Parabolic Field Research Articles

Rate dependency and fragmentation response of phase field models with micro inertia and micro viscosity terms

We present elliptic, parabolic, and hyperbolic phase field (PF) equations, referred to as EPF, PPF, and HPF, for cohesive zone model (CZM) and Linear Elastic Fracture Mechanics (LEFM) PF models. The micro viscosity and micro inertia terms result in PPF and HPF, that can be solved explicitly in time. The additional advantage of the HPF is implying a finite damage propagation speed and a more favorable time advance limit. The desired micro-viscosity for the HPF is shown to correspond to a damping factor of one. The appropriate length and time scales, nondimensional equations, and (asymptotic) strain-stress solutions are provided for each differential equation. Two sources of rate sensitivity are discussed. First, when the fields are spatially uniform (0D solution) the rate effect arises from the form of differential equation. Asymptotic solutions show that at high loading rates, the energy dissipation has the rates of 2/3 and 1 for the PPF and HPF, respectively, with the former matching the Grady’s rate-sensitivity. The dynamic strength and failure strain show half of this rate. The analysis is extended to damage models with stress-based and bounded driving forces. Second, the rate effect arises from the deviation of 1D from 0D solutions as the damage exceeds a critical value and fragments form. Since this critical value tends to zero for quasi-static loading, this rate-effect is mainly present for low loading rates. This results in lower- and upper-shelf energy limits for the EPF at low and high loading rates, resembling some experimentally observed sigmoidal rate models for energy dissipation.

Quantifying the role of electron plateau distribution on the chorus gap formation using one-dimension PIC simulations

Although ubiquitous in the Earth’s inner magnetosphere, how chorus waves with a gap in intensity near half the electron cyclotron frequency, 0.5fce, is formed remains to be understood. One hypothesis invokes the often-observed feature in electron phase space density where two anisotropic populations of warm and hot temperatures, respectively, are separated by a relatively isotropic beam-like component, called the parallel plateau distribution. According to linear theory, its mean velocity is such that the plateau population can lead to a sufficient cyclotron damping in a narrow frequency range near 0.5fce. To test this hypothesis more quantitatively, we carry out a series of one-dimensional particle-in-cell simulations in a parabolic magnetic field with observationally constrained plateau density and mean velocity. We find that even though the chorus generation involves nonlinear physics, the gap formation is largely determined by the linear properties of plateau electrons. In addition, given a sufficiently large plateau density with a right combination of warm and/or hot source populations, our simulations were able to generate lower-band-only chorus, upper-band-only chorus, banded chorus with partial damping at the gap, and banded chorus with a strong power gap. When it comes to comparing with observations, however, we find that (1) the minimum plateau density required to generate banded chorus is somewhat larger than that observed; (2) even with the largest plateau density used, the gap formation in the simulations is not consistent; and (3) most importantly, the dependence of the gap frequency on the plateau velocity in our parametric analysis differs quite substantially from that of the recent statistical results. Provided that the simplifying assumptions in our simulations remain valid, these discrepancies would indicate that the electron plateau distribution is not the major contributor to the chorus gap formation.

On Fully Nonlinear Parabolic Mean Field Games with Nonlocal and Local Diffusions

On Fully Nonlinear Parabolic Mean Field Games with Nonlocal and Local Diffusions

Field measurements reveal insights into the impact of turbulent wind on loads experienced by parabolic trough solar collectors

To ensure efficient and reliable operation of a concentrating solar-thermal power (CSP) plant, its solar collector field needs to accurately focus sunlight. The optical efficiency and structural integrity of the solar collectors is significantly influenced by wind conditions in the field. In this study, we present insights into dynamic wind loading on parabolic trough CSP collectors. We derive novel conclusions by analyzing a first-of-a-kind measurement campaign of wind and structural loads, performed at an operational CSP plant. Previous research primarily relied on wind tunnel tests and simulations, leaving uncertainty about wind loading effects in operational settings. We demonstrate that the parabolic trough field significantly alters the turbulent wind field within the collector field, especially under winds perpendicular to the trough rows. Our measurements within the trough field show reduced wind speeds, changes in wind direction and turbulence properties, and vortex shedding from the trough assemblies. These modifications to the wind field directly impact both static and dynamic support structure loads. Our measurements reveal higher wind loads on trough assemblies compared to those observed previously in wind tunnel tests. The insights from this study offer a novel perspective on our understanding of wind-driven loads on CSP collectors. By informing the development of next-generation design tools and models, this research paves the way for enhanced structural integrity and improved optical performance in future parabolic trough systems.

