3D Heat Research Articles

Three-dimensional modeling of moisture transport in wood using near-infrared hyperspectral imaging and X-ray computed tomography in conjunction with finite element analysis

In past studies, finite element analysis (FEA) methods have been used to simulate the thermal and moisture coupling of wood. However, challenges remain in achieving high-quality three-dimensional (3D) simulations, mainly because of the heterogeneous and complex structure of wood and its difficult-to-detect internal structure, which makes modeling challenging, in addition to the lack of robust experimental techniques to validate simulation results. In this study, the FEA simulation model was refined by combining X-ray computed tomography (CT) and near-infrared hyperspectral imaging (NIR-HSI). CT was used to probe the 3D density of wood, and a novel FEA tetrahedral mesh was constructed based on the results. The NIR-HSI method visualizes the moisture distribution during adsorption and desorption inside the wood. This result is then used to adjust the parameters of the FEA simulation model and as a reference value to evaluate the simulation results. The visualization and simulation results fit well with the theoretical properties. The simulation results can more accurately reflect the spatial distribution and transfer trend of wood moisture at different points in time. Therefore, the CT and NIR-HSI-based 3D heat and moisture-coupled FEA model of wood proposed in this study can be used as a basis for optimizing drying parameters to provide high-quality wood.

Decomposition mechanism of lumpy dolomite particles

Cylindrical cm-sized dolomite particles were decomposed using the two steps decomposition scheme. Several studies on dolomite decomposition have been done using µm- to mm-sized samples over the years. However, these sizes are too small for industrial kiln application and ignore the influence of important parameters, e.g. thermal conductivity, on the decomposition process. In this work, the decomposition behavior of lumpy dolomite particles in each step was investigated. Experimentally measured sample temperature and conversion profiles in steps 1 and 2 are respectively similar to those obtained for the decomposition of pure magnesite and limestone. To measure the core temperature and conversion is to essentially analyze the decomposition behavior. The thermophysical material properties of the decomposition product in each step as a function of temperature was estimated using the shrinking core model. Finite difference method was employed in solving the 1D heat equation describing the process. The reaction rate coefficient in step 1 is shown to have a weak temperature dependence, the thermal conductivity of the product, MgO∙CaCO3, slightly increases with temperature but the permeability is not dependent. In step 2, the reaction coefficient and MgO∙CaO thermal conductivity and pore diffusivity do not show a clear temperature dependence. Finally, the sensitivity of the material and transport properties on the decomposition process was studied under shaft kiln condition. Thermal conductivity is observed to have the highest influence on decomposition with a value of up to 60 % at 1400 °C and for 120 mm particles.

Heat transfer simulation of electric heating fabric system and parametric design: Towards human thermal comfort and energy efficiency

A 3D heat transfer model was developed to optimize the design of electric heating garments (EHGs) towards human thermal comfort and energy efficiency. The model deals with the thermal interaction between human skin, a three-layer electrical heating fabric system, and cold environments, allowing analysis of the heating process and resultant skin temperature. Three heat transfer mechanisms, namely conduction, convection and radiation, were considered. The real-time and final skin temperatures from simulation and measurement exhibited a difference below 4%, demonstrating the efficiency of the model. A steady-state parametric study was conducted using the validated numerical model. Five parameters, such as heating temperature, thermal resistance of the inner and outer fabrics, wind speed, and environmental air temperature, were involved. The results indicate that the heating temperature and thermal resistance of the inner fabric have a more noticeable effect on the temperature of the skin surface below the heated area compared to the other three factors. Apart from the heating temperature, the other four factors have a greater impact on the skin temperature of non-heated regions than that of the heated region. Multiple linear regression analysis was performed to establish a relationship between skin temperature and the five factors. Based on this analysis, a prediction model for skin temperature was developed and subsequently used to determine relevant parameters for EHGs. A method for optimizing the design of electrically heated clothing is proposed, which can achieve the dual goals of human thermal comfort and energy efficiency.

