STOCHASTICALLY GENERATED MULTIGROUP DIFFUSION COEFFICIENTS

Transcription

1 STOCHASTICALLY GENERATED MULTIGROUP DIFFUSION COEFFICIENTS A Thesis Presented to The Academic Faculty by Justin M. Pounders In Partial Fulfillment of the Requirements for the Deree Master of Science in Nuclear Enineerin Georia Institute of Technoloy December 2006 COPYRIGHT 2006 BY JUSTIN POUNDERS

2 STOCHASTICALLY GENERATED MULTIGROUP DIFFUSION COEFFICIENTS Approved by: Dr. Farzad Rahnema, Advisor School of Mechanical Enineerin Georia Institute of Technoloy Dr. Weston Stacey School of Mechanical Enineerin Georia Institute of Technoloy Dr. Nolan Hertel School of Mechanical Enineerin Georia Institute of Technoloy Date Approved: 20 November 2006

3 ACKNOWLEDGEMENTS I would first like to thank my advisor, Dr. Rahnema, for his uidance, instruction and support over the past several years. His direction and insiht have been invaluable. Words can not express my appreciation for my wife, Sarah, for her continual love, support and encouraement in all areas of my life, raduate school bein only one of them. I am also blessed to have had the life-lon support of my parents, Randy and Lynda Pounders. The research for this thesis was performed under appointment to the Naval Nuclear Propulsion Fellowship Proram sponsored by the Naval Reactors Division of the United States Department of Enery. iii

4 TABLE OF CONTENTS Pae ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES SUMMARY iii vi vii viii CHAPTER INTRODUCTION 2 BACKGROUND 2 2. Multiroup Formulation Stochastic Cross Section Generation Transport Cross Section 8 3 DIFFUSION COEFFICIENTS 0 3. Classical Diffusion Theory Flux-limited Diffusion Theory An Exact P Coefficient A Hih Order Diffusion Coefficient 4 4 RESULTS 8 4. Cross Section Validation Fine-Mesh Diffusion 22 5 CONCLUSIONS 36 APPENDIX A: BWR PIN CELL BENCHMARK 37 APPENDIX B: ONE DIMENSIONAL SIMPLIFIED BWR CORE 39 APPENDIX C: ERROR CALCULATIONS AND STATISTICS 42 iv

5 REFERENCES 38 v

6 LIST OF TABLES Table : Analytic vs. Stochastic Cross Sections 9 Table 2: Analytic vs. Stochastic P0 Scatterin Matrices 9 Table 3: MCNP vs. HELIOS Cross Section Differences 20 Table 4: MCNP vs. HELIOS P0 Scatterin Matrix Differences 2 Table 5: Assembly Eienvalue Differences (2G Diffusion v. 2G Transport) 23 Table 5: Assembly Fission Density Error Statistics (2G Diffusion v. 2G Transport) 24 Table 7: Core Eienvalue Differences (2G Diffusion v. 2G Transport) 25 Table 8: Core Fission Density Error Statistics (2G Diffusion v. 2G Transport) 25 Table 9: Eienvalue Differences for Halved Mesh (2G Diffusion v. 2G Transport) 3 Table 0: Fission Density Error Statistics for Halved Mesh (2G Diffusion v. 2G Transport) 3 Table : Eienvalue Differences with Infinite-Medium Diffusion Coefficients (2G Diffusion v. 2G Transport) 33 Table 2: Fission Density Error Statistics with Infinite-Medium Diffusion Coefficients (2G Diffusion v. 2G Transport) 33 Table 3: Eienvalue Differences (2G Diffusion v. 47G Transport) 35 Table 4: Fission Density Error Statistics (2G Diffusion v. 47G Transport) 35 Table 5: Pin Cell Dimensions 38 Table 6: Core Dimensions 40 Pae vi

7 LIST OF FIGURES Pae Fiure : HELIOS vs. MCNP Relative Fine-Group Flux Errors 2 Fiure 2: Fission Density Error Distributions for Confiuration 27 Fiure 3: Fission Density Error Distributions for Confiuration 3 27 Fiure 4: Fast Flux Error Distribution for Core 3 29 Fiure 5: Fast Flux Error Distribution for Core 3 29 Fiure 6: Pin Cell Geometry with Cross Section Meshin 38 Fiure 7: Core Composition and Geometry 48 vii

8 SUMMARY The eneration of multiroup neutron cross sections is usually the first step in the solution of reactor physics problems. This typically includes eneratin condensed cross section sets, collapsin the scatterin kernel, and within the context of diffusion theory, computin diffusion coefficients that capture transport effects as accurately possible. Althouh the calculation of multiroup parameters has historically been done via deterministic methods, it is natural to think of usin the Monte Carlo method due to its eometric flexibility and robust computational capabilities such as continuous enery transport. For this reason, a stochastic cross section eneration method has been implemented in the Mont Carlo code MCNP5 (Brown et al, 2003) that is capable of computin macroscopic material cross sections (includin anular expansions of the scatterin kernel) for transport or diffusion applications. This methodoloy includes the capability of tallyin arbitrary-order Leendre expansions of the scatterin kernel. Furthermore, several approximations of the diffusion coefficient have been developed and implemented. The accuracy of these stochastic diffusion coefficients within the multiroup framework is investiated by examinin a series of simple reactor problems. viii

9 CHAPTER INTRODUCTION The ultimate oal of steady-state reactor physics calculations is to determine the distribution of neutrons in space, anle and enery within a reactor so that key interal quantities such as the power distribution and the neutron multiplication rate can be evaluated. Since solvin the Boltzmann transport equation continuously in all variables is computationally prohibitive, certain simplifyin assumptions must be made. The most common simplification of the transport equation is the multiroup method, where the enery domain is divided into a finite number of contiuous subintervals. A reat deal of care must be iven to collapsin the continuous enery transport equation, however, to ensure that the multiroup equations are enerated in a consistent manner capable of reproducin the true solution with as little approximation as possible. The calculation of multiroup parameters is key to the solution of reactor physics problems. The treatment of the anular dependence of the transport equation, on the other hand, is usually what differentiates one transport approximation from another, ivin rise to diffusion theory, the P N method, the discrete ordinates method, etc. Of all these approximations, diffusion theory is unique in that it does not explicitly treat the anular dependence. Instead, the anular dependence of the problem is wrapped into a sinle parameter the diffusion coefficient. This feature makes the transport problem much easier to solve, but often sacrifices accuracy in the process. The calculation of diffusion coefficients, however, is not as clearly defined as other multiroup parameters. It is usually the case that diffusion coefficients are born out of approximations whose validity larely depends on the type of problem at hand.

