draft: thermal-hydraulics and conjugate heat transfer...

PREPRINT 27/01/2010 Proceedings of ICAPP ‘10

San Diego, CA, USA, June 13-17, 2010 Paper 10174

DRAFT: Thermal-hydraulics and Conjugate Heat Transfer Calculation in a Wire-Wrapped SFR Assembly

C. Péniguel, I. Rupp EDF R&D, 6 quai Watier –78401 Chatou Cedex, France

Email:[email protected]

S. Rolfo School of MACE, The University of Manchester, Manchester M60 1QD, UK

M. Guillaud

INCKA, 85 Avenue Pierre Grenier, 92100 Boulogne-Billancourt, France

Abstract – Fast reactors with liquid metal coolant have recently received a renewed interest owing to a more efficient usage of the primary uranium resources, and they are one of the proposal for the next Generation IV. In order to evaluate nuclear power plant design and safety, 3D analysis of the flow and heat transfer in a wire spacer fuel assembly are ongoing at EDF. The introduction of the wire wrapped spacers, helically wound along the pin axis, enhances the mixing of the coolant between sub-channels and prevents contact between the fuel pins. The mesh generation step constitutes a challenging task if a reasonable amount of cells in conjunction with a suitable spatial discretization is wanted, especially if in the near future, industrial cases with up to 271 pins needs to be tackled as shown in this paper. Quite complex global flow patterns are found using either k-ε or preferably Reynolds Stress turbulent models with a strong influence of the number of pins. Global parameters like friction factor or Nusselt number are compared against experimental correlations. Likewise exploratory conjugated heat transfer calculations using a coupling between the finite element thermal code SYRTHES and the finite volume CFD code Code_Saturne are also shown.

I. INTRODUCTION In France, fast reactors with liquid metal coolant have

recently received a renewed interest due to a more efficient usage of the primary uranium resources, and they are one of the proposals for the Generation IV reactors. In order to evaluate nuclear plant design and safety, three-dimensional numerical studies are on going at EDF.

Fuel bundles of fast reactor are arranged into a triangular configuration and pins are wrapped with wire spacer, which follows a helically pattern around the rod axis. The primary reason of the wire is to avoid collision between adjacent pins. Moreover the presence of the wire is also reducing vibrations and avoiding the trapping of the liquid metal coolant (in general sodium). From the thermal-hydraulic point of view the wire is creating a very complex secondary motion enhancing mixing between sub-channels.

From a historical point of view the effect of the wire via experimental correlations, which provide the friction

factor as function of geometrical and hydraulic parameters. A first example is provided by Novendstern14, where the usual Blasius formula for pipe flow is corrected taking into account several parameters like the number and the hydraulic diameter of the different type of sub-channels which can be found in the fuel assembly. Another famous correlation was given by Rehme18 where a shape factor F, which take into account the pitch-over-diameter ratio P/D and the helix-over-diameter ration H/D, is introduced. A milestone in the experimental evaluation of this type of flow was presented by Cheng and Todreas4 were two sets of correlations were presented. The detailed version are taking into account several geometrical parameters and different hydraulic parameters, making the correlations suitable for many configurations, but also very difficult to use. A simplified version was also presented and the two version are converging toward same values as the Reynolds number increase. Because the interest of this work is toward fully turbulent, relatively high Re number



