A statistical analysis of the double heterogeneity problem

Ann. nucl. Energy, Vol. 18, No. 7, pp. 371 395, 1991 0306-4549/91 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1991 Pergamon Press plc

A STATISTICAL ANALYSIS OF THE DOUBLE HETEROGENEITY PROBLEM

R. SANCHEZt and G. C. POMRANING

School of Engineering and Applied Science, University of California--Los Angeles, Los Angeles, CA 90024-1597, U.S.A.

(Received for publication 15 January 1991)

Abstract--The collision probability treatment for regions comprised of a uniform background medium with a random dispersion of small heterogeneities is analyzed using recently-developed statistical techniques. By assuming that the chord length distributions in such regions follow a renewal process we obtain an exact expression for the equivalent homogeneous cross section. Numerical comparisons show that the collision probability value for this cross section can be significantly in error in most cases, with an error of ~ 8 % for typical PWR poisoned fuel consisting of a mixture of uranium oxide and gadolinia grains. Renewal theory is generalized to the case of materials with internal structure allowing us to justify the central assumption on which the collision probability treatment is based. Also, new formulas are proposed for the calculation of the collision probabilities between matrix and grains in terms of the collision probabilities for the homogenized regions.

NOMENCLATURE

Areas and volumes A = Area of the surface of a region

A g = Area of the surface of a grain of type i V = Volume of a region.

V0 = Volume of the matrix contained in the region V i = Volume of all the grains of type i contained in the

region V g = Volume of one grain of type i Vik = Volume of all the layers ik contained in the region V,gk = Volume of the layer k of grain type i for one grain

Volumetric proportions P0 = Volumetric proportion of the matrix = Vo/V p~ = Volumetric proportion of all the grains of type

i= Vi/V P~k = Volumetric proportion of all the ik layers = V~/V P~k = Volumetric proportion of layer k in grain i =

w, dv~

Linear dimensions L = Maximum chord in a region Li = Maximum chord in a grain of type i 2 = average chord in a region

20 = Average chord in the matrix 2i = Average chord in a grain of type i

l(r, r ) = Length of a trajectory from the surface of a grain to the point r

Statistical quantities f (x) = Distribution for chord lengths in a region f ( x ) = Distribution for chord lengths in a grain

Q(x) = Probability for chords of length greater than x in a region

Qi(x) = Probability for chords of length greater than x in a grain

ti = Relative transition probability from grain i into matrix

Macroscopic cross sections 520 = Total cross section for the matrix I~'0 = 520+ 1/20 Zi = Total cross section for a homogeneous grain

Zig = Total cross section for a layer in a grain Zh = Homogenized cross section for a region

Sources andfluxes So = Angular source intensity in the matrix Si = Angular source intensity in a homogeneous grain

Sik = Angular source intensity in layer ik @1 = Spatial average of the scalar flux in material 1

q/~°~(x) = Ensemble average of the angular flux exiting material I at x

~b~n(x) = Ensemble average of the angular flux entering material I at x

~0i(x ) = Ensemble average of the angular flux in material / a t x

~,out = Contribution to ~/out (x) from the internal sources in the grain

fie = Contribution to ff~ (x) from the internal sources in the grain

qJik(X) = Ensemble average of the angular flux in layer ik at x

~9ik = Contribution to qtik(X) from the internal sources in the grain

]" Permanent address: Service d'Etudes de Rracteurs et de Mathrmatiques Appliqures, Centre d'Etudes Nuclraires de Saclay, 91191 Gif sur Yvette Crdex, France.

371

372 R. SANCI-~Z and G. C. POMRANING

Collision probabilities P00 = Self-collision probability for the matrix

P,k,0 - Collision probability from the matrix to all layers ik

Pgk,o = Collision probability from the matrix to one layer ik

P0,i~ = Collision probability from layer ik to the matrix Pi,,jt = Collision probability from layerfl to all layers ik Pg,,it = Collision probability from layer l to layer k in the

same grain P(~E) = Self-collision probability for the homogenized

region with cross section Z

Escape probabilit&s E 0 = Escape probability from the matrix to the surface

of the region Ei = Escape probability from grain i to the surface of

the region E g = Escape probability from grain i to the surface of

the grain

E: g = Escape probability from grain i with its cross section diminished by Z to the surface of the grain

Eik= Escape probability from layer ik to the surface of the region

E~k = Escape probability from layer ik to the surface of the grain

E~k = Escape probability from layer ik with its cross section diminished by Z to the surface of the grain

E(E) = Escape probability from the homogenized region with cross section Z

Transmission probabilities T = Transmission probability for the heterogeneous

matrix~rain region T(Z) = Transmission probability for the homogenized

region with cross section E T~ = Transmission probability for a grain 7~ = Transmission probability for a grain with its cross

section diminished by Z

1. INTRODUCTION

Collision probability (CP) methods are one of the two basic tools for routine transport calculations in heterogeneous media. The practical implementation of these methods requires the decomposition of the domain under analysis into several homogeneous regions, and then the use of a tracking routine to compute the intersection of linear trajectories or rays with the different regions in the domain (Sanchez and McCormick, 1982). Such a straightforward approach is only possible, however, when the number of the regions is relatively small and when the position of the regions is exactly known. A problem of practical interest that cannot be treated with classical CP methods is that of a medium that contains a random dispersion of grains. This is the case, for instance, in boiling water or pressurized water reactor fuel with grains of burnable poison, and also in high-temperature reactor pebble fuel (Lane et al., 1962). Another example that comes to mind is that of 7- ray propagation in concrete. As opposed to the simple case in which a medium is comprised of a set of homogeneous regions, these problems are characterized by the fact that the regions themselves have heterogeneous components (grains). It is because of this reason that these types of problems are said to have a "double heterogeneity." Basically, what characterizes a geometry with a double heterogeneity is the presence of regions consisting of an otherwise homogeneous material, called matrix, with a large number of very small heterogeneities (the grains) randomly dispersed into it.

The main obstacle to the use of CP methods for the treatment of problems with a double heterogeneity can be overcome by the use of a suitable homogenization together with a prescription for determining grain-matr ix interactions in terms of homogenized quantities. Such a homogenization was initially suggested by Askew (Livolant and JeanPierre, 1974) for the approximate treatment of complicated geometries such as control rod cluster calculations, and recently generalized to the treatment of a double heterogeneity problem. Basically, this treatment of double heterogeneity is founded on a simple assumption that allows one to replace the current of particles exiting the grains by an isotropic source uniformly distributed in the matrix (Hebert, 1980 ; Sanchez et al., 1988). This assumption reduces, on one hand, grain-to-grain and grain-to-matrix collision probabilities to matrix self-collision probabilities, and leads, on the other hand, to a homogenizat ion technique for the calculation of the latter probabilities. The double heterogeneity method has also been extended for use with interface current methods.

The direct application of statistical analysis to linear particle transport in statistical media composed of random mixtures of materials has recently been the object of accruing interest (Adams et al., 1989 ; Levermore et al., 1986, 1988; Malvagi and Pomraning, 1990; Pomraning, 1986, 1989; Sahni, 1989a,b; Sanchez, 1989; Pomraning and Sanchez, 1990; Vanderhaegen, 1986, 1988). The approach here is to derive one or more equations for the ensemble average (expected) flux in terms of the statistical properties of the set of possible physical realizations of the system. For general statistics these equations are not closed, and there is a cascade of equations in which ensemble average fluxes are given in terms of increasingly complicated averages of the angular flux (Sanchez, 1989). For a particular class of statistics, called renewal statistics, it has been demon-

Analysis of the double heterogeneity problem 373

strated, however, that in the absence of scattering the ensemble average fluxes in the different media obey a finite set of integral renewal equations (Pomraning, 1989; Pomraning and Sanchez, 1990). Renewal statistics are characterized by the fact that the chord lengths for any given material along an arbitrary ray follow a prescribed but otherwise arbitrary chord length distribution.

The main purpose of this paper is to use the statistical description based upon renewal equations to analyze the double heterogeneity problem, providing thus a theoretical foundation for the double heterogeneity method. Since the renewal equations are exact when there are no collisiohs, this analysis provides an improved solution to the problem of homogenization. Furthermore, by accepting these equations as an approximate description in the case with collisions, we are able to derive the basic assumption that is at the origin of the double heterogeneity method and thus establish a guideline for the limits of the method and perhaps a tool for its improvement. As an aside, we also generalize the renewal equations to the treatment of materials with internal structure, i.e. to materials whose cross sections and internal sources depend not only on the absolute position along a ray but also on the relative position within the material.

Section 2 deals with the CP description of the double heterogeneity method. Since to our knowledge a detailed description of this method is not available in the literature, this section contains a complete derivation of the basic formulas. The renewal equations are introduced in Section 3, where they are applied to the analysis of a matrix containing homogeneous grains. This section contains our main results for the double heterogeneity problem. An improved solution to the region homogenization is presented, and a new derivation of the relationships between the different probabilities is given. The next section deals with the generalization of the results of Section 3 to the case of heterogeneous grains. This section is very technical and does not bring any new light to the analysis of the double heterogeneity problem. Its interest is in the generalization to renewal processes comprising materials with internal structure. Section 5 gives a comparison between the CP prescription for homogenization and the one derived in Section 3, while conclusions are given in Section 6.

