analisa burnup
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Analisa Burnup
Zaki Su’ud
Pengertian analisa burnup
• Analisa yang berkaitan dengan perubahan jangka panjang (hari-bulan-tahun) komposisi bahan-bahan dalam reaktor akibat berbagai reaksi nuklir yang terjadi saat pengoperasian reaktor nuklir
• Bahan-bahan pecahan reaksi fisi jumlahnya sangat banyak (lebih dari 1200 nuklida) dan karakteristiknya sangat beragam
Analisa burnup secara umum
• Proses burnup merupakan mekanisme yang sangat kompleks yang dipengaruhi berbagai faktor seperti komposisi bahan teras, distribusi fluks netron, temperatur, histori pengoperasian reaktor, dsb.
• Beberapa program analisis burnup telah disiapkan untuk operasi yang bersifat standar misalnya terkait PLTN yang banyak dioperasikan
Analisa Burnup secara umum(2)
• Akan tetapi untuk kasus-kasus khusus misalnya menyangkut advanced NPP yang memiliki skema fuel cycle yang cukup kompleks maka diperlukan program yang lebih komprehensif
• Dalam beberapa kasus program-program analisis yang ada pun perlu dimodifikasi agar cukup akuran dalam menganalisa kasus tersebut
Contoh rantai
burnup
Persamaan Burnup terkait
CONTOH DERET BURNUP YANG DISEDERHANAKAN
Am-241 ^
• Pu-239Pu-240Pu-241Pu-242• ^• Np-239• ^• U-238 U-239
Persamaan Burnup untuk deret yang disederhanakan
88 8
98 8 9 9 9 9
99 9 9 9 9 9
99 9 9 9
UaU U
UcU U U U aU U
NpU U Np Np aNp Np
PuNp Np aPu Pu
dNN
dtdN
N N Ndt
dNN N N
dtdN
N Ndt
Persamaan Burnup untuk deret yang disederhanakan(2)
09 9 0 0
10 0 1 1 1 1
21 1 2 2
11 1 1 1 1 1
PucPu Pu aPu Pu
PucPu Pu aPu Pu Pu Pu
PucPu Pu aPu Pu
AmPu Pu aAm Am Am Am
dNN N
dtdN
N N Ndt
dNN N
dtdN
N N Ndt
Solusi numerik
• Ada sangat banyak metoda yang dapat digunakan untuk memecahkan persamaan burnup
• Di sini diberikan contoh yang bersifat standar diantaranya metoda eksplisit berbasis finite difference dan metoda semi implisit berbasis finite difference juga
• Metoda eksplisit mudah dirumuskan hanyasaja mempunyai tingkat stabilitas yang lebih rendah dari metoda implisit
Solusi Numerik Finite difference Eksplisit
18 8
8 8
18 8 8
19 9
8 8 9 9 9 9
19 8 8 9 9 9
(1 )
(1 )
i iiU U
aU U
i iU aU U
i ii i iU U
cU U U U aU U
i i iU cU U U aU U
N NN
t
N t N
N NN N N
t
N tN t t N
Solusi Numerik Finite difference Eksplisit
19 9
9 9 9 9 9 9
19 9 9 9 9 9
19 9
9 9 9 9
19 9 9 9 9
(1 )
(1 )
i iNp Np i i i
U U Np Np aNp Np
i iNp U U Np aNp Np
i ii iPu Pu
Np Np aPu Pu
i i iPu Np Np aPu Pu
N NN N N
t
N tN t t N
N NN N
dt
N tN t N
Solusi Numerik Finite difference Eksplisit
10 0
9 9 0 0
10 9 9 0 0
11 1
0 0 1 1 1 1
11 0 0 1 1 1 1
(1 )
(1 )
i ii iPu Pu
cPu Pu aPu Pu
i i iPu cPu Pu aPu Pu
i ii i iPu Pu
cPu Pu Pu Pu aPu Pu
i i i iPu cPu Pu Pu Pu aPu Pu
N NN N
t
N tN t N
N NN N N
t
N tN tN t N
Solusi Numerik Finite difference Eksplisit
12 2
1 1 2 2
12 1 1 2 2
11 1
1 1 1 1 1 1
11 1 1 1 1 1
(1 )
(1 )
i ii iPu Pu
cPu Pu aPu Pu
i i iPu cPu Pu aPu Pu
i ii i iAm Am
Pu Pu Am Am aAm Am
i i iAm Pu Pu Am aAm Am
N NN N
t
N tN t N
N NN N N
t
N tN t t N
Metoda Implisit
• Pada metoda implisit ruas kanan diisi dengan kombinasi duku pada iterasi waktu ke i dan i+1 dengan bobot yang dinyatakan dalam parameter tertentu
• Metoda numerik jauh lebih rumit perumusannya dari metoda eksplisit tetapi memiliki keunggulan stabilitas yang jauh lebih tinggi
tt
Solusi Numerik Finite difference Implisit
118 8
8 8 8
18 8 8 8
1 8 88
8
11 19 9
8 8 8 9 9 9 9
19 9
[ (1 ) ]
[1 (1 )] (1 )
(1 )
[1 (1 )]
[ (1 ) ] ( )[ (1 ) ]
[1 (
i ii iU U
aU U U
i iU aU aU U
ii aU UU
aU
i ii i i iU U
cU U U U aU U U
iU U
N NN N
t
N t t N
t NN
t
N NN N N N
t
N
1
9 8 8 8 9 9 9
11 8 8 8 9 9 99
9 9
) (1 )] [ (1 ) ] (1 )
[ (1 ) ] (1 )
[1 ( ) (1 )]
i i iaU cU U U U aU U
i i ii cU U U U aU UU
U aU
t N N t t N
N N t t NN
t
