the solid walls. The finite-volume segregated solver of STAR-CCM+ uses the SIMPLE algorithm [25] to update the velocity and pressure field. The discretization of the conservation equations leads to a sparse system of linear equations that is solved for each variable at each finite-volume

centroid (collocated approach) using an iterative solver. The spatial operators are approximated using second-order accurate numerical schemes: in particular the convection term is discretized with a second-order upwind scheme while Gradients are computed using a second-order accurate hybrid Gauss least square method with Venkatakrishnan limiter [24]. The iterative method for solving the linear system of equations obtained after discretization relies on an algebraic multi grid (AMG) algorithm. One of the main advantages of AMG solvers is that they can be used to solve large sparse linear systems of equations obtained by discretizing elliptic partial differential equations on 2D and 3D unstructured meshes. The solid model has been created using SolidWorks 2013 following the geometrical specifications given in the benchmark documentation. Only the fundamental parts have been modelled, i.e. 1 rod, 1 thimble, 1 SSG, 1 NMV, 1 MV. The original grid geometries have been simplified in order to avoid any contact with the rods for simplifying the treatment of the thermal boundary conditions, also the spring and dimple design have been slightly simplified. The fundamental parts have been imported into STAR-CCM+ and assembled to generate the whole bundle with grids. The enclosing channel that represents the fluid continuum has been modelled in STAR-CCM+ in such a way that 32 axial regions have been created. These regions share internal boundary interfaces where information about the thermal hydraulic variables is exchanged between the adjacent volumes in order to guarantee continuity throughout all the system. This method allows the analyst to perform mesh sensitivity studies only on localized zones (i.e. close to the grids where high temperature and velocity gradients are present) if the foreseen full scale model reaches very large mesh sizes. Secondly, this approach permits local refinement/coarsening depending on the needs by simply introducing the mesh parameters previously identified with the mesh convergence studies. The mesh parameters characterizing the size and the quality of the full length PSBT fuel bundle with grids are reported in Table 3. The mesh convergence study that helped to identify such parameters is presented in Sec. 3. The few cells that present the lowest quality are located in the wall boundary layer where springs and rods surfaces or straps and channel walls are very close. However the chosen wall treatment and the limiters reduce the impact of these low quality cells on the final solution. The number of prismatic layers at the walls as well as the average y+ is comparable with that used in [16][17]. The average number of cells per unit length is also comparable with the average value provided by the MANIVEL benchmark participants. The thermophysical properties of water have been considered dependent on the local value of temperature using polynomial fits to IAPWS-IF97 [26] values that were computed at the system pressure specified by the benchmark documentation. The material used for rods, thimble and grids is Inconel 600. In [27] the average surface roughness ζ was measured for the same application considered here. It was found that 0.5 < ζ < 0.8 [μm] which is consistent with a smooth drawn pipe. For

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simplicity, the walls of the enclosing channel are also assumed to be smooth. 2.2. Sub-channel Model COBRA-TF v3.5 (CTF) solves for the mass, momentum and energy conservation equations of three different phases (liquid, vapour, droplets). Two different partial differential equations are used to describe the evolution of the axial and lateral momentum by taking advantage of the fact that lateral flow momentum is smaller than the axial flow momentum. Turbulent transport terms, but also phaseinteraction terms that are not explicitly resolved. Instead, they are taken into account as distinct source terms and their contribution is modelled following empirical correlations suitable for each different phenomenon and flow regime. For instance, a simple turbulent diffusion model based on a user input that defines the lateral to axial mass flux ratio between adjacent subchannels is introduced [28] (β = 0.05 [21]). Similarly, the viscous stresses are treated as the sum of wall shear and fluid-fluid shear components. The wall shear component of each phase is modelled by wall friction losses and form losses (see Sec. 4). CTF also solves the heat conduction problem inside the heated structures. In contrast to the CFD model, where the thimble tube is considered adiabatic, the inner surface of the thimble tube is instead maintained at a constant temperature that equals the temperature of the coolant at the subchannel inlets. A finite-volume method is employed to discretize the conservation equations of mass, momentum and energy on staggered grids. The SIMPLE algorithm is employed by CTF to solve the conservation equations. A pseudo-steadystate method is used: after every time-step advancement, CTF evaluates a set of simulation convergence criteria to determine if the solution is steady (global energy and mass balance are two quantities monitored). In order to increase stability, smoothing of the solution is performed between the solution at the previous time-step and the current one. The present CTF model employs two different main axial nodalizations, one for the 25 solid structures (heated rods and thimble tube) and one (with its associated staggered grid) for the 36 subchannels that form the heated length of the PSBT fuel rod assembly. The axial nodalization of the subchannels is made of 55 nonuniform elements (see Table 4). The axial cell distribution is dictated by the location of the spacer grids and the location of the temperature and void sensors employed in the actual test facility. 3. CFD SENSITIVITY STUDIES 3.1. Mesh Convergence Convergence of the solution as function of the mesh has been studied for each type of grid using the steady-state solver. Periodic boundary conditions have been applied to the inlet and outlet sections for simulating an infinitely high system with an infinite number of grids of the same type. The size of the periodic domain has been set to be sufficiently large to

