A FINITE VOLUME-BASED NETWORK METHOD FOR THE PREDICTION OF HEAT, MASS AND MOMENTUM TRANSFER IN A PEBBLE BED REACTOR

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1 A FINITE VOLUME-BASED NETWORK METHOD FOR THE PREDICTION OF HEAT, MASS AND MOMENTUM TRANSFER IN A PEBBLE BED REACTOR GP Greyvenstein and HJ van Antwerpen Energy Systems Research North-West University, Private Bag X6001, Potchefstroom 2520, South Africa Corresponding author HJ van Antwerpen: Fax: mgihjva@puk.ac.za 1. ABSTRACT In this paper a new control volume based network scheme for the prediction of heat, mass and momentum transfer in a pebble bed reactor is presented. The discretization scheme takes detail effects such as convection, radiation and conduction heat transfer into account and also provides for several types of reactor flow elements such as pipes, annuli, risers and porous media. The network representation of the reactor is solved as an integral part of the overall thermal-fluid network for the entire power conversion unit. The new control volume based network scheme facilitates the setting up of the reactor network taking detail effects such as complex radiation and effective pebble bed-solid interface conduction into account. The model is used as a benchmark for the current element-based network scheme and it is also used to investigate the effect of more accurate radiation discretization on the temperature profile in the reactor. Excellent agreement has been found with the current element-based reactor model. 2. INTRODUCTION To design a complex thermal-fluid system such as the Pebble Bed Modular Reactor (PBMR) various levels of analysis and simulation are required, from detailed component-level CFD simulations to integrated system-level simulation. In most system-level simulation codes simple one-dimensional models represent components such as pipes, pumps and heat transfer elements. However, nuclear reactor behaviour is too complex to be accurately modelled as a simple one-dimensional element in a system simulation code. Modelling the reactor is not a simple task because of the complexity of the phenomena involved. In the pebble bed, heat is generated in kernels inside the fuel spheres. The heat is then conducted to the sphere surface where it is transferred to the helium by means of convection. In the bed itself, heat is transferred by means of conduction and interstitial radiation in the radial and axial directions. In addition to this, the nuclear reaction that produces the heat is temperature dependent, making the variables in the problem very interrelated. One approach to solve this problem is to couple a systems simulation code with a threedimensional Computational Fluid Dynamics (CFD) code that models the reactor. Classical CFD calculations are indeed very powerful and most commercially available CFD codes are capable of solving the flow in complex three-dimensional geometries. However, these capabilities require large meshes, sophisticated discretization schemes and intricate data management that make CFD calculations computationally very expensive. An additional consideration is that in the case of a pebble bed, even CFD models assume a porous medium, which causes the flow equations to simplify considerably due to the dominance of the bed resistance. Given this simplification and the fact that the cost of the CFD computational overhead remains, the use of classical CFD offers very little benefit in this case. Even if a very simplified CFD model is coupled to a system simulation code, there are still two separate solvers running simultaneously on parts of the same problem, so that information has to be 1

