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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Discrete Fracture Network Modeling to Estimate Upscaled Parameters for the Topopah Spring, Lava Flow, and Tiva Canyon Aquifers at Pahute Mesa, Nevada National Security Site

This report describes the results of Discrete Fracture Network (DFN) simulations for the Topopah Spring Aquifer (TSA), Lava Flow Aquifer, and Tiva Canyon Aquifer (TCA), at Pahute Mesa on the Nevada National Security Site (NNSS), formerly the Nevada Test Site. The research focuses on calculating upscaled groundwater flow and contaminant transport parameters using DFNs generated according to fracture characteristics observed in the TSA, LFA and TCA at Pahute Mesa. The highly fractured and heterogeneous nature of these aquifers makes them candidates for stochastic DFN modeling of radionuclide transport on a small scale with subsequent upscaling. One hundred independent DFN realizations are generated for each aquifer, and the upscaled parameters for continuum simulations of subsurface flow and transport in fractured media at Pahute Mesa are calculated. Our goal is to implement a modeling approach that can translate parameters to larger-scale models that account for local-scale flow and transport processes, such as channelization of flow and transport along a few well connected, large fractures. Additionally, to simulate advective and advective-diffusive transport through the fracture networks, the Time Domain Random Walk (TDRW) approach is applied to account for matrix diffusion into a finite half-space. Moreover, a novel approach to calculate dynamic (active) fracture surface area to reflect flow channeling is implemented. This work will improve the representation of radionuclide transport processes in largescale, regulatory-focused models by providing estimates of hard-to-measure flow and contaminant transport parameters at large scales. In this report, we (1) show recent results of flow and transport simulations on multiple DFN realizations of the TSA, LFA, TCA; (2) discuss the resulting distributions of estimated upscaled parameters; (3) describe the estimation of upscaled parameters for an equivalent parallel-plate continuum model and (4) present a comparison between simulated transport from the equivalent continuum model and an actual DFN.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Quantification of the Impact of Acidified Brine on Fracture-Matrix Transport in a Naturally Fractured Shale Using in Situ Imaging and Modeling

Understanding flow, transport, chemical reactions, and hydromechanical processes in fractured geologic materials is key for optimizing a range of subsurface processes including carbon dioxide and hydrogen storage, unconventional energy resource extraction, and geothermal energy recovery. Flow and transport processes in naturally fractured shale rocks have been challenging to characterize due to experimental complexity and the multiscale nature of quantifying continuum scale descriptions of mass exchange between micrometer-scale fractures and nanometer-scale pores. In this study, we use positron emission tomography (PET) to image the transport of a conservative tracer in a naturally fractured Wolfcamp shale core before and after the core was exposed to low pH brine conditions. Image-based experimental observations are interpreted by fitting an analytical transport model to fracture-containing voxels in the core. Results of this analysis indicate subtle increases in matrix diffusivity and a slightly more uniform fracture velocity distribution following exposure to low pH conditions. These observations are compared with a multicomponent one-dimensional reactive transport model that indicates the capacity for a 10% increase in porosity at the fracture-matrix interface as a result of the low pH brine exposure. This porosity change is the result of the dissolution of carbonate minerals in the shale matrix to low pH conditions. Further, this image-based workflow represents a new approach for quantifying spatially resolved fracture-matrix transport processes and provides a foundation for future work to better understand the role of coupled transport, reaction, and mechanical processes in naturally fractured rocks.

58 GEOSCIENCES↗
DOE OSTI2020

Colloid-facilitated transport of 238 Pu, 233 U and 137 Cs through fractured chalk: Laboratory experiments, modelling, and implications for nuclear waste disposal

