Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “conduction”

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 163 records · Page 9

Proton-conducting oxides for energy conversion and storage

Proton-conducting oxides are a class of solid-state ion-conducting ceramic materials that demonstrate significant hydrogen ion (proton) conductivity at intermediate temperatures (e.g., 300–700 °C). They are garnering significant attention due to several unique characteristics that distinguish them from both higher temperature oxygen ion conducting oxides and lower temperature proton-conducting polymers. By enabling proton-mediated electrochemistry under both dry and wet environments at moderate temperatures, protonic ceramics provide unique opportunities to enhance or synergize a diverse range of complementary electrochemical and thermochemical processes. Because of this potential, significant efforts have been devoted to advancing numerous energy-related applications using these materials. This review aims to comprehensively summarize these applications and analyze the most up-to-date and future developments of proton-conducting oxides. We aim to bring together this diverse subject matter by integrating the fundamentals of proton-conducting oxides with application-oriented insights. We begin with a historical roadmap, followed by a basic overview of the materials, theories and fundamentals, and fabrication and processing technologies underlying the field. The central section of our review summarizes major applications and developments of proton-conducting ceramics, ranging from maturing applications approaching commercialization to embryonic technologies just now emerging from the lab. These include protonic ceramic fuel cells, protonic ceramic electrolysis cells, reversible protonic ceramic electrochemical cells, protonic ceramic membrane reactors, and protonic ceramic electrochemical reactors. For each application, we analyze both the prospects and challenges and offer recommendations for future research directions so that tomorrow's researchers can continue to advance the development and commercialization of these fascinating materials.

Duan, Chuancheng (ORCID:0000000218261415)↗

Anomalous thickness-dependent electrical conductivity in van der Waals layered transition metal halide, Nb 3 Cl 8

Understanding the electronic transport properties of layered, van der Waals transition metal halides (TMHs) and chalcogenides is a highly active research topic today. Of particular interest is the evolution of those properties with changing thickness as the 2D limit is approached. Here, we present the electrical conductivity of exfoliated single crystals of the TMH, cluster magnet, Nb 3 Cl 8 , over a wide range of thicknesses both with and without hexagonal boron nitride (hBN) encapsulation. The conductivity is found to increase by more than three orders of magnitude when the thickness is decreased from 280 µm to 5 nm, at 300 K. At low temperatures and below ~50 nm, the conductance becomes thickness independent, implying surface conduction is dominating. Temperature dependent conductivity measurements indicate Nb 3 Cl 8 is an insulator, however, the effective activation energy decreases from a bulk value of 310 meV to 140 meV by 5 nm. X-ray photoelectron spectroscopy (XPS) shows mild surface oxidation in devices without hBN capping, however, no significant difference in transport is observed when compared to the capped devices, implying the thickness dependent transport behavior is intrinsic to the material. A conduction mechanism comprised of a higher conductivity surface channel in parallel with a lower conductivity interlayer channel is discussed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Ultra-High Temperature Thermal Conductivity Measurements of a Reactive Magnesium Manganese Oxide Porous Bed Using a Transient Hot Wire Method

Pelletized magnesium manganese oxide shows promise for high temperature thermochemical energy storage. It can be thermally reduced in the temperature range between 1250 °C and 1500 °C and re-oxidized with air at typical gas-turbine inlet pressures (1–25 bar) in the temperature range between 600 °C and 1500 °C. The combined thermal and chemical volumetric energy density is approximately 2300 MJ/m3. The rate at which a thermochemical storage module can be charged is limited by heat transfer inside the solid packed bed. Hence, the effective thermal conductivity of packed beds of magnesium-manganese oxide pellets is a crucial parameter for engineering Mg-Mn-O redox storage devices. We have measured the effective thermal conductivity of a packed bed of 3.66 ± 0.516 mm sized magnesium manganese oxide (Mn to Mg molar ratio of 1:1) pellets in the temperature range of 300–1400 °C. Since the material is electrically conductive at temperatures above 600 °C, the sheathed transient hot wire method is used for measurements. Raw data is analyzed using the Blackwell solution to extract the bed thermal conductivity. The effective thermal conductivity standard deviation is less than 10% for a minimum of three repeat measurements at each temperature. Experimental results show an increase in the effective thermal conductivity with temperature from 0.50 W/m °C around 300 °C to 1.81 W/m °C close to 1400 °C. We propose a dual porosity model to express the effective thermal conductivity as a function of temperature. This model also considers the effect of radiation within the bed, as this is the dominant heat transfer mode at high temperatures. The proposed model accounts for microscale pellet porosity, macroscale bed porosity, pellet size, solid thermal conductivity (phonon transport), and radiation (photon transport). The coefficient of determination between the proposed model and the experimental results is greater than 0.90.

