Steady-state Analytical Solutions Research Articles

Neutron Flux-Irrated Hydrodynamics and Transport Coefficients of Fissioning 235UF6 Plasma Laminar Flow

For weakly ionized dense plasma exposed to fission fragment radiation, coupled self-consistent Boltzmann equations for fission fragments and generated primary electrons are defined. The kinetics of rapid particle energy generation in a plasma are researched based on these equations. For the helium-3 plasma exposed to neutron flux, steady-state analytical solutions for fission fragments and the functions of energy distribution of the primary electrons were identified and examined. We compare the outcomes with calculations of energy spectra from Monte Carlo methods.

Nonlinear effects of partitioning and diffusion-limiting phenomena on the response and sensitivity of three-layer amperometric biosensors

The influence of the partitioning and diffusion limitations on the response of amperometric biosensors is investigated analytically and computationally using a three-compartment model: an enzyme layer, a diffusion limiting membrane and an outer diffusion layer. The biosensors are modelled by a system of reaction–diffusion equations involving a nonlinear term related to Michaelis–Menten kinetics. Exact steady state analytical solutions for the substrate and reaction product concentrations and the output current are presented for specific cases of first and zero-order reaction rates. The performance of the biosensor is numerically investigated at the transition conditions. Application of different specific types of the interface conditions, perfect contact and partition conditions, at which the steady state biosensor response is the same for both types of the interface conditions is analysed. The effective diffusion coefficients merging two diffusion layers are defined for reducing the three-compartment to the corresponding two-compartment model. The nonlinear and non-monotonic behaviour of the biosensor response is discussed.

Analytical and Numerical Models of a New Hybrid System of Earth Air Tunnel Integrated Solar Air Heater With Sensible Storage

Abstract A novel combination of an earth air tunnel (EAT) and a sensible thermal storage-aided solar air heater has been proposed when the sensible storage medium is taken to be rocks filled within the air flow passage. Both time-dependent and steady-state models are reported. The former describes the exact thermal performance of the system for a two-month wintertime for the city of Baghdad, Iraq. The latter describes average performance of the system. The implicit finite difference method and method of separation of variables have been used to solve the pertinent equations in the respective models. Acceptable level of accuracy was obtained between the steady-state numerical and analytical solutions as well as between the numerical and the published data. It is revealed that preheating by a sufficiently long EAT module improves the effective power as well as the output temperature from the rock bed solar air heater by about 35% and 9% respectively for the present set of parameters considered. It is also observed that for the present set of parameters considered, the temperature gradients in the direction normal to air flow in the solar air heater are insignificant and may be ignored. A parametric study is also carried out that assesses the impact of system parameters on the quality as well as quantity of the energy extracted from the system. This work is hence the first to couple an EAT with a sensible thermal storage equipped solar air heater and may pave the way for future studies.

A Numerical Investigation of Hydro-Magnetic Mixed Convection Flow of Viscous Dissipative Fluid in a Channel Filled with Porous Material

The purpose of this paper is to inspect the unsteady MHD mixed convective Couette flow of viscous dissipative fluid in a vertical channel filled with porous material. The steady state analytical solutions of temperature, velocity, frictional force and heat transfer rate are obtained using the Homotopy perturbation method (HPM). The implicit finite difference technique is used to solve the transient leading equations numerically. Plots and tables with the dynamic flow parameters and dimensionless variables are displayed. During this computation, it was found that raising the Darcy parameter (Da) and the mixed convection (Gre) parameters both push the fluid flow upwards, but increasing the magnetic field's values had the opposite effect (M). HIGHLIGHTS Effect of viscous Dissipation on unsteady MHD Mixed Convection Flow with Porous Material is investigated Increasing Darcy number improves the flow pattern Rate of heat transfer could be enhance by considering high values of Brinkman number GRAPHICAL ABSTRACT

Seasonal scour and siltation induced spatiotemporal variations in riverbed sediment leakage coefficients as measured via the thermal tracer method