10 Due to constraints in computin power, the calculations of these multiroup parameters (and reactor physics problems in eneral) have normally been carried out via deterministic methods. As computer technoloy has advanced however, stochastic methods have been employed more frequently to enerate hih-accuracy solutions to transport problems. The eometric flexibility and lack of approximation associated with the Monte Carlo method makes it very suitable for the eneration of hihly accurate multiroup cross sections and hih-order diffusion coefficients, althouh it has not historically been used for such applications. This thesis will therefore analyze several definitions of the diffusion coefficient within the framework of stochastic cross section eneration. The oal is to compute diffusion coefficients that maximize the accuracy of traditional fine mesh diffusion methods. The next chapter reviews a methodoloy for computin eneral multiroup parameters within Monte Carlo methods. Subsequently several diffusion coefficients are derived and their implementation in a Monte Carlo settin discussed. Finally, some numerical results are shown and some final conclusions drawn. 2

11 CHAPTER 2 BACKGROUND The oal of many reactor physics problem is solvin the time-independent neutron transport equation which can be written Ω ˆ ψ(, r E, Ω ˆ) + σ(, r E) ψ(, r E, Ω ˆ) =.() de ' dωˆ ' (, ', ˆ σs r E E μ0) + χ( E) ν( E ') σ f ( r, E ') ψ( r, E ', Ω') 4π where ψ is the anular flux density, σ is the macroscopic neutron cross section, and χ and ν are the fission spectrum and yield, respectively, and μ 0 is the scatterin anle between Ωˆ and Ω ˆ '. The phase space has been defined in terms of the neutron position, enery and direction of fliht. Out of convenience, it has been assumed that there is no external source. Deterministic transport methods usually treat the scatterin cross section of Equation () in terms of a Leendre expansion in the scatterin anle, μ ˆ ˆ 0 =Ω Ω ': 2l + σ (, r E' E, μ ) = σ (, r E' E) P( μ ) s 0 s, l l 0 l= 2 (2) where the scatterin moment has been defined σ l(, r E' E) = dμ σ (, r E' E, μ ) Pl ( μ ). (3) s, 0 s 0 0 Recallin the addition theorem for Leendre polynomials in terms of spherical harmonics, l ˆ * ( ˆ l μ0) = lm( Ω) ( Ω') lm m= l P Y Y, and pluin this expansion into Equation () results in the followin expression: 3

12 Ω ˆ ψ(, r E, Ω ˆ) + σ(, r E) ψ(, r E, Ω ˆ) = l 2l + de ' ˆ σs, l( r, E ' E) ψlm( r, E ') Ylm( Ω ) +. (4) l= 0 2 m= l χ( E) de' ν( E') σ (, ') (, ', ˆ f E ψ E Ω') r r where the anular flux moment, ψ l m, has been implicitly defined throuh the interation over anle, Ω ˆ ', with spherical harmonics. 2.. Multiroup Formulation The continuous enery dependence of the transport equation is rarely treated explicitly. Instead it is common to interate the equation over contiuous enery ranes, ivin rise to the multiroup expression of the transport equation: Ω ˆ ψ (, r Ω ˆ) + σ () rψ (, r Ω ˆ) = 2l + () () ( ˆ (5) r r ) () r () r G l ' σs, l ψlm, ' Ylm Ω + χνσ f, ' ϕ' ' = l= 0 2 m= l where there is a total of G enery roups. The fission emission is assumed to be isotropic which enables the fission term to be written in terms of the scalar flux, ϕ(, r E) = ψ(, r E, Ωˆ ). 4π Comparin the terms of Equations (4) and (5) ives rise to the followin definitions of the multiroup constants: 4

13 E ψ (, r Ω ˆ) = de ' ψ(, r E ', Ωˆ) σ σ (, r Ω ˆ ) = νσ ' s, l f, () r = (, r Ω ˆ ) = E + E E + E E + E E + de ' σ( r, E ') ψ( r, E ', Ωˆ ) E E + de ' ψ ( r, E ', Ωˆ ) de ' ν( E ') σ ( r, E ') ϕ( r, E ') E E + f de ' ϕ( r, E ') E ' de de ' σ ( r, E ' E) ψ ( r, E ') Y ( Ωˆ ) E ' + E E + l s, l lm lm m= l l de ' ψ ( r, E ') Y ( Ωˆ ) m= l lm lm (6) where the fission yield and cross section have been treated as a sinle term. The total and scatterin cross sections now have an undesirable anular dependence that most transport methodoloies do not accommodate. It is common practice to weiht the total cross sections with the scalar flux instead. Doin so makes the assumption that the anular and eneretic dependencies of the flux can be separated within a roup, i.e. ψ ( r, E, Ω ˆ) = ϕ( r, E) f( r, Ωˆ), where ˆ ˆ dω' f( r, Ω ) =, thereby 4 causin the anular dependence in the numerator and denominator to cancel. Experience has shown that usin this assumption within reactor contexts to enerate multiroup constants leads to accurate results as lon as spatial homoenization is not used. The common definition of the total cross section, and the one which will be used presently, is therefore π σ () r E E+ = de ' σ( r, E ') ϕ( r, E '). (7) E de ' ϕ( r, E ') E+ Similarly, the condensed scatterin kernel will be defined as E E ' de de ' σ, (, ' ) (, ') ' E E s l r E E ϕ r E + ' + σ s, l () r. (8) E de ' ϕ( r, E ') = E+ 5