Engineering↗

Design of an Out-Of-Pile Experimental Facility to Demonstrate the Feasibility of In Situ Thermal Conductivity Measurements of Nuclear Fuels Under Irradiation

There is substantial merit in quantifying nuclear fuel performance under irradiation. At Oak Ridge National Laboratory (ORNL), the MiniFuel irradiation platform has become the primary test vehicle for conducting separate-effects fuel performance irradiation experiments. The MiniFuel experiment is a passively controlled capsule design deployed in the High Flux Isotope Reactor (HFIR) through which fuel performance data is collected post-irradiation. Separate effects fuels irradiation capabilities are being expanded at ORNL by developing instrumented capsule designs that aim to capture fuel performance phenomena in-situ. One such capsule will specifically target fuel specimen thermal conductivity changes as a function of fuel burnup. Due to the complexity of making this measurement on nuclear fuel in-pile, this paper describes the necessary out-of-pile testing conducted on the thermal conductivity capsule (TCC) design. The measurement is ascertained via a thermopile system with heat transferred unidirectionally through a surrogate fuel specimen sandwiched between two conductive materials. The capsules investigated in this study are representative of the in-pile design, with the primary departure from irradiation conditions being the distribution of heat generation within the capsule. In the out-of-pile experiment, an external heater was used to drive heat through the conductive slug materials and into the specimen. This paper expounds the design of the out-of-pile experimental system and the thermal conductivity measurement technique. Predictive models used to determine the sensitivity of the measurement to variables governing thermal contact conductance between the specimen and slug materials and to predict experimental results are also described. Data from the out-of-pile experiment will be used to validate the readiness of the design for insertion into HFIR for irradiation.

Parker, Trevor [ORNL]↗

Thermal conductivity of 3D-printed block-copolymer-inspired structures

This study primarily focuses on examining the impact that geometric structure has on thermal conductivity of multi-phase constructs in different 3D-printed poly(lactic acid), PLA, samples. The investigated structures are inspired by morphologies formed by diblock copolymers: lamellae, hexagonally packed cylinders, and gyroid. This research also investigates how volume percentage and material combination influence the thermal conductivity of these structures. Further, the samples can be tailored to simulate various thermal management structures observed in practical applications, such as thermal interface materials in electronic devices. Thermal conductivity ratio is controlled using air, the least conductive material at 0.026 W/(m K), PLA at 0.136 W/(m K), and thermal paste at 5.11 W/(m K). Different models were tested against thermal conductivity measurements in order to capture the effect of material type (PLA-Air versus PLA-Thermal Paste), volume percentage, structure, and orientation. Simple, effective medium models were good predictions of thermal conductivity in lamellar structures, but it was necessary to develop models for conduction through cylindrical and gyroid structures. Finally, all results were normalized to find a universal model that is independent of structure and material. This approach provides a simple method to predict how to reduce or enhance transport properties and heat management capabilities of 3D printed objects.

36 MATERIALS SCIENCE↗

Hierarchical microstructural parameters governing electrical and thermal conductivity in Al-based eutectic systems

Enhancing conductivity properties of aluminum alloys can drive superior performance and efficiency across diverse engineering applications, yet currently commercially available Al casting alloys are severely limited in electrical and thermal conductivity. There is a general understanding that the presence of any defects and alloying elements within Al will inevitably lower conductivity properties. However, there is a lack of comparative studies evaluating the intricate interplay between the microstructural parameters and conductivity properties across cast Al-based eutectic systems. Therefore, this study bridges this gap by conducting a comparative study across hypoeutectic alloy compositions consisting of 50 % primary Al and 50 % eutectic region to elucidate the key parameters that dictate conductivity properties. This research contributes to advancing the understanding of alloy design opportunities for high conductivity, high strength castable Al alloys.

