Neutronic Evaluation of GCFR Core Diluents and Reflectors
by
Kun Yu
B.E., Engineering Physics,
Tsinghua University, P.R.China (1998)
Submitted to the Department of Nuclear Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Nuclear Engineering
At The
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
© 2003 Massachusetts Institute of Technology. All rights reserved
Signature of Author
Kun Yu
Department of Nuclear Engineering
June 10, 2003
Certified by
Michael J. Driscoll
Professor Emeritus of Nuclear Engineering
Thesis Supervisor
Certified by
Pavel Hejzlar
Program Director, Center for Advanced Nuclear Energy Systems
Thesis Reader
Accepted by
Jeffrey Coderre
Associate Professor of Nuclear Engineering
Chairman, Department Committee on Graduate Students
Neutronic Evaluation of GCFR Core Diluents and Reflectors
By
Kun Yu
Submitted to the Department of Nulcear Engineering
on June 11, 2003 in Partial Fullfillment of the
Requirements for the Degree of Master of Science in
Nuclear Engineering
ABSTRACT
Materials are evaluated for use as in-core diluents and as peripheral reflectors for
Gas-Cooled Fast Reactor (GFR) service, using coupled Monte Carlo (MCNP) and
isotopics (ORIGEN) codes. The principal performance indices compared were effects on
beginning of irradiation multiplication factor, reactivity-lineated burnup, and coolant
(here CO2) void reactivity.
While low values of the macroscopic absorption cross section, Σa, and slowing
down power, ξΣs, are qualitatively useful predictors of good performance, it was found
that only full scope calculations were valid for quantitative assessment. For example,
several materials (Ni, Nb) having poor performance as in-core diluents proved to be good
reflectors. Many materials which reduced coolant void reactivity also proved detrimental
to reactivity lifetime. Others, mostly the strong moderators, increased initial reactivity,
but decreased reactivity lifetime. Cores fueled with plutonium exhibited a much larger
void reactivity than those started up using U-235 as the fissile material.
While there are no ideal candidates that are superior in all respects, considering
only neutronic performance, the following appear worthy of further investgation:
Metallic fuel diluents or matrices (eg. CERMET or METMET): Zr, Ti, V, Ba2Pb;
High temperature fuel diluents or matrices (eg, CERMET, CERCER): SiC, BaS
Cladding: Fe alloys with Cr, Al (eg ODS)
Reflector: Zr3Si2, Pb, Ba2Pb, ZrS2, MoSi2 plus a variety of sulfides and silicides
Thesis Supervisor: Michael J. Driscoll
Title: Professor Emeritus of Nuclear Engineering
i
ACKNOWLEDGMENTS
The help, patient guidance and generous support of Professor Michael J. Driscoll
and Dr Pavel Hejzlar, my thesis supervisors, are greatly appreciated.
I am also grateful to two members of the Physics and Materials group of the Gas
Cooled Fast Reactor Project at MIT: Pete Yarsky for his discovery of the possible cross
section library deficiency of Potassium, and Mike Pope for beneficial discussions on the
coolant void coefficient.
This work has been funded by the Idaho National Engineering & Environmental
Laboratory (INEEL) under their LDRD program.
ii
TABLE OF CONTENTS
ABSTRACT......................................................................................................................... i
ACKNOWLEDGMENTS .................................................................................................. ii
TABLE OF CONTENTS ................................................................................................ iii
LIST OF TABLES ............................................................................................................ v
LIST OF FIGURES ......................................................................................................... vi
Chapter 1 Introduction ........................................................................................................ 1
1.1 Foreword ................................................................................................................... 1
1.2 Background ............................................................................................................... 1
1.3 Organization of this report ........................................................................................ 8
Chapter 2 Computer Codes and Models ........................................................................... 10
2.1 Introduction............................................................................................................. 10
2.2 MCODE Description .............................................................................................. 10
2.2.1 Introduction...................................................................................................... 10
2.2.2 Normalization .................................................................................................. 11
2.2.3 Predictor-Corrector Algorithm......................................................................... 13
2.2.4 Running MCODE ............................................................................................ 14
2.4 Whole Core Model for matrix and reflector configuration..................................... 16
2.5 Summary ................................................................................................................. 26
Chapter 3 Review of core diluent material candidates ..................................................... 27
3.1 Introduction............................................................................................................. 27
3.2 Review of element properties ................................................................................. 27
3.3 Review of material candidates for matrix core ....................................................... 28
3.3.1 Neutronic Evaluation parameters..................................................................... 28
3.3.2 Results for matrix study ................................................................................... 30
3.3.3 Fissile and fertile properties in the energy range of interest............................ 32
3.3.4 Promising materials ......................................................................................... 34
3.4 Applicability of superposition................................................................................. 42
3.4.1 Non-linearity of neutronic effects as a function of matrix concentration........ 42
3.4.2 Neutronic effects for a compound and its constituents.................................... 43
3.4.3 Relation of reactivity to enrichment ................................................................ 45
3.5 Conclusions............................................................................................................. 47
Chapter 4 Review of reflector material candidates........................................................... 48
4.1 Introduction............................................................................................................. 48
4.2 Review of material candidates for reflector............................................................ 48
4.2.1 Albedo calculation ........................................................................................... 48
4.2.2 General Results ................................................................................................ 50
4.2.3 Detailed evaluation and explanation................................................................ 51
4.2.4 Brief summary ................................................................................................. 56
4.3 Parameter Studies.................................................................................................... 56
4.3.1 Reflector thickness requirement ...................................................................... 56
4.3.2 UPuC fuel – UC fuel........................................................................................ 57
4.3.3 keff – albedo...................................................................................................... 59
4.3.4 Full burnup study of several interesting reflector materials ............................ 60
4.4 Conclusions............................................................................................................. 61
Chapter 5 Summary, Conclusions and Recommendations ............................................... 62
iii
5.1 Summary and Conclusions ..................................................................................... 62
5.2 General Evaluation Results..................................................................................... 62
5.3 Recommendations for future work ......................................................................... 65
References......................................................................................................................... 67
Appendix A Estimate of Gas Produced By Sulfur........................................................... 69
Appendix B Relation of reactivity ρ to enrichment x...................................................... 70
Appendix C Sample input files for matrix material study ............................................... 72
iv
LIST OF TABLES
Table 1-1 Periodic table of the chemical elements showing excluded candidates....... 2
Table 1-2 Footnotes to Table 1–1..................................................................................... 3
Table 1-3 Representative Hard-Spectrum σ Values...................................................... 3
Table 1-4 Roster of Potential Diluent Candidates ......................................................... 8
Table 2-1 Matrix test core model parameters .............................................................. 17
Table 2-2 Initial region–homogenized compositions in matrix test core model........ 18
Table 2-3 Reflector test core model parameters .......................................................... 18
Table 2-4 Initial region homogenized compositions in reflector test core model...... 19
Table 2-5 Matrix material cell 1 homogenized composition for whole core model .. 19
Table 2-6 Description of chosen actinides..................................................................... 22
Table 2-7 Description of chosen fission products......................................................... 23
Table 2-8 Description of chosen matrix materials ....................................................... 25
Table 3-1 Results of matrix comparisons...................................................................... 30
Table 4-1 Neutronic Comparisons of GFR Reflectors................................................. 50
Table 5-1 General Evaluation Results........................................................................... 63
v
LIST OF FIGURES
Figure 2-1 Flow diagram for MCODE.......................................................................... 15
Figure 2-2 Original fuel assembly and core layout of the MFGR-GT [6] ................. 16
Figure 2-3 Final homogenized cylindrical core layout ................................................ 17
Figure 3-1 Definition of B1 ............................................................................................. 29
Figure 3-2 Examples of error of linear extrapolation method.................................... 29
Figure 3-3 Relation of initial multiplication factor and burnup potential ................ 31
Figure 3-4 Relation of multiplication factor and macroscopic absorption................ 32
Figure 3-5 U235 capture, fission and elastic scattering cross sections * ...................... 33
Figure 3-6 U238 fission, elastic scatter, absorption cross sections ............................... 34
Figure 3-7 Map of diluent material performance ........................................................ 35
Figure 3-8 Capture and elastic scattering cross sections for minor Ba isotopes....... 36
Figure 3-9 Comparison of BaS and BaO diluent core spectra.................................... 41
Figure 3-10 Non-linearity of neutronic effects vs. Pb matrix concentration ............. 42
Figure 3-11 ρ vs. compound components...................................................................... 44
Figure 3-12 Relationship between enrichment and keff for a representative core .... 46
Figure 4-1 Variation of albedo with absorption........................................................... 49
Figure 4-2 Map of reflector material performance ..................................................... 52
Figure 4-3 keff versus nickel reflector thickness ........................................................... 57
Figure 4-4 keff (UC fuel) – keff (UPuC) fuel ................................................................... 58
Figure 4-5 Comparison of ∆keff(void) for UPuC fuel and UC fuel ............................. 58
Figure 4-6 The ∆k(void) increase with burnup ............................................................ 59
Figure 4-7 Relationship of multiplication factor and albedo ...................................... 60
Figure 4-8 Full burnup runs for different reflectors ................................................... 61
vi
Chapter 1 Introduction
1.1 Foreword
Gas cooled fast reactors have attracted new interest in the past several years both
within the US and internationally. It is widely recognized, however, that passive postLOCA decay heat removal is a challenge for reactors of this type. One approach to
amelioration is to increase the heat capacity of the fuel assemblies, and thereby store
energy until decay heat power levels decrease sufficiently (approximately as
1/(time)0.3 ) to facilitate energy removal via some combination of convection,
conduction and radiation. This leads to consideration of fuel diluents in the form of
alloys or ceramics in either homogeneous or dispersion form. The latter can be allmetallic (METMET) ceramic (CERCER) or a combination (CERMET). It was the
objective of the work reported here to evaluate various candidate materials primarily
in terms of their effect on core neutronics, as a guide to future studies of specific core
designs. Most of these same considerations apply to the selection of reflector
materials, which are also essential to good neutronic performance. There are,
however, some differences in performance for this application which motivated a
separate set of comparisons.
1.2 Background
If one starts with the full periodic table of the elements (see Table 1.1) and all
possible combinations as chemical compounds, the task faced in any comprehensive
evaluation would be truly daunting. Fortunately preliminary screening according to a
few simple criteria greatly reduces the list of potential candidates; specifically we
exclude at the outset:
•
All inert gases (e.g. He, Ar etc…)
•
Candidates costing more than 200$/kg
•
Heavy nuclei above 220 AMU (which are either unstable or fissionable)
1
•
Species having spectrum-average microscopic absorption cross sections
greater than about 200 mbarn
•
Excessively strong moderators such as H
•
Radioactive materials such as Ra, Po, etc.
Other important criteria such as thermal conductivity, heat capacity, melting point
and corrosion resistance were not explicitly applied at this point, but must be in any
final downselection. In addition, some elements rejected because of their high σa will
find use as control absorbers, for example B and Ta.
Table 1-1 Periodic table of the chemical elements showing excluded candidates
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sr
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc*
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te*
I
Xe
Cs
Ba
Lu
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po*
At*
Rn
Fr*
Ra*
Ac*
Rf
Db
Sg
Bh
Hs
Mt
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Ac
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Key to rating:
1
2
3
4
5
6*
7
1. Strong moderator.
(Atomic weight < 5)
(1)
2. Actinides/fissionable. (Atomic number > = 90)
(20)
3. Expensive or rare.
(Price > 200$/kg)
(30)
4. Inert gas.
(6)
(13)
5. Strong absorber.
(σc > 200 millibarns)
6. Radioactive.
(7)
7. Potentially usable matrix material.
(32)
──────────────────────────────────────────────────
Total
109
2
Table 1-2 Footnotes to Table 1–1
(a) Some elements have more than one reason for exclusion. We assign only one, based
on its most serious shortcoming.
(b) Some elements could only be used in compounds, such as N, O, F, Cl, Br, I.
(c) Metal prices were obtained from ref. [1].
(d) Li-7 and Be are light moderators. But FLiBe molten salt is used as coolant in some
recent concepts (see ref [2]). Beryllium also has a relatively large (n, 2n) cross section,
which improves the neutron economy, so we re-instate these two materials as candidates.
(e) The one group spectrum averaged neutron absorption cross sections of 90 elements
were obtained using the Pb matrix core model discussed in Chapter 2. The results are
shown in Table 1.3. The results are generally consistent with the central worth
measurements in fast critical assemblies compiled in ref [3].
(f) Some strong absorbers could not be used as matrix material but could be used as a
reflector. Thus there are 45 potentially usable reflector elements, 13 more than as matrix
candidates (Li, B, As, Se, Br, Ag, Cd, In, Sb, I, Cs, Ta, W).