Investigation of Solidification Heat Transfer in Slab Continuous Casting Process Based on Different Roll Contact Calculation Methods.

The heat transfer of a slab is significantly influenced by roll contact during the continuous casting process. The influence of roll contact calculation methods on the predicted heat transfer results has not been previously investigated. In this work, the non-uniform solidification of the wide-thick slab was studied with a 2D heat transfer model using real roll contact method (R. method) and equivalent roll contact method (E. method). The predicted slab surface temperature and shell thickness were verified with the measured results of the infrared camera and nail shooting experiments, respectively. Then, the predicted heat transfer results (including the slab surface temperature, mushy region length, and solidification end position) for the wide-thick slab with different thicknesses and different casting speeds were calculated using the E. method and R. method, and the influence of these two roll contact methods on the predicted heat transfer results was discussed for the first time. The results show that both these two roll contact methods could be applied to accurately predict the slab surface temperature without considering the transient temperature dips in the roll-slab contact regions. However, the deviation of the predicted mushy region length and solidification end position using the E. method are obvious. Compared with the R. method, the predicted mushy region length obtained using the E. method is larger and the solidification end obviously subsequently moves along the casting direction.

City-scale heating and cooling with aquifer thermal energy storage (ATES)

Sustainable and climate-friendly space heating and cooling is of great importance for the energy transition. Compared to conventional energy sources, Aquifer Thermal Energy Storage (ATES) systems can significantly reduce greenhouse gas emissions from space heating and cooling. Hence, the objective of this study is to quantify the technical potential of shallow low-temperature ATES systems in terms of reclaimable energy in the city of Freiburg im Breisgau, Germany. Based on 3D heat transport modeling, heating and cooling power densities are determined for different ATES configurations located in an unconsolidated gravel aquifer of varying hydrogeological subsurface characteristics. High groundwater flow velocities of up to 13 m d−1 cause high storage energy loss and thus limit power densities to a maximum of 3.2 W m−2. Nevertheless, comparison of these power densities with the existing thermal energy demands shows that ATES systems can achieve substantial heating and cooling supply rates. This is especially true for the cooling demand, for which a full supply by ATES is determined for 92% of all residential buildings in the study area. For ATES heating alone, potential greenhouse gas emission savings of up to about 70,000 tCO2eq a−1 are calculated, which equals about 40% of the current greenhouse gas emissions caused by space and water heating in the study areas’ residential building stock. The modeling approach proposed in this study can also be applied in other regions with similar hydrogeological conditions to obtain estimations of local ATES supply rates and support city-scale energy planning.

Enhancing the high-temperature thermal evacuation of Cf/C-Mo30Cu joint via grooving 3D heat transfer interface

The joining of Cf/C composite to Mo30Cu alloy plays an important practical role in a bias filter for thermonuclear fusion. In this study, the Cf/C composite is joined to Mo30Cu alloy at 860 °C/10 min using AgCu4.5Ti braze after fabricating the three dimensional (3D) heat transfer interface along the Cf/C interface. The effect of interface grooving on the thermal conductivity of the joint is investigated and compared with the original brazed joint. The results demonstrate that the room temperature thermal conductivity of the joint reached a maximum of 132 W·m−1·K−1 at a grooving depth of 200 μm, which is 17.8 % higher than that of the original joint (112 W·m−1·K−1). The thermal conductivity of the joint is still as high as 100 W·m−1·K−1 at a temperature of 500 ℃. The excellent high-temperature thermal conductivity significantly reduces the Cf/C damage and extends the service life of the joint. And the shear strength of the joint has also been significantly increased by 252 % from 13.1 MPa to 33.1 MPa. Grooves prepared along the Cf/C interface provide channels for braze penetration, increasing the contact area between the brazing seam and the Cf/C matrix. Ultimately, the thermal evacuation performance of the joint is improved by increasing the heat exchange area across the joint. Also, grooves along the Cf/C interface enable mechanical interlocking with the brazing seam, achieving an enhanced joint strength. The heat transfer process of the brazed joint is simulated and analyzed by the finite element method. The analysis indicated that the thermal evacuation across the joint can be enhanced by pre-fabricating a 3D heat transfer interface in the joint. This research can provide a reference for solving the problem of heat accumulation in the first wall materials inside thermonuclear reactors.