The Integration of Renewable Energy into a Fossil Fuel Power Generation System in Oil-Producing Countries: A Case Study of an Integrated Solar Combined Cycle at the Sarir Power Plant

Libya is facing a serious challenge in its sustainable development because of its complete dependence on traditional fuels in meeting its growing energy demand. On the other hand, more intensive energy utilization accommodating multiple energy resources, including renewables, has gained considerable attention. This article is motivated by the obvious need for research on this topic due to the shortage of applications concerning the prospects of the hybridization of energy systems for electric power generation in Libya. The 283 MW single-cycle gas turbine operating at the Sarir power plant located in the Libyan desert is considered a case study for a proposed Integrated Solar Combined Cycle (ISCC) system. By utilizing the common infrastructure of a gas-fired power plant and concentrating solar power (CSP) technology, a triple hybrid system is modeled using the EES programming tool. The triple hybrid system consists of (i) a closed Brayton cycle (BC), (ii) a Rankine cycle (RC), which uses heat derived from a parabolic collector field in addition to the waste heat of the BC, and (iii) an organic Rankine cycle (ORC), which is involved in recovering waste heat from the RC. A thermodynamic analysis of the developed triple combined power plant shows that the global power output ranges between 416 MW (in December) and a maximum of 452.9 MW, which was obtained in July. The highest overall system efficiency of 44.3% was achieved in December at a pressure ratio of 12 and 20% of steam fraction in the RC. The monthly capital investment cost for the ISCC facility varies between 52.59 USD/MWh and 58.19 USD/MWh. From an environmental perspective, the ISCC facility can achieve a carbon footprint of up to 319 kg/MWh on a monthly basis compared to 589 kg/MWh for the base BC plant, which represents a reduction of up to 46%. This study could stimulate decision makers to adopt ISCC power plants in Libya and in other developing oil-producing countries.

Linear Theory Analysis and One‐Dimensional Hybrid Simulations of High‐Frequency EMIC Waves in a Dipole Magnetic Field

AbstractNarrowband (Δf ≲ 0.1fcp), high‐frequency (0.9fcp ≲ f < fcp) electromagnetic ion cyclotron (EMIC) waves, or HFEMIC waves for short, are a relatively new type of EMIC waves, where fcp is the proton cyclotron frequency. Here, we investigate the instability threshold conditions and the nonlinear evolution of HFEMIC waves at parallel propagation. First, linear theory analysis is extended to a regime relevant to HFEMIC waves (parallel proton beta β‖p < 0.01). The instability threshold follows a similar anisotropy‐β‖p relation and requires a large value of anisotropy (T⊥p/T‖p ≳ 10) for wave growth. As a result of decreasing group velocity, the convective growth rate at a fixed threshold exhibits a similar inverse relation with β‖p. Heavy ions affect the instability only weakly, primarily through the introduction of stop bands. Second, we carry out one‐dimensional hybrid simulations in a parabolic background magnetic field, with initial parameters constrained by observation. Despite the narrow source region (within about ±3° latitude), HFEMIC waves can grow well above the thermal noise level due in large part to a small group velocity of HFEMIC waves. The saturation level is well within the range of observational amplitudes, and the quasilinear process primarily determines the wave evolution. Finally, we demonstrate that the present results compare favorably to the recent statistical results, thereby supporting anisotropic low‐energy protons as free energy source for HFEMIC waves.

