Examination of ray-effect using some benchmark problems

EXAMINATION OF RAY-EFFECT USING SOME BENCHMARK

PROBLEMS

S. SAĞLAM

İstanbul University, Faculty of Science Physics Department, Vezneciler-İstanbul, TÜRKİYE

ABSTRACT

The subject o f this study is the ray-effect anomaly observed on the numerical solution o f neutron transport equation. The ray-effect is some unexpected ripples observed in the flux distributions. In recent years, a considerable effort has been expended in deriving some approaches to eliminate or mitigate this anomaly. In this study to observe the ray-effect an important two dimensional transport code has been executed for a well-known Benchmark problem which is given in cartesian geometry. Then, the problem has been adapted to cylindrical geometry with the same data. The flux distributions have been examined by changing the values o f scattering cross-section in two geometries. Finally, the results obtained in each medium have been discussed in detail and tried to choose the “best” medium o f all.

1. INTRODUCTION

The main problem o f reactor physics is the determination o f neutron distribution in a system. To determine this distribution, the neutron transport must be investigated. The neutron transport is given by the linear Boltzmann transport equation [1,2],

L (f>(r,E,n,t) Q ( r , E , n t ) (1 ) —> i / —> -■> —>00 —> —y —* L O. V

+Xr(/‘

f

In nuclear reactor design and analysis, the solution o f transport equation plays very important role. However, it is very difficult to solve this equation except for simple problems; since it has an integro-differential form with seven variables. Other factors have also an effect complicating the solution such as complex energy dependence o f cross-sections and the geometrical arrangement o f the materials in the system. Therefore, the equation with the boundary conditions (and an initial condition for time-dependent problems) is solved usually by deterministic transport codes. This equation and boundary conditions are required for problems o f finite geometric extent. There are two wellknown deterministic methods used in these codes [3].

1. Spherical Harmonics Method (PN method) 2. Discrete Ordinates Method (SN method)

2. THE METHOD

Discrete Ordinates Method that is the main subject o f this study is one of the important deterministic methods as mentioned above and widely used in transport calculations. In the method, a set o f discrete directions, ordinates, for angular direction variable, Ö , is chosen and directional fluxes are evaluated for these directions. The choice of these directions is not arbitrary: they must satisfy some conditions (i.e. some mechanical integration requirements).

The derivatives appearing in the transport equation must be replaced by a corresponding discrete representation by using finite difference techniques (i.e. forward difference, backward difference, diamond difference). The set o f equation seen below gives a diamond difference form of flux according to the space and angular variables [2,4].

2 N = N i+1/2 + N

_ 1/2

2 N - N m + 1/2 + N m - l / 2

The integrals in the transport equation must also be replaced by a standard numerical integration schemes (i.e. standard Gauss quadratures: Gauss-Legendre quadrature, Gauss-Chebyshev quadrature, etc.)

The Gauss - Legendre quadrature form is shown below,

1+1 A

j

F(x)dx = ^ W;F(X;)

i=0 Where, Wi are called the weights o f the quadrature.

In the study, angular flux is expanded in a series o f Legendre polynomials as, ■m İ ^ i i+ D ^ rM

n=0 k=0 and the expansion coefficients are given by,

^ = f | ld +

d<PR + «

This integral is given in the approximated form by the sum,

,

MT

,

Where (pm is defined in the study [4] as,

_ ] / o 2 V / 2

<Pm=tan (i-Mm-rim

)/^Im

»

> 0

|

_

1

/

9

9 V ^

f

cp m — tan \1 — ft m — m / //rl m “t_7i: 5 m ^ 0 J The scattering transfer probability is also be given by a finite Legendre polynomial expansion:

ISCT I s I ^ ± i Pn(M0 ) o « l ( E , - » E ) ' s _n=0o 4rr (8) (9) Where,

Ho = O.Q ' = M-M

- ’+ (l - |u2 )X (l - ft,2 )/2 cos(tp - (p ’)