Solusi Numerik Finite difference Implisit
19 9 1 1
9 9 9 9 9 9 9
1 19 9 9 9 9 9 9 9 9
19 9 9 91
9
[ (1 ) ] ( )[ (1 ) ]
[1 ( ) (1 )] [ (1 ) ] [1 ( ) ]
[ (1 ) ] [1 (
i iNp Np i i i i
U U U Np aNp Np Np
i i i iNp Np aNp U U U Np aNp Np
i iU U U Npi
Np
N NN N N N
t
N t t N N t t N
t N N tN
9 9
9 9
11 19 9
9 9 9 9 9 9
1 19 9 9 9 9 9 9
9 919
) ]
[1 ( ) (1 )]
[ (1 ) ] [ (1 ) ]
[1 (1 )] [ (1 ) ]
[ (
iaNp Np
Np aNp
i ii i i iPu Pu
Np Np Np aPu Pu Pu
i i i iPu aPu Np Np Np aPu Pu
iNp Npi
Pu
t N
t
N NN N N N
t
N t t N N t N
t NN
19 9 9
9
1 ) ]
[1 (1 )]
i iNp aPu Pu
aPu
N t N
t
Solusi Numerik Finite difference Implisit
11 10 0
9 9 9 0 0 0
1 10 0 9 9 9 0 0
11 9 9 9 0 00
0
[ (1 ) ] [ (1 ) ]
[1 (1 )] [ (1 ) ]
[ (1 ) ]
[1
i ii i i iPu Pu
cPu Pu Pu aPu Pu Pu
i i i iPu aPu cPu Pu Pu aPu Pu
i i ii cPu Pu Pu aPu PuPu
aPu
N NN N N N
t
N t t N N t N
t N N t NN
1
1 11 10 0 0 1 1 1 1
1 11 1 1 0 0 0 1 1 1 1
1 01
(1 )]
[ (1 ) ] ( )[ (1 ) ]
[1 ( ) (1 )] [ (1 ) ] (1 )
[
i ii i i iPu Pu
cPu Pu Pu Pu aPu Pu Pu
i i i i iPu Pu aPu cPu Pu Pu Pu Pu aPu Pu
i cPu PuPu
t
N NN N N N
t
N t t N N tN t N
t NN
10 0 1 1 1 1
1 1
(1 ) ] (1 )
[1 ( ) (1 )]
i i i iPu Pu Pu aPu Pu
Pu aPu
N tN t N
t
Solusi Numerik Finite difference Eksplisit
11 12 2
1 1 1 2 2 2
1 12 2 1 1 1 2 2
11 1 1 1 2 22
2
[ (1 ) ] [ (1 ) ]
[1 (1 )] [ (1 ) ]
[ (1 ) ]
[1 (1 )
i ii i i iPu Pu
cPu Pu Pu aPu Pu Pu
i i i iPu aPu cPu Pu Pu aPu Pu
i i ii cPu Pu Pu aPu PuPu
aPu
N NN N N N
t
N t N N N
N N NN
t
1
1 11 11 1 1 1 1 1 1
1 11 1 1 1 1 1 1 1 1
11 1 1 1 11
]
[ (1 ) ] ( )[ (1 ) ]
[1 ( ) (1 )] [ (1 ) ] ( )
[ (1 ) ] (
i ii i i iAm Am
Pu Pu Pu Am aAm Am Am
i i i iAm Am aAm Pu Pu Pu Am aAm Am
i ii Pu Pu Pu AmAm
N NN N N N
t
N t t N N t N
t N NN
1 1
1 1
)
[1 ( ) (1 )]
iaAm Am
Am aAm
t N
t
Metoda semi analitik
• Metoda analitik seperti yang dirumuskan dalam Bateman equation memiliki akurasi yang tinggi
• Kendalanya metoda ini sangat rumit untuk deret yang panjang, hanya dapat diterapkan dalam deret linier, serta tak dapat digunakan untuk rantai siklus
• Solusinya adalah dengan menggunakan metoda semi analitik
Metoda Semi analitik(2)
• Dalam metoda semi analitik maka rantai burnup dipotong-potong dengan panjang potongan yang diatur sesuai dengan kebutuhan/optimasi
• Selanjutnya dilakukan iterasi burnup untuk masing-masing potongan rantai secara pereodik
• Selanjutnya dilakukan updating nilai konsentrasi nuklida untuk tiap jenis nuklida
THEORY
BURN UP EQUATIONAn explicit Burn Up equation for each nuclide is :
whereNi = concentration of ith nuclideλi = decay constant of ith nuclideσa,i = absorb microscopic cross section for ith nuclideФ = neutron flux of nuclideSm,i = production speed of ith nuclide from mth nuclide
BATEMAN SOLUTION• Bateman equation is one of analytic method to solve
transmutation process in linear chain depend on time evolution
• General solution for linear chain of transmutation process
SIMULATION
1 92234 92235 92236 24 94238 92234 92235 47 95242 94242 94243 95243 95244
2 92235 92236 92237 93237 25 94239 94240 94241 48 95242 96242 96243 96244
3 92236 92237 93237 93238 94238 26 94240 94241 94242 49 95242 96242 96243 94239
4 92237 92237 93238 93239 94239 27 94240 94241 95241 50 95242 96242 94238 94239
5 92237 92237 93238 94238 94239 28 94241 94242 94243 95243 51 95242 96242 94238 92234
6 92237 92237 93238 94238 92234 29 94241 95241 95742 52 95243 95244 96244 96245
7 92238 92239 93239 93240 94240 30 94241 95241 95742 96242 53 95243 95244 96244 94240
8 92238 92239 93239 94239 94238 31 94241 95241 95242 94242 54 95244 96244 96244 94246
9 92238 92237 93237 93238 94241 32 94242 95241 93237 55 95244 96244 94240 94241
10 92239 93239 93240 94240 94241 33 94243 94243 95243 95244 96244 56 96242 96243 96244
11 92239 93239 94239 94240 34 94243 95243 95244 96244 96245 57 96242 96243 94239
