Analytic Solution Technique for Solving One-Group Diffusion Equations for Core Simulations

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 20[8] pp. 627~644 (August 1983). 627 Analytic Solution Technique for Solving One-Group Diffusion Equations for Core Simulations Masafumi ITAGAKI Japan Nuclear Ship Research and Development Agency* Received November 15, 1982 Revised June 8, 1983 For the purpose of efficient core simulations, a new iterative method for solving onegroup diffusion equations has been developed on the basis of an analytic solution technique. A type of shooting method is used in determining the critical eigenvalue and the corresponding flux distribution in a one-dimensional system. Extension to two-dimensional problems is performed by a nodewise buckling iteration. Results obtained in some test calculations for LWRs indicate that the method, as a consequence of the analytic flux representation, can achieve a significant reduction in computing time and memory in comparison with the conventional finite difference method. KEYWORDS: core simulation, on-site minicomputer, power distribution, low-cost computing model, FLARE model, one-group diffusion equation, analytic solution, Buckler's method, shooting method, nodewise buckling iteration, one-group collapsed nuclear constants, accuracy, computing time I. INTRODUCTION Control rod pattern exchanges are frequent in a maritime reactor or a light water reactor designed for load-following operation. The power distribution in the reactor core must therefore be analyzed frequently to predict some core performances such as maximum linear heat rate and minimum DNBR (Departure from Nucleate Boiling Ratio). For this reason, it is desirable that the power distribution for such a reactor be calculated by an on-site computer. However, a conventional fine-mesh diffusion code such as CITATION(1) cannot be used for this purpose, because the number of variables required is too large to be stored in the usual on-site minicomputer. On-site computation must therefore conform to the low computing memory capacity and short execution time requirement. To satisfy these requirements, a simplified modelling is necessary for reactor physics computations. Although the onegroup approximation or the omission of reflector calculation is a useful simplification, the best solution to the problem is to develop a low-cost computational method which does not require a large number of variables. The FLARE type nodal method(2) is known as a low-cost one-group computing model. With the FLARE model, however, which is not based on the diffusion theory, it may be difficult to assure consistency with the conventional diffusion methods. On the other hand, most of the coarse mesh methods based on the diffusion theory are still expensive with onsite computation. Among many efforts to reduce the computing cost, an analytic solution method, where the flux distribution is represented by analytic solutions of the diffusion equation, is thought to be one of the goals. * Toranomon, Minato-ku, Tokyo

2 628 J. Nucl. Sci. Technol., Buckler proposed an analytic solution method which describes one-dimensional (1-D) fast neutron flux distribution in multi-layered reflector regions in a diffusion codem. This method is based on the simple algebraic relationship between an unknown coefficient in the flux solution and the logarithmic derivative condition at a region boundary. The simplicity of this enables us to minimize the size of the computer program. This method, however, has not been extended to eigenvalue problems in a multiplying system. The present paper describes a new analytic solution method for solving one-group diffusion equations. The calculation procedure of the method contains the following three key ideas : (1) To connect the 1-D analytic solutions at each region boundary, using Buckler's method (2) To find the critical eigenvalue and the corresponding flux distribution, using a type of shooting method (3) To extend to 2-D problems, using a nodewise buckling iteration technique. These ideas are described in Chaps. II~IV. As shown in Chap. V, the computational efficiency of the method has been tested on some light water reactor problems by comparing the results with those obtained by the conventional finite difference calculations. The results obtained in the 2-D test calculation indicate that the new method can reduce the computing time by a factor of about 100, compared to the finite difference method. Furthermore, the above 2-D test calculation could be performed with a minicomputer of 32K-byte memory capacity.. ONE-GROUP ANALYTIC FLUX IIREPRESENTATION IN MULTI-LAYERED 1-D SLAB SYSTEM The one-group diffusion equation for a 1-D slab system can be written in the form with the usual symbols, DBI being the perpendicular neutron leakage cross section. Over a homogeneous region i in a multi-layered slab, the use of the uniform buckling gives the simplified form of Eq. ( 1 ) ( 1 ) ( 2 ) ( 3 ) The general solution of Eq. ( 3 ) can be expressed by ( 4 ) Table 1 Functions in 1-D solutions for a slab system ( 5 ) where the functions coi(x) and Oi(x) can be defined as shown in Table 1. In general, the unknown coefficients ai and wi in Eqs. ( 4 ) and ( 5 ) can be determined by the conditions of continuity of current and flux. In this paper, to determine the coefficients efficiently, Buckler's method") was adopted, which has been developed to express the fast neutron flux distribution in a multi-layered reflector system. 2

