In typical lattice cells where a highly absorbing, small fuel element is embedded in the moderator, a large weakly absorbing medium, high-order transport methods become unnecessary. In this work we describe a hybrid discrete

ordinates (SN) method for energy multigroup slab lattice calculations. This hybrid SN method combines the convenience of a low-order SN method in the moderator with a high-order SN method in the fuel. The idea is based on the fact that in weakly absorbing media whose physical size is several neutron mean free paths in extent, even the S2 method (P1 approximation), leads to an accurate result. We

use special fuel-moderator interface conditions and the Laplace transform (LTSN) analytical numerical method to calculate the two-energy group neutron flux distributions

and the thermal disadvantage factor. We present numerical results for a range of typical model problems

Barros, R. C.

Segatto, C. F.

Vilhena, M. T.

English

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DOI 10.1393/ncc/i2010-10579-yColloquia: ICTT2009IL NUOVO CIMENTO Vol. 33 C, N. 1 Gennaio-Febbraio 2010A Laplace transform method for energy multigroup hybriddiscrete ordinates slab lattice calculationsC. F. Segatto(1), M. T. Vilhena(1) and R. C. Barros(2)(∗)(1) Universidade Federal do Rio Grande do Sul - Av. Bento Gonalves 950091509-900 Porto Alegre, RS, Brazil(2) Universidade do Estado do Rio de Janeiro - Rua Alberto Rangel s/n28630-050 Nova Friburgo, RJ - Brazil(ricevuto il 30 Ottobre 2009; approvato il 28 Gennaio 2010; pubblicato online il 16 Marzo 2010)Summary. — In typical lattice cells where a highly absorbing, small fuel ele-ment is embedded in the moderator, a large weakly absorbing medium, high-ordertransport methods become unnecessary. In this work we describe a hybrid discreteordinates (SN ) method for energy multigroup slab lattice calculations. This hybridSN method combines the convenience of a low-order SN method in the moderatorwith a high-order SN method in the fuel. The idea is based on the fact that inweakly absorbing media whose physical size is several neutron mean free paths inextent, even the S2 method (P1 approximation), leads to an accurate result. Weuse special fuel-moderator interface conditions and the Laplace transform (LTSN )analytical numerical method to calculate the two-energy group neutron flux distri-butions and the thermal disadvantage factor. We present numerical results for arange of typical model problems.PACS 28.20.Gd – Neutron transport: diffusion and moderation.1. – IntroductionIn typical lattice cells where a highly absorbing, small fuel element is embedded in themoderator, a large weakly absorbing medium, high-order transport methods become un-necessary. This situation was noted by Leslie [1] who devised techniques for grafting diffu-sion theory onto transport theory. Later, Nanneh and Williams [2] developed a diffusion-transport theory hybrid method for calculating neutron flux distributions in slab lattices.Further Segatto [3] offered a hybrid discrete ordinates (SN ) Laplace transform methodfor one-speed slab lattice calculations. In this paper we extend this hybrid SN method toenergy multigroup slab lattice calculations. This hybrid multigroup SN method combinesthe convenience of a low-order SN method in the moderator with a high-order SN method(∗) E-mail: rcbarros@pq.cnpq.brc© Societa` Italiana di Fisica 231232 C. F. SEGATTO, M. T. VILHENA and R. C. BARROSFig. 1. – Unit cell of slab lattice, NF ≥ NM .in the fuel. The idea is based on the fact that in weakly absorbing media whose physicalsize is several neutron mean free paths in extent, even the S2 method (P1 approximation)leads to an accurate result [4]. We use special fuel-moderator interface conditions andthe Laplace transform (LTSN ) analytical numerical method to calculate the neutron fluxdistribution and the thermal disadvantage factor [5], which is defined as the ratio of theaverage scalar flux in the moderator region to the average scalar flux in the fuel rod. Atthis point we give an outline of the remainder of this paper. In sect. 2 we present thegeneral theory. In sect. 3 we give numerical results for a range of typical lattices. Anumber