2 exchanged either with an explicit or semi-implicit interface. This requires considerable programming effort to maintain consistency between the two codes. An explicit interface furthermore severely limits the timestep length in transient simulations [1]. Reactor models for system simulation have previously been developed using the thermal-fluid network approach, in which a network of elements was assembled. Elements are onedimensional entities that represent either flow paths or heat transfer paths. Heat transfer paths include conduction, convection and radiation heat transfer elements. Du Toit et al[3] used such an element-based approach in Flownex[2], the code used for the PBMR plant simulation. Flownex is a systems-cfd code in which the unsteady compressible flow and energy equations are solved for an unstructured network of one-dimensional elements representing components such as pipes, valves, turbomachinery, heat exchangers or heat transfer elements. As described for this context, the fundamental mass, momentum and energy equations governing unsteady compressible flow through the reactor is discretised in nodes representing mass and volume, interconnected by elements representing heat and mass flux between nodes. Phenomena that are taken into account include convection heat transfer between all gas and solid structures as well as solid conduction. The assembled network comprising the reactor sub-network and the rest of the power conversion system is solved as one system using the implicit pressure correction method [4]. In this way, a two-dimensional solution of the reactor is obtained as part of the solution of the overall power conversion system. This has the advantage that the solution is in no way constrained by the coupling interface between a systems-cfd and a classical CFD code, as a fully implicit coupling exists between the reactor and the rest of the system. In this way, a balance is found between accuracy and calculation speed, while taking into account all phenomena that influence the reactor interaction with the rest of the system. However, this element based discretization does not lend itself to easily take detail effects like radiation or surface temperature into account. A need therefore developed to devise an alternative discretization scheme that firstly will facilitate the modelling of certain detail effects and secondly, that can serve as a benchmark for element based discretization model. This paper describes a new control volume based network approach with which the discretization process for complex geometries is eased considerably. Complex effects like pebble bed effective conduction and radiation heat transfer in cavities are easier taken into account. 3. RELATION BETWEEN THE CURRENT AND BENCHMARK REACTOR MODELS The control volume based approach serves the following three purposes: To facilitate the discretization of complex geometries, taking detail effects into account. To serve as benchmark for the element-based reactor model because the discretization approach is totally different than that of the existing model. To investigate the effect of radiation in the reactor as detail radiation is taken into account with the control volume based discretization scheme. 2

3 With the control volume based approach full control volumes are used throughout the discretization, as opposed to half control volumes at the boundaries in case of the elementbased approach. Boundary or interface values are then calculated with massless nodes connected to full control volumes having mass, as shown in FIG 1 and FIG 8. The use of massless nodes is made possible by the implicit solution algorithm, as an explicit time discretization requires nonzero node masses. Zero volume nodes Full volume nodes FIG 1. Element-based model (left) with half control volumes at boundaries and some interfaces, and control volume based model (right) with full control volumes and massless nodes at boundaries. What the control volume based network approach has in common with classical CFD control volume approach is that momentum conservation is solved at control volume interfaces (elements) while mass and energy conservation are solved at nodes. In contrast to the control volumes used in classical CFD, the control volumes in this approach are of a much larger scale, incorporating empirical correlations or analytical models to take smaller-scale detail into account such as flow channels through a solid or conduction shape factors, without losing the simplicity of the large-scale control volumes. To construct the simulation network from a control volume specification chart, five fundamental elements are used, namely gas flow elements, conduction elements, convection heat transfer elements as well as two types of radiation elements: surface elements and spatial elements. The control volume type, as well as the types of its neighbours determine the network topology. The use of these elements at control volume-level is now illustrated with some control volume type examples. 4. SOLID WITH ONE-DIMENSIONAL FLOW A control volume type that illustrate the approach very well is solid with one-dimensional flow. Solid material, with flow passages passing through it, is typically encountered at the coolant riser channels in the side reflector or at control rod channels. Because of the pipegeometry of the flow channels, the Darcy-Weisbach correlation is used for calculating pressure drop, while the Dittus-Boelter correlation or a fixed heat transfer coefficient is used in the convection heat transfer between solid and fluid nodes [2]. For the solid conduction along the flow channel direction, area is reduced with a permeability-factor, while a conduction form factor can be used to account for the flow channel in transverse conduction. In this way, geometrical detail is taken into account without adding complexity to the solution. The network topology for this control volume type is as shown in FIG 2. 3