3 Vol. 20, No. 8 (Aug. 1983) 629 Consider one of the regions, say the i-th, in a multi-layered slab system as shown in Fig. 1. The logarithmic derivative constant at the left boundary (x=xi) in this region =Ji(xi)/pi(xi) can be expressed by ri ( 6 ) from Eqs. ( 4 ) and ( 5 ). Note that the coefficient a, is canceled in this expression. The quantity defined by coefficient is given directly from Eq. ( 6 ). At the right boundary in the region i, the logarithmic derivative constant is Fig. 1 Schematic representation of 1-D calculation method If the ri is given at the left boundary (x=xi), the coefficient w, is obtained from Eq. (7), and in terms of wi the 7,, at the right boundary (x=xi+1) can be obtained from Eq. ( 8 ). Once the boundary condition at the outermost left boundary (x=x1) is given, the above sweep can be repeated through the system slab by slab until the set of wn and rn+1 is obtained at the outermost right-hand region N. The unknown coefficient a, in Eqs. ( 4 ) and ( 5 ) can be determined by the continuity of flux at each region boundary. At the left boundary in region i, there is a relationship (7) ( 8 ) Then, after the pi-1(xi) is known, a, can be determined as follows : ( 9 ) The flux pi(xi+1) can be determined from the known ai, so that the ai+1 in the next region can be obtained in the same manner. To perform the above sweep of (1,, we must choose a reference value for one of the values ai, e.g. let a1 equal unity. In the cases where p(x1)=0 or J(x1)=0 is given as the outermost left boundary condition, the coefficient w1 may be determined as follows : (10) w1= coi(x1)/t1(x1) for 0(x,) =0, (11) w1= coa(x1)/t'0(x1) for J(x1) =0. (12) The above is an outline of Buckler's concept. The following new technique is added to this concept in our method. To avoid the undesirable exponential growth of the hyperbolic basis functions in some problems, we introduce the following modifications : (13) where xi=(xi+xi+1)/2, i.e. the center point of the region i. The new expressions Bi(x-Xi) and li(x-xi) will minimize the absolute values of yg,(x) and 0,(x) in Table 1. There is another advantage in the modifications in (13). There are symmetric relationships 3

4 630 J. Nucl. Sci. Technol., (14) in region i when adopting these modifications, so that the number of unknown functions, the values of which take a lot of computing time to be obtained, can be reduced by half. III. FINDING CRITICAL EIGENVALUE BY SHOOTING METHOD In Chap. the method of connecting analytic flux solutions was developed when the outermost left boundary condition is given. However, the critical eigenvalue Ken, which is unknown at the beginning of computation, is included in the diffusion parameters fg and AL as shown in Eq. ( 2 ). Consequently we must find the Keff that fulfills both the left and right outer boundary conditions. In other words, we must solve an eigenvalue problem. To find the Keff, a type of shooting method"' is used. Although many types of shooting methods have been proposed in the field of mathematics, the main idea in a shooting method is to transform a given boundary value problem into an initial value problem. An outline of the shooting scheme applied to the present method can be given as follows : When the first guess at Keff is made, the sweep described in Chap. II is started at the outermost left boundary. The outermost right boundary value is computed at the end of the sweep and compared with the original set value of the boundary condition. If it does not agree, this procedure is iterated using a new guess value for Ken. The shooting trajectory that fulfills both the left and right outermost boundary conditions is the true flux distribution, and the guess value of Ken at this time is the critical eigenvalue in the system. As an additional condition, the flux must be positive everywhere. There are two ways of updating the guess value of Ken according to whether a negative flux is found on the way to the right outer boundary (see Fig. 4). (1) Where negative flux is found Let K(i-I)be the guess value in this case. Obviously the true eigenvalue is greater than the K(i-1), so that the new guess value K(i)eff is estimated by where K(*)eff is the smallest value among the old Keff's that gave no negative fluxes. procedure is repeated until no negative flux is found anymore. (2) Where negative flux is not found and the right outer boundary value is obtained In this case, Newton's method can be used as follows. When the guess values KA-" and K(i-2)eff were given for (i-1)-th and (i-2)-th iteration, respectively, the new eigenvalue KJ is estimated by using linear extrapolation : (15) This Here fc" is the check value at the outermost right boundary, that is, either the flux value, the current value, or the deviation of the logarithmic derivative value from the set value, according to the type of right outer boundary condition. The shooting iteration is repeated until the fcl' becomes less than the convergence criterion, e.g Figure 4 gives a concrete example showing that Keff and the corresponding flux distribution is being gradually converged. (16) 4

