Multiphysics Coupling for Fast Burst Reactors in Slab Geometry

1. Japan Patel, Chemical and Nuclear Engineering, UNMRyosuke Park, Theoretical Division (T-3), LANLCassiano de Oliveira, Chemical and Nuclear Engineering, UNMDana Knoll, Theoretical Division (T-3), LANL September 19, 2013 Multiphysics Coupling for Fast Burst Reactorsin Slab Geometry

2. Overview • Objective • Basic idea • Coupling scheme • Coupled equations • Neutronics • Linear mechanics • Material energy • Density change • Modeling with diffusion Neutronics • Extension to transport Neutronics • Moment Based Acceleration – The HOLO concept • Extension to scale bridging framework • Summary

3. Objective • Recent development of moment based scale bridging acceleration concept naturally fits into Multiphysics solution algorithm • This project demonstrates the concept using simplified fast burst reactor modeling • This work is a natural extension of the previous work1 where they used diffusion Neutronics (with tight coupling, but no transport) 1. Kadioglu, Knoll, de Oliveira NS&E, 163 p132 (2009)

4. Reactivity insertion/ SLIGHTLY supercritical system Basic Idea • Coupled Physics • Neutronics – diffusion, transport (SN), accelerated SN • Linear Mechanics – linear elastic wave equation • Material Energy – heat equation without conduction • Others – density change, cross section change

5. Coupling Scheme • Previous work by Kadioglu, Knoll, and de Oliveira demonstrated a tightly coupled algorithm for modeling fast burst reactor kinetics. Their algorithm included an implicit block, and an explicit block. • IMplicit block • Neutronics, material energy, density change, cross section change • EXplicit block • Linear mechanics • Why IMEX? • Time steps required to model neutronics are impractically small due to stiffness of the neutronics problem 1 - Implicit neutronics, and directly related physics • The wave equation follows the characteristic wave speed – Explicit linear mechanics • Algorithm • Initialize variables – use the eigenvalue problem to initialize flux profile and amplify it. • March linear mechanics – calculate resultant displacements, densities, cross section changes - we use small time steps (~5 e-9) • March neutronics – calculate new flux profile, flux gradient, temperature gradient • Loop back for next time step. • 1. Kadioglu, Knoll, de Oliveira NS&E, 163 p132 (2009)

6. Coupled Equations – Linear Mechanics • Linear elastic wave equation where, The linear elastic wave equation is discretized using the second order central difference scheme – both in time, and space. Here, u is the material displacement, T is temperature, σ is Poisson’s ratio, α is thermal expansion coefficient, E is Young’s modulus, and ρ is material density. EXPLICIT!

7. Coupled Equations – Density Change • Density Change Here density is updated at every step, and used for updating number densities. These updated number densities are then used for generation of new interaction cross sections for implicit neutronics calculations. Here, ρ0 is the initial density.

8. Coupled Equations - Neutronics • The transient neutron diffusion equation – Finite volume discretization • The transient transport equation – SN in angle, Diamond Difference in space • 0th moment equation (we use diffusion plus drift assumption for current – more on that later) – Centered difference for discrete consistency with transport Here, , and are angular, and scalar flux. , , are macroscopic total, scattering, and fission cross-sections. is the total number of neutrons per fission, and is the average neutron speed. is the current. IMPLICIT!

9. Coupled Equations - Material Energy • Heat (generation) equation Note that here we do not have heat conduction term (no cooling mechanism). That is because the time scales involved in this simulation is too small for conduction to have any significant impact on the simulation. Here, is the specific heat capacity, and ω is the average energy released per fission.

10. Modeling with Diffusion Neutronics • A large enough reactivity insertion leads to “ringing” effect – periodic oscillation in displacement. • The difference in dynamical time-scales between neutronics, and linear elasticity leads to this behavior (Kadioglu et al). • IMplicit-EXplicit (IMEX) time stepping can treat coupling terms accurately – there is no linearization in physics coupling.

11. Why Extension to Transport Neutronics? • The system, here, is small so we see significant transport effects on the neutron flux, and flux gradient. • The temperature is directly proportional to the fission rate. • The forcing term in the elastic wave equation is proportional to temperature gradient • As a result, the overall multiphysics solution differ significantly from the diffusion model when the transport model is employed. (see the next slide) • The second figure on this slide shows how the system is predicted to ring with transport (S12 ) at low reactivity insertion (keff = 1.0001) while the diffusion model predicted the system to go to a steady state without ringing as seen on one of the previous slides. We do, however, see a non ringing steady state for the system for sufficiently low reactivity insertion.