4 Solid with flow passage Solid node Gas node Solid conduction Convection heat transfer element Pipe flow element FIG 2. Nodes and elements used to represent a solid with one-dimensional flow through it. 5. PEBBLE BED REPRESENTATION In a pebble bed zone, heat is generated inside spheres by nuclear reaction, conducted to the sphere surface and transferred to the fluid by convection. When a temperature gradient exists across the bed, heat is transferred between pebbles by means of conduction through contact resistances as well as surface radiation exchange. These complex phenomena are modeled in a similar method to the current reactor model, of which the network topology for the pebble bed was established by Greyvenstein and Van Antwerpen [5]. This topology is based on the assumption of the pebble bed as a homogeneous conducting material of which the effective conductivity can be calculated with the Zehner-Schlünder correlation [2]. Maximum fuel temperature is calculated with a single representative sphere, having the heat generation, conduction resistances and internal node masses equivalent to the sum of all spheres in a control volume. Gas flow pressure drop is calculated with the well-known Ergun equation for a pebble bed, while the convection heat transfer coefficient is calculated with the correlation by Kugeler and Schulten [2]. The network representation of the abovementioned is presented in FIG 3. Packed bed with porous flow Solid node Gas node Representative fuel sphere with internal layers Point of maximum fuel temperature Porous flow elements Convection heat Pebble bed effective conduction and transfer element interstitial radiation heat transfer Fuel sphere surface node FIG 3. Assembly of nodes and elements representing a pebble bed zone. 6. CAVITIES AND RADIATIVE HEAT TRANSFER The surfaces of all solids bordering on cavity zones, exchange heat by means of radiation, which is of particular importance as radiative heat transfer increases with temperature to the fourth power and gas cooled reactors do operate at very high temperatures. Radiation heat exchange has previously not been fully calculated because of the complexity of calculating the 4

5 geometric view factor. In the element-based reactor model, only radiative heat transfer between directly opposing vertical surfaces is taken into account, so that the effect of radiation has not yet been accurately assessed. However, in the benchmark reactor model presented here, this effect has to be ascertained, even though it adds a large number of elements. For that purpose, analytical view factor formulae for discretised surfaces was derived from the formulae Brockmann [6] presented for right-angular cavities in a cylindrical geometry. As shown in FIG 4, the radiation network in a cavity is set up by connecting a radiative surface node to a massless surface node by means of a radiative surface element. The equation for its heat flux is then: A q = σε T T 1 ε 4 4 ( s sp ) (1) Each radiative surface node in a cavity is connected to every other radiative surface node by means of a radiative spatial element, for which the heat flux formula is given by: 4 4 ( ) q = A F σ T T (2) sp, sp1 sp2 Solid Cavity Primary nodes Massless nodes Radiative spatial elements Radiative surface elements Radiative surface nodes FIG 4. Radiation network across a cavity, showing surface and spatial radiation elements. 7. DISCRETIZATION OF REACTOR AND ASSEMBLY OF ZONES With the approach of having different control volumes, giving rise to appropriate network topologies, it is possible to automatically generate complete networks for system simulation. The process is illustrated in FIG 5 for a pebble bed type reactor with a centre column. 5

6 Pebble bed type reactor core assembly 2D Axi-symmetric Control volume type specification System CFD simulation network FIG 5. The discretization process for a pebble bed type reactor with a central column. 8. HEAT GENERATION AND NEUTRONICS Heat generation is managed by distributing a total specified power with normalised distribution curves obtained from detailed neutronics codes. Temperature and fast neutron flux history is not kept track of at present, but it is taken into account by specifying an appropriate temperature-dependent conductivity according to location in the reactor assembly. 9. STEADY-STATE RESULTS Using a pebble bed type reactor operating at full power, full mass flow conditions was simulated in steady state. The figures below present solid temperature profiles in two sections through the bed, axially and radially, with pebble bed surface temperature taken as solid temperature in the pebble bed. FIG 6 shows solid temperature profiles along an axial line at the middle of the pebble bed zone. In the pebble bed zone, sphere surface temperatures are used as solid temperatures. Maximum fuel temperatures differed by less than five percent, while the largest difference on the axial temperature profile is less than ten percent. Considering the difference in discretization methods, this can be considered excellent agreement. 6

