THE DEVELOPMENT OF DEPLETION PROGRAM COUPLED WITH MONTE CARLO COMPUTER CODE

Cuong Kien NGUYEN(1), Nghiem Ton HUYNH(1), Tan Huu VUONG (2) (1)Vietnam Atomic Energy Institute, Dalat Nuclear Research Institute (2)Vietnam Agency for Radiation and Nuclear Safety Abstract The paper presents the development of depletion code for light water reactor coupled with MCNP5 code called the MCDL code (Monte Carlo Depletion for Light Water Reactor). The first order differential depletion system equations of 21 actinide isotopes and 50 fission product isotopes are solved by the Radau IIA Implicit Runge Kutta (IRK) method after receiving neutron flux, reaction rates in one group energy and multiplication factors for fuel pin, fuel assembly or whole reactor core from the calculation results of the MCNP5 code. The calculation for beryllium poisoning and cooling time is also integrated in the code. To verify and validate the MCDL code, high enriched uranium (HEU) and low enriched uranium (LEU) fuel assemblies VVR-M2 types and 89 fresh HEU fuel assemblies, 92 LEU fresh fuel assemblies cores of the Dalat Nuclear Research Reactor (DNRR) have been investigated and compared with the results calculated by the SRAC code and the MCNP_REBUS linkage system code. The results show good agreement between calculated data of the MCDL code and reference codes. Keywords: MCNP5, MCDL, depletion, Implicit Runge Kutta method, HEU and LEU fuels 1. Introduction

Fuel burn-up evaluation is played very important role of reactor because reactor’s

safety, utilization and core life time are depended on fuels burn-up process during operation. The determination of fuel burn-up in a reactor can be achieved by experiments or theory calculations. Many experiment methods have been applied to estimate fuel burn-up such as gamma scanning method, reactivity method…Although experimental burn-up data are very accurate but conducting experiment is complicated and difficult. Calculation method is more simple, economic and good enough for optimization in reactor life time estimating.

Normally, depletion code needs to link or couple with another code such as diffusion code using deterministic method or transport code to obtain reaction rate, neutron flux or power distribution. Depletion equation can be solved by using numerical method together with these data and supplied data that include fission yield fraction of heavy isotopes, half-life of isotopes and reaction types of each considered isotope. The MCNP5 code [1] has been chosen for couple in depletion calculation because the code can be applied for three dimensions true geometry of reactor as well as using continuous energy library. A main problem in using a Monte Carlo code is significantly long calculation time. But high performance computation of personal computer today is more and more powerful so running time is extremely reduced by parallel or vectorized calculation using PC-cluster system. MCNP libraries are also advantage for users to update or modify if necessary by using NJOY code [2] with latest evaluated nuclear data.

MONTEBURN [3], MOCUP [4], MCODE [5] are famous computational codes for burn-up calculation coupling between the ORIGEN [6] code and the MCNP code. The disadvantages of these codes are depending on ORIGEN code and need numerical method like predictor and corrector method to achieve small error of burn-up calculation results such

as in the MCODE code but time step in the code is limited. The SRAC system code [7] using the CITATION code to solve the neutron diffusion equation and diffusion code is only good for simple geometry of reactor core. For depletion calculation, COREBN based on CITATION and burn-up package (DCHAIN: Code for analysis of build-up and decay of nuclides) using analytical method to solve the depletion equation. The main problem here is the lumped fission product. This data is not easy to create by users. The MCNP_REBUS linkage system code [8] is quite good for depletion calculation but the same problem of data libraries is also inconvenient for users. Beside the lumped fission product has been created by WIMS-ANL [9], pseudo libraries for the REBUS code with many isotopes (about 21 isotopes) are also needed to prepare for the MCNP_REBUS system code. Number density of U-235 for each type of fuel also needs to be prepared for interpolation using polynomial equation or spine third or sixth order following burn up time steps. BURNCAL code [10] has been developed by SANDIA National Laboratory (USA) is a simple code for depletion calculation using Euler method and users need to supply power density of depletion cell. This code is simple but provides good idea for improvement in depletion calculation.

