Results for “Coefficient of variation”

DOE OSTI2022

Smoothed aggregation for difficult stretched mesh and coefficient variation problems

Abstract Four adaptations of the smoothed aggregation algebraic multigrid (SA‐AMG) method are proposed with an eye toward improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill‐in within the coarse discretization matrices and can also give rise to suboptimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi‐like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.

Hu, Jonathan J.↗

DOE OSTI2022

Smoothed Aggregation for Difficult Stretched Mesh and Coefficient Variation Problems.

Abstract not provided.

Tuminaro, Raymond↗

DOE OSTI2023

Rigorizing the use of the coefficient of variation to diagnose fracture periodicity and clustering

Not Available

02 PETROLEUM↗

DOE OSTI2023

Sensitivity Analysis of a Polyphase Wireless Power Transfer System under Off-Nominal Conditions

Misaligned and/or variable coil airgaps cause coupling coefficient variation in wireless power transfer (WPT) systems, resulting in a decrease in the system’s transmitted power and efficiency. This paper presents sensitivity analyses of a three-phase, Y-Y connected, series-tuned WPT system in the frequency domain in terms of several different electric vehicle wireless charging off-nominal conditions (misalignments in Δ x and Δ y directions, change of airgap Δ z , and the roll Δψ, pitch Δθ, and yaw ΔΦ angles) as specified in Society of Automotive Engineers (SAE) J2954 Standard. Coil inductance matrices were obtained by measuring the self- and mutual inductances of the primary and secondary coil phase windings at variable airgap classes (from 5 cm to 30 cm) for five different charging positions as identified in SAE J2954. These 6 × 6 inductance matrices were used in sensitivity and Plexim/PLECS simulation analyses. The sensitivity of the WPT system was analyzed analytically using the input impedance and phase angle, voltage gain, current gain, quality factor, and coupling coefficient parameters of the series-tuned WPT system. The results were confirmed experimentally on a 50 kW WPT system.

42 ENGINEERING↗

DOE OSTI2021

Modular Design of Receiver Side Power Electronics for 200 kW High Power Dynamic Wireless Charging System

Dynamic wireless power transfer (DWPT) can provide energy to EVs in motion and extend the drive range. By upscaling the charging power to 200 kW (High Power DWPT), the percentage of electrified roadway reduces and the solution becomes cost-effective. However, coil coupling-coefficient variation during vehicle movement fluctuates the transferred power which is unfavorable for vehicle battery. Secondary regulation design can smooth the power but the converter design becomes very challenging due to requirement in high power, high efficiency, fast control, as well as high power density since the unit will be onboard. This paper provides the modular design approach of a 200 kW secondary side unit to achieve high performance and scalability. The DC/DC converter using SiC devices demonstrated 98.3% efficiency.

Xue, Lincoln↗

DOE OSTI2022

Role of Three-Dimensional Swirl in Forced Convection Heat Transfer Enhancement in Wavy-Plate-Fin Channels

The influence of wall-corrugation-induced swirl flow on enhanced forced convection in wavy-plate-fin cores has been investigated. Three-dimensional computational simulations were carried out for steady-state periodically developed air flow (Pr ~ 0.71; 50 ≤ Re ≤ 4000) with channel walls subject to constant-uniform temperature conditions. The recirculation that develops in the wall troughs and grows to have an axially helical character is scaled by the Swirl number Sw. As Sw increases with higher flowrate and/or corrugation severity, tornado-shaped vortices appear in the wave trough region midway of the interfin channel height, then extend longitudinally to encompass majority of the flow channel. The local wall-shear and heat transfer coefficient variations indicate that boundary-layer thinning upstream of the wave peak aids in intensifying momentum and heat transfer. However, the flow recirculation in wall trough impedes heat transfer at low Sw due to flow stagnation but promotes it at high Sw because of the vortices-induced augmented fluid mixing. The effects of this secondary flow are quantified by Φf(or j), which is seen to increase log-linearly as fin corrugation aspect ratio γ and/or fin spacing ratio ζ increases; the influence of cross section aspect ratio α is marginal. Moreover, the pressure drag penalty due to swirl critically affects overall pressure loss, and its proportion remains nearly constant when α varies, but grows as Sw, γ, and/or ζ increases and can be as much as 80% of the total pressure drop.

