application of the chord method to obtain analytical...

Application of the Chord Method to Obtain Analytical Expressionsfor Dancoff Factors in Stochastic Media

Wei Ji*

Rensselaer Polytechnic InstituteDepartment of Mechanical, Aerospace, and Nuclear Engineering

Troy, New York 12180-3590

and

William R. Martin

University of MichiganDepartment of Nuclear Engineering and Radiological Sciences

Ann Arbor, Michigan 48109-2104

Received October 10, 2010Accepted February 8, 2011

Abstract – In this paper the chord method is applied to the computation of Dancoff factors for doublyheterogeneous stochastic media, characteristic of prismatic and pebble bed designs of the Very HighTemperature Gas-Cooled Reactor (VHTR), where TRISO fuel particles are randomly distributed in fuelcompacts or fuel pebbles that are arranged in a full core configuration. Previous work has shown that achord length probability distribution function (PDF) can be determined analytically or empirically andused to model VHTR lattices with excellent results. The key observation is that once the chord length PDFis known, Dancoff factors for doubly heterogeneous stochastic media can be expressed as closed-formexpressions that can be evaluated analytically for infinite and finite media and semianalytically for acollection of finite media.

Based on the assumption that the chord length PDF in the moderator region between two fuel kernelsin a VHTR compact or pebble is exponential, which was shown to be an excellent approximation inprevious work, closed-form expressions for Dancoff factors are derived for a range of configurations frominfinite stochastic media to finite stochastic media, including multiple finite stochastic media in a back-ground medium (e.g., a pebble bed core). Numerical comparisons with Monte Carlo benchmark resultsdemonstrate that the closed-form expressions for the Dancoff factors for VHTR configurations are accu-rate over a range of packing fractions characteristic of prismatic and pebble bed VHTRs.

I. INTRODUCTION

The first study on the reduction in the resonanceabsorption for a single fuel lump due to neighboring fuellumps was performed by Dancoff and Ginsburg1 in 1944.They derived a formula in terms of multiple integralsover solid angles and surfaces. The reduction in the res-onance absorption was later called the Dancoff factor.Preceding the work by Dancoff and Ginsburg, Dirac2

calculated the neutron multiplication factor for a finitevolume in 1943, where he introduced the chord methodto replace the complicated integrals over angles and sur-faces with one-dimensional ~1-D! integrals over the chordlength probability distribution function ~PDF! for thefinite volume under examination. Current work showsthat the complicated expression for the Dancoff factor asderived by Dancoff and Ginsburg can also be simplifiedconsiderably by defining a chord length PDF for the mod-erator region, resulting in a 1-D integral over the mod-erator chord length PDF. More important, this analysis*E-mail: [email protected]

NUCLEAR SCIENCE AND ENGINEERING: 169, 19–39 ~2011!

19

can be extended to obtain analytical Dancoff factors formore general arrangements of fuel and moderator, in-cluding doubly heterogeneous configurations.

When neutrons are slowed by the moderator into theresonance energy range, they may be absorbed by thefuel before having their next collision with the modera-tor. For a single fuel lump, this resonance absorption isreduced by a certain amount due to the presence of ad-jacent lumps. The neighboring lumps may block the pathof resonance neutrons to the fuel lump in question, re-ducing the probability that those resonance neutrons willreach that fuel lump before colliding with a moderator,hence reducing the “surface” resonance absorption. Thiseffect is equivalent to increased self-shielding of the fueland is called the “shadowing effect” in many referenc-es.3,4 Dancoff and Ginsburg derived a formula to calcu-late the reduction in surface resonance absorption due toneighboring absorbers, and this reduction is known asthe Dancoff-Ginsburg factor, or more commonly, the Dan-coff factor.4,5 The Dancoff factor plays an important rolein calculating collision probabilities and resonance inte-grals for reactor fuel lattices in neutronic analysis.6–8

For the analysis of the Very High Temperature Gas-Cooled Reactor ~VHTR!, it is essential to generate ac-curate few-group cross sections for use in global reactorcalculations.9 In particular, one needs to account for thestochastic distribution and double heterogeneity of theTRISO fuel particle in order to obtain reasonable few-group cross sections that correctly account for resonanceabsorption. Although general-purpose Monte Carlo codessuch as MCNP5 ~Ref. 10! can analyze VHTR configu-rations from unit cell calculations to full-core calcula-tions with excellent results,11–14 the computational timeis prohibitive, especially if one considers depletion andfeedback effects. As a result, the conventional method-ology for VHTR analysis has been based on determinis-tic transport methods such as MICROX-2 ~Ref. 15! thatdepend on user-specified Dancoff factors to account forthe double heterogeneity. These Dancoff factors may becalculated by Monte Carlo or semianalytical0analyticalmethods.16–18 In this paper, analytical formulas for Dan-coff factors based on chord length PDFs are derived.Closed-form expressions for the Dancoff factor are ob-tained for infinite geometry, an infinite height cylinder, afinite height cylinder, and a finite sphere. Closed-formexpressions have also been obtained for finite configu-rations of infinite height cylinders, finite height cylin-ders, and spheres, but these expressions include chordlength PDFs that may need to be evaluated empiricallydue to the complexity of analyzing finite geometries.

In the past decade, three papers16–18 have been pub-lished on analytical calculations of Dancoff factors forpebble bed and prismatic-type reactors. These include~a! infinite medium Dancoff factors, ~b! finite mediumDancoff factors for an isolated fuel pebble or fuel com-pact ~intrapebble or intracompact!, and ~c! finite me-dium Dancoff factors for a collection of fuel pebbles or

fuel compacts ~interpebble and intercompact!, includingaccounting for the coating region in a TRISO fuel parti-cle.18 In all of these works, the analytical expressions forthe Dancoff factors are based on complex derivationsthat involve complicated multiple integrals over sur-faces and solid angles. An alternative derivation of theDancoff factor based on chord length PDFs is developedin this paper that replaces these complicated integralswith 1-D integrals over chord length PDFs.

In this work, a general formulation of the Dancofffactor is given, starting with Dancoff and Ginsburg’soriginal procedure.1 It is shown how the expression forthe Dancoff factor is considerably simplified with thechord length PDF, similar to what Dirac2 did when hestudied neutron multiplication. This expression avoidsthe complicated double integrals over surface and solidangle that were obtained by Dancoff and Ginsburg. How-ever, one needs to know the chord length distributionfunction for the moderator region, which has been ad-dressed in previous works by the authors.19,20

The new expressions for the Dancoff factor are ap-plied to the calculation of Dancoff factors for TRISOfuel kernels in several VHTR configurations. These ex-pressions for the Dancoff factor are based on a “single-sphere” model of the TRISO fuel region, shown inFig. 1a, which corresponds to treating the fuel region asa stochastic mixture of fuel kernels in a backgroundmedium consisting of homogenized matrix and coat-ings. We have also examined an alternative approach,the “dual-sphere” model, shown in Fig. 1b, which mod-els the TRISO fuel region as a stochastic mixture ofmicrospheres including fuel coatings. This model is at-tractive because the stochastic packing is constrainedby the presence of the fuel coatings, which leads to aminimum separation distance between any two kernels.This in turn impacts the chord length PDF and the cal-culation of the Dancoff factors. However, the resultantDancoff factors are not as accurate with the dual-spheremodel as with the single-sphere model. This is cur-rently under examination.

Analytical formulas for the Dancoff factor are ob-tained using the analytical chord length PDFs derived ina previous paper19 for VHTR fuel. These closed-formexpressions are used to compute Dancoff factors for bothinfinite medium and finite medium configurations rep-resentative of pebble bed and prismatic-type VHTRs,and excellent agreement is obtained compared to MonteCarlo benchmark results.

