PARTICLE SIZE DISTRIBUTIONS FROM FUEL RODS FRAGMENTED DURING POWER BURST TESTS IN THE CAPSULE DRIVER CORE J. A. McClure slliils m mmi mm mmm mm- s§Sa»s wBm m s':'50SPaR^ |^fc 'lllfe IDAHO NUCLEAR CORPC^^TION NATIONAL REACTOR TESTING STATION IDAHO FALLS, IDAHO Date Published— October 1970 1 S. ATOMIC ENERGY COMMISSION
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PARTICLE SIZE DISTRIBUTIONS FROM FUEL RODS FRAGMENTED DURING POWER BURST TESTS
IN THE CAPSULE DRIVER CORE
J. A. McClure
slliils
mmmi
mmmmmmm-s§Sa»s
wBmm
s':'50SPaR^
|^fc
'lllfe
IDAHO NUCLEAR CORPC^^TIONNATIONAL REACTOR TESTING STATION
IDAHO FALLS, IDAHO
Date Published— October 1970
1 S. ATOMIC ENERGY COMMISSION
Printed in the United States of America Available from
61,eayitei|3®HSe for Federal Scientific and Technical Information Sft&jMnal Bureau of Standards, 0. S, Department of Commerce
Springfield, Virginia 22151Price: Printed Copy $3*00; Microfiche $0.65
——---- -— LEGAL NOTICE------------——tills report was prepared as an account of Government sponsored work. Neither the United
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IN-1428 Reactor TechnologyTID-4500
-------------------- LEGAL NOTICE----------------------This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights’
PARTICLE SIZE DISTRIBUTIONS FROM FUEL RODS FRAGMENTED DURING POWER BURST TESTS
IN THE CAPSULE DRIVER CORE
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
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-14. Frequency distribution of particles from rod SPXM-C-505 ............... 35
A-15. Particle size distribution obtained from rod SPXM-C-503irradiated at 613 cal/g............................................................................... .. 36
A-16. Frequency distribution of particles from rod SPXM-C-503 ............... 37
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-20. Frequency distribution of particles from rod SPXM-C-554 ............... 41
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-24. Frequency distribution of particles from rod SPXMG-3............... 45
A-25. Particle size distribution obtained from rod SPXMG-2 irradiated at 450 cal/g............................................... .. .............. .. ............................. 46
A-26. Frequency distribution of particles from rod SPXMG-2............... 47
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-38. Frequency distribution of particles from rod SPX-C-024 ............ 59
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-42. Frequency distribution of particles from rod SPX-C-047 ............ 63
A-43. Particle size distribution obtained from rod SPX-C-048 irradiated at 569 cal/g . ............................. ...................................... .. 64
A-44. Frequency distribution of particles from rod SPX-C-048 ............ 65
A-45. Particle size distribution obtained from rod SPX-C-052 irradiated at 388 cal/g............................................................................................. 66
A-46. Frequency distribution of particles from rod SPX-C-052 ............ 67
A-47. Particle size distribution obtained from rod SPX-C-053 irradiated at 498 cal/g ............................................................................................. 68
A-48. Frequency distribution of particles from rod SPX-C-053 ............ 69
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
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
PARTICLE SIZE DISTRIBUTIONS FROM FUEL RODS FRAGMENTED DURING POWER BURST TESTS
IN THE CAPSULE DRIVER CORE
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
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.
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.
%
11
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,
[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.
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.
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
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
Volume-to-Surfacemean diameter (mils) 7.5
MOST PROBRBLE PARTICLE SIZE = 27. MILS
FIG. A-6 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD GEX—SL—3.
GEX-SL-3
PARTICLE SIZE (MILS)
Normalized Data
Size Range (mils)
Percent of Particle WeightMeasured Calculated Chi-Square
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
Volume-to-Surfacemean diameter (mils)
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
PARTICLE SIZE (MILS)
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square< 1.7 0.0431 0.0487 0.001
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
Volume-to-Surfacemean diameter (mils)
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)
Percent of Particle WeightMeasured Calculated Chi-Square
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
Volume-to-Surfacemean diameter (mils) 40.0
FIG. A—12 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM —C—504.
MOST PROBRBLE PARTICLE SIZE = 573. MILS
SPM-50H
PRRTICLE SIZE (MILS]
Normalized Data
Size Range Percent of Particle Weight(mils) Measured< 1.7 0.0608
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
Volume-to-Surfacemean diameter (mils) 14.6
MOST PROBABLE PARTICLE SIZE = 31. MILS
FIG. A—14 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXM—C—505.
SPM-505
2° -
PARTICLE SIZE (MILS)
Normalized Data
Size Range (mils)
Percent of Particle WeightMeasured Calculated Chi-Square
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
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 4
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
PARTICLE SIZE (MILS)
Normalized Data
Size Range (mils)
Percent of Particle WeightMeasured Calculated Chi-Square
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
Volume-to-Surfacemean diameter (mils)
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
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
Volume-to-Surfacemean diameter (mils) 10.5
MOST PROBABLE PARTICLE SIZE = 17. MILS
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-
Fuel Form PelletCladding GlassActive Length (in.) 4.5Test Number 526Period (msec) 3.84Energy (cal/g of UO2) 450Gaussian Distribution Parametersmean 3.2288variance 0.89964degrees-of-freedom 3
Volume-to-surfacemean diameter (mils) 13.3
MOST PROBABLE PARTICLE SIZE = 25. MILS
FIG. A—26 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXMG—2.
