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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 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Entanglement phase transitions in random stabilizer tensor networks

Here we explore a class of random tensor network models with “stabilizer” local tensors which we name random stabilizer tensor networks (RSTNs). For RSTNs defined on a two-dimensional square lattice, we perform extensive numerical studies of entanglement phase transitions between volume-law and area-law entangled phases of the one-dimensional boundary states. These transitions occur when either (a) the bond dimension D of the constituent tensors is varied or (b) the tensor network is subject to random breaking of bulk bonds, implemented by forced measurements. In the absence of broken bonds, we find that the RSTN supports a volume-law entangled boundary state with bond dimension D ≥ 3 where D is a prime number, and an area-law entangled boundary state for D = 2. Upon breaking bonds at random in the bulk with probability p, there exists a critical measurement rate p c for each D ≥ 3 above which the boundary state becomes area-law entangled. To explore the conformal invariance at these entanglement transitions for different prime D, we consider tensor networks on a finite rectangular geometry with a variety of boundary conditions, and extract universal operator scaling dimensions via extensive numerical calculations of the entanglement entropy, mutual information, and mutual negativity at their respective critical points. Our results at large D approach known universal data of percolation conformal field theory, while showing clear discrepancies at smaller D, suggesting a distinct entanglement transition universality class for each prime D. We further study universal entanglement properties in the volume-law phase and demonstrate quantitative agreement with the recently proposed description in terms of a directed polymer in a random environment.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Comprehensive Margin Assessment of the ASME Section III, Division 5, Class A Primary Load Design Rules

This report provides a comprehensive margin assessment of the ASME Section III, Division 5, Subsection HB, Subpart B rules for the design of high temperature reactor components against load controlled stress limits. These rules, often called the Code primary load design rules, provide protection against creep rupture and plastic collapse under steady conditions. The method adopted here is to compare the deterministic ASME design life, according to the primary load design rules, against the expected, statistical service life of the component, considering creep rupture as the relevant failure mode. The report provides the design margin of the ASME rules in terms of the probability of premature failure -- the probability that the component fails in service before reaching the ASME design life. To complete this assessment, the report describes the development of a statistical creep life assessment procedure accounting for variations in the component loading, material creep rate, and the material rupture time. This includes a novel method for correlating creep rupture data using a Gaussian process accounting for heat-to-heat variation in the Larson-Miller time-temperature parameter. The report applies the complete margin assessment process to quantify the design margin inherent in the ASME primary load design rules as a function of temperature for relevant component geometries and materials. The overall conclusion is that the ASME rules are conservative, providing designs with very small probability of premature failure, but some optimization of the Code design rules is possible to provide a more uniform design margin.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

The effect of initial conditions on mixing transition of the Richtmyer–Meshkov instability

We investigate the late-time Richtmyer–Meshkov instability (RMI) growth of sinuous perturbations on an air/sulphur hexafluoride interface (Atwood number, A ~ 0.67 ) subjected to a Mach 1.2 planar shock wave at Los Alamos National Laboratory's vertical shock tube facility. Interface perturbations are established using a novel membraneless technique where cross-flowing air and SF 6 separated by an oscillating splitter plate create a perturbed density interface. The interface formed has multi-modal features and residual small perturbations, however, a dominant mode is still noticeable. The late-time perturbation growths scale with $ka_0$ initial conditions (where $k$ is the wavenumber and $a_0$ is the initial amplitude of the dominant mode) as measured at the pre-shock interface. Past nonlinear models based on potential-flow theory, heuristic/interpolation approaches, Padé approximants and numerical simulations are evaluated against present experimental results. Accounting for an explicit $ka_0$ dependence in Sadot et al.'s model, we propose an empirical rational function that captures the asymptotic behaviour of perturbation growth for a broad range of initial conditions ( 0.30 $≤ ka_0 ≤$ 0.86 ). The onset of mixing transition and its initial condition dependence are investigated with respect to the minimum state criterion ($Re =$ 1.6 $\times$ 10 5 ) for unsteady flows by Zhou. Earlier mixing transitions for higher ka0 initial conditions are noted from local and global Reynolds number estimates which are corroborated by the existence of an inertial sub-range and formation of mixing regions indicating the physical significance of the minimum state criterion in RMI flows. The transition is accompanied by the increasing teapot-like appearance of joint probability density functions of $p–q$ (invariants of the reduced velocity gradient tensor), establishing the technique as a useful tool for turbulence detection in two-dimensional diagnostics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Watch and learn—a generalized approach for transferrable learning in deep neural networks via physical principles

