We develop a numerical model for coupling of neutron diffusion adn termomechanics in order to stimulate transient behavior of a fast burst reactor. The problem involves solving a set of non-linear different equations which approximate neutron diffusion, temperature change, and material behavior. With this equation set we will model the transition from a supercritical to subcritical state and possible mechanical vibration

Kadioglu, Samet Y.

Knoll, Dana A.

Oliveira, Cassiano De

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This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint should not be cited or reproduced without permission of the author. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, or any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for any third party’s use, or the results of such use, of any information, apparatus, product or process disclosed in this report, or represents that its use by such third party would not infringe privately owned rights. The views expressed in this paper are not necessarily those of the United States Government or the sponsoring agency. INL/CON-08-14429PREPRINTA Numerical Model for Coupling of Neutron Diffusion and Thermomechanics in Fast Burst Reactors Nuclear Power – Ready, Steady, Go Samet Y. Kadioglu Dana A. Knoll Cassiano de Oliveira November 2008 A Numerical Model for Coupling of Neutron Diffusion and Thermomechanics inFast Burst ReactorsSamet Y. Kadioglua Dana A. Knolla Cassiano de Oliveiraba Multipysics Methods Group, Idaho National Laboratory, P. O. Box 1625, MS 3840, Idaho Falls, ID 83415b Chemical and Nuclear Engineering Department, The University of New Mexico, Albuquerque, NM 87131-1341e-mails : samet.kadioglu@inl.gov, dana.knoll@inl.gov, cassiano@unm.edu1 IntroductionThere has been a long standing interest in developing nu-merical models for the analysis of fast burst critical ex-periments. For example, in an early model [1] the uncou-pled thermomechanics of fissile materials was studied byinserting prescribed temperature fields into the elastic equa-tions. More recently, a model was developed [4] whichcoupled thermomechanics of a fast burst reactor with pointreactor kinetics. Here, we present a more realistic modelwhich fully couples spatially-dependent neutron diffusionand thermomechanics in order to simulate transient behav-ior of a fast burst criticality excursion.The problem involves solving a set of non-linear differen-tial equations which approximate neutron diffusion, temper-ature change, and material behavior. With this equation setit is possible to model the transition from a supercritical tosubcritical state and corresponding material response, e.g,possible mechanical vibration. For instance, the reactor isput into supercritical state leading to power rise (thereforeresulting temperature rise). The rising temperature causesthe material to expand leading to an increase in neutronleakage and thus to subcriticality.We tested our new approach on a one-group, sphericallysymmetric diffusion model accounting for prompt neutronsonly. We don’t include delayed neutrons, since the timescales involved are too fast in order for delayed neutrons tocontribute to the system.2 Model EquationsOur model equations are formulated in a spherically sym-metric coordinates [1, 4, 2]. In this case, the neutron diffu-sion is governed by1v∂φ∂t− 1r2∂∂r[r213Nσtr∂φ∂r]+ [Nσa − νNσf ]φ = 0. (1)The temperature field is evolved withρcp∂T∂t− ωNσfφ = 0. (2)The material displacement is modeled by the followingelastic wave equation;1c2∂2u∂2t− [∂2u∂2r+2r∂u∂r− 2r2u] +1 + υ1 − υβ∂T∂r= 0. (3)Finally, the material density is computed by considering themass/particle conservation in a spherical domain;ρ = ρ0[rr + u]3. (4)The unknowns and constant parameters that appear in theseequations are defined asφ : Neutron fluxT : Temperatureu : Material displacementρ : Material densityr : Spatial variablet : Time variablev : Average neutron speedN : Number atom densityν : Number of neutron produced per fissionσtr : Microscopic transport cross sectionσa : Microscopic absorption cross sectionσf : Microscopic fission cross sectionω : Amount of average energy released per fissioncp : Specific heatβ : Linear thermal expansion coefficientc = [ (1−υ)ε(1+υ)(1−2υ)ρ ]1/2 : Wave speedυ : Poisson’s ratioε : Young’s modulusWe note that the material density can be written as the prod-uct of the number atom density and the atomic mass, i.e,ρ = NAm.350 Computational Methods: General3 Numerical AlgorithmOur numerical algorithm consists of an explicit and an im-plicit block. The reason why we develop this kind of ap-proach is that the time steps are impractically small(e.g, dueto the stiffness of the problem) if one wants to solve Eq. (1)explicitly. On the other hand, one has to follow character-istic wave speed to solve Eq. (3) due to numerical instabil-ities. Therefore, an explicit scheme is better choice for Eq.(3). We are currently using a first order coupling scheme,but will be moving to a second order predictor correctorscheme.The numerical algorithm is executed as follows. First, Eq.