Table 1-3 Representative Hard-Spectrum σ Values
ZAID
1001.60c
1002.60c
2003.60c
2004.60c
3006.60c
3007.60c
4009.60c
5010.60c
5011.60c
6000.60c
6012.50c
Nuclei
H1
H2
H(nat)*
He3
He4
He(nat)
Li6
Li7
Li(nat)
Be9
B10
B11
B(nat)
C
C12
Abundance
Atom
fraction
0.999885
0.000115
1.37E-06
0.999999
0.0759
0.9241
0.199
0.801
0.9893
3
σ(n,γ)∗∗
σ(n,α)∗∗
σ(n,p)∗∗
σa(total)**
millibarns
0.153
0.003
0.153
0.0
0.0
0.0
0.024
0.032
0.032
0.100
0.278
0.033
0.082
0.002
0.002
millibarns
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3.9
2167.5
0.0
431.3
0.1
0.1
millibarns
0.0
0.0
0.0
2513.2
0.0
0.0
0.3
0.0
0.0
0.0
1.1
0.0
0.2
0.0
0.0
millibarns
0.153
0.003
0.153
2513.4
0.0
0.003
972.7
0.033
73.9
4.03
2173.0
0.036
432.4
0.064
0.071
ZAID
6013.42c
7014.60c
7015.60c
8016.60c
8017.60c
9019.60c
11022.96c
11023.60c
12000.60c
12024.96c
12025.96c
12026.96c
13027.60c
14000.60c
14028.96c
14029.96c
14030.96c
15031.60c
16000.60c
16032.60c
16033.96c
16034.96c
16036.96c
17000.60c
17035.96c
17037.96c
19000.60c
19039.96c
19040.96c
19041.96c
20000.60c
20040.21c
21045.60c
22000.60c
22046.96c
22047.96c
22048.96c
22049.96c
22050.96c
23000.60c
23051.96c
24000.50c
24052.60c
24053.60c
24054.60c
Nuclei
C13
N14
N15
N(nat)
O16
O17
F19
Na22
Na/Na23
Mg
Mg24
Mg25
Mg26
Al27
Si
Si28
Si29
Si30
P31
S
S32
S33
S34
S36
Cl
Cl35
Cl37
K
K39
K40
K41
Ca
Ca40
Sc45
Ti
Ti46
Ti47
Ti48
Ti49
Ti50
V
V51
Cr
Cr52
Cr53
Cr54
Abundance
Atom
fraction
0.0107
0.99632
0.00368
0.99757
0.00038
0.7899
0.1
0.1101
0.922297
0.046832
0.030872
0.9493
0.0076
0.0429
0.0002
0.7578
0.2422
0.932581
0.000117
0.067302
0.0823
0.0744
0.7372
0.0541
0.0518
0.83789
0.09501
0.02365
4
σ(n,γ)∗∗
σ(n,α)∗∗
σ(n,p)∗∗
σa(total)**
millibarns
0.278
0.049
0.012
0.048
0.000
0.061
2.71
8.2
1.9
0.7
1.6
2.1
0.3
2.5
2.5
0.8
2.6
10.9
4.2
2.3
2.8
1.1
0.3
0.4
3.9
5.9
1.6
11.3
11.1
12.1
25.2
3.8
3.8
40.4
12.6
12.5
31.9
17.4
8.6
1.1
17.3
20.3
16.3
5.7
24.6
5.3
millibarns
0.0
8.4
0.004
8.4
0.724
30.2
0.91
70.7
0.0
0.2
0.3
1.1
0.0
0.0
0.2
0.1
0.4
0.0
0.1
12.4
8.2
172.6
0.2
0.0
0.9
3.7
0.1
1.5
2.8
41.0
0.1
5.7
0.0
0.0
0.1
0.1
0.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
millibarns
0.0
14.2
0.001
14.1
0.001
0.0
0.06
1318.1
0.1
0.1
0.1
0.1
0.0
0.3
0.4
0.3
0.2
0.0
2.8
5.2
5.0
10.7
0.0
0.0
3.5
13.9
0.0
7.3
11.0
13.7
0.1
8.2
0.0
4.3
0.2
0.7
2.0
0.0
0.1
0.0
0.0
0.0
0.3
0.1
0.0
0.0
millibarns
0.280
22.7
0.020
22.6
0.725
30.3
3.68
1397.2
2.0
1.0
2.0
3.3
0.4
2.8
3.1
1.3
3.1
10.9
7.2
19.8
16.0
184.3
0.6
0.4
8.3
23.6
1.7
20.1
24.9
66.8
25.4
17.8
3.8
44.8
12.9
13.2
34.2
17.4
8.7
1.1
17.4
20.3
16.5
5.8
24.7
5.3
ZAID
25055.60c
26000.55c
26054.60c
26056.60c
26057.60c
26058.60c
27058.96c
27059.60c
28000.50c
28058.60c
28059.96c
28060.60c
28061.60c
28062.60c
28064.60c
29000.50c
29063.60c
29065.60c
30000.62c
30064.96c
31000.60c
32072.96c
32073.96c
32074.96c
32076.96c
32072.96c
37085.96c
37086.96c
37087.96c
38084.96c
38086.96c
38087.96c
38088.96c
38089.96c
38090.96c
39088.35c
39089.60c
39090.96c
39091.96c
40000.60c
40090.86c
40091.96c
40092.86c
40093.86c
Nuclei
Mn55
Fe
Fe54
Fe56
Fe57
Fe58
Co58
Co/Co59
Ni
Ni58
Ni59
Ni60
Ni61
Ni62
Ni64
Cu
Cu63
Cu65
Zn
Zn64
Ga
Ge72
Ge73
Ge74
Ge76
Ge(but no Ge70)
Rb85
Rb86
Rb87
Rb(nat)
Sr84
Sr86
Sr87
Sr88
Sr89
Sr90
Sr
Y88
Y/Y89
Y90
Y91
Zr
Zr90
Zr91
Zr92
Zr93
Abundance
Atom
fraction
0.05845
0.91754
0.02119
0.00282
0.680769
0.262231
0.011399
0.036345
0.009256
0.6917
0.3083
0.3479
0.09765
0.45831
0.09614
0.7217
0.2783
0.0056
0.0986
0.07
0.8258
0.5145
0.1122
0.1715
5
σ(n,γ)∗∗
σ(n,α)∗∗
σ(n,p)∗∗
σa(total)**
millibarns
24.0
11.0
16.9
7.7
20.1
11.3
37.4
32.0
20.3
23.2
58.3
17.5
40.1
28.0
12.1
44.4
51.2
26.3
34.8
50.5
71.4
53.0
197.6
32.9
11.6
53.9
131.4
98.8
10.7
97.8
165.0
45.9
79.9
1.1
19.0
13.1
11.9
49.7
16.5
107.8
32.5
22.8
16.6
44.0
30.5
56.0
millibarns
0.0
0.0
0.1
0.0
0.1
0.0
0.5
0.0
0.3
0.6
8.9
0.1
0.2
0.0
0.0
0.0
0.0
0.0
3.7
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
millibarns
0.0
0.4
6.0
0.1
0.0
0.0
709.7
0.1
5.8
8.6
43.3
0.1
0.2
0.0
0.0
1.1
1.9
0.0
2.4
3.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
412.8
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
millibarns
24.0
11.4
22.9
7.8
20.2
11.3
747.7
32.1
26.4
32.4
110.4
17.7
40.5
28.0
12.1
45.5
53.1
26.4
40.8
53.8
71.5
53.0
197.6
32.9
11.6
53.9
131.4
98.8
10.7
97.8
165.0
45.9
79.9
1.1
19.0
13.1
11.9
462.5
16.6
107.8
32.5
22.8
16.6
44.0
30.5
56.0
ZAID
40094.86c
40095.60c
41093.60c
41094.96c
41095.96c
42000.60c
42092.96c
42094.96c
42095.50c
42096.96c
42097.60c
42098.50c
42099.60c
42100.96c
46102.96c
46104.96c
46105.50c
46106.96c
46107.96c
46108.50c
46110.96c
50000.42c
50112.96c
50114.96c
50115.96c
50116.96c
50117.96c
50118.96c
50119.96c
50120.96c
50122.96c
50123.96c
50124.96c
50125.96c
50126.96c
52120.96c
52122.96c
52123.96c
52124.96c
52125.96c
52126.96c
52127.96c
52128.96c
52129.96c
52130.96c
Nuclei
Zr94
Zr95
Nb/Nb93
Nb94
Nb95
Mo
Mo92
Mo94
Mo95
Mo96
Mo97
Mo98
Mo99
Mo100
Pd102
Pd104
Pd105
Pd106
Pd107
Pd108
Pd110
Pd(nat)
Sn(nat)
Sn112
Sn114
Sn115
Sn116
Sn117
Sn118
Sn119
Sn120
Sn122
Sn123
Sn124
Sn125
Sn126
Te120
Te122
Te123
Te124
Te125
Te126
Te127
Te128
Te129
Te130
Abundance
Atom
fraction
0.1738
0.1484
0.0925
0.1592
0.1668
0.0955
0.2413
0.0963
0.0102
0.1114
0.2233
0.2733
0.2646
0.1172
0.0097
0.0066
0.0034
0.1454
0.0768
0.2422
0.0859
0.3258
0.0463
0.0579
0.0009
0.0255
0.0089
0.0474
0.0707
0.1884
0.3174
0.3408
6
σ(n,γ)∗∗
σ(n,α)∗∗
σ(n,p)∗∗
σa(total)**
millibarns
20.4
117.6
170.0
172.6
263.6
110.8
49.6
65.8
236.2
74.7
227.4
76.6
398.8
65.7
132.5
250.3
708.1
210.0
763.0
194.3
121.2
310.4
54.0
238.9
224.2
33.0
45.5
163.0
89.5
40.0
33.5
21.8
87.3
13.2
127.0
7.2
311.8
256.8
495.6
196.7
256.5
84.5
271.2
82.9
84.0
11.6
millibarns
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
millibarns
0.0
0.0
0.0
0.0
0.0
0.0
0.7
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
millibarns
20.4
117.6
170.0
172.6
263.6
110.8
50.2
65.8
236.2
74.7
227.4
76.6
398.8
65.7
132.5
250.3
708.1
210.0
763.0
194.3
121.2
310.4
54.0
238.9
224.2
33.0
45.5
163.0
89.5
40.0
33.5
21.8
87.3
13.2
127.0
7.2
311.8
256.8
495.6
196.7
256.5
84.5
271.2
82.9
84.0
11.6
ZAID
Nuclei
σ(n,γ)∗∗
Abundance
Atom
fraction
σ(n,α)∗∗
σ(n,p)∗∗
σa(total)**
millibarns millibarns millibarns millibarns
52132.96c
Te132
0.4
0.0
0.0
0.4
Te(nat)
84.9
0.0
0.0
84.9
56130.96c
Ba130
0.00106 598.6
0.0
0.0
598.6
56132.96c
Ba132
0.00101 361.2
0.0
0.0
361.2
56134.96c
Ba134
0.02417 87.9
0.0
0.0
87.9
56135.96c
Ba135
0.06592 231.1
0.0
0.0
231.1
56136.96c
Ba136
0.07854 33.9
0.0
0.0
33.9
56137.96c
Ba137
0.11232 38.1
0.0
0.0
38.1
56138.60c
Ba138
0.71698 4.7
0.1
0.0
4.9
56140.60c
Ba140
9.5
0.0
0.0
9.5
Ba(nat)
28.7
0.1
0.0
28.8
57138.96c
La138
0.0009 204.0
0.0
0.0
204.0
57139.60c
La139
0.9991 28.1
0.0
0.0
28.1
57140.60c
La140
156.7
0.0
0.0
156.7
La(nat)
28.2
0.0
0.0
28.2
58140.96c
Ce140
0.88837 17.3
0.0
0.0
17.3
58141.60c
Ce141
108.2
0.0
0.0
108.2
58142.96c
Ce142
0.11163 31.9
0.0
0.0
31.9
58143.60c
Ce143
120.0
0.0
0.0
120.0
58144.96c
Ce144
28.9
0.0
0.0
28.9
Ce(nat)
19.0
0.0
0.0
19.0
82000.50c
Pb(nat)
3.6
0.0
0.0
3.6
82206.86c
Pb206
0.241 9.6
0.0
0.0
9.6
82207.60c
Pb207
0.221 7.2
0.0
0.0
7.2
82208.60c
Pb208
0.524 0.7
0.0
0.0
0.7
83209.60c
Bi209
4.1
0.0
0.0
4.1
*some cross sections of natural materials are obtained by abundance weighted summation
We now have in all 33 matrix candidate elements and 45 reflector candidate elements
remaining.(including Li-7) Based on their distinctive properties, we can classify them
under 4 main categories; see Table 1.4.
7
Table 1-4 Roster of Potential Diluent Candidates
Possible Form of Use
Elements
In Ceramics (6)
C, N, O, P, S, Si
(including CERCER, CERMET)
As metals and alloys (19)
Mg, Ca, Sr, Ba, Ti, V, Cr, Mn, Fe, Co, Ni,
(including CERMET, METMET)
Cu, Zn, Al, Sn, Zr, Nb, Mo
As liquid metal (5)
Na, K, Pb, Bi, Hg
(coolants, pools)
In molten salts (3 + 1)**
F, Cl, Be*
(coolants, pools)
(also the separated isotope Li-7) [2]
*Be could also be used in metallic form and as BeO ceramic.
** see ref [2]
1.3 Organization of this report
Chapter 2 describes the computer codes employed and the whole-core model used to
evaluate important neutronic parameters such as multiplication factor, its rate of change
with burnup, initial conversion ratio and spectrum-averaged cross sections. This degree
of sophistication is necessary because a priori judgments are unreliable for hard spectrum
fast reactors in view of the influence of less familiar phenomena such as (n,p) (n,α) and
(n,2n) reactions, the effect of scattering resonances on leakage and inelastic scattering on
moderation.
In chapter 3 results for matrix studies are shown and analyzed. Issues such as the nonlinearity of neutronic effects vs. diluent concentration and the failure of the superposition
principle in predicting the effect of compounds based on their individual components are
discussed.
Chapter 4 reports a detailed study for reflector material candidates. The candidate
material range is broadened and more compounds are included. Materials good as in-core
8
matrix diluents are not necessarily good as reflectors. The different demands for different
functions are discussed.
Chapter 5 presents a summary, principal overall conclusions, and recommendations
for follow-on work.
An appendix is included discussing the potential problem due to helium production
via (n, α) reactions in sulfur.
9
Chapter 2 Computer Codes and Models
2.1 Introduction
In this chapter descriptions are presented of the computer codes employed in the
evaluation of core diluents in whole core models. Sufficient descriptive information and
data are provided that others could reproduce or extend the results to be presented later in
chapters 3 and 4. Appendices to this report provide sample copies of code input and
output in further fulfillment of this goal.
2.2 MCODE Description
2.2.1 Introduction
MCODE (MCNP-ORIGEN Depletion program)[4] is a linkage program (~3000
lines of ANSI C), which uses MCNP and ORIGEN to do burnup calculations for
arbitrarily-defined MCNP regions[5]. MCNP is used to calculate neutron flux and from it
determine the effective one-group cross sections for materials in different MCNP-defined
regions. ORIGEN, in turn, can carry out depletion calculations for each region and output
time-dependent isotopic composition. MCODE serves as a console program to control the
data flow between MCNP and ORIGEN as well as the alternate running of these two
codes.
MCNP-4c, the latest MCNP version, was used, which is a general purpose,
generalized geometry, continuous energy, time-dependent, Monte Carlo transport code
for neutrons/photons/electrons developed at the Los Alamos National Laboratory
(LANL)[5]. The Monte Carlo method is employed in MCNP, which sets up a virtual
world analog to reality to solve neutron transport problems. It follows each of many
particles from a source to their death in some terminal category (absorption, escape, etc.).
Probability distributions are randomly sampled to determine the outcome at each step. In
MCODE burnup calculations, three kinds of data are needed from MCNP:
1. criticality or eigenvalue, keff,
2. effective one-group cross sections,
10
3. one-group neutron flux data.
Specifically, the effective one-group cross sections of fission products and actinides are
needed. For fission products, only neutron capture cross sections are calculated. For
actinides, four types of cross sections are considered including capture, fission, (n, 2n),
and (n, 3n) reactions. Although not all nuclides and all reactions are calculated, the
representation of fission products and actinides is quite complete for burnup calculations
(i.e. altogether the chosen isotopes account for more than 99% of neutron absorption). In
addition to the effective one-group cross sections, the one-group flux value in each
MCNP depletion cell is needed.
ORIGEN (version 2.1) is a one-group depletion and radioactive decay computer
code developed at the Oak Ridge National Laboratory (ORNL)[6]. Given appropriate
one-group cross sections and decay constants, ORIGEN 2.1 uses a matrix exponential
method to solve a large system of coupled, linear, first-order ordinary differential
equations with constant coefficients. Both nuclear reactions and isotope decay are
considered. Several generic reaction specific cross section and fission product yield data
libraries are available with ORIGEN 2.1. For cross sections not provided from MCNP,
ORIGEN uses library values, which are fairly representative of a given type of reactor.
The cross section data used in our work is from the fast flux test facility core library
(FFTFC.LIB).
2.2.2 Normalization
Since there are two modes of depletion in ORIGEN, constant power or constant
flux, there are two corresponding ways to do depletions. In burnup calculations, the total
power of the reactor is usually assumed to be known and maintained constant. However,
the power fractions among different zones vary. Therefore, the two options should not
affect final results if small time steps are used. MCODE provides the user with both of
the above options to run depletion calculations. The flux values from MCNP are in units
of number of neutrons per fission source neutron per cm2, which must be multiplied by an
appropriate factor to convert into n/cm2 per second if an actual flux value is wanted.
11
For power normalization, the power of each cell is determined and fixed in each
time step. It is not necessary to normalize relative flux values from MCNP because the
power fractions for each cell can be obtained using only these relative values:
∑ N {∫ σ (E )φ (E )dE}⋅V ⋅ R
mi
fi =
j
j =1
mi
i
i
j
k
j
k, f
k
j
i
∑∑ N {∫ σ (E )φ (E )dE}⋅V
n
k =1 j =1
where
j
i, f
i
k
,
(2-1)
⋅R
j
k
fi is the power fraction of cell i,
N i j is the number density of isotope j in cell i,
Vi is the volume of cell i,
Ri j is the recoverable energy of isotope j in cell i,
σ i,j f (E ) is the fission cross section at energy E for isotope j in cell i,
φi (E ) is the neutron flux at energy E,
The j summation is over all actinides,
and the k summation is over all depletion cells.
Then, the power of each cell can be determined by multiplying the fraction factor fi by the
given total power.
For flux normalization, the absolute flux value for each depletion cell is needed.
Therefore, the relative flux values from MCNP are multiplied by a constant factor. This
flux multiplication factor (FMF) in units of fission neutrons per second can be calculated
by either of the following two ways:
FMF =
where
P ⋅ν
,
Q ⋅ keff
(2-2)
P is the total power of the modeled system (watts),
ν is the average number of neutrons per fission,
Q is the average recoverable energy per fission (Joules/fission),
keff is the eigenvalue of the system;
FMF =
P
∑∑ N {∫ σ (E )φ (E )dE}⋅V ⋅ R
n
mi
i =1 j =1
j
i
j
i, f
12
i
i
i
.
j
(2-3)
Equation (2-2) has a simpler form but with some ambiguities in its quantities. For
instance, the average recoverable energy per fission needs to be computed carefully. One
can imagine that for different kinds of fuel Q can be very different. For a relevant
discussion see Ref. [16]. Equation (2-3) appears complicated, but has a very clear
meaning and no ambiguities with regard to its quantities. However, both Eq. (2-2) and
Eq. (2-3) give an instantaneous flux multiplying factor only. For the real situation in each
depletion cell, the flux level changes continuously with burnup. The time step average
flux should be used instead of beginning-of-time-step instantaneous flux. This might be
done by the internal “predictor-corrector”, namely after the first trial ORIGEN depletion
gives an average flux to satisfy given energy production, the second ORIGEN depletion
uses the average flux (corrector).
In the ideal case, the two ways of normalization produce identical results. But
when the time step is long, power normalization assumes constant power in each cell,
which is incorrect; flux normalization assumes constant flux in each cell, which is also
incorrect. Hence the specified time step length must be sufficiently short such that the
two approaches give comparable results.
2.2.3 Predictor-Corrector Algorithm
The coupling of MCNP and ORIGEN requires careful attention to detail. Because
the cross sections, flux and power fraction in each depletion cell are varying during
reactor operation, it is not valid to use beginning-of-time-step values to represent the
entire time step. A better estimate of time step average value is required.