Transfer learning refers to the use of knowledge gained while solving a machine learning task and applying it to the solution of a closely related problem. Such an approach has enabled scientific breakthroughs in computer vision and natural language processing where the weights learned in state-of-the-art models can be used to initialize models for other tasks which dramatically improve their performance and save computational time. Here we demonstrate an unsupervised learning approach augmented with basic physical principles that achieves fully transferrable learning for problems in statistical physics across different physical regimes. By coupling a sequence model based on a recurrent neural network to an extensive deep neural network, we are able to learn the equilibrium probability distributions and inter-particle interaction models of classical statistical mechanical systems. Our approach, distribution-consistent learning, DCL, is a general strategy that works for a variety of canonical statistical mechanical models (Ising and Potts) as well as disordered interaction potentials. Using data collected from a single set of observation conditions, DCL successfully extrapolates across all temperatures, thermodynamic phases, and can be applied to different length-scales. This constitutes a fully transferrable physics-based learning in a generalizable approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Stochastic Home Energy Management Systems with Varying Controllable Resources

This paper studies the performance of a model predictive control (MPC) algorithm in a home energy management system (HEMS) as the set of controllable resources varies and under both a constant and a time-of-use (TOU) electricity price structure. The set of controllable resources includes residentially-owned photovoltaic (PV) panels, a home battery system (HBS), an electric vehicle (EV), and a home heating, ventilation, and air conditioning (HVAC) system. The HEMS optimally schedules the set of controllable resources given user preferences such as indoor thermal comfort and electricity cost sensitivity. The home energy management system is built on a chance constrained, MPC-based algorithm, where the chance constraint ensures the indoor thermal comfort is satisfied with a high probability given uncertainty in the outdoor temperature and solar irradiance forecasts. Simulation results for varying sets of controllable resources under two different electricity price structures demonstrate the variation in the HEMS control with respect to HBS operation, electricity cost, and grid power usage.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2023

Insights into the physiological and genomic characterization of three bacterial isolates from a highly alkaline, terrestrial serpentinizing system

The terrestrial serpentinite-hosted ecosystem known as “The Cedars” is home to a diverse microbial community persisting under highly alkaline (pH ~ 12) and reducing (Eh < -550 mV) conditions. This extreme environment presents particular difficulties for microbial life, and efforts to isolate microorganisms from The Cedars over the past decade have remained challenging. Herein, we report the initial physiological assessment and/or full genomic characterization of three isolates: Paenibacillus sp. Cedars (‘Paeni-Cedars’), Alishewanella sp. BS5-314 (‘Ali-BS5-314’), and Anaerobacillus sp. CMMVII (‘Anaero-CMMVII’). Paeni-Cedars is a Gram-positive, rod-shaped, mesophilic facultative anaerobe that grows between pH 7–10 (minimum pH tested was 7), temperatures 20–40°C, and 0–3% NaCl concentration. The addition of 10–20 mM CaCl 2 enhanced growth, and iron reduction was observed in the following order, 2-line ferrihydrite > magnetite > serpentinite ~ chromite ~ hematite. Genome analysis identified genes for flavin-mediated iron reduction and synthesis of a bacillibactin-like, catechol-type siderophore. Ali-BS5-314 is a Gram-negative, rod-shaped, mesophilic, facultative anaerobic alkaliphile that grows between pH 10–12 and temperatures 10–40°C, with limited growth observed 1–5% NaCl. Nitrate is used as a terminal electron acceptor under anaerobic conditions, which was corroborated by genome analysis. The Ali-BS5-314 genome also includes genes for benzoate-like compound metabolism. Anaero-CMMVII remained difficult to cultivate for physiological studies; however, growth was observed between pH 9–12, with the addition of 0.01–1% yeast extract. Anaero-CMMVII is a probable oxygen-tolerant anaerobic alkaliphile with hydrogenotrophic respiration coupled with nitrate reduction, as determined by genome analysis. Based on single-copy genes, ANI, AAI and dDDH analyses, Paeni-Cedars and Ali-BS5-314 are related to other species (P. glucanolyticus and A. aestuarii, respectively), and Anaero-CMMVII represents a new species. The characterization of these three isolates demonstrate the range of ecophysiological adaptations and metabolisms present in serpentinite-hosted ecosystems, including mineral reduction, alkaliphily, and siderophore production.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Data‐Driven Safety Risk Prediction of Lithium‐Ion Battery