of concluding remarks and suggestions for future work are given in sect. 4.2. – General theoryLet us consider the energy multigroup SN equations in slab geometry with isotropicscattering(1) μmddxψm,g(x) + σtg(x)ψm,g(x) =12G∑g′=1σsg′→g(x)N∑n=1ψn,g′(x)wn + Qg(x),m = 1 : N , g = 1 : G. Here the notation is standard for a given angular quadratureset [6], where G is the number of energy groups. To describe the technique, we considerthe slab lattice represented in fig. 1. In eq. (1) if x belongs to the fuel region of thicknessa, then we assume the interior neutron source as uniform Q1 = 1, Qg = 0 for g = 1and N = NF ; if x belongs to the moderator region of thickness b, then we set the groupneutron interior source Qg = 0 for g = 1 : G and N = NM , with NF ≥ NM . Periodic orreflective boundary conditions apply in accordance with the domain we consider withinthe slab lattice. At this point we remark that we assume in this model that the promptfission neutrons are released in the fast energy group (g = 1) in the fuel pin and they arebrought down to lower energy groups as they migrate to the moderator region. This isthe reason why we set Q1 = 1 in the fuel region, i.e. to model the prompt fission neutrons.To describe the present hybrid SN method, which we refer to as the LTSNF ,NMmethod, we apply the Laplace transform in space to eq. (1) in the fuel and moderatorregions, as represented in fig. 1. It is well known that the SN equations (1) are equivalentto the one-dimensional PN−1 equations [7]. Therefore, Barros [8] considered specialfuel-moderator interface conditions for the neutrons migrating from the fuel region toA LAPLACE TRANSFORM METHOD FOR ENERGY MULTIGROUP HYBRID DISCRETE ETC. 233the moderator, by considering PNF−1 expansion of the angular flux to evaluate theangular flux in the NM/2 discrete directions entering the moderator. A similar procedureapplies when neutrons leave behind the moderator, to evaluate the angular flux in theNF /2 discrete directions entering the fuel region. To apply the multigroup LTSNF ,NMmethod, as described in this paper, we use an alternate special fuel-moderator interfaceconditions based on an approximate angular flux interpolation analytical method [9],that we describe further in this section. As it is, eq. (1) can be written in matrix form as(2)ddxΨ(x)−AΨ(x) = S,where we have defined the N ×G dimensional vectors(3) Ψ(x) = (ψ1,1, ψ2,1, . . . , ψN,1, ψ1,2, ψ2,2, . . . , ψN,2, . . . . . . . . . , ψ1,G, ψ2,G, . . . , ψN,G)T ,and(4) S =(Q1μ1,Q1μ2, . . . ,Q1μN,Q2μ1,Q2μ2, . . . ,Q2μN, . . . . . . . . . ,QGμ1,QGμ2, . . . ,QGμN)T,for N = NF or N = NM with NF ≥ NM , Q1 is the fast group fission neutron interiorsource in the fuel pin as shown in fig. 1. Therefore, we apply the Laplace transform ineq. (2) to obtain the analytical general solution, that appears as(5) Ψ(x) = B(x)Θ + H,with(6) B(x) = XE(x).Here E(x) is a diagonal matrix whose entries are(7) ei,j(x) ={exp[λix], if λi < 0,exp[λi(x− )], if λi > 0,where  = a or b, cf. fig. 1. The constants λi, i = 1 : G×NF or NM , are the eigenvaluesof matrix A; X is a matrix whose columns are eigenvectors of A; and H is the particularsolution vector which is determined by solving the system(8) −AH = S.In order to describe the special interface conditions that we use in the present LTSNF ,NMmethod, we consider the multigroup slab-geometry neutron transport equation withisotropic scattering(9) μ∂ψg(x, μ)∂x+ σtgψ(x, μ)=12G∑g′=1σsg′→g(x)∫ +1−1ψg′(x, μ′)dμ′+Qg(x), g=1 : G,234 C. F. SEGATTO, M. T. VILHENA and R. C. BARROSwhere x belongs to the fuel or moderator region in the repeating slab lattice. Now weapproximate the integral term of the scattering source by appropriate Gauss-Legendreangular quadrature. Then, by considering that the angular quadrature order is distinctin the fuel and in the moderator regions, we solve the resulting approximate transportequation analytically to evaluate the angular fluxes at the fuel-moderator interface inthe necessary discrete angular directions. In other words, for x in the moderator region,the macroscopic cross-sections in eq. (9) are the total and scattering cross-sections ofthe moderator and Qg = 0, g = 1 : G, as in fig. 1. Therefore, we first approximate theintegral term by the Gauss-Legendre quadrature with N = NM(10)∫ +1−1ψg(x, μ′)dμ′ =NM∑n=1ψn,g(x)wn, g = 1 : G,where ψn,g(x) is the LTSNM solution [10], and then we solve the resulting transportequation so approximated to evaluate the neutron angular flux in the NF /2 angulardirections entering the fuel region. A similar procedure is carried out for x in the fuelregion, where the macroscopic cross-sections in eq. (9) are the total and scattering cross-sections of the fuel and Qg = 0 unless g = 1, for which we set Q1 = 1, as shown infig. 