Riverbed scour and siltation can change the hydraulic conductivity and thickness of sediments, leading to notable spatiotemporal variations of groundwater–surface water interaction. However, correlated quantitative research on the dynamic leakage coefficient (LC) remains lacking under riverbed scour and siltation. Based on dynamic monitoring and analysis of sediment medium properties, river-groundwater levels, and temperature profiles in the section of Liao River, this study compared the applicable conditions of different thermal analytical models and identified spatiotemporal variations in the LC values induced by riverbed scour and siltation. The results showed that: (1) The thermal tracer method combined with river-groundwater level monitoring can effectively quantify spatiotemporal variations in the LCs of riverbed sediment. Based on the sediment temperature presents diurnal fluctuations and gradient characteristics in different seasons, the LCs in summer, spring, and autumn compared to winter in seasonally frozen regions should be calculated using transient and steady-state analytical solution models, respectively. (2) The LCs ranged from 0.47 to 24.90 × 10-6 s−1, showed larger temporal variability than spatial, increased from the river bank to the center, and had larger values in autumn and winter. (3) Distinct from previous findings, the riverbed underwent a scouring process associated with reductions in the incoming sediment coefficient as the river stage shifted from high to low, eroding the surface 4–8 cm of sediment. During this process, the LC increased by 40 to 420%. Further, the influence of riverbed scour and siltation on the LC varied by river section. Near the riverbank, the sediment LC was primarily controlled by hydraulic conductivity related to the coarsening of sediment lithology; however, it was influenced by variations in both sediment thickness and hydraulic conductivity near the riverbed center. Collectively, these results further understand the mechanisms underlying changes in the sediment LC induced by riverbed scour and siltation, as well as provide a scientific basis for ensuring the sustainable development of riverside water sources under such changes.

Analytical Model for the Design of Permeable Reactive Barriers Considering Solute Transport in a Dual-Domain System

A permeable reactive barrier (PRB) is an effective groundwater in situ remediation technology, and the design methods used for PRBs are significant in ensuring that they meet remediation goals. Steady-state analytical solutions are an effective tool to provide conservative and simple design methods. A steady-state analytical solution is proposed to describe organic contaminant transport through the PRB and aquifer in a PRB and cut-off wall system. The proposed analytical solution may serve as an effective tool to provide conservative and simple design methods. The shape factor (S) is introduced at the PRB-aquifer interface to investigate the effects of a PRB’s layout forms on its performance. The results show that the relative contaminant concentration at the point of compliance for a PRB with S=8 is 11 orders of magnitude larger than that without considering the shape factor. Effects of degradation, dispersion, and advection on PRB design are subjected to dimensionless analysis. Dimensionless analysis shows that degradation plays a key role in decreasing contaminant concentration in the PRB. In addition, increasing advection may promote contaminant transport from the source to the aquifer. Simplified solutions to estimate PRB thickness and source remediation time are derived for the practical design and performance evaluation of the PRB system.

Study on the Effect of Anti-galloping Device with Cubic Stiffness and Particle Damping

The galloping of iced conductors is a common disaster of overhead conductors. Most traditional anti-galloping devices lack a nonlinear dynamic vibration absorber; hence, they have a limited anti-galloping effect. In this study, we designed, optimized, and tested an electric power fitting and an anti-galloping device with cubic stiffness and particle damping for ultra-high-voltage (UHV) transmission lines. We built a nonlinear dynamic model of the coupling galloping system comprising split conductors and anti-galloping devices. The harmonic balance method obtained the steady-state analytical solutions and corresponding averaged equations. The test spans in the laboratory were used to design the test program and equipment. We verified the accuracy of the nonlinear dynamic model and the harvesting effect of the anti-galloping device with cubic stiffness and particle damping. The theoretical and experimental results were highly consistent, and in the range of (0, 2] Hz, the anti-galloping device reduced the galloping amplitude. Therefore, cubic stiffness and particle damping can effectively improve the anti-galloping ability of UHV transmission lines, prolonging their service period.

