Power-law Model Research Articles

Comparative Studies of Nonlinear Models and Their Applications to Magmatic Evolution and Crustal Growth of the Huai’an Terrane in the North China Craton

Power-law, inverse exponential and logarithmic models are widely used as empirical tools to describe anomalies in spatial and temporal geodynamic processes. However, the lack of clear interpretation of the relationships and distinctions among these models often makes their selection challenging, leaving them as empirical tools to be validated by data. This paper introduces these nonlinear functions derived from a unified differential equation, with parameters that reflect their relative nonlinearities and singularities, enabling their comparative application. By applying these functions to analyze magmatic events of the Huai’an Terrane, this study reveals two major crustal growth and reworking events between 2.6 and 1.7 Ga, each exhibiting distinctive nonlinear characteristics. The power-law function highlights strong nonlinearity and singularity during phases of intense magmatic activity, while logarithmic and exponential functions effectively characterize transitions between different tectonic processes. Geochemical data, including U-Pb zircon dating and Lu-Hf isotopic analyses, further validate the models by delineating distinct phases of crustal growth and reworking within the Trans-North China Orogen. The findings help connect the anomalies of frequency of magmatic events with the tectonic processes, providing important insights into the evolution processes of the North China Craton.

Constraining the Gamma-Ray Burst Jet Opening Angle Based on the Very Steep Decay Phase

Abstract Thanks to the rapid follow-up observations by the Swift X-Ray Telescope (XRT), a good part of gamma-ray bursts' (GRBs) high-latitude emission have been observed in the X-ray band. Some even show a dropdown decay after this period, which strongly indicates the edge of the jet is corresponding to the breaking time. This study constrains the jet opening angles of GRBs by analyzing the very steep decay phase in the early X-ray afterglow. Using data from Swift/XRT, we identified GRBs with significant breaks in their light curves and applied a broken power-law model to describe the decay phases. Assuming a spherical and isotropic emitting surface, we set constraints on the radiation radius (R γ ) to estimate jet opening angles (θ jet) from the breaking time. Our results indicate that jet opening angles can be constrained, although they are sensitive to the assumed radiation radius. This approach provides yet another method for estimating GRB jet opening angles.

Probing Cosmic Isotropy with the FAST All Sky H i Survey

Abstract This paper leverages the first released catalog from the FAST All Sky H i Survey (FASHI) to examine the hypothesis of cosmic isotropy in the local Universe. Given the design of the overall FAST survey, the inhomogeneous detection sensitivity of FASHI is likely to introduce significant biases in the statistical properties of the catalog. To mitigate the potential influence of spurious clustering effects due to these sensitivity variations, we focus on extragalactic H i sources within the sensitivity range of [0.65, 1.0]. This refined subsample is divided into 10 distinct sky regions, for which we compute the two-point angular correlation functions (2PACF) over angular scales of 0 . ° 5 < θ < 10°. We apply the Markov Chain Monte Carlo method to fit these 2PACFs with a power-law model and assess the statistical significance of the best-fit parameters for the 10 FASHI sky regions by comparing them against results from mock catalogs generated under the assumptions of homogeneity and isotropy. Our findings indicate that the local Universe, as traced by the H i sources in the FASHI survey, aligns with the cosmic isotropy hypothesis within a 2σ confidence level. We do not detect any statistically significant deviations from cosmic isotropy in the FASHI survey data.