avoid the generation of numerical artefacts in the solution (Table 5) by allowing the flow to dissipate the perturbations introduced by the grids and return to a fully developed turbulent profile before encountering the next grid. For this study also inner interfaces were generated to take into account their influence. The initial and boundary conditions are representative of the test 7T (same mass flow rate and system pressure and temperature at the inlet) with the exception of the rod heating, indeed isothermal flow has been assumed. To judge the mesh convergence the pressure loss coefficient [-] has been used:

=

−

=

Δ〈

〉 − Δ〈 1 〈 〉〈 2

〉

〉 (1)

Eq. (1) is consistent with [20] and the CTF model for form pressure losses [28]. The subscript “G+B” means that the pressure drop is computed in a model made of one grid (G) and the rod bundle (B) enclosed by four walls, whereas the subscript “B” implies that the pressure drop is computed in a corresponding model where the grid has been removed and only the bundle (B) enclosed by fours vertical walls remains. Mesh convergence is shown in Table 6 for the tree types of spacers. Because the spacer grids present very narrow gaps between solid surfaces, in order to achieve a reasonably good quality mesh a minimum number of cells is always needed. On the other hand, in order to be able to simulate the full length assembly, very fine numerical grids are not suitable and of interest because of the computational constraints. For such reasons numerical solutions are considered converged if the relative error compared to the result obtained with the finest grid is less than 2.5%. The optimal parameters are reported in the table as Normal (while Coarse and Fine are selfexplanatory). 3.2. Sensitivity to turbulence models For the sake of brevity, in Table 7 only the values of are provided for models with grids built with the same , mesh parameters identified as optimal in Sec. 3.1. It is important to notice that differences between solutions obtained with different turbulence models have to be expected in complex geometries (spacer grids) and in the absence of accurate experimental measurements one cannot identify the most appropriate RANS turbulent model. It has been found in [16][17] that numerically predicted pressure loss coefficients are affected by turbulence models depending on the geometrical features, too (SS are affected while MV are less affected). In the present study, given the pressure loss coefficients specified in the benchmark, k − ω SST was preferred because it provides lower pressure loss coefficients (i.e. more in line with values given in the benchmark documentation) than the k − ϵ model with a quadratic constitutive law. A similar behaviour is reported in [16][17], i.e. the MV grids are less affected by the turbulent model than the grids without vanes. Considering that the largest discrepancies between the two turbulence models is