Experimental Study on Performance of a Solar Thermal-Driven Vapor Absorption System Integrated with Hot Thermal Energy Storage for Milk Chilling

Abstract Solar-powered vapor absorption system has proven to be a feasible and viable cooling source. However, most reported installations for milk chilling applications are equipped with an auxiliary heater that consumes significant electricity/gas, making it economically unviable. In this study, the experimental investigation of the performance of a solar-powered vapor absorption chiller has been reported for milk chilling applications as per standard ISO 5708 – 2 II. It has been identified that the performance of the vapor absorption chiller is quite uncertain and underperforming while operated with the heat directly fed through the evacuated tube compound parabolic concentrator solar field due to diurnal and seasonal variations of solar radiation intensity. Therefore, a hot thermal energy storage integration has been studied and analyzed in this study. The performance of the vapor absorption chiller has improved significantly with the use of hot thermal energy storage in the solar circuit as the coefficient of performance of the vapor absorption chiller improved up to 0.4, which was earlier around 0.25. Further, hot thermal energy storage provides better thermal management to increase the productivity and performance of the vapor absorption chiller, and the cooling time for the first milking is 2 hours and 45 minutes. The energy efficiency ratio has a maximum value of 6.1 with an average of 4.3, whereas the thermal COP has an average of 0.35 and a maximum value of 0.52 when run with thermal energy storage alone.

Design of a Parabolic Trough Collector Solar Field for a Pilot CO2 Capture Plant from Natural Gas Combined Cycle Power Plant Flue Gas.

A parabolic trough collector solar field was designed to supply the heat needed to regenerate the CO2-rich monoethanolamine in a solar-assisted carbon capture system. Process design modeling was performed for 90% of the CO2 removal from 1% of the flue gas produced by a 255 MWe natural gas combined cycle power plant. Calculations with and without 24 h of thermal energy storage by increasing the solar collector size needed and providing a buffer vessel to store hot heat transfer fluid were performed. A dynamic analysis of the solar field using the hourly solar forecast was performed to investigate how heat transfer fluid mass flow changes during January and June to maintain the desired parabolic solar field outlet temperature needed for CO2 reboiler operation. The calculations provided here present an explicit method to calculate relevant solar field design parameters that can be scaled up and used in solar energy-assisted gas capture processes.

Equality of the Wobbly and Shaky Loci

Abstract Let $X$ be a smooth complex projective curve of genus $g\geq 2$, and let $D\subset X$ be a reduced divisor. We prove that a parabolic vector bundle ${\mathcal{E}}$ on $X$ is (strongly) wobbly, that is, ${\mathcal{E}}$ has a non-zero (strongly) parabolic nilpotent Higgs field, if and only if it is (strongly) shaky, that is, it is in the image of the exceptional divisor of a suitable resolution of the rational map from the (strongly) parabolic Higgs moduli to the vector bundle moduli space, both assumed to be smooth. This solves a conjecture by Donagi–Pantev [ 14] in the parabolic and the vector bundle context. To this end, we prove the stability of strongly very stable parabolic bundles, and criteria for very stability of parabolic bundles.

Dynamic fracture investigation of concrete by a rate-dependent explicit phase field model integrating viscoelasticity and micro-viscosity

To investigate dynamic fracture mechanisms of quasi-brittle materials, this work proposes a rate-dependent phase field model that integrates both macroscopic viscoelasticity and micro-viscosity to reflect the rate effects by free water and unhydrated inclusions. Based on the unified phase field theory, the model introduces a linear viscoelastic constitutive relation in effective stress space to consider the macro-viscosity of the bulk material. Additionally, the micro-force balance concept is utilized with the micro-viscosity to derive a parabolic phase field evolution law that accurately describes the dynamic micro-crack development. Explicit numerical solution schemes are established for the governing equations by developing VUEL and VUMAT subroutines in ABAQUS. This eliminates the convergence issue in implicit phase field modelling. Four typical benchmarks are investigated to validate the proposed model for macroscale and mesoscale heterogeneous problems. It is found that the proposed model can well capture the crack branching, delaying characteristic of micro-crack growth, and increase of macroscopic strength under higher strain rates. Using real meso‑structures from CT images, the complicated dynamic behaviour of concrete is investigated which yields deeper insight into stress wave propagation, crack evolution and load-carrying capacities.