Effect of Beverage Solutions on Flexural Strength of PEEK CAD –CAM, 3D Printing Denture Base and Heat Cure Acrylic Resin Denture Base Materials

This research was to evaluate the effect of distilled water (D.W.), tea, coffee, and cola on flexural strength of PEEK CAD-CAM, 3D printing and heat cure acrylic resin denture base at different durations (6, 12and 24 days). Material used in this research are PEEK, 3D printing and Heat cure acrylic resin with dimension (64 × 10 × 3.3 ±0.2 mm, ISO 20795- 1:201320) for each tested material. Total number of samples are n= (195) samples, Control samples n=15, experimental samples n=180.The control samples were non immersed in beverage media. The experimental samples were immersed in beverage media (distilled water, tea, coffee, and cola) for evaluating the flexural strength at different time which are (6, 12 and 24 days), then all collecting data were statically analyzed by spss version 25, by means of Descriptive statistics, analysis of Variance (ANOVA) and Duncan multiple range tests at p ≤ 0.05. The beverage solutions had been non significantly decreased the flexural strength and this decreasing worst as the immer sion periods increased, the most decreasing in flexural strength o f PEEK denture base was achieved in 24 immersion days in cola (252.0960 MPa ±5.19825) followed by coffee (253.4360 MPa ± 6.60975) followed by tea(253.7780 MPa ± 6.44697) followed by distilled water (254.1420 MPa ±7.41318) as comparing with control(260.2420MPa ±5.08412), the most decreasing in flexur al strength of 3D printing denture base was achieved in 24 immersion days in cola (103.1680 MPa ±12.26532) followed by coffee (105.4360 MPa ±12.36951) followed by tea (10 7.4530MPa ±11.47399) followed by distilled water (109.1320 MPa ±11.66156) as comparing with control(119.7140 MPa ± 12.25421), the most decreasing in flexural strength of heat cure acrylic resin denture base was achieved in 24 immersion days in cola (118.5640MPa ±13.06764) follow ed by coffee (122.3640MPa ± 9.78213) followed by tea( 123.5600MPa ±14.05144) followed by distilled water (126.3620 MPa ± 11.35254) as comparing with control (134.6420 MPa ± 9.79330) at p ≤ 0.05. All beverage media had been negatively aff ected the flexural strength of all experimental materials and as t he immersion periods increased the affect became worst. The aggressive effect was in cola followed by coffee followed by tea and the least effect was in D.W. beverage media. The worst immersion period was 24 then 12 and 6 days. for all tests use but this affect was non significantly.

Effect of Beverage Solutions on Surface Roughness of PEEK CAD –CAM, 3D Printing Denture Base and Heat Cure Acrylic Resin Denture Base Materials

The Aim: The aim of this research was to investigate how distilled water, tea, coffee, and cola affected the surface roughness of poly ether ether ketone CAD-CAM, 3D printing, and heat cure acrylic resin denture bases at specific intervals (6 ,12days). Materials and Methods: Disk shape 10 mm (diameter) ×2 m m (thickness), manufactured from materials used which were PEEK, 3D printing and heat cure acrylic resin. Total number of samples are n= (75) samples, control samples not immersed in beverage media n= 15, experimental samples immersed in beverage solutions (distilled water, tea, coffee and cola) n=60 for evaluating the surface roughness at (6, 12 and 24 days) .Then all collecting data were statically analyzed by spss v ersion 25, by means of descriptive statistics, analysis of variance (ANOVA) and duncan multiple range tests at p ≤ 0.05. The Results: The results revealed a significant increase in surface roughness in all beverage solutions at p ≤ 0.0 5. Conclusions: All beverage solutions had a negative effect on surface roughness of all experimental materials, the surface roughness increased. cola had the most aggressive effect followed by coffee then tea and distilled water have the least ef fect.

