Equation For Electric Potential Research Articles

Minimum energy principles in electrostatics

Abstract We employ the principle of minimum energy of the electrostatic field to derive Laplace's equation for electric potential in charge-free space. We also apply the principle of minimum electrical potential energy of a system of discrete charges to explain the possibility of the existence of their stable equilibrium configuration on the surface of a conductor.

Numerical Calculations of Liquid-Metal Magnetohydrodynamic Flows in Rectangular Ducts with Sudden Contractions

For the design of the liquid-metal blanket in a fusion reactor, numerical calculations have been carried out on liquid-metal magnetohydrodynamic flows in rectangular ducts with sudden contractions. Conservation equations of fluid mass and fluid momentum and the Poisson equation for electrical potential have been solved numerically. The numerical calculations have been conducted for a Hartmann number of ~10 000; a Reynolds number of ~10 000; and contraction ratios (CRs) of 2, 3, and 4. The pressure loss through the contraction has been estimated by the loss coefficient ζ divided by the interaction parameter N, i.e. ζ/N. The loss coefficient ζ/N through the contraction parallel to the magnetic field is much larger than that through the corresponding contraction perpendicular to the magnetic field. The loss coefficient ζ/N increases consistently with the CR and does not change very much with N. While ζ/N also does not change very much with the wall conductance ratio for the contraction parallel to the magnetic field, ζ/N increases gradually with the wall conductance ratio for the contraction perpendicular to the magnetic field.

Numerical treatment of entropy generation and Bejan number into an electroosmotically-driven flow of Sutterby nanofluid in an asymmetric microchannel

This article aims to investigate the electrokinetic peristaltic flow of the Sutterby nanofluid through an asymmetric microchannel with a porous medium and the Joule heating parameter. The magnetic field is applied in the transverse direction and the electric field on the flow direction. The flow model consists of the continuity, momentum, heat, and electric potential equations, with appropriate boundary conditions. The Debye-Hückel linearization approximation is considered. Under the long wavelength and small Reynolds number approximation, the lengthy equations were reduced. The resultant coupled equations are numerically solved by the renowned NDSolve coding using Mathematica software. Entropy and Bejan number are also incorporated in this study. The velocity, temperature, and the phenomenon of entrapment are analyzed in depth with the aid of graphical representations. Streamlines of various waveforms are also discussed. The analysis discovered that the velocity of the fluid enhanced for porosity parameter and reverse effects is observed on the quantity of the magnetic parameter.

Deriving an equation of electric potential of a point P separated from a charged flat plate

Deriving an equation of electric potential of a point P separated from a charged flat plate

Electrohydrodynamics of diffuse porous colloids.

The present article deals with the electrohydrodynamic motion of diffuse porous particles governed by an applied DC electric field. The spatial distribution of monomers as well as the charge distribution across the particle are considered to follow sigmoidal distribution involving decay length. Such a parameter measures the degree of inhomogeneity of the monomer distribution across the particle. The diffuse porous particles resemble several colloidal entities which are often seen in the environment as well as in biological and pharmaceutical industries. Considering the impact of bulk pH and ion steric effects, we modelled the electrohydrodynamics of such porous particulates based on the modified Boltzmann distribution for the spatial distribution of electrolyte ions and the Poisson equation for electric potential as well as the conservation of mass and momentum principles. We adopt regular perturbation analysis with weak field assumption and the perturbed equations are solved numerically to calculate the electrophoretic mobility and neutralization fraction of the particle charge during its motion as well as fluid collection efficiency. We further deduced the closed form relation between the drag force experienced by the charged porous particle and the fluid collection efficiency. In addition to the numerical results, we further derived the closed form analytical results for all the intrinsic parameters indicated above derived within the Debye-Hückel electrostatic framework and homogeneous distribution of monomers within the particle for which the decay length vanishes. The deduced mathematical results as indicated above will be useful to analyze several electrostatic and hydrodynamic features of a wide class of porous particulate and environmental entities.

Numerical Calculations of Liquid-Metal Magnetohydrodynamic Flows in Rectangular Ducts with Sudden Expansions

Relating to the design of liquid-metal blankets in a fusion reactor, numerical calculations have been performed on liquid-metal magnetohydrodynamic (MHD) flows in rectangular ducts with sudden expansions. Conservation equations of fluid mass and fluid momentum, together with the Poisson equation for electrical potential, have been solved numerically. The numerical calculations have been performed for Hartmann (Ha) numbers up to the order of 10000 and expansion ratios up to 4. The pressure loss through the expansion has been estimated by the loss coefficient ζ divided by the interaction parameter N, i.e., ζ/N. The loss coefficient ζ/N through the expansion parallel to the magnetic field is much larger than that through the expansion perpendicular to the magnetic field. The loss coefficient ζ/N increases consistently with the expansion ratio. The loss coefficient ζ/N does not change very much with the interaction parameter N and the wall conductance ratio.

