PNNL-13501 - Pacific Northwest National Laboratory

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Invariant Discretization Methods for n-Dimensional Nonlinear Computational Reactive Transport Models Study Control Number: PN99034/1362 Joseph S. Oliveira The objective of this research has been to investigate, develop, and apply a new set of scale invariant, discrete algebraic and geometric symmetry preserving algorithms to the problem of effectively defining and remapping multiscale, multidimensional hybrid computational meshes. This effort provides a critical part of the applied mathematical infrastructure needed to solve complex computational problems. The primary goal is to develop new combinatorial methods for generating and preserving computationally, symmetry-invariant, coordinate-free, three-dimensional and n-dimensional spatial discretizations, on mixed (structured and/or unstructured), hybrid, moving, and stochastic grids. Project Description We are building the mathematical tools to create invariant grid discretizations of adaptive moving hybrid grids that are coordinate free. We are using algebraic combinatorial, combinatorial geometric, and combinatorial topological invariants to remap, reconnect, and renormalize multiscale, multidimensional, adaptive, moving hybrid grids. In each of these cases, scalars, vectors, and tensors are remapped by sets of coordinatefree transformations between logical and real physical nspace. This emerging computational technology implies that the boundaries (external and internal) of the computational domain must be canonically based on the invariance of the solution(s) to the partial differential equations that model the physical problem domain. These may include, but are not limited to: material interfaces, liquid-gas interfaces, contact discontinuities, and shock fronts. Furthermore, the grid covering the computational domain (subdomains) has to be constructed using information about the position of the free moving boundary and the information about the invariance of the solution to the system as a whole. Lastly, finite difference schemes have to be reformulated in such a way that they do not depend on the components of vectors and tensors, but rather, the discrete analogs of coordinate-free differential forms. The coefficients of the finite difference equations should depend only on the coordinate-free characteristics of the grid, lengths of the edges, nearest neighbor point distributions, areas of faces, volumes of mesh cells, and angles between edges, and not specific vector coordinate representations. 132 FY 2000 <strong>Laboratory</strong> Directed Research and Development Annual Report Approach The violation of physical symmetry in the discrete geometric representation of the partial differential equations is a principal cause of the numerical instabilities that lead to large numerical errors. If this type of problem is to be eliminated, we must generate computationally invariant hybrid discretization methods (threedimensionally, mixed [structured and unstructured] grids, and stable three-dimensional numerical schemes built on these grids) for obtaining stable, high-accuracy numerical solutions to the partial differential equations that preserve the inherent symmetry of the physical domain space. The overall goal is to generalize these methods to ndimensions, multiscale, hydro-transport physics models. The extension to n-dimensions will allow us to more accurately model physical parameter space by representing each algebraic parameter by a corresponding geometric dimension. The geometric invariance that is preserved by the grid discretization of the physical continuum is spatial symmetry, which is realized as algebraic invariant symmetry. For example, the preservation of physically based fluid flow symmetries—associated with multiscale, multiphase advective and reactive diffusive fluid flows in subsurface transport, magnetohydrodynamic flow, combustion systems, reactive atmospheric transport chemistry, and intra-inter cellular communications—must remain scale invariant. The divergence from spherical and rotational symmetries due to discretization errors can lead to large errors in the discretized systems of ordinary differential equations with large convergence ratios. The
loss of local and global curvature data will introduce large scaling errors. Also, the uncertainty as to whether a nonsymmetric result is due to numerical errors or to the physical continuum model severely limits our predictive capabilities and our basic understanding of the dynamics of coupled transport phenomena. For the linearized equations of mathematical physics, there exists a well-developed theory of finite difference schemes. These schemes are based on three fundamental concepts: 1) truncation error analysis, 2) numerical stability analysis, and 3) convergence, which follows from an estimate of the truncation error and the stability of the numerical scheme. It is possible to extend truncation error measures to nonlinear equations, but investigating the numerical stability of methods for solving such equations is much more complex. A general mathematical theory of numerical stability for threedimensional advection-diffusion equations does not currently exist. The hope of obtaining an n-dimensional characterization requires that we first determine the “observability” and identification of the symmetry invariants of the partial differential equations state variable representation, given as an n-dimensional manifold. By carefully examining the parameter space data of both the time and phase space domains of quasilinear and nonlinear equations in terms of their linear equation analogs, we can begin to develop identifying measures of stability and instability. We will begin to develop n-dimensional grids to discretize the multiscale, multidimensional energy surfaces of phase space representations of these equations. Given that there is currently no mathematical theory for obtaining quantitative a priori estimates for the accuracy of finite difference approximations to full three-dimensional (and generalized to n-dimensional) nonlinear hydro-transport equations, we will initially use qualitative multidimensional grid methods to derive approximate measures of the time and phase space domains of multiscale hydro-transport physics domains. Therefore, it is natural to require that the invariant grid discretized models should preserve the physical symmetry of the physical continuum model by maintaining both accurate measures of local and global curvature, together with the local algebraic symmetry groups that provide us with a Lie Group symmetry invariant based definition of “adequate numerical approximations,” where “adequate” still needs to be defined. Preservation of spatial symmetry is one of the manifestations of the more general notion of invariance of finite difference methods, related to the preservation of group properties (up to isometric-isomorphisms) of the continuum equations and both local and global curvature symmetry. The theory of Lie Groups, which has been a classical tool for the analysis of the fundamental symmetries of continuum ordinary differential equations for over 100 years, has been used recently to construct finite difference schemes with symmetry preserving group invariance in one-dimensional (Vorodnitsyn 1994). A number of results also are available for three-dimensional advection-diffusion equations in Eulerian form (Leutoff et al. 1992). Extending these approaches to two- and three-dimensional Lagrangian advection-diffusiontransport equations will lead to significant improvements in our simulation