7 Solid temperature, axial 1.2 Normalised temperature Normalised height Bencmark Current FIG 6. Axial solid temperature profile, in the middle of the pebble bed zone. 1.2 Solid temperature,radial Normalised temperature Normalised radius Benchmark Current FIG 7. Radial solid temperature profile, at the middle of the pebble bed zone. The radial temperature profile in FIG 7 has a large temperature percentage difference at a normalised radius of 0.7, just next to the pebble bed in the side reflector. Referring to FIG 8, this temperature difference can be explained as a direct result of discretization difference between the two topologies. The temperature of surface nodes in the pebble bed are largely determined by the heat transfer coefficient and the heat production rate in the pebbles, so that in the case of the element-based (current) model, the pebble bed boundary node effectively fixes the side reflector inner temperature at the pebble bed surface temperature. The control volume based (benchmark) model has no heat generation (or pebble bed surface) nodes in direct contact with the side reflector, but rather has the pebble bed surface temperature separated from the side reflector by an effective conductivity element. Because of the low effective conductivity of the pebble bed compared to the side reflector, there is a large thermal resistance between the pebble bed surface temperature and the side reflector. This highlights the impact of discretization at interfaces on the temperature profile in the reactor. 7

8 Pebble bed/side reflector interface Current model: element-based discretisation Benchmark model: volume-based discretisation Pebble bed Side reflector Pebble bed Side reflector Full mass node in reflector Pebble bed/reflector lumped mass Pebble bed Massless node surface node at interface Pebble bed effective conduction FIG 8. Different network topologies at the pebble bed/side reflector interface. 10. EFFECT OF RADIATION The current model only takes radiation in vertical cavity zones into account, which are typically used to model cavities between the core assembly and radiative heat shields and between radiative heat shields and the pressure vessel. The control-volume based benchmark model has a detailed radiation model. The only effect of taking radiation into account during steady-state full power operation is that the reactor assembly at the top and bottom has lower temperatures. 1.2 Solid temperature, axial Normalised temperature Normalised height Bench,Rad Current,Rad Bench,No Rad FIG 9. Axial solid temperature profile, in the middle of the pebble bed zone. One Benchmark case without radiative heat transfer in cavities is repeated as reference, the other cases with radiation. 11. CONCLUSION With the control-volume based network approach, the reactor can be modelled in greater detail, so that it can be used to validate other reactor models. Excellent agreement has been found with the current element-based reactor model. 8

9 12. ACKNOWLEGDEMENT The authors wish to acknowledge PBMR (Pty.) Ltd and the Technical Human Resources for Industry Programme (THRIP) of the National Research Foundation (NRF) who partially funded this research. 13. NOMENCLATURE A = area F 12 = radiation view factor Q = heat transfer T = temperature ε = emissivity σ = Stefan-Boltzmann constant 14. REFERENCES 1. Aumiller, D.L., Tomlinson, E.T., Bauer, R.C., A coupled RELAP5-3D/CFD methodology with a proof-of-principle calculation. Nuclear Engineering and Design 205, p83-90, Van der Merwe, J. and Van Ravenswaay, J.P., Flownex Version 6.4 User Manual, M-Tech Industrial, Potchefstroom, South Africa, Du Toit, C.G., Greyvenstein, G.P., Rousseau, P.G. (2003) A Comprehensive reactor model for the integrated network simulation of the PBMR power plant. Proceedings of the 2003 International Congress on Advances in Nuclear Power Plants (ICAPP 03). Cordoba, Spain, 4-7 May, Greyvenstein, G.P., An implicit method for the analysis of transient flow in pipe networks, Int. J. Numer. Meth. Engrng., 53, , Greyvenstein, G.P., Van Antwerpen, H.J., Rousseau P.G. The system CFD approach applied to a pebble bed reactor core. International Journal of Nuclear Energy Science and Technology. Inaugural Issue, Brockmann, H. Analytic angle factors for the radiant interchange among the surface elements of two concentric cylinders. Int. J. Heat Mass Transfer. Vol.37, No. 7. pp

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