Effects of anisotropic permeability on EMHD nanofluid flow and heat transfer in porous microchannel with wavy rough walls

This theoretical study thoroughly examines the complexities of fluid flow and heat transfer within microfluidic devices by employing anisotropic porous media with wavy walls. The effect of anisotropic permeability and electromagnetohydrodynamics (EMHD) on nanofluid transport is examined in a wavy microchannel under a constant pressure gradient. The nonlinear coupled governing equations for electric double-layer potential, velocity, temperature, and nanoparticle volume fraction are solved numerically using shooting techniques with the Runge–Kutta method. The numerical results are validated with the asymptotic analytical solutions. We explore the impact of parameters like the Hartmann number, Darcy number, and joule heating on nanofluidic flow. The interplay between anisotropic permeability ratio and angle significantly influences flow, temperature profiles, and nanoparticle volume fraction. Reduction in nanoparticle concentration is attributed to constrained entry into the porous medium due to elevated anisotropic permeability. Frictional heating, influenced by the Forchheimer inertial effect, enhances temperature and induces slug flow at low Darcy numbers. The Nusselt number peaks at low Darcy numbers with an anisotropic angle of φ=π/2, experiencing a minimum when φ=0. These findings deepen our understanding of the role of anisotropic permeability in shaping fluid flow and heat transfer in microfluidic systems influenced by EMHD effects.

Features of ion exchange between the electrodes in metal-ion batteries during discharge

The importance and relevance of the storage of electrical energy is confirmed by events in the world and trends in the development and use of various electrical energy systems, household appliances, computer equipment, communication devices, etc. In addition to the growth of the metal-ion battery markets, there are trends towards a search for metals that in the future will be inexpensive and will have characteristics required for storage systems. This paper considers ion exchange between the electrodes of metal-ion batteries whose charge carriers are metal ions, which diffuse in the process of discharge from the negative electrode to the positive one. A mathematical model was developed and tested. The model is based on a system of diffusion transport equations with the Nernst–Planck–Poisson potential equation replaced by an equivalent conductivity potential equation. Quasi-equilibrium regimes are considered. The entire working area consists of a pore electrode space and a neutral separator. The mathematical model employed consists of potential distribution equations and an electrolyte concentration distribution equation supplemented by the dependence of the electrode surface current on the overvoltage and equations that determine the electrode pore structure depending on the masses transferred inside the electrode. The electric potential and diffuse component mass transfer equations are written within the framework of the modern theory of effective electrical conductivity in batteries with account for current exchange between the solid electrodes and the liquid electrolyte. The research results showed the following. A change in the resistance of the separator (a change in porosity) has little effect, if any, on the electrode current densities, but it causes some change in the potentials themselves. A change in the resistance of the electrolyte affects both the electrode potentials and the internal current distribution between the electrodes and the electrolyte.

Proton exchange membrane water electrolysis at high current densities: Response time and gas‐water distribution

AbstractUnderstanding the distribution of oxygen in proton exchange membrane water electrolysis (PEMWE) is crucial for improving electrolysis efficiency and gas removal. In this study, we developed a two‐dimensional (2D) transient model that couples the Euler–Euler multiphase model with electric potential equations to investigate two‐phase flow in PEMWE. Our simulation reveals that the system's response time initially decreases and then increases with current density, indicating longer response times at high current densities. Modifying the wetting properties of the porous transport layer (PTL) affects gas removal at low gas holdup, resulting in a maximum 15% decrease in gas holdup. However, at high gas holdup, the flow field in the channel predominantly governs bubble removal, making changes in PTL wetting properties less influential. With increasing gas production rate, an inverse gradient distribution of gas saturation appears, leading to uneven gas saturation and hindering efficient oxygen removal. This non‐uniform gas saturation adversely affects electrolysis performance.