The remainder of this paper is organized as follows.Section II includes a derivation of the analytical formulafor the Dancoff factor for an infinite medium of fuellumps, with numerical results for VHTR configurations.Section III extends the infinite medium analysis to ob-tain an expression for the average Dancoff factor for afinite stochastic medium of fuel lumps, where an addi-tional chord length PDF is defined for the finite body~e.g., a pebble! to account for neutron escape. Section IV

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NUCLEAR SCIENCE AND ENGINEERING VOL. 169 SEP. 2011

further extends the analysis to obtain an analytical ex-pression for the average Dancoff factor for multiple fi-nite media, each consisting of fuel lumps, such as a pebblebed core. This requires definition and determination of athird chord length PDF for the background region thatcontains the multiple finite media ~e.g., the chord lengthPDF between pebbles in an annular core!. Conclusionsare given in Sec. V.

II. DANCOFF FACTOR FOR AN INFINITEMEDIUM OF FUEL LUMPS

In this section, the general formulation of the Dan-coff factor for a single fuel lump in an infinite medium isgiven. The derivations are applicable to fuel lumps witharbitrary shapes that are uniformly dispersed in a back-ground moderator with arbitrary packing schemes. Thesingle-sphere model is used, and results are comparedwith benchmark results.

II.A. General Analytical Derivation

Consider the situation in which fuel lumps with ar-bitrary ~nonreentrant! shapes are dispersed randomly ina background moderator. Fission neutrons are generatedinside the fuel lumps and are slowed by the moderator.After several scatterings in the moderator, some of theneutrons may enter the resonance energy range and beabsorbed. To model this, a uniform, isotropic source ofresonance energy neutrons is assumed in the moderator.

Given the source density Q neutrons0cm3{s at apoint r ' in the moderator, the incremental uncollidedflux of neutrons entering a fuel lump at the surfacepoint r due to that source is

dC~r ' r r! � ~Qd3r '!{e�6r�r ' 60l

4p6r � r ' 62, ~1!

where l is the mean free path of resonance neutrons inthe moderator and it is assumed that no other fuel lumpis between r ' and r; i.e., r ' and r can see each other onlythrough the moderator region. The incremental rate thatneutrons enter the fuel lump through a small surfaceelement dA at r is

dJ ~r ' r r! � dC~r ' r r!cos udA

� ~Qd3r '!{e�6r�r ' 60l

4p6r � r ' 62cos udA , ~2!

where u is the angle between r � r ' and the outer normalof dA. The total rate that neutrons enter a fuel lumpwithout colliding with the moderator is

J � ��dJ ~r ' r r!

� Q�dA cos u�d 3r 'e�6r�r ' 60l

4p6r � r ' 62. ~3!

Now using a spherical coordinate transformationd 3r ' � s2dsdV, where r is the origin, s � 6 r � r ' 6, anddV is a solid angle element about direction ~u, w!, J maybe expressed as

J �Q

4p�dA�dV cos u�

0

l

e�s0l ds

�Ql

4p�dA�dV cos u~1 � e�l0l ! , ~4!

Fig. 1. VHTR fuel particle models. The dual-sphere model is defined here, but no results are reported for this model.

DANCOFF FACTORS IN STOCHASTIC MEDIA 21


where l is the length of the chord through the point r inthe direction ~u, w! to the surface of any other fuel lumpsthat can “see” the fuel lump in question. After simplemanipulation, Eq. ~4! can be written as

J �QlA

4 �1 �

�dA�dV cos u~e�l0l !

�dA�dV cos u � , ~5!

where A is the total surface area of the recipient fuellump. This is the equation Dancoff and Ginsburg ob-tained in their paper.1 Now it can be easily seen that thefirst part in Eq. ~5! is the total rate that resonance neu-trons enter the fuel lump if only one fuel lump exists inthe moderator. The second part is the reduction in theentering rate due to neighboring lumps. The complicatedsecond term in the bracket is called the Dancoff factor C,which is the fractional reduction in surface resonanceabsorption due to the presence of the neighboring fuelregions:

C �

�dA�dV cos u~e�l0l !

�dA�dV cos u

. ~6!

Next, introduce the PDF for the distribution of chordlengths in the moderator region, as Dirac did when hestudied neutron multiplication for a solid of arbitraryshape.2 According to Dirac, the chord length distributionPDF f ~l ! is defined such that for any function g~l !,

� f ~l !g~l ! dl �

�dA�dV cos u@g~l !#

�dA�dV cos u

. ~7!

Then Eq. ~6! becomes

C � � f ~l !e�l0l dl . ~8!

The relative simplicity of Eq. ~8! compared to Eq. ~6!is clear—the four-dimensional ~4-D! integral over sur-face and angle has been replaced by a 1-D integral overthe chord length distribution. This expression for the Dan-coff factor in terms of the chord length PDF was firstpublished by Sauer21 in 1963, but with somewhat differ-ent notation. Also, Sauer did not generalize his result tofinite media, as will be discussed later in this paper. Asimilar result for the first flight escape probability wasgiven by Case et al.22 in 1953, but no mention was madeof a corresponding expression for the Dancoff factor. Inspite of these early references, there has been essentiallyno mention of the use of chord length PDFs for the cal-

culation of Dancoff factors in stochastic media in thepast 40 years, even though many papers have been pub-lished on the calculation of Dancoff factors for TRISOfuel kernels.7,16–18,23

As a result of Eq. ~8!, the calculation of the Dancofffactor is reduced to determining the chord length PDFbetween fuel lumps in the moderator. However, earlierwork by the authors yielded an analytical expression forthe chord length PDF ~Ref. 19! in a stochastic medium,and this will be used to determine the Dancoff factor.

In addition to deriving Eq. ~4!, Dancoff performedthe double integral over two fuel lump surfaces to eval-uate C. However, the chord method expression for C inEq. ~8! is not only a far simpler equation, it can be eval-uated easily, avoiding the substantial computational costsof evaluating the double integral, and it can be applied tostochastic media such as the distribution of TRISO fuelkernels in a VHTR configuration.

Also, Eq. ~8! admits another physical explanationfor the Dancoff factor. The original derivation was basedon the physical fact that resonance neutrons are createduniformly in the moderator and reach a fuel lump with-out collision. However, Eq. ~8! suggests another physi-cal interpretation. If resonance neutrons are emitted fromthe surface of a fuel lump and travel through the moder-ator toward another fuel lump, then f ~l !dl is the proba-bility that the distance to the next fuel lump is within~l, l � dl ! and e�l0l is the probability that the neutrontraverses the distance l without a collision. In this sense,the Dancoff factor C can be defined as an average prob-ability that resonance neutrons escaping from a fuel lumpwill reach another fuel lump without experiencing anycollisions with the moderator in between. This definitioncan be found in many references; it is equivalent to theoriginal definition and is a consequence of the reciproc-ity theorem.4,5

II.B. Dancoff Factors for Fuel Kernelsin an Infinite Medium

In this section, Eq. ~8! is used to derive an analyticalformula for the Dancoff factor for an infinite medium offuel kernels in VHTR configurations. Numerical resultsusing this expression are obtained and compared to bench-mark Monte Carlo results.

II.B.1. Simplified Physical Models andAssociated Mathematical Models in VHTR

In VHTR designs, either prismatic type or pebblebed type, TRISO fuel particles are randomly distributedin fuel compacts or fuel pebbles, respectively. Previousanalysis has shown that homogenizing the four coatinglayer regions with the graphite matrix region does notaffect the neutronic analysis11 of the microsphere cell,which consists of the microsphere and its associated ma-trix region. Although the four coating regions and the

22 JI and MARTIN


graphite matrix region are distinct regions for the Dan-coff factor calculation, these regions have identical crosssections obtained by homogenizing the materials in thesefive regions.

