COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF THE FLUID FLOW AND HEAT TRANSFER IN THE CORE BYPASS REGION OF A PWR

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1 COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF THE FLUID FLOW AND HEAT TRANSFER IN THE CORE BYPASS REGION OF A PWR Cliord I, Vasiliev A, Zerkak O, Ferroukhi H and Pautz A Laboratory or Reactor Physics and Systems Behaviour (LRS) Paul Scherrer Institut 5232 Villigen PSI, Switzerland ivor.cliord@psi.ch, alexander.vasiliev@psi.ch, omar.zerkak@psi.ch, hakim.erroukhi@psi.ch, andreas.pautz@psi.ch ABSTRACT The development o analysis models or the Swiss reactors is a key objective o the STARS project at the Paul Scherrer Institut (PSI). Within this context there is a need or the development o computational luid dynamics (CFD) models o the Swiss reactors in support o uture high idelity investigations o steadystate and transient scenarios. This article presents initial results or the CFD analysis o a Siemens KWU PWR with a ocus on the low behavior and heat transer in the gap between the core shroud and core barrel. Temperatures and densities in this region o the reactor are important or accurate estimations o ast neutron luence and activation in the steel structures o the core shroud, core barrel and reactor pressure vessel. The low behavior in this region may also be relevant in the understanding o ex-core detector responses. The low conditions in the core bypass region were ound to be in the transition-toturbulence regime, with vortex shedding taking place downstream o the core ormers as a result o low instabilities. As a consequence, steady-state simulations could not be used or the heat transer simulation. Time-dependent simulations were also deemed uneasible or obtaining converged solid material temperatures. To reduce the computational size o the problem, the core bypass region was subdivided into several axial slices with periodic boundaries in the axial direction. T irst approach assumes time-averaged mass lux and thermal diusivity ields as inputs to the coupled steady-state temperature solution. In the second approach, time-dependent simulations were run assuming ixed wall temperatures to obtain time-averaged heat transer coeicients, which were then applied as boundary conditions to the solid heat conduction problem. Results or both approaches are presented. Additional sensitivity tests were conducted using approximated gamma heating maps. This work conirms that heat conduction through the core shroud on its own has minimal impact on the core bypass temperatures and that uture work should ocus on gamma heating eects in the core bypass region. KEYWORDS Computational luid dynamics, Core bypass, Heat transer 1. INTRODUCTION The topic o reactor lie extension has gained interest over the past years as many reactors approach their 40 year design lie span. There is renewed interest in understanding the behavior o the structural materials in the reactor core, in particular the reactor pressure vessel, core barrel and core shroud which are under high neutron luence and potentially high thermal stress due to their proximity to the reactor core. These operating conditions lead to localized embrittlement o the steel and welds in these