12 93237 93238 93239 94239 35 95241 95243 95244 96244 94240 58 96242 94238 94239
13 93237 93238 94238 94239 36 95241 95742 95243 59 96242 94238 92234
14 93237 93238 94238 92234 37 95241 95742 95242 94242 60 96243 96244 96245
15 93238 93239 93240 94240 38 95241 95742 95242 96242 61 96243 96244 94240
16 93238 93239 94239 94240 39 95241 95242 94242 94243 95243 62 96243 94239 94240
17 93238 94238 94239 94240 40 95241 95242 96242 96243 63 96244 96245 96246
18 93238 94238 92234 92235 41 95241 95242 96242 94238 64 96244 94240 94241
19 93239 93240 94240 94241 42 95241 93237 93238 94238 65 96245 96246 96247
20 93239 94239 94240 94241 43 95742 95243 95244 96244 66 96246 96247 96748
21 93240 94240 94241 94242 44 95742 95242 94242 94243 95243 67 96247 96248 96749
22 93240 94240 94241 95241 45 95742 95242 96242 96243 68 96248 96249
23 94238 94239 94240 46 95742 95242 96242 96243 69 96249
Linear series for analytical method
Burnup chain1 92234 92235 92236
2 92235 92236 92237 93237
3 92236 92237 93237 93238 94238
4 92237 92237 93238 93239 94239
5 92237 92237 93238 94238 94239
6 92237 92237 93238 94238 92234
7 92238 92239 93239 93240 94240
8 92238 92239 93239 94239 94238
9 92238 92237 93237 93238 94241
10 92239 93239 93240 94240 94241
Burnup chain(2)
11 92239 93239 94239 9424012 93237 93238 93239 9423913 93237 93238 94238 9423914 93237 93238 94238 9223415 93238 93239 93240 9424016 93238 93239 94239 9424017 93238 94238 94239 9424018 93238 94238 92234 9223519 93239 93240 94240 9424120 93239 94239 94240 94241
Burnup chain (3)
21 93240 94240 94241 9424222 93240 94240 94241 9524123 94238 94239 9424024 94238 92234 9223525 94239 94240 9424126 94240 94241 9424227 94240 94241 9524128 94241 94242 94243 9524329 94241 95241 9574230 94241 95241 95742 96242
Burnup chain (4)
31 94241 95241 95242 9424232 94242 95241 9323733 94243 94243 95243 95244 9624434 94243 95243 95244 96244 9624535 95241 95243 95244 96244 9424036 95241 95742 9524337 95241 95742 95242 9424238 95241 95742 95242 9624239 95241 95242 94242 94243 9524340 95241 95242 96242 96243
Burnup chain (5)
41 95241 95242 96242 9423842 95241 93237 93238 9423843 95742 95243 95244 9624444 95742 95242 94242 94243 9524345 95742 95242 96242 9624346 95742 95242 96242 9624347 95242 94242 94243 95243 9524448 95242 96242 96243 9624449 95242 96242 96243 9423950 95242 96242 94238 94239
Burnup chain (6)
51 95242 96242 94238 9223452 95243 95244 96244 9624553 95243 95244 96244 9424054 95244 96244 96244 9424655 95244 96244 94240 9424156 96242 96243 9624457 96242 96243 9423958 96242 94238 9423959 96242 94238 9223460 96243 96244 96245
Burnup chain (7)
61 96243 96244 9424062 96243 94239 9424063 96244 96245 9624664 96244 94240 9424165 96245 96246 9624766 96246 96247 9674867 96247 96248 9674968 96248 9624969 96249
0 200 400 600 800 1000 1200 1400 1600 18000
0.5
1
1.5
2
2.5x 10
22
time time0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 106
0
5
10
15x 10
21
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 106
0
2
4
6
8
10
12x 10
20
timetime0 2 4 6 8 10 12
x 105
0
2
4
6
8
10
12
14x 10
22
BEBERAPA HAL PENTING TERKAIT ANALISA BURNUP
• Untuk reaktor cepat maka efek self shielding pada perubahan cross section microscopic tidak terlalu besar sehingga analisa burnup berbasis microscopic cross section dapat diterapkan
• Untuk reaktor thermal efek self shielding pada perubahan cross section microscopic cukup besar sehingga analisa burnup harus dilakukan dalam sel bahan bakar
BEBERAPA HAL PENTING TERKAIT ANALISA BURNUP(2)
• FP berjumlah lebih dari 1200 nuklida dan karakteristiknya bergantung jenis reaktor nuklir yang digunakan
• Untuk reaktor thermal ada beberapa FP yang sangat dominan sehingga dapat mewakili keseluruhan FP yang ada: misal Xenon, Sm, dll.