II.B.2. Analytical Derivations

The single-sphere model chord length PDF f ~l ! waspreviously derived by the authors19 and is given here:

f ~l ! �1

^l &{e�l0^l & , 0 � l � ` , ~9!

where

^l& [4r

3{

1 � frac'

frac'

is the mean chord length between two fuel kernels andfrac '� the ratio of total fuel kernel volume to the wholemedium volume.

Substituting f ~l ! into Eq. ~8!, the infinite mediumDancoff factor with the single-sphere model is obtained:

C` ��0

`

e�x0l{1

^l &{e�l0^l &{dl �

1

1 �1

l{^l &

�1

1 �1

l{

4r

3{

1 � frac '

frac '

. ~10!

It should be noted that a critical VHTR design pa-rameter, known as the “volume packing fraction” forfuel particles in VHTR, is normally used. It is defined asfrac � the ratio of total microsphere fuel particle volumeto the whole medium volume and is related to frac ' byfrac ' � frac{~r0R!3. Unless specified, the volume pack-ing fraction mentioned in this paper refers to frac.

II.B.3. Early Calculations of Dancoff Factorsfor Infinite Stochastic Media

The earliest papers that discussed the evaluation ofthe Dancoff factor for an infinite medium with grainstructure are those by Lane et al.24 and Nordheim25 in1962. These papers gave a result similar to Eq. ~10!, butwith a different mean chord length ^l & � 4r03{~10frac ' !. Since they used a classical atomic structuremodel to obtain the mean chord length, which assumesthat the fuel kernels are infinitesimal, the mean chordlength was overestimated. Their derivations miss the fac-tor ~1 � frac ' ! in the numerator, which accounts for thefinite size of the grains, hence leading to an incorrecthomogeneous limit.25,26 Although Nordheim had pointedout this homogeneous limit inconsistency, he did not givea solution to solve it. Lewis and Connolly26 also noted

this and used an alternative result due to Bell27 to obtainthe correct homogeneous limit. It is clear that this incon-sistency was due to the incorrect evaluation of the meanchord length in the moderator. This has been corrected inour expression for the infinite medium Dancoff factor inEq. ~10!, which yields the correct homogeneous limit forthe escape probability.

Also, Lane et al.24 suggested a method to accountfor the coating region by changing the lower integrallimit from 0 to 2d in Eq. ~10!. However, this treatmentgave poor results since the chord length PDF should alsobe changed if a coating region is added to the fuel kernel,but this was not done by Lane. One needs to simulta-neously change the PDF to account for the coating re-gion as well as modify the lower limit of the integral.This is what the dual-sphere model accomplishes and isthe motivation for continuing our examination of thismodel.

II.C. Monte Carlo Benchmark Calculation

Equation ~10! is the analytical formula for infinitemedium Dancoff factors for the single-sphere model andis based on Eq. ~8!. To verify the accuracy of this for-mula, Monte Carlo benchmark simulations were per-formed to calculate the following quantities:

1. Dancoff factors for fuel kernels in an infinitestochastic medium as a function of volume packing frac-tion. These results are compared directly with the pre-dictions from Eq. ~10!.

2. empirical chord length PDFs between fuel ker-nels in an infinite stochastic medium. These empiricalPDFs are used for a self-consistent verification of Eq. ~8!.

The benchmark Monte Carlo simulations employconventional ray-tracing methods to directly track neu-tron trajectories in explicit stochastic mixtures of ker-nels. The stochastic mixtures are generated with amodified version of the fast random sequential addition~RSA! algorithm28,29 to generate multiple realizationsof the stochastic media at different volume packingfractions.

A Monte Carlo code was written to calculate Dancofffactors of fuel kernels in an infinite medium. Dependingon the packing fraction, approximately 5 million to 10 mil-lion TRISO fuel particles are dispersed randomly in agraphite background region. Neutrons are emitted uni-formly from the surface of fuel kernels with a cosine cur-rent distribution. The number of neutrons that successfullyreach another fuel kernel without having a collision in thegraphite region is tallied, and the Dancoff factor is com-puted as the ratio of this number to the total number ofneutrons emitted. A total of 100 physical realizations ofthe stochastic medium are performed, and 10 million neu-trons are emitted per realization. The final Dancoff factoris ensemble averaged over the 100 realizations.



Geometry and composition quantities used for theDancoff factor calculations and Monte Carlo simula-tions are given in Table I. The geometry data correspondto the NGNP Point Design,30 and cross-section data arefrom the Brookhaven National Laboratory website31 eval-uated at the 6.67-eV resonance of 238U.

To compute the actual chord length PDF betweenfuel kernels at different volume packing fractions, thesame geometry and source distribution are used as in thebenchmark calculation for the Dancoff factors, exceptthe number of fuel particles is increased to 10 million to20 million, and 500 000 neutrons are emitted for each ofthe 100 realizations. Figure 2 shows the results as a func-tion of volume packing fraction from 5.76% ~typical of apebble bed reactor! to 28.92% ~typical of a prismaticreactor!. All the PDFs are nonzero beginning with thechord length 2d � 2~R � r! � 0.043 cm, as expected.The PDFs quickly increase to a peak value and decayexponentially for long chord lengths. The log-linear scaleplot verifies that the chord length PDF is well approxi-mated by an exponential distribution and may be usedfor the calculation of Dancoff factors for this range ofpacking fractions. Similar results have been obtained byother researchers.32

II.D. Numerical Results

Table II compares Dancoff factors predicted withEq. ~10! with the benchmark Monte Carlo results. The

results are excellent, within 1.1% for all volume packingfractions, with somewhat better results for higher pack-ing fractions. Interestingly, all the predicted Dancoff fac-tors are higher than the benchmark results.

The introduction of the chord length PDF makes theDancoff factor calculation succinct and mathematicallysimple and yields excellent results for the single-spheremodel. Equation ~8! is a fundamental equation for theDancoff factor not only in an infinite medium but also ina finite medium, as will be seen in later sections. Be-cause of its importance, an additional comparison is madeto verify its self-consistency; i.e., if the exact chord lengthPDF is used in Eq. ~8!, the resultant Dancoff factor shouldbe exact. This provides additional assurance of the meth-odology that employs Eq. ~8!. To do this, Dancoff fac-tors are calculated using an empirical chord length PDFas shown in Fig. 2 and compared with the Monte Carlobenchmark results. As discussed earlier, the empiricalchord length PDF for the fuel kernels was based on anexhaustive set of Monte Carlo simulations in randommixtures of fuel particles and may be considered close toexact. Therefore, using this empirical PDF in Eq. ~8!essentially constitutes a Monte Carlo benchmark calcu-lation for the Dancoff factor and should yield the sameresults. Table III compares the Dancoff factors calcu-lated with the empirical PDF in Eq. ~8! with the MonteCarlo benchmark results, and the agreement is within0.3%, with no observable trend with packing factor. Weconclude that our methodology for estimating Dancofffactors based on chord length PDFs is accurate andself-consistent.

III. DANCOFF FACTOR FOR A FINITEMEDIUM OF FUEL LUMPS

In the previous section, analytical formulas ofDancoff factors for an infinite stochastic medium wereshown to be accurate for packing factors representativeof prismatic and pebble bed VHTRs. Although this is an

TABLE I

Values of Parameters

Parameter Value Unit

r 0.0175 cmR 0.0390 cm2d � 2~R � r! 0.043 cm10l 0.4137 cm�1

Fig. 2. Chord length PDFs between two fuel kernels.

24 JI and MARTIN


interesting theoretical result, the practical impact is notsignificant when one is faced with analyzing a finite me-dium, such as a cylindrical fuel compact or sphericalfuel pebble that is filled with TRISO fuel particles. Inthis case the average Dancoff factor for a TRISO fuelkernel in the finite medium is needed to determine space-dependent few-group cross sections.