3

about 15% for the SS grid and much lower for the other two grid types, this sensitivity study to the turbulence closure models gives sufficient confidence that the numerically computed values are representative of the actual grid pressure losses, however experimental measurements would be very helpful. 3.3. Sensitivity to grid geometrical details As previously stated, the original grid geometries have been simplified in order to avoid any contact with the rods for simplifying the treatment of the thermal boundary conditions as well as the spring and dimple design. Because of such simplifications it is important to check and quantify their impact by comparing also the pressure loss coefficients of the original geometries using CFD (Table 8). It is clear that while the geometrical simplifications introduced in the SS type of grid have reduced the pressure drop coefficients (mainly because of an increased flow area), this is not the case for the NMV. One hypothesis is that the larger pressure drop coefficient is caused by the increased turbulence intensity generated by the interaction of the flow with the numerous sharp edges of the simplified geometries. The modified MV geometry only shows small discrepancies due to the changed geometrical details mainly because a large part of the pressure loss is determined by the vanes and not by the straps or springs. Overall, we conclude that the simplified geometries represent fairly well the original grid design that is reported with drawings in the PSBT benchmark specifications.

4.

Inputting the obtained mass flow rate distribution to CTF input file.

The resulting velocity profiles calculated with STAR-CCM+ and the obtained boundary condition for CTF are shown in Figure 3. It has been found that a flat, uniform inlet mass flow rate profile leads to higher pressure drop in comparison to the fully-developed inlet mass flow rate profile (for the case under consideration the difference is about 2.0 [kPa], which roughly corresponds to 1% of the total pressure losses). Moreover, by using the fully-developed inlet mass flow rate profile, the rate of convergence of CTF solver has increased. 4.2. Single-Phase Rod Friction Correlation CTF allows the user to change three parameters of the following power law formula to approximate the Darcy friction factor

=

+

1. 2.

3.

Solving for a fully-developed turbulent flow in a periodic domain representing an infinite bare bundle using CFD; Computing the area averaged1 velocity distribution 〈u 〉 , using the CFD solution by grouping the cells surfaces in the corresponding CTF subchannel radial discretization ( , , ) ≡ ( , , ); Applying the following equation to calculate the subchannel mass flow rate

̇ =〈 〉 〈

〉

,

(2)

1 While formally a surface averaging operator should be employed, in the present analysis arithmetic averaging has been used instead.

(3)

For the sake of later comparison it has been decided to impose = 0 and, by a least squares fitting procedure, the duplet ( , ) = (0.188145, −0.202683) has been identified using 1000 points computed with the Colebrook-White empirical correlation (for turbulent flow in smooth pipes, i.e. = 0) in the range of 1.0E + 04 ≤ ≤ 1.0E + 06

1

ζ 3.7

= −2 log

4. RESULTS 4.1. Inlet Velocity Profile Among its features, CTF allows the user to define the inlet mass flow rate at subchannel level. Following the indications of [16][18] a four-step procedure has been implemented that computes the fully-developed CTF subchannel inlet flow distribution based on CFD simulation results:

.

+

2.51

.

(4)

It has to be said that the fitting proposed is very close to McAdam’s correlation also available in CTF, but it differs from the original/default correlation, i.e. max(1.691Re . , 0.117Re . ) [28]. In Table 9, the pressure loss coefficients due to wall friction and the theoretical value (κ , ), computed using the ColebrookWhite equation of the Darcy friction factor f for smooth pipes, are presented for each grid (simulation domain of height H [m]). Following the quantities introduced in Eq. (1), the pressure drop coefficient of the bare bundle due to wall friction is

=

Δ〈 1 〈 〉〈 2

〉 〉

=

(5)

As a matter of fact, all the simulations of the bare bundle predict rather well the expected pressure losses due to friction at the walls. 4.3. Grid Pressure Loss Coefficients The hydraulic resistance of the three different types of grids has been characterized using Eq. (1) and in Table 10 it is compared with the pressure loss coefficients suggested in the