capabilities. Results and Accomplishments The principal investigators have continued to investigate and apply the Fowler, Oliveira, and Trease (Millennium) remapping algorithm to three-dimensional particle transport in a fluid that interacts with a visco-elastic membrane. (See examples in figures below.) We have demonstrated that the Millennium algorithm has a nearoptimal space and time complexity, that is O (n log (n)) and preserves the underlying invariant SO(3) symmetry of this hybrid grid by faithfully remapping coordinate-free representations of sets of scalars, vectors, and tensors. We can show that each of the measure-preserving maps are symmetry invariant with respect to the SO(3) and the O(3) groups and that the computational efficiency of the intersection algorithm depends on the SO(n) invariance of the spherically bounded search space of polyhedral face intersections. The Malard-Oliveira remapping algorithm has a conjectured computational amortized complexity of O (n). Significant improvements over the original algorithm design have been made, which is described as follows: During the first year of this project, we developed a novel remapping algorithm, whose expected run-time was potentially linear in the number of pairs of overlapping elements. Our second year goal for refining and implementing the algorithm was to initially develop and implement an abstract data type for dynamic list representations that would be more computationally effective on nonuniform memory access multiprocess architectures. Figure 1 shows the remapping time for all three variants. The remapping time excludes both the input time and the time needed to compute adjacency relations between elements of the same grid. The runtime of the bridge variant appears more uniform for large problems than the other two variants. A novel remapping algorithm has been simplified, implemented using abstract data types built on top of freely available software, and benchmarked on a Silicon Computational Science and Engineering 133
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Laboratory Directed Research and De
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Laboratory Directed Research and De
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Computational Science and Engineeri
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Policy and Management Sciences Asse
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Introduction Department of Energy O
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5. After reviews by the Technical a
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• The Technical Council peer revi
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methods applicable to combustion, s
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Summary Each national laboratory mu
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Study Control Number: PN0004/1411 A
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Table 1. Positive and negative ions
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m/z Arbitrary Units Figure 1. Mass
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introduce low-volatility compounds
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arrangement, but the charge is reta
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two liquid chromatographic gradient
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Integrated Microfabricated Devices
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Matrix Assisted Laser Desorption/Io
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Molecular Beam Synthesis and Charac
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temperature distribution of the des
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expected to result in significant a
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Study Control Number: PN99069/1397
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Seuss DT and KA Prather. 1999. “M
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Analysis of Receptor-Modulated Ion
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laboratory (Lu and Xie 1997; Lu et
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Automated DNA Fingerprinting Microa
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Comparison of 4 Xanthomonas Isolate
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Approach Hematopoietic (bone marrow
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Anti-Human lgG Human lgG Protein G
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Characterization of Global Gene Reg
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PCR products for immobilization on
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identify novel proteins of importan
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Immunoprecipitaton of tyrosinephosp
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Coupled NMR and Confocal Microscopy
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In the optical microscopy image, th
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Development and Applications of a M
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Figure 3. Hepa-1 cells undergoing a
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cellular processing (such as post-t
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8. Initial demonstration of an off-
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expression systems for several year
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accumulation of protein products, i
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Fungal Conversion of Agricultural W
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Fungal Conversion of Waste Glucose/
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Summary and Conclusions We demonstr
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are shown in Figure 1(a) and 1(b),
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MARC Input initial m esh and result
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(Johnson 1999; Colligan 1999; Gould
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Modeling of High-Velocity Forming f
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Global divergence error 1 10 -14 0
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that users were unlikely to appreci
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Earth System Science
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the lateral extent of the plume so
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Biogeochemical Processes and Microb
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Analyses of populations of viable a
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The first task was to establish fiv
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purged for 2 minutes. The gas sampl
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developing code architecture to min
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Berkowitz, CM. June 2000. “Atmosp
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environmental monitoring, 2) portab
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corresponded to a current density o
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Enhancing Emissions Pre-Processing
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Environmental Fate and Effects of D
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nitrogen and unpredictable eutrophi
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Figure 1. Ordinary kriging of 2 m d
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precipitation of calcite occurred i