The general geometry that we want to consider in this paper is that of a multiregion system with randomly distributed grains in some or all regions. We assume that a given region can contain different kind of grains (for instance grains of different composition or of different sizes), each one present in a large number, and also that a grain can have one or several internal layers. As a consequence the notation is quite involved. The matrix in a region is denoted with the index 0, the indexes i and j are used to distinguish each type of grain, and the indexes k and I to indicate layers inside the grains. The group of indexes ik indicates, for example, the kth layer in a grain of type i. In general we will treat only a single region and will not use, therefore, an index to differentiate the regions. However, when necessary an upper index ct or fl will be used to this end. An upper index g will indicate a quantity relative to a single grain as opposed to the set of grains of the same type contained in the matrix. For instance, V g indicates the volume of a grain of type i, while Vi = niVgi is the volume of the totality of the n~ grains of type i contained in the matrix. Finally, we will use the capital indexes ! and J to establish formulas or equations that are valid for both the matrix and the grains. These indexes can represent either the matrix (I = 0) or a grain (I = i). A short definition for most of the symbols used in the equations is given in the Nomenclature at the beginning of the paper.

2. THE CP DESCRIPTION

The problem we want to address here is that of the interaction between media containing randomly-dispersed grains. Each medium can be thought of as composed of a homogeneous matrix containing a large number of small heterogeneities (the grains) randomly distributed. Our basic assumption is that the grains are very small and their number per unit volume is very large. The example we have in mind is that of a poisoned fuel pin which contains microscopic gadolinia grains immersed in a matrix of uranium dioxide. The facts that are known about the grains are their isotopic composition and a manufacturer's curve giving the probability density for the diameter of the grains. Since we cannot cope with a continuous distribution of grain sizes, we choose to classify the grains in a finite number of different types, each type corresponding to a given value of the diameter. We will use the index 0 to denote the matrix and denote by nl the total number of grains of type i (i = 1, N) contained in the matrix. For very absorbant grains self-shielding effects can be very important and, when doing burnup studies, it is necessary to distinguish several internal layers in each grain. We will use the double index ik to denote the kth layer in a grain of type i.

As a prelude to our discussion we give first some definitions and basic properties of first-collision probabilities (Sanchez and McCormick, 1982). We will use essentially three different probabilities: collision, escape and

ANE 18:7-B

374 R. SANCHEZ and G. C. POMRANING

transmission probabilities. The collision probability P Jl is the probability for particles born uniformly and isotropically in a zone I to have their first collision in zone J, while the escape probability El is the probability for those same particles to cross an external surface F without undergoing any collisions. Here I and J are two arbitrary homogeneous zones. The transmission probability T is the probability for particles entering uniformly and isotropically through F to exit through the same surface without suffering any collisions. An associated quantity PI is the probability for a first collision in zone I for the particles entering surface F with the assumed isotropic and uniform distribution. If we let N be the number of regions within the surface F, then these probabilities obey the following reciprocity and conservation relations :

VjEjPH = V I • I P J I , (1)

PI = J.PIXlE,, (2) N

l = E,+ y~ Pj,, (3) I - - I

N

1 = T + 2 ~ piE,E1, (4) 1 = 1

where 2 = 4 VIA is the average chord length through surface F of area A, V is the total volume contained within the surface and Pl = VI/V is the volumetric proportion of zone L In the following we will use these formulas when the surface F is the surface of the heterogeneous region containing the matrix-grain mixture, or when it is the surface of one of the grains.

2.1. The one-region problem

For simplicity we consider first the problem of the interaction of the matrix with the grains that it contains and of the interactions between these grains. That is, we consider a single region of arbitrary shape and for the time being we disregard the interaction of the region with its surroundings. Because of the large number of grains present in the region and also because their positions are not known, this general problem cannot be treated by classical deterministic methods. Besides, it does not make any sense to want to ascertain the reaction rates in each one of the grains. A more sensible question is to ask what would be the average reaction rate of a grain at a given position or, more generally, what would be the average reaction rate in a grain, regardless of its position. In this section we show that an answer to the second question can be formulated in the framework of the collision probability method by using the very facts that make the general problem unsolvable. Indeed, because of the large number of grains present in any sizable volume of material one can make the following assumption (Hebert, 1980; Sanchez et al., 1988) :

Particles leaving the grains due to internal sources or to collision may be considered as being uniformly and isotropically distributed in the matrix.

We now use this basic assumption to calculate the average fluxes in the matrix and in the grains. We have :

VoE0qb0 = Poo(VoFo+J+), (5)

V~k Eik Oik = P~k,o ( VoFo + J+ ) + ~ Pgk,u Vg Fil, (6) l

where, with S representing the external source and (I) the average flux, F = EsO + S is the density of emission, and the upper index g denotes quantities specific to a single grain, as opposed to quantities relative to the set of all grains of the same type. For instance, P00 is the matrix self-collision probability, while P$,,0 is the probability for particles produced in the matrix to have their first collision in the layer k of one grain of type i. Also, in these equations J+ is the total current emerging from all the grains contained in the matrix. According to our basic assumption this current is given by :

J+ = ~n l Z Eg~k VgkFik, (7) i k

where the sum over k is for all the layers in a grain of type i and E~ is the escape probability through the surface of grain i for particles produced in region k. Using the last equation in equations (5) and (6) we obtain :


VoZ o*o ---- Poo VoFo + ~. Po.~ V~F,k, (8) i,k

V~Z~P~ = e~.o VoFo + ~Pa.jtVj~Fjt, (9) j , t

where Vjk = n~V~ is the total volume of regions/k in the matrix, and

Po.~, = PooE~, (10)

Pile,jr = 3,jP$,,u + P~,oE$. (11)

The other probability in equation (9), P~.0 = niP$,,o, is obtained by invoking reciprocity, given by :

V~Xu, P~,o - VoEo Po.~. (12)

From equations (8-12) we see that the problem of determining the average fluxes in the matrix and in the different layers of each type of the grains has been reduced to calculating P00, and then solving the linear system of equations (8) and (9).

2.2. Homogenization

We show now that the calculation of the matrix self-collision probability P0o is equivalent to a homogenization of the region containing the matrix. We let T be the transmission probability through the surface F of the region, and denote by Eo and E~ the escape probabilities associated with the same surface F. Then, by using the conservation equation (3) together with equations (10) and (11), and then the conservation equation (4), we obtain :

E~ = EoE~, (13)

T = 1--2Y.Eo, (14)

where

Y. = P o ~ 0 + ~ i ( 1 - - T~). (15)

In this expression, :.~ = 4V~/A~ is the mean chord length in grain i, and T~ is the transmission probability through grain i, given by :

2i T~ = 1-- , . ~ E ~ V ~ Y ~ . (16)

v~k

Also, by using the conservation equation (3) for the matrix and equations (12) and (11) we obtain :

PoEo .... P0o = ~ - ( l --/~0), (17)

which together with equation (14) gives a relation between the self-collision and the transmission probabilities, namely •

P ° E ° [ 1 ] ~ ( 1 -- Poo = ~ - 1 -- T) . (18)

We have thus reduced the problem of determining Poo to that of calculating the transmission probability T or, equivalently, of calculating the total number of first collisions C = 1 - T resulting from a unit angular flux entering uniformly and isotropically the surface of the region. Assume now that we replace the mixture matrix- grains with a homogeneous material, and let Y'h be the total macroscopic cross section of this material that would preserve the value of the transmission probability. The total number of first collisions in this case can be written as :

C = V~h(I)hom(~"~h), (19)


where Ohom(•h) is the average uncollided flux due to the aforementioned incoming angular flux. In order to write the corresponding expression for the matrix-grain mixture, we make the assumption that on average the angular flux entering the grains is uniformly and isotropically distributed. As an aside we note that this approximation is consistent with our main assumption. Then, the average number of first collisions in a grain of type i is simply the product of the total number of incoming particles nAg~i, where ~9i is the value of the average incoming angular flux, and the probability for a first collision 1 - T~. Adding the total number of collisions in all the grains and in the matrix we can write :

where Ohet is the average uncollided flux in the matrix, and the argument (I~) = (Y~0, {Z~k, for all ik}) indicates the dependence of this average flux on the cross sections of all the materials present in the mixture. Also in this formula Ki represents the self-shielding ratio 4n0/Ohet. The cross section E h of the equivalent homogeneous medium can be calculated by equating expressions (19) and (20), but this leads us to the equivalent problem of determining the average fluxes Ohom, Oho, and the grain self-shielding factors {Ki}. However, in the limit when all the cross section values are close to each other we can approximately replace these latter quantities by their values calculated when all the cross sections have an equal value that we choose to be 2;0. That is, we compute the average fluxes and the self-shielding factors in the limiting situation when all the cross sections are equal to that of the matrix. In this limit Ohom(Eh) and Ohet(l~) can be replaced by Ohom(l~0), and Ki is calculated with all the grains homogeneous with cross section E 0. Since in this case the average number of first collisions in grains of type i is simply VpiEoOhom(EO) we have :

Vp~E 0(I)hom(~ 0) Ohom(]~o) ~ke(Z) ~ ~9~(Z0) - z~A/g[1_ Tg(y.0)] - 4rcEg(Z0-- ~ , (21)

where the argument Z0 in T~ and E~ indicates that these probabilities are computed for homogeneous grains of cross section Z0. The above expression gives the approximate value K~(E) ~ 1/E~(Zo) which together with the previously mentioned approximations for the fluxes results in the value :

_ ~ p i ( 1 - - T~) Y~h ~ P 0 2 ; 0 + 2 7 / 2 - , E ~ ' (22)

for the equivalent homogeneous cross section. Thus, one can use the previous formula to calculate Zh and then compute the corresponding transmission probability T which, when replaced in equation (18), gives the sought after value for P00. We can also make use of the conservation relations for the homogenized region and rewrite equation (18) as:

P°Z° {1 ~ [1--P(Zh)]}, (23) P00 = ~ - --

where P(E~) is the self-collision probability for the homogenized region.

