mm- IDAHO NUCLEAR CORPC^^TION 1 S. ATOMIC ENERGY

PARTICLE SIZE DISTRIBUTIONS FROM FUEL RODS FRAGMENTED DURING POWER BURST TESTS

IN THE CAPSULE DRIVER CORE

J. A. McClure

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IDAHO NUCLEAR CORPC^^TIONNATIONAL REACTOR TESTING STATION

IDAHO FALLS, IDAHO

Date Published— October 1970

1 S. ATOMIC ENERGY COMMISSION

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IN-1428 Reactor TechnologyTID-4500

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BV

J. A. McClure

IDAHO NUCLEAR CORPORATION

A Jointly Owned Subsidiary of

AEROJET GENERAL CORPORATION ALLIED CHEMICAL CORPORATION PHILLIPS PETROLEUM COMPANY

Date Published — October 1970

PREPARED FOR THE U. S. ATOMIC ENERGY COMMISSION IDAHO OPERATIONS OFFICE

UNDER CONTRACT NO. AT(10-1)-1230

DISTRIBUTION OF Tills DOCUMENT IS UNLIM1TJ

ABSTRACT

Particle distributions obtained from individual fuel rods fragmented during rapid power bursts in the Capsule Driver Core (CDC) are shown to follow a lognormal frequency distribution in weight percent versus the particle size. The parameters for each distribution are given and their correlation with energy deposition in the fuel and with rod design are examined. Evidence is given to show that the distributions do not shift indefinitely toward smaller particle sizes with increasing energy density but approach an asymptotic distribution. The fuel rod design influences the particle distributions near the fragmentation threshold.

ii

CONTENTS

ABSTRACT ...................................................................... ii

I. INTRODUCTION ................................................................... 1

II. CHARACTERISTICS OF PARTICLE DISTRIBUTIONS.............. 3

1. MEASUREMENT PROCEDURE ......................... 3

2. PARTICLE SIZE DISTRIBUTION MODEL ..................................... 4

3. MEAN PARTICLE DIAMETER ......................... 5

III. TYPICAL PARTICLE DISTRIBUTION................................................. 8

IV. SUMMARY AND CONCLUSIONS. .................................................................. 12

V. REFERENCES............ .. ..................................... 16

APPENDIX A — TEST DATA AND PARTICLE DISTRIBUTIONS ....... 17

APPENDIX B — PARTICLE DISTRIBUTION MODEL .............. 73

APPENDIX C — CHI-SQUARE FITTING PROGRAM..................... .. 81

APPENDIX D — PARTICLE DISTRIBUTION FROM FIVE-ROD CLUSTER . 107

FIGURES

1. Particle size distribution obtained from rod SPXM-C-513irradiated at 620 cal/g............................. .. ................. .. ............................... .. 9

2. Frequency distribution of particles from rod SPXM-C-513 ...... 10

3. Variation of the volume-to-surface mean diameter with energydensity in the fuel......................................... .. .................... ......................... 14

A-l. Particle size distribution obtained from rod GEX-SE-35irradiated at 342 cal/g ....................................................................................... 22

A-2. Frequency distribution of particles from rod GEX-SE-35 ...... 23

A-3. Particle size distribution obtained from rod GEX-SB-6irradiated at 414 cal/g.............................................. .. ................ .................... 24

A-4. Frequency distribution of particles from rod GEX-SB-6 ...... 25

A-5. Particle size distribution obtained from rodGEX-SL-3 irradiated at 427 cal/g ................................................................................................ 26

iii

A-6. Frequency distribution of particles from rod GEX-SL-3 ..... 0 27

A-l. Particle size distribution obtained from rod GEX-SE-17 irradiated at 342 cal/g ................................... .. .......................................................... 28

A-8. Frequency distribution of particles from rod GEX-SE-17............... 29

A-9. Particle size distribution obtained from rodGEX-SL-4 irradiated at 460 cal/g................................................................................................ 30

A-10. Frequency distribution of particles from rod GEX-SL-4............... 31

A-ll. Particle size distribution obtained from rod SPXM-C-504irradiated at 378 cal/g........................................................................................ 32

A-12. Frequency distribution of particles from rod SPXM-C-504 ............... 33

A-13. Particle size distribution obtained from rod SPXM-C-505irradiated at 490 cal/g............ .. ......................... .............................................. 34


A-15. Particle size distribution obtained from rod SPXM-C-503irradiated at 613 cal/g............................................................................... .. 36


A-17. Particle size distribution obtained from rod SPXM-C-513irradiated at 620 cal/g........................................................................................ 38

A-18. Frequency distribution of particles from rod SPXM-C-513............... 39

A-19. Particle size distribution obtained from rod SPXM-C-554irradiated at 655 cal/g........................................................................................ 40


A-21. Particle size distribution obtained from rod SPXMG-4 irradiated at 390 cal/g................................................................................................. 42

A-22. Frequency distribution of particles from rod SPXMG-4............... 43

A-23. Particle size distribution obtained from rod SPXMG-3 irradiated at 410 cal/g ................................................................... ............................. 44


A-25. Particle size distribution obtained from rod SPXMG-2 irradiated at 450 cal/g............................................... .. .............. .. ............................. 46


A-27. Particle size distribution obtained from rod SPXMG-6 irradiated at 510 cal/g................................................................................................. 48

iv

J

A-28. Frequency distribution of particles from rod SPXMG-6 ..... 49

A-29. Particle size distribution obtained from rod SPXMG-1 irradiated at 550 cal/g................................ ............................... 50

A-30. Frequency distribution of particles from rod SPXMG-1 ..... 51

A-31. Particle size distribution obtained from bare pellets inalumina crucible irradiated at 605 cal/g................. ......................... 52

A-32. Frequency distribution of particles from bare pellets inalum ma crucible 53

A-33. Particle size distribution obtained from rod SPX-C-022 irradiated at 370 cal/g .......a.#..®..,......., o4

A-34. Frequency distribution of particles from rod SPX-C-022 ............ 55

A-35. Particle size distribution obtained from rod SPX-C-023 irradiated at 477 cal/g ............................................ .. ............................ 56

A-36. Frequency distribution of particles from rod SPX-C-023 ..... 57

A-37. Particle size distribution obtained from rod SPX-C-024 irradiated at 572 cal/g ............................. ................................................ 58


A-39. Particle size distribution obtained from rod SPX-C-046 irradiated at 388 cal/g ....................................................................................... 60

A-40. Frequency distribution of particles from rod SPX-C-046 ..... 61

A-41. Particle size distribution obtained from rod SPX-C-047 irradiated at 469 cal/g....................... ...................... ................... ........................ 62


A-43. Particle size distribution obtained from rod SPX-C-048 irradiated at 569 cal/g . ............................. ...................................... .. 64


A-45. Particle size distribution obtained from rod SPX-C-052 irradiated at 388 cal/g............................................................................................. 66


A-47. Particle size distribution obtained from rod SPX-C-053 irradiated at 498 cal/g ............................................................................................. 68


v

A-49. Particle size distribution obtained from rod SPX-C-054 irradiated at 590 cal/g . .............................................................................................. 70

A-50. Frequency distribution of particles from rod SPX-C-054 ............... 71

360 FORTRAN Listing .................................................................................. .. . 85

D-l. Frequency distribution of particles from five-rod cluster ...... Ill

TABLES

I. Summary of Particle Distribution Data...................................... .. .................... 13

A-I. Fuel Rod Characteristics .................... .................................................... 21

D-I. Analysis of Five-Rod Cluster............................................................................ 110

vi



J . A. M'C CL.URE

I. INTRODUCTION

The effects of transient overpower conditions on reactor fuel have been investigated in experiments carried out in the Capsule Driver Core (CDC) as a part of the Subassembly Test Program!1] at SPERT. Results from these tests have provided information on the thresholds, mechanisms, and consequences of fuel failure in a water environment. In general, the observed consequences of failure have tended to increase in severity with increasing energy deposition in the fuel. The failure threshold is taken as the lowest energy deposition that caused the rod cladding to be breached. Failures near the threshold for unirradiated fuel were of the thermal overload (DNB) ta] type with quiescent melting of the fuel rod cladding. For energy depositions greater than about 300 cal/g of UO2, which is into the melting range of the UO2, the test fuel rods disassembled wholly or in part with the UO2 fragmenting into many small pieces. This report is concerned specifically with this latter aspect of fuel rod failure for unirradiated rods, ie, the analysis according to size of the particulate residue from the fragmented rods.

Comprehensive models of overpower conditions that include fuel failures must describe the transition region between the attainment of melting temperatures in the fuel and its dispersal into the surrounding media. The particle distributions discussed in this report represent the asymptotic state of such a dispersal and provide information for testing model predictions. Secondly, the particle distributions are one controlling factor in determining the severity of incidents involving fuel dispersal since the surface area available for interaction and, hence, reaction rates, are inversely proportional to the square of the particle diameters. The specific surface (area per unit volume) of the distributions does not increase indefinitely with energy deposition but asymptotically approaches an upper bound. This implies that rates of reactions such as heat transfer must also approach an upper bound.

Particle distributions were obtained as a function of the energy deposited in the fuel from rods of several designs. Design variations included cladding material, physical form of the fuel, enrichment, and dimensions. A summary description of each rod type is included with the particle distribution data in Appendix A. A complete description of each rod type, tests performed on the rods, and other results have been reported elsewhere [2-4].

Section II presents a discussion of the method used in measuring the particle distributions, a description of the distribution model, and a discussion of the volume -to -surface mean particle diameter. In Section III, a typical particle distribution is displayed, illustrating the method of data presentation and results.

[a] Departure from Nucleate Boiling.

1

The major part of the detailed particle data is contained in Appendix A, In Section IV, the results obtained from fitting the model to the data are summarized and discussed. Appendix B contains a description of the chi-square minimum fitting procedure and a detailed derivation of the volume-to-surface mean diameter. Finally,, Appendix C contains a FORTRAN listing of the computer program used to perform the calculations.

2

II. CHARACTERISTICS OF PARTICLE DISTRIBUTIONS

1. MEASUREMENT PROCEDURE

The particle distributions discussed in this report were generated during destructive transient irradiations (>300 cal/g of UO2) of single, zero burnup reactor fuel rods in the test space of the CDC. Following each such irradiation, the fragmented remains of the rod were collected from the containment capsule and sifted through a set of standard Tyler ta] sieves. Since the rod fragments were already immersed in a water media, a wet sieving technique was used in which the particles were washed, rather than shaken, through the series of sieves stacked according to increasing sieve size. The material remaining on each screen was then dried and weighed, providing a differential weight versus particle size distribution for the fragments.

Normally, eight sieves ranging in mesh opening from 1.7 to 312 mils were used to separate the fragments into the discrete groups. All fragments larger than 312 mils were excluded from the distribution analyses since these pieces were primarily the fuel rod end plugs and possibly other irregularly shaped nonparticle-like fragments. In certain cases, one or more of the groups having the largest diameter particles were also excluded from the analysis because either the group contained a considerable amount of material whose origin was not the fuel rod, or the group contained an insufficient number of particles to provide a good estimate of the group weight. A good estimate of the weight of a group requires a sufficient number of particles in the group such that each particle contributes only a small percentage to the total group weight. Usually there were a sufficient number of particles in each group to provide a good estimate of the group weight. However, in some instances, there were few particles in the largest diameter group; this led to some doubt as to whether the measured group weight was truely representative of the group and whether the data from this group should or should not be included in the analysis. The retention of such a group was determined by performing the analysis with and without the particular group. The particular group was retained if its inclusion did not significantly alter the results of the analysis.

In general, the total weight of the fragments recovered from the containment capsule was less than the original rod weight and the difference varied from test to test. Mechanisms contributing to this variation in recoverable weight were: (a) plate-out and freezing of molten or vaporized material on the containment capsule and/or the fuel rod support structure, (b) changes in the stoichiometry of the UO2, (c) formation of metal oxides during cl adding-water reactions, and (d) fragmented instrument leads and portions of the support structure which were included in the residue. For this last mechanism, corrections to the group weight were applied only if the group contained a clearly identifiable object of known weight. In general, the effects of these four mechanisms on the particle size distribution could not be estimated quantitatively.

[a] A sieve series conforming to the Tyler Standard Screen Scale that is based on the size of openings in cloth woven with 0.0021-inch diameter wire and having 200 such openings per inch. The width of the openings on adjacent screens throughout the series maintain the constant ratio of J2.

3

2. PARTICLE SIZE DISTRIBUTION MODEL

A review of the literature [aJ on the fracture of (usually) crystalline materials [5,6] indicated that the number of particles produced of a given diameter versus the diameter of the particles is generally distributed in a log-normal manner for those materials in which the direction of successive fractures is randomly oriented with respect to the crystal axes. More precisely, the probability of a particle having a diameter within the range y to y + dy is a gaussian or normal function of particle diameter and is given by

dy = 7s/5¥ exp [“^lo§e y " m)2/2s2] dy (!)

where

y = particle diametern(y)dy = number of particles in size range y to y + dy

N = total number of particles2S = estimator of the variance of the distribution of log y©

m ~ estimator of the mean of the distribution of log y.©

The distribution mean, m, is_also the natural logarithm of the most probable particle size y, ie, m = loge y. Equation (1) is known as the frequency function for log-normal distribution.

Equation (1) could not be fitted directly to the particle data from the fragmented fuel rods because (a) it relates the number of particles to particle diameters, whereas the sieve analyses yielded weight versus particle diameter range; and (b) the definition of the distribution by a small number (< 8) of groups made ambiguous the proper selection of a representative diameter for each group to use in the fitting procedure.

In general, weight distributions and number distributions do not obey the same probability distribution law, making the transition from one to the other difficult. However, the log-normal distribution has the property that all moments of the number distribution [of which weight is the third moment, ie, w(y) is proportional to y3 n(y)] are also distributed in a log-normal manner. Hence, we can immediately replace n(y) in Equation (1) with w(y) and N by W, where

w(y)dy = weight of particles in the size range y to y + dyW = total weight in distribution.

The parameters, m and S^, can be reinterpreted as the mean and variance estimators, respectively, of the weight distribution of log y.

The data were fit to a distribution function (integral to the frequency function) rather than to the frequency function itself. In this technique, only the end points of the range of particle diameters within each group entered into the analysis. These end points were determined by the sieve openings and there was

[a] References cited are representative, not exhaustive.

4

no necessity for selecting representative particle diameters for each group. Additional improvements in the fitting technique realized by using the distribution function were less manipulation of the raw data and the use of all the information available. In a frequency analysis, no frequency could be assigned to the group passing the smallest sieve opening since its size range was undefined. This group could not, therefore, be included in such an analysis. No such problem existed in using the distribution function.

The form of the distribution function used in the fit was■X.

where

W. = ai/ 1+1

exp [-3(t-m) ] dt (2)

W. =iX. = 1m =

a,3 = a =

percent of weight in ith groupnatural logarithm of smallest particle diameter in ith group natural logarithm of most probable particle diameter parameters related to the distribution variance, ie,C/S72i 1/2 S2

where2S = variance estimator C = normalization factor.

In order for Equation (2) to represent a probability distribution, its value summed over all values oft < c°) must be unity (100 if data are in percent).This constraint is simply a normalization of the data but one which could be applied only after the parameters in Equation (2) were determined. Hence, the weight data for each size group were first normalized to a percentage of the total weight in the measured distribution and the three parameters, ot, g, and m, treated as independent variables to determine the shape (3) and displacement (m) of the gaussian curve.

„'+coThe value of d

m)2] dt = 100.

/■+»(or equivalently C) was then adjusted so that 9 / exp [-3 (t-

CO

3. MEAN PARTICLE DIAMETER

During the previous discussion the characteristic length associated with the particles has been referred to as “diameter”. This term requires more explication since the word “diameter” is precisely understood only in reference to shapes having circular cross sections, whereas particles produced by fragmentation are generally of an irregular shape. A complicating factor in making a

5

suitable definition of diameter for such particles is that the method of measurement influences to some extent the outcome of the measurement. Hence, both the working definition and measurement prescription must be given.

Sieving is a member of the class of size measurements based on the principle of geometric similarity. In this method, particles whose projected areas or whose three -dimens ional forms are of the same magnitude, are said to be of the same size. The diameter is then defined as an average measure of the distance across particles of the same size assuming they are resting in a position of stability. In sieving, the particles present themselves to the sieve openings in relatively stable positions and the particle diameters are determined from the sieve apertures. The maximum diameter of particles passing through a given sieve is defined to be the same as that of spheres which just pass through the apertures.

Additional quantities related to shape that are relevant to the particle distributions are volume (weight) and surface area. For individual particles, these quantities can be related by shape factors to the cube and square, respectively, of the particle diameter. It has been found empirically that these shape factors remain sensibly constant for all particles in a distribution consisting of the same material, thus permitting the definition of an average particle shape for the distribution. Thus, the interchange ability of numbers and weight as a means of describing the particle distributions, as was done in Section B, is justified not only from the mechanics of the assumed distribution law but also because of the constancy of the particle shape.

The existence of an average particle shape also makes possible a meaningful definition of a characteristic or mean particle diameter for the ensemble. This quantity can be defined in several ways, depending upon which aspect of the distribution is being emphasized. The definition selected for the distributions discussed in this report was the surf ace -are a-weighted size often referred to as the volume-to-surface mean diameter. This quantity is defined for a discrete distribution by the following equation

dsv

Si dl fBWj)Si v4> "

n. r

i(3)

where

dj = diameter of particle in ith groupn. = number of particles in ith group 21

f (d.) = function relating average surface area of a particle in ith group to particle diameter.

The volume -to -surf ace mean diameter has the dual property that shape factors do not appear explicitly in its definition, and secondly, that it is inversely proportional to the surface area per unit volume of the distribution. This latter quantity is known as the specific surface.