If this expansion is inserted in the eg. 2 and using the addition theorem, we expand the Pn(p 0) polynomial, then using the cp symmetry o f flux and after some algebra we obtain the X s equation as,

ISCT

X > = X (2 n + l ) a sn ( E ' - > E ) X

n=0 k=0

(11)

where ISCT is the scattering order. And total source S is given by the sum of,

IGM ISCT n k k ( S S ) gijm

= (Scatteringsource)gijm =

X Z

(2n

l)osnh^ g Z

^

|

h = l n = 0 k = 0

IQAN n k k

( IS )gijm = (In h om ogen eou s source ) gijm = Z ( 2 n + 1) X R nm Q n gli n =0 k = 0

(12)

Where,

IGM: number o f groups used in the study IQAN: distributed sources anisotropy order.

In the total source equation, fission source is assumed to be negliable according to the problems used in the study.

Although the method has some advantages to solve transport equation, sometimes some unexpected distortions in the flux have been met. These distortions that plague the method are called as ray-effects. The term ray-effect refers to the anomalous scalar flux distortions that appear when traditional discrete ordinate methods are applied to certain problems having strong absorbers and localized (or isolated) neutron sources in two or three dimensional geometries. These effects arise because the SN equations allow propagation o f the solution only along a discrete set of directions rather than the continuum o f directions associated with the analytical transport equation [1,3,5].

In recent years a considerable effort to be spend to eliminate or mitigate that anomaly [5,6,7,8]. It seems that the practical remedy for the ray-effect is to increase the order o f SN method (to increase the number o f directions used). Because the low order SN approximations (i.e. S2 approximation) give qualitatively incorrect results because of the reason mentioned above. The higher order SN calculations more closely exhibit the expected behavior except that they contain unphysical oscillations in the solution. These nonphysical bumps can be confused with physical ones, especially in deep penetration applications [1]. In general it is said that, when high order SN approximations are used, the frequency o f oscillations usually becomes higher and the magnitude becomes smaller. Also, the ripples are often remarkably persistent, even for high order discrete ordinate approximations (even when 144 directions for Sj6 approximation) are used [1,9].

3. THE RESULTS OF BENCHMARK PROBLEMS

In this study, to examine the ray-effect two different Benchmark problems are used. The first problem is a wellknown Benchmark problem that appears in many papers in the literature. It is given in (x-y) two dimensional cartesian geometry. In the problem, there is a flat isotropic source at the left bottom comer o f a square region (Fig. 1). The region is 2.0 cm in lenght on each side. The boundary conditions are given as reflective boundaries on the four sides o f the region.

Reflective boundary

Fig. 1. Problem 1 given in (x-y) geometry

Then, problem 1 has been adapted to (r-z) two dimensional cylindrical geometry (Fig. 2). Table 1 gives the data o f the problems. The boundary conditions are the same for this geometry.

z

A

Fig. 2. Problem 2 given in (r-z) geometry

i . Medium S-1.00 1.00 4 Z f = 0 . 0 0 0.75 1 0.50 2 i r '- 0 0 0.25 3 0.00 5

Table 1. Data for sample problems

In this study, to demonstrate the effects o f scattering cross-section on the ray-effect, one of the important two dimensional transport codes, TWOTRAN - II has been chosen and executed for two problems with the data given in table 1. First, the code has been executed for the problem 1 and to observe the flux distortions, different SN approximations have been used. In each SN approximation, changing the values o f scattering cross-section, the flux distributions have been examined as seen in the figures (Fig.3 - Fig.5). Then, the same operations have been repeated for problem 2 with the same data. The flux distributions o f problem 2 can be seen in the figures 6-8.