Abstract Inevitable safety issues have pushed battery engineers to become more conservative in battery system design; however, battery‐involved accidents still frequently are reported in headlines. Identifying, understanding, and predicting safety risks have become priorities to further accelerate technology and industry development. However, diverse loading scenarios, significantly varied stress‐induced short circuit mechanisms, and highly coupled mechanical–electrochemical safety behaviors have remained grand challenges. Herein, the safety risk is termed as the probability of the mechanical triggering of an internal short circuit, to reflect the safety related behaviors of lithium‐ion batteries. Based on a mechanical model and experimental results, a sufficient dataset is generated consisting of strain states and their corresponding safety risks, covering both cylindrical and pouch cells, various states of charges, and loading conditions. Machine‐learning tools combined with the established finite element mechanical model are applied to predict the safety risks of the cells. The results achieve a high level of accuracy on the test data (the relative error of the average short circuit prediction deviation is less than 6.2%.). This work underpins the safety risk concept and highlights the promise of physics combined with data‐driven modeling methodology to predict the safety behaviors of energy storage systems.

Jia, Yikai↗
DOE OSTI2025

The no boundary density matrix

We discuss a no-boundary proposal for a subregion of the universe. In the classical approximation, this density matrix involves finding a specific classical solution of the equations of motion with no boundary. Beyond the usual no boundary condition at early times, we also have another no boundary condition in the region we trace out. We can find the prescription by starting from the usual Hartle-Hawking proposal for the wavefunction on a full slice and tracing out the unobserved region in the classical approximation. We discuss some specific subregions and compute the corresponding solutions. These geometries lead to phenomenologically unacceptable probabilities, as expected.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Ride-hailing and taxi versus walking: Long term forecasts and implications from large-scale behavioral data

Introduction: Although ride-hailing and taxi trips can potentially reduce single-occupant vehicle trips and auto ownership, they can also replace pedestrian trips. Because physical activity is associated with improved health outcomes, the extent to which ride-hailing and taxi travel captures walking's mode share is of interest to policymakers. Methods: Based on large-scale behavioral data from the 2017 U.S. National Household Travel Survey, this paper reports on the development of a full Bayesian logistic regression model for determining the mode split between (1) ride-hailing and taxi and (2) walk while accounting for unobserved heterogeneity. The results from the stand-alone model inform two longer-term travel forecasting scenarios: a) higher risk of walk trips converting to ride-hailing and taxi, specifically in the future with high prevalence of automated vehicles, b) higher probability of such trips remaining as walking. Results: The results revealed that some of the important characteristics that increase the likelihood of a traveler using the ride-hailing and taxi mode versus walking include having a longer trip, using a smartphone to access the internet, having an interest in technologies, having a medical condition, and living in a metropolitan area with rail access. Further, the results from the first scenario suggest that an overall increase of up to 2.9% in the ride-hailing and taxi mode share may be expected. The second scenario shows that between 68% and 76% of ride-hailing and taxi trips could be diverted to walking if supportive pedestrian infrastructure were provided in the case study locations. The planning process can be adapted to consider not only congestion, crash, and emissions impacts of such shifts but also the effects of a loss of physical activity. Conclusions: The study findings show how the ride-hailing and taxi mode competes with walking. Further, the findings enable planners to update their regional travel forecasting models; policy makers can thus encourage active travel by prioritizing pedestrian infrastructure investments that may divert ride-hailing and taxi trips to walking. However, equity should be a key consideration to ensure that addressing the competition between these two modal choices does not hinder the provision of pedestrian facilities in communities that depend on walking.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2024

