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International Youth Nuclear Congress 2016, IYNC2016, 24-30 July 2016, Hangzhou, China

Finite volume method based neutronics solvers for steady and The 15th International Symposium on District Heating and Cooling transient-state analysis of nuclear reactors Assessing theHufeasibility the Cao heat aa aa a, a,*,demand-outdoor Tianliang , Hongchunof Wuusing , Liangzhi Zhifeng Liaa temperatureSchool function forandatechnology, long-term district heat demand forecast School of of nuclear nuclear science science and technology, Xi’an Xi’an Jiaotong Jiaotong University, University, Xi’an Xi’an 710049, 710049, China China aa

Abstract Abstract a

I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc

IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal b

& Innovation, 291roles Avenue Limay, France modelling of nuclear Coupling more and important in recent years for accurately Coupling canalysis analysis plays playsVeolia moreRecherche and more more important roles inDreyfous recent Daniel, years 78520 for the the accurately modelling of nuclear Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France reactors. reactors. But But with with the the different different discretization discretization methods methods adopted adopted in in different different physical physical phenomena, phenomena, it it is is difficult difficult to to couple different kinds of codes. In this paper, steady and transient-state neutronics solvers have been couple different kinds of codes. In this paper, steady and transient-state neutronics solvers have been developed developed based based on on the the finite finite volume volume method method which which is is widely widely used used for for the the simulation simulation of of fluid fluid flow flow and and heat heat and and mass mass transfer. transfer. With the adoption of this discretization method, mesh mapping technique and data exchange become much Abstract With the adoption of this discretization method, mesh mapping technique and data exchange become much easier easier for for the the coupling coupling codes codes compared compared with with traditional traditional methods. methods. The The developed developed neutronics neutronics solvers solvers are are verified verified by by several several District heating networks are commonly literature as one of the most effective solutionsto the benchmark problems. Numerical results indicate that developed neutronics solvers are applied benchmark problems. Numerical resultsaddressed indicate in thatthethe the developed neutronics solvers are reliable reliable toforbe bedecreasing applied for for greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heat the steady-state and transient-state neutronics analysis of nuclear reactors. the steady-state and transient-state neutronics analysis of nuclear reactors.

Due Authors. to the changed climate conditions ©sales. 2016 The Published by Elsevier Ltd. and building renovation policies, heat demand in the future could decrease, © 2016 Published by Elsevier ©prolonging 2017 The The Authors. Authors. Published byperiod. Elsevier Ltd. Ltd. the investment return Peer-review under responsibility of the organizing committee of IYNC2016. Peer-review under responsibility of the organizing committee of Peer-review under responsibility of the organizing committee of IYNC2016. IYNC2016 The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand forecast.finite The volume districtmethod; of Alvalade, located solver; in Lisbon was used as aOpenFOAM case study. The district is consisted of 665 Keywords: neutron steady and analysis; Keywords: finite volume method; neutron diffusion diffusion solver; steady(Portugal), and transient-state transient-state analysis; OpenFOAM buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district renovation scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were with results from a dynamic heat demand model, previously developed and validated by the authors. 1. Introduction 1.compared Introduction The results showed that when only weather change is considered, the margin of error could be acceptable for some applications (the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation In years, coupling neutronics and thermal-hydraulics analysis becomes more and important, because In recent recentthe years, neutronics thermal-hydraulics becomes morescenarios and more more important,considered). because scenarios, error coupling value increased up to and 59.5% (depending on the analysis weather and renovation combination it helps to identify the most relevant safety issues without conservative assumptions. In addition, such coupled it The helps to identify the most relevant safety issues without conservative assumptions. In addition, such coupled value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the simulations essential the IV nuclear such molten salt (MSR) simulations are essentialoffor for the generation generation IV advanced advanced nuclear power systems suchonas asthe molten salt reactor reactor (MSR) decrease in are the number heating hours of 22-139h during the heatingpower seasonsystems (depending combination of weather and [1], because the presence of strong coupling between different phenomena, like the transport of delayed neutron [1], becausescenarios the presence of strong coupling between different phenomena, the transport of delayed neutron renovation considered). On the other hand, function intercept increased forlike 7.8-12.7% per decade (depending on the precursors by [2,3]. Similarly, supercritical light water reactor fact density precursors by the the fluid fluid flow [2,3]. Similarly, for the supercritical light water parameters reactor (SCWR), (SCWR), the fact that that density of of coupled scenarios). Theflow values suggested couldfor bethe used to modify the function for the the scenarios considered, and improve the accuracy of heat demand estimations. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and * Corresponding *Cooling. Corresponding author. author. Tel.: Tel.: +862982663285; +862982663285; fax: fax: +862982667802. +862982667802.