36 MATERIALS SCIENCE↗

Evaluating the Lightning Strike Damage Tolerance for CFRP Composite Laminates Containing Conductive Nanofillers

Conductive nanofillers, such as carbon nanotube, graphene nanoplatelets, and carbon black particles (with diameters in nanometers) have been shown to enhance the electrical conductivity of fiber reinforced polymer matrix composites in many existing studies. The motivation is primarily for lightning strike protection, electromagnetic interference shielding, de-icing, and the manufacturing of lightweight electronic components. Here, we evaluate the lightning strike damage tolerance of carbon fiber reinforced polymer (CFRP) matrix composite laminates containing conductive nanofillers with varying weight fractions, including carbon black (CB), carbon nanotubes (CNT), and a mix of CB and CNT, through simulated lightning strike tests, followed by both non-destructive ultrasonic inspection and destructive sectioning to characterize the damage inflicted by the simulated lightning strike. Three-point flexural tests are performed to evaluate the residual strength retained by all CFRP specimens. Results show that lightning strike damage experienced varying levels of reduction for CFRP composite specimens containing conductive fillers in comparison to the baseline specimen without fillers. Notably, the delamination only penetrated to the interface between the 1st and 2nd layer for the specimen with 0.25 wt.% CNT in comparison to the baseline CFRP specimen for which the delamination penetrated to the interface between the 5th and 6th layer. Moreover, the retention of the flexural modulus increased from 26.5% to a maximum of 95.0% for the specimen with 0.25 wt.% hybrid CB and CNT. Yet, we show that using our chosen conductive fillers cannot fully eliminate lightning strike damage. Additionally, adding conductive fillers could compromise the flexural properties. We provide discussions on future recommendations on using conductive fillers for the lightning strike protection of CFRP composites.

36 MATERIALS SCIENCE↗

Competing conduction mechanisms in high performance carbon nanotube fibers

The performance of carbon nanotube (CNT) cables, a contender for copper-wire replacement, is tied to its metallic and semi-conducting-like conductivity responses with temperature; the origin of the semi-conducting-like response however is an underappreciated incongruity in literature. With controlled aspect-ratio and doping-degree, over 61 unique cryogenic experiments including anisotropy and Hall measurements, CNT cable performance is explored at extreme temperatures (65 mK) and magnetic-fields (60 T). A semi-conducting-like conductivity response with temperature becomes temperature-independent approaching absolute-zero, uniquely demonstrating for the first time the necessity of heterogeneous fluctuation induced tunneling; complete de-doping leads to localized hopping, contrasting graphite's pure metallic-like response. High-field magneto-resistance (including novel +22 % longitudinal magneto-resistance near room-temperature) is analyzed with hopping and classical two-band models, both yielding a similar parameter useful for conductor development. Varying field-orientation angle uncovers significant two- and four-fold symmetries that are shown to be from Aharonov-Bohm-like corrections to the curvature-induced bandgap, a first for macroscale CNT fibers. Tight-binding calculations using Green's Function formalism model the largest, coherent transport to-date in commensurate CNT bundles in magnetic-field, revealing non-uniform transmission across bundle cross-sections with doping restoring uniformity; independent of doping, transport in bundle-junction-bundle systems are predominantly from CNTs adjacent to the other bundle—demonstrating that smaller bundles are more efficient for electronic transport. The final impact is predicting the ultimate conductivity of heterogeneous CNT cables using temperature and field-dependent transport, surpassing conductivity of traditional metals.