14 It is immediately evident from their definitions that the multiroup parameters can not be computed without knowlede of the continuous-enery neutron spectrum. Since this is usually not known a priori, most current cross section eneration methods either substitute a weihtin function that has approximately the same spectral shape as the neutron flux or use a fine-roup or continuous enery transport method to compute an accurate flux solution, often with approximate boundary conditions or simplified eometry. The former option is limited by the approximations used to enerate the weihtin function. Since the validity of these approximations can vary reatly dependin on their application, this option will not currently be considered any further. The second approach is limited by the computational constraints of the transport method used to collapse the cross sections. Most deterministic transport methods, for example, are limited to one or two spatial dimensions and are incapable of utilizin continuous enery cross section libraries. Stochastic methods such as Monte Carlo, however, do not have these limitations. It is therefore natural to think of usin such methods to enerate multiroup cross sections Stochastic Cross Section Generation A methodoloy for stochastically computin multiroup cross sections based on track-lenth estimates of neutron reaction rates was proposed by Redmond (990). This methodoloy reconizes that the multiroup cross sections for direct reactions (i.e. capture, fission, total, etc.) are simply the reaction rate density divided by the roup flux. The reaction rate density for reaction x can be computed by the Monte Carlo tally (Briesmeister, 2000) R = E wt E < E < E [ σ ( ) ], for +. (9) x, x i i i i NV n i 6

15 where E i, w i and T i are the enery, weiht and path lenth of track i of particle n; N is the total number of particles recorded and V is the volume of the cell associated with this tally. Likewise the scalar neutron flux tally has the form F = wt E < E < E [ ], for +. (0) x, i i i NV n i The collapsed cross section of type x is therefore Rx, σ x, =. () F The scatterin kernel requires slihtly more attention since it must be tallied as a function of incomin and outoin eneries. In this case, the numerator of Equation () must be replaced by a tally matrix, R x,, defined by E < E < E Rs, ' = s Ei Ei wt i i NV n i E < E ' < E x, + i [ σ ( ' ) ], for. (2) ' + i ' Computin the Leendre moments of the scatterin kernel requires only a sliht modification to Equation (2). Recallin Equations (3) and (8), the collapsed scatterin kernel (i.e. roup-to-roup scatterin matrix) can be written E E ' de de ' dμσ 0 (, ', 0) ( 0) (, ') ' ' s r E E μ P μ ϕ E E+ E + l r σ s, l () r. (3) E de ' ϕ( r, E ') = E+ All interations in the above expressions are performed implicitly, by virtue of the fact that the variables of interation are sampled and tallied over their entire domains throuh out the course of the simulation. It is therefore straihtforward to see that the numerator of Equation (3) can be calculated by only a sliht modification to the scatterin matrix: E + < Ei < E Rsl, ' = [ σs( Ei' Ei) Pl ( μi) wt i i], for. (4) NV n i E ' + < Ei' < E' It should be noted that since the precedin tallies are based on neutron track lenths, they are accrued (i.e. interated) over a volumetric sub-reion of the system, but 7

16 the analytic expressions have been written as point-wise continuous in space. To be consistent, the multiroup equation iven by Equation (5) should also be interated over the tally volume, V T, which yields σ E VT E+ = dv ' de ' σ( r, E ') ϕ( r, E ') VT E dv ' de ' ϕ( r, E ') E+ (5) where r VT. If V T contains multiple homoenous sub-reions, then Equation (5) is a spatially homoenized cross section. Even if Equation (5) is evaluated over a spatially homoenous reion, there still may be homoenization effects due to the joint variation of the flux in space and enery. Caution must therefore be exercised when collapsin cross sections to avoid unintentional homoenization. Redmond implemented the above methodoloy in MCNP version 4C (Briesmeister, 2000). Modifications to the code are necessary to support the matrix tallies required for the Leendre scatterin moments. Furthermore, a mechanism for samplin a scatterin anle and the associated chane in neutron enery at every tally point (whether or not the physical event occurred) was added to ensure that the cross sections were fully interated and had reasonable statistics. The present author applied this methodoloy to the latest release of MCNP version 5 (Brown et al, 2003) Transport Cross Section One issue not addressed by Redmond is how to stochastically compute an accurate transport cross section for diffusion applications. Ilas, G. and Rahnema (2003) presented a method for tallyin the transport cross section defined by σ = σ μ σ, (6) tr, 0, s, where μ 0, is the averae scatterin anle in roup. The scatterin anle was estimated by tallyin the product of the incident and emerent directional cosine vectors at each tally point in MCNP: 8

17 σ = {[ σ( E ) ( uu' + vv' + ww') σ ( E )] wt}, for E + < E < E (7) tr, i i i i i i NV n i where Ω= ˆ [ u v w] T is the direction vector. This transport cross section was developed and initially tested within the framework of spent fuel storae lattices, but has showed promise for reactor applications (Pounders et al 2005). It is believed, however, that the robustness and accuracy of the Monte Carlo method can be further exploited to yield a transport cross section capable of producin hihly accurate diffusion solutions. 9

18 CHAPTER 3 DIFFUSION COEFFICIENTS Several definitions of the diffusion coefficient will now be discussed, as well as their implementation in the Monte Carlo code MCNP5. MCNP5 was chosen since it is the latest release of one of the most popular Monte Carlo codes within the nuclear enineerin community, but the followin methods are applicable to any Monte Carlo implementation. 3.. Classical Diffusion Theory Classical diffusion theory is based on the P approximation to the transport equation. The time-independent multiroup formulation of these equations can be written (Bell and Glasstone, 979) G ' σ ϕ σs,0 ϕ χνσ f, ' ϕ' ' = J () r + () r () r = () r () r + () r () r (8) G ' s, ' ' = ϕ () r + 3 σ () r J () r = 3 σ () r J () r. (9) where Jr () = Ωˆψ (, r Ωˆ)d Ωˆ is the neutron current. The only approximation made in 4π these equations is that the anular flux is linearly anisotropic. Traditionally, Equation (9) has been recast in a form similar to Fick s law, but this requires the additional approximation that G G ' ' σs, ' = σs, ' = ' = () rj () r () rj () r (20) (Stamm ler and Abbate, 983). This equation is a form of detailed balance which is a ood approximation in the presence of heavy scatterin. In a purely scatterin medium, in fact, this equation is exact. Usin this expression in Equation (9) results in 0

