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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 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A Robust Numerical Treatment of Solid-Phase Diffusion in Pseudo Two-Dimensional Lithium-Ion Battery Models

Solid-phase diffusion in active materials of lithium-ion batteries significantly affects charging and safety-related behavior of lithium-ion batteries. Therefore, it is essential to develop an efficient and robust numerical algorithm for solving solid-phase diffusion equations in physics-based battery models. In this work, we discuss the origins of numerical instabilities that can occur when solving the solid-phase diffusion equations using iterative methods. Then, in order to resolve such issues, we propose a simple numerical treatment to the surface flux term of discretized solid-phase diffusion equations. To demonstrate its numerical robustness, the proposed method is implemented into a pseudo two-dimensional (P2D) physics-based battery model and simulations are conducted at wide ranges of operating conditions. Even with extremely poor initial guesses for the Li+ concentrations of the active materials, computations using the proposed method do not diverge and the their computational speeds are comparable to those with conventional initial guesses. Comprehensive tests of the proposed method are also performed with a dynamic current profile based on US06 driving profile and a multi-stage charging profile with very high initial C-rate (12C).

battery modeling↗
DOE OSTI2020

Simulation Analysis of Geometry and Material Effects for Dropkinson Bar

The reported research is motivated by the need to address a key issue affecting the Dropkinson bar apparatus. This unresolved issue is the interference of the stress wave reflected from the bar-beam boundary with the measurement of the stress-strain response of a material tested in the apparatus. The purpose of the wave beam that is currently connected to the bar is to dissipate the stress wave, but the portion of the wave reflected from the bar-beam boundary is still significant. First, we focused on understanding which parameters affect the reflected wave's arrival time at a strain gauge. Specifically, we used finite-element numerical simulations with the Sierra/SM module to study the effects of various bar-beam connection fixities, alternative wave beam materials, and alternative geometries of the Dropkinson bar system based on a monolithic design. The conclusion of this study is that a partial reflection always occurs at the bar-beam boundary (or, for a monolithic design, at a point where the bar geometry changes). Therefore, given a fixed total length of the bar, it is impossible to increase the reflected wave's arrival time by any significant amount. After reaching this conclusion, we focused instead on trying to minimize the energy of the reflected stress wave circulating up and down through the bar over a relatively long period of time (10 ms). Once again, we used numerical simulations with the Sierra/SM module to investigate the effects of various bar-beam connection fixities, alternative wave beam materials, and parameters of an asymmetric monolithic design of the bar-and-beam system. This study demonstrated that various parameters can significantly affect the energy of the wave reflections, with the difference between best and worst configurations being about one order of magnitude in terms of energy. Based on the obtained results, we conclude with concrete takeaways for Dropkinson bar users and propose potential directions for future research and optimization.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Challenges and Opportunities in Magnetospheric Space Weather Prediction

Abstract Space Weather is the study of the dynamics of the coupled Solar‐Terrestrial environment, as these dynamics impact technological systems and human activity. This paper reviews a selection of the advances, challenges, and new opportunities for magnetospheric space weather, and while the focus is on specific phenomena, many other aspects of space weather will have similar challenges and needs. As a scientific field with direct applications, the field of space weather is partly driven by imperatives from both policy and operations. We provide an introduction to some of the context in which the field exists and discuss how this might shape future developments and norms within the space weather enterprise. We briefly examine benchmarking, as a policy and operationally driven activity, as it provides immediate societal relevance and an opportunity to stretch scientific understanding. As numerical space weather prediction now becomes routine, and exascale computing is in the near future, we identify challenges relating to computational expense and big data, capturing and accounting for uncertainties, and specification of boundary conditions. Here, as with the observations supporting numerical space weather prediction, the key challenge lies in extending the lead time of predictions. We also discuss the role of data, particularly in regard to model validation and empirical modeling. Due to the growing societal impact of space weather, we also examine the relationships between space weather and its terrestrial counterpart and look at the importance of continuous evaluation, monitoring progress in predictive capability, and communication with researchers, forecasters, and end users.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Intrinsic Elastic Anisotropy of Westerly Granite Observed by Ultrasound Measurements, Microstructural Investigations, and Neutron Diffraction