III.A. General Derivations

The key observation is that Eq. ~8! can be used tocalculate the Dancoff factor for a single fuel lump in afinite medium if the upper limit is set to a finite valuerelated to the position of the fuel lump in the mediumand the direction of neutron travel, as shown in Fig. 3.Moreover, this is essentially the same situation that isencountered when computing the average escape proba-bility for a point source in a finite medium, which wassolved using Dirac’s chord method. A similar analysisleads to another analytical formula for the average Dan-coff factor in a finite stochastic medium of fuel lumps, asdiscussed below.

To obtain the average Dancoff factor for all the fuellumps in the volume, the integral in Eq. ~8! needs to beperformed over the surface area of each fuel lump and

over the entire finite medium, accounting for every fuellump. This would be a cumbersome integration. How-ever, if the chord method is employed to perform thecalculation as was done in Sec. II, the calculation is much

TABLE II

Analytical Results with Single-Sphere Model Compared to Benchmark Results

VolumePackingFraction

AnalyticalFormula

Result C`

@Eq. ~10!#

Monte CarloBenchmarkResult C B,`

~1s!Difference

C` � C B,`

Relative Error~C` � C B,` !0C B,`

~%!

0.0576 0.3515 0.3477 ~0.0002! 0.0038 1.090.10 0.4857 0.4820 ~0.0002! 0.0037 0.770.15 0.5873 0.5841 ~0.0002! 0.0032 0.550.20 0.6559 0.6534 ~0.0001! 0.0025 0.380.25 0.7054 0.7029 ~0.0001! 0.0025 0.350.2892 0.7353 0.7331 ~0.0001! 0.0021 0.30

TABLE III

Analytical Results from Eq. ~8! Using Simulated PDF Compared to Benchmark Results

VolumePackingFraction

AnalyticalFormula

Result C A,`

@Eq. ~8!#

Monte CarloBenchmarkResult C B,`

~1s!Difference

C A,` � C B,`

Relative Error~C A,` � C B,` !0C B,`

~%!

0.0576 0.3471 0.3477 ~0.0002! �0.0006 �0.170.10 0.4810 0.4820 ~0.0002! �0.0010 �0.210.15 0.5827 0.5841 ~0.0002! �0.0014 �0.240.20 0.6516 0.6534 ~0.0001! �0.0018 �0.270.25 0.7014 0.7029 ~0.0001! �0.0015 �0.210.2892 0.7316 0.7331 ~0.0001! �0.0015 �0.20

Fig. 3. Fuel lumps in finite volume.



simpler. Here one takes advantage of the fact that thefuel lumps are randomly distributed in the finite me-dium. The physical process to be simulated is summa-rized as follows: A source of resonance energy neutronsis uniformly distributed on the surface of a fuel lump,and a resonance neutron is emitted outward with a co-sine current angular distribution. The probability that aneutron enters another fuel lump without colliding withthe intervening moderator is calculated and averaged overthe surface neutron emissions. This average probabilityis only for a specific fuel lump at a specific locationinside the solid body. Since the fuel lump could be any-where in the volume, we need to average this to accountfor fuel lumps at different locations in the volume. How-ever, it should be noted that a uniform distribution offuel lumps with uniform cosine current surface sourcesis equivalent to a uniform, isotropic volumetric source inthe moderator because the surface source can appear any-where in the volume with equal probability. At this point,the mathematical derivation becomes equivalent to thatperformed by Case et al.22 and Bell and Glasstone5 whenthey calculated escape probabilities for finite volumes.This derivation is given below.

In the moderator with a uniform isotropic volumet-ric source, a resonance energy neutron is generated indVdV with probability dVdV04pV, traveling in the di-rection dV about V. Since fuel lumps are randomly lo-cated in the moderator, this neutron may be regarded asescaping the fuel lump with a cosine current distribution.The maximum distance along the neutron trajectory toenter another fuel lump within the finite medium is de-noted and is determined by the distance to the boundaryof the finite medium along the chord determined by ~r, V!.Strictly speaking, this distance should be reduced by theaverage chord length in the fuel lump since a fuel lumpcannot overlap the outer boundary, but this has not beendone in this work. Since the probability of traversingdistance l without a collision in the moderator is givenby e�l0l , the following expression for the Dancoff factoris obtained:

C~l ! � �min_d

l

f ~l ' !e�l '0l dl ' , ~11!

where min_d is a problem-dependent lower limit ~suchas the minimum distance between two TRISO fuel ker-nels due to the coatings! and l is the distance to theboundary of the finite medium along the neutron trajec-tory. This is the Dancoff factor for a specific neutronemitted randomly from the surface of a random fuel lumpwithin the finite medium. The probability of choosingthis specific neutron is dVdV0~4pV !; hence, the aver-age Dancoff factor over all fuel lumps in the volume isgiven by

C intra � �� 1

4pVC~l ! dV dV . ~12!

Note that l is a function of ~r, V! and min_d is a constantdepending on the TRISO fuel dimensions. Equation ~12!is a complicated expression for the average Dancoff fac-tor due to the implicit dependence of l on ~r, V!. FromFig. 4, dV � ~n{V!dAdl, so Eq. ~12! becomes

C intra �1

4pV�dA�dV cos u�

min_d

L

C~l ! dl , ~13!

where min_d has the same meaning as in Eq. ~11! and Lis now the maximum distance to the boundary. Equation~13! is just as complex to evaluate as Eq. ~12!, but it is ina form where the chord method can again be used toadvantage. Define

G~L! � �min_d

L

C~l ! dl

and use

�dA�dV cos u � pA ,

where A is the total surface area of the finite medium,yielding

C intra �pA

4pV

�dA�dV cos uG~L!

�dA�dV cos u

�1

^L&

�dA�dV cos uG~L!

�dA�dV cos u

, ~14!

Fig. 4. Volume element division.

26 JI and MARTIN


where ^L& � 4V0A, the mean chord length for the finitemedium. As was done earlier with Eq. ~7! in Sec. II.A,introduce the chord distribution function F~L! for thefinite medium, leading to the following simple expres-sion for C intra :

C intra �1

^L&�F~L!G~L! dL , ~15!

or a more explicit form,

C intra �1

^L&�dLF~L!�

min_d

L

dl�min_d

l

dl ' f ~l ' !e�l '0l .

~16!

The complicated expression for the average Dancoff fac-tor in a finite medium of fuel lumps, which consists ofmultiple 4-D integrals over surface and angle domains,is reduced to a straightforward multiple integral by usingtwo chord length distribution functions: ~a! the PDF f ~l !for the distribution of chord lengths between fuel lumpsand ~b! the PDF F~L! for the distribution of chord lengthsinside the finite medium.

III.B. Dancoff Factor Calculationsfor a Finite Medium

In this section, average Dancoff factors for finiteVHTR geometries, including fuel compacts and fuel peb-bles, will be calculated. For a single-sphere model, sub-stituting f ~l ! from Eq. ~9! into Eq. ~16!, the intravolumeDancoff factor with the single-sphere model is obtained:

C intra �1

^L&�dLF~L!�

0

L

dl�0

l

dl '1

^l &e�l '0^l &e�l '0l ,

~17!

where we have set min_d � 0 due to the single-spheremodel assumption.

Now define an effective cross section S* :

S* �1

l*�

1

l�

1

^l &, ~18!

which leads to the following expression after severalstraightforward manipulations:

C intra �l*

^l &�1 �

l*

^L&�~1 � e�L0l* !F~L! dL� . ~19!