Ground Heat Flux Reconstruction Using Bayesian Uncertainty Quantification Machinery and Surrogate Modeling

Ground heat flux (G 0 ) is a key component of the land-surface energy balance of high-latitude regions. Despite its crucial role in controlling permafrost degradation due to global warming, G 0 is sparsely measured and not well represented in the outputs of global scale model simulation. In this study, an analytical heat transfer model is tested to reconstruct G 0 across seasons using soil temperature series from field measurements, Global Climate Model, and climate reanalysis outputs. The probability density functions of ground heat flux and of model parameters are inferred using available G 0 data (measured or modeled) for snow-free period as a reference. When observed G 0 is not available, a numerical model is applied using estimates of surface heat flux (dependent on parameters) as the top boundary condition. These estimates (and thus the corresponding parameters) are verified by comparing the distributions of simulated and measured soil temperature at several depths. Aided by state-of-the-art uncertainty quantification methods, the developed G 0 reconstruction approach provides novel means for assessing the probabilistic structure of the ground heat flux for regional permafrost change studies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Similarity properties in capacitive radio frequency plasmas with nonlinear collision processes

Abstract Similarity laws (SL) are essential for correlating the characteristics of plasmas at different dimensional scales, which have been validated for radio frequency (rf) discharges at low pressure but under limited conditions. In this work, we evaluate the effects of nonlinear collisions (e.g. stepwise ionization) on the similarity properties in capacitive rf discharges in argon across a wide range of pressure regimes via fully kinetic particle-in-cell simulations. The SL scalings of fundamental discharge parameters, e.g. the electron power absorption and electron energy probability function, are examined with and without nonlinear collisions, respectively. Without the nonlinear collisions, the similarity scalings are found to be rigorously valid. When the nonlinear collisions are considered, the similarity properties in rf discharges still exist approximately, which indicates that the violations caused by the nonlinear collisions are still minor in the studied cases. The reasons for the effectiveness of SL scalings with nonlinear collision processes are also discussed. The results from this study confirmed the validity of similarity transformations with more complex reaction kinetics, which may promote the applicability of the SL scalings for the design and fabrications of plasma devices.

Physics↗
DOE OSTI2020

Particle-number distribution in large fluctuations at the tip of branching random walks

Here, we investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times t , focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position, but still within a distance smaller than the diffusion radius ~$\sqrt{t}$. Our approach consists in a study of the generating function $G_{Δx}(λ) = Σ_n$ ${λ^n}p_n(Δx)$ for the probabilities $p_n(Δx)$ of observing $\textit{n}$ particles in an interval of given size $Δ\textit{x}$ from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-$Δ\textit{x}$ limits, we find that the mean number of particles in the interval grows exponentially with $Δ\textit{x}$, and that the generating function obeys a nontrivial scaling law, depending on $Δ\textit{x}$ and λ through the combined variable $[Δx — f(λ)]^3 / Δx^2$, where $\textit{f}$(λ) ≡ – ln(1 – λ) – ln [– ln(1 – λ)]. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large Δ$\textit{x}$. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Applying Quantum Tomography to Hadronic Interactions

A proper description of inclusive reactions is expressed with density matrices. Quantum tomography reconstructs density matrices from experimental observables. We review recent work that applies quantum tomography to practical experimental data analysis. Almost all field-theoretic formalism and modeling used in a traditional approach is circumvented with great efficiency. Tomographically-determined density matrices can express information about quantum systems which cannot in principle be expressed with distributions defined by classical probability. Topics such as entanglement and von Neumann entropy can be accessed using the same natural language where they are defined. A deep relation exists between separability, as defined in quantum information science, and factorization, as defined in high energy physics. Factorization acquires a non-perturbative definition when expressed in terms of a conditional form of separability. An example illustrates how to go from data for momentum 4-vectors to a density matrix while bypassing almost all the formalism of the Standard Model.