A Pore-Scale Investigation of Oil Contaminant Remediation in Soil: A Comparative Study of Surfactant- and Polymer-Enhanced Flushing Agents

Pore-scale remediation investigation of oil-contaminated soil is important in several environmental and industrial applications, such as quick responses to sudden accidents. This work aims to investigate the oil pollutant removal process and optimize the oil-contaminated soil remediation performance at the pore scale to find the underlying mechanisms for oil removal from soil. The conservative forms of the phase-field model and the non-Newtonian power-law fluid model are employed to track the moving interface between two immiscible phases, and oil pollutant flushing removal process from soil pores is investigated. The effects of viscosity, interfacial tension, wettability, and flushing velocity on pore-scale oil pollutant removal regularity are explored. Then, the oil pollutant removal effects of two flushing agents (surfactant system and surfactant–polymer system) are compared using an oil content prediction curve based on UV-Visible transmittance. The results show that the optimal removal efficiency is obtained for a weak water-wetting system with a contact angle of 60° due to the stronger two-phase fluid interaction, deeper penetration, and more effective entrainment flow. On the basis of the dimensionless analysis, a relatively larger flushing velocity, resulting in a higher capillary number (Ca) in a certain range, can achieve rapid and efficient oil removal. In addition, an appropriately low interfacial tension, rather than ultra-low interfacial intension, contributes to strengthening the oil removal behavior. A reasonably high viscosity ratio (M) with a weak water-wetting state plays synergetic roles in the process of oil removal from the contaminated soil. In addition, the flushing agent combined with a surfactant and polymer can remarkably enhance the oil removal efficiency compared to the sole use of the surfactant, achieving a 2.5-fold increase in oil removal efficiency. This work provides new insights into the often-overlooked roles of the pore scale in fluid dynamics behind the remediation of oil-contaminated soil via flushing agent injection, which is of fundamental importance to the development of effective response strategies for soil contamination.

Exploring novel aspects of designed macromolecular structures on their interactions with cement and nano/microsilica particles and rheological properties

Various copolymers and novel branched terpolymers with different sulfonate ion and amine groups were prepared and characterized. Zeta potentials of cement slurries increased following the addition of synthesized polymers, indicating the effects of molecular weight, sulfonate ion and polyethylene glycol (PEG) branch. The rheology experiments revealed that it is possible to control the viscosity of cement slurries by varying the molecular structures of the resulting polymers. The formation of different networks through interactions between nano/microsilica and various polymer chains was accounted for the diversity in viscosity. The viscosity behavior of polymer-modified cement slurries was fit by the Generallized Newtonian Power-Law model. The results of SEM showed that polymer structures have an effect on the microstructure and morphology of cement, such as the dispersion, shape and size of hydrated calcium silicate (CSH) gels, as well as the dispersion of nano/microsilica particles. The presence of nano/microsilica particles along with various copolymers increased the hardness of the 1 and 7-day hydration cement. These results provide new ideas to control cement slurries properties using novel functional and branched polymers and nano/micro silica.

Hydrogen Production from Ethanol Steam Reforming by Stable LaNixCu1−xO3−λ Perovskite-Type Catalysts

Hydrogen production from ethanol steam reforming (ESR) was performed using the synthesized LaNixCu1−xO3−λ perovskite-type catalysts in a continuous two-stage fixed-bed reactor from 450 to 700 °C under atmospheric pressure. The elemental analysis (EA), XRD, SEM, BET, and TGA-DTG technologies were used to characterize the structures and properties of the synthesized catalysts. The thermodynamic equilibrium model, based on the minimization of Gibbs free energy using a non-stoichiometric methodology, was carried out and compared with experimental data. The results demonstrated that the catalytic activity of the perovskite-type catalysts for ESR can be improved after modification with a certain amount of copper (about 0.67 mmol/g) and decreased further with an increase in copper content (about 3.41 mmol/g). The most active catalyst was found to be LaNi0.9Cu0.1O3−λ, with an ethanol conversion value of 96.0% and hydrogen selectivity of 71.3%. The perovskite-type catalysts with an appropriate amount of Cu promoter improved coking resistance and presented excellent stability with no loss of activity over 101 h at 700 °C. Based on the power-law kinetic model with the first reaction order, the activation energy and the frequency factor for ethanol steam reforming by perovskite-type catalysts were calculated. Our studies indicated the enhanced effects of Ni and Cu on the small Ni-Cu bimetallic particles in the water gas shift (WGS) reaction, which could also contribute to the activity and stability of the LaNixCu1−xO3−λ perovskite-type catalysts in hydrogen production.