The MCDL code is developed with more convenient for users in which the isotope Pr-141 is used to replace the lumped fission product because its cross section is nearly the same with the lumped fission product. Also, 21 actinide and 50 fission product isotopes are considered to suffice for depletion calculation. Reaction chains including reaction types used in the MCDL code have been taken from the SRAC system. The library of the MCDL code contains data of isotope label in the MCNP5 code, user’s defined name of 71 isotopes, decay constants and fission yield products of 21 actinide isotopes to 50 fission product isotopes. Radau IIA method [11] has been applied for solving burn-up equation to assure accuracy of calculation results with long time step about 50 to 100 days.

Verification and validation of the MCDL code has been carried out by applying it for burn-up analysis of the HEU, LEU fuels and fresh HEU, LEU cores of the DNRR. For whole core calculation, active length 60 cm of each fuel assembly is divided to 5 equal volumes following axial direction with 12.0 cm height in each node.

2. MCDL depletion code introduction 2.1. MCDL structure

The MCDL code includes three parts: main control of the code, receiving calculated results from MCNP running and depletion calculation. The data library for depletion is taken in SRAC code and WIMSD-5 [12] codes for all isotopes. Only five reaction types are considered for depletion including (n, fission), nu-bar reaction to calculate for number of neutron released per fission of 21 heavy isotopes, (n, 2n), (n, ) and (n, p) for Sm149 only. In beryllium poisoning calculation [13], three reactions (n, α), (n, p) and (n, t) have been used. FORTRAN90 programming language is chosen for composing the code and allocation array mode is applied. So the limit of running memory of the code is only depended on memory of user’s computer. The MCDL code can be run both on Linux or Windows operating systems with MPI environment for parallel calculation of the MCNP5 code.

The MCDL code structure is described in Figure. 1 including 5 important subroutines for depletion cycle calculation. Detail function of each subroutine is as follows:

1. Main subroutine is to control all depletion calculation following time steps and read all input and library files: Input for depletion, modified input for MCNP, library data.

2. RunMCNP subroutine is to submit running case of the MCNP5 code with the modified input. The modified input is added data for tallying neutron flux, reaction rate of each active cell in original MCNP input file. The MCDL code will be executed after the MCNP5 code is finished.

3. ReadMCTAL subroutine is used to read tallied results from MCNP and then transfer all these information to the depletion subroutine.

4. Depletion subroutine carries out depletion calculation and writes burn-up time, number densities to a temporary file called burnup.dat.

5. Update subroutine has main function for updating calculation number densities of each isotope to create new input file for the MCNP5 code for the next running step. Number density of all isotopes in depletion node and total number densities isotopes in each material card are updated after having new calculated number densities from “depletion” subroutine.

Figure 1. General structure of the MCDL code

MCNP-input

Input-DepletionLibrary-Depletion

MAIN-Reading MCNP-input

- Reading Input-Depletion- Reading Library-Depletion

Running MCNP

Reading MCTAL

MCTAL file

Library-MCNP NJOY+ENDF

Depletion calculation

TEMP file (time, # density)