2.3. The mul t i region case

We consider now the case of a domain containing several regions in mutual interaction, where each individual region can be either homogeneous or can consist of a stochastic matrix-grain mixture (Hebert, 1980; Sanchez et al., 1988). In order to be able to discuss this problem we have to extend our notation slightly. We will use a new index to differentiate matrices and grains belonging to different regions. Then we can rewrite our basic equations (5) and (6) as:

V0Z000 ~ =P ~ ~ ~ (24) ~ ~ = P o o ( V o F o + J + ) ,

: : ~ ~ o ~ + V # ~ ' # + j ~ ) + ~ p ~ k , u V , t F ~ t , (25) VikZik (~ ik = . ~ X i k , O k O x O g . . . . t

where we have already introduced total grain volumes in the second equation. The total uncollided current leaving the grains in region fl is given by equation (7) with terms specific to the region. By replacing this


expression for the current in equations (24) and (25) we get the multiregion version of equations (8) and (9), namely :

VoZ o~o = Y', (P'o{ VoFo + ~ Po,,~ V~kF,k), (26) fl i , k

V~kZ~k~k = ~ (P~,o V~oF~o + Z P~;' V~F~), (27) fl j , l

where now

and, by reciprocity,

p'o~,ik o,a ~'g.a (28) x O O * - ' i k ,

= g,o~ ~ g, f l P~i~fl 6ij3o, fP ik,it+ Pi'~,oE~l , (29)

eio~,0 ~kz~ _~ (30) = VPoXg ro,~.

Therefore the whole problem of calculating the average fluxes reduces to that of determining the matrix-to- matrix collision probabilities P~P0. We have already seen that the matrix self-collision probabilities P ~ are uniquely determined by a homogenization procedure that does not involve the other regions present in the domain. For the off-diagonal terms we adopt the decomposition :

e~go = (P')'oT°~EPo. (31)

Here Eg is the escape probability for particles born uniformly and isotropically in the matrix of region fl, T ~a is the transmission probability for particles escaping the surface of region fl to enter the surface of region ct without suffering any collisions, and (P')~ is the first collision probability in the matrix of region ct for particles entering the surface of this region. We know from our previous discussion on homogenization that E~0 is related to the total transmission probability for region ft. According to equation (14), we have :

E0 ~ = (1 - Ta)/(2~,#). (32)

On the other hand, the first collision probability (P'); depends on the space and angular distribution of the uncollided particles entering region ct after being uniformly and isotropically produced in the matrix of region ft. Since this distribution is not known we will introduce a factor x ~ such that (P'); = x~aP~o, where P~ corresponds to particles entering uniformly and isotropically through the surface. This latter probability can be obtained from E~ by reciprocity, so equation (31) becomes:

P~o~o = x ~ T~ E~oE~o. (33)

We now consider an equivalent multiregion problem in which each region has been homogenized by preserving its transmission probability, i.e. region ct has the constant cross section Z~,. For this equivalent problem we can also write an expression similar to equation (33), namely :

P~'~ = x~T~t~ E~'E~, (34)

and, by comparing with equation (33) and making use of relation (32), we obtain:

~c~ T~B ~ P ~ (35) P~0 -- tc~T~ ~ ~Z~ -

We finally assume that the homogenization approximately preserves the distribution of uncollided particles on the surfaces of the regions so that the precedent equation simplifies to :

P~0 = ~ ! P~ (36)

Another way to look at this result is to say that the multiregion problem is solved with an interface current method in which entering and exiting angular fluxes are assumed to be uniform and isotropic on the surface of each region. The multiregion procedure can be summarized as follows: first, a region-by-region homo-

378 R. SANCrmZ and G. C. POMRANING

genization that preserves the region transmission probability is done to obtain a set of equivalent homogeneous cross sections {E~}. Second, these cross sections are used in a multiregion collision probability calculation to obtain the P~a. And third, equations (23), (36) and (28-30) are used to obtain matrix-to-matrix, matrix-to- grain and grain-to-grain collision probabilities.

3. AN ANALYSIS BASED ON THE RENEWAL EQUATIONS

Lately there has been considerable interest in the study of particle linear transport in statistical media composed of random mixtures of materials (Adams et al., 1989; Levermore et al., 1986, 1988; Malvagi and Pomraning, 1990 ; Pomraning, 1986, 1989; Sahni, 1989a,b ; Sanchez, 1989 ; Pomraning and Sanchez, 1990 ; Vanderhaegen, 1986, 1988). By a statistical medium we mean a medium that can have different physical realizations, each of which has a given probability of occurrence. For our heterogeneous matrix-grain problem each physical realization corresponds to a well-defined spatial distribution of the grains in the matrix. Whereas deterministic transport theory is concerned with the calculation of the particle flux for a given physical realization, the statistical approach seeks to derive a direct description of the ensemble average (expected) flux. Of special interest to us are the so-called renewal statistics in which the chord length for any given material along any ray follows a prescribed, but otherwise arbitrary, chord length distribution. The statistics are said to be homogeneous and isotropic if the prescriptions are independent of the relative position along the ray and are also independent of the position and orientation of the ray. For this type of statistics it is possible to obtain an exact description for the ensemble average fluxes ~b~Ut(x) and ~bt(x) in the absence of scattering. Here g,~U~(x) is the average flux exiting material I at position x along a given ray and ~l(x) is the average flux in material I at position x. The first average is done over all physical realizations in which the point x is an exiting (interface) point in material I, while the second average is done for all realizations in which the point x is in material L In general these two averages are different along a given ray. For homogeneous statistics and homogeneous materials containing constant sources one has the so-called renewal equations :

O~UL(x) = Ql(x)O~(O)e-LX + dy[Ql (y )S l+ f~(y)Oip(x- y)]e ~,Y, (37)

ff,(x) = Ql(x)~b~(0) e - s , / + fo x dy [O_.,(y)Sl+f~(y)~bi?(x-y)] e -~,y, (38)

where ¢~"(x) denotes the average angular flux entering material I at position x, f~(y) = - d Q t ( y ) / d y is the density of probability for a chord of length y in material I and Qt(y) is the probability for a chord of length greater or equal than y. Also for the present case of homogeneous statistics, f~(y) = - d O l ( y ) / d y = Qz(y)/2z and therefore QI(y) = (1/21) S~,' dx xfl(x). Here LI is the maximum chord length in material I and )-i is the average chord length :

~1 = d x x f , ( x ) . (39/

A derivation and discussion of these renewal equations is available in the literature (Vanderhaegen, 1988 ; Levermore et al., 1988 ; Pomraning, 1989; Pomraning and Sanchez, 1990). In the absence of localized sources on the material interfaces the angular fluxes are continuous and we can write :

~,~" (x) = ~ p,Ak~ ut (x), (40) J ~ l

where Pu is the relative probability for a transition from material J into material I at position x along the particle trajectory. These probabilities are normalized to one according to :

Pu = 1. (41)

The precedent equations will now be used to treat the interactions between grains and matrix in a single region containing a matrix with a random distribution of homogeneous grains. For this we assume that the distribution of chord lengths along any ray in the region follows homogeneous renewal statistics. Accordingly,


we will use the renewal equations just described for both the matr ix (I = 0) and the grains (I = i). We will also neglect grain collapsing; that is, we assume that the probabi l i ty for two or more grains to be in contact is very small. Then, if we denote by t~ the relative transit ion probabi l i ty from grain i into the matrix, we have the relations :

20 2oPi Po 2o+ ~t i2~ ' ti p02i ' (42)

i

where the Po and the {Pi} are volumetric fractions. Since ~ ti = 1, the sum over i in the second equality in (42) i

gives an equat ion for the average chord length 20 in the matrix, namely :

P0 Pi - Z , ( 4 3 )

20 i 2i

and therefore 2 o and the transit ion probabil i t ies (t~} can be determined from the known volumetric fractions Po and {Pl} and the grain average chord lengths {2~}.

3.1. Average uncollided fluxes

A general solution to the system of equations (37) and (38) (with I = {0, 1 , . . . , N} can be obtained using s tandard Laplace t ransform techniques but the result is usually untractable. Here we will use the fact that the grains have a very small size compared with the average chord length through the region to obtain an analytical solution. This solution is valid over the entire length of the ray with the exception of a small initial boundary layer whose size is the maximum chord length of the grains Lma x. TO do this we have to assume also that the chord length in the matrix follows a Poisson distribution, i.e. the matrix chord length is exponentially distributed. In this case we have :

fo(Y) = fo(Y) = (1/20) e x/Ao, (44)

Qo(y) = (~0(y) = e-X/z°. (45)

Then taking x = 0 as the point at which the ray enters the region and prescribing at this point a known incoming angular flux ~0(0) = ~bg(0) = ~ we can rewrite equation (37) for the m a t r i x ( / = 0) and for the grains (! = i) as :

~,~Ut(x) = ~ e x p ( - l ~ o x ) + dy[S~+(1/2o)o~n(x-y )]exp( -Zoy) , (46)

fO L O~t(x) = dy [Qi(y)S~+f(y)O°oU'(x-y)]exp(-Z,y) , x >~ L~, (47)

where I~ o ---- Z o + 1/2o and in equation (46) :

I~i0n (X) = E til/l°ut (X). (48) i

Also in writing equat ion (47) we have used the assumption that the grains do not collapse, ~Oln(x) = q/~"~(X) and, in order to neglect the boundary contribution, we have considered only positions at which the grains are not in contact with the incoming boundary. Outside this small boundary layer of dimension Lm,x = max~{Li}, equations (46) and (47) admit the analytical solution :