2 components. The low behavior in this region may also be relevant in the understanding o ex-core detector responses. Some general inormation on luence and dose in PWR components is available in literature. Reerence [1], or example, contains luence, dose and operating temperature values or selected steel components directly adjacent to the core or French, British and German PWRs. While useul, this inormation is not suiciently detailed to build up a picture o the luence distribution in these components. Altstadt et al. [2] obtained inite-element solutions or the temperatures and stresses in a German PWR core bale, using heat sources derived rom Monte Carlo simulations. Their analysis assumed a simple one-dimensional approach to modeling the luid low. Rupp et al. [3] perormed detailed CFD and inite element stress analysis o the core bale structure o a French PWR, with heat sources similarly derived rom Monte Carlo simulations. The development o analysis models or the Swiss reactors is a key objective o the STARS project at the Paul Scherrer Institut (PSI) [4]. Within this context there is a need or the development o computational luid dynamics (CFD) models o the Swiss reactors in support o uture high idelity investigations o steady-state and transient scenarios. This article presents initial results or the CFD analysis o a Siemens KWU PWR. The low behavior and heat transer in the gap between the core shroud and core barrel are analyzed using the open-source CFD sotware OpenFOAM [5]. Version o OpenFOAM has been used. This analysis orms the irst step towards coupled CFD / Monte Carlo simulations o the core bypass region, in support o ageing and lie extension studies in STARS. The paper is organized as ollows; Section 2 contains a description o the models and approaches used or analyzing the core bypass low. Section 3 provides the analysis results and urther discussion. Conclusions and recommendations or uture work are given in Section MODELING METHODOLOGY The core bypass region is relatively complex with many heat transer suraces and, at the start o this work, relatively little was known about the low characteristics in the core bypass o German KWU reactors. The geometry o the core ormers diers rom that o the French PWRs. In particular, more perorations are present, which is expected to aect the low velocities, pressure drop and heat transer characteristics in this region. In their work, Rupp et al. [3] initially subdivided the core bypass region into multiple axial regions to reduce the problem complexity. This approach is attractive since it allows solutions to be obtained in a reasonable amount o time with ewer computational resources. Rather than simply meshing the entire core bypass region and obtaining a limited number o solutions or the heat transer in the core bypass region, the choice was made to simpliy the problem using a similar axial slice approach. In this way more simulations could be run, the results could be analyzed in more detail, and a better understanding o the behavior o the core bypass lows could be obtained. A side view o the core bypass region and the vertical spacing o the core ormers is given in Figure 1(a) and (b). The spacing between the core ormers varies rom 425 mm to 535 mm. While the spacing is not constant, it does not dier signiicantly rom an average value o 0.5 m over a total height o approximately 4 m. Each core ormer acts as a throttle or obstruction to the core bypass low and disturbances upstream o each core ormer will likely not inluence the low beyond the obstruction. This eectively decouples the low solution in the axial direction. We make the assumption that the axial slices are o equal height, and that the low is ully developed in each o the axial slices. We lastly assume that the low is incompressible with constant density. Under these assumptions the low is periodic in the axial direction. Our approach, thereore, is to set up a periodic model or the luid low and heat transer in the core bypass region as shown in Figure 1(c)

3 (a) (b) (c) Figure 1. (a) Core bypass region, (b) axial locations o the core ormers and (c) the axial slice model or heat transer in the core bypass region 2.1. Flow Conditions Beore discussing the details o the axial slice model, however, we consider the expected low characteristics in the core bypass region. Under nominal conditions the Reynolds number is 5 5 Re through the perorations o the core ormers ( D H 40 mm ), and Re in the open regions between the core shroud and core barrel ( D H 300 mm ). These values suggest turbulent low but are misleading since the low through each o these cross-sections never reaches a ully developed state. In reality the low is in a transitional state as can be seen by considering the Strouhal number ( St ) suggested by Castro [6] or low through perorated plates with 25% low area. This geometry is approximately equivalent to the core ormers. Based on this value the expected requency o vortex shedding is 0.5 Hz. While this Strouhal number is or air lows in dierent geometries, and is thereore highly approximate, the low shedding requency suggests that the low or this complex model may be in the transitional turbulence regime and that steady-state solutions may not be easible. This is indeed the case, since initial steady-state RANS simulations ailed to ully converge and subsequent timedependent simulations exhibited vortex shedding downstream o the perorations in the core ormers with a shedding requency o approximately 0.3 Hz. This inherent non-stationary low behavior makes the modelling o heat transer in the core bypass region challenging. Firstly, the time-scales associated with the solid and luid components are considerably dierent. Simulations o the temperatures in the steel structures, with time scales o hours, become uneasible when the luid velocity and temperature exhibit luctuations with time scales in the order o seconds. This issue is urther discussed in Section 2.3. A second complication stems rom the inherent inability o RANS turbulence models to accurately model very low Reynolds lows. Traditional RANS models (e.g. k- ) are optimal or highly turbulent lows. Large eddy simulation (LES) is thereore more appropriate or this analysis, but LES is even more prohibitive since it requires more reined meshes and even smaller time steps. We have proceeded using traditional RANS turbulence models in order to simpliy the modelling complexity and to reduce computational requirements, with the knowledge that these models are not strictly appropriate or our case