The predictor-corrector algorithm is the standard algorithm to solve depletion
problems. For each burnup step the depletion is calculated twice, first using the spectra at
the start of the step and then, after a new spectrum calculation, using the spectra at the
end of the step. Average number densities from these two calculations are used as start
values for the next burnup step. This algorithm has proven to be efficient and useful to
solve depletion problems, especially in poisoned assemblies [4]. It has been implemented
in MCODE, which distinguishes MCODE from other MCNP-ORIGEN linkage codes,
such as MOCUP, MONTEBURNs, etc.
13
2.2.4 Running MCODE
One of the best features of MCODE is its user-friendly interface. Users need a
minimal amount of time to learn and initiate MCODE runs. Only three input files are
needed:
•
initial MCNP input,
•
MCODE input file,
•
MCNP source file (optional).
Users have many options to run the code, such as the predictor-corrector option,
normalization option, etc. The flow chart is shown in Figure: 1.
The default and recommended settings are to employ the predictor-corrector, plus
flux normalization. Power normalization is usually used to check the result. When time is
limiting, the predictor-corrector can be turned off: this reduces overall time per step by
approximately a factor of two.
14
Parse MCODE input and initialize variables
Initial run?
NO (restart)
YES
Preprocess initial mcnp input and run MCNP
Loop through all timesteps
Extract beginning-of-timestep cross-sections and flux values
Run ORIGEN depletions for all active cells
Update MCNP input based on ORIGEN output
material composition (predictor), and run MCNP
Predictor-Corrector?
NO
YES
Extract end-of-timestep cross-sections and flux values
Re-run ORIGEN depletions for all active cells
Average the predictor and corrector material,
update MCNP input, and re-run MCNP
NO
Finish all timesteps?
YES
END
Figure 2-1 Flow diagram for MCODE
15
2.4 Whole Core Model for matrix and reflector configuration
A simplified matrix core model was developed from the homogenization of the
hexagonal cell core developed in ref[9]. See Figure: 2.3. Axial leakage is assumed to be
zero. The extruded coolant tubes and the cladding of the assembly are made of the same
material as the matrix. These two parts are included in the calculated matrix volume
fraction. The core parameters for matrix tests are given in tables 2.5 and 2.6. Similarly,
the parameters for reflector tests are given in tables 2.7 and 2.8.
2
)
Gas coolant (CO2)
D = 1.2cm, 106holes/cell
CERMET or METMET
fuel in matrix
extruded sheath
matrix metal
matrix metal
serves as clad
active core
reflector
36 cm
Figure 2-2 Original fuel assembly and core layout of the MFGR-GT [6]
16
Figure 2-3 Final homogenized cylindrical core layout
For matrix tests, the reflector is always Nickel and the core diameter 300cm; for reflector
tests, the matrix is always Lead and the core diameter 180cm. The reflector thickness is
always 90cm.
Table 2-1 Matrix test core model parameters
Parameters
Values
Fuel*
UC, UPuC
Fuel temperature (ºK)
773.15
Fuel percent of theoretical density
100.00
Fuel enrichment (%)
13.00
core diameter (cm)
300.00
core height (m)
1.00
Parameters
Coolant
reflector thickness (cm)
volume percent of fuel (%)
volume percent of coolant (%)
volume percent of matrix (%)**
Power density (kW/l)
Values
CO2
90.00
26.92
10.28
62.80
10.61
*
We are mainly using UC fuel. The UPuC fuel with matrix study is limited.
**
volume fraction of matrix material is kept the same for performance comparisons.
17
Table 2-2 Initial region–homogenized compositions in matrix test core model
Fuel
U238
Weight
percent
(w/o)
-
(UC+matrix+CO2)
Cell1*
Fuel
U235
C
O
U238
-
1.163166E-03
9.041788E-03
3.853585E-04
7.685943E-03
(US+matrix+CO2)
U235
-
1.163166E-03
Cell1*
Reflector
C
O
S
U238
Pu238
Pu239
Pu240
Pu241
Pu242
C
O
Ni
9.995943E+01
1.926790E-04
3.853585E-04
8.849109E-03
7.685943E-03
1.170402E-05
7.342604E-04
3.365826E-04
1.155801E-05
6.906098E-05
9.041788E-03
3.853585E-04
8.898912E-02
(Ni+CO2)
Cell2
C
O
1.107238E-02
2.949893E-02
4.816981E-05
9.633962E-05
Nuclide
Fuel
(UPuC+matrix+CO2)
Cell1*
Number
density
(#/barn.cm)
7.685943E-03
* Since UC/US/UPuC and CO2 keep their same volume percentages when matrix
material changes, the homogenized atom number densities of uranium carbide and
carbon dioxide in the core cell are always the same. The parameters for the reflector
cell are fixed. The weight percent of UC/US/UPuC and CO2 depend on the density
and formula weight of the specified matrix.
Table 2-3 Reflector test core model parameters
Parameters
Fuel
Fuel temperature (ºK)
Fuel percent of theoretical density
Fuel enrichment (%)
core diameter (cm)
core height (m)
*
Values
UC
773.15
100.00
13.00
180.00
1.00
Parameters
Coolant
reflector thickness (cm)
volume percent of fuel (%)
volume percent of coolant (%)
volume percent of matrix (%)*
Power density (kW/l)
Values
CO2
90.00
26.92
10.28
62.80
10.61
Volume fraction of reflector material is kept the same for performance
comparisons.
18
In the matrix model, the core diameter is set to 300.0cm, which makes the core’s leakage
negligible. When assessing radial reflector performance, we need larger leakage to get
accurate performance comparisons. The 180cm(d) × 100cm(h) cylinder bare(unreflected)
core has keff = 1.02368. Thus keff – 1.02368 can be considered as mainly gains
attributable to the reflector.
Table 2-4 Initial region homogenized compositions in reflector test core model
Fuel
U238
Weight
percent
(w/o)
28.10
(UC+Pb+CO2)
Cell1
U235
C
O
Pb
4.20
1.67
0.09
65.94
1.163166E-03
9.041788E-03
3.853585E-04
2.071776E-02
reflector
C
-
4.816981E-05
(reflector+CO2)
Cell2
O
-
9.633962E-05
Nuclide
Number density
(#/barn.cm)
7.685943E-03
Since CO2 keeps the same volume percentage when reflector material changes, the
homogenized atom number densities of carbon dioxide in the reflector cell are always the
same. The parameters for the matrix/fuel cell are fixed.
Table 2-5 Matrix material cell 1 homogenized composition for whole core model
matrix
component
Al
Al
Al4C3
Al
C
Ba130
Ba132
Ba134
Ba135
Ba136
Ba137
Ba138
Ba130
Ba132
Ba134
Ba135
Ba
BaO
weight percent
(%)
100.00
numberdensity
(#/barn.cm)
3.783206E-02
74.97
25.03
0.10
0.10
2.36
6.48
7.77
11.20
72.00
0.09
0.09
2.18
6.00
2.480135E-02
1.860101E-02
1.024670E-05
9.763390E-06
2.336450E-04
6.372300E-04
7.592240E-04
1.085766E-03
6.930846E-03
1.495680E-05
1.425130E-05
3.410430E-04
9.301440E-04
19
matrix
BaS
BeO
Bi
C
Ca
component
Ba136
Ba137
Ba138
O
Ba130
Ba132
Ba134
Ba135
Ba136
Ba137
Ba138
S
Be
O
Bi
C
Ca
weight percent
(%)
7.20
7.26
66.73
10.43
0.08
0.08
1.91
5.25
6.30
9.08
58.37
18.93
36.03
63.97
100.00
100.00
100.00
numberdensity
(#/barn.cm)
1.108214E-03
1.108214E-03
1.011672E-02
1.363355E-02
1.055130E-05
1.005360E-05
2.405890E-04
6.561700E-04
7.817900E-04
1.118037E-03
7.136843E-03
9.954034E-03
4.536367E-02
4.536367E-02
1.769952E-02
8.344854E-02
1.462735E-02
CaC2
Ca
C
62.52
37.48
1.311143E-02
2.622286E-02
CeO2
Ce
O
Co
Cr
Cu
Fe
81.41
18.59
100.00
100.00
100.00
100.00
1.566749E-02
3.133498E-02
7.539229E-02
5.193445E-02
5.325470E-02
5.282486E-02
Fe
C
Fe
Ni
Cr
Mo
Si
V
W
C
Mn
K
Mg
Mn
Mo
Na
Nb
Ni
P
Pb
Pb
93.31
6.69
84.7
0.5
12
1
0.2
0.3
0.5
0.2
0.6
100
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
92.83
4.862416E-02
1.620805E-02
4.474266E-02
2.513063E-04
6.808213E-03
3.074843E-04
2.100731E-04
1.737289E-04
8.023293E-05
4.912298E-04
3.221816E-04
8.280259E-03
2.704544E-02
5.142513E-02
4.052822E-02
1.592461E-02
3.488987E-02
5.740094E-02
2.225978E-02
2.071776E-02
1.633518E-02
Co
Cr
Cu
Fe
Fe3C
HT9
K
Mg
Mn
Mo
Na
Nb
Ni
P
Pb
PbO
20
matrix
component
PbS
O
Pb
S
Ba2Pb
S
Si
SiC
Sn
Sr
SrO
SrS
Sr2Pb
Ti
TiC
TiN
TiN15
U238
V
VC
weight percent
(%)
7.17
86.60
13.40
numberdensity
(#/barn.cm)
1.633518E-02
1.201358E-02
1.201358E-02
Ba130
Ba132
Ba134
Ba135
Ba136
Ba137
Ba138
Pb
S
Si
Si
C
Sn
Sr84
Sr86
Sr87
Sr88
Sr84
Sr86
Sr87
Sr88
O
Sr84
Sr86
Sr87
Sr88
S
0.06
0.06
1.34
3.69
4.43
6.38
41.04
43.00
100.00
100.00
70.05
29.95
100.00
0.54
9.67
6.94
82.85
0.45
8.18
5.87
70.06
15.44
0.39
7.08
5.08
60.65
26.79
8.172440E-06
7.786950E-06
1.863470E-04
5.082330E-04
6.055320E-04
8.659700E-04
5.527809E-03
3.854926E-03
2.311814E-02
3.137716E-02
3.034421E-02
3.034421E-02
2.328939E-02
6.35777E-05
0.001119422
0.000794721
0.00937544
9.61E-05
0.001691917
0.001201158
0.014170237
1.72E-02
6.55E-05
0.001152786
0.000818408
0.009654873
0.011691539
Sr84
Sr86
Sr87
Sr88
Pb
Ti
Ti
C
Ti
N
Ti
N15
U238
V
V
C
0.11
1.95
1.40
16.68
79.86
100.00
79.94
20.06
77.36
22.64
77.36
22.64
100.00
100.00
80.92
19.08
4.90E-05
0.000862324
0.000612197
0.007222178
1.47E-02
3.56E-02
3.118990E-02
3.118990E-02
3.184884E-02
3.184884E-02
3.184884E-02
3.184884E-02
3.027119E-02
4.536256E-02
3.466758E-02
3.466758E-02
21
matrix
component
W
Zn
ZnC
Zr
ZrO2
ZrC
Void
W
Zn
Zn
C
Zr
weight percent
(%)
100.00
100.00
84.48
15.52
100.00
numberdensity
(#/barn.cm)
3.960215E-02
4.129666E-02
3.288503E-02
3.288503E-02
2.696982E-02
Zr
O
Zr
C
-
74.03
25.97
88.37
11.63
-
1.743353E-02
3.486706E-02
2.465768E-02
2.465768E-02
-
A total of 39 actinides and 100 fission products (including some excited states as
different nuclides) have been tracked in MCODE burnup runs. 34 elements are used
singly or in combination as matrix material. See tables 2.10, 2.11 and 2.12.