Updating MCNP input

Running MCNP for next time step

END

Report Bu distribution - keff

The “be-poisoning” subroutine is also implemented to evaluate number densities of 9Be, 6Li, 4He and 3H isotopes after finishing the burn-up calculation. In case cooling calculation, all reaction rates have to be assigned equal zero; this means that only decay is considered. 2.2. Input and output of MCDL code Input files for the MCNP5 and the MCDL codes need to be prepared by users. Name of the input file for the MCDL code has to be set as INPUT.inp and arbitrary for name of MCNP input. The order of depletion and beryllium poisoning cells in MCDL code must be matched with depletion cells from the top to the bottom in input for MCNP code. Material card in the MCNP input file needs to put all 71 number densities for each depletion cell and other materials like aluminum, silicon, and oxygen. This means the MCDL code can be used for many kinds of fuel types with different enrichments or material compositions. Name of cells and material cards must be used 5 digits and in formatted form for easily to assess in reading and writing data by the FORTRAN programming language. In the input file of the MCDL code, required data need to supply including: label of depletion cells as the same in MCNP input, volume of depletion cells, number of burn-up steps, power for burn up in MW unit, time steps in day unit, surface cards of changing control rod positions following burn up steps, name of the MCNP input file, directory of MCNP and output file. All declared data will be read in free format by the MCDL code. Notice that the last step for burn-up calculation is only input for next step in MCNP code but not running at all. After running the MCNP code, the MCTAL file recorded main data that needed for solving depletion equation of each isotope and each active cell. This MCTAL file is kept after each burn-up calculation step.

Output file of MCDL code contains number density of each isotope, burn-up of each depletion cell, multiplication factor of all time steps. Information about the standard deviation of multiplication factor and the relative error of reaction rates after reading MCTAL file is also printed in the mctest file. One temporary file records data about the mass of uranium and plutonium isotopes. 2.3. Algorithm for solving depletion equation The time rate change of concentration or number density in each specific isotope is equal to its production rate per unit volume less its removal rate per unit volume. Production can be by: fission, neutron reactions of parent isotope, decay of parent isotope. Removal can be by: fission if interested isotope is fissionable isotope, neutron reactions and self-decay to a daughter isotope. A system linear first-order differential equations can be developed for all isotopes that be considered in the burn up problem. Because a large number of coupled equations of interested isotopes need to be solved, numerical method is only chosen practical solution method. Depletion equation of each specific isotope in the burn-up problem has the form:

)( ,,,, iiixiifj k l

lilkikxjjfjii NNNNNN

dtdN (1.1)

where

dtdNi is the time rate of change in number density of isotope i,

j

jjfji N , is the production rate of isotope i from fission of all fissionable nuclides,

k

kikx N , is the production rate of isotope i from neutron transmutation of all isotopes

including (n, 2n), (n,), (n, p),

l

lil N is the production rate of isotope i from decay of all isotopes including α, β+, β-,

iif N, is the removal rate of isotope i by fission,

Nix, is the removal rate of isotope i by all reactions except fission,

iiN is the removal of isotope i by decay. In the equation, Ni is number density of isotope i in atoms/cm3, t is time in seconds,ji

is the fission yield fraction for the production of isotope i from fission of isotope j, f,i is the microscopic fission cross section for isotope i in cm2, x,i is the microscopic capture cross section (all reactions minus fission) for isotope i in cm2, is the neutron flux in n/cm2.s, and i is the decay constant for isotope i in 1/s. Neutron flux, reaction rates of 4 reactions and multiplication factor are calculated by using the MCNP code. One group neutron flux in active cell is provided by tally F4 of the MCNP code:

dEEmm )( (1.2)

Neutron flux has to be normalized to get absolute value of neutron flux under thermal power of reactor. The absolute neutron flux can be obtained as follows [14]:

mefff

m kwvP 1106022.1 13

(1.3)

where P is power of reactor (MW), v is average number of neutrons released per fission, wf is effective energy released per fission (MeV/fission) and keff is the effective multiplication factor.

Actually, getting data including neutron flux, reaction rate and effective multiplication factor from MCTAL file is easier than reading from output file of MCNP code. Reaction rates R are received by MCNP code in tally as:

dEEER mnxmnx )()( (1.4)

where Rmnx is reaction rate type x of nuclide n in cell m, micro cross section has unit barns. Library of the MCDL code supplies name of each isotope in the MCNP code, defined users’ names of each isotope, decay constants and fission yield fraction of 21 actinide isotopes to 50 fission product isotopes. Reaction types of burn up chain model of the SRAC and MVP-BURN code systems [6, 15] are applied for the MCDL code and the lumped fission product is replaced by the Pr-141 isotope. Burn-up chain of actinide and fission product isotopes is presented in Fig. 2.

