Research Article
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The Slab Critical Thickness Problem with Reflecting Boundary Condition for the Anlı-Güngör Scattering Function

Year 2023, Volume: 44 Issue: 4, 775 - 784, 28.12.2023

Demet Gülderen , R. Gökhan Türeci

https://doi.org/10.17776/csj.1240161
https://izlik.org/JA38KT26TF

Abstract

The criticality equation, which defines the relation between the secondary neutron number and the thickness of the slab, and the numerical solutions of this equation are investigated with reflecting boundary condition for the recently studied the Anlı-Güngör (AG) scattering function. The analytical calculations are performed by HN method. The numerical results are calculated with Wolfram Mathematica software for the varying secondary neutron number, the varying scattering parameter, and the varying reflection coefficient. The critical slab thickness values decrease for increasing reflection coefficient as expected.

Keywords

Legendre expansion of scattering function Anlı-Güngör scattering function HN method The critical slab problem Reflecting boundary condition.

References

  • [1] Anli F., Güngör S., Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry, Applied Mathematical Modelling, 31 (2007) 727-733.
  • [2] Mitsis G.J., Transport Solutions to the One-Dimensional Critical Problem, Nuclear Science and Engineering, 17 (1963) 55-64.
  • [3] Case K.M., Elementary Solutions of the Transport Equation and Their Applications, Annals of Physics, 9 (1960) 1-23.
  • [4] Case K.M., Zweifel P.F., Linear Transport Theory, Massachusetts, (1967).
  • [5] Carlvik I., Monoenergetic Critical Parameters and Decay Constants for Small Homogeneous Spheres and Thin Homogeneous Slabs, Nuclear Science and Engineering, 31 (1968) 295-303.
  • [6] Sahni D.C., Sjöstrand N.G., Criticality and Time Eigenvalues for One-Speed Neutron Transport, Progress in Nuclear Energy, 23 (1990) 241-289.
  • [7] Sahni D.C., Some New Results Pertaining to Criticality and Time Eigenvalues of One-Speed Neutron Transport Equation, Progress in Nuclear Energy, 30 (1996) 305-320.
  • [8] Sahni D.C., Sjöstrand N.G., Non-monotonic variation of the criticality factor with the degree of anisotropy in one-speed neutron transport, Transport Theory and Statistical Physics, 20 (1991) 339-349.
  • [9] Sahni D.C., Sjöstrand N.G., Criticality eigenvalues of the one-speed transport equation with strong forward-backward scattering relationship with rod model, Transport Theory and Statistical Physics, 27 (1998) 137-158.
  • [10] Dahl E.B., Sjöstrand N.G., Eigenvalue Spectrum of Multiplying Slabs and Spheres for Monoenergetic Neutrons with Anisotropic Scattering, Nuclear Science and Engineering, 69 (1979) 114-125.
  • [11] Garis N.S., One-Speed Neutron Transport Eigenvalues for Reflected Slabs and Spheres, Nuclear Science and Engineering, 107 (1991) 343-358.
  • [12] Garis N.S., Sjöstrand N.G., Eigenvalues for reflecting boundary conditions in one-speed neutron transport theory, Annals of Nuclear Energy, 21 (1994) 67-80.
  • [13] Atalay M.A., The Critical Slab Problem for Reflecting Boundary Conditions in One-Speed Neutron Transport Theory, Annals of Nuclear Energy, 23 (1996) 183-193.
  • [14] Türeci R.G., Güleçyüz M.Ç., Kaşkaş A., Tezcan C., Application of the HN Method to the Critical Slab Problem for Reflecting Boundary Conditions, Journal of Quantitative Spectroscopy and Radiative Transfer, 88 (2004) 499-517.
  • [15] Kavenoky A., The CN Method of Solving the Transport Equation: Application to Plane Geometry, Nuclear Science and Engineering, 65 (1978) 209-225.
  • [16] Grandjean P., Siewert C.E., The FN Method in Neutron-Transport Theory. Part II: Applications and Numerical Results, Nuclear Science and Engineering, 69 (1979) 161-168.