6

Combining the log-normal distribution function given by Equation (2) with Equation (3) leads to the following expression for the volume -to-surf ace mean diameter in terms of the distribution parameters:

dsv2* SUM

aVI “p tm ' 4I1

1 + ERF | /g[log (D ) - m] + e max(4)

where

D = largest particle diameter in the measured distribution max ^SUM = value of the distribution function in percent at DM AX

a,3 ,m = parameters as defined in Equation (2)ERF = error function.

A complete derivation of Equation (4) from Equation (3) is given in Appendix B.

The total surface area of the particles in a distribution can be obtained from the mean diameter, dgv, if some assumptions are made about the magnitude of the shape factors. The relation between the area and the mean diameter is given by

AM Ys/^

P dsv(5)

where

M = weight of material in distribution

p - density of particles

A = surface area of ensembleratio of surface shape factor to volume shape factor. For spheres Y = F, Y = it/6, and yq/ytt = 6- Hence, in general, y/y^T > 6.

o V o V o V

7

ANALYSIS OF ROD SPXM-C-513 (contd.)

Fuel Form PelletCladding Zircaloy-2Active Length (in.) 5Test Number 536Period (msec) 3.28Energy (cal/g of UO2) 620Initial Rod Weight (g) 48.1Gaussian Distribution Parametersmean 2.3394variance 0.73184degrees-of-freedom

Volume-to-Surface4

mean diameter (mils) 7.2

MOST PROBABLE PARTICLE SIZE * 10. MILS

FIG. 2 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM — C—513.

SPM'513

z 30

10*

PARTICLE SIZE (MILS)

Normalized Data

Size Range(mils)

Percent ofMeasured

Particle WeightCalculated Chi-Square

< 1.7 1.0865 1.7241 0.2361.7 - 3.5 8.7347 8.4762 0.0083.5 - 7.0 22.6069 22.0767 0.0137.0 - 13.9 33.9208 31.1017 0.256

13.9 - 27.8 20.1079 24.1582 0.67927.8 - 55.5 10.1384 9.9648 0.00355.5 - 111 3.1247 2.2182 0.370

99.7200 99.7200 1.565

10

experimental data did not completely span the size range of the parent distribution.

The last column in the “Normalized Data” table contains the value of chi- square for each group. Its value is given by

2 (W, - W!)2*1 - w. <6>

1

where

W. = calculated probability for ith group

W.' = measured probability for ith group.

The value of chi-square for the distribution is the sum of the individual values and is given by the last number in the column. This value provides a measure of the goodness-of-fit of the data to the distribution function. This point will be discussed further in the next section.

%

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IV. SUMMARY AND CONCLUSIONS

Data from the analysis of each particle distribution are summarized in Table I. Included are the rod number; type of cladding; the energy deposition; the mean, variance, and most probable particle diameter of the fitted distribution function; the volume-to-surface mean diameter; and the goodness- of-fit criterion.

Fuel rod design variations (eg, fuel form, cladding material, length, enrichment) were expected to affect the posttest particle distributions. The number of tests performed for each of six designs was insufficient to draw general conclusions related to the effects of design variations; however, the distribution data qualitatively indie ate that:

(1) Measured distributions were independent of fuel form (powder or pellet).

(2) Ensembles having the smallest variance, or spread, in the particle size range were those from short rods in which the fuel was heated uniformly over its length and for which the cladding material did not form composite particles with the UC>2. These ensembles were from the SPXMG rods with glass cladding and the SPXM rods in which essentially all the Zircaloy-2 cladding reacted with the water. Deviations from these conditions increased the variance of the resulting distributions as indicated for both the SPX and GEX-SL ensembles.

(3) Metal cladding, if unoxidized, also tended to increase the mean size as well as the variance of the particle distributions. Examination of particles from these ensembles showed that cladding and UC^ particles had formed larger composite particles. At higher energies, where essentially all of the cladding reacted with the water, the metal oxides and UO2 did not form any composite particles.

(4) Decreasing the enrichment, thereby flattening the radial fission density profile, tended to shift the particle distributions to smaller particle sizes for a given average energy deposition, as indicated by the GEX data.

The goodness-of-fit criterion in Table I was derived from the value of chi-square for the distributions and indicated how well the data are described by the theoretical distribution mode. Specifically, the criterion is the probability in percent that another sample drawn at random from the parent distribution would have a larger value of chi-square than the given distribution. For a given number of degrees-of-freedom, sufficiently large values of chi-square lead to the rejection of the hypothesis that the data follow the assumed distribution law. A reasonable rejection level for the type of data contained in this report is say, 10%, which implies that if the goodness-of-fit criteria are less than 10%, the distribution law does not adequately describe the data. As shown in Table I, only one of the distributions fell into this c ategory,

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TABLE I

SUMMARY OF PARTICLE DISTRIBUTION DATA

Rod r 1 Number a

CladdingType

Energy (cal/g UCO

Log-Normal Mean (m) Variance

GEX-SE-35 Zr-2 342 3.89 1.50GEX-SB-6 Zr-2 414 3.83 1.63GEX-SL-3 Zr-2 427 3.29 2.44GEX-SE-17 | GEX-SL-4

Zr-2 342 6.38 3.15Zr-2 460 3.76 2.07

SPXM-C-504 Zr-2 378 6.35 3.49SPXM-C-505 Zr-2 490 3.44 1.42SPXM-C-503 Zr-2 613 2.23 0.90SPXM-C-513 Zr-2 620 2.34 0.73SPXM-C-554 Zr-2 655 1.91 0.69

SPXMG-4 Glass 390 2.83 0.80SPXMG-3 Glass 410 3.87 1.09SPXMG-2 Glass 450 3.23 0.90SPXMG-6 Glass 510 2.72 0.91SPXMG-1 Glass 550 2.13 0.89912 Bare 605 1.87 1.12

SPX-C-022 ss^ 370 6.52 3.21SPX-C-023 SS 477 3.78 1.20SPX-C-024 SS 572 3.29 1.74

SPX-C-046 Zr-2[a] [b] [c] [d] [e] 388 7.16 3.95SPX-C-047 Zr-2 469 4.61 1.31SPX-C-048 Zr-2 569 2.78 2.23

SPX-C-052 Zr-2^ 388 7.13 3.67SPX-C-053 Zr-2 498 4.91 1.63SPX-C-054 Zr-2 590 3.59 1.60

Most Probable >ize [mils (y)]

Volume-to- Surface Mean

Diameter [mils (d )1Goodness-of-Fit Probability (%)

49 21.6 76.746 19.1 84.127 7.5 67.1

592 47.9 100.043 14.0 99.9

573 40.0 99.831 14.6 73.39 5.9 98.3

10 7.2 81.57 4.8 4.1

17 10.5 95.448 17.7 98.525 13.3 90.115 8.9 56.78 5.3 84.16 3.6 94.2

680 49.8 61.644 23.3 96.027 10.6 59.1

1190 39.4 99.9100 44.4 89.716 5.1 79.7

1240 53.2 97.2135 45.7 99.036 15.1 95.7

[a] Rod design details are presented in Appendix A.[b] Powder fuel.[c] Stainless steel (304) cladding.[d] Annealed cladding.[e] Cold worked cladding.

70

60

50

40

30

20

0 —300

AA

□ A

AA

Legend

O GEX Rods-Pellet Fuel @ GEX Rods — Powder Fuel □ SPXM Rods ■ SPXMG Rods A SPX Rods A SPXM Rod - Bore

OO

A

□ A

A

A □ □□

400 500 600 700Energy Deposition in Test Rod (cal/g) INC-A-15473

FIG. 3 VARIATION OF THE VOUUM E-TO—SURFACE MEAN DIAMETER WITH ENERGY DENSITY IN THE FUEU,

ie, that from Rod SPXM-C-554^. All of the other distributions had goodness- of-fit probabilities exceeding 55%. We pan therefore conclude that:

The particle ensembles obtained from the thermal fracture of reactor fuel rods containing zero-burnup UO2 follow a lognormal distribution law in common with the residue from fracture of other brittle materials.

A second characteristic common to all the distributions was their trend toward smaller particle sizes with increasing energy deposition in the fuel. This trend is clearly shown by the volume-to-surface mean diameters in Table I, as well as the most probable sizes, particularly when the data from similar rod types are examined. The trend is illustrated graphic ally in Figure 3, which shows the volume-to-surface mean diameter as a function of energy deposition. The data in Figure 3 also indie ate that the trend toward smaller particle sizes does not continue indefinitely but asymptotically approaches a mean diameter of about 5 mils. This asymptotic behavior is consistent with brittle fracture theory, which implies that the probability of fracturing a [a]

[a] This distribution contained a significant amount of weight in one group in excess of that predicted by the distribution function. It is possible that the sieve analysis was in error by not getting an adequate separation of the fine particles. Based on this assumption, the three groups having the smallest particle sizes were pooled and the data refitted to the log-normal distribution. The goodness-of-fit probability for the pooled data was 96%.

14

particle decreases rapidly with particle size. Finally, therefore, it may be concluded that:

The mean particle diameter of the residue resulting from the thermal fracture of zero-burnup UO2 fuel rods does not decrease indefinitely as a function of energy deposition, but asymptotically approaches a lower limit of about 5 mils at energy depositions exceeding about 500 cal/g of UOg.

15

V. REFERENCES

1. J. E. Grund et al, Subassembly Test Program Outline forFY-1969andFY- 1970, IN-1313 (IDO-17277) (August 1969).

2. J. A. McClure and L. J. Siefken, Transient Irradiation of 1/4-Inch OD Stainless Steel Clad Oxide Fuel Rods to 570 cal/g UO2, IDO-ITR-100 (October 1968).

3. Z. R. Martinson and R. L. Johnson, Transient Irradiation of 1/4-Inch ODZircaloy-2 Clad Oxide Fuel Rods to 590 cal/g UO2, IDO-ITR-102 (November 1968). ~ ^ 1 -----

4. T. G. Taxelius et al. Annual Report SPERT Project, October 1968 - September1969, IN-1370 (June 1970). ~ " “

5. G. Herdan, Small Particle Statistics, Elsevier Publishing Company, Houston Texas, 1953.

6. C. Orr, Jr. and J. M. Dalleralle, Fine Particle Measurement, The MacMillian Company, New York, N. Y., 1959.

16

APPENDIX ATEST DATA AND PARTICLE DISTRIBUTIONS

17

APPENDIX ATEST DATA' AND PARTICLE DISTRIBUTIONS

Particle distributions were obtained for four basic fuel rod designs. The characteristics of each design are presented in Table A-I. The experimental sieve screen data, the frequency histograms, and the fitted gaussian functions for each rod analyzed are presented in this appendix.

1. GEX RODS

Fragments from five GEX rods were analyzed. Three had an active fuel length of about 5.2 in. and the other two had an active fuel length of about 24.2 in. One each of the short and long rods contained powder fuel; the others contained pellet fuel. Distribution data, frequency histograms, and fitted gaussian functions are given in Figures A-l through A-10.

The powder rods were fabricated with UO2 powder of the following size distribution:

Size Range Percent of(mils) Fuel Weight<1.7 20

5.9 - 8.3 1533 - 47 65

The posttest particle distributions obtained from the above rods are shown in Figures A-7 and -9, respectively. Theseparticleshad a considerably different size distribution than did the original powder and were essentially indistinguishable from distributions obtained from rods containing UO2 pellets. The foreign material evident on screens 42, 24, and 12 in Figure A-9 is mainly A-LP [a] ceramic cement which coated a stainless steel screen surrounding the fuel rod. Because of the relatively large difference in density between the cement and UO2, the effect of the cement on the distribution data is small.

2. SPXM RODS

Distributions were analyzed for five SPXM rods. The distribution data, frequency histograms, and fitted gaussian functions are shown in Figures A-ll through -20. The larger fragments were removed from the first three distributions prior to screening.

[a] A-LP (Type A, low porosity Astroceran) is the trade name of a Zr02- ZrSiOq base, high temperature cement manufactured by the American Thermocatalytic Corporation.

19

3. SPXMG RODS

Particle distributions for five SPXMG rods and an unclad stack of pellets in an alumina crucible were analyzed. The results are given in Figures A-21 through -32. Only particle groups with sizes smaller than 55.5 mils were analyzed. The larger groups were comprised mostly of glass and alumina. 4

4. SPX RODS

Nine particle distributions were analyzed from SPX rods. Results are presented in Figures A-34 through -50. Particles larger than 223 mils were excluded from the distributions shown in Figures A-39 and -43 because of extraneous debris. The spring weight was subtracted from the particle weight on the No. 3-1/2 screen in Figure A-41. The low weight fraction collected on the No. 80 screen in Figure A-45 caused convergence difficulties in the fitting technique. This was overcome by pooling the particles from the No. 170, 80, and 42 screens.

TABLE A-I

FUEL ROD CHARACTERISTICS

GEX SPXM SPXMG SPX

Cladding Material Zircaloy-2 Zircaloy-2 Flint Glass Zircaloy-2 or Stainless Steel

Cladding Heat Treatment ~ 10% Cold-Worked - 10% Cold-Worked — Annealed or = 10% Cold- Worked

Cladding OD (in.) 5/16 1/4 0.315 1/4Cladding Thickness (in.) 0.020 0.014 0.0433 0.014Fuel Material U02 uo2 U°2 uo2

Fuel Form Pellet or Powder Pellet Pellet PelletFuel Density (g/cc) 10.3 or 9.21 10.4 10.4 10.4Fuel Enrichment 7 10.5 10.5 10.5Fuel Stack Length (in.) 5.2 or 24.2 5.0 4.5 5.0 or 18.0Pellet Diameter (in.) 0.268 0.220 0.220 0.220Diametral Gas Gap (in.) 0.003 0.002 0.008 0.002

ANALYSIS OF ROD GEX-SE-35

FIG. A-l PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD GEX-SE-35 IRRADIATED AT342 CAL/G.

Screen Size (meshes/inch)

Particle Size Range (mils)

Weight(grams)

Percent of Rod Wt

CumulativePercent

PAN < 1.7 0.0432 0.051 0.051325 1.7 - 3.5 0.1120 0.132 0.183170 3.5 - 7.0 0.7673 0.907 1.090

80 7.0 - 13.9 2.3073 2.728 3.81842 13.9 - 27.8 3.2277 3.816 7.63424 27.8 - 55.5 3.2066 3.792 11.4312 55.5 - 223 7.1510 8.455 19.88

3h 223 - 312 0.6664 0.788 20.672k > 312 None —

22

ANALYSIS OF ROD GEX-SE-35 (contd.)

*

Fuel Form CladdingActive Length (in.) Test Number Period (msec)Energy (cal/g UOg) Initial Rod Weight (g) Gaussian Distributior mean variancedegrees-of-freedom

Volume-to-Surfacemean diameter (mils)

Pellet MOST PROBABLE PARTICLE SIZE = M9. MILS

FIG. A—2 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD GEX—SE—35.

Normalized Data

Size Range (mils)

Percent of Measured

Particle WeightCalculated Chi-Squari

< 1.7 0.2310 0.3096 0.0201.7 - 3.5 0.5989 1.2745 0.3583.5 - 7.0 4.1033 4.0853 0.0007.0 - 13.9 12.3388 9.6275 0.764

13.9 - 27.8 17.2609 17.0298 0.00327.8 - 55.5 17.1480 21.8603 1.01655.5 - 223 38.2416 35.0494 0.291223 - 312 3.5637 4.2500 0.111

93.4864 93.4864 2.563

%[a] Chi-Square - the square of the difference between the measured

and calculated weights divided by the calculated weight. The sum is a measure of the goodness-of-fit, small numbers indicating a good fit.

23

ANALYSIS OF ROD GEX-SB-6

FIG. A—3 PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD GEX-SB-6 IRRADIATED AT



Weight(grams)

Percent of Rod Weight

CumulativePercent

PAN < 1.7 0.3535 0.422 0.422325 1.7 - 3.5 0.7736 0.924 1.346170 3.5 - 7.0 2.7643 3.301 7.41580 7.0 - 13.9 6.2104 7.514 12.0642 13.9 - 27.8 10.112 12.07 24.1324 27.8 - 55.5 14.479 17.29 41.4212 55.5 - 223 17.126 20.45 61.873% 223 - 312 3.567 4.26 66.132% >312 37.117 44.32 110.45

24

ANALYSIS OF ROD GEX-SB-6 (contd.)

Fuel Form PelletCladding Zircaloy-2Active Length (in.) 5Test Number 479Period (msec) 2.95Energy (cal/g of UO ) 414Initial Rod Weight (g) 83.75Gaussian Distribution Parametersmean 3.8349variance 1.6322degrees-of-freedom 5

Volume-to-Surfacemean diameter (mils) 19.1

MOST PROBABLE PARTICLE SIZE = 46. MILS

FIG. A—4 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD GEX—SB—6.

GEX-SB-6

PARTICLE SIZE (MILS1

Normalized Data

Size Range (mils)

Percent of Particle WeightMeasured Calculated Chi-Square

< 1. 7 0.59511.7 - 3.5 1.30223.5 - 7.0 4 .'65327.0 - 13.9 10.4542

13.9 - 27.8 17.025527.8 - 55.5 24.372155.5 - 223 28.8279223 - 312 6.0078

93.2350

0.4849 0.0251.6786 0.0844.7997 0.00410.3566 0.00117.1727 0.00121.1565 0.48933.4288 0.6334.1574 0.823

93.2350 2.060

25

ANALYSIS OF ROD GEX-SL-3

h

2k

12 7 3ig

FIG. A—5 PARTICLE SIZE DISTRIBUTION 427 CAL /G.

Screen Size Particle Size(meshes/inch) Range (mils)

OBTAINED FROM ROD GEX-SL.-3 I

Weight Percent of (grams) Rod Weight

RRADIATED AT

CumulativePercent

PAN < 1.7 3.3424 1.173 1.17325 1.7 - 3.5 9.7296 3.414 4.59170 3.5 - 7.0 21.7873 7.645 12.23

80 7.0 - 13.9 23.3074 8.178 20.4142 13.9 - 27.8 26.1075 9.161 29.5724 27.8 - 55.5 24.0092 8.424 38.0012 55.5 - 111 18.2270 6.395 44.39

7 111 - 223 16.0265 5.623 50.013k 223 - 312 7.4047 2.598 52.612k > 312 40.9618 14.373 66.98

26

ANALYSIS OF ROD GEX-SL-3 (contd.)