19 G ' ϕ() r = 3 σ() r σs, () r J() r (2) ' = The term in brackets is typically defined as the transport cross section, σ, tr. This definition of the transport cross section will hereafter be referred to as the classical definition and denoted by a C superscript: G C ' tr, = s, ' = σ () r σ () r σ () r. (22) By lettin G ' s, σs, ' = σ l = l () r and definin the averae scatterin anle as μ = 0, μ0σs ( μ0) dμ0, Equation (22) can be cast into a more common form: σ ( μ ) dμ s 0 0 σ () r = σ () r μ σ (23) C tr, 0, s,0 Equation (2) is in the form of Fick s law, J = D ϕ, implyin the followin definition of the diffusion coefficient: D C = (24). 3σ C tr, The advantae of havin the first moment of the P equations in the form of Fick s law is, of course, that it can be substituted into the zero th moment equation to eliminate the neutron current, resultin in an equation in terms of the scalar flux only. The transport cross section and diffusion coefficient will therefore be treated as equivalent parameters related by D ( 3σ ) =. tr, It is the classical definition of the transport cross section that Ilas, G. and Rahnema estimated for spent fuel lattices. Given the tallyin capabilities described Section 2.2, however, it can be seen that a Monte Carlo estimate of the classical diffusion coefficient can also be obtained by simply tallyin the first Leendre moment of the scatterin kernel and subtractin this from the total cross section.

20 3.2. Flux-Limited Diffusion Theory Levermore and Pomranin (98) proposed a flux-limited diffusion theory in which the particle current can not exceed the scalar flux (this is not the case in classical diffusion theory.) Flux-limited diffusion theory has been successfully applied to problems in the fields of radiative transfer and radiation hydrodynamics (Olson et al, 2000; Turner and Stone, 200), but its relevance to reactor physics problems has yet to be investiated. Throuh his analysis, Pomranin (984) derived a new formulation of the transport cross section which may improve upon classical diffusion results. The essence of his formulation is the assumption that the anular flux can be decomposed into a scalar component and normalized anular factor, and furthermore, the normalized anular factor is independent of the enery roup. Quantitatively, ψ (, r Ω ˆ) = ϕ () r Ψ(, r Ωˆ) (25) where ˆ (, ˆ dωψ r Ω ) =. This assumption implies that 4π ϕ '() r J' () r = J() r ϕ () r (26) Upon pluin this separation into the second P equation [Equation (9)], the followin definition of the transport cross section arises, which will hereafter be denoted by the superscript FL: σ FL tr, G = ' = () r σ () r σ () r ϕ () r. (27) ϕ () r ' s, ' 3.3. An Exact P Diffusion Coefficient Althouh the P approximation to the transport equation can often lead to very ood results for nuclear reactor cores, the detailed balance approximation of Equation 2

21 (20) is dubious, especially in reions of the reactor with stronly absorbin materials. Writin Equation (9) in the form of Fick s law without this approximation yields G ' σ s, () rj' () r ' = σ J () r ϕ () r = 3 () r J () r. (28) Unless this equation is written for only one spatial dimension, the second term in brackets is undefined because of the vector division. Stamm ler (983) points out that this term should be seen as a tensor which is necessary in order for the spatial components of the current to vary independently in enery. He further states that this additional complexity is unnecessary in liht of the already simplified framework of P theory, and indicates that it is reasonable to replace the current by its manitude, J () r = J () r. With this approximation (or the equivalent assumption of a D problem), a transport cross section can be defined that avoids the detailed balance approximation of Equation (20). Since, in one dimension, this cross section leads to the exact solution of the P equations, it will subsequently be denoted by a P superscript: σ P tr, () r = σ () r G ' σ s, r J ' r ' = J () () () r. (29) In order to implement this definition, it is necessary to calculate the multiroup neutron current, J (r). Within the current context of cross section eneration by Monte Carlo methods, this implies calculatin a volume-averae of the neutron current over the tally cell. This capability is not available in MCNP, but such a tally can be created by recallin the definition of the neutron current: 3

22 J = dvj () r V T = dv dωω ˆˆψ (, r Ωˆ ) V T 4π = dv d ˆ Ω ˆ ˆ ˆ (, ˆ) V 4π x Ωy Ω Ω i+ j+ zk ψ r Ω T = dv d ˆ Ωψ (, ˆ) ˆ (, ˆ) ˆ (, ˆ) ˆ V 4 x Ωψ π y Ωψ Ω r Ω i+ r Ω j+ z r Ω k T (30) where Ωx, Ωy and Ω z are the directional cosines in the x, y and z directions, respectively, (i.e. Ω =Ω i ˆ ˆ, etc.). MCNP calculates and stores the directional cosines for each track, x therefore a tally can be created for each spatial direction which has the form: for ξ {, x yz,}. J = wt E < E < E (3) ξ, i i i, for i NV Ω ξ, + n i Now that a current tally has been developed, it is also possible to calculate the current-weihted condensation of the first moment of the scatterin kernel consistent with the multiroup derivation [c.f. Equation (6)]. This is an improvement over the fluxweihtin approximation of Equation (8). With the current-weihted scatterin moment and the avoidance of the detail balance approximation, this definition of the transport cross section has bypassed all the approximations normally associated with classical diffusion theory and should produce the same level of accuracy as direct solution of the P transport equations A Hih Order Diffusion Coefficient All previous definitions of the diffusion coefficient have truncated the P N equations at N =, nelectin hiher order terms, and used the first moment equation to express the neutron current in terms of the flux radient (an expression of Fick s Law). These expressions are of course only approximate closures to the transport equation since they nelect information carried by the hiher order moments. It seems unduly wasteful, however, to be limited to the assumption of linear anisotropy when workin within the 4

23 context of Monte Carlo methods. The hih-order accuracy of the Monte Carlo method should be exploited as much as possible when computin a diffusion coefficient. Consider instead an exact solution of Fick s Law as a closure to the transport equation. That is, instead of postulatin an expression of Fick s Law based on the P equation, assume that Fick s Law is an a priori valid relationship and solve for the diffusion coefficient in terms of a stochastically computed current and flux radient. Interatin Fick s Law over a tally cell and dividin by the volume, V T, one has dv () dv D () () V J r = ϕ V T V r r. (32) VT T T If it is assumed that V T is a small and homoenous cell in which the diffusion coefficient is constant, Equation (32) can be written V T VT D dv J() r = dv ϕ() V r. (33) VT T The left-hand side of this expression is the averae current within the cell, which can be tallied by Equation (3). The riht-hand side can be simplified usin a special case of the diverence theorem to yield where J ˆ V = T D ndsϕ() r (34) V T VT is the boundin surface of V T containin the unit area s with outward normal ˆn. The riht-hand side is the surface-interated flux for all boundin surfaces, a value which can be tallied usin a surface-flux tally that is a standard feature in MCNP. Since Equation (34) is a vector equation, special care must be iven to the calculation of D. If the diffusion coefficient is forced to be a scalar constant then Equation (34) will be exact only if the ratio of the current to the flux radient is the same for each spatial direction an assumption will almost certainly be untrue in three dimensions. One option which will preserve the ratio in all dimensions is to take D to be the diaonal matrix 5