Changes in Functional Properties and In Vitro Digestibility of Black Tartary Buckwheat Starch by Autoclaving Combination with Pullulanase Treatment.

The processing properties of resistant starch (RS) and its digestion remain unclear, despite the widespread use of autoclaving combined with debranching in its preparation. In this study, the physicochemical, rheological and digestibility properties of autoclaving modified starch (ACB), autoclaving-pullulanase modified starch (ACPB) and native black Tartary buckwheat starch (NB) were compared and investigated. The molecular weight and polydispersity index of modified starch was in the range of 0.15 × 104~1.90 × 104 KDa and 1.88~2.82, respectively. In addition, the SEM results showed that both modifications influenced the morphological characteristics of the NB particles, and their particles tended to be larger in size. Autoclaving and its combination with pullulanase significantly increased the short-range ordered degree, resistant starch yield and water- and oil-absorption capacities, and decreased the syneresis properties with repeated freezing/thawing cycles. Moreover, rheological analysis showed that both ACB and ACPB exhibited shear-thinning behavior and lower gel elasticity as revealed by the power law model and steady-state scan. The degradation of starch chains weakened the interaction of starch molecular chains and thus changed the gel network structure. The in vitro digestion experiments demonstrated that ACB and ACPB exhibited greater resistance to enzymatic digestion compared to the control, NB. Notably, the addition of pullulanase inhibited the hydrolysis of the ACB samples, and ACPB showed greater resistance against enzymatic hydrolysis. This study reveals the effects of autoclaving combined with debranching on the processing properties and functional characteristics of black Tartary buckwheat starch.

Mathematical Modelling of Multiphase Hybrid Gyro-tactic Nanofluid Flow Through Porous Convergent Pipe with Injection and Suction Using BVP4c

This research focuses on enhancing fluid mobility by optimizing heat transfer, a crucial aspect in various industrial applications, including oil recovery. The study introduces an innovative framework that integrates microorganisms, hybrid nanoparticles, non-Newtonian fluid properties, a power law model, and inclined magnetic fields. The underlying dynamics are described by nonlinear partial differential equations, which are converted to ordinary differential equations using similarity transformation and subsequently solved through the BVP4c method. Key results demonstrate that fluid velocity increases with higher Reynolds, Hartman, Thermal Grashof, and Mass Grashof numbers due to factors such as reduced viscous drag, the Lorentz force’s acceleration effect, and enhanced buoyancy. On the other hand, a higher Prandtl number slightly reduces velocity, while an increased Schmidt number raises it by steepening the velocity gradient. Regarding temperature, higher Reynolds and Prandtl numbers, along with increased Eckert and Radiation parameters, result in elevated fluid temperatures due to enhanced convective heat transfer, decreased thermal diffusivity, viscous dissipation, and radiative heat effects. The insights gained from this study are valuable for improving oil extraction efficiency by identifying and manipulating key parameters that affect fluid behavior.

A deep study confirms diffuse nonthermal X-ray emission from the globular cluster Terzan 5