Westerly granite (WG) has been generally accepted as an isotropic homogeneous rock. Here, we return to WG and observe significant elastic anisotropy using multidirectional ultrasonic sounding on spherical samples at pressures up to 400 MPa. Thermal treatment of WG leads to formation of microcracks that reduce elastic wave velocities and increase its elastic anisotropy. The 3D distribution of P-wave velocities at low pressure is close to orthorhombic symmetry. Application of hydrostatic pressure closes most of thermally induced microcracks and decreases elastic anisotropy of WG, but at high pressure the anisotropy is practically reversed compared to low pressure: maximum P-wave velocity direction at low pressures is near minimum velocity direction at high pressure and vice versa. To understand this effect, microstructures of the rock were investigated by optical and scanning electron microscopy. Preferred orientations of four major rock-forming minerals—quartz, orthoclase, plagioclase, and biotite—were measured by time-of-flight neutron diffraction, which confirms significant crystal alignment. All these data were used to numerically model anisotropic elastic properties of WG. It is shown that WG possesses weak intrinsic elastic anisotropy related mainly to the preferred orientation of feldspars formed during igneous crystallization. Observed microcracks are mostly related to the cleavage planes of feldspars and biotite, and thus also demonstrate preferred orientation. Higher preheating temperatures produce larger quantity of longer microcracks. A numerical model shows that these microcracks act against the weak intrinsic elastic anisotropy of WG, and define the elastic anisotropy at low pressures.

58 GEOSCIENCES↗
DOE OSTI2021

Inclusive prompt photon-jet correlations as a probe of gluon saturation in electron-nucleus scattering at small x

We compute the differential cross-section for inclusive prompt photon+quark production in deeply inelastic scattering of electrons off nuclei at small x (e + A DIS) in the framework of the Color Glass Condensate effective field theory. The result is expressed as a convolution of the leading order (in the strong coupling α s ) impact factor for the process and universal dipole matrix elements, in the limit of hard photon transverse momentum relative to the nuclear saturation scale Q s,A (x). We perform a numerical study of this process for the kinematics of the Electron-Ion Collider (EIC), exploring in particular the azimuthal angle correlations between the final state photon and quark. We observe a systematic suppression and broadening pattern of the back-to-back peak in the relative azimuthal angle distribution, as the saturation scale is increased by replacing proton targets with gold nuclei. Our results suggest that photon+jet final states in inclusive e + A DIS at high energies are in general a promising channel for exploring gluon saturation that is complementary to inclusive and diffractive dijet production. They also provide a sensitive empirical test of the universality of dipole matrix elements when compared to identical measurements in proton-nucleus collisions. However because photon+jet correlations at small x in EIC kinematics require jet reconstruction at small k ⊥ , it will be important to study their feasibility relative to photon-hadron correlations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Multi-fidelity modeling to predict the rheological properties of fiber suspensions

Unveiling the rheological properties of fiber suspensions is of paramount interest to many industrial applications like biofuel production. The 3D numerical simulations of the suspension of fibers are often computationally expensive and time-consuming. Machine learning methods such as neural networks can simplify the prediction of rheological behavior; however, they require a relatively large training data set. Multi -fidelity models, which combine high-fidelity data from numerical simulations and less expensive lower fidelity data from resources such as simplified physical equations, can lead to optimized predictions. Here, we focus on a neural network with two levels of fidelity, i.e., high and low fidelity networks. To produce high-fidelity data, we perform direct numerical simulations to model the fibers as one-dimensional inextensible slender bodies that obey the Euler- Bernoulli beam equation. The Navier-Stokes equations govern the suspended fluid, and an immersed boundary method is used to couple the fluid and solid motion. The low-fidelity data is produced by using constitutive equations. Noticeable improvements have been observed in the accuracy of predicting the rheological behavior when a multi-fidelity network is used compared to the single-fidelity network.

