Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “thermodynamic separation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 181 records · Page 10

Understanding Cation Selectivity in Carbon Nanopores with Hybrid First-Principles/Continuum Simulations: Implications for Water Desalination and Separation Technologies

Understanding ion adsorption in porous carbons is crucial for a range of technologies, including water desalination and energy storage. In this work, we combined density functional theory with a continuum solvation model to investigate thermodynamics and kinetics of the adsorption process of alkali metal ions from aqueous solutions into carbon nanopores with different sizes and geometries. We found that cations with a larger ionic radius are more favorable to enter the nanopores because of a lower energy penalty of dehydration. In addition, the pore size and geometry were found to have a significant impact on the ion–pore interaction under confinement and cation selectivity. Our study highlights a complex interplay among nanopore geometry, ion size, and hydration on the cation adsorption selectivity, suggesting that tuning the porosity could represent a general strategy for improving ion separations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Air separation and N 2 purification with Ba 0.15 Sr 0.85 FeO 3-$δ$ via a two-step thermochemical process

Thermochemical air separation to produce high-purity N 2 was demonstrated in a vertical tube reactor via a two-step reduction–oxidation cycle with an A-site substituted perovskite Ba 0.15 Sr 0.85 FeO 3–δ (BSF1585). BSF1585 particles were synthesized and characterized in terms of their chemical, morphological, and thermophysical properties. A thermodynamic cycle model and sensitivity analysis using computational heat and mass transfer models of the reactor were used to select the system operating parameters for a concentrating solar thermal-driven process. Thermal reduction up to 800 °C in air and temperature-swing air separation from 800 °C to minimum temperatures between 400 and 600 °C were performed in the reactor containing a 35 g packed bed of BSF1585. The reactor was characterized for dispersion, and air separation was characterized via mass spectrometry. Gas measurements indicated that the reactor produced N 2 with O 2 impurity concentrations as low as 0.02 % for > 30 min of operation. Additionally, a parametric study of air flow rates suggested that differences in observed and thermodynamically predicted O 2 impurities were due to imperfect gas transport in the bed. Temperature swing reduction/oxidation cycling experiments between 800 and 400 °C in air were conducted with no statistically significant degradation in N 2 purity over 50 cycles.

14 SOLAR ENERGY↗

Molecular Simulations Probing the Adsorption and Diffusion of Ammonia, Nitrogen, Hydrogen, and Their Mixtures in Bulk MFI Zeolite and MFI Nanosheets at High Temperature and Pressure

Recent advances in the synthesis of MFI zeolite nanosheets have led to highly selective membranes that are promising candidates for small-scale ammonia separation from ammonia/nitrogen/hydrogen mixtures in distributed green ammonia production plants. Using force-field-based molecular simulations, we evaluate the performance of bulk all-silica MFI zeolite and a 3 nm thick all-silica MFI nanosheet for ammonia/nitrogen/hydrogen separations at moderate temperature and pressure conditions (T = 373 K, p = 5 bar) as well as conditions relevant for a membrane-based reactor–separator process (T = 523 or 623 K, p = 80 bar). Isobaric–isothermal Gibbs ensemble Monte Carlo (GEMC) simulations were carried out to understand the selective adsorption behavior for these mixtures. Molecular dynamics simulations in the canonical ensemble were performed to examine the transport behavior of ammonia in a mixture with nitrogen or hydrogen at loadings obtained from the GEMC simulations. Our results show that both bulk MFI and the nanosheet with explicit surface silanols are highly selective toward ammonia adsorption, but the adsorption selectivity decreases by factors of 4 and 10 for ammonia/nitrogen and ammonia/hydrogen mixtures as the temperature is increased from 373 to 623 K. Conversely, the diffusion selectivities toward ammonia are more favorable at process-relevant temperatures and for the MFI nanosheet. Here, at T = 523 K and p = 80 bar, our simulations predict overall separation factors of 3.8 ± 0.2 and 4.5 ± 0.3 for ammonia/nitrogen and ammonia/hydrogen mixtures, respectively, in the bulk MFI zeolite, and separation factors of 8.6 ± 0.4 and 12.5 ± 0.8, respectively, for the MFI nanosheet.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Design and optimization of flexible decoupled high-temperature gas-cooled reactor plants with thermal energy storage