Diffuse X-ray emission has been detected from a few Galactic globular clusters (GCs), but its nature remains largely unclear. The GC Terzan 5 was previously found to show a significant diffuse thermal X-ray excess from its field, likely contributed by the Galactic background, and a nonthermal component described by a power-law model with photon index $ With over 16 times the accumulated exposure time compared to a prior study, we reexamined and verified the diffuse X-ray emission from the field of which enabled us to place constraints on its nature. We analyzed all available useful observations of including 18 observations over a span of 13 years, with a total exposure time of 641.6 ks. To study the diffuse X-ray emission, we focused on four annular regions with an equal width of 0.72 arcmin centered on (0.72--3.60 arcmin), from which we extracted and analyzed the X-ray spectra after removing point sources and instrumental backgrounds. We confirm a significant diffuse X-ray excess from the field of in the band 0.8--3 keV. After constraining the contribution from the local X-ray background, we find a diffuse X-ray component that is genuinely associated with and can be well described by a power-law model. More interestingly, the fitted photon indices show a significant increase from $ 0.18$ in the inner region to $ 0.71$ in the outer region. The diffuse X-rays are also well fit by a thermal bremsstrahlung model, with plasma temperatures declining from $kT 3$ keV to $kT 1$ keV. We suggest that synchrotron radiation from the combined pulsar winds of millisecond pulsar population is a possible origin of the observed diffuse X-ray emission but that the sharp steepening in the spectra cannot be produced solely by synchrotron cooling. Other radiation processes, like thermal bremsstrahlung, may also contribute to the diffuse X-rays.

Effects of γ-polyglutamic acid on the rheological, microstructural and sensory properties of low-fat yogurt.

While low-fat yogurt offers numerous health benefits, its texture and sensory qualities are poor. This study aimed to investigate the effects of γ-polyglutamic acid (γ-PGA) on the rheological, microstructural and sensory properties of low-fat yogurt using rheological tests, scanning electron microscopy (SEM) and sensory evaluation. The results showed that the syneresis of low-fat yogurt added with 0.15% γ-PGA was significantly (P < 0.05) reduced to 23.03%, compared to that of low-fat yogurt without γ-PGA (42.87%). An improvement in storage and loss moduli (G', G") and apparent viscosity was also observed. The power law model fitted to rheological data indicated that γ-PGA enhanced the viscoelasticity and strength of the gel network though interaction with casein gel, which might lead to changes in the moduli of yogurts. Microstructural observation via SEM confirmed the enhanced crosslinking between casein micelles and the filling effect of γ-PGA. Sensory liking scores were strongly correlated with rheological properties and apparent viscosity. The results of principal component analysis (PCA) confirmed the positive effects of γ-PGA on quality and rheological attributes of low-fat yogurt. The results of this study provided valuable references for the potential use of γ-PGA as a fat replacer to stabilize the sensory quality of low-fat yogurt. © 2024 Society of Chemical Industry.

Optimization of levulinic acid esterification via surface response methodology using single metal oxide nanoparticle as heterogeneous catalyst

Increasing energy demands have led to the depletion of fossil fuels, instigating the need for alternative and greener energy sources. The present study focuses on the esterification of levulinic acid into value-added products like butyl Levulinate using heterogeneous sulfated-Al2O3 nanocatalyst. The synthesis method used is very simple and robust and the catalyst possesses high catalytic activity. Its physicochemical properties were studied using various characterization techniques, which indicated its nanocrystalline morphology and the presence of sulfonated active sites. The esterification reaction parameters such as mole ratio, catalyst loading, and time were optimized using JMP PRO 17 focused on the Box-Behnken design. The optimum values were found to be 1:8 (LA: BOH), 1.32 wt.% catalyst loading, and 6 h with 81.04% levulinic acid conversion for the reaction with sulfated-Al2O3 catalyst. Response surface plots were used for evaluating the experimental data, which was subsequently plugged into a complete quadratic regression model. After conducting kinetic investigations, the results of the experiment were found to suit the pseudo-second-order power law model, yielding an activation energy of 26.24 ± 1.1 kJ/mol.

Detection of Diffuse Hot Gas around the Young, Potential Superstar Cluster H72.97–69.39

We present the first Chandra X-ray observations of H72.97–69.39, a highly embedded, potential superstar cluster in its infancy located in the star-forming complex N79 of the Large Magellanic Cloud. We detect particularly hard, diffuse X-ray emission that is coincident with the young stellar objects identified with JWST, and the hot gas fills cavities in the dense gas mapped by the Atacama Large Millimeter/submillimeter Array. The X-ray spectra are best fit with either a thermal plasma or power-law model, and assuming the former, we show that the X-ray luminosity of L X = (1.0 ± 0.3) × 1034 erg s−1 is a factor of ∼20 below the expectation for a fully confined wind bubble. Our results suggest that stellar wind feedback produces diffuse hot gas in the earliest stages of massive star cluster formation and that wind energy can be lost quickly via either turbulent mixing followed by radiative cooling or by physical leakage.