When we examine the figures obtained it is seen that, the medium 4 does not exhibit any ray-effect because it represents a purely scattering medium. But, the other mediums suffer from ray-effect as expected. Among the others, the medium 1 seems better. Comparing the graphs obtained in medium 1 according to the SN approximations, it is observed that the S4 approximation exhibits more ripples than the others. This is the ray- effect. In the low order SN approaches, such as S2 and S4 approximations, angular integrals usually give incorrect results as mentioned before. Because angular fluxes include streaming contributions only if isolated (or localized) sources are intercepted by the directions o f the discrete set used. But, as the order o f SN approximations is getting higher a considerable mitigation is obtained. Although the graph magnitude is reduced, the Sj6 approximation seems better in all approximations. Because in this approximation the number o f discrete directions used is 144 (or 36 directions per octant). These directions are calculated by the relation given below [4],

M M - (ISN)* (ISN+2) / 8 (13) Where,

MM: number o f directions per octant ISN: order o f SN approximation

Besides, there is purely absorbing medium that is pointed as medium 5. This medium gives the worst solutions in all approximations as expected. Because, in such medium a poor value o f the scalar flux is obtained no matter how many discrete directions are used.

The results o f the study mentioned above seem in an agreement with the studies appear in the literature [5]. Fig. 9 shows like a study taken from literature.

4. CONCLUSION

In this study, we have tried to observe the ray-effect changing the values o f scattering cross-section. We have used two Benchmark problems given in (x-y) cartesian and (r-z) cylindrical two dimensional geometries. To demonstrate the effects o f scattering cross-sections on the ray-effect, in each medium we have examined different order SN approximations (such as S2; S4, S6, S8, Si0, Si6). Observing the ray-effect distortions we have chose the best and the worst medium o f all used in the study. Also, we have compared these results with the other studies coming from the literature. At the end o f the study, observing the ray-effect distortions we have tried to make a conclusion about “the best” and “the worst” medium. But the study mentioned above has not been completed yet. This article is only one chapter o f the whole study. After that, we will continue to examine the ray-effect according to the other parameters which have also an effect complicating the solution. Then, we will recommend some remedies to eliminate or strongly mitigate the ray-effect anomaly. In this study we have not used any acceleration method to speed up convergence for the iteration count comparisons.

r-z geometry

Z

Fig.8 . Scalar flux distribution (Sı6 approximation)

Transport equation Discrete Ordinates Method Gauss-Legendre quadrature Ray-effect

5. REFERENCES

1. E.E. Lewis and W.F. Miller Jr., Computational Methods o f Neutron Transport, John Wiley and Sons Publ., (1984).

2. G. Arfken, Mathematical Methods fo r Physicists, Academic Press, 3th ed. Orlondo, (1985).

3. R.D. O’dell and R.E. Alcouffe, Transport Calculations fo r Nuclear Analysis: Theory and Guidelines fo r Effective Use o f Transport Codes, Los Alamos Lab. Report, LA -10983, (1987).

4. K.D. Lathrop and F.W. Brinkley, TWOTRAN-II: An Interfaced Exportable Version o f the Twotran Codes fo r Two Dimensional Transport, Los Alamos Scientific Lab. Report, (1973).

5. J.E. Morel, T.A. Wareing, R.B. Lowrie and R.K. Parsons, Analysis o f Ray-Effect Mitigation Techniques, Nucl. Sci. Eng., 144, (2003), 1.

6. G.Bal and W.Xavier, Discrete Ordinates Methods in x-y Geometry With Spatially Varying Angular Discretization, Nucl. Sci. Eng., 127, (1997), 169-181.

7. K.A. Mathews, On the Propagation o f Rays in Discrete Ordinates, Nucl. Sci. Eng., 132 (1999), 155-180. 8. K.D. Lathrop, A Comparison o f Angular Difference Schemes fo r One Dimensional Spherical Geometry

SNEquations, Nucl. Sci. Eng., 134, (2000), 239-264.[9] L.L. Briggs, W.F. Miller Jr., and E.E. Lewis, Ray-Effect Mitigation in Discrete Ordinate-Like Angular Finite Element Approximations in Neutron