only the simplified version is considered. Last correlation, which will be used herein, was presented by Engel et al. 5. In this work a modified version by Bubelis and Schikorr2 is used. This last work present a very wide and methodic comparison of several well accepted correlations with some available experimental and numerical data. This paper is a perfect introduction of one of the key issue encountered during this work: which is the accuracy of the results? Experimental correlations have validity ranges depending on the experiments used for the definition. Pure experimental data are more reliable but very scarce. In this contest CFD can play a role in supplying a vast and very specific amount of data. An example is again present in Bubelis and Schikorr2 were experimental correlations are compared with RANS (Reynolds Average Navier-Stokes equations) turbulence models from Gajapathy et al8. Now a problem arises in Gajapathy et al8 the CFD results are validated against correlations finding good agreement and in Bubelis and Schikorr2 the same correlations are validated against the CFD: but now which are the data to be trusted and used as reference? A possible solution could be provided by LES (Large Eddy Simulation) and DNS (Direct Numerical Simulation) as the one presented by7. LES and DNS are able to provide a very broad and very accurate, if a proper code is used, amount of data, which can be very difficult to obtain with experimental techniques. Instantaneous flow field and extensive average results (for example Nusselt distribution along the fuel rod) will be available, making more rigorous the validation of RANS models. Because of the extremely time consuming and cost (very powerful High Parallel Computing, HPC, facilities are necessary) of LES and DNS, they are still limited to reduced geometry and moderate Reynolds number, whereas usual RANS will be devoted to the study of more industrial cases.

Some RANS study are also starting to appear like Raza and Kim17 and Smith et al22. In general those type of paper present studies carried out with commercial unstructured codes, employing reduced geometries with a limited number of pins and results compared with well establish experimental correlations.

A even more difficult task is to find heat transfer correlations for the evaluation of the Nusselt number. Several studies were conducted during the sixty and the seventy for several projects. Pfrang and Struwe15 presents a review of the outcome of several studies. In this work the Nusselt correlations of the EUTATOM project9, of Karim and Carelli12 and Mikityuk13.

The aim of this work is firstly to investigate the ability

of Code_Saturne to study these types of flow, underlining which are the important parameters to take into consideration (mesh configuration, turbulence models, etc…). This validation is carried out using reduced

geometries composed by only seven and nineteen pins with only one helix pass. Consequently periodicity is used in the streamwise direction, in order to reduce further more the domain. Global results of pressure drop and global Nusselt number are compared with the previous mentioned correlations for both friction factor and Nusselt number.

A second kind of objective, more industrially related, has also begun and is briefly discussed. The first aspect concerns the feasibility of applying the CFD approach to real configurations counting up to 271 pins. This leads to huge meshes for which HPC aspects are required. The second aspects which needs to be tackled concerns the coupling of Code_Saturne with the thermal code SYRTHES, in order to study the heat deposit in the solid part of the domain. In this case fuel an inlet/outlet problem formulation need to be employed and the computational domain requires several helix passes.

For both aims the mesh generation step constitutes a fairly challenging task. The wire induces a very large number of singularities in the geometry due to the fact that the wire attached to each pin is almost in contact with the surrounding pins. After several attempts with commercial mesh generators, the homemade procedure described in Peniguel et al16 is followed. Several variants regarding the way to handle the connection between the wire and the pin have been investigated. It leads to an almost structured mesh with a very good control on the number of cells across two adjacent pins. Here all meshes used have at least 8 cells between two pins.

II. NUMERICAL METHODS

In this simulation, conduction, convection heat transfer must be solved simultaneously. The model used and equations solved are described thereafter.

II A. Code_Saturne In this study the CFD code Code_Saturne1, developed by EDF. The code is able to solve steady or transient, single phase, incompressible, laminar or turbulent flows. The Reynolds Average Navier-Stokes equations are discretized using a co-located finite volume approach. Velocity and pressure coupling is insured by a prediction/correction method with a SIMPLEC algorithm6 and the Poisson equation is solved with a conjugate gradient method. A Rhie and Chow interpolation19 in the correction step is also used in order to stabilize the solution. The code is completely unstructured and able to handle any type of cell including polyhedral and embedded refinements. Since 2006 Code_Saturne has become open source and a very extensive work of Validation and Verification (V&V) has been carried out. A very good overview about V&V and why it is extremely important in nuclear engineering is givem by Chabard and Laurence 3



Governing equations for periodic calculations. In a fully developed turbulent flow pressure and temperature can be decomposed as and

with , x3 being the

periodic direction. The terms and represent respectively the pressure drop and the rise of enthalpy in the periodic direction. The equation can then be written as:

(1)

(2)

The parameter is recomputed at every time step in order to keep constant the mass flow rate, is instead obtained from a simple energy balance:

(3)

where is the wall heat flux, is the wall heat surface and the mass flow rate.