Fuel Form PelletCladding Zircaloy-2Active Length (in.) 24Test Number 491Period (msec) 2.96Energy (cal/g of U0„) 427Initial Rod Weight (g) 285Gaussian Distribution Parametersmean 3.2945variance 2.4444degrees-of-freedom 6


MOST PROBRBLE PARTICLE SIZE = 27. MILS

FIG. A-6 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD GEX—SL—3.

GEX-SL-3


Normalized Data

Size Range (mils)


< 1.7 2.0984 3.8546 0.8001.7 - 3.5 6.1083 5.7246 0.0263.5 - 7.0 13.6781 9.8399 1.4977.0 - 13.9 14.6325 14.1668 0.015

13.9 - 27.8 16.3904 17.1942 0.03827.8 - 55.5 15.0730 17.0069 0.22055.5 - 111 11.4430 13.9422 0.448111 - 223 10.0615 9.4409 0.041223 - 312 4.6487 2.9636 0.958

94.1338 94.1338 4.043

27

ANALYSIS OF ROD GEX-SE-17

FIG. A-7 PARTICL.E 342 CAL/G.

SIZE DISTRIBUTION OBTAINED FROM ROD GEX-SE-17 IRRADIATED AT

Screen Size Particle Size Weight Percent of Cumulative(meshes/inch) Range (mils) (grams) Rod Weight Percent

PAN < 1.7 0.0467 0.057 0.057325 1.7 - 3.5 0.1330 0.163 0.220170 3.5 - 7.0 0.4483 0.549 0.76980 7.0 - 13.9 1.3888 1.700 2.46942 13.9 - 27.8 2.5957 3.180 5.6524 27.8 - 55.5 5.2359 6.41 12.0612 55.5 - 223 21.7182 26.59 38.65

3k 223 - 312 7.3572 9.01 47.662h > 312 None

28

Lli.

ANALYSIS OF ROD GEX-SE-17 (contd.)

Fuel Form CladdingActive Length (In.)Test Number Period (msec)Energy (cal/g of U0„)Initial Rod Weight (g)Gaussian Distribution Parameters mean variancedegrees-of-freedom


PowderZircaloy-2

5480

3.88342

81.6776.38423.1545

547.9

HOST PROBABLE PARTICLE SIZE = 592. MILS

IG. A—8 FREQUENCY DISTRIBUTION OF PARTICLES ROM ROD GEX—SE~17.

GEX-SE-17


Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square< 1.7 0.0431 0.0487 0.001

1.7 - 3.5 0.1227 0.1441 0.0033.5 - 7.0 0.4135 0.4298 0.0017.0 - 13.9 1.2811 1.1086 0.027

13.9 - 27.8 2.3943 2.5184 0.00627.8 - 55.5 4.8297 4.8743 0.00055.5 - 223 20.0334 19.9887 0.000223 - 312 6.7865 6.7919 0.000

35.9043 35.9043 0.038

29

ANALYSIS OF ROD GEX-SL-4

FIG. A_9 PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD GEX-SL-4 IRRADIATED AT 460 CAL /G.



Weight(grams)


CumulativePercent

PAN < 1. 7 1.6659 0.584 0.58325 1.7 - 3.5 6.3705 2.235 2.82170 3.5 - 7.0 13.7874 4.838 7.6680 7.0 - 13.9 21.9732 7.710 15.3742 13.9 - 27.8 31.6104 11.091 26.4624 27.8 - 55.5 35.5350 12.469 38.9312 55.5 - 111 36.5121 12.811 51.747 111 - 223 24.9630 8.759 60.503% 223 - 312 8.3408 2.927 63.42

> 312 29.4771 10.343 73.77

V

30

ANALYSIS OF ROD GEX-SL-4 (contd.)

Fuel Form CladdingActive Length (in.)Test Number Period (msec)Energy (cal/g of IK^)Initial Rod Weight (g)Gaussian Distribution Parameters mean variancedegrees-of-freedom


PowderZircaloy-2

24492

3.05460285

75970732

614.0

MOST PROBRBLE PARTICLE SIZE = H3. MILS

FIG. A—10 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD GEX-SL-4.

Normalized Data

Size Range (mils)


< 1. 7 0.84411.7 - 3.5 3.22763.5 - 7.0 6.98537.0 - 13.9 11.1326

13.9 - 27.8 16.015227.8 - 55.5 18.003655.5 - 111 18.4986111 - 223 12.6474223 - 312 4.2258

91.5801

1.2457 0.1302.8371 0,0546.3053 0.073

11.2839 0.00216.4637 0.01218.9371 0.04617.4524 0.06312.8464 0.0034.2084 0.00091.5801 0.383

31

ANALYSIS OF ROD SPXM-C-504

TvsPr-ar'/V- PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD SPXM —C—504 IRRADIATED AT

Screen Size Particle Size Weight Percent of Cumulative(meshes/inch) Range (mils) (grams) Rod Weight PercentPAN < 1.7 0.0553 0.11 0.11325. 1.7 - 3.5 0.1561 0.32 0.43170 3.5 - 7.0 0.5141 1.07 1.5080 7.0 - 13.9 1.5708 3.27 4.7742 13.9 -27.8 3.0094 6.26 11.0324 27.8 - 55.5 3.9644 8.24 19.2712 55.5 - Ill 7.4847 15.56 34.837 111 - 223 11.5547 24.02 58.8535s 223 - 312 5.5490 11.54 70.392% > 312 None

32


Fuel Form PelletCladding Zircaloy-2Active Length (in.) 5Test Number 506Period (msec) 4.93Energy (cal/g of UO2) 378Initial Rod Weight (g) 48.1Gaussian Distribution Parametersmean 6.3517variance 3.4859degrees-of-freedom 6


FIG. A—12 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM —C—504.


SPM-50H

PRRTICLE SIZE (MILS]

Normalized Data

Size Range Percent of Particle Weight(mils) Measured< 1.7 0.0608

1.7 - 3.5 0.17163.5 - 7.0 0.56517.0 - 13.9 1.7267

13.9 - 27.8 3.308227.8 - 55.5 4.358055.5 - 111 8.2278111 - 223 12.7018223 - 312 6.0999

37.2199

Calculated Chi-Square

0.0907 0.0100.2246 0.0130.5986 0.0021.4023 0.0752.9332 0.0485.3004 0.1688.4048 0.004

11.6914 0.0876.5738 0.034

37.2199 0.441

33


12 7 3ia

FIG. A—13 PARTICLE 490 CAL / G.


SIZE DISTRIBUTION OBTAINED FROM ROD SPXM—C—505 IRRADIATED AT

Particle Size Weight Percent of CumulativeRange (mils) (grams) Rod Weight Percent

PAN < 1.7 0.1188 0.25 0.25325 1.7 - 3.5 0.5120 1.06 1.31170 3.5 ~ 7.0 2.5868 5.38 6.6980 7.0 - 13.9 6.0920 12.67 19.3642 13.9 - 27.8 6.9569 14.46 33.8224 27.8 - 55.5 6.2177 12.93 47.7512 55.5 - Ill 5.3715 11.17 58.927 111 - 223 3.7406 7.78 66.703*5 223 - 312 2.3023 4.79 71.492h > 312 None

34


Fuel Form Pellet Cladding Zircaloy-2 Active Length (in.) 5 Test Number 507 Period (msec) 4.10 Energy (cal/g of UO2) 490 Initial Rod Weight (g) 48.1Gaussian Distribution Parametersmean 3.4354variance 1.4154degrees-of-freedom 5



FIG. A—14 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM—C—505.

SPM-505

2° -


Normalized Data

Size Range (mils)


< 1. 7 0.3577 0.7309 0.1911.7 - 3.5 1.5415 2.5970 0.4293.5 - 7.0 7.7881 7.2005 0.0487.0 - 13.9 18.3412 14.4425 1.052

13.9 - 27.8 20.9452 21.3328 0.00727.8 - 55.5 18.7197 22.4301 0.61455.5 - 111 16.1720 17.0563 0.046111 - 223 11.2619 9.3372 0.379

95.1272 95.1272 2.784

35


FIG. A—15 FARTICL.E SIZE DISTRIBUTION OBTAINED FROM ROD SPXM-C-503 613 CAL. /G. RRADIATED AT

Screen Size Particle Size(meshes/inch) Range (mils)

PAN < 1.7325 1.7 - 3.5170 3.5 - 7.080 7.0 - 13.942 13.9 - 27.824 27.8 - 55.512 55.5 - Ill7 111 - 2233h 223 - 312

Weight-(grams)


CumulativePercent

1.5714 3.27 3.273.9473 8.21 11.488.7856 18.27 29.75

11.3547 23.61 53.368.1250 16.89 70.253.3426 6.95 77.200.8509 1.77 78.970.3029None

0.63 79.60

36


Fuel Form PelletCladding Zircaloy-2Active Length (in.) 5Test Number 505Period (msec) 3.27Energy (cal/g of UO2)Initial Rod Weight (g)Gaussian Distribution Parameters

61348.1

mean 2.2322variance 0.90175degrees-of-freedom

Volume-to-Surface5

mean diameter (mils) 5.9HOST PROBABLE PART:

FIG. A—16 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM—C—503.

§£

Normalized Data

SPM-503

PARTICLE SIZE ' (NILS)

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squan< 1. 7 4.1030 3.6575 0.054

1.7 - 3.5 10.3065 11.4596 0.1163.5 - 7.0 22.9395 23.0350 0.0007.0 - 13.9 29.6545 28.1563 0.080

13.9 - 27.8 21.2146 21.2014 0.00027.8 - 55.5 8.7276 9.4770 0.05955.5 - 111 2.2217 2.5588 o.ooi[a111 - 223 0.7909 0.4129

99.9584 99.9584 0.310

[a] x value calculated from pooled data.

37


FIG. A—1 7 PARTICLE 620 CAL. /G.

SIZE DISTRIBUTION OBTAINED FROM ROD SPXM—C—513 IRRADIATED AT


PAN < 1.7 0.4053 8.43 8.43325 1.7 - 3.5 3.2583 6.77 15.20170 3.5 - 7.0 8.4330 17.53 32.7380 7.0 - 13.9 12.6534 26.31 59.0442 13.9 - 27.8 7.5008 15.59 74.6324 27.8 - 55.5 3.7819 7.86 82.3912 55.5 - 111 1.1656 2.42 84.917 111 - 223 0.6356 1.32 86.233% 223 - 312 1.4996 3.12 89.352% > 312 8.2784 17.21 106.56

38



Volume-to-Surfacemean diameter (mils) .7.2

HOST PROBABLE PARTICLE SIZE = 10. MILS

FIG. A—18 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM—C—513 .

SPM-513


Normalized Data

Size Range (mils)


< 1.7 1.0865 1.7241 0.2361.7 - 3.5 8.7347 8.4762 0.0083.5 - 7.0 22.6069 22.0767 0.0137.0 - 13.9 33.9208 31.1017 0.256

13.9 - 27.8 20.1079 24.1582 0.67927.8 - 55.5 10.1384 9.9648 0.00355.5 - 111 3.1247 2.2182 0.370

99.7200 99.7200 1.565

39


FIG, A—19 PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD SPXM—C—554 IRRADIATED AT 655 CAL./G,


Particle SizeRange (mils)

Weight(grams)


CumulativePercent

PAN ' < 1.7 1.4500 3.01 3.01325 1.7 - 3.5 5.1088 10.62 13.63170 3.5 - 7.0 15.8198 32.89 46.5280 7.0 - 13.9 8.2180 17.09 63.6142 13.9 - 27.8 4.4278 9.21 72.8224 27.8 - 55.5 1.8615 3.87 76.6912 55.5 - 111 0.5647 1.17 77.867 111 - 223 0.4033 0.84 78.7031$ 223 - 312 1.4172 2.95 81.6521s > 312 8.5025 17.68 99.33

40

ANALYSIS OF ROD SPXM-C-554 (contd.)Fuel Form CladdingActive Length (ix>.) Test Number Period (msec)Energy (cal/g of U0„) Initial Rod Weight fg) Gaussian Distribution mean variancedegrees-of-freedom


PelletZircaloy-2

5549

3.01 655

48.1Parameters

1.90540.68753

44.8

MOST PROBfiBLE PARTICLE SIZE = 7. MILS

FIG. A—20 FREQUENCY DISTRIBUTION OF PARTICL.ES FROM ROD SPXM—C —554.

PARTICLE SIZE (MILS!

Normalized Data

Size Range Percent of Particle Weight(mil s ) Measured Calculated Chi-Square

< 1.7 3.8704 4.8652 0.2031.7 - 3.5 13.6365 16.6945 0.5603.5 - 7.0 42.2265 30.3863 4.6147.0 - 13.9 21.9356 29.0051 1.723

13.9 - 27.8 11.8188 14.7043 0.56627.8 - 55.5 4.9687 3.7993 0.36055.5 - 111 1.0573 0.5091 1.957

99.9638 99.9638 9.983

41

ANALYSIS OF ROD SPXMG-4

FIG. A—21 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPXMG—4 IRRADIATED AT 390 CAL / G .



Weight(grams)


CumulativePercent

PAN < 1.7 0.1148 0.31 0.31325 1.7 - 3.5 0.7656 2.06 2.37170 3.5 - 7.0 3.6225 9.76 12.1380 7.0 - 13.9 6.6239 17.84 29.9742 13.9 - 27.8 7.5591 20.36 50.3324 27.8 - 55.5 5.4727 14.74 65.0712 55.5 - 111 5.8999 15.89 80.967 111 - 223 6.7418 18.16 99.123% 223 - 312 0.3245 0.87 100.0

42

ANALYSIS OF ROD SPXMG-4 (contd.)

Fuel Form PelletCladding GlassActive Length (in.) 4.5Test Number 531Period (msec) 4.56Energy (cal/g of UO2) 390Gaussian Distribution Parametersmean 2.8332variance 0.80391degrees-of-freedom 3



FIG. A—22 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXMG—4.

PARTICLE SIZE (MILS

Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squar-

< 1.7 0.4308 0.5110 0.0131.7 - 3.5 2.8728 3.3861 0.0783.5 - 7.0 13.5928 12.2199 0.1547.0 - 13.9 24.8550 24.9985 0.001

13.9 - 27.8 28.3642 29.7171 0.06227.8 - 55.5 20.5354 19.8184 0.026

90.6510 90.6510 0.334

43


FIG, A-23 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPXMG-3 IRRADIATED AT 410 CAL. /G.



Weight(grams)


CumulativePercent

PAN <1.7 0.0372 0.10 0.10325 1.7 - 3.5 0.1827 0.49 0.59170 3.5 - 7.0 0.6990 1.89 2.4880 7.0 - 13.9 2.9090 7.86 10.3442 13.9 - 27.8 5.8959 15.93 26.2724 27.8 - 55.5 8.3089 22.44 48.7112 55.5 - 111 11.1788 30.19 78.907 111 - 223 7.5327 20.35 99.253^ 223 - 312 0.2786 0.75 100.0

44


Fuel Form PelletCladding GlassActive Length (in.) 4.5Test Number 529Period (msec) 4.29Energy (cal/g of UO2)Gaussian Distribution Parameters

410


Volume-to-Surface3




SPXMG-3


Normalized Data

Size Range (mils)


< 1.7 0.11461.7 - 3.5 0.56303.5 - 7.0 2.15417.0 - 13.9 8.9647

13.9 - 27.8 18.169527.8 - 55.5 25.6056

55.5716

0.0642 0.0400.5177 0.0042.5984 0.0768.4449 0.032

18.3178 0.00125.6285 0.00055.5716 0.153

45


FIG. A—25 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPXMG—2 IRRADIATED AT 450 CAU /G.



Weight(grams)


CumulativePercent

PAN < 1.7 0.1403 0.37 0.37325 1.7 - 3.5 0.4430 1.16 1.53170 3.5 - 7.0 1.8356 4.81 6.34 .80 7.0 - 13.9 5.9309 15.53 21.8742 13.9 - 27.8 8.2553 21.61 43.4824 27.8 - 55.5 7.9021 20.63 64.1112 55.5 - 111 8.6384 22.61 86.72

7 111 - 223 5.0533 13.23 99.953*5 223 - 312 none

46



Volume-to-surfacemean diameter (mils) 13.3



SPXMG-2

PARTICLE SIZE (MILS1

Normalized Data

Size Range Percent of Particle Weight(mils) Measured

< 1.7 0.45621.7 - 3.5 1.44033.5 - 7.0 5.96827.0 - 13.9 19.2834

13.9 - 27.8 26.840827.8 - 55.5 25.6925

777EEI3

Calculated Chi-Square0.2220 0.2471.6385 0.0246.9483 0.138

17.6461 0.15227.5836 0.02025.6427 0.00079.6813 0.581

47


FIG. A—27 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPXMG—6 IRRADIATED AT 510 CAU / G.


PAN < 1.7 0.2369 0.63 0.63325 1.7 - 3.5 1.0678 2.84 3.47170 3.5 - 7.0 4.9574 13.21 16.6880 7.0 - 13.9 7.3152 19.49 36.1742 13.9 - 27.8 6.4403 17.16 53.3324 27.8 - 55.5 5.5577 14.61 67.9412 55.5 - 111 7.2486 19.31 87.257 111 - 223 3.2790 8.74 95.993% 223 - 312 1.4305 3.81 99.80

48



Volume-to-Surfacemean diameter 8.9



SPXMG-6

10 10 PARTICLE SIZE (MILS!