The influence of montmorillonite colloids on the mobility of 238 Pu, 233 U and 137 Cs through a chalk fracture was investigated to assess the transport potential for radioactive waste. Radioisotopes of each element, along with the conservative tracer tritium, were injected in the presence and absence of montmorillonite colloids into a naturally fractured chalk core. In parallel, batch experiments were conducted to obtain experimental sorption coefficients (K d , mL/g) for both montmorillonite colloids and the chalk fracture material. Breakthrough curves were modelled to determine diffusivity and sorption of each radionuclide to the chalk and the colloids under advective conditions. Uranium sorbed sparingly to chalk (log K d = 0.7 ± 0.2) in batch sorption experiments. 233 U(VI) breakthrough was controlled primarily by the matrix diffusion and sorption to chalk (15 and 25% recovery with and without colloids, respectively). Cesium, in contrast, sorbed strongly to both the montmorillonite colloids and chalk (batch log K d = 3.2 ± 0.01 and 3.9 ± 0.01, respectively). The high affinity to chalk and low colloid concentrations overwhelmed any colloidal Cs transport, resulting in very low 137 Cs breakthrough (1.1–5.5% mass recovery). Batch and fracture transport results, and the associated modelling revealed that Pu migrates both as Pu (IV) sorbed to montmorillonite colloids and as dissolved Pu(V) (7% recovery). Transport experiments revealed differences in Pu(IV) and Pu(V) transport behavior that could not be quantified in simple batch experiments but are critical to effectively predict transport behavior of redox-sensitive radionuclides. Finally, a brackish groundwater solution was injected after completion of the fracture flow experiments and resulted in remobilization and recovery of 2.2% of the total sorbed radionuclides which remained in the core from previous experiments. In general, our study demonstrates consistency in sorption behavior between batch and advective fracture transport. Finally, the results suggest that colloid-facilitated radionuclide transport will enhance radionuclide migration in fractured chalk for those radionuclides with exceedingly high affinity for colloids.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2020

Iodine effective diffusion coefficients through volcanic rock: Influence of iodine speciation and rock geochemistry

Accurate prediction of the subsurface transport of iodine species is important for the assessment of long-term nuclear waste repository performance, as well as monitoring compliance with the Comprehensive Nuclear-Test-Ban Treaty, given that radioiodine decays into radioxenon. However, the transport of iodine through intact geologic media is not well understood, compromising our ability to assess risk associated with radioiodine migration. The current study's goal is to quantify the matrix diffusion of iodine species through saturated volcanic rock, with particular attention paid to the redox environment and potential speciation changes. Here, diffusion experiments were run for iodide through lithophysae-rich lava, lithophysae-poor lava, and welded tuff, whereas iodate diffusion was studied through welded tuff. Iodine transport was compared with a conservative tracer, HDO, and effective diffusion coefficients were calculated. Likely due to a combination of size and anion exclusion effects, iodine species diffused more slowly than the conservative tracer through all rock types tested. Furthermore, oxidation of iodide to iodate was observed in the lithophysae-poor lava, affecting transport. Results provide much needed data for subsurface transport models that predict radioiodine migration from underground sources, and indicate the pressing need for geochemical and redox interactions to be incorporated into these models.

58 GEOSCIENCES↗
DOE OSTI2022

Closures for multicomponent reacting flows based on dispersion analysis

This work presents algebraic closure models associated with advective transport and nonlinear reactions in a Reynolds-averaged Navier-Stokes context for a system of species subject to binary reactions and transport by advection and diffusion. Expanding upon analysis originally developed for non-reactive transport in the context of Taylor dispersion of scalars, this work extends the modified gradient diffusion model explicated by Peters [N. Peters, Turbulent Combustion, Cambridge Monographs on Mechanics (Cambridge University Press, Cambridge, 2000)] and based on work by Corrsin [S. Corrsin, The reactant concentration spectrum in turbulent mixing with a first-order reaction, J. Fluid Mech. 11, 407 (1961)] beyond single-component transport phenomena and involving nonlinear reactions. The presented model forms, from this weakly nonlinear extension of the original dispersion theory, lead to an analytic expression for the eddy diffusivity matrix that explicitly captures the influence of the reaction kinetics on the closure operators. Furthermore, we demonstrate that the derived model form directly translates between flow topologies through a priori and a posteriori testing of a binary species system subject to homogeneous isotropic turbulence. Using two- and three-dimensional direct numerical simulations involving laminar and turbulent flows, it is shown that this framework improves prediction of mean quantities compared to previous results. Lastly, the presented model form, collapses to the earlier gradient diffusion and its modified version derived by Corrsin in the limits of nonreactive species and linear reactions, respectively.