5 Vol. 20, No. 8 (Aug. 1983) 631 IV. EXTENSION TO 2-D PROBLEMS BY NODEWISE BUCKLING ITERATION The 1-D method described in Chaps. II and III can be easily extended to 2-D problems using a nodewise buckling iteration technique(5)~(8), which has the advantage of minimizing the computing memory required. Assuming Cartesian geometry, we can simplify the one-group 2-D diffusion equation as follows : (17) where.b2m=(vsf/keff-s'a)/d. The core is divided into homogeneous rectangular nodes as shown in Fig. 2. Integrating Eq. (17) over a node results in Fig. 2 Partition of two-dimensional (18) plane for nodewise buckling iteration where Dx and Dy are the x- and y-directional length of a node. Using the following notations (19) Eq. (18) can be written as or (20) Consequently, the relationship (21) gives the following two 1-D equations : (22) As described in Chap. II, px and py are represented as a linear combination of either trigonometric or hyperbolic functions, so that the bucklings B1 and are nodewise constants. To assure consistency between Eq. (17) and the set of Eq. (22), the following two conditions must be simultaneously fulfilled : (23) (24) Only the quantities.b2x and B2v connect the set of Eq. (22), so that Eq. (22) can be solved by updating and alternately (a nodewise buckling iteration procedure). 5

6 632 J. Nucl. Sci. Technol., The fact that B2x and B2y are given algebraically and are nodewise constants makes the procedure simple (other kinds of nodewise buckling iterations based on the finite difference scheme need numerical integrals of flux and leakage to estimate the nodewise averaged bucklings")). The buckling iterations can be accelerated using a linear extrapolation technique. Although the procedure described above is similar to the "leakage iterative method" proposed by Naito et al.'', it differs from their method in that Eq. (22) have no source term which appears in a finite difference source iteration (S") in the k-th source iteration of the form Ap(k)=S(k)). Consequently, special consideration must be paid to the normalization of 2-D flux distribution in our method. The normalization method adopted here is described as follows. The core is divided into IMAX xjmax rectangular nodes as shown in Fig. 2. In this case, JMAX arrays of x-directional 1-D flux distributions and IMAX arrays of y-directional 1-D flux distributions are computed. Two types of averaged neutron fluxes in node (i, j) are defined as (25) where pjx(x) and piy(y) are the x- and y-directional 1-D flux distributions, respectively. Using the coefficients (26) we can calculate two types of nodewise averaged neutron fluxes which are normalized in a two-dimensional plane : (27) where C is the normalization factor deterimined so that the core averaged power is unity. If the buckling iteration is fully-converged, the following relationship is found : V. TEST CALCULATION RESULTS AND DISCUSSION Two test programs SICO 1D for 1-D calculations and SICO 2D for 2-D ones have been developed based on the new method described above. To study the computational efficiency and the reliability of the method, the results obtained by SICO 1D and SICO 2D calculations were compared with those by the conventional finite difference code CITATION"' for some light water reactor problems. The test calculations were made using the large-scale computer HITAC H-8680 and the minicomputer PANAFACOM U-100. This minicomputer, which has only a 32K-byte memory, was used only for test runs of SICO 1D and SICO 2D while the CITATION run was impossible using this computer D Test Calculations The 1-D example chosen for calculation is the reactor core of the Nuclear Ship "Mutsu". We deal with axial power distribution of the fuel assembly which is located in the central part of the core. This assembly is surrounded by two cruciform type control rods identified as G1 and G2. The axial core configuration of the problem is shown in Fig. 3. The active height of the core (104 cm) is divided into 19 regions according to the heterogeneity (presence of control rods, grid spacers etc. and the axial distributions of coolant and fuel temperatures etc.). In this problem both G1 and G2 control rods are assumed to be withdrawn by 512 mm from the core bottom. The three-group nuclear constants have been previously given for the 19 6