Engineering↗

DOE OSTI2020

A Comparison of Groundwater Sampling Technologies, Including Passive Diffusion Sampling, for Radionuclide Contamination - 20205

Using traditional high-flow purge methods for long-term water quality monitoring of deep groundwater wells can be expensive, affect contaminant migration, and produce excessive volumes of discharge water that can be difficult to manage. The use of low-flow pumping methods and depth discrete bailers (DDBs) can reduce the cost of sampling deep groundwater wells. In general, using different pumping methods to obtain reproducible and representative groundwater can be challenging. Passive diffusion samplers (PDSs) have successfully been used in long-term monitoring for volatile organic compounds, major ions, and trace-elements, but application has been limited for stable and radioactive isotopes. Beginning in 2018, the United States Geological Survey (USGS) conducted three sampling events to test the ability of PDSs and DDBs to obtain reproducible and representative groundwater samples. The sampled well is completed in a regional, permeable carbonate aquifer and is one in a set of wells that have historically been used for tritium tracer testing. All samples were obtained at 180 m (590 ft) below land surface (bls) in a 13.97 cm (5.5 inch) uncased well that has a total depth of 202 m (662 ft) bls. The first sampling event deployed a regenerated cellulose dialysis membrane (RCDM) PDS for 14 days with deionized water as the blank. The second sampling event deployed a RCDM for 27 days also with deionized water as the blank. The third sampling event deployed a Dual Membrane (DM) PDS for 65 days using a blank of tritium-dead carbonate water. The DM PDS was used to assess the effect of longer term deployment on tritium concentrations and address whether or not the PDSs reached equilibrium with ambient groundwater. The day after each of the three passive samplers were retrieved a DDB was used to obtain discrete non-integrated groundwater samples. For each DDB sampling day, the bailer was lowered into the well 10 consecutive times to determine if the water chemistry changed from the first to the last bailed sample. Quality assurance samples including blanks and duplicates were obtained during all three sampling events. All blank waters had tritium concentrations less than 21±33 pCi/L. Major ion (e.g. calcium, chloride, sodium, and sulfate) results were compared between all samples obtained with RCDM and DDB. Major ion concentrations showed a coefficient of variation of less than 6% between all RCDM and DDB samples; however, the coefficient of variation between the different deployment times and the two different methods for trace-element concentrations was much larger, particularly for manganese (82%), lead (41%), and zinc (33%). Stable isotope values were compared between the RCDM and DDB samples. The DDB sample results all fell within analytical uncertainty and were considered representative of the formation groundwater. The stable isotope values from the RCDM samples indicated that a longer deployment time was necessary to gain equilibrium and to obtain representative groundwater samples. Tritium results from groundwater samples obtained from the RCDM, DM, and DDB indicate that groundwaters obtained with PDSs produced tritium concentrations 2 to 3 times higher (between 1,426 and 3,060±87 pCi/L) than groundwaters obtained with a DDB (479 to 1,219±51 pCi/L). The longer the passive diffusion samplers were deployed, the higher the tritium concentration, suggesting that equilibrium with tritium was not reached within a 27-day deployment. DDB samples showed tritium results declining from the first to the last bailed sample for all three sampling events. This research suggests that tritium results from groundwater samples obtained from PDSs are more reproducible than samples obtained from DDBs. Also, PDSs likely do not accumulate isotopes of water but rather equilibrate with the ambient groundwater. On the other hand, both PDSs and DDBs were able to provide representative groundwater samples for major ions and have the potential to produce representative groundwater samples for stable isotopes of water (δD and δ{sup 18}O). Utilizing the most effective method for sampling groundwater is important for long-term groundwater monitoring. Quantifying the ability of passive diffusion groundwater samplers to obtain representative radioactive chemistry samples will help determine if these methods can be a useful alternative to pumping and purging wells for long-term monitoring. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2023

An Experimental Comparison of Cyclic Variations in Diesel–Natural Gas and POMDME–Natural Gas Dual Fuel Combustion

Abstract Cyclic variations in internal combustion engines are caused by various factors, including combustion mixture stratification, in-cylinder flows, local fluctuations in air-fuel ratio, etc. Cyclic variations have a profound impact on engine performance and emissions. In this study, cyclic variations in dual fuel combustion are analyzed, comparing diesel–natural gas (NG) and polyoxymethylene dimethyl ether (POMDME)-NG dual fuel combustion. Cyclic variability was initially quantified using the coefficient of variation of gross indicated mean effective pressure (IMEPg) computed from experimental cylinder pressure data. The cases analyzed in this study had a coefficient of variation (COV) of IMEPg greater than or around 5%, which was the lower limit of onset of instability for this engine. Experiments were performed at two fixed start of injection (SOI) of high-cetane fuel: 310 CAD and 350 CAD. For all experiments, a constant load of 5 bar IMEPg was maintained, and the intake boost pressure and rail pressure were fixed at 1.5 bar and 500 bar, respectively. For each case, 1000 cycles of cylinder pressure data were recorded, filtered, and processed using an in-house heat release analysis code for each cycle. A comparison between individual cycles and the “ensemble averaged cycle” was made for both diesel–NG and POMDME–NG combustion. For the early SOI of 310 CAD, the peak cylinder pressure fluctuations of individual cycle were found to be ± 15 bar for both fuel combinations, compared to the ensemble averaged cycle, and < 1/10th of the cycles had an IMEPg lower than 0.05 bar of the ensemble averaged cycle. However, the peak pressure fluctuations were found to be lower for POMDME–NG (±3 bar) than diesel–NG dual fuel combustion at 350 CAD SOI, indicating lower cyclic variations. The higher reactivity of POMDME helped reduce fluctuations in combustion phasing at the retarded SOI. The presence of cycles of deterioration and cycles of recovery were also observed with diesel–NG combustion for 310 CAD SOI, and the scatter in the IMEPg return map was similar for both fuel combinations. The IMEPg return map for POMDME–NG combustion was less scattered at the 350 CAD SOI.