E-mail E-mail address: address: [email protected] [email protected] Keywords: Heat demand; Forecast; Climate change 1876-6102 1876-6102 © © 2016 2016 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. Peer-review Peer-review under under responsibility responsibility of of the the organizing organizing committee committee of of IYNC2016. IYNC2016.

1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling.

1876-6102 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of IYNC2016 10.1016/j.egypro.2017.08.102

Tianliang Hu et al. / Energy Procedia 127 (2017) 275–283 Author name / Energy Procedia 00 (2016) 000–000

276 2

supercritical water undergoes a dramatic change in the reactor, resulting in strong effect of neutronics/thermalhydraulics coupling [4]. Due to the different numerical discretization methods adopted in reactor physics analysis and thermal-hydraulics analysis, it is hard to couple different kinds of codes. Nodal methods are widely used in the neutronics analysis of nuclear reactors because of its higher computational efficiency than the conventional discretization methods like finite difference method and finite element methods. The nodal methods use very coarse spatial mesh (usually in assembly scale) so that its computational efficiency is high. But most of the nodal methods are based on the rectangular or hexagonal geometry, which makes it difficult to handle complex geometries. As far as the thermalhydraulic analysis is concerned, the traditional method is to use the system codes with two-fluid approach, such as RELAP-5[5]. However with the advent of modern computational resources, computational fluid dynamics (CFD) method has also been developed to be an essential tool in nuclear engineering. This difference leads to problems when coupling the solvers with different discretization method. Often complex interface code needs to be developed for the mesh mapping and data exchange. This work presents an attempt to solve the multi-group neutron diffusion equation with the finite volume method which is quite widely used for the numerical simulation of a variety of applications involving fluid flow and heat and mass transfer. The unstructured finite volume method (FVM) based on open source C++ library OpenFOAM was chosen as the development platform [6,7]. OpenFOAM is distributed with a large set of precompiled applications mainly about the CFD, but no neutron diffusion solver included, so our present work focuses on the implementation of the neutronics solvers. And because of the same discretization method applied to the neutronics analysis, the coupling between the developed code and CFD code will be much easier than the traditional method. The objective of this paper is to concisely present the main features of the developed codes in terms of the theoretical background and numerical approach and as can be seen in the following sections, the calculation results obtained by the NDSFoam which solves the steady-state neutron diffusion equation and NDTFoam which solves the transient-state neutron diffusion equation are discussed and compared to the benchmark problems. 2. Methodology 2.1. Neutronics modelling The multi-group neutron diffusion approximation is adopted for the modelling of the reactor neutronics. The steady-state equation for neutron and delayed neutron precursors can be written as:

Dg g (r)  t , gg (r) 

G

 g 1

 (r)

g  g g 

G I 1  (1   0 )  p , g  ( f ) g g  (r)    d ,i , g i Ci (r) k g  1 i 1 eff

  (UC  i (r)) where

g

represents the neutron flux, cm-2s-1;

i

keff

G

 ( g 1

f

) g g  (r)  i Ci (r)

(1)

(2)

t , g represents the total macroscopic cross section, cm-1;  g  g

represents the scattering macroscopic cross section from group g’ to group g, cm-1; keff represents the effective multiplication factor;  p , g represents the spectrum of the prompt neutron;  represents number of prompt neutrons for a fission;

 f is macroscopic fission cross section, cm-1;  d ,i , g is neutron spectra for delayed neutrons emitted

by precursors in family i;

i represents the radioactive decay constant of family i; Ci represents the concentration

of delayed neutron precursors of family I; U represents the fluid velocity;  i is the fraction of delayed neutron precursors of family i.