Advanced nuclear power plants are well-positioned for future zero-carbon grids, however, the need for flexible power generation will be required over the traditional emphasis on baseload generation for meeting historical demands. To achieve such flexibility, this work examines viable configurations for coupling nuclear energy production with thermal energy storage. Previous designs on nuclear-thermal energy storage configurations for advanced reactor designs, which utilized reactor steam as the heat source for charging the thermal energy storage, are restricted by the heat diversion ratio and efficiency losses, thus their impacts can be limited. In this context, this study proposes configurations for fully decoupling the nuclear reactor from the power cycle and positioning the storage as an intermediate loop, thereby achieving an unconstrained heat diversion ratio and improved efficiency. Compared with a standard high-temperature gas-cooled reactor’s power cycle, steady-state thermodynamic modeling and dispatch optimizations quantify the benefits of a steam reheat cycle within the fully decoupled thermal energy system to separate the plant cycle from the high-pressure primary side. These benefits are further detailed, compatible with required high-temperature and high-pressure conditions, through (1) open-source dynamic transient models that examine the impact of off-design operation on the systems, (2) the investigation of components design and costing and finally (3) sizing and dispatch optimization. The fully-decoupled design achieves a cycle efficiency of 43.1%, an enhancement over the vendor’s standard efficiency of 42.2% (Xe-100 design). Here, the proposed design offers strengthened physical barriers from the nuclear island as well as superior operational flexibility and power boosting. Dispatch optimization and market analysis reveal that thermal energy storage size is highly dependent on the peak patterns of electricity prices and the minimum generation level constraint imposed on the balance of plant. Evaluation of off-design operation demonstrates that the full decoupling design with the suggested fail-safe control mechanisms ensures a minimal impact on reactor parameters, even during rapid power ramping.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Nanodomain Formation and Temperature-Dependent Diffusion in Deep Eutectic Solvents Revealed by Single-Molecule Tracking

Deep eutectic solvents (DESs) are typically regarded as homogeneous liquids; however, recent work shows that many exhibit nanoscale structural heterogeneity. Most studies attribute these nanoscale features to short-range chemical interactions. It is still unclear whether a long-range physical mechanism also plays a role. Here, in this study, we examined the nanoscale structure in two hydrophobic DESs, 1:3 tetrabutylammonium bromide: l-menthol (DES-butyl) and 1:3 tetraoctylammonium bromide: l-menthol (DES-octyl). The notation 1:3 represents the molar ratio of the hydrogen bond acceptors to hydrogen bond donors used in the synthesis of the DESs. Single-molecule tracking (SMT) coupled with maximum entropy method (MEM) analysis was used to measure the number of diffusion populations of a dilute concentration of an added fluorescent probe. The presence of more than one population of diffusion coefficients indicates the existence of multiple local environments for the fluorescent probe (i.e., nanoscale structures in the DES). DES-butyl showed a relatively narrow diffusion coefficient distribution centered at 0.55 μm 2 /s, whereas DES-octyl displayed two distinct diffusing populations at 20 °C, with diffusion coefficients of 0.12 μm 2 /s and 0.53 μm 2 /s for the slow and fast populations, respectively. As DES-octyl was heated, the slow-diffusing population steadily diminished and disappeared above ∼30 °C, indicating that the nanodomains present at lower temperatures collapse as the liquid becomes more thermodynamically mixed. This temperature-dependent homogenization is consistent with a physical mechanism of nanostructure formation, for example, liquid–liquid phase separation (LLPS), wherein the structure is not driven solely by specific chemical interactions. The SMT-MEM results suggest that a long-range physical mechanism is the most plausible origin of the measured nanoscale structure in DES-octyl.

