analisa burnup

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

analisa burnup

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