Analisa Burnup - WordPress.com
advertisement
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 dNU 8 aU 8 NU 8 dt dNU 9 cU 8 NU 8 U 9 NU 9 aU 9 NU 9 dt dN Np 9 U 9 NU 9 Np 9 N Np 9 aNp 9 N Np 9 dt dN Pu 9 Np 9 N Np 9 aPu 9 N Pu 9 dt Persamaan Burnup untuk deret yang disederhanakan(2) dN Pu 0 cPu 9 N Pu 9 aPu 0 N Pu 0 dt dN Pu1 cPu 0 N Pu 0 aPu1 N Pu1 Pu1 N Pu1 dt dN Pu 2 cPu1 N Pu1 aPu 2 N Pu 2 dt dN Am1 Pu1 N Pu1 aAm1 N Am1 Am1 N Am1 dt 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 i 1 U8 i U8 i 1 U9 i U9 N i aU 8 NU 8 t i 1 i NU 8 (1 aU 8t ) NU 8 N N i i i cU 8 NU 8 U 9 NU 9 aU 9 NU 9 t i 1 i i NU 9 cU 8tNU 8 (1 U 9 t aU 9t ) NU 9 N Solusi Numerik Finite difference Eksplisit N i 1 Np 9 N i Np 9 U 9 N i U9 Np 9 N i Np 9 aNp 9 N i Np 9 t i i 1 N Np 9 U 9 tNU 9 (1 Np 9 t aNp 9t ) N Np 9 i 1 Pu 9 N i i Np 9 N Np 9 aPu 9 N Pu 9 dt i i i 1 N Pu 9 Np 9 tN Np 9 (1 aPu 9t ) N Pu 9 N i Pu 9 Solusi Numerik Finite difference Eksplisit i 1 Pu 0 i Pu 0 i 1 Pu1 i Pu1 N i i cPu 9 N Pu 9 aPu 0 N Pu 0 t i 1 i i N Pu 0 cPu 9tN Pu 9 (1 aPu 0t ) N Pu 0 N N i i i cPu 0 N Pu 0 Pu1 N Pu1 aPu1 N Pu1 t i 1 i i i N Pu1 cPu 0tN Pu 0 (1 Pu1tN Pu1 aPu1t ) N Pu1 N Solusi Numerik Finite difference Eksplisit i 1 Pu 2 i Pu 2 i 1 Am1 i Am1 N i i cPu1 N Pu1 aPu 2 N Pu 2 t i 1 i i N Pu 2 cPu1tN Pu1 (1 aPu 2t ) N Pu 2 N N i i i Pu1 N Pu1 Am1 N Am1 aAm1 N Am1 t i 1 i i N Am1 Pu1tN Pu1 (1 Am1t aAm1t ) N Am1 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 t t Solusi Numerik Finite difference Implisit NUi 81 NUi 8 aU 8[ NUi 8 (1 ) NUi 81 ] t NUi 81[1 aU 8t (1 )] (1 aU 8t ) NUi 8 i 1 U8 N (1 aU 8t ) NUi 8 [1 aU 8t (1 )] NUi 91 NUi 9 cU 8[ NUi 8 (1 ) NUi 81 ] (U 9 aU 9 )[ NUi 9 (1 ) NUi 91 ] t NUi 91[1 (U 9 aU 9 )t (1 )] cU 8[ NUi 8 (1 ) NUi 81 ] (1 U 9t aU 9t ) NUi 9 i 1 U9 N cU 8[ NUi 8 (1 ) NUi 81 ] (1 U 9t aU 9t ) NUi 9 [1 (U 9 aU 9 )t (1 )] Solusi Numerik Finite difference Implisit i 1 i N Np 9 N Np 9 i i 1 U 9 [ NUi 9 (1 ) NUi 91 ] (Np 9 aNp 9 )[ N Np 9 (1 ) N Np 9 ] t i 1 i i 1 i N Np [1 ( ) t (1 )] t [ N (1 ) N ] [1 ( t t ) ] N 9 Np 9 aNp 9 U9 U9 U9 Np 9 aNp 9 Np 9 i i 1 i t [ N (1 ) N ] [1 ( t t ) ] N U9 U9 U9 Np 9 aNp 9 Np 9 i 1 N Np 9 [1 (Np 9 aNp 9 )t (1 )] i 1 i N Pu i i 1 i i 1 9 N Pu 9 Np 9 [ N Np 9 (1 ) N Np 9 ] aPu 9 [ N Pu 9 (1 ) N Pu 9 ] t i 1 i i 1 i N Pu 9[1 aPu 9t (1 )] Np 9 t[ N Np 9 (1 ) N Np 9 ] aPu 9t N Pu 9 N i 1 Pu 9 i i 1 i Np 9 t[ N Np 9 (1 ) N Np 9 ] aPu 9t N Pu 9 [1 aPu 9t (1 )] Solusi Numerik Finite difference Implisit i 1 i N Pu i i 1 i i 1 0 N Pu 0 cPu 9[ N Pu 9 (1 ) N Pu 9 ] aPu 0 [ N Pu 0 (1 ) N Pu 0 ] t i 1 i i 1 i N Pu 0[1 aPu 0t (1 )] cPu 9t[ N Pu 9 (1 ) N Pu 9 ] aPu 0t N Pu 0 N i 1 Pu 0 i i 1 i cPu 9t[ N Pu 9 (1 ) N Pu 9 ] aPu 0t N Pu 0 [1 aPu 0t (1 )] i 1 i N Pu i i 1 i i 1 1 N Pu1 cPu 0[ N Pu (1 ) N ] ( )[ N (1 ) N 0 Pu 0 Pu1 aPu1 Pu1 Pu1 ] t i 1 i i 1 i i N Pu 1[1 (Pu1 aPu1 ) t (1 )] cPu 0t[ N Pu 0 (1 ) N Pu 0 ] (1 Pu1tN Pu1 aPu1t ) N Pu1 N i 1 Pu1 i i 1 i i cPu 0t[ N