24 D D 0 0 x, = 0 Dy, D z, (35) where the diaonal elements are directional diffusion coefficients for the three spatial dimensions x, y and z. One particularly nice aspect of this formulation is that it allows one to calculate an independent axial diffusion coefficient for 3D systems. This would be particularly useful for cases where streamin alon a particular direction is of concern (i.e. dominant) such as the CANDU reactor with voided channels, for which the axial streamin is known to be poorly treated by classical diffusion theory. Many diffusion codes, however, do not make allowances for directional diffusion coefficients, so some sort of averain must be performed to combine two or three of the directions. In one dimension, however, the solution of Equation (34) is trivial. As a preliminary investiation, therefore, the focus will be on the one dimensional case, which can be written D S = J V T ( ϕ + ϕ ) A T (36) where ϕ + and ϕ are the surface-averaed fluxes averaed over the riht and left boundaries of the tally cell, respectively, and the superscript S on the diffusion coefficient denotes the fact this definition relies only on the stochastic solutions of the current and flux. There is one final caveat about this formulation. The tally cell volume, V T, must be small in the sense that the flux radient should be approximately linear across it. If this is not the case then the denominator of Equation (36) will be a poor estimate of the actual flux radient. An alternative way of lookin at this requirement is to notice that when proceedin from Equation (32) to Equation (33) the assumption was made that the diffusion coefficient was constant within the tally cell. In actuality there will be 6

25 variations in the diffusion coefficient due to radient variations within the cell. Comparin Equations (32) and (33) it can be seen that the constant diffusion coefficient (denoted here as D ) is implicitly defined as D = V T dv D() r ϕ() r. (37) dv ϕ () r V T If the spatial flux distribution is approximately linear (i.e. constant radient) then the flux terms in the numerator and denominator cancel and the definition results in a simple averae. Otherwise the diffusion coefficient will be weihted with the flux radient over the cell volume, which will lead to inconsistent results. 7

26 CHAPTER 4 RESULTS Several confiurations of a one dimensional reactor core were solved usin the diffusion coefficients described in the previous section. These problems exhibit a varyin deree of heteroeneity that will aid in determinin the accuracy and applicability of each diffusion method. Before proceedin to the results, however, a validation of the stochastic cross section eneration routines will be shown. 4.. Cross Section Validation The methods described in Section 2.2 for eneratin stochastic cross sections were implemented in MCNP5. In order to verify that the implementation was functionin properly, 3-roup cross sections were enerated from a fine-roup library for a 2D BWR pin cell. The condensed cross section set tallied by the modified version of MCNP was checked aainst an analytic calculation of the cross sections usin the fineroup cross sections and MCNP output fluxes. Next, as a point of reference and comparison, the collision probability code HELIOS (Casal et al, 2003) was used to enerate the same cross section set as an external validation. Both MCNP and HELIOS used the standard 47-roup HELIOS library for the condensation to ensure a consistent comparison. The benchmark is a multiroup version of the problem previously published by Rahnema et al (997). The problem is a.6 cm square BWR pin cell consistin of fuel, clad and moderator. The fuel reion is subdivided into 5 annular cross section reions to avoid spectral homoenization (see Section 2.2). The problem eometry is shown in Appendix A. 8

27 Tables and 2 compare the MCNP-collapsed cross section aainst those computed analytically outside of MCNP. The RMS averae of the errors in the five fuel reions have been shown for brevity. It can be seen that there are no statistically sinificant (within 3σ) errors. This indicates that MCNP has been successfully modified to accurately tally condensed multiroup cross sections. RMS Fuel PD** Clad PD Table : Analytic vs. Stochastic Cross Sections Group Σ tr Σ ab Σ fi νσ fi 0.0% (0.02%) 0.02% (0.03%) 0.04% (0.04%) 0.05% (0.04%) % (0.03%) 0.02% (0.02%) 0.0% (0.02%) 0.0% (0.02%) % (0.02%) 0.0% (0.02%) 0.0% (0.02%) 0.0% (0.03%) 0.0% (0.02%) 0.0% (0.02%) % (0.02%) 0.0% (0.02%) % (0.02%) 0.00% (0.02%) 0.0% (0.06%) -0.06% (0.44%) % (0.03%) 0.02% (0.05%) Non-fissionable materials Water PD % (0.04%) 0.00% (0.04%) *The parenthetical values are the statistical uncertainties correspondin to one standard deviation **Percent difference = Σ MCNP Σ EXACT / Σ EXACT Table 2: Analytic vs. Stochastic P0 Scatterin Matrices Outoin roups RMS Fuel % (0.02%) 0.% (.34%) 0.00% (0.00%) % (0.00%) 0.00% (0.02%) 0.% (0.09%) % (0.00%) 0.9% (0.3%) 0.00% (0.02%) Incomin roups Clad % (0.02%) 0.4% (2.30%) 0.00% (0.00%) % (0.00%) 0.00% (0.02%) 0.0% (0.2%) % (0.00%) -0.03% (0.4%) 0.00% (0.02%) Water % (0.08%) 0.02% (0.%) 0.00% (0.00%) % (0.00%) 0.00% (0.03%) 0.03% (0.07%) % (0.00%) -0.05% (0.59%) 0.00% (0.04%) *The parenthetical values are the statistical uncertainties correspondin to one standard deviation **Error = Σ MCNP Σ EXACT / Σ EXACT 9