Opare-Addo, Jemima [Ames Laboratory (AMES), Ames, ↗

Thermodynamics of grain-boundary phases

The past decade has seen a significant increase in research efforts aimed at understanding the thermodynamics of low-dimensional phases existing in many materials systems, ranging from two-dimensional materials to core regions of extended defects in crystalline solids. We review the current status of theoretical, computational, and experimental research on the “defect phases,” focusing on grain boundaries (GBs) in elemental and multicomponent polycrystalline materials. After reviewing the generalized concept of a phase of any dimensionality, we discuss recent progress in atomistic computer simulations of GB phase transformations and phase coexistences, including the observation of one-dimensional defects separating GB phases (defects in defects). Computational predictions compare well with experimental observations of multiple GB phases and segregation-induced phase transformations. An intriguing open question of GB thermodynamics is whether the GB free energy can be driven to a zero value by increasing solute segregation. We review recent efforts to understand this ultimate thermodynamic stabilization of GB phases and the possible polycrystalline microstructures that may arise. An outlook for future research in the field is discussed.

Materials science↗

Uncovering the True Active Sites in Ni–N–C Catalysts for CO 2 Electroreduction

Understanding and designing active sites in single-atom catalysts (SACs) requires going beyond static models to capture their dynamic evolution under realistic electrochemical conditions. Here, in this work, we develop an integrated theoretical framework that accounts for operational conditions, by combining grand canonical density functional theory (GC-DFT) with machine-learning-accelerated sampling, to uncover structure–activity–stability relationships in Ni–N–C SACs for the CO 2 reduction reaction (CO 2 RR). A library of NiN x C 4–x (x = 0–4) motifs─representing coordination defects likely formed during high-temperature synthesis─was systematically evaluated. Under working conditions, these sites were found to undergo hydrogenation, and NiN 3 C 1_ H 1 was identified as the most probable active site. At reducing potentials, hydrogen adsorbs spontaneously at C–Ni bridge sites rather than Ni top sites, while subsurface hydrogen facilitates bent CO 2 adsorption crucial for activation. High CO 2 RR selectivity toward CO arises from site separation: Ni centers drive CO2RR, while the hydrogen evolution reaction (HER) occurs at the C–Ni bridge or N sites and from thermodynamic suppression of HER at moderate hydrogen coverage. At more negative potentials, a shift in the CO 2 RR rate-determining process (RDP) and Ni out-of-surface displacement induced by coadsorption of H and H 2 O jointly reduce activity and selectivity. Thus, both the high CO2RR selectivity of Ni–N–C catalysts and its reversal with more negative potentials can be rationalized by accounting for hydrogenated surfaces. This highlights the necessity of modeling realistic; in situ conditions. This framework provides generalizable insights into the dynamic behavior of active sites in SACs, offering guidance for the rational design of active and robust catalysts for a wide range of electrochemical reactions.

25 ENERGY STORAGE↗

Mapping causal patterns in crystalline solids

The evolution of the atomic structures of the combinatorial library of Sm-substituted thin film BiFeO 3 along the phase transition boundary from the ferroelectric rhombohedral phase to the non-ferroelectric orthorhombic phase is explored using scanning transmission electron microscopy. Localized properties, including polarization, lattice parameter, and chemical composition, are parameterized from atomic-scale imaging, and their causal relationships are reconstructed using a linear non-Gaussian acyclic model. This approach is further extended to explore the spatial variability of the causal coupling using the sliding window transform method, which revealed that new causal relationships emerged at both the expected locations, such as domain walls and interfaces, and at additional regions forming clusters in the vicinity of the walls or spatially distributed features. While the exact physical origins of these relationships are unclear, they likely represent nanophase-separated regions in the morphotropic phase boundaries. Overall, we posit that an in-depth understanding of complex disordered materials away from thermodynamic equilibrium necessitates understanding not only the generative processes that can lead to observed microscopic states but also the causal links between multiple interacting subsystems.