Martens, J. C.↗
DOE OSTI2023

Modeling Exhaust-Generated Aerodynamic Pressure Loads on Airfield Matting Repair Systems

Airfield matting systems are commonly used for rapid repair of damaged runways to facilitate continuity of critical operations. Under normal service conditions, matting repair systems are subject not only to wheel loads exerted by airfield traffic but also to aerodynamic pressure loads resulting from high-speed, turbulent exhaust plumes produced by jets during taxi and take-off. Matting systems employed by the U.S. Air Force have been tested with respect to wheel loads, but probabilities of failure because of pressure loads produced by jet exhaust have not yet been established. This study presents a numerical approach for preliminary estimation of worst-case matting anchor forces resulting from exhaust-generated pressure loads. Using a two-dimensional computational fluid dynamics model, system behavior is evaluated by means of a parametric study of six system variables, of which the most significant are: (1) distance between engine and matting, (2) depth of cavity openings at matting edges, and (3) engine exhaust velocity. The results demonstrate that matting systems are likely to experience net uplift in typical service scenarios, driven by the combined effects of flow separation and cavity pressurization. Worst-case anchor pull-out forces, computed according to a tributary-area approach, are estimated to fall in the range of 130 lb (581 N) to 979 lb (4,350 N), depending on assumed load-sharing behavior among anchors and the size of the repair site. Field testing of instrumented matting systems during jet taxi and take-off sequences is recommended as the best next step toward understanding system behavior.

Engineering↗
DOE OSTI2023

Stochastic Framework for Optimal Control of Planetary Reentry Trajectories Under Multilevel Uncertainties

We present a novel stochastic optimal control framework that accounts for various types of uncertainties, with application to reentry trajectory planning. The formulation of the optimal trajectory control problem is presented in the context of an indirect method where a functional objective associated with the terminal vehicle speed is to be minimized. Uncertain input parameters in the optimal trajectory control model, including aerodynamic parameters and initial and terminal conditions, are modeled as aleatory random variables, while the statistical parameters of these aleatory distributions are themselves random variables. The parametric and model uncertainties are simultaneously propagated through an extended polynomial chaos expansion (EPCE) formalism. Several metrics are described to evaluate response statistics and presented as insightful tools for robust decision making. Specifically, the response probability density function (PDF) reflecting influence of both epistemic and aleatory uncertainties is obtained. By sampling over the random variables representing model error, an ensemble of response PDFs is generated and the associated failure probability is estimated as a random variable with its own polynomial chaos expansion. Besides, the sensitivity index functions of response PDF with respect to the statistical parameters are evaluated. Coupling parametric and model uncertainties within the EPCE framework leads to a robust and efficient paradigm for multilevel uncertainty propagation and PDF characterization in general optimal control problems.