  • [17] Tezcan C., Kaşkaş A., Güleçyüz M.Ç., The HN method for solving linear transport equation: Theory and applications, Journal of Quantitative Spectroscopy and Radiative Transfer, 78 (2003) 243-254.
  • [18] İnönü E., A Theorem on anisotropic scattering, Transport Theory and Statistical Physics, 3 (1973) 137-146.
  • [19] Siewert C.E., Williams M.M.R., The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel, Journal of Physics D: Applied Physics, 10 (1977) 2031-2040.
  • [20]Mika J.R., Neutron Transport with Anisotropic Scattering, Nuclear Science and Engineering, 11 (1961) 415-427.
  • [21] Türeci R.G., Güleçyüz M.Ç., Kaşkaş A., Tezcan C. The singular eigenfunction method: the critical slab problem for linearly anisotropic scattering, Kerntechnik, 70 (5-6) (2005) 322-326.
  • [22] Güleçyüz M.Ç., Türeci R.G., Tezcan C., The Critical Slab Problem for Linearly Anisotropic Scattering and Reflecting Boundary Conditions with the HN Method, Kerntechnik, 71 (2006) 149-154.
  • [23]Türeci R.G., Güleçyüz M.Ç., The Slab Albedo and Criticality Problem for the Quadratic Scattering Kernel with the H-N Method, Kerntechnik, 73 (2008) 171-175.
  • [24]Türeci R.G., Solving the criticality problem with the reflected boundary condition for the triplet anisotropic scattering with the modified FN method, Kerntechnik, 80 (6) (2015) 583-591.
  • [25]Türeci R.G., Türeci D. The critical slab problem for pure-triplet anisotropic scattering by singular eigenfunction method, Kerntechnik, 82 (6) (2017) 693-699.
  • [26] Koklu H., Ozer O., Critical thickness problem for tetra-anisotropic scattering in the reflected reactor system. Pramana - J Phys, 95 (2021) 190
  • [27] Gülderen D., Sahni D.C., Türeci R.G., Aydιn A., The Milne Problem for Linear-Triplet Anisotropic Scattering with HN Method, Journal of Computational and Theoretical Transport, 51 (2022) 329-353.
  • [28]Köklü H., Özer O., Analyzing of the Scattering Coefficients in the Neutron Transport Equation for Critical Systems, Journal of Computational and Theoretical Transport, 51 (2022) 112-138.
  • [29]Bülbül A., Anli F., Criticality calculations with PN approximation for certain scattering parameters of Anlı-Güngör and Henyey-Greenstein phase functions in spherical geometry, Kerntechnik, 80 (2015) 161-166.
  • [30]Bozkır A.Z., Türeci R.G., Sahni D.C., Half-space albedo problem for the Anlı-Güngör scattering function, Kerntechnik, 87 (2022) 237-248.
  • [31] Türeci R.G., The Milne Problem with the Anlı-Güngör Scattering. Journal of Computational and Theoretical Transport, 51 (6) (2022) 354-371.
  • [32]Türeci R.G., The Slab Albedo Problem with the Anlı-Güngör Scattering Function. Indian Journal of Physics, 97 (2023) 2483-2506.
  • [33]Türeci R.G., The Critical Slab Problem with The Anlı-Güngör Scattering Function. Nuclear Engineering and Technelogy, 95 (2023) 2864-2872.
  • [34]Maleki B.R., Using the Monte Carlo method to solve the half-space and slab albedo problems with Inönü and Anlı-Güngör strongly anisotropic scattering functions, Nuclear Engineering and Technology, In Press. Available online: 27 Sep 2022.
  • [35]Türeci R.G., Bülbül A., Case’s Method for Anlı-Güngör Scattering Formula, Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi, 17 (2022) 1-8.
  • [36]Türeci, R.G. Dağıstanlı H., Çakır İ.T. Fizik ve Mühendislikte Python, Ankara, (2021) 210.
  • [37] Türeci R.G. Fizik ve Mühendislikte Wolfram Mathematica, Ankara, (2020) 176.
  • [38]Burden A., Burden R., and Faires J., Numerical Analysis, 10th Edition, Cenveo, (2016).
  • [39]Wolfram Research, Inc., Mathematica, Version 12.2, Champaign, IL (2023).

Year 2023, Volume: 44 Issue: 4, 775 - 784, 28.12.2023

Demet Gülderen , R. Gökhan Türeci

https://doi.org/10.17776/csj.1240161
https://izlik.org/JA38KT26TF

Abstract

Keywords

Mika scattering function Anlı-Güngör scattering function HN method the critical slab problem reflecting boundary condition.