Normalized Data

Size Range (mils)


< 1.7 0.84531.7 - 3.5 3.81003.5 - 7.0 17.68867.0 - 13.9 26.1015

13.9 - 27.8 22.979827.8 - 55.5 19.8305

91.2557

1.0655 0.0465.0596 0.309

14.5789 0.66325.4463 0.01727.4463 0.72717.6591 0.26791.2557 2.029

49


12 7

FIG. A-29 PARTICUE 550 CAU /G.

SIZE DISTRIBUTION OBTAINED iFROM ROD SPXMG—1 IRRADIATED AT


PAN < 1.7 1.0982 3.07 3.07325 1.7 - 3.5 3.6278 10.13 13.20170 3.5 - 7.0 7.0631 19.72 32.9280 7.0 - 13.9 7.8265 21.85 54.7742 13.9 - 27.8 4.5103 12.59 67.3624 27.8 - 55.5 2.5715 7.18 74.5412 55.5 - Ill 4.6484 12.98 87.527 111 - 223 4.4759 12.50 100.03*5 223 - 312 none

50





FIG. A—30 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXMG-1.

SPXMG-1


Normalized Data

Size Range (mils)


< 1.7 4.02131.7 - 3.5 13.28403.5 - 7.0 25.86307.0 - 13.9 28.6584

13.9 - 27.8 16.515427.8 - 55.5 9.4161

97.7582

4.4716 0.04513.1233 0.00224.7043 0.05428.0614 0.01319.4611 0.4467.9364 0.276

97.7582 0.836

51

ANALYSIS OF BARE PELLETS IN ALUMINA CRUCIBLE

9 ip mPAN 325 170

12

FIG. A—31 PARTICLE SIZE DISTRIBUTION OBTAINED FROM BARE PELLETS IN ALUMINA CRUCIBLE IRRADIATED AT 605 CAL /G.


PAN < 1.7 2.2536 £aJ 7.83 7.83325 1.7 - 3.5 3.4590 12.02 19.85170 3.5 - 7.0 5.2811 18.35 38.2080 7.0 - 13.9 4.7617 16.55 54.7542 13.9 - 27.8 3.4730 12.07 66.8224 27.8 - 55.5 1.4495 5.04 71.8612 55.5 - 223 3.0966 10.76 82.623^ 223 - 312 None

[a] Based on fuel weight of 28.775 g.

52

ANALYSIS OF BARE PELLETS IN ALUMINA CRUCIBLE (contd.)

Fuel Form PelletCladding NoneActive Length (in.) 4.5Test Number 481Period (msec) 2.97Energy (cal/g of UO2) 605Gaussian Distribution Parametersmean 1.8687variance 1.1231degrees-of-freedom 3


MOST PROBABLE PARTICLE SIZE ■ 6. MILS

FIG. A—32 FREQUENCY DISTRIBUTION OF PARTICLES FROM BARE PELLETS IN ALUMINA CRUCIBLE.

ALUMINA CRUCIBLE


Normalized Data

Size Range (mils)

Percent of Measured


0.023< 1.7 10.8239 10.33631.7 - 3.5 16.6133 17.7186 0.0693.5 - 7.0 25.3647 24.8484 0.0117.0 - 13.9 22.8701 23.5242 0.018

13.9 - 27.8 16.6805 15.1030 0.16527.8 - 55.5 5.5118 6.3338 0.107

97.8642 97.9642 0.393

53

ANALYSIS OF ROD SPX-C-022

FIG. A—33 PARTICUE 3 70 CAL / G.

SIZE DISTRIBUTION OBTAINED FROM ROD SPX—C—022 IRRADIATED AT


PAN < 1.7 0.020 0.013 0.013325 1.7 - 3.5 0.142 0.092 0.105170 3.5 - 7.0 0.353 0.229 0.33480 7.0 - 13.9 1.770 1.150 1.48442 13.9 - 27.8 6.281 4.08 5.5624 27.8 - 55.5 15.776 10.24 15.8012 55.5 - 223 32.730 21.26 37.063:-2 223 - 312 20.590 13.38 50.442Jg > 312 72.65 47.20 97.6

54

ANALYSIS OF ROD SPX-C-022 (contd.)

Fuel Form Cladding Active Length (in.)Test Number Period (msec)Energy (cal/g of UO2)Initial Rod Weight (g)Gaussian Distribution Parameters mean variancedegrees-of-freedom


Pellet Stainless Steel

18 326 4.6 370

153.936.52153.2131

549.8


FIG. A—34 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C-022.

SPX-C-022


Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squi

< 1.7 0.0086 0.0412 0.0261.7 - 3.5 0.0607 0.1229 0.0313.5 - 7.0 0.1508 0.3701 0.1307.0 - 13.9 0.7569 0.9660 0.045

13.9 - 27.8 2.6858 2.2267 0.09527.8 - 55.5 6.7392 4.3849 1.26455.5 - 223 13.9957 18.5955 1.138223 - 312 8.8045 6.4949 0.821

33.2022 33.2022 3.550

55


12 3% 2Jg

FIG, A-35 PARTICLE 477 CAL/G,



PAN < 1.7 0.2786 0.181 0.181325 1.7 - 3.5 1.1531 0.749 0.930170 3.5 - 7.0 3.9130 2.542 3.47280 7.0 -13.9 9.9418 6.459 9.93142 13.9 - 27.8 19.7722 12.845 22.77624 27.8 - 55.5 30.8017 20.03 42.8112 55.5 - 223 35.1956 22.86 65.673^2 223 - 312 4.1764 2.71 68.382^> > 312 31.1370 20.23 88.61

56


Fuel FormCladding StainlessActive Length (in.)Test Number Period (msec)Energy (cal/g of UO2)Initial Rod Weight (g)Gaussian Distribution Parameters mean variancedegrees-of-freedom

PelletSteel

183273.8477

153.93

3.78351.1989

5Volume-to-Surfacemean diameter (mils) 23.3

MOST PROBRBLE PARTICLE SIZE 44. MILS

SPX-C-023

FIG. A—36 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX-C-023.


Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square

< 1.7 0.2550 0.1483 0.0771.7 - 3.5 1.0554 0.8923 0.0303.5 - 7.0 3.5819 3.6239 0.0007.0 - 13.9 9.1007 9.9804 0.078

13.9 - 27.8 18.0970 19.1255 0.05527.8 - 55.5 28.1936 24.6501 0.50955.5 - 223 32.2175 34.6737 0.174223 - 312 3.8226 3.2294 0.109

96.3236 96.3236 1.032

57


FIG. A—37 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPX-C—024 IRRADIATED AT 572 CAU / G.



Weight(grams)


CumulativePercent

PAN < 1.7 3.3730 2.191 2.191325 1.7 - 3.5 4.5872 2.980 5.171170 3.5 - 7.0 9.3483 6.073 11.24480 7.0 - 13.9 16.6603 10.82 22.0642 13.9 - 27.8 22.5094 14.62 36.6824 27.8 - 55.5 29.6838 19.28 55.9612 55.5 - 223 24.5604 15.96 71.923^5 223 - 312 0.4974 0.32 72.24

> 312 14.8680 9.66 81.90

58


Fuel Form CladdingActive Length (in.) Test Number Period (msec)Energy (cal/g of UO2) Initial Rod Weight (g)

Pellet Stainless Steel

18 328

3.15 572

153.93Gaussian Distribution Parametersmeanvariancedegrees-of-freedom

Volume-to-Surface

3.28581.7407

4

mean diameter (mils) 10.6MOST PROBRBLE PARTICLE SIZE = 27. MILS

FIG. A—38 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C—024.

SPX-C-024

PARTICLE SIZE (MILSI

Normalized Data


< 1.7 2.8822 1.8383 0.5931.7 - 3.5 3.9195 4.3279 0.0393.5 - 7.0 7.9877 9.3249 0.1927.0 - 13.9 14.2357 15.5158 0.106

13.9 - 27.8 19.2344 20.1785 0.04427.8 - 55.5 25.3610 19.8262 1.54555.5 - 223 20.9861 23.5952 0.289

94.6066 94.6066 2.808

59


PAN 325 170

80 42 24

12 '2k

FIG. A—39 PARTICLE SIZE DISTRIBUTION OBTAINED FROM ROD SPX—C—046 IRRADIATED AT388 CAU / G.


PAN < 1.7 0.1941 0.131 0.131325 1.7 - 3.5 0.4997 0.338 0.469170 3.5 - 7.0 1.0472 0.708 1.17780 7.0 - 13.9 2.4667 1.668 2.84542 13.9 - 27.8 5.5436 3.750 6.59524 27.8 - 55.5 12.342 « 8.348 14.9412 55.5 - 223 50.511 34.16 49.10

3k 223 - 312 67.626 45.74 94.842% > 312 7.789 ' 5.27 100.56

60


Fuel Form PelletCladding Zircaloy-2Active Length (in.) 18Test Number 437Period (msec) 5.02Energy (cal/g of UOo) 388Initial Rod Weight (g) 147.85Gaussian Distribution Parametersmean 7.1628variance 3.9485degrees-of-freedom 4


MOST PROBRBLE PARTICLE SIZE = 1291.MILS

FIG, A—40 FREQUENCY DISTRIBUTION OF FARTICL.es FROM ROD SPX-C-046.


Normalized Data


< 1.7 0.0504 0.0418 0.0021.7 - 3.5 0.1297 0.1046 0.0063.5 - 7.0 0.2718 0.2859 0.0017.0 - 13.9 0.6403 0.6971 0.005

13.9 - 27.8 1.4390 1.5421 0.00727.8 - 55.5 3.2038 2.9941 0.01555.5 - 223 13.1120 13.1815 0.000

18.8471 18.8471 0.036

61


12 $5 2%

FIG. A—41 PARTICUE 469 CAU /G,



PAN < 1.7 0.2285 0.155 0.155325 1.7 - 3.5 0.3367 0.223 0.378170 3.5 - 7.0 1.4293 0.967 1.34580 7.0 - 13.9 4.4349 3.000 4.34542 13.9 - 27.8 10.9360 7.397 11.74224 27.8 - 55.5 25.0330 16.93 28.6712 55.5 - 223 68.6549 46.44 75.11

3k 223 - 312 10.6814 7.22 82.332k > 312 5.2983 3.58 85.91

62




MOST PROBABLE PfiRTICLE SIZE = 100. MILS

FIG. A—42 FREQUENCY DISTRIBUTION OFPARTICLES FROM ROD SPX-C-047,

I01 to* to5PARTICLE SIZE (MILSI

Normalized Data


< 1.7 0.1586 0.0187 1.0441.7 - 3.5 0.2336 0.1537 0.0423.5 - 7.0 0.9918 0.8456 0.0257.0 - 13.9 3.0776 3.2428 0.008

13.9 - 27.8 7.5889 8.9485 0.20727.8 - 55.5 17.3714 17.1710 0.00255.5 - 223 47.6423 45.4130 0.109223 - 312 6.8908 8.1616 0.198

83.9550 83.9550 1.635

63


FIG. A—43 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPX-C—048 569 CAU / G .



Weight(grams)


CumulativePercent

PAN < 1.7 6.9909 4.798 4.79325 1.7 - 3.5 15.8790 10.74 15.53170 3.5 - 7.0 19.2150 13.00 28.5380 7.0 - 13.9 23.3641 15.80 44.3342 13.9 - 27.8 23.8581 16.14 60.4724 27.8 - 55.5 18.6149 12.59 73.0612 55.5 - 223 24.7393 16.73 89.703% 223 - 312 11.4860 7.77 97.5623g > 312 None — ----- —

64





FIG. A—44 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C—048 .

SPX-C-ORS

10* io' io2 PARTICLE SIZE (MILSI

Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squar-

< 1.7 5.0644 6.6149 0.3631.7 - 3.5 11.5031 8.7437 0.8713.5 - 7.0 13.9198 13.5289 0.0127.0 - 13.9 16.9255 17.2495 0.006

13.9 - 27.8 17.2834 18.2011 0.04627.8 - 55.5 13.4851 15.3575 0.22855.5 - 223 17.9217 16.4075 0.140

96.1030 96.1030 1.666

65


12 33s h\?CM

FIG, A--45 PARTICUE 388 CAU/G,

Screen Size (meshes/ inch)

SIZE DISTRIBUTION


OBTAINED FROM

Weight(grams)

ROD SPX—C—052


IRRADIATED AT

CumulativePercent

PAN < 1.7 0.2818 0.191 0.191325 1.7 - 3.5 0.3487 0.236 0.427170 3.5 - 7.0 2.5510 1.725 2.15280 7.0 - 13.9 1.5793 1.068 3.22042 13.9 - 27.8 2.5999 1.758 4.97824 27.8 - 55.5 9.3773 6.34 11.3212 55.5 - 223 47.692 32.26 43.58

3h 223 _ 312 15.601 10.55 54.132h > 312 68.660 46.44 100.57

66



388147.85


Volume-to-Surface3


HOST PROBABLE PARTICLE SIZE = 1205.MILS

FIG, A—46 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C-052,

SPX-C-052


Normalized Data

Size Range (mils)

Percent of Measured


< 1.7 0.0828 0.0282 0.1061.7 - 3.5 0.1025 0.0794 0.0073.5 - 27.8 1.9777 2.2502 0.033

27.8 - 55.5 2.7556 2.8635 0.00455.5 - 223 14.0148 13.2564 0.043223 - 312 4.5844 5.0402 0.041

23.5179 23.5179 0.234

67


FIG. A—47 PARTICUE SIZE DISTRIBUTION OBTAINED FROM ROD SPX—C—053 IRRADIATED AT 498 CAU / G.

Screen Size Particle Size Weight Percent of Cumulative(meshes/inch) Ranee (mils) (grams) Rod Weight Percent

PAN < 1.7 0.1440 0.097 0.097325 1.7 - 3.5 0.2682 0.181 0.278170 3.5 - 7.0 1.1983 0.810 1.08880 7.0 - 13.9 4.2235 2.857 3.94542 13.9 - 27.8 9.3944 6.354 10.29924 27.8 - 55.5 17.9847 12.164 22.4612 55.5 - 223 61.9743 41.92 64.383^2 223 - 312 11.5043 7.78 72.162h > 312 35.1328 23.76 95.92

68



498147.85


Volume-to-Surface5

mean diameter (mils) 45.7HOST PROBRBLE PRRTICLE SIZE * 135. MILS

FIG, A—48 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C—053 .


Normalized Data

Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squar<

< 1.7 0.1004 0.0304 0.1611.7 - 3.5 0.1871 0.1811 0.0003.5 - 7.0 0.8357 0.8126 0.0017.0 - 13.9 2.9456 2.7284 0.017

13.9 - 27.8 6.5520 7.0452 0.03527.8 - 55.5 12.5433 13.5243 0.07155.5 - 223 43.2234 40.9656 0.124223 - 312 8.0236 9.1236 0.133

74.4111 74.4111 0.542

69


FIG. A—49 PARTICLE SIZE DISTRIBUTION OBTAINED FROM 590 CAL. /G.

ROD SPX-C-054 IRRADIATED AT



Weight(grams)


CumulativePercent

PAN < 1.7 1.5803 1.069 1.060325 1.7 - 3.5 2.8670 1.939 3.008170 3.5 - 7.0 7.7835 5.265 8.27380 7.0 - 13.9 14.9030 10.080 18.3542 13.9 - 27.8 24.6866 16.70 35.0524 27.8 - 55.5 30.1290 20.38 55.4312 55.5 - 223 35.9361 24.31 79.74

3k 223 - 312 None ------------- —

2k > 312 12.2027 8.25 87.99

70


Fuel Form PelletCladding Zircaloy-2Active Length 18Test Number 457Period (msec) 3.3Energy (cal/g of UO2) 590Initial Rod Weight (g) 147.85Gaussian Distribution Parametersmean 3.5896variance 1.5995degrees-of-freedom 4


MOST PROBRBLE PRRTICLE SIZE = 36. MILS

FIG. A—50 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX”C—054.

SPX-C-054


Normalized Data

Size Range (mils)

Percent of Measured

Particle WeightCalculated Chi-Squan

< 1.7 1.2395 0.7785 0.2731.7 - 3.5 2.2488 2.4534 0.0173.5 - 7.0 6.1052 6.4539 0.0197.0 - 13.9 11.6895 12.7590 0.090

13.9 - 27.8 19.3635 19.2703 0.00027.8 - 55.5 23.6323 21.4963 0.21255.5 - 223 28.1873 29.2546 0.039

92.4660 92.4660 0.650

71

APPENDIX B

PARTICLE DISTRIBUTION MODEL

73

APPENDIX BPARTICLE DISTRIBUTION MODEL

1. CHI-SQUARE MINIMUM FITTING METHOD

In order to arrive at a “best” set of values for the adjustable parameters in an overdetermined system, some assumptions must be made about the errors associated with each measurement. Different assumptions lead, generally, to different criteria for quantitatively defining “best”. In particular, the method of least-squares is based on the assumptions that the errors in each measurement are normally distributed and that all the errors are statistically independent of each other. Kottler [a] has shown that this latter assumption is invalid for distributions such as those considered in this report since the weight percent in each particle group is a fraction of the total rather than being drawn independently. Hence, if 8 j is the error in the ith weight percent, Wj, then

I. 5. = 0 (B-l)

since

Ei Wi = 100 (B-2)

before normalization. Kottler then developed the chi-square minimum method which includes Equation (B-l) as a constraint on the system in addition to the P parameters in the fitting function. Hence, the number of degrees -of-freedom for this method with N data points is

D. F. = N - P - 1. (B-3)The quantity which is minimized in the chi-square minimum method is

defined as2

X Ei 6i/fi (B-4)

where

§., = calculated wt% in group i - measured wt% in group i L = calculated wt% in group i.

2 2The minimum in y is found by setting the partial derivatives of y with respectto each of the parameters equal to zero; ie.

8 28 2 = _3_9a. x 9a i f3 j i

A!!i £± 9. 0 1, .. . , P (B-5)

[a] F. Kottler, “The Goodness of Fit and the Distribution of Particle Sizes”, Journal of the Franklin Institute, 251, p 499 (1951).

75

where = jth parameter.