• Untuk reaktor cepat tak ada Fp yang terlalu dominan sehingga secara keseluruhan harus diperhitungkan
BEBERAPA HAL PENTING TERKAIT ANALISA BURNUP(3)
• Untuk reaktor cepat metoda yang biasa digunakan adalah menggunakan lumped FP atau menggunakan beberapa puluh nuklida FP dan sisanya menggunakan lumped FP
• Untuk perhitungan conversion/breeding ratio maka perlu dilakukan kalibrasi cross section fisi dan nilai v untuk masing-masing bahan fisil dominan
• Dalam hal digunakan sejumlah bahan fisil secara serempak maka dilakukan kalibrasi FP
Senstivitas Burnup pada Cross section
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 37
Code Modification
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 38
Parameter Parameter Value/description
SPINNOR A SPINNOR B VSPINNOR
Installed capacity 55 MWth / 20 MWe
27.5 MWth/10 MWe
17.5 MWth/6.25 MWe
Operation life time (without refueling and fuel shuffling)
15 years 25 years 35 years
Mode of operation Basic/load follow (selectable)Beyond 95% *
Load factor
Summary of major design characteristics- type of fuel- fuel enrichment- type of coolant/moderator- type of structural material
UN-PuN**10 – 12.5%Pb-Bi eutecticStainless
UN-PuN**10 – 12.5%
Pb-Bi eutecticStainless
UN-PuN**10 – 12.5%Pb-Bi eutecticStainless
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 39
B1
B2 B2
B2 C1
C1 C1 C1
C1
C2
C2 C2 C2
C2
C2
C2
R R
S
RRR
R
R
R
R
SS
S
S
S
S
S
S S S
Radial direction
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 40
outlet
Tostack
Figure 1. Reactor assembly of SPINNOR AND VSPINNOR
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 41
Burnup parametric study results: U238 fission
105%102.5%
100% 97.5% 95%
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 42
Burnup parametric study results:Pu-239 fission
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 43
Burnup parametric study results:Pu-241 fission
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 44
Burnup parametric study results: U-238 capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 45
Burnup parametric study results: Pu-239 capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 46
Burnup parametric study results: Pu-240 capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 47
Burnup parametric study results: FP capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 48
Burnup parametric study results: Pb capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 49
Burnup parametric study results: Bi capture
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 50
Burnup parametric study results: Pb transport
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 51
Burnup parametric study results: Bi transport
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 52
Burnup parametric study results: FP scattering
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 53
Burnup parametric study results: Pb scattering
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 54
Burnup parametric study results: Bi scattering
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 55
Burnup parametric study results: Pu-239 fission conversion ratio
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 56
Burnup parametric study results: U-238 capture conversion ratio
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 57
Burnup parametric study results: FP capture conversion ratio
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 58
Burnup parametric study results: Pu239 fission coolant void coefficient
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 59
Burnup parametric study results: U-238 capture coolant void coefficient
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 60
Burnup parametric study results: FP capture coolant void coefficient
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 61
Burnup parametric study results: Pb scattering coolant void coefficient
04/21/23 IAEA CRP RCM 21-25 Nov. 2005 62
Conclusion for Burnup parametric survey
• From the parametric survey results, we find that FP cross section is important to be considered to get reliable neutronic analysis results.
• Some other cross section is also critical such as U-238 capture cross section and main fissile fission cross section, and Pb and Bi transport and scattering cross section.
• FP cross section is important to be treated in more accurate way to get better accuracy especially at the end of life.
INTRODUCTION:Background• Small and very small nuclear power plant with
moderate economical aspect is an important candidate for electric power generation in many part of the third world countries including outside Java-Bali area in Indonesia.
• The nuclear energy system with the range of 5-50 Mwe match with the necessity and planning of many cities and provinces outside Java-Bali islands.
• In addition to electricity, desalination plant or cogeneration plant is a good candidate for nuclear energy application
INTRODUCTION:Background• Due to the difference of the load between afternoon
and night the use of fast reactors is a better choice due to capability to follow the load.
• Lead and lead bismuth cooled nuclear power reactors is now considered as potential candidate of next generation nuclear power reactors in the 21th centuries.
• Various versions of lead cooled nuclear power reactors have been analyzed and safety analysis also have been applied to them.
• Accuracy of the simulation system need to be tested through international benchmark program under IAEA.
Introduction: ObjectiveSolving FP treatment group constant with the following
approach:• First alternative: Rigorous treatment : We cover 165 nuclides
with other relevant FP nuclides in direct individual burnup calculation. This method will give rigorous results but with considerable calculation time. However this method is important to test other simpler methods.
• Second alternative: Lumped FP treatment : We just build best FP lumped cross section for many general condition and use this FP group constant in burnup calculation. This method can give accurate results if the spectrum is same or near the spectrum to build the lumped FP cross section.
Introduction: Objective• Third alternatives : Combination method: We treat some
most important nuclides individually and treat the rest FP using lumped FP cross section. This method seems to be good alternative for general usage.
• Forth alternative : Lumped FP cross section with many interpolable parameter: We develop the concept similar to the back ground cross section in the Bondanrenko based cell calculation libraries. This will improve Lumped FP cross section results for general usage.
• Fifth alternative : We develop the few group effective FP similar to that in reactor kinetic problem. If we can get reasonable good few group effective FP then we can solve for all type of the core generally
METHODOLOGY• Identifying the important FP nuclides which have
strong influence to the overall FP cross section• Identifying important FP decay chains relevant the
important nuclides• Analyzing the contribution of each FP nuclides to
the overall FP crosssection based on the equilibrium model
• Analyzing the contribution of each FP nuclides to the overall FP crosssection based on the time dependent model
Identifying the important FP nuclides which have strong influence to the overall FP cross section
• Based on the study of Shiro TABUCHI and Takafumi AOYAMA we select 50 most important nuclides for fast reactors.
• Based on this selection we then identify relevant and important decay chains which should be considered.
• The 118 nuclides which has the contribution to the total FP cross section more than 0.01% are shown in the following table.