SPXMG-2
PARTICLE SIZE (MILS1
Normalized Data
Size Range Percent of Particle Weight(mils) Measured
Fuel Form PelletCladding GlassActive Length (in.) 4.5Test Number 533Period (msec) 3.42Energy (cal/g of UO2) 510Gaussian Distribution Parametersmean 2.7240variance 0.90752degrees-of-freedom 3
Volume-to-Surfacemean diameter 8.9
MOST PROBABLE PARTICLE SIZE = 15. MILS
FIG. A—28 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXMG—6.
SPXMG-6
10 10 PARTICLE SIZE (MILS!
Normalized Data
Size Range (mils)
Percent of Particle WeightMeasured Calculated Chi-Square
Fuel Form PelletCladding GlassActive Length (in.) 4.5Test Number 517Period (msec) 3.11Energy (cal/g of UO2) 550Gaussian Distribution Parametersmean 2.1278variance 0.88534degrees-of-freedom 3
Volume-to-Surfacemean diameter (mils) 5.3
MOST PROBABLE PARTICLE SIZE = 8. MILS
FIG. A—30 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPXMG-1.
SPXMG-1
PARTICLE SIZE (MILS)
Normalized Data
Size Range (mils)
Percent of Particle WeightMeasured Calculated Chi-Square
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
Volume-to-Surfacemean diameter (mils) 3.6
MOST PROBABLE PARTICLE SIZE ■ 6. MILS
FIG. A—32 FREQUENCY DISTRIBUTION OF PARTICLES FROM BARE PELLETS IN ALUMINA CRUCIBLE.
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
Volume-to-Surfacemean diameter (mils)
Pellet Stainless Steel
18 326 4.6 370
153.936.52153.2131
549.8
MOST PROBABLE PARTICLE SIZE = 680. MILS
FIG. A—34 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C-022.
SPX-C-022
PARTICLE SIZE (MILS)
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squi
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.
PARTICLE SIZE (MILS)
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square
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
Volume-to-Surfacemean diameter (mils) 39.4
MOST PROBRBLE PARTICLE SIZE = 1291.MILS
FIG, A—40 FREQUENCY DISTRIBUTION OF FARTICL.es FROM ROD SPX-C-046.
PARTICLE SIZE (MILS)
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square
Fuel Form PelletCladding Zircaloy-2Active Length (in.) 18Test Number 438Period (msec) 4.1Energy (cal/g of UO2) 469Initial Rod Weight (g) 147.85Gaussian Distribution Parametersmean 4.6050variance 1.3143degrees-of-freedom 5
Volume-to-Surfacemean diameter (mils) 44.4
MOST PROBABLE PfiRTICLE SIZE = 100. MILS
FIG. A—42 FREQUENCY DISTRIBUTION OFPARTICLES FROM ROD SPX-C-047,
I01 to* to5PARTICLE SIZE (MILSI
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Square
Fuel Form PelletCladding Zircaloy-2Active Length (in.) 18Test Number 439Period (msec) 3.4Energy (cal/g of UO2) 569Initial Rod Weight (g) 147.85Gaussian Distribution Parametersmean 2.7766variance 2.2269degrees-of-freedom 4
Volume-to-Surfacemean diameter (mils) 5.1
MOST PROBRBLE PARTICLE SIZE = 16. MILS
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-
Fuel Form PelletCladding Zircaloy-2Active Length (in.) 18Test Number 455Period (msec) 5.1Energy (cal/g of UO2)Initial Rod Weight (g)Gaussian Distribution Parameters
388147.85
mean 7.1251variance 3.6655degrees-of-freedom
Volume-to-Surface3
mean diameter (mils) 53.2
HOST PROBABLE PARTICLE SIZE = 1205.MILS
FIG, A—46 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX—C-052,
Fuel Form PelletCladding Zircaloy-2Active Length (in.) 18Test Number 456Period (msec) 4.0Energy (cal/g of UO2)Initial Rod Weight (g)Gaussian Distribution Parameters
498147.85
mean 4.9051variance 1.6308degrees-of-freedom
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 .
PARTICLE SIZE (MILS)
Normalized Data
Size Range Percent of Particle Weight(mils) Measured Calculated Chi-Squar<
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
Volume-to-Surfacemean diameter (mils) 15.1
MOST PROBRBLE PRRTICLE SIZE = 36. MILS
FIG. A—50 FREQUENCY DISTRIBUTION OF PARTICLES FROM ROD SPX”C—054.
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
Data Type 3 (8F10.6)
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
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)
Data Type 5 (3F10.6)
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.
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 ^OR 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
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
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
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
- 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)
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
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
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
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
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
.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))
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) )
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
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
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
Screen Size Particle Size Weight Percent of Cumulative(meshes/inch) Range (mils) (grams) Rod Weight Percent