Please cite this article in press as: Liang, C., et al. Effects of fuel particle size distribution on neutron

transport in stochastic media. Ann. Nucl. Energy (2014).

http://dx.doi.org /10.1016/j.anucene.2014.01.005

Effects of Fuel Particle Size Distributions on Neutron Transport in Stochastic Media

Chao Liang, Andrew T. Pavlou and Wei Ji*

Department of Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic Institute

110 8th Street, Troy, NY 12180

*Corresponding author, email address jiw2@rpi.edu

Abstract

This paper presents a study of the fuel particle size distribution effects on neutron transport in three-

dimensional stochastic media. Particle fuel is used in gas-cooled nuclear reactor designs and innovative

light water reactor designs loaded with accident tolerant fuel. Due to the design requirements and fuel

fabrication limits, the size of fuel particles may not be perfectly constant but instead follows a certain

distribution. This brings a fundamental question to the radiation transport computation community: how

does the fuel particle size distribution affect the neutron transport in particle fuel systems? To answer this

question, size distribution effects and their physical interpretations are investigated by performing a series

of neutron transport simulations at different fuel particle size distributions. An eigenvalue problem is

simulated in a cylindrical container consisting of fissile fuel particles with five different size distributions:

constant, uniform, power, exponential and Gaussian. A total of 15 parametric cases are constructed by

altering the fissile particle volume packing fraction and its optical thickness, but keeping the mean chord

length of the spherical fuel particle the same at different size distributions. The tallied effective

multiplication factor (keff) and the spatial distribution of fission power density along axial and radial

directions are compared between different size distributions. At low packing fraction and low optical

thickness, the size distribution shows a noticeable effect on neutron transport. As high as 1.00% relative

difference in keff and ~1.5% relative difference in peak fission power density are observed. As the packing

fraction and optical thickness increase, the effect gradually dissipates. Neutron channeling between fuel

particles is identified as the effect most responsible for the different neutronic results. Different size

distributions result in the difference in the average number of fuel particles and their average size. As a

result, different degrees of neutron channeling are produced. The size effect in realistic reactor unit cells is

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also studied and, from the predicted values of infinite multiplication factors, it is concluded that the fuel

particle size distribution effect are not negligible.

Key Words: size distribution effects; eigenvalue problem; radiation transport; Monte Carlo method;

stochastic media

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1. Introduction 3-D particle systems, characterized by the stochastic distribution of spherical inclusions in a background

material, are typical radiation transport media encountered in many scientific and engineering fields. For

example, in the area of atmospheric science, solar radiation transports through clouds consisting of tiny

water droplets, ice crystals and dust aerosols. These cloud particles have sizes ranging from 105 m to 103

m and their distribution in the cloud can influence the solar energy radiative transfer to the surface of the

earth (Breon et al., 2002). In the area of nuclear engineering, some advanced nuclear reactor designs, such

as the Very High Temperature Gas-Cooled Reactors (VHTR) (Ji et al., 2005), the Fort Saint Vrain (FSV)

reactor (Pavlou et al., 2012) or innovative light water reactor designs (LWRs) loaded with fully ceramic microencapsulated (FCM) fuel (Brown et al., 2013; Liang and Ji, 2013; Sen et al., 2013; Snead et al.,

2011), utilize unique fuel elements called TRISO fuel particles that are fabricated to different fuel types

(fissile or fertile) and different sizes (to achieve high packing fractions). These fuel particles are randomly

packed in the reactor core at volume packing fractions ranging from 5% to 60%. To provide reliable

predictions of solar energy transfer through the atmosphere or neutronic safety analysis in nuclear

reactors, one needs to model not only the stochastic distribution of particles but also the size distribution

of each type of particle in the system, which presents a significant computational challenge to the study of

radiation transport in 3-D particle systems. Although radiation transport computation in stochastic media,

specifically in 3-D particle systems, has been an actively researched topic for a long time (Donovan and

Danon, 2003; Ji et al., 2005; Ji and Martin, 2007, 2008; Liang and Ji, 2011; Liang et al., 2013; Liang et

al., 2012; Murata et al., 1996; Murata et al., 1997; Reinert et al., 2010), much of the focus is on the

methodology development that accounts for the spatial and size distribution of particles in the media. Less

attention has been given to the study of particle spatial or size distribution effects on radiation transport,

especially the size distribution effects. Recently, Brantley and Martos (Brantley and Martos, 2011) have

studied a general radiation transport problem to calculate reflection, transmission and absorption rates

when radiation penetrates a cubic 3-D particle system, consisting of optically thick spherical particles and

an optically thin background at the particle volume packing fractions ranging from 5% to 30%. Spherical

particles were assumed to follow three different distributions in radius: constant, uniform and exponential.

They found a weak dependence of the predicted radiation transport rates on particle size distribution,

provided different distribution functions give the same mean chord length of spherical particles. Earlier

work by Olson (Olson, 2007) studied photon transport in 2-D disk and 3-D spherical particle systems and

found that very different disk/sphere sizes produced similar predictions of photon transport as long as the

total volume packing fractions were the same. Recent work by the University of Michigan (Burke et al.,

2012; Pavlou et al., 2012) on the Fort Saint Vrain reactor has shown the effects of six different

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distribution functions on the multiplication factors in a densely packed particle system. The radius size

distribution range was fixed and different distributions were applied to generate a fuel compact consisting

of fuel particles in a packing range of 58% to 60%. No significant differences in multiplication factor

were found.

Previous studies primarily focused on how the particle size distribution affects the integral physical

quantities that depend on the global distribution of particles, such as total absorption rate or multiplication

factor. Computational results at different particle size distributions were reported. Little or no physical

interpretation, however, was provided to explain differences in the simulation results. The lack of

theoretical analysis makes such research remain incomplete. Also, quantities that depend on the local

properties of particle distributions, such as fission density distribution, have not been studied yet. The

fission density distribution is important in reactor analysis because it can be used to identify a hot spot

location in the reactor fuel pins that are filled with TRISO fuel particles. In this paper, we first study the

size distribution effect on a simplified cylindrical fission system that consists of randomly distributed

fissile fuel particles in one-group eigenvalue problems. Five different size distributions are assumed for

the fissile particles and their impacts on the multiplication factor, volume-average flux in fuel particles

and background media, and spatial distribution of fission density in the system are studied. To ensure that

the size distribution is the only parametric effect, the parameters of each distribution function are

optimized to produce exactly the same mean chord length in the spherical particles. Physical

interpretations are provided to explain the differences in the above calculated quantities at different

particle size distributions. Then, we pack the same fuel particles used in the simplified system in three

realistic reactor design unit cells. Continuous energy neutronic analysis for eigenvalue problems is

performed at different particle size distributions.