It is easy to see from Eq. ~10! that l*0^l & is just C` .It is also found with more careful observation that thesecond term in the bracket is just the first flight escapeprobability for a finite medium with a total cross sectionS* ~see page 22 in Ref. 22!. The effective cross sectionS* is identical to the modified total cross section for a

two-region lattice in equivalence theory when the “es-cape” cross section 10^l & is added to the total crosssection.5

This leads to a very concise result:

C intra � C` @1 � Pesc* # . ~20!

Thus, the average intramedium Dancoff factor is theproduct of the infinite medium Dancoff factor and thefirst flight collision probability for the finite medium.Bende et al.16 and Talamo17 also found a similar expres-sion for C intra for fuel pebbles and fuel compacts, respec-tively. However, their analyses are somewhat morecomplicated due to a different definition of S* and thecomplicated integral expression for Pesc

* .The introduction of the chord length PDF F~L! in

the expression of C intra in Eq. ~19! makes the formulamore flexible, in that it can handle different finite mediawith arbitrary ~nonreentrant! shapes filled with ran-domly distributed fuel particles. Next, as an applicationto VHTR analysis, Eqs. ~19! and ~20! are used to deriveintrapebble and intracompact Dancoff factors in terms ofbasic geometry parameters.

A fuel pebble is composed of two concentric spheres:an inner sphere with a typical radius R1 � 2.5 cm and anouter sphere with a typical radius R2 � 3.0 cm. The innersphere is the fuel zone with fuel particles randomly dis-tributed within it, and the outer spherical shell is graphite.

The chord length PDF for a sphere of radius R1 iswell known ~see Appendix of Ref. 33!:

F~L! �L

2R12

, 0 � L � 2R1 . ~21!

Substituting this into Eq. ~19!, the following expressionfor Pesc

* is found after several algebraic manipulations:

Pesc* �

3

4 � l*

R1��

3

4 � l*

R1�2

e�2~R1 0l* !

�3

8 � l*

R1�3

@1 � e�2~R1 0l* ! # . ~22!

This leads to the following equation for the Dancoff factor:

C intra � C` 1 �3

4 � l*

R1��

3

4 � l*

R1�2

e�2~R1 0l* !

�3

8 � l*

R1�3

@1 � e�2~R1 0l* ! # . ~23!

As verification, when R1 r `, C intra r C` .In the prismatic VHTR, the fuel compacts are ar-

ranged in cylindrical columns of 793-cm height. Al-though these are essentially infinite cylinders, anexpression for C intra will be derived for both infiniteand finite cylinder cases.



For an infinite cylinder region, the chord length PDFwas derived by Case et al.22:

F~L! �16Rc

2

pL3 �0

x0 x 4 dx

M~L204Rc2! � x 2M1 � x 2

,

x0 � �1 L � 2Rc

L

2Rc

L � 2Rc .~24!

Using the above PDF and results from Corngold34 forPesc* in terms of Bessel functions, one finds

Pesc* �

2

3$2t 2 @I0~t !K0~t ! � I1~t !K1~t !#

� t @I0~t !K1~t ! � I1~t !K0~t ! � 2#

� I1~t !K1~t !% , ~25!

where t � Rc 0l* . Substituting this into Eq. ~20!, the fol-lowing result is obtained:

C intra � C` �1 �2

3$2t 2 @I0~t !K0~t ! � I1~t !K1~t !#

� t @I0~t !K1~t ! � I1~t !K0~t ! � 2#

� I1~t !K1~t !% . ~26!

As verification, when Rc r`, t r`, so Pesc* � ~102! �

~10t ! � ~3032! � ~10t 3 ! � {{{ r 0; hence, C intra r C` .For a finite cylinder, the analytical derivation of F~L!

has been studied by many researchers.35,36 Formulas forright circular cylinders and general cylinders have beenfound, but they are given in complicated forms. How-ever, by using some simple mathematical transforma-tions, researchers37,38 have derived an expression for theescape probability Pesc

* for a finite cylinder, which can beused directly to obtain C intra . According to Marleauet al.,37 the following results for the escape probabilitieshave been obtained:

Pesc, t&b* �

l*

H�E3~0! � E3� H

l*��

4

pD 2

l*

H

� �0

DE3� t

l*�� E3� ~t 2 � H 2 !102

l*�

� ~D 2 � t 2 !102 dt , ~27!

Pesc, cyl* �

4

pD 2H�

0

D

t 2 dt�0

H

~H � u!

�

exp�� ~t 2 � u2 !102

l*�~D 2 � t 2 !102

~t 2 � u2 !302 du ,

~28!

Pesc, cyl* �

4

pD 2H�

0

D

t 2 dt�0

H

~H � u!

�

exp�� ~t 2 � u2 !102

l*�~D 2 � t 2 !102 � D

~t 2 � u2 !302 du

�4

pD 2H

D

2

� H 2 ln� D � ~D 2 � H 2 !102

H�

� D~D 2 � H 2 !102 � D 2 , ~29!

where Pesc, t&b* is the axial escape probability across the

top and bottom surfaces, Pesc, cyl* is the radial escape

probability across the cylindrical surface, and D � 2Rc .Adding these two escape probabilities yields the totalescape probability for a finite cylinder:

Pesc* � Pesc, t&b

* � Pesc, cyl* . ~30!

So the final result is

C intra � C`~1 � Pesc* ! . ~31!

III.C. Monte Carlo Benchmark Calculation

Monte Carlo benchmark simulations are performedto verify the accuracy of the analytical formulas for av-erage Dancoff factors of fuel kernels in fuel pebbles andfuel compacts. The fuel compact and fuel pebble dimen-sions correspond to the NGNP Point Design.30

For Dancoff factors in a pebble bed reactor, as shownin Fig. 5, the fuel pebble is modeled as two concentricspheres. The inner sphere has a radius R1 � 2.5 cm andrepresents the fuel zone filled with a stochastic distribu-tion of TRISO fuel particles. The outer spherical shellrepresents the moderator region with outer radius R2 thatis typically 3.0 cm but may change depending on thefuel0moderator ratio in a pebble bed reactor. This geom-etry is used to calculate both intrapebble and interpebbleDancoff factors. The latter will be discussed in Sec. IV.

To estimate intrapebble Dancoff factors, from 5000to 70 000 fuel particles are packed inside the fuel zone

28 JI and MARTIN


using the RSA algorithm, corresponding to a range ofpacking fractions from 2% to 25%. Neutrons are emitteduniformly from the surface of the fuel kernels with acosine current distribution. The number of neutrons thatsuccessfully reach another fuel kernel without having acollision is tallied, and the Dancoff factor is the ratio ofthis number to the total number of neutrons emitted.

To estimate Dancoff factors in a prismatic-type re-actor, a hexagonal lattice structure is set up to model afuel compact cell ~Fig. 5!. The fuel compact has radiusRc � 0.6225 cm and the outer hexagonal graphite regionhas a flat-to-flat ~one side to opposite side of hexagon!distance P � 2.196 cm. Both infinite cylinders and finitecylinders are modeled.

For the infinite cylinder, a height H � 200 cm waschosen with reflecting boundaries on the top and bottom.Fuel particles are randomly packed inside the cylinder

with packing fractions from 2% to 28.92%. There are100 realizations with 10 million neutrons emitted perrealization, similar to the pebble bed analysis.

For the finite cylinder, the packing fraction is fixedat 28.92% and H is adjusted from 2 to 100 cm for theintracompact Dancoff factor computations. The other set-tings are the same as for the infinite cylinder case.

III.D. Numerical Results

In this section, numerical results for average Dan-coff factors for fuel kernels in fuel pebbles and fuel com-pacts from the analytical formulas given earlier arecompared with the Monte Carlo benchmark results.

For intrapebble Dancoff factors, results are com-pared as a function of volume packing fraction rangingfrom 2% to 25%. Results for analytical results using thesingle-sphere model are plotted in Fig. 6, along with theMonte Carlo benchmark results. Figure 6 shows thatthe single-sphere model gives excellent results over therange of volume packing fractions, especially at higherpacking fractions.