Boodaghidizaji, Miad↗
DOE OSTI2021

On the Use of Smart Meter Data to Estimate the Voltage Magnitude on the Primary Side of Distribution Service Transformers

This paper develops a novel method to estimate the voltage magnitude on the primary side of distribution service transformers. The proposed method relies exclusively on smart meters, and therefore it is fully data-driven. This is an important feature because electric utilities have detailed models of only the primary network - that is, the network between the distribution substation and the primary side of service transformers that are installed closer to end-customer sites. The network that connects the secondary side of service transformers to end-customer sites, referred to as the secondary network, is simply represented by a lumped load. For each secondary network, the proposed method uses data acquired from only 2 smart meters: the closest and the farthest-in the sense of electrical distance - from the service transformer. As a reference to this feature, the proposed method is named SM2Vp. To our knowledge, this is the first time a method is shown to provide actionable information for realtime operation and control of power distribution grids using only two smart meters per secondary network. This is important because utilities have experienced barriers in managing and using large data sets for real-time operation and control. SM2Vp is primarily intended to provide pseudo-measurements for distribution system state estimation, but it can also be used directly for voltage control schemes. The performance of SM2Vp is demonstrated by numerical simulations carried out on three secondary network synthetic models and by using field data provided by a utility partner serving customers in southwestern California. A maximum relative error of approximately 3.9% or less is observed for the primary voltage magnitude estimates in all numerical experiments.

distribution service transformer↗
DOE OSTI2021

On the Use of Smart Meter Data to Estimate the Voltage Magnitude on the Primary Side of Distribution Service Transformers: Preprint

This paper develops a novel method to estimate the voltage magnitude on the primary side of distribution service transformers. The proposed method relies exclusively on smart meters, and therefore it is fully data-driven. This is an important feature because electric utilities have detailed models of only the primary network--that is, the network between the distribution substation and the primary side of service transformers that are installed closer to end-customer sites. The network that connects the secondary side of service transformers to end-customer sites, referred to as the secondary network, is simply represented by a lumped load. For each secondary network, the proposed method uses data acquired from only 2 smart meters: the closest and the farthest--in the sense of electrical distance--from the service transformer. As a reference to this feature, the proposed method is named SM2Vp. To our knowledge, this is the first time a method is shown to provide actionable information for real-time operation and control of power distribution grids using only two smart meters per secondary network. This is important because utilities have experienced barriers in managing and using large data sets for real-time operation and control. SM2Vp is primarily intended to provide pseudo-measurements for distribution system state estimation, but it can also be used directly for voltage control schemes. The performance of SM2Vp is demonstrated by numerical simulations carried out on three secondary network synthetic models and by using field data provided by a utility partner serving customers in southwestern California. A maximum relative error of approximately 3.9% or less is observed for the primary voltage magnitude estimates in all numerical experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

We derive a Kubo-like formula for the thermal Hall conductance of a 2d lattice systems which is free from ambiguities associated with the definition of energy magnetization. We use it to define a relative topological invariant of gapped 2d lattice systems at zero temperature. Up to a numerical factor, it can be identified with the difference of chiral central charges for the corresponding edge modes. This establishes the bulk-boundary correspondence for the chiral central charge. Here, we also show that for any local commuting projector Hamiltonian, the relative chiral central charge vanishes, while for free fermionic systems, it is related to the zero-temperature electric Hall conductance via the Wiedemann-Franz law.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Simulated CO 2 storage efficiency factors for saline formations of various lithologies and depositional environments using new experimental relative permeability data