Steady-state Analytical Reservoir Pressure and Temperature Solutions during Hydrogen Refueling to Environment-friendly Ships from Offshore Underground Storage

As hydrogen (H2) conserves intermittent electricity from renewable energies, H2-fueled ships are overlooked for carbon emission reduction. H2-fueled ships require offshore refueling for long-distance shipping routes. At offshore refueling stations, H2 is expected to be generated by renewable energies, stored in depleted offshore reservoirs, and then extracted for refueling. Temperature and pressure variations near the wellbore during H2 extraction impact the severity of hydrogen-induced corrosion. Therefore, it is necessary to conduct hydrothermal analysis for offshore platforms’ front-end engineering design and operation strategy. This paper proposes an analytical reservoir pressure and temperature solution for steady-state offshore H2 extraction from depleted reservoirs for refueling. We derive the radial analytical solution for a two-phase flow (water-H2) and validate them with a numerical model. We apply the derived solution to sensitivity analysis with respect to water depth, reservoir depth, H2 saturation, permeability, reservoir thickness, and total extraction rate. Due to the negative water thermal gradient, deep water depth leads to pressurization and cooling near the wellbore. Critical H2 saturation is derived as a criterion of transition between Joule-Thomson heating and cooling. The higher flow mobility of H2 ensures less depressurization during extraction than water. Higher H2 saturation over the critical saturation causes less cooling due to the decreasing pressure gradient.

Model-Based Detection and Estimation of DC Offset of Phase Current Sensors for Field Oriented PMSM Drives

Failure of a phase current sensor in Permanent Magnet Synchronous Machine (PMSM) drive can highly impact the performance of the drive. Especially, offset faults can cause torque oscillations, which can lead to an unacceptable deterioration of the drive's performance and mechanical damages. The diagnosis of this fault condition is of fundamental importance in many applications and it is often solved by means of highly computational state estimation techniques. This paper presents a new model-based approach to systematically and accurately detect and estimate phase current sensor offsets in PMSM drives. The steady-state analytical solution of the PMSM drive model, including the control loop of the Field Oriented Control (FOC) and the sensor offset disturbance, has been found, allowing for estimating the effect of the fault on the electric drive performance. Then, a simple and effective diagnostic algorithm has been developed to detect, isolate and accurately estimate the location and the magnitude of the fault eliminating any need for state estimation or observers. Numerical and experimental results have been reported to validate the proposed model and diagnostic algorithm even in case of motor parameters variation.

Modeling and Harmonic Analysis of a Fractional-Order Zeta Converter

The Zeta converter is an essential and widely used high-order converter. The current modeling studies on Zeta converters are based on the model that devices, such as capacitors and inductors, are of integer order. For this reason, this paper takes the Zeta converter as the research object and conducts an in-depth study on its fractional-order modeling. However, the existing modeling and analysis methods have high computational complexity, the analytical solutions of system variables are tedious, and it is difficult to describe the ripple changes of state variables. This paper combines the principle of harmonic balance with the equivalent small parameter method (ESPM); the approximate analytic steady-state solution of the state variable can be obtained in only three iterative steps in the whole solving process. The DC components and ripples of the state variables obtained by the proposed method were compared with those obtained by the Oustaloup’s filter-based approximation method; the symbolic period results obtained by ESPM had sufficient precision because they included more combinations of higher harmonics. Finally, the influence of fractional order on harmonics were analyzed. The obtained results show that the proposed method has the advantage of being less computational and easily describing changes in the ripple of the state variables. The simulation results are provided for validity of the theoretical analysis.

