Effects of Fuel and Coolant Temperatures and Neutron Fluence on ...

Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 43, No. 2, p. 189–197 (2006)

TECHNICAL REPORT

Effects of Fuel and Coolant Temperatures and Neutron Fluence on CANDLE Burnup Calculation Hiroshi SEKIMOTO and Yutaka UDAGAWAy Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152-8550, Japan (Received September 13, 2005 and accepted November 17, 2005) A new method is developed to analyze CANDLE (Constant Axial shape of Neutron flux, nuclide densities and power shape During Life of Energy producing reactor) burnup, where the microscopic group cross-sections are evaluated at every space mesh by TLLI (Table Look-up and Linear Interpolation) method, and used to analyze a fast reactor with natural uranium as a fresh fuel. The results are compared with the conventional method, where only one set of the microscopic group cross-sections is employed, to investigate the effects of space-dependency of the microscopic group cross-sections and feasibility of the old method. The differences of the effective neutron multiplication factor, burning region moving speed, spent fuel burnup and spatial distributions of nuclide densities, neutron fluence and power density may be considerable from the reactor designer point. However, they are small enough when we study about the characteristics of CANDLE burnup for different designs. KEYWORDS: CANDLE burnup, fast reactor, group constants, effective neutron multiplication factor, burnup, power density, nuclide density, neutron fluence, fuel temperature, coolant temperature

I. Introduction

spent fuel

fresh fuel

2. Principles of CANDLE Burnup CANDLE burnup strategy can be realized, when the infinite-medium neutron multiplication factor, kinf , satisfies some characteristics. The value of fresh fuel should be less than unity. Otherwise the power profile will extend so much to the fresh fuel region or make the reactor super-critical. After certain amount of burnup it takes more than unity to keep the reactor critical by the transmutation of fertile materials to fissile materials. Finally it becomes again less than unity y

Corresponding author, E-mail: [email protected] Present address: Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan

spent fuel

refueling burnup burning region

burning region

burnup burning region

fresh fuel

Fig. 1 Burnup and refueling scheme of the CANDLE burnup strategy

Infinite Medium Neutron Multiplication Factor

1. CANDLE Burnup Strategy A new reactor burnup strategy CANDLE (Constant Axial shape of Neutron flux, nuclide densities and power shape During Life of Energy producing reactor) was proposed,1) where neutron flux shape, nuclide density distribution shapes and power profile remain constant but move to an axial direction with a same speed. Figure 1 shows the schematic view of this burnup strategy, and also refueling scheme for this strategy. Here important points are that solid fuel elements are fixed at their corresponding positions in a core and that any movable burnup reactivity control mechanisms such as control rods are not required. Namely the abovementioned motion of the shapes is autonomous. This burnup strategy can be realized, if the infinite-medium neutron multiplication factor, kinf , satisfies some characteristics. The principle of this burnup will be mentioned in the next section.

burning region

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

200

400 Axial Position (cm)

600

800

Fig. 2 Change of kinf along core axis

caused by the accumulation of fission products (FPs) and consumption of fissile materials. A typical kinf change along the core axis is shown in Fig. 2. The left side of the core is the fresh fuel region. At the area where the fresh fuel region changes to the burning region, kinf increases with time. On