Fig.2. Burn-up chain model of actinide and fission products isotopes in the MCDL code

Implicit Runge-Kutta (IRK) [16] with the Radau IIA method [11] is applied for solving depletion equations of each isotope in each active cell. In the IRK method, we consider the numerical solution of ordinary differential equation system with the form:

Pr141

))(,()( tytfdt

tdy for ],( 0 fttt , and 00 )( yty (2.1)

where y is unknown functions, ftt 00 , y0 is initial values and f is a given vector of unknown functions. Denoting yn is a numerical approximation of y(tn) for n=0, 1, 2, …, m with m is positive integer while tn being the mesh points satisfying t0 < t1 < . . . <tm-1 < tm = tf. And hn= tn+1 − tn is called the step size. For simplicity, a fixed step size, h = (tf − t0)/m, is used. The IRK method is defined by

),(1

1

s

iiininn uhctfbhyy (2.2)

s

jjjnijni siuhctfahyu

1

,...,1),,( (2.3)

where n=0, 1, 2, …m-1, tn=t0+nh, s is a positive integer called stage number, and aij, bi, ci are IRK parameters. The condition is aij 0 and satisfies the requirements:

s

iiij ca

1for si ,...,2,1 and

s

iib

1

1

Different selections these parameters give different IRK method which may have different solutions and stable properties. In the Radau IIA method, the parameter ci for i=1, 2,…, s are

selected as zero of Radau polynomial ))1(( 11

1ss

s

s

ttdd

, aij for i,j=1, 2, …, s are solution of

linear system:

s

i

qiq

jij qcca

1

1 with i, q=1, 2, …, s

and bi=as,i for i=1, 2, …, s. In Radau IIA method, s=5 is chosen. In each IRK step, a system (2.3) of sN equations is needed to solve for stage value s

iiu 1 by iterative algorithm. Normally, we can reduce equation (2.3) as:

)(zhAFz (2.4) where z = (z1, z2, …, zs)t with zi = ui - yn

ssss

s

s

aaa

aaaaaa

A

.....

.

.

21

22221

11211

and

),(.

),(),(

)( 22

11

nssn

nn

nn

yzhctf

yzhctfyzhctf

zF (2.5)

Superscript t denotes the transpose matrix. If inverse matrix A-1 exists that is true in Radau IIA method, equation (2.4) can be

written as zAzhF 1)( so the update of equation (2.2) can be shown as below:

s

iiinsnsnn zdyzAbbbyzhFbbbyy

1

121211 ),...,,()(),...,,(

Where (d1, d2, …, ds) = (b1, b2, …, bs)A-1. In k-th iterate z(k) of an iterative algorithm is selected as a numerical solution of (2.4), the IRK iterative sequence ny is calculated as:

s

i

kiinn zdyy

1

)(1 for n=0, 1, 2, …, m-1 (2.6)

Where tks

kkk zzzz ),...,,( )()(2

)(1

)( with Nki Rz )( . Using equation (2.6) can reduce the risk of

amplifying numerical errors from solving the nonlinear system (2.4). It is confirmed that Radau IIA method is efficient for solving nonlinear differential

system equations with error smaller than 10-9 and completely good application for depletion calculation [17]. 3. Calculation results of fuels and reactor cores of the DNRR 3.1. Fuel assembly calculation Burn-up calculation of the HEU and LEU VVR-M2 fuel assemblies of the DNRR has been carried out using the MCDL code and the results were compared with those calculated by the PIJ module in the SRAC code. Detailed characteristics of HEU and LEU fuels are described in Table. 1 and Fig. 3. Table. 1 Characteristics of HEU and LEU fuel VVR-M2 types