277 3

In equation (2) the convection term represents the effect of delayed neutron precursors transported by the fluid flow. This term is very important in the reactors with fluid fuel like molten salt reactors. The power iteration method is adopted for the steady state calculation and the steady-state calculation results serve as the initial value for the transient-state calculation. The transient-state equation for neutron and delayed neutron precursors can be written as: G 1 g (r, t)  Dg g (r, t)  t , gg (r, t)    g  gg  (r, t) t vg g 1

G I 1  (1   0 )  p , g  ( f ) g g  (r, t)    d ,i , g i Ci (r, t) keff  g  1 i 1

 Ci (r, t) i    (UC  i (r, t)) t keff

G

 ( g 1

f

) g g  (r, t)  i Ci (r, t)

(3)

(4)

2.2. Numerical method In this work, the multi-group neutron diffusion equations are discretized spatially using the finite volume method (FVM) [9]. Finite volume method is a numerical technique that transforms the partial differential equations representing conservation laws over differential volumes into discrete algebraic equations over finite volumes. The neutron balance over any control domain in energy group g can be written in the symbolic form as below:  Dg (r)   gg (r)  Qg (r) (5) By integrating the above equation over the control volume; Equation (5) is transformed to:



V

  Dg (r)dV    gg (r)dV   Qg (r)dV V

V

(6)

Replacing the volume integrals of the diffusion terms by surface integrals through the use of Gauss theorem, thus equation (6) becomes:



V

 Dg (r)dS    gg (r)dV   Qg (r)dV V

V

(7)

Applying equation (7) to each control volume, the discretization equation on each given cell is:

  D  S f

f

g

f

  ggVg  QgVg

(8)

For the calculation of diffusion coefficient at the cell surface, the conservative interpolation, based on a harmonic mean, should be adopted for the diffusion coefficients to guarantee the continuity of the neutron flux and the neutron current. In the case of two control volume P and E, each with homogeneous diffusion coefficient DP and DE, the harmonic mean of the diffusion coefficient can be characterized as Eqn. (9):

Df 

P

 PE

DP

P



E

DE

E

(9)


278 4

Fig. 1. Diffusion coefficient calculation

The delayed neutron precursors balance equation over any control domain of family i can be written as:

  (UCi (r))  iCi (r)  Si (r)

(10)

By doing the same discretization process, Equation (10) finally becomes:

 UC (r)  S i

f

f

 i CiV  SiV

(11)

In OpenFOAM several divergence schemes can be selected including upwind, linearUpwind, QUICK and so on. The time-dependent kinetic equations are discretized in both space and time, and fully implicit scheme is applied for time discretization of neutron flux equations and delayed neutron precursor equations:

g (t )

|t



g (tn )  g (tn 1 )

(12)

t tn n Ci (t ) C (t )  Ci (tn 1 )  i n t tn tn

|

(13)

3. Verification and discussion The total suite of tests conducted for the verification of the solvers ranges from two dimensional problems to there-dimensional problems from steady-state problems to transient-state problems. The test cases included herein are 2D-IAEA, 3DIAEA, 2D-LRA, 3DLRA for the steady state calculation, and 2D-TWIGL seed Blanket, 3D-LMW for the transient state calculation. 3.1. Steady-state benchmarks A steady-state neutronics solver NDSFoam has been successfully programed based on the proposed model in section 2, benchmark calculation in reference are employed for verification. The following four problems were analyzed: 2D-IAEA; 3D-IAEA; 2D-LRA; 3D-LRA. The numerical results of the benchmark problems [10] are summarize in Table 1. In the calculation, the convergence criterion is set to 10-5 for the node-wise fission sources and 10-6 for the reactor eigenvalues. In table 1 ∆Pmax and ∆Paverage are the maximum and average percentage errors of assembly power. Table 1.Results of steady-state benchmark problems. Benchmark problems

Keff error(10-5)

∆Pmax(%)

∆Paverage(%)

2D-IAEA

1.5

0.318

0.103

2D-LRA

4.7

1.13

0.374

3D-IAEA

1.0

1.79

0.194

3D-LRA

14.0

1.40

0.562

The contours of the fast and thermal neutron flux distributions of 2D-IAEA are depicted in Fig 2. The normalized power distributions of the four benchmark problems are given in Fig 3. It can be seen that the results show good agreement with the reference. The maximal keff error is within 20 pcm and the maximal relative difference in power distribution is within 2%.