42 ENGINEERING↗
DOE OSTI2017

Assessing Electrolyte Transport Properties with Molecular Dynamics

Here in this work we use estimates of ionic transport properties obtained from molecular dynamics to rank lithium electrolytes of different compositions. We develop linear response methods to obtain the Onsager diffusivity matrix for all chemical species, its Fickian counterpart, and the mobilities of the ionic species. We apply these methods to the well-studied propylene carbonate/ethylene carbonate solvent with dissolved LiBF 4 and O 2 . The results show that, over a range of lithium concentrations and carbonate mixtures, trends in the transport coefficients can be identified and optimal electrolytes can be selected for experimental focus; however, refinement of these estimation techniques is necessary for a reliable ranking of a large set of electrolytes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Regional-scale analysis of karst underground flow deduced from tracing experiments: examples from carbonate aquifers in Malaga province, southern Spain

Tracer concentration data from field experiments conducted in several carbonate aquifers (Malaga province, southern Spain) were analyzed following a dual approach based on the graphical evaluation method (GEM) and solute transport modeling to decipher flow mechanisms in karst systems at regional scale. The results show that conduit system geometry and flow conditions are the principal factors influencing tracer migration through the examined karst flow routes. Solute transport is mainly controlled by longitudinal advection and dispersion throughout the conduit length, but also by flow partitioning between mobile and immobile fluid phases, while the matrix diffusion process appears to be less relevant. The simulation of tracer breakthrough curves (BTCs) suggests that diffuse and concentrated flow through the unsaturated zone can have equivalent transport properties under extreme recharge, with high flow velocities and efficient mixing due to the high hydraulic gradients generated. Tracer mobilization within the saturated zone under low flow conditions mainly depends on the hydrodynamics (rather than on the karst conduit development), which promote a lower longitudinal advection and retardation in the tracer migration, resulting in a marked tailing effect of BTCs. The analytical advection-dispersion equation better approximates the effective flow velocity and longitudinal dispersion estimations provided by the GEM, while the non-equilibrium transport model achieves a better adjustment of most asymmetric and long-tailed BTCs. The assessment of karst underground flow properties from tracing tests at regional scale can aid design of groundwater management and protection strategies, particularly in large hydrogeological systems (i.e. transboundary carbonate aquifers) and/or in poorly investigated ones.

58 GEOSCIENCES↗
DOE OSTI2016

Investigating the age distribution of fracture discharge using multiple environmental tracers, Bedrichov Tunnel, Czech Republic

We investigated the transit time distribution (TTD) of discharge collected from fractures in the Bedrichov Tunnel, Czech Republic, using lumped parameter models and multiple environmental tracers. We then utilize time series of δ 18 O, δ 2 H and 3 H along with CFC measurements from individual fractures in the Bedrichov Tunnel of the Czech Republic to investigate the TTD, and the uncertainty in estimated mean travel time in several fracture networks of varying length and discharge. We also compare several TTDs, including the dispersion distribution, the exponential distribution, and a developed TTD which includes the effects of matrix diffusion. The effect of seasonal recharge is explored by comparing several seasonal weighting functions to derive the historical recharge concentration. We identify best fit mean ages for each TTD by minimizing the error-weighted, multi-tracer χ2 residual for each seasonal weighting function. We use this methodology to test the ability of each TTD and seasonal input function to fit the observed tracer concentrations, and the effect of choosing different TTD and seasonal recharge functions on the mean age estimation. We find that the estimated mean transit time is a function of both the assumed TTD and seasonal weighting function. Best fits as measured by the χ2 value were achieved for the dispersion model using the seasonal input function developed here for two of the three modeled sites, while at the third site, equally good fits were achieved with the exponential model and the dispersion model and our seasonal input function. The average mean transit time for all TTDs and seasonal input functions converged to similar values at each location. The sensitivity of the estimated mean transit time to the seasonal weighting function was equal to that of the TTD. These results indicated that understanding seasonality of recharge is at least as important as the uncertainty in the flow path distribution in fracture networks and that unique identification of the TTD and mean transit time is difficult given the uncertainty in the recharge function. But, the mean transit time appears to be relatively robust to the structural model uncertainty. The results presented here should be applicable to other studies using environmental tracers to constrain flow and transport properties in fractured rock systems.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