4 2.2. Modelling Equations The unsteady incompressible Reynolds-averaged Navier Stokes (RANS) equations are solved in the luid and solid regions. The conservation equations or mass, momentum and energy are given below. Conservation o mass in the luid: u = 0 (1) Conservation o momentum in the luid: u t + u u + τ e = P (2) Conservation o energy in the luid: Conservation o energy in the solid: where T Q (3) + u T α e T = t ( C ) ( ρc P ) ρ P T s s κ s Ts = Q (4) s t the subscripts and s reer to the luid and solid regions respectively ρ, u, P and T are the density, velocity, pressure and temperature respectively κ and C P are the thermal conductivity and speciic heat capacity respectively Q is a heat source per unit volume τ e is the luid shear stress tensor including turbulent mixing eects α is the luid eective thermal diusivity including turbulent mixing eects e In OpenFOAM, the system o equations is discretized using standard inite-volume techniques [7] and solved iteratively using segregated algorithms. For steady-state solutions, the algorithm is based on the SIMPLE algorithm o Patankar [8]. For time-dependent problems, the algorithm is based on the PISO algorithm o Issa [9] Quasi-Static Solution Algorithms In order to obtain converged temperatures in the steel structures, two quasi-static approaches are proposed: 1. Transient simulations or the low in the core bypass region, with adiabatic boundary conditions, are time-averaged to obtain quasi-static mass lux and eective thermal diusivity ields. A steady-state ully coupled conjugate heat transer simulation is run thereater using this static low-ield inormation. This approach assumes that the time evolution o temperature, mass lux and the

5 eective thermal diusivity can be separated in time. This assumption has no physical or mathematical basis. I the time-variation o low quantities is small, however, the error introduced by this assumption is expected to be small. The algorithm is outlined below. 1: Transient isothermal CFD solution in luid u, P, α e 2: Calculate time-average o ields u, P, α e 3: Steady-state conjugate heat transer solution in luid and solid (static lowield) T, T s 2. Time-dependent solutions or the low in the core bypass region, with ixed wall temperatures, are obtained and the luid temperature and wall heat luxes are time-averaged. The time-average wall heat luxes are used to derive approximate heat transer coeicients on the walls in terms o the bulk luid temperature and wall temperature. The temperature in the steel structures is calculated thereater by setting the wall heat transer coeicient and iterating to obtain the converged bulk luid temperature in the core bypass. This approach is mathematically consistent, but assumes that the wall temperature is constant over time and that the heat transer can be characterized by a simple heat transer coeicient and luid bulk temperature. There are uncertainties associated with obtaining values or both o these quantities. The algorithm is outlined below. 1: Speciy ixed wall temperature in luid T w 2: Transient CFD solution in luid u, P, T 3: Calculate heat transer coeicient h w = T w q w T 4: Calculate time-average heat transer coeicient h w 5: Update wall heat transer coeicient in solid 6: FOR user-speciied count 1: Initialize/update bulk luid temperature T proile in core bypass 2: Solve steady-state heat conduction in solid T s 3: Calculate bulk temperature rise in core bypass T 7: END FOR 8: Calculate solid wall temperature T w In the above algorithms, ϕ reers to time-averaging o the ield over a period o approximately 40 s and ϕ reers to volume averaging over a single axial slice as deined in the equations below. ϕ = 1 Δ Δ 0 ϕ dt (5)

6 1 (6) ϕ = ϕ dxdydz V V Convergence o the inner iteration o algorithm 2 is determined by considering the bulk temperature rise in the core bypass. The inner iteration is considered converged when the dierence in bulk temperature alls below 0.01 K rom one iterate to the next. It is worthwhile mentioning at this point that the CFD solution or the luid low, which is common to both approaches, uses the axial slice model o Figure 1(c) in both algorithms. The solid and coupled luid/solid solutions, however, use ull-height models o the core bypass region. This circumvents the need to repeatedly update the boundary conditions in the axial direction. Despite the larger mesh size required and the need to map ield data between meshes, this was ound to be a more practical approach CAD Geometry A three-dimensional CAD model o the core barrel, core ormers and core barrel was developed using the commercial sotware SolidWorks [10]. The geometry or this model is based on original technical drawings or the reactor with a number o simpliications: The bolts holding the core shroud and bypass assembly together and the associated holes, brackets, etc. have been neglected. The slots in the core barrel or locating the core ormers axially have been neglected. These were ound to cause problems during meshing. The geometry o several holes/perorations in the core ormers was adjusted according to Figure 2(a), since the original geometry was problematic or meshing. In all cases, the low area was preserved. The resulting CAD geometry or the solid and luid regions is shown in Figure 2. (a) (b) Figure 2. (a) Core ormer geometry as modelled and (b) CAD geometry or the luid and solid regions