Although our study focuses specifically on neutron transport in 3-D particle systems, it may be of interest

to the general radiation transport community. First, it can provide a more insightful understanding of the

radiation transport phenomena in stochastic media by quantifying the uncertainties in the radiation

transport computation due to the uncertainties in particle size. This is fundamental to radiation transport

research. Second, from a more practical perspective, it can provide pragmatic guidelines on how to

simplify the realistic physical model for routine basis analysis by understanding the size distribution

effects. For example, an equivalent single size particle can be used in modeling systems that consist of

poly-dispersed particles. This equivalent size is determined by finding an optimal size that produces

similar solutions as poly-dispersed distributions with acceptable errors in practice. Fuel particles with a

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single size in a reactor system are easy to model and efficient to simulate. These considerations have

motivated our present work.

The remainder of the paper is outlined as follows: In Section 2, the geometry and material configurations

of a simplified 3-D particle fission system are described. Specifically, the size distribution of fuel

particles and the determination of distribution functions are thoroughly addressed. In Section 3, one-group

eigenvalue problems are solved in the simplified system. Numerical solutions of effective multiplication

factor, volume-average flux in fuel particles and background material, and spatial distributions of fission

density along axial/radial directions are presented for each studied scenario. Comparisons and

interpretations of the results are provided. A thorough investigation of the size distribution effect on the

neutronic behavior is performed and physical reasons for the effect are identified. In Section 4, fissile fuel

particles with the same distributions as in the simplified system are loaded into unit cells of three types of

reactor designs. Realistic material composition and density are applied and continuous energy eigenvalue

problems are solved for infinite multiplication factors. The size distribution effect is shown. In Section 5,

final conclusions and possible future work are presented.

2. Configurations of simplified 3-D particle fission systems 2.1. Geometric configuration We construct a series of simplified 3-D particle fission systems in a cylindrical container of radius 2cm

and height 4cm. Fissile fuel particles are randomly packed in the cylindrical container with five different

radii distributions: constant, uniform, power, exponential and Gaussian. In each distribution, the fuel

particle radius is distributed over the same range (except for the constant distribution) and the mean chord

length in fuel particles, denoted as , is fixed at 0.05cm by adjusting distribution function parameters.

Monte Carlo neutron transport simulations are performed in the system assuming vacuum boundary

conditions for each size distribution and for five different fuel particle volume packing fractions, denoted

as frac, of 5%, 15%, 30%, 45% and 60%. A total of 25 geometric configurations are constructed for the

study. Fig. 1 shows one physical realization of the stochastic packing of fuel particles with constant size

in a cylindrical container at frac=45%. Once a geometry realization of a 3-D particle system is generated,

Monte Carlo radiation transport simulations are performed in the system. How the fuel particle size

distribution affects the neutronic behavior in a stochastic medium system, such as the reaction rates in

fuel and moderator, leakage rate, etc. can be quantitatively studied. Physical reasons for the differences in

the final solutions, if they exist, can be explored. The simplified system is a typical configuration that

represents the stochastic medium region seen in practical reactor designs, such as the fuel compact region

in the prismatic VHTR design or the FSV design or the fuel pin region in recently LWR designs loaded

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with FCM. In reality, there are moderators around the simplified system shown in Fig. 1. If the neutronic

behavior in this simplified system can be fully understood, it can be naturally extended to understand the

neutronic behavior in realistic configurations subject to the random distribution of fuel particles at

different size distributions inside the stochastic fuel region.

H=4cm

R=2cm

H=4cm

R=2cm

Fig. 1. One physical realization of a 3-D particle fission system. Fuel particles with 0.075 cm diameter

are randomly packed in a cylinder at the volume packing fraction of 45%. The cylinder is 4 cm high and

has a diameter of 4 cm.

In our size distribution effect study, we impose two constraints for different fuel particle size distributions:

both the size distribution range (except for constant size) and the mean chord length in fuel particles are

fixed. These constraints are necessary and meet the pragmatic needs. For example, fabrication of TRISO

fuel particles for Fort Saint Vrain reactor has shown that the fuel particle size has a distribution within a

certain range (Burke et al., 2012; Pavlou et al., 2012). Meanwhile, fixing the mean chord length in fuel

particles can exclusively allow us to study the size distribution effects only on the radiation transport

since neutrons on average see the same optical thickness in fuel and background materials (measured by

the product of mean chord length and total cross section in each material). To meet these two constraints,

we start with the constant size distribution to determine its fuel particle size and mean chord length (as

references for other size distributions). Then the size distribution range is determined for other

distribution functions by enforcing the range to cover the reference constant size and producing the same

mean chord length. We next give a specific description of each radii distribution.

2.1.1. Determination of particle size distribution functions

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It was shown by Olson et al. (Olson et al., 2006) that the probability density function (PDF) for chords in

a sphere is given by

2 /2( ) , 0 2 ,

( ) 20, 2 ,

R

l

l f r dr l Rd l r

R l

(1)

where f(r) is the radii PDF, l is the chord length and R is the maximum value of sphere radius. The value

is defined in terms of f(r) as

2 2

0

( ) .r r f r dr

(2)

a) Constant distribution

For constant size distribution of radii,

( ) ( ),f r r R (3)

and we obtain

2 2/2( ) , 0 2 ,

( ) 2 20, 2 .

