GENERALIZED ENERGY CONDENSATION THEORY

Transcription

1 GENERALIZED ENERGY CONDENSATION THEORY A Thesis Pesented to The Academic Faculty by Steven James Douglass In Patial Fulfillment of the Requiements fo the Degee Maste of Science in Nuclea Engineeing Geogia Institute of Technology Decembe 27

2 GENERALIZED ENERGY CONDENSATION THEORY Appoved by: D. Fazad Rahnema, Adviso School of Mechanical Engineeing Geogia Institute of Technology D. Weston Stacey School of Mechanical Engineeing Geogia Institute of Technology D. Ronaldo Szilad Diecto of Nuclea Science and Engineeing Idaho National Laboatoy Date Appoved: Novembe 8, 27

3 ACKNOWLEDGEMENTS I would fist like to thank D. Rahnema, whose guidance, motivation, and tieless patience has been invaluable in the last couple of yeas. I would also like to thank Dingkang Zhang, fo offeing valuable insight in helping develop this wok, as well as D. Stacey and D. Szilad, fo seving on my eading committee. I would like to thank my paents, who have always been suppotive of my educational effots, and who have always been willing to help me in any way I need fo as long as I have lived. I could not have achieved anything without the foundation upon which I was aised. Lastly, I would like to thank my bothes. They have been a constant souce of encouagement in my gaduate endeavos, and without them, I would likely not have gone to gaduate school at Geogia Tech. iii

4 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES SUMMARY iii v vi vii CHAPTER 1 INTRODUCTION 1 2 BACKGROUND 3 3 METHOD Deivation Othogonal Expansion Legende Expansion Application in 1-D Discete Odinates 17 4 EXAMPLE PROBLEMS Single Assembly Veification Whole Coe Veification 28 5 TIME COMPARISON AND ERROR ANALYSIS Solution Time Compaison Eo Analysis 34 6 CONCLUSIONS AND FUTURE WORK 37 APPENDIX A: BENCHMARK PROBLEM MATERIAL DEFINITION 4 REFERENCES 42 iv

5 LIST OF TABLES Page Table 1: 47 Goup Eigenvalues 2 Table 2: Selected Regions, Assemblies 21 Table 3: Selected Regions, Whole Coe 28 Table 4: Computation Times (coupled) 33 Table 5: Solution Times fo Legende Polynomials (decoupled) 34 Table 6: Mateial Definitions, Densities in 1 24 /cm 3 4 v

6 LIST OF FIGURES Page Figue 1: Sample Poblem Stuctue 2 Figue 2: Fine Goup Specta 22 Figue 3: Standad 2-Goup Collapsed Specta 24 Figue 4: 1 st, 3 d, and 5 th Ode Appoximations in Assembly 1, Region 1 26 Figue 5: Twentieth Ode Appoximation fo Assembly 1, Region 1, Assembly 2, Region 3, and Assembly 3, Region Figue 6: Coe 3 Specta 29 Figue 7: Goup Flux Eo vs. Expansion Ode fo a Homogeneous Slab 35 Figue 8: Goup Flux Eo vs. Expansion Ode fo Coe 3 36 vi

7 SUMMARY A genealization of multigoup enegy condensation theoy has been developed. The new method geneates a solution within the few-goup famewok which exhibits the enegy spectum chaacteistic of a many-goup tanspot solution, without the computational time usually associated with such solutions. This is accomplished by expanding the enegy dependence of the angula flux in a set of geneal othogonal functions. The expansion leads to a set of equations fo the angula flux moments in the few-goup famewok. The th moment geneates the standad few-goup equation while the highe moment equations geneate the detailed spectal esolution within the fewgoup stuctue. It is shown that by caefully choosing the othogonal function set (e.g., Legende polynomials), the highe moment equations ae only coupled to the th -ode equation and not to each othe. The decoupling makes the new method highly competitive with the standad few-goup method since the computation time associated with detemining the highe moments become negligible as a esult of the decoupling. The method is veified in seveal 1-D benchmak poblems typical of BWR configuations with mild to high heteogeneity. vii

8 CHAPTER 1 INTRODUCTION Multigoup teatment of the enegy vaiable is extensively used in solving the tanspot equation o its diffusion appoximation in eacto physics poblems. Fo solving fixed souce o eigenvalue poblems in lage complicated systems such as eactos, it is common pactice, fo the sake of efficiency and pacticality, to condense the coss section data fom an ulta-fine-goup fomat to a set that is manageable (e.g., fine o coase goup) in tems of computational esouce (memoy and time) limitation and the desied accuacy. The condensation pocedue equies the exact enegy spectum of the flux as a weighting function, which is not known a pioi. As a esult, appoximate flux specta ae obtained fo smalle subegions of the system (e.g., lattice cell) with appoximate bounday conditions (e.g., full specula eflection), which ae used to condense the coss sections into a smalle numbe of goups. Clealy, this condensation pocedue esults in loss of enegy esolution in addition to accuacy. Recoveing the enegy esolution while maintaining the computational efficiency is highly desiable in both eigenvalue (citicality) and fixed souce (shielding) calculations. In this pape, we develop a new method to ecove (unfold) the enegy spectum to any desied esolution (e.g., fom coase to fine o ulta-fine, fine to ultafine). This is achieved by genealizing the standad condensation pocedue, assuming that the enegy dependence of the neuton flux (spectum) may be expanded in a set of othogonal basis functions, and folding this dependence into the coss section condensation pocess. It will be shown that the standad condensation pocedue is 1

9 contained within this genealized method as a th -ode appoximation, and that though implementing this method, the computation time is educed to that of standad coasegoup computations, but with the detail usually associated with much fine goup solutions. The validity of the new method will be demonstated by application to seveal one-dimensional poblems with vaying degees of heteogeneity. The method is deived and tested in tanspot theoy. Its extension to its diffusion appoximation is staightfowad. 2