279 5

Fig. 2. (a) Fast flux distribution of 2D-IAEA problem (b) Thermal flux distribution of 2D-IAEA problem

Reference NDSFoam Error

0.7456 0.7439 -0.228%

1.4351 1.4349 -0.014% 1.3097 1.31 0.023%

1.4694 1.4695 0.007% 1.4799 1.4797 -0.014% 1.4537 1.4535 -0.014%

1.1929 1.1936 0.059% 1.3451 1.3455 0.030% 1.3149 1.3153 0.030% 1.2107 1.2123 0.132%

0.4706 0.4699 -0.150% 0.967 0.9681 0.110% 1.1792 1.18 0.068% 1.0697 1.071 0.122% 0.61 0.6092 -0.131%

0.5849 0.5833 -0.270% 0.6856 0.6862 0.088% 0.9064 0.9068 0.044% 1.0705 1.0711 0.056% 1.0361 1.0371 0.097% 0.9351 0.9365 0.150%

0.5972 0.5953 -0.318% 0.8461 0.8449 -0.140% 0.9752 0.9754 0.021% 0.9504 0.9507 0.032% 0.9343 0.9347 0.043%

0.6921 0.69 0.300% 0.7358 0.7346 -0.163% 0.7549 0.7538 -0.146%

Fig. 3. Compassion of assembly power for 2D-IAEA problem

The calculated results and benchmark results show good agreement in eigenvalues as well as the power distribution. It could be concluded that the finite volume method based steady-state neutronics solver implemented in NDSFoam is reasonable and valid. 3.2. Transient-state benchmarks For the transient-state solver NDTFoam, two benchmark problems are performed, including both 2D and 3D problems.  2D-TWIGL seed Blanket Two transient problems are included in 2D-TWIGL benchmark problem, namely the step perturbation and the ramp perturbation transient problem. The layout of the 2D-TWIGLE benchmark can be seen in Figure 4.In the step perturbation transient, the macroscopic absorption cross section of thermal neutrons in region 1 decreases to 0.0035 cm-1 at time 0 s and stays constant. While in the ramp perturbation transient, the macroscopic absorption cross section of thermal neutrons in region 1 decreases linearly in the first 0.2 s and then stays constant. The total time interval of the analyzed problem is 0.5s.


280 6

Step perturbation in region 1:

Ramp perturbation in region 1:

a 2 (t)  0.97666  (t 0) a 2 (0)  a 2 (t)  1  0.11667t  a 2 (0)

 a 2 (t)  0.97666  a 2 (0)

(14)

(0s  t  0.2s) (15)

(t  0.2s)

Fig. 4. Layout of the 2D-TWIGLE benchmark

The step perturbation transient is solved with time step 0.001, the normalized power predictions during transient process by DNTFoam are given in Figure 5 and Table 2, and the maximal error is within 0.3%.

Fig. 5. The relative power for the 2D-TWIGEL step perturbation problem Table 2.Comparison of the relative power for 2D-TWIGLE step perturbation problem. Time (s)

Reference

NDTFoam

Error (%)

0.1

2.061

2.056

-0.2431

0.2

2.078

2.073

-0.2243

0.3

2.095

2.091

-0.2115


0.4

2.113

2.108

-0.2404

0.5

2.131

2.126

-0.2487

281 7

The ramp perturbation transient is solved with time step 0.01, because of the relatively slow change of power by time. The normalized power prediction during transient process are given in Figure 6 and Table 3, and the maximal error is within 0.5%.

Fig. 6. The relative power for the 2D-TWIGEL ramp perturbation problem Table 3.Comparison of the relative power for 2D-TWIGLE ramp perturbation problem. Time(s)

Reference

NTDFoam

Error(%)

0.05

1.125

1.123

-0.1173

0.1

1.307

1.307

0.0031

0.2

1.957

1.957

-0.0001

0.3

2.074

2.070

-0.2141

0.4

2.096

2.087

-0.4455

0.5

2.109

2.104

-0.2360

 3D-LMW The Langenbuch–Maurer–Werner (LMW) problem simulates an operational transient involving control rods (CRs) movement and is useful for the verification of space–time kinetics codes [11]. The model is a simplified pressurized water reactor (PWR) core and is composed of two kinds of fuel assemblies. The number of neutron energy groups is two and the number of delayed precursors groups is six. Detailed macroscopic cross sections and kinetic parameters of each region can be found in the reference [12]. This problem simulates the motion of 2 groups of CRs, and the transient is initiated by withdrawing group 1 at 3.0 cm/s. CR group 1 is withdrawn to the top in 26.666 s, and group 2 is inserted to 120 cm during the time interval from 7.5 s to 47.5 s. The transient process lasts 60s. The radial and axial layout of the LMW problems are shown in Fig 7 and Fig 8.