1.3. – Numerical resultsTo illustrate the accuracy of the LTSNF ,NM hybrid method, we consider two typicalmodel problems. Model problem No. 1 is a one-speed model (G = 1) and correspondsto enriched uranium-water lattice, whose dimensions and material parameters are givenin table I [2]. Table II shows the disadvantage factor for this monoenergetic problem, asgenerated by neutron diffusion theory, DIFTRAN numerical method [2], Gauss-LegendreS16 and the present hybrid LTS16,2 method. Comparing these results, we see that thepresent hybrid method leads to accurate solutions for the disadvantage factor and forsuch thin lattices, diffusion theory is clearly inadequate.Model problem No. 2 is a two-group model (G = 2) whose dimensions and materialparameters are also given in table I. Table III shows the thermal disadvantage factor forthis energy-dependent problem, as generated by the present hybrid LTSNF ,NM method,where we fixed NF = 20 with NM = 2 : 20. From these results, we see that the presenthybrid method leads to accurate solutions for the thermal disadvantage factor yieldingmaximum relative deviation with respect to the non-hybrid result LTS20,20 that did notexceed 2% for the LTS20,2 hybrid method. Moreover, fig. 2 shows how the fast flux(g = 1) profile is depressed in the highly diffusive moderator region using the LTS20,4method. Figure 3 shows the thermal flux (g = 2) profile in the highly absorbing fuelregion also using the LTS20,4 hybrid method.4. – Concluding remarksWe described a hybrid discrete ordinates (SN ) method for energy multigroup slablattice calculations. This hybrid SN method combines the convenience of a low-orderSN method in the moderator with a high-order SN method in the fuel. We use specialfuel-moderator interface conditions based on an approximate angular flux interpolationanalytical method and the Laplace transform (LTSN ) [9] numerical method to calculatethe group neutron flux distribution and the disadvantage factor.A LAPLACE TRANSFORM METHOD FOR ENERGY MULTIGROUP HYBRID DISCRETE ETC. 235Table I. – Data for model problems.Enriched UData + Two-energy group model(∗∗)Water(∗)(one-speed model)Total cross-section of fuel 1.648 σt1 = 1.4, σt2 = 1.2(cm−1)Scattering cross-section of fuel 0.7278 σs11 = 0.99, σs12 = 0.30(cm−1) σs21 = 0.10, σs22 = 0.75Total cross-section of moderator 3.691 σt1 = 1.0, σt2 = 1.2(cm−1)Scattering cross-section of moderator 3.6713 σs11 = 0.90, σs12 = 0.70(cm−1) σs21 = 0.05, σs22 = 0.80a 0.1016 2.5098(cm)b various 15.7326(cm)(∗) Model problem 1.(∗∗) Model problem 2.Table II. – Numerical results to model problem No. 1. b/a = ratio of moderator length to fuellength.Disadvantage factorb/a Diffusion theory(∗) DIFTRAN(∗) S16 LTSNF ,NMNF = 16, NM = 23 1.032 1.116 1.112 1.096 (1.4%)(∗∗)4 1.039 1.125 1.125 1.109 (1.4%)5 1.048 1.133 1.137 1.121 (1.4%)6 1.056 1.142 1.149 1.132 (1.5%)7 1.065 1.151 1.160 1.142 (1.6%)8 1.074 1.160 1.170 1.152 (1.5%)(∗) See [2].(∗∗) Relative deviation with respect to the S16 result.236 C. F. SEGATTO, M. T. VILHENA and R. C. BARROSTable III. – Numerical results to model problem No. 2.Hybrid angular quadratures Thermal disadvantageFuel S20 and Moderator S2–S20 factorLTS20,2 1.074923 (1.73%)(∗)LTS20,4 1.060138 (0.15%)LTS20,6 1.059213 (0.06%)LTS20,8 1.058898 (0.03%)LTS20,10 1.058753 (0.02%)LTS20,12 1.058674 (0.01%)LTS20,14 1.058625 (0.007%)LTS20,16 1.058593 (0.004%)LTS20,18 1.058571 (0.002%)LTS20,20 1.058555(∗) Relative deviation with respect to the non-hybrid LTS20,20 result.Fig. 2. – Fast group flux profile (g = 1) for model problem No. 2.Fig. 3. – Thermal group flux profile (g = 2) for model problem No. 2.A LAPLACE TRANSFORM METHOD FOR ENERGY MULTIGROUP HYBRID DISCRETE ETC. 237Table IV. – Numerical results to model problem No. 2 as generated by using Wynn-epsilonacceleration technique [11].Hybrid angular quadratures Thermal disadvantageFuel S20 and Moderator S2–S20 factorLTS20,2 1.074923(∗) (1.55%)(∗∗)LTS20,4 1.060138 (0.15%)LTS20,6 1.059151 (0.06%)LTS20,8 1.058736 (0.02%)LTS20,10 1.058625 (0.01%)LTS20,12 1.058556 (0.005%)LTS20,14 1.058532 (0.003%)LTS20,16 1.058513 (0.001%)LTS20,18 1.058505 (0.0005%)(∗) Wynn-epsilon acceleration technique [11].