Field experiments of surface water to groundwater recharge to characterize the mobility of uranium and vanadium at a former mill tailing site

Characterizing the mobility of uranium and vanadium in groundwater with a hydraulic connection to surface water is important to inform the best management practices of former mill tailing sites. In this study, the recharge of river water to the unsaturated and saturated zones of a uranium-contaminated alluvial aquifer was simulated in a series of forced-gradient single- and multi-well injection-extraction tests. The injection fluid (river water) was traced with natural and artificial tracers that included halides, fluorobenzoates, lithium, and naphthalene sulfonate to characterize the potential mass transport mechanisms of uranium and vanadium. The extraction fluid (river water/groundwater mixture) was analyzed for the tracers, uranium, and vanadium. The results from the tracers indicated that matrix diffusion was likely negligible over the spatiotemporal scales of the tests as evident by nearly identical breakthrough curves of the halides and fluorobenzoates. In contrast, the breakthrough curves of lithium and naphthalene sulfonate indicated that sorption by cation exchange and sorption to organic matter, respectively, were potential mass transport mechanisms of uranium and vanadium. Uranium was mobilized in the saturated zone containing gypsum (gypsum-rich zone), the vadose zone (vadose-rich zone), and the saturated zone containing organic carbon (organic-rich zone) whereas vanadium was mobilized only in the saturated gypsum-rich zone. The mechanisms responsible for the mobilization of uranium and vanadium were likely dissolution of uranium- and vanadium-bearing minerals and/or desorption from the gypsum-rich zone, flushing of uranium from the vadose-rich zone, and desorption of uranium from the organic-rich zone due to the natural contrast in the geochemistry between the river water and groundwater. The experimental design of this study was unique in that it employed the use of multiple natural and artificial tracers coupled with a direct injection of native river water to groundwater. Here, these results demonstrated that natural recharge and flooding events at former mill tailing sites can mobilize uranium, and possibly vanadium, and contribute to persistent levels of groundwater contamination.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

A comparative study of discrete fracture network and equivalent continuum models for simulating flow and transport in the far field of a hypothetical nuclear waste repository in crystalline host rock

One of the major challenges of simulating flow and transport in the far field of a geologic repository in crystalline host rock is related to reproducing the properties of the fracture network over the large volume of rock with sparse fracture characterization data. Various approaches have been developed to simulate flow and transport through the fractured rock. The approaches can be broadly divided into Discrete Fracture Network (DFN) and Equivalent Continuum Model (ECM). The DFN explicitly represents individual fractures, while the ECM uses fracture properties to determine equivalent continuum parameters. In this paper, we compare DFN and ECM in terms of upscaled observed transport properties through generic fracture networks. The major effort was directed on making the DFN and ECM approaches similar in their conceptual representations. This allows for separating differences related to the interpretation of the test conditions and parameters from the differences between the DFN and ECM approaches. The two models are compared using a benchmark test problem that is constructed to represent the far field (1 × 1 × 1 km 3 ) of a hypothetical repository in fractured crystalline rock. The test problem setting uses generic fracture properties that can be expected in crystalline rocks. The models are compared in terms of the: 1) effective permeability of the domain, and 2) nonreactive solute breakthrough curves through the domain. The principal differences between the models are mesh size, network connectivity, matrix diffusion and anisotropy. We demonstrate how these differences affect the flow and transport. Finally, we identify the factors that should be taken in consideration when selecting an approach most suitable for the site-specific conditions.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2018

PLUMECALC, Version 2.3.x

PLUMECALC is a post processor to random walk particle tracking simulations of subsurface solute transport. PLUMECALC takes results of a single particle tracking run without retention processes and incorporates the effects of matrix diffusion or other retention processes for a user-specified solute source.