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Tianliang et al. Procedia / Energy Procedia (2017) 275–283 Author name Hu / Energy 00 (2016)127 000–000

Fig. 7. Radial layout of the LMW benchmark problem

Fig. 8. Axial layout of the LMW benchmark

The normalized power predictions during transient process by NDTFoam are given in Figure 9 and Table 4. The results of NTDFoam are compared to the result of SPANDEX with fine time step mesh. It can be seen that the results show good agreement with the reference, and the maximal relative difference is within 1.0%.


283 9

Fig. 9. The relative power for the 3D-LMW problem Table 4.Comparison of the relative power for 3D-LMW problem. Time(s)

Reference

NTDFoam

Error (%)

0

1.000

1.000

0.0

10

1.341

1.341

0.0

20

1.713

1.704

-0.525

30

1.373

1.366

-0.510

40

0.810

0.808

-0.247

50

0.503

0.505

0.398

60

0.386

0.388

0.518

4. Conclusions In this paper, two new neutronics solvers with finite volume method have been developed to perform the reactor physics analysis. The solvers are based on OpenFOAM, and have been tested with several steady-state and transientstate benchmark problems. By verifications of the codes, it is proved that the development of NDSFoam and NDTFoam are correct. In our future work, both of these two solvers will be coupled with the precompiled CFD solvers in OpenFOAM to do the coupled analysis of nuclear reactors like molten salt reactors. Acknowledgements The authors would like to thank the Natural Science Foundation of China (Grant Nos.91226106). References [1] Serp J, Allibert M, Beneš O, et al. The molten salt reactor (MSR) in generation IV: Overview and perspectives[J]. Progress in Nuclear Energy, 2014, 77: 308-319 [2] Zhuang K, Cao L, Zheng Y, et al. Studies on the molten salt reactor: code development and neutronics analysis of MSRE-type design[J]. Journal of Nuclear Science and Technology, 2015, 52(2): 251-263. [3] Aufiero M, Brovchenko M, Cammi A, et al. Calculating the effective delayed neutron fraction in the Molten Salt Fast Reactor: Analytical, deterministic and Monte Carlo approaches[J]. Annals of Nuclear Energy [4] Zhao C, Cao L, Wu H, et al. Conceptual design of a supercritical water reactor with double-row-rod assembly[J]. Progress in Nuclear Energy, 2013, 63: 86-95. [5] Ransom V H, Wagner R J, Trapp J A, et al. RELAP5/MOD2 code manual, Volume 1: Code structure, system models, and solution methods[J]. Report Nos. NUREG/CR-4312 and EGG-2796, 1985. [6] Jasak H, Jemcov A, Tukovic Z. OpenFOAM: A C++ library for complex physics simulations[C]//International workshop on coupled methods in numerical dynamics. 2007, 1000: 1-20. [7] Aufiero M, Cammi A, Geoffroy O, et al. Development of an OpenFOAM model for the Molten Salt Fast Reactor transient analysis[J]. Chemical Engineering Science, 2014, 111: 390-401. [8] 2014, Clifford I. Block-Coupled Simulations Using OpenFOAM[C]//6th OpenFOAM workshop, PennState University. 2011. 65: 78-90. [9] Versteeg H K, Malalasekera W. An introduction to computational fluid dynamics: the finite volume method[M]. Pearson Education, 2007 [10] Muller EZ, Weiss ZJ. Benchmarking with the Multigroup Diffusion High-Order Response Matrix-Method[J]. Annals of Nuclear Energy, 1991, 18 (9): 535-544. [11] He M, Wu H, Zheng Y, et al. Beam transient analyses of Accelerator Driven Subcritical Reactors based on neutron transport method[J]. Nuclear Engineering and Design, 2015, 295: 489-499. [12] Ban Y, Endo T, Yamamoto A. A unified approach for numerical calculation of space-dependent kinetic equation[J]. Journal of nuclear science and technology, 2012, 49(5): 496-515