Assuming the ratio Sj/fi« 2 leads to the set of equations

z 0. (B-6)

If the above assumption is not true, then the value will be large, indicating a poor fit to the data. Since the unknown function, fj, appears in both numerator and denominator of Equation (B-6), a nonlinear fitting technique was required to solve the system of equations. Consequently the function, fj, was expanded as a Taylor series in the unknown parameters and truncated after the first order terms. This system of equations could then be solved uniquely for the parameter corrections and iterated upon to arrive at the converged solution, provided the iterations converged. Within its region of convergence, the Taylor series iterations converge rapidly, but if the initial estimates of the parameter values are grossly in error or if the error surface in parameter space is highly irregular, then the method may not converge and another technique is required. This latter difficulty was encountered with a few of the particle distributions.

The alternate approach^ used in these cases was a variational one in which the quantity gj as defined by

8j

was varied. This method was iterated Taylor series was located.

ix3a j = 1,• • •, P (B-7)j

until the region of convergence for the

2. DERIVATION OF MEAN PARTICLE DIAMETER

Particles derived from fracture of brittle materials are often of an irregular shape. For such particles, the size must be defined in a consistent manner, and a prescription given for obtaining related quantities (surface area, volume, etc). The particle distributions discussed in this report were obtained by sieve analysis where the sieve mesh size is given as the diameter of spheres which will just pass through the apertures. It has been found that the ratios V/d^ and S/d^ where

V - average volume of particles of size d S = average surface area of particles of size d d = size of particle

[a] W. H. Southwell, “Fitting Experimental Data”, Journal of Computational Physics, 4, pp 465-474 (1969).

76

are approxim ately constant for irregularly shaped particles of the same material. Hence, these ratios can be used to define volume and surface shape factors,Yv andys; ie,

Yv =V, = S/d2.

The mean size for the ensemble can be defined a number of ways, depending upon which aspect of the distribution is being emphasized. Possible definitions are number-averaged mean size, surface-area-weighted size, weight-averaged size, etc. The definition selected for calculating the mean particle size of the distributions discussed in this report was the surface -are a-we ighte d size often referred to as the volume -to -surface diameter. This quantity is defined as

dsvE. d. f (d.) n.i i s i i

E. f (d?) n. x s i i(B-8)

where

d^ = particle diameter of ith groupn. = number of particles in ith group 21

f (d.) = surface area of particle in ith group.S 1

The mean diameter is related to the frequency function given in Equation (1) as follows. Let

w(y)dy = weight percent in size interval y to y + dy

n(y)dy = number percent or probability of a particle having diameter between y and y + dy.

Then

w(y)dy = fv(y3) n(y)dy (B-9)

or

n(y)dy = - W *(7^7100 p fv(y3)

where

p = density of material 3

f (y ) = volume of particleW = total weight of particles.

As a continuous function, Equation (B-8) becomes^/yfg (y2) n(y)dy

SV /fs(y2) n(y)dy

77

Substituting Equation (B-9) for n(y)dy gives

/yfs (y ) w w(y)

100 p fv(y3)

sv/f (y ) W w(y)

3

100 p f (yj)

dy

dyv

Introducing the shape factor discussed above, ie, assuming that

fs(y2) = V2

, , 3. 3fv(y > = v

gives finally

/ (y)dy

sv /7 w(y)dy(B-10)

The integration is carried out to the maximum particle size. Equation (B-10) is the continuous equivalent of the Sauter Mean Diameter quoted for TREAT (ANL) particle distributions CaJ.

Equation (B-10) can be integrated directly by substituting in the frequency function for the particle sizes expressed in the real size domain. Converting the frequency function into the particle size domain can be accomplished through the distribution function, Equation (2), and the rule for an integral of a function of a function. Let

x = loge y

where y = particle diameter (mils).

Then

(B-ll)

tL = weight percent in [x1’ X2] J f (x)dx. (B-12)

But this is also given by

[a] R. O. Ivins, R. C. Liimatainen, and F. J. Testa, “Reaction of Uranium with Water as Initiated by a Power Excursion in a Nuclear Reaction (TREAT)”, Nuclear Science and Engineering, 25, No. 2, pp 131-140 (June 1966).

78

(B-13)W.T2 rI f(x)dx = /J x, *' y

f(log y) - dy. y

Hence

, . , a — 3(log y - m) w(y)dy = — e °e ^ dy (B-14)

and Equation (B-10) becomes

SUMsv /*D

yo y

tv> '/ niaxa_ -g(loge y - m)J 0 v2 dy

(B-15)

where

SUM J maXw(y)dy = total wt% in distribution

D = maximum particle size in distribution, max ^The integration in Equation (B-15) can be carried out by making a change

of variables. Let

z = /g (log y - m) °ethen

sv SUM* /g"/ -(Z + •l/Mdz

-1(B-16)

or

_1 r -zd = SUM* /g / 4g f e-(z + z//B)dz sv a J

-1(B-17)

where £ = /6 (loge Dmax - m|.

Equation (B-17) is in the form of the error function given by

-Kr’

Let

iERF (t) = / e”U du.

u = z + 27g

79

then

umax z + 1I7f

and

d = 2 * S™ exp (m - yr-) [1.0 + ERF (umax) ] sv a f it 4p (B-18)

80

APPENDIX C

CHI-SQUARE FITTING PROGRAM

81

APPENDIX C

CHI-SQUARE FITTING PROGRAM

This appendix contains the instructions for using the Chi-Square Fitting Program used to obtain the distribution parameters in this report. The complete 360 FORTRAN listing is included. Four data card groups are required and, depending upon the amount of information available about the distribution, a fifth data group may be added. Several problems can be run consecutively, each problem requiring a complete data set. The end of the data is signaled by a card having END in Columns 5-8.

Data Type 1 (I1,I3,19A4)

Column1 (LC0N)

4 (NUM)

5-80

Column

Data Type 2 (8F10.6)

Screen Sizes


Particle weights in grams or percent in order corresponding to the above screen sizes as the upper size limit.Ex: Weight in pan has smallest screen as its upper size limit.

Data Type 4 (I5,5X,2F10.6)

Column Data5 Weighting Factor Control

-1 - Unity Weighting FactorStandard Least-Squares Fit

0 - Weighting Factor = l/Frequency 21 - Weighting Factor = 1/ (Frequency)2 - Weighting Factor = l/wt%

Chi-Square Fit

DataInitial Estimate for Parameter Values0 value - program calculates values1 - initial values read in

program iterates2 - initial values read in

no normalization

Number of data points in distribution negative value implies weights given in percent, not grams

TitleTitle will fit graphics if restricted to Column 5-44.

In ascending order until list is complete

83

11-20 (VGINT)

21-30 (VC0R)


Weight percent sum if different from 100

Maximum fractional magnitude allowed for correction vectors. Program uses 0.1 if none given.

Use only if LC0N >0

Estimates of Gaussian Parameters - ALPHA, BETA, M.

In addition to the printed output, plots of both frequencies and distribution functions can be obtained. These plots can be obtained either on paper or microfilm by designating the proper device in the JCL statements.

84

360 FORTRAN LISTING

DECK GAUSIMPLICIT REALMS IA-HtO-ZI GAUS0010COMMON DLSS, DMT, HEAD, NN, XXN(85I I, Eli (851), YY, DD, SQPI, ILGAUS0020 RFAL*4 DDX,XX(401),FX(4D1),FMIN ,DDF,X,Y1,HEAD,XMI,DX(30),DI(30) GAUS0030 REAL*4 DEMIN,DDD,DM,PNUM,AF,TODAY,DIA,FINI,YY(30),DD( 30)REAL*4 XMAX, YMAX, XSUF, YSIZEDIMENSION DLSS(30), DWT(30), DCXC30), SIZFC30), GRAMS(30),

1 HF ADI 10), FREQ(30 J, WT(30), TODAY(2), AA(30I, ALF(301,2 DELTA(30), CHI(30), CWT(30) , EXA(30),EXB(30),EXC(30),ANC0(3,3)

CCccccccccr.ccccccccccccccccccccccccccccc

DIMENSION CONV(lOl) LOGICAL 01V, GOIV*******

P A R T I C L S I Z F.H I - S 0

S I s 0 I s

M 0 D I F I E I R U T I 0 N

THIS PROGRAM CALCULATES SUM(DELTA(I)**2/CWT(I))PROGRAM INFORMATIONGRAMS( It SIZEd! AA (I )YY (I )01 (I 1ocxmrx (i)XXII )oxn)DEGF SQPI El K I) XXN(I) DDd) DWTJI) DLSSII ) HE AD d ) FRFOtI) WT (I ) LCON

VGINTVC OP NUM

THE MINIMUM WHERE F(x) =

I

VALUE OFALRH*EXP(-BETA*(X-AM)**2)

GRAMS IN SIZE RANGE SCREEN size in mils NATURAL LOG OF SCREEN SIZESARGUMENT OF NORMAL INTEGRAL SUCH THAT WE IGHTFRACTION EQUALS NORMAL INTEGRALWEIGHT PERCENT FREQUENCIESCENTER POINT OF LOG(SIZE) INTERVALSCALCULATED VALUES OF FITTED FUNCTIONX VALUES FOR FXSCALED VALUES OF DLSSDEGREES OF FREEDOM IN CHI-SQUARED DISTRIBUTION l.O/SQRTI?» 0*P I)VALUE OF NORMAL INTEGRAL FOR ARGUMENT XXNARGUMENT ÔR NORMAL INTEGRALCUMULATIVE WFIGHT FRACTIONWEIGHT PERCENT IN SIZE RANGENATURAL LOG OF SCREEN SIZESTITLE TO PROBLEMWEIGHT PERCENT/(DELTA X! FREQUENCIESWEIGHTING FACTORPARAMETER INITIALIZATION CONTROL0 -= INITIAL VALUES CALCULATED BY PROGRAM1 = INITIAL VALUES READ IN - PROGRAM ITERATES2 = FINAL VALUES READ IN - NO ITERATION WEIGHT PERCENT SUM IF DIFFERENT FROM 100 (NORMALLY NEEDED ONLY IF LC0N=2)MAXIMUM FRACTIONAL MAGNITUDE OF CORRECTION VECTOR NUMBER OF DATA POINTS IN DISTRIBUTION

GAUS0040 GAUS0050 GAUS0Q60 GAUS0070 GAUSOOBO GAUSOQPO GAUS0100 GAUS0110 GAUS0120 GAUS0130 GAUS0140 GAUS0150 GAUS0160 GAUS0170 GAUS0180 GAUS0190 GAUS0200 GAUS0210 GAUS022 0 GAUS0230 GAUSQ240 GAUS0250 GAUS0260 GAUS0270 GAUS0280 G4US0290 GAUS0300 GAUS0310 GAUS0320 GAUSQ330 GAUS0340 GAUS0350 GAUS0360 GAUS0370 GAUS0380 GAUSOacO GAUS0400 G4US0410 GAUSQ420 GAUS0430 GAUS0440 GAUS0450 GAUS0460 GAUS0470 GAUS0480 GAUS0490

NEGATIVE VALUE IMPLIES GRAMS GIVEN IN WEIGHT PERCENTGAUS0500

85

nnnn

nnan

nnnn

nnn

DECK GAUSIWT

1 01

WEIGHTING FACTOR CONTROL—1=UNITY WTS, 0=1/FREQ» 1=1/FRE0*«2LOG OF FREQUENCIESVALUES OF DELTA**2/WT PERCENTCALCULATED WT PERCENT - MEASURED WTCALCULATED WT PERCENT IN SIZE GROUP

100

12

ALF(T)CHim DELTA(I)CWTII )DATA CAROSCARD 1 LCON,NUM,HE AD { 11,13,1RA4)SCREEN SIZES (8F10.6) IN AC ENDING ORDER UNTIL PARTICLE WTS (8F10.6I IN ORDER CORRESPONDING

AS UPPER LIMITIWT,VGINT,VCOR (I5,5X,2F10.6IGAUSSIAN PARAMETERS I FOR LC0N>01 ALPH,BETA,AMEND COLUMNS 5-8 END OF DATA

DATA DT A/Z3DOOOOOO/,FIN T/'END •/CALL DATE(TODAY)READ (5,100) LCON,NUM,(HEAD(11,1=1,19)IF(HEAOU) .FQ.FTNI ) CALL EXIT FORMAT(I 1,13,19A4)VCORM=1,OD-1CALL PLOTS(1.00,-11.0,-3)LNG=0GD IV=» FALSE,KNUM=0 DIV=.FALSE.UU=0.OD+O UUP=0.OD+O vv=o.oo+oVVP=0.OD+O WW=0.OD+Owwp=o.on+oEl 1(851)=0.0 IPNT=1NN= I AS S(NUM)IU=NNREAD SCREEN SIZESREAD (5,101) (SIZE!T),1=1,NN)FORMAT(8F10. 6)READ PARTICLE WEIGHTSREAD (5,101) (GRAMS(I),1=1»NN)CONVERT SIZES TO LOG DIAMETERDO 12 1=1,NNDL SS(I)=DLOG(SIZE( I))AA (I )=DLSS (I )CONTINUESIJMV=1.0D+2CONVERT GRAMS TO WEIGHT PERCENT SUM=0, OD+0 DO 15 1=1,NN

, 2=1/DWT

PERCENTT

GAUS0510 GAUS0520 GAUS0530 GAUS0540 GAUS0550 GAUS0560 GAUS0570 GAUS0580

LIST IS COMPLETFGAUS0590 TO SCREEN SIZE ASGAUS0600

GAUS0610 GAUS0620

(3F10.6) GAUS0630GAUS0640 GAUS0650 GAUS0660 GAUS0670 GAUS0680 G6US0690 GAUS0700 GAUS0710 GAUS0720 GAUS0730 GAUS0740 GAUS0750 GAUS0760 GAUS0770 GAUS0780 GAUS0790 GAUS0800 GAUS0810 GAUS0820 GAUS0830 GAUS0840 GAUS0850 GAUS0860 GAUS0870 GAUS08R0 GAUS0890 GAUS0900 GAUS0910 GAUS0920 GAUS0930 GAUS0940 GAUS0950 GAUS0960 GAUS0970 GAUS0980 GAUS0990 GAUS1000

86

DECK GAUSSUM=SUM+ GRAMSm GAUSI010

15 CONTINUE GAUS1020IF(NUM.LT.0) SUMV=SUM GAUS1030

21 IF(NUM,LT.0) SUM=1•00+ 2 GAUS1040IL=1 GAUS1050FREQ (11=0.004-0 GAUS1060DO 20 1=1,NN GAUS10T0DWT(I)=1.00+2*GRAMS(II/SUM GAUS1080IFIDWTIII.EQ.O.OD+OI IL=TL*1 GAUS1090

20 CONTINUE GAUS1100COMPUTE CENTRAL X POINTS AND EXPERIMENTAL FREQUENCIES GAUSUiODO 25 1=2,NN GAUS1120OCXS I) = «DLSS«I M-DLSS(1 — 111/2.00*0 GAUS!130FREQt I ) = DWTU )/( DLSSI I )-DlSS( 1-1 )) GAUS1140

25 CONTINUE GAUS1150CALCULATE WEIGHTING FACTOR GAUS1160READ (5,102) IWT, VGINT, VCOR GAUS1170

102 FORMAT(15, 5X,2F10.6) GAUS1180IF(VCOR.GT.O.OD+O) VCORM=VCOR GAUS1190IFtVGINT.FQ.O.OD+O) GO TO 29 GAUS1200CORR=VGINT/SUMV GAUS121000 28 1=1,NN GAUS1220DWT(I)=DWT(I)*CORR GAUS1230FREQ(I)=FREQ(I)*CORR GAUS1240

28 CONTINUE GAUS125029 IWT*IWT+2 GAUS1260

DO 42 1=2,NN GAUS1270GO TO (38,39,40,41), IWT GAUS1280

38 WT(I)=1.0D+0 GAUS1290GO TO 42 GAUS1300

39 WT (I > = 1.00+0/FREQ(I) GAUS1310GD TO 42 GAUS1320

40 WT U) = 1.0D+0/FRFQ(I)**2 GAUS1330GO TO 42 GAUS134QWEIGHTING FACTOR FOP CHI-SQUARED DISTRIBUTION GAUS1350

41 WT (I)=1.OD+O/DWT(I) GAUS136042 CONTINUE GAUS1370

Ic(IWT.FQ.4.AND.IL.EQ.1) WT(1)=1.CD+O/DWT(1) GAUS1380PRINT DATA GAUS1390

35 WRITE (6,201) (TODAY(I),1 = 1,2), (HEAD(J),J = 1,19) GAUS1400201 FORMAT(1 Ml,T21,'WEIGHT-SIZE PARTICLE DISTRIBUTI0NS*,T61, 'RUN MADEGAUS1410

1 ON ',2A4//Tli,lQA4 //) GAUS1420IF(NUM.LT.O) WRITE (6,210) GAUS1430

210 FORMAT(Til,* MASS DISTRIBUTION DATA INITIALLY GIVEN IN WEIGHT PERCGAUS14401ENT •//I GAUS1450WRITE (6,211 ) GAUS1460

211 FORMAT(120H SCREEN SIZE-MILS NATURAL LOG GRAMS IN SIZE GAUS14701 WEIGHT PERCENT MEAN LOG SIZE FREQUENCY WE IGHTINGGAUS14S03 /120H MAX PARTICLE SIZE SCREEN SIZE RANGE < MAX GAUS14904 IN SIZES < MAX FACTOR GAUS1500

87

o o

no

in 26ocxm ,frfq( n ,wimSI7F(I).nLSS(n .GRAMS l I» ,DWTm

RFC K GAUS5 //)

00 30 1 = 1,NN TFt T.EQ.l) GO WRTTF (6,202)

26 WRITE (6,203!30 CONTINUE

202 FO RM AT(67X,2(4X,1P014.6S,?X,1PD12.6)203 PORMAT(3X,4( lP014.6,4Xn

IFIKNUM.E0.0) GO TO 44 KNUM=0GO TO 51CALCULATE INITIAL ESTIMATE OF PARAMETERS CALCULATE LOG OF FREQUENCIES

44 DO 45 1=2,NNALF(I)=DLOG(FRFO{I 1 )

45 CONTINUECHECK METHOD OF GENERATING INITIAL ESTIMATES OF PARAMETERS VALUES GENERATED INTERNALLY IF(LCON.EO.O) GO TO 46

: VALUES READ IN - FINAL VALUE IF LC0N=2READ (5,101) ALPH,BETA,AM AL pHI= AL PH 8ETAI=BETA AM 1= AM IPNT=2 GO TO 51

: CALCULATE THE LEAST SQUARES FACTORS,46 SUMX=0.0n+0

SUMY=0,0D4-0 SUMX2=0.OD+O SUMX3=0«OD+O SUMX4=0« OD+O SUMXY=0,OD+O SUMXXY=0,OD+O no so 1=2,NN SUMX = SUMX + OCX( I)SUMXY= SUMXY + ALF(I)*DCX(I)dc=dcx(i)**?SUMX2 = SUMX 2+ DC SU MX X Y= S UM XX Y + ALF(n*OC DC=DCX(I)*DC SUMX3=SUMX3+DC DC =DCX(I)*DC SUMX4=SUMX4+DC SUMY = SUMY

50 CONTINUEAU=DFL0AT(NN-1)DEN= AU* (Rl)MX2 *

1 SUMX2 * SUMX3) +CA =(SUMY*(SUMX?

+ ALFt!)