Causal inference↗

Separation, speciation, and mechanism of astatine and bismuth extraction from nitric acid into 1-octanol and methyl anthranilate

We report a detailed study of At and Bi extraction from nitric acid media into conventional solvents, namely 1-octanol and methyl anthranilate, has been performed. The analysis includes a mathematical modeling which allows the fitting of experimental data and determination of extraction constants of the two above mentioned elements. Also, this approach helped to estimate a stability constant of a weak AtO(NO3) complex along with thermodynamic constants describing the redox process of At species in the acidic solution and formation of an adduct of Bi in the presence of methyl anthranilate. The results of the fitting have been used to calculate corresponding separation factors of Bi and At as well. Moreover, a computational study has been performed to evaluate At interaction with the above mentioned solvents.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Computational investigation of water glasses using machine-learning potentials

The molecular origins of water’s anomalous properties have long been a subject of scientific inquiry. The liquid–liquid phase transition hypothesis, which posits the existence of distinct low-density and high-density liquid states separated by a first-order phase transition terminating at a critical point, has gained increasing experimental and computational support and offers a thermodynamically consistent framework for many of water’s anomalies. However, experimental challenges in avoiding crystallization near the postulated liquid–liquid critical point have focused attention to water’s canonical glassy states: low-density and high-density amorphous ice. Here, we use two Deep Potential machine-learning models, trained on the Strongly Constrained and Appropriately Normed density functional and the highly accurate Many-Body Polarizable potential, to conduct an investigation of water’s glassy phenomenology based on quantum mechanical calculations. Despite not being explicitly trained on amorphous ices, both models accurately capture the structure and transformation of the water glasses, including their interconversion along different thermodynamic paths. Isobaric quenching of liquid water at various pressures generates a continuum of intermediate amorphous ices and density fluctuations increase near the liquid–liquid critical pressure. The glass transition temperatures of the amorphous ices produced at different pressures exhibit two distinct branches, corresponding to low-density and high-density amorphous ice behaviors, consistent with experiment and the liquid–liquid transition hypothesis. Extrapolating transformation pressures from isothermal compressions to experimental compression rates brings our simulations into excellent agreement with data. Our findings demonstrate that machine-learning potentials trained on equilibrium phases can effectively model nonequilibrium glassy behavior and pave the way for studying long-timescale, out-of-equilibrium processes with quantum mechanical accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Superconductivity from Luttinger surfaces: Emergent Sachdev-Ye-Kitaev physics with infinite-body interactions

The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature (T = 0). The equivalent question of pairing instability on a Luttinger surface (LS)—190a contour of zeros of the propagator—instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the q → ∞ SYK model with q-body interactions close to the LS, and (ii) the enforcement of gauge invariance results in a $\frac{1}{√τ}$ (τ ~ T -1 ) behavior of the fluctuation propagator near the QCP, as is the feature of SYK conformal Green functions, but leaves the overall form of the free energy map robust.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Laser-induced quenching of metastability at the Mott insulator to metal transition

There is growing interest in strongly correlated insulator thin films because the intricate interplay of their intrinsic and extrinsic state variables causes memristive behavior that might be used for biomimetic devices in the emerging field of neuromorphic computing. Here, in this study, we find that laser irradiation tends to drive V 2 ⁢O 3 from supercooled/superheated metastable states toward thermodynamic equilibrium, most likely in a nonthermal way. We study thin films of the prototypical Mott-insulator V2⁢O3, which show spontaneous phase separation into metal-insulator herringbone domains during the Mott transition. Here, we use low-temperature microscopy to investigate how these metal-insulator domains can be modified by scanning a focused laser beam across the thin film surface. We find that the response depends on the thermal history: When the thin film is heated from below the Mott transition temperature, the laser beam predominantly induces metallic domains. On the contrary, when the thin film is cooled from a temperature above the transition, the laser beam predominantly induces insulating domains. Very likely, the V 2 ⁢O 3 thin film is in a superheated or supercooled state, respectively, during the first-order phase transition, and the perturbation by a laser beam drives these metastable states into stable ones. This way, the thermal history is locally erased. Our findings are supported by a phenomenological model with a laser-induced lowering of the energy barrier between the metastable and equilibrium states.