10 CHAPTER 2 BACKGROUND Methods fo teating the enegy dependence of the neuton flux in a eacto ae many and vaied. The usual multi-goup fomulation with few-goup condensation is by fa the most common, but a geat deal of wok has been done to find impoved methods fo teating the enegy spectum. Wok by M. L. Williams and M. Asgai [1] fomulated a combination of multi-goup theoy and continuous-enegy theoy to impove the calculation of the enegy spectum in the esonance egion. Thei wok takes advantage of a Legende Sub-moment Expansion in the scatteing tansfe function, and beaks the enegy spectum into thee egions, using multi-goup theoy in the fast and themal anges, and a point-wise solution in the esonance egion, all within a one-dimensional discete-odinates famewok. Wok has also been done in eacto analysis using point-wise enegy lattice methods. M. L. Zekle has developed methods fo solving the neuton tanspot equation using a nea-continuous enegy point-wise solution method which collapses the enegy dependence to a small numbe of goups fom point-wise data [2,3]. These methods have been applied in the RAZOR lattice code by Zekle, Abu-Shumays, Ott, and Winwood. Wok by M. L. Williams has also povided a solution fo themal neutons in a eacto using continuous enegy methodology implemented in the CENTRM solution module fo the SCALE code system [4]. These techniques wee all developed to impove the esonance and enegy teatments within the multi-goup methodology. 3

11 Most wok focusing on enegy dependence has thus fa been towads impoving the accuacy of the multigoup appoximation to the solution within single assemblies. Thee has not been much wok towads peseving the spectal infomation duing the condensation pocedue and whole coe calculation, whee the detailed spectum infomation is lost. Wok by Silve, Roede, Vote, and Kess addesses a method of Kenel Polynomial Appoximation fo spectal functions, using applications of polynomial expansion in the computation of the density of enegy states within electonic stuctues [5]. This allows fo teatment of spectal functions with polynomial expansions in a staightfowad and accuate way. Thei wok fully teats the method fo Chebyshev Polynomials, and it has simila applications to wok that can be used in nuclea eacto computations, paticulaly in the teatment of the angula dependence of the scatteing kenel. 4

12 CHAPTER 3 METHOD The genealized enegy condensation theoy epesents a method wheeby the enegy spectum of the neuton flux is poduced to a high degee of accuacy duing a few-goup calculation. This method begins by geneating fine-goup coss sections, as well as a fine-goup tanspot solution fo the individual lattice cells (e.g., fuel assemblies) which make up the system. The method of fine-goup coss section geneation is entiely independent of the genealized theoy, and should be done in whateve manne the use deems appopiate. Fo example, lattice depletion codes geneate fine o ulta-fine-goup coss sections by popely accounting fo esonance smeaing and tempeatue effects. Similaly, fine o ulta-fine-goup tanspot solutions within each assembly may be obtained using any computational method appopiate fo the desied application (e.g., discete odinates, collision pobability, etc). In addition, the method is independent of the teatment of the angula dependence of the scatteing kenel, and the use may use any desied technique. The fine-goup tanspot solution within each lattice cell (fuel assembly) is then used as a weighting function in the geneation of othogonal expansion moments fo the enegy dependence of the coss sections and eaction ates fo each egion of the assembly fo a set of coase-goups. This eplaces the standad condensation pocedue, which uses the ulta-fine o fine-goup tanspot solution to geneate fine o coase-goup coss sections that ae constant in enegy within each coase goup. Using the expansion moments of the coss sections and eaction ates, the poblem is then solved via a coupled 5

13 set of modified tanspot equations fo the whole coe. The esultant seies of flux moments within each coase-goup can then be used to constuct the fine goup enegy spectum of the neuton distibution in the entie coe. The coe enegy spectum is not accessible using the cuent standad condensation methods, but obtained using the genealized method pesented hee. The genealized method is entiely independent of the tanspot solution methodology (e.g., spatial diffeencing and angula appoximation schemes), and theefoe allows fo a high degee of flexibility in application. It expands the standad few-goup equations into a new set of expansion equations, which ae then solved in any manne desied. The standad condensation method is a special case (the th ode) of the geneal theoy. The additional moment equations coupled to the th ode epesent coection tems to the enegetically flat flux assumed in the standad few-goup model. It is noted that with a popely chosen expansion basis, this method can be used to geneate vey high ode expansions with negligible computation time due to the decoupling of the set of equations, to be descibed in a late section. The new method is tested by consideing some 1-D example poblems. Legende polynomials ae used as the basis function and the lattice depletion code HELIOS [8] is used to geneate the fine-goup coss sections in these examples. Fine-goup tanspot solutions, which ae geneated using a discete-odinates code witten fo the pupose of testing the new method, ae then used as the weighting function fo the geneation of expansion moments. 6

14 3.1 Deivation Within a eacto of abitay geomety, the balance of neutons at position with lethagy u and moving in a diection Ω ˆ ( θ, ϕ) is descibed by the tanspot equation in its intego-diffeential fom (Eq. (1)). ˆ ˆ (,, ) (, ) (, ˆ, ) ' ˆ ' (, ˆ ' ˆ Ω ψ Ω u + σ u ψ Ω u = du dω σ Ω Ω, u ' u) ψ (, Ω ˆ, u ') 4π χ( u) ' ˆ + du dω' νσ (, ') (, ˆ f u ψ Ω, u ') k 4π s (1) whee Ψ(, u, Ωˆ ) is the angula flux and σ (, u) epesents the total macoscopic eaction coss section at position fo neutons with lethagy u. The function σ (, ˆ ' ˆ s Ω Ω, u' u) is the macoscopic scatteing coss section at position with incoming lethagy u and angle ˆ Ω and outgoing lethagy u and angle ˆΩ. The system multiplication constant is epesented by k and νσ (, u ') and χ ( u) ae the fission neuton poduction coss section and fission spectum, espectively. The lethagy integals on the ight hand side of Eq. (1) can be boken up into smalle egions epesenting the enegy intevals of the few-goup stuctue chosen. (The numbe of few-goups G is abitay.) By applying the segmentation to the lethagy integal tems in Eq. (1) and beaking the spectum into G (coase) goups, Eq. (1) becomes f G 1 s g = ug 4π Ωˆ ψ (, Ω ˆ, u) + σ (, u) ψ (, Ω ˆ, u) = du ' dωˆ ' σ (, Ωˆ ' Ωˆ, u ' u) ψ (, Ω ˆ, u ') G 1 χ( u) ' ˆ + du dω' νσ (, ') (, ˆ f u ψ Ω, u ') k g = ug 4π (2) 7