7 Vol. 20, No 8 (Aug. 1983) 633 regions and for the upper and lower reflector. Figure 3 also shows the CITATION mesh arrangement. The one-group nuclear constants required in the SICO 1D calculation were collapsed from the above three-group constants by using the asymptotic spectra defined only by the material compositions (see APPENDIX-3). These one-group collapsed nuclear constants are listed in Table 2. Table 2 One-group collapsed nuclear constants used in SICO 1D calculation Fig. 3 Vertical core configuration for 1-D test problem The logarithmic derivative conditions at the upper and lower reflector boundaries required in the SICO 1D calculation were adjusted to rl= and rr=0.185 respectively, in order to make the power distribution and critical eigenvalue produced by SICO 1D consistent with those of the three-group CITATION calculation. This adjustment is similar to the concept of albedo-parameter adjustment used in FLARE-type calculations (see APPENDIX-1) (2). Figure 4 shows the convergence characteristics of the shooting procedure in this case. The shooting iterations were repeated until the deviation of the outermost right boundary value for the logarithmic derivative from its set value rr=0.185 became less than 10-s. The number of iterations required for the convergence was 12 in this problem, and at this time the eigenvalue convergence was In contrast, in the case of the three-group CITATION calculation, the 32 source iterations were required for the convergence criterion Fig. 4 Convergence behavior of shooting iteration 7

8 634 J. Nucl. Sci, Technol., Figure 5 shows a comparison between the power distribution obtained from SICO 1D and that from the three-group CITATION calculation. The discontinuity of power distribution obtained by the heterogeneity cross sections. by SICO 1D is caused of fission The maximum relative power error of 20.2% is found near the lower reflector boundary, where the flux information is not important (the maximum linear heat rate and minimum DNBR do not occur in this region due to the low power generation). Except for the region up to 3.4 cm from the core bottom, however, the error in the power distribution is within a 4.6% deviation. It is thought that the power deviation near the lower reflector boundary is largely dependent upon the asymptotic spectrum approximation in the one-group collapsing. This is confirmed by comparison between the power distributions obtained from the SICO 1D calculation and from the one-group CITATION calculation using the same core-reflector boundary conditions (see Fig. 5). In this case, the deviations between the power distributions obtained by the two methods remain less than 0.8% over all the regions of the core. Results of these 1-D test calculations are summarized in Table 3. Comparison between Cases B and C shows that the computing time required for the SICO 1D calculation is less than 1/10 of that required for the three-group CITATION calculation. The short computing time required for a SICO 1D calculation is due to the analytic flux representation and the high convergence in the shooting method. Fig 5 Comparison of axial power distributions obtained by SICO 1D and by three-group and one-group CITATION calculations for 1-D test problem Table 3 Summary of 1-D test calculation results 2. 2-D Test Calculations The 2-D IAEA benchmark problem") (8) (see Fig. 6) was taken up to verify the SICO 2D program. The problem is a simplified model of a pressurized water reactor. The two-zone core consists of 177 square fuel assemblies with a dimension of 20 cm on a side and is reflected by 20 cm of water. Each of the nine fully-inserted control rods are represented as 8

9 Vol. 20, No. 8 (Aug. 1983) 635 smeared absorbers of a single fuel assembly. The reference two-group calculation was performed using the CITATION code. In each fuel assembly, 12 x 12 meshes were taken (Dx=Dy=1.67 cm). The one-group nuclear constants required for the SICO 2D calculations were collapsed from the two-group data shown in Fig. 6 by using the asymptotic spectra defined only by the material compositions (see APPENDIX-3). Let rp be the logarithmic derivative on a plane reflector boundary and rl the one on an "L-shaped" reflector boundary which shares faces with two fuel assemblies, as shown in Fig. 9. The simple relationship rl=rp/r2 (see APPENDIX-2) was assumed and rp=0.089 was set to make the power distribution and critical eigenvalue produced by SICO 2D consistent with those of the two-group CITA- TION calculation. The reliability of calculation results due to the adjustment is discussed in APPENDIX-1. Figure 7 shows that the maximum and minimum Keff, which appeared in the 1-D Fig. 6 2-D IAEA benchmark problem specifications Fig. 7 Behavior of maximum and minimum eigenvalues according to number of buckling iterations 9