I~Ut(x) = ~e-~'X+~O,a~(1--e z~), (49)

I//°Ut(x) = TgGe z~-t-lP0,as[T~--]rg e zx]-t-~°ut. (50)

Here ~b0.,~ is the asymptot ic solution for the matrix and g,o,~ is the average angular flux leaving the surface of grain i due to the internal sources in the grain as given by :

" 0o,,~ = (poSo+ Y~ p~S~E~)/Z, (51) i

f0 O~ ut = dyQi(y)S~exp(-Y~iy) = ,~SiE~, (52)

380 R. SANCr~Z and G. C. POMRANING

where T~ is the transmission probability for a grain of type i and 2?~ is the transmission probability for the grain when its cross section is diminished by the value Z. We have :

f0 T~ = dyf (y ) exp ( - Z~y), (53)

if'~ = dy f ( y ) exp [ - ( E i - l~)y] (54)

and 2~Z~E~ = 1 - T~. The cross section l~ is the one given in (15), namely:

Z/Po = Z o - ~o ~ t~Tg' (55)

and E is a new cross section determined by the non-linear equation :

1 ~ ti2?~. (56)

It remains to calculate the average internal angular fluxes given by equation (38). Since the matrix has Poisson statistics, the equations for ~b~Ut(x) and ~b0(x) are identical so that ~k0(x ) = ~O~Ut(x), regardless of the chord statistics for the other components of the mixture. Also, outside the incoming boundary layer, x/> L~, the equation for ~,~(x) is similar to equation (47) for qjout (x) with the exception that Q~ and f must be replaced by (~i and fi, respectively. Using equation (49) one obtains :

~,i (x) = / ~ ~ e - zx + ~k 0,,s [E/g - E~ e - ~x ] + ~,~, (57)

where ~be is the contribution from the internal sources :

~1 i = dyO~(y)Sie z,y = S~P~/Z,, (58)

with Pig,. = 1 - E ~ . Finally, from the internal solutions for the angular fluxes one can calculate the average flux at any point in the region containing the mixture according to :

¢(x) = po~Oo(X) + Z p~tp,(x) = I]/0.as(P0-[- ZpiEgi) ~- (~-- I/t0.as ) (P0-[- ~P~E~) e -zx + ~pi~k,. (59) i i i i

3.2. Homogenization revisited

The transmission probability T for the mixture matrix-grains is obtained by calculating the total uncollided current leaving the region due to a unit angular flux entering uniformly and isotropically the surface F of the region. We have :

T = [ d2r I d"lf~ 'nl~k(r ,f~) , (60) J F d(2n)ou t

where ~b(r, ~ ) is the uncollided angular flux exiting at r in direction f~. However, according to equation (59), in the absence of internal sources the average uncollided angular flux exiting the region after having crossed a chord of length x is :

~k(x) = ee-~/(zcA), x >- L . . . . (61)

where we have used q7 = 1~(hA) and

= Po + ~ pgE~. i

Then, using equation (61) into equation (60) we can write :

(62)

T = v T ( Z ) , (63)


where T(Y~) is the transmission probability of the region with a homogeneous material of total cross section 22 according to :

T(Z) = dx J(x) e ,-x. (64)

Here L is the maximum chord length through the region, and the density of probability for a chord of length f (x ) has been taken proportional to I~" nl according to the prescription :

f(x) = ~ I d2r I d~ln'nl6ix-/(r,~)], ~A £ £~->0°,

(65)

where/(r, ~ ) is the length of the chord exiting at r in direction ~. Finally, from equations (63) and (64) we conclude that the cross section 57 h of the equivalent homogeneous material that preserves the value of the transmission probability of the region is given by the non-linear equation :

T(Eh) = 7T(Y~). (66)

Before discussing this result, we want first to comment on its general validity. Although it seems reasonable to use a renewal process to describe the statistical distribution of matrix and grains along any ray across the region, it is also clear that these distributions cannot be independent. Indeed, one has only to think of all the rays that intersect the same grain in a given physical realization of the system, to realize that the statistical distributions along the rays must be correlated. Note that this correlation has been neglected in deriving equation (63) because the statistical average has been done for each ray independently of the other rays. Another minor objection to equation (63) arises from the fact that expression (61) is only valid outside the boundary layer, and therefore is not exact for chords of length less than the maximum dimension of the grains Lm~x. However, for regions with smooth surfaces (without edges) the density of probability for small chord lengths vanishes with the chord length f(0) = 0, which means that the contribution from short chords is very small in equation (63). A particular case for which this error completely disappears is that of slab region of width greater than Lma x. On the other hand, this error can be considerable for regions of small size and one should abstain completely from using equation (66) in such cases. Observe that both effects, ray correlation and short chords, diminish with the size of the grains so one might expect equation (66) to account correctly for the statistical nature of the problem when the sizes of the grains are much smaller than the size of the region. Finally, it could also be argued, since 7 can be much greater than 1, that for small systems the value of Tas given in equation (66) can exceed 1 so this equation would predict a negative value for the homogenized cross section. By writing the expression for/~g as :

Eg = exp (~L,) dy Qi (y) exp [ - Z,y - Z(L~-y)], (67)

and by re-examining equations (61) and (62) it is easily seen that O(x) can be unphysically greater than 1 only for chords of length smaller than the maximum grain size. Consequently, for regions much larger than the grain size, formula (66) will give a positive homogenized cross section, even though 7 can be much greater than 1.

We now turn to an analysis of equation (66). The right-hand-side of this equation is known in terms of the shape of the region, as given by f (x) , and of the parameters 7 and ~, which depend on the cross sections and on the statistics of the mixture matrix grains. We notice first that, regardless of the size of the region, the homogenized cross section 2h is greater, equal or smaller than the Z of equation (56) whenever the parameter 7 is smaller, equal or greater than 1. This parameter, which does not depend on the size or shape of the system, is always greater than P0 but can be also much greater than 1. As shown in the Appendix, the dependence of 7 on the cross sections of the grains is quite involved for the general case when there are several types of grains, so we will consider here the simple case of a single type of grain. Then 7 monotonously diminishes as the cross section of the grain ~, increases, having a value 1 when ~ = Z0, while the behavior of Z is just the opposite.

ANE 18:7-C

382 R. SANCHEZ and G. C. POrm),ANING

6 I I I I I I

- - - R / R = 5 e.-"--"°---"---'---'-

4 . . . . . R ; , . > , 0 , .

/ 2 7

0 i i i i

o.o ; t ,0 ' T'grain 1195 cm 1

Fig. 1. Behavior of 7 and Z vs the total macroscopic cross section of the grain Y~g~a~, for a mixture of uranium dioxide and spherical gadolinia grains of radius R e = 200/~. The volumetric proportion of UO2 in the mixture is P0 = 0.9055. The dashed fines represent the exact homogenized cross section T~ h for a

cylindrical region of radius R,.

Figure 1 illustrates these facts for spherical grains of radius Rg~in = 200/~ and for Po = 0.9055. These numbers are representative of a uranium dioxide pin with a 7% in mass of gadolinium oxide Gd203. The cross section values for the calculations in the figure, Z0 = 0.95 cm- ~ and Z1 = 1195 cm- ], are typical thermal cross sections at 1/40 eV. The figure gives also the values of Zh for cylindrical regions of different radii. We observe, in particular, that as the size of the region increases, the value of Zh reaches an asymptotic value. As shown in the Appendix, when the characteristic dimension of the system increases without bound, 2Z >> 1, then :

I: h ~ E/V/?, (one surface), (68)

Eh -~ E, (two surfaces), (69)

where these results hold for regions with smooth surfaces. The second limiting value applies to planar regions and hollow cylinders or spheres among others. Thus, for most practical applications one can avoid the extra iterations required to solve equation (66) and use instead the limiting values in equations (68) and (69), that require only the value of E from equation (56) which is much easier to calculate. On the other hand, when the optical size of the region is very small one has"

Zh ~ (1 - -7 ) /2 - -?Z, 2Z << 1. (70)

A more detailed analysis of the behavior of Z and Zh is given in the Appendix. Also, a comparison between equation (66) and the Po prescription of equation (22) is given, together with numerical examples, in Section 5. Here we simply want to point out that equation (66) gives the correct values in the following limiting situations : (a) in the homogeneous limit when all the cross sections became the same Eh, Z ~ Z0 ; and (b) in the so-called atomic-mix limit when all the average chord lengths became negligibly small Zh, Z --* Zmix, where ] ~ m i x = P0Z0 + ~ PiEi is the volumetric average of the cross sections.

i

3.3. Evaluation of escape and collision probabilities

The renewal equations (37) and (38) are only exact for collisionless transport. However, in the case of Markovian statistics for both the matrix and the grains, these equations have been shown to be reasonably accurate for nonpurely absorbing binary mixtures (Adams et al., 1989). In the absence of a better formulation, we adopt equations (37) and (38) as an approximate model for the analysis of matrix-grain interactions in the case of collisions, and with arbitrary homogeneous renewal statistics describing the grains. Then, the general solution in equation (59) can be used to calculate the average leakage rate through the surface of the region F, while equations (49) and (57) can be used to evaluate the average collision rates in the different materials composing the region. Escape and collision probabilities are obtained by normalizing these rates to the sources in a given material. The current of uncollided particles leaving surface F due to the sources in material I is given by :


J = f dz r I d~111"nlq//(r, 11), (71) ~(2~)out

where q/i(r, 11) is the uncollided angular flux exiting at r in direction 11. Since the average total number of particles emitted in a material is 4~Slpl V = ~A~.plSl we have, after use of equation (65) :

if] E1 = 2~tSl dx f (x)q/t(x), (72)

where q/l(x) is given by equations (59) with ~ = 0 and all the sources except Sl equal to zero. Thus :