R

l

l lr R dr l Rd l R R

R l

(4)

The mean chord length is defined by 0

( )l ld l dl

. Then we have 2

22

0

1 4 .2 3

R Rl l dlR

We set

R=0.0375cm for constant size distribution and this corresponds to = 0.05cm.

b) Uniform distribution

Radii sampling is performed with a uniform distribution with lower endpoint a and upper endpoint b such

that the mean chord length matches the constant size case. The radii distribution PDF for the uniform

distribution is written as

1( ) , .f r a r b

b a

(5)

The chord length PDF for uniform size distribution fuel particles can be written based on Eq. (1),

1 2

2 2/2

( ) ( ) , 0 2 ,2

( )( ) ( ) , 2 2 .

2

b

ab

l

ld l f r dr l ar

d lld l f r dr a l br

(6)

The mean chord length is

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2 2 4 4

1 2 3 30 2

( ) ( ) .a b

a

b al ld l dl ld l dlb a

(7)

We fix a = 0.02cm and then find b = 0.0478cm produces a mean chord length of 0.05cm. The range of

radius size [a, b] is used for all other size distribution functions.

c) Power distribution

We choose a power law to sample radii of the form

1 2( ) (1 ) ( ) ( ),f r p f r pf r (8)

where 2

1 3

3( )( ) , ,( )

r af r a r RR a

2

2 3

3( )( ) , ,( )

b rf r R r bb R

and p is the probability of

sampling a sphere radius in the range [R, b]. The value of p is dependent on the values of a and b. Here,

we want to have the sphere mean chord length at 0.05cm by fixing R=0.0375cm and the values of a and b

the same with uniform distribution. The distribution of chord length can be written based on Eq. (1):

1 1 22

2 1 22/2

3 22/2

( ) (1 ) ( ) ( ) , 0 2 ,2

( ) ( ) (1 ) ( ) ( ) , 2 2 ,2

( ) ( ) , 2 2 .2

R b

a R

R b

l R

b

l

ld l p f r dr p f r dr l ar

ld l d l p f r dr p f r dr a l Rr

ld l p f r dr R l br

(9)

The mean chord length is then determined by .)()()(2

23

2

22

2

01

b

R

R

a

a

dlllddlllddllldl To keep the

mean chord length at 0.05cm, p is determined to be 0.47967. Thus,

1 2( ) 0.52033 ( ) 0.47967 ( ).f r f r f r (10)

d) Exponential distribution

The radii distribution function is expressed as

( ) ,rf r e (11)

where the unknown parameters and are directly related to each other because the distribution function

must satisfy the normalization condition. Using a similar approach as the other sampling schemes, a

mean chord length of 0.05cm is achieved for =36.009 and =0.034.

e) Gaussian distribution

The radii distribution function is expressed in the form

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2

2 ( )1( ) (1/ ) .

c r Rf r c e (12)

A similar procedure as before is performed to determine the mean chord length and the values of c1 and c2

that give a mean chord length of 0.05cm: c1 = 0.0278 and c2 = 0.3333.

Table 1 summarizes the distribution functions and the value of each parameter.

Table 1 Summary of size distribution functions.

Size distribution functions Values of parameters

( ) ( )f r r R R = 0.0375 cm

1( ) , .f r a r bb a

a = 0.02 cm, b = 0.0478 cm

1 2( ) (1 ) ( ) ( )f r p f r pf r 2

1 3

3( )( ) , .( )

r af r a r RR a

2

2 3

3( )( ) , .( )

b rf r R r bb R

p = 0.47967

( ) , .rf r e a r b = 36.009, = 0.034

22 ( )

1( ) (1 / ) , .c r Rf r c e a r b c1 = 0.0278, c2 = 0.3333

2.1.2. Construction of 3-D particle systems After the PDFs for these five fuel particle size distribution functions are determined by fixing the sphere

mean chord length at 0.05cm, we can sample the sphere radii and pack them in the container. Due to the

difficulty of determining the cumulative distribution function (CDF) of the PDFs, Monte Carlo rejection

sampling is performed to accept or reject spheres from the radii distribution functions. If the sampled [r, y]

is under the corresponding point [r, f(r)], then r is accepted as the spheres radius; otherwise, it is rejected.

This procedure continues until the total volume of the sampled spheres reaches the prescribed volume

packing fraction in the cylinder.

For the fuel particle system with volume packing fractions (frac) of 5%, 15% and 30%, a fast algorithm

based on the Random Sequential Addition (RSA) method (Widom, 1966) is used to pack particles into the

cylindrical container one-by-one with no overlapping. For the particle system with frac= 45% and 60%, a

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Quasi-Dynamic Method (QDM) (Li and Ji, 2012; Li and Ji, 2013) is used to generate a high packing system. This method first generates N spatial points that are uniformly distributed within the container for

each known sphere center. Then, a normal contact force model is employed to eliminate sphere overlaps

while constraining all the spheres within the container boundary.

2.2. Material configuration

For the studied fuel particle system, we assume the background material is purely scattering with total

cross section tb=0.4137cm-1, which represents the graphite matrix material in gas-cooled reactors.

Spherical fuel particles are composed of fissile materials. For a complete study, three sets of cross section

data are assumed in fuel particles. The total cross section of fissile material and the corresponding optical

thickness at each cross section set are summarized in Table 2.

Table 2 Fissile material properties in fuel particles.

frac (cm) tf (cm-1) tf sf (cm-1) ff (cm-1) v* 5%

15% 30% 45% 60%

0.05

2 0.1 1 0.4

2.8 20 1.0 10 4

200 10.0 100 40

*v is the the mean number of neutrons produced per fission.

It should be noted that if is the same at all distributions, the mean chord length in the background

material within the cylindrical container, , is also the same. This can be verified by using the Dirac

formula (Dirac, 1943) to calculate as follows,

4 /b b bl V S , (13)

where Vb=V(1-frac), V is the total volume of the container and Sb is the total surface area of the

background material, calculated as the sum of two areas: the area of the container surface, Scylinder, and the

area of surfaces of all the fuel particles in the container, . is the average number

of fuel particles and can be calculated by Vfrac/. and are the average surface

area and volume of a fuel particle. The mean chord length in the background material can be written as

4 (1 ) / ( )b cylinder spheresphere

V fracl V frac S SV

(14)

Dividing the numerator and denominator by 4V and recognizing that the mean chord length in the

cylinder is given by =4V/Scylinder and the mean chord length in a sphere is =4/, we

obtain

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1 1(1 ) / ( ).bl frac fracL l

(15)

From Eq. (15), it can be seen clearly that the mean chord length in the background material in the

container is independent of the fuel particle size distributions as long as the mean chord length in the fuel

particles is the same.