Table 2-6 Description of chosen actinides
Number Actinides
ZAID
Library Name Source Temperature (°C)
1
Th-232 90232.60c
endf60
B-V.0
294
2
Pa-231 91231.60c
endf60
B-VI.0
294
3
Pa-233 91233.50c
endf5u
B-V.0
294
4
U-232 92232.60c
endf60
B-VI.0
294
5
U-233 92233.60c endf60[14]
B-VI.0
294
6
U-234 92234.60c
endf60
B-VI.0
294
7
U-235 92235.60c
endf60
B-VI.2
294
8
U-236 92236.60c
endf60
B-VI.0
294
9
U-237 92237.50c
endf5p
B-VI.0
294
10
U-238 92238.60c
endf60
B-VI.2
294
11
Np-236 93236.35c
endl85
LLNL
0
12
Np-237 93237.60c
endf60
B-VI.1
294
13
Np-238 93238.35c
endl85
LLNL
0
14
Np-239 93239.60c
endf60
B-VI.0
294
15
Pu-238 94238.60c
endf60
B-VI.0
294
16
Pu-239 94239.60c
endf60
B-VI.2
294
17
Pu-240 94240.60c
endf60
B-VI.2
294
18
Pu-241 94241.60c
endf60
B-VI.1
294
19
Pu-242 94242.60c
endf60
B-VI.0
294
20
Pu-243 94243.60c
endf60
B-VI.2
294
21
Am-241 95241.60c
endf60
T-2
300
22
Am-242 95242.50c
endf5u
B-V.0
294
23
Am-242 95242.51c
rmccs
B-V.0
294
24
Am-243 95243.60c
endf60
B-VI.0
294
25
Am-244 95244.96c
hfirxs1
INEEL
300
22
Number Actinides
ZAID
Library Name Source Temperature (°C)
26
Cm-242 96242.60c
endf60
B-VI.0
294
27
Cm-243 96243.60c
endf60
B-VI.0
294
28
Cm-244 96244.60c
endf60
B-VI.0
294
29
Cm-245 96245.60c
endf60
B-VI.2
294
30
Cm-246 96246.60c
endf60
B-VI.2
594
31
Cm-247 96247.60c
endf60
B-VI.2
294
32
Cm-248 96248.60c
endf60
B-VI.0
294
33
Cm-249 96249.96c
hfirxs1
INEEL
300
34
Bk-249 97249.60c
endf60
B-VI:XTM
294
35
Bk-250 97250.96c
hfirxs1
INEEL
300
36
Cf-249 98249.60c
endf60
B-VI:XTM
294
37
Cf-250 98250.60c
endf60
B-VI.2
294
38
Cf-251 98251.60c
endf60
B-VI.2
294
39
Cf-252 98252.60c
endf60
B-VI.2
294
Table 2-7 Description of chosen fission products
Number Actinides
ZAID
Library Name
Source Temperature (°C)
1
Br-81 35081.55c miscSxs[6,8]
T-2
294.0
2
Kr-82 36082.50c
rmwsa
ENDF/B-V.0
294.0
3
Kr-83 36083.50c
rmccsa
ENDF/B-V.0
294.0
4
Kr-84 36084.50c
rmccsa
ENDF/B-V.0
294.0
5
Rb-85 37085.55c miscSxs[6,8]
T-2
294.0
6
Rb-87 37087.55c Misc5xs[6,8]
T-2
294.0
7
Sr-90 38090.96c
hfirxs1
INEEL
300.0
8
Y-89
39089.60c
endf60
ENDF/B-VI.0
294.0
9
Zr-91 40091.96c
hfirxs1
INEEL
300.0
10
Zr-92 40092.62c
Zr92.300
UTXS
300.0
11
Zr-93 40093.50c
kidman
ENDF/B-v.0
294.0
12
Zr-94 40094.62c
Zr92.300
UTXS
300.0
13
Zr-96 40096.62c
Zr92.300
UTXS
300.0
14
Nb-95 41095.96c
hfirxs1
INEEL
300.0
15
Mo-95 42095.50c
kidman
ENDF/B-V:0
294.0
16
Mo-96 42096.96c
hfirxs1
INEEL
300.0
17
Mo-97 42097.60c
mason1
INEEL
294.0
18
Mo-98 42098.50c
mason1
INEEL
294.0
19
Mo-100 42100.50c
mason1
INEEL
294.0
20
Tc-99 43099.50c
kidman
ENDF/B-V.0
293.6
21
Ru-100 44100.96c
hfirxs1
INEEL
300.0
22
Ru-101 44101.50c
kidman
ENDF/B-V.0
293.6
23
Ru-102 44102.60c
mason1
INEEL
293.6
24
Ru-103 44103.50c
kidman
ENDF/B-V.0
293.6
25
Ru-104 44104.96c
ornlxsb1
INEEL
300.0
26
Rh-103 45103.50c
rmccsa
ENDF/B-V.0
293.6
27
Rh-105 45105.50c
kidman
ENDFIB-V.0
293.6
28
Pd-104 46104.96c
ornlxs1
INEEL
300.0
23
Number Actinides
ZAID
Library Name
29
Pd-105 46105.50c
kidman
30
Pd-106 46106.96c
ornlxs1
31
Pd-107 46107.96c
ornlxs1
32
Pd-108 46108.50c
kidman
33
Pd-110 46110.96c
ornlxs1
34
Ag-109 47109.60c
endf60
35
Cd-110 48110.62c Cd110.300
36
Cd-111 48111.62c Cd111.300
37
Cd-112 48112.62c Cd112.300
38
Cd-113 48113.60c
mason1
39
Cd-114 48114.62c Cd114.300
40
In-115 49115.60c
mason1
41
Sb-121 51121.96c
ornlxsb1
42
Sb-123 51123.96c
ornlxsb1
43
Te-128 52128.96c
ornlxsa1
44
I-127
53127.60c endf60[121
45
I-129
53129.60c
endf60
46
Xe-131 54131.50c
kidman
47
Xe-132 54132.62c Xe132.300
48
Xe-133 54133.60c
mason1
49
Xe-134 54134.62c Xe134.300
50
Xe-135 54135.50c endf5mttll
51
Xe-136 54136.62c Xe136.300
52
Cs-133 55133.60c
endf60
53
Cs-134 55134.60c
endf60
54
Cs-135 55135.60c
endf60
55
Cs-137 55137.60c
endf60
56
Ba-134 56134.62c Ba134.300
57
Ba-137 56137.62c Ba136.300
58
Ba-138 56138.60c
endf60
59
La-139 57139.60c
mason1
60
Ce-140 58140.96c
ornlxsb1
61
Ce-141 58141.60c
mason1
62
Ce-142 58142.96c
ornlxsb1
63
Ce-144 58144.96c
ornlxsb1
64
Pr-141 59141.50c
kidman
65
Pr-143 59143.60c
mason1
66
Nd-142 60142.96c
ornlxsb1
67
Nd-143 60143.50c
kidman
68
Nd-144 60144.96c
ornlxsb1
69
Nd-145 60145.50c
kidman
70
Nd-146 60146.96c
ornlxsb1
71
Nd-147 60147.50c
kidman
72
Nd-148 60148.50c
kidman
73
Nd-150 60150.96c
ornlxsb1
74
Pm-147 61147.50c
kidman
24
Source Temperature (°C)
ENDF/B-V.0
293.6
INEEL
300.0
INEEL
300.0
ENDF/B-V.0
293.6
INEEL
300.0
ENDF/B-VI.0
293.6
INEEL
300.0
INEEL
300.0
INEEL
300.0
INEEL
293.6
INEEL
300.0
INEEL
293.6
INEEL
300.0
INEEL
300.0
INEEL
300.0
LANL/T-2
293.6
ENDF/B-VI.0
293.6
ENDF/B-V.0
293.6
INEEL
300.0
INEEL
293.6
INEEL
300.0
ENDFIB-V
293.6
INEEL
300.0
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
INEEL
300.0
INEEL
300.0
ENDF/B-VI.0
293.6
INEEL
293.6
INEEL
300.0
INEEL
293.6
INEEL
300.0
INEEL
300.0
ENDF/B-V.0
293.6
INEEL
293.6
INEEL
300.0
ENDF/B-V.0
293.6
INEEL
300.0
ENDF/B-V.0
293.6
INEEL
300.0
ENDFIB-V.0
293.6
ENDF/B-V.0
293.6
INEEL
300.0
ENDF/B-V.0
293.6
Number Actinides
ZAID
Library Name
75
Pm-148 61148.50c
kidman
76
Pm-148 61148.60c
mason1
77
Pm-149 61149.50c
kidman
78
Sm-147 62147.50c
kidman
79
Sm-148 62148.96c
ornlxsa1
80
Sm-149 62149.50c
endf5u
81
Sm-150 62150.50c
kidman
82
Sm-151 62151.50c
kidman
83
Sm-152 62152.50c
kidman
84
Sm-153 62153.60c
mason1
85
Sm-154 62154.96c
ornlxsa1
86
Eu-151 63151.60c
endf60
87
Eu-153 63153.60c
endf60
88
Eu-154 63154.50c
endf5u
89
Eu-155 63155.50c
kidman
90
Eu-156 63156.60c
mason1
91
Gd-154 64154.60c
endf60
92
Gd-155 64155.60c
endf60
93
Gd-156 64156.60c
endf60
94
Gd-157 64157.60c
endf60
95
Gd-158 64158.60c
endf60
96
Tb-159 65159.96c
ornlxsb1
97
Dy-160 66160.96c
ornlxsa1
98
Dy-161 66161.96c
ornlxsa1
99
Dy-162 66162.96c
ornlxsa1
100
Dy-163 66163.96c
ornlxsa1
Source Temperature (°C)
ENDF/B-V.0
293.6
INEEL
293.6
ENDF/B-V.0
293.6
ENDFfB-V.0
293.6
INEEL
300.0
ENDF/B-V.0
293.6
ENDF/B-V.0
293.6
ENDP/B-V.0
293.6
ENDF/B-V.0
293.6
INEEL
293.6
INEEL
300.0
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-V.0
293.6
ENDF/B-V.0
293.6
INEEL
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
INEEL
300.0
INEEL
300.0
INEEL
300.0
INEEL
300.0
INEEL
300.0
Table 2-8 Description of chosen matrix materials
Number Actinides
ZAID
Library Name
1
Li7
3007.60c
endf60
2
Be
4009.60C
endf60
3
C
6000.60c
endf60
4
N
7014.60c
endf60
5
O
8016.6OC
endf60
6
F
9019.6Oc
endf60
7
Na
11023.60c
endf60
8
Mg
12000.60c
endf60
9
Al
13027.60c
endf60
10
Si
14000.60c
endf60
11
P
15031.60c
endf60
12
S
16000.60c
endf60
13
Cl
17000.60C
endf60
14
K
19000.60c
endf60
15
Ca
20000.60c
endf60
16
Ti
22000.60c
endf60
25
Source
Temperature (°C)
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.1
293.6
LANL/T-2
293.6
ENDFIB-VI.0
293.6
ENDFIB-VI.0
300
ENDF/B-VI.1
293.6
ENDFIB-VI.0
293.6
ENDFIB-VI.0
293.6
ENDF/B-VI.0
293.6
ENDF/B-VI.0
293.6
ENDFIB-VI.0
293.6
ENDFIB-VI.0
293.6
ENDFIB-VI.0
293.6
ENDFIB-VI.0
293.6
ENDF/B-VI.0
293.6
Number Actinides
ZAID
Library Name
17
V
23000.60c
endf60
18
Cr
24000.50c
mlccs
19
Mn
25055.60c
endf60
20
Fe
26000.55c
rmccs
21
Co
27059.6Oc
endf60
22
Ni
28000.50c
rmccs
23
Cu
29000.50c
mccs
24
Zn
30000.42c
end192
25
Sr
38088.96c
ornlxs1
26
Zr
40000.60c
endf60
27
Nb
41093.60c
endf60
28
Mo
42000.60c
endf60
29
Sn
50000.42c
end192
30
Te
52129.96c
ornlxsa1
31
Ba
56138.60c
endf60
32
Hg
80000.42c
end192
33
Pb
82000.50c
mccs
34
Bi
83209.60c
endf60
Source
Temperature (°C)
ENDF/B-VI.0
293.6
ENDF/B-V.0
293.6
ENDFfB-VI.0
293.6
LANL/T-2
293.6
ENDF/B-VI.2
293.6
ENDF/B-V.0
293.6
ENDF/B-V.0
293.6
LLNL:XCI
300
INEEL
300
ENDFfB-VI.1
293.6
ENDF/B-VI.1
293.6
ENDF/B-VI.0
293.6
LLNL:XCI
300
INEEL
300
ENDF/B-VI.0
293.6
LLNL:xCI
300
ENDF/B-V.0
293.6
ENDFIB-VI.0
293.6
2.5 Summary
In this chapter, we have described the computer code MCODE and set up whole
core models for matrix and reflector tests. The region-wise (MCNP cell) configurations
are documented. The roster and cross section libraries of all constituents are also
specifically identified.
26
Chapter 3 Review of core diluent material candidates
3.1 Introduction
In the present work candidate materials for gas-cooled fast reactor core design
were evaluated using static beginning-of-life reactivity calculations and fuel burnup
analyses. MCODE and MCNP were executed using the core model and regional
compositions given in Chapter 2. We first review the material candidate properties in
section 3.2. The importance of σa as an evaluation parameter is also discussed in
section 3.2. In section 3.3 burnup and k calculations for the matrix core model are
shown and compared with other material properties. In section 3.4 we discuss the
non-linearity of neutronic effects vs. diluent concentration and the failure of the
superposition principle in predicting the effect of compounds based on their
individual components. Then we discuss conclusions for the final selection of matrix
material in section 3.5.
3.2 Review of element properties
The most important neutronic property relative to use as a core diluent is a
material’s macroscopic absorption cross section Σa, the product of the microscopic value
and the nuclei’s number density, since this defines its tendency to consume neutrons
unproductively. A second, less easily quantified effect is the change in Σa of other
materials and core leakage due to changes induced in Φ(E). Considering that good heat
storage capacity (ρCp) is expected for matrix material, the ratio of Σa to ρCp is a useful
index of diluent suitability.
To be certain that no innovative option escaped it was decided to carry out a set of
very fundamental studies. These involved calculation, using MCNP, of the spectrum
average microscopic absorption cross section of all of the elements in the periodic table
in a representative GFR spectrum. In addition to the obvious goal of avoiding materials
having a σa even 10% of that of U-235 ( for which σa is roughly 2 barns), σa is also a good
index of the ability of a core diluent to store energy in a transient without excessive
neutron loss. Recall the law of Dulong and Petit, namely that solid elements have a heat
27
capacity close to 25 J/mol⋅K[1,2], thus the ratio of macroscopic absorption cross section
to volumetric heat capacity is just
Σa
ρ N Aσ a
N
N
=
= σa A = σa A ∼ σa
Aρ C p
AC p
ρC p
25
(3-1)
Here NA is Avogadro’s Number, A is the atomic weight, and Cp is the heat capacity in the
units of J/g.K. Since all solid materials have very similar values of ACp, the molecular
heat capacity, the performance index reduces to only one variable, σa. Hence the σa
values displayed in Table 1.3 are a good preliminary indicator of potential suitability. The
heat capacity table for solid and liquid elements in ref(18) shows the systematic behavior
of molecular heat capacity which allows this simplification.
3.3 Review of material candidates for matrix core
3.3.1 Neutronic Evaluation parameters
Together with the inherent properties such as cross sections, we use keff(BOL),
∆keff(void), and B1 as the final evaluation parameters. High keff(BOL) increases the need
for compensatory control, but it will give a higher burnup potential. Negative or small
positive ∆keff(void) is desired for dynamic stability. The linearly extrapolated burnup
potential B1 is defined as B1 = (k – 1) / (∆k/∆B). It is mainly determined by the beginning
of life keff and conversion ratio. In our investigation, since a full burnup whole core
simulation is very time consuming, we use the first 3 keff – burnup points to linearly
extrapolate to the just-critical point. See fig3.1.
Linear extrapolation does not give a highly accurate estimate of the true B1, but
only an indicative trend, as shown in fig3.2 for two extreme cases, the moderator Al4C3
and Ba which has a low slowing down power as diluent.
28
Definition of B 1
1.5
k0
1.4
k1
k eff
1.3
k2
1.2
1.1
B1
1
0
10
20
30
40
50
60
70
80
90
Burnup (MWd/kg)
Figure 3-1 Definition of B1
1.15
Al4C3(actual) B1=95
Ba(actual) B1=200
Al4C3(linear) B1=89
Ba(linear) B1=424
1.1
keff
1.05
1
0.95
0.9
0
50
100
150
200
250
300
350
400
burnup (MWd/kg IHM)
Figure 3-2 Examples of error of linear extrapolation method
29
450
3.3.2 Results for matrix study
All the materials in the candidate list were tested in single element form except
for some less common elements for which there is a lack of cross section libraries.
Several compounds of interest are also studied. Their density, heat capacity, and melting
point are also listed in table3.1. The descending slope of the keff vs B1 curve as well as the
internal conversion ratio is listed as well.
Table 3-1 Results of matrix comparisons
Matrix
Al
Al4C3
AlN
Ba
ρ(g/cc)
2.70
2.36
3.255
3.51
ρCp(J/ccK)
2.44
2.78
2.39
0.72
Tmelt( C)
660
2100
3000
727
Ba2Pb
BaO
BaS
BeO
Bi
C
Ca
CaC2
CeO2
Co
Cr
Cu
Fe
Fe3C
HT9
K
Mg
Mn
Mo
Na
Nb
Ni
P
Pb
PbO
PbS
S
Si
SiC
4.91
5.72
4.30
3.01
9.78
2.60
1.55
2.22
7.13
8.90
7.14
8.92
7.80
7.69
7.69
0.86
1.74
7.47
10.28
0.97
8.57
8.91
1.82
10.43
9.64
7.60
1.96
2.33
3.22
1.78
1.30
3.08
1.19
1.52
0.98
2.17
2.55
3.75
3.20
3.41
3.50
4.54
4.54
0.65
1.78
3.58
2.58
1.19
2.27
3.96
1.40
1.33
1.98
1.57
1.38
1.66
2.67
928
1973
2229
2508
271
3527
842
2300
2400
1495
1907
1358
1538
1227
63
650
1246
2623
98
2477
1455
44
328
888
1118
115
1414
2700
keff
∆keff(void) (pcm) ∆k/∆B (pcm) Β1 (MWd/kg)
1.10184
-47
54
166
1.13060
81
156
84
1.02780
23
174
16
1.1085
275
24
424
ICR
0.76
0.71
0.68
0.72
1.11638
1.04289
1.04882
1.14914
1.15112
1.13513
1.10582
1.10741
1.07003
0.87239
1.03173
0.81876
1.01428
1.00979
1.00773
1.16061
1.13686
0.94022
0.6403
1.16882
0.60907
0.9236
1.1235
1.14233
1.12172
1.09198
1.05176
1.12424
1.12578
0.72
0.75
0.73
0.65
0.73
0.69
0.70
0.83
0.74
0.75
0.76
0.75
0.77
0.77
0.76
0.70
0.74
0.79
0.73
0.71
0.74
0.72
0.74
0.75
0.74
0.73
0.73
0.69
30
129
53
-43
-105
288
46
127
103
46
90
190
103
98
-16
72
150
-1
214
-180
148
224
9
76
268
60
53
120
123
32
28
54
27
320
43
249
46
146
127
30
43
50
43
119
51
22
73
38
73
64
29
28
90
35
12
46
180
419
78
183
49
351
54
232
74
55
74
-366
34
8
15
441
188
230
-120
258
424
136
261
414
234
70
Matrix
Sn
Sr
Sr2Pb
SrO
SrS
Ti
TiC
TiN
TiN15
U238
V
VC
Void
Zn
Zr
Zr90*
ZrC
ZrO2
Zrot*
ρ(g/cc)
7.31
2.63
ρCp(J/ccK)
1.59
0.78
Tmelt( C)
232
777
4.7
3.7
4.51
4.93
5.21
5.21
19.05
6.11
5.77
7.14
6.51
6.51
6.73
5.68
6.51
1.20
1.51
2.35
2.78
2.22
2.99
2.96
2.77
1.81
1.81
2.47
2.59
1.81
2430
2227
1668
3067
2950
2950
1132
1910
2810
420
1852
1852
3532
2677
1852
keff
∆keff(void) (pcm) ∆k/∆B (pcm) Β1 (MWd/kg)
0.97004
94
34
-89
1.15638
257
37
425
1.12379
255
28
445
1.09621
41
82
117
1.07972
183
37
217
1.09353
124
116
80
1.09154
18
239
38
0.99394
12
184
-3
1.08545
25
191
45
0.71207
123
1.04644
36
142
33
1.05622
86
235
24
1.20575
222
28
736
0.83196
-36
71
-237
1.04468
107
26
173
1.09325
134
42
219
1.03446
232
145
24
1.0457
-58
128
36
1.01598
388
41
39
*
Zrot: is natural Zr with Zr-90 removed. SDM of keff = ±0.0002 ≡ 20pcm, hence
CO2 voiding comparisons are only qualitative.