Revisiting the conundrum of the sub-Jovian and Neptune desert

Context. The search for exoplanets has led to the identification of intriguing patterns in their distributions, one of which is the so-called sub-Jovian and Neptune desert. The occurrence rate of Neptunian exoplanets with an orbital period P ≲ 4 days sharply decreases in this region in period-radius and period-mass space. Aims. We present a novel approach to delineating the sub-Jovian and Neptune desert by considering the incident stellar flux F on the planetary surface as a key parameter instead of the traditional orbital period of the planets. Through this change of perspective, we demonstrate that the incident flux still exhibits a paucity of highly irradiated Neptunes, but also captures the proximity to the host star and the intensity of stellar radiation. Methods. Leveraging a dataset of confirmed exoplanets, we performed a systematic analysis to map the boundaries of the sub-Jovian and Neptune desert in the (F, Rp) and (F, Mp) diagrams, with Rp and Mp corresponding to the planetary radius and mass, respectively. By using statistical techniques and fitting procedures, we derived analytical expressions for these boundaries that offer valuable insights into the underlying physical mechanisms governing the dearth of Neptunian planets in close proximity to their host stars. Results. We find that the upper and lower bounds of the desert are well described by a power-law model in the (F, Rp) and (F, Mp) planes. We also obtain the planetary mass-radius relations for each boundary by combining the retrieved analytic expressions in the two planes. This work contributes to advancing our knowledge of exoplanet demographics and to refining theoretical models of planetary formation and evolution within the context of the sub-Jovian and Neptune desert.

The Statistical Analysis of GeV Spectral Breaks in Bright Gamma-Ray Flat-spectrum Radio Quasars

We present the statistical results of GeV spectral breaks of bright gamma-ray flat-spectrum radio quasars (FSRQs) in the energy range of 0.1–10 GeV based on New Pass 8 data of the Large Area Telescope on board Fermi Gamma-Ray Space Telescope. We have fitted the 15 yr average gamma-ray spectra of 755 FSRQs by using both the broken power law (BPL) and logarithmic parabola (LP) models, and obtained 87 bright gamma-ray FSRQs with their integrated photon fluxes greater than 2.16 × 10−8 cm−2 s−1. From our results, the FSRQ population shows similar preferences for both the BPL and LP models in gamma-ray spectral fitting, and the clustering analysis suggests that BPL-preferred and LP-preferred FSRQs belong to the same category. Our results indicate that GeV spectral breaks in bright gamma-ray FSRQs are located at 2.90 ± 1.92 GeV in the rest frame, and the observed change in photon index is Δγ = 0.45 ± 0.19, which is consistent with the expected value for a cooling break of electrons scattering seed photons.

Investigation of the Nonthermal X-Ray Emission from the Supernova Remnant CTB 37B Hosting the Magnetar CXOU J171405.7-381031

