NUMERICAL INTEGRATION OF THE SPHERICAL-HARMONICS EQUATIONS

Journal Article · Wed Apr 01 04:00:00 EST 1964 · Nuclear Science and Engineering (U.S.)

OSTI ID:4028245

Waldinger, H; Agresta, J; Goertzel, G

A method is formulated for numerical integration of the spherical- harmonics in the case of cylindrical geometry. This method avoids many of the difficulties of the usual analytical techniques and allows space-varying sources as well as regions of low neutron cross section and large physical size. The usual spherical-harmonic equations (truncated) are presented in cylindrical geometry. To obthin a set of equations that (because they are more intuitive in form) lead to readily manageable numerical solution, the equations are converted to the discrete ordinate form in cylindrical geometry. From the discrete- ordinate equations, one may readily discuss inward- and outward-going neutrons. Based on this, reflection matrices are introduced at each radius r, one describing the reflection of inwardly directed neutrons by the medium outward of r and the other describing the reflection of outwardly directed neutrons by the medium outward of r. The complete source-independent properties of the medium are described by these reflection matrices. Furthermore, the matrices can be obtained by numerical integration in a single pass, one by integrating from the center out and the other by integrating from the outside in. The source can be treated by considering at each radius r the flux that escapes outward due to sources inward of r and by considering separately the flux that goes inward due to sources outward of r. The first of these escape fluxes is obtained by integrntion outward from the origin, using the corresponding reflection matrix, the second by integration inwards. Once the above quantities are found, the fluxes are obtained by solution of simultaneous algebraic equations (no further integrations). Numerical results necessary for the use of this method in the P/ sub 3/ approximation are also given. (auth)

• Research Organization: Polytechnic Inst. of Brooklyn
• NSA Number: NSA-18-019320
• OSTI ID: 4028245
• Journal Information: Nuclear Science and Engineering (U.S.), Journal Name: Nuclear Science and Engineering (U.S.) Vol. Vol: 18; ISSN NSENA
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

CROSS SECTIONS
CYLINDERS
DIFFERENTIAL EQUATIONS
MATHEMATICS
MATRICES
NEUTRON FLUX
NEUTRON SOURCES
NEUTRONS
NUMERICALS
REACTOR TECHNOLOGY
REFLECTION
SPHERICAL HARMONICS
ZONES