Saline formations are attractive geologic reservoirs for permanent carbon dioxide (CO 2 ) storage. Here, the U.S. Department of Energy's National Energy Technology Laboratory (DOE-NETL) has worked to develop and refine methods and tools for the calculation of CO 2 storage potential in subsurface reservoirs. DOE-NETL's CO 2 -SCREEN provides an online tool for executing these storage methods. CO 2 storage efficiency terms are input parameters in DOE-NETL's methods and equations embedded in the CO 2 -SCREEN, which assesses pore space available for CO 2 storage. In this work, a modeling workflow was initiated to refine two CO 2 storage efficiency terms - volumetric displacement (E V ) and microscopic displacement (E d ). The models are based on new experimental relative permeability data that are specific to homogenous lithology and depositional environments of key subsurface saline formations targeted for CO 2 storage. In future work, heterogenous features will be added to this initial modeling effort to update efficiency factors as described in DOE-NETL's methods and CO 2 -SCREEN tool. E V accounts for the volume utilized in the reservoir under the areal plume, while E d accounts for saturation values in the plume to assess efficiency of CO 2 storage at the pore scale. The results of this work are significant in that prior values were based on a limited geologically non-specific relative permeability data set that were collected prior to 2009. Specifically, we applied numerical simulations using TOUGH3 models to update CO 2 storage efficiency values for supercritical CO 2 injection into brine-saturated reservoirs for three lithologies (clastics, limestone, dolomite) and six depositional environments (Marginal Marine, Strand Plain, Deltaic Complex Fluvial, Aeolian, Shallow Marine, and Reef) that have a high potential for geologic CO 2 storage. Experimental relative permeability data in cores from these environments were utilized in the models with corresponding rock type/sedimentary environment. Results of this study showed that dolomite followed by limestone generated higher ranges of storage efficiency compared to clastics. The updated values provided a tighter efficiency range for clastics, lower P 10 but higher P 90 range for limestone, and higher P 10 and P 90 for dolomite. In general, tighter reservoirs with relatively low permeability and porosity were associated with higher E V and E d , showing efficient reservoir and pore utilization in these scenarios. High reservoir pressure and temperature associated with increasing depth increased the E V , and high CO 2 injection rates resulted in increases in E V and E d , while the impact of permeability anisotropy was minimal after the 30-year injection period.

03 NATURAL GAS↗
DOE OSTI2023

Earthquake response of head-mounted equipment in advanced nuclear reactors

The seismic response of safety-related equipment mounted on the head of an advanced reactor, including pumps, control rod drive mechanisms, and reactor monitoring devices, will affect the design and layout of many advanced reactors. High earthquake-induced accelerations in such equipment may challenge their seismic qualification and trigger the need for additional support framing on the reactor head. Base isolation is a design solution that can drastically reduce seismic demands on equipment. This article describes a set of earthquake-simulator experiments conducted on a scale-model of a base-isolated reactor vessel including four representations of head-mounted equipment, with frequencies spanning from 4.5 to 27 Hz. Dynamic responses of the head-mounted equipment, including displacements, accelerations, and strains, were measured in the experiments for three support conditions: conventional, and seismically isolated using single concave Friction Pendulum (SFP) bearings and triple Friction Pendulum (TFP) bearings. Seismic isolation was effective at reducing equipment responses (accelerations, displacements, and strains) with respect to those in the conventionally supported vessel across a range of seismic inputs. Companion numerical studies highlight the accuracy to be expected in the calculation of different response quantities for lightly damped equipment. The importance of characterizing damping in head-mounted, safety-related equipment through physical experiments to support design and risk assessment is made clear through the numerical simulations.

Engineering↗
DOE OSTI2021

Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators

Here, we present exact results for the classical version of the out-of-time-order commutator (OTOC) for a family of power-law models consisting of N particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form ∝ $Σ_{i,j=(i≠j)}^{N}$ |x i -x j | -k for every k>1 where x i is the position of the i th particle. We present numerical results for the OTOC for finite N at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many-body ground state. In the large-N limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the “light cone.” We find remarkably distinct features in OTOC below and above k=3 in terms of going from non-Airy behavior (1 < k < 3) to an Airy universality class (k>3). We present certain additional rich features for the case k=2 that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatiotemporal spread of perturbations in long-range interacting systems.