1.20575
600
500
S
400
Sr 2 Pb K
BaBa2PbPb Sr
Bi
300
PbS P
Ca Si
SrSZr90
B1
200
BaS
Zr
100
0
0.7
Al
SrO
0.8
PbO
Ti CaC2Al4 C3
Cr
SiC
CeO2 15
C BeO
TiN
Zrot
TiC
ZrO2
Fe ZrC
V VC
HT9
Fe3 CAlN
TiN
1
1.1
0.9
ZnC
-100
Na
Mg
1.2
1.3
Sn
Ni
-200
Zn
-300
Cu
-400
keff
Figure 3-3 Relation of initial multiplication factor and burnup potential
31
ICR
0.72
0.71
0.74
0.74
0.73
0.68
0.67
0.67
0.68
3.41
0.69
0.82
0.69
0.74
0.75
0.75
0.79
0.74
0.76
As shown in Fig3.3, there is no useful correlation of initial multiplication factor
and burnup potential except for a rough positive trend.
Cu
1.22
Zn
1.17
Co
1.12
1/keff = 1 - ρ
Ni
1.07
Mn
Sn
1.02
Fe
0.97
Zr
0.92
0.87
Al
Si
Pb
Ca
Ba
P
Cr
V
Ti
K Sr
No diluent
0.82
0.00E+00
5.00E-01
1.00E+00
1.50E+00
Σ a, cm
2.00E+00
2.50E+00
-1
Figure 3-4 Relation of multiplication factor and macroscopic absorption
Figure 3.4 shows that for metal matrix materials, 1/keff is linearly proportional to
the matrix material’s macroscopic absorption cross section.
We have
ρ=
k −1
1 νΣ f − Σ a
Σ
= 1− =
= 1− ∑ a i
νΣ f
k
k
i νΣ f
(3-2)
For different metal matrix cores, the spectrum average fission cross section is
close in magnitude. The main difference of reactivity comes from the different neutron
absorption ability of matrix materials. As expected, the inverse of multiplication factor
and matrix absorption is linearly correlated.
3.3.3 Fissile and fertile properties in the energy range of interest
For the 13% enriched Uranium carbide fueled diluent core defined in Chapter 2,
the energy spectrum is usually concentrated to the range between 1kev and 1Mev. The
32
presence of moderators can extend this energy range to lower energies. Neutron
absorption reactions and capture peaks will change the shape of the spectrum, but the
spectrum energy range is around the same. Figure3.5 and figure3.6 show the fissile and
fertile capture and fission cross sections from 1kev to 10Mev.
Figure 3-5 U235 capture, fission and elastic scattering cross sections *
*
The uppermost curve is the elastic scattering cross section; the middle curve is the
fission cross section. And the lowest curve is the capture cross section.
The figures show that the capture and fission cross section of U235 both decrease
with energy but the fission to capture ratio increases with energy. The U238 fission cross
section is threshold type, rising abruptly at ∼ 1Mev. Spectrum hardening will therefore
lead to an increase of reactivity in U235 and U238 mixtures.
33
Figure 3-6 U238 fission, elastic scatter, absorption cross sections
*
The uppermost curve is the elastic scattering cross section.The curve having
resonances at lower energies but decreasing smoothly at higher energies is the capture
cross section. The threshold type curve is the fission cross section.
3.3.4 Promising materials
Since the test core volume is relatively large, ∆keff(void) is not very sensitive to
leakage changes. Since the volume fraction of coolant is relatively small (~10%), the
∆keff(void) is also not very sensitive to coolant capture changes. However, ∆keff(void)
would be much larger for more realistic values (eg 25-50%), thus ∆keff(void) is
nevertheless a key criteria. Note that the amplitude of ∆keff(void) for the diluent cores is
somewhat smaller than that for the reflector comparisons of chapter 4. Thus the selection
of neutronically attractive materials is mainly based on their burnup potential (see fig3.7)
and melting point.
34
500
UNACCEPTABLE
PERFORMANCE
400
Zrot
300
Bi
Ba
Pb
Sr
Sr2Pb
ZrC
200
∆k(void)
Cr
100
SrS
Ti
CaC2
Al4C3
Fe
VC
HT9
CeO2 BaO
C
TiN15 SiC
TiC
0
0
Fe3C
50
100
Zr
SrO
P
PbS
PbO
150
Al
ZrO2
K
Ba Pb
S 2
Na
Zr90
Ca
Si
Mg
200
BaS
250
300
350
400
450
500
ATTRACTIVE PERFORMANCE
-100
BeO
-200
B1 MWd/kg IHM
Figure 3-7 Map of diluent material performance
Figure3.7 shows that, several materials are of potential interest, such as Sr2Pb,
Ba2Pb, K, S, Ba, Bi, P, PbS, Si, Ca, Na, Mg, BaS, Zr, Al. Sr, Ba and Pb have very close
∆keff(void) and B1; as expected, their compounds Ba2Pb and Sr2Pb, give even better
performance: higher B1, lower ∆keff(void). These materials will be discussed in detail in
the following sections.
3.3.4.1 Barium-2 Lead(Ba2Pb)
From table3.2, solid barium-2 lead has the longest burnup potential. The Ba2Pb
melting point is 928ºC, higher than both barium and lead. This gives Ba2Pb another
advantage over the individual constituents.
The diluent atom density of Ba2Pb is much less than that of BaS but a bit more
than Barium metal. Elimination of light moderators keeps the spectrum hard. The reduced
amount of barium and the small lead capture cross section lead to small diluent
absorption. The higher total diluent atom density and higher average lead scattering cross
section increase the interaction with neutrons hence reduce leakage. Thus, the overall
keff(BOL) is much higher than for the Ba diluent core.
35
1.00E+03
1.00E+02
1.00E+01
cross section (barn)
1.00E+00
1.00E-01
1.00E-02
Ba137_elscatter
Ba137_capture
Ba136_elscatter
Ba136_capture
Ba135_elscatter
Ba135_capture
Ba134_elscatter
Ba134_capture
1.00E-03
1.00E-04
1.00E-05
1.00E-06
0.001
0.01
0.1
1
10
energy (Mev)
Figure 3-8 Capture and elastic scattering cross sections for minor Ba isotopes
One thing to note is that among the 7 siblings, Ba-138 is the most abundant
naturally occurring isotope. Its cross section data is well studied and recorded in detail.
But for the other 6 isotopes, it appears that 1/v behavior is assumed to estimate the cross
sections: See fig3.7. Since over 70% of natural Barium is Ba-138, the 1/v estimation of
other isotopes is probably not too detrimental.
Ba is the heaviest non-radioactive alkaline earth metal element. Hence its slowing
down power is quite small. Even though the microscopic average scattering cross section
is around the same as aluminum, the spectrum softening effect of barium is much less
than for aluminum. Lead is much heavier than barium, hence the overall impact of Ba2Pb
on the core spectrum is very small. However the total absorption by barium is not
negligible, due to which the keff(BOL) of the Ba2Pb diluent core is lower than for the Pbonly core.
Due to their 1/v approximation, the Ba scattering and capture cross sections
change smoothly with increasing energy. Spectrum hardening reduces Ba’s capture, and
increases the fission to capture ratio. Since the fissile and fertile capture and fission are
36
very sensitive to spectrum hardening, ∆keff(void)leakage is less than ∆keff(void)spectrum. Since
∆keff(void)leakage is the only term which gives rise to negative total ∆keff(void), this will
lead to a positive ∆keff(void). Lead has smaller capture microscopic cross section and
smaller slowing down power than most other materials. It also has a lower atom density
in the core. The impact of lead on ∆keff(void) is much smaller than barium. Hence,
∆keff(void) is positive. The amplitude is a little smaller than pure barium metal but the
difference is whithin the standard error range.
The burnup potential of Ba2Pb is much larger than for the BaS core and is close to
that of the Ba core. This is mainly because of their similar hard spectra and almost the
same internal conversion ratio.
From the discussion above, if the library cross sections for Ba are valid, Ba2Pb
appears to be a good diluent candidate from a neutronic point of view. Its void coefficient
is a little bit high, but it is endurable.
3.3.4.2 Strontium-2 Lead (Sr2Pb)
Analogous to Ba2Pb, use of Sr2Pb can apply the small absorption of Sr and lead
while avoiding a low melting point. The absorption cross section of Sr is much lower
than barium, except for a few capture resonances below 0.2Mev.
Because of the smaller capture, Sr2Pb has a little higher keff at beginning of life
compared to the barium-2 lead core, consequently its burnup potential is close to that of
Ba2Pb. The difference of ∆keff(void) is inside the standard error range. The melting point
of Sr2Pb is a little higher than Ba2Pb, which gives it another advantage. It is a qualified
candidate, as good as Ba2Pb.
3.3.4.3 Potassium (K)
Na and K are an interesting pair in table3.2. The microscopic capture cross
section of K in a Pb matrix spectrum is around 20.1 mbarns, around 10 times that of Na
(2.0 mbarns), yet the B1 for K is almost twice that of Na. In the ENDF cross section set
used in MCNP, there is a resonance region in the epithermal range for Na, while for K,
37
the capture cross section curve is almost logarithmically linear, which probably indicates
1/v estimation. Although the integrated average total absorption cross section of Na is
much lower than that of K, Na’s resonance absorption is far stronger than K’s continuous
and flat curve. As will be seen in the later reflector study (chap 4), the difference between
Na and K is almost negligible when they are positioned peripherally as reflectors.
However, they both have very low melting points, not far above room temperature. Thus
it is not feasible to use them as a fuel matrix. However, both can be used as a fuel-to-clad
thermal bonding agent; and both are suitable LMR coolants.
3.3.4.3 Lead Sulfide (PbS)
Lead and Bismuth are good materials from the neutronic point of view. They are
heavy which makes the spectrum hard. Their absorption cross sections are small which
helps the neutron economy. No (n,α), or (n,p) reactions create annoying gas generation
problems. The lead matrix core has a positive void coefficient because of its greater
sensitivity to spectrum hardening and less sensitivity to increased leakage, but the
magnitude is acceptable. Pb and Bi would be the best choice if they had a high enough
melting point; thus different compounds of lead and bismuth are evaluated to exploit their
advantages.
S is a very interesting material based on our results. Except for the resonance
peaks, the absorption cross section of S is rather smooth. It also has a steep rise very
close to 1Mev caused by (n, α) reaction. That offsets the decrease of neutron capture as
the spectrum hardens and explains S matrix’s negative void response. The disadvantage
of using S as a matrix is obvious: it has a very low melting point (115.21ºC), a very low
density, and essentially no structural strength. Furthermore, it undergoes (n,α) reactions
to produce He, and (n,p) reactions to produce H. He, H2S, or H2 gas will be generated as
a result, thus additional internal pressure will be produced. On the other hand, S seems to
help improve internal conversion ratio: Almost all sulphur compounds have a low ∆k/∆B,
hence a bigger B1. The (n, α) cross section for natural S is 12.4 millibarns and its (n,p)
cross section is 5.2 millibarns based on the Pb matrix core spectrum. Considering the fact
that there is over 60% volume percentage of S in the core, using S will create a
significant amount of internal gas pressure. See Appendix A.
38
Accordingly we tried the compound PbS to avoid some of the above
disadvantages. PbS has a melting point of 1118ºC. The burnup potential is 258MWd/kg.
But its void response is bigger than both the Pb and S cases. The reactivity of a
compound diluent is not the simple summation of reactivities for that compound’s
components. This is explained in section 3.4.
3.3.4.4 Calcium (Ca)
Calcium is one of the alkaline earth elements. It is the fifth in abundance in the
earth's crust, of which it forms more than 3%. It undergoes (n, γ), (n, p), and (n, α)
reactions but the average total cross section is not very large. The elastic scattering cross
section has resonances in the range of interest. The atomic number of calcium is
intermediate. Its atomic density is lower than BaO, BeO and AlN, but higher than all the
other materials listed in the table. Less moderation makes its keff(BOL) higher than for
aluminum. More moderation and absorption make its keff(BOL) lower than for the barium
diluent core. For the ∆keff(void), since there is no significant Ca – U235 or Ca – U238
cross section coupling, the ∆keff(void) value is positive. The analysis is analogous to the
discussion in section 3.3.2.4. Considering that calcium’s total absorption increases at high
energy, the diluent absorption increases upon voiding. This compensates for the positive
∆keff(void) and the amplitude of ∆keff(void) is less than for a barium containing core.
The spectrum softening and larger absorption make the calcium core conversion
ratio lower than for barium. Thus the burnup curve is steeper and the burnup potential is
less. Overall considering the ∆keff(void) benefit, calcium is a usable material.
3.3.4.5 Silicon (Si)
Crystaline Si is expensive and its strength as a matrix is questionable. The
neutronic performance of Si is also mediocre. Thus we are more interested in its
frequently used compound, SiC. From our study, SiC is detrimental because of its carbon
content. Carbon down-scatters the neutrons and softens the spectrum significantly.
Together with several absorption peaks of Si, neutron economy is worsened. Even though
a softened spectrum can achieve a very high multiplication factor at the beginning of life,
the internal conversion ratio is very low. In addition, the generated fissile plutonium has a
39
poorer fission to capture ratio, thus keff drops down very quickly with the increase of
burnup. The extrapolated burnup B1 is only 70 MWd/kg. This performance of SiC is
much worse than pure Si.
3.3.4.6 Barium Sulfide (BaS)
Barium sulfide is a white crystal with the high melting point of 2229ºC. Its heat
capacity is around the same as BaO. Since sulfur has twice the atomic number of oxygen,
it is anticipated that a BaS diluent core will have a harder spectrum and inherit the high
burnup potential of Ba metal. BaS has some disadvantages. First sulfur has a relatively
large (n,α) cross section, so that the presence of sulfur will enhance gas generation and
increase the pressure inside the cladding (see appendix A). Second, some structural
materials may corrode in contact with barium sulfide (see ref[10]). Thus, a BaS diluent
core has more restrictions on material selection.
The atom density of BaS is less than BaO but they are of the same scale. Sulfur
has more and sharper elastic scattering resonances which begin at low energies. But
sulfur’s slowing down power is much smaller than oxygen; So their overall impact on the
neutron spectrum is around the same, except that the BaS core spectrum has fewer
moderated neutrons below ~50kev and has a deeper valley in the middle of the spectrum
peak compared with the BaO core(see Figure: 3.9). Thus the BaS core has more high
energy neutrons. The spectrum contribution to keff is higher. For the same reason, the
leakage of the BaS core is larger than for the BaO core, which is a not very important
drawback. Sulfur has 27 times the average total absorption cross section of oxygen in a
hard GFR spectrum. Hence the diluent absorption in BaS matrix cores is much more than
for BaO cores. This tends to depress its keff. Among the 3 factors, the spectrum impact is
the dominant one. Thus the BaS core has a slightly higher keff(BOL) than BaO.
40
3.50E-02
3.00E-02
BaS_spectrum
BaO_spectrum
2.50E-02
fraction
2.00E-02
1.50E-02
1.00E-02
5.00E-03
0.00E+00
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
energy (Mev)
Figure 3-9 Comparison of BaS and BaO diluent core spectra
When void is introduced, the spectrum turns harder, and leakage increases. Since
sulfur has an increasing (n, p) cross section and (n, γ) capture resonances at relatively
high energy, the total absorption by sulfur increases a little. This leads to a small
∆keff(void).
Even though the BaS core has a lower conversion ratio than BaO, its burnup
potential is higher. The reason is a much harder spectrum (analogous to section 3.3.2.2).
For the same reason, BaS has lower burnup potential than for a Ba metal matrix core.
3.3.4.7 Zirconium (Zr)
Zirconium alloys are very popular in LWRs because of the small absorption cross
section of Zr in the thermal energy range. But at high energies, the significant resonance
absorption reduces its advantage. This also helps to increase the ∆keff(void). The poorer
neutron economy reduces the core’s burnup potential. Nevertheles, compared with strong
41
moderators such as carbide compounds and strong absorbers such as copper, zirconium is
a material worthy of consideration.
3.4 Applicability of superposition
There are many many more compounds than pure elements. If we could predict
the performance of compounds by combining results for their constituent elements,
considerable work would be avoided. However, our results show that the reactivity effect
of compounds can not be simply expressed as a weighted function of individual
constituents because of changes in spectrum. The results for different amounts of the
same matrix also show non-linearity (see fig3.10), and thus accurate extrapolation or
interpolation for a given volume of matrix material is possible only over a narrow range.
However, results for different fuel enrichments for fixed matrix material show that the
reactivity-enrichment curve can be fit to a simple derivable function.
3.4.1 Non-linearity of neutronic effects as a function of matrix
concentration
1.32
1.3
Pb(perfectly reflected)
Pb(with leakage)
1.28
1.26
keff
1.24
1.22
1.2
1.18
1.16
1.14
0
50
100
150
200
volume ratio (matrix to fuel)
Figure 3-10 Non-linearity of neutronic effects vs. Pb matrix concentration
42
250
The non-linearity of core characteristics with diluent concentration was not
unexpected, in view of their tendency to soften the reactor spectrum. To show why this is
the case, and also to call attention to a way to take this effect into account, the approach
introduced by Sheafer [13] is noted. He showed that fast reactor-spectrum-averaged cross
sections can be correlated in the form:
σ = σ 1S g
(3-3)
where σ1 and g are constants for a given nuclide.