Eo = (Po + ~ p,E~ - T)/(2~.), (73) i

' Ei = Eo E~ + P~/(2~i), (74)

where T is the transmission probability given in equation (63). In order to compute collision probabilities we have to calculate collision rates of the form :

Cu=PiYv fvdr I , , )d11¢ ,u ( r ,11 ) , (75)

where q/u(r, 11) is the average uncollided angular flux in material I due to sources in material J. In order to obtain a more tractable expression, we convert the integration over angle into an integration over the surface F, i.e. we use d11 = d z r l ~ ' n l / x : , where x is the distance in direction 11 from r to the surface. Next, we interchange the integrations in dr and d2r and express the volume integration as an integration over the angle and an integration along the trajectory, dr = df l dx x z. The final result is :

Cij =piZ, [ d2r [ dnln'nl dxq/u(r+xn,11) . (76) l ( rS l )

Jr d(2n) i , d o

We now introduce the probability Q(x) for chords of length greater than x, Q(x) = S~ dy f(y) . Using equation (65) we can write :

Q(x) = ~ d2r d11 If~" nlH[l(r,11) - x ] , (77) ~(2n)out

where H stands for Heaviside's step function. Using this last expression in equation (76) and normalizing the result to the total average source in material J yields a formula for the collision probability, namely :

PlEl ~o ~ P1J = ~ dx Q(x)@H(x). (78)

The average flux q/ij(x) is then obtained by setting ~ and all the sources except Ss to zero, and by using q/0(x) = q/~"t(x) and equation (49) for the matrix I = 0, or equation (57) for the grains I = i. The result is :

= p~E~ P00 = P ~ 2 P(E), P~o ~ 7 - [E~ - E~E(E)], (79)

g g Poj = Poo Eg, Pv = fi~JP,, + P, oE~i. (80)

In these equations, P(E) and E(E) are the self-collision and the escape probabilities for an equivalent homogeneous region with the cross section of equation (56). These quantities can be obtained from the transmission probability T(Z) by the conservation relations 2EE(X) = 1 - T(E) and P(E) = 1 - E(E).

We observe that the expressions for P0~ and P~j in equations (79) and (80) are identical to the P~j formulas (10) and (11) obtained in Section 2, and therefore confirm the main assumption on which the CP formalism is based. This can also be seen from equation (51) which shows that the uncollided current exiting the grains,

p~S~E~, is directly added to the uniform and isotropic source in the matrix poSo. However, from the formula i

for P~j and by comparing the expressions for P~0 and P0~ as given in equations (79) and (80) we find :


PiZi PiZi Pij = p ~ j P j i+ ~(E~E~-E~E~)E(Z), (81)

PiEi PiZi P~o = P0Z0 P0i + ~ - (E~ - E~)E(Z), (82)

which shows that reciprocity is violated. We can also prove that conservation is not preserved either. By using equation (3) to estimate escape probabilities from the collision probabilities given by equations (79) and (80) we get :

Eo = 7(2/Z)E(Z), El = EoE~, (83)

which are different from the values given in equations (73) and (74). These discrepancies can be traced back to the errors originated by the improper use of the solutions in equations (49) and (57) in the boundary layer, errors that are compounded because the boundary layer appears in each ray. Indeed, for large regions 2~,

and T(Z), E(Z)--+0 and consequently equations (81) and (82) predict reciprocity, while the escape probabilities in equations (83), (73) and (74) all vanish.

On the other hand, the escape probabilities given by equations (73) and (74) are consistent with the total transmission probability of equation (63) in the sense that they respect the conservation equation (4). This is due to the fact that here the boundary layer will cause an error only for rays of very small length, which are usually of very small probability. Also this very small error contributes in similar fashion to escape and transmission probabilities. For all practical purposes the transmission probability of equation (63) and the escape probabilities of equations (73) and (74) can be taken as exact, whereas the set of collision probabilities in equations (79) and (80) are affected by a boundary layer error. In order to preserve reciprocity and conservation we choose to correct these collision probabilities by adding small, boundary-layer terms to Poi and to P0 according to :

Po~ = Poo Eg + ~e/2, (84)

P~j = {)ijP g. + PsoE~ + flo/,~, (85)

while Pio and P00 will be obtained from reciprocity and conservation, respectively. Also, in equation (85) PJ~/flo = PiY~iflji . Then, use of the conservation equation for E0 gives a relation for P00, namely :

Poo ~ 1 -- Pg -- - ~ - + 2Z ~p~ PoZ0JJ'

and the conservation equations for the {Ei} yield the following relations for the as and fls :

O~i ~ -E flji = -- Pgii/Zi" (87) /

There are of course a variety of ways to choose the c~s and fls. Here we make the simple choice fl~j = 0 and ei = - Pg /2i which gives :

Po, = P o o E ~ - P~/ (2Z, ) , (88)

Poo ~, 1 -- (89) -- 2Z + 2 Z ~ p~Pg l+p0Eo "

4. THE CASE OF HETEROGENEOUS GRAINS

One of the assumptions that makes the renewal equations so easy to handle is that the cross sections and the sources of the material are taken to depend only on the absolute position along every ray. This assumption is invalid when one wants to include materials, such as grains, that have a well-defined internal structure and for which the cross sections and sources depend on the relative position within the material. In this case, a knowledge of the chord length distribution is not sufficient to determine cross sections and sources in the materials. Rather, what is required is to know the density of probability for each possible trajectory through the material. We observe that a trajectory (or chord) is uniquely specified by its exiting point r and its exiting


direction ~2, where r is a point on the surface F~ of the grain and ~ is an exiting direction in (2~)ou, at the point. Note that r and ~ are defined in local grain coordinates and therefore do not depend on the position of the grain along the ray. Hence all that is needed to define a renewal process is to assign a density of probability f ) ( r , ~ ) at every point x along a ray. This density of probability may depend on the specific ray and on the position x along the ray, but here we consider only homogeneous and isotropic statistics for which f ) (r, ~ ) depends only on r and ~ . We will assume the normalization :

fr d z r l d ~ l ~ ' n l f A ( r , ~ ) = l , (90) ~g 2n)ou t

where the integration is over the surface of the grain Fi g .

4.1. The interface flux Sincef~(r , f~) is the density of probability for the trajectory (r, ~ ) and since the transport within the grain

is deterministic, we can write :

O°Ut(x) = fr d 2 r ~ d~l, 'nl fA(r, , )[exp{--z~[r, , , l (r , , )]}~°o~t[x-- l(r , , )]+~°m(r, , )] . (91) 2~)ou,

Here l(r, f~) is the length of the chord exiting at (r, f~), I//°~t(r, ~'~) is the exiting flux due to the internal sources along the trajectory (r, f~) :

I I(r,I '~)

qj o~t (r, f~) = dz exp [ - % (r, ~ , z)] Si(r - z~, ~), (92) do

and r~(r, ~ , z) is the optical distance from the surface point r to the interior point r - z ~ . Note also that the source term depends on local (grain) coordinates, while the average transition flux I ] / ~ u t depends on the absolute position on the ray.

We now introduce the length chord variable y by simply inserting the identity ~, dy 6[y-l(r, ~)] inside the d 2 r df~ integration in equation (91). Then, by changing the orders of integration we obtain :

f0' ~,~t(x) = dy {(exp [-~,( r , ~ , y)] 8 [ y - / ( r , n)])A0~ut(x--y)

+ (exp [-- r,(r, ~ , y)] S, ( r - - y~ , ~ ) H [l(r,~) --y])A}, (93)

where the ( )A indicates multiplication by I~" nlfA(r, ~ ) and integration over the surface F~ g and over (2~) .... We are now able to show that our generalization of renewal processes contains as a limiting case the previous formulation. Indeed, if we assume that the optical thickness and the source in the material depend only on the position along the ray, i.e.

r~(r, ~ , y) = z~(x-y, x), & ( r - y f L ~ ) = &(x-y ) , (94)

then equation (93) reduces to the more familiar form :

0,-°Ut(x) = dyexp[-r~(x-y,x)][f(y)O°oUt(x-y)+Q~(y)&(x-y)], (95)

where

f (Y) = - dyQi(y) = (6[y - l(r, ~)] )A, (96)

is the density of probability for a chord of length y. This analysis shows also how to proceed in the more general case when the grains have internal structure. In fact, it suffices to define appropriate averages to reduce equation (93) to its by now familiar form. For instance, we may associate an average optical thickness ri(Y) to chords of length y by the expression :

f ( y ) exp [ - "t-i(y)] = (exp [ -- zi(r, ~ , y)] c5[y - l(r, ~)])A, (97)

and an average source &(y) to chords of length greater than y according to :

386 R. SANCrmZ and G. C. POMRANING

Q,(y) exp [-z , (y)] S~(y) = (exp [-z+(r, ll, y)] S~(r-yl l , I'I)H [ y - / ( r , II)])A , (98)

which effectively allows to rewrite equation (93) in the form of equation (95). However, in order to deal directly with each different layer in the grain, we choose to write the source contribution in terms of the "escape probability" as :

E/gk = 7tA/g ~ d2r f d~l~'nlfA(r,f~)~, ,k(r ,~) drf d(2n)ou ,

= gA~(g'~k(r, II))A, (99)

where

1 f l(r,fl) ~b+(r, 11) = 4-~.~ Jo dyexp [-z , ( r , ll , y)] x+( r -y [ l ) , (100)

is the uncollided flux exiting the grain at (r, fl) due to a unit isotropic and uniform source in layer k. The characteristic function x~(r), equal to 1 for r in layer k and 0 otherwise, identifies the particles coming exclusively from layer k. We recognize that when f~(r, 11) is constant, equation (99) gives the usual escape probability.