Engineering↗
DOE OSTI2024

DECOVALEX-2023: Task A Final Report

Task A, also known as HGFrac, examines the fracturing processes that may occur in the Callovo-Oxfordian claystone (COx) in the context of the high-level (HLW) and intermediate-level long-lived (ILW-LL) radioactive waste repository in France. Understanding these processes and improving numerical models to reproduce them will aid in the design, optimization, and safety of the repository. Heat and gas fracturing are studied in two independent subtasks following a stepwise approach: laboratory tests/benchmark exercises, in-situ experiment, and finally, an application case. The in-situ heater experiment aimed to thermally induce a hydraulic fracture; temperature and pore pressure were monitored to detect any evidence of fracturing. The in-situ gas injection experiment aimed to study the effect of the stress orientation and gas injection kinetics on the gas fracturing process. The occurrence of fracturing was monitored by gas pressure measured in the injection interval. In both experiments, the excavation-induced fracture network around the heater/injection boreholes played an important role in the reduction of the compressive stress state, leading to both a tensile and shear failure response of the COx. During the first two years of the project, the research teams working on each task developed and/or proposed numerical approaches for reproducing the occurrence of fracturing in the in-situ experiments. The failure criteria were defined by reproducing the measurements from laboratory extension tests for the heat fracturing subtask. In the gas fracturing subtask, their approaches were used for simulating several benchmark exercises, and an inter-comparison between models was carried out. In both tasks, the developed approaches were compared with a simplified approach considering poro-elasticity for the mechanical behaviour of the COx. Most of the developed approaches are based on a continuous medium that takes into account variations in hydraulic properties due to mechanical degradation, such as plastic deformation or damage. Other approaches implicitly modelled weak planes or embedded discontinuities to reproduce fracture propagation. The potential for fracture initiation was also studied through of a discrete approach. In the second half of the project, the research teams mainly focused on interpretative modelling of two in-situ experiments and a blind prediction exercise to test their respective approaches. The models developed by the research teams were also applied at the repository scale to evaluate fracture initiation in a case study under vi unfavourable conditions, particularly in terms of spacing between High-Level Waste cells. The results showed that the poro-elasticity approach could be an efficient tool for understanding the main processes occurring in the COx. One example is the explicit representation of the excavation-induced fracture network around the boreholes, which yielded acceptable results compared to the measurement data. However, advanced approaches were needed to evaluate the potential increase of the excavation-induced fracture network extend and better understand fracture initiation. The stress analyses carried out by the teams revealed that hydraulic boundary conditions had a strong impact on fracture initiation in the heater experiment. Furthermore, in most cases, the results required higher pore pressure increments to reach fracturing than those measured in the experiment. This implies that the measurements may have been biased by the packer’s capacity to fully isolate the piezometric chambers, leading to lower pressures. On the contrary, there was no agreement on the fracturing mode, as some reported either shear or tensile fracturing, while others reported a combination of the two modes. In the case of gas fracturing, the research teams were limited to the comparison of a single point, which complicated their task. Nonetheless, the numerical results were able to reproduce the measurements and capture processes such as longitudinal gas flow through the excavation-induced fracture network, as suggested by some evidence in the observation piezometric chambers. The numerical models also agreed with the measurements in the sense of higher probability of developing along the injection borehole than radially towards the sound rock. The approaches developed by the research teams showed that they are capable of analysing and reproducing fracture initiation in the COx. However, areas of future work should focus on the fracture propagation and fracture aperture, which were out of the scope of this task. To this end, additional data must be gathered for the parameter characterisation and validation of the numerical models. Nonetheless, various approaches showed promising results as they were able to reproduce fracture development under certain conditions.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2023

Development of sulfonated carbon-based solid-acid catalysts derived from biorefinery residues and biomass ash for xylan hydrolysis

In this study, biorefinery residues and biomass ash were used to develop sulfonated carbon-based solid-acid catalysts to enhance xylan hydrolysis. Here, metals extracted from the biomass ash were impregnated into a carbon support prepared from biorefinery residues to synthesize sulfonated ash-impregnated carbon-based catalyst. Metals present in the ash most probably are responsible for the increased acid density in the catalyst. Using this catalyst, 11.4 % xylose yield was produced, an increase of 171 % over the yield obtained from a catalyst without ash impregnation. Using a xylan-to-catalyst ratio of 1:2, 72.9 % xylose yield was obtained, comparable to using 0.5 wt% sulfuric acid under the same reaction condition. Moreover, this catalyst significantly inhibited xylose dehydration, only forming 11.3 % furfural compared to that generated using dilute acid. Hence, these results show the potential for using biomass ash as a metal source to increase the number of acid sites in solid-acid catalysts used for biomass pretreatment.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