Pu 0 (1 ) N Pu 0 ] (1 Pu1tN Pu1 aPu1t ) N Pu1 [1 (Pu1 aPu1 )t (1 )] Solusi Numerik Finite difference Eksplisit i 1 i N Pu i i 1 i i 1 2 N Pu 2 cPu1[ N Pu 1 (1 ) N Pu1 ] aPu 2 [ N Pu 2 (1 ) N Pu 2 ] t i 1 i i 1 i N Pu 2[1 aPu 2t (1 )] cPu1 [ N Pu1 (1 ) N Pu1 ] aPu 2 N Pu 2 N i 1 Pu 2 i i 1 i cPu1[ N Pu (1 ) N ] N 1 Pu1 aPu 2 Pu 2 [1 aPu 2t (1 )] i 1 i N Am i i 1 i i 1 1 N Am1 Pu1[ N Pu 1 (1 ) N Pu1 ] ( Am1 aAm1 )[ N Am1 (1 ) N Am1 ] t i 1 i i 1 i N Am 1[1 ( Am1 aAm1 ) t (1 )] Pu1t[ N Pu1 (1 ) N Pu1 ] ( Am1 aAm1 ) t N Am1 N i 1 Am1 i i 1 i Pu1t[ N Pu 1 (1 ) N Pu1 ] ( Am1 aAm1 ) t N Am1 [1 (Am1 aAm1 )t (1 )] 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 EQUATION An explicit Burn Up equation for each nuclide is : where Ni = concentration of ith nuclide λi = decay constant of ith nuclide σa,i = absorb microscopic cross section for ith nuclide Ф = neutron flux of nuclide Sm,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 Linear series for analytical method 1 92234 92235 92236 24 94238 92234 92235 47 95242 94242 94243 95243 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 6 92237 92237 93238 94238 94239 28 94241 94242 94243 51 95242 96242 94238 92234 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 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 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 95243 95243 95243 95244 Burnup chain 1 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 12 13 14 15 16 17 18 19 20 92239 93237 93237 93237 93238 93238 93238 93238 93239 93239 93239 93238 93238 93238 93239 93239 94238 94238 93240 94239 94239 93239 94238 94238 93240 94239 94239 92234 94240 94240 94240 94239 94239 92234 94240 94240 94240 92235 94241 94241 Burnup chain (3) 21 22 23 24 25 26 27 28 29 30 93240 93240 94238 94238 94239 94240 94240 94241 94241 94241 94240 94240 94239 92234 94240 94241 94241 94242 95241 95241 94241 94241 94240 92235 94241 94242 95241 94243 95742 95742 94242 95241 95243 96242 Burnup chain (4) 31 32 33 34 35 36 37 38 39 40 94241 94242 94243 94243 95241 95241 95241 95241 95241 95241 95241 95241 94243 95243 95243 95742 95742 95742 95242 95242 95242 93237 95243 95244 95244 95243 95242 95242 94242 96242 94242 95244 96244 96244 94242 96242 94243 96243 96244 96245 94240 95243 Burnup chain (5) 41 42 43 44 45 46 47 48 49 50 95241 95241 95742 95742 95742 95742 95242 95242 95242 95242 95242 93237 95243 95242 95242 95242 94242 96242 96242 96242 96242 93238 95244 94242 96242 96242 94243 96243 96243 94238 94238 94238 96244 94243 96243 96243 95243 96244 94239 94239 95243 95244 Burnup chain (6) 51 52 53 54 55 56 57 58 59 60 95242 95243 95243 95244 95244 96242 96242 96242 96242 96243 96242 95244 95244 96244 96244 96243 96243 94238 94238 96244 94238 96244 96244 96244 94240 96244 94239 94239 92234 96245 92234 96245 94240 94246 94241 Burnup chain (7) 61 62 63 64 65 66 67 68 69 96243 96243 96244 96244 96245 96246 96247 96248 96249 96244 94239 96245 94240 96246 96247 