Engineering↗

DOE OSTI2022

Uncertainty quantification in elastic constants of SiC f /SiC m tubular composites using global sensitivity analysis

Silicon carbide fiber and silicon carbide matrix (SiC f /SiC m ) tubes produced through the chemical vapor infiltration process have become a candidate cladding material in nuclear applications. The performance of this composite is influenced by many variables such as braiding angle, porosity, material properties, etc., which vary over a range of values due to the inherent fluctuations in the manufacturing process. In this study, the variability in elastic constants of SiC f /SiC m composite has been quantified through multiscale finite element (FE) simulations, variable screening, and high-fidelity surrogate modeling. The key variables dominantly affecting the elastic constants of SiC f /SiC m tubes were identified using global sensitivity analysis. A surrogate to the high-fidelity FE-based model was used in Monte Carlo simulations to generate a hundred thousand samples from which the uncertainty in elastic constants was assessed. It turned out that the coefficient of variation was less than 10%.

Materials Science↗

DOE OSTI2024

In‐mold rheology and automated process control for injection molding of recycled polypropylene

Abstract Manufacturing plastic parts with secondary feedstocks has risen to the forefront of importance in recent years. However, the variation in molecular weight and rheology of secondary feedstock can lead to inconsistent part quality. This work evaluates the effectiveness of a novel closed‐loop adaptive process control system that adjusts nozzle pressure in response to in‐mold pressure data. Five different recycled polypropylene blends, with a broad distribution of flow properties, were evaluated to determine the effectiveness of the control system at reducing processing variation. The experimental results show that the process control strategy reduced the variation within the mold, as seen by in‐mold pressure curves and calculated in‐mold viscosity values. Additionally, the parameters that control the automated process adjustments were investigated, showing the importance of optimization. The analysis of the correlation between in‐mold rheology and mechanical properties showed a slight variation in the mechanical properties and parts weight with a coefficient of variation of under 5%. Overall, the results demonstrate the ability of pressure‐controlled molding and automated viscosity adjustment to reduce the variability when molding a secondary feedstock. Highlights Pressure‐controlled injection molding of recycled polypropylene. Automated closed‐loop adaptive process control methodology. Methodology resulted in a reduction in pressure variation during molding. Changes in mechanical properties and in‐mold viscosity were investigated. Results show the potential of pressure‐controlled molding at reducing variation.

Krantz, Joshua↗

DOE OSTI2023

Synthesis of high-entropy carbides from multi-metal polymer precursors

Three compositions of high-entropy carbides, (TiHfVNbTa)C, (TiZrHfNbTa)C and (TiZrNbTaW)C were synthesized via a modified Pechini process, in which citric acid served as both a cation chelating agent and a carbon source. Through pyrolysis and spark plasma sintering, single phase high-entropy carbides were formed from the homogeneous precursors at a relatively low temperature of 1800 °C. The dispersion of cations in the polymer precursor facilitated shorter diffusion distances in polymer-derived materials, and thus compositional homogeneity was significantly improved relative to materials produced by a solid-state method, as quantified by a defined coefficient of variation applied to energy-dispersive X-ray spectroscopy. Finally, a finer microstructure in polymer-derived materials results in improved fracture toughness with a K IC value of 4.29 MPa·m 1/2 achieved for (TiHfVNbTa)C.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Variability and associated uncertainty in image analysis for soiling characterization in solar energy systems