15 whee ug is the lethagy inteval of coase-goup g. The infinite-lethagy bound on the integal in Eq. (1) has also been chosen to be some value lage enough to admit few neutons beyond it. In the example poblems pesented late, the uppe lethagy limit is chosen to be 26.22, coesponding to an enegy of.1 ev. This value was chosen to coespond to the enegy bounds of the 47-goup coss sections obtained fom the lattice depletion code HELIOS, and could be changed to fit othe applications, such as adiation detection o shielding poblems. As has been peviously mentioned, the angula dependence of the scatteing kenel can be teated in its most geneal fom. Fo the deivations that follow, the angula dependence of the scatteing kenel is teated with an expansion in angle using Spheical Hamonics [6], and fission is teated as isotopic: ˆ Ωˆ ψ (, Ω ˆ, u) + σ (, u) ψ (, Ω ˆ, u) = du ' σ (, u ' u) φ (, u ') G 1 1 Ylm ( Ω) m sl l g = l = m= 1 4π ug + G 1 χ( u) 4π k g = ug du ' νσ (, u ') φ(, u ') f (3) whee φ(, u ') is the scala flux at position and lethagy u, and σ sl (, u ' u) and m φ l (, u ') epesent angula moments of the scatteing kenel and angula flux: 1 1 σ (, u ' u) = dµ σ (, u ' u, µ ) P( µ ) sl o s o l o 2 1 m (, ') ˆ ( ˆ φ u = dω Y Ω ) Ψ(, u ', Ωˆ ) l 4π lm (4) (5) wheey ( ˆ ) lm Ω ae the nomalized spheical hamonics, and µ ˆ ˆ o = Ω Ω is the cosine of the scatteing angle. The function P l ( µ ) epesents the l th ode Legende polynomials. 8

16 In ode to expand the angula flux in a paticula lethagy egion, we must fist ensue that the basis functions chosen ae othogonal ove it. Since this is to be done fo an abitay set of othogonal basis functions, the lethagy vaiable is changed to a scaled vaiable within each goup to align the inteval of that goup with the inteval of othogonality fo the basis set. Theefoe, a new vaiable is defined in each goup: u u i ug = u + u i u f u i (6) whee u i and u f ae the bounds of ug u f ui =, the lethagy inteval of the coasegoup, and ui and u f ae the bounds of u = u f ui, the inteval of othogonality of the basis functions. To peseve the neuton distibution unde this tansfomation, we enfoce the balance conditions (,, ˆ Ψ u Ω ) du = Ψ(, u, Ωˆ ) du g g and σ (, u) = σ (, u ) x x g (7) whee σ (, u ) epesents the coss sections of the system. This allows the ight hand x side (RHS) of Eq. (3) to be witten as: ( ˆ Y Ω) χ(, u) RHS = du (, u u) (, u ) + du (, u ) (, u ) (8) G 1 1 G 1 lm m gσ sl g φl g gνσ f g φ g g l m 1 4π = = = g 4π k u = u As a consequence of the balance condition above, the following tansfomations esult: (,, ˆ ) (,, ˆ u u Ψ u Ω = Ψ uh Ω ), χ(, u) = χ(, uh ), u u h u σ sl (, u g u) = σ sl (, u g uh) u h h (9) With the assumption that the tanspot equation is valid fo all values of the lethagy u, Eq. (3) can be split into G coupled equations, each descibing the neuton balance within its own goup h, with the lethagy vaiable scaled using the above tansfomations. 9

17 Theefoe, neuton balance within a coase-goup h, with lethagy u h, position, and diection ˆΩ is given by Eq. (1). ( ˆ ˆ ˆ ˆ Y Ω) Ω ψ (, Ω, ) + σ (, ) ψ (, Ω, ) = σ (, ) φ (, ) G 1 1 lm m uh uh uh du g sl u g uh l u g g= l= m= 1 4π u + G 1 χ(, uh) 4π k g= u du νσ (, u ) φ(, u ) g f g g (1) Othogonal Expansion Assume a set of othogonal functions of lethagy within coase-goup h: ξ ( u ), which obey the othogonality condition on u : δij duhw( uh) ξi( uh) ξ j ( uh) = (11) α u whee w( u h) is a weighting function, δ ij is the Konecke Delta, and α j is a nomalization constant detemined by the choice of ξ ( u ). Any function f ( u ) on u can be then witten accoding to the expansion: i j h h i h h = i i i h whee fi duh f ( uh) w( uh) ξi ( uh) i= u f ( u ) α f ξ ( u ) =. (12) Multiplying Eq. (1) by w( u h) ξi ( uh) epesent the lethagy bounds of goup h, one obtains: and integating ove the othogonality limits, which ˆ ˆ (, ) (, ˆ Ω Ψ Ω + σ Ω) Ψ (, Ωˆ ) ih ih ih Y ( Ωˆ ) = ( ) ( ) (, ) (, ) G 1 1 lm m duhw uh ξi uh dugσ sl ug uh φl ug g l m 1 4π = = = u u + G 1 χih( ) 4π k g= u du νσ (, u ) φ(, u ) g f g g (13) whee the following enegy (lethagy) moments have been intoduced: 1

18 (, ˆ Ψ Ω ) = du Ψ(, Ωˆ, u ) w( u ) ξ ( u ) ih h h h i h u σ (, Ω ˆ ) = ih u du σ (, u ) Ψ(, Ωˆ, u ) w( u ) ξ ( u ) u h h h h i h du Ψ(, Ωˆ, u ) w( u ) ξ ( u ) h h h i h (, ˆ χ Ω ) = du χ(, Ωˆ, u ) w( u ) ξ ( u ) ih h h h i h u (14) (15) (16) The moments σ (, Ω ˆ ) and χ (, Ω ˆ ) ih ih ae computed numeically, with σ (, Ω ˆ ) ih weighted with the flux distibution obtained in a fine-goup calculation (e.g., fo a single assembly). The collision tem is modified to expand not the total eaction ate, as in Eq. (15), but athe to expand the deviation of the total collision eaction ate fom the mean within each goup, based on standad petubation techniques. Thus, the total coss section within goup h is ewitten as: ˆ (,, ) (, ˆ σ u Ω = σ Ω ) + δ (, u, Ω ˆ ) h whee δ (, u, Ωˆ ) is the petubation of the coss section fom the spectal mean, and (17) σ (, Ω ˆ ) h is the standad fom of the flux-weighted coss-section in coase goup h, as defined in Eq. (18). σ (, Ω ˆ ) = h u du σ (, u ) Ψ(, Ωˆ, u ) u h h h (18) du Ψ(, Ωˆ, u ) h h Thus, when multiplying Eq. (1) by w( u h) ξ i( uh) and integating, as done befoe, the collision tem changes, and Eq. (13) becomes: 11