Figures 7 and 8 compare benchmark Monte Carloresults with the results of the analytical expressions foraverage Dancoff factors in fuel compacts modeled ascylinders, including both infinite and finite height com-pacts at packing fractions ranging from 2% to 28.92%.

For the infinite cylinder, it is interesting to see that forall packing fractions, the single-sphere model overesti-mates the Dancoff factors, compared to the benchmarkMonte Carlo results. This was also observed for the infi-nite medium Dancoff factors. The agreement is better atlow packing fraction than high packing fraction but all inthe acceptable range. In general, the single-sphere model

Fig. 5. Fuel pebble and fuel compact cells in Monte Carlosimulation.

Fig. 6. Comparison between analytical and Monte Carlo average intrapebble Dancoff factors.



gives reasonable results for the entire range of packingfractions.

For the finite cylinder, the volume packing fractionof fuel particles is fixed at the nominal value 28.92%within the compact and the height of the hexagonal cell~H ! is varied from 2 � D to 100 � D, where D is thediameter of the fuel compact. Figure 8 shows the resultsfor the intracompact Dancoff factors as a function of thecompact height. Similar to the Dancoff factor for an in-finite medium, the single-sphere model overestimates theresults over the range of heights but overall gives verygood acceptable results.

IV. AVERAGE DANCOFF FACTORS FORMULTIPLE FINITE REGIONS

The previous sections derived analytical expres-sions for average Dancoff factors for infinite and finiteregions comprised of a stochastic mixture of fuel lumps,characteristic of an infinite medium or a finite region ofTRISO fuel such as a fuel compact or fuel pebble, re-spectively. This section extends the analysis to includemultiple finite regions, such as a prismatic core com-prised of many fuel compacts or a pebble bed core con-sisting of many fuel pebbles. When the Dancoff factor is

Fig. 7. Comparison between analytical and Monte Carlo average intracompact Dancoff factors for infinite height compacts.

Fig. 8. Comparisons between analytical and Monte Carlo average intracompact Dancoff factors for finite height compacts.

30 JI and MARTIN


calculated for a fuel volume, one needs to account notonly for the intravolume contribution discussed in theprevious section but also for the intervolume contribu-tion since a neutron might exit a finite volume and streamto another finite volume without a collision. Figure 9shows this general configuration, where the average Dan-coff factor will depend on the intervolume contributions.

IV.A. General Analytical Derivations

The intervolume Dancoff factor is defined as theprobability that a neutron escaping from a fuel lump in afinite volume enters another fuel lump in a different fi-nite volume. This probability can be expressed in termsof several basic probabilities:

P1 � average probability that a neutron escaping froma fuel lump in a finite volume escapes the volume with-out entering another fuel lump or colliding with the mod-erator. This is identical to the first flight escape probabilityPesc* .

P2 � average probability that a neutron escaping froma finite volume enters another finite volume without acollision.

Ptr � average probability that a neutron incident on afinite volume traverses it without entering any fuel lumpor having a collision with the moderator.

P3 � average probability that a neutron incident on afinite volume enters a fuel lump within that volume.

Assuming that the finite volumes are uniformly dis-tributed in an infinite background medium, and assum-

ing that average probabilities can be used for the actualprobabilities describing specific neutron trajectories, theintervolume Dancoff factor can be expressed as

C inter � P1 P2 P3 � P1 P2 Ptr P2 P3 � P1~P2 Ptr !2P2 P3

� P1~P2 Ptr !3P2 P3 � {{{

� P1 P2 P3 (i�0

`

~P2 Ptr !i . ~32!

The series can be summed, yielding

C inter � P1 P2 P3

1

1 � P2 Ptr

. ~33!

The chord method can be used to derive expressions forthese probabilities.

To obtain an expression for P1, one proceeds in amanner similar to the derivation of C intra that led toEq. ~16!. The average probability that a neutron in dVdVescapes a finite volume along V without entering an-other fuel lump or colliding with the moderator is givenby ~see Fig. 4!

P1~l ! �dVdV

4pVe�l0l�

l

`

f ~l ' ! dl ' , ~34!

where l is the distance along the neutron trajectory fromdVdV to the exiting point of the volume. Integratingover dVdV, one obtains

Fig. 9. Distribution of finite volumes containing fuel lumps.



P1 � �� dVdV

4pVe�l0l�

l

`

f ~l ' ! dl ' �

�dA�dV cos u

4pV

�dA�dV cos u�min_d

L

dle�l0l�l

`

f ~l ' ! dl '

�dA�dV cos u

. ~35!

Using the chord length distribution function in a finite body F~L! that was introduced in the previous section inEq. ~15!, the final expression for P1 is obtained:

P1 �1

^L&�� dVdV

4pVe�l0l�

l

`

f ~l ' ! dl ' �1

^L&�dLF~L!�

min_d

L

dle�l0l�l

`

f ~l ' ! dl ' . ~36!

Assuming H~S! as the chord length PDF between two volumes in an infinite background medium of volumes, P2is readily found to be

P2 � �H~S!e�S0l dS , ~37!

where H~S! is assumed to be known, perhaps empirically, for the given geometry.To determine Ptr , begin with the probability that a neutron incident on a finite volume through a small element

dAdV along V crosses the finite volume without hitting any fuel lump or colliding with the moderator:

Ptr ~L! � e�L0l�1 ��0

L

f ~l ' ! dl '�� e�L0l�L

`

f ~l ' ! dl ' , ~38!

where L is the chord length from dAdV to the exit point along V. The first term is the probability that no collisionswith the moderator happen, and the second term is the probability that no fuel lumps are hit. Integrating Ptr ~L! overall possible chords in the finite volume assuming that incident neutrons have a cosine current distribution, anexpression for Ptr is found:

Ptr �

��dAdV cos uPtr ~L!

��dAdV cos u

��dLF~L!e�L0l�L

`

f ~l ' ! dl ' . ~39!

The probability P3 may be viewed as the Dancoff factor for a neutron incident on the boundary of the finitevolume. Therefore, using our previous results, the average probability that a neutron entering a finite volume willenter a fuel region within that volume without colliding with the moderator is

P3~L! � �min_d

L

f ~l !e�l0l dl , ~40!

where L is the chord length from the incident point to the exit point along V. This is then averaged over all possiblechords in the finite volume:

P3 �

��dAdV cos uP3~L!

��dAdV cos u

��dLF~L!�min_d

L

f ~l !e�l0l dl . ~41!

Now all of the terms P1, P2, P3, and Ptr have been expressed in terms of chord length distribution functions.Substituting Eqs. ~36!, ~37!, ~39!, and ~41! into Eq. ~33!, one obtains the final analytical formula for C inter :

C inter �

� 1

^L&�dLF~L!�

min_d

L

dle�l0l�l

`

f ~l ' ! dl '�{��H~S!e�S0l dS�{��dLF~L!�min_d

L

f ~l !e�l0l dl�1 � ��H~S!e�S0l dS�{��dLF~L!e�L0l�

L

`

f ~l ' !dl '� . ~42!

32 JI and MARTIN


Equation ~42! is an analytical expression for the Dancofffactor in a collection of multiple finite stochastic re-gions. It depends on three chord length PDFs, althoughone of these, H~S!, may need to be determined empiri-cally. This expression, consisting of three-dimensional~or lower dimensionality! integrals over chord lengthPDFs, represents a considerable reduction in complexityfrom what in principle would be a 12-dimensional inte-gral: multiple two-dimensional integrals both in surfaceand solid angle for each of three imbedded geometries—TRISO particles, finite volumes of TRISO particles ~e.g.,compacts or pebbles!, and the outer region consisting ofmultiple finite volumes ~e.g., pebble bed or prismaticreactor core!.