Investigating steady unconfined groundwater flow using Physics Informed Neural Networks

A deep learning technique called Physics Informed Neural Networks (PINNs) is adapted to study steady groundwater flow in unconfined aquifers. This technique utilizes information from underlying physics represented in the form of partial differential equations (PDEs) alongside data obtained from physical observations. In this work, we consider the Dupuit–Boussinesq equation, which is based on the Dupuit–Forchheimer approximation, as well as a recent, more complete model derived by Di Nucci (2018) as underlying models. We then train PINNs on data obtained from steady-state analytical solutions and laboratory based experiments.Using PINNs, we predict phreatic surface profiles given different input flow conditions and recover estimates for the hydraulic conductivity from the experimental observations. We show that PINNs can eliminate the inherent inability of the Dupuit–Boussinesq equation to predict flows with seepage faces. Moreover, the inclusion of physics information from the Di Nucci and Dupuit–Boussinesq models constrains the solution space and produces better predictions than training on data alone. PINNs based predictions are robust and show a little effect from added noise in the training data. Furthermore, we compare the PINNs solutions obtained via the Di Nucci and Dupuit–Boussinesq flow models to examine the effects of higher order flow terms that are included in the Di Nucci formulation but are neglected by the Dupuit–Boussinesq approximation. Lastly, we discuss the effectiveness of using PINNs for examining groundwater flow.

Mathematical Modeling and Analysis of the Steady Electro-Osmotic Flow of Two Immiscible Fluids: A Biomedical Application

The in vitro fabrication of big osteoarticular implants integrating biomaterials and cells is of tremendous interest because these tissues have a limited ability to regenerate. However, the growth of such cells in vitro is highly problematic, especially later in the culture, when the extracellular matrix has almost filled the initial porous network. Thus, the fluid flow required to properly perfuse the sample cannot be obtained by the hydraulic driving force alone. Fluid pumping is a central concern of a microfluidic system and electro-osmotic pumps (EOPs) are commonly employed for this purpose. Using electro-kinetic equations as a basis, this study analyzed the variations of a two-fluid electro-osmotic flow of viscoelastic fluid flow through a channel. The behavior of the fluid was studied through the Ellis equation. This is how the electro-osmotic pump functions, as demonstrated in the literature that it electrically drags a conducting fluid across a non-conducting fluid through interfacial dragging force along the channel. A steady-state analytical solution for the system in a conducting fluid channel was studied by undertaking an interface planner for fluids exhibiting Newtonian rheological properties. The pumping characteristics were studied in detail by using the Ellis model’s parameters. The fluid rheology was studied, which showed the viability of this technique.

Extreme Energy Spectra of Relativistic Electron Flux in the Outer Radiation Belt.

Electron diffusion by whistler-mode chorus waves is one of the key processes controlling the dynamics of relativistic electron fluxes in the Earth's radiation belts. It is responsible for the acceleration of sub-relativistic electrons injected from the plasma sheet to relativistic energies as well as for their precipitation and loss into the atmosphere. Based on analytical estimates of chorus wave-driven quasi-linear electron energy and pitch-angle diffusion rates, we provide analytical steady-state solutions to the corresponding Fokker-Planck equation for the relativistic electron distribution and flux. The impact on these steady-state solutions of additional electromagnetic ion cyclotron waves, and of ultralow frequency waves are examined. Such steady-state solutions correspond to hard energy spectra at 1-4MeV, dangerous for satellite electronics, and represent attractors for the system dynamics in the presence of sufficiently strong driving by continuous injections of 10-300keV electrons. Therefore, these analytical steady-state solutions provide a simple means for estimating the most extreme electron energy spectra potentially encountered in the outer radiation belt, despite the great variability of injections and plasma conditions. These analytical steady-state solutions are compared with numerical simulations based on the full Fokker-Planck equation and with relativistic electron flux spectra measured by satellites during one extreme event and three strong events of high time-integrated geomagnetic activity, demonstrating a good agreement.