SUMX4 - SUMX3 * SUMX3) - SUMX *(SUMX * SUMX4 - SUMX2 *(SUMX * SUMX3 - SUMX2 * SUMX2)* SUMX4 - SUMX3 * SUMX3) - SUMXY *(SUMX * SUMX4

GAUS15L0 GAUS1520 GAUS1530 GAUS1540 GAUS1550 GAUS1560 GAUSl5TO GAUS15B0 GAUS15D0 GAUS1600 GAUS1610 GAUS1620 GAUS1630 GAUS1640 GAUS1650 GAUS1660 GAUS1670 GAUS1680 GAUS1690 GAUS1700 GAUS1T10 GAUS1720 GAUS1730 GAUS1740 GAUS1750 GAUS1760 GAUSI770 GAUS17R0 GAUS1790 GAUSI 800 GAUS1B10 GAUS1820 GAUS1830 GAUS1840 GAUS1850 GAUS1860 GAUS1870 GAUS1880 GAUS1890 GAUS1900 GAUS1910 GAUS1920 GAUS1930 GAUS1940 GAUS1950 GAUS1960 GAUS1970 GAUS1980 GAUS1990 GAUS2000

88

DECK GAUS1 - SUMX2 * SUMX3) + SUMXXY *f SUMX * SUMX3

CB = t AU*(SUMXY * SUMX4 - SUMX3 * SUMXXYI 1 SUMX2 * SUMXXYI + SUMX2 *1 SUMY * SUMX3 -

CC =(AU*{SUMX2 * SUMXXY - SUMXY * SUMX3I

- SUMX2 * SUMX?)I/OEN- SUMX *!SUMY * SUMX4 - SUMX2 * SUMXY))/DEN- SUMX *(SUMX * SUMXXY

SUMX? *1SUMX * SUMXY CB*CB/(4.on-K)*CC) I

- SUMX2 * SUMY!)/OFN1 SUMY * SUMX3) +ALPH = OEXPtCA BETA = -CCAM = -CB/(2.0D+0*CC)IF(CC.LT.O.On+0.AND.AM.LT.7.5D+0) GO TO 52 USE FIT TO INTEGRAL VALUES CALL ERFIT(ALPH,BETA,AM,OSV,IU)

52 AL PHI = ALPH BETAI=BETA AM I*AMCALCULATE VALUE OF CHI-SQUARED = (DELTA**2/DWT)

51 SQFA=0.OD+O SUMD=0.OD+OAL=AM-DSQRTIDLOG(ALPH*1.0D+3)/BETA)SUMC = O.OD+O XY=DMIN1(AL,O.OD+O)SUM= O.OD+ODO 55 1=1,NNAU-DLSS(I)CALL ADGAS(AL,AU,ALPH,BETA,AM,ANS,1)DELTA(T)=ANS-DWT(IS CHI!I) = DELTA!I>**2/ANS SUMC = SUMC + CHT(I)AL=AUSQEA=SQE A + ANS SUM=SUM + DWT(I)CWT( n=ANSSUMD= SUMD + DELTA!!)

55 CONTINUE I0F=NN-4

- SA = (l.OD+2/ALPH1**2/(2.0D+0*3.14159265D+0)SB=5.OD-1/BETAIF(DABS!(SA-SB)/(SA*SB)).LE.l.OD-4 ) IDF=NN-4+l DEGF=DFLOAT(IOF)/2.0D+0 AL=0.OD+O AU=SUMC/2.0n+0CALL AOGAS(AL,AU,DEGF,BFTA,AM,ANS,5)PROB=(1.OD+O - ANS/DGAMMA(DEGF))*l.0D+2 UMIN=(5.OD-1 + BFTA*(XY-AM))/OSQRT(BETA)UM AX=(5.OD-1+BETA*(DLSS(NN)-AM))/DSQPT(BFT A)ZM AX=DA8 S(UMAX)ZMIN=DABSCUM INICORR=(2.OD+O/(DSIGN(1.OD+O,UMAX)*DERF(ZMAX)-DSIGN(1.OD+O,UMIN)*

1 DERF(ZMIN)))DSV = SQEA*USQRT(8FTA/3.14150265D+01*CORR^DEXP(AM-2. 5D-1/BETA)/

1 ALPH

GAUS2010GAUS2020 GAUS2030

-GAUS2040 GAUS2050 GAUS2060 GAUS2070 GAUS20B0 GAUS2090 GAUS2100 GAUS21l0 GAUS2120 GAUS2130 GAUS2140 GAUS2150 GAUS2150 GAUS2170 GAUS2180 GAUS2190 GAUS2200 GAUS2210 GAUS2220 GAUS2230 GAUS2240 GAUS2250 GAUS2260 GAUS2270 GAUS22B0 GAUS2290 GAUS2300 GAUS2310 GAUS2320 GAUS2330 GAUS2340 GAUS2350 GAUS2360 GAUS2370 GAUS23B0 GAUS2390 GAUS2400 GAUS2410 GAUS242 0 GAUS2430 GAUS2440 GAUS2450 GAUS2460 GAUS2470 GAUS24B0 GAUS2490 GAUS2500

89

n n

PERCENT

GAUS2510 GAUSS 520 GAUS2530 GAUS2540 GAUS2550 GAUS2560

CALCULATGAUS2570 6AUS2580 GAUS2590 GAUS2600 GAUS2610 GAUS2620 GAUS2630 GAUS2640

DECK GAUSVGINT=ALPH*DSORT(3®14159265/BETA I CONV (KNUM-s-l) =DABS ( SUMO )WRITE 16,2065 KNUM, (HEAD!15,1=1,19)

206 FORMATIlHl,Til,'COMPARISON OF CALCULATED AND MEASURED DATA',T62,A ’KNUM = ',I3//T11,19A4//1 T10,•MAX PARTICLE SIZE LOG MAX SIZE WEIGHT2 DIFFERENCE CHI SQUARED*/T50MEASURED3ED *//)

DO 122 1=1,NNWRITE 16,207 5 SIZE! I), DLSSI I) , DWTI I) , CWT(I) , DELTA! I ) , CHIU)

122 CONTINUE207 FORMAT 110X, 1 !,6D16)

WRITE (6,208) SUM, SQEA, SUMD, SUMC, IDF, PROS208 F0RMAT(1H0,41X,1P4D16.6//T11,'DEGREES OF FREEDOM FOR CHI-SQUARED

1TEST = NN - NO. OF PARAMETERS - 1 = ',I2//T11,’PROBABILITY THAT THGAUS2650 2E RANDOM VARIABLE <CH!-SQUARE> EXCEEDS CALCULATED VALUE =•,0PF7.2,GAUS2660 3T95,,PFRCENT,//T11,'XF ABOVE PROBABILITY IS < 10 PERCENT, THEN DISGAUS2670 4TRI BUTTON FUNCTION IS A POOR FIT TO DATA* 5 GAUS2680

IFI.NOT.DIV) WRITE 16,2125 GAUS2690IF IDIV 5 WRI T"E (6,2135 GAUS2700

212 FORMATUHO.Tll,** * * * * **//Tll ,,nARAMETFPS AND CORRECTION VECTGAUS271010R COMPONENTS FOR PARTICLE DISTRIBUTION*/Til,'CORRECTIONS CALCULATGAUS2720 2ED FROM A FIRST ORDER TAYLOR SERIFS EXPANSION OF FITTING FUNCTI ON*GAUS2730 3//I GAUS2740

213 FORMAT(1 HO,T11,* * * * * * **//Tl1,*PARAMETERS AND CORRECTION VECTGAUS2750 10R COMPONENTS FOR PARTICLE DISTRIBUTION*/T11,*CORRECT IONS CALCULATGAUS2760 2ED FROM THE GRADIENT OF THE FITTING FUNCTION IN PAR AMFTER SPACE* GAUS27703 //) GAUS2780

WRITE 16,205 5 At PH,UU,UUP,BETA,VV,VVP,AM,WW,WWP,DSV,SA,SB,VGINT GAUS279020 5 FORMAT CT17,» PARAMETERS*,T60,•CORRECT IONS'/T51,'USED*,T74,'CALCULAGAUS2800

1TED*//T13,'ALPHA = •,3(1PD14.6,11X)/T14,* BETA = ' , 3(1PD14.6,1IX ) / GAUS2810 2T17 , •M = ',3(1PD14.6,11X5//Til,'SAUTER MEAN DIAMETER = ', 1PD14.6, GAUS2820 AT51,'MILS'/ GAUS28307 T15, * < VOLUME— TO—SUP F ACE RATIO!*// TU,'VARIANCE CALCULATED FROM AGAUS2840 41 PH A AND BFTA*/T11,'IF THR TWO ARE EQUAL, THEN DISTRIBUTION IS LOGOAUS 2850 5-NORMAI '//T15, • SA = (100/ALPHA5**2/2*PI = • 1PD14.6/T15,'SB = 1/(2#0AUS2860 6BFTA1 = ' 1 t>D 14.6//T11 , 'VALUF OF GAUSSIAN INTFGRAL OVER T0GAUS28707TAL RANGE = *1PD14.6)

IF(IPNT.FQ.?» GO TO 77 WRITE (6,204! CA,C.B,CC

204 FORMATUHOZ/Tll,'COEFFICIENTS TO PARABOLIC 10G FRFQUENCIE5'//T16,'A0 = ', 1PD14.6.T36,*2 • CO = ',1PD14.6//)

PLOT CURVES AND DATA nDX=0.025 xxm=o.5 DO 65 1=2,220xx(i)=xx(T-i)+nnx

65. CONTINUE

LEAST SOUARFS FIT TO BO = • , IPD14.6.T56,

GAUS2880 GAUS2890 GAUS2900

LGAUS2 910 GAUS2920 GAUS2930 GAUS2940 GAUS2950 GAUS2960 GAUS2970 GAUS2980 GAUS2990 GAUS3000

90

DECK

C

f.

6?

1

70

75

77

90

81

85

82

87

GAUSDO 67 1=1,220 GAUS3010FX m = CA +CB*XX( I) * CC*XX.( I )*XX( I ) GAUS3020CONTINUE GAUS3030CALL PL0T(2.»0,1. 0,-3) GAUS3040CALL AXTSIO.0,0.0,32HX = LOG(SCREEN MESH SIZE (MILS)),32,12.0,0.0GAUS3050,0.0,0.5,3)CALL SCAL E(FX,210,9.0,FM IN,DOF,1)DO 70 J=1,220 XX (J ) =2. 0 * XX ( J )CONTINUECALL AXIS(0.0,0.0,14HLOG(FREQUENCY),14,8.00,90.0, FMIN, DDF, 3) CALL LINE(XX,FX,210,1)DO 75 1=2,NNX=DCX( n*2.0D+0Yl=(SNGL( ALFtI))-FMIN)/DDFCALL SYM8L4(X,Y1,0.42,DTA,0.0,1)CONTINUECALL SYMBL4(0.2,9.7,0.21,HEAD,0.0,76)CALL PLOTS(13.0,-11.0,-3)XX(1)=9MIN1(0.OD+O,XY)DDX=(AM—D8LE(XX(1))1/2.0D+2 DO 80 1=1,400FX(l)=ALPH*DEXP(-BETA*(OBLE(XXm )-AM)**2)XX(I+1)=XX(I)+ODXCONTINUEK= 1IF (XY.GT.0.0D4-0) K=-l CALL PLOT (7.0, 1.0,-3)XM I=XX(1)DDF=ALPH/7.0D+0 DDX=400.0*DD X/5.5 XMAX=XX(400)SCALE THE FX AND XX VALUESDO 81 1=1,400XX(I)=(XX(I)—XMIS/DDXFX(I) = FX(I I/DDFCONTINUESCALE FREQUENCIESnpn=o.oDO 85 1=1,NN DI (I ! = FR EO (I )DX(I)=(SNGL(DLSS(I))-XMI)/DDX IF (Did l.GT.DDD) DDD=D I ( I)CONTINUE DDD= DDD/7.0 DO 82 1=1,IU DI(I)=DI(I)/DDD CONTINUEIF(DDF-DDD) 87,95,P0 AF=DDF/DDD

GAUS3060 GAUS3070 GAUS3080 GAUS3090 GAUS3100 GAUS3110 GAUS3120 GAUS3130 GAUS3140 GAUS3150 GAUS3160 GAUS3170 GAUS3180 GAUS3190 GAUS3200 GAUS3210 GAUS3220 GAUS3230 GAUS 3240 GAUS3250 GAUS3260 GAUS3270 GAUS3280 GAUS3290 GAUS3300 GAUS3310 GAUS3320 GAUS3330 GAUS3340 GAUS3350 GAUS3360 GAUS3370 GAUS3380 GAUS3390 GAUS3400 GAUS3410 GAUS3420 GAUS3430 GAUS3440 GAUS3450 GAUS3460 GAUS3470 GAUS 3480GAUS3490 GAUS3500

91

DECK GAUSDO 88 1=1,400

88 ^XU )=FXU )*AF 0DF=00nGO TO 95

90 AF=nDD/DDF DO 92 1 = 1,IU

92 Dtm = Dim*AFC DRAW AXIS AND LABELS

95 CALL PL0T(5«5,0»0,2)AF=AINT(XMAX)J= 0

83 X=(A F—XMI)/DDX J=J+1ALF{ J)=XCALL PLni(X,0.0,3)CALL PLOT I X,0»125,2)X=X-0,05IFIAF.LT.O.O.OR.AF.GE.10.0) X=X-0.05 CALL NUMBERIX,-0,25,0.10,AF,0.0,-l) AF=AF-2.0IF(AF.GE.ATNT(XMn ) GO TO 83 CALL PL01(0,0,0.0,3)CALL PLOT!0.0,8,0,2)CALL PLOT(5.5,8.0,2)DO 84 1=1,J X= ALF(I)CALL PLOT(X,8.0,3)CALL PL0T(X,7.875,2)

84 CONTINUECALC PLOTIO.0,0.0,3)AL F(1)=0.0 X=0.0CALL NUMBER(-0.2,0.0,0.10,X,0.0,-1)YMAX=7.0*DDFY1=5.0IF(YMAX.GT.3 0.0) Yl=10.0 1=1

89 X= X+Yl 1=1+1AL F( I )=X/90FIP (ALF( I ).GT.8.0) GO TO 93 AF =—0,3IF (X.LT. 10.0) AF=-0.2 DF MIN=ALF(I)-0.05 CALL PLOT(0.0,ALF(I),3)CALL PLOT (0. 125, ALF( I) , 2 )CALL NUMBER!AF,0FMIN,0.10,X,0.0,-1) GO TO 39

93 J=I-1CALL PLOT(5.9,8.0,3)

GAUS3510 GAUS3520 GAUS3530 GAUS3540 GAUS3550 GAUS3560 GAUS3570 GAUS3580 GAUS3590 GAUS3600 GAUS3610 GAUS3620 GAUS3630 GAUS3640 GAUS3650 GAUS3660 GAUS3670 GAUS3680 GAUS3690 GAUS3700 GAUS3710 GAUS3720 GAUS3730 GAUS3740 GAUS3750 GAUS3760 GAUS3770 GAUS 3780 GAUS3790 GAUS3800 GAUS3810 GAUS3820 GAUS3830 GAUS3840 G4US3350 GAUS3860 GAUS 3870 GAUS3880 GAUS38Q0 GAUS3900 GAUS 39)0 GAUS3920 GAUS3930 GAUS3940 GAUS3950 GAUS3960 GAUS3970 GAUS3980 GAUS3990 GAUS4000

92

o o

c

r.