Dash, Zora↗
DOE OSTI2019

Numerical Evaluation of Unsaturated-Zone Flow and Transport Pathways at Rainier Mesa, Nevada

Sixty-one underground nuclear tests were conducted at Rainier Mesa at the Nevada National Security Site between 1957 and 1992. The mesa includes the highest-elevation areas at the Nevada National Security Site and experiences some of the highest estimated infiltration rates. Perched water is ubiquitous in many of the tunnels used for testing. The presence of sloping layers with strongly varying degrees of fracturing and overall permeability suggests that some infiltrating water may be diverted laterally along higher-permeability layers, thereby reducing overall percolation flux and radionuclide transport through the low-permeability testing horizons but also potentially transporting radionuclides laterally away from the test locations. A high-resolution, two-dimensional, cross-sectional model was created to simulate water and particle movement along potential lateral flow paths and to investigate flow and transport paths within and surrounding the perched water bodies. The model results indicate that particles with starting locations within the low-permeability perching layers will follow dominantly vertical trajectories through the perching horizons and reach the water table in the underlying carbonate aquifer only after many thousands of years. In contrast, particles starting near ground surface above the testing horizon are mostly diverted laterally before reaching the testing horizon, except where nearby faults through the perching layers are present. Where the rock above the perching horizon is fractured, particles originating near ground surface can move laterally with relatively high transport velocities. Transport calculations for 3 H show that transport extent is limited by slow matrix flow, radioactive decay, and matrix diffusion in the fractured units.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Dissolved Organic Carbon 14C in Southern Nevada Groundwater and Implications for Groundwater Travel Times

Dissolved inorganic carbon (DIC) carbon-14 ( 14 C) ages must be corrected for complex chemical and physical reactions and processes that change the amount of 14 C in groundwater as it flows from recharge to downgradient areas. Because of these reactions, DIC 14 C can produce unrealistically old ages and long groundwater travel times that may, or may not, agree with travel times estimated by other methods. Dissolved organic carbon (DOC) 14 C ages are often younger than DIC 14 C ages because there are few chemical reactions or physical processes that change the amount of DOC 14 C in groundwater. However, there are several issues that create uncertainty in DOC 14 C groundwater ages including limited knowledge of the initial (A 0 ) DOC 14 C in groundwater recharge and potential changes in DOC composition as water moves through an aquifer. This study examines these issues by quantifying A 0 DOC 14 C in recharge areas of southern Nevada groundwater flow systems and by evaluating changes in DOC composition as water flows from recharge areas to downgradient areas. The effect of these processes on DOC 14 C groundwater ages is evaluated and DOC and DIC 14 C ages are then compared along several southern Nevada groundwater flow paths. Twenty-seven groundwater samples were collected from springs and wells in southern Nevada in upgradient, midgradient, and downgradient locations. DOC 14 C for upgradient samples ranged from 96 to 120 percent modern carbon (pmc) with an average of 106 pmc, verifying modern DOC 14 C ages in recharge areas, which decreases uncertainty in DOC 14 C A 0 values, groundwater ages, and travel times. The HPLC spectra of groundwater along a flow path in the Spring Mountains show the same general pattern indicating that the DOC compound composition does not change along this flow path. Although DOC concentration decreases from recharge-area to downgradient groundwater, the organic compounds are similar, indicating that DOC 14 C is unaffected by other processes such as microbial degradation. A small amount of organic carbon was leached from crushed volcanic and carbonate aquifer outcrop rock in rock-leaching experiments. The leached DOC was high in 14 C (75 pmc carbonate rocks, 91 pmc volcanic) suggesting that the leached DOC likely came from microbes in the rock samples. The small amount of DOC and high 14 C indicates that the amount of old organic carbon in these rocks is low so there should be minimal impact on groundwater DOC 14 C ages. Based on the results from this study, DOC 14 C ages do not require additional corrections. Several correction models were applied to DIC 14 C ages to correct for water-rock reactions along two carbonate and two volcanic flow paths and the corresponding travel times were compare to DOC 14 C travel times. The DOC 14 C travel times were hundreds to thousands of years shorter than uncorrected and corrected DIC 14 C travel times except for the upper section of one carbonate flow path. DOC 14 C travel times ranged from 400 to 5,400 years as compared to DIC 14 C that ranged from modern to 20,900 years. The DIC 14 C ages are greatly influenced by carbonate mineral and gas reactions and other processes such as matrix diffusion, isotope exchange, or adsorption, which are not always adequately accounted for in DIC 14 C groundwater age correction models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