19 ˆ ˆ ˆ (, ) (, ) (, ˆ ) (, ˆ Ω Ψ Ω + σ Ω Ψ Ω + δ Ω) Ψ (, Ωˆ ) ih h ih ih h Y ( Ωˆ ) = ( ) ( ) (, ) (, ) G 1 1 lm m duhw uh ξi uh dugσ sl ug uh φl ug g l m 1 4π = = = u u + G 1 g = χih( ) 4π k u du νσ (, u ) φ(, u ) g f g g (19) whee the moment of the total coss-section petubation is defined as δ (, Ω ˆ ) = ih u du δ (, u ) Ψ(, Ωˆ, u ) w( u ) ξ ( u ) u h h h h i h du Ψ(, Ωˆ, u ) w( u ) ξ ( u ) h h h h (2) In this manne, the only moment in the denominato of the petubation is the thode flux moment, which is typically the lagest, and theefoe least likely to be too small. This technique geatly educes the likelihood of numeical issues due to dividing by nea-zeo flux moments. Teating the enegy dependence of the ight hand side of Eq. (13) is complicated by the desie to ensue that the total neuton eaction ates on the ight hand side ae peseved fo expansions of abitay ode. To ensue that this is the case, athe than condensing the coss sections diectly, the eaction ate enegy density is expanded in othogonal functions. Let R (, u ) = νσ (, u ) φ(, u ) f g f g g (21) m m Rs, l (, u g uh) = σ sl (, u g uh) φl (, u g ). (22) The ight hand side (RHS) of Eq. (13) can then be witten as: ( ˆ Y Ω) χ ( ) RHS = du w( u ) ( u ) du R (, u u ) + du R (, u ) (23) G 1 1 G 1 lm m ih h h ξi h g sl g h g f g g l m 1 4π = = = g 4π k u u = u This leads us to an expansion of the incoming enegy dependence of the eaction ate densities, eithe fission o scatteing, in the chosen othogonal basis: 12

20 R (, u ) = α R ( ) ξ ( u ) g j jg j g j = (24) whee R ( ) = du R(, u ) w( u ) ξ ( u ). (25) jg g g g j g u This esults in the following fom fo the RHS of Eq. (13): α RHS = Y ( Ω) du w( u ) ( u ) R (, u ) du ( u ) G 1 1 j ˆ m lm h h ξi h sljg h gξ j g g l m 1 j 4π = = = = u u χ ( ) + R ( ) ( ) G 1 ih α j fjg dugξ j ug g j 4π k = = u (26) The fom of Eq. (26) is then modified by defining the moments of the fission poduction coss section and the scatteing kenel in the following manne: νσ fjg R ( ) = ( ) du ξ ( u ) fjg g j g u φ ( ) jg σ R (, u ) = (, u ) du ξ ( u ) m sljg h g j g m u sljg h m ljg (27) φ ( ) Substitute Eqs. (26) in Eq. (27) and then the esulting equation in Eq. (13) to get the geneal condensed fom of the tanspot equation given in Eq. (28). ˆ ˆ ˆ ˆ ˆ ˆ α Ω Ψ (, Ω ) + (, Ω) Ψ (, Ω ) + (, Ω) Ψ (, Ω ) = ( Ωˆ ) ( ) ( ) G 1 1 j m m ih σ h ih δih h Ylm φljg σ slijg h g = l = m= 1 j = 4π G 1 χih( ) + α j νσ fjg ( ) φ jg ( ), h =,1, K G, i =,1, K 4π k g = j = (28) whee G is the numbe of coase-goups the spectum has been divided into, and i epesents the expansion ode of the moment this equation is used to calculate. At this point, no appoximations have been made, and Eq. (28) fully descibes the enegy dependence of the system. By tuncating the expansion afte I tems, a solution can be found with accuacy detemined only by the ode of the appoximation chosen. 13

21 Though this method, the enegy dependence of the tanspot equation has been completely eliminated by folding it into the moments of the coss sections and fission distibution, and then allowing fo the solution of each moment individually. This amounts to a seies of ( I + 1) G coupled equations which can be numeically solved fo the flux moments, which can then be used to constuct an appoximation of the angula flux by the Eq. (29). I Ψ ( u ) = α Ψ ξ ( u ) (29) h i i i h i= Legende Expansion Equation (28) was deived fo an abitay othogonal basis; howeve, fo the emainde of this pape, shifted Legende Polynomials have been chosen as the expansion basis [6]. This has seveal benefits that seve to geatly simplify the condensed fom of the tanspot equation and the definitions of the coss section moments of the system. Fist, the weighting function, w( u g ), is equal to unity, which simplifies all the moment definitions in the deivation. In addition, the definition of the coss section moments ae simplified by the popety of the Legende polynomials, as well as most standad othogonal polynomials, 1 du ξ ( u ) = du P( u ) = δ (3) u g i g g i g i This elation, when applied in Eq. (27), causes the fission and scatteing coss sections to vanish fo all expansion odes except the th -ode, which seves to uncouple the equations in Eq. (28) such that all odes ae coupled to the th -ode, but not to any othes. This simultaneously inceases the efficiency of the solution method and emoves 14

22 the dependence of the eigenvalue and conveged eaction ates on the ode of the expansion. The Legende application of the Genealized Enegy Condensation Theoy can then be seen as an unfolding of the enegy spectum, as all the integal popeties ae encompassed in the th -ode calculation, and the detailed shape is ecoveed fom that solution by highe ode computations. The conveged eigenvalue will then be the same as the eigenvalue computed with the th -ode appoximation in Eq. (31). ( ˆ ˆ ˆ ˆ ˆ Y Ω) χ ( ) Ω Ψ (, Ω ) + (, Ω) Ψ (, Ω ) = ( ) ( ) + ( ) ( ) (31) G 1 1 G 1 lm m m h h σ h h φgl σ slg h νσ fg φg g = l = m= 1 4π g = 4π k Examining this appoximation, as well as the definitions of the coss sections, it is appaent that the th -ode is nothing moe than the standad few-goup condensation, as was desied. The petubation tem is suppessed because definition of the coss-section in Eq. (17) leads to the definition of the th ode petubation tem in Eq. (32). δ (, Ω ˆ ) = h u ˆ ( (,, ) (, ˆ du σ u Ω σ Ω)) Ψ(, Ωˆ, u ) w( u ) ξ ( u ) h h h h h u du Ψ(, Ωˆ, u ) w( u ) ξ ( u ) h h h h (32) This tem is clealy equal to zeo fo any coase goup h. Unde the Legende application, then, we have the following equations to solve: ˆ ˆ ˆ ˆ (, ) ˆ Ω Ψ ˆ ih Ω + σ h(, Ω) Ψih(, Ω ) + δih(, Ω) Ψh(, Ω) Y ( ˆ Ω) χ ( ) = ( ) ( ) + ( ) ( ) G 1 1 G 1 lm m m ih φgl σ slig h νσ fg φg g = l = m= 1 4π g = 4π k (33) h =,1, KG i =,1, KI whee the j tems have been suppessed, since all non-zeo moments ae eliminated by the Legende application, and 15