Materials Science↗

Fermions, quantum gravity, and holography in two dimensions

We study a model comprising N flavors of Kähler Dirac fermion propagating on a triangulated two-dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit N → ∞ that the partition function is dominated by a regular triangulation of two-dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However, we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a nonholographic phase. In contrast, for massless fermions, we find evidence that the boundary is conformal even for finite N . This is consistent with theoretical results in quantum Liouville theory. Published by the American Physical Society 2024

Astronomy & Astrophysics↗

Weak scaling of the contact distance between two fluctuating interfaces with system size

A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers-of-mass remain separated by a nonzero distance ℓ. An example of such a situation is the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Here we present a general approach to calculate the probability distribution of the contact distance ℓ and determine how its most likely value ℓ* depends on the surfaces' lateral size L. Using the Edward-Wilkinson equation as a model for interfaces, we demonstrate that ℓ* scales weakly with system size, i.e., the dependence of ℓ* on L for both (1+1)- and (2+1)-dimensional interfaces is such that lim L→∞ (ℓ*/L) = 0. In particular, for (2+1)-dimensional interfaces ℓ* is an algebraic function of logL, a result that is confirmed by computer simulations of slab-shaped domains formed under periodic boundary conditions. Overall, this weak scaling implies that such domains remain topologically intact until ℓ becomes very small compared to the lateral size of the interface, contradicting expectations from equilibrium thermodynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Measurement of Three Transport Coefficients and the Thermodynamic Factor in Block Copolymer Electrolytes with Different Morphologies

The design and engineering of composite materials is one strategy to satisfy the materials needs of systems with multiple orthogonal property requirements. In the case of rechargeable batteries with lithium metal anodes, the system requires a separator with fast lithium ion transport and good mechanical strength. In this work, we focus on the system polystyrene-block-poly(ethylene oxide) (SEO) with bis(trifluoromethane)sulfonimide lithium salt (LiTFSI). Ion transport occurs in the salt-containing poly(ethylene oxide)-rich domains. Mechanical rigidity arises due to the glassy nature of polystyrene (PS). If we assume that the salt does not interact with the PS-rich domains, we can describe ion transport in the electrolyte by three transport parameters (ionic conductivity, κ, salt diffusion coefficient, D, and cation transference number, t + 0 ) and a thermodynamic factor, T f . By systematically varying the volume fraction of the conducting phase, Φ c between 0.29 and 1.0, and chain length, N between 80 and 8000, we elucidate the role of morphology on ion transport. We find that κ is the strongest function of morphology, varying by three full orders of magnitude, while D is a weaker function of morphology. To calculate t + 0 and T f , we measure the current fraction, ρ + , and the open circuit potential, U, of concentration cells. We find that ρ + and U follow universal trends as a function of salt concentration, regardless of chain length, morphology, or Φ c , allowing us to calculate t + 0 for any SEO/LiTFSI or PEO/LiTFSI mixture when κ and D are known. Finally, the framework developed in this paper enables predicting the performance of any block copolymer electrolyte in a rechargeable battery.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Hydrogen Blending into Natural Gas Pipeline Infrastructure: Review of the State of Technology