12. Extension to transport Neutronics - Results • Same reactivity insertion leads to different amplitude, and phase in surface oscillations. • This is attributed to differing forcing term generated by different approximations. • Like for the diffusion case, however, we do observe ringing effect with transport neutronics as well. • For this example, keff was 1.00052. • The third figure (bottom right) shows how the system may achieve different steady states depending on reactivity insertion.

13. The Moment Based Acceleration – The Scale Bridging HOLO Concept • Transport equation has large dimensions (7D). So we want to isolate the transport solver from the coupled non linear system. • By introducing a “discretely consistent” Lower Order system, that looks a lot like the diffusion equation, we can separate the expensive HO solver from the non linear system. • The moment based algorithm bridges transport and diffusion regimes using a diffusion plus drift assumption for the current () in the first moment equation. • Angular dependence of the transport equation is integrated out by taking a moment w.r.t. angle which then results in 0th moment equation. • The transport equation forms what is called the higher order (HO) system, while the 0th moment equation and the closure term form the lower order (LO) system. Here, denotes the LO scalar flux and denotes the HO scalar flux. HO system: LO system: ; Same HO scalar flux

14. The HOLO Concept – Diffusion plus Drift Assumption for Current (J) • Then, say - That is, current is calculated by adding a consistency term to the Fick’s law to obtain consistency with transport. • Thus the LO system now becomes: HO system: LO system: ; Note that the LO system depends scalar flux alone. We see how with minor modifications to the diffusion code – addition of the consistency term (plus a transport sweep for calculating the consistency term, outside the coupled system) – we can smoothly transition from diffusion to transport.

15. Standard SNvs. Accelerated SN (Eigenvalue problem – vacuum boundary on both faces)

16. Extension to Scale Bridging Framework Clearly, the simulations show that surface flux, and displacement profiles over time from stand alone SN and HOLO SN agree quite well (relative error ~1e-6 for surface displacement) thus proving the applicability of the scale bridging algorithm in simplified, but true multiphysics setting.

17. Summary • The diffusion model has smaller leakage compared to the transport model – that results in different forcing term in the wave equation producing different surface displacements for different neutronics models. • For high (relatively) reactivity insertion, the power level rises relatively fast, and the system reaches equilibrium state where it rings. For low (enough) reactivity insertion, the material expands to a certain degree and then stays in that expanded state (without ringing) – this was demonstrated using both transport, and diffusion neutronics models. • It was also observed that for the same low reactivity insertion, where the diffusion model predicted the system to go into a steady expanded, non vibrational state, the transport model resulted in vibration if system with small amplitude. (magnitudes of displacement were of the same order). • We have demonstrated the applicability of scale-bridging concept.

18. Future Work • Introduction of conduction into the material energy equation, and delayed neutrons into the neutronics model (multiple time scales) • Multigroupneutronics • Multidimensional geometries with moving mesh or multidimensional, mesh-free modeling of this system • Higher order time integration (NOT Crank Nicolson!) for higher order convergence • Error propagation analysis for this system • Application of HOLO concept to atmospheric radiation dispersal problems • Things I haven’t thought of yet…

19. References • S. Kadioglu, D. Knoll, and C. de Oliveira, “Multi-physics Analysis of Spherical Fast Burst Reactors”, Nuclear Science and Engineering: 163, 1-12, March 2009. • K. Smith, J. Rhodes III, Full core, 2D LWR core calculations with CASMO-4E, 2002, PHYSOR 2002, Seoul, Korea • D. Knoll, H. Park, K. Smith, Application of the Jacobian-free Newton-Krylov method to non linear acceleration of transport source iteration in slab geometry, Nuclear Science and Engineering: 167, (2011) 122-132. • G. Bell, and S. Glasstone, Nuclear Reactor Theory, New York: Von Nostrand Reinhold Company, 1979. Print.Alcouffe, R. E. (1977) Nuclear Science and Engineering 66, 344–355 • Smith, K. S. and Rhodes III, J. D. (2002) In Proc. Int. Conf. New Frontiers of Nuclear Technology: Reactor Physics, Safety and High-Performance Computing (PHYSOR 2002) Seoul, Korea: American Nuclear Society. • Park, H., Knoll, D. A., Rauenzahn, R. M., Wollaber, A. B., and Densmore, J. D. (2012) Transport Theory and Statistical Physics 41, 284–303

20. Thank You 