36 MATERIALS SCIENCE↗

Modeling the impact of high thermal conductivity paper on the performance and life of power transformers

Degradation of insulation paper is a key contributor to the failure of power transformers. Insulation degradation accelerates at elevated temperatures, which highlights the potential for better thermal management to prolong life. While several studies have analyzed the benefits of high thermal conductivity oil for reducing temperatures inside a transformer, this study is an initial assessment of the benefits of high thermal conductivity paper on transformer life. Blending particulates with cellulosic fibers offers a pathway for high thermal conductivity paper (with good dielectric properties), which can reduce internal temperatures. Presently, life extensions that can be achieved by the use of such thermally conducting papers were estimated, with the thermal conductivity of the paper being the key parameter under study. The analytical-numerical thermal model used in this study was validated against experimental measurements in a distribution transformer, adding confidence to the utility of the model. This model was then used to provide estimates of hot-spot temperature reduction resulting from the use of papers with higher thermal conductivity than baseline. Transformer life was predicted conventionally by tracking the degree of polymerization of paper over time, based on an Arrhenius model. Results indicate that increasing the thermal conductivity of paper from 0.2 W/mK (baseline) to 1W/mK reduces the hot spot temperature by 10°C. While degradation significantly depends on the moisture and oxygen content, the model shows that such a temperature reduction can increase life for all conditions, by as much as a factor of three.

42 ENGINEERING↗

Bimodal particle distributions with increased thermal conductivity for solid particles as heat transfer media and storage materials

Solid particles are being considered in several high temperature thermal energy storage systems and as heat transfer media in a variety of advanced power generation systems, particularly in concentrated solar power plants. The downside of such an approach is the low overall heat transfer coefficients caused by the inherently low thermal conductivity values of the low-cost solid media when coupled to heat exchanger for the power cycle working fluid. Choosing the right particle size distribution, emittance, and material of the solid media can all make a substantial difference in packed bed thermal conductivity. Current research though exclusively focuses on continuous unimodal distributions of particles. Here, in this work, we propose the use of a binary particle system with a bimodal size distribution to significantly increase packed bed thermal conductivity by reducing packed bed porosity. This is the first study related to ceramic solid particle heat transfer that has considered the thermal conductivity of non-unimodal size distributions at room and elevated temperatures. The following study found that for the binary particle system using Carbo particles CP 16/30 – CP 70/140 where the large particle volume fraction was 50% there was an 17–47% increase in packed bed thermal conductivity when compared with a nearly unimodal particle size distribution of CP 16/30 between 50 and 300 °C. Two different porosity and effective thermal conductivity models were studied, with one providing better prediction of porosity but both effective thermal conductivity models providing less predictive capacity. Importantly this approach can have a substantial impact of thermal performance, with little to no impact on the particle cost.

36 MATERIALS SCIENCE↗

Solid and gas thermal conductivity models improvement and validation in various porous insulation materials

In the past few decades, significant efforts have been made to improve the theoretical understanding of thermal transport mechanisms in thermal insulation materials and push the thermal conductivity's lower limits. However, most works focused singularly on specific types of materials, and the models used for thermal conductivity predictions are diverse - a model that fits one material might not fit others. Here, we improve and unify the gas and solid thermal conductivity models for porous materials. Through experimental characterization of several different materials as well as literature data for other materials, these models are validated. Further, we have also found that the pressure-dependent gas thermal conductivity of most materials can be well fitted by using one or two pore sizes without using a complex pore size distribution. With the refined models, we decompose the effective thermal conductivity of several thermal insulation materials into gas, solid, and radiation contributions. For cellular (polystyrene and polyurethane) foams, the relative contributions from air, solid, and radiation are 58–75%, 3–11%, 16–38%, respectively. For granular porous materials (polyurethane and silica in this work), the contributions from air, solid, and radiation are 45–66%, 34–46%, and 0–8%, respectively. This work is expected to provide guidance on the design and optimization of the next generation of thermal insulation materials, for example, through the effort of reducing gas conduction and radiation in foams and suppressing gas and solid conduction in aerogels.

36 MATERIALS SCIENCE↗

Three-dimensional microstructural characterization of FBR MOX fuel and the contribution of microstructural features to the thermal conductivity of the fuel