Table 1 118 Important FP Nuclides
No Z A %X-sect Symbol
1 44 101 8.93 Ru
2 46 105 8.93 Pd
3 43 99 7.06 Tc
4 45 103 6.02 Rh
5 55 133 5.72 Cs
6 46 107 4.65 Pd
7 42 97 4.54 Mo
8 62 149 4.39 Sm
9 61 147 3.77 Pm
10 60 145 3.37 Nd
11 55 135 2.74 Cs
12 60 143 2.64 Nd
13 54 131 2.38 Xe
14 44 102 2.21 Ru
15 62 151 2.19 Sm
16 42 95 2.15 Mo
17 42 98 1.89 Mo
18 47 109 1.80 Ag
19 44 104 1.69 Ru
20 42 100 1.58 Mo
21 63 153 1.56 Eu
22 40 93 1.27 Zr
23 44 103 1.19 Ru
24 59 141 1.03 Pr
25 53 129 0.97 I
26 40 95 0.88 Zr
27 40 96 0.75 Zr
28 60 146 0.70 Nd
29 54 132 0.69 Xe
30 46 108 0.68 Pd
31 41 95 0.67 Nb
32 58 141 0.62 Ce
33 40 91 0.61 Zr
34 40 92 0.48 Zr
35 54 134 0.48 Xe
36 44 106 0.48 Ru
37 62 152 0.48 Sm
38 60 148 0.46 Nd
39 48 111 0.44 Cd
40 37 85 0.43 Rb
41 53 127 0.42 I
42 57 139 0.42 La
43 46 106 0.41 Pd
44 63 155 0.35 Eu
45 40 94 0.32 Zr
46 62 147 0.31 Sm
47 58 142 0.29 Ce
48 60 150 0.28 Nd
49 60 147 0.26 Nd
50 55 137 0.25 Cs
51 39 91 0.20 Y
52 60 144 0.19 Nd
53 36 83 0.19 Kr
54 58 144 0.18 Ce
55 64 157 0.18 Gd
56 46 110 0.14 Pd
57 42 99 0.14 Mo
58 64 156 0.13 Gd
59 48 113 0.11 Cd
60 55 134 0.11 Cs
61 63 154 0.10 Eu
62 58 140 0.10 Ce
63 51 125 0.10 Sb
64 65 159 0.10 Tb
65 62 154 0.10 Sm
66 38 90 0.10 Sr
67 53 131 0.09 I
68 39 89 0.09 Y
69 56 138 0.08 Ba
70 59 143 0.08 Pr
71 35 81 0.08 Br
72 52 130 0.08 Te
73 49 115 0.08 In
74 52 128 0.07 Te
75 48 112 0.07 Cd
76 52 129m 0.07 Te
77 37 87 0.06 Rb
78 36 84 0.06 Kr
79 54 133 0.05 Xe
80 51 121 0.05 Sb
81 52 127m 0.05 Te
82 61 148m 0.05 Pm
83 34 79 0.05 Se
84 45 105 0.05 Rh
85 62 150 0.04 Sm
86 51 123 0.04 Sb
87 64 155 0.03 Gd
88 50 117 0.03 Sn
89 61 149 0.03 Pm
90 54 136 0.03 Xe
91 46 104 0.03 Pd
92 64 158 0.03 Gd
93 44 100 0.03 Ru
94 36 85 0.03 Kr
95 38 89 0.03 Sr
96 48 114 0.02 Cd
97 38 88 0.02 Sr
98 50 119 0.02 Sn
99 62 148 0.02 Sm
100 34 82 0.02 Se
101 56 136 0.02 Ba
102 47 110m 0.02 Ag
103 34 77 0.01 Se
104 36 86 0.01 Kr
105 63 156 0.01 Eu
106 34 80 0.01 Se
107 63 151 0.01 Eu
108 48 116 0.01 Cd
109 50 118 0.01 Sn
110 48 110 0.01 Cd
111 34 78 0.01 Se
112 54 130 0.01 Xe
113 56 137 0.01 Ba
114 64 160 0.01 Gd
115 56 140 0.01 Ba
116 50 126 0.01 Sn
117 52 125 0.01 Te
118 50 120 0.01 Sn
Identifying important FP decay chains relevant the important nuclides
(1) 84mBr 6.0m
84Ga 84Ge 84As 84Se 84Kr 0.085s 0.95s 3.2s 3.1m stable
84Br 31.8m
(2) 85mKr 4.48h
85Ga 85Ge 85As 85Se 85Br 85Rb(0.09s) 0.54s 2.02s 31.7s 2.90m stable
85Kr 10.77y
II.3 Analyzing the contribution of each FP nuclides to the overall FP crosssection based on the equilibrium
model
• Based on the relevant and important decay chains, differential equation for the model can be derived.
• And using equilibrium approximation model we can obtain the formula for the contribution of each nuclide for certain flux level.
• Detail process will be discussed in the next part.
Analyzing the contribution of each FP nuclides to the overall FP cross section based on the time dependent
model • To see the process toward equilibrium, the
time dependent change of each important nuclides is calculated.