ÓAtomic Energy Society of Japan 189

190 the other hand at the area where the burning region changes to the spent fuel region, it decreases with burnup. Therefore, as the bunup succeeds, the burning region moves to the fresh fuel region. At the equilibrium state, the shape of power density does not change along burnup. 3. Merits and Demerits of CANDLE Burnup CANDLE burnup strategy has the following general merits: (1) Burnup reactivity control mechanism is not required: The reactor control becomes simpler and easier. The excess burnup reactivity is zero, and the reactor is free from reactivity-induced accidents. Neutrons are efficiently utilized. (2) Reactor characteristics do not change with burnup: The reactor characteristics such as power peaking and power coefficient of reactivity do not change with burnup. The estimation of core condition becomes very reliable. The reactor operation strategy remains unchanged for different burnup stage. The inaccuracy of present burnup calculation is much less important for this reactor compared with for conventional reactors. Therefore, the reactor control is much simpler, easier and more reliable than the conventional ones. (3) Orifice control along burnup is not required: Since the radial power profile does not change with burnup, the required flow rate for each coolant channel does not change. Therefore, the orifice control along burnup is not required. The operational mistakes are avoided. (4) Radial power distribution can be optimized more thoroughly: Since the radial power distribution does not change with time, it can be optimized more thoroughly than conventional reactors. The optimization method is much simpler. (5) By simply increasing the core height, the reactor life can be elongated: Design of long-life reactor becomes easier. Even a very long-life reactor does not require refueling during its life. Such a reactor is well suited to the case that high infrastructures cannot be expected. (6) kinf of fresh fuel after the second core is less than unity: The risk for criticality accident is small. The transportation and storage of fresh fuels become simple and safe. Some general demerits of this burning strategy may be considered. Coolant pressure drop becomes larger, since the core becomes higher. However, the core height is a function of axial power profile half-width and drift speed of burning region. Both of these values become small for many designs. Therefore, the pressure drop is usually manageable. Freedom of optimization of axial power distribution becomes smaller. However, the total power distribution is optimized to a high level, since the radial power distribution is optimized thoroughly. Therefore, CANDLE burnup does not show any fatal demerits, even though it shows many considerable merits. The above-mentioned characteristics are only general characteristics of CANDLE. Some more merits will appear when it is applied to certain reactors. Some of these characteristics are outstandingly excellent. They will be discussed in the next section.

H. SEKIMOTO and Y. UDAGAWA

4. Equilibrium State of CANDLE Burnup in Neutron Rich Fast Reactors When CANDLE burnup strategy is applied to neutron rich fast reactors (FRs), the following outstandingly excellent merits will be expected:1,2) (1) Enriched fuels are not required after the second core: Only natural or depleted uranium is enough to be charged to the core after the second core. Namely, if the fuel for the first core or external neutron source3) is available, neither enrichment nor reprocessing plant is required. It is excellent from a physical protection point of view. (2) The burnup of the spent fuel is about 40% in typical cases: This value is competitive to the value of the presently expected FR system with reprocessing plant. It means that the 40% of natural uranium burns up without enrichment or reprocessing. (3) Long-life reactor can be designed easily, since the burning region drift speed is only about 4 cm/yr for typical cases: Even the reactor with 30 years life can be designed simply by adding 1.2 m to the initial core height. The present once-through fuel cycle of 4% enriched uranium in light water reactor (LWR) performs the burnup of about 4% of the inserted fuel, and it corresponds to the utilization of 0.7% of natural uranium. For this case 87% of the original natural uranium is left as depleted uranium. If this depleted uranium is utilized as the fuel for CANDLE reactor, 35% (¼0:870:4) of the original natural uranium is utilized. Therefore, if the LWR has already produced energy of X Joules, the CANDLE reactor can produce 50X Joules from the depleted uranium stored at the enrichment facility for the LWR fuel. If LWRs have already produced energy sufficient for full 20 years and the nuclear energy production rate will not change in the future, we can produce the energy for 1,000 years by using the CANDLE reactors with the depleted uranium. We need not mine any more uranium ore, and do not need reprocessing facility. The burnup of spent fuel becomes 10 times. Therefore, the spent fuel amount per produced energy is also reduced to be one-tenth. They are outstandingly good characteristics. However, for this case it meets the following severe problems: (1) The fuel material should maintain integrity performance for very high burnup. (2) The equilibrium state should be realized from initial core, which can employ only easily available material. For the first problem, development of fuel cladding material, which wears well for 50% burnup, is required but it may take too much time for research and development. However, though the presently attainable burnup of cladding is much less than 50%, employing simple reprocessing, where only the cladding is replaced by new one, can realize the CANDLE burnup. For the second problem, we have already found several good initial core configuration using only available neuclides.4,5) 5. Calculation of CANDLE Burnup As mentioned in Sec. I-2, CANDLE burnup can be performed only by the proper change of infinite neutron multiJOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

