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The choice of Eqs.(7.3.4-24), (7.3.4-26) or (7.3.4-27) depends on the values of Z 1 , Z 2 on which the P ij (X 1 , X 2 ) is to be interpolated. (3) Conservation and reciprocity of collision probabilities The first-flight collision probability must satisfy the two important relations, that is, the conservation law ∑ Pij =1 j for all i (7.3.4-28) and the reciprocity relation 61) V Σ P = V Σ P for all i, j. (7.3.4-29) i i ij j j ji In the routine ‘PEACO’, at first, the values of P ij are calculated only for i ≤ j by Eqs.(7.3.4-15) and (7.3.4-27). Then, the collision probabilities satisfying Eqs.(7.3.4-28) and (7.3.4-29) are successively obtained by the following equations starting from i=1: * 1 − β1 * 1 − β1 Pij = Pij , Pji = Pji ( j = β β 0 0 i, J ) , (7.3.4-30) J where β = P and β = , (7.3.4-31) j= 1 j−1 * ij 1 P ij i= 1 0 ∑ ∑ with P = V Σ P ( V Σ ) for i j / . (7.3.4-32) ij j j ji i i < The collision probabilities P * ij given by Eq.(7.3.4-30) will be readily known to satisfy Eqs.(7.3.4-28) and (7.3.4-29). Using the interpolation and the asymptotic expansion, combined with the methods mentioned above, we can guarantee the accuracy of 0.1% for the calculation of the collision probability, including the one resonance-absorbing mixture problem. Furthermore it should be emphasized that most of practical problems can be executed in a computing time of the same order as the convenient method based on the IRA. 7.3.5 Resonance Absorption in Doubly Heterogeneous System The HTTR (High Temperature Engineering Test Reactor): an HTGR, constructed at JAEA, uses fuel in the form of small spherical coated particles. A coated particle consists of a fuel kernel of UO 2 with an ∼600 μm diameter and several layers of pyrolytic carbon and SiC of ∼150 μm thickness. Such coated particles, together with graphite diluent, are formed into hollow annular fuel compacts that are packed in a graphite sleeve, and then inserted into a graphite block (See Fig.7.3.5-1). 264
One of the physical problems associated with this type of fuel is a double heterogeneity through the self-shielding of the grain and also of the lattice configuration of fuel rods on the resonance absorption. Here, the “Accretion” method by Leslie & Jonsson 70) to calculate collision probabilities in a cluster-type fuel element is applied to evaluate the resonance absorption in the doubly heterogeneous system in the fuel block of the HTTR. The details will be found in the reference 26) together with some typical numerical examples. We assume that the fuel grains in a fuel compact are uniformly distributed so that each coated particle, together with graphite diluent, forms a two-region spherical cell (microscopic cell) containing only a fuel grain (region f) and the associated amount of graphite diluent (region m). The neutron slowing-down equations can be written by using the collision probabilities under the assumption that the neutron flux is flat in each spatial region and the neutron scattering is isotropic and elastic: V jΣ j ( u) ϕ j ( u) = ∑ Pij ( u) Vi Si ( u) , i (7.3.5-1) S ( u) = i ∫ u 0 Σ ( u' → u) ϕ ( u' ) du' i i , (7.3.5-2) where V i , Σ i , ϕ i , Σ i (u’→u), and S i are the volume, the total cross-section, the neutron flux, the differential scattering cross-section, and the slowing down source of the i-th region, respectively. The quantity P ij is the collision probability that a neutron emitted in the region i has the next collision in the region j, evaluated by assuming the flat flux in each region. BURNABLE POISON ROD (BPR) FUEL PELLET COATED PARTICLE HANDLING HOLE GRAPHITE BINDER DETAILED FUEL PELLET GRAPHITE SLEEVE GRAPHITE BLOCK SPARE HOLE FOR BPR Fig.7.3.5-1 Horizontal cross-section of the standard fuel block of the VHTRC VHTRC: Very High Temperature Reactor Critical Assembly, decommissioned 265
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JAEA-Data/Code 2007-004 SRAC2006 :
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Contents 1. General Descriptions ..
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5.5 BWR Fuel Assembly Calculation (
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目次 1. 概要 ................