A nanofluid-based hybrid photovoltaic-thermal -thermoelectric generator system for combined heat and power applications

This study investigates the use of nanofluid-cooled Thermoelectric Generators (TEGs) integrated with a Photovoltaic (PV) panel to enhance the overall energy efficiency of the system. Three cooling fluids — Water (W), CuO/W single Nanofluid (NF), and CuO-Fe/W hybrid NF — are examined at various flow rates for cooling the PV panel. The research studies the PVT-TEG-NF system considering both normal and concentrated solar irradiation in a 3D parallelly-cooled heat sink setup and comparing its electrical and thermal performance, which has not been previously addressed in the literature. The results demonstrate that by utilizing CuO-Fe/W, the PV temperature at irradiation of 935 W/m2 in the summer can be maintained at 31.7 °C, which is 8.8 % and 13 % lower than those achieved by using CuO/W and water, respectively. For 0.08 m/s of CuO-Fe/W inlet velocity, the total output power PV-TEG is about 9.5 W, with a heat removal rate of ∼37.2 W. Additionally, by utilizing the CuO-Fe/W compared to water, the combined efficiency of the system (the ratio of electrical and thermal power output to solar input) increases from 66.7 % to 76.7 %. In the second part, the study introduces a new approach to maintain a steady PV efficiency while increasing solar concentration by adjusting the cooling fluid flow rate. By increasing solar concentration from 1.5 suns to 2.5 suns, the PV temperatures could be maintained at 44.5 °C and 51.5 °C using CuO-Fe/W and water with 0.12 m/s of inlet velocity, respectively. This made it possible to enhance the power output of the PV and TEGs by 65.9 % and 187 %, respectively, by using CuO-Fe/W as coolant, while the temperature of the panel could be maintained at a safe level.

Generalized frequency shift and attenuation in simply-supported micro/nano-beam resonators with thermoelastic dissipation

The low frequency shift, weak frequency attenuation, and large quality factor (Q) are strongly required for micro/nano-resonators with excellent operation performance. Therefore, the accurate predictions of frequency shift, frequency attenuation, and Q based on thermoelastic dissipation (QTED ) are significant when optimally designing micro/nano-resonators. In this work, employing the modified-couple-stress (MCS) and nonlocal-single-phase-lag (NSPL) models, the generalized analytical expressions of the frequency-shift ratio, frequency-attenuation ratio, and QTED are first proposed for the simply-supported micro/nano-beam resonator with two-dimensional (2D) heat conduction. In the modeling procedure, utilizing the Galerkin approach, the function of 2D fluctuation temperature is solved from the governing equation of NSPL thermoelasticity. For simplicity, the function of one-dimensional (1D) fluctuation temperature is presented directly. Various expressions of complex natural frequency are obtained from the governing equation of MCS vibration with thermoelastic coupling, and then the models of the frequency-shift ratio, frequency-attenuation ratio, and QTED are developed based on the relevant expressions of complex natural frequency. It is confirmed that the present developed models with generality can effectively degenerate into the classical models. In the simulation cases, silicon (Si) that is a representative semiconductor material is selected. Results reveal that the MCS size-dependence effect can affect the frequency shift more markedly than the NSPL and heat-conduction-dimension (HCD) effects. Moreover, for the beam with a small thickness and operated in a high-order mode, the NSPL and HCD effects will significantly determine behaviors of the frequency shift, frequency attenuation, and QTED .