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Effects of Fuel and Coolant Temperatures and Neutron Fluence on CANDLE Burnup Calculation

plication factor along burnup. This behavior can be realized mainly by neutron reactions starting from proper values of the nuclide densities of fresh fuel. For confirming the performance of CANDLE burnup, accurate neutronics calculation should be employed. In the present paper, the accuracy of the neutronics calculation of CANDLE burnup at the steady state is investigated. The calculation system is nonlinear one of neutron flux and nuclide densities, and requires time consuming iterations for obtaining solutions. Therefore, in the previous calculations1,2,4,5) the temperature and burnup effects on microscopic group cross-sections were ignored, and constant microscopic group cross-sections were employed. It may cause a considerable effect on the CANDLE burnup performances, since the burnup of spent fuel is quite large. This effect is investigated in the present paper. For this purpose a new computer code is developed for steady state CANDLE burnup analysis, which can evaluate the effects of fuel and coolant temperatures and fuel burnup on group constants by table look-up and linear interpolation (TLLI) method. Obtained results are compared with those of the previous method, and accuracy of the previous method is investigated.

S;n;g : Slowing-down cross-section of n’th nuclide from g’th energy group fn;g0 !g : Element of slowing-down matrix of n’th nuclide from (g0 )’th to g’th energy group n0 !n;g : Transmutation cross-section of (n0 )’th to n’th nuclide for g’th energy group n : Decay constant of n’th nuclide n0 !n : Decay constant of (n0 )’th to n’th nuclide. In the present study, the neutron flux level is normalized by the total fission rate P as written by Z XX Nn ðrÞF;n;g g ðrÞdr ¼ P; ð3Þ g

n

where r is the space coordinate vector representing (r; z) and integration is performed over the whole core. The production of FP can be evaluated using n0 !n;g given by n0 !n;g ¼ F;n0 ;g n0 !n ; where

II. Calculation Method 1. Fundamental Equations In the present paper a cylindrical core is considered. For r–z coordinate system, the neutron balance equation and nuclide balance equation in the core can be written as the following equations, respectively: 1 @ @ @ @ rDg g þ Dg g r @r @r @z @z X X X Nn R;n;g g þ Nn fn;g0 !g S;n;g0 g0 n

n

g0

g X X þ Nn F;n;g0 g0 ¼ 0 keff g0 n ! X X @Nn A;n;g g þ Nn0 n0 !n ¼ Nn n þ @t g n0 X X þ Nn0 n0 !n;g g ; n0

where

ð1Þ

R;n;g ¼ A;n;g þ S;n;g and the absorption cross-section is the sum of fission and capture cross-sections. The distributions of neutron flux and nuclide densities move with burnup. As already mentioned their relative shapes are not changed and their positions move with a constant speed V along z axis for the CANDLE burnup. For this burnup scheme, when the following Galilean transformation given by r0 ¼ r z0 ¼ z þ vt t0 ¼ t

ð2Þ

g

g ¼g ðr; z; tÞ: Neutron flux in g’th energy group Nn ¼Nn ðr; z; tÞ: Nuclide number density of n’th nuclide Dg ¼Dg ðr; z; tÞ: Diffusion coefficient for g’th energy group g : Probability that fission neutron will be born in g’th energy group keff : Effective neutron multiplication factor R;n;g : Removal cross-section of n’th nuclide for g’th energy group A;n;g : Absorption cross-section of n’th nuclide for g’th energy group F;n;g : Fission cross-section of n’th nuclide for g’th energy group