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5.4 三次元拡散計算 (CI
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1. General Descriptions 1.1 Functio
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essential programs of the integrate
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Sphere (Pebble, HTGR) 1D-Plate (JRR
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ecause the scattering cross-section
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As shown in Fig.1.4-1, one PDS file
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1.5.1 Fast Fission Energy Range (10
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i n m≠n i n i n i n i 1 i i g( C
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mixture(s) having resonant nuclides
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thermal energy range. In the fixed
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1.10 Calculation Scheme In Fig.1.10
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3 2 1 CALL MICREF [20] CALL REACT C
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homogenized P 1 spectrum obtained a
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Consequently next step ’CALL MACR
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1.11 Output Information Major calcu
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(4) A floating number may be entere
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2.2 General Control and Energy Stru
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cell. = 2 The PEACO routine (hyperf
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should be avoided for the core wher
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Note: Usually IC15=1 is used in FBR
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Behrens’ term of the Benoist mode
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Only the first character (capital l
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NET ∑ i= 1 NEGT( i ) =(Total numb
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Block-1 Control integers /18/ 1 IGT
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asymmetric cell is surrounded by en
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= 0 Isotropic (white) reflection =
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degree for the numerical angular in
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3 EPSG Extrapolation criterion 4 R
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Block-10’ Required if IGT=11, or
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= 2 Neutron from moderator = 3 Neut
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Unit Cell Periodic B.C. 4 2 3 1 RX(
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NX=4 NTPIN=6 NAPIN=1 NPIN(1)=6 6 ID
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RDP(1)=0.0 RDP(2) RDP(3) 14 13 12 R
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2.5 ANISN ; One-dimensional S N Tra
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7 IBR Right boundary condition, sam
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22 IPM Angular dependent incident s
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EV = best guess for α or 0.0 when
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Block-04* Positions of interval bou
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2.6 TWOTRAN ; Two-dimensional S N T
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Note: The original TWOTRAN uses two
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= 0 (internally set) 19 IHM Total n
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= 3 R-θ 31 IEDOPT Edit options. =
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= 1 Yes Block-4 Control floating po
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Repeat Block-20 through Block-21, N
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2.7 TUD ; One-dimensional Diffusion
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= 1 Monitor print at each inner ite
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equation is applied to meshes, the
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Pin Plate D2 D2 D1 D1 In cylindrica
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XYZ(1,2) Y-abscissa of the top side
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READ(9) (WORK(J+I),I=1,NDATA) END D
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absorption cross-section, each of w
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ITMX19 The upper limit of CPU time
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corner and there are the same numbe
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NUAC19 Override use of Chebychev po
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NUAC17 > 0 The constant for all gro
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ottom starting with a new card. For
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Card-006-3 Specification of meshes
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4 SIG3 Absorption cross-section (
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2 NFX2 Optional print control = 0 N
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Region 1 Region 2 Left Right X/R/R
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X T 22*11 Meshes 1 Mesh = 1 Region
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2.9 Material Specification ’Mater
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fuel pin: MU08F0U2 and those in MOX
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hydrogen, add character ‘0’ to
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Usually, enter IRES=2 for the heavy
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2.10 Reaction Rate Calculation The
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Block-1 Control integers for reacti
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3 U238 The position of the second n