Preliminary Interpretations from Fall 2018 Field Test at Ne’ot Hovav

Presentation conclusions: Conservative tracer transport was reasonably consistent with earlier tests between RH11c and RH11a; No attempt to model bentonite colloids because of minimal elevation of Al concentrations over background; Modeling confirms what we knew already for Cs and Ce; Cs transports as solute that sorbs in matrix (after diffusion into matrix); Ce transports as colloid that is rapidly filtered; Sr not modeled, but qualitatively appears consistent with combination of colloid and solute transport

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1972

The effect of alloy composition on the mechanism of stress-corrosion cracking of titanium alloys in aqueous environments

A detailed study was made of the relation between the size distribution of Ti3Al particles in a Ti-8Al alloy and the tensile properties measured in air and in saltwater. The size distribution of Ti3Al was varied by isothermal aging for various times at temperatures in the range 770 to 970 K (930 to 1290 F). The aging kinetics were found to be relatively slow. Quantitative measurements of the particle coarsening rate at 920 K (1200 F) showed good agreement with the predicted behavior for coarsening controlled by matrix diffusion, and suggested that the specific free energy of the Ti3Al alpha interface in negligible small. In all cases, the Ti3Al particles were sheared by the glide dislocations. It was concluded that there is a definite correlation between the presence of deformable Ti3Al particles and an alloy's susceptibility to aqueous stress corrosion cracking. Furthermore, the appearance of the surface slip lines and the dislocation substructure in deformed specimens suggest that the specific effect of the Ti3Al particles is to cause a nonhomogeneous planar slip character and an enhanced chemical potential of the slip bands.

Wood, R. A.↗
NASA NTRS1988

Analytical solutions of a class of multidimensional Fokker-Planck equations

Analytical representations of the solutions of Kolmogorov's forward (i.e., Fokker-Planck) and backward partial differential equations are obtained by employing Girsanov's (1960) transformation. The equations considered are restricted to those with the drift vector being the gradient of a function and the diffusion matrix being nonsingular and dependent upon time only.

Zhang, Weijian↗
NASA NTRS2006

Fuel Cells Utilizing Oxygen From Air at Low Pressures

A fuel cell stack has been developed to supply power for a high-altitude aircraft with a minimum of air handling. The fuel cell is capable of utilizing oxygen from ambient air at low pressure with no need for compression. For such an application, it is advantageous to take oxygen from the air (in contradistinction to carrying a supply of oxygen onboard), but it is a challenging problem to design a fuel-cell stack of reasonable weight that can generate sufficient power while operating at reduced pressures. The present fuel-cell design is a response to this challenge. The design features a novel bipolar plate structure in combination with a gas-diffusion structure based on a conductive metal core and a carbon gas-diffusion matrix. This combination makes it possible for the flow fields in the stack to have a large open fraction (ratio between open volume and total volume) to permit large volumes of air to flow through with exceptionally low backpressure. Operations at reduced pressure require a corresponding increase in the volume of air that must be handled to deliver the same number of moles of oxygen to the anodes. Moreover, the increase in the open fraction, relative to that of a comparable prior fuel-cell design, reduces the mass of the stack. The fuel cell has been demonstrated to operate at a power density as high as 105 W/cm2 at an air pressure as low as 2 psia (absolute pressure 14 kPa), which is the atmospheric pressure at an altitude of about 50,000 ft ( 15.2 km). The improvements in the design of this fuel cell could be incorporated into designs of other fuel cells to make them lighter in weight and effective at altitudes higher than those of prior designs. Potential commercial applications for these improvements include most applications now under consideration for fuel cells.

Cisar, Alan↗
NASA NTRS1956

Extrapolation techniques applied to matrix methods in neutron diffusion problems

A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.

Mccready, Robert R↗