VOL. 43, NO. 2, FEBRUARY 2006

n0 !n : Yield of n’th nuclide (FP) from neutron-induced fission by (n0 )’th nuclide (actinide). The spontaneous fission is neglected in the present study. The removal cross-section is the sum of absorption and slowing-down cross-sections:

is applied to Eqs. (1) through (3), then they can be rewritten as 1 @ 0 0 @ 0 @ @ 0 r Dg 0 g þ 0 D0g 0 0g 0 r @r @r @z @z X X 0 0 Nn R;n;g g þ Nn0 n;g1!g 0g1 n

n

g X X 0 þ N F;n;g0 0g0 ¼ 0 keff g0 n n

ð4Þ

! X @Nn0 @Nn0 0 0 ¼ v 0 Nn n þ A;n;g g @t0 @z g X X X Nn0 0 n0 !n þ Nn0 0 n0 !n;g 0g þ n0

and

n0

g

ð5Þ

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H. SEKIMOTO and Y. UDAGAWA

Z XX g

Nn ðr0 ÞF;n;g g ðr0 Þdr0 ¼ P;

ð6Þ

n

where v is the relative speed of the transformed coordinate system to the original system. If v¼V, the distributions of nuclide densities and neutron flux stand still for the transformed coordinate system, and Eq. (5) becomes ! X @Nn0 V 0 Nn0 n þ A;n;g 0g @z g X X X þ Nn0 0 n0 !n þ Nn0 0 n0 !n;g 0g ¼ 0: ð7Þ n0

n0

g

It can be expected from this equation that the speed of burning region, V, is proportional to the flux level, if the effects of radioactive decay of nuclides can be neglected. 2. Iteration Scheme of the Original Calculation Method We have two equations, Eqs. (4) and (7), which should be solved simultaneously with the flux normalization condition (6) for obtaining the equilibrium CANDLE burnup state. Hereafter only transformed equations will be treated, and the original equations will not. Then the prime used for the Galilean transformed variables will be omitted for simplicity in the following discussions. We introduce an iteration scheme to solve these equations. From a given flux distribution nuclide density distributions are obtained using Eq. (7), and then from these nuclide density distributions a more accurate distribution of neutron flux is obtained using Eq. (4) with the normalization condition. This procedure is repeated until it converges. However, in usual cases, the exact value of V is unknown. Then an initial guess is introduced, and this value should be improved at each iteration stage. If the employed value of V is not correct, the distributions of neutron flux and nuclide densities are expected to move along z-axis. Therefore, the value of V can be modified from the value of distance by which those distributions move per each iteration stage. In order to define the position of these distributions, the center of neutron flux distribution is introduced, which is defined as Z ðrÞrdr rC ¼ Z ; ðrÞdr where the meaning of integration is the same as for Eq. (3) and is the energy integrated neutron flux. In the present paper we consider our problem more theoretical than practical. Though the height of the core is finite for the practical case, the infinite length is taken as the core

height for the present study, since the height is an artificial parameter but a uniquely determined value is preferable. For this condition, CANDLE burnup can be operated during infinite past to infinite future in the strict meaning. The discussion on the core height for the actual calculation is given in the reference.1) Details of the iteration scheme and algorism to find the burning region speed V is omitted in this paper but can be found in the reference.1) 3. Modification for Improvement of Microscopic Group Cross-sections (1) Reevaluation of Microscopic Group Cross-sections In the original calculations, microscopic group cross-sections are evaluated at the beginning of each analysis and treated constant over the whole core. However, the temperature and exposed neutron fluence change for different positions in the reactor. The change of fluence causes the change of nuclide densities in the fuel. Both changes of temperature and nuclide densities cause the change of neutron spectrum. The change of neutron spectrum causes the changes of microscopic group cross-sections. The change of microscopic group cross-sections affects the CANDLE burnup characteristics. In the present study, these effects are evaluated by calculating the microscopic group cross-sections at each spatial mesh through outermost iteration loop. This outermost iteration is repeated until these cross-sections are converged as well as reactor characteristics. Since direct evaluation of the microscopic group cross-sections by using cell calculation program requires too much time, we employ the table look-up and linear interpolation (TLLI) method. Details of this method are described in the following. (2) TLLI Method The microscopic group cross section evaluation by using an interpolation table is proper for many cases in reactor physics calculation, and many methods have been proposed.6) In the TLLI method, the linear interpolation is employed from the simplicity of calculation. In the TLLI method, for the first step an interpolation table should be prepared. At the beginning the microscopic group cross-sections X;n;g (ordinates) are evaluated for typical values of fuel temperature TF , coolant temperature TC and neutron fluence F (abscissas) and prepared as a form of the interpolation table. The details of method employed in the present work for preparation of the table are described in Chap. IV. At the step for microscopic group cross-sections reevaluation in the outermost iteration of steady state CANDLE burnup calculation, the cross-sections are calculated by using linear interpolation using two values for each abscissa as the followings:

For TF;m TF TF;mþ1 ; TC;m TC TC;mþ1 and Fm F Fmþ1 X;n;g ðTF ; TC ; Fmþ1 Þ X;n;g ðTF ; TC ; Fm Þ ðF Fm Þ; Fmþ1 Fm X;n;g ðTF ; TC;lþ1 ; Fmþ1 Þ X;n;g ðTF ; TC;l ; Fm Þ ðTC TC;l Þ where X;n;g ðTF ; TC ; Fm Þ ¼ X;n;g ðTF ; TC;l ; Fm Þ þ TC;lþ1 TC;l X;n;g ðTF ; TC ; FÞ ¼ X;n;g ðTF ; TC ; Fm Þ þ

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where X;n;g ðTF ; TC;l ; Fm Þ ¼ X;n;g ðTF;k ; TC;l ; Fm Þ þ

X;n;g ðTF;kþ1 ; TC;l ; Fm Þ X;n;g ðTF;k ; TC;l ; Fm Þ ðTF TF;k Þ TF;kþ1 TF;k

where X;n;g ðTF;k ; TC;l ; Fm Þ is looked up on the interpolation table. The size of each abscissa is determined from preparatory calculations. These calculations are performed with increasing one abscissa while the other two abscissas are fixed. The size of abscissa is determined so that the ordinate saturates at a certain value. (3) Temperature Calculation The coolant temperature TC is calculated by solving the following equation: CG

@TC ¼ QC ; @z

where C, G and QC are the thermal capacity, coolant mass flow rate and heat production rate per unit cell volume, respectively. The mass flow rate is set to equalize the coolant pressure at each outlet of the coolant channel. The fuel temperature TF is calculated by solving the following equation: 1 @ @ rk TF ¼ 0; QF þ ð8Þ r @r @r where k and QF are the thermal conductivity of the fuel and the heat production rate per unit fuel pellet volume, respectively. The cladding temperature is calculated by solving the equation similar to Eq. (8) where QF ¼0. The heat transfer through the gap between fuel pellet and cladding is treated by using the gap conductance model. For the calculation of heat transfer from the coolant to the cladding is performed by using the following Nusselt number:

Table 1 Reactor core design parameters Thermal rate Core height Core radius Reflector thickness Coolant channel diameter Cladding thickness Coolant channel pitch Material Fuel Cladding Coolant Coolant core-inlet temperature Coolant core-outlet maximum temperature

3,000 MW 800 cm 200 cm 50 cm 0.668 cm 0.035 cm 1.132 cm Nitride (99% enriched 15 N, 81%TD) HT-9 Pb–Bi (44.5%, 55.5%) 600 K 770 K

reflective and vacuum boundary conditions at the top and bottom boundaries give almost the same results. Therefore, the core height can be considered equivalent to infinity. It can provide an ideal CANDLE burnup, for which the burning region moves from minus infinity to plus infinity of the core axial coordinate without any changes of distributions of neutron flux, nuclide densities and power density. The neutron energy group structure is given in Table 2. The microscopic group cross-sections are calculated by using a part of SRAC code system11) with JENDL-3.2 nuclear data library.12) The thermal hydraulic data of lead bismuth are obtained from the references.7,13)

Nu ¼ 3:0 þ 0:014ðRe PrÞ0:8 ; where Re and Pr are Reynolds number and Prandtl number, respectively.7) Here detailed description of thermal analysis is omitted, but can be found in the references.8,9) (4) Fluence Calculation The neutron fluence is calculated by solving the following equation: Fðz þ dzÞ ¼ FðzÞ þ

dz ðzÞ; V

where dz is the axial space mesh width of the calculation.

