Isoconcentration Contours Research Articles

On thermodynamics and heat and mass transfer aspects of CH4 adsorption onto coconut shell-based carbonaceous material

The current study investigates the equilibrium methane (CH4) adsorption capacity and the adsorption kinetics of a locally produced coconut shell-derived activated carbon (AC CARB 6X12 60) for pressure and temperatures ranging from 0 to 55 bar and -20 to 70 °C, respectively. The adsorption isotherms and kinetics are determined using an in-house developed adsorption test rig. A maximum methane adsorption uptake of 0.140 kg/kg is obtained at -20 °C and 40 bar pressure. The adsorption equilibrium isotherm data are fitted using Toth, Dubinin-Astakhov (DA), and Modified-DA isotherm models. Afterwards, the kinetics study is done considering pseudo first order, second order, Elovich, and intra-particle diffusion models. In continuation, based on the DA isotherm model, Clausius Clapeyron relation, and Maxwell's relations, the expressions for the crucial thermodynamic parameter of adsorbed phase, i.e., isosteric heat of adsorption, specific heat capacity, enthalpy, and entropy is derived, and same is estimated using DA isotherm parameters. Additionally, the activated carbon's thermophysical properties (i.e., thermal conductivity, thermal diffusivity, and specific heat) are determined using the TPS 2500 S analyzer, which works on the hot-disk principle. Using the most suitable isotherm (DA) and kinetic model (pseudo second order), heat and mass transfer analysis is carried out on novel reactor configuration with an external cooling jacket, double row of cooling pipes with external longitudinal fins for methane storage using a 3D transient model in Comsol Multiphysics. The charge characteristics are studied for gas charging pressure varying from 1 to 50 bar and cooling fluid temperature of -20 to 70 °C. Iso-concentration contours are plotted to identify the pressure-temperature combination resulting in identical storage capacity (i.e., including both gaseous and adsorbed phases). Further simulations are carried out for constant pressure charging and constant flow discharge conditions to expound upon the effects of the dormancy period, gas charging pressure, and cooling/heating fluid temperature. 14% improvement in total discharge amount is obtained with a dormancy period between charging and discharging compared to no dormancy case.

Influence of magnetic field inclination angles on hybrid nanofluids: Enhanced MHD double-diffusive mixed convection and entropy generation in a trapezoidal enclosure

This current numerical investigation is focused on analyzing the influences of various magnetic field inclination angles on mixed convective heat and mass transfer along with entropy generation in a lid-driven trapezoidal enclosure with double rotating cylinders inside by highlighting significant correlations between system’s input & output parameters. In this study, the effects of three types of SWCNT-Cu-–water hybrid-nanofluids composed of various ratios of nanoparticles in water are observed along with the individual impacts of SWCNT-water, Cu-water, and –water nanofluids. Galerkin weighted residual finite element method is used to solve the governing Navier-Stokes, thermal energy, and mass conservation equations numerically to get the results in forms of average Nusselt number, average Sherwood number, average entropy generation, and average temperature. The optimum conditions are identified to get the best results and these results are also represented in the forms of streamlines, isotherm, and isoconcentration contours for several parameters to get a better insight into heat and mass transfer. In most cases, maximum heat, and mass transfer along with maximum average total entropy generation occur if the magnetic field is assigned at an inclined angle with respect to the horizontal axis for each type of fluid in this current framework.

Variations of MHD double-diffusive mixed convection and entropy generation in various nanofluids and hybrid nanofluids due to the deviation of the spinning of double rotating cylinders

The current numerical study aims to investigate the effects of various rotating speeds of rotating cylinders on mixed convection and entropy generation in a lid-driven trapezoidal enclosure for fixed magnetic field inclination angle. Several water-based hybrid nanofluids are used as governing fluids. In this work, the impacts of SWCNT-water, Cu-water, and Al2O3-water nanofluids individually, as well as the effects of three different types of SWCNT-Cu-Al2O3-water hybrid nanofluids are examined. The average Nusselt number, average Sherwood number, average temperature, and average entropy generation because of heat transfer, diffusion, fluid friction, and the applied magnetic field as well as total entropy generation and Bejan number are evaluated as output parameters for different values of governing parameters. The Galerkin weighted residual finite element method is used to numerically solve the governing Navier-Stokes, thermal energy, and mass conservation equations. The optimal values of each governing parameter are established in order to produce the best results for the output parameters. To inspect the thermal and flow field within the trapezoidal enclosure as well as the variations in mass concentrations, isotherm contours, streamlines and isoconcentration contours are studied. Better performance is found for higher rotating speed regardless of other parameters.