References

  • [1] Anli F., Güngör S., Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry, Applied Mathematical Modelling, 31 (2007) 727-733.
  • [2] Mitsis G.J., Transport Solutions to the One-Dimensional Critical Problem, Nuclear Science and Engineering, 17 (1963) 55-64.
  • [3] Case K.M., Elementary Solutions of the Transport Equation and Their Applications, Annals of Physics, 9 (1960) 1-23.
  • [4] Case K.M., Zweifel P.F., Linear Transport Theory, Massachusetts, (1967).
  • [5] Carlvik I., Monoenergetic Critical Parameters and Decay Constants for Small Homogeneous Spheres and Thin Homogeneous Slabs, Nuclear Science and Engineering, 31 (1968) 295-303.
  • [6] Sahni D.C., Sjöstrand N.G., Criticality and Time Eigenvalues for One-Speed Neutron Transport, Progress in Nuclear Energy, 23 (1990) 241-289.
  • [7] Sahni D.C., Some New Results Pertaining to Criticality and Time Eigenvalues of One-Speed Neutron Transport Equation, Progress in Nuclear Energy, 30 (1996) 305-320.
  • [8] Sahni D.C., Sjöstrand N.G., Non-monotonic variation of the criticality factor with the degree of anisotropy in one-speed neutron transport, Transport Theory and Statistical Physics, 20 (1991) 339-349.
  • [9] Sahni D.C., Sjöstrand N.G., Criticality eigenvalues of the one-speed transport equation with strong forward-backward scattering relationship with rod model, Transport Theory and Statistical Physics, 27 (1998) 137-158.
  • [10] Dahl E.B., Sjöstrand N.G., Eigenvalue Spectrum of Multiplying Slabs and Spheres for Monoenergetic Neutrons with Anisotropic Scattering, Nuclear Science and Engineering, 69 (1979) 114-125.
  • [11] Garis N.S., One-Speed Neutron Transport Eigenvalues for Reflected Slabs and Spheres, Nuclear Science and Engineering, 107 (1991) 343-358.
  • [12] Garis N.S., Sjöstrand N.G., Eigenvalues for reflecting boundary conditions in one-speed neutron transport theory, Annals of Nuclear Energy, 21 (1994) 67-80.
  • [13] Atalay M.A., The Critical Slab Problem for Reflecting Boundary Conditions in One-Speed Neutron Transport Theory, Annals of Nuclear Energy, 23 (1996) 183-193.
  • [14] Türeci R.G., Güleçyüz M.Ç., Kaşkaş A., Tezcan C., Application of the HN Method to the Critical Slab Problem for Reflecting Boundary Conditions, Journal of Quantitative Spectroscopy and Radiative Transfer, 88 (2004) 499-517.
  • [15] Kavenoky A., The CN Method of Solving the Transport Equation: Application to Plane Geometry, Nuclear Science and Engineering, 65 (1978) 209-225.
  • [16] Grandjean P., Siewert C.E., The FN Method in Neutron-Transport Theory. Part II: Applications and Numerical Results, Nuclear Science and Engineering, 69 (1979) 161-168.
  • [17] Tezcan C., Kaşkaş A., Güleçyüz M.Ç., The HN method for solving linear transport equation: Theory and applications, Journal of Quantitative Spectroscopy and Radiative Transfer, 78 (2003) 243-254.
  • [18] İnönü E., A Theorem on anisotropic scattering, Transport Theory and Statistical Physics, 3 (1973) 137-146.
  • [19] Siewert C.E., Williams M.M.R., The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel, Journal of Physics D: Applied Physics, 10 (1977) 2031-2040.
  • [20]Mika J.R., Neutron Transport with Anisotropic Scattering, Nuclear Science and Engineering, 11 (1961) 415-427.
  • [21] Türeci R.G., Güleçyüz M.Ç., Kaşkaş A., Tezcan C. The singular eigenfunction method: the critical slab problem for linearly anisotropic scattering, Kerntechnik, 70 (5-6) (2005) 322-326.
  • [22] Güleçyüz M.Ç., Türeci R.G., Tezcan C., The Critical Slab Problem for Linearly Anisotropic Scattering and Reflecting Boundary Conditions with the HN Method, Kerntechnik, 71 (2006) 149-154.