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benchmark documentation. It is important to note that the pressure drops measured in the CFD models are caused by the actual surface forces acting on the solid walls (that are explicitly modelled) and volume forces present in the simulated flow. Clearly the suggested values are far too small , when compared to those computed by CFD. Further investigation was performed by simulating the flow of the same assembly which is not enclosed by a channel but it is surrounded by an infinite number of similar assemblies using periodic boundary conditions. This configuration well represents the behavior of central assemblies inside a core. It has been found that the pressure loss coefficients predicted by CFD for such periodic domain match those suggested in [1] (only SS and MV are calculated because of their number and since they account for the largest part of the total pressure drop along the assembly height). This implies that the values of are realistic for a central core assembly but not for the , actual PSBT experimental facility closely surrounded by four vertical walls and it would justify the fact that the PSBT pressure measurements have not been released as well as that the pressure benchmark was defined only as a code-to-code comparison. 4.4. Overall Pressure Drop Before presenting the results along the axial direction, it is useful to introduce a simple analytical model that can be used to estimate the expected pressure drop over the heated length of the PSBT fuel bundle for single-phase flow. To derive it, the integral form of the momentum conservation equation is re-written in vector notation as [29]

( ⋅ )

=

Ω+

.

(6)

After few assumptions and simplifications are introduced, the overall pressure drop over the bundle can be written as

1 Δ〈 〉 ≈ 〈 〉 〈 2

〉 +〈 〉

1 + 〈 〉 〈 2

〉

+ (7)

̇

1 〈 〉

−

1 〈 〉 =Δ

+Δ

+Δ

+Δ

pressure conditions, which are computed iteratively by solving the macroscopic energy balance, the mass balance and Eq. (7) until the latter reaches convergence. Table 11 includes the results for total pressure drop calculated over the heated length. CTF results obtained with the PSBT suggested values for the grid pressure loss coefficients are given and the analytical values are reported for the sake of comparison. CFD and CTF show a rather good agreement with a 4.7% relative error (here the CFD value is considered the reference), the analytical model is also very close to the CFD predicted value (1.2%, with the analytical value considered the reference) when using the grid pressure loss coefficients computed by the CFD model. However, a large difference (more than 30% relative error if CFD is considered the reference) is seen when CFD is compared with numerical models (CTF or analytical solutions) that use the benchmarkprovided loss coefficients. In Figure 4, all the available results (CFD, CTF, Eq. (7), PSBT Pressure Benchmark) have been plotted for the sake of completeness. 4.5. Axial Pressure Distribution at Subchannel Level In order to give a picture of the detail that is captured by the CFD solution, a three dimensional representation of the flow is presented in Figure 5. Velocity magnitude is shown in the cross-section where a MV grid is placed. High velocities can be seen due to the restriction of the cross-sectional area induced by the grid. The local temperature on the central heated rods is also shown. There, hot streaks are clearly visible. These hot streaks are not straight because of the mixing vanes that deflect the flow inducing a swirling motion (see streamlines in magenta-green colors). Since in the CFD model the rod heat flux is imposed, the surface temperature of the rods is directly related to the local velocity and thickness of the turbulent boundary layer: the thinner the layer the lower the surface temperature. To compare local quantities obtained by CTF and CFD in a strictly consistent way, volume-averaging must be applied to the CFD results using a coarse grid that overlaps on the CTF axial and radial nodalization. A tool was developed to group all the CFD cells that belong to a certain CTF node (using the so called point in polyhedron algorithm) by knowing the CFD cells centroid coordinates (point) and the CTF node boundaries (polyhedron). After grouping the CFD cells (with cell volumes and cell scalar values with = 1, … , ) in the corresponding CTF node , the discrete volume averaging operator is employed

.

Eq. (7) can also be used to estimate the contribution of each phenomenon to the overall pressure drops (wall friction at the bundle, pressure loss due to the grids, etc.). To keep the analytical model as simple as possible, the volume averaged density 〈 〉 and velocity 〈 〉 are evaluated using the IAPWS-IF97 table at the bundle average temperature and

=

∑ ∑

.

