Variational Nodal PerturbationCalculations Using Simplified Spherical HarmonicsO

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1 Paper to be submitted to the ANS/ENS 1996 International Meeting, November 1015, 1996, Washington, D.C. Variational Nodal PerturbationCalculations Using Simplified Spherical HarmonicsO ST/ K. LawinKovitz and G. Palmiotti Reactor Analysis Division Argonne National Laboratory Argonne, IL and E. E. Lewis NorthwesternUniversity Department of MechanicalEnginneering Evanston,IL n e submitted manuscript has been created by the University of Chicago as Operator of Argome National Laboratory ( Argome ) under Contract No. W31109ENG38 with the U.S. Department of Energy. The US.Government retains for itself, and others acting on its behalf, a paidup, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform I *Work supported by the US. Department of Energy, Nuclear Energy Programs under Contract W31 109ENG38 n

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4 Variational Nodal Perturbation Calculations Using Simplified Spherical Harmonics K. LaurinKovitz and G. Palmiotti Argonne National Laboratory Reactor Analysis Division Argonne, IL E.E. Lewis Northwest ern University Department of Mechanical Engineering Evanston, IL Introduction The simplified spherical harmonics (SPN) method has been used as an approximation to the transport equation in a number of situations.1~2~3 Recently, the SPN method has been formulated within the framework of the variational nodal method (VNM). Implementation in the VARIANT4 code indicated that for many two and three dimensional problems, near PN accuracy can be obtained at a fraction of the cost.5 Perturbation methods offer additional computational cost reduction for reactor core calculations and are indispensable for performing a variety of calculations including sensitivity studies and the breakdown by components of reactivity worths. Here, we extend the perturbation method developed for the VNM in the full PN approximation6 to treat simplified spherical harmonics. The change in reactivity predicted by both first order and exact perturbation theory using the SPN approximation is demonstrated for a benchmark problem and compared to diffusion and full PN estimates.

5 Theory The evenparity form of the slab geometry PNequations may be written in terms of even and oddparity vectors of the Legendre flux moments: 6, = +2m2 and x, = $2m1. Since the forward and adjoint equations differ only in the source and by a sign change in the oddparity term, they may be expressed in the k paired form d i d EETc dx dx 0 + d,= b(os$, + s') and where E is a twostripe coefficient matrix, E, = (2m1)6m1+2m6m,ml+ d To obtain the SPNequations, we simply replace & by 9. The evenparity flux +and source become functions in three dimensions: c(x)c(?) and s'+(x). s( r ). The oddparity flux is a vector in three dimensions: x(x) x ( r ). 4 Expressing the resulting SPNequation in terms of variational functionals which employ Lagrange multipliers to enforce nodal balance, we obtain V V Requiring the functional to be stationary with respect to 5 yields the SPN generalization of Eq. (1)within the node and Eq. (2) on the interfaces. Requiring the functional to be stationaq with respect to x yields the continuity condition on 5 across the interface. In the W M, the flux and source moments are approximated by known spatial trial functions with unknown coefficients, and the Ritz procedure is applied. The reduced functionals have the form

6 where the elements of the A (symmetric) and M matrices contain integrals of the spatial trial functions. The forward and adjoint equations are obtained by requiring stationarity with respect t o C, and x: We form global multigroup equations by combining the vectors of fluxes and sources from the corresponding nodal vectors for all groups. The matrices A and M are now block diagonal combinations of the nodal matrices for all nodes and groups. Expanding the source terms into fission and scattering, the unperturbed adjoint equations become AM [M? 0 I[,.]=d C,* 1 FT [,. <* while the perturbed forward equations are where the F matrix consists of the fission spectrum and cross sections, C contains the grouptogroup scattering cross sections and A' = A + 6A, etc. Performing the prescribed operations,6 we obtain the exact change in reactivity: Irk' c*tf 5 The corresponding first order approximation results from replacing by C,, 0 and k c, 0'and k' Since M contains no material cross sections, SM = 0 vanishes from the perturbation expressions.

7 processor to VARIANT. Results of perturbations applied to the Takeda Benchmark7 I Models 1 and 4 are given in Figure 1. For the Takeda 1 problem, the standard rodsin problem is used as the base state. The applied perturbations consist of increasing the thermal capture cross section of the control rod material. For the Takeda 4 problem, the base state is the standard rodsout problem. Initially the control rod channel contains sodium. The perturbations consist of sodium voiding of the channel. I In both problems, the SP3 approximation produces improved eigenvalues compared to diffusion theory. However, the corresponding improvement in the change in reactivity estimated by the SP3 perturbation theory is highly problem dependent. In the Takeda 1 problem, SP3 estimates of I Gkkk I are nearly identical to the P3 predictions. The base and perturbed cases can be accurately modeled using the SP3 approximation. For the Takeda 4 problem, SP3 theory fails to greatly improve the estimated change in reactivity because full PNexpansions are required to accurately model the flux distribution about the small, nearlyvoided control rod channel. References 1. K.S. Smith, Multidimensional Nodal Transport Using the Simplified PL Method, Proc. ANS Topical Mtg., Reactor Physics and Safety, Saratoga Springs, NYp 223, Sept Y.H. Liu and E.M. Gelbard, Accuracy of Nodal Transport and Simplified P3 Fluxes in Benchmark Tests, Trans. Am. Nucl. SOC.52,430 (1986). 3. R.G. Gamino, Threedimensional Nodal Transport Using the Simplified PL Method, Proc. A N S Topical Mtg., Advances in Mathematics, Computations, and Reactor Physics, April 29May 2, 1991, Pittsburgh, Vol. 2, Sec 7.1 p 3 (1991).

8 4. G. Palmiotti, E.E. Lewis and C.B. Carrico, VARIANT: VARIational Anisotropic Nodal Transport for Multidimensional Cartesian and Hexagonal Geometry Calculation, ANL95/40, Argonne National Laboratory, E.E. Lewis and G. Palmiotti, Comparison of Simplified and Standard Spherical Harmonics in the Variational Nodal Method, Trans. Am. Nucl. SOC. 73,200 (1995). 6. Kirsten F. LaurinKovitz and E.E. Lewis, Perturbation Theory Based on the Variational Transport Nodal Method in XYZ Geometry, Proc. International Conf: Mathematics and Computations, Reactor Physics and Environmental Analyses, April 30May4, 1995, Portland, Oregon, Vol2, p. 970 (1995). 7. T. Takeda et al., 3D Neutron Transport Benchmarks, NEACRPL330, Organization for Economic Cooperation and Deve1opmentlNuclea.r Energy Agency (1991).

9 D first order, P1 exact, PI e first order, SP3 0 exact, SP3 ofirst order, P3 0 exact, P ,I otot Fig. l a Takeda Benchmark Model 1 Perturbation U exact, P1 0 exact, SP3 0 exact, P3 c.r( % sodium void Fig. l b Takeda Benchmark Model 4 Perturbation 50