We present a detailed X-ray investigation of a region (S1) exhibiting nonthermal X-ray emission within the supernova remnant (SNR) CTB 37B hosting the magnetar CXOU J171405.7−381031. Previous analyses modeled this emission with a power law (PL), inferring various values for the photon index (Γ) and absorbing column density (N H). Based on these, S1 was suggested to be an SNR shell, a background pulsar wind nebula, or an interaction region between the SNR and a molecular cloud. Our analysis of a larger data set favors a steepening (broken or curved PL) spectrum over a straight PL, with the best-fit broken power-law (BPL) parameters of Γ = 1.23 ± 0.23 and 2.24 ± 0.16 below and above a break at 5.57 ± 0.52 keV, respectively. However, a simple PL or srcut model cannot be definitively ruled out. For the BPL model, the inferred N H = (4.08 ± 0.72) × 1022 cm−2 towards S1 is consistent with that of the SNR, suggesting a physical association. The BPL-inferred spectral break ΔΓ ≈ 1 and hard Γ can be naturally explained by a nonthermal bremsstrahlung (NTB) model. We present an evolutionary NTB model that reproduces the observed spectrum, which indicates the presence of subrelativistic electrons within S1. However, alternate explanations for S1, an unrelated PWN or the SNR shock with unusually efficient acceleration, cannot be ruled out. We discuss these explanations and their implications for gamma-ray emission from CTB 37B and describe future observations that could settle the origin of S1.

Study on the Micromechanical Interface Response Behavior of Propellants Based on Nano-Impact Testing

Abstract This study utilizes a nano-impact experimental platform to investigate the stress-strain response of propellant interfaces with two different binders, HTPE and GAP. The mechanical behavior of HTPB-AP and GAP-AP interfaces was examined at the nano-scale under varying strain rates, with experiments conducted at rates up to 100 s−1. These experiments successfully captured the strain rate-dependent mechanical properties of both propellant components and their interfaces. The experimental results align well with the rate-dependent power-law viscoplastic constitutive models developed for HTPE, HTPE/AP, GAP, and GAP/AP interfaces, validating the model’s effectiveness in describing the viscoplastic behavior of these materials and their interfaces. The study demonstrates that interfaces generally exhibit lower stress responses compared to bulk materials. HTPE shows higher initial stress responses and more pronounced strain hardening than GAP.

Rheology of polyacrylamide-based fluids and its impact on proppant transport in hydraulic fractures

Rheological experiments and measurements of particle settling velocity under static conditions were conducted for two types of polyacrylamide (PAA-based) fluids. The flow behavior of the viscoelastic fluids is described by a simplified, three-parameter model that integrates a constant viscosity at low shear rates with a power-law viscosity dependency at higher shear rates. The estimated dependencies of the rheological parameters and particle settling velocity on PAA concentration align with the classical power-law scaling for semi-dilute polymer solutions. The identified scaling offers valuable insight for optimizing the chemical composition of fracturing fluids, thereby improving the efficiency of hydraulic fracturing technology. By applying the lubrication approximation to the Navier–Stokes equations, a set of equations for the suspension flow in a vertical plane channel, characterized by the three-parameter rheology model, was derived. In typical fracturing conditions, the conventional power-law viscosity model used in current fracturing simulators significantly overestimates the gap-averaged apparent fluid viscosity compared to the proposed model. The novel model of the suspension flow aims to enhance the accuracy of proppant transport models applied in hydraulic fracturing simulators. Based on the three-parameter rheology model, a new model of particle settling in the studied PAA-based fluids is developed. It is based on smart Stokes' law, where viscosity is calculated according to either the plateau or power-law region, depending on the self-induced shear rate. The new model more accurately approximates experimentally determined settling velocity of proppant under static conditions across a broad range of PAA concentrations than existing models.

Utilising terahertz pulsed imaging to analyse the anhydrous-to-hydrate transformation of excipients during immediate release film coating hydration.

Pharmaceutical tablets are routinely film-coated to improve appearance, reduce medication errors and enhance storage stability. Terahertz pulsed imaging (TPI) can be utilised to study the liquid penetration into the porous tablet matrix in real time. Using polymer-coated flat-faced tablets with anhydrous lactose or mannitol, we show that when the tablet matrix contains anhydrous material, the anhydrous form transforms to the solid-state hydrate form in the tablet core while the immediate release coating dissolves. The process starts at the interface between the coating and the core. It commences inward for more than 7min, resulting in a shell of hydrate form on the order of a few hundred microns thicknesses immediately underneath the dissolving coating. TPI is sensitive enough to detect and spatially resolve this process based on the subtle change in the refractive index of the two materials. TPI can further resolve tablets with different components using the time-domain waveforms in reflection and provide insights into the presence of hydrate material (lactose monohydrate) in altering the water mass transport mechanism. The kinetics data obtained from TPI fit the power law model (y=ktm), whose constants enable us to infer the different mass transport mechanisms.