7 2.5. Material Thermodynamic Properties We assume subcooled water at a temperature o C and pressure o 155 bar. The thermodynamic properties o subcooled water at these conditions are obtained using IAPWS IF-97 [11]. A turbulent Prandtl number o 1.0 is assumed. The thermodynamic properties o the steel structures are taken rom the IAEA on-line nuclear materials properties database (THERPRO) [12] Mesh Generation Dierent mesh generation techniques were used or the solid and luid regions. In the luid region, a hexdominant polyhedral mesh was used. This mesh was generated using the open-source sotware cmesh [13], which uses an octree-based meshing algorithm. Three dierent meshes were considered, starting with an initial coarse mesh o 2.4 M cells. Boundary layers were then added to improve the y + values. The inal mesh (4 M cells) used urther reinement in the center o the core bypass region. A snapshot o this mesh is shown in the Figure 3(a). The maximum aspect ratio, skewness and non-orthogonality or this mesh are 10.8, 2.6 and 67 respectively. The mathematical deinitions o these quality parameters are given in [14]. The solid mesh (1.4 M cells), which is shown in Figure 3(b), is a block structured mesh containing a mixture o hex and triangular prism elements, generated using Trelis CFD [15]. (a) (b) Figure 3. (a) Fluid and (b) solid meshes 2.7. Boundary Conditions Periodic low behavior is ensured by mapping values at the axial midpoint o the mesh to the inlet and outlet suraces. The velocity is mapped to the inlet and normalized to obtain the nominal mass low rate. The turbulence parameters are mapped to the inlet without adjustment. The pressure is mapped to the outlet and adjusted to obtain a ixed average outlet pressure. No periodic mapping o the temperature is perormed, i.e. the temperature is ixed at the inlet. Heat transer rom the core barrel to the downcomer, and rom the core shroud to the core, are speciied in terms o simple heat transer coeicients and bulk luid temperatures. Heat transer coeicients are estimated using the traditional Dittus-Boelter correlation. The bulk temperature in the downcomer is assumed constant and equal to the reactor inlet temperature. The bulk temperature o coolant in the core, directly adjacent to the shroud, was taken rom subchannel analysis results

8 Gamma heating In the absence o accurate gamma and ast neutron heat sources in the steel structures and core bypass region, heating in the steel structures was approximated based on a combination o inormation rom several sources. In the axial and azimuthal directions the gamma heating is assumed proportional to the ast neutron luence. The ast luence proiles are taken rom Vasiliev et al. [16]. The proiles were normalized and discrete Fourier transorms used to obtain curve its as shown in Figure 4. (a) (b) Figure 4. (a) Axial and (b) azimuthal approximations or the gamma heat sources in the steel structures In the radial direction, an exponential attenuation rom the core shroud outwards is assumed. We assume a ast neutron attenuation coeicient o 0.1 cm -1 at 20 C corrected or the density o water at reactor inlet conditions. The resulting heat source distribution is given by q 6 ( 1. r ) ( r, z, ) = ( z) ( θ ) e θ (7) The actor was chosen to obtain a maximum internal heat generation o 5 MW/m 3 in the core shroud, which is consistent with results presented in [2]. This yields a net internal heat generation o 3 MW in the steel structures o the core shroud, core ormers and core barrel (0.1% o thermal power). The resulting power distribution is shown in Figure CFD Solution Control Multiple simulations using both lower order (LO) and higher order (HO) dierencing schemes were run. The actual dierencing schemes chosen are summarized in Table I. For the low order solutions, a constant time step o 5 ms was used, while a time step o 2 ms was used or the higher order solutions. The linear solvers employed or each transport equation are summarized in Table II. 3. RESULTS AND DISCUSSION Transient simulations were run or the luid region or 40 s or nominal operation, assuming a ixed wall temperature o 308 C. In each case, a 10 s initialization period was included to ensure that the low was suiciently developed at the start o the transient. The reerence model simulation, representing a quarter height o the bypass region, took approximately 8 hours to run on 64 Intel Haswell 2.40 GHz cores