Combination of microstructural characterization, property measurements, and phase field modeling is used to investigate fast breeder reactor (FBR) mixed oxide (MOX) fuel irradiated to burnup of 13.7% fissions per initial metal atom (FIMA). Here, the fuel was characterized at different radial locations, which revealed that grey phase can be present in the central region if it nucleates on five metal precipitates (FMPs). In addition, in the mid-radial region FMPs do not diffuse out of the region once formed and the size of Pd–Te precipitates is dictated by the porosity present in the region. Thermal conductivity measurements were conducted as a function of radial location and the microstructure of the fuel was correlated with the observed trend. Reconstructions of the 3D solid and gaseous fission product structures in different regions of the fuel were used to simulate the effective thermal conductivity (ETC) of the respective regions and determine which microstructural feature has the strongest impact on thermal conductivity. Based on conducted assessment, FMPs and Pd-Te precipitates improve local conductivity of central and mid-radial regions even in the presence of grey phase, but defects are primary contributor to the degradation of thermal conductivity on the periphery of the fuel.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Implications of point defect accumulation on UO 2 thermal conductivity and fission gas release under accelerated fuel irradiation

Evaluation of thermal properties is a crucial factor for nuclear fuel performance. During reactor operation, the accumulation of fission products and irradiation-induced lattice defects are responsible for degradation in thermal conductivity. Consequently, it affects fuel temperature and fission gas release (FGR) among other Multiphysics processes important for economics and safety analysis. We analyze the implications of point defects (PD) accumulation described using a rate theory (RT) Model on lattice thermal conductivity of UO 2 . Here, we demonstrate that fission rate-dependent point defect concentrations have the largest impact on in-pile thermal conductivity in the periphery of light water reactor fuels below a temperature threshold governed by the migration barrier of defects. Our analysis provides a mechanistic description of this phenomena which current fuel performance codes treat empirically. The reduction of thermal conductivity in the low -temperature rim region acts as additional thermal resistance and leads to a temperature notably larger than suggested by Lucuta thermal conductivity correlation. These effects are anticipated to have notable impacts when fuels are exposed to accelerated radiation. The impact of such point defect-informed treatment of thermal conductivity on fuel performance is evaluated by a detailed analysis of fission gas behavior and its release. We consider several models capturing different stages of fission gas bubble evolution and fission gas release (FGR). Finally, a new fission rate-dependent correction to the Lucuta correlation is proposed. The results show a significant reduction in thermal conductivity at the fuels’ periphery and an increase in fuel centerline temperature specifically at low burnups. Ultimately a modified LC shows a higher FGR compared to the original LC, while the acceleration process results in a reduction in overall FGR.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Accurately measuring the ionic conductivity of membranes via the direct contact method

Accurately measuring the ionic conductivity of membranes is important both for assessing the performance of new membranes as well as for advancing the fundamental understanding of electric field-driven ion transport in membranes. Yet, despite a considerable number of reports on the topic, there is no standardized method for ionic conductivity measurements. One of the main challenges with ionic conductivity measurements is isolating the membrane resistance from the other resistances in the electrochemical cell. In this study, we present a detailed discussion of factors that affect ionic conductivity measurements with the commonly used direct contact method, in which the membrane contacts the electrodes. Such factors include external resistances due to imperfect contact between the membrane and electrodes, residual impedances due to the sample holder and wiring, and fringe effects. Further, we also present and validate simple methods for isolating the true membrane ionic conductivity in both the through-plane and in-plane orientations. These methods can be implemented with virtually any membrane to obtain accurate and reproducible ionic conductivity values. Finally, we outline recommendations for performing ionic conductivity measurements with the direct contact method and a set of parameters to verify when reporting these results.

36 MATERIALS SCIENCE↗

Ultrahigh thermal conductivity in three-dimensional covalent organic frameworks

Covalent organic frameworks (COFs) are a new, growing class of crystalline polymers with exceptional characteristics such as low density, mechanical strength, and flexibility. Taking three-dimensional (3D) COF-300 derivatives as an example, we apply molecular dynamics simulations to show that ultrahigh thermal conductivity (>15 W/(mK)) is achieved at a pore size of d = 0.63 nm. This is the first observation of such an outstanding intrinsic thermal conductivity of a 3D polymer in all directions. Further temperature-dependent thermal conductivity calculations and X-ray diffraction (XRD) patterns highlight the importance of maintaining microscopic order for a large thermal conductivity. We also demonstrate that the tunability and anisotropy in COFs provide sufficient room to engineer and obtain desired thermal conductivity. These findings provide a fundamental understanding of structure-thermal conductivity relation in COFs and shine a light on high-thermal-conductivity COFs for thermal management applications.