Thermally developing combined electroosmotic and pressure-driven flow of Phan–Thien–Tanner fluids in a microchannel

We present semi-analytical solutions for the hydrodynamically developed and thermally developing flow of a non-Newtonian fluid through an isothermal rectangular microchannel. The fluid motion is actuated by the combined consequences of the electroosmotic and pressure-gradient forces. For the rheological behavior of the non-Newtonian fluid, we have used the simplified Phan–Thien–Tanner viscoelastic model. Going beyond the Debye Hückel linearization approximation, we have used the full-scale solution for the electrical double-layer potential equation to obtain the exact analytical solutions for the velocity, flow rate, and shear rate parameters. In contrast, the temperature distribution and heat transfer for the thermally developing flow have been obtained by solving the energy equation numerically considering the effects of volumetric heat generation due to Joule heating and viscous dissipation. We find that a larger value of the viscoelastic set ε̃Wĩk2 contributes toward the net gain in flow rate. Both the normal and shear stress increase for increasing ε̃Wĩk2, while the shear viscosity reduces with a degree of surface charging. The average shear viscosity reduces with the degree of surface charging and at higher ε̃Wĩk2 values. The heat transfer is enhanced for augmenting ε̃Wĩk2, although the thermal entrance region gets contracted for a pure electroosmotic flow at higher Peclet numbers. Our study reveals that the heat transfer rate can be amplified by effectively modulating the degree of surface charging and ε̃Wĩk2. We have also carried out an entropy generation analysis, which shows the dominance of heat transfer irreversibility over fluid friction irreversibility. We believe that the present research will offer essential approaches for designing advanced energy-efficient microchannels appropriate to modern industrial applications using viscoelastic fluids.

Efficient algorithm based on two-grid method for semiconductor device problem

We study a three-step two-grid algorithm for solving semiconductor device equations that is discretized by the standard finite element method for the concentration equation and the mixed finite element method for the electric potential equation. The main procedures involve solving the original nonlinear equations on the coarse grid to obtain the initial approximate solution, as well as the linearized equations on the fine grid to obtain a modified solution; furthermore, the calculation on the fine grid is corrected. Error estimation for the method shows that the coarse grid can be much coarser than the fine grid, and the two-grid method still preserves asymptotically optimal approximation. Numerical experiments validate the reliability and effectiveness of the algorithm.

3D Multi-Ion Corrosion Model in Hierarchically Structured Cementitious Materials Obtained from Nano-XCT Data.

Corrosion of steel reinforcements in concrete constructions is a worldwide problem. To assess the degradation of rebars in reinforced concrete, an accurate description of electric current, potential and concentrations of various species present in the concrete matrix is necessary. Although the concrete matrix is a heterogeneous porous material with intricate microstructure, mass transport has been treated in a homogeneous material so far, modifying bulk transport coefficients by additional factors (porosity, constrictivity, tortuosity), which led to so-called effective coefficients (e.g., diffusivity). This study presents an approach where the real 3D microstructure of concrete is obtained from high-resolution X-ray computed tomography (XCT), processed to generate a mesh for finite element method (FEM) computations, and finally combined with a multi-species system of transport and electric potential equations. This methodology allows for a more realistic description of ion movements and reactions in the bulk concrete and on the rebar surface and, consequently, a better evaluation of anodic and cathodic currents, ultimately responsible for the loss of reinforcement mass and its location. The results of this study are compared with a state-of-the-art model and numerical calculations for 2D and 3D geometries.

Active vortex generation and enhanced heat transfer in a 3D minichannel by Onsager–Wien effect

A 3D numerical analysis of conjugate heat transfer in a rectangular minichannel in the presence of vortices generated by electric field induced Onsager–Wien effect has been performed. Fully coupled governing equations for flow, heat transfer, electric potential and charge transport are solved using the finite-volume framework of OpenFOAM®. Present study proposes a novel approach to increase mixing and heat transfer in a minichannel using the vortices generated by Onsager–Wien effect. Effects of the flow Reynolds number Re and electric Reynolds number ReEL on the thermo-hydraulic performance of the minichannel have been analyzed. The Onsager–Wien effect produces small flow vortices in the vicinity of thin plate electrodes flushed to the top and bottom walls of the channel. These vortices enhance the fluid mixing, deplete the thermal boundary layer and increase the heat transfer. The Onsager–Wien effect produces stable and prominent vortex structures at low Reynolds numbers. Whereas, the inertial force begins to dominate at higher Re and the vortices get smeared in the positive flow direction. In general, the heat transfer enhancement is directly proportional to the electric Reynolds number ReEL. A maximum of 48.25% increase in mean Nusselt number is observed within the parameter space considered herein. Results of this study indicate that, thermal boundary layer depletion and mixing by electric-field-induced Onsager–Wien effect substantially enhances the heat transfer performance of a minichannel, with a trivial additional electric power consumption. Results of the present numerical study will serve as a benchmark to design minichannel heat sinks with enhanced heat transfer based on thermal boundary layer depletion by application of weak/medium electric fields.