28 Tables 3 and 4 compare the cross sections enerated by MCNP with the cross sections enerated HELIOS. Overall there is very ood cross section areement, but there are sinificant differences in the fast roup fuel cross sections. Upon closer examination these cross section differences can be attributed to differences in the calculated flux spectra for these reions, which are plotted in Fiure. These flux differences arise from the difficulty of accurately treatin specularly reflective boundary conditions with the collision probability method for such a small system. Since the Monte Carlo method does not have this limitation (associated with numerical interations), it is capable of producin a more accurate flux spectrum for collapsin the cross sections. Table 3: MCNP vs. HELIOS Cross Section Differences Group Σ tr Σ ab Σ fi νσ fi 0.23% (0.05%)* 0.34% (0.02%) 0.59% (0.02%) 0.63% (0.02%) RMS Fuel PD** % (0.02%) 0.0% (0.02%) 0.0% (0.0%) 0.0% (0.0%) % (0.0%) 0.03% (0.0%) 0.03% (0.0%) 0.03% (0.0%) 0.04% (0.02%) 0.04% (0.02%) Clad PD % (0.0%) 0.02% (0.0%) % (0.0%) 0.03% (0.0%) 0.06% (0.06%) 0.49% (0.44%) % (0.03%) 0.00% (0.04%) Non-fissionable materials Water PD % (0.03%) 0.06% (0.04%) *The parenthetical values are the statistical uncertainties correspondin to one standard deviation **Percent difference = Σ HELIOS Σ MCNP / Σ MCNP 20

29 Table 4: MCNP vs. HELIOS P0 Scatterin Matrix Differences Outoin roups RMS Fuel % (0.0%)* 0.60% (0.%) 0.00% (0.00%) % (0.00%) 0.00% (0.0%) 0.06% (0.05%) % (0.00%) 0.22% (0.09%) 0.00% (0.0%) Incomin roups Clad % (0.02%) 0.27% (0.22%) 0.00% (0.00%) % (0.00%) 0.00% (0.0%) 0.05% (0.0%) % (0.00%) 0.06% (0.2%) 0.00% (0.0%) Water % (0.08%) 0.% (0.%) 0.00% (0.00%) % (0.00%) 0.00% (0.03%) 0.00% (0.07%) % (0.00%) 0.23% (0.58%) 0.05% (0.04%) *The parenthetical values are the statistical uncertainties correspondin to one standard deviation **Error = Σ HELIOS Σ MCNP / Σ MCNP.00% 0.50% Flux Error = (HELIOS-MCNP)/MCNP 0.00% -0.50% -.00% -.50% -2.00% -2.50% Enery roup Fiure : HELIOS vs. MCNP Relative Fine-Group Flux Errors fuel fuel 2 fuel 3 fuel 4 fuel 5 clad water 2

30 4.2. Fine-Mesh Diffusion Attention will now be turned to examinin the various diffusion coefficients defined in Chapter 3. A simplified D BWR reactor core was chosen as a benchmark problem so that the underlyin properties of the diffusion methods can be examined without unnecessary complexity. The benchmark problem was taken from Ilas, D. and Rahnema (2003) and fundamentally consists of four unique assembly types. Each assembly consists of 4 homoeneous fuel reions sandwiched between two homoenous coolant/clad channels. These assemblies are arraned in three different 7-assembly confiurations to form cores of varyin heteroeneity. The assembly and core eometries are shown in Appendix B. Core confiuration is the simplest core, with only slihtly varyin enrichment and no burnable absorber elements. Core confiurations 2 and 3 are more difficult cases, containin increasin amounts of adolinium in alternatin assemblies. This test suite allows the diffusion coefficients to be tested in a rane of problems with varyin difficulty. Diffusion Without Group Condensation In order to capture only the errors associated with usin diffusion theory instead of transport theory, a number of problems were solved in order of increasin complexity. First, the four assemblies with specularly reflective boundary conditions were solved in two roups. The transport and diffusion solutions were both in two roups to ensure that there were no errors arisin from roup condensation. After HELIOS was used to enerate a 2-roup material library, MCNP enerated solutions for all four fuel assemblies and tallied diffusion coefficients for each material reion. The diffusion code NESTLE (Turinksy et al, 994) was then used to solve the 22

31 four assemblies with the same 2-roup library while varyin the diffusion coefficients over the rane of definitions previously discussed. Table 5 displays the reference eienvalues and the eienvalue errors associated with the different diffusion theories for all four assemblies. Since there was no cross section collapsin, these errors are a result only of usin the diffusion approximation instead of full transport theory. In all cases the stochastic diffusion coefficient, D S, outperforms the others by a sinificant marin. The flux-limited coefficient seems to fare the worst. Table 5: Assembly Eienvalue Differences (2G Diffusion v. 2G Transport) D k * eff [mk] Assembly k eff ± 0.03 mk D C D FL D P D S *D k eff = [k DIFFUSION k MCNP ]*000 mk Table 6 displays the averae, root-mean-square, mean relative and maximum errors for the reion-wise fission densities for the four assemblies (see Appendix C for specific definitions of these statistics.) The classical, flux-limited and P coefficients all perform comparably. The stochastic diffusion coefficient exhibits the lowest fission errors in all cases. 23

32 Table 6: Assembly Fission Density Error Statistics (2G Diffusion v. 2G Transport) Assembly Method AVG RMS MRE MAX D C 0.0% 0.0% 0.0% 0.2% D FL 0.9% 0.20% 0.9% 0.24% D P 0.30% 0.33% 0.29% 0.44% D S 0.0% 0.0% 0.0% 0.2% D C 0.26% 0.3% 0.20% 0.42% 2 D FL 0.09% 0.% 0.07% 0.6% D P 0.3% 0.35% 0.24% 0.47% D S 0.06% 0.06% 0.06% 0.06% D C.52%.53%.60%.69% 3 D FL 2.2% 2.4% 2.26% 2.42% D P.54%.55%.62%.72% D S.0%.05%.5%.30% D C 0.29% 0.34% 0.28% 0.47% 4 D FL 0.5% 0.7% 0.4% 0.24% D P 0.28% 0.32% 0.27% 0.44% D S 0.2% 0.5% 0.2% 0.2% *Averae σ uncertainty associated with reference fission densities ~0.02% Next, the three core confiurations were examined. Aain, a two-roup library was enerated by HELIOS, then MCNP was used to enerate a transport solution and compute diffusion coefficients. NESTLE then solved the cores usin the various diffusion coefficients. All diffusion methods do relatively well on confiuration, since it contains no stron absorbers and has slowly varyin material properties. The averae error increases as expected for confiurations 2 and 3, but the hih order method usin D S does well for all three confiurations an order of manitude better for confiuration 3, in fact. 24