(8)

Subchannel axial pressure distributions appear in Error! Reference source not found.. CFD and CTF perform very well all along the assembly heated length and very minor discrepancies are recorded. As it can be seen a consistent

5

definition of grid pressure loss coefficients is of extreme importance in order to be able to correctly model the effect of grids in CTF. Given the CFD calculated κ values of Error! Reference source not found. a remarkable agreement of the pressure jumps across SS and MV spacer grids between the two numerical solutions is obtained. As expected at each elevation all subchannels share the same pressure. The small discrepancies could be further decreased by improving the CTF modelling of pressure drop coefficients (per-channel/per-gridtype) and by introducing a better inter-subchannel mixing. Indeed, by a careful examination a rather clear trend can be seen in the curves: after each MV grid the slope of the CTF pressure profile is lower than the one computed with CFD. However, for the conditions analysed in the report, such detail is of secondary importance for the pressure estimation. 4.6. Axial Temperature Distribution at Subchannel Level The axial liquid temperature profiles in each subchannel are shown in Error! Reference source not found.. A very good agreement is obtained along all the heated length. At about z = 1.5 [m] the CTF solution deviates from the CFD one because of evaporation and re-condensation phenomena that increase the enthalpy level of the water contained in the CTF cell volumes, however at the exit the two models predict very similar temperatures. In Table 12 the outlet temperatures computed with CFD and CTF are compared with the analytical model briefly described in Sec. 4.4. It can be seen that, in comparison to a reference temperature jump of about 52 [K] between inlet and outlet, the discrepancies from the analytical solution are 1.20% and 1.39% for CFD and CTF respectively. A remarkable spread between subchannel temperature profiles is seen in the CFD solution (Error! Reference source not found.) while, basically, only 4 families of curves characterize the CTF solution. The four families correspond to subchannels at the corners of the assembly, at edges of the assembly, central subchannels surrounded by four fuel rods and subchannels adjacent to the thimble. The explanation for such behaviour has to be found in the presence of a non-uniform radial mass flux distribution and the explicit modelling of grids with mixing vanes with alternating orientations (0, /2,0, /2, … [rad]) along the heated length in the CFD model. Such details are lost in the CTF model where an over-prediction of mixing has been identified. CTF shows a large spread of the liquid temperature 1.25 ≤ ≤ 2.00 [m], i.e. around the maximum of the power profile of the heating rods where boiling is present. 5. CONCLUSIONS A verification exercise has been performed by comparing CFD (STAR-CCM+ v10.04.009) and subchannel (CTF v3.5) simulations of the full heated length of PSBT 5x5 fuel rod bundle. CFD and CTF codes present different approaches for solving similar conservation equations at very different scales. For this reason, one of the goals of the study was to build consistent models of the same test facility using the current code capabilities and approaches that are commonly

employed in the community. In order to generate a numerical model in CTF that is consistent with the CFD model, particular care has been taken in describing the grid pressure loss coefficients, the wall friction factor and the fully developed turbulent velocity inlet profile by extracting information directly from the CFD solution. CFD solutions have undergone mesh convergence studies, a parametric study of the turbulence closure models and parametric study of the geometrical simplifications of the original grid design. In support to the presented findings, all the abovementioned studies have shown common trends and features in the solutions that have been reported in recent published work [16][17] where a different assembly and grid design have been analysed. A simple analytical model based on first principles (global energy balance, conservation of mass and integral form of the momentum equation) has been used to support the numerical solutions. This exercise has verified the consistency in the definition and the information extracted from the CFD solution and simultaneously has supported the validity of the CFD solution from an integral point of view. It has been found that CTF and CFD provide very close results (within 5%) for the overall pressure drop along the rod bundle heated length. On the other hand, it has been found that the grid pressure loss coefficients suggested in the benchmark well approximate the value of a system (central assembly in a core) that is different from the PSBT facility where indeed the effect of the enclosing channel walls is strong and present. Pressure and temperature have been compared for each subchannel along the bundle axial elevation. In order to be able to compare CTF and CFD solutions, the latter have been volume-averaged accordingly to CTF nodalization. The pressure and temperature profiles have shown a very close match between the two codes. This result stresses the importance of using a consistent value for the pressure loss coefficient associated to each type of grid. The small discrepancies could be further decreased by improving the CTF modelling of pressure drop coefficients (per-channel/per-gridtype) and by introducing a better inter-subchannel mixing. On the other hand the CFD model could be improved by using Eulerian two-phase flow model and conjugate heat transfer. ACKNOWLEDGMENTS This work was partly funded by the Swiss Nuclear Safety Inspectorate ENSI (H-101230) and by PreussenElektra GmbH and was conducted within the framework of the STARS program (http://www.psi.ch/stars). The authors would like to acknowledge the Swiss National Supercomputing Centre (CSCS) for their assistance with computing resources. NOMENCLATURE p Pressure u Velocity component k Turbulent kinetic energy y Distance from the wall V Volume  General scalar field