A numerical solver for investigating the space charge effect on theelectric field in liquid argon time projection chambers

This paper reports the development of a numerical solver aimed to simulate the interaction between the space charge (i.e. ions) distribution and the electric field in liquid argon time projection chamber (LArTPC) detectors. The ion transport equation is solved by a time-accurate, cell-centered finite volume method and the electric potential equation by a continuous finite element method. The electric potential equation updates the electric field which provides the drift velocity to the ion transport equation. The ion transport equation updates the space charge density distribution which appears as the source term in the electric potential equation. The interaction between the space charge distribution and the electric field is numerically simulated within each physical time step. The convective velocity in the ion transport equation can include the background flow velocity in addition to the electric drift velocity. The numerical solver has been parallelized using the Message Passing Interface (MPI) library. Numerical tests show and verify the capability and accuracy of the current numerical solver. It is planned that the developed numerical solver, together with a Computational Fluid Dynamics (CFD) package which provides the flow velocity field, can be used to investigate the space charge effect on the electric field in large-scale particle detectors.

Thermal boundary layer depletion in minichannels by electrohydrodynamic conduction pumping

The article presents a numerical analysis of two-dimensional laminar flow and heat transfer in a minichannel with flat plate electrode pairs periodically flushed to the bottom wall. The two-way coupled governing equations of flow, heat transfer, electric potential and charge conservation are solved using the opensource finite-volume framework of OpenFOAM. Two cases of the minichannel at completely horizontal (case A) and 90° bent (case B) configurations are considered. The inlet laminar flow with Reynolds number varying from 10 to 1000 has been taken into account. The applied electric potential is varied in three steps 0, 0.5 and 1 kV. Electrohydrodynamic (EHD) conduction pumping due to the residual conductivity of the working fluid and the electric-field enhanced dissociation of free charges generate vortices in the proximity of electrodes. The flow and thermal performance of the minichannel in the presence of EHD conduction induced vortices are quantified in terms of pressure drop, Nusselt number and a performance factor (which is a combined function of pressure drop and Nusselt number). The vortices generated by EHD conduction pumping phenomenon deplete the thermal boundary layer, aid mixing and enhance the heat transfer. The mean and local Nusselt numbers are notably increased with only a minimal raise in pressure drop. In both the configurations of the minichannel, the performance factor is greater than 1 for the entire parameter space considered in this study. However, the thermal boundary layer depletion and heat transfer enhancement is prominent at low Reynolds number flows. The maximum drop in performance factor from case A to case B is only 18.5%. Results of this study indicate that EHD conduction pumping is a potential technique to enhance heat transfer in minichannels with minimum additional power consumption.

Melting performance enhancement in a thermal energy storage unit using active vortex generation by electric field

Latent heat thermal energy storage (LHTES) devices aid in efficient utilization of alternate energy systems and improve their ability to handle supply–demand fluctuations. A numerical analysis of melting performance in a shell-and-tube LHTES unit in the presence of a direct current (DC) electric field has been performed. The governing equations of fluid flow, heat transfer, electric potential and charge conservation are solved using a customized finite-volume solver built in the open-source framework of OpenFOAM. Enthalpy-porosity method based fixed grid approach is used to track the melt interface. Primary objective of the study is to highlight the interface and flow morphology evolution in the presence of electric field induced flow and to evaluate the melting performance of the LHTES unit. The transient evolution of the melting process in the presence of electric field has been mapped in terms of total liquid fraction, kinetic energy density and mean Nusselt number. The charge injection from the tube surface generates multiple electrohydrodynamic (EHD) flow vortices in the liquid region. Thus, the inherent uni-cellular flow structure of the natural convection driven melting is disrupted. The multi-cellular flow structure with stronger velocity distribution enhances mixing and heat transfer. Melting performance at various levels of applied voltages (0≤V≤10kV) in both vertical and horizontal orientations of the LHTES unit has been quantified in terms of charging time and total power storage. The charging time gets shorter and total power storage gets higher with increasing applied voltages. In the vertical orientation, a maximum 82.52% reduction in charging time and 80.85% increase in net power storage is achieved. In the horizontal orientation, weaker buoyancy force leads to stronger influence of the electric field. A maximum of 89.61% reduction in charging time and 88.35% increase in power storage is achieved in the horizontal orientation. The results of this study aid in understanding the mechanism of EHD flow assisted melting and provide a reference for design of a shell-and-tube LHTES unit with improved performance.