33 Table 7: Core Eienvalue Differences (2G Diffusion v. 2G Transport) D k eff * [mk] Confiuration k eff ± 0.03 mk D C D FL D P D S *D k eff = [k DIFFUSION k MCNP ]*000 mk Table 8 displays the fission density error statistics of the three cores. In spite of the hih order method producin more accurate eienvalues, it is not discernibly better than any other method with respect to fission densities. In fact, the maximum error for confiuration 3 is markedly hiher at 9.3%. Table 8: Core Fission Density Error Statistics (2G Diffusion v. 2G Transport) Confiuration Method AVG RMS MRE MAX D C 0.55% 0.82% 0.35% 2.33% D FL 0.53% 0.88% 0.28% 2.7% D P 0.64% 0.97% 0.40% 2.90% D S 0.64%.04% 0.38% 3.42% D C.32%.6% 0.74% 3.25% 2 D FL.64% 2.05% 0.85% 4.0% D P.39%.77% 0.77% 3.94% D S 0.92%.40% 0.59% 4.4% D C 2.08% 2.3%.4% 4.24% 3 D FL 2.5% 2.7%.23% 5.37% D P 2.09% 2.49%.42% 5.06% D S 2.40% 3.72%.43% 9.25% *Averae σ uncertainity associated with reference fission densities ~0.02% The fission density error distributions for confiurations and 3 are shown in Fiures 2 and 3. The classical, flux-limited and P diffusion theories all exhibit a similar 25

34 error trend. The stochastic diffusion method has a distinctly different distribution that is more accurate for the central assemblies, but deviates sinificantly towards the boundary. In spite of havin lare maximum fission density errors, the stochastic diffusion method has an order of manitude better estimate of the eienvalue. It accurately predicts the fission densities for the central hih-power assemblies, even thouh it deviates towards the extremities. These different tendencies sinify that the stochastically defined diffusion coefficient is fundamentally different than the other three. The classical, flux-limited and P diffusion coefficients are all stron functions of the cross sections, namely the total and the first scatterin moment cross sections. With no collapsin the classical diffusion coefficient is a function of only these cross sections with no flux dependency, while the flux-limited and P coefficients can be seen as bein similar to the classical coefficient, but with various spectrum-dependent adjustments. These coefficients are therefore stronly tied to the material. The stochastic diffusion coefficient, on the other hand, is not explicitly dependent on the system cross sections; the dependence is implicit throuh the current and flux radient within a cell. This coefficient is a stron function of the local flux distribution. The benefit of this stron dependency is that it carries more information about the neutron flow to the diffusion calculation. Caution must be taken in reions with sharp flux radients, however, since the tallies of the current and flux radient represent averaes and are only first order accurate by definition (see Section 3.4). 26

35 Classical Flux-limited P Gradient Relative Error Position [cm] Fiure 2: Fission Density Error Distributions for Confiuration Classical Flux-limited P Stochastic Relative Error Position [cm] Fiure 3: Fission Density Error Distributions for Confiuration 3 27

36 Since the fission density errors are stronly influenced by errors in the thermal flux, they are not a ood indicator of how the different diffusion coefficients perform at hiher eneries. The roup flux errors for confiuration 3 will therefore be examined in more depth. Confiuration 3 will be examined since the reions with heavy adolinium additions exhibit harder spectra and should hihliht errors associated with the fast flux. Fiure 4 shows the fast flux error distribution. The stochastic diffusion coefficient produces a solution that deviates sharply towards the boundary, but is otherwise very accurate. The classical diffusion coefficient produces the best answer in the fast roup in the assembly adjacent to the boundary. Beyond the first assembly, however, the flux-limited and P coefficients perform better. Similar trends are observed near the boundary in the thermal roup in Fiure 5. Away from the boundary, the coefficients perform more or less comparably, but the fluxlimited errors peak substantially in the water reions at assembly boundaries. 28

37 Classical Flux-limited P Stochastic Fast Flux Error, % Position [cm] Fiure 4: Fast Flux Error Distribution for Core Classical Flux-limited P Stochastic 0.07 Thermal Flux Error, % Position [cm] Fiure 5: Thermal Flux Error Distribution for Core 3 29

38 To investiate how sensitive these results are to the mesh size (i.e. the reion volumes which are considered to have linear flux radients and flat currents), the calculation for confiuration 3 was repeated with each material reion divided in two tally reions. The results follow in Tables 9 and 0. It can be seen that the eienvalue error for the stochastic coefficient is approximately halved and there was a moderate decrease in error for the fission densities. This indicates that decreasin the cross section mesh does improve the solution, but the stochastic method still strules more than the others near boundaries due to its stron dependence on spatial variations in the flux. The other coefficients did not chane appreciably, which was expected because of their stron material dependence. 30

39 Table 9: Eienvalue Differences for Halved Mesh (2G Diffusion v. 2G Transport) D k * eff [mk] k eff ± 0.03 mk D C D FL D P D S *D k eff = [k DIFFUSION k MCNP ]*000 mk Table 0: Fission Density Error Statistics for Halved Mesh (2G Diffusion v. 2G Transport) Confiuration Method AVG RMS MRE MAX D C 0.55% 0.82% 0.36% 2.29% D FL 0.53% 0.88% 0.28% 2.63% D P 0.63% 0.94% 0.40% 2.59% D S 0.63%.5% 0.3% 3.93% D C.32%.6% 0.75% 3.29% 2 D FL.66% 2.09% 0.86% 4.8% D P.44%.83% 0.80% 3.99% D S.7%.80% 0.65% 5.3% D C 2.09% 2.33%.42% 4.39% 3 D FL 2.4% 2.72%.22% 5.5% D P 2.2% 2.56%.44% 5.5% D S 2.6% 3.40%.20% 8.87% *Averae σ uncertainity associated with reference fission densities ~0.02% 3