The spectral index S, the ratio of average neutron energy to fission neutron
energy, is given by:
ν ∑f
E
=S=
Ef
ν ∑ f + ξ el ∑ TR
(3-4)
in which
ΣTR and Σf = transport and fission macroscopic cross sections, respectively
ξel = logarithmic mean energy decrement for neutron scattering, (approximated
as that due to elastic scattering alone)
ν = mean neutron yield per fission
Since S is less than 1.0 and g typically a negative quantity, σ values increase as
the spectrum softens (S decreases), and by a different amount since different species have
different g values.
Sheafer studied a wide variation of oxide, carbide and metal fueled cores and
critical assemblies. He found that k could be calculated within ±0.59%
Even better results should result if one confines interest to a restricted range of
compositions, or focuses on relative comparisons.
3.4.2 Neutronic effects for a compound and its constituents
From Sheafer’s method, we would expect that since the spectrum of Al4C3 is softer
than that for Al, the average capture cross section of Al in an Al4C3 matrix is greater then
43
that in an Al only metal matrix. Thus a metal carbide matrix should always have a lower
keff than pure metal matrix.
In reality the opposite is true: the keff of cores with pure metal matrices are mostly
lower than their carbides. The reason is mainly because there are more reduced energy
neutrons contributing to total fission rate at the beginning of life in a softened spectrum.
Also, carbon has almost no neutron absorption cross section. That makes the neutron
utilization factor much larger than a metal core with the same diluent atom density.
We can approximate kinf by requiring linear addition of reactivity losses relative to
a no diluent (i.e. void in place of diluent) reference core
ρ (void ) − ρ (compound) = ∑ ( ρ (void ) − ρ (component i only) )
(3-5)
i
where component i is present at the some number density as in the compound.
0.25
0.2423
ρ(void) - ρ(Al)
=0.078679
0.2
ρ(void) -ρ(C)
=0.046076
0.1962
ρ(void) - ρ(Al4C3)
=0.094821
0.1636
0.1475
0.15
2ρ(void) − ρ(C) + ρ(Al)
=0.124755
ρ
0.1176
0.1
0.05
0
Void
Al only
C only
material
Al4C3 actual Predicted by Superposition
= ρ(Al) + ρ(C) -ρ(void)
Figure 3-11 ρ vs. compound components
In the comparison shown in Fig 3.11 note that
•
The maximum standard error of keff is 0.00086
44
•
“Al only” is a matrix with the same Al number density as in Al4C3 but without the
C component; similarly for “C only”.
•
All the core models are perfectly reflected, hence leakage is not relevant.
Thus, the reactivity of Al4C3 could be expressed as
ρ (Al4 C3 ) = ρ (Al only) + ρ (C only) − ρ (void )
(3-6)
The standard error of kinf is 80pcm, thus the estimation of the Al4C3 core’s reactivity
should be within ±240pcm. Figure3.11 shows that the deviation of ρ from the linear
approximation to the real MCNP simulation is 2990pcm, far beyond the standard error
change. This demonstrates the non-linearity relationship between multiplication factor in
the compound containing core and that inferred from its single component cores.
3.4.3 Relation of reactivity to enrichment
Reactivity is defined as
ρ=
ν ∑ f − ∑a k −1
=
ν ∑f
k
(3-7)
For a mixture containing U-235, U-238 and diluent materials, it is not difficult
(see Appendix B) to show that
ρ=
[η 25 − 1 − λ (η28 − 1)]x + [λ (η28 − 1) − γ ]
(η25 − λη 28 ) x + λη28
(3-8)
where
η = neutrons produced per absorption
λ=
σ a 28
σ a 25
 ∑ a , diluent & other absorbers 

∑ a ,U − 235


γ = 
Hence if spectrum averaged cross sections remain unchanged, one expects a relation of
the form:
45
k eff =
ax + b
x+c
(3-9)
From curve fitting for a mirror-reflected infinite cylinder core with pure UO2 fuel, we
get:
a = 2.5098
b = 0.042
c = 0.1235
where fractional enrichment x ∈ [0, 1]
This relation is plotted in Fig3.12, and tracks the calculated points quite well. It is
anticipated that for the same diluent material, the relationship between enrichment and
keff is the same but with different values of a, b, and c.
Figure 3-12 Relationship between enrichment and keff for a representative core
46
3.5 Conclusions
We have compared approximately 50 materials as core diluents in this chapter. As
would be expected, strong moderators such as C and BeO are detrimental because they
soften the spectrum, reducing fissile η and increasing parasitic absorption. The metal Zr,
so useful in thermal reactors, is here a mediocre performer; nevertheless it has a high
volumetric heat capacity and better structural properties than most other metals with
higher B1. The alkaline earth metals (Mg, Ca, Sr, Ba) are relatively benign diluents, as
predictable from their relatively small absorption cross sections. Al and Ni confer a
negative coolant void coefficient by virtue of their relatively large (n,α) and/or (n,p )
threshold reactions. As expected, Pb excels. However, because of its low melting point, it
could only be employed in exotic concepts such as molten matrix cermet fuel (see
ref[15]) or perhaps as its oxide or sulfide compounds. SiC, which has favorable material
properties, is at best average with respect to neutronics, but should not be ruled out at this
point if ceramic cercer or cermet fuel is preferred. To summarize, we found some good
materials such as Pb, Bi, Ba, but they all have low melting points. Use of a molten salt
matrix or a liquid metal coolant design could be a feasible solution. There are also good
candidates such as Sr2Pb, Ba2Pb, PbS, if the requirement of void coefficient or burnup
potential is not too restrictive. What material is the best one depends on specific design
considerations. If a metal matrix (e.g. cermet fuel) is preferred, then Zr, V and Ti should
be evaluated.
47
Chapter 4 Review of reflector material candidates
4.1 Introduction
As for the evaluation of matrix materials in the core, the evaluation of reflector
candidate materials for gas-cooled fast reactor core design was based on static beginningof-life reactivity calculations and fuel burnup analyses. MCODE and MCNP were
executed using the core models and regional compositions given in Chapter 2. Since the
reflector acts more to set boundary conditions and has less impact on the core neutron
energy spectrum compared to matrix materials and since reflectors can tolerate more
neutron absorption, there are more reflector choices than for matrix use. In section 4.2 we
will present data for all the reflector candidates and then discuss them in groups.
Parameter studies are included in section 4.3. Conclusions drawn are presented in section
4.4.
4.2 Review of material candidates for reflector
4.2.1 Albedo calculation
Reflector performance is often characterized by the albedo values at core-reflector
boundary surfaces. The tabulated values of outgoing and return current at the radial
periphery in MCNP permit inference of albedo for the materials under study from the
relation:
α=
J−
J+
(4-1)
For example, for Pb, one has α ≈ 89%. This shows the inferior nature of fast spectrum
reflector performance if one recalls that good thermal spectrum reflectors such as D2O,
Be and C have albedos of 95% and higher. Since thermal hydraulic and fuel economic
considerations favor radial and axial power flattening, it is also difficult to offset this
inherent shortcoming even by significantly increasing core size (hence plant power
rating).
48
Theory provides only rough and potentially misleading guidance in reflector
selection. In particular, simple one group theory provides an expression for the albedo of
a thick weakly absorbing slab:
α = 1−
4 σa
3 σs
(4-2)
Since at high neutron energies the scattering cross-section, σs, varies only slowly and
systematically with nuclide mass (roughly as A to the 2/3 power), a low value of the fast
spectrum average absorption cross section, σa, is a first order indicator of suitability. This
criterion is useful for initial screening purposes, but in reality the situation is more
complex since moderation also plays a role. Degradation in neutron energy causes a loss
of neutron worth (which varies roughly as k(E)), hence the effects of both elastic and
inelastic downscatter must also be taken into account. Figure4.1 plots α vs σa:
100.00%
BeO
B11
90.00%
Bi
Zr90 PbS ZrSi
2
SiC
Na
Si
albedo
TiSi2
FeSi2
Al
80.00%
Fe
NaCl
70.00%
Co
CoS
BaO
Cu FeS
Mn FeS
BaS 2
TiN Zn
Ba
Sn
ZnS
Mo
Nb
ZrH
Rb
H 2O
60.00%
S
Ca
50.00%
K
40.00%
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
σa (mbarn)
Figure 4-1 Variation of albedo with absorption
The lack of coherent trend is obvious.
49
160.0
180.0
4.2.2 General Results
Similar to the evaluation of matrix materials, aspects compared were beginningof-life multiplication factor, k, coolant void coefficient and the linearly extrapolated
burnup potential, B1=(k-1)/(∆k/∆B). Note that the core diameter was reduced to 180cm to
increase sensitivity to leakage. Table 4.1 summarizes the results.
Table 4-1 Neutronic Comparisons of GFR Reflectors
Species
bare
Al
AlN
B11
Ba
Ba2Pb
BaO
BaS
BeO
Bi
C
Ca
Co
CoS
Cr
Cr3Si
Cu
Eu
Fe
FeS
FeS2
FeSi2
H2O
Hg
K
Mg
Mn
MnS
Mo
MoSi2
Na
NaCl
natUC
Nb
Ni
keff
1.02370
1.07453
1.07197
1.22993
1.06329
1.09198
1.07023
1.06625
1.25383
1.10477
1.24346
1.04402
1.07561
1.07021
1.07297
1.07528
1.06921
1.03602
1.06906
1.06730
1.06738
1.07603
1.09463
1.05132
1.03877
1.09846
1.06678
1.06372
1.06154
1.06338
1.07148
1.06556
1.05606
1.05723
1.07301
∆kvoid, pcm
21
345
183
294
50
141
86
18
262
392
451
352
304
196
335
176
399
-14
386
-1
102
190
-17
365
206
384
315
138
172
21
288
407
-198
139
98
50
B1,MWd/kg
115
231
157
83
247
395
217
326
83
326
87
193
346
423
247
299
154
255
249
267
254
45
231
147
136
298
262
258
283
158
216
290
239
234
Albedo
0
80.65%
77.20%
91.76%
74.09%
85.89%
78.67%
76.36%
93.50%
88.70%
93.03%
54.11%
80.97%
78.90%
78.75%
80.13%
77.70%
42.42%
77.79%
77.26%
77.06%
80.72%
62.90%
66.98%
46.56%
85.48%
77.45%
75.40%
73.55%
75.29%
77.40%
75.70%
70.24%
70.11%
78.66%
Species
NiS
P
Pb
PbO
PbS
Rb
S
Sc
Si
SiC
Sn
Sr
Ti
TiN
TiSi2
Ti5Si3
V
V3Si
Zn
ZnS
Zr
Zr90
ZrC
ZrH
ZrNi
ZrS2
ZrSi2
Zr3Si2
mirror
*
keff
1.07012
1.07274
1.10855
1.11636
1.09047
1.04239
1.04421
1.06058
1.06382
1.10772
1.06149
1.07046
1.07224
1.06582
1.08096
1.08253
1.07563
1.07781
1.06459
1.06363
1.08799
1.09148
1.09773
1.09921
1.07863
1.07190
1.08877
1.08925
1.19830
∆kvoid, pcm
442
247
198
342
390
-31
82
31
415
376
391
376
265
63
6
83
365
182
375
-12
118
19
134
97
106
55
1
90
310
B1,MWd/kg
212
183
317
170
359
168
282
239
232
105
317
303
226
175
193
237
297
267
287
265
144
53
264
403
233
252
778
Albedo
78.13%
77.77%
89.51%
89.10%
85.63%
56.68%
73.68%
74.06%
85.17%
72.43%
77.84%
77.92%
74.75%
81.62%
82.08%
79.96%
80.99%
75.19%
75.02%
85.09%
85.54%
85.03%
69.27%
81.14%
84.98%
85.25%
100%
The standard deviation of keff is ±30pcm, hence CO2 coolant voiding comparisons
are only qualitative.
**
The uranium carbide reflector has a natural U235 enrichment.
4.2.3 Detailed evaluation and explanation
Similar to the selection of diluent material, materials with small ∆keff(void) and
large B1 are preferable. Since the reflector could be liquid in cans, there is no melting
temperature restriction on reflectors. Hence, the feasible choices are much more
numerous than for diluent candidates. The best reflectors are Zr3Si2, Ba2Pb, ZrS2, and
51
BaS. Many other sulfide and silicide compounds are also good candidates. The next
section will introduce these candidates in groups.
500
C
NiS
NaClSi
400
SiC
Cu SnBi
ZnSr
Fe
Mg
UNSUITABLE
PbO
Ca
HgV
Al
PbS
Cr
Mn
300
B11
BeO
Ti
P
K
200
∆k(void)
Nb
ZrC
ZrH
bare
0
HO
50 2
100
150
Eu
ZrNi
FeS2
S
Ba
Sc
TiSi2
Rb
200
ZrSi2FeS
250
CoS
Ba2Pb
MnS
BaO
TiN
0
Pb
FeSi2
Mo
AlN
100
Co
Na
Zr
ZrS2
Zr90 MoSi2
ZnS
300
BaS
350
400
450
Ni
-100
AREA OF INTEREST
natUC
-200
B1 MWd/kg IHM
Figure 4-2 Map of reflector material performance
4.2.3.1 Zirconium sulfide (ZrS2) and other sulfide compound reflectors
Except for CoS, the zirconium sulfide reflector system has the longest burnup
potential among all the materials tested. Since Co-59 produces Co-60 by (n, γ) reaction,
ZrS2 is preferable because of its lower induced radioactivity. Note that many other sulfide
compound reflectors (eg, BaS, ZnS, FeS2, MnS, FeS, and NiS), lead to reasonably high
burnup potential. This is mainly caused by the high reflectivity value at the core –
reflector surface for these sulfide compound systems.
Sulfur has a high resonance scattering microscopic cross section in the energy
range 0.1Mev ~ 1Mev. This reduces the core neutron leakage. But the slowing down
power of sulfur is not that large. This effectively changes the neutron’s direction without
softening the core neutron spectrum significantly. The hard spectrum leads to a high
burnup potential.
52
Comparing the scattering cross section of zirconium and sulfur, one finds that
Zr’s scattering resonances end at around 0.1Mev, which is at the beginning of sulfur’s
scattering resonances. Thus, almost all the important region of a GCFR neutron spectrum
is covered by the strong reflecting scattering in Zr or S. One thing to note is that even
though ZrS2 is a good reflector, it is not a good candidate as a diluent. This is because of
the absorption cross section resonances of Zr and S which increase in magnitude at lower
energies.
The other advantage of a sulfide reflector is that sulfur helps to depress the
coolant void coefficient because its microscopic absorption cross section increases at
high energy. For almost all sulfide compounds and sulfur itself, whether they are used as
diluent or reflector, the system’s void coefficients are always small and endurable
compared to most other materials.
4.2.3.2 2-Barium Lead (Ba2Pb)
As shown in chapter 3, Ba2Pb is a good candidate material as a diluent. It is also a
good candidate for reflector service. The relatively large average scattering cross section
of Pb helps reflect outgoing neutrons. Its large atomic number leads to a hard core
neutron spectrum and helps burnup potential. The void reactivity coefficient for Ba2Pb is
a little larger than for BaS and lead. It is caused by the descending slope of barium’s
absorption cross section as energy increases. Given an appropriate arrangement of
additional reflector layers, for example, if we add a reflector layer which enhances the
negative void coefficient outside of the Ba2Pb layer, the overall performance could
potentially be improved.
4.2.3.3 Molybdenum disilicide (MoSi2) and other silicide reflectors
Silicides attract attention because of their potential to withstand high operating
temperatures. Among them, MoSi2 is one of the best performers according to our study.
The MoSi2 reflector system’s burnup potential is around 300MWd/kg with a ∆keff(void)
which is negligible considering the estimated standard deviation of the MCNP runs.
The high average microscopic scattering cross section of Mo helps to reflect the
outgoing neutrons back into the core. Silicon does not have as high an average
53
microscopic scattering cross section as molybdenum. However, its scattering resonances
begin at around 0.3Mev, extending to over 3Mev, hence covering the fast neutron
spectrum. Even though the average absorption cross section of Mo is quite large at
energies of interest, it does not affect the core’s neutron economy significantly. This
shows that good peripheral reflector materials are not necessarily good in-core fuel
diluents. Co, Ni, Cu, Zn, Mn, Nb, Mo, Sn are inferior diluents due to their large
absorption cross sections, but fairly good reflectors, considering only burnup potential.