(∗∗) Relative deviation with respect to the non-hybrid LTS20,20 result displayed in table III.Based on the numerical results shown in sect. 3, we conclude that the present hybridSN method generates accurate numerical results. It seems clear that the special fuel-moderator interface conditions based on the approximate angular interpolation of theneutron flux lead to more accurate results than the continuity interface conditions basedon PN expansion [12]. Table IV shows the numerical results generated by Barry Ganapol(University of Arizona, Tucson, USA) and presented at the 21st International Conferenceon Transport Theory (21 ICTT) to test problem No. 2. It seems that the LTSNF ,NM withthe Wynn-epsilon technique [13] should be suitable to applications involving hybrid SNmethods as described in this paper, since it generated numerical results for the thermaldisadvantage factor [11] with smaller relative deviations.To conclude we remark that the main goal of the present work with isotropic scatteringapproximation and slab geometry is to analyze the numerical results generated by thehybrid LTSN method in energy multigroup calculations. As future work, we intend toconsider linearly anisotropic scattering approximation since scattering in the fuel regionis definitely not isotropic, and such an approximation in the moderador may be also poorin some cases. Bearing in mind that the LTSN method is not an approximate numericaltechnique for problems in curvilinear geometry, we intend to solve more realistic fuelcell problems by the suitable Hankel Transform (HTSN ) [14] approach in cylindricalgeometry.∗ ∗ ∗This work was supported by Universidade Federal do Rio Grande do Sul (UFRGS),Universidade do Estado do Rio de Janeiro (UERJ), Conselho Nacional de Desenvolvi-mento Cient´ıfico e Tecnolo´gico (CNPq—Brazil) and Fundac¸a˜o Carlos Chagas Filho deAmparo a` Pesquisa do Estado do Rio de Janeiro (FAPERJ—Brazil). It is part of theproject of the National Institute for Science and Technology (INCT) on Innovative Nu-clear Reactors.238 C. F. SEGATTO, M. T. VILHENA and R. C. BARROSREFERENCES[1] Leslie D. C., J. Nucl. Energy, 17 (1963) 293.[2] Nanneh M. M. and Williams M. M. R., Kerntechnik, 47 (1985) 221.[3] Segatto C. F., Vilhena M. T., Zani J. H. and Barros R. C., International Conferenceon the Physics of Reactors in PHYSOR 2008, September 14-19 (Interlaken, Switzerland)2008.[4] Alcouffe R. E. and O’Dell R. D., CRC Handbook of Nuclear Reactor Calculations,Vol. 1 (CRC Press, Inc.) 1986, p. 341.[5] Dudersatdt J. J. and Hamilton L. J., Nuclear Reactor Analysis (John Wiley & Sons,Inc) 1976.[6] Barros R. C., Ann. Nucl. Energy, 24 (1997) 1013.[7] Lewis E. E. and Miller W. F. jr., Computational Methods of Neutron Transport (JohnWiley & Sons, New York) 1984.[8] Barros R. C., Yavuz M., de Abreu M. P., Alves Filho H. and Mello J. A. M.,Prog. Nucl. Energy, 33 (1998) 117.[9] Segatto C. F., Vilhena M. T. and Leite S. Q. B., JQSRT, 95 (2005) 415.[10] Segatto C. F., Vilhena M. T. and Gomes M. G., Ann. Nucl. Energy, 26 (1999) 925.[11] Ganapol B. D., paper presented at this Conference.[12] Barros R. C., Silva F. C. and Alves Filho H., Prog. Nucl. Energy, 35 (1999) 293.[13] Segatto C. F., Vilhena M. T. and Tavares L. S. S., JQSRT, 70 (2001) 227.[14] Gonc¸alves G. A., Bogado Leite S. Q. and Vilhena M. T., Ann. Nucl. Energy, 36(2009) 98.

A Laplace transform method for energy multigroup hybrid

discrete ordinates slab lattice calculations

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A Laplace transform method for energy multigroup hybrid discrete ordinates slab lattice calculations

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