Calculation model of fuel is the same between the MCDL and the SRAC codes. This

means that the active volume taking the same with outermost layer having hexagonal shape. The ENDF/B-VII.0 was applied for the MCNP and SRAC calculation. Total burn-up of HEU and LEU fuel are about 36% U-235 and 25% U-235 respectively. Output parameters including neutron flux, reaction rate and multiplication factor were obtained errors under 0.1% and standard deviations under 0.005% with total particles in the problem about 12 millions. Difference of multiplication factor between MCDL code and SRAC code is smaller than 120 pcmk/k for both cases (see Table. 2). Maximum difference of atom densities of all

Parameter HEU Fuel LEU Fuel

Thickness, mm

Fuel element (fuel meat and cladding) 2.5 2.5 Fuel meat (Al-U alloy ) 0.7 0.94

Cladding (Al) 0.9 0.78

Spaces for water flow 2.5-3 2.5-3

Cross section area, cm2

Fuel cell 10.61 10.61

Water flow 5.85 5.85

Length, mm

Total fuel assembly 865 865

Active height (fueled part) 600 600

U-235 content

Enrichment, % 36 19.75

Weight, g 40.2 (approx.) 49.7 (approx.) Fig. 3. HEU and LEU fuels

isotopes is under 10% to actinide isotopes Cm-243, Cm-245 and Cm-246. Figs. 4 and 5 depict calculated results of MCDL and SRAC codes of infinite multiplication factor and atom density of heavy isotopes at last burn up-step.

Table. 2. Infinite multiplication factor HEU and LEU fuel depending on burn-up

Fig. 4. Infinite multiplication factor of HEU fuel depending on burn-up steps and atom

density of actinide isotopes at the end of burn-up step (~ 36% burn up of U-235)

Fig. 5. Infinite multiplication factor of HEU fuel depending on burn-up steps and atom

density of actinide isotopes at the end of burn-up step (~ 29% burn up of U-235)

HEU fuel LEU fuel

BU (%) MCDL SRAC Difference (pcm) BU (%) MCDL SRAC Difference

(pcm) 0.00 1.64156 1.638661 107.77 0.00 1.63462 1.631742 107.90 1.28 1.60500 1.602053 114.61 1.02 1.60514 1.602441 104.93 2.56 1.59709 1.594228 112.41 2.03 1.59819 1.595491 105.85 5.12 1.58523 1.582706 100.60 4.05 1.58836 1.586178 86.61 7.67 1.57437 1.571904 99.65 6.06 1.57994 1.578034 76.45 10.20 1.56334 1.561076 92.77 8.06 1.57212 1.570020 85.08 15.25 1.54062 1.538884 73.22 12.03 1.55588 1.553888 82.39 17.76 1.52874 1.527383 58.12 14.01 1.54743 1.545684 73.00 20.26 1.51669 1.515543 49.90 15.97 1.53911 1.537218 79.97 22.76 1.50415 1.503187 42.59 17.92 1.53037 1.528687 71.94 25.24 1.49119 1.490513 30.46 19.87 1.52171 1.519988 74.45 27.71 1.47798 1.477389 27.07 21.80 1.51241 1.511100 57.32 30.17 1.46391 1.463778 6.16 23.73 1.50348 1.502018 64.74 32.63 1.44936 1.449649 13.75 25.64 1.49368 1.492728 42.70 33.85 1.44202 1.442379 17.26 26.60 1.48922 1.488003 54.92 36.29 1.42661 1.427405 39.04 28.50 1.47926 1.478384 40.06