(3) is advanced in time to obtain a new density. Then theupdated density is inserted into the implicit loop to advanceEqs. (1) and (2). Our explicit block is based on a secondorder centered in time and space scheme. Our implicit blockuses a Crank-Nicolson type time and space discretization.The non-linear solver within the implicit block is based onthe Jacobian-Free Newton Krylov method [3].4 Numerical Results4.1 Results of a simplified modelIn this section, we present the thermomechanical behaviorof a simplified model. Here, the results are based on a sim-plified temperature model, i.e,∂T∂t= P (t)f(r), (5)where P (t) represents a prescribed power, and f(r) corre-sponds to the first spatial mode of the eigensolution of thelinearized diffusion equation (e.g, the diffusion coefficientfor Eq. (1) is assumed to be constant in space).We tested three different power pulses and observed the ma-terial response accordingly. We note that the time scale forthe power pulses is given by τ = [ 1P (t)∂P∂t ]−1. And the elas-tic wave time scale is τ elastic = R/c with R being the ra-dius of the sphere. Figure 1 represents the material displace-ment resulting from a fast power pulse ( e.g, τ < τ elastic).In this case, the material doesn’t have enough time to re-spond to the fast temperature rise. Consequently, it vibratesafter expanding to a certain level. Figure 2 is the result of arelatively slower power pulse. We notice that the vibrationis reduced significantly. Our last test (Figure 3) correspondsto a much slower power pulse (e.g, τ > τ elastic). This time,we don’t see vibrations since the material can respond to theslow temperature rise with non-vibrating expansion.0 0.5 1 1.5 2 2.5 3 3.5 4x 10−300.20.40.60.811.21.4x 10−4 Surface Displacement−−−> timeFigure 1: Displacement resulting from a fast power pulse.0 0.5 1 1.5 2 2.5 3 3.5 4x 10−300.20.40.60.811.2x 10−4 Surface Displacement−−−> timeFigure 2: Displacement resulting from a medium pulse.0 0.5 1 1.5 2 2.5 3 3.5 4x 10−3012x 10−4 Surface Displacement−−−> timeFigure 3: Displacement resulting from a slower pulse.351Computational Methods: General0 0.2 0.4 0.6 0.8 1x 10−301234567x 107 Power−−−> timeFastSlowFigure 4: Power pulses from the coupled model.4.2 Preliminary results of our coupled modelIn this section, we present our preliminary results from ourcoupled model. We intend to see the similar mechanicalbehavior as we have seen in the simplified model. Figure4 shows two different power pulses one being faster thanthe other. The corresponding material response is shown inFigure 5. We notice that material vibration reduces when wehave slow temperature rise (slower pulse). This is consistentwith our simplified model. Again we observe an increase invibration when the temperature rises faster.We would like to make some initial remarks about the per-formance of our algorithm. Our implicit solver convergesto a given tolerance with on average three Krylov iterationsand one Newton step in smooth regions. On the other hand,in high gradient regions (e.g, where there is a steep powerrise), the convergence takes on average 35 Krylov iterationsand two Newton steps. We note that the code performancecan be improved by preconditioning the Krylov block.5 Conclusion and Future WorkWe presented a preliminary study for the coupling ofneutron diffusion and thermomechanics. We illustratedthe mechanical response of the material to different power(temperature) settings. We showed that if there is a fasttemperature increase in the system, then the materialexpands to a certain level and starts vibrating. On theother hand, if the temperature slowly increases, then thematerial expands with significantly less vibration. Ourfuture work is to eliminate the time splitting error byintroducing a predictor corrector methodology. Also, weare developing a semi-analytical solution to the coupledsystem for benchmarking purposes.0 0.2 0.4 0.6 0.8 1x 10−300.10.20.30.40.50.60.70.80.91x 10−3 Surface Displacement−−−> timeFastSlowFigure 5: Displacements corresponding to the power pulsesin Figure 4.AcknowledgementThis work has been carried out for the U.S. Department ofEnergy Office of Nuclear Energy Under DOE Idaho Oper-ations Office Contract DE-AC07-05ID14517 (INL/CN-08-14429).References[1] D. Burgreen. Thermoelastic dynamics of rods, thinshells, and solid spheres. Nuclear Science and Engi-neering, 12:203–217, 1962. 1, 2[2] J. J. Duderstadt and L. J. Hamilton. Nuclear ReactorAnalysis. John Wiley & Sons, Inc, 1976. 2[3] D. A. Knoll and D. E. Keyes. Jacobian-free newtonkrylov methods: a survey of approaches and applica-tions. J. Comput. Phys., 193:357–397, 2004. 3[4] S. C. Wilson and S. R. Biegalski. Computational mod-eling of coupled thermomechanical and neutron trans-port behavior in a godiva-like nuclear assembly. Nu-clear Science and Engineering, 157:344–353, 2007. 1,2352 Computational Methods: General

A Numerical Model for Coupling of Neutron Diffusion and Thermomechanics in Fast Burst Reactors

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A Numerical Model for Coupling of Neutron Diffusion and Thermomechanics in Fast Burst Reactors

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