96248 96249 94240 94240 96246 94241 96247 96748 96749 21 22 15 2 10 1.5 1 5 0.5 0 0 200 400 600 800 Nuclide concentration (NPu240 ) 1000 1200 1400 1600 1800 x 10 0 0.2 0.4 0.6 0.8 time1 22 14 10 x 10 1.2 1.4 1.6 1.8 2 6 x 10 Nuclide concentration (N) 12 10 8 6 4 2 0 0 time 20 12 x 10 Nuclide concentration (NPu239 ) x 10 Nuclide concentration (NU8 ) 2.5 0 0.2 0.4 0.6 0.8 time 1 1.2 1.4 1.6 1.8 8 6 4 2 2 6 x 10 0 0 2 4 time6 8 10 12 5 x 10 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 Code Modification 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 37 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 4/13/2015 UN-PuN** 10 – 12.5% Pb-Bi eutectic Stainless IAEA CRP RCM 21-25 Nov. 2005 UN-PuN** 10 – 12.5% Pb-Bi eutectic Stainless UN-PuN** 10 – 12.5% Pb-Bi eutectic Stainless 38 S S S S S S R R R R R S C2 C2 C2 C2 R S C1 C1 C1 C2 R S B2 B2 C1 C2 R S B1 B2 C1 C2 R S Radial direction 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 39 To stack outlet 4/13/2015 Figure 1. Reactor assembly of SPINNOR AND VSPINNOR IAEA CRP RCM 21-25 Nov. 2005 40 Burnup parametric study results: U238 fission 105% 102.5% 100% 97.5% 95% 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 41 Burnup parametric study results:Pu-239 fission 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 42 Burnup parametric study results:Pu-241 fission 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 43 Burnup parametric study results: U-238 capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 44 Burnup parametric study results: Pu-239 capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 45 Burnup parametric study results: Pu-240 capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 46 Burnup parametric study results: FP capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 47 Burnup parametric study results: Pb capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 48 Burnup parametric study results: Bi capture 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 49 Burnup parametric study results: Pb transport 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 50 Burnup parametric study results: Bi transport 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 51 Burnup parametric study results: FP scattering 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 52 Burnup parametric study results: Pb scattering 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 53 Burnup parametric study results: Bi scattering 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 54 Burnup parametric study results: Pu-239 fission conversion ratio 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 55 Burnup parametric study results: U-238 capture conversion ratio 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 56 Burnup parametric study results: FP capture conversion ratio 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 57 Burnup parametric study results: Pu239 fission coolant