Most silicide reflectors exhibit a small ∆keff(void). This is partly caused by the
increase of absorption in silicon at higher energies via (n,p) and (n,α) threshold reactions.
4.2.3.4 Zirconium (Zr)
Zirconium is used in LWRs in alloy form as cladding material. Our result shows
that it is also useful as a GFR reflector. The relative high atomic number, and lower
energy absorption resonances help maintain a hard spectrum. The relatively large
scattering cross section reduces leakage. Loss of coolant sends more neutrons into the
reflector region and increases capture in zirconium. As for silicon, Zr also has an increase
of neutron capture at high energy. Although the ascending slope appears at an energy
higher than that of silicon and with smaller amplitude, this still helps reduce the positive
∆keff(void). We also tested use of separated Zr-90, but the advantage of less absorption is
not very large.
4.2.3.5 Nickel (Ni)
Nickel is one of the best reflectors from the coolant void coefficient point of view.
It is the reflector material with a very low void coefficient in uranium cores. Nickel’s
atom density is among the highest among our test materials; nickel also has a higher
average scattering cross section. This assures that a nickel reflector will have a high
albedo. Nickel’s absorption is stronger than zirconium and the high energy end total
absorption cross section increase caused by its (n,α) threshold reaction is much larger
than for zirconium in amplitude, (its threshold energy is also smaller). All of the above
leads to a reduced void coefficient.
54
However, nickel’s relatively low atomic number makes the reflected spectrum a
little softer than that for heavy metal compound reflectors such as Ba2Pb. It also is
sensitive to fissile component changes. Thus the burnup potential of a nickel reflected
system is not as high as that for ZrS2. If we give high priority of consideration to void
coefficient, nickel is a suitable choice. Again mixtures or layers combining nickel with
other good reflectors may be an alternative.
4.2.3.6 Nb, Ti, Rb, Eu, Sc, etc
Because of the less restrictive penalties of absorption, the choices of reflector are
broadened to a large extent. Nb, Ti, Rb, Eu, Sc, and their high melting point compounds
could all be considered as reasonable candidates. However, with the exception of Ti,
higher cost would undoubtedly rule them out.
4.2.3.7 Blanket (natural Uranium carbide)
A conventional blanket was also investigated as a reflector. Results show that the
reflecting capability of natural uranium carbide is much worse than lead(the albedo is
much smaller). A uranium blanket leads to a negative core coolant void coefficient.
Although blankets are not preferred because of non-proliferation concerns, at this stage
uranium carbide should be carried forward as a potential candidate – especially for axial
blankets, where they are an integral part of the fuel pin.
4.2.3.8 Trizirconium disilicide (Zr3Si2)
Zr3Si2 is recommended by the French GFR research group at CEA. It has a high
melting point of over 2000ºC, which is an additional advantage. Silicide is one of the best
low average absorption materials except for Na, Mg and some strong moderaters. It also
doesn’t have dense absorption resonances in the fast spectrum. The small scattering cross
section helps to reduce the overall contribution by silicon to core spectrum softening. The
relatively low zirconium scattering cross section and its high atomic number gives even
less contribution of slowing down. Small absorption and moderation lead to a long B1.
55
Figure4.8 shows that the B1 of Zr3Si2 determined by a full core lifetime burnup
calculation is longer than strontium and barium sulfide.
Silicon undergoes (n,α) and (n,p) reactions, which helps reduce the coolant void
coefficient.
4.2.4 Brief summary
From table(4.1) and the discussion above, several conclusions can be drawn. Strong
moderators such as BeO and C increase beginning-of-life reactivity, but significantly
decrease reactivity-limited burnup capability. Ni confers a reduced coolant void reactivity
in part because of its (n,α) threshold reaction, but some otherwise good reflectors such as
Cu cause significant increases. U-238 does not, in this example, breed sufficient
plutonium to confer a larger reactivity-limited burnup than many non-multiplying
reflectors. It does however contribute a significant negative ∆k void. Good peripheral
reflector materials are not necessarily good in-core fuel diluents. For example, Ni, Nb,
Eu, Rb, Sc are inferior diluents due to their large σa, but fairly good reflectors.
4.3 Parameter Studies
4.3.1 Reflector thickness requirement
As Figure 4.3 shows, for a nickel reflector, the multiplication factor reaches its
saturation value at around 25cm. Hence, 20 ~ 30cm of nickel (or other candidates)
suffices for the purpose of reflection. Our test cores use 90cm thick reflectors, which are
far more than necessary. The material beyond 25cm is necessary, however, to reduce fast
reactor fluence on the reactor vessel, and in fact should be optimized to best satisfy
shielding requirements.
56
1.146
1.144
1.142
1.14
keff
1.138
Pb matrix
Ni reflector
std err = 20pcm
1.136
1.134
1.132
1.13
1.128
1.126
0
10
20
30
40
50
60
reflector thickness (cm)
Figure 4-3 keff versus nickel reflector thickness
4.3.2 UPuC fuel – UC fuel
Figure 4.4 shows that the beginning of life multiplication factors of UC fuel and
UPuC fuel for the various reflector materials are roughly proportional to each other, since
average capture and fission cross sections in a fast spectrum for U-235 and fissile Pu are
of the same magnitude. For both core types the fissile enrichment is 13% of heavy metal;
but the Pu has the isotopic composition of typical PWR spent fuel.
Unlike the case for uranium, for plutonium fueling the void coefficient does not
differ significantly among the different diluents. This is determined by the detailed cross
section energy variation of U-235 and Pu-239. For the latter, the capture and fission cross
sections are more sensitive to spectrum changes, and thus the void coefficient for UPuC
fuel is larger than for UC fuel. This is significant because even if one starts with U-235
enriched uranium, it is eventually replaced by plutonium, at which point any beginning of
life advantage is lost. Thus one must plan to accommodate the positive coolant void
reactivity under the worst case, namely end of a core burnup cycle. One thing to note is
that sulfur has a relatively small void coefficient for a plutonium core because of its large
(n,α) cross section. But its advantage is not inherited by its compounds.
57
UPuC_UC
1.16
1.14
1.12
keff (UPuC)
1.1
1.08
1.06
1.04
1.02
1
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
keff (UC)
Figure 4-4 keff (UC fuel) – keff (UPuC) fuel
800
700
600
500
∆kvoid, pcm
400
300
P
200
100
0
1 Rb
H2OEu
Sc
BaS
FeS
6
Co Mn
Na B11
BeO
Ca
Cr Al
Hg
Si
NaCl
PbSSn Bi Cu
SiCMg Fe
NiSC
UPuC
UC
K
CoS
FeSi
2
Mo AlN
Nb Ba2Pb
MnS
Ni FeS2
S BaO
Ba ZrS2
11
16
21
26
31
36
41
-100
-200
reflector
Figure 4-5 Comparison of ∆keff(void) for UPuC fuel and UC fuel
Figure 4.6 shows an example of ∆k(void) increase with burnup for an initially
uranium carbide fuel and nickel reflector.
58
600
500
∆kvoid (pcm)
400
300
∆k(void)(Ni)
200
100
0
0
20
40
60
80
100
120
140
160
180
200
burnup (MWd/kg IHM)
Figure 4-6 The ∆k(void) increase with burnup
4.3.3 keff – albedo
As stated in 4.2.1, albedo is closely related to the multiplication factor since this
parameter characterizes the leakage at the periphery. Figure 4.7 shows that this
relationship is monotonically increasing, except for some strongly moderating
reflectors.The difference between 13w/o U-235 and Pu fueling appears to be constant and
due to the lower fissile Pu content.
59
100.00%
CBeO
B11
90.00%
Pb
Bi
Al
BaO
AlN
BaS
MnS
Ba
Mo
Ni
UC
Nb
albedo
70.00%
C
BeO
Bi
Ba2Pb
SiC
SiC
80.00%
B11
TiSi2
albedo_keff_UPuC
albedo_keff_UC
BaO
BaS
Ba
Sn
UC
ZrH
ZrH
Hg
Hg
Rb
H2 O
H2 O
60.00%
S
S
Ca
Ca
50.00%
K
K
Eu
Eu
40.00%
1
1.05
1.1
1.15
1.2
1.25
1.3
keff
Figure 4-7 Relationship of multiplication factor and albedo
4.3.4 Full burnup study of several interesting reflector materials
Even though the linear extrapolation method gives a rough idea of multiplication
factor changes during burnup, the estimation has significant uncertainty. In light water
reactors, the reactivity (or multiplication factor) is a nearly linear function of burnup.
(ref[19]) But for fast reactors, the conversion ratio is high. As the U-235 burns out, more
and more plutonium is produced, and more and more additional reactivity is contributed
to the total. Hence the reactivity – burnup curve is a convex function instead of a linear
function. Thus for accurate comparisons a full burnup study is necessary. Because
MCNP/ORIGEN burns are very time consuming, only several materials of interest were
studied.
60
1.12
Zr3Si2
Pb
Ni
Sr
BaS
Ba2Pb
1.1
1.08
1.06
keff
1.04
1.02
1
0.98
0.96
0.94
0
20
40
60
80
100
120
140
160
180
200
burnup (MWd/kg)
Figure 4-8 Full burnup runs for different reflectors
Figure 4.8 gives the multiplication factor – burnup curves for several reflector
materials for uranium carbide fuel. It shows that Pb is the best reflector from neutronic
point of view, and Ba2Pb, Zr3Si2 maybe the best realistic choices since their melting point
is over 900ºC: much better than nickel and strontium, even though Zr3Si2’s linearly
extrapolated B1 in Table4.1 is lower than strontium. Hence all the materials in the
acceptable region in Figure 4.2 should be studied in detail. The results in figure 4.8 are
also interesting that BOL keff is a farely good performance index if one confines attention
to the best performers.
4.4 Conclusions
According to the discussion above, there are many choices of reflector material.
They all have their competing advantages. Among them, trizirconium disilicide,
zirconium sulfide, barium-2 lead and nickel appear to be the best four materials
considering both burnup potential and void coefficient. A natural uranium blanket does
not increase the core’s leakage significantly; thus it is also a feasible choice if nonproliferation concerns are not a disqualifying issue. It has the (beginning of life)
advangage of a large negative coolant void contribution.
61
Chapter 5 Summary, Conclusions and Recommendations
5.1 Summary and Conclusions
The work documented in this report had as its objective a broad ranging
evaluation of potential materials for use in GFR service. The principal criteria were
neutronic, but qualitative consideration was given to thermal and mechanical properties.
In addition, the evaluation was conducted with specific reference to the proposed use of
CO2 as the coolant/working fluid, in a direct or indirect Brayton cycle. Finally our
concern was mainly with non-fuel constituents, hence UC was specified as the fuel phase
throughout.
The methodology employed involved use of Monte Carlo and burnup isotopics
codes: MCNP and ORIGEN, coupled by the in-house program MCODE. A simplified
standard whole core model was defined, consisting of two regions, a homogenized core
and a reflector, and individual constituents were tested one-by-one to generate
performance data: initial multiplication factor, coolant void reactivity, linearlyextrapolated reactivity-limited burnup potential, and for reflector candidates, their albedo.
5.2 General Evaluation Results
Materials cause spectrum changes and absorb neutrons to an extent which differs
when they are used for different functions, e.g. diluent, cladding, coolant, and reflector.
Based on our results, the best of the candidate materials can be grouped into several
categories:
62
Table 5-1 General Evaluation Results
Element
Al
Ba
Bi
C
Ca
Co
Cr
Si
Cu
Fe
K
Mn
Possible
Use
REF
DIL REF
COO REF
DIL REF
DIL REF
REF
REF CLA
REF DIL
REF
REF CLA
COO DIL
REF CLA
Mo
REF CLA
Na
COO DIL
Usable
Forms
SNG, ALY
SNG, COM
ALY
SNG, COM
SNG
SNG, ALY
SNG ALY
SNG COM
SNG ALY
ALY
ALY SNG
ALY SNG
ALY SNG
COM
SNG ALY
Element
Ni
P
Pb
S
Sn
Ti
U
V
Possible
Use
REF CLA
DIL
COO REF
DIL
REF DIL
REF
REF CLA
DIL
REF
REF CLA
DIL
REF
Zn
Zr
REF CLA
DIL
Usable
Forms
SNG ALY
COM
SNG ALY
COM
SNG COM
SNG ALY
SNG ALY
COM
COM ALY
SNG ALY
SNG ALY
COM
ALY COM
SNG
KEY: CLA = cladding,
REF = reflector,
COO = coolant,
DIL = diluent, cermet or metmet matrix,
ALY = alloy,
SNG = single element, principal constituent of alloy
COM = chemical compound, eg. sulfide, silicide, etc.
Criteria leading to the classification in table 5.1 are as follows:
CLA: Cladding. Requires low neutron absorption cross section, low neutron
scattering cross section, small slowing down power, adequate strength, adequate
resistance to radiation damage, high thermal conductivity, high melting point and high
corrosion resistance.
COO: Coolant. Requires low neutron absorption, low neutron scattering, small
slowing down power, high thermal conductivity, low melting point, and high boiling
point.
DIL:
Diluent. Requires small neutron absorption, low neutron scattering, small
slowing down power, high thermal conductivity, large heat capacity, high melting point
and adequate strength.
63
REF: Reflector. Requires no more than moderate neutron absorption, high
neutron scattering cross section but small slowing down power, melting point above
normal operating temperature and adequate strength.
Also, because of their different chemical properties and manufacturing procedures,
including the objective of combining the neutronic properties of different materials, these
materials may appear in 3 forms:
SNG: Single element or major alloy constituent
ALY: minor alloy constituent (if minor, can have larger σa)
COM: use in chemical compound
To evaluate the overall performance of a certain material, we need to consider its
unalloyed properties, potential of alloying, fabrication strength and resistance to
corrosion, in addition to its neutronic properties. A Tmelt ≥ 1000ºC is probably needed. As
noted earlier, we mainly focus on a discussion of a material’s physical properties as a
matrix, cladding or reflector. Materials given serious further consideration have test cores
with a beginning-of-life multiplication factor, k bigger than 1. For matrix studies, the
coolant void coefficient of all materials should be less than β of Pu (~350pcm). A
negative ∆kvoid is a significant benefit, although hard to obtain. As an important
consideration in assessing fuel cycle performance, the linearly extrapolated burnup
potential, B1 varies significantly among materials. This is caused in part by the sensitivity
of the conversion ratio to spectrum hardness. For the diluent cases, B1≥150MWd/kg is a
reasonable requirement. For reflectors, since many reflector candidates give a
B1≥150MWd/kg, we give materials with B1≥200MWd/kg higher priority of
consideration. Rarely are materials advantageous for all three evaluation parameters.
Hence one must settle for a reasonable compromise. An important observation is that to
explain minor differences between material performance of interest, spectrum weighted
cross sections based on a standard reference spectrum can not necessarily indicate
neutron behavior accurately. It is necessary to use case-specific neutronic spectra.
64
5.3 Recommendations for future work
All things considered, the following materials appear best suited for further
consideration in specific GFR core designs:
Metallic fuel diluents or matrices (eg. CERMET or METMET): Zr, Ti, V, Ba2Pb;
High temperature fuel diluents or matrices (eg, CERMET, CERCER): SiC, BaS
Cladding: Fe alloys with Cr, Al (eg ODS)
Reflector: Zr3Si2, Pb, Ba2Pb, ZrS2, MoSi2 plus a variety of sulfides and silicides
Future work also needs more attention to the interaction of other core materials
with fuel type and composition. The present work was almost exclusively focused on UC
and U-235 enrichment. However enough was done to show that Pu-239 induces a
significantly different behavior – for example, a much higher coolant void reactivity,
which is less suspectible to mitigation by selection of other core or reflector constituents.
Future work should involve repeating tests for fuels other than UC, for example, UO2,
U10Zr, and fissile other than U-235, for example, plutonium fuel with representative
isotopic compositions.
The present work also used a block type fuel with a very low CO2 coolant volume
fraction. Since coolant void ∆k is of paramount importance in LOCA accidents, future
work should investigate its behavior at higher volume percent, means for positive void
∆k reduction, and the relative behavior of CO2 and He in this regard. In particular we
need to increase the volume fraction of coolant to 25%~50% to cover the parameter space
representative of pin-type cores. This will increase ∆kvoid by 2-4 times. A study of this
type is currently underway at MIT.
Another task left for future study is the optimization of radial reflector/shield
composition as a function of pressure vessel fluence. Only about 25cm are needed to
65
realize the maximum albedo. Thus one can modify outboard configuration to reduce
fluence on the reactor vessel.
In view of apparent cross section library differences, results should be compared
using different available libraries(JEF, JENDL). For some nuclei, (for example, K and
Ba,) their absorption cross section data from ENDF is suspicious since their cross section
vs. energy curves appear to be artificially smoothed at high energy.