1-dimensional systems↗
DOE OSTI2025

Ten-moment fluid modeling of the Weibel instability

We investigate the one-dimensional non-relativistic Weibel instability through the capture of anisotropic pressure tensor dynamics using an implicit 10-moment fluid model that employs the electromagnetic Darwin approximation. The results obtained from the 10-moment model are compared with an implicit particle-in-cell simulation. The linear growth rates obtained from the numerical simulations are in good agreement with the theoretical fluid and kinetic dispersion relations. The fluid dispersion relations are derived using Maxwell’s equations and the Darwin approximation. We also show that the magnetohydrodynamic approximation can be used to model the Weibel instability if one accounts for an anisotropic pressure tensor and unsteady terms in the generalised Ohm’s law. In addition, we develop a preliminary theory for the saturation magnetic field strength of the Weibel instability, showing good agreement with the numerical results.

Kuldinow, D. A. (ORCID:0000000319730196)↗
DOE OSTI2020

Calibration of CDM-Based Creep Constitutive Model Using Accelerated Creep Test (ACT) Data

In conventional creep testing (CCT) a specimen is subject to constant load and temperature for a long-duration until creep rupture occurs. Conventional testing can be costly when considering the number of experiments needed to characterize the creep response of a material over a range of stress and temperature. To predict long-term creep-rupture properties, the time-temperature-stress superposition principle (TTSSP) approach has been employed where stress and/or temperature is applied at an elevated level; the result of which are extrapolated down to low stress and/or temperature conditions. These methods have been successful in predicting minimum-creep-strain-rate (MCSR) and stress-rupture (SR) but suffer from an inability to predict the creep deformation curve or account for changes in deformation mechanisms or aging that occurs at long-duration. An accelerated technique, termed the Stepped isostress method (SSM) allows the accelerated testing of materials to determine their creep deformation response. Unlike TTSSP tests, the SSM test employs a single specimen where the stress is periodically step increased until rupture. The SSM creep deformation curve is processed (time and strain shifted) to produce an accelerated creep deformation curve that represent the creep deformation curve at the initial stress level in SSM. A processing procedure for metals has yet to be developed. The research objective of this study is to develop a processing procedure for SSM test data using a creep-damage constitutive model. Triplicate SSM tests were conducted on Ni-based superalloy Inconel 718 at 650°C with stress being periodically increased until rupture. Triplicate CCT tests were conducted at the initial stress level of the SSM tests. The Sine-hyperbolic (Sinh) creep-damage model was employed in this study. The Sinh creep-damage constitutive model is based on coupled creep strain rate and damage evolution equations; where both rates are dependent on the current state of damage. Calibration is two-step: analytical and numerical optimization. Each stepped creep deformation curve is tackled quasi-analytically to determine MCSR and SR related material constants and accumulated damage. The damage accumulated at the end of each step was then passed onto subsequent steps to calibrate the MCSR, rupture prediction, and damage evolution. Numerical optimization was applied to optimize model constants involved in the creep strain constitutive equations in order to generate best-fitted Sinh creep deformation curves. The Sinh model predictions were compared to the SSM and CCT data. The Sinh model satisfactorily predicts the SSM data and thus the calibrated material constants provides a good estimate of rupture found in the CCT data. Calibration using SSM data reduces the number of tests needed to calibrate a model; significantly reducing costs. A single SSM test replaces numerous creep tests at different stresses.

36 MATERIALS SCIENCE↗
DOE OSTI2024

The AGORA High-resolution Galaxy Simulations Comparison Project. V. Satellite Galaxy Populations in a Cosmological Zoom-in Simulation of a Milky Way–Mass Halo