3. Numerical results and analyses In this section, we present numerical results for eigenvalue problems in simplified 3-D particle fission

systems described in Section 2. An in-house Monte Carlo transport code is written to solve one-group

eigenvalue problems. Standard variance reduction methods, such as absorption suppression and splitting

and Russian Roulette, are used. Isotropic scattering are assumed in the simulations. Table 3 summarizes

the simulation cases for each fuel particle size distribution: five volume packing fractions combined with

three sets of cross sections in fuel particles give a total of 15 cases studied. For each case, a total of 100

independent realizations are generated for radiation transport simulation and final solutions are averaged

over 100 realizations. In the simulation of each realization, a total of 2000 fission cycles with 1000

inactive cycles are performed. In each cycle, a total of one million neutron histories are tracked. The

effective multiplication factor, volume-average flux in fuel and moderator, and axial/radial fission

reaction density distributions in the cylindrical container are estimated using collision estimators. These

predictions will be compared between different fuel particle size distributions. Then the impact of the

spherical size distribution can be studied from comparisons.

Table 3 Cases for eigenvalue problems.

frac Case tf

5% 15% 30% 45% 60%

a 0.10

b 1.00

c 10.0 3.1. Effective multiplication factor Table 4 shows values of effective multiplication factor keff for 15 cases in each size distribution. The

standard deviation is kept within 0.0001 for all cases. The numbers in parentheses show the relative

difference of keff from the value of constant distribution for the same case.

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Table 4 Effective multiplication factor keff (1 = 0.0001).

Case Constant Uniform Power Exponential Gaussian

5% a 0.1430 0.1444 (.98%) 0.1434 (.28%) 0.1444 (.98%) 0.1444 (.98%)

b 0.7340 0.7367 (.37%) 0.7350 (.14%) 0.7367 (.37%) 0.7367 (.37%)

c 1.0850 1.0864 (.13%) 1.0855 (.05%) 1.0862 (.11%) 1.0864 (.13%)

15% a 0.3248 0.3257 (.28%) 0.3251 (.09%) 0.3260 (.37%) 0.3256 (.25%)

b 0.9969 0.9970 (.01%) 0.9969 (.00%) 0.9969 (.00%) 0.9969 (.00%)

c 1.1133 1.1134 (.01%) 1.1134 (.01%) 1.1134 (.01%) 1.1134 (.01%)

30% a 0.5212 0.5221 (.17%) 0.5216 (.07%) 0.5221 (.17%) 0.5221 (.17%)

b 1.0797 1.0798 (.01%) 1.0798 (.01%) 1.0798 (.01%) 1.0798 (.01%)

c 1.1185 1.1186 (.01%) 1.1186 (.01%) 1.1186 (.01%) 1.1186 (.01%)

45% a 0.6688 0.6696 (.12%) 0.6694 (.09%) 0.6697 (.13%) 0.6696 (.12%)

b 1.1020 1.1021 (.01%) 1.1020 (.00%) 1.1021 (.01%) 1.1021 (.01%)

c 1.1195 1.1195 (.00%) 1.1196 (.01%) 1.1196 (.01%) 1.1195 (.00%)

60% a 0.7592 0.7595 (.04%) 0.7594 (.03%) 0.7595 (.04%) 0.7595 (.04%)

b 1.1093 1.1094 (.01%) 1.1094 (.01%) 1.1094 (.01%) 1.1094 (.01%)

c 1.1198 1.1198 (.00%) 1.1198 (.00%) 1.1198 (.00%) 1.1198 (.00%) From Table 4, the value of keff increases with packing fraction and optical thickness in fuel particles for all

the radii distributions. The size distribution effect on keff can be found by comparing the values of the five

distributions. Several interesting phenomena are observed: 1) the values of keff from uniform, exponential

and Gaussian distributions are almost the same for all the cases; 2) the constant distribution always gives

the smallest value of keff. The values for the power distribution are the second smallest; 3) the relative

difference in keff among the distributions becomes smaller as the packing fraction and optical thickness

increase. For cases b and c at volume packing fractions of 15% to 60%, the values of keff among the

distributions have no obvious difference. In summary, there is a noticeable difference of keff (as large as

1.00% in relative difference) in the cases with low packing fractions and low optical thickness. In these

cases, the system is more optically thin and more transparent to neutrons.

We can qualitatively interpret the differences in keff from the following perspective. Although the same

mean chord length in spheres is applied for different distributions, the average size and number of

sampled spheres are different at the same volume packing fraction. The average number of fuel particles

at different size distributions is shown in Table 5. The calculation is performed by using the formula

Vfrac/, where 3(4 / 3) ( )b

sphere aV r f r dr . The difference in the average size and number of

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fuel particles can result in different degrees of neutron channeling when a neutron travels through the

gaps (background material) between fuel particles at different size distributions. More fuel particles of a

smaller size (corresponding to the uniform, exponential, and Gaussian distributions) means less

channeling effect since fuel particles can better shield each other, and neutrons leaving a fuel particle and

traveling in the background can have higher probabilities to enter another fuel particle before leaking out

of the system. A direct neutronic effect is that the average neutron leakage rate at different distributions

becomes different, leading to the difference in keff. However, the channeling effect is significant (in

affecting the particle shadowing) only when the fuel particle volume packing fraction and/or optical

thickness are low. At these physical situations, neutrons have a high probability of escaping from fuel

particles, streaming into the background gaps (channels), and leaking out of the system. The channeling

effect (and fuel particle shadowing) is very sensitive to the average size and number of sparsely packed

fuel particles, which are different at different size distributions. Uniform, exponential and Gaussian

distributions produce more fuel particles with smaller average size than constant and power distributions.

These fuel particles can shield each other more significantly than bigger size fuel particles. The higher

particle shadowing effect results in a lower channeling effect and a reduction in leakage and thus higher

keff. As the packing fraction increases, neutrons that escape from the fuel particles see smaller gaps.