40 Infinite-Medium Cross Sections The previous results provide valuable insiht into the behavior of the different diffusion coefficients, but have limited usefulness for practical reactor physics problems since it has been assumed that the full-core transport solution is known. It is common practice to sub-divide reactor cores into repeatin cells (usually fuel assemblies) and seek a solution for each unique assembly with approximate boundary conditions that simulate the boundary conditions it will see inside the core (specularly reflective, for example). The current core confiurations were analyzed in a similar manner. Each core contains two unique assembly types. These sinle assemblies were solved with specular reflection in MCNP to enerate material-wise diffusion coefficients, then the entire cores were solved usin the infinite-medium diffusion coefficients. Tables and 2 summarize the errors associated with these calculations. As was expected, since the classical diffusion coefficient does not depend on the flux spectrum or spatial distribution, there are no additional errors associated with the infinite medium diffusion coefficients. The flux-limited and P coefficients do not produce sinificantly different results in this case either, indicatin that the enery spectrum is sufficiently conserved when the infinite medium calculations are performed. The stochastic diffusion coefficient does much more poorly in this case, however. This is to be expected since it so stronly depends on the spatial distribution of the flux. Chanin the boundary conditions on these small assemblies sinificantly alters local flux radients near the boundaries. This in turn creates diffusion coefficients that do not perform well in the full-core calculations. 32

41 Table : Eienvalue Differences with Infinite-Medium Diffusion Coefficients (2G Diffusion v. 2G Transport) D k * eff [mk] Confiuration k eff ± 0.03 mk D C D FL D P D S *D k eff = [k DIFFUSION k MCNP ]*000 mk Table 2: Fission Density Error Statistics with Infinite-Medium Diffusion Coefficients (2G Diffusion v. 2G Transport) Confiuration Method AVG RMS MRE MAX D C 0.90%.26% 0.49% 3.25% D FL.04%.56% 0.53% 4.0% D P.04%.46% 0.57% 3.94% D S.38%.88%.04% 5.58% D C.43%.72% 0.83% 3.36% 2 D FL.83% 2.8%.09% 4.0% D P.47%.8% 0.83% 3.94% D S 7.94% 0.39% 7.34% 9.67% D C 2.25% 2.48%.58% 4.37% 3 D FL 2.4% 2.85%.59% 5.7% D P 2.2% 2.88%.33% 5.79% D S 5.29% 6.03% 2.30% 2.97% *Averae σ uncertainity associated with reference fission densities ~0.02% 33

42 Diffusion With Group Condensation We will now briefly look at the above diffusion methods coupled with roup condensation. The same three core confiurations analyzed above were solved aain, but this time the 47-roup library from HELIOS was used. Condensed 2-roup cross sections and diffusion coefficients were tallied for each material reion in the three cores. The two-roup libraries and diffusion coefficients were then used to solve the three confiurations with NESTLE, and the results were compared aainst the 47-roup reference solutions. Table 3 shows the diffusion eienvalue errors associated with the four diffusion coefficients, followed by Table 4 showin the fission density error statistics. In the context of roup condensation, the P and stochastic methods seem to perform much better. The classical and flux-limited coefficients most likely do so poorly because of spectral homoenization errors, where the enery spectrum was collapsed (tallied) over too broad an area (see Section 2.2). The stochastic diffusion coefficient performs better perhaps because it relies more on the spatial distribution of neutrons rather than the enery distribution and avoids errors associated with collapsin cross sections. 34

43 Table 3: Eienvalue Differences (2G Diffusion v. 47G Transport) D k * eff [mk] Confiuration k eff ± 0.03 mk D C D FL D P D S *D k eff = [k DIFFUSION k MCNP ]*000 mk Table 4: Fission Density Error Statistics (2G Diffusion v. 47G Transport) Confiuration Method AVG RMS MRE MAX D C 3.5% 4.7% 2.3% 0.5% D FL 3.30% 4.35% 2.44% 0.85% D P 0.99%.23% 0.78% 2.76% D S 0.8%.22% 0.53% 3.85% D C 8.02% 9.86% 6.55% 7.97% 2 D FL 8.37% 0.39% 6.88% 8.6% D P 2.08% 2.36%.86% 4.06% D S.49% 2.07%.03% 5.06% D C 2.8% 5.07% 0.77% 25.77% 3 D FL 3.36% 5.69%.24% 26.74% D P.94% 2.59%.84% 5.22% D S 2.93% 3.97%.82% 8.76% *Averae σ uncertainity associated with reference fission densities ~0.02% 35

44 CHAPTER 5 CONCLUSIONS Several definitions of the diffusion coefficient have been analyzed in an attempt to ascertain the most accurate way of computin a diffusion solution for reactor physics problems. The classical, flux-limited and P diffusion coefficients performed comparably for all test problems. This indicates that the primary source of diffusion error is the limitation imposed by the P equations that the flux is linearly anisotropic. The intermediate assumptions associated with the classical or flux-limited derivation of the transport cross section seem to be secondary to this approximation. These approximations, however, tended to improve the performance of diffusion methods at fast eneries and in hihly absorbin media. The stochastic diffusion coefficient represents a fundamentally different form of closure of the transport equation. Overall this coefficient performed much better, but its stron dependence on the spatial flux distribution makes it harder to implement in a manner which is practical for reactor physics problems. There is much work that remains in this area. This thesis has revealed that there exists a diffusion methodoloy which is capable of supersedin P theory, but a practical implementation requires more investiation. Simulatin leakae may allow hih order diffusion coefficients to be calculated, similar to performin bucklin searches in B N theory, but it is advantaeous to make the most of the Monte Carlo Method, avoidin unnecessary approximations or series truncations that frequently accompany deterministic methods. 36

45 APPENDIX A BWR PIN CELL BENCHMARK The BWR pin cell benchmark was taken from Rahnema, Ilas and Sitaraman (997). The problem consists of a square cell containin 2.80% enriched U0 2 pin with natural zirconium claddin surroundin by moderator. The moderator is H 2 0 at 300 K. The eometry is shown in Fiure 4, with dimensions followin in Table 3. The fuel reion was subdivided into five annular reions only for the purpose of eneratin cross sections. The materials and densities in these reions are constant. 37

46 Fiure 6: Pin Cell Geometry with Cross Section Meshin Table 5: Pin Cell Dimensions Cell Pitch, P.6256 cm Fuel Outer Radius, RO cm Fuel Inner Radius, RI cm 38