Heat transfer optimization of uo2-mo fuel using genetic algorithms

Two genetic algorithm (GA) methods were applied to thermal finite element models to optimize the heat transfer efficacy of a UO2-Mo composite fuel pellet with typical pressurized water reactor fuel geometry. Mo additions to UO2 have been shown to increase the thermal conductivity, thus reducing centerline temperatures and temperature gradients. Previous studies evaluated uniformly dispersed Mo or continuous Mo internal geometries (e.g., fins, plates, discs) that were selected using engineering intuition. The current study uses two different implementations of the same GA to optimize Mo placement and minimize the fuel temperature with the only constraint being a maximum 10% Mo volume fraction. One approach superimposed Mo line elements onto the monolithic UO2 pellet model, and the other converted entire UO2 volume elements to Mo. The former method generated 1D heat transfer connections between nodes, whereas the latter method allowed for the formation of 3D structures. Features of the optimal fuel design produced by the GAs included dispersed Mo near the centerline that shifted the peak fuel temperature outward by 0.6 mm, Mo chains in the high-heat-flux region in the mid-to-outer radial zone, and a large continuous structure that spanned the full radius and height of the pellet and accounted for 87.7 % of the total Mo in the pellet. Analysis of this design indicates that the optimal Mo configuration is a balance between creating continuous heat transfer pathways and optimally dispersing Mo to minimize the heat transfer distance through UO2. This architecture ultimately produced an effective thermal conductivity of 11.3 W/m·K under the assumed boundary conditions. This result is higher than any previous values from the literature. Potential fabrication methods and challenges are discussed in addition to the implications on fuel performance.

Using a 3D heat map to explore the diverse correlations among elements and mineral species

This paper presents an enhanced 3D heat map for exploratory data analysis (EDA) of open mineral data, addressing the challenges caused by rapidly evolving datasets and ensuring scientifically meaningful data exploration. The Mindat website, a crowd-sourced database of mineral species, provides a constantly updated open data source via its newly established application programming interface (API). To illustrate the potential usage of the API, we constructed an automatic workflow to retrieve and cleanse mineral data from it, thus feeding the 3D heat map with up-to-date records of mineral species. In the 3D heat map, we developed scientifically sound operations for data selection and visualization by incorporating knowledge from existing mineral classification systems and recent studies in mineralogy. The resulting 3D heat map has been shared as an online demo system, with the source code made open on GitHub. We hope this updated 3D heat map system will serve as a valuable resource for researchers, educators, and students in geosciences, demonstrating the potential for data-intensive research in mineralogy and broader geoscience disciplines.

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

Variational Approach to 2D and 3D Heat Conduction Modeling

Variational Approach to 2D and 3D Heat Conduction Modeling

Morphology and novel classification of proximal humeral fractures.

Background: The morphology of proximal humeral fractures (PHFs) is complex, and the fixation and selection of implants need to be guided by the fracture type and classification, which requires an accurate understanding of the fracture line. This study had three purposes. 1) Define and analyze the fracture lines and morphological features of all types PHFs by three-dimensional (3D) mapping technology. 2) Determine the osteotomy position of the biomechanical model of the PHFs according to the fracture heat map. 3) Based on the analysis of the pathological morphology and distribution of a large number of consecutive cases of PHFs, propose a novel classification of PHFs. Methods: We retrospectively collected 220 cases of PHFs and generated a 3D fracture map and heat map based on computed tomography (CT) imaging. Through analysis of the fracture morphology of the 220 PHFs, a novel classification was proposed. The primary criterion for staging was the continuity between the humeral head and the greater tuberosity and lesser tuberosity, and the secondary criterion was the relationship between the humeral head segment and the humeral shaft. Results: The fracture line was primarily found around the metaphyseal zone of region of the surgical neck, with the most extensive distribution being below the larger tuberosity and on the posterior medial side of the epiphysis. We suggest that the osteotomy gap should be immediately (approximately 5-10mm) below the lower edge of the articular surface. The most common type of fracture was type I3 (33 cases, 15.0%), followed by type IV3 fracture (23 cases, 10.4%), and type III2 fracture (22 cases, 10.0%). Interobserver and intraobserver reliability analysis for the fracture classification revealed a k value (95% confidence interval) of 0.639 (0.57-0.71) and 0.841, P < 0.01, respectively. Conclusion: In this study, the fracture line and morphological characteristics of PHFs were clarified in detail by 3D mapping technique. In addition, a new classification method was proposed by analysis of the morphological characteristics of 220 PHFs, A two-part fracture model for PHFs is also proposed.