The accumulation of soiling on photovoltaic modules and on the mirrors of concentrating solar power systems causes non-negligible energy losses with economic consequences. These challenges can be mitigated, or even prevented, through appropriate actions if the magnitude of soiling is known. Particle counting analysis is a common procedure to characterize soiling, as it can be easily performed on micrographs of glass coupons or solar devices that have been exposed to the environment. Particle counting does not, however, yield invariant results across institutions. The particle size distribution analysis is affected by the operator of the image analysis software and the methodology utilized. The results of a round-robin study are presented in this work to explore and elucidate the uncertainty related to particle counting and its effect on the characterization of the soiling of glass surfaces used in solar energy conversion systems. An international group of soiling experts analyzed the same 8 micrographs using the same open-source ImageJ software package. The variation in the particle analyses results were investigated to identify specimen characteristics with the lowest coefficient of variation (CV) and the least uncertainty among the various operators. The mean particle diameter showed the lowest CV among the investigated characteristics, whereas the number of particles exhibited the largest CV. Additional parameters, such as the fractional area coverage by particles and parameters related to the distribution's shape yielded intermediate CV values. These results can provide insights on the magnitude inter-lab variability and uncertainty for optical and microscope-based soiling monitoring and characterization.

14 SOLAR ENERGY↗

DOE OSTI2023

Acute stress reduces population-level metabolic and proteomic variation

Abstract Background Variation in omics data due to intrinsic biological stochasticity is often viewed as a challenging and undesirable feature of complex systems analyses. In fact, numerous statistical methods are utilized to minimize the variation among biological replicates. Results We demonstrate that the common statistics relative standard deviation (RSD) and coefficient of variation (CV), which are often used for quality control or part of a larger pipeline in omics analyses, can also be used as a metric of a physiological stress response. Using an approach we term Replicate Variation Analysis (RVA), we demonstrate that acute physiological stress leads to feature-wide canalization of CV profiles of metabolomes and proteomes across biological replicates. Canalization is the repression of variation between replicates, which increases phenotypic similarity. Multiple in-house mass spectrometry omics datasets in addition to publicly available data were analyzed to assess changes in CV profiles in plants, animals, and microorganisms. In addition, proteomics data sets were evaluated utilizing RVA to identify functionality of reduced CV proteins. Conclusions RVA provides a foundation for understanding omics level shifts that occur in response to cellular stress. This approach to data analysis helps characterize stress response and recovery, and could be deployed to detect populations under stress, monitor health status, and conduct environmental monitoring.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Impedance Response Influenced by Variability in the Random Distribution of Physical Properties of Coated Materials in Two-Dimensional Space

Heterogeneous physical characteristics of a system featuring a single-layer film on a metallic surface have been explored via its impedance response. The Nyquist plot showed a distorted semicircle, indicative of the system's unique distribution characteristics. Utilizing a copula-based probability method, a two-dimensional deterministic impedance model was successfully integrated, accounting for spatial physical properties such as permittivity and electrical conductivity. This strategy enabled in-depth exploration and mechanistic quantification of a broad spectrum of properties. A quantitative understanding of impedance signal alterations, characterized by normally or log-normally correlated variables, was achieved through the variation in aspect ratio and characteristic frequency of the impedance spectra. Log-normally distributed electrical properties provided a superior representation of the distorted impedance spectra. As coefficient of variation (CV) values fluctuated, the aspect ratio and characteristic frequency showed heightened sensitivity to log-normal permittivity compared to log-normal electrical conductivity. Notably, a marked positive linear correlation between electrical properties resulted in an impedance response that approximated perfect semicircular spectra. The variability in the electrical properties' distribution was demonstrated by considering the correlation coefficient between electrical conductivity and the z-direction position. Furthermore, the highest aspect ratio of the impedance spectra was observed when the electrical conductivity was randomly distributed across the z-direction space.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

In-Situ Magnetic Field Reconstruction in the MAGIS-100 Experiment

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.; Ferm↗

DOE OSTI2022

A matrix completion algorithm for efficient calculation of quantum and variational effects in chemical reactions

This work examines the viability of matrix completion methods as cost-effective alternatives to full nuclear Hessians for calculating quantum and variational effects in chemical reactions. The harmonic variety-based matrix completion (HVMC) algorithm, developed in a previous study (https://doi.org/10.1063/5.0018326), exploits the low-rank character of the polynomial expansion of potential energy to recover, using a small sample, vibrational frequencies (square roots of nuclear Hessian eigenvalues) constituting the reaction path. Furthermore, these frequencies are essential for calculating rate coefficients using variational transition state theory with multidimensional tunneling (VTST-MT). HVMC performance is examined for four SN2 reactions and five hydrogen transfer reactions, with each H-transfer reaction consisting of at least one vibrational mode strongly coupled to the reaction coordinate. HVMC is robust and captures zero-point energies, vibrational free energies, zero-curvature tunneling, and adiabatic ground state and free energy barriers as well as their positions on the reaction coordinate. For medium to large reactions involving H-transfer, with the exception of the most complex Ir catalysis system, less than 35% of total eigenvalue information is necessary for accurate recovery of key VTST-MT observables.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