Dynamic optimization and thermo-economic assessment of a solar parabolic trough combined heat and power system with thermal storage: A case study

The solar-driven combined heat and power system provides an efficient and green approach for the energy demand in remote areas. In this study, a two-tank molten salt thermal storage system is coupled with solar parabolic trough collectors and feeds an organic Rankine unit that produces electricity and heating. Specially, the organic Rankine unit can operate in off-design conditions following the end-user electricity load demand. The configuration of the complete system is parametrically investigated and dynamically optimized aiming to identify the operating conditions for which the net present value is maximized. The optimal arrangement for Lhasa is found to be a 1175-m2 solar parabolic trough field and two 50-m3 molten salt storage tanks. The monthly results show that this optimal case is able to run uninterruptedly the whole month in November and December with a 100% monthly electricity supply reliability rate. Furthermore, the application of this system for typical cities in western China is optimized and compared. The best economical location for the apparatus is in Ejinaqi, with a maximum net present value of 1471 thousand dollars, of which carbon reduction benefit account for 8.8%, and the payback period is only 5.4 years. Moreover, the yearly energy efficiency, exergy efficiency, electricity supply reliability rate and solar abandonment rate for optimal case of Ejinaqi are 55.0%, 20.2%, 93.9% and 25.4%, respectively. The study demonstrates that the proposed apparatus can be a promising option to provide energy for remote areas faced with poorly power grid.

Charged particles and quasiperiodic oscillations around magnetized Schwarzschild black holes

We study the motion of charged particles around Schwarzschild black holes immersed in external (i) asymptotically uniform, (ii) dipolar, and (iii) parabolic-like magnetic fields. The effect of the different magnetic-field configurations on the position of innermost stable circular orbits (ISCOs) for test-charged particles is analyzed. Furthermore, we investigated frequencies of radial and vertical oscillations of the charged particles along their stable circular orbits together with the Keplerian one. As an astrophysical application, we explore quasiperiodic oscillations (QPOs) observed in microquasars sourced by black hole candidates in the frame of the relativistic precession (RP) model. In order to obtain constraints on the values of the magnetic parameter and black hole mass for the microquasars GRO J1655-40 and GRS 1915-105, we use the method so-called chi ^2 in the Bayesian approach. Also, we get constraints on the magnetic field around the black hole in the microquasars by treating electrons and protons as oscillating test-charged particles in the accretion disc. Our performed analyses show that the masses of black hole candidates in the above-mentioned objects and magnetic parameters are different for the uniform and dipolar magnetic field configurations. However, no constraints on the magnetic field and black hole masses are obtained in the case of a parabolic magnetic field configuration. The obtained results on the black hole masses are compared with the measurements in independent astrophysical observations of these black hole masses.

Charged particle dynamics in parabolic magnetosphere around Schwarzschild black hole

The study of charged particle dynamics in the combined gravitational and magnetic field can provide important theoretical insight into astrophysical processes around black holes. In this paper, we explore the charged particle dynamics in parabolic magnetic field configuration around Schwarzschild black hole, since the paraboloidal shapes of magnetic field lines around black holes are well motivated by the numerical simulations and supported by observations of relativistic jets. Analysing the stability of bounded orbits and using the effective potential approach, we show the possibility of existence of stable circular off-equatorial orbits around the symmetry axis. We also show the influence of radiation reaction force on the dynamics of charged particles, in particular on the chaoticity of the motion and Poincaré sections, oscillatory frequencies, and emitted electromagnetic spectrum. Applied to Keplerian accretion disks, we show that in parabolic magnetic field configuration, the thin accretion configurations can be either destroyed or transformed into a thick toroidal structure given the radiation reaction and electromagnetic-disk interactions included. Calculating the Fourier spectra for radiating charged particle trajectories, we find that the radiation reaction force does not affect the main frequency peaks, however, it lowers the higher harmonics making the spectrum more flat and diluted in high frequency range.