Fewer neutrons can stream in the background and leak out of the system. Neutron channeling effects

become weaker and the variations in the average number of fuel particles between distributions are

insignificant in affecting the neutronic behavior. Similarly, as the fuel particle optical thickness increases,

neutrons have a lower probability of escaping from fuel particles and streaming in the background. The

channeling effect also becomes weaker and therefore the size distribution effect is smaller.

Table 5 Average number of fuel particles with different size distributions. This is calculated based on the formula Vfrac/, where 3(4 / 3) ( )

b

sphere aV r f r dr .

frac Constant Uniform Power Exponential Gaussian

5% 11377 13182 11858 13180 13182

15% 34133 39547 35576 39542 39547

30% 68266 79095 71152 79085 79095 45% 102400 118642 106729 118627 118643

60% 136533 158190 142305 158170 158191 To further understand the neutronic effects due to the different particle size distribution, we select Case a

at 5% packing fraction for a neutronic investigation by calculating various total reaction rates in the

system. These reaction rates are listed in Table 6, including the absorption reaction rate, fission reaction

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rate and leakage rate. We can see that the sum of absorption and leakage rate is always 1, since a fission

neutron will either be absorbed by the fuel particle or leak out of the system. The leakage rates for the

uniform, exponential and Gaussian size distributions are lower than those for the constant and power size

distributions. This is consistent with our qualitative analysis in the previous paragraph. The absorption

rate and fission rate show an opposite tendency, compared to the leakage rate. Based on the multiplication

factor formula:

,fisseffabs leak

Rk

R R

(16)

a higher fission reaction rate means a higher value of keff . Note that Rabs+Rleak=1. For Case a at 5%

packing fraction, leakage is the dominant reaction in the system. A small change (-0.0012 from constant

to uniform as shown in Table 6) in the leakage rate due to the channeling effect at different size

distributions can lead to the same amount of change in the absorption rate (0.0012 from constant to

uniform as shown in Table 6). Due to the small value of absorption, the relative difference change in

absorption rate becomes noticeable (.94% from constant to uniform as shown in Table 6). This

subsequently leads to a similar relative difference change in fission rate and keff (.98% from constant to

uniform as shown in Table 4).

Table 6 Total rates (#/sec/source particle) of absorption, fission and leakage in fission systems for Case a at 5% packing fraction.

Quantity Constant Uniform Power Exponential Gaussian

Rabs 0.1277 0.1289 (.0012) 0.1281 0.1289 0.1289

Rfiss 0.0511 0.0516 (.0005) 0.0512 0.0516 0.0516

Rleak 0.8723 0.8711 (-.0012) 0.8719 0.8711 0.8711 When the optical thickness in fuel particles increases, for example Case c at the 5% packing fraction, the

absorption reaction becomes the dominant reaction, as shown in Table 7. We can see a similar change

(.0012 as shown in Table 7) in leakage rate and absorption rate from constant to uniform as the Case a at

the 5% packing fraction. However, due to the large absorption rate (close to 1), the relative difference

change in absorption rate becomes very small (.12% from constant to uniform as shown in Table 7). As a

result, the relative difference change in fission rate and keff also becomes very small (.13% from constant

to uniform as shown in Table 4). This means that the channeling effect is minor in affecting the keff at a

high optical thickness.

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Table 7 Total rates (#/sec/source particle) of absorption, fission and leakage in fission systems for Case c at 5% packing fraction.

Quantity Constant Uniform Power Exponential Gaussian

Rabs 0.9688 0.9700 (.0012) 0.9692 0.9700 0.9700 Rfiss 0.3875 0.3880 (.0005) 0.3877 0.3880 0.3880

Rleak 0.0312 0.0300 (-.0012) 0.0308 0.0300 0.0300 When the volume packing fraction increases, we see a similar tendency as seen when optical thickness

increases. To show this, we calculate the absorption, fission and leakage rate for Case a at 60% packing

fraction, which is shown in Table 8. At high packing fraction, the channeling effect becomes weaker,

which can be seen from the difference in the leakage rates at the different size distributions (-.0002 from

constant to uniform as shown in Table 8), compared to the 5% packing fraction (-.0012 from constant to

uniform as shown in Table 8). This is intuitive since fuel particles are densely packed and gaps between

fuel particles are smaller at all size distributions, therefore it becomes less possible that a neutron streams

through gaps to leak out compared with the lower packing fractions. Meanwhile, as the packing fraction

changes from 5% to 60%, the dominant reaction changes from leakage to absorption (absorption rate from

0.1277 to 0.6779). Due to a small change in leakage rate, which leads to a small change in the absorption

rate (0.0002 from constant to uniform as shown in Table 8), the relative change in the absorption rate as

well as the fission rate also becomes small. Therefore, a small relative difference change in keff is

expected.

Table 8 Total rates (#/sec/source particle) of absorption, fission and leakage in fission systems for Case a at 60% packing fraction.

Quantity Constant Uniform Power Exponential Gaussian

Rabs 0.6779 0.6781 (.0002) 0.6780 0.6781 0.6781

Rfiss 0.2711 0.2713 (.0002) 0.2712 0.2713 0.2713 Rleak 0.3221 0.3219 (-.0002) 0.3220 0.3218 0.3219

3.2. Volume average flux in fuel particles and background moderator The multiplication factor formula Eq. (16) can be further written as:

,

,

,fiss f F F Feffabs leak a F F F leak

R Vk

R R V R

(17)

16/28

where F is the volume-average flux in fuel particles defined as 1 ( )

FF V

F

dVV

r . For any fission

neutron, it will either be absorbed in the system (by the fuel) or leak out of the system. Therefore, the

denominator in Eq. (17) is always 1. Then,

, , ( ) .F

eff f F F F f F Vk V dV r (18)

It is clear that the multiplication factor is related to the flux integral in the fuel region, i.e. the average

total track length in the fuel particles per one fission source neutron. Different values of keff at different

size distributions indicate that the average neutron track length in fuel particles is affected by the size

distribution. By calculating the volume average flux in fuel particles, we can study how the flux is

affected by the size distribution. Moreover, the effect on the flux can reinforce the interpretation of the

size effect on keff.