66
References
[1] http://minerals.usgs.gov/minerals/pubs/metal_prices/
[2] G.J. Janz, “Molten Salts Handbook”, Academic press, (1967).
[3] ANL-5800, Reactor Physics Constants, U.S. Atomic Energy Commission, Division
of Technical Information , Washington, (1963).
[4] Zhiwen Xu, Pavel Hejzlar, Michael J. Driscoll, and Mujid S. Kazimi, An Improved
MCNP-ORIGEN Depletion Program (MCODE) and Its Verification For High-Burnup
Applications, PHYSOR, Seoul, Korea, (2002).
[5] Judith F. Briesmeister, MCNP TM — A General Monte Carlo N-Particle Transport
Code, Version 4C, LA-13709-M, Los Alamos National Laboratory, (2000).
[6] Allen G. Croff, A User’s Manual for the ORIGEN2 Computer Code, ORNL/TM7175, Oak Ridge National Laboratory, (1980).
[7] Xianfeng Zhao, Pavel Hejzlar, M.J. Driscoll, Comparison of Code Results for PWR
Thorium/Uranium Pin Cell Burnup, MIT-NFC-TR-027, Center for Advanced Nuclear
Energy Systems, MIT (2000).
[8] C.M. Kang, R.O. Mosteller, Incorporation of a Predictor-Corrector Depletion
Capability into the CELL-2 Code, Trans. Am. Nucl. Soc., (1983), vol. 45, pp. 729-731.
[9] Hejzlar P., Driscoll M.J., and Todreas N.E., A Modular, Gas Turbine Fast Reactor
Concept (MFGR-GT), Trans. Am. Nucl. Soc.Vol. 84, Milwaukee, June 17-21, p. 242,
(2001).
[10] John A. Dean, Lange’s Handbook of Chemistry, McGRAW-HILL, New York,
(1999)
[11] Corrosion Survey Database (COR·SUR), NACE and NIST, Gaithersburg, MD,
(2002)
[12] Eugene A. Avallone, Theodore Baumeister III, Marks' Standard Handbook for
Mechanical Engineers, 10th ed., McGRAW-HILL, New York, (1996), pp. 6-82
[13] Charles A. Harper, Handbook of Materials for Product Design, McGRAW-HILL,
New York, (2001), ch7, pp 7.41-7.42
67
[14] Richard P. Pohanish, Sittig's Handbook of Toxic and Hazardous Chemicals and
Carcinogens, 4th ed. Noyes Publications, Norwich, NY, (2002)
[15] L. Biondi, Research and Development Proposal for a Fuel Element Made up with
Uranium Oxide Grains and a Lead Mixture Contained in a SAP Tube in Fuel Element
Fabrication with Specific Emphasis on Cladding Materials(Proceedings of IAEA
Symposium, Vienna May 10-13, 1960), Academic Press, (1961), vol. 2
[16] M. K. Sheaffer, M. J. Driscoll, I. Kaplan, A one-group method for fast reactor
calculations Nucl. Sci. Eng. 48, P459(1972)
[17] National Research Council of USA, International Critical Tables of Numerical
Data, Physics, Chemistry and Technology, 1st ed., Knovel, Norwich, NY, (2003), vol. 5,
pp. 92
[18] Michael de Podesta, Understanding the properties of matter, Taylor & Francis,
Washington, DC (1996), pp.178
[19] M. J. Driscoll, T.J. Downar, E.E.Pilat, The linear reactivity model for nuclear fuel
management, American Nuclear Society, La Grange Park, IL (1990)
68
Appendix A Estimate of Gas Produced By Sulfur
I
Sulfur in the fuel
For a 13wt% enriched US fuel, the gas produced by sulfur is estimated as
following:
S-32 (n, α) gas production in US relative to fission
R = y(
Ns
σ ( n, α )
)
NU χ 25 ⋅ g ⋅ (1 + δ 28 )σ f 25
(B-1)
where
y = abundance of S-32 in S
g = gas atom yield per fission (Kr + Xe)
δ28 = ratio of U-238 to U-235 fissions
σf25 = U-235 fission cross section
σ(n,α) = S-32 (n, α) cross section
χ25 = enrichment
(Ns/Nu) = atom ratio of sulfur to uranium
= 0.95
= 0.30
= 0.41
= 1525mb
= 12.5mb
= 0.13
= 1.0 for US
Thus R(n,α) = 0.14 which is significant. We also have production by (n,p) of H2: 0.5
molecules per reaction, thus:
1  σ (n, p ) 
(B-2)
R ( n, p ) = 
 • R (n, α )
2  σ (n, α ) 
where σ(n,p) of S-32 = 5.2mb
Thus R(n,p) = 0.030, and Rgas(total) = 0.17. This is probably tolerable, but if we also use a
sulfur compound for the matrix, the added gas would be quite significant.
II
Sulfur in the matrix
For a pure natural sulfur matrix and 13wt% enriched UC fuel, the gas produced
by sulfur can still be calculated by equation (B-1), but the parameters change to:
y = abundance of S-32 in S
g = gas atom yield per fission (Kr + Xe)
δ28 = ratio of U-238 to U-235 fissions
σf25 = U-235 fission cross section
σ(n,α) = S-32 (n, α) cross section
χ25 = enrichment
(Ns/Nu) = atom ratio of sulfur to uranium
= 0.95
= 0.30
= 0.174
= 1685mb
= 13.6mb
= 0.13
= 2.61 for S matrix, UC fuel
Thus R(n,α) = 0.36 which is more than twice that of the US fuel case. Taking the H2
generation into consideration, R(n,p) = 0.089, one obtains Rgas(total) = 0.45. This is a
quite large number, and would be even larger (~0.55) if US fuel is employed.
69
Appendix B Relation of reactivity ρ to enrichment x
ρ=
νΣ f − Σ a
νΣ f
ρ = 1−
Σa
νΣ f
ρ = 1−
Σ a 25 + Σ a 28 + Σ ad
νΣ f 25 +νΣ f 28
ρ = 1−
Σ a 025 x + (1 − x)Σ a 028 + Σ ad
νΣ f 025 x + (1 − x)νΣ f 028
Σ a 028 Σ ad
+
Σ a 025 Σ a 025
ρ = 1−
νΣ
η25 x + (1 − x) f 028
Σ a 025
x + (1 − x)
Let
λ=
σ a 28
σ a 25
γ=
Σ ad
Σ a 025
Then
ρ = 1−
x + (1 − x)λ + γ
η25 x + (1 − x)η28λ
η 25 x + (1 − x)η 28λ − x − (1 − x)λ − γ
η25 x + (1 − x)η28λ
(η − 1) x + (1 − x)(η 28 − 1)λ − γ
ρ = 25
η 25 x + (1 − x)η 28λ
η −1
≈ −1.17
η28 ≈ 0.46, for x → 0, ρ = 28
η 28
ρ=
In a fission spectrum, η25 = 2.46. Omit the γ term, then
ρ≅
1.46 x + (1 − x)λ (−0.54)
2.46 x + (1 − x)λ (0.46)
ρ≅
1.46 [ x − 0.37(1 − x)λ ]
2.46 [ x + 0.19(1 − x)λ ]
Furthermore, λ =
σ a 28 0.21
≈
= 0.13 for a very hard spectrum
σ a 25 1.57
70
If so, ρ =
0.64 x − 0.029
x + 0.025
0.60 x − 0.032
. Comparing
x + 0.017
each term in the two equations, we can see that the theoretical deduction gives a fairly
good explanation and estimation.
The least square curve fit to MCNP calculation gives ρ =
71
Appendix C Sample input files for matrix material study
Uranium carbide fuel, Ba2Pb matrix, nickel reflector
1. Beginning of life keff calculation, mcnp input:
MCNP INPUT DECK FOR MFGR YK_01
c cell cards
1 1 2.984103E-02 -1 2 -3 imp:n=1 tmp= 6.662234E-08
2 2 8.913363E-02 -1 2 3 -4 imp:n=1 tmp= 6.662234E-08
99 0 1:-2:4 imp:n=0
c end of cell cards
c surface cards
*1 pz 50
*2 pz -50
3
cz 150
4
cz 240
c end of surface cards
awtab 34079 78.240500 38089 88.143700 38090 89.135400
44105 104.007000 46107 105.987000
47111 109.953000 48115 113.919000 50123 121.850000
50125 123.835000 50126 124.826000 51124 122.842000
51125 123.832000 51126 124.826000 52127 125.815000
52129 127.800000 53130 128.791000 53131 129.781998
54133 131.764008 58141 139.697998
58144 142.677000 59142 140.691000
59143 141.682999 61151 149.625000
62153 151.608002 63156 154.585007 63157 155.577000
96249 246.936000 97250 247.930000
72
c Material cards
c Material 1: inner core,Material 2: reflector
c Material 3: reflecter, Material 4: cladding
m1 6000.60c 9.041788E-03 $C
8016.60c 3.853585E-04 $O
c
56138.60c 7.709851E-03 $Ba
82000.50c 3.854926E-03 $Pb
c
92235.60c 1.163166E-03 $U235
c
92238.60c 7.685943E-03 $U238
35081.55c 1.0000e-24 $ begin_mcode_FP
c
36082.50c 1.0000e-24
36083.50c 1.0000e-24 36084.50c 1.0000e-24 37085.55c 1.0000e-24
37087.55c 1.0000e-24 38090.96c 1.0000e-24 39089.60c 1.0000e-24
40090.62c 1.0e-24
40091.96c 1.0000e-24 40092.62c 1.0000e-24 40093.50c 1.0000e-24
40094.62c 1.0000e-24 40096.62c 1.0000e-24 41095.96c 1.0000e-24
42095.50c 1.0000e-24 42096.96c 1.0000e-24 42097.60c 1.0000e-24
42098.50c 1.0000e-24 42100.50c 1.0000e-24 43099.50c 1.0000e-24
44100.96c 1.0000e-24 44101.50c 1.0000e-24 44102.60c 1.0000e-24
44103.50c 1.0000e-24 44104.96c 1.0000e-24 45103.50c 1.0000e-24
c
45105.50c 1.0000e-24
46104.96c 1.0000e-24 46105.50c 1.0000e-24 46106.96c 1.0000e-24
46107.96c 1.0000e-24 46108.50c 1.0000e-24 46110.96c 1.0000e-24
47109.60c 1.0000e-24 48110.62c 1.0000e-24 48111.62c 1.0000e-24
48112.62c 1.0000e-24 48113.60c 1.0000e-24 48114.62c 1.0000e-24
49115.60c 1.0000e-24 50117.96c 1.0e-24
51121.96c 1.0000e-24 51123.96c 1.0000e-24 52125.96c 1.0e-24
52128.96c 1.0000e-24 52130.96c 1.0e-24
53127.60c 1.0000e-24 53129.60c 1.0000e-24
54128.62c 1.0e-24
54130.62c 1.0e-24
54132.62c 1.0000e-24
73
54131.50c 1.0000e-24
c
54133.60c 1.0000e-24
54134.62c 1.0000e-24
c
54135.50c 1.0000e-24
54136.62c 1.0000e-24 55133.60c 1.0000e-24
55134.60c 1.0000e-24 55135.60c 1.0000e-24 55137.60c 1.0000e-24
56136.96c 0.000605532
56130.96c 8.17244E-06 56132.96c 7.78695E-06 56135.96c 0.000508233
56134.62c 0.000186347 56137.62c 0.00086597 56138.60c 0.005527809
57139.60c 1.0000e-24 58140.96c 1.0000e-24
c
58141.60c 1.0000e-24
58142.96c 1.0000e-24 58144.96c 1.0000e-24 59141.50c 1.0000e-24
c
59143.60c 1.0000e-24
60142.96c 1.0000e-24 60143.50c 1.0000e-24 60144.96c 1.0000e-24
60145.50c 1.0000e-24 60146.96c 1.0000e-24
c
60147.50c 1.0000e-24
60148.50c 1.0000e-24 60150.96c 1.0000e-24 61147.50c 1.0000e-24
c
61148.50c 1.0000e-24
61148.60c 1.0000e-24 $ ORIGEN_ID 611481
c
61149.50c 1.0000e-24
62147.50c 1.0000e-24 62148.96c 1.0000e-24 62149.50c 1.0000e-24
62150.50c 1.0000e-24 62151.50c 1.0000e-24 62152.50c 1.0000e-24
c
62153.60c 1.0000e-24
62154.96c 1.0000e-24 63151.60c 1.0000e-24 63152.50c 1.0e-24
63153.60c 1.0000e-24 63154.50c 1.0000e-24 63155.50c 1.0000e-24
c
63156.60c 1.0000e-24
64154.60c 1.0000e-24 64155.60c 1.0000e-24 64156.60c 1.0000e-24
64157.60c 1.0000e-24 64158.60c 1.0000e-24 65159.96c 1.0000e-24
66160.96c 1.0000e-24 66161.96c 1.0000e-24
66162.96c 1.0000e-24 $ end_mcode_FP
c
66163.96c 1.0000e-24
c
90232.60c 1.0000e-24
74
c
91231.60c 1.0000e-24
c
91233.50c 1.0000e-24
c
92232.60c 1.0000e-24
c
92233.60c 1.0000e-24
92234.60c 1.0000e-24
$ begin_mcode_ACT
92235.60c 1.163166E-03
$ fuel u-235
92236.60c 1.0000e-24 92237.50c 1.0000e-24
92238.60c 7.685943E-03
c
$ fuel u-238
93236.35c 1.0000e-24
93237.60c 1.0000e-24
c
93238.35c 1.0000e-24
93239.60c 1.0000e-24 94238.60c 1.0000e-24 94239.60c 1.0000e-24
94240.60c 1.0000e-24 94241.60c 1.0000e-24 94242.60c 1.0000e-24
c
94243.60c 1.0000e-24
95241.60c 1.0000e-24
c
95242.50c 1.0000e-24
95242.51c 1.0000e-24 $ ORIGEN_ID 952421
95243.60c 1.0000e-24 $ end_mcode_ACT
m2 6000.60c 4.816981E-05 $C
8016.60c 9.633962E-05 $O
28000.50c 8.898912E-02 $Ni
c
ksrc 0 0 0
mode n
kcode 10000 1 10 220
prdmp 220 220 220
print
75
2. Beginning of life ∆kvoid calculation, mcnp input:
MCNP INPUT DECK FOR MFGR YK_01
c cell cards
1 1 2.926299E-02 -1 2 -3 imp:n=1 tmp= 6.662234E-08
2 2 8.898912E-02 -1 2 3 -4 imp:n=1 tmp= 6.662234E-08
99 0 1:-2:4 imp:n=0
c end of cell cards
c surface cards
*1 pz 50
*2 pz -50
3
cz 150
4
cz 240
c end of surface cards
c Material cards
c Material 1: inner core,Material 2: reflector
c Material 3: reflecter, Material 4: cladding
m1 6000.60c 8.849109E-03 $C
56136.96c 0.000605532
56130.96c 8.17244E-06 56132.96c 7.78695E-06 56135.96c 0.000508233
56134.62c 0.000186347 56137.62c 0.00086597 56138.60c 0.005527809
82000.50c 3.854926E-03 $Pb
92235.60c 1.163166E-03
92238.60c 7.685943E-03
m2
28000.50c 8.898912E-02 $Ni
c tally materials follows (39 ACT + 100 FP)
c
Tally Materials
c
----------------------------------------------------------------------
76
c
100 fission products, m701 to m800
c
ksrc 0 0 0
mode n
kcode 3000 1.107 5 120
prdmp 120 120 120
print
77
3. Burnup study, mcode input:
$ MCODE, UC fuel GCR, metal matrix, CO2 coolant, cold condition
TTL test case
$ defines title
MCD 1 mcnp.exe Ba2Pb.i ykm.src $ MCNP files def.
$ mcnp cells def.: cell-number type(1=delp.,2=actv.) act-mass(g) vol.(cm3) flux-t#
cross-t#
ORG /usr/local/bin/origen22/origen22 /usr/local/bin/origen22/LIBS
DECAY.LIB
GXUO2BRM.LIB
CEL 1 1 1 2.751209912E+07 7.06858E+06 FFTFC.LIB
$ total volume of modeling system (cm3)
VOL 7.06858E+06
$ ORIGEN files def.
$ normalization method, 1=flux, 2=power
NOR 2
$ predictor-corrector (OFF)
COR 0
$ power density, opt: WGU=W/gIHM, KWL=kW/(liter core)
PDE 10.61033475 KWL
$points 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
DEP E 0 5 10 15 20 30 40 50 60 70 80 90 100 120 140 160 180 200
NMD
40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
STA 0 $ starting point
END 17 $ ending point
78