NUMERICAL study of MAGNETO convective Buongiorno nanofluid flow in a rectangular enclosure under oblique magnetic field with heat generation/absorption and complex wall conditions

A mathematical model is presented to analyze the magnetohydrodynamic (MHD) convective flow in a rectangular enclosure. Earlier studies on the Tiwai-Das volume fraction nanofluid model did not consider the Buongiorno nanofluid model. This is the focus of the present analysis which examines the laminar mixed convection magnetohydrodynamic flow of a nanofluid in a differentially heated rectangular enclosure with complex boundary conditions under an inclined magnetic field. Buongiorno's two-component nanofluid model is employed, which incorporates the effects of Brownian motion and thermophoretic diffusion of nanoparticles. Magnetic nanofluids have considerable potential for enhancing transport processes in energy systems such as hybrid fuel cells. The study is essential with heat generation/absorption effects. Additionally, the work is highlighted by the general case of an oblique (inclined) magnetic field. The conservation equations for mass, primary and secondary momentum, energy, and nanoparticle concentration with wall boundary conditions are dimensionless using appropriate scaling transformations. A finite-difference computational scheme known as the Harlow-Welch Marker and Cell (MAC) method is employed to solve the dimensionless nonlinear coupled boundary value problem. A mesh independence study is included. Graphical plots are presented for the impact of key control parameters on streamline contours, isotherm contours, iso-concentration (nanoparticle mass) contours, and local Nusselt number. With heat sink (absorption), the Nusselt number is enhanced in magnitude whereas it is suppressed with heat generation since there is a heat reduction transmitted to the boundary. The configurations of streamlines, isotherms, and iso-concentrations are mostly invariant to magnetic field direction changes. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve the effect of Brownian motion and thermophoresis parameters, as well as unsteady regimes, depending on specific values of the Schmidt number, Richardson number, and Prandtl numbers. Increasing the Schmidt number induces a contraction in the central cooler zone in the enclosure and also reduces the iso-concentration magnitudes in the central region across the enclosure. The core region of the enclosure heats up as the thermophoresis and Brownian motion parameters rise, pushing the previously cooler top and bottom wall zones further away from the center. There is also a decrease in iso-concentration magnitudes in particular at the upper and lower boundaries at higher values of Brownian motion and thermophoresis parameters. At decreasing buoyancy ratios, the left vortex cell first decelerates while the right vortex cell accelerates. However, when the buoyancy ratio increases, the left vortex cell streamlines magnitudes increase with a contraction in vortex size, while the right cell develops. A Very minor alteration is observed in the isotherm and iso-concentration contours with an increasing buoyancy ratio. When the Richardson number increases, the vortex cell structures shift from a strong circulation cell on the left to a weaker cell on the right, resulting in reverse distribution. With the rising Richardson number, significant cooling is also caused in the core zone, as well as a drop in iso-concentrations, with the original dual low-concentration upper and lower zones merging into a single center zone. The original symmetric left and right vortex cells are gradually twisted diagonally towards the right wall as the magnetic field increases, yet the stronger right cell and the weaker left cell are maintained.

Entropy optimization of MHD non-Newtonian fluid in a wavy enclosure with double diffusive natural convection

This study investigates the impact of an inclined magnetic field (MHD) on entropy generation in double diffusive natural convective flow in a wavy enclosure filled with a non-Newtonian Casson fluid. The Galerkin Finite Element Method (GFEM) is employed to numerically solve the standard formulation, utilizing quadratic polynomials for momentum interpolation and a linear interpolating function for model approximation. The discretized system is resolved using Newton’s approach and PARDISO's matrix factorization. Through simulations of varying ranges of Rayleigh numbers ( 1 e 3 ≤ Ra ≤ 1 e 5 ) , Casson parameter ( 0.1 ≤ β ≤ 10 ) , Hartmann numbers ( 0 ≤ Ha ≤ 40 ) , Lewis numbers ( 0.1 ≤ Le ≤ 5 ) , and inclined angle gamma ( 0 ≤ γ ≤ 60 o ) , the study provides valuable insights into the behavior of double diffusive natural convection in the wavy enclosure. Isotherms, iso-concentration contours, and streamlines are analyzed to assess different input distributions, and the study presents graphical representations and tabular data on heat transfer, mass transfer rate, and entropy production.