• The calculation is performed based on the most important equation using analytical method or numerical methods
MATHEMATICAL MODEL DESCRIPTION AND THE
METHODOLOGY OF SOLUTION
1. Simplification of Decay Scheme and Mathematical Model
Differential Equation
(2.c) )1(
(2.b)
(2.a) *85
55555525
555525
555
RbaRbKrKrmKrmKrKr
KrKrmKrmKrKr
mKrmKrmKr
NNNfdt
dN
NNfdt
dN
NFydt
dN
(9) *92
(8.b)
(8.a) *91
11222
11111
111
ZrcZrZraZrZr
ZraZrYYZr
YYY
NNFydt
dN
NNdt
dN
NFydt
dN
(11) *94
(10.b)
(10.a) *93
33444
33333
2233333
ZrcZrZraZrZr
NbaNbZrZrNb
ZrcZrZraZrZrZrZr
NNFydt
dN
NNdt
dN
NNNFydt
dN
(15) *98
(14) *97
(13) *96
(12.c)
(12.b) -
(12.a) *95
77888
66777
55666
55555
5555555
4455555
MocMoMoaMoMo
ZrcZrMoaMoMo
ZrcZrZraZrZr
NbaNbNbNbMo
MoMoNbaNbZrZrNb
ZrcZrZraZrZrZrZr
NNFydt
dN
NNFydt
dN
NNFydt
dN
NNdt
dN
NNNdt
dN
NNNFydt
dN
(18) *101
(17)
(16.b)
(16.a) *99
00111
9900000
55999
8899999
NbcNbRuaRuRu
TccTcNbaNbNbNbMo
NbaNbRuRuRu
NbcNbTcaTcTcTcTc
NNFydt
dN
NNNdt
dN
NNdt
dN
NNNFydt
dN
(20.b)
(20.a) *103
(19) *102
33333
33333
222
RhaRhRuRuRh
RuaRuRuRuRu
RuaRuRu
NNdt
dN
NNFydt
dN
NFydt
dN
(22) N *105
(21) N*104
Ru44555
Ru33444
cRuPdaPdPd
cRuRuaRuRu
NFydt
dN
NFydt
dN
(24) *107
(23.b) N
(23.a) *106
66777
Pd5cPd566666
66666
PdcPdPdaPdPd
PdaPdRuRuPd
RuaRuRuRuRu
NNFydt
dN
NNdt
dN
NNFydt
dN
(25) *108 77888
PdcPdPdaPdPd NNFydt
dN
(26) *109 88999
PdcPdAgaAgAg NNFydt
dN
(28) *111 111
CdaCdCd NFydt
dN
(30) *127 777
IaII NFy
dt
dN
(32) *129 999
IaII NFy
dt
dN
(34) *131 111
XeaXeXe NFydt
dN
(35) *132 11222
XecXeXeaXeXe NNFydt
dN
(36) *133 22333
XecXeCsaCsCs NNFydt
dN
(37) *134 444
XeaXeXe NFydt
dN
(38.b)
(38.a) *135
55555
44555
BaaBaCsCsBa
XecXeCsCsCs
NNdt
dN
NNFydt
dN
(40.b)
(40.a) *137
77777
777
BaaBaCsCsBa
CsCsCs
NNdt
dN
NFydt
dN
(42) *139 999
LaaLaLa NFydt
dN
(44.b)
(44.a) *141
1Pr1Pr111Pr
111
NNdt
dN
NFydt
dN
aCeCe
CeCeCe
(45) *142 11222
CecCeCeaCeCe NNFydt
dN
(46) *143 33223
NdaNdCecCeNd NNFydt
dN
(48) *145 555
NdaNdNd NFydt
dN
(49.b)
(49.a) *146
66666
55666
PmaPmNdNdPm
NdcNdNdNdNd
NNdt
dN
NNFydt
dN
(50.b) N-
(50.a) *147
Pm7Pm777777
66777
PmaPmNdNdPm
NdcNdNdNdNd
NNdt
dN
NNFydt
dN
(50.c) 77777
SmaSmPmPmSm NNdt
dN
(51) *148 77888
NdcNdNdaNdNd NNFydt
dN
(52) *149 88999
NdcNdSmaSmSm NNFydt
dN
(53) *150 000
NdaNdNd NFydt
dN
(54.b)
(54.a) *151
11111
00111
EuaEuSmSmEu
NdcNdSmSmSm
NNdt
dN
NNFydt
dN
(55) *152 11222
SmcSmSmaSmSm NNFydt
dN
(56) *153 22333
SmcSmEuaEuEu NNFydt
dN
(58.b)
(58.a) N-*155
55555
Eu5aEu5555
GdaGdEuEuGd
EuEuEu
NNdt
dN
NFydt
dN
Table 2 Cumulative fission yield(Form JNDC)
_______________________________
Kr-85m 6.10677000000000025E-1Y -91 2.43774999999999986E+0Zr-92 2.95633999999999997E+0Zr-93 3.67079000000000022E+0Zr-94 4.26259000000000032E+0Zr-95 4.70092999999999961E+0Zr-96 4.78516399999999997E+0Mo-97 5.27359000000000044E+0Mo-98 5.62816999999999990E+0Tc-99 5.98852000000000029E+0
Mo-100 6.58037000000000027E+0Ru-101 6.54110999999999976E+0Ru-102 6.63984000000000041E+0Ru-103 6.83164999999999978E+0Ru-104 6.51982000000000017E+0Pd-105 5.41333999999999982E+0Ru-106 4.36779000000000028E+0Pd-107 3.05134600000000011E+0Pd-108 1.90365600000000001E+0Ag-109 1.92017700000000002E+0Cd-111 3.55362000000000011E-1I -127 5.52984999999999949E-1I -129 1.63166999999999995E+0Xe-131 3.86864000000000008E+0Xe-132 5.30914999999999981E+0Cs-133 6.88192000000000004E+0
Xe-134 7.37063999999999986E+0Cs-135 7.45038000000000000E+0Cs-137 6.58718100000000018E+0La-139 5.61065699999999978E+0Ce-141 5.23207999999999984E+0Ce-142 4.77627000000000024E+0Nd-143 4.30201999999999973E+0Nd-145 2.96883600000000003E+0
Nd-146 2.43299999999999983E+0Nd-147 1.97354680000000005E+0Nd-148 1.63632099999999991E+0Sm-149 1.23951699999999998E+0Nd-150 9.80944000000000038E-1Sm-151 7.76606000000000018E-1Sm-152 6.06010999999999966E-1Eu-153 4.34675499999999992E-1Eu-155 2.26013600000000009E-1
Results of EQUILIBRIUM APPROACH
Nuclide Equilibrium atomic density 10 years fission yieldsKr-85m 4.44770531791907562E+14 6.01822183500000051E+18Kr-85 1.97633360887029606E+18 6.01822183500000051E+18Rb-85 4.07118000000000076E+22 6.01822183500000051E+18Y -91 5.56515074783236992E+17 2.40240262499999990E+19Zr-91 1.87519230769230774E+23 2.40240262499999990E+19Zr-92 2.69704499999999973E+24 2.91347307000000020E+19Zr-93 7.22362078298686804E+23 3.61756354500000031E+19Zr-94 1.44399416962568053E+25 4.20078244500000031E+19Zr-95 4.42046908461249792E+18 4.63276651499999969E+19Nb-95 2.41666241370500864E+18 4.63276651499999969E+19Mo-95 4.77426312204802589E+23 4.63276651499999969E+19Zr-96 5.52377933331738450E+24 4.71577912199999980E+19Mo-97 7.48809471931197233E+23 5.19712294500000072E+19Mo-98 2.84272507230397735E+24 5.54656153500000010E+19
Tc-99 5.80448830818055222E+23 5.90168646000000041E+19
Mo-100 6.50652098679982160E+23 6.48495463500000051E+19Ru-101 1.64415151553121916E+23 6.44626390499999990E+19Ru-102 1.11917766324970256E+24 6.54356232000000082E+19Ru-103 4.07353193643529267E+18 6.73259107499999969E+19Rh-103 3.62142075730733155E+23 6.73259107499999969E+19Ru-104 1.90874718849906334E+24 6.42528261000000061E+19Pd-105 3.71900383008765652E+23 5.33484656999999980E+19Ru-106 6.23076879994554204E+19 4.30445704500000031E+19Pd-106 2.80060091023820422E+24 4.30445704500000031E+19Pd-107 7.33297125020930985E+23 3.00710148300000010E+19Pd-108 3.10105667293295228E+24 1.87605298800000000E+19Ag-109 1.16346325998803663E+24 1.89233443350000026E+19Cd-111 4.33608363654999232E+21 3.50209251000000000E+18I -127 8.31849101789200961E+21 5.44966717499999949E+18I -129 3.96245109681555547E+22 1.60801078499999990E+19Xe-131 1.21112624246693271E+23 3.81254472000000000E+19Xe-132 1.20288420958816868E+24 5.23216732500000031E+19