10 636 J. Nucl. Sci. Technol., calculations described in Eq. (22), gradually approach each other as the buckling iteration proceeds. Owing to this tendency, the average number of shooting iterations required for the 1-D calculations decreases gradually as the buckling iteration proceeds, as shown in Fig. 8. After the 63 buckling iteration, each of the shooting calculations is converged in three or fewer iterations. The 110 buckling iterations were required to fulfill the convergence criterion Fig. 8 Behavior of average number of shooting iterations according to number of buckling iterations At this time, it was shown that max Keff-min Keff< 2.3x10-5, max (P(110)-P(109)/p(109) <9.5x10-5, where Pc" is the assembly power at n-th buckling iteration. On the contrary, the reference two-group CITATION calculation required 96 counts of source iterations for the criteria Keff/Keff <10-5 and DP/P<10-4. D Figure 9 shows a comparison between power distributions obtained from the two-group CITATION calculation and from the SICO 2D calculation. The SICO 2D calculation is shown to have the maximum and mean relative errors of 12.5 and 3.3%, respectively, in the assembly. The maximum error occurs at the assembly adjacent to the water reflector. The accuracy described above is quite similar to what we often observe in the FLARE-type one-group calculations using cell-averaged Ifc.o's and migration areas. It is thought that the above deviations in assembly powers are largely dependent upon the one-group collapsed nuclear constants used. This is confirmed by a comparison between power distributions obtained from the SICO 2D calculation and from the one-group CITATION calculation using the same reflector boundary condition rp=il=0.080, as shown in Fig. 10. It is thought that the small deviation of each assembly power in Fig. 10 is caused only by the assumption of nodewise separable buckling described in Eq. (23). The results of these 2-D calculations are summarized in Table 4. Comparison between Cases C and E shows that the computing time required for the SICO 2D calculation is about 1/100 of that required for the two-group CITATION calculation. The contribution to the above magnification consists of (1) using analytic flux solutions, (2) adopting the one-group approximation and (3) omitting reflector calculations. It is noted that the above SICO 2D calculations could also be performed with the minicomputer U-100 with 32K-byte memory, which is smaller in scale than the usual on-site computer. 10

11 Vol. 20, No. 8 (Aug. 1983) 637 Fig. 9 Comparison of assembly power distributions obtained by SICO 2D and by two-group CITATION calculations for 2-D IAEA benchmark problem Fig. 10 Comparison of assembly power distributions obtained by SICO 2D and by one-group CITATION calculations for 2-D IAEA benchmark problem 11

12 638 Nucl. Sci. Technol., Table 4 Summary of test calculation results for 2-D IAEA benchmark problem 3. More Accurate 1-D and 2-D Test Calculations The above SICO 1D and SICO 2D calculations are based on the use of one-group constants collapsed under the assumption of asymptotic spectra (see APPENDIX-3). The accuracy of power distributions obtained under this assumption is similar to that observed in FLAREtype calculationsm, so the level of accuracy is thought to be adequate for usual incore fuel management. If desired, however, more accurate results may be obtained by introducing corrections to account for the spectral effects between heterogeneous material zones. To confirm this, additional test calculations were carried out as follows. Firstly, to improve the accuracy of the 1-D test calculation results shown in Fig. 5, only the one-group constants in the region up to 3.4 cm from the core bottom were replaced by the ones that had been collapsed using the actual spectrum obtained by the reference three-group CITATION calculation. The new constants are D=0.939 cm, Sa = cm-1 and nsf = cm'. The same constants listed in Table 2 were used again for the remaining 18 rerzions and the new log-derivative boundary conditions were set to 1= and rr_= r The new calculation reduces the maximum power error tr 3.9%, as shown in Fig. 11. Fig. 11 Modified results of axial power distribution calcu - lated by SICO 1D with comparison of three-group CITATION calculation for 1-D test problem Secondly, the accuracy of the SICO 2D results shown in Fig. 9 for the IAEA 2-D problem was improved by taking four nodes per fuel assembly (i.e. Dx =Dy=10 cm). In order to account for the spectral effects between heterogeneous zones, the one-group constants were replaced by the ones which were collapsed using the actual spectra obtained from the re- 12