IV.B. Dancoff Factor Expressions for FiniteVHTR Configurations

The results of the previous section will be applied toboth prismatic and pebble bed VHTR configurations withmultiple fuel compacts and multiple fuel pebbles, respec-tively. Substituting Eq. ~9! into Eqs. ~36!, ~39!, and ~41!and using the effective cross section S* defined byEq. ~18!, the following expressions are obtained:

P1 �1

^L&�dLF~L!�

0

L

dle�l0l�l

` 1

^l &{e�l '0^l & dl '

�l*

^L&�~1 � e�L0l* !F~L! dL � Pesc

* , ~43!

Ptr ��dLF~L!e�L0l�L

` 1

^l &e�l '0^l & dl '

��dLF~L!e�L0l* � 1 �^L&

l*Pesc* , ~44!

P3 ��dLF~L!�0

L 1

^l &{e�l0^l &e�l0l dl

� C s,`�~1 � e�L0l* !F~L! dL

� C s,`^L&

l*Pesc* . ~45!

Substitute these results into Eq. ~33! to find

C inter �

Pesc* {��H~S!e�S0l dS�{�C`

^L&

l*Pesc* �

1 � ��H~S!e�S0l dS�{�1 �^L&

l*Pesc* � .

~46!

Again, it can be seen that the average intervolumeDancoff factor for a fuel kernel is a relatively simpleexpression involving the first flight escape probabilityPesc* , the effective cross section S* , and the chord length

PDF H~S! between the finite volumes.In the following sections, Eq. ~46! will be applied to

compute the interpebble and intercompact Dancoff fac-tors for pebble bed and prismatic VHTR configurations,respectively.

IV.B.1. Interpebble Dancoff Factors forPebble Bed Configurations

Equation ~22! gives the first flight escape probabil-ity Pesc

* for a spherical fuel pebble. Substituting this intoEqs. ~43!, ~44!, and ~45!, explicit expressions for P1, Ptr ,and P3 can be found for a fuel kernel in an infinite me-dium of pebbles:

P1 �3

4 � l*

R1��

3

4 � l*

R1�2

e�2~R1 0l* !

�3

8 � l*

R1�3

@1 � e�2~R1 0l* ! # , ~47!

Ptr � �� l*

R1�e�2~R1 0l

* ! �1

2 � l*

R1�2

@1 � e�2~R1 0l* ! # ,

~48!

and

P3 � C`�1 � � l*

R1�e�2~R1 0l

* !

�1

2 � l*

R1�2

~1 � e�2~R1 0l* ! !� . ~49!

For P2, the chord length PDF between two sphericalfuel zones of fuel pebbles in an infinite medium is needed.Assuming that the spherical fuel pebbles are randomlydistributed in the background moderator, H~S! is ex-pressed as an exponential PDF:

H~S! �1

^S&e�S0^S& , 0 � S � ` , ~50!

where

^S& �4R1~1 � FRAC!

3FRAC

and FRAC is the volume packing fraction of the spheri-cal fuel zone in the whole medium. Substituting Eq. ~50!into the expression for P2, we find



P2 � �0

` 1

^S&e�S '0^S&e�S '0l dS ' �

1

1 � ^S&1

l

. ~51!

It should be noted that the model presented in Eqs. ~50! and ~51! to calculate P2 is a crude approximation. Topredict accurate interpebble Dancoff factors, a Monte Carlo simulation is needed to obtain P2. This simulation hasbeen found to be relatively inexpensive to obtain a “1s” ~one relative standard deviation! result accurate to within10�4. In the results comparison section later in this paper, a comparison will be made between this crude model andthe more precise Monte Carlo simulation for calculating P2.

At this point, expressions for P1, Ptr , P3, and P2 have been obtained. Introducing these quantities into Eq. ~33!,C inter is obtained in terms of basic geometry quantities for fuel pebbles:

C inter � C`

3

4

l*

R1�1 � � l*

R1�e�2~R1 0l

* ! �1

2 � l*

R1�2

@1 � e�2~R1 0l* ! #�2

{P2

1 � P2{�� l*

R1�e�2~R1 0l

* ! �1

2 � l*

R1�2

@1 � e�2~R1 0l* ! #� , ~52!

where P2 is obtained from a simple Monte Carlocalculation.

IV.B.2. Intercompact Dancoff Factors forPrismatic Configurations

To obtain intercompact Dancoff factors for the pris-matic VHTR, the fuel compacts will be modeled as bothinfinite cylinders and finite cylinders.

For both cases, Eqs. ~25! and ~30! give the first flightescape probabilities respectively. Substituting these for-mulas into Eqs. ~43!, ~44!, and ~45!, it is straightforwardto find P1, Ptr , and P3 for both infinite and finite cylin-ders. Note that the average chord length is ^L& � 2Rc foran infinite cylinder and ^L& � 2Rc H0~Rc � H ! for afinite cylinder.

For P2, the chord length PDF between two cylindri-cal fuel compacts is needed. At this point, looking at thedefinition of P2 in Eq. ~37!, it can be seen that this isactually the Dancoff factor definition for finite volumesdistributed in a moderator background. So the evaluationof P2 becomes a classical problem: the calculation of theDancoff factor for cylindrical fuel rods in a hexagonallattice, which was the original application studied byDancoff and Ginsburg1 in their 1944 report.

In a lattice structure, an accurate expression for thePDF H~S! is difficult to derive analytically, especiallyfor finite cylinders. However, lattice structure, whethersquare or hexagonal, is a relatively simple geometry forcalculating Dancoff factors. Taking advantage of modern-day computing capability, this work can be done in a fewseconds using direct Monte Carlo methods, yielding ex-cellent results with a relative standard deviation ,10�4.Accordingly, Monte Carlo simulation is used to obtainP2 for infinite and finite cylindrical fuel compacts.

Analytical expressions for P1, Ptr , and P3 and numer-ical values for P2 have now been obtained. Introducingthese terms into Eq. ~33!, either an analytical formula or

a numerical estimate for C inter in terms of basic geom-etry quantities of fuel compacts can be found.

IV.C. Monte Carlo Benchmark Computation

As for Dancoff factors for an infinite and finite me-dium of fuel kernels, the average Dancoff factors for fuelkernels between fuel pebbles and fuel compacts are cal-culated by Monte Carlo simulation for comparisons withthe results from analytical formulas.

For the interpebble Dancoff factor, the same config-uration is set up as for the intrapebble calculation, exceptthat a white boundary condition is set on the outer spher-ical surface. The packing fraction is fixed at 5.76%, atypical design parameter for VHTR pebble bed reactors,but R2 is adjusted from 3.0 to 6.0 cm. Neutrons are emit-ted from the surface of fuel kernels and travel throughthe fuel zone and moderator shell region. Only thoseneutrons that reenter the fuel zone at least once and en-counter another fuel kernel without colliding with a mod-erator are tallied. The ratio of this number to the totalnumber of neutrons emitted is the interpebble Dancofffactor.

For the intercompact Dancoff factor, the same hex-agonal lattice structure is set up as for the intracompactcalculations. For the infinite cylinder, the packing frac-tion 28.92% is chosen for the intercompact Dancoff fac-tors. The flat-to-flat distance P is adjusted from 2.196 to4.0 cm. For the finite cylinder, the packing fraction isfixed at 28.9% and H is adjusted from 2 to 100 cm.

IV.D. Numerical Results

In this section, analytical interpebble and intercom-pact Dancoff factors are compared with Monte Carlobenchmark simulations.