FLOW ANALYSIS IN A ROLLER DRILL BIT DURING DRILLING USING WATER-BASED AND HYDROCARBON-BASED MUD

Drilling is the main type of increase in hydrocarbon production. Bits of various types are used as a rock-destroying tool during drilling. When drilling any wells for oil and gas, drilling fluids are used as a working fluid. The flow of these types of liquids differs from the flow of water, which is an incompressible medium. The purpose of this work is to study the flow of the Newtonian fluid of water and two types of water-based drilling fluids, which is described by the power-law model of a non-Newtonian fluid and the hydrocarbon-based fluid of the Herschel-Bulkley type. In the work, the construction of the geometric model of the square bit was carried out, and the estimated unstructured grid of the liquid volume filling the internal area of the bit and the space behind the bit was constructed. Calculations of the three-dimensional flow of water, drilling fluids on water and hydrocarbon bases were carried out using the open platform OpenFOAM. It was found that during the flow of liquids described by non-Newtonian models, the kinematic viscosity of the liquid changes depending on the velocities and shear stresses. Another important factor in the use of non-Newtonian fluids when drilling wells is the reduction of hydraulic losses during their flow. This is achieved due to the presence of a certain structure of the liquid, non- zero values of shear stresses, lubricating properties even with their viscosity, which is ten times higher than the viscosity of water. Visualization of the flow of three types of fluids: Newtonian, non-Newtonian power-law and non-Newtonian Herschel-Bulkley type is presented. The use of non- Newtonian fluids makes it possible to reduce the formation of vortices and, as a result, also affects the amount of hydraulic losses in the direction of their reduction.

Accretion disc dynamics in extragalactic black hole X-ray binaries: a comprehensive study of M33 X–7, NGC 300 X–1, and IC 10 X–1

ABSTRACT Extragalactic black hole X-ray binaries (BH-XRBs) are the most intriguing X-ray sources as some of them are ‘home’ to the most massive stellar-mass BHs ever found. In this work, we conduct a comprehensive study of three massive, eclipsing extragalactic BH-XRBs i.e. M33X-7, NGC300X-1, and IC10X-1 and using entire X-ray observations available from XMM–Newton and NuSTAR till date. Preliminary analysis using diskbb and power-law models shows that the sources have steep spectra and sub-Eddington luminosities (L&amp;lt;0.69 L$_{\mathrm{ Edd}}$), with major flux contribution from non-thermal component, resembling the relatively uncharted steep power-law state (SPL). To understand the accretion disc properties in this state, we explore alternate modelling scenario that reveals the presence of a ‘hot’ ($kT_{\mathrm{ in}}=1\!-\!2$ keV) slim-disc (diskpbb) with radial temperature profile $T(r)\propto r^{-p}$ ($p=0.5\!-\!0.66$), along with a cooler ($kT_{\mathrm{ in}}=0.1\!-\!0.2$ keV) standard thermal disc (diskbb). We carry out the continuum-fitting method using relativistic slim-disc model (slimbh) and estimate the mass range of M33 X–7, NGC300X-1, and IC10X-1 is to be 9–15 M$_{\odot }$, 9–28 M$_{\odot }$, and 10–30 M$_{\odot }$, respectively. Further, eclipse periods are determined by modelling the light curve, using which we estimate the size of the eclipsing bodies. Modelling of the eclipse spectra revealed the complete obscuration of soft spectral component during eclipse, implying the emission of hard component from an extended accretion region. Based on our findings, we provide an inference on geometry of accretion disc in these wind-fed systems and compare their properties with the other two extragalactic BH-XRBs.