OFCK GAUSCALL PLOT(S.5 > O.0» 2) GAUS4010AF-5.375 GAUS402000 oi 1=2,J GAUS4030Vl=ALFtl ) GAUS4040CALL PLOT(5.5 »Y1,3) GAUS4050CALL PLOT(AF,Y1,2) GAUS4060

91 CONTINUE GAUS4070CALL SYM3L4(1,095,-0.5,0.14,'LOG(SCREEN MFSH SIZE (MILS) )',0.0, GAUS4080

l 28) GAUS4090CALL SYMBL4(-0.5,2.56,0.14,*WFIGHT PERCENT/(DELTA X)•,90.0,24) GAUS4100 CALL LINF(XX,Fx,400,1) GAUS4110CALL SYM314(0.5 ,7.5, 0.14,HEAD,0.0,381 GAUS4120CALL SYM3L4(0.5,8.3,0.14,39HM0ST PROBABLE PARTICLE SIZE = MIGAUS4130

1L S,0.0,39) GAUS4140DM=orXP(AM) GAUS4150CALL NUMBER (4. 1 ,'8. 3,0, 14,DM, 0.0, 01 GAUS4160CALL SYMB14(0.5,B.7,0.14,7HKNUM = ,0.0,7) GAUS4170PNUM=KNUM GAUS41R0CALL NUMBER(1.4,8.7,0.14,PNUM,0.C,-l) GAUS4190CALL SYMBL4(3.0,8.7,0.14,TODAY ,0.0,8 ) GAUS4200DI(11=0.0 GAUS42101=0 GAUS4220FX (1 ) = 0.0 GAUS4230

86 1=1+1 GAUS4240CALL °LnT( ox m ,nm ),3) GAUS4250CALL PinT(DX(II,DI(1+1 ) ,1) GAUS4260CALL PLOT ( DX U+l) ,01(1+1 ),1 ) GAUS4270IFd.LT. (TU-1 ) ) CALL PLOT ( DX ( I+1 ) , DI (I+2 ) , 1) GAUS4280TFfl.LT.(IU-1)) GO TO 36 GAUS42O0CALL PLOT (DX U+l), Did ),1) GAUS4300CALL PLOTSdO.0,-11.0,-3) GAUS4310IF(KNUM.LT.5.AND.DABS(SUMD).GT.1.0D-7) GO TO 121 GAUS4320IF(DABS!SUMP).LF.l.00-7) GO TO 96 GAUS4330CHFCK FOP TAYLOR SFRIES OP GRADIENT METHOD OF ITERATION GAUS4340If(DIV) GO TO 145 GAUS4350CHFCK FOR CONVERGENCE OF ITERATIONS GAUS4360KL=KNUM+! GAUS4370IT(CONVfKL).LT.CONV(KNUM)) GO TO 94 GAUS4380TAYLOR SERIES DIVERGES, SWITCH TO GRADIENT METHOD GAUS4390CHECK GR AD IFNT METHOD FOR DIVERGENCE GAUS4400

145 IF(GDIV) GO TO 5 GAUS4410LNG= LNG+1 GAUS4470IF(LNG.lE,5) GO TO 150 GAUS4430DTV=.FALSP. GAUS4440GO TO 94 GAUS4450

150 UU=VCORM GAUS4460AL = AMT-nSQRT(DLOG(ALPHI*1.0D+3 )/8ETAI) GAUS4470XY=D MINI(AL,0.OD+O) GAUS4480CALL COMPY(ALPHI,BET AT,AM I,XY,UU,VV,WW,UUP,VVR,WWP,GOIV,KNUM,01V)GAUS4490 AL PH=ALPHI GAUS4500

93

DECK GAUSBE TA=BET4! GAUS4510AM=AMT GAUS4520on rn 51 GAUS4530

04 IF (KNUM.GE,100) GO TO 8 GAUS4540GO rn 121 GAUS4550

96 CALL PL0T(2.0,1.0,-31 GAUS4560CALL PLOT!0.0,5.5,21 GAUS4570AF=AINT(XMAX) GAUS4580J=0 GAUS4590

98 X=(AF-XMT1/DOX GAUS4600J = J+1 GAUS4610ALF<J)=X GAUS4620CALL PLOT!0.0,X,31 GAUS4630CALL PLOT!0.125,X,2) GAUS4640Yl=-0.2 GAUS4650TF(4F.GE.10.0.0R.AF.LT.O.O) Yl=-0.3 GAUS4660X=X-0,05 GAUS467 0CALL NUMBER(Y1,X,0.10,AF,0.0,-11 GAUS4680AF =AF-2.0 GAUS4690IF<AF.GE.AINT(XMI)) GO TO 93 GAUS4700CALL PLOTtR.0,0.0,31 GAUS4710CALL ffL07(8.0,5.5,21 GAUS4720DO 90 1=1,J G4US4730X=ALF(I 1 GAUS4740CALL PLOT ( 8. 0, X, 31 GAUS4750CALL PLOT(7.375,X,2! GAUS476 099 CONTINUE GAUS4770CALL SYMBL4(-0.4,1.095,0,14,'LOG(SCREEN MESH SIZE (MILS))',90.0, GAUS4780

1 28 ) GAUS4790XXU > = XX(1 >*DDX +XMI GAUS4800DO 97 1=2,400 GAUS4810XXU ) = XXU ) *DDX + XMI GAUS4820t=(xx(n +xxt i-i) )/2.o GAUS4330fx (i )=fx (i-i )4-alphôexp(-bfta*(t-ami **?}*( xx( n-xx u-i n GAUS4340

97 CONTINUE GAUS4850CALL PLOT(0.0,0.0,31 GAUS4360XS IZF= 8.0 GAUS4370YS I Z E= 5, 5 GAUS4380CALL PAXIS(0o0,0.0,XSIZE,YSIZF,ALPH,BFTA,AM,XX,FX,IK) GAUS4R90DO HO 1 = 1,400 GAUS490Oxx m = (xxc i )-xmi i /nnx 0AUS4910

110 CONTINUE GAUS4920NK=0 GAUS4930DO 115 1=1,400 GAUS4940IF UK. EO.l) GO TO 112 GAUS4950K= I—1 + T K GAU$4960IF (K.GT.400) GO TO 118 GAUS4970FX m = FX(K 1 GAUS4930XXU 1 = XX (K 1 GAUS4990

112 IF (FX( D.GF. 7. 11 GO TO US GAUS5000

r> o

DFCK GAUSNK=NK+1

115 CONTINUEIIP CALL LINE(FX,XX,NK,t J

NK=NN-1IF (YY (NN)aLEoT»l) NK=NNCALL SYMBL4(1.688,5. 0,0.14,HEAD,O.O, 33)CALL SYM3L4? 0.5,8,5,0.14,7HKNUM = ,0.0,7)CALL NUMBER(1.4,8.5,0.14,PNUM,0.0,-1)CALL SYMBL4(3.0,3.5,0.14,TODAY,0.0,8)

.DO 120 I=IL,NK Y1=DX(n X= YY 1 T }CALL SYMBL4(X,Y1,0.42,0IA,0.0,1)

120 CONTINUECALL PLOTSdO.0,-11.0,-3)IP(LCON,FQ.2) GO TO 5IF(DA3S(VGINT-1.0D+2).LE.1.0D-6) GO TO 5

C ADJUST WRIGHT PERCENTS TO MAKE GAUSSIAN INTEGRAL SUM TO 100CnRR=1.0D+2/VGINT DO 105 I=1,NN PVJTt I ) = DW‘r (I ) *COPR FR EO(I) = EREO(I)*CORR WT(I)=WT(I)/CORR

105 CONTINUEALPH = ALPH*C,nPR KNUM=—1ElT(851)=-2.OD-O GO TO 35MAKE CORRECTIONS IN FITTED CONSTANTS CONSTRUCT NORMAL EQUATIONS

121 A11=0.OD+O B11=0.OD+O Cl 1 = C.OD+O AS 3=0.OD+O AB1=0.OD+O AC 1=0.OD+O RSQ= O.OD+O PC 1=0.OD+O CS0=0. OD+O LNG=0 AL=XYDO 125 1=1,NN AU=DLSS(I)CALL ADGAS(AL,AU,ALPH,BETA,AM,ANS,2)EX A!I)=A NSCALL ADGASIAL,AU,ALPH,BETA,AM,ANS, 3)EXB!I)= -ALPH*ANSCALL ADGASIAL ,'AU , ALPH , BETA , A M, ANS, 4 )EXC(I)= 2.0D+0*ALPH*BETA*ANS IE(IWT.E0.4) WT(T)=1.OD+O/(ALPH*EXA(I))

GAUS5010 GAUS5020 GAUS5030 GAUS5040 GAUS5050 GAUS5060 GAUS50T0 GAUS5080 GAUS5090 GAUS5100 GAUSS 110 GAUSS 120 GAUS5130 GAUS5140 GAUS5150 GAUS5160 GAUSS 170 GAUSS180 GAUSS 190 GAUS5200 GAUSS210 GAUS5220 GAUS5230 GAUS5240 G6USS2S0 GAUS5260 GAUS5270 GAUS5280 GAUSS290 GAUS5300 GAUS5310 GAUS5320 GAUS5330 GAUS5340 GAUS5350 GAUS 5360 GAUSS370 GAUS5380 GAUS5390 GAUS5400 GAUS5410 GAUS5420 GAUS5430 GAUS5440 GAUSS450 GAUS5460 GAUS5470 GAUS5480 GAUS5490 GAUS5500

95

orcK GAUSAB1 = AB1 + EXA(I)*EXB!I)^WTtI 5 AC 1=AC1 + EXAt n*EXC(T)*WTU !BC1 = BC1 + F^B( I)*EXC(I )*WT(T }AS 0= ASQ+ EX At I)*EXA(I)*WTtI )BSO= BSO+ FX B(I)*FXBfI ) *WT(I )CSQ=CSO+EXC( I ) *EXC t T )*WTU )A1i = Al1 + EX AtI)*{ DMT(I)-ALPH*FXA( I ) )*WT(I)B11= B11+ FXB(T )*( DWT(I)-AL PH*EX A t I ) )*WT( I )Cl1=C11 + EXC( !)*( DWT(l)-ALPH*EXAt T ) )«WTtI)AL = AU

125 CGNTIMUF ASSIGN 1?0 TO KP ASSIGN 127 n imk 1 = 1

126 AMCD(1,1) = ASQ .ANCD (1 » ? ) = AB 1 ANCDt1,2) = AC 1 ANn(2,l) = ARl ANCO(2 f 2) = BS 0 AN CD(2 > 3)=BC 1 ANCnt3,l)=ACl ANCDt 3»2) = BC1 ANCDt3,3)=CSQGD TO IMK» (127,131)127 DU M= ANCDU, 1 )*{ANCO(2,2)*ANCD( 3,3) - ANC0(3,2)* ANCDt?,3) )

1 t ANCDt1,2)*(ANCO(2,3>*ANCO(3,1) - ANCD(3,3)* ANCn(2,lS)2 + ANCTin,31*(ANCOt 2,1)*ANCD(3,2 ) - ANCn(3,l!* ANC0!2,2)1

GD TO KR, (170,132,136,140)130 DE MOM=0UM

ASSIGN 132 TO KD131 ANCDt1,L ) = A11

ANCD(2,L)=B11 ANCO(3,L)=C11 GO TO 127

132 UU =0U N / D E N ON ASSIGN 131 TO I MX ASSIGN 136 TO KRL = 2GO TO 126

136 VV=OUN/DENnMASSIGN 140 TP KR L= 3GO TO 126

140 WW=OUN/DENOM UUP=UU VVP=VV WWP= WWC CHFCK MAGNITUDE OF CORRECTION VECTORSIF(DABS(UU/AL PH).GT.VCORM) UU=DSIGN(VCORM ,UU)*ALPH IF(DABS(VV/BETA).GT.VCORM) VV=DSIGN(VCORM ,VV)3<BFTA

GAUS5510 GAUS5520 GAUS5530 GAUS5540 GAUS5550 GAUS5560 GAUS5570 GAUS5580 GAUS5590 GAUS5600 GAUS5610 GAUS5620 GAUS5630 GAUS5640 GAUSS 65 0 GAUS5660 GAUS5670 GAUS56B0 GAUS5690 GAUS5700 GAUS5710 GAUS5720 GAUS5730 GAUS5740 GAUS575 0 GAUS 5760 GAUS5770 GAUS57B0 GAUS5790 GAUS5800 GAUSS 810 GAUS5820 GAUS5830 GAUS5840 GAUS5850 GAUS5860 GAUS5870 GAUS5880 GAUS5890 GAUS5900 GAUS5910 GAUS5920 GAUS5930 GAUS5940 GAUS5950 GAUS5960 GAUS5970 GAUSS980 GAUS5990 GAUS6000

96

OFCK GAUSIF(DABS<WW/AM).GT.VCORM ) WW=DSIGN(VCORM ,WW)«AM ALPH=UU-*-AL PH BE TA=BFT A+VV A M=A M+ WW

C CALCULATE NEW VALUES OF CHI-SQUAREKNUM=KNUM4-1 IF{KNUM.GF c1) IPNT=2 GO TO 51 FNO

GAUS5010GAUS6020GAUS6030GAUS60A0GAUS6050GAUS6060GAUS6070GAUS60R0GAUS6090

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DECK ERF I: PRINT DATA ERFI0510

WRITE (6,2011 (HEADJ J) ,J=1,19) ERf10520201 FORMAT(1H1//10X,19A4//10X,35HCUMULATIVE SIZE-WEIGHT PERCENT 0ATA/ERFI0530

1/10 X,46HDAT A IS PERCENT SMALLER THAN GIVEN SCREEN SIZE//11X,10HL0GERFI0540

2020.2

55

60

75

203

204

2(SCREEN,5X,10HWT PERCENT/10X,11HSIZEIMILS)I//J DO 20 1=1,NM F2=100.0*DD(I)WRITE (6,202) DX(I ),F2 CONTINUEFORMAT(10X,1P2E14.6)DO 55 1=1,NN IF (YYm.NE.O.O) IU=IU+1 CONTINUE 00 60 1= IL , IU SUMX= SUMX+YY(I)SUMXX= SUMXX+YY(I)**2 SUMY = SUMY + DX(I)SUMXY=SUMXY+DX(I)*YY( I )CONTINUEAAN= FLOAT(IU-IL + 1)DE N=AAN*SUMXX -SUMX**2 CA=(SUMXX * SUMY - SUMX * SUMXY)/DEN CB=(AAN * SUMXY - SUMX * SUMY)/DEN WRITE CALCULATED NUMBERS WRITE (6,204)DO 75 1=1,NNWRITE (6,202) DX(I),YY(I)CONTINUEWR ITF (6,703 ! CA,CB FORMAT(1H0,10X,19HGAUSSIAN 1PE12.6)FORMATdHO, 10X,45HCALCUL ATED VALUES

1 10X,9HL0G(SIZE),7X,3HARGUMENT//)S = CBAL pH=l00.0*S0PI/S BETA=1.0/(2.0TS*S)X1=AINT(YY(IL)-1.0)X2=AINT(YY(IU)+1.0)Z1=C A +C3*X1 Z2=C.A +CB * X2 ZM IN = AINTS Z1-!.0)DZ=(AINT(Z2*1.0-ZMIN))/B,0 DDX=(X2-X11/10.0 CALL PLOT (1.0,0.6,-3)CALL AXTStO.0,0.0,22HYY = NORMAL CALL AXIS(0.0,0.0,33HXX

1 90.0,ZMIN,DZ,3?Z1=(Z1-ZMIN)/DZ Z2=(Z2-ZMIN)/DZ CALL PLOT (0. ,Z1,3)

ERE 10550 E RE 10560 ERF 105 70 ERF 10580 ERFI0590 ERF 10600 ERF 10610 ERF 10620 ERF 10630 ERF 10640 ERF 10650 FRF 10660 ERFI0670 ERFI0680 ERF 10690 ERF 10700 ERF10710 ERF 10720 ERF 10730 ERF 10740 ERF 10750 ERFI0760 ERF 10770 FRF 10780 ERF 10790 ERF 10800 ERF I 0810

ARGUMENT//ERF 10820 ERF 10830 ERFI0840 FRF I 0850 ERF 10860 ERFI0870 ERF 10880 ERF I 0890 ER F 10900 ERFI0910 ER FI 0920 FRF I 0930 FRF I 0940

COORD!NATF,2 2,10.0,0.0,XI,DDX,3 ) ERF I 0950

PARAMETERS//20X,4HM = 1PF12.6, 6H S =OF NORMAL INTEGRAL

= LOG(SCREEN MFSH SIZE (MILS) 1,33,8.0, ERE 10960 ERF I 0970 ERF I 0980 ERF 10990 ERFI1000

99

DECK ERFICALL PLOT (10.0,Z2,2) ERF 11010DO 80 T= IL , IU ERF 11020X3=(TV(IS-XlS/DDX ERFI1030Z3=( DX( n-ZMINJ/DZ ERF 11040, CALL SYMBLMX3,Z3,0. 21,SCJ, 0.0,1) ERF 1105080 COMTINUE ERFT1040CALL PLOTS(12.0,-11.0,-3) ERF 11070CALL SYMBL4(0.2,9.7,0.21,HEAO,0.0,76S ERFI1080C SCALP YY VALUES FOR PROBABILITY PLOT ERFI1090DO PC 1=1,NN ERFI1100YY S f ) = YY(IJ+4.0 ERF Ii11090 CONTINUE ERF 11120RETURN ERF 11130END ERFI1140

100

o u o

DECK ADGflSUBROUTINE ADGAS(AtB,ALPHtBETAfAM,ANS.K)IMPLICIT REALMS (A-H,0-Zl

GAUSSIAN INTEGRATION SUBROUTINE

DIMENSION A %{6411 ZI64)DATA AA/O.049690057,0.048575467,0.048344762,0.047999389,

1 0.047540166,0.046968183,0.046284797,0.045491628,0.044590558,2 0.0435 8372 5, 0.042473515,0.041262563,0.039953741,0. 03 8550153,3 0.037055129,0.035472213,0.033905162,0.032057928,0.030234657,4 0.028339673,0.026377470,0.024352703,0.022270174,0.020134823,5 0. 017951716,0.015726030,0.013463048,0.011168139,0. 0088467598,6 0. 00650445 8D + 00,0.004147 0333,0.0017832807,32*0.0/,Z/0.024350293,7 0.072993122,0.12146282,0.16964442,0.21742364,0.26468716,8 0. 31132287,0.35722 016,0.40227016,0.44636 602,0.48940315,9 0. 5312 7946,0.57189565,0.611155 36,0.64896 547,0.68523631 ,A 0.71988185,0.75281991,0.78397236,0.81326532,0.840629300+00,B 0.865999400+00,0.88931545,0.91052214,0.92956917,0.94641137,C 0.9%>1 008800+00,0.97332683,0.98333625,0.9 9101337,0,99634012,0 0.99930504,32*0.0/,TSET/1/

