Accurate spherical harmonics solutions for neutron transport problems in multi-region spherical geometry
Introduction
Transport problems in spherical geometry occur in many areas. In reactor physics, there are subjects of study that require solving the neutron transport equation in spherical geometry. One could mention, for example, critical systems such as Godiva [1], microspherical fuel particles [2], and spherical fuel elements used in pebble-bed reactors [3]. In this context, the interesting problem of a double heterogeneity [4] arises when the last two appear combined [5]. Other examples of nuclear applications that involve dealing with spherical geometry are space radiation shielding [6], neutron transmission experiments through spherical shells [7], radiation transport simulations of inertial confinement fusion experiments [8], and radiative transfer in a reactor core under severe accident conditions [9]. Radiative transfer in spherical geometry is also important to other areas of application, as discussed next.
In studies of atmospheric radiative transfer, many existing codes either ignore the Earth's sphericity completely or use approximations such as the pseudo-spherical model [10], where the uncollided component of the radiance is treated exactly in spherical geometry and the collided one only approximately, using plane geometry. Both of these approaches may lead to substantially large errors when the sun is positioned at a large zenithal angle in the sky [11], more so when a sub-horizon position is involved. In mechanical engineering, modeling of high-temperature industrial processes often require solving the radiative heat transfer equation in spherical cavities and cavities delimited by concentric spherical surfaces [12], usually coupled with other forms of heat transfer (conduction, convection). Moreover, in an astrophysical setting, solving the radiative transfer equation in spherical geometry may be necessary in many situations, such as when studying infrared objects, stellar atmospheres, planetary nebulae, quasars, and so on [13].
In this work, a new method is developed for solving accurately the problem of a multi-region sphere with internal sources subject to an externally incident angular flux. The method makes use of two recently reported solutions in spherical geometry [14], [15] and consists in solving the problem locally (on a region by region basis) and coupling the local solutions iteratively. For this purpose, a series of spatial sweeps is performed back and forth across the regions to connect and update the local solutions, until global convergence is attained. Solving a multi-region problem this way may be more efficient, especially when the number of regions characterizing the problem is large. In addition, it allows the use of different orders of the approximation in the regions. Although not explored in this work, this may be an interesting option for some problems [16].
With the usage of the stable solutions reported in Refs. [14] and [15], the well-known numerical instability of the standard method in spherical geometry [17] is avoided. Our solution does not involve spatial and angular discretizations and, after being post-processed by means of an integral equation derived from the technique of integration along the neutron path, is capable of reproducing accurately the angular flux discontinuities that result from shadowing effects in spherical geometry [18].
High-quality numerical results are reported for a three-region, isotropically scattering case originally proposed and solved by Williams [19], for a second case identical to the first but with a different driving term and for a variant of the first case that includes anisotropic scattering. We hope that these results can be useful as benchmarks for verification of general purpose codes and newly developed solution methods.
Section snippets
Formulation of the problem
We treat in this work the problem of neutron transport in a multi-region sphere, as described by the one-speed spherically symmetric transport equation for regions , where K is the total number of regions. The angular flux depends on r, the dimensional radial distance from the origin (measured in cm), and μ, the cosine between the direction of neutron travel
The solutions
In this section, we discuss briefly the local solutions for odd N upon which our approach is based. For details, see previous works [14], [15]. We also discuss the spatial sweep scheme that we use for finding the global solution of the problem.
We begin by expressing the complete solution for region k as a combination of , which denotes a solution of the homogeneous transport equation obtained by setting in Eq. (6) and , a particular solution of that equation. We thus
Post-processed solutions
An undesirable feature of the method is that it cannot represent well the angular flux at (and in the neighborhood of) a discontinuity. A remedy for this difficulty is to use the method only for computing the Legendre moments of the angular flux and then use a post-processing technique to find an improved expression for the angular flux in terms of the computed Legendre moments. This idea was introduced in plane geometry by Kourganoff [31] and is known as the source-function integration
Numerical results and discussion
Table 1 reproduces the three-region basic data (dimensions and cross sections) of Williams [19] that were used to define all of the test cases studied in this work. The values of some parameters that are specific to our formulation are also given in the table.
Case 1 in this work is the same as Example 5 of Williams [19], while Case 2 changes the driving term from internal sources to an externally incident angular flux, according to Table 2. Scattering is isotropic, i.e. in all regions for
Final comments and conclusions
We have discussed in this work an accurate method for solving multi-region neutron transport problems in spherical geometry that is free from spatial and angular discretizations. The method makes use of previously reported spherical harmonics solutions for a sphere [14] and for a spherical shell [15] to solve the problem locally, on a region-by-region basis, and then constructs the global solution by performing sweeps across the regions until convergence is achieved. Converged results thought
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) of Brazil under grant 306231/2014-0. The author is indebted to M. M. R. Williams for communicating the improved integral equation results reported in Table 9. Thanks are also due to L. N. F. Guimarães for making available computational resources used to generate part of the numerical results reported in this work.
References (42)
- et al.
A statistical analysis of the double heterogeneity problem
Ann. Nucl. Energy
(1991) - et al.
Extension of double heterogeneity treatment method for coated TRISO fuel particles
Ann. Nucl. Energy
(2017) - et al.
3DHZETRN: shielded ICRU spherical phantom
Life Sci. Space Res.
(2015) - et al.
Variance reduction method for particle transport equation in spherical geometry
J. Comput. Phys.
(2018) Thermal radiation from nonisothermal spherical particles of a semitransparent material
Int. J. Heat Mass Transf.
(2000)A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell
J. Comput. Phys.
(2020)- et al.
Particle flux in an annular gap about a sphere
Ann. Nucl. Energy
(2005) - et al.
A comparison of transport methods for the solution of a problem with shadowing effects in spherical geometry
Ann. Nucl. Energy
(2019) Integral transport in a three region sphere with associated benchmark problems
Ann. Nucl. Energy
(2005)- et al.
A particular solution for the method in spherical geometry
J. Quant. Spectrosc. Radiat. Transf.
(1991)
A particular solution for the radiative transfer equation in spherical geometry
J. Quant. Spectrosc. Radiat. Transf.
On computing the Chandrasekhar polynomials in high order and high degree
J. Quant. Spectrosc. Radiat. Transf.
The method for radiative transfer problems with reflective boundary conditions
J. Quant. Spectrosc. Radiat. Transf.
Multislab multigroup transport theory with Lth order anisotropic scattering
J. Comput. Phys.
Computing the angular dependence of the radiation of a planetary atmosphere
J. Quant. Spectrosc. Radiat. Transf.
On intensity calculations in radiative transfer
J. Quant. Spectrosc. Radiat. Transf.
An analysis of the source-function integration technique for postprocessing angular fluxes
Ann. Nucl. Energy
On the integral form of the equation of transfer for a homogeneous sphere
J. Quant. Spectrosc. Radiat. Transf.
Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series
Comput. Phys. Rep.
An unreflected U-235 critical assembly
Nucl. Sci. Eng.
Coated particle fuel for high temperature gas cooled reactors
Nucl. Eng. Technol.
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