13 Vol. 20, No. 8 (Aug. 1983) 639 ference two-group CITATION calculation, but only for the nodes adjacent to the peripheral reflector and for the nodes where control rods are inserted. In the remaining nodes we again used the one-group constants collapsed by using the asymptotic spectra. The logderivative conditions were newly adjusted to rp=0.145 and rl=rp/r2. Figure 12 compares the results of the new SICO 2D calculation with that of the two-group CITATION calculation. It is observed that the new calculation reduces the maximum and mean relative power errors to 5.6 and 2.1%, respectively. Fig. 12 Modified results of assembly power distribution calculated by SICO 2D with comparison of two-group CITATION calculation for 2-D IAEA benchmark problem It is concluded that the test programs, SICO ID and SICO 2D, can give excellent results, introducing the spectral corrections. The new SICO 2D computation required CPU seconds using the H-8680 computer. This is about 6 times slower than the one-node-perassembly computation (Case C listed in Table 4) due to the new noding (for nodes per assembly), however, it is still more than 16 times faster than the reference CITATION calculation. It is possible to choose between the two (the fast computation with one-node-perassembly or the accurate computation with four-node-per-assembly) according to the purpose of the computing. VI. CONCLUSIONS A new method for solving the one-group diffusion equation has been presented on the basis of the analytic flux representation and of a shooting method. The results obtained from a series of test calculations demonstrate that the new method fulfills the two requirements for on-site computations-little use of computer memory and high-speed computing. The advantages of the method are summarized as follows : 13

14 640 J. Nucl. Sci. Technol., (1) The analytic flux representation does not require a large number of variables, with the result that both computing time and memory are substantially reduced compared with the conventional fine mesh difference methods. (2) The analytic flux representation helps us to understand direct images in reactor physics phenomena. (3) The method, which is based on the diffusion theory, is a convenient way of assuring consistency with two- or three-group diffusion calculations. (4) For the purpose of on-site DNB calculations, the integral contained in a DNB-correlation, known as the F-factor, can be performed analytically due to the analytic representation of axial power distribution. Although the discussion in the present paper is limited to 1-D and 2-D problems, 3-D calculations can be easily performed by a simple extension of the 2-D buckling iteration procedure. ACKNOWLEDGMENT The author wishes to express his thanks to Dr. Y. Futamura and Mr. Y. Harayama of the Engineering Division, Japan Nuclear Ship Research and Development Agency, for their continuous encouragement. REFERENCES (1) FOWLER, T.B., et al.: ORNL-TM-2496, (1971). (2) DELP, D.L., et al.: GEAP-4598, (1964). (3) BUCKLER, A.N. : AEEW-R 1242, (1979). (4) AKTAS, Z., et al.: Int. J. Num. Meth. Eng., 11, 771 (1977). (5) NAITO, Y., et al,: Nucl. Sci. Eng., 58, 182 (1975). (6) BONALUMI, R., et al.: Trans. Am. Nucl. Soc., 20, 362 (1975). (7) MAKAI, M., et al.: EIR-401, (1980). (s) LAWRENCE, R.D., et al.: Proc. OECD-Meeting Calculation of 3-Dimensional Rating Distributions in Operating Reactors, Paris, p. 383 (19 [APPENDIX] 79). 1. Adjustment of Core-reflector Boundary Conditions This one-group method requires the adjustment of the logarithmic derivative conditions at the core-reflector boundaries of light water reactors. This adjustment is similar to the concept of the 'albedo-parameter adjustment' encountered in the FLARE type one-group calculations(2). Several calculations must be made to find the logarithmic derivative conditions that make the power distribution and eigenvalue produced by SICO 1D or SICO 2D consistent with those produced by few-group diffusion calculations or with actual reactor operating data. Once adjusted, however, these boundary conditions ensure that results of SICO 1D or SICO 2D calculations are in good agreement with those of more precise few-group calculations for different reactor operating conditions. This is confirmed by the following test calculations. Firstly, one-demensional calculations were made for the "Mutsu" core under various control rod patterns. The series of SICO 1D calculations based on the constant boundary conditions ri= and rr=0.185, which are the same values that are used in the 1-D calculation described in Chap. V, are compared with the three-group finer mesh calculations utilizing the CITATION code. Table A1 summarizes the results. The relative error of 4