Numerical analysis of thermal differential distribution in Bobbin tool friction stir welding of 7075-AZ31B dissimilar alloy

The theoretical model of bobbin tool friction stir welding (BT-FSW) structure and dynamic process is established. Because of the great differences in properties of different materials, the mathematical relationship between material stress and deformation will be expressed based on 3D transient heat conduction equation and J-C constitutive model. The BT-FSW process of 7075 and AZ31B was analyzed by finite element method, and the field distribution, thermal cycle curve and stress distribution characteristics of different welding areas of the welded material were simulated. To predict and control the non-equilibrium distribution of welding temperature and stress field, the influences of different Rotating speed, welding speed, offset and other variables on the BT-FSW process of 7075/AZ31B aluminum magnesium dissimilar alloy were studied comprehensively. Finally, the reliability of the model is proved by fitting the experimental and numerical analysis results. The research results show that the temperature of BT-FSW presents an “hourglass” symmetrical distribution, and the stress distribution on both sides of the plate is uniform when the aluminum alloy is placed on the advancing side, the stirring tool is offset to the aluminum alloy by 2mm, and the rotating speed and welding speed are 700r/min and 180mm/min, respectively.

A LOCALIZED MESHLESS METHOD FOR TRANSIENT HEAT CONDUCTION WITH APPLICATIONS

The localized radial basis function (RBF) meshless approach is well suited for modeling transient heat conduction. The advantages of meshless methods, such as ease of discretization, are well known. However, there are still few examples of the method extended to three-dimensional (3D) transient heat conduction for geometries of practical engineering importance; in particular, with respect to graphics processing units (GPUs). In this study, we investigated the localized RBF meshless method in 3D transient heat conduction and explored its application for GPUs. Numerical examples are provided with an application to modeling chip cooling and friction welding.

A COMBINED DESALINATION AND REFRIGERATION SYSTEM BASED ON SOLAR EJECTOR TECHNOLOGY

A solar ejector technology-based system that combines refrigeration and desalination was investigated for the present study. The proposed model combined a conventional ejector refrigeration system with a desalination unit to examine its ability to achieve cooling as well as produce clean water. An analytical model of the ejector was developed using 1D compressible flow equations based on mass, momentum, and energy conservation. The output from the ejector was then fed to a 1D heat exchanger model to compute the clean water production. The analytical model was implemented using the Matlab platform. A 2D axisymmetric numerical simulation of the ejector system was also performed to comprehend the internal flow structures. It has been observed that the entrainment ratio, which is the ratio of the vapor refrigerant's mass flow rate to the motive steam's mass flow rate, falls as the stagnation temperature of the motive steam increases. It was noted that the coefficient of performance (COP) rises as the evaporator temperature rises, but it is seen to decline with the rise in generator temperature. The amount of desalinated water that can be produced with the system was also explored. It was observed that the production of desalinated water increased proportionally with the rise in generator temperature. At a generator temperature of 140&amp;#176;C, the system obtained clean water at a rate of about 2.9 g/s, which corresponds to a 24.5&amp;#37; mass flow rate of the input steam.

A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations

In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more difficult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using both the obtained solution of the conformable time-derivative heat equation and the inverse Cole-Hopf transformation. This approach shows its efficiency to deal with this class of fractional nonlinear problems. Some numerical experiments are displayed to consolidate our approach

Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions

&lt;p&gt;In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.&lt;/p&gt;