We analyze and compare the satellite halo populations at z ~ 2 in the high-resolution cosmological zoom-in simulations of a 10 12 M ⊙ target halo (z = 0 mass) carried out on eight widely used astrophysical simulation codes (Art-I, Enzo, Ramses, Changa, Gadget-3, Gear, Arepo-t, and Gizmo) for the AGORA High-resolution Galaxy Simulations Comparison Project. We use slightly different redshift epochs near z = 2 for each code (hereafter "z ~ 2") at which the eight simulations are in the same stage in the target halo's merger history. After identifying the matched pairs of halos between the CosmoRun simulations and the DMO simulations, we discover that each CosmoRun halo tends to be less massive than its DMO counterpart. When we consider only the halos containing stellar particles at z ~ 2, the number of satellite galaxies is significantly fewer than that of dark matter halos in all participating AGORA simulations and is comparable to the number of present-day satellites near the Milky Way or M31. The so-called "missing satellite problem" is fully resolved across all participating codes simply by implementing the common baryonic physics adopted in AGORA and the stellar feedback prescription commonly used in each code, with sufficient numerical resolution (≲100 proper pc at z = 2). We also compare other properties such as the stellar mass–halo mass relation and the mass–metallicity relation. Our work highlights the value of comparison studies such as AGORA, where outstanding problems in galaxy formation theory are studied simultaneously on multiple numerical platforms.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Risk-Averse Bi-Level Stochastic Network Interdiction Model for Cyber-Security Risk Management

Security of cyber networks is crucial; recent severe cyber-attacks have had a devastating effect on many large organizations. The attack graph, which maps the potential attack paths of a cyber network, is a popular tool for analyzing cyber system vulnerability. In this study, we propose a bi-level stochastic network interdiction model on an attack graph to enable a risk-averse, resource constrained cyber network defender to optimally deploy security countermeasures that protect against attackers with an uncertain budget. This risk- averse conditional-value-at-risk (CVaR) model minimizes a weighted sum of the expected maximum loss over all scenarios and the expected maximum loss from the most damaging attack scenarios. We develop a customized constraint and column generation algorithm to solve our model as well as several acceleration techniques to improve the computational efficiency. Numerical experiments demonstrate that the acceleration techniques enable the solution of relatively large problems within a reasonable amount of time: applying all the acceleration techniques also reduces the average computation time of the basic algorithm by 71% for 100-node graphs. Using metrics called mean-risk value of stochastic solution and value of risk-aversion, computational results suggest that our stochastic risk-averse model significantly outperforms deterministic and risk-neutral models when 1) the distribution of attacker budget is heavy-right-tailed and 2) the defender is highly risk-averse.

Bhuiyan, Tanveer H.↗
DOE OSTI2021

A screening model for predicting injection well pressure buildup and plume extent in CO 2 geologic storage projects

We present a new screening model for predicting injection well pressure buildup and CO 2 plume migration for CO 2 geologic sequestration projects. The model requires only limited information and is quite accurate when compared to detailed simulation results. Such models can assist project developers during the early days of project planning (e.g., for 45Q related projects), and also help regulators perform simple checks against detailed numerical models. The screening model consists of two correlations: one for injectivity index in terms of the slope of the CO 2 fractional flow curve, and the second for total storage efficiency within the footprint of plume as a function of gravity number and slope of the CO 2 fractional flow curve. Using these two correlations, as well as a knowledge of some basic reservoir characteristics and estimates of fluid properties from standard correlations, the injection-well pressure buildup and CO 2 plume extent in the formation can be readily estimated. Finally, the new correlations show a good match with the results of the underlying simulations, and also provide good agreement with independent calculations for two example problems.

42 ENGINEERING↗
DOE OSTI2023

Assessing the source of error in the Thomas–Fermi–von Weizsäcker density functional

We investigate the source of error in the Thomas–Fermi–von Weizsäcker (TFW) density functional relative to Kohn–Sham density functional theory (DFT). In particular, through numerical studies on a range of materials, for a variety of crystal structures subject to strain and atomic displacements, we find that while the ground state electron density in TFW orbital-free DFT is close to the Kohn–Sham density, the corresponding energy deviates significantly from the Kohn–Sham value. Here, we show that these differences are a consequence of the poor representation of the linear response within the TFW approximation for the electronic kinetic energy, confirming conjectures in the literature. In so doing, we find that the energy computed from a non-self-consistent Kohn–Sham calculation using the TFW electronic ground state density is in very good agreement with that obtained from the fully self-consistent Kohn–Sham solution.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