Spatio-temporal synchronization in complex ore-forming systems: Mesoscale case study in south China

Synchronization is an important current frontier area of nonlinear sciences. In order to explore the spatio-temporal evolution of ore-forming elements in complex system at mesoscale, the principles, promises, methods and procedures of a spatio-temporal synchronization method were defined.Based on the phase conversion equation, the phase data and the phase difference were calculated in two directions, and the phase difference constant was set as 0.1 as a critical value to extract the plateau length in two directions. After the plateau length in both directions was superimposed, we got the synchronization threshold. Through its spatio-temporal extension, the synchronized space-time domain was determined. Then, we utilized geostatistics and Kriging method to generate the map of synchronized space-time domain, which includes ore formation, time and space. Applying the developed method and using local stream sediment geochemical data in Dajishan tungsten deposit (China), we have carried out the research on the synchronization of pairwise matching elements between ore-forming and ore-transporting elements in two directions at mesoscale, besides pairwise matching elements between ore-forming elements.Results indicate that the mineralization time-space domains of W, Sn, Mo, Bi, Cu, Pb, Zn, Au, Ag, and Be elements in the study area and their isoconcentration contours show a good coincident correspondence. For pairwise matching elements of ore-forming element matching with ore-transporting element, the high synchronization threshold area appears on the edge of the element mineralization center in the map of space-time domain for synchronization (or mineralization), which reveals the onset of ore formation; For pairwise matching elements between ore-forming elements, it reveals the mechanism of ore formation. Furthermore, this theory and method can be also applied to other fields of geochemistry, such as environmental geochemistry.

Layer-adapted meshes for solute dispersion in a steady flow through an annulus with wall absorption: Application to a catheterized artery

This paper describes the longitudinal dispersion of passive tracer molecules injected in a steady, fully developed, viscous, incompressible, laminar flow through an annular pipe with a first order heterogeneous boundary absorption at the outer wall, numerically using layer-adapted meshes. The model is based on steady advection-diffusion equation with Dirichlet and Robin boundary conditions. The solutions are discussed in the form of iso-concentration contours of the tracer molecules in the vertical plane. An artanh transformation is used to convert the infinite domain into a finite one. A combination of central finite difference and 2-point upwind scheme is adopted to solve the governing advection-diffusion equation. It is shown that how the mixing of tracers is affected by the shear flow, aspect ratio and the first-order boundary absorption. When the flow becomes convection dominated, the monotone finite difference on a uniform mesh does not work properly, so a layer-adapted mesh, namely a “Shishkin” mesh, is used to capture the layer phenomena at the different downstream stations. The present results are compared with existing experimental and numerical data and we have earned an excellent agreement with them. It is observed that, due to the use of layer adapted mesh, we have achieved a better agreement with the experimental data than some other previous results available in the literature, especially in the closest downstream location. The results of this study are likely to be of interest to understand the basic mechanism of dispersion process of solute in blood through a catheterized artery with an absorptive arterial wall.

Lattice Boltzmann simulation of double-diffusive natural convection of viscoplastic fluids in a porous cavity

In this paper, a two-dimensional double diffusive natural convection in a porous cavity filled with viscoplastic fluids is simulated. The dimensional and non-dimensional macroscopic equations are presented, employing the Papanastasiou model for viscoplastic fluids and the Darcy–Brinkman–Forchheimer model for porous media. An innovative approach based on a modification of the lattice Boltzmann method is explained and validated with previous studies. The effects of the pertinent dimensionless parameters are studied in different ranges. The extensive results of streamlines, isotherms, and isoconcentration contours, yielded/unyielded regions, and local and average Nusselt and Sherwood numbers are presented and discussed.

Modeling of Al Doping During 4H-SiC Chemical-Vapor-Deposition Trench Filling

Aluminum doping during 4H-SiC chemical-vapor-deposition (CVD) trench filling was numerically modeled toward precise design of high-voltage superjunction devices. As a first-order approximation, growth-rate- and surface-normal-scaling functions were determined based on the reported experimental results. Simulated isoconcentration contours of aluminum were confirmed to qualitatively agree with the reported imaging of doping in SiC by scanning spreading resistance microscopy. Improvement of the proposed models based on additional experiments should contribute to reducing the development time for 4H-SiC superjunction devices fabricated using CVD trench filling.

Formation and evolution of target patterns in Cahn-Hilliard flows

We study the evolution of the concentration field in a single eddy in the two-dimensional (2D) Cahn-Hilliard system to better understand scalar mixing processes in that system. This study extends investigations of the classic studies of flux expulsion in 2D magnetohydrodynamics and homogenization of potential vorticity in 2D fluids. Simulation results show that there are three stages in the evolution: (A) formation of a "jelly roll" pattern, for which the concentration field is constant along spirals; (B) a change in isoconcentration contour topology; and (C) formation of a target pattern, for which the isoconcentration contours follow concentric annuli. In the final target pattern stage, the isoconcentration bands align with stream lines. The results indicate that the target pattern is a metastable state. The band merger process continues on a time scale exponentially long relative to the eddy turnover time. The band merger process resembles step merger in drift-ZF staircases; this is characteristic of the long-time evolution of phase-separated patterns described by the Cahn-Hilliard equation.