IF(I SET.EO.O ) GO TO 20 IS ET=0DO 10 1=1,32 Z( l+321 = -zm AA(1+32)= AA(I)

10 continue20 DIF=(R-A)*0.50+0

SUM=(B+A)*0.50+0 SUMM=0.00+0 00 101 1=1 ,64 SX=DIF*Z(t ) + SUMIF(K.NE.51 TP=OEXP(-BETA«{SX-AM)**2)GO TO (25,80,35,40,45,50,55), K

25 FCN=AIPH*TP GO TO 70

30 FCN=TP GO TO 70

35 FCN=TP*( SX-A*M**2 GO TO 70

40 FCN=TP*(SX-AM)GO TO 70

45 ECN=DEXP(-SX1*SX**(ALPH-1.00+0)GO TO 70

50 FCN=TP*(SX-AMt**7 GO TO 70

55 FCN=TP*(SX-AM)**4 70 SUMM=SUMM + AA(I)*ECN

101 CONTINUEANS=DIF*SUMM RE TURN END

A0GA0010 A0GA0020 A0GA0030 A0GA0040 A0GAQ050 ADGA0060 A0GA0070 A0GA0080 ADGA0090 AOGAOIOO ADGA0110 A0GA0120 ADGA0130 ADGA0140 ADGA0150 A0GA0160 A0GA017 0 ADGA0180 A0GA0190 A0GA0200 A0GA0210 A0GA0220 A0GA0230 A0GA0240 ADGA0250 ADGA0260 A0GA0270 ADGA0280 A0GA0290 ADGA0300 ADGA0310 ADGA0320 ADGA0330 ADGA0340 ADG40350 ADGA0360 ADGA0370 A0GA0380 ADGA0390 ADGA0400 ADGA0410 ADGA0420 ADGA0430 ADGA0440 A0GA0450 A0GA0460 ADGA0470 A0GA0480 ADGA0490 ADGA0500 ADGA0510 ADGA0520

101

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DECK AXISAXIS0010 AXIS0020 AXTS003D AXIS0040 AXIS0050 AXIS0060 AXIS0070 AXISOORO AXISOOPOAXISOlOnAX I SOI10

DATA WC/O. 01,0.05,0.1,0.2t0.5,1.Ot2.Ot5.0,10.0,20.0,30.0,4 0.0, AXISO120 1 50,0,60. 0, 70.0,30. 0,00.0,0 5.0,98.0,90.0,99.5,99. 9,9<?. 99,0.0,0.0/ AXIS01302 ,PX/C,275,0,7,0.90,1.12,1.42,1.672,1.946,2.355,2.718,3.158, AXIS01403 3.478,3.754,4,00,4.246,4.522,4.842,5.282,5.645,6.054,6.328, AXIS01504 6.576,7.10,7.725,0.0,0,0/ AXIS0160

AXIS0170CHECK IF NORMAL INTEGRAL HAS BEEN CALCULATED AXIS0180IF(EII(851).EQ.-l.OD-O 1 GO TO 52 AXIS0190DD(1)=DWT{1)/1.0D+2 AXIS0200DO 99 1=2,NN AXIS0210no ? i) =oo? i-i n-nwT ? n/i .on+2 AXIS0220

99 CONTINUE AXIS0230IF?EII(851).FQ.-7.00-0 ) GO TO 15 AXIS0240DX 1=1.00-2 AXIS0250G=l.OD+O/nsoRT? 2.0D+0) AXIS0260SOPI=1.OD+O/DSORT(2.00+0*3.14159265) AXIS0270CALCULATE NORMAL INTEGRAL AXIS0280XXN( n=-4, 25D+0 AXIS029000 10 1=1,850 AXIS0300Z= DABS(XXN(I))*G AXIS0310ElKI)=5•0 D—1 + OS IGN(5.0D-1,XXN(I))*DERF(7) AXIS0320XXN?I+1)=XXN(I) + DXI AXIS0330

10 CONTINUE AXIS034015 ElK851)=~1.OD-O AXIS0350

DETERMINE VALUES OF ARGUMENT OF PROBABILITY FUNCTION (YY) AXIS0360SUCH THAT WEIGHT PERCENT AT XXN = El I(YY) AND SCALE VALUES AXIS0370

20 DO 25 1=1,NN AXIS0380YY(I)=8.50+0 AXIS0390

25 CONTINUE AXISO400IL =1 AXTS041000 50 1=1,NN AXIS0420AA=DD(I! AXIS0430IF(AA.GT.0.00+0) GO TO 42 AXIS0440YY (I) = -4.2 5+ 4.0 AXIS0450IL=IL+1 AXIS0460GO TO 50 AXIS0470

42 DO 45 L=1,8 50 AXIS0480IE(AA-EII(L)) 48,47,45 AXIS0490

47 YY(I) = XXN(t' + 4.0 AXIS0500

SUBROUTINE PAXIS(A,B,XSIZE,YSIZE,ALPH,BETA,AM,XX,FX,IK)THIS SUBPOUT INF CONVERTS WEIGHT PERCENT INTO PROBABILITY COORDINATPS AND PLOTS THE PROBABILITY AXIS

COMMON DLSS, DWT, HEAD, NN, XXN, Eli, YY, DD, SOP I, IL REAL*8 DL SS(30)» DWT? 30 S ,ALPH, BETA, AM P E AL*R XXN, F11 , SOPI , DX I, G, Z, AA, CC , BB DIMENSION HE AD ?19!, XXN? 851 ) , Eli<851) , YY130), 00(301,

X XXI401), EX(401) , b X(25 ) , WE<25)

102

DECK AXISGO TO 50

48 K=L-lCC* -XXN(K) ^XXN(K)*0.5n+0 BB=DEXP(CC)*SQPIYY(I)=XXN(K) * (AA-EIIIK))/B8 + 4.0 GO TO 50

45 CONTINUE 50 CONTINUE 5? IK=1

DO 60 1=1,400 AA=EX(I S/100.0 IF(AA.GT.5.0D-4) GO TO 55 FX (I )=-4«25 + 4.O'IK=IK+1 GO TO 60

55 DO 59 1=1,850IF (AA-EI I (U S 58,57,59

57 FX <1)=XXN< LJ + 4.00+0GO TO 60

58 K=L-1CC=-XXN(K)=FXXN(KS ♦0.50+0 BS=DEXP(CCS*SQPIFX(I ) = XXN(K) + (AA-EIIIKS)/BB +4.0D+0 GO TO 60

59 CONTINUE60 CONTINUE

C DRAW PROBABILITY AXISCALL PLOT I A,8,3!X=A+XSIZE CALC PLOT(X,8,21 P=B+0. 1 00 65 1=1,23 K=23-1+1 C=PX(KJCALL PL0T(C,n,3>CALL PLOT(C,0,2)X* C-0.05 Y=B-0.25 N= -1IF(K.GT.5,AND.K.LE.20) GO TO 64 N=2X= C-0.125

64 CALL NUMBER(X,Y,0.10,WE(K),0.0,N)65 CONTINUE

Y= B-0.5 X= 2. 5CALL SYMBL 4(X,Y,0.14,2 5HCUMULATIVE WEIGHT PEPCENT,0.0,25» X = AY=8+YSIZE CALL PLOT(X,Y,3)

AXIS0510 AXIS0520 AXIS0530 AXIS0540 AXTS0550 AXIS0560 AXIS0570 AXIS0580 AXIS0590 AXIS0600 AXIS0610 AX1S0620 AXIS0630 AXIS0640 AXIS0650 AXIS0660 AXIS0670 AXIS0680 AXIS0690 AXIS0700 AXIS0710 AXIS0720 AXIS0730 AXIS0740 AXIS0750 AXIS0760 AXIS0770 AXIS0780 AXIS0790 AXIS0800 AXIS0810 AXIS0820 AXIS0830 AXIS0840 AXIS0850 AXIS0860 AXIS0870 AXIS0880 AXIS0890 AXIS0900 AXIS0910 AXIS0920 AXIS0930 AXIS0940 AXIS0950 AXIS0960 AXIS0970 AXIS0980 AXIS0990 AXISI 000

103

DFCK 4X1SX=X+XSTZF AXIS1010CALL PLnT(X, Y, ?) AXIS1020D=Y-P.l AXIS1030DO 70 1 = 1,?3 AXISI040K= 23-T * 1 AX1S1050C=PX(K» AXIS1060CALL PLOT!C,Y,31 AXIS1070CALL PLOT(C,0,2) AXIS1080CONTINUE AXIS1090RETUPN AX I SI100END AX I SI110

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DfCK COMPSUBROUTINE COMPVI ALPH, BETA,AM,XV,UU,VV,WW,UUP,VVP ,WWP,GDIV,

1 KNUM,01V)THIS SUBROUTINE CALCULATES THE CORRECTIONS TO A NONLINEAR FITTING FUNCTION BY VARYING PARTIAL DERIVATIVESIMPLICIT RFAL*8 (A-H,0-Z)COMMON DLSS, DWT, HEAD, NN, XXN, Eli, YY, DD, SOPI, IL DIMENSION DL SSI 30 5 , DWTI30), HEADI19), XXNI851), FII(851),

1 YY(30), DDI30), CWT(30), R(3,3), B(3)LOGICAL GDIV, DIV REAL*A HEAD, YY, DD

CD! V=•TRUE.

C CALCULATE THE PARTIAL DERIVATIVESDO 10 1=1,3 Bt I ) = 0.0D«-0 DO 10 J=1,3 R(I,J»=0.0D*0

10 CONTINUE CONS=UU AL = XYDO 20 1=1,NN AU=DLSS(I)CALL ADGAS(AL,AU,ALPH,BETA,AM,ANS,1!CWT{I I=ANSCALL AOGASIAL,AU,ALPH,BETA,AM,ANS,2)PA RTDO = ANSCALL ADGAS(AL,AU,ALPH,BETA,AM, ANS,4)PARTD1=ANSCALL ADGASCAL,AU,ALPH,BETA,AM,ANS,3)PARTD2=ANSCALL AOGASIAL,AU,ALPH,BETA,AM,ANS,61 PARTP3=ANSCALL ADGAS!AL,AU,ALPH,B^TA,AM,ANS,7)PARTD4=ANS RATI 0=DWTI I)/CWT{I)TEMP=(1.OD+D-PATTO)*(1.OD+O+RATIO)BI I(= B 11) + PARTDO*TEMPBI 2) = B(2) -PARTD2*ALPH*TFMPB( 3) = B( 3 )+2o0D+0’!'ALPH*BFTA*PAPT01*TEMPP(l,n = R (1 ,1M- 2.0D4-O*(RATIO*PARTD0)**2/CWT( I )R( 1,2)=R ( 1,2 )-(1.0D+0-RATI0**2 )*PARTD2 - ALPH*2,00•»-0*RATI0**2*

1 PARTDO*PARTD?/CWT(I IR11,3)=P(1,3)+(1,0D*0-RATI0**2)*2.0D*0*BETA*PAPT01 * 4,00 + 0*

1 ALPH*BETA*PATIO**2*PARTno*PARTDl/CWT(I)PI 2, 2)=RI 2,2)+ (1.0D+0-RATIO**2)*ALPH*PARTD4 + 2,00+0*I RATIO*

I ALPH*PARTD2)**2/CWT(I)RI2,3)=R(2,3)+ (1.OD+O-RATI0**2)*2.00+0*ALPH*(PARTD1 - BETA*

1 PARTD3) - 4.OD + 0*BETA*(ALPH*RATIO)**2*PA RTD1*PAR TD2/CWT 11 )

C0MP0010 COMP0020 C0MP0030 C 0MP0040 C0MP0050 COMP0060 COMP0070 C0MP0080 CDMP0090 C0MP0100 COMP011O COMP0120 C0MP0130 C0MP0140 C 0MP0150 C0MP0160 C0MP0170 COMP0180 C0MP0190 C0MP0200 C0MP0210 C0MP0220 C0MP0230 C0MP0240 C0MP0250 C0MP0260 C0MP0270 C0MP0280 C0MO02P0 C0MP0300 C0MP0310 C0MP0320 C0MP0330 C0MP0340 COMP0330 C0MP0360 COMP0370 C0MP0380 C0MP0390 C0MP0400 C0MP0410 C0MP0420 C0MP0430 C0MP0440 COMP0450 C0MP046O C0MP0470 C OMP0480 C0MP04P0 COMPOSOO

105

DECK COMPR< 3,3)=R(3,3) +{1.OD+O-RAT 10**2)*2.00+0*ALPHABETA*(2.OD+O^BFTA* COMPOS 10

1 PARTD2 - PAR TOO) + 2.OD+O*(RATI 0*2.0 0+0*ALPH*BETA*PART01)**2/ C0MP05202 CWTU ) C0MP053O

20 CONTINUE C0MP0540R(2,1)=R(1,2) COMPO550R(3,1)=R(1,3) C0MP0560R(3,2)=R(2,3) C0MP0570INVERT THE MATRIX C0MP0580CALL MAIND(P,3,1,3,B,1 ,0ET, IFS ) C 0MP0590CHECK FOR SINGULAR MATRIX COMP0600IF(IES.NE.O) GO TO 25 C0MP0610WRITE (6,201) C0MP0620

201 FORMATdHO, Til,'PARTIAL DERIVATIVE MATRIX IS SINGULAR - PROBLEM TCOMP0630lERMINATED*) C0MP0640

GO I V=. TP UE. C0MP0650RETURN C0MP066025 UU=-8(1! C0MP06T0VV =-B(2) CQMP0680

WW =-B(3) C0MP0690UUP=UU C0MP0700VVP=VV COMP0710WWP=WW COMP0720IF(DABS(UU/ALPH).GT.CONS! UU=OSIGN(CONS,UU)*ALPH C0MP0730IF(DABS(VV/BETA).GT.CONS) VV-OSIGNtCONS,VV)*BETA COMP0740IF(DABS(WW/AM ).GT.CONS ) WW=DSIGNICONS,WW)*AM COMP0750AL PH = ALPH+UU C0MP0760BETA = BET A+VV C0MP07T0AM=AM+WW C0MP0780KNUM=KNUM+1 COMP07R0IF(KNUM.GE.100) GDIV=.TRUE. C0MP0800RETURN C0MP081 0ENO COMP0820

106

1

1

APPENDIX DPARTICLE DISTRIBUTION FROM FIVE-ROD CLUSTER

107

APPENDIX DPARTICLE DISTRIBUTION FROM FIVE-ROD CLUSTER

The particle data from a destructive test in the CDC involving a five-rod cluster of SPXM rods [a3 became available after this report was written. These data were considered of sufficient importance to warrant their inclusion here not only because they are the only particle data available from rod clusters but also because they represent a composite ensemble from rods receiving different energy depositions.

The cluster was of an open lattice type and had a pitch (center-to-center spacing) of 0.328 inch and a ratio of pitch-to-diameter of approximately 1.3. The center rod in the cluster received an energy deposition of 315 cal/g of UO2 and each of the four outer rods received 383 cal/g. All five rods failed during the test and a sieve analysis was made of the debris. The data from this experiment are presented in the same format as the data in Appendix A except that a photograph of particle ensemble is not given. The goodness-of-fit probability for this distribution was 99.9% and the volume-to-surface mean diameter 25.2 mils.

As discussed in Section IV of this report, the rod design parameters were of sufficient importance in determining the details of the particle distributions that only those ensembles from rods of similar design followed any consistent pattern. An interesting hypothesis, and one that would be quite important if it could be proven, is that the effects on the resultant particle ensembles attributable to each of the design parameters are separable. Ensembles could then be constructed which represented the composite interaction of all parameters. The excellent fit to the log-normal distribution obtained for this ensemble from a five- rod cluster gives support to this hypothesis. Considerably more analysis and probably more data will be required to establish such a hypothesis.

[a] L. J. Siefken, The Response of Fuel Rod Clusters to Power Bursts, IN-ITR- 116 (May 1970).

109

TABLE D-I

ANALYSIS OF FIVE-ROD CLUSTER

Particle Size Distribution Obtained fromFive-Rod Cluster of SPXM Rods


PAN < 1.7 0.442 0.18 0.18325 1.7 - 3.5 2.446 1.02 1.20170 3.5 - 7.0 6.178 2.56 3.7680 7.0 - 13.9 10.597 4.41 8.1742 13.9 - 27.8 17.586 7.30 15.4724 27.8 - 55.5 29.601 12.30 27.7712 55.5 - 111 35.846 14.90 42.677 111 - 223 33.034 13.72 56.393-1/2 223 - 312 14.916 6.20 62.592-1/2 > 312 27.767 11.53 74.12

Cladding Zircaloy-2Active Fuel Length (in.) 5Test Number 710Period (msec) 3.04Energy (cal/g of UOo)

center rod 315outer rods 383

Initial Cluster Weight (g) 240.5Gaussian Distribution Parametersmean 4.6605variance 2.4169degrees-of-freedom 6

Normalized Data


1.7 0.2226 0.3945 0.0751.7 - 3.5 1.2321 1.0242 0.0423.5 - 7.0 3.1120 2.6204 0.0927.0 - 13.9 5.3289 5.5573 0.009

13.9 - 27.8 8.8585 9.9199 0.11427.8 - 55.5 14.9107 14.4157 0.01755.5 - 111 18.0564 17.3261 0.031111 - 223 16.6370 17.1909 0.018223 - 312 7.3271 7.2293 0.001

75.6843 75.6843 0.399

110

KNUM . 0 08/24/70MOST PROBABLE PARTICLE SIZE = 106. MILS

FIG. D-1 FREQUENCY DISTRIBUTION OF PARTICLES FROM FIVE-ROD CLUSTER.

5 ROD CLUSTER

10'1 10° ID1 10* 10s io'

PARTICLE SIZE (MILS!

Ill

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mm- IDAHO NUCLEAR CORPC^^TION 1 S. ATOMIC ENERGY