Cs-133 3.62009300606139853E+23 6.78213216000000000E+19Xe-134 4.91376000000000031E+24 7.26376572000000000E+19Cs-135 7.45522158674253426E+23 7.34234948999999980E+19Cs-137 2.82069197535739773E+20 6.49166687550000005E+19La-139 1.65677159309021128E+24 5.52930247350000026E+19Ce-141 1.51566891983954208E+17 5.15621484000000000E+19Pr-141 9.13746420803279551E+22 5.15621484000000000E+19Ce-142 2.51802498411755389E+23 4.70701408500000031E+19Nd-143 4.34924587526784595E+23 4.23964071000000020E+19Nd-145 5.88221448186497744E+22 2.92578787800000020E+19Nd-146 7.71690857142856989E+23 2.39772150000000000E+19Nd-147 1.02188351991170944E+17 1.94493037139999990E+19Pm-147 8.89640490784821760E+18 1.94493037139999990E+19Sm-147 2.20304355074652252E+22 1.94493037139999990E+19Nd-148 1.32707108147550356E+23 1.61259434550000005E+19Sm-149 1.43957988001329279E+22 1.22154400350000005E+19Nd-150 3.06545000000000014E+22 9.66720312000000000E+18Sm-151 8.09183452634436700E+15 7.65345213000000000E+18Sm-152 1.42246774052948772E+22 5.97223840499999949E+18Eu-153 4.44932410294246792E+22 4.28372705250000026E+18Eu-155 1.53124322481422464E+18 2.22736402800000026E+18
Equilibrium results analysis• Not all of the nuclides can be treated properly using
equilibrium approach.• The nuclides which need long time to reach the equilibrium
are not appropriate for this approach. • To investigate this we also show the yields of 10 years of
burn-up using 100 W/cc power density and fission macroscopic cross section 0.01 cm-1.
• The equilibrium approach will be useful for nuclides in which equilibrium atomic density is much larger than the corresponding yields in the right column.
• Therefore we can find that Y-91, Zr-95, Nb-95, Ru-103, Ru-106, Ce-141, and Nd-147 are nuclides which can be treated collectively using equilibrium approach.
• The verification of this can be found in the next session.
DIRECT NUMERICAL SOLUTION RESULTS
Series1Series2Series3Series4
Kr-85
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5.5E+18
5E+18
4.5E+18
4E+18
3.5E+18
3E+18
2.5E+18
2E+18
1.5E+18
1E+18
5E+17
Series1Series2Series3Series4
Rb-85
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+19
3.5E+19
3E+19
2.5E+19
2E+19
1.5E+19
1E+19
5E+18
Series1Series2Series3Series4
Nb-95
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.4E+18
2.2E+18
2E+18
1.8E+18
1.6E+18
1.4E+18
1.2E+18
1E+18
8E+17
6E+17
4E+17
2E+17
Series1Series2Series3Series4
Y-91
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.2E+18
2E+18
1.8E+18
1.6E+18
1.4E+18
1.2E+18
1E+18
8E+17
6E+17
4E+17
2E+17
Series1Series2Series3Series4
Zr-91
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Zr-92
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.2E+20
2E+20
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Zr-93
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.8E+20
2.6E+20
2.4E+20
2.2E+20
2E+20
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Zr-94
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Zr-95
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4.5E+18
4E+18
3.5E+18
3E+18
2.5E+18
2E+18
1.5E+18
1E+18
5E+17
Series1Series2Series3Series4
Zr-96
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Mo-95
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Mo-97
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Mo-98
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Mo-100
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Tc-99
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Ru-101
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Ru-102
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Ru-103
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+18
3.5E+18
3E+18
2.5E+18
2E+18
1.5E+18
1E+18
5E+17
Series1Series2Series3Series4
Ru-104
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Ru-106
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.4E+19
2.2E+19
2E+19
1.8E+19
1.6E+19
1.4E+19
1.2E+19
1E+19
8E+18
6E+18
4E+18
2E+18
Series1Series2Series3Series4
Rh-103
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Pd-105
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Pd-106
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3.2E+203E+20
2.8E+202.6E+202.4E+202.2E+20
2E+201.8E+201.6E+201.4E+201.2E+20
1E+208E+19
6E+194E+192E+19
Series1Series2Series3Series4
Pd-107
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.2E+20
2E+20
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Pd-108
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.5E+20
1.4E+20
1.3E+20
1.2E+20
1.1E+20
1E+20
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Ag-109
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.5E+20
1.4E+20
1.3E+20
1.2E+20
1.1E+20
1E+20
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Cd-111
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.6E+19
2.4E+19
2.2E+19
2E+19
1.8E+19
1.6E+19
1.4E+19
1.2E+19
1E+19
8E+18
6E+18
4E+18
2E+18
Series1Series2Series3Series4
I-127
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+19
3.5E+19
3E+19
2.5E+19
2E+19
1.5E+19
1E+19
5E+18