15 Vol. 20, No. 8 (Aug. 1983) 641 eigenvalue is within 0.2% in each case. Although the maximum power error of about 20% is found near the core bottom adjacent to the lower reflector in each case, except for the region up to 3.4 cm from the core bottom, the relative errors in the power distributions are within a 6% deviation for all the cases. The level of agreement in both eigenvalue and power distribution in each case is quite similar to that of Case A which is the same as that described in Chap. V. The range of control rod patterns considered in these cases is broader than will be encountered in the operation of the "Mutsu" reactor. Table Al One-dimensional comparisons between SICO 1D and CITATION calculations for various control rod patterns of Mutsu core Secondly, the following three perturbations to the IAEA 2-D benchmark problem are considered as two-dimensional test cases. Case (a) : The control rod of region 1 shown in Fig. 6 is withdrawn (the nuclear constants of region 1 are changed from No. 3 to No. 2). Case (b) : The control rod of region 18 is inserted (the nuclear constants of region 18 are changed from No. 2 to No. 3). Case (c) : The control rod of region 31 is withdrawn (the nuclear constants of region 31 are changed from No. 3 to No. 1). The results obtained by the SICO 2D calculations based on the constant boundary conditions rp=0.089 and rl=rp/r2, which are the same as those used in the 2-D calculation described in Chap. V, are compared with those of the two-group CITATION calculations for the three types of 'perturbed' IAEA 2-D problems. The results are summarized in Fig. Al (a)~(c). It is found that the level of agreement in both eigenvalue and power distribution in each case is quite similar to that observed in Fig. 9 (the original IAEA-2D problem). Based on these 1-D and 2-D simple tests, it appears that it may be possible to obtain results that are consistent with the few-group finer calculations for the broad range of reactor conditions using the adjusted constant core-reflector boundary values. 2. Derivation of Relationship rl=lpr2 in Chap. V The source-free 2-D diffusion equation for reflector regions is described in the form (Al) We adopt the following assumptions : (1) The perpendicular neutron leakage can be ignored at the plane reflector boundaries, i.e. p2pp/py2=-0, 15

16 642 J. Nucl. Sci. Technol., Fig. Al (a) (c) Comparison of assembly power distributions obtained by SICO 2D and by two-group CITATION calculations for ' perturbed ' IAEA 2 -D benchmark problems 16

17 Vol. 20, No. 8 (Aug. 1983) 643 Fig. Al (c) Control rod of region 31 is withdrawn. (2) The Laplacian in Eq. (Al) is evenly split in the x- and y-directions at the L-shaped reflector boundaries, i. e. p2pl/px2=p2pl/py2. The use of these assumptions give the following two types of equations as the simplified forms of Eq. (A1) : (A2) The solutions of (A2) under the conditions pp(x->oo)=0 and pl(x->oo)=0 can be obtained as follows: The logarithmic derivative values for the two-type boundaries are therefore given as Consequently, the approximated relationship rl=rp/r2 is obtained. A similar relationship of reflector albedos is adopted by Bonalumi et al. in their one-group diffusion model"). 3. Asymptotic Spectra Used in One-group Collapsing in Chap. V The asymptotic spectra defined only by the material compositions were used for producing one-group nuclear constants required in the test calculations in Chap. V. These spectra can be expressed with the usual symbols as follows : for the two-group model, (A3) 17

18 644 J. Nucl. Sci. Technol., for the three-group model. (A4) The quantity B2 is the energy-independent material buckling derived from the 'bare reactor concept' and the quantity x=-b2 is the least real root of the characteristic equation corresponding to the two- or three-group diffusion equation. The spectra defined by Eqs. (A3) and (A4) are thought to be equivalent to those obtained directly from the cell calculation in a homogeneous region. 18