3.2. HEU and LEU cores calculation For burn up distribution calculation of whole core, all fuel assemblies and beryllium rods or blocks are divided to 5 nodes with the same volumes. Error of neutron flux and reaction rates data from MCNP code is under 0.12% and 2.1%, respectively and standard deviation of multiplication factor is under 0.008% with total particles 12 millions in one running case. Depletion calculation of fresh core using 89 HEU fuel assemblies and 15 beryllium rods of the DNRR at start up working core in 1984 was implemented by MCDL code and MCNP_REBUS code. 538 full power days (FPD) continuous operating of the DNRR under nominal power 500 kW and 100 days for cooling are modeled in both codes. The maximum discrepancies of active nodes and cells are under 2.0% and 1% respectively. The excess reactivity at the end of fuel cycle of results of MCDL code is higher than MCNP_REBUS code about 0.215%k/k. Fig. 6 shows calculated results of both codes. The main reason of difference of multiplication value between two codes is different used library data for MCNP code. In MCNP_REBUS code, ENDF/B6.0 is applied and calculated multiplication factor is smaller. Library ENDF/B6.0 of MCNP code has coarse meshes of energy while ENDF/BVII has fine meshes of energy and evaluated data in ENDF/BVII is updated with latest experimental data. Multiplication calculation for benchmark problems of both libraries is shown in [18].

Fig. 6. Burn-up (% U-235) distribution of fresh HEU core after 538 FPDs operation

(MCNP_REBUS code at upper values – MCDL code at lower values)

Fresh LEU core using 92 fuel assemblies and 12 poisoned beryllium rods is calculated for burn up distribution. Total time for depletion scheme is 600 full power days and 100 days for cooling. Maximum difference between two codes in active nodes and cells is about 5% and 2.0% respectively. The different of excess reactivity at the end of cycle is about 0.30%k/k. Fig.7 presents calculated results of both codes.

SR

SR

ShR

ShR

ShR

ShR

RgR 9.04 9.12

9.45 9.52

9.37 9.44

10.00 10.07

10.01 10.09

9.44 9.51

9.61 9.69

9.72 9.79

9.63 9.69

10.17 10.24

10.12 10.19

9.37 9.44

9.43 9.51

8.99 9.05

9.02 9.09

9.35 9.44

9.23 9.29

9.05 9.12

8.45 8.53

8.46 8.53

8.44 8.50

8.27 8.31

9.14 9.20

9.16 9.22

8.23 8.28

12.88 12.94

11.47 11.55

12.84 12.92

9.09 9.14

9.38 9.42

10.27 10.30

10.41 10.46

9.62 9.68

9.40 9.44

8.30 8.34

8.09 8.13

8.27 8.31

8.71 8.75

8.06 8.10

7.75 7.81

7.81 7.86

8.22 8.26

7.87 7.93

8.67 8.73

8.52 8.57

9.09 9.12

8.02 8.07

8.52 8.59

9.68 9.73

8.78 8.85

9.58 9.64

9.87 9.90

8.91 8.97

8.39 8.46

8.33 8.39

8.36 8.42

7.63 7.71

8.07 8.13

7.60 7.66

7.60 7.65

8.77 8.81

8.33 8.38

9.62 9.64

8.77 8.80

8.17 8.22

9.21 9.24

8.63 8.68

8.56 8.61

9.34 9.41

9.20 9.26

8.27 8.33

8.09 8.14

8.11 8.16

9.86 9.91

10.06 10.10

9.97 10.03

9.53 9.59

9.27 9.32

9.17 9.23

12.97 13.04

11.48 11.56

12.94 13.02

11.44 11.51

13.05 13.12

11.63 11.70

13.07 13.13

11.51 11.58

11.66 11.74

10.73 10.75

Fig. 7. Burn-up (U-235%) distribution of fresh LEU core after 600 FPDs operation (MCNP_REBUS code at upper values – MCDL code at lower values)