Upstream Shift of Generation Region of Whistler‐Mode Rising‐Tone Emissions in the Magnetosphere

AbstractWe have performed a series of simulation runs for whistler‐mode wave‐particle interaction in a parabolic magnetic field with 12 different frequencies of triggering waves and 3 different plasma frequencies specifying cold plasma densities. Under a given plasma condition, a specific frequency range of the triggering wave exists that can generate rising‐tone emissions. The generation region of rising‐tone emission shifts upstream. The velocity of the wave generation region is dependent on duration of the subpacket, which is controlled by the formation of the resonant current in the generation region. When the source velocity, which is a sum of the resonance and group velocities, is approximately the same as the velocity of the wave generation region, a long‐sustaining rising‐tone emission is generated. When the spatial and temporal gap between subpackets exists due to damping phase of short subpacket generation, resonant electrons in the gap of the subpackets are carried at the resonance velocity to the upstream region, and the velocity of the wave generation region becomes large in magnitude. When formation of resonant currents is delayed, the velocity of the generation region becomes smaller than the source velocity in magnitude. Below one quarter of the cyclotron frequency, coalescence of subpackets takes place, suppressing formation of the resonant current in the generation region. Since gradual upstream shift of the generation region is necessary for the wave to grow locally, the source velocity should be a small negative value.

On the fully coupled quasi-static equations for the thermoelastic halfspace

Influence functions for fully coupled quasi-static thermoelasticity are presented. They can be used to calculate displacements, stresses as well as temperature distributions within a halfspace for arbitrarily shaped and time dependent heat sources or pressure distributions on the surface. To this end, the underlying equations are solved for a mechanically or thermally loaded rectangular element on the surface by potential theory. The solution procedure is described in detail. The obtained results are verified and may easily be transferred to the theory of poroelasticity where the underlying equations are structurally equal. Example calculations for a parabolic pressure field, heat flux and temperature field are performed and simulation results are discussed particularly in comparison to the uncoupled case. Estimations of the extent to which the Gough–Joule effect and its consequences can be neglected are given.

Fast SEMS size distribution measurements under down-scan operation

Particle size distributions obtained with a Scanning Electrical-Mobility Spectrometer (SEMS) are based on the response of charged particles in an applied electric field. A complete operation of SEMS instrument for particle sizing involves the exposure of charged particles to an exponentially increasing voltage (up-scan) followed by an exponentially decreasing voltage (down-scan). Accurate calculation of particle size distributions from SEMS measurements requires knowledge of the instrument transfer function or Kernel as a function of its operating conditions and subsequently, the inversion of an integral equation. A semi-theoretical transfer function calculation approach based on particle arrival-times was shown to accurately characterize fast up-scan SEMS operation. In this paper, we extend the arrival-time calculation approach for SEMS down-scan operation. The combination of a decreasing voltage over time and a parabolic velocity field results in particle trajectories that are fundamentally different from the up-scan operation. However, when smearing due to downstream transport of particles to the detector is considered, the down-scan transfer functions are seen to be a mirror image of the up-scan transfer functions in the mobility space. This suggests that the inversion algorithms optimized for fast up-scan operation could also be applicable for fast down-scan operation. The combination of near real-time up-scan and down-scan transfer functions should enable fast, and accurate size distribution measurements under a range of operating conditions.

Nonlinear Electron Phase‐Space Dynamics in Spontaneous Excitation of Falling‐Tone Chorus

AbstractWhistler mode chorus, characterized by frequency chirping, is an important type of waves in planetary magnetospheres. To investigate the role of nonlinear wave particle interactions in excitation of chorus, analysis of electron phase space dynamics is required. While electron phase hole associated with rising tone chorus has been well studied, the phase space structure corresponding to falling tone chorus is less understood. Here, we investigate in detail the electron phase space dynamics in a spontaneously generated falling tone chorus with a parabolic type magnetic field, where field strength decreases away from the equator. We demonstrate the formation and evolution of electron phase space clump from downstream to upstream regions, and show that the variation of frequency chirping rate is caused by the movement of the source region. The results are consistent with the recently proposed Trap‐Release‐Amplify model, and should be useful to understand the mechanism of chorus frequency chirping.