FerroX: A GPU-accelerated, 3D phase-field simulation framework for modeling ferroelectric devices

We present a massively parallel, 3D phase-field simulation framework for modeling ferroelectric materials based scalable logic devices. This code package, FerroX, self-consistently solves the time-dependent Ginzburg Landau (TDGL) equation for ferroelectric polarization, Poisson's equation for electric potential, and charge equation for carrier densities in semiconductor regions. The algorithm is implemented using the AMReX software framework [1], which provides effective scalability on manycore and GPU-based supercomputing architectures. We demonstrate the performance of the algorithm with excellent scaling results on NERSC multicore and GPU systems, with a significant (15×) speedup on the GPU using a node-by-node comparison. We further demonstrate the applicability of the code in simulations of ferroelectric domain-wall induced negative capacitance (NC) effect in Metal-Ferroelectric-Insulator-Metal (MFIM) and Metal-Ferroelectric-Insulator-Semiconductor-Metal (MFISM) devices. The charge (Q) v.s. voltage (V) responses for these 3D structures clearly indicate stabilized negative capacitance with multidomain formation, which is corroborated by amplification of the voltage at the interface between the ferroelectric and dielectric layers.

PO-03-096 DEFINING RECORDING ANTENNA FOR ATRIAL ELECTRODES BY ENDOCARDIAL AND EPICARDIAL COMPUTER MAPPING

Mapping arrhythmias and concretely atrial fibrillation (AF) is complicated by our inability to determine local from far field electrograms, as far field interfere with the identification of local activity in the specific electrode location. In particular, how “local” is local field on any specific electrode and catheter is not yet biophysically defined. We tested the hypothesis that EGM conformation on specific electrode configurations can be explained by modelling the biophysical relation between the electrode size and separation in active myocardial tissue. We modelled laminar conduction in atrial tissue of 30x30x3 mm on a 1 mm electrode, discretized into 100.000 and 100 nodes respectively (fig. A) and simulating propagation waves with constant propagation direction at different conduction velocities. The EGM was constructed using the electric potential transfer equation between the tissue and the electrode, and this equation was discretized for each tissue node to evaluate individual contributions of atrial regions to construct unipolar (fig B) and bipolar (fig C) EGM signals. This allowed to unify the near field definition for all types of electrodes as regions contributing at least 30% respect to the maximal tissue contribution. In Figure B, the unipolar electrode recorded near field from a circle of radius 8 mm endocardially, and 2.7 mm transmurally with a 1.9 ratio of endo-to-epicardial amplitude. Bipolar electrodes showed directional antennas for near field aligned with the propagation direction (fig. C) with radii of 1.7, 5.7 and 8.3 mm for 2, 5, and 10 mm spacing (fig. C). Bipolar electrodes showed less transmural variability on the near field antenna, from 2 to 2.7 mm (fig. D). These data show that local signals on clinical catheters can represent near field activity in 1.7 to ∼8 mm radius spheres. Studies are required to define optimal electrode configurations to map AF, ventricular fibrillation, atrial and ventricular tachycardia.

A simplified lattice Boltzmann model for two-phase electro-hydrodynamics flows and its application to simulations of droplet deformation in electric field

This paper proposes a numerical scheme within the lattice Boltzmann framework for effective simulations of two-phase electro-hydrodynamic (EHD) flows, in which the simplified multiphase lattice Boltzmann method serves as the flow solver and a leaky dielectric model is utilized to recover electric potential equation. The resultant formulations present direct evolution of macroscopic variables, due to which the present method outperforms the conventional lattice Boltzmann method in terms of memory cost and boundary treatment. Numerical validations are first carried out under the same parametric conditions as the benchmarking experimental studies with the conductivity ratio ranging from 0.03 to 1000, demonstrating that the present method is a practical and accurate tool to study large-conductivity-ratio two-phase EHD flows in scientific applications. Through the comparison with previous theoretical and numerical studies, the performance of the present method is further evaluated by varying physical parameters such as electric capillary number, electric field strength, interfacial tension, and electrical properties of liquids. The good agreements with the reference data confirm the accuracy and robustness of the proposed method. Additional studies of the dynamics of a deformed droplet in uniform electric field are carried out using the validated method. Specifically, we conduct a close numerical examination on the effect of the electrical conductivity ratio and permittivity ratio on the electric forces, and highlight the role of the dielectric electrophoretic force and the Coulomb force on the electric forces the deformation of a leaky dielectric droplet in the electrical conductivity ratio and the permittivity ratio phase diagram.

A mass conservative characteristic finite element method with unconditional optimal convergence for semiconductor device problem

We consider a mass conservative type method for semiconductor device problem by employing mixed finite element method (FEM) for electric potential equation and mass conservative characteristic FEM for both electron and hole density equations. The boundedness of numerical solution without certain time‐step restriction and optimal error estimates of full discrete scheme are proved. Numerical experiment is presented to verify the effectiveness and unconditional stability of the proposed method.