For interpebble Dancoff factors comparisons, the vol-ume packing fraction is fixed at 5.76%. The outer radius

34 JI and MARTIN


R2 of the fuel pebble is varied by changing the ratioR20R1 from 1.2 to 2.4. The results are plotted in Fig. 10.It can be seen that the analytical model gives excellentresults, with good agreement with Monte Carlo resultsfor different values of R2 0R1. It should be noted that thisagreement is partly due to an accurate calculation of P2,which is the average probability that a neutron escapingfrom the fuel zone in a pebble enters the fuel zone inanother pebble without collision. A direct Monte Carlosimulation is used to estimate P2 rather than using theapproximate expression given in Eq. ~51!, which as-sumes that H~S! is described by an exponential PDF. Wehave found that Eq. ~51! yields poor results for P2 com-pared to direct Monte Carlo simulation, as can be seenfrom Table IV.

Although a white boundary is set on the outer sur-face of the fuel pebble, which seems expedient for a

stochastic distribution of the fuel zones, the assumptionof an exponential PDF for H~S! does not give a correctvalue of P2 using Eq. ~51!. When one studies a stochasticdistribution of microspheres in a moderator, the assump-tion of an exponential chord length PDF between micro-spheres yields excellent results for infinite mediumDancoff factors. However, when one uses the same as-sumption for fuel zones of pebbles within a full core,inaccurate results are obtained, indicating that the as-sumption of an exponential chord length PDF betweentwo fuel zones is not appropriate, at least for the problemdiscussed in this paper.

Another analysis can also be given to explain thehigh accuracy of the single-sphere model results. WhenR2 0R1 � 1.0, this corresponds to the infinite mediumcase; i.e., fuel particles are distributed in an infinite back-ground medium. In this case the sum of the intrapebbleand interpebble Dancoff factors should equal the infinitemedium Dancoff factor, or ~C intra � C inter !0C`�1. Thesingle-sphere model predicts this value exactly when oneintroduces expressions for C intra, C inter , and C` into theabove relation. Looking at the definitions of C intra0C ,̀P1, P2, P3, and Ptr , some relations among them shouldhold. In particular,

C intra

C`� 1 � P1 ~53!

and

P3

C`� 1 � Ptr . ~54!

Fig. 10. Comparisons between analytical and Monte Carlo average interpebble Dancoff factors at volume packing fraction5.76%.

TABLE IV

Comparison of P2 from Eq. ~51! and Monte Carlo Method

R2 0R1~R1 � 2.5 cm!

P2 fromEq. ~51!

P2 fromMonte Carlo

1.2 0.4990 0.44061.4 0.2937 0.19961.6 0.1898 0.09451.8 0.1305 0.04662.0 0.0939 0.02382.2 0.0699 0.01252.4 0.0535 0.0067



The single-sphere model preserves these relations,as shown by inspection of Eqs. ~20!, ~43!, ~44!, and ~45!.This provides additional assurance for the accuracy ofthe single-sphere model.

Next, intercompact Dancoff factors are calculatedby analytical formulas and compared with the MonteCarlo simulation results. As before, results are given forinfinite cylinder and finite cylinder configurations.

IV.D.1. Infinite Cylinder

The volume packing fraction of fuel particles is fixedat 28.92% for the intercompact Dancoff factor computa-tion while the flat-to-flat distance of the hexagonal mod-erator cell is varied. The results are presented in Fig. 11.

Similar to the interpebble Dancoff factor results, theintercompact Dancoff factors are predicted with excel-lent accuracy. However, it should be noted that the ana-lytical results calculate P2 from an inexpensive MonteCarlo simulation.

IV.D.2. Finite Cylinder

For Dancoff factor calculations, the volume packingfraction of fuel particles is fixed at 28.92% within thecompact and the height of the hexagonal cell ~H ! is var-ied from 2 � D to 100 � D, where D is the diameter ofthe fuel compact.

Results are shown in Fig. 12. It can be seen thatwhen the ratio of the height to the diameter is small,approximately ,10, the analytical model gives very poor

results, significantly overestimating the intercompact Dan-coff factor. This is mainly due to the neutron leakage atthe top and bottom surfaces of the fuel compacts. Atlower H0D ratios, the leakage is very high, and the neu-trons escaping through top and bottom surfaces will notcontribute to the intercompact Dancoff factors. How-ever, the analytical model assumes these neutrons canstill enter another fuel compact, which overestimates thefinal results. Once the H0D ratio becomes .20, less leak-age occurs, and excellent agreement is obtained withMonte Carlo simulations. In practice, when one analyzesthe fuel compact, the ratio is normally about 64 and withinthe range where the predictions are excellent.

IV.E. Final Remarks

The general methodologies presented in previousSecs. II.A, III.A, and IV.A to calculate Dancoff factorsfor infinite, finite, and multiple media of fuel lumps havebroad application to general reactor physics analysis butare specifically intended for application to the neutronicanalysis of TRISO fuel particles in VHTRs. These meth-odologies can be applied to the analysis of both pebblebed and prismatic VHTR designs because the chord lengthPDF F ~L! is known analytically for spheres33 andcylinders.22,34–37 Also, the chord length PDFs H~S! be-tween fuel compacts and between fuel pebbles are wellapproximated as exponential distributions.21,39–43 As longas we are able to determine the chord length PDF f ~l !between two fuel kernels in an infinite medium, the av-erage Dancoff factor for fuel kernels in either VHTRdesign is readily obtained by Eqs. ~16! and ~42!.

Fig. 11. Comparisons between analytical and Monte Carlo average intercompact Dancoff factors at volume packing fraction28.92% for infinite height compacts.

36 JI and MARTIN


In Secs. II.D, III.D, and IV.D, numerical results basedon the analytical formulas for Dancoff factors are com-pared with Monte Carlo benchmark results for stochasticconfigurations generated using the RSA algorithm. Thosecomparisons were obtained for both pebble bed and pris-matic VHTRs.

V. CONCLUSIONS

The chord method was first proposed by Dirac forthe calculation of escape probabilities and related quan-tities more than 60 years ago. Since that time, the chordmethod has not received much attention outside this spe-cialized area of reactor analysis. The advent of the VHTRas a viable nuclear reactor concept has underscored theimportance of accurate and efficient analysis of TRISOfuel configurations characteristic of the prismatic andpebble bed VHTR concepts. The stochastic nature ofTRISO fuel coupled with the presence of the double het-erogeneity presents substantial computational chal-lenges. In particular, the computation of Dancoff factorsfor both prismatic and pebble bed reactors is needed forroutine neutronic analysis of TRISO fuel configurations.This work demonstrates that the chord method can beused to obtain analytical expressions for Dancoff factorsfor both prismatic and pebble bed VHTRs.

In this paper, the derivation of analytical expres-sions for Dancoff factors in stochastic media with dou-ble heterogeneities is addressed. General derivations forDancoff factors in infinite and finite media are obtained

by introducing the chord length distribution PDF. Thechord length PDF considerably simplifies the mathemat-ical formulas and enables a computationally efficientand accurate approach to calculate Dancoff factors. Theo-retical expressions for the chord length PDFs were shownearlier to be accurate and consistent with empirical PDFsobtained with benchmark Monte Carlo simulations. Usingthese chord length PDFs, the Dancoff factor expres-sions were applied to infinite and finite VHTR config-urations characteristic of both pebble bed and prismaticdesigns with excellent results, giving results for a rangeof infinite medium and finite medium Dancoff factorsthat agreed with benchmark Monte Carlo simulations towell under 1% for pebble bed and prismatic VHTRconfigurations.

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of En-ergy NERI Project Grant DE-FC07-06ID14745 and the U.S.Nuclear Regulatory Commission Faculty Development GrantNRC-38-08-950.

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Fig. 12. Comparisons between analytical and Monte Carlo average intercompact Dancoff factors at volume packing fraction28.92% for finite height compacts.



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