void coefficient 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 58 Burnup parametric study results: U-238 capture coolant void coefficient 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 59 Burnup parametric study results: FP capture coolant void coefficient 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 60 Burnup parametric study results: Pb scattering coolant void coefficient 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 61 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 U238 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. 4/13/2015 IAEA CRP RCM 21-25 Nov. 2005 62 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 JavaBali 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: Objective Solving 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 40 37 85 0.43 Rb 21 63 153 1.56 Eu 41 53 127 0.42 I 22 40 93 1.27 Zr 42 57 139 0.42 La 23 44 103 1.19 Ru 43 46 106 0.41 Pd 24 59 141 1.03 Pr 44 63 155 0.35 Eu 25 53 129 0.97 I 45 40 94 0.32 Zr 26 40 95 0.88 Zr 46 62 147 0.31 Sm 27 40 96 0.75 Zr 47 58 142 0.29 Ce 28 60 146 0.70 Nd 48 60 150 0.28 Nd 29 54 132 0.69 Xe 49 60 147 0.26 Nd 30 46 108 0.68 Pd 50 55 137 0.25 Cs 31 41 95 0.67 Nb 51 39 91 0.20 Y 32 58 141 0.62 Ce 52 60 144 0.19 Nd 33 40 91 0.61 Zr 53 36 83 0.19 Kr 34 40 92 0.48 Zr 54 58 144 0.18 Ce 35 54 134 0.48 Xe 55 64 157 0.18 Gd 36 44 106 0.48 Ru 56 46 110 0.14 Pd 37 62 152 0.48 Sm 57 42 99 0.14 Mo 38 60 148 0.46 Nd 58 64 156 0.13 Gd 39 48 111 0.44 Cd 59 48 113 0.11 Cd 60 55 134 0.11 Cs 80 51 121 0.05 Sb 61 63 154 0.10 Eu 81 52 127m 0.05 Te 62 58 140 0.10 Ce 82 61 148m 0.05 Pm 63 51 125 0.10 Sb 83 34 79 0.05 Se 64 65 159 0.10 Tb 84 45 105 0.05 Rh 65 62 154 0.10 Sm 85 62 150 0.04 Sm 66 38 90 0.10 Sr 86 51 123 0.04 Sb 67 53 131 0.09 I 87 64 155 0.03 Gd 68 39 89 0.09 Y 88 50 117 0.03 Sn 69 56 138 0.08 Ba 89 61 149 0.03 Pm 70 59 143 0.08 Pr 90 54 136 0.03 Xe 71 35 81 0.08 Br 91 46 104 0.03 Pd 72 52 130 0.08 Te 92 64 158 0.03 Gd 73 49 115 0.08 In 93 44 100 0.03 Ru 74 52 128 0.07 Te 94 36 85 0.03 Kr 75 48 112 0.07 Cd 95 38 89 0.03 Sr 76 52 129m 0.07 Te 96 48 114 0.02 Cd 77 37 87 0.06 Rb 97 38 88 0.02 Sr 78 36 84 0.06 Kr 98 50 119 0.02 Sn 79 54 133 0.05 Xe 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 84mBr (1) 6.0m 84Ge 84As 84Se 0.085s 0.95s 3.2s 3.1m 84Ga 84Kr stable 84Br 31.8m (2) 85mKr 4.48h 85Ga 85Ge 85As 85Se 85Br (0.09s) 0.54s 2.02s 31.7s 2.90m 85Kr 10.77y 85Rb stable 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 dN Kr 5m y85 * F Kr 5m N Kr 5 m dt dN Kr 5 f 2 Kr 5m N Kr 5m Kr 5 N Kr 5 dt dN Kr 5 (1 f 2 ) Kr 5 m N Kr 5 m Kr 5 N Kr 5 aRb5N Rb5 dt (2.a) (2.b) (2.c) dNY 1 y91* F Y 1 N Y 1 dt dN Zr 1 Y 1 N Y 1 aZr 1N Zr 1 dt dN Zr 2 y92 * F aZr 2N Zr 2 cZr1N Zr 1 dt dNZr 3 y93* F Zr 3 N Zr 3 aZr 3N Zr 3 cZr 2N Zr 2 dt dN Nb 3 Zr 3 N Zr 3 aNb 3N Nb 3 dt dNZr 4 y94 * F aZr 4N Zr 4 cZr 3N Zr 3 dt (8.a) (8.b) (9) (10.a) (10.b) (11) dN Zr 5 y95 * F Zr 5 N Zr 5 aZr 5N Zr 5 cZr 4N Zr 4 (12.a) dt dN Nb 5 Zr 5 N Zr 5 aNb 5N Nb 5 - Mo 5 N Mo 5 (12.b) dt dN Mo 5 Nb 5 N Nb 5 aNb 5N Nb 5 (12.c) dt dN Zr 6 y96 * F aZr 6N Zr 6 cZr 5N Zr 5 (13) dt dN Mo 7 y97 * F aMo 7N Mo 7 cZr 6N Zr 6 (14) dt dN Mo 8 y98 * F aMo 8N Mo 8 cMo7N Mo 7 (15) dt dNTc 9 y99 * F Tc 9 N Tc 9 aTc 9N Tc 9 cNb8N Nb 8 (16.a) dt dN Ru9 Ru9 N Ru9 aNb 5N Nb 5 (16.b) dt dNMo 0 Nb 0 N Nb 0 aNb 0N Nb 0 cTc 9N Tc 9 (17) dt dN Ru1 y101* F aRu1N Ru1 cNb0N Nb 0 (18) dt dN Ru 2 y102* F aRu 2N Ru 2 dt dN Ru3 y103* F Ru3 N Ru3 aRu 3N Ru3 dt dN Rh3 Ru3 N Ru3 aRh3N Rh3 dt (19) (20.a) (20.b) dN Ru 4 y104* F aRu 4N Ru 4 cRu3N Ru3 dt dN Pd 5 y105* F aPd 5N Pd 5 cRu4N Ru4 dt dN Ru 6 y106* F Ru 6 N Ru 6 aRu 6N Ru 6 dt dN Pd 6 Ru 6 N Ru 6 aPd 6N Pd 6 cPd5N Pd5 dt dN Pd 7 y107* F aPd 7N Pd 7 cPd 6N Pd 6 dt (21) (22) (23.a) (23.b) (24) dN Pd 8 y108* F aPd 8N Pd 8 cPd 7N Pd 7 dt dN Ag 9 y109* F aAg 9N Ag 9 cPd8N Pd 8 dt dNCd 1 y111* F aCd 1N Cd 1 dt dN I 7 y127* F aI 7N I 7 dt dN I 9 y129* F aI 9N I 9 dt dN Xe1 y131* F aXe1N Xe1 dt dN Xe2 y132* F aXe 2N Xe2 cXe1N Xe1 dt dNCs 3 y133* F aCs 3N Cs 3 cXe2N Xe2 dt (25) (26) (28) (30) (32) (34) (35) (36) dN Xe4 y134 * F aXe4N Xe4 dt dNCs 5 y135* F Cs 5 N Cs 5 cXe4N Xe4 dt dN Ba5 Cs 5 N Cs 5 aBa5N Ba5 dt dNCs 7 y137* F Cs 7 N Cs 7 dt dN Ba7 Cs 7 N Cs 7 aBa 7N Ba7 dt dN La 9 y139* F aLa 9N La 9 dt (37) (38.a) (38.b) (40.a) (40.b) (42) dNCe1 y141* F Ce1 N Ce1 dt dN Pr 1 Ce1 N Ce1 a Pr 1N Pr 1 dt dNCe 2 y142* F aCe 2N Ce 2 cCe1N Ce1 dt dN Nd 3 y143* F cCe 2N Ce 2 aNd 3N Nd 3 dt dN Nd 5 y145* F aNd 5N Nd 5 dt dN Nd 6 y146* F Nd 6 N Nd 6 cNd 5N Nd 5 dt dN Pm6 Nd 6 N Nd 6 aPm 6N Pm6 dt (44.a) (44.b) (45) (46) (48) (49.a) (49.b) dN Nd 7 dt dN Pm7 dt dN Sm 7 dt dN Nd 8 dt dN Sm 9 dt dN Nd 0 dt y147* F Nd 7 N Nd 7 cNd 6N Nd 6 (50.a) Nd 7 N Nd 7 aPm7N Pm7 - Pm7 N Pm7 (50.b) Pm7 N Pm7 aSm 7N Sm 7 (50.c) y148* F aNd 8N Nd 8 cNd 7N Nd 7 (51) y149* F aSm 9N Sm 9 cNd 8N Nd 8 (52) y150* F aNd 0N Nd 0 (53) dN Sm1 dt dN Eu1 dt dN Sm 2 dt dN Eu3 dt dN Eu5 dt dNGd 5 dt y151* F Sm1 N Sm1 cNd 0N Nd 0 (54.a) Sm1 N Sm1 aEu1N Eu1 (54.b) y152* F aSm 2N Sm 2 cSm1N Sm1 (55) y153* F aEu3N Eu3 cSm2N Sm 2 (56) y155* F Eu5 N Eu5 - aEu5N Eu5 (58.a) Eu5 N Eu5 aGd 5N Gd 5 (58.b) Table 2 Cumulative fission yield (Form JNDC) _______________________________ Kr-85m 6.10677000000000025E-1 Y -91 2.43774999999999986E+0 Zr-92 2.95633999999999997E+0 Zr-93 3.67079000000000022E+0 Zr-94 4.26259000000000032E+0 Zr-95 4.70092999999999961E+0 Zr-96 4.78516399999999997E+0 Mo-97 5.27359000000000044E+0 Mo-98 5.62816999999999990E+0 Tc-99 5.98852000000000029E+0 Mo-100 Ru-101 Ru-102 Ru-103 Ru-104 Pd-105 Ru-106 Pd-107 Pd-108 Ag-109 Cd-111 I -127 I -129 Xe-131 Xe-132 Cs-133 6.58037000000000027E+0 6.54110999999999976E+0 6.63984000000000041E+0 6.83164999999999978E+0 6.51982000000000017E+0 5.41333999999999982E+0 4.36779000000000028E+0 3.05134600000000011E+0 1.90365600000000001E+0 1.92017700000000002E+0 3.55362000000000011E-1 5.52984999999999949E-1 