23 1 (, ˆ Ψ Ω ) = du Ψ(, Ωˆ, u ) P( u ) ih h h i h 1 (, ˆ χ Ω ) = du χ(, Ωˆ, u ) P( u ) ih h h i h (34) σ 1 du σ (, u ) Ψ(, Ωˆ, u ) h h h ˆ h(, Ω ) = 1 du Ψ(, Ωˆ, u ) h h νσ ( ) = fg 1 du νσ (, u ) φ(, u ) g f g g 1 du φ(, u ) g g (35) δ (, Ω ˆ ) = ih 1 du ( σ (, u ) σ (, u )) Ψ(, Ωˆ, u ) P( u ) h h h h i h 1 du Ψ(, Ωˆ, u ) P( u ) h h i h (36) σ m slig h ( ) = 1 1 m duhpi uh du gσ sl u g uh φl u g 1 m du gφl (, u g ) ( ) (, ) (, ) (37) With the elimination of the j th moments in the desciption of the souce tem on the ight side of Eq. (28), the geneal enegy condensation method can be viewed as solving fo the enegy spectum of neutons enteing goup h. The j th moment epesents the neuton spectum of the neutons leaving vaious othe goups to ente goup h, which is not paticulaly impotant, as long as we know the spectum they ente goup h with, and this is accounted fo in the i th moments of the scatteing kenel and fission spectum distibution. This demonstates the value of the decoupling. Because only the total (enegy integated) eaction ate is impotant in the souce tems, the detailed spectum of the flux is not needed to detemine the spectum of the souce tem. This allows the poblem to be solved only fo the th ode, and the est of the moments to be geneated fom the th ode solution. Fo shifted Legende polynomials, the nomalization constant of the othogonality condition (Eq. (11)) takes the value: 16

24 and the expansion of the angula flux becomes: α = (2i + 1) (38) i I Ψ ( x, u ) = (2i + 1) Ψ ( x) P( u ) (39) h i i h i = 3.2 Application in 1-D Discete Odinates As an initial veification of the method and stating point fo futhe development in moe obust applications, the genealized condensation pocedue is applied in a onedimensional discete odinates fomulation. Within slab geomety, Eq. (33) can be ewitten as: Ωˆ Ψ ( x, Ω ˆ ) + σ ( x, Ωˆ ) Ψ ( x, Ω ˆ ) + δ ( x, Ωˆ ) Ψ ( x, Ωˆ ) ih h ih ih h (2l + 1) P ( µ ) χ ( x) = ( x) ( x) + ( x) ( x) G 1 G 1 l ih φgl σ slig h νσ fg φg g = l = 4π g = 4π k (4) Hee, the m subscipt in the scatteing tem has been suppessed as it is equal to zeo fo slab geomety. Eq. (4) is assumed to be valid fo N distinct values of the diection cosine µ, as in the standad discete odinates fomulation of the tanspot equation. The scala flux and any othe angulaly integated values (such as angula cuent) ae eplaced with a Gauss-Legende quadatue fomulation [6]. The one-dimensional, discete odinates tanspot equations, with genealized lethagy collapse, ae thus: G 1 (2l + 1) n Ψ ih n + h n Ψ ih n + ih n Ψ h n = l n gl slig h x g= l = 2 µ ( x, µ ) σ ( x, µ ) ( x, µ ) δ ( x, µ ) ( x, µ ) P( µ ) φ ( x) σ ( x) + G 1 g = χih( x) νσ fg ( x) φg ( x) 2k (41) h =,1, KG i =,1, KI n = 1,2, KN whee the moments have been tuncated at the I th ode. 17

25 In the example poblems, fo the sake of simplicity, we make the usual appoximation of neglecting the angula dependence of the enegy moment of the total coss section (both the standad and petubation tem). 18

26 CHAPTER 4 EXAMPLE PROBLEMS As an example of the application of the genealized enegy condensation theoy to actual poblems, seveal 1-D eacto poblems typical of boiling wate eacto (BWR) coe configuations ae chosen, each composed of seven fuel assemblies. The coes epesent a vaiety of situations, including both supe-citical and sub-citical systems with vaying amounts of highly absobing mateial (gadolinium mixed in the fuel). Each fuel assembly (see Figue 1) is based on a simplified fesh GE9 assembly design containing fou egions of fuel and wate mixtue, each cm thick, suounded by cm of wate. In assembly 1, the two inteio egions have diffeent enichment than the oute fuel egions. The enichment in assembly 2 is unifom. Gadolinium is added to the two inne most egions of assembly 3 while all of the fuel egions in assembly fou contain gadolinium. Appendix A contains the mateial definitions and densities fo each mateial pesent in the system. See Figue 1 fo coe and assembly geometies. 19

27 Coe 1 Coe 2 Coe 3 Assembly 1 Assembly 2 Assembly 3 Assembly 4 Wate Fuel I Fuel II Fuel + Gd Figue 1: Sample Poblem Stuctue The standad discete odinates method using S 16 appoximation is used to calculate the individual assembly and the whole coe efeence solutions in 47 goups. The fine-goup coss sections at the hot opeating condition fo each assembly wee geneated using the diect collision pobability method in HELIOS with its 47-goup poduction libay. The single assembly 47 goup solutions wee used to geneate the Legende moments. The bounday conditions fo the single assembly and the coe calculations wee specula eflective and vacuum, espectively. Table 1 shows the eigenvalue of each assembly and coe, obtained though a full 47 goup tanspot calculation, using a one-dimensional discete odinates code. Table 1: Foty-Seven Goup Eigenvalues Stuctue k Assembly Assembly Assembly Assembly Coe