III. Calculation Conditions We will analyze a 3,000 MWt fast reactor with nitride fuel and lead-bismuth coolant. Its design parameters are shown in Table 1. The present design employs tube-in-shell type fuel unit10) in order to increase fuel-to-coolant volume ratio under a given cooling ability to enhance neutron economy. The core height is chosen to be 8 m. The core is surrounded by reflector of lead-bismuth eutectic. For this core height both VOL. 43, NO. 2, FEBRUARY 2006

IV. Preparation of Interpolation Table 1. Choice of Proper Abscissa Size of the Table Before evaluating the elements of interpolation table, proper number of values for each abscissa of the table is necessary to be investigated and fixed. At first the minimum and maximum values for each abscissa are chosen as its boundary values. They are 450 and 2,100 K for the fuel temperature, 450 and 1,900 K for the coolant temperature, and 2:11017 /cm2 and 3:31024 /cm2 for the neutron fluence, respectively. The proper number of values for each abscissa of the table is determined independently. It is performed by preparing one-dimensional interpolation tables for each abscissa. At first the proper number of values for fuel temperature abscissa is tried to be determined. An interpolation table is prepared, which has two elements. The maximum and minimum values, 450 and 2,100 K, are chosen as the fuel temperature abscissa, and the minimum values, 450 K and 2:1 1017 /cm2 , are chosen as the other abscissas, coolant temperature and neutron fluence, respectively. The microscopic group cross-sections for these two conditions are calculated

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H. SEKIMOTO and Y. UDAGAWA Table 2 Neutron energy group structure

multiplication factor as a good estimate of the converged value. For the other abscissas, coolant temperature and neutron fluence, the similar procedure is repeated by using corresponding proper one-dimensional tables, where the number of investigated abscissa is increased from two (the maximum and minimum values), and the numbers of the other abscissas are fixed to ones (the minimum values). The proper number of values for each abscissa is determined by the same way as for the fuel temperature. The obtained effective neutron multiplication factors for different interpolation tables are shown in Fig. 3. Each line corresponds to the investigation for each abscissa, fuel temperature, coolant temperature and neutron fluence. From this figure and computer performance, the numbers of values of abscissa are chosen 6, 2 and 8 for the fuel temperature, coolant temperature and neutron fluecne, respectively. The actual values of each abscissa are shown in Table 3. The total number of elements of interpolation table is 628¼96. The microscopic group cross-sections (ordinates) are calculated for each of these 96 cases, and the interpolation table is completed.

Energy boundaries (eV)

Group index

Upper

Lower

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1.00E+07 6.07E+06 3.68E+06 2.23E+06 1.35E+06 8.21E+05 3.88E+05 1.83E+05 8.65E+04 4.09E+04 1.93E+04 9.12E+03 4.31E+03 2.03E+03 9.61E+02 4.54E+02 2.14E+02 1.01E+02 2.38E+00 4.14E01 6.40E02

6.07E+06 3.68E+06 2.23E+06 1.35E+06 8.21E+05 3.88E+05 1.83E+05 8.65E+04 4.09E+04 1.93E+04 9.12E+03 4.31E+03 2.03E+03 9.61E+02 4.54E+02 2.14E+02 1.01E+02 2.38E+00 4.14E01 6.40E02 1.00E05

2. Effects of Different Burnup Histories on the Relation between Group Cross-sections and Fluence As already mentioned in Sec. II-3 (1) the microscopic group cross-sections depends on the nuclide densities. Since

Effective Neutron Multiplication Factor

by using SRAC code system, and the interpolation table is completed for the first case. By using this table the effective neutron multiplication factor is calculated. In the next step the number of fuel temperature abscissa is increased to four, where additional values of the temperature are selected between the boundary values already given. By using the newly prepared table next effective neutron multiplication factor is calculated. This procedure is repeated by increasing the number of values of fuel temperature of the table until the effective neutron multiplication factor converges to a certain value. The proper number of values of fuel temperature abscissa is determined as the minimum number to be able to give a value of the effective neutron