Emissions of volatile organic compounds during the ship-loading of petroleum products: Dispersion modelling and environmental concerns

Emissions due to ship-loading of hydrocarbons are currently not addressed neither by the Directive on the integrated pollution prevention or by other environmental regulations. The scope of this study is to point towards the environmental and safety concerns associated with such emissions, even if proper attention has not been given to this issue until now. In order to achieve this goal, the modelling of the emission volatile organic compounds (VOC), due to ship-load operations at refineries has been made by means of the definition of a simulation procedure which includes a proper treatment of the hours of calm. Afterwards, a quantitative analysis of VOC dispersion for an Italian case-study is presented with the primary aims: (i) to develop and verify the validity of the approach for the modelling of the emission sources and of the diffusion of these contaminants into the atmosphere by a proper treatment of the hours of calm and (ii) to identify their contribution to the total VOC emitted in a typical refinery. The calculated iso-concentration contours have also been drawn on a map and allowed the identification of critical areas for people protecting by the adoption of abatement solutions.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

PurposeThe purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically.Design/methodology/approachNon-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations.FindingsBoth the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer.Originality/valueCalculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

Free convection in wavy porous enclosures with non-uniform temperature boundary conditions filled with a nanofluid: Buongiorno’s mathematical model

In the present work, the influence of the amplitude ratio, phase deviation, and undulation number on natural convection in a wavy-walled enclosures differentially heated and filled with a water based nanofluid is studied. The upper and bottom walls are wavy with several undulations. The sinusoidal distribution of temperature is imposed at the vertical walls. The flow, heat, and mass transfer are calculated by solving governing equations for embody the conservation of total mass, momentum, thermal energy, and nanoparticles, taking into account the Darcy-Boussinesq-Buongiorno approximation with second order finite difference method in ?stream function-temperature-concentration? formulation. Results are presented in the form of streamlines, isotherm, and isoconcentration contours, and distributions of the average Nusselt number for the different values of the amplitude ratio of the sinusoidal temperature on the right side wall to that on the left side wall (? = 0-1), phase deviation (? = 0-?), and undulation number (? = 1-4). It has been found that variations of the undulation number allow to control the heat and mass transfer rates. Moreover, an increase in the undulation number leads to an extension of the non-homogeneous zones.

An Unsteady Double-Diffusive Natural Convection in an Inclined Rectangular Enclosure with Different Angles of Magnetic Field

An unsteady double-diffusive natural convection flow in an inclined rectangular enclosure subject to an applied magnetic field and heat generation parameter is studied. The enclosure is heated and concentrated from one side and cooled from the adjacent side. The other two sides are adiabatic. The governing equations are solved numerically using a staggered grid finite-difference method to determine the streamline, isotherm and iso-concentration contours. We have further obtained the average Nusselt numbers and average Sherwood numbers for various values of buoyancy ratio and different angles of the magnetic field by considering three different inclination angles of the enclosure while keeping the aspect ratio fixed. The results indicate that the flow pattern, temperature and concentration fields are significantly dependent on the buoyancy ratio and the magnetic field angles. It is found that different angles of the magnetic field suppress the convection flow and its direction influences the flow patterns. This leads to the appearance of inner loop and multiple eddies.

The spatial distribution of human exposure to PCBs around a former production site in Slovakia.

We evaluated concentrations of 15 PCB congeners in blood serum of 2047 adults, 431 8-9-year old children and 1134 mother-child pairs born in 2001-2003. These subjects were long-standing residents living up to 70 km (to the north) and up to 50 km (to the south) of the former Chemko Strážske PCB production facility in the Michalovce district of Slovakia. We plotted serum concentration against distance from the plant both with and without consideration of the direction of their homes from the site. The decrease in exposure with distance could be described by an exponential function which was dependent on direction and climatic parameters. By kriging we created maps depicting predicted isoconcentration contours for sex- and age-adjusted serum concentration of ∑PCBs for the same group of children, adults and mothers. The principle of our risk analysis was to relate serum concentration data, reflecting PCB body burden, using the critical concentrations established by the French Agency for Food, Environmental and Occupational Health & Safety (ANSES 2010) as thresholds below which the probability of effects on health is regarded as negligible. We conclude that 10 years ago, around 200,000 residents were at risk in this densely populated area. Exposure has since decreased but the mechanism for this has not yet been studied.