9 Figure 5. Assumed gamma heat source distribution in the steel structures Table I. Summary o discretization schemes Equation operator Low order solution Higher order solution Time derivative Euler implicit 2 nd order backward Convection Upwind Limited linear TVD Diusion Gaussian integration with linear dierencing Gaussian integration with linear dierencing Pressure gradient Gaussian integration with linear interpolation 2 nd order least squares Table II. Summary o linear solvers Equation (Solution Linear solver Comments variable) Momentum (u ) Bi-conjugate Gradient DILU preconditioner Continuity (p) Geometric Algebraic Multigrid Gauss Seidel smoother Turbulence (k,, ) Bi-conjugate Gradient DILU preconditioner Energy (T) Bi-conjugate Gradient DILU preconditioner Time-averaged values or the major low parameters were calculated. These are summarized in Table III below or several RANS turbulence models and mesh reinements. Independent simulations were also run using the commercial sotware Star-CCM+ [18]. In general the pressure drop is approximately 200 Pa/m and the average wall heat transer coeicient is consistently in the order o 5-6 kw/m 2 K. A value o 8.5 kw/m 2 K is proposed in [2]. The maximum heat transer coeicient predicted by Star-CCM+, is signiicantly larger than that by OpenFOAM. In all simulations, this peak value is located on the lower edges o the perorations in the core ormers where the heat transer characteristics are highly sensitive to the local mesh size and boundary layer thickness. This is a localized eect, which has minimal impact on the overall heat transer behavior o the model

10 Table III. Comparison o solution parameters or dierent meshes, turbulence models and solution methods + yavg y + max ΔP Coarse mesh (2.4 M cells) k- Realizable k- k- SST k- (HO) Realizable k- (HO) 233 / / / / / havg hmax 5153 / / / / /15575 Boundary reined (3 M cells) k- 86 / / Reined mesh (4 M cells) k- k- (HO) Realizable k- (HO) 98/ / / / / /16722 STAR-CCM+ (4.8 M cells) Realizable k- 124 / / For the remainder o this work, the reined mesh was chosen using the standard k- turbulence model and higher order discretization schemes (see Table I). The time-variation o pressure drop and heat lux due to vortex shedding downstream o the core ormers is shown in Figure 6. These vortices are clearly visible in the top-let igure o Figure 7. The time-averaged velocity and wall heat transer coeicient are similarly included in this igure. Figure 6. Time-dependent variation in pressure drop and wall heat lux or the reerence solution Using this time-averaged inormation, heat transer simulations or the ull height o the core bypass were conducting using both approaches proposed in Section 2.3. Figure 8 shows the shroud outer surace temperature or both approaches assuming no gamma heating. The irst approach yields results with sharper, more resolved eatures, which is likely an artiact o the method. The second approach shows more diuse eatures, which is an expected consequence o the time-averaging, but overpredicts the temperatures on the periphery (let and right sides) o the model. This is a result o the simple model

11 employed or the heat transer, which only takes into account the axial variation in bulk coolant temperature and assumes a constant low distribution. As expected, there is a marked dierence in steel structure temperatures i gamma heating is included (see Figure 9). There is distinct peaking in shroud temperatures in the corners closest to the core, coincident with the gamma heat source maxima. The temperatures shown in Figure 9 are comparable with those o [2], which show a peak temperature o 329 C. Figure 7. Instantaneous (top-let) and time-averaged velocity (bottom-let) and heat transer coeicient (right) or the reerence solution (a) (b) Figure 8. Unolded shroud outer surace temperatures, neglecting the gamma heat sources in the core bypass region, obtained using the (a) irst and (b) second approaches Far less noticeable is the bulk temperature rise in the core bypass (Figure 10). I gamma heat sources are excluded, the primary source o heating in the core bypass is by direct conduction rom the core through the core shroud. In this case the temperature rise in the core bypass is very small. I gamma heating is