Turbulence modeling. Two different turbulence models have been used: the two-equation model of Jones and Launder11 and the Reynolds Stress model of Speziale, Sarkar and Gatski (SSG)23. 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 resolved and used to compute the turbulent viscosity as:

(4)

where is a constant equal to 0.09. One reminds that the value of the constant has been tuned in order to match experimental results in simple cases. This model has many advantages from the user point of view: it is stable and easy to implement. The model is quite accurate in a wide range of simple flows, but for more complex cases it might be not accurate enough because of its inability to take into account the anisotropy of the Reynolds stress tensor. In the SSG the closure is obtained by solving a transport equation for every component of the Reynolds Stress Tensor. The equations assume the following formulation:

(5)

Where is the viscous diffusion, is the turbulent

diffusion, is the pressure strain rate correlation term, is the production term and is the turbulent

dissipation rate term. The turbulent dissipation, the pressure strain rate correlation and the third order correlations in the turbulent diffusion require further modelling. In the SSG model a non-linear quadratic model is used for pressure strain rate correlation. The wall modeling is based on the so-called scalable wall functions10. The main disadvantage of the standard wall function, apart the non-general validity of the log law, is the difficulty to place the in a specific range

( ), as required for a good wall modeling. This problem is even more challenging when the geometry is quite complex and characterized by a huge variation of the wall distance. In the scalable wall function the minimum value is limited to a value of 11.06, so the value of the velocity gradient at the first cell will be the same as if it were at the edge of the viscous sub-layer. The turbulent heat fluxes are modeled using a gradient diffusion hypothesis and the eddy diffusivity employ a turbulent Prandtl number analogy:

(6)

II.B Heat Transfer at the Fluid/Solid Interface At the interface, every time step, the thermal coupling is performed. Let Ts be the temperature of an internal solid node, Tw the temperature at a node that belongs to the interface, and Tf the temperature of a fluid point (located generally in the log layer). At time t(n), the CFD tool Code_Saturne provides after calculation:

• h(n): the local heat exhange coefficient at time t(n) • Tf

(n): the local inside fluid temperature at time t(n) using these data, the flux to be applied to the solid is :

(8) (5)

Then, using this flux or the exchange conditions h(n) (calculated by a log law) and Tf

(n), SYRTHES21 can solve the heat conduction equation inside the solid. At the same time, a similar procedure is used in the fluid to update the fluid temperature field.



III PERIODIC FLUID CALCULATIONS. Periodic boundary conditions are in general used for refined turbulent calculation in order to reduce the domain and keep the mesh size reasonable. The Reynolds number is varied between 5000 and 50000 (based on the hydraulic diameter and bulk velocity). The wall heat flux is constant and equal to 6x105 W/m2 which makes the Peclet number ( ) ranging from 25 till 400. Two different helix-to-diameter ratios H/D are used, which take the value of 22 and 17.As expected, the flow field presents no symmetry in the plane perpendicular to the stream-wise direction. The presence of the wire is inducing a global swirling motion on the edge and corner sub-channels. Moreover the location of the maximum of the stream wise velocity is rotating following the pattern of the helices. In all wall channels, characterized by a high axial velocity, a big secondary vortex is also visible. It is interesting to notice that the wall sub-channel, characterized by the big bulk velocity, has also the maximum of the swirl flow velocity. On the opposite wall sub-channels, where the velocity is lower, this secondary structure is not so obvious. Central sub-channels are characterized by a secondary vortex for all the wire angles. For those sub-channels the unbalance of the stream-wise velocity, typical of the wall sub-channels, doesn’t appear clearly.