When the sources within the grain are uniform and isotropic in each layer, then we can use equations (99) and (100) to write the source contribution in equation (91) as :

~o~., = 2i ~ Sa~p/gkE~, (101) k

where we have used the decomposition of unity as 1 = ~ Zik(r) which is valid in any point within the grain, k

and where p$ = V~/V~ is the volumetric proportion of layer k in grain i. Using equation (97) to calculate the transition term gives the result :

f0' ~b?m(x) = dyexp[-zi(y)]f(y)O°oUt(x-y)+O °"t. (102)

This equation is very similar to equation (47) and again the solution for the average exiting fluxes is given by equation (49) and (50) with :

+o.+ = + >

However, all the coefficients depending on the cross section of the grain must be modified to account for the internal structure. In particular, equations (53) and (54) are replaced by :

T/g = dy f ( y ) exp [ - zi(Y)] = (exp [ - ~,(r,~)])A, (104)

f, i~ = dy f~(y) exp [ - ~,(y) + Eyl = (exp [ - ~i(r, ll) + El(r, II)])A, (105)

while equations (55) and (56) remain valid. In both of these equations ~(r, ll) = ~[r, ~, l(r, 12)] denotes the optical thickness of the chord (r, ll). Finally, we can use the relation :

+,o, ), 1 - T+ g = exp [ - z,(r, [1, y)] Y.(r-y[~) , (106)

\dO

to show that the escape probabilities E/gk and the transmission probability T/g satisfy the usual conservation equation given by :


1 = T~ + ~, ~ p ~ Z a E ~ . (107) k

4.2. The interior flux The equation for the interior flux ~ki(x) is also obtained in a similar manner but now we need to know the

density of probability for a point x in a ray to be somewhere within the grain. We let fV(r, fl) be the relative density of the probability that, given that the point x is inside a grain of type i, it is at relative position r and the direction of the ray coincides with the relative direction II. Relative here refers to local grain coordinates. As before, for homogeneous and isotropic statistics we take this quantity to be independent of the point x and of the ray and adopt the normalization :

f v d r ~ d~fV(r , . )=l . (108) 4n)

By using this density of probability we can write an equation for the average flux at x when this point is in layer k of a grain of type i. This is given by :

p~kJ/,,(x) = fvrdrx,,(r) f~4~)d.O fV(r,l'Z)texp(-z,[r, FZ, l(r,~)l}~U*Ix-l(r,~)]+¢,(r, ll)], (109)

where l(r, II) is the distance in direction - ~ from the point r to the surface of the grain (distance to the point at which the trajectory enters the surface of the grain), ~k~(r, II) is the contribution of the internal sources to the flux at (r, I lL namely :

~i (r,fl) ~b,.(r, fl) = dz exp [ - x,(r, ~ , z)] S , ( r - z~, 11), (110)

and x~(r, 1"1, z) is the optical thickness from point r to point r - z f l . The characteristic function X~ in equation (109) selects those points for which x is in layer k. Also in equation (109) we have:

p~, = fv drz~(r) I d , fV(r, £1). (111) /8 4~:)

We will show later that for homogeneous and isotropic renewal statistics fV(r, f~) is constant and consequently this coefficient is nothing else but the volumetric proportion of layer k in grain i. As before we introduce the length along the trajectory y by inserting So L, dy 3[y- l ( r , II)] inside the dr dI l in equation (109). After pulling out the integration in dy we obtain an expression very similar to equation (93), namely :

p~O~(x) = dy{()~e(r)exp[-~,(r, ll, y)16[y-l(r, ll)])vO~=(x-y)

+ (exp [ - ~,(r,l~, y)lS~(r-yf~, ll)H[/(r, ll) -ylx~(r))v}, (112)

but now ( )v indicates multiplication byfV(r, f~) and integration over the grain volume V~ and over (4r 0. To account for the contribution from the sources in each layer we define the '"collision probability" according to :

P-~,,, = 4xVrE~ I drza,(r) f df~fV(r,n)~bit(r,n) = 4nV~X;,-k(Z,k(r)fu(r, f l))v, (113) d~ J4

where ~z(r, ~ ) is the unco]lided flux at (r, ~ ) due to a unit isotropic and uniform source in layer L This flux is obtained from equation (100) by replacing ik with i j . The difference now is that (r, ~ ) is an interior point in the grain and that l(r, ~ ) is the length of the trajectory from the entering point on surface F~ to point r. For a constant density of probability we have f v (r, ~ ) = 1/(4x Fig) and equation (113) reduces to the classical collision probability. Use of this definition allows us to write the volumetric source contribution to ~ ( x ) in equation (112) in the form:


1 0,, - Zg ,% ~, vg, s , PL , . (114)

To handle the contr ibut ion from the interface flux ~p°oU¢(x-y) we introduce an average optical path for each layer as :

p,.g~ (y) exp [ - z~k(y)l = (Zik(r) exp [-- ze(r, KL Y)] 6[y-- l(r, £t) l )v, (115)

wheref//(y) is the density of probabi l i ty that, given that the point x is inside grain i, the distance of the trajectory from the entering point to x is y, i.e.

f~(y) = (6[y-- l (r , ~)]>v- (116)

Putting the two contributions (volumetric and surface) together we have :

f0 LI Oik(x) (Pgk) 1 dyexp [-z ik(y)] - out = f(Y)~Po ( x - y ) + ~b,k. (117)

Therefore, use of solution (49) yields the result:

ff~k(x) =/~gk q7 exp ( - Zx) + ~b 0,as[Egk --/~gk exp ( -- Ex)] + ~b~k. (118)

The "escape probabil i t ies" in this formula are given in terms of the d e n s i t y f a s :

f0 L Egk = (Pgk) 1 dy f(y)exp[-ri~(y)] = <Z~k(r) exp [-z~(r,f~)]>v, (119)

L'

Egk = (.Pgk) 1 dyf~(y)exp[--z~k(y)+Ey] = < Z ~ ( r ) e x p [ - ~ , ( r , n ) + Y 4 ( r , n ) l > v , (120)

where z~(r, f~) = z~[r, g~, l(r, f~)] is the optical distance in direction - f ~ from point r to the surface Fg. The fact that we have used the same symbol to denote the escape probabi l i ty of equation (99), defined by

a surface average, and the escape probabi l i ty of equation (l 19), defined by a volume average, is not fortuitous. In fact, as we now prove, for homogeneous and isotropic renewal statistics these two quantities are identical. First we use the same change of variables, i.e. dr d ~ = d2r df~ [f~'nl, that was used when transforming equation (75) into (76) to establish the general formula :

f i/(r,f~) (g(r ,g~)>v = , d2r d ~ l f ~ ' r [ d y f V ( r - y l L l ' Z ) g ( r - y ~ , ~ ) , (121) dr~ 2~)oo, d o

val id for an arbi t rary funct ion g(r, ~ ) . We now recall that the relat ion f , (y) = (I/2~) ~#, dz f ( z ) holds for homogeneous and isotropic statistics. Using equations (96) and (116) we obtain, after having used equation (121) to convert volume integration into surface integrat ion:

fr d z r I dg~lf~ 'r[ H[l(r,n)-y][fV(r-yn, a)--fA(r,n)/2,] = 0. (122) ,J(2n)o.,

Since this identity must be true for any point r - y ~ in the trajectory (r, ~ ) , we conclude that f V ( r - y ~ , f~) depends only on the surface point (r, ~ ) . Moreover, since an interior point lies on trajectories connecting it with any surface point, one concludes that fV(r, I'~) must be constant, i.e.

1 1 f V ( r ' a ) - 4nVg ' f f f ( r , ~ ) = gAg. (123)

g g This proves that the Pgk defined in equation (111) is indeed the volumetric propor t ion V~k/V, of layer ik in the grain. Another way of looking at equation (123) is to realize that homogeneous and isotropic statistics means that an arbi t rary translat ion or rotat ion of a physical configuration is again a physical configuration with the same probabi l i ty as the first one. This of course means that all interior points (r, ~ ) inside a given grain are equiprobable. Finally, using (123) in equation (121) we have :


1 / pt(,,a) \ <g(r,a)>v = Z / j 0 y(r--yl'~, [~)/A, (124)

which shows that equations (99) and (119) are identical.

4.3. Homogenization, escape and collision probabilities

Let us now briefly extend the analysis of Sections 3.2 and 3.3 to the present case involving heterogeneous grains. The interior solution for the matrix ~0(x) = ~b0(x) °~t can again be written as in equation (49) with ~k0,~s as in equation (51) but with the definition :

g g V~S,E g = ~ V,kS~kE g. (125) k

Also, the interior solution in a grain i becomes the average :

~li(X ) = E p ~ l i k ( X ) , (126) k

where ~kik(X) is given in equation (118). Hence, equation (57) is still valid but now:

Eg~ = ~ p $ E~k, E~ = ~,p~kE~k (127) k k

and

p g

= E PitSilE v-" (128) k l k z"qk

Thus, with these modifications, the expression for the average flux in equation (59) is also valid. All of this means that all the conclusions and formulas that were obtained when discussing region homo-

genization in Section 3.2 apply for the case of heterogeneous grains. Moreover, carrying out the calculations of Section 3.3 for the present case results in the formulas :

E o = ( P o ' ~ - ~ i P i E g - - T I / ( ~ ) ,

l Pil, E~ = EoEgik + ~

"~ l ~ i l '

(129)

(130)

for the escape probabilities, and

P0Zo . . . . pikZ/~ Poo = - ~ - r t ' o , P,k.o = ~ f f - [E~, - E~,E(E)], (131)

- - g g Po.,t = PooE~, ei~,jz - 61jfktPikJ + Pik.oE~, (132)

where p~ =PtP$ is the volumetric fraction for all layers ik in the region. A direct comparison with the corresponding formulas for homogeneous grains show that equations (129-132) are straightforward gener- alizations of those obtained in Section 3.3. Again the escape and transmission probabilities are conservative, and the collision probabilities are affected by a boundary layer error. This means that the conclusions in Section 3.3 apply also to the present case.