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 H A D f g  Re a b c ̇  i

Science and Technology of Nuclear Installations, pp. 1-19, Vol. 2014, 2014

Density Bundle height Bundle cross-sectional area Diameter Darcy friction factor Gravitational acceleration Pressure loss coefficient Reynolds number Coefficient Coefficient Coefficient Mass flow rate Pipe roughness index

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Table 1: PSBT test 7T: Assembly specifications. Assembly Rod array Number of heated rods Number of thimble rods Heated rod outer diameter [mm]

B7 5x5 24 1 9.50

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12.24 12.60 3658.00 64.90 B Cosine 7 2 (only 1 within the heated length) 8 471, 925, 1378, 1832, 2285, 2739, 3247 2.5, 3755 237, 698, 1151, 1605, 2059, 2512, 2993, 3501 1.0 0.7 0.4

[mm] Thimble rod outer diameter Rod pitch [mm] Axial heated length [mm] Flow channel inner width [mm] Radial power shape Axial power shape Number of MV spacer grids Number of NMV spacer grids Number of SS grids MV spacer grid locations [mm] NMV spacer grid locations [mm] SS grid locations [mm] Pressure loss coefficient MV Pressure loss coefficient NMV Pressure loss coefficient SS

Table 2: Rated conditions employed for initiating PSBT test 7T (power increase) and pressure benchmark code-to-code comparison. Time [s] Outlet Pressure [MPa] Inlet Mass Flux [kg m-2 s-1] Total Power [MW] Inlet Temperature [K]

0 – 32 15.5141203 3338.889 2.500 565.05

Table 3: Summary of mesh parameters and quality measures employed in the full-length PSBT fuel bundle CFD model. Quantity Value Quantity Value 122’997’403 Min./Ave./Max. cell aspect 0.01/0.40/1.0 Tot. num. cells [-] ratio Min./Ave./Max. cell 0.00/5.25/135.00 Linear average num. of 33’624’221.71 skewness angle [°] cells [1/m] 0.4 Min./Ave./Max. cell quality 1.0E-5/0.950/1.00 Cell base size [mm] [-] Num. wall boundary 5 prism layers Prism layer thickness 0.2 [mm] Prism layer stretching 1.0 factor [-] 1.61/40.81/135.55 Min./Ave./Max. y+ [-] Medium Volume grow rate Table 4: Axial elevation of faces delimiting CTF nodes. Node z [m] Node z [m] Node z [m] 1 0.000 15 1.000 29 2.059 2 0.003 16 1.076 30 2.206 3 0.081 17 1.151 31 2.226 4 0.159 18 1.227 32 2.246 5 0.237 19 1.302 33 6 0.315 20 1.378 34 7 0.393 21 1.454 35 8 0.471 22 1.529 36 9 0.547 23 1.605 37 10 0.622 24 1.681 38 11 0.698 25 1.756 39

Node 43 44 45 46 2.265 2.285 2.361 2.436 2.512 2.659 2.679

z [m] 2.824 2.908 2.993 3.167 47 3.187 48 3.207 49 3.227 50 3.247 51 3.332 52 3.416 53 3.501