1.63166999999999995E+0 3.86864000000000008E+0 5.30914999999999981E+0 6.88192000000000004E+0 Xe-134 Cs-135 Cs-137 La-139 Ce-141 Ce-142 Nd-143 Nd-145 Nd-146 Nd-147 Nd-148 Sm-149 Nd-150 Sm-151 Sm-152 Eu-153 Eu-155 7.37063999999999986E+0 7.45038000000000000E+0 6.58718100000000018E+0 5.61065699999999978E+0 5.23207999999999984E+0 4.77627000000000024E+0 4.30201999999999973E+0 2.96883600000000003E+0 2.43299999999999983E+0 1.97354680000000005E+0 1.63632099999999991E+0 1.23951699999999998E+0 9.80944000000000038E-1 7.76606000000000018E-1 6.06010999999999966E-1 4.34675499999999992E-1 2.26013600000000009E-1 Results of EQUILIBRIUM APPROACH Nuclide Equilibrium atomic density Kr-85m 4.44770531791907562E+14 Kr-85 1.97633360887029606E+18 Rb-85 4.07118000000000076E+22 Y -91 5.56515074783236992E+17 Zr-91 1.87519230769230774E+23 Zr-92 2.69704499999999973E+24 Zr-93 7.22362078298686804E+23 Zr-94 1.44399416962568053E+25 Zr-95 4.42046908461249792E+18 Nb-95 2.41666241370500864E+18 Mo-95 4.77426312204802589E+23 Zr-96 5.52377933331738450E+24 Mo-97 7.48809471931197233E+23 Mo-98 2.84272507230397735E+24 Tc-99 5.80448830818055222E+23 10 years fission yields 6.01822183500000051E+18 6.01822183500000051E+18 6.01822183500000051E+18 2.40240262499999990E+19 2.40240262499999990E+19 2.91347307000000020E+19 3.61756354500000031E+19 4.20078244500000031E+19 4.63276651499999969E+19 4.63276651499999969E+19 4.63276651499999969E+19 4.71577912199999980E+19 5.19712294500000072E+19 5.54656153500000010E+19 5.90168646000000041E+19 Mo-100 6.50652098679982160E+23 Ru-101 1.64415151553121916E+23 Ru-102 1.11917766324970256E+24 Ru-103 4.07353193643529267E+18 Rh-103 3.62142075730733155E+23 Ru-104 1.90874718849906334E+24 Pd-105 3.71900383008765652E+23 Ru-106 6.23076879994554204E+19 Pd-106 2.80060091023820422E+24 Pd-107 7.33297125020930985E+23 Pd-108 3.10105667293295228E+24 Ag-109 1.16346325998803663E+24 Cd-111 4.33608363654999232E+21 I -127 8.31849101789200961E+21 I -129 3.96245109681555547E+22 Xe-131 1.21112624246693271E+23 Xe-132 1.20288420958816868E+24 6.48495463500000051E+19 6.44626390499999990E+19 6.54356232000000082E+19 6.73259107499999969E+19 6.73259107499999969E+19 6.42528261000000061E+19 5.33484656999999980E+19 4.30445704500000031E+19 4.30445704500000031E+19 3.00710148300000010E+19 1.87605298800000000E+19 1.89233443350000026E+19 3.50209251000000000E+18 5.44966717499999949E+18 1.60801078499999990E+19 3.81254472000000000E+19 5.23216732500000031E+19 Cs-133 Xe-134 Cs-135 Cs-137 La-139 Ce-141 Pr-141 Ce-142 Nd-143 Nd-145 Nd-146 Nd-147 Pm-147 Sm-147 Nd-148 Sm-149 Nd-150 Sm-151 Sm-152 Eu-153 Eu-155 3.62009300606139853E+23 4.91376000000000031E+24 7.45522158674253426E+23 2.82069197535739773E+20 1.65677159309021128E+24 1.51566891983954208E+17 9.13746420803279551E+22 2.51802498411755389E+23 4.34924587526784595E+23 5.88221448186497744E+22 7.71690857142856989E+23 1.02188351991170944E+17 8.89640490784821760E+18 2.20304355074652252E+22 1.32707108147550356E+23 