5. N U M E R I C A L C O M P A R I S O N S

In this section we investigate the accuracy of the different prescriptions for region homogenization. More precisely we compare the values of the homogenized cross section Eh, given by the CP formula (22) and by the asymptotic expression in equation (68), to the exact value defined in equation (66). In order to establish the ability of these two approximations to give an accurate estimation of the homogenized cross section we consider the simplest possible setting consisting of a matrix with a single type of spherical, homogeneous grain. Also, the number of parameters entering the problem is minimized by using dimensionless variables : cross sections are measured in units of the grain cross section Z 1, and distances in terms of the grain optical size

ANE 18:7-D


Tgrain = ~, iRgrain, where Rgrain is the radius of the grain. Hence, the parameters defining the mixing are the grain optical size Zgr~, the ratio of the matrix cross section to the grain cross section So = Eo/E~ and the volumetric proport ion of the matrix Po. Denoting by E* the CP prescription for Eh in equation (22), and by E~ the asymptotic expression (68) given by renewal theory, we have :

[ 1--Te l E g E * = P o + P , I _ T g ( E o ) E o = P 0 E 0 + p , E , E g ( E o ~ , (133)

~as = (1/x/V)E = (1/x/~) PoXo +/91 ~1/~g p0+pj j~g , (134)

where for a homogeneous grain :

7 = P o E , - Y , " (135)

In these formulas, the transmission and escape probabilities for the grain, T g and E g, depend only on zvam, while Tg(E0) and Eg(Zo) depend on the product Zgrai.S o. Also the quantity J~g depends on '~grain (1--S) with s = E/El.

Although we want to make a large parametric study, the practical application that we have in mind is that of a PWR poisoned fuel pin containing gadolinia grains in a uranium dioxide matrix. The material composing the grains is gadolinium oxide, Gd203, with the main absorbing isotopes, 153Gd and ~57Gd, present in isotopic proportions of approx. 15% each. The size of the grains is typically of 200 M or, for ground grains (powder) 5 #. Using typical values for the grain density, the proport ion in weight of the grains in the UO2 matrix, and the average volumetric mass of the mixture, we obtain at thermal energies (1/40 eV) E0 ~ 0.95 cm - ~ and E l ~ 1195 cm- i. Thus, in the thermal range So ~ 0.001 with Zgr~in ranging from 0.6 for powder to 24 for grains. Also, a common value for the volumetric proport ion of the uranium dioxide matrix is P0 ~ 0.95. We have first investigated the behavior of the exact cross section, given by equation (66), in terms of the shape and size of the region containing the mixture matrix-grains. Figure 2 shows the values of Eh for a spherical and for a cylindrical region vs the relative ratio of the region r = radius region/Rgrain. The calculations here correspond to powered grains, Zgra~n = 0.6, with a cross section ratio of So = 0.01 and with P0 = 0.95, i.e. for powered grains that absorb 100 times more than the matrix. The results in this figure show that, as expected, the homogenized cross section converges rapidly towards an asymptotic value that is independent of the shape of the region. A similar beha;eior has also been obtained for other values of rgr,i,, So and P0. Therefore, in the analysis to follow we will use as reference the value of Eh for a cylindrical pin of relative radius r = 50.

Figures 3-5 present the relative errors in percent :

e* = 100(E*/Eh-- 1), ~ = 100(EaJEh-- 1), (136)

VS the optical size of the grain 27grain for P0 = 0.95 and for three different values of the cross section ratio,

0.0475

Eh 0.045 -

0.0425

I I

i'o 160 i o'oo R I R

rogion grain

Fig. 2. Homogenized cross section Zh for a cylindrical and for a spherical region vs the relative size of the region r = Rr~on/Rg,ai,. The calculations were done for a mixture of uranium dioxide with gadolinia grains

with p0 = 0.95, E0/Et = 10 -2 and Zgr~in = 0.6.


I I I y I I I

0.

- 4 -

- 8 -

- 1 0 - , , , , , , 1 6 " 4 0 . 0 1 1 100

'l:grain

Fig. 3. Relative errors in percent for two different estimations of the homogenized cross section vs the optical radius of the grains for a mixture withp0 = 0.95 and E0/l~l = 10 -3. Here ~* corresponds to the CP estimate given by equation (22), and e is the error of the asymptotic formula given by equation (68). The

value of the coefficient 7 is also shown.

2 i I I I I I

7 o

- 2 ~ . . - 4 - E - 8 i i i i i

1 0 -4 0 . 0 1 1 100

~ " g r a i n

Fig. 4. As in Fig 3 but forp0 = 0.95 and E0/Z1 = l0 2.

1 0 s

1 0 0 0 - -

0 . 1 -

0 . 0 0 1 ~

10-s- 1 0 -4

I I I L I

J

E ', ~'.. ,, ,,

i i i i i i 0 . 0 1 1 1 0 0

' ~g ra in

Fig. 5. As in Fig. 3 but forp0 = 0.95 and Eo/2 ~ = 102.

So = 0.001, 0.01 and 100. The value of the parameter 7 in equation (135) is also given. F r o m these figures we see that E,~ gives a very good approximation for the value of the homogenized cross section Eh, with the difference between the two vanishing at the limits when Zgrain goes to 0 or oo. We also see that the error is much smaller when the matrix is more absorbant than the grain as shown in Fig. 5. Notice that for ~gra~, ~ 0, the quantity/~g in equation (134) goes to 1, and therefore Z ~ ~'~mix = P0Y0+P~E1 and 7 --* 1, which explains why the error vanishes at this limit. The reason why e. also vanishes at the limit Tgrain ---400 is due to the fact that, as shown in equation (A14) of the Appendix, the error of the asymptotic approximation is of the order of the inverse of the optical size of the system, and the latter goes to oo as Z~ain --" oe. This also explains why the precision of equation (134) is better for high values of the cross section ratio so. Indeed, the greater is so, the greater will be the ratio E/E 1 and therefore the greater will be the optical size of the system. The erratic behavior of e in Fig. 5 is due to the fact that the iterations for both E and Eh were considered converged when a relative precision of 10 5 was achieved. The asymptotic behavior of 7 in these figures can also be explained by recalling that E lies between E0 and Z 1. That is, we may have E/E~ smaller or greater than 1 according to whether So is smaller or greater than 1. Accordingly, as "(grain --4 OO, the optical path Tgr,i,(1 - -E/£1) will go to minus or plus oo, corresponding to/~g going to oo or 0. Summarizing, for so < 1 we have E g ~ 0 SO that E --. E0 and 7 --*P0, whereas for So > 1 we have Eg ~ oo which implies that t2 ~ E ~ and 7 -* oo.

The behavior of the CP estimate of equation (133) is given by the curves labelled e* in Figs 3-5. For Zgrai. 0, both 12g and Eg(120) go to 1, and equation (133) shows that 12*goes to the correct limit value Zm~x- However, for large values of Zgr,in we have T g, Tg(120) ~ 0 and equation (133) gives the wrong behavior E* ~ 120. The curves in the figures shown that the CP formula for 12h is less precise than the asymptotic value of equation

392 R. SANC~mZ and G. C. POMRANING

1 0 0 0 - I I t t t t

/ 100-

lo- ~ T .

1 h

0.1 " - .

0 . 0 1 , , , , ,

1 O" 4 0 . 0 1 1 1 O 0

~ g r a i n

Fig. 6. The CP estimate (E~ and the exact homogenized cross sections vs the optical radius of the grains for a mixture with p0 = 0.95 and E0/Z, = 102. The value of coefficient 3' is also shown.

(134), and also that the CP prediction is completely erroneous for large values of the cross section ratio s0, This last fact is best appreciated in Fig. 6 in which we have shown the actual values of the CP estimate lg* and the exact expression for Zh for the parameters of Fig. 3. In this case Z0 > El and therefore TS(Eo) < T ~ which forces the CP value to be always greater than P0E0. In contrast, the exact homogenized cross section vanishes for large values of rgra~n.

Given that the CP formula was derived as a perturbation around a state with so ~ 1, it is surprising to find out that the CP formula is relatively accurate for small values of s0. Still, we observe in Fig. 3 that the relative error of the CP estimation for a representative gadolinia poisoned fuel pin (So ~ 0.001, P0 ~ 0.95 and Zgra~o ~ 0.6-- 24) is approx. 8%, whereas the asymptotic formula exhibits only a 3% error. The behavior of Z* and of Za~ shown in the figures is characteristic of a large range of values of the parameters So and P0. As an example we show in Fig. 7 the corresponding relative errors for the case So = 0.001 and P0 = 0.5. A comparison with Fig. 3 shows that, while the precision of the asymptotic formula has not changed, the error of the CP estimation is much larger. From the numerical examples shown here we may conclude that the CP expression usually gives very large errors with the exception of mixtures with a small volumetric proportion of grains and for grains more absorbant than the matrix.