Backward-propagating source as a component of rising tone whistler-mode chorus generation

Whistler-mode chorus waves in the magnetosphere play a crucial role in space weather via wave–particle interactions. The past two decades have observed tremendous advances in theory and simulations of chorus generation; however, several details of the generation mechanism are still actively contended. To simulate chorus generation, a new envelope particle-in-cell code is introduced. The model produces a rising tone chorus element in a parabolic geomagnetic field. The initial chorus element “embryo” frequency is shown to initialize near the equator at the frequency of maximum linear growth. A backward resonant current is then observed to propagate upstream of the equator. The trajectory of the backward current follows that of a freely falling electron that has been de-trapped at the equator superimposed with forward motion at the group velocity. The backward current iteratively radiates a rising tone element where the highest frequency components are generated furthest upstream. The work provides new advancements in modeling chorus and corroborates other recent work that has also demonstrated a backward-moving source during the generation of coherent whistler-mode waves.

Energy, exergy, environmental impact, and economic analyses of evacuated tube compound parabolic concentrator–powered solar thermal domestic water heating system

In the reported study, a dynamic analytical model is developed to propose the energy, exergy, environmental impact, and economic analyses of the water heating system at Jaipur (India) with an evacuated tube compound parabolic concentrator field of a total area of 81 m2. Consequently, the model is used to perform parametric studies to report the effect of operating and meteorological parameters on the productivity and performance of the system. Moreover, the system's performance, environmental impact, and economic aspects have been investigated and compared under different meteorological conditions at four different Rajasthan (India) locations using TMY2 weather data files. Results clarified that Jodhpur receives the highest solar radiation intensity from these four locations. The model results were validated with the experimental data, and a good agreement has prevailed. Consequently, the results indicate the highest annual energy and exergy gain for Jodhpur with 79.72 MWh and 9.311 MWh, respectively, followed by Jaisalmer, Barmer, and Jaipur. The economic analysis results clarified that the simple payback period ranged from 4.5 to 4.75 years and the discounted payback period ranged from 6.6 to 7 years based on a 6% discount rate. At the same time, the levelized cost of heating for the given system is around 0.023 $/kWh which is very economical closest to that of CNG as a fuel which costs around 0.059 $/kWh. The internal rate of return is reported to be 16.76, 16.82, 16.77, and 16.75% for Barmer, Jodhpur, Jaipur, and Jaisalmer, respectively, and savings of 74.4, 78.1, 75.4, and 73.8 tonnes of CO2 emission to the environment.

Modeling, Validation, and Analysis of a Concentrating Solar Collector Field Integrated with a District Heating Network

In recent years, concentrating solar collectors have been integrated with several district heating systems with the aim of taking advantage of their low heat losses. The present study investigates the Brønderslev combined heat and power plant, which consists of a 16.6 MW parabolic trough collector field, two biomass boilers, and an organic Rankine cycle system. The study focuses on the solar collector field performance and integration with the district heating network. An in situ characterization of the parabolic solar collector field using the quasi-dynamic test method found that the field had a peak efficiency of 72.7%. Furthermore, a control strategy for supplying a constant outlet temperature to the district heating network was presented and implemented in a TRNSYS simulation model of the solar collector field. The developed simulation model was validated by comparison to measurement data. Subsequently, the simulation model was used to conduct a sensitivity analysis of the influence of the collector row spacing and tracking axis orientation. The results showed that the current suboptimal tracking axis rotation, made necessary by the geography of the location, only reduced the annual power output by 1% compared to the optimal configuration. Additionally, there were only minor improvements in the annual heat output when the row spacing was increased past 15 m (ground cover ratio of 0.38).