1.43957988001329279E+22 3.06545000000000014E+22 8.09183452634436700E+15 1.42246774052948772E+22 4.44932410294246792E+22 1.53124322481422464E+18 6.78213216000000000E+19 7.26376572000000000E+19 7.34234948999999980E+19 6.49166687550000005E+19 5.52930247350000026E+19 5.15621484000000000E+19 5.15621484000000000E+19 4.70701408500000031E+19 4.23964071000000020E+19 2.92578787800000020E+19 2.39772150000000000E+19 1.94493037139999990E+19 1.94493037139999990E+19 1.94493037139999990E+19 1.61259434550000005E+19 1.22154400350000005E+19 9.66720312000000000E+18 7.65345213000000000E+18 5.97223840499999949E+18 4.28372705250000026E+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 burnup 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, Ru106, 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 Kr-85 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Rb-85 Series1 Series2 Series3 Series4 4E+19 3.5E+19 3E+19 2.5E+19 2E+19 1.5E+19 1E+19 5E+18 2 4 6 8 10 12 Time (years) 14 16 18 20 Nb-95 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Y-91 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-91 Series1 Series2 Series3 Series4 1.8E+20 1.6E+20 1.4E+20 1.2E+20 1E+20 8E+19 6E+19 4E+19 2E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-92 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-93 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-94 Series1 Series2 Series3 Series4 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-95 Series1 Series2 Series3 Series4 4.5E+18 4E+18 3.5E+18 3E+18 2.5E+18 2E+18 1.5E+18 1E+18 5E+17 2 4 6 8 10 12 Time (years) 14 16 18 20 Zr-96 Series1 Series2 Series3 Series4 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Mo-95 Series1 Series2 Series3 Series4 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Mo-97 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Mo-98 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Mo-100 Series1 Series2 Series3 Series4 5E+20 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Tc-99 Series1 Series2 Series3 Series4 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Ru-101 5E+20 Series1 Series2 Series3 Series4 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Ru-102 Series1 Series2 Series3 Series4 5E+20 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Ru-103 Series1 Series2 Series3 Series4 4E+18 3.5E+18 3E+18 2.5E+18 2E+18 1.5E+18 1E+18 5E+17 2 4 6 8 10 12 Time (years) 14 16 18 20 Ru-104 Series1 Series2 Series3 Series4 5E+20 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Ru-106 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Rh-103 Series1 Series2 Series3 Series4 5E+20 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Pd-105 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Pd-106 3.2E+20 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 Series1 Series2 Series3 Series4 2 4 6 8 10 12 Time (years) 14 16 18 20 Pd-107 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Pd-108 1.5E+20 