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2.11 Cell Burn-up Calculation The i
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= 2 Information for debugging = 3 D
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chains under consideration is enlar
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MACROWRK file. 2 INTSTP Step number
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Block-8-1 Required if IBC6=1 /1/ NM
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e avoided. Because U08W and U080 ar
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2.12 PEACO ; The hyperfine Group Re
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3. I/O FILES We shall describe the
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(E(g),g=1,NGF+1) structure. Boundar
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LTH(1) = (LD(1)+1)*LA(1), ordered a
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Background cross-sections Member Yz
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INT(2) = 0 K-member only without sc
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Member name Contents **************
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(WEIGHT(g),g=1,NGF) Lethargy widths
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3.1.5 Fine Group Macroscopic Cross-
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Member mmmmebfM or caseebxM /10*ng+
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Member caseBNUP Material-wise burn-
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Initial inventory in the restart ca
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(YDIOX(j),j=1,NOWSTP) Fission yield
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(n/cm 2 /sec*cm 3 ), NRR is number
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3.2.1 Common PS Files The following
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3.2.5 PS files for TUD The PS files
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at the first column. Block-1-1 Numb
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= ‘DAYS’ days = ‘YEARS’ yea
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GD156 XGD60001 641560 154.660 0 0 0
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: ZZ050 1.50080E+00 *FP-Yield-Data
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Kr83 Zr95 fission β + decay, EC (n
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Ge73 Ge74 As75 Ge76 fission β + de
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4. Job Control Statements In this c
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~SRAC/tool/lmmake/lmmk/lmmk.sh, whi
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5. Sample Input Several typical exa
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613.0 650.0 760.0 769.0 & cooling 7
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5 5 5 5 5 5 & 5 5 5 5 5 5 & 1 2 3 4
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5.3 S N Transport Calculation (PIJ,
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XFEN0001 2 0 5.78720E-02 XNIN0001 2
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Water reflector Extrapolated B.C. B
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1 1 1 1 1 1 1 2 3 008 -2 1 1 999 /
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Note: This sample is time consuming
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XCRN0008 0 0 1.55546E-02 XNIN0008 0
Page 225 and 226: ****** Input for material specifica
Page 227 and 228: T-Region R-Region X-Region Reflecti
Page 229 and 230: 6. Utility for PDS File Management
Page 231 and 232: To read in the microscopic cross-se
Page 233 and 234: 7. Mathematical Formulations 7.1 Fo
Page 235 and 236: scattering kernel at point r’ fro
Page 237 and 238: The integration by R between R j- a
Page 239 and 240: G j Pij ( lattice) = Pij ( isolated
Page 241 and 242: We, however, should take care of th
Page 243 and 244: 2 = ρ 2 + x r sin β = ρ R = x
Page 245 and 246: 1 i ∑ − 1 k = j+ 1 λ ij = λ k
Page 247 and 248: where λ λ 1 is 2 is = = N ∑ k k
Page 249 and 250: 2π ri −1 Pij = − Σ ∫ ρdρ
Page 251 and 252: In Fig.7.1-5, the line PQ’ define
Page 253 and 254: pin rod are divided into four regio
Page 255 and 256: the arrays T and II, respectively.
Page 257 and 258: Note that among four terms appearin
Page 259 and 260: If a group-dependent form of Fick
Page 261 and 262: 7.3 Optional Processes for Resonanc
Page 263 and 264: 7.3.2 Table-look-up Method of f-tab
Page 265 and 266: has been introduced for the correct
Page 267 and 268: ϕ i ( 0 u ) = ∑ Pij ( u) W j ( u
Page 269 and 270: Geometry b i m f b f −( γ + γ )
Page 271 and 272: will be seen in the reference 65) .
Page 273 and 274: The lattice cell under study may co
Page 275: (2) Two resonance-absorbing composi
Page 279 and 280: and V = V + V , F f m where Σ m ,
Page 281 and 282: We shall discuss the following two
Page 283 and 284: where X = 2R p ( Σ C = 2R πρ R p
Page 285 and 286: • the reduced collision probabili
Page 287 and 288: The root mean square (RMS) residual
Page 289 and 290: group flux and for the fast group f
Page 291 and 292: Table 7.5-1 (1/2) Nomenclature Symb
Page 293 and 294: 7.5.1 Smearing Smearing or spatial
Page 295 and 296: 7.5.2 Spectrum for Collapsing The f
Page 297 and 298: Here, an equivalent relation holds
Page 299 and 300: 1 F = K ∑∫ g * χ g ( r) ϕ g (
Page 301 and 302: M dN i ( t) M = ∑ f j→iλ j N i
Page 303 and 304: 8. Tables on Cross-Section Library
Page 305 and 306: Table 8.1-2 (2/2) Element index (zz
Page 307 and 308: JEFF-3.0 are based on other nuclear
Page 309 and 310: Table 8.2-1 (1/8) List of SRAC publ
Page 317 and 318: 8.3 Energy Group Structure The ener
Page 319 and 320: (4) Upper Boundary of PEACO Routine
Page 321 and 322: References 1) K. Tsuchihashi, H. Ta
Page 323 and 324: Experiment,” J. Nucl. Sci. Techno
Page 325: 75) K. Tasaka : “DCHAIN: Code for
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