Series1Series2Series3Series4
I-129
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.2E+20
1.1E+20
1E+20
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Xe-131
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3E+20
2.8E+20
2.6E+20
2.4E+20
2.2E+20
2E+20
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Xe-132
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Xe-134
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5.5E+20
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Cs-133
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Cs-135
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5.5E+20
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Cs-137
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
La-139
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Ce-141
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.8E+17
1.6E+17
1.4E+17
1.2E+17
1E+17
8E+16
6E+16
4E+16
2E+16
Series1Series2Series3Series4
Ce-142
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
7.5E+20
7E+20
6.5E+20
6E+20
5.5E+20
5E+20
4.5E+20
4E+20
3.5E+20
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Pr-141
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.8E+19
2.6E+19
2.4E+19
2.2E+19
2E+19
1.8E+19
1.6E+19
1.4E+19
1.2E+19
1E+19
8E+18
6E+18
4E+18
2E+18
Series1Series2Series3Series4
Nd-143
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3E+20
2.5E+20
2E+20
1.5E+20
1E+20
5E+19
Series1Series2Series3Series4
Nd-145
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.2E+20
2E+20
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Nd-146
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.8E+20
1.6E+20
1.4E+20
1.2E+20
1E+20
8E+19
6E+19
4E+19
2E+19
Series1Series2Series3Series4
Nd-147
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
3E+17
2.5E+17
2E+17
1.5E+17
1E+17
5E+16
Series1Series2Series3Series4
Nd-148
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.2E+20
1.1E+20
1E+20
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Nd-150
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
7.5E+19
7E+19
6.5E+19
6E+19
5.5E+19
5E+19
4.5E+19
4E+19
3.5E+19
3E+192.5E+19
2E+19
1.5E+19
1E+19
5E+18
Series1Series2Series3Series4
Pm-147
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
2.8E+19
2.6E+19
2.4E+19
2.2E+19
2E+19
1.8E+19
1.6E+19
1.4E+19
1.2E+19
1E+19
8E+18
6E+18
4E+18
2E+18
Series1Series2Series3Series4
Sm-147
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.2E+20
1.1E+20
1E+20
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Sm-149
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
9E+19
8E+19
7E+19
6E+19
5E+19
4E+19
3E+19
2E+19
1E+19
Series1Series2Series3Series4
Sm-151
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
1.4E+16
1.3E+16
1.2E+16
1.1E+16
1E+16
9E+15
8E+15
7E+15
6E+15
5E+15
4E+15
3E+15
2E+15
1E+15
Series1Series2Series3Series4
Sm-152
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
4.5E+19
4E+19
3.5E+19
3E+19
2.5E+19
2E+19
1.5E+19
1E+19
5E+18
Series1Series2Series3Series4
Eu-153
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3) 3E+19
2.5E+19
2E+19
1.5E+19
1E+19
5E+18
Series1Series2Series3Series4
Eu-155
Time (years)2018161412108642
Ato
mic
de
nsi
ty(n
ucl
ide
s/cm
3)
5.5E+18
5E+18
4.5E+18
4E+18
3.5E+18
3E+18
2.5E+18
2E+18
1.5E+18
1E+18
5E+17
Analysis• The first pattern is about nuclides which soon reach
asymptotic value, such as Nb-95, Y-91, Zr-95, Ru-103, Ru-106, Ce-141, Nd-147,and Sm-151.
• Such nuclides can be grouped together with certain weight which ma depend on some parameters such as flux, power density, etc.
• This results are also inline with the equilibrium model. The Ru-106 is may be in the boundary between first pattern and second pattern.
• The second pattern includes nuclides which change during burn-up include non-linear pattern. Such nuclides includes Kr-85, Pd-106, Cs-137, Ce-142, Pm-147, Sm-147, and Eu-155. Such nuclides can be combined into one group or more with non linear wight (quadratic, cubic, quartic, etc.)
Analysis• The third pattern is about nuclides which change
almost linear during burnup. • Such nuclides includes Rb-85, Zr-91, Zr-92, Zr-93, Zr,
94, Zr-96, Mo-95, Mo-97, Mo-98, Mo-100, Tc-99, Ru-101, Ru-102, Ru-104, Rh-103, Pd-105, Pd-107, Pd-108, Ag-109, Cd-111, I-127, I-129, Xe-131, Xe-132, Xe-134, Cs-133, Cs-135, La-139, Pr-141, Nd-143, Nd-145, Nd-146, Nd-148, Nd-150, Sm149, Sm152, and Eu153.
• Such nuclides can be grouped into two or more group constants with flux level, power level and time.
CONCLUSION AND RECOMENDATION
• In this study we focus on the FP group constant treatment by considering around 50 most important nuclides. We then calculate the fission product effective yield for each modified chains and also generating one group constants using SRAC code system and other method (Origen etc.).
• We use two approach for investigating the important FP nuclides: using equilibrium model and using numerical solution for time dependent model. We found that we can separate the FP nuclides into three groups: which soon reach asymptotic value, which have non linear pattern and which have linear pattern
CONCLUSION AND RECOMENDATION
• In he future work we will complete the detail lumped FP model and include this in the full core benchmark calculation
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