1.4E+20 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Ag-109 1.5E+20 1.4E+20 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Cd-111 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 I-127 Series1 Series2 Series3 Series4 4E+19 3.5E+19 3E+19 2.5E+19 2E+19 1.5E+19 1E+19 5E+18 2 4 6 8 10 12 Time (years) 14 16 18 20 I-129 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Xe-131 3E+20 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Xe-132 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Xe-134 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Cs-133 Series1 Series2 Series3 Series4 5E+20 4.5E+20 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Cs-135 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Cs-137 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 La-139 Series1 Series2 Series3 Series4 4E+20 3.5E+20 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Ce-141 1.8E+17 Series1 Series2 Series3 Series4 1.6E+17 1.4E+17 1.2E+17 1E+17 8E+16 6E+16 4E+16 2E+16 2 4 6 8 10 12 Time (years) 14 16 18 20 Ce-142 7.5E+20 7E+20 6.5E+20 6E+20 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Pr-141 2.8E+19 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-143 Series1 Series2 Series3 Series4 3E+20 2.5E+20 2E+20 1.5E+20 1E+20 5E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-145 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-146 Series1 Series2 Series3 Series4 1.8E+20 1.6E+20 1.4E+20 1.2E+20 1E+20 8E+19 6E+19 4E+19 2E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-147 Series1 Series2 Series3 Series4 3E+17 2.5E+17 2E+17 1.5E+17 1E+17 5E+16 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-148 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Nd-150 7.5E+19 7E+19 6.5E+19 6E+19 5.5E+19 5E+19 4.5E+19 4E+19 3.5E+19 3E+19 2.5E+19 2E+19 1.5E+19 1E+19 5E+18 Series1 Series2 Series3 Series4 2 4 6 8 10 12 Time (years) 14 16 18 20 Pm-147 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Sm-147 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Sm-149 Series1 Series2 Series3 Series4 9E+19 8E+19 7E+19 6E+19 5E+19 4E+19 3E+19 2E+19 1E+19 2 4 6 8 10 12 Time (years) 14 16 18 20 Sm-151 1.4E+16 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 Sm-152 Series1 Series2 Series3 Series4 4.5E+19 4E+19 3.5E+19 3E+19 2.5E+19 2E+19 1.5E+19 1E+19 5E+18 2 4 6 8 10 12 Time (years) 14 16 18 20 Eu-153 Series1 Series2 Series3 Series4 3E+19 2.5E+19 2E+19 1.5E+19 1E+19 5E+18 2 4 6 8 10 12 Time (years) 14 16 18 20 Eu-155 Series1 Series2 Series3 Series4 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 2 4 6 8 10 12 Time (years) 14 16 18 20 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 Kr85, 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, Ru101, Ru-102, Ru-104, Rh-103, Pd-105, Pd-107, Pd108, Ag-109, Cd-111, I-127, I-129, Xe-131, Xe-132, Xe-134, Cs-133, Cs-135, La-139, Pr-141, Nd-143, Nd145, 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