28 Coe Coe Full solutions ae obtained fo each egion of each assembly; howeve, fo bevity, we pick seveal egions which ae epesentative of the mateials pesent in the system. The selected egions ae pesented in Table 2. Table 2: Selected Regions, Assemblies Assembly Region Mateial 1 1 Wate 2 3 Fuel (Low Enichment) 3 3 Fuel (High Enichment) All solutions ae egion-aveages (ove the mateial/laye), nomalized to the total numbe of neutons within the egion, and integated ove all enegies. Figue 2 contains the fine-goup efeence solution fo each egion in table 2. 21

29 Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 Enegy (ev) 1E+5 1E+6 1E+7 1E+8. (a) (b).2 Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (c) Figue 2: Fine-goup Specta (fo (a) Assembly 1, Region 1 (wate), (b) Assembly 2, Region 3 (fuel I), and (c) Assembly 3, Region 3 (fuel II)) 4.1 Single Assembly Veification Fo initial veification of the geneal condensation method, the 47 goup flux fom a fine-goup tanspot solution pefomed on a single assembly is used as the weighting function duing the condensation pocedue. The 47 goup mateial coss sections ae condensed down to two goup expansion moments of the coss sections. These ae used in Eq. (41), which is used to povide an appoximate solution (flux moments) fo that assembly. The spectum poduced fom these flux moments should, fo high ode, epoduce the fine goup efeence spectum vey accuately, since the exact solution is used as the weighting function. 22

30 Fo all calculations pefomed, in ode to maintain consistency with HELIOS, tanspot-coected coss sections ae used. Also, as discussed ealie, the angula dependence of the total coss section is emoved by weighting the fine-goup tanspot coss section with the scala flux as opposed to the angula flux. In addition, fo the example poblems, the angula dependence of the scatteing kenel is teated as linealy isotopic by applying the tanspot coection as descibed below (Eq. 42). In this case Eq. (41) takes the following fom. 1 χ ( x) µ Ψ ( x, µ ) + σ ( x, µ ) Ψ ( x, µ ) + δ ( x, µ ) Ψ ( x, µ ) = φ ( x) % σ ( x) + νσ ( x) φ ( x) (42) x G 1 G 1 t ih ih h n ih n ih n h n g sig h fg g g= 2 g= 2k t whee σ ( x ) is the moment of the tanspot-coected coss section, ih t σ ( x) = h 1 du σ ( x, u ) Φ( x, u ) P ( u ) t h h h h 1 du Φ( x, u ) P ( u ) h h h and % σ ( ) is the enegy moment of the scatteing kenel, in which the within goup x sig h elements have been eplaced using: G 1 iso t iso sg g = sg g ag + g sg h h= % σ ( x) σ ( x) σ ( x) σ ( x) σ ( x) (43) t whee σ ( x) is the standad multigoup tanspot coss section. Moments of the g tanspot-coected scatteing coss section ae then condensed fom the coss sections computed in Eq. (43). The petubation moment is computed in the same manne as befoe, using the tanspot coss-section instead of the total coss section. t (, ) (,, ˆ t δ u = σ u Ω) σ (, Ωˆ ) h (44) 23

31 The two-goup bounday used in these examples is.625 ev. Standad two-goup appoximations ( th ode) fo the egions in table 2 ae pesented in Figue 3, ovelaid on the 47 goup efeence solution fom Figue 2. Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (a) Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (b).2 Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (c) Figue 3: Standad 2-Goup Collapsed Specta (fo (a) Assembly 1, Region 1 (wate), (b) Assembly 2, Region 3 (fuel I), and (c) Assembly 3, Region 3 (fuel II)) Figue 3 clealy demonstates the infomation loss in the condensation fom many/fine-goups to a few/coase-goups. This loss of infomation is quantified in the eo analysis section of this pape. Next, to demonstate the genealized condensation method, the same systems ae solved using Eq. (42) fo vaious odes of expansion i. As discussed peviously, the eigenvalues of the expansion calculation ae identical to the two-goup, as a esult of the popeties of the Legende polynomial set. In addition, total 24

32 fission densities and othe integal popeties ae also completely equivalent to two goup values, since pesevation of the neuton eaction ates in the souce distibution is stictly enfoced duing the condensation pocedue. Due to this, if one is only inteested in enegy integated quantities (such as the total fission eaction ate), than this method, in its cuent fom, does not povide an impovement; howeve, Two applications ae immediately appaent: shielding and detection [11,12] and eacto physics, in which one would be inteested in impoving the esults (eigenvalue, powe distibutions, etc.) by econdensation of the coss sections iteatively within the whole coe calculation. Though the eigenvalue may not impove duing this initial step, the highe the ode of the expansion, the moe accuate the spectum that will be poduced, as is evident in Figue 4, which shows the pogession fom a fist ode appoximation to a thid and fifth ode appoximation in the fast and themal egions of Assembly 1, Region 1 (wate), ovelaid with the 47 goup solution. Fom Figue 4, it is evident that even at low ode we have specta that ae much close to the many-goup solution than the standad two-goup solution. Also appaent, paticulaly when the flux is vey nea zeo, is the issue of negative flux in the polynomial appoximation. This is an inheent esult of appoximating a highly vaying function with a tuncated expansion, paticulaly when tuncating at low ode. This does not, howeve, impact the integal quantities, which ae maintained in the th tem of the expansion. Pesently, it is sufficient that the distibution is appoaching the actual flux spectum, with much geate detail as one goes to highe and highe ode, and that negative flux values become negligible at high enough ode. Figue 5 shows a twentieth ode expansion appoximation fo the selected egions. 25

33 Themal Region Fast Region Relative Flux E-4 1E-3 1E-2 1E-1 1E+ Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) Enegy (ev) (a) Relative Flux Relative Flux E-4 1E-3 1E-2 1E-1 1E+ 1E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) Enegy (ev) (b) Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (c) Figue 4: 1 st (a), 3 d (b), and 5 th (c) ode appoximations in Assembly 1, Region 1 (wate). 26

34 Themal Region Fast Region Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (a) Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (b) Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (c) Figue 5: Twentieth Ode Appoximation fo (a) Assembly 1, Region 1 (wate), (b) Assembly 2, Region 3 (fuel I), and (c) Assembly 3, Region 3 (fuel II). 27