4. Conclusions and future works

The depletion calculation code system called MCDL has been developed for evaluation, analysis of fuel burn-up and beryllium poisoning as well as cooling depending on control rod positions in light water reactors. Tally results from MCNP code including neutron flux, reaction rates in one neutron energy group and multiplication factor to depletion module that applied the Radau IIA method. Burn-up calculation scheme in the MCDL code allows large time step about 100 full power days. The MCDL code has been used to investigate burn up of HEU and LEU fuel VVR-M2 types that used for the DNRR and burn up distribution of whole fresh HEU, LEU cores. Calculated results show good agreement with calculated results by reference codes SRAC and MCNP_REBUS linkage. The code is a good tool for core and fuel management of the DNRR using HEU and LEU fuels. However, the code needs to use for burn-up calculation of nuclear power plant or benchmark problems to confirm its accuracy. In the next study, the full fuel using cycle of HEU and new LEU working core of the DNRR will be considered with detailed operating time and core configurations. 5. References [1]. X-5 Monte Carlo Team, “MCNP — A General Monte Carlo N-Particle Transport Code, Version 5”, Los Alamos national laboratory, Apr. 2005. [2]. Kahler, Albert C. III, MacFarlane, Robert, “The NJOY Nuclear Data Processing System, Version 2012”, Los Alamos, USA, 2012. [3]. Poston, D.I, Trellue, H.R., “User’s Manual, Version 1.00 for MONTEBURNS, Version 3.01”, LA-UR-98-2718, Los Alamos National Laboratory, USA, June 1998. [4]. Moor, R.L. et al., “MOCUP: MCNP–ORIGEN2 Coupled Utility Program”, INEL-95/0523, Idaho National Laboratory, USA, 1995.