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Interrelationships Between Processe
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phenanthrene resistant fraction con
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injection of an immiscible gas phas
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still some key kinetic issues that
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Application of a Crop Model The res
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The Nature, Occurrence, and Origin
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Particle number concentration, 1/cm
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geometry optimized AlOOH and FeOOH
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allowable parameters using thermody
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TCE + 3H + + 3Fe 2+ Fe)=> acetylene
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Use of Shear-Thinning Fluids for In
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concentration of 0.5% AMX was the m
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Roberts AL, LA Totten, WA Arnold, D
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tetra-scale computing, envisioned a
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Energy Technology and Management
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Figure 2. Setup for the second expe
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phase. The algorithms will be teste
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[Pb-212]/[Pb-212] final 100 10 1 0.
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Plasma Particle Dielectric Fiber El
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(species, sub-species, and sex), sp
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Figure 3. Relative risk of bone and
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Development of a Preliminary Physio
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Development of a Virtual Respirator
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Figure 3. Use of the chest cavity a
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Examination of the Use of Diazolumi
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Indoor Air Pathways to Nuclear, Che
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Source Building Release Observed re
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To illustrate the potential impact
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Non-Invasive Biological Monitoring
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Materials Science and Technology
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Figure 1. (a) - (c) Successive magn
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Component Fabrication and Assembly
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Development of a Low-Cost Photoelec
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elaxations, indicating the adequacy
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Promising chemical durability, as m
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Study Control Number : PN98028/1274
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High Power Density Solid Oxide Fuel
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Hybrid Plasma Process for Vacuum De
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Matson DW, PM Martin, WD Bennett, J
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We showed the proof-of-principle ap
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Bandgap Energy (eV) 2.9 2.85 2.8 2.
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Ultrathin Electrolyte Test Results
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Detailed Investigations on Hydrogen
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identified low energy step structur
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Development of Novel Photo-Catalyst
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-2 -1 Energy (eV) 0 1 2 Cu2O SrTiO3
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Here, we perform adsorption and des
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exceed those in the all-metal devic
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Intensity (a.u.) 0 100 200 300 400
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integrated voltage (V) 0.10 0.08 0.
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The first step was to implement thi
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Spontaneous Formation of Cu2O Nano-
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Actinide Chemistry at the Aqueous-M
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Figure 5. Close-up atomic force mic
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to 300°C and 400°C. Unfortunately
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Summary and Conclusions Transmutati
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Advanced Calibration/Registration T
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Figure 1a. Violet filter Figure 1b.
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Autonomous Ultrasonic Probe for Pro
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containing nitrous oxide, an optica
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Chemical Detection Using Cavity Enh
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Figure 3 is a color rendered simula
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appropriate study has been performe
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Exploitation of Conventional Chemic
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Figures 3 and 4 illustrate the impa
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enefits of using a modified spread-
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Single Channel Signal 0.0025 0.0020
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Therefore, in order to extract the
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A next-generation technology is nee
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Wind Dir. Figure 2. Negative ion co
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Volumetric Imaging of Materials by
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amplitude and peak frequency change
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Figure 3. 13 C NMR spectrum of corn
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Approach Two flow loops, one that o
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Modification of Ion Exchange Resins
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Permanganate Treatment for the Deco
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minimized by medium exchange. After
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Figure 1. Thermogravimetric analysi
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Validation and Scale-Up of Monosacc
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Concentration (g/100g solution) Con
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Analysis of High-Volume, Hyper-Dime
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ecent figure of merit values. Shoul
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Probabilistic Methods Development f
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a) Traditional PCA-based process co
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Acronyms and Abbreviations 2-D two-
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PNNL-13501 - Pacific Northwest National Laboratory

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