Convective heat and solute transfer in Newtonian fluid saturated inclined porous cavity

This paper summarizes a numerical study of double-diffusive natural convection in square inclined cavity filled with fluid saturated porous media. Transverse gradients of heat and solute are applied on the two horizontal walls of the cavity, while the other two walls are impermeable and adiabatic. The Darcy model with the Boussinesq approximation is used to solve the governing equations. The flow is driven by a combined buoyancy effect due to both temperature and concentration variations. A finite volume approach has been used to solve the non-dimensional governing equations. The results are presented in streamline, isothermal, iso-concentration, Nusselt and Sherwood contours for different values of the non-dimensional governing parameters. Keywords: Boussinesq Approximation, Darcy Model, Finite Volumes, Inclined Cavity, Natural Convection, Porous Media.

Onset of double-diffusive convection in horizontal cavity with Soret and Dufour effects

Abstract An unsteady numerical model based on thermosolutal buoyancies with Soret and Dufour effects for double-diffusive convection is developed. The thermosolutal model is discretized by the finite volume method and solved numerically using the SIMPLE algorithm with QUICK scheme in non-uniform staggered mesh. The flow field, temperature and concentration distributions for different aspect ratios, buoyancy ratios, Rayleigh numbers, Soret and Dufour coefficients are investigated systematically. The results show that the flow structure of different aspect ratios develops from conduction-dominated to steady convection-dominated, and finally evolves into periodic oscillatory convection as buoyancy ratio or Rayleigh number increases. The vortex number of flow structure, recirculation zones of isotherm and isoconcentration contours reduce along the transition route while both of them increase as aspect ratio decreases. The average Nusselt and Sherwood numbers keep constants during the conduction-dominated stage, but increase with increasing Rayleigh number, increasing buoyancy ratio or decreasing aspect ratio during the convection-dominated stage.

Ground Water Conditions and Spatial Distribution of Lead and Cadmium in the Shallow Aquifer at Effurun- Warri Metropolis, Nigeria

A water table head distribution map of the shallow Benin Formation aquifer in the Effurun-Warri area has been drawn from dug well data and used to define groundwater gradients as well as identify directions of groundwater movement in this densely populated urban setting. Water samples from forty dug wells were also screened for the presence of lead and cadmium and results showed a variation in concentration from not detectable to 0.04mg/l for each metal. Iso-concentration contours for lead in groundwater suggest that enrichment may be from two sources: wastes from the refinery and petrochemical industrial complex on the northwestern edge of the city and secondly from leachates associated with the many unregulated waste dumpsites. Lead appears to be constrained from spreading eastwards from the industrial complex area by the south and westwards trending groundwater gradient. The city wide prevalence of elevated levels of cadmium is also probably due to leachates from unregulated dumpsites as well as the mixing of groundwater as suggested by existing gradients. Potential implications of the findings for public health, local and regional water quality monitoring are discussed.

Two-Dimensional Modeling of Aluminum-Ion Implantation into 4H-SiC

A two-dimensional model of aluminum-ion implantation into 4H-SiC at moderate doses (1011 to 1013 cm-2) has been developed. The model is based on a Monte-Carlo simulation using a binary-collision approximation. This simulation reveals that iso-concentration contours are independent of the orientation of the masking edge. Lateral range straggling is extracted by expressing the lateral-concentration profiles as a one-dimensional dual-Pearson-distribution function multiplied by a Gauss-distribution function. Compared to vertical straggling, the lateral straggling is found to be more weakly dependent on projected range.

Two-Dimensional Analytical Model for Concentration Profiles of Aluminum Implanted Into 4H-SiC (0001)

A 2-D model of aluminum-ion implantation into 4H-SiC (0001) was developed and assessed through reverse current I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -voltage V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> characteristics of p-n diodes. The model was based on a Monte Carlo simulation using a binary-collision approximation. For a moderate dose (10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sup> - 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> ), simulated isoconcentration contours were independent of the orientation of the masking edge. This condition allowed us to extract lateral straggling by expressing the lateral-concentration profiles as a 1-D dual-Pearson distribution function multiplied by a Gaussian distribution function. To demonstrate its applicability to higher doses, the model was applied in the simulation of a 4H-SiC p-n diode whose anode was formed by a 4 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">14</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> aluminum implant into a moderately n-type doped (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> = 9 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">15</sup> cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> ) drift layer. The I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> characteristics calculated with the model were found to agree with the measured ones, suggesting that the developed model is appropriate for designing 4H-SiC power devices, including not only a p-type region with moderate aluminum implant but also a p-type region with a heavy aluminum implant on the condition that N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> is sufficiently high.