SR

SR

ShR

ShR

ShR

ShR

RgR 9.21 9.19

10.69 10.74

11.29 11.29

11.31 11.31

10.95 10.88

10.91 10.92

11.00 11.00

11.06 11.00

11.41 11.41

11.32 11.32

10.62 10.67

10.62 10.63

9.33 9.37

9.35 9.39

9.60 9.57

9.52 9.55

9.28 9.31

7.37 7.32

7.62 7.59

7.77 7.80

7.37 7.39

8.35 8.34

8.24 8.21

7.28 7.30

7.79 7.75

7.81 7.77

9.01 9.18

9.07 9.26

7.43 7.41

7.00 6.97

7.02 6.98

7.85 7.83

8.68 8.83

7.99 8.15

7.30 7.30

7.87 7.85

7.21 7.18

6.82 6.85

6.94 6.96

7.52 7.48

7.43 7.39

7.01 7.03

7.03 7.05

7.55 7.49

6.99 6.95

7.43 7.37

7.01 6.96

7.68 7.63

7.69 7.62

7.15 7.12

7.90 7.92

8.95 8.89

7.95 7.88

7.96 7.93

8.40 8.33

9.18 9.10

8.09 8.02

8.07 8.03

7.51 7.49

6.85 6.79

7.30 7.26

7.63 7.67

6.72 6.70

6.84 6.80

6.76 6.79

6.89 6.86

8.09 8.11

7.42 7.38

8.14 8.06

8.06 7.99

7.36 7.37

7.94 7.87

7.21 7.22

7.57 7.59

8.49 8.48

8.25 8.22

7.22 7.24

6.70 6.74

6.91 6.89

8.23 8.19

7.54 7.52

8.16 8.32

8.79 8.94

7.90 7.88

7.02 6.99

6.97 6.96

7.33 7.32

9.31 9.49

9.10 9.28

7.86 7.82

7.81 7.78

10.67 10.68

[5]. Xu, Zh., “Design Strategies for Optimizing High Burn up Fuel in Pressurized Water Reactors”, PhD Dissertation, Massachusetts Institute of Technology, USA, January 2003. [6]. A. G. Croff, “A User’s Manual for The ORIGEN2 Computer Code”, Oak Ridge National Laboratory, Tennessee USA, July 1980. [7]. Keisuke OKUMURA, Teruhico KUGO, Kunio KANEKO and Keichiro TSUCHIHASHI, “SRAC2006: A Comprehensive Neutronics Calculation Systems”, JAEA, Japan, Feb. 2007. [8]. John G. Stevens, “The REBUS-MCNP Linkage”, Argonne National Laboratory, Apr. 2008. [9]. J.R. Deen et al., “WIMS-ANL User Manual, Rev. 5,” ANL/TD/TM99-07, Feb. 2003. [10]. Parma, E.J., “BURNCAL: A Nuclear Reactor Burn up Code Using MCNP Tallies, Sandia National Laboratories”, SAND2002-3868, SANDIA NL, USA, 2002. [11]. Dexuan Xie, “A New Numerical Algorithm for Efficiently Implementing Implicit Runge Kutta Method”, pantherfile.uwm.edu/dxie/www/Papers/radau_new.pdf. [12]. WIMS Library Update Project WLUP, http://www.nds.iaea.org/wimsd/, 2001. [13]. Teresa Kulikowska et al.,”He-3 and Li-6 Poisoning of The Maria Reactor Beryllium Matrix”, Raport IAE-40/A, 1999. [14] Snoj, L., Ravnik, M.,” Calculation of power density with MCNP in TRIGA reactor”, In: Proceedings of the International Conference Nuclear Energy for New Europe 2006, Portoroz, Slovenia, p. 102. [15]. Keisuke OKUMURA, Yasunobu NAGAYA and Takamasa MORI, “MVP-BURN: Burn-up Calculation Code Using A Continuous-energy Monte Carlo Code MVP”, JAEA, Japan, Jan. 2005. [16]. William H. Press et.al, “Numerical Recipes in Fortran 90. The Art of Parallel Scientific Computing Second Edition”, NY USA, 2002. [17]. A. Stankovskiy and G. Van den Eynde, “ Advanced Method for Calculation of Core Burn-up, Activation of Structural Materials, and Spallation Products Accumulation in Accelerator-Driven Systems”, Hindawi Publishing Corporation, Science and Technology of Nuclear Installations, Volume 2012. [18]. http://www.nndc.bnl.gov/endf/b7.0/doc/b7-keff_kalc.txt

PHÁT TRIỂN CHƯƠNG TRÌNH TÍNH CHÁY KẾT HỢP VỚI

CHƯƠNG TRÌNH MONTE CARLO Nguyễn Kiên Cường(1), Huỳnh Tôn Nghiêm(1), Vương Hữu Tấn (2)

(1)Viện Năng lượng Nguyên tử Việt Nam, Viện Nghiên cứu Hạt nhân Đà Lạt (2)Cục An toàn Bức xạ và Hạt nhân

Tóm tắt Bài báo trình bày việc phát triển chương trình tính toán cháy cho các Lò Phản ứng nước nhẹ kết hợp với chương trình tính toán MCNP5 và được đặt tên là chương trình MCDL (Monte Carlo Depletion for Light Water Reactor). Hệ phương trình vi phân bậc nhất của 21 đồng vị nặng và 50 đồng vị sản phẩm phân hạch được giải bằng phương pháp Radau IIA sau khi có thông lượng neutron, tốc độ phản ứng một nhóm năng lượng và hệ số nhân tính từ chương trình MCNP5. Quá trình tính toán nhiễm độc berily và làm nguội cũng được tích hợp trong chương trình. Để thẩm định và hiệu lực chương trình tính cháy MCDL, nhiên liệu độ giàu cao và độ giàu thấp cùng với vùng hoạt ban đầu dùng 89 bó nhiên liệu độ giàu cao và 92 bó nhiên liệu độ giàu thấp được khảo sát, kết quả tính toán được so sánh với kết quả tính toán từ hệ

chương trình SRAC và MCNP_REBUS. Kết quả cho thấy có sự phù hợp tốt giữa kết quả tính toán từ chương trình MCDL và các chương trình tính toán cháy nhiên liệu này. Từ khóa: MCNP5, MCDL, tính cháy, phương pháp Radau IIA, nhiên liệu độ giàu cao và thấp