36 MATERIALS SCIENCE↗

Improved Single-Ion Conductivity of Polymer Electrolyte via Accelerated Segmental Dynamics

Single-ion conducting polymer electrolytes (SICPEs) have many advantages for solid-state battery applications, while low conductivities remain the bottleneck for their practical applications. In this paper, a strategy that can significantly improve the ionic conductivity of SICPEs based on the concept of accelerated segmental dynamics was demonstrated by the evaluation of the thermal property, rheological behavior, morphology analysis, molecular dynamics simulation, and conductivity performance. With both “soft” poly(dimethylsiloxane) (PDMS) backbone and poly(ethylene glycol) (PEG) side chains, the obtained SICPE possesses faster segmental relaxation and higher lithium-ion conductivity. With lithium “transference” number close to unity, the obtained SICPE also shows excellent electrochemical stability against lithium metal electrodes. The clear relationship established between the segmental dynamics of polymer electrolyte and its ionic conductivity should contribute to achieve a solid electrolyte with improved ionic conductivity toward the next-generation solid-state battery.

Segmental Dynamics↗

Deconvolution of Water-Splitting on the Triple-Conducting Ruddlesden–Popper-Phase Anode for Protonic Ceramic Electrolysis Cells

Triple-conducting materials have been proved to improve the performance of popular protonic ceramic electrolysis cells. However, partially because of the complexity of the water splitting reaction involving three charge carriers, that is, oxygen (O 2– ), proton (H + ), and electron (e – ), the triple-conducting reaction mechanism was not clear, and the reaction conducting pathways have seldom been addressed. In this study, the triple conducting Ruddlesden–Popper phase Pr 1.75 Ba 0.25 NiO 4+δ as an anode on the BaCe 0.7 Zr 0.1 Y 0.1 Yb 0.1 O 3–δ electrolyte was fabricated and its electroresponses were characterized by electrochemical impedance spectroscopy with various atmospheres and temperatures. The impedance spectra are deconvoluted by means of the distribution of the relaxation time method. The surface exchange rate and chemical diffusivity of H + and O 2– are characterized by electrical conductivity relaxation. The physical locations of electrochemical processes are also identified by atomic layer deposition with a surface inhibitor. A microkinetics model is proposed toward conductivities, triple-conducting pathways, reactant dependency, surface exchange and bulk diffusion capabilities, and other relevant properties. Lastly, the rate-limiting steps and suggestions for further improvement of electrode performance are presented.

25 ENERGY STORAGE↗

Molecular-Level Control over Ionic Conduction and Ionic Current Direction by Designing Macrocycle-Based Ionomers

Poor ionic conductivity of the catalyst-binding, submicrometer- thick ionomer layers in energy conversion and storage devices is a huge challenge. However, ionomers are rarely designed keeping in mind the specific issues associated with nanoconfinement. Here, we designed nature-inspired ionomers (calix-2) having hollow, macrocyclic, calix[4]arene-based repeat units with precise, sub-nanometer diameter. In ≤100 nm-thick films, the in-plane proton conductivity of calix-2 was up to 8 times higher than the current benchmark ionomer Nafion at 85% relative humidity (RH), while it was 1-2 orders of magnitude higher than Nafion at 20-25% RH. Confocal laser scanning microscopy and other synthetic techniques allowed us to demonstrate the role of macrocyclic cavities in boosting the proton conductivity. The systematic self-assembly of calix-2 chains into ellipsoids in thin films was evidenced from atomic force microscopy and grazing incidence small-angle X-ray scattering measurements. Moreover, the likelihood of alignment and stacking of macrocyclic units, the presence of one-dimensional water wires across this macrocycle stacks, and thus the formation of long-range proton conduction pathways were suggested by atomistic simulations. We not only did see an unprecedented improvement in thin-film proton conductivity but also saw an improvement in proton conductivity of bulk membranes when calix-2 was added to the Nafion matrices. Nafion-calix-2 composite membranes also took advantage of the asymmetric charge distribution across calix[4]arene repeat units collectively and exhibited voltage-gating behavior. The inclusion of molecular macrocyclic cavities into the ionomer chemical structure can thus emerge as a promising design concept for highly efficient ion-conducting and ion-permselective materials for sustainable energy applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