  • [23]Türeci R.G., Güleçyüz M.Ç., The Slab Albedo and Criticality Problem for the Quadratic Scattering Kernel with the H-N Method, Kerntechnik, 73 (2008) 171-175.
  • [24]Türeci R.G., Solving the criticality problem with the reflected boundary condition for the triplet anisotropic scattering with the modified FN method, Kerntechnik, 80 (6) (2015) 583-591.
  • [25]Türeci R.G., Türeci D. The critical slab problem for pure-triplet anisotropic scattering by singular eigenfunction method, Kerntechnik, 82 (6) (2017) 693-699.
  • [26] Koklu H., Ozer O., Critical thickness problem for tetra-anisotropic scattering in the reflected reactor system. Pramana - J Phys, 95 (2021) 190
  • [27] Gülderen D., Sahni D.C., Türeci R.G., Aydιn A., The Milne Problem for Linear-Triplet Anisotropic Scattering with HN Method, Journal of Computational and Theoretical Transport, 51 (2022) 329-353.
  • [28]Köklü H., Özer O., Analyzing of the Scattering Coefficients in the Neutron Transport Equation for Critical Systems, Journal of Computational and Theoretical Transport, 51 (2022) 112-138.
  • [29]Bülbül A., Anli F., Criticality calculations with PN approximation for certain scattering parameters of Anlı-Güngör and Henyey-Greenstein phase functions in spherical geometry, Kerntechnik, 80 (2015) 161-166.
  • [30]Bozkır A.Z., Türeci R.G., Sahni D.C., Half-space albedo problem for the Anlı-Güngör scattering function, Kerntechnik, 87 (2022) 237-248.
  • [31] Türeci R.G., The Milne Problem with the Anlı-Güngör Scattering. Journal of Computational and Theoretical Transport, 51 (6) (2022) 354-371.
  • [32]Türeci R.G., The Slab Albedo Problem with the Anlı-Güngör Scattering Function. Indian Journal of Physics, 97 (2023) 2483-2506.
  • [33]Türeci R.G., The Critical Slab Problem with The Anlı-Güngör Scattering Function. Nuclear Engineering and Technelogy, 95 (2023) 2864-2872.
  • [34]Maleki B.R., Using the Monte Carlo method to solve the half-space and slab albedo problems with Inönü and Anlı-Güngör strongly anisotropic scattering functions, Nuclear Engineering and Technology, In Press. Available online: 27 Sep 2022.
  • [35]Türeci R.G., Bülbül A., Case’s Method for Anlı-Güngör Scattering Formula, Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi, 17 (2022) 1-8.
  • [36]Türeci, R.G. Dağıstanlı H., Çakır İ.T. Fizik ve Mühendislikte Python, Ankara, (2021) 210.
  • [37] Türeci R.G. Fizik ve Mühendislikte Wolfram Mathematica, Ankara, (2020) 176.
  • [38]Burden A., Burden R., and Faires J., Numerical Analysis, 10th Edition, Cenveo, (2016).
  • [39]Wolfram Research, Inc., Mathematica, Version 12.2, Champaign, IL (2023).
There are 39 citations in total.

Details

Primary Language English
Subjects Classical Physics (Other)
Journal Section Research Article
Authors

Demet Gülderen 0000-0001-8038-219X

R. Gökhan Türeci 0000-0001-6309-6300

Submission Date January 20, 2023
Acceptance Date October 9, 2023
Publication Date December 28, 2023
DOI https://doi.org/10.17776/csj.1240161
IZ https://izlik.org/JA38KT26TF
Published in Issue Year 2023 Volume: 44 Issue: 4

Cite

APA Gülderen, D., & Türeci, R. G. (2023). The Slab Critical Thickness Problem with Reflecting Boundary Condition for the Anlı-Güngör Scattering Function. Cumhuriyet Science Journal, 44(4), 775-784. https://doi.org/10.17776/csj.1240161

Cited By

Monte Carlo solution of the reflected systems’ criticality problems with Anlı–Güngör, Inönü and tetra-anisotropic kernels for both fission and scattered neutrons
Pramana
https://doi.org/10.1007/s12043-024-02781-8

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