Fig. 1. Velocity in the streamwise direction and secondary flow for seven pin configuration P/D = 1.1 H/D = 22 and Re=10000. Results obtained with a k-ε model Fig. 1 and Fig. 2 present the comparison between mean streamwise direction velocity and secondary motion for the two turbulence models employed herein. Both flows look very similar even in their quantitative comparison. This could lead us to the wrong conclusion that the turbulence model has a minor effect on the results. As a matter of fact if the velocity at the wall is compared (Fig. 3 and Fig. 4) the two models are showing quite a substantial difference, making the estimation of the pressure drop quite different.

Fig. 2. Velocity in the streamwise direction and secondary flow for seven pin configuration P/D = 1.1 H/D = 22 and Re=10000. Results obtained with a Rij model.

Fig. 3. Contours of the norm of the velocity at the wall for P/D = 1.1, H/D = 22 and Re=10000. Results obtained with a k-ε model.

Fig. 4. Contours of the norm of the velocity at the wall for P/D = 1.1 H/D = 22 and Re=10000. Results obtained with a Rij model. The variation of the helix pass has a great influence on the solution as can be appreciated in Fig. 5, where results are obtained using a k-ε model. The helix-to-diameter ratio H/D = 22 correspond to a new EDF design, whereas H/D = 17 is the old design used in the SuperPhenix reactor. The flow features are still the same, but a higher velocity and a stronger secondary motion can be appreciated. If now the now the k-ε is compared with the second moment closure Rij of Fig 8, a very large difference can be appreciated. The k-ε model is giving an almost 20% underestimation of the maximum axial velocity respect to the Rij, and also the secondary motion is less strong.



Fig. 5. Velocity in the streamwise direction and secondary flow for seven pin configuration P/D = 1.1 H/D = 17 and Re=10000. Results obtained with a k-ε model.

Fig. 6. Velocity in the streamwise direction and secondary flow for seven pin configuration P/D = 1.1 H/D = 17 and Re=10000. Results obtained with a Rij model. As the number of pin increase the future of the flow does not change as can be establish from Fig. 7. It is interesting to notice that as the pins number increase the edge and corner sub-channels start to loose their predominance and the flow field is more homogeneous.

Fig. 7. Velocity in the streamwise direction and secondary flow for nineteen pins configuration P/D = 1.1 H/D = 22 and Re=10000. Results obtained with a Rij model.

Fig. 8. Dimensionless temperature field for seven pins configuration P/D = 1.1 H/D = 22 and Pe=55. Results obtained with a Rij model. The temperature field has also a very complex pattern as can be seen from Fig. 8. On the fuel rod surfaces a simplified boundary condition (constant temperature) is imposed (Dirichlet BC), whereas walls of the external case are adiabatic (Neumann BC with wall normal gradient equal to zero). In reality of course, as explained in the last part of the paper, only the pin itself contains fuel and therefore heat deposit. The heat flux is taken into account using a sink term into the temperature equation as described in eq. (2). Also in this case the field can be divided into central and side sub-channels. The first-ones are characterized by higher temperatures and in particular in the gap region between two adjacent pins and by a quite uniform temperature distribution. Side channels are more influenced by the location of the streamwise velocity maximum, which correspond to the temperature minimum. The influence of the turbulence model in the heat transfer is very limited probably due to the use of scalable wall functions. Movies about the results presented in this sections can be found in20.