Analytical solutions for seawater intrusion and the maximum pumping rate of a well in unconfined coastal aquifers bounded by L-shaped coastlines

Determining the maximum pumping rate (i.e., the maximum allowable pumping rate that does not cause seawater intrusion into the well under the steady-state condition) of a well is the key for sustainable management of coastal aquifers, as overexploitation of coastal aquifers could lead to the widespread occurrence of aquifer salinization (i.e., seawater intrusion). In this study, we develop steady-state analytical solutions to describe seawater intrusion and the maximum pumping rate of a well located in unconfined coastal aquifers bounded by L-shaped coastlines. Rectangular coastal aquifers bounded by two orthogonal constant-head boundaries (i.e., two coastlines) and two orthogonal no-flow inland boundaries are assumed in conceptual models, where replenishment of aquifers is via surface recharge. Analytical solutions are derived by the potential theory and the conformal mapping method, and validated by variable-density flow numerical simulations. It is found that analytical solutions based on the sharp-interface approximation overestimate seawater intrusion and underestimate the maximum pumping rate. An acceptable, conservative maximum pumping rate is obtained by using the correction factor proposed by Lu and Werner (2013). Moreover, the sensitivity of the well location on the maximum pumping rate is investigated, showing that an optimal well location exists. Analytical solutions developed in the current study provide a convenient tool for assessing pumping effects in coastal aquifers bounded by L-shaped coastlines, servicing to various management objectives.

Well hydraulics in dual-zone heterogeneous wedge-shaped aquifers under an ambient flow from stream-wise varying boundary heads

• Closed-form analytical solution for flow net considering non-uniform ambient flow. • Generality of solution in choosing hydrological input parameters. • A framework for assessment of mass balance at aquifer and sub-aquifer scales. • Uncertainty analysis to quantify model response to randomness in input parameters. A steady-state analytical solution based on the method of Green’s function is presented for the analysis of hydraulic interference between pumping wells and ambient flow in a heterogeneous wedge-shaped aquifer. Two intersecting streams with spatially variable head form the boundaries of the wedge and establish non-uniform distribution of ambient flow across it. The dual-zone aquifer consists of two porous media of constant but contrasting transmissivities and refraction conditions are exactly satisfied along their common interface. The method of Green’s function affords an integral representation of boundary heads which is readily evaluated in terms of Lerch transcendent function. Integration of Cauchy-Riemann equations gives the associated stream function in closed-form which in turn completes analytical description of the flow net. A systematic way for the assessment of mass balance at both aquifer and sub-aquifer (well capture zone) scales is discussed which also helps identifying whether a certain reach of the boundary stream is losing or gaining water to or from the adjoining aquifer. The solution is general in the sense that the wedge angle, location and discharge of the wells, dual-zone transmissivities and slopes of boundary head variation can be arbitrary chosen. The solution encompasses the case of stream-barrier wedge as a subset. Hypothetical examples demonstrate the complex interaction between surface–subsurface waters showing formation of stagnation point on the capture zone envelopes. In each case, head profiles are in excellent agreement with numerical counterparts from adaptive finite element method. Finally, uncertainty analysis quantifies the extent to which stochastic nature of dual-zone transmissivities and boundary head slopes (model inputs) affects hydraulic head at specific observation points (model output).

Droplet dynamics under an impinging air jet

Partially wetting droplets under an airflow can exhibit complex behaviours that arise from the coupling of surface tension, inertia of the external flow and contact-line dynamics. Recent experiments by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020) revealed that a millimetric partially wetting water droplet under an impinging jet can oscillate in place, split or depin away from the jet, depending on the magnitude (i.e. $5\unicode{x2013}20\ {\rm m}\ {\rm s}^{-1}$ ) and position of the jet. To rationalise the experimental observations, we develop a two-dimensional lubrication model of the droplet that incorporates the external pressure of the impinging high-Reynolds-number jet, in addition to the capillary and hydrostatic pressures of the droplet. Distinct from the previous model by Hooshanginejad et al. (J. Fluid Mech., vol. 901, 2020), we simulate the motion of the contact line using precursor film and disjoining pressure, which allows us to capture a wider range of droplet behaviours, including the droplet dislodging to one side. Our simulations exhibit a comparable time-scale of droplet deformations and similar outcomes as the experimental observations. We also obtain the analytical steady-state solutions of the droplet shapes and construct the minimum criteria for splitting and depinning.