35 It has thus been shown that fo a single assembly, using fine-goup coss sections and a fine-goup solution to condense the coss sections into expansion moments, a lage amount of spectal infomation can be obtained, in contast with standad few-goup condensation methods. The solution was compaed ove the entie assembly, and shown to have the same accuacy as in the selected egions. 4.2 Whole Coe Veification Fo whole coe veification, the pocess is as peviously discussed. Single assembly, 47 goup, tanspot calculations ae pefomed fo each unique assembly in the eacto coe. The single-assembly solutions ae then used to weight the fine-goup coss sections fo that assembly, geneating egion-specific, two-goup, coss section moments. These moments ae then used in Eq. (41) to compute two-goup, coe-level flux moments, which ae used to poduce the coe-level enegy spectum of the neuton flux. As in the single assembly veification, compaisons ae pefomed in a few epesentative egions. We pesent the esults fo Coe 3 since this is the most heteogeneous and theefoe most challenging geomety. The selected egions ae shown in Table 3. Figue 6 contains the 47 goup whole coe efeence solution, the two-goup th -ode appoximation fo the selected coe egions, as well as the twentieth ode appoximate solutions. Table 3: Selected Regions, Whole Coe Coe Region Mateial 3 7 Wate 3 15 Fuel (High Enichment) 3 21 Fuel + Gd 28

36 Themal Region Fast Region Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (a) Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) (b) Relative Flux c 1E-4 1E-3 1E-2 1E-1 1E+ Relative Flux E-1 1E+ 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Enegy (ev) Enegy (ev) (c) Figue 6: Coe 3 Specta ((Fine, th ode, and 2 th ode) fo egions (a) 7 (wate), (b) 15 (eniched fuel), and (c) 21(fuel + gd)) 29

37 The genealized method has thus poduced, within each coase-goup h, a vey accuate appoximation to the enegy spectum of the neuton distibution thoughout the whole coe, without eve having to solve the whole coe using a fine goup tanspot solution method. Aveaging ove the entie coe also poduces figues simila to that of Figue 6, and has the same accuacy as the selected egions above. 3

38 CHAPTER 5 TIME COMPARISON AND ERROR ANALYSIS 5.1 Solution Time Compaison The goal of the development of a genealized enegy condensation theoy was to poduce accuate spectal infomation fo the neuton flux duing a few-goup computation, with significant speedup compaed to a many-goup tanspot calculation. As peviously discussed, fo cetain othogonal polynomial sets, namely those that ae defined such that duξi ( u) = δi (45) u the computation time can be educed even futhe. The selection of Legende polynomials (and most othe standad othogonal polynomials), fo instance, decouples Eq. (41) in spectal expansion ode. This leaves I few-goup equations, each coupled though the souce tem to the th -ode solution, as discussed in the Legende Expansion section. One of the advantages of this decoupling is that due to the discete natue of the fine-goup stuctue, the computation time fo the coss section moments is dependent almost solely on the goup stuctue chosen. Since each coss section is constant ove a fine goup, the only computationally expensive opeation in condensation is computing the integals of the basis function ove the fine goups. These integals depend only on the goup stuctue (and not on the geomety o mateial composition), and theefoe ae calculated only once. 31

39 In fact, if one has aleady solved the poblem with the standad two-goup method, meaning they would aleady have fine-goup solutions fo the individual assemblies as well as the coase-goup solution fo the whole coe, they could poduce moments of abitay ode in negligible time, as long as they have computed the integals of the Legende polynomials ove the fine-goup enegy intevals fo that ode. This is the geatest advantage of using basis functions that satisfy Eq. (45). In ode to compae the solution times fo a geneal case (fully coupled), athe than a simplified one (decoupled), the Legende moments of the flux ae solved as though the equations ae coupled. Fo fai compaison, the flux moments fo evey ode wee conveged to the same citeia. Fo all calculations pesented in this pape, the flux was conveged to within 1-4, and the eigenvalue was conveged to within 1-6. Solution Time efes to the time it takes to solve fo the flux moments. Pe-Computation is also equied to geneate the integals of the basis function (Legende Polynomials) ove the enegy ange of each fine goup. This takes appoximately 2 seconds fo each expansion ode, when going fom 47 goups to 2 goups. The pe-computation only needs to be done once, howeve, fo each specific goup stuctue, and fom that, the coss-sections fo any mateial specifications o geomety can be condensed fo abitay ode in negligible time (less than 3 seconds fo up to 2 th ode). The solution time, in seconds, fo convegence ae pesented in Table 4. These times wee computed by solving the equations as though they wee coupled, as would be necessay fo an abitay set of othogonal functions. Single assembly values ae aveaged ove all fou of the assembly types. Coe values ae aveaged ove all thee coe types (see Figue 1). 32

40 Table 4: Computation Times (coupled) 47g Single Assembly g Full Coe g, th Ode Single Assembly.3 2g, th Ode Full Coe 2.1 2g, 4 th Ode Single Assembly 1.9 2g, 4 th Ode Full Coe g, 1 th Ode Single Assembly 6.3 2g, 1 th Ode Full Coe 69.2 Table 4 does not take advantage of the decoupling of spectal moments, and theefoe the times poduced ae chaacteistic of those that would appea fo any choice of basis function. Even without the decoupling, the impovement in computation time it is evident fom Table 4, which shows a significant speedup fo the 1 th ode. As will be seen late (figue 8), the 1 th ode full coe solution is off by less than 3% RMS. This is because pefoming a 1 th -ode calculation, without decoupling, equies the solution of 22 equations (11 expansion odes * 2 coase-goups), wheeas solving with the finegoup method equies the solution of 47 equations. The standad two-goup method gives one the ability to solve the system in appoximately % of the time it takes to solve the fine-goup, howeve, one loses detailed spectal infomation. The new method, in its coupled fom, howeve, poduces that infomation, and still only equies (fo 1 th -ode) about 3% of the computation time of the fine-goup calculation. Fo lage eactos o highly complex shielding poblems which can take hous o days to solve, this can educe the needed time by a significant amount. In this manne, the new method, even without decoupling, is faste than pefoming fine-goup, whole-coe tanspot calculations. The use of Legende Polynomials, as discussed ealie, decouples the spectal moments in the ight hand side of Eq. (41), which geatly speeds up the solution. When taking advantage of the decoupling of the spectal moments by solving the system in the 33