Friction factor and Nusselt profiles. In order to compare the results with available and well-accepted correlations global parameters have to be evaluated. The friction factor f is defined as:

(9)

where is the hydraulic diameter the periodic length in the streamwise direction the bulk velocity and the pressure drop. The evaluation of the pressure drop presents a very delicate situation because several option are available and the results can vary considerably. For example the pressure drop can be evaluated from the wall

shear as: being the wall shear and

the total wall surface. The wall shear can be estimated from the friction velocity given by the wall function as:



(10)

with I’ being the intersection between the wall normal direction through the wall face centre F and the projection of the cell centre on that line. K and C are two constant in equal to 0.42 and 5.2 respectively. The wall dimensionless distance is evaluated from a turbulent velocity

as

(11)

where d is the distance between I’ and F. Another possibility could be to use the imposed pressure drop β of eq. (1) obtaining:

�

Δ p = βLz (12)

Fig. 9. Friction factor f for the seven pins configuration, P/D = 1.1 H/D = 22. In the first case also pressure redistribution on the cross section are taken into account. The value are quite different, in particular at low Reynolds number, as can be seen in Fig. 9 for the seven pins bundle and in Fig. 10 for the nineteen. The method of eq. (10) is labelled as

, whereas the one of eq. (12) as QDM.

Fig. 10. Friction factor f for the nineteen pins configuration, P/D = 1.1 H/D = 22.

In general the Reynolds stress model is giving higher value of f respect to the eddy viscosity model, but value tend to converge as the Reynolds number increase. Indeed the same effect can be seen as the number of pins increase because of the minor importance of the edge channels on the global flow field. The data are in the range given by the experimental correlations and they seems to agree better with the one of Cheng and Todreas4.

Fig. 11. Nusselt number 7 pins configuration, P/D = 1.1 H/D = 22. The Nusselt number is instead evaluated as:

The comparison against the experimental correlation is plotted in Fig. 15 for the seven pins configuration. In this case the difference between the two turbulence models is almost negligible, but slightly increase with the Peclet number. The reason could be the wall treatment; consequently more accurate modeling is required. CFD profiles are increasing with a relatively steep slope. On the other hand the experimental correlations have almost the same and not very rapid grade of growing, although they have different starting points.

Effect of the mesh configuration. Despite the fact that the geometry can be described using few geometrical parameter, the model is quite difficult and very difficult to mesh as described in Peniguel et al.16. The wire is introducing in the geometry many singularities that are skewing the mesh. Now an obvious question arises: with which grade of accuracy the mesh has to reproduce the geometry? Can a simply geometry still produce the same flow features and accurate global flow parameters?



Fig. 12. Base mesh used for the study. In blue the mesh for the fluid, others colors represent the different solid part of the pin

Mesh with high blending of the wire

Improved mesh with triangular elements around the wire tips

Fig. 13. Different meshes approach to handle the wire

The base mesh used in this study is visualized in Fig. 12. An easy way to improve the orthogonality is to blend the wire with the relative pin creating what is shown in Fig. 13 left. A more accurate solution is to hybrid hexahedral and prism with triangular base around the wire obtaining a better geometrical description (Fig. 13 right). Fig. 14 shows a visual comparison between the blended and the triangular (around the wire) meshes. The flow features seems pretty similar, the triangular mesh is only showing very low velocities at the tips of the wire, whereas these areas are chopped out in the blended mesh. Table I, that reports the value of the friction factor f for the different meshes and the perceptual difference with respect to the base mesh, also confirms this fact. The difference between the models is around 11%, while between the base and the blended mesh is less than 5%.

TABLE I

Comparison of f for different mesh at Re=25000 using k-ε.

Mesh Type f eq. () Difference [%] Base 0.287 e -1 0.00 Base Rij 0.319 3-1 11.1 Triangular 0.283 e-1 1.39 Blended 0.274 e-1 4.53

However this conclusion is valid mainly for global dynamic aspects. It is likely that from the local thermal point of view the hybrid approach and its better geometry approximation may have an influence that needs to be investigated.

Fig. 14. Comparison of the axial velocity for the configuration P/D = 1.1 H/D = 22 and Re=25000. Results obtained with k-ε model.