9

12 13 14

0.774 0.849 0.925

26 27 28

1.832 1.908 1.983

40 41 42

2.699 2.719 2.739

54 55 56

3.553 3.606 3.658

Table 5: Grid heights and CFD periodic domain simulated sizes. Grid Type Grid Height [mm] Periodic Domain Size [mm] SS 12.8 256.00 NMV 41.148 411.48 MV 58.928 468.65 Table 6: Mesh convergence study Grid Type Num. of Cells [1E+06] Grid Zone Channel Zone SS Coarse 2.43 0.38 SS Normal 3.35 0.59 SS Fine (Reference) 5.94 2.85 MV Coarse 5.24 0.80 MV Normal 6.67 1.46 MV Fine (Reference) 12.0 6.71 NMV Coarse 2.91 0.77 NMV Normal 4.12 1.01 NMV Fine (Reference) 7.29 5.91

Pressure Loss Coefficient [−] [−] 0.747 0.366 0.781 0.380 0.772 0.382 1.728 0.662 1.752 0.682 1.770 0.690 1.546 0.591 1.514 0.623 1.490 0.620

Rel. change to reference [%] Grid Zone Channel Zone 2.90 4.29 1.48 0.63 2.35 4.05 1.02 1.15 4.09 3.90 1.95 1.30 -

Table 7: Comparison of grid pressure loss coefficients obtained with a non-isotropic turbulence model (k − ϵ model with a quadratic constitutive law, KEQ) and an isotropic one (k − ω SST, KO). SS NMV MV 0.932 1.600 1.760 0.781 1.514 1.752 κ Table 8: Calculated grid pressure loss coefficients (CFD) for the simplified and the original geometries. Grid type / , , SS 0.781/0.954 NMV 1.514/1.222 MV 1.752/1.791 Table 9: Calculated bare bundle pressure loss coefficients (CFD) and theoretical one (TH). Grid type / , , SS 0.380/0.382 NMV 0.623/0.615 MV 0.682/0.700

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Table 10: Calculated and suggested grid pressure loss coefficients for the modified geometries with and without the presence of the channel (the latter indicated with the subscript “PER”). Subscript “MOD” refers to the modified geometry and “PSBT” to the recommended values. Grid Type , / , , // , SS 0.781/0.51/0.4 NMV 1.514/-/0.7 MV 1.752/1.05/1.0 Table 11: Calculated pressure loss over the heated length for PSBT test 7T benchmark initial conditions using CFD, CTF and Eq. (7). In brackets values calculated using PSBT loss coefficients. CTF 〈 〉 [kPa] Analytical 〈 〉 [kPa] CFD 〈 〉 [kPa] 237.330 248.442 (166.993) 234.396 (160.205) Table 12: Comparison of average outlet temperatures based on subchannel values computed with CFD, CTF and the analytical model. CFD 〈 〉 [K] CTF 〈 〉 [K] Analytical 〈 〉 [K] 616.17 616.07 616.79 1.6

1.4

Power coefficient [-]

1.2

1

0.8

0.6

0.4 0

0.5

1

1.5 2 Axial coordinate [m]

2.5

3

Figure 1: Axial profile of relative power coefficients.

Figure 2: PSBT full length fuel bundle CAD geometry assembled in STAR-CCM+.

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3.5

-1

CTF Mass Flow Rate Inlet Distribution [kg s ]

6

0.3

5 0.25

4

0.2

3

0.15

2

1 0.1

1

2

3

4

5

6

Figure 3: On the left: CFD fully-developed turbulent flow inlet boundary condition. On the right: CTF mass flow rate inlet distribution derived from the CFD model (one square corresponds to one subchannel).

Figure 4: Comparison of PSBT pressure benchmark results and present study.

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Figure 5: 3-D representation of the CFD solution of PSBT 7T in a region of fluid downstream a MV spacer grid (transparency is used to show the grid, 16 peripheral rods and the thimble have been hidden as well). The arrow indicates the flow direction.

Figure 6: On the left: comparison of CTF (dashed lines) and volume averaged CFD (solid lines) axial pressure profiles in the 36 subchannels. On the right: Comparison of CTF (colored dashed lines) and volume averaged CFD (colored solid lines) axial liquid temperature profiles in the 36 subchannels. In black the volume averaged saturation temperature computed in CFD. Vertical lines show the position (lower surface) of SS grids (magenta) and MV spacer grids (red).

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