The flux integral at different volume packing fractions (5%, 30% and 60%), optical thicknesses and

different fuel particle size distributions is calculated. Table 9 shows the flux integral in fuel particles.

Table 9 Flux integral in fuel particles in the cylindrical container [cm/sec/source particle]. The numbers in parentheses show the relative difference of flux integral from the value of constant distribution for the same case.

Case Constant Uniform Power Exponential Gaussian

5%

a 0.1277 0.1289 (.94%) 0.1281 (.31%) 0.1289 (.94%) 0.1289 (.94%) b 0.0655 0.0658 (.46%) 0.0656 (.15%) 0.0659 (.61%) 0.0658 (.46%)

c 0.0097 0.0097 (.00%) 0.0097 (.00%) 0.0097 (.00%) 0.0097 (.00%)

30%

a 0.4653 0.4662 (.19%) 0.4657 (.09%) 0.4662 (.19%) 0.4662 (.19%) b 0.0964 0.0964 (.00%) 0.0964 (.00%) 0.0964 (.00%) 0.0964 (.00%)

c 0.0099 0.0099 (.00%) 0.0099 (.00%) 0.0099 (.00%) 0.0099 (.00%)

60%

a 0.6779 0.6781 (.03%) 0.6780 (.01%) 0.6781 (.03%) 0.6781 (.03%) b 0.0991 0.0991 (.00%) 0.0991 (.00%) 0.0991 (.00%) 0.0991 (.00%)

c 0.0100 0.0100 (.00%) 0.0100 (.00%) 0.0100 (.00%) 0.0100 (.00%) From Table 9, some general conclusions can be drawn. For different size distributions at the same

packing fraction, as optical thickness increases, the flux integral in fuel particles decreases. At the same

optical thickness, as the packing fraction increases, the average number of fuel particles increases,

resulting in a larger track length in fuel particles. In terms of size distribution effect, a similar

phenomenon can be seen for the calculated flux integral in fuel particle as was found for keff. The

17/28

following phenomena are observed: 1) the values of flux integral from uniform, exponential and Gaussian

distributions are almost the same for all the cases, 2) the constant distribution always gives the smallest

flux integral in fuel particle; the values for the power distribution are the second smallest, 3) the

difference among the distributions is noticeable only for cases with low packing fractions and low optical

thickness. The difference becomes much smaller as the packing fraction and optical thickness increase.

Because the value of keff is proportional to the tallied flux integral in fuel particle as shown in Eq. (18), the

same changing tendency for keff has been observed in Section 3.1. In the size distributions with a larger

average number of fuel particles (uniform, exponential and Gaussian), fuel particles are more effective to

shadow each other. Neutrons on average travel more frequently in the fuel particles for these size

distributions than the constant and power distributions. This leads to a larger tallied track length during

each neutrons history.

Flux integral in the moderator is also tallied and is shown in Table 10. In Section 3.1, we have used the

channeling effect to interpret the size distribution effect on the multiplication factor. A higher value of the

flux integral implies a longer path length a neutron generates when it travels through the moderator. More

gaps between the fuel particles can provide a wider channel that neutrons fly through and cause the larger

flux integral. Therefore, the flux integral in the moderator can be regarded as a parameter that

quantitatively measures the level of channeling effect.

Table 10 Flux integral in the background moderator in the cylindrical container [cm/sec/source particle].

Case Constant Uniform Power Exponential Gaussian

5%

a 1.9204 1.9180 1.9195 1.9182 1.9179 b 0.9996 0.9906 0.9969 0.9888 0.9902

c 0.1267 0.1217 0.1252 0.1217 0.1217

30%

a 1.0387 1.0370 1.0380 1.0370 1.0370 b 0.2158 0.2148 0.2154 0.2148 0.2148

c 0.0223 0.0221 0.0222 0.0221 0.0221

60%

a 0.4242 0.4241 0.4242 0.4240 0.4241 b 0.0605 0.0605 0.0605 0.0605 0.0605

c 0.0061 0.0061 0.0061 0.0061 0.0061 From Table 10, it can be seen that for each case that is studied, constant and power size distributions

generate a larger flux integral in moderator than the other distributions. This indicates a larger channeling

effect in the fuel particle system packed with constant and power size distributions, where neutrons are

more likely to stream into the background gaps and have collisions with the moderator. At the same

18/28

packing fraction, as the optical thickness increases, more neutrons are absorbed within fuel particles and

fewer neutrons can stream into the background, resulting in a smaller flux integral. For the same optical

thickness, as the packing fraction increases, the moderator volume decreases, resulting in a smaller flux

integral in the matrix background. The decrease of flux integral with the increase of packing fraction or

optical thickness implies that the channeling effect decreases, so the size impact on the neutronics in the

system becomes small.

3.3. Fission density distribution in axial and radial directions

Fission density in a finite volume V is defined as 1 ( )fiss fVF dVV

r . The spatial distribution of

fission density is a measure of the fission power density distribution, which directly affects the heating

source distribution in the analysis of reactor thermal hydraulic behavior. Therefore, it is necessary to

investigate the fuel particle size distribution effect on the fission density distribution. To do this, we solve

the same eigenvalue problems in the cylindrical container as in Sections 3.1 and 3.2. The container is

uniformly divided into 10 axial layers or 10 radial cells with the same volume in each cell. The fission

density averaged to each neutron in the unit of [# of fission/sec/cm3] in each axial layer or radial cell is

calculated and a standard deviation of

19/28

30% packing fraction w/ optical thickness of 0.1 30% packing fraction w/ optical thickness of 10

60% packing fraction w/ optical thickness of 0.1 60% packing fraction w/ optical thickness of 10

Fig. 2. Axial fission density distributions at different size distributions over packing fraction of 5%, 30%,

60% and optical thickness of 0.1, 10.