12 included the bulk temperature rise is more signiicant since a large raction o the gamma heating is removed directly by the core bypass low. The absolute temperature rise in this case is highly approximate because the true gamma heating in the steel structures is at this stage not known. However, based on the simulation results one can conclude that approximately 60% o the gamma heating in the steel structures is removed by convection in the core bypass region and the remainder is removed in the downcomer and core regions. Assuming a uniorm gamma heat source o 0.2 MW/m 3 in the coolant [17] increases the bypass temperature at the top o the core by a urther 2 C. Figure 9. Steel structure temperatures obtained using the second approach with approximated heating in the steel structures (a) (b) Figure 10. Bulk luid temperature in the core bypass using (a) no gamma heat source and (b) gamma heat sources in the steel structures In both approaches the most signiicant portion o the required computational eort is in obtaining the initial transient CFD solution or the luid. Consequently, neither approach is computationally more

13 eicient. While conceptually very simple, the irst approach assumes that the low inormation is separable in time, which has no real physical or mathematical justiication. As an independent veriication this approach has proven useul in this work, however uture work will likely ocus on the second approach, since it has a more physically consistent basis. 4. CONCLUSIONS Initial results have been presented or the CFD analysis o a Siemens KWU PWR with a ocus on the low behavior and heat transer in the gap between the core shroud and core barrel. Temperatures and densities in this region o the reactor are important or accurate estimations o ast neutron luence and activation in the steel structures o the core shroud, core barrel and reactor pressure vessel. First simulations o the low in this region showed that the low is in a transitional state, with vortex shedding taking place downstream o the core ormers. The non-stationary nature o the low presented a challenge in terms o obtaining a solution within a reasonable time period. Two approaches were proposed to address this challenge; Time-averaging o the low-ield inormation beore solving the conjugate heat transer problem and time-averaging o surace heat luxes in order to derive detailed surace heat transer coeicients. Both approaches yielded similar results with similar computational eort, although the second approach has a more physically consistent basis. This work conirms that, on its own, the heat transer rom the core to the core bypass has a negligible impact on the core bypass temperature. Simulations using an approximated gamma heating map have conirmed that the gamma heating within the steel structures increases the temperatures in these structures noticeably. It is thereore important that accurate values or the gamma heating in the steel structures, obtained using Monte Carlo analysis or other methods, be known. This work is a irst step in the STARS project towards coupled CFD / Monte Carlo simulations or the accurate prediction o temperatures and ast neutron luence in the steel structures o the core. 5. ACKNOWLEDGMENTS This work was partly unded by the Swiss Federal Nuclear Saety Inspectorate ENSI (Eidgenössisches Nuklearsicherheitsinspektorat), within the ramework o the STARS project ( 6. REFERENCES 1. P. Petrequin, R. Pelli, P. Soulat and A.A. Tavassoli, Eect o Irradiation on Water Reactors Internals (Including Russion WWER), Study Contract COSU CT94-074, CEA (France), TECNATOM (Spain) and VTT (Finland) (1997). 2. E. Altstadt, H. Kump, F.P. Weiss, E. Fischer, G. Nagel, G. Sgarz, Analysis o a PWR core bale considering irradiation induced creep, Annals o Nuclear Energy, 31(7), pp (2004). 3. I. Rupp, P. Christophe and M. Tommy, Large Scale Finite Element Thermal Analysis o the Bolts o a French PWR Core Internal Bale Structure, Nuclear Engineering and Technology, 41(9), pp (2009). 4. Steady-state and Transient Analysis Research or the Swiss Reactors, (under maintenance) (2015). 5. H.G. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to computational continuum mechanics using object-oriented techniques, Computers in physics, 12, pp. 620 (1998). 6. I.P. Castro, Wake characteristics o two-dimensional perorated plates normal to an air stream, Journal o Fluid Mechanics, 6(3), pp (1971). 7. H.K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd Ed., Prentice Hall (1995) 8. S.V. Patankar and D.B. Spalding, A calculation procedure or heat, mass and momentum transer in three-dimensional parabolic lows, Int. J. o Heat and Mass Transer, 15(10), pp (1972)

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