Table 3 Values of abscissa of interpolation table Fuel temperature (K)

Coolant temperature (K)

Neutron fluence (/cm2 )

450 550 650 750 850 1,900

450 1,900

2.1E+17 6.4E+23 1.1E+24 1.7E+24 2.0E+24 2.4E+24 2.9E+24 3.3E+24

1.007 Neutron Fluence 1.006

1.005

1.004 Fuel Temperature 1.003

1.002

Coolant Temperature

1.001 0

2

4

6

8

10

12

14

16

Number of Values of Investigated Abscissa Fig. 3 Effective neutron multiplication factor for different number of each abscissa of the interpolation table

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The fast reactor core, whose design parameters are given in Chap. III, is analyzed by the present method (new method), where the microscopic group cross-sections are evaluated at every space mesh by TLLI method. The preparation of interpolation table is given in Chap. IV. The same core is also analyzed by the previous method (old method), where only one set of group constants are employed. The group constants for old method are calculated for the fuel and coolant temperatures of 763 and 725 K, respectively, and for the neutron fluence of zero. These temperatures are not average but arbitrarily chosen by considering previous calculations. The zero fluence is apparently a bad value and gives a softer neutron spectrum than the exact one, but chosen as a reference. These two calculation results are compared to investigate the effects of space dependency of microscopic group cross-sections on the CNDLE burnup characteristics. The obtained distributions of fuel and coolant temperatures, neutron fluence, nuclide densities, and power density are shown in Figs. 5 through 11. In each figure, the curves obtained by both new and old methods are shown. Of course the temperatures and fluence are entirely different between the values for microscopic group cross-sections and the obtained outputs for the old method, but same for the new method. The temperature difference is very small between the new and old methods for both fuel and coolant temperatures. It is attributed to the condition that the coolant core-outlet maximum temperature is a design input and the same for both cases. The neutron fluence obtained by the VOL. 43, NO. 2, FEBRUARY 2006

Peripheral

2

Central

1.5

1 0

0.5

1

1.5

2

2.5

3

3.5

3

3.5

3

3.5

(×10 24n/cm 2)

(a) 238U Nuclide Number Density ( ×10 22n/cm 3)

V. Results Obtained by the Present Method and Discussions on the Effects of Space Dependent Microscopic Group Cross-sections

2.5

Neutron Fluence

2.5

Central 2

Peripheral

1.5 1 0.5 0 0

0.5

1

1.5

Neutron Fluence

2

2.5

(×1024n/cm 2)

(b) 239Pu Nuclide Number Density ( ×10 22n/cm 3)

the fresh fuel is only one kind, the natural uranium, in the present study, the number density of each nuclide in the fuel is expected to be determined almost uniquely for a given neutron fluence. However, if the burnup history is changed, the density also changes even for the same fluence, since the ratio of neutron reaction to natural decay changes. In the present study the nuclide densities for a given neutron fluence are evaluated for the bunup history given on the central core axis, and the microscopic cross-section set is generated for these nuclide densities. We should confirm this procedure proper. The burnup history changes for different core radial position. In the present study the radial distance is divided into 40 meshes. The nuclide densities along neutron fluence are compared for these radial positions. The comparisons for 238 U, 239 Pu and 240 Pu are shown in Fig. 4. The maximum attained neutron fluence depends on the radial position, and generally decreases for larger radial distance. However, the agreements of these curves are good. Only 2 curves out of 40 curves are departed from the others for 238 U and 239 Pu, and 3 curves for 240 Pu. The other nuclides also show similar behaviors. The fission and capture contributions of these three nuclides are 95 and 67%, respectively. It confirms that our procedure, employment of only neutron fluence for the evaluation of burnup effects on microscopic group cross-sections, is proper.