5% packing fraction w/ optical thickness of 0.1 5% packing fraction w/ optical thickness of 10

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30% packing fraction w/ optical thickness of 0.1 30% packing fraction w/ optical thickness of 10

60% packing fraction w/ optical thickness of 0.1 60% packing fraction w/ optical thickness of 10

Fig. 3. Radial fission density distributions at different size distributions over packing fraction of 5%,

30%, 60% and optical thickness of 0.1, 10. From Fig. 2, for the low packing fraction and low cross section cases, a noticeable difference in the

spatial distribution of fission density can be observed among the size distributions. At the packing

fraction of 5% and the optical thickness of 0.1, the peak value of fission density with uniform distribution

is 1.29% larger than that with constant distribution in axial direction. However, the difference diminishes

as packing fraction and optical thickness increase. At the packing fraction of 60% and optical thickness of

10, the difference decreases to 0.03% in axial direction. This phenomenon is the same as was observed for

the multiplication factor.

Fig. 3 shows the fission density distribution in radial direction over different packing fraction and optical

thickness. The same with the axial situation, for the low packing fraction and low cross section cases, a

similar difference in the spatial distribution of fission density can be observed among the size

distributions. At the packing fraction of 5% and the optical thickness of 0.1, the peak value of fission

density with uniform distribution is 1.51% larger than that with constant distribution, and diminishes as

21/28

packing fraction and optical thickness increase. At the packing fraction of 60% and optical thickness of

10, the difference decreases to 0.05% in axial direction.

4. Fuel particle size distribution effect in unit cells of realistic reactor designs The study of size effect in one-group eigenvalue problems in Section 3 has shown noticeable differences

in neutronic solutions. The benefit of using one-group simulation in simplified geometry is that it can help

pinpoint the key transport phenomena that are directly affected by the fuel particle size distribution, and

provide a convenient way to understand major reasons behind the observed different results from the

perspective of neutron transport process. It should be emphasized that the effect of the fuel particle size

distribution varies not only with the volume packing fraction, optical thickness in the fuel particle, but

also with the system size, the background material absorption, and possible moderator/coolant region

around the stochastic fuel region. In practical reactor designs, configurations are more complicated than a

simplified stochastic fuel region studied in our one-group simulations. To further study the size

distribution effect, we utilize existing reactor designs and pack the fuel particles in unit cells of these

designs for a more realistic investigation by continuous energy simulations.

In this section, unit cells of three realistic reactor designs are packed with fuel kernels that have the same

size distributions as assumed in the one-group simulations. The size range is adjusted to be compliant

with the nominal design of TRISO fuel particles in each design. The three designs are based on prismatic

type VHTR, pebble-bed type VHTR, and innovative LWR loaded with FCM fuel. Only unit cells, i.e. a

fuel compact cell, a fuel pebble cell and a fuel pin cell, are simulated. The fuel compact cell consists of a

cylindrical fuel zone packed with TRISO fuel particles in a graphite matrix at 28.92% packing fraction.

The cylindrical fuel zone has a radius of 0.6225 cm and a height of 4.9276 cm, surrounded by a hexagonal

graphite region with reflecting boundary conditions on each side. The flat-to-flat distance of the hexagonal

graphite region is 2.1958 cm. The fuel pebble cell consists of a 2.5 cm radius spherical fuel zone and a 0.5

cm thick graphite shell. TRISO fuel particles are packed in the graphite matrix at 5.76% packing fraction.

A white boundary condition is applied to the outer graphite shell surface. The FCM fuel cell consists of a

cylindrical fuel zone 0.41 cm in radius and 4.0 cm in height, and the outer zirconium cladding has a radius

of 0.475 cm. The pins are surrounded by coolant water. The pitch of the fuel lattice is 0.61 cm. TRISO

fuel particles are packed in the fuel zone at 45% packing fraction. In the models of unit cells, material

compositions of the moderator and the background matrix in the original design are selected. Only fuel

kernels (assuming coating regions are homogenized with the background material) are randomly

distributed in the matrix background. Since only fuel kernels are packed, the volume packing fraction

becomes 2.613%, 0.988%, and 12.483% for the three unit cells, respectively. These values are based on

22/28

the nominal design of the VHTR designs and FCM fueled LWR designs. In fuel compact and fuel pebble

unit cells, nominal sizes of fuel kernel are .0175 cm and .025 cm in radius, which are selected as the

parameters in the constant size distribution functions. The parameters for other distribution functions are

determined following the same procedure as done in Section 2.1.1. Since the nominal size of fuel kernel

used in FCM fueled LWR unit cell is about 0.0375 cm in radius, we use the same parameter values as

used in one-group simulation study in Section 3. Table 11 summarizes the size distribution functions used

for the unit cell study. Fig. 4 shows the geometrical configurations of the three fuel cells.

Table 11 Summary of size distribution functions.

Size distribution functions Values of parameters

Fuel compact cell Fuel pebble cell Fuel pin cell

( ) ( )f r r R R = 0.0175 cm R = 0.025 cm R = 0.0375 cm

1( ) , .f r a r bb a

a = 0.01 cm b = 0.0221 cm a = 0.02 cm

b = 0.0289 cm a = 0.02 cm

b = 0.0478 cm

1 2( ) (1 ) ( ) ( )f r p f r pf r 2

1 3

3( )( ) , .( )

r af r a r RR a

2

2 3

3( )( ) , .( )

b rf r R r bb R

p = 0.48045 p = 0.48843 p = 0.47967

( ) , .rf r e a r b = 82.705 = 0.034 = 112.123 = 0.034

= 36.009 = 0.034

22 ( )

1( ) (1 / ) , .c r Rf r c e a r b c1 = 0.0121 c2 = 0.3333

c1 = 0.00893 c2 = 0.3333

c1 = 0.0278 c2 = 0.3333

(a) Compact cell at frac=28.92% (b) Pebble cell at frac=5.76% (c) LWR FCM cell at frac=45%

Fig. 4. Illustration of unit cell configurations.

23/28

The material composition of fuel kernel and graphite matrix/shell is shown in Table 12, adopted from

Idaho National Laboratory (MacDonald et al., 2003; Terry et al., 2006) and Oak Ridge National

Laboratory reports (Powers, 2013). For the fuel compact cell, the graphite matrix and hexagonal region

contain a 6.9 ppm natural boron impurity. For the fuel pebble cell, TRISO fuel particles with 4 ppm

natural boron concentration in the kernels are packed in the graphite matrix. The natural boron content of

the graphite matrix/shell is assumed to be 1.3 ppm. For LWR FCM fuel cell, 1000 ppm natural boron

concentration is assumed in the coolant water.