Tree hydrodynamic modelling of the soil–plant–atmosphere continuum using FETCH3

Abstract. Modelling the water transport along the soil–plant–atmosphere continuum is fundamental to estimating and predicting transpiration fluxes. A Finite-difference Ecosystem-scale Tree Crown Hydrodynamics model (FETCH3) for the water fluxes across the soil–plant–atmosphere continuum is presented here. The model combines the water transport pathways into one vertical dimension, and assumes that the water flow through the soil, roots, and above-ground xylem can be approximated as flow in porous media. This results in a system of three partial differential equations, resembling the Richardson–Richards equation, describing the transport of water through the plant system and with additional terms representing sinks and sources for the transfer of water from the soil to the roots and from the leaves to the atmosphere. The numerical scheme, developed in Python 3, was tested against exact analytical solutions for steady state and transient conditions using simplified but realistic model parameterizations. The model was also used to simulate a previously published case study, where observed transpiration rates were available, to evaluate model performance. With the same model setup as the published case study, FETCH3 results were in agreement with observations. Through a rigorous coupling of soil, root xylem, and stem xylem, FETCH3 can account for variable water capacitance, while conserving mass and the continuity of the water potential between these three layers. FETCH3 provides a ready-to-use open access numerical model for the simulation of water fluxes across the soil–plant–atmosphere continuum.

An Analytical Heat Transfer Model in Oil Reservoir during Long-Term Production

Contrary to the assumption of previous researchers, the radial temperature in the petroleum reservoir during production is non-isothermal because several heat transfer mechanisms change the radial temperature in reservoirs. As there have been few studies, especially after long-term production, this work derives steady-state analytic solutions considering the long-term production. It also presents sensitivity analysis with the various production conditions to investigate heat transfer between the producing fluids and surrounding formations during fluids flowing (hereafter, system heat transfer) in a steady-state. For oil production, the system heat transfer induces a cooling effect on the radial temperature in the reservoir, reducing the temperature rise due to the Joule–Thomson (J–T) heating. This cooling effect increases with the larger Peclet number, however, the relative contribution of the cooling effect to the radial temperature change diminishes. The equations explain that the cooling effect is proportional to the temperature increase due to J–T heating. With a larger permeability, a more convective-dominant phase causes more heat transfer actively. Although the cooling effect itself is amplified with the larger permeability, its relative contribution to the temperature change decreases. From the analysis, the cooling effect of system heat transfer is significant in the low-permeability reservoirs with large drawdown. The system heat transfer is confirmed to be an essential factor in measuring the accurate productivity index of unconventional reservoirs.

New analytical solution of the fractal anharmonic oscillator using an ancient Chinese algorithm: Investigating how plasma frequency changes with fractal parameter values

This paper uses the two-scale fractal dimension transform and He’s formula derived from the ancient Chinese algorithm Ying Bu Zu Shu to find the approximate frequency–amplitude expression of the fractal and forced anharmonic oscillator that can be used to study the nonlinear oscillations produced by the plasma physics fractal structures. The results show how the electron frequency and wavelength change as a function of the plasma physics fractal structure. In fact, if the value of the fractal parameter is decreased, the wavelength increases, and consequently, the system frequency decreases. The introduced solution procedure sheds a bright light on the easy-to-follow steps to obtain an accurate steady-state analytical solution of fractal anharmonic nonlinear oscillators.

Analytic steady state solution for interfacial overpotential distribution in Li-ion battery and degradation prognosis

Overpotential distribution of lithium ion battery shows the force is applied to lithium ion over intercalation and deintercalation between active materials and electrolyte. This distribution determines irreversible degradation chemical reactions, lithium plating and solid electolyte interphase layer growing, occur or not for each interfacial areas. However, the calculation of the distribution requires large memory space, so most of industrial battery management system board cannot fit its requirement. To overcome this limitation, we introduce new analytic steady state solution for overpotential distribution so that online prognosis of lithium ion battery can be provided with industrial battery management system. Also, we estimate electrochemical parameters of end-of-life lithium ion batteries with different operated condition and apply it to our model to confirm that our model follows other reported degradation prognosis.