IV. PERPECTIVES The previous calculations and tests have indicated that a reasonable confidence could be placed in Code_Saturne results. Moreover, it is also clear that the number of pins has a very strong influence on the global behavior. Therefore EDF is interested to know if it is possible to tackle real cases with up to 271 pins and understand what happens for example when local hot spot or degraded inlet velocity occur. Several meshes have been generated counting 61 pins and several helices and 271 pins on 1 helix and five helices so far. The latest meshes counts above 100 millions cells and needs to be handled thanks to EDF’s mainframe with 256 or 511 processors to get reasonable CPU times.

Fig. 15: Domain decomposition of a 271pins and 1 helix case in 256 independent domains.



As shown on figure 16 (corresponding to only one helix case) the velocity field confirm that the edge and corner sub-channels start to loose their predominance and the flow field is more homogeneous for the inner pins.

Fig. 16 : Velocity norm for the 271 pins case The second type of test currently done at EDF corresponds to the investigation of the influence of the wire wounded around each pin on a small spot (here a scalar set initially to zero and set to one in a small portion of the inlet). One clearly note on Fig 17 and 18, both a deviation (even after only one helix) and a diffusion of the scalar field. Similar calculations on a five helixes case are in progress. The second industrial aspect that EDF wants to investigated in the near future concerns the possibility to tackle the conjugated heat transfer taking place between the solid and the fluid to have access to the local cladding or fuel temperatures as well as the sodium temperature. In order to do that the solid parts, ie pins (with fuel and cladding), wires as well as the hexagonal can are explicitly taken into account. The heat deposit is set only on the central part where fuel is located. For the time being the procedure is tested with success on the 7 pins case and should be extended in the near future to assemblies with a larger number of pins (up to 271).

Fig. 17: Scalar field of a small spot (set to 1 at the inlet)

Fig 18: Scalar field of a small spot (set to 1 at the inlet)

Fig. 19: Solid and fluid meshes used for conjugate heat transfer Fig 19 presents a view where both fluid and solid meshes are gathered. One underlines that in order to provide more flexibility, meshes at the interface do not need to coincide. The preliminary results obtained (The configuration presented on Fig. 20 corresponds to a 3 helices case) suggest a strong local influence of the spacer wire.



Fig. 20: Cladding and wire temperature (7 pins case )

V. CONCLUSIONS

In this paper thermal-hydraulic of wire wrapped fuel bundle was investigated. Two different configurations, with seven and nineteen fuel rods, were taken into account, finding that the main flow features remain unchanged as the number of pins increase. Two different turbulence models were tested finding good agreement with experimental correlations. On the other hand the evaluation of the heat transfer requires more investigation. Experimental correlations for Nusselt number are quite scattered, making difficult the assessment of the CFD. Indeed the wall modeling for the heat transfer is only fist order accuracy, which can make quite dubious the Nusselt evaluation. Better modeling employing low Reynolds models could be required and comparison with high Reynolds approach is on going. Another solution to the problem, which can avoid the use of refined mesh in the near wall region, could be the use of more advance wall functions as the one presented in24. As the number of pins increase the influence of side and corner sub-channels is becoming less important and flow is more homogeneous. In general it was difficult to find data to compare with. In this contest refined LES and DNS could play a big role providing a large and reliable amount of data for RANS modeling evaluation.

ACKNOWLEDGMENTS Authors express their thanks to D. Laurence (EDF) for

fruitful discussions on turbulent modeling and D. Monfort for advices in Code_Saturne usage. S. Rolfo is grateful to the UK Engineering and Physical Sciences Research Concil for funding under grant EP/C549465/1 “Keeping the Nuclear Option Open”.

NOMENCLATURE ρ : density Ui : velocity component k : turbulent energy ε : turbulent dissipation rate term Cµ : constant (0.09) PrT : turbulent Prandtl number p* : pressure g : gravity xi : coordinates t : time µ : viscosity Cp : specific heat ks : thermal conductivity φ : volume heat source qw: wall heat flux Swh : wall heat surface h : heat transfer coefficient T : temperature β : coefficient for periodic model γ : coefficient (energy balance) m : mass flow rate

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