Table 12 Material composition in the three fuel cells.

Fuel compact cell Isotope and atom density (atom/barn-cm)

Fuel kernel 235U: 2.4750E-3; 238U: 2.1142E-2;

C: 1.1809E-2; O: 3.5427E-2 Graphite matrix/shell C: 8.6738E-2; 10B: 1.1910e-7; 11B: 4.7939e-7

Fuel pebble cell Isotope and atom density (atom/barn-cm)

Fuel kernel 235U: 3.9922E-3; 238U: 1.9245E-2; O: 4.6475E-2;

10B: 4.0641E-7; 11B: 1.6358E-6 Graphite matrix/shell C: 8.6738E-2; 10B: 2.2439e-8; 11B: 9.0320e-8

LWR FCM fuel cell Isotope and atom density (atom/barn-cm)

Fuel kernel 234U: 3.75478E-5; 235U: 4.94050E-3; 238U: 2.79586E-2;

N: 3.26073E-2; C: 3.29367E-5 Graphite matrix C: 4.80439E-2; Si: 4.80439E

Gap He: 1.88065E-4 Clad Zr: 4.33047E-2

Water Box H: 4.42252E-2; O: 2.21126E-2; 10B: 1.31990E-5; 11B: 5.31276E-5 A continuous energy Monte Carlo transport code MCNP5 (X-5, 2003) is used to simulate an eigenvalue

problem in three unit cells, where the stochastic distribution of fuel kernels is explicitly modeled. The

infinite multiplication factors, denoted as kinf, with the five different size distributions, are calculated by

averaging kinf from 100 independent physical realizations. In each realization, a total of 200 cycles with 50

inactive are simulated. In each cycle, a total of 100,000 histories are tracked. The standard deviation of

the ensemble-average kinf is less than 1E-5. Table 13 shows kinf values and the reactivity change in the

unit of dollar for different fuel particle distributions compared with the constant size distribution.

24/28

Table 13 Infinite multiplication factor kinf (1 = 1E-5) and reactivity change relative to the constant size distribution.

Fuel Configuration Constant Uniform Power Exponential Gaussian

Fuel compact cell 1.56020 1.56187 1.56078 1.56187 1.56200 Reactivity change (pcm) 0.00 167 58 167 180

Fuel pebble cell 1.63170 1.63191 1.63180 1.63183 1.63189 Reactivity change (pcm) 0.00 21 10 13 19

LWR FCM fuel cell 1.11854 1.12139 1.11865 1.12123 1.12141 Reactivity change (pcm) 0.00 285 11 269 287

The same tendency as in the one-group energy simulation is observed for the continuous energy

simulation: constant and power distributions generate smaller values than the other three distributions.

The maximum reactivity change reaches 180 pcm in the fuel compact cell, 21 pcm in the fuel pebble cell

and 287 pcm in the LWR fuel cell. Fuel particle size distribution effect is still noticeable in realistic

configuration designs except the fuel pebble unit cell. It should be noted that these three realistic

configurations still belong to low volume packing configurations studied in one-group problems in

Sections 2 and 3 because the volume packing fraction of fuel kernels (not the fuel particles) is low.

Therefore the neutron channeling effect is still appreciable. However, different from the simplified one-

group problem configuration, realistic unit cells include moderator regions around the stochastic fuel

region. The size of the stochastic fuel region and the range of the fuel kernel size are also different. These

differences complicate the level of the size distribution effect. Hence the magnitude of differences in

neutronic results among different size distributions becomes different and problem-dependent, which

implies a need for more future research in order to fully understand the complications.

5. Conclusions In this paper, we investigate the fuel particle size distribution effects on the neutronic behavior in a

stochastic medium. A series of radiation transport scenarios in 3-D stochastic particle systems has been

constructed. In these stochastic media, particles are composed of fissile material with a specific size

distribution. Five radii distributions are adopted for observation in this paper: constant, uniform, power,

exponential and Gaussian distributions, by keeping the mean chord length in sampled sphere particles the

same. Effective multiplication factors and fission density distributions are tallied in one-group energy

Monte Carlo simulations with variations of two factors: (1) volume packing fraction of the fuel particles

in container and (2) optical thickness within the fuel particles. After a thorough comparison of the tallied

results, it is found that the channeling effect of the fuel particles dominates the sensitivity of neutron

transport in stochastic particle systems.

25/28

It is determined that the extent of the channeling between the fuel particles is highly dependent on the

number of fuel particles. Even with the same volume packing fraction, different size distributions can

generate significantly different numbers of fuel particles in the stochastic region. When neutrons transport

within a stochastic medium, the case with a higher number of fuel particles has a stronger channeling

effect. With the same optical thickness for the five distributions at low packing fraction, neutron

channeling out of the stochastic system without interaction with the fuel particles is highly dependent on

the fuel particle number. Thus, the tallied results are very sensitive to the size distribution. The variation

in size distributions can generate as high as ~1.00% relative difference in keff value and ~1.50% relative

difference in peak fission power. As the volume packing fraction increases, the probability of neutrons

channeling out the system decreases. At very high packing fractions (e.g. 60%), this probability is

insignificant.

Similarly, with the same volume packing fraction for the five distributions and at low optical thickness,

the size distribution cases with a higher number of fuel particles lead to a much higher interaction

probability between neutrons and fuel particles. As optical thickness increases, the interaction probability

between neutrons and fuel particles increases. Because of the large number of total particles at high

packing fractions, variations in particle number between size distributions become less important to the

neutronic behavior.

Three realistic fuel cells including two VHTR designs and innovation LWR fuel ceramic design are

modeled with a continuous energy MC transport code. In the simulation, the fuel kernels are sampled

from five size distributions with kernel range covering realistic manufactured kernel sizes. The infinite

multiplication factors for the five size distributions are tallied for each fuel cell. Recognizable reactivity

changes over the constant size distribution from the four poly-sized distributions are observed as large as

180 pcm in the fuel compact cell, 21 pcm in the fuel pebble cell and 287 pcm in the LWR fuel cell,

indicating that the fuel particle size distribution still has a noticeable effect in realistic configuration

designs except the fuel pebble unit cell.

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