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Title: (U) Analytic First and Second Derivatives of the Uncollided Leakage for a Homogeneous Sphere

Abstract

The second-order adjoint sensitivity analysis methodology (2nd-ASAM), developed by Cacuci, has been applied by Cacuci to derive second derivatives of a response with respect to input parameters for uncollided particles in an inhomogeneous transport problem. In this memo, we present an analytic benchmark for verifying the derivatives of the 2nd-ASAM. The problem is a homogeneous sphere, and the response is the uncollided total leakage. This memo does not repeat the formulas given in Ref. 2. We are preparing a journal article that will include the derivation of Ref. 2 and the benchmark of this memo.

Authors:
 [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1356096
Report Number(s):
LA-UR-17-23453
DOE Contract Number:
AC52-06NA25396
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats

Favorite, Jeffrey A. (U) Analytic First and Second Derivatives of the Uncollided Leakage for a Homogeneous Sphere. United States: N. p., 2017. Web. doi:10.2172/1356096.
Favorite, Jeffrey A. (U) Analytic First and Second Derivatives of the Uncollided Leakage for a Homogeneous Sphere. United States. doi:10.2172/1356096.
Favorite, Jeffrey A. 2017. "(U) Analytic First and Second Derivatives of the Uncollided Leakage for a Homogeneous Sphere". United States. doi:10.2172/1356096. https://www.osti.gov/servlets/purl/1356096.
@article{osti_1356096,
title = {(U) Analytic First and Second Derivatives of the Uncollided Leakage for a Homogeneous Sphere},
author = {Favorite, Jeffrey A.},
abstractNote = {The second-order adjoint sensitivity analysis methodology (2nd-ASAM), developed by Cacuci, has been applied by Cacuci to derive second derivatives of a response with respect to input parameters for uncollided particles in an inhomogeneous transport problem. In this memo, we present an analytic benchmark for verifying the derivatives of the 2nd-ASAM. The problem is a homogeneous sphere, and the response is the uncollided total leakage. This memo does not repeat the formulas given in Ref. 2. We are preparing a journal article that will include the derivation of Ref. 2 and the benchmark of this memo.},
doi = {10.2172/1356096},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2017,
month = 4
}

Technical Report:

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  • The Second-Level Adjoint Sensitivity System (2nd-LASS) that yields the second-order sensitivities of a response of uncollided particles with respect to isotope densities, cross sections, and source emission rates is derived in Refs. 1 and 2. In Ref. 2, we solved problems for the uncollided leakage from a homogeneous sphere and a multiregion cylinder using the PARTISN multigroup discrete-ordinates code. In this memo, we derive solutions of the 2nd-LASS for the particular case when the response is a flux or partial current density computed at a single point on the boundary, and the inner products are computed using ray-tracing. Both themore » PARTISN approach and the ray-tracing approach are implemented in a computer code, SENSPG. The next section of this report presents the equations of the 1st- and 2nd-LASS for uncollided particles and the first- and second-order sensitivities that use the solutions of the 1st- and 2nd-LASS. Section III presents solutions of the 1st- and 2nd-LASS equations for the case of ray-tracing from a detector point. Section IV presents specific solutions of the 2nd-LASS and derives the ray-trace form of the inner products needed for second-order sensitivities. Numerical results for the total leakage from a homogeneous sphere are presented in Sec. V and for the leakage from one side of a two-region slab in Sec. VI. Section VII is a summary and conclusions.« less
  • The first and second pressure derivatives of the adiabatic elastic constants of CsBr single crystal were measured at 300, 298, 258.2, and 222/sup 0/K using pulse interference techniques. The corresponding quantities of the bulk modulus of CsBr are also calculated. The maximum pressures used in the experiments ranged from 6 kbars to 11 kbars. B/sub 0//sup T'/ is measured to an accuracy of approximately 5 percent, and B/sub 0//sup S''/ to approximately 15 to 30 percent, assuming the systematic error is absent. It is found, however, the value of B/sub 0//sup S''/ is strongly affected by the systematic error arisingmore » from the choice of model in fitting the data. The widely used quadratic fit is found inadequate for a maximum pressure of 10 kbars. The contribution from the third-order term /sup 1///sub 6/ B''/sub 0/'P/sup 3/ can not be neglected. The quasi-harmonic approximation is used to derive the theoretical equation of state. Several two- or three-parameter empirical equations of state are also discussed. Using the long-wave method and the present data, the phonon dispersion relation of CsBr is calculated and compared with the experimental results of neutron scattering. 14 tables, 34 figures, 74 references.« less
  • The local packing, occurring in cylindrical containers when randomly stacked with homogeneous spheres, is examined over a range of 4 to 24 of the ratio of container diameter to sphere diameter. Sphere packings are examined both as a function of radial and vertical position in the bed. It is concluded that only one kind of close packing exists or tend to exist in random packing. i.e.. rhombohedral. Some understanding of the packing mechanism has evolved for the largest cylindrical container. i.e.. container to sphere diameter ratio of 24. A three-dimensional picture is presented of the heat transfer over the surfacemore » area of a sphere when immersed in a packed bed of ""infinite'' dimensions arranged in rhombohedral No. 6 blocked passage array for a range of Reynolds numbers of 8,000 to 60.000. where Reynolds number is based on sphere diameter, average velocity of the coolant across the cross-section of the bed in the absence of the spheres. and fluid properties evaluated at the mean film conditions. (auth)« less
  • The leakage neutron spectrum from a chain reaction in an aqueous solution of uranyl nitrate has been determined for the purpose of calibrating indium foil to be used for personnel monitoring of a possible accidental critical reaction at K-25. Approximately 75% of the neutrons were found to be slow, and experimental results are in reasonable agreement with a semi-theoretical treatment of the problem. The overall cross section of indium for neutrons of all energies in this leakage spectrum was found to be about 60 +/- 4 barns. This is of the same order of magnitude as the value of 126more » barns which was determined from the calculated energy spectrum using weighted averages of directly measured cross sections.« less