Nuclide Number Density (×1022n/cm 3)


9 8 7

Peripheral

6 5

Central

4 3 2 1 0 0

0.5

1

1.5

2

2.5

Neutron Fluence (×10 24n/cm 2)

(c) 240Pu Fig. 4 Number densities of important nuclides changing along neutron fluence for each radial mesh

new method is higher than the old method over whole region. It enhances the burnup, and causes the larger depression of 238 U and increase of 239 Pu, though the nuclide densities are sensitive to not only neutron fluence but also corre-

196

H. SEKIMOTO and Y. UDAGAWA 1000 Nuclide Number Density (n/cm3)

3.E+22

Fuel Temperature (K)

900

800

700 Old Method New Method

600

2

3

4 Core Axial Position (m)

5

1.E+22

2

6

Fig. 5 Distributions of fuel temperature calculated by the new and old methods

3


Fig. 8 Distributions of nuclide density of new and old methods

238

5

6

U calculated by the

2.5E+21 Nuclide Number Density (n/cm3)

800

Coolant Temperature (K)

New Method

2.E+22

0.E+00

500


600

2.0E+21

1.5E+21

1.0E+21 Old Method New Method

5.0E+20

0.0E+00 2

500 2

3


5

3

6

Fig. 6 Distributions of coolant temperature calculated by the new and old methods


Fig. 9 Distributions of nuclide density of new and old methods

4.E+24

239

5

6

Pu calculated by the

1.E+21 Nuclide Number Density (n/cm3)

Neutron Fluence (n/cm2)

Old Method

3.E+24

2.E+24 Old Method New Method

1.E+24

0.E+00

8.E+20

6.E+20 Old Method New Method

4.E+20

2.E+20

0.E+00

2

3


5

6

2

3

4 Core Axial Position (cm)

5

6

Fig. 7 Distributions of neutron fluence calculated by the new and old methods

Fig. 10 Distributions of nuclide density of 240 Pu calculated by the new and old methods

sponding microscopic cross-sections. The power density obtained by the new method is higher than the old method and shifts to the fresh fuel region. It is consistent with the characteristics mentioned above.

The differences of some characteristic values of CANDLE burnup are shown in Table 4. The reactor characteristics obtained by the old method, lower effective neutron multiplication factor and lower spent fuel burnup, show worse neutron JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

197


factor, burning region moving speed, spent fuel burnup and spatial distributions of nuclide densities, neutron fluence and power density may be considerable from the reactor designer point. However, they are small enough when we study about the characteristics of CANDLE burnup for different designs.

Power Density (W/cm3)

400


200

Acknowledgments 100

The authors wish to thank Mr. S. Miyashita for preparation of the figures.

0 2

3


5

6

References

Fig. 11 Distributions of power density calculated by the new and old methods

Table 4 Main characteristics values of CANDLE burnup obtained by using the new and old methods

Effective neutron multiplication factor Burning region moving speed (cm/yr) Axial power density half width (cm) Spent fuel burnup (%)

Previous method

New method

1.0035 3.17 62.6 42.3

1.0082 3.10 63.1 43.2

economy than by the new method. It is attributed to the zero fluence used for the group cross-section preparation for the old method. It makes neutron spectrum softer and neutron economy worse. The deference between the results obtained by the old and new methods may be considerable, if we design the core of reactor which will be actually built. However, the difference is small enough when we study about the characteristics changes of CANDLE burnup for different designs.

VI. Conclusions Conventional calculations of the CANDLE (Constant Axial shape of Neutron flux, nuclide densities and power shape During Life of Energy producing reactor) burnup has been performed by using the old method, where only one set of the microscopic group cross-sections. In the present study the new method is developed, where the microscopic group cross-sections are evaluated at every space mesh by TLLI (Table Look-up and Linear Interpolation) method, and used to analyze a fast reactor with natural uranium as a fresh fuel. The results are compared with the old method to investigate the effects of space-dependency of the microscopic group cross-sections and feasibility of the old method. The differences of the effective neutron multiplication

VOL. 43, NO. 2, FEBRUARY 2006

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