Hydrogen is an energy carrier that could play an important role in reducing emissions associated with difficult-to-decarbonize sectors including peaking and load-following electricity and industrial heating. Blending hydrogen into the natural gas pipelines has been proposed as an approach for achieving near-term emissions reductions and early-market access for hydrogen technologies such as electrolyzers. Numerous challenges and uncertainties complicate this approach to natural gas decarbonization, however, and this review summarizes current research on the material, economic, and operational factors that must be considered for hydrogen blending. First, this review explores previous research regarding the effects of blending hydrogen on gas mixture fluid and thermodynamic properties, pipeline materials and equipment performance within transmission and distribution networks, and supporting facilities such as underground storage and end-use hydrogen separation. We also investigate and summarize studies that developed mathematical models of natural gas pipeline networks with hydrogen blending, and the operational and techno-economic findings of these network studies. Finally, we discuss notable hydrogen blending demonstrations and their key outcomes. Many blending demonstrations internationally have proven that low hydrogen percentage blending is feasible under very specific scenarios with limited end-usage applications on both high-pressure transmission lines and low-pressure distribution lines. This report summarizes findings from literature into key areas of consensus and disagreement. Research gaps and disagreements between the literature are highlighted to provide directions for future hydrogen blending research.

03 NATURAL GAS↗

Laboratory Development of a Pot Calcination Process for Converting Liquid Wastes to Solids

Simulated Darex, Purex, Thorex, and TBP-25 wastes were batch-evaporated and calcined to solids. The weight of residue was decreased to ~ 8% of the original waste for Darex, to ~ 4% for Purex, to ~ 2% for Thorex, and ~ 7% for TBP-25 by calcination to 1200°C. Semicontinuous evaporation and calcination to solids at 900°C of Darex and Purex wastes gave volume reduction factors of ~ 8 and 7 to 10, respectively. The nitrate contents of the residues from batch calcination to 800°C were from ~ 0.07 to 0.5 wt % while the nitrate contents of the residues from semicontinuous calcination to 900°C were from ~ 0.1 to 0.7 wt %. Sodium, calcium, and magnesium additives to Purex waste decreased the percent of sulfate volatilized into the condensate from ~ 30% to <0.5%. Replacement of the atmosphere above the waste with nitric oxide decreased ruthenium volatility from 50 to 60% of that originally present in Purex waste to 0.5 to 3.5%. The thermal conductivities for the calcined wastes, measured in situ, were all >0.1 Btu/hr ft °F at ~ 400°F and increased almost linearly with increasing temperature to >0.3 Btu/hr ft °F at ~ 1600°F in all cases studied.

052001* -- Nuclear Fuels-- Waste Processing↗

In Situ Neutron Scattering Study of the Structure Dynamics of the Ru/Ca 2 N:e – Catalyst in Ammonia Synthesis

NH 3 synthesis is one of the most critical industrial processes. Compared to commercial iron catalysts, Ru catalysts show high intrinsic activity in this reaction but suffer from hydrogen poisoning. By loading Ru onto supports such as electrides and hydrides, the hydrogen poisoning problem can be significantly alleviated. However, relevant studies on the structural dynamics of the Ru/electride catalysts under reaction conditions are very scarce. Taking advantage of the high sensitivity to hydrogen species, it is possible to obtain insights into the structural changes during the reaction using in situ neutron techniques. In this study, we have investigated the structural evolution of the Ru/Ca 2 N:e – catalyst during the ammonia synthesis reaction by in situ neutron scattering (inelastic neutron scattering, INS) technique. In situ INS experiments suggest that Ca 2 N:e – is likely converted to the Ca 2 NH phase during the reaction. Unlike the previously known structure where H and N atoms are intermixed, the formed Ca 2 NH exhibits a segregated structure where the H and N atoms are located in different layers separated by the Ca layer. In conclusion, density functional theory calculations of the reaction energetics reveal that there are minor changes in the barriers and thermodynamics of the first N hydrogenation step between the two phases (Ca 2 NH phase with segregated H/N layers and intermixed Ca 2 NH phase), suggesting the impact of the phases on the reaction kinetics to be relatively minimal.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