6. CONCLUDING REMARKS

The CP treatment of the double heterogeneity problem consists of three steps. First, the matrix-grain mixture in any region containing grains is replaced by a homogenized material that preserves the transmission probability of the region. Second, a traditional collision probability method is used to obtain a set of collision probabilities for the homogenized regions. Third, all the collision probabilities between grains and matrices

2 0 i i i i i i

Y o ..... ~-

-2o. - 4 0 -

- 6 o - ~ - ~

-a °O" l ' o ; 1 ' 1 ' ' 1 0 0

~ g r a i n

Fig. 7. As in Fig. 3 bu t fo rp0 = 0.5 and E0/Z1 = 10 -3.


are obtained from algebraic combinations of the collision probabilities for the homogenized regions. This last operation requires only the extra calculation of the escape probabilities for single typical grains in each mixture. The formulas used in the last step are based on the assumption that particles leaving the grains appear as uniformly and isotropically distributed in the matrix containing the grains.

In this paper we have investigated the CP formalism for the double heterogeneity problem using statistical techniques that have been recently developed for the treatment of particle transport in stochastic media. By assuming that the matrix-grain mixture could be represented as a renewal process along any trajectory, we have been able to derive an exact formula for the homogenized cross section of step 1. The evaluation of this formula requires the iterative solution of two nonlinear problems. A first set of iterations, involving repeated evaluations of grain transmission probabilities, yields an intermediary cross section 1~ that can itself be used to obtain an accurate asymptotic expression for the homogenized cross section. The exact value for this cross section ]~h results from a second set of iterations requiring repeated evaluations of the transmission probability for the region itself. Numerical tests have shown that the asymptotic formula for the homogenized cross section exhibits a small error for a large range of values of the parameters defining the mixture. It has also been shown that not only is the error from the CP formula always greater than that from the asymptotic formula, but also that the CP formula fails to give an acceptable value in many instances. In particular, for parameters representative of a poisoned fuel pin containing gadolinia grains, the CP formula exhibits errors of about 8%, while the asymptotic expression gives an error of 3%. It remains, however, to estimate the impact of this error in typical burnup calculations. These evaluations require large multigroup transport-diffusion depletion calculations that are beyond the scope of the present work. In this regard, we want to point out that the fractional additional computational effort required to evaluate either the asymptotic approximation or the exact formula is very small, and thus does not constitute a significant cost burden if greater precision is required in the homogenization aspect of depletion calculations.

The renewal equations used to evaluate the correct value of the homogenized cross section are only exact for collisionless transport. However, by using these equations as an approximate description in the case of collisions, we have also been able to analyze the formulas used to relate the different probabilities in the CP formalism. We have shown that, except for a boundary layer of size that of the grains, the assumption on which these formulas are based is quite compatible with statistical renewal theory. We also proposed some modifications of these formulas to make them more consistent with the results obtained from our renewal analysis. Again, the interest of such formulas should be evaluated in the context of practical burnup calculations.

Finally, in order to treat multilayered grains, we have generalized renewal processes to the treatment of materials with internal structure. These new renewal equations have been shown to reduce to the previous renewal equations in the proper limit.

Acknowledgements--This work was partially supported by the U.S. Department of Energy under Grant FG03-89ER14016. One of the authors (RS) would like to thank Anna K. for helpful suggestions.

REFERENCES

Adams M. L., Larsen E. W. and Pomraning G. C. (1989) J. Quant. Spectros. Radiat. Transfer 42, 253. Hebert A. (1980) Personal Communication. Lane R. K., Nordheim L. W. and Sampon J. B. (1962) NucL Sci. Engng 14, 390. Livolant M. and JeanPierre F. (1974) Report CEA-R-4533, Commissariat fi l'Energie Atomique. Levermore C. D., Pomraning G. C., Sanzo D. L. and Wong J. (1986) J. Math. Phys. 27, 2526. Levermore C. D., Pomraning G. C. and Wong J. (1988) J. Math. Phys. 29, 995. Malvagi F. and Pomraning G. C. (1990) J. Math. Phys. 31, 892. Pomraning G. C. (1986) Transp. Theor. Statis. Phys. 15, 773. Pomraning G. C. (1989) J. Quant. Spectros. Radiat. Transfer 42, 279. Pomraning G. C. and Sanchez R. (1990) J. Quant. Spectros. Radiat. Transfer 43, 267. Sahni D. C. (1989a) Ann. nucL Energy 16, 397. Sahni D. C. (1989b) J. Math. Phys. 30, 1554. Sanchez R. (1989) J. Math. Phys. 30, 2498. Sanchez R. and McCormick N. J. (1982) Nucl. Sci. Engng 80, 481. Sanchez R., Mondot J., Stankovski Z., Cossic A. and Zmijarevic I. (1988) Nucl. Sci. Engng 100, 352. Vanderhaegen D. (1986) J. Quant. Spectros. Radiat. Transfer 36, 557. Vanderhaegen D. (1988) J. Quant. Spectros. Radiat. Transfer 39, 333.


A P P E N D I X

Analysis of Z, and 2h

The behavior o f the cross section E can be explained by observing that the "transmission probability" if "g is a monotonously increasing function of E. This "probabili ty" has its min imum value T~ for E = 0, the value 1 at 2 = Zi and increases without bound as Z --* oo. In view of equations (55) and (56) this means that:

E <~ f?/Po <~ '2,o. (A1)

Moreover, the value of Z lies between the min imum and the max imum values, Emin and Em,x of the cross sections of matrix and grains, i.e.

Emi n ~ ~, ~ min (~,0, Emax)- (A2)

Indeed, assuming that Z < Em~n, then all the T,g are smaller than 1 and equation (56) shows that E > Z 0 which is contradictory to our assumption. Therefore we mus t have E ~> Emin. The superior bound by Era, x is proved in a similar manner. We now analyze the case when Z reaches its limiting values. The conditions for E = Z 0 are either that all the Tf are zero and that 2o ¢ 0 or that the ~ are finite and 2o = 0. The first case happens for infinitely absorbing grains, whereas the second case means that there are no grains present and ~,o = Z0. Similarly, E will be zero for Z0 = 0 and either 2o = oo or the ibf = 1. Both cases correspond to a vacuum.

Next we consider three important limits. In the homogeneous limit, when all cross sections are the same, the solution to equation (56) is E = Eo which ensures i?~ = 1 since Ej = E0. When there is only one type of grain present E = E0 implies that El = E0, but this is not necessarily true for several types of grains, in which case the value Z = Z0 can be reached several times as the cross section o f a given type of grain changes. In the pure matrix limit, Po ~ 1, we obtain again the correct physical value Z = E0. Finally, when the size of the grains goes to zero, one can use a small argument expansion of the exponential in equation (50) to obtain :

7"gi ~ 1 ~- (Z-Zi)~i , (A3)

which gives the volume averaged cross section :

E ~ Zmi x = P 0 E 0 ~- '~. P i Z i . (A4) i

Also, in view of equation (42), this will happen either for vanishingly small grain proportions or in the so-called atomic mix limit when also 2o ~ 0.

The behavior of E and y in terms of the cross section values can be characterized from the expressions for the corresponding derivatives given by :

8 E / O E , : B , / ( 2 o + ~ B j ) , (A5)

8E/SEo=I/(2o+~Bj), (A6)

a?/ax, = (az/aE,) 52 B j - (1 - az/az,)B,, (AV) j ~ i

07/3Eo = (SZ/SNo) Z/~J, (A8) J

where

B, = t, d x f ( x ) exp [ - x ( Z , - Z ) ] , (A9) 0

L

B~ = ti dxQi(x) exp [ - x ( E i - E ) ] . (A10)

Equations (A5) and (A6) show that 0Z/OE, and 0E/BE 0 are positive and vanish for Z~ ~ oo. Furthermore OZ/SE~ is always less than 1. On the other hand, the behavior of 7 is much more complicated, as shown in equations (AV) and (A8). The derivative ~?y/3E0 is positive but the sign of OV/0Xi depends on whether the positive contribution arising from the sum of j dominates the main negative term. However, as E~ --* oo this derivative becomes negative. Also, in the particular case in which there is only one type o f grain, the derivative is always negative. Our conclusion is that Z increases with positive changes o f the cross sections, whereas y increases with Y'0- The behavior of 7 for changes in El is complicated for small values of E1 but for sufficiently large values of E~ the value of 7 will decrease when Z~ increases.

We analyze now the behavior of the homogenized cross section Eh as a function of the size of the region. We will assume that the shape of the region remains unchanged as the region changes size. If we denote by f ( x ; 2) the chord length distribution for a region whose mean chord length is 2, then the assumption that the shape does not change can be expressed a s :


f (x; 2) = g(x/2)/2, (A11)

where g(x) is the reference chord length distribution when 2 = 1. By using equation (A 11) we can rewrite equation (64) as :

f/' T(Z) = dx g(x) e -~zx, (A12) 0

where l0 and l~ are the minimum and maximum chord lengths for the region when 2 = 1. An asymptotic expression can now be obtained by successive integration by parts. This yields :

T(Z) = ~ g(lo)+ 2Z-- + 0 (,tZ)2 . (A13)

For a region with a single smooth surface l0 = 0 and 9(0) = 0, so using the asymptotic expansion of equation (A13) to calculate T(Zh) and T(Z) in equation (66) we obtain :

x Fl g"(o) . F ] ~]h ~ b + ~ ( x / 7 - 1 ) ' 2 ~ 0 o , (114)

while, for a region with two separate surfaces (10 :# 0) we have :

/ ' ln2"~ Zh ~ Z | I - - \ ~.Z/o],/ 2 ~ oo. (A15)

A statistical analysis of the double heterogeneity problem

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