41 th -ode, and then unfolding the spectal dependence fom that solution, the computation times educe significantly, as seen in Table 5. Table 5: Solution Times fo Legende Polynomials (decoupled) 47g Single Assembly g Full Coe g, th Ode Single Assembly.25 2g, th Ode Full Coe 2.1 2g, 4 th Ode Single Assembly.33 2g, 4 th Ode Full Coe 2.2 2g, 1 th Ode Single Assembly.33 2g, 1 th Ode Full Coe 2.3 In this manne, the Legende genealized equations ae solved in a time compaable to that of the standad two-goup solution ( % of the fine-goup computation time), but with a lage amount of spectal infomation poduced. Each additional ode adds the computational time of a single oute iteation, which is negligible given the numbe of inne iteations pefomed in the coe calculation. 5.2 Eo Analysis We have shown that the genealized condensation theoy does an effective job of peseving the fine goup spectum duing a coase goup whole coe tanspot solution. It emains howeve, to quantify the impovement this allows ove standad two goup solutions. The total flux in each coase goup, which is the most impotant quantity, has been peseved by the method, but if one wants to unfold the spectum and detemine an appoximate whole coe flux fo a subinteval of the coase goup, such as to compae detecto esponse, thee is no eadily available technique fo use in neuton tanspot poblems. It is hee that the genealized condensation theoy becomes vey useful. To demonstate, a solution was obtained fo seveal systems using genealized expansion theoy. This solution was then integated ove the fine-goup limits, and the RMS eo fom a fine-goup tanspot solution was computed fo seveal odes of expansion. The fist system tested in this manne was a simple, homogeneous 1-D slab 34

42 composed of the eniched fuel mateial used in Assembly 2 in the pevious section, with specula eflective conditions on both sides. A 47-goup solution was obtained, and this solution was used to geneate 2-goup moments. The expansion equations (Eq. 4) wee then solved fo a seies of inceasing odes. Fom this, the econstucted 47-goup flux was computed by integating the flux ove the fine-goup limits. How close the47 goup flux is to the econstucted flux is a good measue of how effectively the high ode effects ae peseved. The RMS eo of these goup fluxes, as a function of expansion ode ae plotted in figue 7. As seen in figue 7, the flat flux ( th ode) has an RMS eo of 23.9%. It is also clea fom figue 7 that fo high expansion ode, the expanded flux is appoaching the fine-goup flux. Figue 7: Goup Flux Eo vs. Expansion Ode fo a Homogeneous Slab Figue 7 demonstates that the genealized condensation theoy is able to poduce the fine-goup spectum quite well. It does not, howeve, demonstate how this will affect whole-coe poblems when the assembly level flux is used to condense, as is the case in pactical application. 35

43 One of the most useful aspects of the genealized theoy is that it esults in a solution that is much close to the coe level efeence than to the assembly level flux that was used to weight the moments. To show how effective the genealized method is in poducing the coe-level spectum, the same computations wee done as in the homogenous slab, aveaged ove the entiety of coe 3. The expansion solution was computed fo a seies of odes, as in the slab poblem, and the esultant goup flux was compaed with the whole-coe fine-goup efeence solution. These esults ae pesented in figue 8. 1E+2 1E+1 1E E Nomalized RMS Eo (%) Expansion Ode Figue 8: Goup Flux Eo vs. Expansion Ode fo Coe 3. As is evident in figue 8, the expansion solution conveges to about 2% away fom the coe-level efeence solution (RMS). The expansion solution is, by compaison, 15.3% away fom the assembly-level, fine-goup tanspot solution. The expansion solution on the coe-level is theefoe significantly close to the actual coelevel flux, even at elatively low ode. This implies that an iteative condensation pocedue has meit (see the summay and futue wok section). 36

44 CHAPTER 6 CONCLUSIONS AND FUTURE WORK The standad multigoup theoy has been genealized in this pape. The new enegy condensation theoy geneates detailed (many-goup) spectal esolution within the few-goup stuctue without the computational expense associated with solving the many-goup tanspot equations. The genealization, which contains the standad multigoup theoy as a special ( th -ode) case, is achieved by using an enegy expansion of the angula flux in an abitay set of othogonal functions. This expansion leads to a set of equations fo the enegy moments of the flux with a coupling chaacteistic that diectly depends on the choice of the expansion functions. It was shown that the highe moment equations ae only coupled to the th -ode moment equation and not to each othe if Legende polynomials ae used as the expansion set. As a esult of this decoupling, the computational time associated with the solution of the highe ode moments become negligible as compaed to that fo the th -ode solution. This desiable featue makes the new theoy vey attactive fo application to eacto coe simulations since the few-goup calculations can now poduce the enegy esolution of the manygoup method with negligible computational penalty. The method was developed in the famewok of tanspot theoy. Howeve, its extension to diffusion theoy is staightfowad. The new theoy, which is completely independent of the tanspot solution method, was veified and tested in a few 1-D BWR benchmak poblems within the discete odinates appoximation. As expected, it was shown that the enegy esolution 37

45 within a two-goup stuctue is inceased with inceasing expansion ode close to that of the efeence (47) fine-goup spectum with computational expense competitive to that of the standad two-goup solution. One of the issues with condensation is that the data we have to input into the poblem is discete, and anytime a continuous expansion basis is being used to appoximate a discete set of points, numeical issues aise, such as Gibbs oscillations at enegy goup boundaies. These ae mostly small effects, but in cases whee it is desiable to emove such effects, one possible way to coect this issue is to implement a discete fom of Legende Polynomials [1]. The method pesented in this pape (with continuous Legende Polynomials) can aleady poduce the fine-goup spectum to a high degee of accuacy, but because they do not suffe these numeical effects, using discete Legende Polynomials could possibly poduce the same esults moe efficiently. Discete Legende polynomials might make it possible to poduce high accuacy solutions at lowe odes, and cut down on pe-computation time. We note that the genealized method is not limited to eacto eigenvalue poblems as applied in this pape. This in geneal is an enegy unfolding method and theefoe should find applications to detection and shielding poblems as futue wok. The most common use of spectal unfolding simila to this method is in detection, eithe fo neutons, photons, o electons. One poblem inheent in detection schemes is the inheent impefections of spectometes. Detecto esponses must be passed though an unfolding scheme to detemine actual incident neuton specta. This is due to the individual detecto mechanics causing the incidence of neutons at vaious enegies to bin themselves into a discete numbe of channels. The standad method (FERDoR 38